LMFDB - Embedded newform 39.5.g.a.31.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,5,Mod(31,39)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("39.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	39.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.03142856027$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.9
Root			$-4.67933 + 4.67933i$ of defining polynomial
Character	$\chi$	$=$	39.31
Dual form			39.5.g.a.34.9

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.67933 - 4.67933i) q^{2} -5.19615 q^{3} -27.7923i q^{4} +(13.1418 - 13.1418i) q^{5} +(-24.3145 + 24.3145i) q^{6} +(-30.2175 - 30.2175i) q^{7} +(-55.1800 - 55.1800i) q^{8} +27.0000 q^{9} -122.989i q^{10} +(58.8012 + 58.8012i) q^{11} +144.413i q^{12} +(120.640 + 118.351i) q^{13} -282.795 q^{14} +(-68.2867 + 68.2867i) q^{15} -71.7343 q^{16} +257.703i q^{17} +(126.342 - 126.342i) q^{18} +(336.208 - 336.208i) q^{19} +(-365.240 - 365.240i) q^{20} +(157.015 + 157.015i) q^{21} +550.300 q^{22} -486.231i q^{23} +(286.724 + 286.724i) q^{24} +279.587i q^{25} +(1118.32 - 10.7102i) q^{26} -140.296 q^{27} +(-839.813 + 839.813i) q^{28} -771.373 q^{29} +639.072i q^{30} +(-1304.18 + 1304.18i) q^{31} +(547.211 - 547.211i) q^{32} +(-305.540 - 305.540i) q^{33} +(1205.88 + 1205.88i) q^{34} -794.223 q^{35} -750.392i q^{36} +(1053.22 + 1053.22i) q^{37} -3146.45i q^{38} +(-626.864 - 614.971i) q^{39} -1450.33 q^{40} +(-1126.29 + 1126.29i) q^{41} +1469.45 q^{42} -1930.35i q^{43} +(1634.22 - 1634.22i) q^{44} +(354.828 - 354.828i) q^{45} +(-2275.24 - 2275.24i) q^{46} +(-65.7645 - 65.7645i) q^{47} +372.742 q^{48} -574.806i q^{49} +(1308.28 + 1308.28i) q^{50} -1339.06i q^{51} +(3289.25 - 3352.86i) q^{52} -5040.23 q^{53} +(-656.492 + 656.492i) q^{54} +1545.50 q^{55} +3334.80i q^{56} +(-1746.99 + 1746.99i) q^{57} +(-3609.51 + 3609.51i) q^{58} +(-206.279 - 206.279i) q^{59} +(1897.84 + 1897.84i) q^{60} +6998.83 q^{61} +12205.4i q^{62} +(-815.872 - 815.872i) q^{63} -6268.91i q^{64} +(3140.77 - 30.0794i) q^{65} -2859.44 q^{66} +(-939.648 + 939.648i) q^{67} +7162.15 q^{68} +2526.53i q^{69} +(-3716.43 + 3716.43i) q^{70} +(-3049.34 + 3049.34i) q^{71} +(-1489.86 - 1489.86i) q^{72} +(-840.175 - 840.175i) q^{73} +9856.69 q^{74} -1452.78i q^{75} +(-9343.97 - 9343.97i) q^{76} -3553.65i q^{77} +(-5810.95 + 55.6521i) q^{78} +4044.54 q^{79} +(-942.716 + 942.716i) q^{80} +729.000 q^{81} +10540.5i q^{82} +(2060.82 - 2060.82i) q^{83} +(4363.80 - 4363.80i) q^{84} +(3386.67 + 3386.67i) q^{85} +(-9032.76 - 9032.76i) q^{86} +4008.17 q^{87} -6489.30i q^{88} +(6032.18 + 6032.18i) q^{89} -3320.71i q^{90} +(-69.1630 - 7221.71i) q^{91} -13513.5 q^{92} +(6776.74 - 6776.74i) q^{93} -615.468 q^{94} -8836.73i q^{95} +(-2843.39 + 2843.39i) q^{96} +(5575.10 - 5575.10i) q^{97} +(-2689.71 - 2689.71i) q^{98} +(1587.63 + 1587.63i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) $ 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	4.67933	−	4.67933i	1.16983	−	1.16983i	0.187584	−	0.982249i	$-0.439934\pi$
$2$	4.67933	−	4.67933i	1.16983	−	1.16983i	0.982249	−	0.187584i	$-0.0600658\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$		−	27.7923i		−	1.73702i
$5$	13.1418	−	13.1418i	0.525671	−	0.525671i	−0.393608	−	0.919279i	$-0.628773\pi$
$5$	13.1418	−	13.1418i	0.525671	−	0.525671i	0.919279	+	0.393608i	$0.128773\pi$
$6$	−24.3145	+	24.3145i	−0.675403	+	0.675403i
$7$	−30.2175	−	30.2175i	−0.616684	−	0.616684i	0.327996	−	0.944679i	$-0.393627\pi$
$7$	−30.2175	−	30.2175i	−0.616684	−	0.616684i	−0.944679	+	0.327996i	$0.893627\pi$
$8$	−55.1800	−	55.1800i	−0.862187	−	0.862187i
$9$	27.0000			0.333333
$10$		−	122.989i		−	1.22989i
$11$	58.8012	+	58.8012i	0.485960	+	0.485960i	0.907029	−	0.421069i	$-0.138345\pi$
$11$	58.8012	+	58.8012i	0.485960	+	0.485960i	−0.421069	+	0.907029i	$0.638345\pi$
$12$			144.413i			1.00287i
$13$	120.640	+	118.351i	0.713846	+	0.700303i
$13$	120.640	+	118.351i	0.713846	+	0.700303i
$14$	−282.795			−1.44283
$15$	−68.2867	+	68.2867i	−0.303496	+	0.303496i
$16$	−71.7343			−0.280212
$17$			257.703i			0.891706i	0.895106	+	0.445853i	$0.147100\pi$
$17$			257.703i			0.891706i	−0.895106	+	0.445853i	$0.852900\pi$
$18$	126.342	−	126.342i	0.389944	−	0.389944i
$19$	336.208	−	336.208i	0.931323	−	0.931323i	−0.0664659	−	0.997789i	$-0.521172\pi$
$19$	336.208	−	336.208i	0.931323	−	0.931323i	0.997789	+	0.0664659i	$0.0211724\pi$
$20$	−365.240	−	365.240i	−0.913100	−	0.913100i
$21$	157.015	+	157.015i	0.356042	+	0.356042i
$22$	550.300			1.13698
$23$		−	486.231i		−	0.919151i	−0.888139	−	0.459576i	$-0.848001\pi$
$23$		−	486.231i		−	0.919151i	0.888139	−	0.459576i	$-0.151999\pi$
$24$	286.724	+	286.724i	0.497784	+	0.497784i
$25$			279.587i			0.447340i
$26$	1118.32	−	10.7102i	1.65432	−	0.0158436i
$27$	−140.296			−0.192450
$28$	−839.813	+	839.813i	−1.07119	+	1.07119i
$29$	−771.373			−0.917209			−0.458604	−	0.888641i	$-0.651650\pi$
$29$	−771.373			−0.917209			−0.458604	+	0.888641i	$0.651650\pi$
$30$			639.072i			0.710080i
$31$	−1304.18	+	1304.18i	−1.35711	+	1.35711i	−0.479654	+	0.877458i	$0.659238\pi$
$31$	−1304.18	+	1304.18i	−1.35711	+	1.35711i	−0.877458	+	0.479654i	$0.840762\pi$
$32$	547.211	−	547.211i	0.534386	−	0.534386i
$33$	−305.540	−	305.540i	−0.280569	−	0.280569i
$34$	1205.88	+	1205.88i	1.04315	+	1.04315i
$35$	−794.223			−0.648345
$36$		−	750.392i		−	0.579006i
$37$	1053.22	+	1053.22i	0.769333	+	0.769333i	0.977989	−	0.208656i	$-0.0669091\pi$
$37$	1053.22	+	1053.22i	0.769333	+	0.769333i	−0.208656	+	0.977989i	$0.566909\pi$
$38$		−	3146.45i		−	2.17898i
$39$	−626.864	−	614.971i	−0.412139	−	0.404320i
$40$	−1450.33			−0.906454
$41$	−1126.29	+	1126.29i	−0.670010	+	0.670010i	−0.957718	−	0.287708i	$-0.907107\pi$
$41$	−1126.29	+	1126.29i	−0.670010	+	0.670010i	0.287708	+	0.957718i	$0.407107\pi$
$42$	1469.45			0.833020
$43$		−	1930.35i		−	1.04400i	−0.852946	−	0.521999i	$-0.825186\pi$
$43$		−	1930.35i		−	1.04400i	0.852946	−	0.521999i	$-0.174814\pi$
$44$	1634.22	−	1634.22i	0.844121	−	0.844121i
$45$	354.828	−	354.828i	0.175224	−	0.175224i
$46$	−2275.24	−	2275.24i	−1.07525	−	1.07525i
$47$	−65.7645	−	65.7645i	−0.0297712	−	0.0297712i	0.692064	−	0.721836i	$-0.256701\pi$
$47$	−65.7645	−	65.7645i	−0.0297712	−	0.0297712i	−0.721836	+	0.692064i	$0.756701\pi$
$48$	372.742			0.161781
$49$		−	574.806i		−	0.239403i
$50$	1308.28	+	1308.28i	0.523313	+	0.523313i
$51$		−	1339.06i		−	0.514827i
$52$	3289.25	−	3352.86i	1.21644	−	1.23996i
$53$	−5040.23			−1.79432			−0.897158	−	0.441710i	$-0.854372\pi$
$53$	−5040.23			−1.79432			−0.897158	+	0.441710i	$0.854372\pi$
$54$	−656.492	+	656.492i	−0.225134	+	0.225134i
$55$	1545.50			0.510910
$56$			3334.80i			1.06339i
$57$	−1746.99	+	1746.99i	−0.537699	+	0.537699i
$58$	−3609.51	+	3609.51i	−1.07298	+	1.07298i
$59$	−206.279	−	206.279i	−0.0592586	−	0.0592586i	0.676856	−	0.736115i	$-0.263342\pi$
$59$	−206.279	−	206.279i	−0.0592586	−	0.0592586i	−0.736115	+	0.676856i	$0.763342\pi$
$60$	1897.84	+	1897.84i	0.527178	+	0.527178i
$61$	6998.83			1.88090			0.940450	−	0.339932i	$-0.110404\pi$
$61$	6998.83			1.88090			0.940450	+	0.339932i	$0.110404\pi$
$62$			12205.4i			3.17519i
$63$	−815.872	−	815.872i	−0.205561	−	0.205561i
$64$		−	6268.91i		−	1.53050i
$65$	3140.77	−	30.0794i	0.743377	−	0.00711939i
$66$	−2859.44			−0.656438
$67$	−939.648	+	939.648i	−0.209322	+	0.209322i	−0.803979	−	0.594657i	$-0.797288\pi$
$67$	−939.648	+	939.648i	−0.209322	+	0.209322i	0.594657	+	0.803979i	$0.297288\pi$
$68$	7162.15			1.54891
$69$			2526.53i			0.530672i
$70$	−3716.43	+	3716.43i	−0.758456	+	0.758456i
$71$	−3049.34	+	3049.34i	−0.604908	+	0.604908i	−0.941611	−	0.336703i	$-0.890688\pi$
$71$	−3049.34	+	3049.34i	−0.604908	+	0.604908i	0.336703	+	0.941611i	$0.390688\pi$
$72$	−1489.86	−	1489.86i	−0.287396	−	0.287396i
$73$	−840.175	−	840.175i	−0.157661	−	0.157661i	0.623869	−	0.781529i	$-0.285560\pi$
$73$	−840.175	−	840.175i	−0.157661	−	0.157661i	−0.781529	+	0.623869i	$0.785560\pi$
$74$	9856.69			1.79998
$75$		−	1452.78i		−	0.258272i
$76$	−9343.97	−	9343.97i	−1.61772	−	1.61772i
$77$		−	3553.65i		−	0.599367i
$78$	−5810.95	+	55.6521i	−0.955121	+	0.00914728i
$79$	4044.54			0.648059			0.324030	−	0.946047i	$-0.394962\pi$
$79$	4044.54			0.648059			0.324030	+	0.946047i	$0.394962\pi$
$80$	−942.716	+	942.716i	−0.147299	+	0.147299i
$81$	729.000			0.111111
$82$			10540.5i			1.56760i
$83$	2060.82	−	2060.82i	0.299147	−	0.299147i	−0.541533	−	0.840680i	$-0.682156\pi$
$83$	2060.82	−	2060.82i	0.299147	−	0.299147i	0.840680	+	0.541533i	$0.182156\pi$
$84$	4363.80	−	4363.80i	0.618452	−	0.618452i
$85$	3386.67	+	3386.67i	0.468744	+	0.468744i
$86$	−9032.76	−	9032.76i	−1.22130	−	1.22130i
$87$	4008.17			0.529551
$88$		−	6489.30i		−	0.837977i
$89$	6032.18	+	6032.18i	0.761543	+	0.761543i	0.976601	−	0.215059i	$-0.0689942\pi$
$89$	6032.18	+	6032.18i	0.761543	+	0.761543i	−0.215059	+	0.976601i	$0.568994\pi$
$90$		−	3320.71i		−	0.409965i
$91$	−69.1630	−	7221.71i	−0.00835202	−	0.872082i
$92$	−13513.5			−1.59658
$93$	6776.74	−	6776.74i	0.783529	−	0.783529i
$94$	−615.468			−0.0696546
$95$		−	8836.73i		−	0.979139i
$96$	−2843.39	+	2843.39i	−0.308528	+	0.308528i
$97$	5575.10	−	5575.10i	0.592528	−	0.592528i	−0.345785	−	0.938314i	$-0.612387\pi$
$97$	5575.10	−	5575.10i	0.592528	−	0.592528i	0.938314	+	0.345785i	$0.112387\pi$
$98$	−2689.71	−	2689.71i	−0.280061	−	0.280061i
$99$	1587.63	+	1587.63i	0.161987	+	0.161987i
$100$	7770.37			0.777037
$101$			14591.5i			1.43040i	0.698920	+	0.715200i	$0.253665\pi$
$101$			14591.5i			1.43040i	−0.698920	+	0.715200i	$0.746335\pi$
$102$	−6265.92	−	6265.92i	−0.602261	−	0.602261i
$103$			13071.3i			1.23210i	0.787708	+	0.616048i	$0.211267\pi$
$103$			13071.3i			1.23210i	−0.787708	+	0.616048i	$0.788733\pi$
$104$	−126.298	−	13187.5i	−0.0116770	−	1.21926i
$105$	4126.90			0.374322
$106$	−23584.9	+	23584.9i	−2.09905	+	2.09905i
$107$	−14255.0			−1.24509			−0.622545	−	0.782584i	$-0.713901\pi$
$107$	−14255.0			−1.24509			−0.622545	+	0.782584i	$0.713901\pi$
$108$			3899.15i			0.334289i
$109$	−4886.68	+	4886.68i	−0.411302	+	0.411302i	−0.882192	−	0.470890i	$-0.843933\pi$
$109$	−4886.68	+	4886.68i	−0.411302	+	0.411302i	0.470890	+	0.882192i	$0.343933\pi$
$110$	7231.92	−	7231.92i	0.597680	−	0.597680i
$111$	−5472.67	−	5472.67i	−0.444174	−	0.444174i
$112$	2167.63	+	2167.63i	0.172802	+	0.172802i
$113$	−3433.45			−0.268890			−0.134445	−	0.990921i	$-0.542925\pi$
$113$	−3433.45			−0.268890			−0.134445	+	0.990921i	$0.542925\pi$
$114$			16349.4i			1.25804i
$115$	−6389.94	−	6389.94i	−0.483171	−	0.483171i
$116$			21438.2i			1.59321i
$117$	3257.28	+	3195.48i	0.237949	+	0.233434i
$118$	−1930.50			−0.138645
$119$	7787.14	−	7787.14i	0.549900	−	0.549900i
$120$	7536.11			0.523341
$121$		−	7725.84i		−	0.527685i
$122$	32749.8	−	32749.8i	2.20034	−	2.20034i
$123$	5852.36	−	5852.36i	0.386831	−	0.386831i
$124$	36246.3	+	36246.3i	2.35733	+	2.35733i
$125$	11887.9	+	11887.9i	0.760825	+	0.760825i
$126$	−7635.47			−0.480944
$127$		−	21948.2i		−	1.36079i	−0.732846	−	0.680395i	$-0.761808\pi$
$127$		−	21948.2i		−	1.36079i	0.732846	−	0.680395i	$-0.238192\pi$
$128$	−20578.9	−	20578.9i	−1.25604	−	1.25604i
$129$			10030.4i			0.602753i
$130$	14555.9	−	14837.4i	0.861298	−	0.877955i
$131$	−1658.15			−0.0966231			−0.0483116	−	0.998832i	$-0.515384\pi$
$131$	−1658.15			−0.0966231			−0.0483116	+	0.998832i	$0.515384\pi$
$132$	−8491.65	+	8491.65i	−0.487354	+	0.487354i
$133$	−20318.7			−1.14866
$134$			8793.84i			0.489744i
$135$	−1843.74	+	1843.74i	−0.101165	+	0.101165i
$136$	14220.0	−	14220.0i	0.768817	−	0.768817i
$137$	−21604.3	−	21604.3i	−1.15106	−	1.15106i	−0.986340	−	0.164724i	$-0.947327\pi$
$137$	−21604.3	−	21604.3i	−1.15106	−	1.15106i	−0.164724	−	0.986340i	$-0.552673\pi$
$138$	11822.5	+	11822.5i	0.620798	+	0.620798i
$139$	−4658.17			−0.241094			−0.120547	−	0.992708i	$-0.538465\pi$
$139$	−4658.17			−0.241094			−0.120547	+	0.992708i	$0.538465\pi$
$140$			22073.3i			1.12619i
$141$	341.723	+	341.723i	0.0171884	+	0.0171884i
$142$			28537.8i			1.41528i
$143$	134.587	+	14053.0i	0.00658157	+	0.687220i
$144$	−1936.83			−0.0934040
$145$	−10137.2	+	10137.2i	−0.482150	+	0.482150i
$146$	−7862.91			−0.368874
$147$			2986.78i			0.138219i
$148$	29271.3	−	29271.3i	1.33634	−	1.33634i
$149$	393.889	−	393.889i	0.0177420	−	0.0177420i	−0.698180	−	0.715922i	$-0.746007\pi$
$149$	393.889	−	393.889i	0.0177420	−	0.0177420i	0.715922	+	0.698180i	$0.246007\pi$
$150$	−6798.04	−	6798.04i	−0.302135	−	0.302135i
$151$	−7620.89	−	7620.89i	−0.334235	−	0.334235i	0.519957	−	0.854192i	$-0.325948\pi$
$151$	−7620.89	−	7620.89i	−0.334235	−	0.334235i	−0.854192	+	0.519957i	$0.825948\pi$
$152$	−37103.8			−1.60595
$153$			6957.98i			0.297235i
$154$	−16628.7	−	16628.7i	−0.701159	−	0.701159i
$155$			34278.6i			1.42679i
$156$	−17091.4	+	17422.0i	−0.702311	+	0.715893i
$157$	25587.3			1.03807			0.519034	−	0.854754i	$-0.326292\pi$
$157$	25587.3			1.03807			0.519034	+	0.854754i	$0.326292\pi$
$158$	18925.7	−	18925.7i	0.758121	−	0.758121i
$159$	26189.8			1.03595
$160$		−	14382.7i		−	0.561822i
$161$	−14692.7	+	14692.7i	−0.566826	+	0.566826i
$162$	3411.23	−	3411.23i	0.129981	−	0.129981i
$163$	−16600.8	−	16600.8i	−0.624820	−	0.624820i	0.321940	−	0.946760i	$-0.395665\pi$
$163$	−16600.8	−	16600.8i	−0.624820	−	0.624820i	−0.946760	+	0.321940i	$0.895665\pi$
$164$	31302.1	+	31302.1i	1.16382	+	1.16382i
$165$	−8030.67			−0.294974
$166$		−	19286.6i		−	0.699904i
$167$	−37894.4	−	37894.4i	−1.35876	−	1.35876i	−0.875451	−	0.483307i	$-0.839435\pi$
$167$	−37894.4	−	37894.4i	−1.35876	−	1.35876i	−0.483307	−	0.875451i	$-0.660565\pi$
$168$		−	17328.1i		−	0.613950i
$169$	547.013	+	28555.8i	0.0191524	+	0.999817i
$170$	31694.7			1.09670
$171$	9077.60	−	9077.60i	0.310441	−	0.310441i
$172$	−53648.9			−1.81344
$173$			1712.90i			0.0572321i	0.999590	+	0.0286161i	$0.00911002\pi$
$173$			1712.90i			0.0572321i	−0.999590	+	0.0286161i	$0.990890\pi$
$174$	18755.6	−	18755.6i	0.619486	−	0.619486i
$175$	8448.43	−	8448.43i	0.275867	−	0.275867i
$176$	−4218.06	−	4218.06i	−0.136172	−	0.136172i
$177$	1071.86	+	1071.86i	0.0342130	+	0.0342130i
$178$	56453.1			1.78176
$179$		−	2700.43i		−	0.0842803i	−0.999112	−	0.0421401i	$-0.986582\pi$
$179$		−	2700.43i		−	0.0842803i	0.999112	−	0.0421401i	$-0.0134176\pi$
$180$	−9861.48	−	9861.48i	−0.304367	−	0.304367i
$181$			31325.6i			0.956186i	0.878309	+	0.478093i	$0.158672\pi$
$181$			31325.6i			0.956186i	−0.878309	+	0.478093i	$0.841328\pi$
$182$	−34116.4	−	33469.1i	−1.02996	−	1.01042i
$183$	−36367.0			−1.08594
$184$	−26830.2	+	26830.2i	−0.792481	+	0.792481i
$185$	27682.3			0.808832
$186$		−	63421.2i		−	1.83320i
$187$	−15153.2	+	15153.2i	−0.433333	+	0.433333i
$188$	−1827.75	+	1827.75i	−0.0517131	+	0.0517131i
$189$	4239.40	+	4239.40i	0.118681	+	0.118681i
$190$	−41350.0	−	41350.0i	−1.14543	−	1.14543i
$191$	52778.0			1.44672			0.723362	−	0.690469i	$-0.242596\pi$
$191$	52778.0			1.44672			0.723362	+	0.690469i	$0.242596\pi$
$192$			32574.2i			0.883632i
$193$	1405.79	+	1405.79i	0.0377403	+	0.0377403i	0.725725	−	0.687985i	$-0.241504\pi$
$193$	1405.79	+	1405.79i	0.0377403	+	0.0377403i	−0.687985	+	0.725725i	$0.741504\pi$
$194$		−	52175.5i		−	1.38632i
$195$	−16319.9	+	156.297i	−0.429189	+	0.00411038i
$196$	−15975.2			−0.415847
$197$	10175.2	−	10175.2i	0.262188	−	0.262188i	−0.563755	−	0.825942i	$-0.690644\pi$
$197$	10175.2	−	10175.2i	0.262188	−	0.262188i	0.825942	+	0.563755i	$0.190644\pi$
$198$	14858.1			0.378995
$199$		−	7439.54i		−	0.187862i	−0.995579	−	0.0939312i	$-0.970057\pi$
$199$		−	7439.54i		−	0.187862i	0.995579	−	0.0939312i	$-0.0299434\pi$
$200$	15427.6	−	15427.6i	0.385691	−	0.385691i
$201$	4882.55	−	4882.55i	0.120852	−	0.120852i
$202$	68278.5	+	68278.5i	1.67333	+	1.67333i
$203$	23308.9	+	23308.9i	0.565627	+	0.565627i
$204$	−37215.6			−0.894263
$205$			29602.8i			0.704410i
$206$	61165.0	+	61165.0i	1.44135	+	1.44135i
$207$		−	13128.2i		−	0.306384i
$208$	−8654.02	−	8489.84i	−0.200028	−	0.196233i
$209$	39538.8			0.905172
$210$	19311.1	−	19311.1i	0.437895	−	0.437895i
$211$	−45876.3			−1.03044			−0.515221	−	0.857057i	$-0.672290\pi$
$211$	−45876.3			−1.03044			−0.515221	+	0.857057i	$0.672290\pi$
$212$			140080.i			3.11676i
$213$	15844.9	−	15844.9i	0.349244	−	0.349244i
$214$	−66704.1	+	66704.1i	−1.45655	+	1.45655i
$215$	−25368.3	−	25368.3i	−0.548800	−	0.548800i
$216$	7741.54	+	7741.54i	0.165928	+	0.165928i
$217$	78818.4			1.67382
$218$			45732.8i			0.962309i
$219$	4365.67	+	4365.67i	0.0910255	+	0.0910255i
$220$		−	42953.1i		−	0.887460i
$221$	−30499.4	+	31089.3i	−0.624464	+	0.636541i
$222$	−51216.9			−1.03922
$223$	26187.3	−	26187.3i	0.526601	−	0.526601i	−0.392956	−	0.919557i	$-0.628548\pi$
$223$	26187.3	−	26187.3i	0.526601	−	0.526601i	0.919557	+	0.392956i	$0.128548\pi$
$224$	−33070.7			−0.659094
$225$			7548.86i			0.149113i
$226$	−16066.3	+	16066.3i	−0.314556	+	0.314556i
$227$	−20418.0	+	20418.0i	−0.396244	+	0.396244i	−0.876906	−	0.480662i	$-0.840396\pi$
$227$	−20418.0	+	20418.0i	−0.396244	+	0.396244i	0.480662	+	0.876906i	$0.340396\pi$
$228$	48552.7	+	48552.7i	0.933993	+	0.933993i
$229$	−52109.2	−	52109.2i	−0.993672	−	0.993672i	0.00630803	−	0.999980i	$-0.497992\pi$
$229$	−52109.2	−	52109.2i	−0.993672	−	0.993672i	−0.999980	+	0.00630803i	$0.997992\pi$
$230$	−59801.3			−1.13046
$231$			18465.3i			0.346045i
$232$	42564.3	+	42564.3i	0.790806	+	0.790806i
$233$		−	45782.3i		−	0.843308i	−0.906757	−	0.421654i	$-0.861450\pi$
$233$		−	45782.3i		−	0.843308i	0.906757	−	0.421654i	$-0.138550\pi$
$234$	30194.6	−	289.177i	0.551439	−	0.00528119i
$235$	−1728.53			−0.0312997
$236$	−5732.97	+	5732.97i	−0.102933	+	0.102933i
$237$	−21016.0			−0.374157
$238$		−	72877.2i		−	1.28658i
$239$	−54954.6	+	54954.6i	−0.962073	+	0.962073i	−0.999307	−	0.0372339i	$-0.988145\pi$
$239$	−54954.6	+	54954.6i	−0.962073	+	0.962073i	0.0372339	+	0.999307i	$0.488145\pi$
$240$	4898.50	−	4898.50i	0.0850433	−	0.0850433i
$241$	23474.0	+	23474.0i	0.404159	+	0.404159i	0.879696	−	0.475537i	$-0.157746\pi$
$241$	23474.0	+	23474.0i	0.404159	+	0.404159i	−0.475537	+	0.879696i	$0.657746\pi$
$242$	−36151.8	−	36151.8i	−0.617304	−	0.617304i
$243$	−3788.00			−0.0641500
$244$		−	194513.i		−	3.26716i
$245$	−7553.98	−	7553.98i	−0.125847	−	0.125847i
$246$		−	54770.3i		−	0.905054i
$247$	80350.6	−	769.526i	1.31703	−	0.0126133i
$248$	143930.			2.34017
$249$	−10708.4	+	10708.4i	−0.172713	+	0.172713i
$250$	111255.			1.78008
$251$		−	51668.4i		−	0.820120i	−0.912059	−	0.410060i	$-0.865508\pi$
$251$		−	51668.4i		−	0.820120i	0.912059	−	0.410060i	$-0.134492\pi$
$252$	−22674.9	+	22674.9i	−0.357063	+	0.357063i
$253$	28591.0	−	28591.0i	0.446671	−	0.446671i
$254$	−102703.	−	102703.i	−1.59190	−	1.59190i
$255$	−17597.7	−	17597.7i	−0.270629	−	0.270629i
$256$	−92288.7			−1.40821
$257$		−	87980.2i		−	1.33204i	−0.745932	−	0.666022i	$-0.767996\pi$
$257$		−	87980.2i		−	1.33204i	0.745932	−	0.666022i	$-0.232004\pi$
$258$	46935.6	+	46935.6i	0.705120	+	0.705120i
$259$		−	63651.1i		−	0.948869i
$260$	−835.976	−	87289.1i	−0.0123665	−	1.29126i
$261$	−20827.1			−0.305736
$262$	−7759.03	+	7759.03i	−0.113033	+	0.113033i
$263$	113585.			1.64214			0.821068	−	0.570831i	$-0.193379\pi$
$263$	113585.			1.64214			0.821068	+	0.570831i	$0.193379\pi$
$264$			33719.4i			0.483806i
$265$	−66237.6	+	66237.6i	−0.943220	+	0.943220i
$266$	−95077.9	+	95077.9i	−1.34374	+	1.34374i
$267$	−31344.1	−	31344.1i	−0.439677	−	0.439677i
$268$	26114.9	+	26114.9i	0.363596	+	0.363596i
$269$	5647.94			0.0780523			0.0390262	−	0.999238i	$-0.487574\pi$
$269$	5647.94			0.0780523			0.0390262	+	0.999238i	$0.487574\pi$
$270$			17254.9i			0.236693i
$271$	−3722.39	−	3722.39i	−0.0506854	−	0.0506854i	0.681310	−	0.731995i	$-0.261411\pi$
$271$	−3722.39	−	3722.39i	−0.0506854	−	0.0506854i	−0.731995	+	0.681310i	$0.761411\pi$
$272$		−	18486.1i		−	0.249867i
$273$	359.382	+	37525.1i	0.00482204	+	0.503497i
$274$	−202187.			−2.69310
$275$	−16440.1	+	16440.1i	−0.217389	+	0.217389i
$276$	70218.1			0.921787
$277$			31673.1i			0.412792i	0.978469	+	0.206396i	$0.0661736\pi$
$277$			31673.1i			0.412792i	−0.978469	+	0.206396i	$0.933826\pi$
$278$	−21797.1	+	21797.1i	−0.282039	+	0.282039i
$279$	−35213.0	+	35213.0i	−0.452371	+	0.452371i
$280$	43825.2	+	43825.2i	0.558995	+	0.558995i
$281$	−10897.2	−	10897.2i	−0.138007	−	0.138007i	0.634728	−	0.772736i	$-0.281112\pi$
$281$	−10897.2	−	10897.2i	−0.138007	−	0.138007i	−0.772736	+	0.634728i	$0.781112\pi$
$282$	3198.07			0.0402151
$283$			30994.4i			0.387000i	0.981100	+	0.193500i	$0.0619839\pi$
$283$			30994.4i			0.387000i	−0.981100	+	0.193500i	$0.938016\pi$
$284$	84748.2	+	84748.2i	1.05074	+	1.05074i
$285$			45917.0i			0.565306i
$286$	66388.2	+	65128.7i	0.811632	+	0.796233i
$287$	68067.2			0.826369
$288$	14774.7	−	14774.7i	0.178129	−	0.178129i
$289$	17110.2			0.204861
$290$			94870.7i			1.12807i
$291$	−28969.1	+	28969.1i	−0.342096	+	0.342096i
$292$	−23350.4	+	23350.4i	−0.273860	+	0.273860i
$293$	−113536.	−	113536.i	−1.32251	−	1.32251i	−0.911741	−	0.410765i	$-0.865262\pi$
$293$	−113536.	−	113536.i	−1.32251	−	1.32251i	−0.410765	−	0.911741i	$-0.634738\pi$
$294$	13976.1	+	13976.1i	0.161694	+	0.161694i
$295$	−5421.75			−0.0623011
$296$		−	116233.i		−	1.32662i
$297$	−8249.58	−	8249.58i	−0.0935231	−	0.0935231i
$298$		−	3686.28i		−	0.0415103i
$299$	57546.0	−	58658.9i	0.643684	−	0.656133i
$300$	−40376.0			−0.448623
$301$	−58330.4	+	58330.4i	−0.643817	+	0.643817i
$302$	−71321.3			−0.781998
$303$		−	75819.7i		−	0.825842i
$304$	−24117.6	+	24117.6i	−0.260968	+	0.260968i
$305$	91977.0	−	91977.0i	0.988735	−	0.988735i
$306$	32558.7	+	32558.7i	0.347716	+	0.347716i
$307$	75222.4	+	75222.4i	0.798124	+	0.798124i	0.982800	−	0.184675i	$-0.0591233\pi$
$307$	75222.4	+	75222.4i	0.798124	+	0.798124i	−0.184675	+	0.982800i	$0.559123\pi$
$308$	−98764.0			−1.04111
$309$		−	67920.5i		−	0.711351i
$310$	160401.	+	160401.i	1.66910	+	1.66910i
$311$			150426.i			1.55525i	0.628726	+	0.777627i	$0.283577\pi$
$311$			150426.i			1.55525i	−0.628726	+	0.777627i	$0.716423\pi$
$312$	656.265	+	68524.4i	0.00674171	+	0.703941i
$313$	126604.			1.29229			0.646144	−	0.763216i	$-0.276381\pi$
$313$	126604.			1.29229			0.646144	+	0.763216i	$0.276381\pi$
$314$	119732.	−	119732.i	1.21437	−	1.21437i
$315$	−21444.0			−0.216115
$316$		−	112407.i		−	1.12569i
$317$	−42350.8	+	42350.8i	−0.421447	+	0.421447i	−0.885702	−	0.464255i	$-0.846322\pi$
$317$	−42350.8	+	42350.8i	−0.421447	+	0.421447i	0.464255	+	0.885702i	$0.346322\pi$
$318$	122551.	−	122551.i	1.21189	−	1.21189i
$319$	−45357.6	−	45357.6i	−0.445727	−	0.445727i
$320$	−82384.6	−	82384.6i	−0.804537	−	0.804537i
$321$	74071.4			0.718853
$322$			137504.i			1.32618i
$323$	86641.7	+	86641.7i	0.830466	+	0.830466i
$324$		−	20260.6i		−	0.193002i
$325$	−33089.5	+	33729.4i	−0.313273	+	0.319332i
$326$	−155362.			−1.46187
$327$	25391.9	−	25391.9i	0.237465	−	0.237465i
$328$	124297.			1.15535
$329$			3974.48i			0.0367188i
$330$	−37578.2	+	37578.2i	−0.345070	+	0.345070i
$331$	23593.9	−	23593.9i	0.215349	−	0.215349i	−0.591186	−	0.806535i	$-0.701340\pi$
$331$	23593.9	−	23593.9i	0.215349	−	0.215349i	0.806535	+	0.591186i	$0.201340\pi$
$332$	−57275.0	−	57275.0i	−0.519624	−	0.519624i
$333$	28436.8	+	28436.8i	0.256444	+	0.256444i
$334$	−354641.			−3.17904
$335$			24697.3i			0.220069i
$336$	−11263.3	−	11263.3i	−0.0997674	−	0.0997674i
$337$		−	114863.i		−	1.01139i	−0.862711	−	0.505697i	$-0.831235\pi$
$337$		−	114863.i		−	1.01139i	0.862711	−	0.505697i	$-0.168765\pi$
$338$	136182.	+	131062.i	1.19202	+	1.14721i
$339$	17840.7			0.155244
$340$	94123.4	−	94123.4i	0.814216	−	0.814216i
$341$	−153375.			−1.31900
$342$		−	84954.2i		−	0.726328i
$343$	−89921.4	+	89921.4i	−0.764319	+	0.764319i
$344$	−106517.	+	106517.i	−0.900122	+	0.900122i
$345$	33203.1	+	33203.1i	0.278959	+	0.278959i
$346$	8015.22	+	8015.22i	0.0669520	+	0.0669520i
$347$	93409.6			0.775769			0.387885	−	0.921708i	$-0.373206\pi$
$347$	93409.6			0.775769			0.387885	+	0.921708i	$0.373206\pi$
$348$		−	111396.i		−	0.919839i
$349$	−14479.3	−	14479.3i	−0.118877	−	0.118877i	0.645166	−	0.764043i	$-0.276788\pi$
$349$	−14479.3	−	14479.3i	−0.118877	−	0.118877i	−0.764043	+	0.645166i	$0.776788\pi$
$350$		−	79066.0i		−	0.645437i
$351$	−16925.3	−	16604.2i	−0.137380	−	0.134773i
$352$	64353.3			0.519380
$353$	127078.	−	127078.i	1.01981	−	1.01981i	0.0200140	−	0.999800i	$-0.493629\pi$
$353$	127078.	−	127078.i	1.01981	−	1.01981i	0.999800	−	0.0200140i	$-0.00637109\pi$
$354$	10031.2			0.0800469
$355$			80147.6i			0.635966i
$356$	167648.	−	167648.i	1.32281	−	1.32281i
$357$	−40463.2	+	40463.2i	−0.317485	+	0.317485i
$358$	−12636.2	−	12636.2i	−0.0985939	−	0.0985939i
$359$	40565.6	+	40565.6i	0.314752	+	0.314752i	0.846747	−	0.531995i	$-0.178558\pi$
$359$	40565.6	+	40565.6i	0.314752	+	0.314752i	−0.531995	+	0.846747i	$0.678558\pi$
$360$	−39158.8			−0.302151
$361$		−	95750.0i		−	0.734724i
$362$	146583.	+	146583.i	1.11858	+	1.11858i
$363$			40144.7i			0.304659i
$364$	−200708.	+	1922.20i	−1.51482	+	0.0145076i
$365$	−22082.8			−0.165755
$366$	−170173.	+	170173.i	−1.27037	+	1.27037i
$367$	−40747.7			−0.302532			−0.151266	−	0.988493i	$-0.548335\pi$
$367$	−40747.7			−0.302532			−0.151266	+	0.988493i	$0.548335\pi$
$368$			34879.4i			0.257557i
$369$	−30409.8	+	30409.8i	−0.223337	+	0.223337i
$370$	129534.	−	129534.i	0.946198	−	0.946198i
$371$	152303.	+	152303.i	1.10652	+	1.10652i
$372$	−188341.	−	188341.i	−1.36100	−	1.36100i
$373$	−44353.1			−0.318791			−0.159396	−	0.987215i	$-0.550955\pi$
$373$	−44353.1			−0.318791			−0.159396	+	0.987215i	$0.550955\pi$
$374$			141814.i			1.01386i
$375$	−61771.3	−	61771.3i	−0.439262	−	0.439262i
$376$			7257.77i			0.0513367i
$377$	−93058.4	−	91292.8i	−0.654746	−	0.642324i
$378$	39675.1			0.277673
$379$	−27066.7	+	27066.7i	−0.188433	+	0.188433i	−0.795018	−	0.606585i	$-0.792539\pi$
$379$	−27066.7	+	27066.7i	−0.188433	+	0.188433i	0.606585	+	0.795018i	$0.292539\pi$
$380$	−245593.			−1.70078
$381$			114046.i			0.785652i
$382$	246966.	−	246966.i	1.69243	−	1.69243i
$383$	−166419.	+	166419.i	−1.13451	+	1.13451i	−0.145087	+	0.989419i	$0.546346\pi$
$383$	−166419.	+	166419.i	−1.13451	+	1.13451i	−0.989419	+	0.145087i	$0.953654\pi$
$384$	106931.	+	106931.i	0.725174	+	0.725174i
$385$	−46701.2	−	46701.2i	−0.315070	−	0.315070i
$386$	13156.3			0.0882997
$387$		−	52119.5i		−	0.347999i
$388$	−154945.	−	154945.i	−1.02923	−	1.02923i
$389$			16474.9i			0.108874i	0.998517	+	0.0544371i	$0.0173364\pi$
$389$			16474.9i			0.108874i	−0.998517	+	0.0544371i	$0.982664\pi$
$390$	−75634.9	+	77097.6i	−0.497271	+	0.506888i
$391$	125303.			0.819613
$392$	−31717.8	+	31717.8i	−0.206410	+	0.206410i
$393$	8616.00			0.0557854
$394$		−	95226.6i		−	0.613431i
$395$	53152.4	−	53152.4i	0.340666	−	0.340666i
$396$	44123.9	−	44123.9i	0.281374	−	0.281374i
$397$	−155869.	−	155869.i	−0.988957	−	0.988957i	0.0109822	−	0.999940i	$-0.496504\pi$
$397$	−155869.	−	155869.i	−0.988957	−	0.988957i	−0.999940	+	0.0109822i	$0.996504\pi$
$398$	−34812.1	−	34812.1i	−0.219768	−	0.219768i
$399$	105579.			0.663181
$400$		−	20056.0i		−	0.125350i
$401$	−142185.	−	142185.i	−0.884230	−	0.884230i	0.109732	−	0.993961i	$-0.465001\pi$
$401$	−142185.	−	142185.i	−0.884230	−	0.884230i	−0.993961	+	0.109732i	$0.965001\pi$
$402$		−	45694.2i		−	0.282754i
$403$	−311689.	+	2985.07i	−1.91916	+	0.0183800i
$404$	405531.			2.48463
$405$	9580.35	−	9580.35i	0.0584079	−	0.0584079i
$406$	218141.			1.32338
$407$			123861.i			0.747730i
$408$	−73889.5	+	73889.5i	−0.443877	+	0.443877i
$409$	9949.25	−	9949.25i	0.0594763	−	0.0594763i	−0.676743	−	0.736219i	$-0.736609\pi$
$409$	9949.25	−	9949.25i	0.0594763	−	0.0594763i	0.736219	+	0.676743i	$0.236609\pi$
$410$	138521.	+	138521.i	0.824042	+	0.824042i
$411$	112259.	+	112259.i	0.664567	+	0.664567i
$412$	363282.			2.14017
$413$			12466.5i			0.0730876i
$414$	−61431.4	−	61431.4i	−0.358418	−	0.358418i
$415$		−	54165.8i		−	0.314506i
$416$	130779.	−	1252.48i	0.755701	−	0.00723742i
$417$	24204.6			0.139195
$418$	185015.	−	185015.i	1.05890	−	1.05890i
$419$	−17793.2			−0.101351			−0.0506753	−	0.998715i	$-0.516137\pi$
$419$	−17793.2			−0.101351			−0.0506753	+	0.998715i	$0.516137\pi$
$420$		−	114696.i		−	0.650204i
$421$	−123242.	+	123242.i	−0.695334	+	0.695334i	−0.963400	−	0.268066i	$-0.913615\pi$
$421$	−123242.	+	123242.i	−0.695334	+	0.695334i	0.268066	+	0.963400i	$0.413615\pi$
$422$	−214671.	+	214671.i	−1.20545	+	1.20545i
$423$	−1775.64	−	1775.64i	−0.00992373	−	0.00992373i
$424$	278120.	+	278120.i	1.54704	+	1.54704i
$425$	−72050.5			−0.398896
$426$		−	148287.i		−	0.817114i
$427$	−211487.	−	211487.i	−1.15992	−	1.15992i
$428$			396180.i			2.16274i
$429$	−699.332	−	73021.3i	−0.00379987	−	0.396767i
$430$	−237413.			−1.28401
$431$	−81585.7	+	81585.7i	−0.439197	+	0.439197i	−0.891742	−	0.452545i	$-0.850516\pi$
$431$	−81585.7	+	81585.7i	−0.439197	+	0.439197i	0.452545	+	0.891742i	$0.350516\pi$
$432$	10064.0			0.0539268
$433$		−	60799.1i		−	0.324281i	−0.986768	−	0.162141i	$-0.948160\pi$
$433$		−	60799.1i		−	0.324281i	0.986768	−	0.162141i	$-0.0518398\pi$
$434$	368817.	−	368817.i	1.95809	−	1.95809i
$435$	52674.5	−	52674.5i	0.278369	−	0.278369i
$436$	135812.	+	135812.i	0.714439	+	0.714439i
$437$	−163475.	−	163475.i	−0.856027	−	0.856027i
$438$	40856.9			0.212969
$439$			275044.i			1.42716i	0.700572	+	0.713581i	$0.252928\pi$
$439$			275044.i			1.42716i	−0.700572	+	0.713581i	$0.747072\pi$
$440$	−85280.9	−	85280.9i	−0.440500	−	0.440500i
$441$		−	15519.8i		−	0.0798010i
$442$	2760.06	+	288194.i	0.0141278	+	1.47516i
$443$	−8716.50			−0.0444155			−0.0222077	−	0.999753i	$-0.507070\pi$
$443$	−8716.50			−0.0444155			−0.0222077	+	0.999753i	$0.507070\pi$
$444$	−152098.	+	152098.i	−0.771539	+	0.771539i
$445$	158547.			0.800642
$446$		−	245078.i		−	1.23207i
$447$	−2046.71	+	2046.71i	−0.0102433	+	0.0102433i
$448$	−189431.	+	189431.i	−0.943832	+	0.943832i
$449$	246638.	+	246638.i	1.22340	+	1.22340i	0.966416	+	0.256982i	$0.0827280\pi$
$449$	246638.	+	246638.i	1.22340	+	1.22340i	0.256982	+	0.966416i	$0.417272\pi$
$450$	35323.6	+	35323.6i	0.174438	+	0.174438i
$451$	−132454.			−0.651197
$452$			95423.5i			0.467066i
$453$	39599.3	+	39599.3i	0.192971	+	0.192971i
$454$			191086.i			0.927078i
$455$	−95815.0	−	93997.2i	−0.462819	−	0.454038i
$456$	192797.			0.927195
$457$	55246.3	−	55246.3i	0.264527	−	0.264527i	−0.562363	−	0.826890i	$-0.690108\pi$
$457$	55246.3	−	55246.3i	0.264527	−	0.264527i	0.826890	+	0.562363i	$0.190108\pi$
$458$	−487672.			−2.32486
$459$		−	36154.7i		−	0.171609i
$460$	−177591.	+	177591.i	−0.839277	+	0.839277i
$461$	−31473.0	+	31473.0i	−0.148093	+	0.148093i	−0.777266	−	0.629172i	$-0.783394\pi$
$461$	−31473.0	+	31473.0i	−0.148093	+	0.148093i	0.629172	+	0.777266i	$0.283394\pi$
$462$	86405.2	+	86405.2i	0.404815	+	0.404815i
$463$	134703.	+	134703.i	0.628371	+	0.628371i	0.947658	−	0.319287i	$-0.103443\pi$
$463$	134703.	+	134703.i	0.628371	+	0.628371i	−0.319287	+	0.947658i	$0.603443\pi$
$464$	55333.9			0.257013
$465$		−	178117.i		−	0.823757i
$466$	−214231.	−	214231.i	−0.986529	−	0.986529i
$467$			106801.i			0.489714i	0.969559	+	0.244857i	$0.0787410\pi$
$467$			106801.i			0.489714i	−0.969559	+	0.244857i	$0.921259\pi$
$468$	88809.7	−	90527.2i	0.405479	−	0.413321i
$469$	56787.6			0.258171
$470$	−8088.34	+	8088.34i	−0.0366154	+	0.0366154i
$471$	−132956.			−0.599329
$472$			22765.0i			0.102184i
$473$	113507.	−	113507.i	0.507342	−	0.507342i
$474$	−98341.0	+	98341.0i	−0.437701	+	0.437701i
$475$	93999.4	+	93999.4i	0.416618	+	0.416618i
$476$	−216422.	−	216422.i	−0.955186	−	0.955186i
$477$	−136086.			−0.598105
$478$			514301.i			2.25093i
$479$	69279.4	+	69279.4i	0.301949	+	0.301949i	0.841776	−	0.539827i	$-0.181510\pi$
$479$	69279.4	+	69279.4i	0.301949	+	0.301949i	−0.539827	+	0.841776i	$0.681510\pi$
$480$			74734.4i			0.324368i
$481$	2410.64	+	251709.i	0.0104194	+	1.08795i
$482$	219685.			0.945597
$483$	76345.4	−	76345.4i	0.327257	−	0.327257i
$484$	−214719.			−0.916599
$485$		−	146533.i		−	0.622950i
$486$	−17725.3	+	17725.3i	−0.0750448	+	0.0750448i
$487$	334191.	−	334191.i	1.40908	−	1.40908i	0.644372	−	0.764713i	$-0.277119\pi$
$487$	334191.	−	334191.i	1.40908	−	1.40908i	0.764713	−	0.644372i	$-0.222881\pi$
$488$	−386195.	−	386195.i	−1.62169	−	1.62169i
$489$	86260.5	+	86260.5i	0.360740	+	0.360740i
$490$	−70695.1			−0.294440
$491$		−	204363.i		−	0.847696i	−0.905733	−	0.423848i	$-0.860679\pi$
$491$		−	204363.i		−	0.847696i	0.905733	−	0.423848i	$-0.139321\pi$
$492$	−162650.	−	162650.i	−0.671932	−	0.671932i
$493$		−	198785.i		−	0.817880i
$494$	372386.	−	379588.i	1.52595	−	1.55546i
$495$	41728.6			0.170303
$496$	93554.7	−	93554.7i	0.380279	−	0.380279i
$497$	184287.			0.746074
$498$			100216.i			0.404090i
$499$	263314.	−	263314.i	1.05748	−	1.05748i	0.0592381	−	0.998244i	$-0.481133\pi$
$499$	263314.	−	263314.i	1.05748	−	1.05748i	0.998244	−	0.0592381i	$-0.0188671\pi$
$500$	330391.	−	330391.i	1.32157	−	1.32157i
$501$	196905.	+	196905.i	0.784479	+	0.784479i
$502$	−241773.	−	241773.i	−0.959403	−	0.959403i
$503$	120971.			0.478131			0.239065	−	0.971003i	$-0.423159\pi$
$503$	120971.			0.478131			0.239065	+	0.971003i	$0.423159\pi$
$504$			90039.6i			0.354464i
$505$	191758.	+	191758.i	0.751920	+	0.751920i
$506$		−	267573.i		−	1.04506i
$507$	−2842.36	−	148380.i	−0.0110577	−	0.577244i
$508$	−609990.			−2.36371
$509$	−50199.3	+	50199.3i	−0.193759	+	0.193759i	−0.797318	−	0.603559i	$-0.793749\pi$
$509$	−50199.3	+	50199.3i	−0.193759	+	0.193759i	0.603559	+	0.797318i	$0.293749\pi$
$510$	−164691.			−0.633182
$511$			50775.9i			0.194454i
$512$	−102587.	+	102587.i	−0.391337	+	0.391337i
$513$	−47168.6	+	47168.6i	−0.179233	+	0.179233i
$514$	−411688.	−	411688.i	−1.55827	−	1.55827i
$515$	171780.	+	171780.i	0.647677	+	0.647677i
$516$	278768.			1.04699
$517$		−	7734.07i		−	0.0289352i
$518$	−297845.	−	297845.i	−1.11002	−	1.11002i
$519$		−	8900.49i		−	0.0330430i
$520$	−174967.	−	171648.i	−0.647068	−	0.634792i
$521$	−307242.			−1.13189			−0.565946	−	0.824443i	$-0.691489\pi$
$521$	−307242.			−1.13189			−0.565946	+	0.824443i	$0.691489\pi$
$522$	−97456.7	+	97456.7i	−0.357660	+	0.357660i
$523$	271249.			0.991664			0.495832	−	0.868418i	$-0.334863\pi$
$523$	271249.			0.991664			0.495832	+	0.868418i	$0.334863\pi$
$524$			46083.7i			0.167836i
$525$	−43899.3	+	43899.3i	−0.159272	+	0.159272i
$526$	531501.	−	531501.i	1.92102	−	1.92102i
$527$	−336092.	−	336092.i	−1.21014	−	1.21014i
$528$	21917.7	+	21917.7i	0.0786189	+	0.0786189i
$529$	43420.3			0.155161
$530$			619895.i			2.20682i
$531$	−5569.54	−	5569.54i	−0.0197529	−	0.0197529i
$532$			564703.i			1.99525i
$533$	−269173.	+	2577.89i	−0.947494	+	0.00907424i
$534$	−293339.			−1.02870
$535$	−187337.	+	187337.i	−0.654508	+	0.654508i
$536$	103699.			0.360950
$537$			14031.8i			0.0486593i
$538$	26428.6	−	26428.6i	0.0913081	−	0.0913081i
$539$	33799.3	−	33799.3i	0.116340	−	0.116340i
$540$	51241.7	+	51241.7i	0.175726	+	0.175726i
$541$	184394.	+	184394.i	0.630017	+	0.630017i	0.948072	−	0.318055i	$-0.103030\pi$
$541$	184394.	+	184394.i	0.630017	+	0.630017i	−0.318055	+	0.948072i	$0.603030\pi$
$542$	−34836.6			−0.118587
$543$		−	162773.i		−	0.552054i
$544$	141018.	+	141018.i	0.476515	+	0.476515i
$545$			128439.i			0.432419i
$546$	177274.	+	173911.i	0.594648	+	0.583366i
$547$	373290.			1.24759			0.623795	−	0.781588i	$-0.285590\pi$
$547$	373290.			1.24759			0.623795	+	0.781588i	$0.285590\pi$
$548$	−600433.	+	600433.i	−1.99942	+	1.99942i
$549$	188968.			0.626967
$550$			153857.i			0.508619i
$551$	−259341.	+	259341.i	−0.854217	+	0.854217i
$552$	139414.	−	139414.i	0.457539	−	0.457539i
$553$	−122216.	−	122216.i	−0.399647	−	0.399647i
$554$	148209.	+	148209.i	0.482898	+	0.482898i
$555$	−143841.			−0.466979
$556$			129461.i			0.418784i
$557$	200107.	+	200107.i	0.644987	+	0.644987i	0.951777	−	0.306790i	$-0.0992550\pi$
$557$	200107.	+	200107.i	0.644987	+	0.644987i	−0.306790	+	0.951777i	$0.599255\pi$
$558$			329546.i			1.05840i
$559$	228460.	−	232878.i	0.731115	−	0.745254i
$560$	56973.0			0.181674
$561$	78738.5	−	78738.5i	0.250185	−	0.250185i
$562$	−101983.			−0.322891
$563$			488676.i			1.54172i	0.637007	+	0.770858i	$0.280172\pi$
$563$			488676.i			1.54172i	−0.637007	+	0.770858i	$0.719828\pi$
$564$	9497.25	−	9497.25i	0.0298566	−	0.0298566i
$565$	−45121.7	+	45121.7i	−0.141348	+	0.141348i
$566$	145033.	+	145033.i	0.452725	+	0.452725i
$567$	−22028.6	−	22028.6i	−0.0685204	−	0.0685204i
$568$	336525.			1.04309
$569$		−	562947.i		−	1.73877i	−0.494132	−	0.869387i	$-0.664514\pi$
$569$		−	562947.i		−	1.73877i	0.494132	−	0.869387i	$-0.335486\pi$
$570$	214861.	+	214861.i	0.661314	+	0.661314i
$571$		−	183359.i		−	0.562380i	−0.959652	−	0.281190i	$-0.909271\pi$
$571$		−	183359.i		−	0.562380i	0.959652	−	0.281190i	$-0.0907291\pi$
$572$	390564.	−	3740.47i	1.19371	−	0.0114323i
$573$	−274242.			−0.835267
$574$	318509.	−	318509.i	0.966713	−	0.966713i
$575$	135944.			0.411173
$576$		−	169261.i		−	0.510165i
$577$	196407.	−	196407.i	0.589938	−	0.589938i	−0.347677	−	0.937614i	$-0.613029\pi$
$577$	196407.	−	196407.i	0.589938	−	0.589938i	0.937614	+	0.347677i	$0.113029\pi$
$578$	80064.2	−	80064.2i	0.239653	−	0.239653i
$579$	−7304.69	−	7304.69i	−0.0217894	−	0.0217894i
$580$	281736.	+	281736.i	0.837503	+	0.837503i
$581$	−124546.			−0.368958
$582$			271112.i			0.800391i
$583$	−296372.	−	296372.i	−0.871966	−	0.871966i
$584$			92721.6i			0.271866i
$585$	84800.7	−	812.145i	0.247792	−	0.00237313i
$586$	−1.06254e6			−3.09422
$587$	−127057.	+	127057.i	−0.368742	+	0.368742i	−0.867018	−	0.498276i	$-0.833966\pi$
$587$	−127057.	+	127057.i	−0.368742	+	0.368742i	0.498276	+	0.867018i	$0.333966\pi$
$588$	83009.5			0.240089
$589$			876953.i			2.52782i
$590$	−25370.2	+	25370.2i	−0.0728818	+	0.0728818i
$591$	−52872.1	+	52872.1i	−0.151374	+	0.151374i
$592$	−75551.7	−	75551.7i	−0.215576	−	0.215576i
$593$	168617.	+	168617.i	0.479503	+	0.479503i	0.904973	−	0.425470i	$-0.139891\pi$
$593$	168617.	+	168617.i	0.479503	+	0.479503i	−0.425470	+	0.904973i	$0.639891\pi$
$594$	−77205.0			−0.218813
$595$		−	204674.i		−	0.578133i
$596$	−10947.1	−	10947.1i	−0.0308181	−	0.0308181i
$597$			38657.0i			0.108462i
$598$	−5207.65	−	543761.i	−0.0145626	−	1.52057i
$599$	−22624.2			−0.0630550			−0.0315275	−	0.999503i	$-0.510037\pi$
$599$	−22624.2			−0.0630550			−0.0315275	+	0.999503i	$0.510037\pi$
$600$	−80164.3	+	80164.3i	−0.222679	+	0.222679i
$601$	621958.			1.72192			0.860958	−	0.508675i	$-0.169865\pi$
$601$	621958.			1.72192			0.860958	+	0.508675i	$0.169865\pi$
$602$			545895.i			1.50632i
$603$	−25370.5	+	25370.5i	−0.0697741	+	0.0697741i
$604$	−211802.	+	211802.i	−0.580572	+	0.580572i
$605$	−101531.	−	101531.i	−0.277389	−	0.277389i
$606$	−354786.	−	354786.i	−0.966097	−	0.966097i
$607$	−600769.			−1.63054			−0.815268	−	0.579084i	$-0.803410\pi$
$607$	−600769.			−1.63054			−0.815268	+	0.579084i	$0.803410\pi$
$608$		−	367953.i		−	0.995371i
$609$	−121117.	−	121117.i	−0.326565	−	0.326565i
$610$		−	860782.i		−	2.31331i
$611$	−150.525	−	15717.1i	−0.000403204	−	0.0421009i
$612$	193378.			0.516303
$613$	−230276.	+	230276.i	−0.612812	+	0.612812i	−0.943678	−	0.330866i	$-0.892659\pi$
$613$	−230276.	+	230276.i	−0.612812	+	0.612812i	0.330866	+	0.943678i	$0.392659\pi$
$614$	703981.			1.86734
$615$		−	153821.i		−	0.406691i
$616$	−196090.	+	196090.i	−0.516767	+	0.516767i
$617$	34766.2	−	34766.2i	0.0913244	−	0.0913244i	−0.659969	−	0.751293i	$-0.729430\pi$
$617$	34766.2	−	34766.2i	0.0913244	−	0.0913244i	0.751293	+	0.659969i	$0.229430\pi$
$618$	−317823.	−	317823.i	−0.832162	−	0.832162i
$619$	−420362.	−	420362.i	−1.09709	−	1.09709i	−0.994750	−	0.102339i	$-0.967367\pi$
$619$	−420362.	−	420362.i	−1.09709	−	1.09709i	−0.102339	−	0.994750i	$-0.532633\pi$
$620$	952680.			2.47836
$621$			68216.3i			0.176891i
$622$	703892.	+	703892.i	1.81939	+	1.81939i
$623$		−	364555.i		−	0.939262i
$624$	44967.6	+	44114.5i	0.115486	+	0.113295i
$625$	137714.			0.352547
$626$	592423.	−	592423.i	1.51176	−	1.51176i
$627$	−205450.			−0.522601
$628$		−	711130.i		−	1.80314i
$629$	−271417.	+	271417.i	−0.686018	+	0.686018i
$630$	−100344.	+	100344.i	−0.252819	+	0.252819i
$631$	93922.3	+	93922.3i	0.235890	+	0.235890i	0.815146	−	0.579256i	$-0.196657\pi$
$631$	93922.3	+	93922.3i	0.235890	+	0.235890i	−0.579256	+	0.815146i	$0.696657\pi$
$632$	−223177.	−	223177.i	−0.558748	−	0.558748i
$633$	238380.			0.594926
$634$			396347.i			0.986045i
$635$	−288438.	−	288438.i	−0.715327	−	0.715327i
$636$		−	727875.i		−	1.79946i
$637$	68029.0	−	69344.6i	0.167654	−	0.170897i
$638$	−424487.			−1.04285
$639$	−82332.3	+	82332.3i	−0.201636	+	0.201636i
$640$	−540888.			−1.32053
$641$		−	123299.i		−	0.300084i	−0.988680	−	0.150042i	$-0.952059\pi$
$641$		−	123299.i		−	0.300084i	0.988680	−	0.150042i	$-0.0479409\pi$
$642$	346605.	−	346605.i	0.840938	−	0.840938i
$643$	−280732.	+	280732.i	−0.678999	+	0.678999i	−0.959774	−	0.280774i	$-0.909409\pi$
$643$	−280732.	+	280732.i	−0.678999	+	0.678999i	0.280774	+	0.959774i	$0.409409\pi$
$644$	408343.	+	408343.i	0.984586	+	0.984586i
$645$	131817.	+	131817.i	0.316850	+	0.316850i
$646$	810850.			1.94301
$647$		−	45435.6i		−	0.108539i	−0.998526	−	0.0542697i	$-0.982717\pi$
$647$		−	45435.6i		−	0.108539i	0.998526	−	0.0542697i	$-0.0172831\pi$
$648$	−40226.2	−	40226.2i	−0.0957986	−	0.0957986i
$649$		−	24258.9i		−	0.0575946i
$650$	2994.45	+	312668.i	0.00708746	+	0.740042i
$651$	−409552.			−0.966379
$652$	−461375.	+	461375.i	−1.08532	+	1.08532i
$653$	−378723.			−0.888168			−0.444084	−	0.895985i	$-0.646471\pi$
$653$	−378723.			−0.888168			−0.444084	+	0.895985i	$0.646471\pi$
$654$		−	237635.i		−	0.555590i
$655$	−21791.0	+	21791.0i	−0.0507920	+	0.0507920i
$656$	80793.4	−	80793.4i	0.187745	−	0.187745i
$657$	−22684.7	−	22684.7i	−0.0525536	−	0.0525536i
$658$	18597.9	+	18597.9i	0.0429548	+	0.0429548i
$659$	289116.			0.665735			0.332867	−	0.942974i	$-0.391984\pi$
$659$	289116.			0.665735			0.332867	+	0.942974i	$0.391984\pi$
$660$			223191.i			0.512375i
$661$	−168774.	−	168774.i	−0.386281	−	0.386281i	0.487078	−	0.873359i	$-0.338063\pi$
$661$	−168774.	−	168774.i	−0.386281	−	0.386281i	−0.873359	+	0.487078i	$0.838063\pi$
$662$		−	220807.i		−	0.503845i
$663$	158480.	−	161545.i	0.360534	−	0.367507i
$664$	−227433.			−0.515842
$665$	−267024.	+	267024.i	−0.603819	+	0.603819i
$666$	266131.			0.599994
$667$			375065.i			0.843054i
$668$	−1.05317e6	+	1.05317e6i	−2.36019	+	2.36019i
$669$	−136073.	+	136073.i	−0.304033	+	0.304033i
$670$	115567.	+	115567.i	0.257444	+	0.257444i
$671$	411539.	+	411539.i	0.914043	+	0.914043i
$672$	171840.			0.380528
$673$			163719.i			0.361467i	0.983532	+	0.180734i	$0.0578472\pi$
$673$			163719.i			0.361467i	−0.983532	+	0.180734i	$0.942153\pi$
$674$	−537482.	−	537482.i	−1.18316	−	1.18316i
$675$		−	39225.0i		−	0.0860906i
$676$	793630.	−	15202.7i	1.73670	−	0.0332681i
$677$	462025.			1.00806			0.504032	−	0.863685i	$-0.331849\pi$
$677$	462025.			1.00806			0.504032	+	0.863685i	$0.331849\pi$
$678$	83482.8	−	83482.8i	0.181609	−	0.181609i
$679$	−336931.			−0.730805
$680$		−	373753.i		−	0.808290i
$681$	106095.	−	106095.i	0.228771	−	0.228771i
$682$	−717693.	+	717693.i	−1.54301	+	1.54301i
$683$	−213724.	−	213724.i	−0.458155	−	0.458155i	0.439895	−	0.898049i	$-0.355016\pi$
$683$	−213724.	−	213724.i	−0.458155	−	0.458155i	−0.898049	+	0.439895i	$0.855016\pi$
$684$	−252287.	−	252287.i	−0.539241	−	0.539241i
$685$	−567838.			−1.21016
$686$			841544.i			1.78825i
$687$	270767.	+	270767.i	0.573697	+	0.573697i
$688$			138473.i			0.292541i
$689$	−608054.	−	596517.i	−1.28087	−	1.25656i
$690$	310737.			0.652671
$691$	−344817.	+	344817.i	−0.722159	+	0.722159i	−0.969045	−	0.246886i	$-0.920593\pi$
$691$	−344817.	+	344817.i	−0.722159	+	0.722159i	0.246886	+	0.969045i	$0.420593\pi$
$692$	47605.4			0.0994132
$693$		−	95948.5i		−	0.199789i
$694$	437094.	−	437094.i	0.907520	−	0.907520i
$695$	−61216.6	+	61216.6i	−0.126736	+	0.126736i
$696$	−221171.	−	221171.i	−0.456572	−	0.456572i
$697$	−290248.	−	290248.i	−0.597452	−	0.597452i
$698$	−135507.			−0.278132
$699$			237892.i			0.486884i
$700$	−234801.	−	234801.i	−0.479186	−	0.479186i
$701$		−	83587.3i		−	0.170100i	−0.996377	−	0.0850500i	$-0.972895\pi$
$701$		−	83587.3i		−	0.170100i	0.996377	−	0.0850500i	$-0.0271050\pi$
$702$	−156896.	+	1502.61i	−0.318374	+	0.00304909i
$703$	708198.			1.43299
$704$	368619.	−	368619.i	0.743760	−	0.743760i
$705$	8981.68			0.0180709
$706$		−	1.18928e6i		−	2.38602i
$707$	440919.	−	440919.i	0.882104	−	0.882104i
$708$	29789.4	−	29789.4i	0.0594285	−	0.0594285i
$709$	507737.	+	507737.i	1.01006	+	1.01006i	0.999949	+	0.0101092i	$0.00321793\pi$
$709$	507737.	+	507737.i	1.01006	+	1.01006i	0.0101092	+	0.999949i	$0.496782\pi$
$710$	375037.	+	375037.i	0.743973	+	0.743973i
$711$	109203.			0.216020
$712$		−	665711.i		−	1.31318i
$713$	634135.	+	634135.i	1.24739	+	1.24739i
$714$			378681.i			0.742809i
$715$	186450.	+	182912.i	0.364711	+	0.357792i
$716$	−75051.0			−0.146396
$717$	285552.	−	285552.i	0.555453	−	0.555453i
$718$	379640.			0.736415
$719$			50596.5i			0.0978729i	0.998802	+	0.0489364i	$0.0155832\pi$
$719$			50596.5i			0.0978729i	−0.998802	+	0.0489364i	$0.984417\pi$
$720$	−25453.3	+	25453.3i	−0.0490998	+	0.0490998i
$721$	394982.	−	394982.i	0.759814	−	0.759814i
$722$	−448046.	−	448046.i	−0.859505	−	0.859505i
$723$	−121974.	−	121974.i	−0.233341	−	0.233341i
$724$	870610.			1.66091
$725$		−	215666.i		−	0.410304i
$726$	187850.	+	187850.i	0.356400	+	0.356400i
$727$		−	891410.i		−	1.68659i	−0.537453	−	0.843294i	$-0.680613\pi$
$727$		−	891410.i		−	1.68659i	0.537453	−	0.843294i	$-0.319387\pi$
$728$	−394678.	+	402310.i	−0.744697	+	0.759099i
$729$	19683.0			0.0370370
$730$	−103333.	+	103333.i	−0.193906	+	0.193906i
$731$	497458.			0.930939
$732$			1.01072e6i			1.88629i
$733$	−439182.	+	439182.i	−0.817404	+	0.817404i	−0.985731	−	0.168327i	$-0.946164\pi$
$733$	−439182.	+	439182.i	−0.817404	+	0.817404i	0.168327	+	0.985731i	$0.446164\pi$
$734$	−190672.	+	190672.i	−0.353912	+	0.353912i
$735$	39251.6	+	39251.6i	0.0726579	+	0.0726579i
$736$	−266071.	−	266071.i	−0.491182	−	0.491182i
$737$	−110505.			−0.203445
$738$			284595.i			0.522533i
$739$	83855.7	+	83855.7i	0.153548	+	0.153548i	0.779701	−	0.626153i	$-0.215371\pi$
$739$	83855.7	+	83855.7i	0.153548	+	0.153548i	−0.626153	+	0.779701i	$0.715371\pi$
$740$		−	769353.i		−	1.40495i
$741$	−417514.	+	3998.57i	−0.760387	+	0.00728230i
$742$	1.42535e6			2.58890
$743$	−153273.	+	153273.i	−0.277643	+	0.277643i	−0.832167	−	0.554524i	$-0.812900\pi$
$743$	−153273.	+	153273.i	−0.277643	+	0.277643i	0.554524	+	0.832167i	$0.312900\pi$
$744$	−747881.			−1.35110
$745$		−	10352.8i		−	0.0186529i
$746$	−207543.	+	207543.i	−0.372933	+	0.372933i
$747$	55642.3	−	55642.3i	0.0997157	−	0.0997157i
$748$	421143.	+	421143.i	0.752708	+	0.752708i
$749$	430752.	+	430752.i	0.767827	+	0.767827i
$750$	−578096.			−1.02773
$751$		−	208551.i		−	0.369771i	−0.982760	−	0.184886i	$-0.940808\pi$
$751$		−	208551.i		−	0.369771i	0.982760	−	0.184886i	$-0.0591915\pi$
$752$	4717.57	+	4717.57i	0.00834225	+	0.00834225i
$753$			268477.i			0.473497i
$754$	−862640.	+	8261.59i	−1.51735	+	0.0145319i
$755$	−200304.			−0.351395
$756$	117822.	−	117822.i	0.206151	−	0.206151i
$757$	−762290.			−1.33024			−0.665118	−	0.746739i	$-0.731619\pi$
$757$	−762290.			−1.33024			−0.665118	+	0.746739i	$0.731619\pi$
$758$			253308.i			0.440870i
$759$	−148563.	+	148563.i	−0.257886	+	0.257886i
$760$	−487610.	+	487610.i	−0.844201	+	0.844201i
$761$	790313.	+	790313.i	1.36468	+	1.36468i	0.867850	+	0.496826i	$0.165501\pi$
$761$	790313.	+	790313.i	1.36468	+	1.36468i	0.496826	+	0.867850i	$0.334499\pi$
$762$	533659.	+	533659.i	0.919082	+	0.919082i
$763$	295326.			0.507287
$764$		−	1.46682e6i		−	2.51299i
$765$	91440.2	+	91440.2i	0.156248	+	0.156248i
$766$			1.55746e6i			2.65436i
$767$	−472.140	−	49298.9i	−0.000802565	−	0.0838005i
$768$	479546.			0.813033
$769$	170652.	−	170652.i	0.288574	−	0.288574i	−0.547942	−	0.836516i	$-0.684589\pi$
$769$	170652.	−	170652.i	0.288574	−	0.288574i	0.836516	+	0.547942i	$0.184589\pi$
$770$	−437061.			−0.737158
$771$			457158.i			0.769056i
$772$	39070.1	−	39070.1i	0.0655555	−	0.0655555i
$773$	238774.	−	238774.i	0.399602	−	0.399602i	−0.478490	−	0.878093i	$-0.658816\pi$
$773$	238774.	−	238774.i	0.399602	−	0.399602i	0.878093	+	0.478490i	$0.158816\pi$
$774$	−243885.	−	243885.i	−0.407101	−	0.407101i
$775$	−364634.	−	364634.i	−0.607090	−	0.607090i
$776$	−615268.			−1.02174
$777$			330741.i			0.547830i
$778$	77091.7	+	77091.7i	0.127365	+	0.127365i
$779$			757333.i			1.24799i
$780$	4343.86	+	453567.i	0.00713981	+	0.745509i
$781$	−358610.			−0.587923
$782$	586335.	−	586335.i	0.958810	−	0.958810i
$783$	108221.			0.176517
$784$			41233.3i			0.0670836i
$785$	336263.	−	336263.i	0.545682	−	0.545682i
$786$	40317.1	−	40317.1i	0.0652596	−	0.0652596i
$787$	−460701.	−	460701.i	−0.743823	−	0.743823i	0.229488	−	0.973311i	$-0.426295\pi$
$787$	−460701.	−	460701.i	−0.743823	−	0.743823i	−0.973311	+	0.229488i	$0.926295\pi$
$788$	−282793.	−	282793.i	−0.455424	−	0.455424i
$789$	−590204.			−0.948087
$790$		−	497435.i		−	0.797044i
$791$	103750.	+	103750.i	0.165820	+	0.165820i
$792$		−	175211.i		−	0.279326i
$793$	844339.	+	828320.i	1.34267	+	1.31720i
$794$	−1.45872e6			−2.31383
$795$	344181.	−	344181.i	0.544568	−	0.544568i
$796$	−206762.			−0.326320
$797$		−	940096.i		−	1.47998i	−0.672618	−	0.739989i	$-0.734830\pi$
$797$		−	940096.i		−	1.47998i	0.672618	−	0.739989i	$-0.265170\pi$
$798$	494039.	−	494039.i	0.775811	−	0.775811i
$799$	16947.7	−	16947.7i	0.0265471	−	0.0265471i
$800$	152993.	+	152993.i	0.239052	+	0.239052i
$801$	162869.	+	162869.i	0.253848	+	0.253848i
$802$	−1.33066e6			−2.06880
$803$		−	98806.5i		−	0.153234i
$804$	−135697.	−	135697.i	−0.209922	−	0.209922i
$805$			386176.i			0.595928i
$806$	−1.44453e6	+	1.47246e6i	−2.22359	+	2.26660i
$807$	−29347.6			−0.0450635
$808$	805159.	−	805159.i	1.23327	−	1.23327i
$809$	612686.			0.936140			0.468070	−	0.883691i	$-0.344950\pi$
$809$	612686.			0.936140			0.468070	+	0.883691i	$0.344950\pi$
$810$		−	89659.3i		−	0.136655i
$811$	431732.	−	431732.i	0.656406	−	0.656406i	−0.298121	−	0.954528i	$-0.596360\pi$
$811$	431732.	−	431732.i	0.656406	−	0.656406i	0.954528	+	0.298121i	$0.0963601\pi$
$812$	647809.	−	647809.i	0.982505	−	0.982505i
$813$	19342.1	+	19342.1i	0.0292633	+	0.0292633i
$814$	579585.	+	579585.i	0.874719	+	0.874719i
$815$	−436329.			−0.656900
$816$			96056.8i			0.144261i
$817$	−648999.	−	648999.i	−0.972300	−	0.972300i
$818$		−	93111.7i		−	0.139155i
$819$	−1867.40	−	194986.i	−0.00278401	−	0.290694i
$820$	822730.			1.22357
$821$	−193284.	+	193284.i	−0.286754	+	0.286754i	−0.835795	−	0.549041i	$-0.814993\pi$
$821$	−193284.	+	193284.i	−0.286754	+	0.286754i	0.549041	+	0.835795i	$0.314993\pi$
$822$	1.05060e6			1.55486
$823$			547967.i			0.809011i	0.914536	+	0.404506i	$0.132556\pi$
$823$			547967.i			0.809011i	−0.914536	+	0.404506i	$0.867444\pi$
$824$	721275.	−	721275.i	1.06230	−	1.06230i
$825$	85425.1	−	85425.1i	0.125510	−	0.125510i
$826$	58334.8	+	58334.8i	0.0855003	+	0.0855003i
$827$	−859953.	−	859953.i	−1.25737	−	1.25737i	−0.952343	−	0.305029i	$-0.901334\pi$
$827$	−859953.	−	859953.i	−1.25737	−	1.25737i	−0.305029	−	0.952343i	$-0.598666\pi$
$828$	−364864.			−0.532194
$829$		−	726172.i		−	1.05665i	−0.849043	−	0.528324i	$-0.822821\pi$
$829$		−	726172.i		−	1.05665i	0.849043	−	0.528324i	$-0.177179\pi$
$830$	−253460.	−	253460.i	−0.367919	−	0.367919i
$831$		−	164578.i		−	0.238326i
$832$	741933.	−	756282.i	1.07181	−	1.09254i
$833$	148129.			0.213477
$834$	113261.	−	113261.i	0.162835	−	0.162835i
$835$	−995999.			−1.42852
$836$		−	1.09887e6i		−	1.57230i
$837$	182972.	−	182972.i	0.261176	−	0.261176i
$838$	−83260.3	+	83260.3i	−0.118563	+	0.118563i
$839$	−507389.	−	507389.i	−0.720804	−	0.720804i	0.247965	−	0.968769i	$-0.420238\pi$
$839$	−507389.	−	507389.i	−0.720804	−	0.720804i	−0.968769	+	0.247965i	$0.920238\pi$
$840$	−227722.	−	227722.i	−0.322736	−	0.322736i
$841$	−112265.			−0.158728
$842$			1.15338e6i			1.62685i
$843$	56623.5	+	56623.5i	0.0796786	+	0.0796786i
$844$			1.27501e6i			1.78990i
$845$	382462.	+	368085.i	0.535642	+	0.515507i
$846$	−16617.6			−0.0232182
$847$	−233456.	+	233456.i	−0.325415	+	0.325415i
$848$	361557.			0.502789
$849$		−	161052.i		−	0.223434i
$850$	−337148.	+	337148.i	−0.466641	+	0.466641i
$851$	512107.	−	512107.i	0.707133	−	0.707133i
$852$	−440365.	−	440365.i	−0.606643	−	0.606643i
$853$	192677.	+	192677.i	0.264808	+	0.264808i	0.827004	−	0.562196i	$-0.190043\pi$
$853$	192677.	+	192677.i	0.264808	+	0.264808i	−0.562196	+	0.827004i	$0.690043\pi$
$854$	−1.97924e6			−2.71383
$855$		−	238592.i		−	0.326380i
$856$	786593.	+	786593.i	1.07350	+	1.07350i
$857$			96833.2i			0.131845i	0.997825	+	0.0659223i	$0.0209990\pi$
$857$			96833.2i			0.131845i	−0.997825	+	0.0659223i	$0.979001\pi$
$858$	−344963.	−	338419.i	−0.468596	−	0.459705i
$859$	248211.			0.336383			0.168192	−	0.985754i	$-0.446207\pi$
$859$	248211.			0.336383			0.168192	+	0.985754i	$0.446207\pi$
$860$	−705042.	+	705042.i	−0.953275	+	0.953275i
$861$	−353687.			−0.477104
$862$			763533.i			1.02757i
$863$	1.02273e6	−	1.02273e6i	1.37321	−	1.37321i	0.517571	−	0.855640i	$-0.326836\pi$
$863$	1.02273e6	−	1.02273e6i	1.37321	−	1.37321i	0.855640	−	0.517571i	$-0.173164\pi$
$864$	−76771.6	+	76771.6i	−0.102843	+	0.102843i
$865$	22510.5	+	22510.5i	0.0300853	+	0.0300853i
$866$	−284499.	−	284499.i	−0.379355	−	0.379355i
$867$	−88907.1			−0.118276
$868$		−	2.19054e6i		−	2.90745i
$869$	237824.	+	237824.i	0.314931	+	0.314931i
$870$		−	492962.i		−	0.651291i
$871$	−224567.	+	2150.70i	−0.296013	+	0.00283494i
$872$	539294.			0.709239
$873$	150528.	−	150528.i	0.197509	−	0.197509i
$874$	−1.52990e6			−2.00282
$875$		−	718444.i		−	0.938376i
$876$	121332.	−	121332.i	0.158113	−	0.158113i
$877$	263404.	−	263404.i	0.342470	−	0.342470i	−0.514825	−	0.857295i	$-0.672143\pi$
$877$	263404.	−	263404.i	0.342470	−	0.342470i	0.857295	+	0.514825i	$0.172143\pi$
$878$	1.28702e6	+	1.28702e6i	1.66954	+	1.66954i
$879$	589949.	+	589949.i	0.763549	+	0.763549i
$880$	−110866.			−0.143163
$881$			580702.i			0.748171i	0.927394	+	0.374086i	$0.122043\pi$
$881$			580702.i			0.748171i	−0.927394	+	0.374086i	$0.877957\pi$
$882$	−72622.2	−	72622.2i	−0.0933538	−	0.0933538i
$883$			1.28367e6i			1.64638i	0.567763	+	0.823192i	$0.307809\pi$
$883$			1.28367e6i			1.64638i	−0.567763	+	0.823192i	$0.692191\pi$
$884$	864042.	+	847649.i	1.10568	+	1.08470i
$885$	28172.2			0.0359695
$886$	−40787.4	+	40787.4i	−0.0519587	+	0.0519587i
$887$	166700.			0.211879			0.105940	−	0.994373i	$-0.466215\pi$
$887$	166700.			0.211879			0.105940	+	0.994373i	$0.466215\pi$
$888$			603964.i			0.765923i
$889$	−663219.	+	663219.i	−0.839176	+	0.839176i
$890$	741894.	−	741894.i	0.936617	−	0.936617i
$891$	42866.1	+	42866.1i	0.0539956	+	0.0539956i
$892$	−727805.	−	727805.i	−0.914714	−	0.914714i
$893$	−44221.1			−0.0554532
$894$			19154.5i			0.0239660i
$895$	−35488.4	−	35488.4i	−0.0443037	−	0.0443037i
$896$			1.24369e6i			1.54916i
$897$	−299018.	+	304801.i	−0.371631	+	0.378818i
$898$	2.30820e6			2.86234
$899$	1.00601e6	−	1.00601e6i	1.24475	−	1.24475i
$900$	209800.			0.259012
$901$		−	1.29888e6i		−	1.60000i
$902$	−619796.	+	619796.i	−0.761791	+	0.761791i
$903$	303094.	−	303094.i	0.371708	−	0.371708i
$904$	189458.	+	189458.i	0.231833	+	0.231833i
$905$	411674.	+	411674.i	0.502639	+	0.502639i
$906$	370596.			0.451486
$907$			1.06525e6i			1.29490i	0.762108	+	0.647450i	$0.224164\pi$
$907$			1.06525e6i			1.29490i	−0.762108	+	0.647450i	$0.775836\pi$
$908$	567464.	+	567464.i	0.688282	+	0.688282i
$909$			393971.i			0.476800i
$910$	−888194.	+	8506.32i	−1.07257	+	0.0102721i
$911$	−794671.			−0.957526			−0.478763	−	0.877944i	$-0.658915\pi$
$911$	−794671.			−0.957526			−0.478763	+	0.877944i	$0.658915\pi$
$912$	125319.	−	125319.i	0.150670	−	0.150670i
$913$	242358.			0.290747
$914$		−	517031.i		−	0.618905i
$915$	−477927.	+	477927.i	−0.570846	+	0.570846i
$916$	−1.44823e6	+	1.44823e6i	−1.72603	+	1.72603i
$917$	50105.1	+	50105.1i	0.0595859	+	0.0595859i
$918$	−169180.	−	169180.i	−0.200754	−	0.200754i
$919$	514196.			0.608833			0.304416	−	0.952539i	$-0.401539\pi$
$919$	514196.			0.608833			0.304416	+	0.952539i	$0.401539\pi$
$920$			705193.i			0.833168i
$921$	−390867.	−	390867.i	−0.460797	−	0.460797i
$922$			294545.i			0.346489i
$923$	−728766.	+	6979.46i	−0.855431	+	0.00819254i
$924$	513193.			0.601086
$925$	−294466.	+	294466.i	−0.344153	+	0.344153i
$926$	1.26064e6			1.47018
$927$			352925.i			0.410699i
$928$	−422104.	+	422104.i	−0.490143	+	0.490143i
$929$	−746377.	+	746377.i	−0.864823	+	0.864823i	−0.991894	−	0.127071i	$-0.959442\pi$
$929$	−746377.	+	746377.i	−0.864823	+	0.864823i	0.127071	+	0.991894i	$0.459442\pi$
$930$	−833468.	−	833468.i	−0.963658	−	0.963658i
$931$	−193254.	−	193254.i	−0.222961	−	0.222961i
$932$	−1.27240e6			−1.46484
$933$		−	781635.i		−	0.897926i
$934$	499758.	+	499758.i	0.572884	+	0.572884i
$935$			398281.i			0.455582i
$936$	−3410.05	−	356063.i	−0.00389233	−	0.406420i
$937$	379998.			0.432815			0.216408	−	0.976303i	$-0.430566\pi$
$937$	379998.			0.432815			0.216408	+	0.976303i	$0.430566\pi$
$938$	265728.	−	265728.i	0.302017	−	0.302017i
$939$	−657854.			−0.746103
$940$			48039.7i			0.0543681i
$941$	−906449.	+	906449.i	−1.02368	+	1.02368i	−0.0239676	+	0.999713i	$0.507630\pi$
$941$	−906449.	+	906449.i	−1.02368	+	1.02368i	−0.999713	+	0.0239676i	$0.992370\pi$
$942$	−622143.	+	622143.i	−0.701114	+	0.701114i
$943$	547636.	+	547636.i	0.615841	+	0.615841i
$944$	14797.3	+	14797.3i	0.0166050	+	0.0166050i
$945$	111426.			0.124774
$946$		−	1.06227e6i		−	1.18701i
$947$	290385.	+	290385.i	0.323798	+	0.323798i	0.850222	−	0.526424i	$-0.176468\pi$
$947$	290385.	+	290385.i	0.323798	+	0.323798i	−0.526424	+	0.850222i	$0.676468\pi$
$948$			584083.i			0.649918i
$949$	−1923.03	−	200794.i	−0.00213527	−	0.222956i
$950$	879709.			0.974747
$951$	220061.	−	220061.i	0.243323	−	0.243323i
$952$	−859388.			−0.948234
$953$		−	1.13841e6i		−	1.25347i	−0.779234	−	0.626733i	$-0.784392\pi$
$953$		−	1.13841e6i		−	1.25347i	0.779234	−	0.626733i	$-0.215608\pi$
$954$	−636793.	+	636793.i	−0.699683	+	0.699683i
$955$	693596.	−	693596.i	0.760501	−	0.760501i
$956$	1.52731e6	+	1.52731e6i	1.67114	+	1.67114i
$957$	235685.	+	235685.i	0.257341	+	0.257341i
$958$	648362.			0.706459
$959$			1.30566e6i			1.41968i
$960$	428083.	+	428083.i	0.464500	+	0.464500i
$961$		−	2.47827e6i		−	2.68350i
$962$	1.18911e6	+	1.16655e6i	1.28491	+	1.26053i
$963$	−384886.			−0.415030
$964$	652395.	−	652395.i	0.702032	−	0.702032i
$965$	36949.1			0.0396780
$966$		−	714491.i		−	0.765672i
$967$	19218.1	−	19218.1i	0.0205521	−	0.0205521i	−0.696756	−	0.717308i	$-0.745374\pi$
$967$	19218.1	−	19218.1i	0.0205521	−	0.0205521i	0.717308	+	0.696756i	$0.245374\pi$
$968$	−426312.	+	426312.i	−0.454964	+	0.454964i
$969$	−450203.	−	450203.i	−0.479470	−	0.479470i
$970$	−685678.	−	685678.i	−0.728747	−	0.728747i
$971$	416812.			0.442080			0.221040	−	0.975265i	$-0.429055\pi$
$971$	416812.			0.442080			0.221040	+	0.975265i	$0.429055\pi$
$972$			105277.i			0.111430i
$973$	140758.	+	140758.i	0.148678	+	0.148678i
$974$		−	3.12758e6i		−	3.29679i
$975$	171938.	−	175263.i	0.180868	−	0.184366i
$976$	−502056.			−0.527051
$977$	394416.	−	394416.i	0.413205	−	0.413205i	−0.469649	−	0.882853i	$-0.655619\pi$
$977$	394416.	−	394416.i	0.413205	−	0.413205i	0.882853	+	0.469649i	$0.155619\pi$
$978$	807283.			0.844011
$979$			709399.i			0.740159i
$980$	−209942.	+	209942.i	−0.218599	+	0.218599i
$981$	−131940.	+	131940.i	−0.137101	+	0.137101i
$982$	−956284.	−	956284.i	−0.991663	−	0.991663i
$983$	−159479.	−	159479.i	−0.165043	−	0.165043i	0.619754	−	0.784796i	$-0.287233\pi$
$983$	−159479.	−	159479.i	−0.165043	−	0.165043i	−0.784796	+	0.619754i	$0.787233\pi$
$984$	−645866.			−0.667041
$985$		−	267441.i		−	0.275649i
$986$	−930181.	−	930181.i	−0.956783	−	0.956783i
$987$		−	20652.0i		−	0.0211996i
$988$	−21386.9	−	2.23313e6i	−0.0219095	−	2.28770i
$989$	−938598.			−0.959593
$990$	195262.	−	195262.i	0.199227	−	0.199227i
$991$	869851.			0.885722			0.442861	−	0.896590i	$-0.353964\pi$
$991$	869851.			0.885722			0.442861	+	0.896590i	$0.353964\pi$
$992$			1.42733e6i			1.45044i
$993$	−122597.	+	122597.i	−0.124332	+	0.124332i
$994$	862340.	−	862340.i	0.872782	−	0.872782i
$995$	−97768.8	−	97768.8i	−0.0987539	−	0.0987539i
$996$	297610.	+	297610.i	0.300005	+	0.300005i
$997$	798123.			0.802934			0.401467	−	0.915874i	$-0.368501\pi$
$997$	798123.			0.802934			0.401467	+	0.915874i	$0.368501\pi$
$998$		−	2.46427e6i		−	2.47415i
$999$	−147762.	−	147762.i	−0.148058	−	0.148058i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.5.g.a.31.9	✓	20
3.2	odd	2		117.5.j.b.109.2		20
13.8	odd	4	inner	39.5.g.a.34.9	yes	20
39.8	even	4		117.5.j.b.73.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.9	✓	20	1.1	even	1	trivial
39.5.g.a.34.9	yes	20	13.8	odd	4	inner
117.5.j.b.73.2		20	39.8	even	4
117.5.j.b.109.2		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.67933 - 4.67933i) q^{2} -5.19615 q^{3} -27.7923i q^{4} +(13.1418 - 13.1418i) q^{5} +(-24.3145 + 24.3145i) q^{6} +(-30.2175 - 30.2175i) q^{7} +(-55.1800 - 55.1800i) q^{8} +27.0000 q^{9} -122.989i q^{10} +(58.8012 + 58.8012i) q^{11} +144.413i q^{12} +(120.640 + 118.351i) q^{13} -282.795 q^{14} +(-68.2867 + 68.2867i) q^{15} -71.7343 q^{16} +257.703i q^{17} +(126.342 - 126.342i) q^{18} +(336.208 - 336.208i) q^{19} +(-365.240 - 365.240i) q^{20} +(157.015 + 157.015i) q^{21} +550.300 q^{22} -486.231i q^{23} +(286.724 + 286.724i) q^{24} +279.587i q^{25} +(1118.32 - 10.7102i) q^{26} -140.296 q^{27} +(-839.813 + 839.813i) q^{28} -771.373 q^{29} +639.072i q^{30} +(-1304.18 + 1304.18i) q^{31} +(547.211 - 547.211i) q^{32} +(-305.540 - 305.540i) q^{33} +(1205.88 + 1205.88i) q^{34} -794.223 q^{35} -750.392i q^{36} +(1053.22 + 1053.22i) q^{37} -3146.45i q^{38} +(-626.864 - 614.971i) q^{39} -1450.33 q^{40} +(-1126.29 + 1126.29i) q^{41} +1469.45 q^{42} -1930.35i q^{43} +(1634.22 - 1634.22i) q^{44} +(354.828 - 354.828i) q^{45} +(-2275.24 - 2275.24i) q^{46} +(-65.7645 - 65.7645i) q^{47} +372.742 q^{48} -574.806i q^{49} +(1308.28 + 1308.28i) q^{50} -1339.06i q^{51} +(3289.25 - 3352.86i) q^{52} -5040.23 q^{53} +(-656.492 + 656.492i) q^{54} +1545.50 q^{55} +3334.80i q^{56} +(-1746.99 + 1746.99i) q^{57} +(-3609.51 + 3609.51i) q^{58} +(-206.279 - 206.279i) q^{59} +(1897.84 + 1897.84i) q^{60} +6998.83 q^{61} +12205.4i q^{62} +(-815.872 - 815.872i) q^{63} -6268.91i q^{64} +(3140.77 - 30.0794i) q^{65} -2859.44 q^{66} +(-939.648 + 939.648i) q^{67} +7162.15 q^{68} +2526.53i q^{69} +(-3716.43 + 3716.43i) q^{70} +(-3049.34 + 3049.34i) q^{71} +(-1489.86 - 1489.86i) q^{72} +(-840.175 - 840.175i) q^{73} +9856.69 q^{74} -1452.78i q^{75} +(-9343.97 - 9343.97i) q^{76} -3553.65i q^{77} +(-5810.95 + 55.6521i) q^{78} +4044.54 q^{79} +(-942.716 + 942.716i) q^{80} +729.000 q^{81} +10540.5i q^{82} +(2060.82 - 2060.82i) q^{83} +(4363.80 - 4363.80i) q^{84} +(3386.67 + 3386.67i) q^{85} +(-9032.76 - 9032.76i) q^{86} +4008.17 q^{87} -6489.30i q^{88} +(6032.18 + 6032.18i) q^{89} -3320.71i q^{90} +(-69.1630 - 7221.71i) q^{91} -13513.5 q^{92} +(6776.74 - 6776.74i) q^{93} -615.468 q^{94} -8836.73i q^{95} +(-2843.39 + 2843.39i) q^{96} +(5575.10 - 5575.10i) q^{97} +(-2689.71 - 2689.71i) q^{98} +(1587.63 + 1587.63i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{5} - 24 q^{7} + 540 q^{9} + 372 q^{11} - 224 q^{13} + 480 q^{14} - 252 q^{15} - 2328 q^{16} - 840 q^{19} + 228 q^{20} + 936 q^{21} + 3536 q^{22} - 1404 q^{24} - 828 q^{26} - 1984 q^{28} - 5064 q^{29}+ \cdots + 10044 q^{99}+O(q^{100}) \) 20 * q + 24 * q^5 - 24 * q^7 + 540 * q^9 + 372 * q^11 - 224 * q^13 + 480 * q^14 - 252 * q^15 - 2328 * q^16 - 840 * q^19 + 228 * q^20 + 936 * q^21 + 3536 * q^22 - 1404 * q^24 - 828 * q^26 - 1984 * q^28 - 5064 * q^29 + 1712 * q^31 - 7260 * q^32 - 540 * q^33 + 8040 * q^34 + 5976 * q^35 - 868 * q^37 - 2952 * q^39 + 6336 * q^40 - 6000 * q^41 + 648 * q^42 + 20868 * q^44 + 648 * q^45 - 1992 * q^46 + 1692 * q^47 + 2592 * q^48 + 8952 * q^50 - 12416 * q^52 - 18816 * q^53 - 3280 * q^55 - 9360 * q^57 + 1256 * q^58 - 6588 * q^59 + 9036 * q^60 - 5512 * q^61 - 648 * q^63 + 33048 * q^65 + 13464 * q^66 - 4704 * q^67 - 28296 * q^68 - 31096 * q^70 + 9756 * q^71 + 13692 * q^73 - 28296 * q^74 - 17072 * q^76 - 23868 * q^78 - 5056 * q^79 + 42228 * q^80 + 14580 * q^81 + 24708 * q^83 + 26496 * q^84 - 4752 * q^85 - 77424 * q^86 + 5256 * q^87 + 36000 * q^89 + 19104 * q^91 + 12912 * q^92 - 18432 * q^93 + 112832 * q^94 - 52344 * q^96 + 50012 * q^97 + 11544 * q^98 + 10044 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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