LMFDB - Embedded newform 117.5.j.b.73.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (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:	$12.0942856808$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.2
Root			$-4.67933 - 4.67933i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.5.j.b.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.67933 - 4.67933i) q^{2} +27.7923i q^{4} +(-13.1418 - 13.1418i) q^{5} +(-30.2175 + 30.2175i) q^{7} +(55.1800 - 55.1800i) q^{8} +122.989i q^{10} +(-58.8012 + 58.8012i) q^{11} +(120.640 - 118.351i) q^{13} +282.795 q^{14} -71.7343 q^{16} +257.703i q^{17} +(336.208 + 336.208i) q^{19} +(365.240 - 365.240i) q^{20} +550.300 q^{22} -486.231i q^{23} -279.587i q^{25} +(-1118.32 - 10.7102i) q^{26} +(-839.813 - 839.813i) q^{28} +771.373 q^{29} +(-1304.18 - 1304.18i) q^{31} +(-547.211 - 547.211i) q^{32} +(1205.88 - 1205.88i) q^{34} +794.223 q^{35} +(1053.22 - 1053.22i) q^{37} -3146.45i q^{38} -1450.33 q^{40} +(1126.29 + 1126.29i) q^{41} +1930.35i q^{43} +(-1634.22 - 1634.22i) q^{44} +(-2275.24 + 2275.24i) q^{46} +(65.7645 - 65.7645i) q^{47} +574.806i q^{49} +(-1308.28 + 1308.28i) q^{50} +(3289.25 + 3352.86i) q^{52} +5040.23 q^{53} +1545.50 q^{55} +3334.80i q^{56} +(-3609.51 - 3609.51i) q^{58} +(206.279 - 206.279i) q^{59} +6998.83 q^{61} +12205.4i q^{62} +6268.91i q^{64} +(-3140.77 - 30.0794i) q^{65} +(-939.648 - 939.648i) q^{67} -7162.15 q^{68} +(-3716.43 - 3716.43i) q^{70} +(3049.34 + 3049.34i) q^{71} +(-840.175 + 840.175i) q^{73} -9856.69 q^{74} +(-9343.97 + 9343.97i) q^{76} -3553.65i q^{77} +4044.54 q^{79} +(942.716 + 942.716i) q^{80} -10540.5i q^{82} +(-2060.82 - 2060.82i) q^{83} +(3386.67 - 3386.67i) q^{85} +(9032.76 - 9032.76i) q^{86} +6489.30i q^{88} +(-6032.18 + 6032.18i) q^{89} +(-69.1630 + 7221.71i) q^{91} +13513.5 q^{92} -615.468 q^{94} -8836.73i q^{95} +(5575.10 + 5575.10i) q^{97} +(2689.71 - 2689.71i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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	−0.982249	−	0.187584i	$-0.939934\pi$
$2$	−4.67933	−	4.67933i	−1.16983	−	1.16983i	−0.187584	−	0.982249i	$-0.560066\pi$
$3$		0			0
$3$		0			0
$4$			27.7923i			1.73702i
$5$	−13.1418	−	13.1418i	−0.525671	−	0.525671i	0.393608	−	0.919279i	$-0.371227\pi$
$5$	−13.1418	−	13.1418i	−0.525671	−	0.525671i	−0.919279	+	0.393608i	$0.871227\pi$
$6$		0			0
$7$	−30.2175	+	30.2175i	−0.616684	+	0.616684i	−0.944679	−	0.327996i	$-0.893627\pi$
$7$	−30.2175	+	30.2175i	−0.616684	+	0.616684i	0.327996	+	0.944679i	$0.393627\pi$
$8$	55.1800	−	55.1800i	0.862187	−	0.862187i
$9$		0			0
$10$			122.989i			1.22989i
$11$	−58.8012	+	58.8012i	−0.485960	+	0.485960i	−0.907029	−	0.421069i	$-0.861655\pi$
$11$	−58.8012	+	58.8012i	−0.485960	+	0.485960i	0.421069	+	0.907029i	$0.361655\pi$
$12$		0			0
$13$	120.640	−	118.351i	0.713846	−	0.700303i
$13$	120.640	−	118.351i	0.713846	−	0.700303i
$14$	282.795			1.44283
$15$		0			0
$16$	−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	$0.852900\pi$
$18$		0			0
$19$	336.208	+	336.208i	0.931323	+	0.931323i	0.997789	−	0.0664659i	$-0.0211724\pi$
$19$	336.208	+	336.208i	0.931323	+	0.931323i	−0.0664659	+	0.997789i	$0.521172\pi$
$20$	365.240	−	365.240i	0.913100	−	0.913100i
$21$		0			0
$22$	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	$-0.151999\pi$
$24$		0			0
$25$		−	279.587i		−	0.447340i
$26$	−1118.32	−	10.7102i	−1.65432	−	0.0158436i
$27$		0			0
$28$	−839.813	−	839.813i	−1.07119	−	1.07119i
$29$	771.373			0.917209			0.458604	−	0.888641i	$-0.348350\pi$
$29$	771.373			0.917209			0.458604	+	0.888641i	$0.348350\pi$
$30$		0			0
$31$	−1304.18	−	1304.18i	−1.35711	−	1.35711i	−0.877458	−	0.479654i	$-0.840762\pi$
$31$	−1304.18	−	1304.18i	−1.35711	−	1.35711i	−0.479654	−	0.877458i	$-0.659238\pi$
$32$	−547.211	−	547.211i	−0.534386	−	0.534386i
$33$		0			0
$34$	1205.88	−	1205.88i	1.04315	−	1.04315i
$35$	794.223			0.648345
$36$		0			0
$37$	1053.22	−	1053.22i	0.769333	−	0.769333i	−0.208656	−	0.977989i	$-0.566909\pi$
$37$	1053.22	−	1053.22i	0.769333	−	0.769333i	0.977989	+	0.208656i	$0.0669091\pi$
$38$		−	3146.45i		−	2.17898i
$39$		0			0
$40$	−1450.33			−0.906454
$41$	1126.29	+	1126.29i	0.670010	+	0.670010i	0.957718	−	0.287708i	$-0.0928932\pi$
$41$	1126.29	+	1126.29i	0.670010	+	0.670010i	−0.287708	+	0.957718i	$0.592893\pi$
$42$		0			0
$43$			1930.35i			1.04400i	0.852946	+	0.521999i	$0.174814\pi$
$43$			1930.35i			1.04400i	−0.852946	+	0.521999i	$0.825186\pi$
$44$	−1634.22	−	1634.22i	−0.844121	−	0.844121i
$45$		0			0
$46$	−2275.24	+	2275.24i	−1.07525	+	1.07525i
$47$	65.7645	−	65.7645i	0.0297712	−	0.0297712i	−0.692064	−	0.721836i	$-0.743299\pi$
$47$	65.7645	−	65.7645i	0.0297712	−	0.0297712i	0.721836	+	0.692064i	$0.243299\pi$
$48$		0			0
$49$			574.806i			0.239403i
$50$	−1308.28	+	1308.28i	−0.523313	+	0.523313i
$51$		0			0
$52$	3289.25	+	3352.86i	1.21644	+	1.23996i
$53$	5040.23			1.79432			0.897158	−	0.441710i	$-0.145628\pi$
$53$	5040.23			1.79432			0.897158	+	0.441710i	$0.145628\pi$
$54$		0			0
$55$	1545.50			0.510910
$56$			3334.80i			1.06339i
$57$		0			0
$58$	−3609.51	−	3609.51i	−1.07298	−	1.07298i
$59$	206.279	−	206.279i	0.0592586	−	0.0592586i	−0.676856	−	0.736115i	$-0.736658\pi$
$59$	206.279	−	206.279i	0.0592586	−	0.0592586i	0.736115	+	0.676856i	$0.236658\pi$
$60$		0			0
$61$	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
$63$		0			0
$64$			6268.91i			1.53050i
$65$	−3140.77	−	30.0794i	−0.743377	−	0.00711939i
$66$		0			0
$67$	−939.648	−	939.648i	−0.209322	−	0.209322i	0.594657	−	0.803979i	$-0.297288\pi$
$67$	−939.648	−	939.648i	−0.209322	−	0.209322i	−0.803979	+	0.594657i	$0.797288\pi$
$68$	−7162.15			−1.54891
$69$		0			0
$70$	−3716.43	−	3716.43i	−0.758456	−	0.758456i
$71$	3049.34	+	3049.34i	0.604908	+	0.604908i	0.941611	−	0.336703i	$-0.109312\pi$
$71$	3049.34	+	3049.34i	0.604908	+	0.604908i	−0.336703	+	0.941611i	$0.609312\pi$
$72$		0			0
$73$	−840.175	+	840.175i	−0.157661	+	0.157661i	−0.781529	−	0.623869i	$-0.785560\pi$
$73$	−840.175	+	840.175i	−0.157661	+	0.157661i	0.623869	+	0.781529i	$0.285560\pi$
$74$	−9856.69			−1.79998
$75$		0			0
$76$	−9343.97	+	9343.97i	−1.61772	+	1.61772i
$77$		−	3553.65i		−	0.599367i
$78$		0			0
$79$	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
$81$		0			0
$82$		−	10540.5i		−	1.56760i
$83$	−2060.82	−	2060.82i	−0.299147	−	0.299147i	0.541533	−	0.840680i	$-0.317844\pi$
$83$	−2060.82	−	2060.82i	−0.299147	−	0.299147i	−0.840680	+	0.541533i	$0.817844\pi$
$84$		0			0
$85$	3386.67	−	3386.67i	0.468744	−	0.468744i
$86$	9032.76	−	9032.76i	1.22130	−	1.22130i
$87$		0			0
$88$			6489.30i			0.837977i
$89$	−6032.18	+	6032.18i	−0.761543	+	0.761543i	−0.976601	−	0.215059i	$-0.931006\pi$
$89$	−6032.18	+	6032.18i	−0.761543	+	0.761543i	0.215059	+	0.976601i	$0.431006\pi$
$90$		0			0
$91$	−69.1630	+	7221.71i	−0.00835202	+	0.872082i
$92$	13513.5			1.59658
$93$		0			0
$94$	−615.468			−0.0696546
$95$		−	8836.73i		−	0.979139i
$96$		0			0
$97$	5575.10	+	5575.10i	0.592528	+	0.592528i	0.938314	−	0.345785i	$-0.112387\pi$
$97$	5575.10	+	5575.10i	0.592528	+	0.592528i	−0.345785	+	0.938314i	$0.612387\pi$
$98$	2689.71	−	2689.71i	0.280061	−	0.280061i
$99$		0			0
$100$	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	$0.746335\pi$
$102$		0			0
$103$		−	13071.3i		−	1.23210i	−0.787708	−	0.616048i	$-0.788733\pi$
$103$		−	13071.3i		−	1.23210i	0.787708	−	0.616048i	$-0.211267\pi$
$104$	126.298	−	13187.5i	0.0116770	−	1.21926i
$105$		0			0
$106$	−23584.9	−	23584.9i	−2.09905	−	2.09905i
$107$	14255.0			1.24509			0.622545	−	0.782584i	$-0.286099\pi$
$107$	14255.0			1.24509			0.622545	+	0.782584i	$0.286099\pi$
$108$		0			0
$109$	−4886.68	−	4886.68i	−0.411302	−	0.411302i	0.470890	−	0.882192i	$-0.343933\pi$
$109$	−4886.68	−	4886.68i	−0.411302	−	0.411302i	−0.882192	+	0.470890i	$0.843933\pi$
$110$	−7231.92	−	7231.92i	−0.597680	−	0.597680i
$111$		0			0
$112$	2167.63	−	2167.63i	0.172802	−	0.172802i
$113$	3433.45			0.268890			0.134445	−	0.990921i	$-0.457075\pi$
$113$	3433.45			0.268890			0.134445	+	0.990921i	$0.457075\pi$
$114$		0			0
$115$	−6389.94	+	6389.94i	−0.483171	+	0.483171i
$116$			21438.2i			1.59321i
$117$		0			0
$118$	−1930.50			−0.138645
$119$	−7787.14	−	7787.14i	−0.549900	−	0.549900i
$120$		0			0
$121$			7725.84i			0.527685i
$122$	−32749.8	−	32749.8i	−2.20034	−	2.20034i
$123$		0			0
$124$	36246.3	−	36246.3i	2.35733	−	2.35733i
$125$	−11887.9	+	11887.9i	−0.760825	+	0.760825i
$126$		0			0
$127$			21948.2i			1.36079i	0.732846	+	0.680395i	$0.238192\pi$
$127$			21948.2i			1.36079i	−0.732846	+	0.680395i	$0.761808\pi$
$128$	20578.9	−	20578.9i	1.25604	−	1.25604i
$129$		0			0
$130$	14555.9	+	14837.4i	0.861298	+	0.877955i
$131$	1658.15			0.0966231			0.0483116	−	0.998832i	$-0.484616\pi$
$131$	1658.15			0.0966231			0.0483116	+	0.998832i	$0.484616\pi$
$132$		0			0
$133$	−20318.7			−1.14866
$134$			8793.84i			0.489744i
$135$		0			0
$136$	14220.0	+	14220.0i	0.768817	+	0.768817i
$137$	21604.3	−	21604.3i	1.15106	−	1.15106i	0.164724	−	0.986340i	$-0.447327\pi$
$137$	21604.3	−	21604.3i	1.15106	−	1.15106i	0.986340	−	0.164724i	$-0.0526732\pi$
$138$		0			0
$139$	−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
$141$		0			0
$142$		−	28537.8i		−	1.41528i
$143$	−134.587	+	14053.0i	−0.00658157	+	0.687220i
$144$		0			0
$145$	−10137.2	−	10137.2i	−0.482150	−	0.482150i
$146$	7862.91			0.368874
$147$		0			0
$148$	29271.3	+	29271.3i	1.33634	+	1.33634i
$149$	−393.889	−	393.889i	−0.0177420	−	0.0177420i	0.698180	−	0.715922i	$-0.253993\pi$
$149$	−393.889	−	393.889i	−0.0177420	−	0.0177420i	−0.715922	+	0.698180i	$0.753993\pi$
$150$		0			0
$151$	−7620.89	+	7620.89i	−0.334235	+	0.334235i	−0.854192	−	0.519957i	$-0.825948\pi$
$151$	−7620.89	+	7620.89i	−0.334235	+	0.334235i	0.519957	+	0.854192i	$0.325948\pi$
$152$	37103.8			1.60595
$153$		0			0
$154$	−16628.7	+	16628.7i	−0.701159	+	0.701159i
$155$			34278.6i			1.42679i
$156$		0			0
$157$	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
$159$		0			0
$160$			14382.7i			0.561822i
$161$	14692.7	+	14692.7i	0.566826	+	0.566826i
$162$		0			0
$163$	−16600.8	+	16600.8i	−0.624820	+	0.624820i	−0.946760	−	0.321940i	$-0.895665\pi$
$163$	−16600.8	+	16600.8i	−0.624820	+	0.624820i	0.321940	+	0.946760i	$0.395665\pi$
$164$	−31302.1	+	31302.1i	−1.16382	+	1.16382i
$165$		0			0
$166$			19286.6i			0.699904i
$167$	37894.4	−	37894.4i	1.35876	−	1.35876i	0.483307	−	0.875451i	$-0.339435\pi$
$167$	37894.4	−	37894.4i	1.35876	−	1.35876i	0.875451	−	0.483307i	$-0.160565\pi$
$168$		0			0
$169$	547.013	−	28555.8i	0.0191524	−	0.999817i
$170$	−31694.7			−1.09670
$171$		0			0
$172$	−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	$0.990890\pi$
$174$		0			0
$175$	8448.43	+	8448.43i	0.275867	+	0.275867i
$176$	4218.06	−	4218.06i	0.136172	−	0.136172i
$177$		0			0
$178$	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	$-0.0134176\pi$
$180$		0			0
$181$		−	31325.6i		−	0.956186i	−0.878309	−	0.478093i	$-0.841328\pi$
$181$		−	31325.6i		−	0.956186i	0.878309	−	0.478093i	$-0.158672\pi$
$182$	34116.4	−	33469.1i	1.02996	−	1.01042i
$183$		0			0
$184$	−26830.2	−	26830.2i	−0.792481	−	0.792481i
$185$	−27682.3			−0.808832
$186$		0			0
$187$	−15153.2	−	15153.2i	−0.433333	−	0.433333i
$188$	1827.75	+	1827.75i	0.0517131	+	0.0517131i
$189$		0			0
$190$	−41350.0	+	41350.0i	−1.14543	+	1.14543i
$191$	−52778.0			−1.44672			−0.723362	−	0.690469i	$-0.757404\pi$
$191$	−52778.0			−1.44672			−0.723362	+	0.690469i	$0.757404\pi$
$192$		0			0
$193$	1405.79	−	1405.79i	0.0377403	−	0.0377403i	−0.687985	−	0.725725i	$-0.741504\pi$
$193$	1405.79	−	1405.79i	0.0377403	−	0.0377403i	0.725725	+	0.687985i	$0.241504\pi$
$194$		−	52175.5i		−	1.38632i
$195$		0			0
$196$	−15975.2			−0.415847
$197$	−10175.2	−	10175.2i	−0.262188	−	0.262188i	0.563755	−	0.825942i	$-0.309356\pi$
$197$	−10175.2	−	10175.2i	−0.262188	−	0.262188i	−0.825942	+	0.563755i	$0.809356\pi$
$198$		0			0
$199$			7439.54i			0.187862i	0.995579	+	0.0939312i	$0.0299434\pi$
$199$			7439.54i			0.187862i	−0.995579	+	0.0939312i	$0.970057\pi$
$200$	−15427.6	−	15427.6i	−0.385691	−	0.385691i
$201$		0			0
$202$	68278.5	−	68278.5i	1.67333	−	1.67333i
$203$	−23308.9	+	23308.9i	−0.565627	+	0.565627i
$204$		0			0
$205$		−	29602.8i		−	0.704410i
$206$	−61165.0	+	61165.0i	−1.44135	+	1.44135i
$207$		0			0
$208$	−8654.02	+	8489.84i	−0.200028	+	0.196233i
$209$	−39538.8			−0.905172
$210$		0			0
$211$	−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
$213$		0			0
$214$	−66704.1	−	66704.1i	−1.45655	−	1.45655i
$215$	25368.3	−	25368.3i	0.548800	−	0.548800i
$216$		0			0
$217$	78818.4			1.67382
$218$			45732.8i			0.962309i
$219$		0			0
$220$			42953.1i			0.887460i
$221$	30499.4	+	31089.3i	0.624464	+	0.636541i
$222$		0			0
$223$	26187.3	+	26187.3i	0.526601	+	0.526601i	0.919557	−	0.392956i	$-0.128548\pi$
$223$	26187.3	+	26187.3i	0.526601	+	0.526601i	−0.392956	+	0.919557i	$0.628548\pi$
$224$	33070.7			0.659094
$225$		0			0
$226$	−16066.3	−	16066.3i	−0.314556	−	0.314556i
$227$	20418.0	+	20418.0i	0.396244	+	0.396244i	0.876906	−	0.480662i	$-0.159604\pi$
$227$	20418.0	+	20418.0i	0.396244	+	0.396244i	−0.480662	+	0.876906i	$0.659604\pi$
$228$		0			0
$229$	−52109.2	+	52109.2i	−0.993672	+	0.993672i	−0.999980	−	0.00630803i	$-0.997992\pi$
$229$	−52109.2	+	52109.2i	−0.993672	+	0.993672i	0.00630803	+	0.999980i	$0.497992\pi$
$230$	59801.3			1.13046
$231$		0			0
$232$	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	$-0.138550\pi$
$234$		0			0
$235$	−1728.53			−0.0312997
$236$	5732.97	+	5732.97i	0.102933	+	0.102933i
$237$		0			0
$238$			72877.2i			1.28658i
$239$	54954.6	+	54954.6i	0.962073	+	0.962073i	0.999307	−	0.0372339i	$-0.0118546\pi$
$239$	54954.6	+	54954.6i	0.962073	+	0.962073i	−0.0372339	+	0.999307i	$0.511855\pi$
$240$		0			0
$241$	23474.0	−	23474.0i	0.404159	−	0.404159i	−0.475537	−	0.879696i	$-0.657746\pi$
$241$	23474.0	−	23474.0i	0.404159	−	0.404159i	0.879696	+	0.475537i	$0.157746\pi$
$242$	36151.8	−	36151.8i	0.617304	−	0.617304i
$243$		0			0
$244$			194513.i			3.26716i
$245$	7553.98	−	7553.98i	0.125847	−	0.125847i
$246$		0			0
$247$	80350.6	+	769.526i	1.31703	+	0.0126133i
$248$	−143930.			−2.34017
$249$		0			0
$250$	111255.			1.78008
$251$		−	51668.4i		−	0.820120i	−0.912059	−	0.410060i	$-0.865508\pi$
$251$		−	51668.4i		−	0.820120i	0.912059	−	0.410060i	$-0.134492\pi$
$252$		0			0
$253$	28591.0	+	28591.0i	0.446671	+	0.446671i
$254$	102703.	−	102703.i	1.59190	−	1.59190i
$255$		0			0
$256$	−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	$-0.232004\pi$
$258$		0			0
$259$			63651.1i			0.948869i
$260$	835.976	−	87289.1i	0.0123665	−	1.29126i
$261$		0			0
$262$	−7759.03	−	7759.03i	−0.113033	−	0.113033i
$263$	−113585.			−1.64214			−0.821068	−	0.570831i	$-0.806621\pi$
$263$	−113585.			−1.64214			−0.821068	+	0.570831i	$0.806621\pi$
$264$		0			0
$265$	−66237.6	−	66237.6i	−0.943220	−	0.943220i
$266$	95077.9	+	95077.9i	1.34374	+	1.34374i
$267$		0			0
$268$	26114.9	−	26114.9i	0.363596	−	0.363596i
$269$	−5647.94			−0.0780523			−0.0390262	−	0.999238i	$-0.512426\pi$
$269$	−5647.94			−0.0780523			−0.0390262	+	0.999238i	$0.512426\pi$
$270$		0			0
$271$	−3722.39	+	3722.39i	−0.0506854	+	0.0506854i	−0.731995	−	0.681310i	$-0.761411\pi$
$271$	−3722.39	+	3722.39i	−0.0506854	+	0.0506854i	0.681310	+	0.731995i	$0.261411\pi$
$272$		−	18486.1i		−	0.249867i
$273$		0			0
$274$	−202187.			−2.69310
$275$	16440.1	+	16440.1i	0.217389	+	0.217389i
$276$		0			0
$277$		−	31673.1i		−	0.412792i	−0.978469	−	0.206396i	$-0.933826\pi$
$277$		−	31673.1i		−	0.412792i	0.978469	−	0.206396i	$-0.0661736\pi$
$278$	21797.1	+	21797.1i	0.282039	+	0.282039i
$279$		0			0
$280$	43825.2	−	43825.2i	0.558995	−	0.558995i
$281$	10897.2	−	10897.2i	0.138007	−	0.138007i	−0.634728	−	0.772736i	$-0.718888\pi$
$281$	10897.2	−	10897.2i	0.138007	−	0.138007i	0.772736	+	0.634728i	$0.218888\pi$
$282$		0			0
$283$		−	30994.4i		−	0.387000i	−0.981100	−	0.193500i	$-0.938016\pi$
$283$		−	30994.4i		−	0.387000i	0.981100	−	0.193500i	$-0.0619839\pi$
$284$	−84748.2	+	84748.2i	−1.05074	+	1.05074i
$285$		0			0
$286$	66388.2	−	65128.7i	0.811632	−	0.796233i
$287$	−68067.2			−0.826369
$288$		0			0
$289$	17110.2			0.204861
$290$			94870.7i			1.12807i
$291$		0			0
$292$	−23350.4	−	23350.4i	−0.273860	−	0.273860i
$293$	113536.	−	113536.i	1.32251	−	1.32251i	0.410765	−	0.911741i	$-0.365262\pi$
$293$	113536.	−	113536.i	1.32251	−	1.32251i	0.911741	−	0.410765i	$-0.134738\pi$
$294$		0			0
$295$	−5421.75			−0.0623011
$296$		−	116233.i		−	1.32662i
$297$		0			0
$298$			3686.28i			0.0415103i
$299$	−57546.0	−	58658.9i	−0.643684	−	0.656133i
$300$		0			0
$301$	−58330.4	−	58330.4i	−0.643817	−	0.643817i
$302$	71321.3			0.781998
$303$		0			0
$304$	−24117.6	−	24117.6i	−0.260968	−	0.260968i
$305$	−91977.0	−	91977.0i	−0.988735	−	0.988735i
$306$		0			0
$307$	75222.4	−	75222.4i	0.798124	−	0.798124i	−0.184675	−	0.982800i	$-0.559123\pi$
$307$	75222.4	−	75222.4i	0.798124	−	0.798124i	0.982800	+	0.184675i	$0.0591233\pi$
$308$	98764.0			1.04111
$309$		0			0
$310$	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	$0.716423\pi$
$312$		0			0
$313$	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
$315$		0			0
$316$			112407.i			1.12569i
$317$	42350.8	+	42350.8i	0.421447	+	0.421447i	0.885702	−	0.464255i	$-0.153678\pi$
$317$	42350.8	+	42350.8i	0.421447	+	0.421447i	−0.464255	+	0.885702i	$0.653678\pi$
$318$		0			0
$319$	−45357.6	+	45357.6i	−0.445727	+	0.445727i
$320$	82384.6	−	82384.6i	0.804537	−	0.804537i
$321$		0			0
$322$		−	137504.i		−	1.32618i
$323$	−86641.7	+	86641.7i	−0.830466	+	0.830466i
$324$		0			0
$325$	−33089.5	−	33729.4i	−0.313273	−	0.319332i
$326$	155362.			1.46187
$327$		0			0
$328$	124297.			1.15535
$329$			3974.48i			0.0367188i
$330$		0			0
$331$	23593.9	+	23593.9i	0.215349	+	0.215349i	0.806535	−	0.591186i	$-0.201340\pi$
$331$	23593.9	+	23593.9i	0.215349	+	0.215349i	−0.591186	+	0.806535i	$0.701340\pi$
$332$	57275.0	−	57275.0i	0.519624	−	0.519624i
$333$		0			0
$334$	−354641.			−3.17904
$335$			24697.3i			0.220069i
$336$		0			0
$337$			114863.i			1.01139i	0.862711	+	0.505697i	$0.168765\pi$
$337$			114863.i			1.01139i	−0.862711	+	0.505697i	$0.831235\pi$
$338$	−136182.	+	131062.i	−1.19202	+	1.14721i
$339$		0			0
$340$	94123.4	+	94123.4i	0.814216	+	0.814216i
$341$	153375.			1.31900
$342$		0			0
$343$	−89921.4	−	89921.4i	−0.764319	−	0.764319i
$344$	106517.	+	106517.i	0.900122	+	0.900122i
$345$		0			0
$346$	8015.22	−	8015.22i	0.0669520	−	0.0669520i
$347$	−93409.6			−0.775769			−0.387885	−	0.921708i	$-0.626794\pi$
$347$	−93409.6			−0.775769			−0.387885	+	0.921708i	$0.626794\pi$
$348$		0			0
$349$	−14479.3	+	14479.3i	−0.118877	+	0.118877i	−0.764043	−	0.645166i	$-0.776788\pi$
$349$	−14479.3	+	14479.3i	−0.118877	+	0.118877i	0.645166	+	0.764043i	$0.276788\pi$
$350$		−	79066.0i		−	0.645437i
$351$		0			0
$352$	64353.3			0.519380
$353$	−127078.	−	127078.i	−1.01981	−	1.01981i	−0.999800	−	0.0200140i	$-0.993629\pi$
$353$	−127078.	−	127078.i	−1.01981	−	1.01981i	−0.0200140	−	0.999800i	$-0.506371\pi$
$354$		0			0
$355$		−	80147.6i		−	0.635966i
$356$	−167648.	−	167648.i	−1.32281	−	1.32281i
$357$		0			0
$358$	−12636.2	+	12636.2i	−0.0985939	+	0.0985939i
$359$	−40565.6	+	40565.6i	−0.314752	+	0.314752i	−0.846747	−	0.531995i	$-0.821442\pi$
$359$	−40565.6	+	40565.6i	−0.314752	+	0.314752i	0.531995	+	0.846747i	$0.321442\pi$
$360$		0			0
$361$			95750.0i			0.734724i
$362$	−146583.	+	146583.i	−1.11858	+	1.11858i
$363$		0			0
$364$	−200708.	−	1922.20i	−1.51482	−	0.0145076i
$365$	22082.8			0.165755
$366$		0			0
$367$	−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
$369$		0			0
$370$	129534.	+	129534.i	0.946198	+	0.946198i
$371$	−152303.	+	152303.i	−1.10652	+	1.10652i
$372$		0			0
$373$	−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
$375$		0			0
$376$		−	7257.77i		−	0.0513367i
$377$	93058.4	−	91292.8i	0.654746	−	0.642324i
$378$		0			0
$379$	−27066.7	−	27066.7i	−0.188433	−	0.188433i	0.606585	−	0.795018i	$-0.292539\pi$
$379$	−27066.7	−	27066.7i	−0.188433	−	0.188433i	−0.795018	+	0.606585i	$0.792539\pi$
$380$	245593.			1.70078
$381$		0			0
$382$	246966.	+	246966.i	1.69243	+	1.69243i
$383$	166419.	+	166419.i	1.13451	+	1.13451i	0.989419	+	0.145087i	$0.0463461\pi$
$383$	166419.	+	166419.i	1.13451	+	1.13451i	0.145087	+	0.989419i	$0.453654\pi$
$384$		0			0
$385$	−46701.2	+	46701.2i	−0.315070	+	0.315070i
$386$	−13156.3			−0.0882997
$387$		0			0
$388$	−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	$0.982664\pi$
$390$		0			0
$391$	125303.			0.819613
$392$	31717.8	+	31717.8i	0.206410	+	0.206410i
$393$		0			0
$394$			95226.6i			0.613431i
$395$	−53152.4	−	53152.4i	−0.340666	−	0.340666i
$396$		0			0
$397$	−155869.	+	155869.i	−0.988957	+	0.988957i	−0.999940	−	0.0109822i	$-0.996504\pi$
$397$	−155869.	+	155869.i	−0.988957	+	0.988957i	0.0109822	+	0.999940i	$0.496504\pi$
$398$	34812.1	−	34812.1i	0.219768	−	0.219768i
$399$		0			0
$400$			20056.0i			0.125350i
$401$	142185.	−	142185.i	0.884230	−	0.884230i	−0.109732	−	0.993961i	$-0.534999\pi$
$401$	142185.	−	142185.i	0.884230	−	0.884230i	0.993961	+	0.109732i	$0.0349991\pi$
$402$		0			0
$403$	−311689.	−	2985.07i	−1.91916	−	0.0183800i
$404$	−405531.			−2.48463
$405$		0			0
$406$	218141.			1.32338
$407$			123861.i			0.747730i
$408$		0			0
$409$	9949.25	+	9949.25i	0.0594763	+	0.0594763i	0.736219	−	0.676743i	$-0.236609\pi$
$409$	9949.25	+	9949.25i	0.0594763	+	0.0594763i	−0.676743	+	0.736219i	$0.736609\pi$
$410$	−138521.	+	138521.i	−0.824042	+	0.824042i
$411$		0			0
$412$	363282.			2.14017
$413$			12466.5i			0.0730876i
$414$		0			0
$415$			54165.8i			0.314506i
$416$	−130779.	−	1252.48i	−0.755701	−	0.00723742i
$417$		0			0
$418$	185015.	+	185015.i	1.05890	+	1.05890i
$419$	17793.2			0.101351			0.0506753	−	0.998715i	$-0.483863\pi$
$419$	17793.2			0.101351			0.0506753	+	0.998715i	$0.483863\pi$
$420$		0			0
$421$	−123242.	−	123242.i	−0.695334	−	0.695334i	0.268066	−	0.963400i	$-0.413615\pi$
$421$	−123242.	−	123242.i	−0.695334	−	0.695334i	−0.963400	+	0.268066i	$0.913615\pi$
$422$	214671.	+	214671.i	1.20545	+	1.20545i
$423$		0			0
$424$	278120.	−	278120.i	1.54704	−	1.54704i
$425$	72050.5			0.398896
$426$		0			0
$427$	−211487.	+	211487.i	−1.15992	+	1.15992i
$428$			396180.i			2.16274i
$429$		0			0
$430$	−237413.			−1.28401
$431$	81585.7	+	81585.7i	0.439197	+	0.439197i	0.891742	−	0.452545i	$-0.149484\pi$
$431$	81585.7	+	81585.7i	0.439197	+	0.439197i	−0.452545	+	0.891742i	$0.649484\pi$
$432$		0			0
$433$			60799.1i			0.324281i	0.986768	+	0.162141i	$0.0518398\pi$
$433$			60799.1i			0.324281i	−0.986768	+	0.162141i	$0.948160\pi$
$434$	−368817.	−	368817.i	−1.95809	−	1.95809i
$435$		0			0
$436$	135812.	−	135812.i	0.714439	−	0.714439i
$437$	163475.	−	163475.i	0.856027	−	0.856027i
$438$		0			0
$439$		−	275044.i		−	1.42716i	−0.700572	−	0.713581i	$-0.747072\pi$
$439$		−	275044.i		−	1.42716i	0.700572	−	0.713581i	$-0.252928\pi$
$440$	85280.9	−	85280.9i	0.440500	−	0.440500i
$441$		0			0
$442$	2760.06	−	288194.i	0.0141278	−	1.47516i
$443$	8716.50			0.0444155			0.0222077	−	0.999753i	$-0.492930\pi$
$443$	8716.50			0.0444155			0.0222077	+	0.999753i	$0.492930\pi$
$444$		0			0
$445$	158547.			0.800642
$446$		−	245078.i		−	1.23207i
$447$		0			0
$448$	−189431.	−	189431.i	−0.943832	−	0.943832i
$449$	−246638.	+	246638.i	−1.22340	+	1.22340i	−0.256982	+	0.966416i	$0.582728\pi$
$449$	−246638.	+	246638.i	−1.22340	+	1.22340i	−0.966416	+	0.256982i	$0.917272\pi$
$450$		0			0
$451$	−132454.			−0.651197
$452$			95423.5i			0.467066i
$453$		0			0
$454$		−	191086.i		−	0.927078i
$455$	95815.0	−	93997.2i	0.462819	−	0.454038i
$456$		0			0
$457$	55246.3	+	55246.3i	0.264527	+	0.264527i	0.826890	−	0.562363i	$-0.190108\pi$
$457$	55246.3	+	55246.3i	0.264527	+	0.264527i	−0.562363	+	0.826890i	$0.690108\pi$
$458$	487672.			2.32486
$459$		0			0
$460$	−177591.	−	177591.i	−0.839277	−	0.839277i
$461$	31473.0	+	31473.0i	0.148093	+	0.148093i	0.777266	−	0.629172i	$-0.216606\pi$
$461$	31473.0	+	31473.0i	0.148093	+	0.148093i	−0.629172	+	0.777266i	$0.716606\pi$
$462$		0			0
$463$	134703.	−	134703.i	0.628371	−	0.628371i	−0.319287	−	0.947658i	$-0.603443\pi$
$463$	134703.	−	134703.i	0.628371	−	0.628371i	0.947658	+	0.319287i	$0.103443\pi$
$464$	−55333.9			−0.257013
$465$		0			0
$466$	−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	$0.921259\pi$
$468$		0			0
$469$	56787.6			0.258171
$470$	8088.34	+	8088.34i	0.0366154	+	0.0366154i
$471$		0			0
$472$		−	22765.0i		−	0.102184i
$473$	−113507.	−	113507.i	−0.507342	−	0.507342i
$474$		0			0
$475$	93999.4	−	93999.4i	0.416618	−	0.416618i
$476$	216422.	−	216422.i	0.955186	−	0.955186i
$477$		0			0
$478$		−	514301.i		−	2.25093i
$479$	−69279.4	+	69279.4i	−0.301949	+	0.301949i	−0.841776	−	0.539827i	$-0.818490\pi$
$479$	−69279.4	+	69279.4i	−0.301949	+	0.301949i	0.539827	+	0.841776i	$0.318490\pi$
$480$		0			0
$481$	2410.64	−	251709.i	0.0104194	−	1.08795i
$482$	−219685.			−0.945597
$483$		0			0
$484$	−214719.			−0.916599
$485$		−	146533.i		−	0.622950i
$486$		0			0
$487$	334191.	+	334191.i	1.40908	+	1.40908i	0.764713	+	0.644372i	$0.222881\pi$
$487$	334191.	+	334191.i	1.40908	+	1.40908i	0.644372	+	0.764713i	$0.277119\pi$
$488$	386195.	−	386195.i	1.62169	−	1.62169i
$489$		0			0
$490$	−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	$-0.139321\pi$
$492$		0			0
$493$			198785.i			0.817880i
$494$	−372386.	−	379588.i	−1.52595	−	1.55546i
$495$		0			0
$496$	93554.7	+	93554.7i	0.380279	+	0.380279i
$497$	−184287.			−0.746074
$498$		0			0
$499$	263314.	+	263314.i	1.05748	+	1.05748i	0.998244	+	0.0592381i	$0.0188671\pi$
$499$	263314.	+	263314.i	1.05748	+	1.05748i	0.0592381	+	0.998244i	$0.481133\pi$
$500$	−330391.	−	330391.i	−1.32157	−	1.32157i
$501$		0			0
$502$	−241773.	+	241773.i	−0.959403	+	0.959403i
$503$	−120971.			−0.478131			−0.239065	−	0.971003i	$-0.576841\pi$
$503$	−120971.			−0.478131			−0.239065	+	0.971003i	$0.576841\pi$
$504$		0			0
$505$	191758.	−	191758.i	0.751920	−	0.751920i
$506$		−	267573.i		−	1.04506i
$507$		0			0
$508$	−609990.			−2.36371
$509$	50199.3	+	50199.3i	0.193759	+	0.193759i	0.797318	−	0.603559i	$-0.206251\pi$
$509$	50199.3	+	50199.3i	0.193759	+	0.193759i	−0.603559	+	0.797318i	$0.706251\pi$
$510$		0			0
$511$		−	50775.9i		−	0.194454i
$512$	102587.	+	102587.i	0.391337	+	0.391337i
$513$		0			0
$514$	−411688.	+	411688.i	−1.55827	+	1.55827i
$515$	−171780.	+	171780.i	−0.647677	+	0.647677i
$516$		0			0
$517$			7734.07i			0.0289352i
$518$	297845.	−	297845.i	1.11002	−	1.11002i
$519$		0			0
$520$	−174967.	+	171648.i	−0.647068	+	0.634792i
$521$	307242.			1.13189			0.565946	−	0.824443i	$-0.308511\pi$
$521$	307242.			1.13189			0.565946	+	0.824443i	$0.308511\pi$
$522$		0			0
$523$	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
$525$		0			0
$526$	531501.	+	531501.i	1.92102	+	1.92102i
$527$	336092.	−	336092.i	1.21014	−	1.21014i
$528$		0			0
$529$	43420.3			0.155161
$530$			619895.i			2.20682i
$531$		0			0
$532$		−	564703.i		−	1.99525i
$533$	269173.	+	2577.89i	0.947494	+	0.00907424i
$534$		0			0
$535$	−187337.	−	187337.i	−0.654508	−	0.654508i
$536$	−103699.			−0.360950
$537$		0			0
$538$	26428.6	+	26428.6i	0.0913081	+	0.0913081i
$539$	−33799.3	−	33799.3i	−0.116340	−	0.116340i
$540$		0			0
$541$	184394.	−	184394.i	0.630017	−	0.630017i	−0.318055	−	0.948072i	$-0.603030\pi$
$541$	184394.	−	184394.i	0.630017	−	0.630017i	0.948072	+	0.318055i	$0.103030\pi$
$542$	34836.6			0.118587
$543$		0			0
$544$	141018.	−	141018.i	0.476515	−	0.476515i
$545$			128439.i			0.432419i
$546$		0			0
$547$	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
$549$		0			0
$550$		−	153857.i		−	0.508619i
$551$	259341.	+	259341.i	0.854217	+	0.854217i
$552$		0			0
$553$	−122216.	+	122216.i	−0.399647	+	0.399647i
$554$	−148209.	+	148209.i	−0.482898	+	0.482898i
$555$		0			0
$556$		−	129461.i		−	0.418784i
$557$	−200107.	+	200107.i	−0.644987	+	0.644987i	−0.951777	−	0.306790i	$-0.900745\pi$
$557$	−200107.	+	200107.i	−0.644987	+	0.644987i	0.306790	+	0.951777i	$0.400745\pi$
$558$		0			0
$559$	228460.	+	232878.i	0.731115	+	0.745254i
$560$	−56973.0			−0.181674
$561$		0			0
$562$	−101983.			−0.322891
$563$			488676.i			1.54172i	0.637007	+	0.770858i	$0.280172\pi$
$563$			488676.i			1.54172i	−0.637007	+	0.770858i	$0.719828\pi$
$564$		0			0
$565$	−45121.7	−	45121.7i	−0.141348	−	0.141348i
$566$	−145033.	+	145033.i	−0.452725	+	0.452725i
$567$		0			0
$568$	336525.			1.04309
$569$		−	562947.i		−	1.73877i	−0.494132	−	0.869387i	$-0.664514\pi$
$569$		−	562947.i		−	1.73877i	0.494132	−	0.869387i	$-0.335486\pi$
$570$		0			0
$571$			183359.i			0.562380i	0.959652	+	0.281190i	$0.0907291\pi$
$571$			183359.i			0.562380i	−0.959652	+	0.281190i	$0.909271\pi$
$572$	−390564.	−	3740.47i	−1.19371	−	0.0114323i
$573$		0			0
$574$	318509.	+	318509.i	0.966713	+	0.966713i
$575$	−135944.			−0.411173
$576$		0			0
$577$	196407.	+	196407.i	0.589938	+	0.589938i	0.937614	−	0.347677i	$-0.113029\pi$
$577$	196407.	+	196407.i	0.589938	+	0.589938i	−0.347677	+	0.937614i	$0.613029\pi$
$578$	−80064.2	−	80064.2i	−0.239653	−	0.239653i
$579$		0			0
$580$	281736.	−	281736.i	0.837503	−	0.837503i
$581$	124546.			0.368958
$582$		0			0
$583$	−296372.	+	296372.i	−0.871966	+	0.871966i
$584$			92721.6i			0.271866i
$585$		0			0
$586$	−1.06254e6			−3.09422
$587$	127057.	+	127057.i	0.368742	+	0.368742i	0.867018	−	0.498276i	$-0.166034\pi$
$587$	127057.	+	127057.i	0.368742	+	0.368742i	−0.498276	+	0.867018i	$0.666034\pi$
$588$		0			0
$589$		−	876953.i		−	2.52782i
$590$	25370.2	+	25370.2i	0.0728818	+	0.0728818i
$591$		0			0
$592$	−75551.7	+	75551.7i	−0.215576	+	0.215576i
$593$	−168617.	+	168617.i	−0.479503	+	0.479503i	−0.904973	−	0.425470i	$-0.860109\pi$
$593$	−168617.	+	168617.i	−0.479503	+	0.479503i	0.425470	+	0.904973i	$0.360109\pi$
$594$		0			0
$595$			204674.i			0.578133i
$596$	10947.1	−	10947.1i	0.0308181	−	0.0308181i
$597$		0			0
$598$	−5207.65	+	543761.i	−0.0145626	+	1.52057i
$599$	22624.2			0.0630550			0.0315275	−	0.999503i	$-0.489963\pi$
$599$	22624.2			0.0630550			0.0315275	+	0.999503i	$0.489963\pi$
$600$		0			0
$601$	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
$603$		0			0
$604$	−211802.	−	211802.i	−0.580572	−	0.580572i
$605$	101531.	−	101531.i	0.277389	−	0.277389i
$606$		0			0
$607$	−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
$609$		0			0
$610$			860782.i			2.31331i
$611$	150.525	−	15717.1i	0.000403204	−	0.0421009i
$612$		0			0
$613$	−230276.	−	230276.i	−0.612812	−	0.612812i	0.330866	−	0.943678i	$-0.392659\pi$
$613$	−230276.	−	230276.i	−0.612812	−	0.612812i	−0.943678	+	0.330866i	$0.892659\pi$
$614$	−703981.			−1.86734
$615$		0			0
$616$	−196090.	−	196090.i	−0.516767	−	0.516767i
$617$	−34766.2	−	34766.2i	−0.0913244	−	0.0913244i	0.659969	−	0.751293i	$-0.270570\pi$
$617$	−34766.2	−	34766.2i	−0.0913244	−	0.0913244i	−0.751293	+	0.659969i	$0.770570\pi$
$618$		0			0
$619$	−420362.	+	420362.i	−1.09709	+	1.09709i	−0.102339	+	0.994750i	$0.532633\pi$
$619$	−420362.	+	420362.i	−1.09709	+	1.09709i	−0.994750	+	0.102339i	$0.967367\pi$
$620$	−952680.			−2.47836
$621$		0			0
$622$	703892.	−	703892.i	1.81939	−	1.81939i
$623$		−	364555.i		−	0.939262i
$624$		0			0
$625$	137714.			0.352547
$626$	−592423.	−	592423.i	−1.51176	−	1.51176i
$627$		0			0
$628$			711130.i			1.80314i
$629$	271417.	+	271417.i	0.686018	+	0.686018i
$630$		0			0
$631$	93922.3	−	93922.3i	0.235890	−	0.235890i	−0.579256	−	0.815146i	$-0.696657\pi$
$631$	93922.3	−	93922.3i	0.235890	−	0.235890i	0.815146	+	0.579256i	$0.196657\pi$
$632$	223177.	−	223177.i	0.558748	−	0.558748i
$633$		0			0
$634$		−	396347.i		−	0.986045i
$635$	288438.	−	288438.i	0.715327	−	0.715327i
$636$		0			0
$637$	68029.0	+	69344.6i	0.167654	+	0.170897i
$638$	424487.			1.04285
$639$		0			0
$640$	−540888.			−1.32053
$641$		−	123299.i		−	0.300084i	−0.988680	−	0.150042i	$-0.952059\pi$
$641$		−	123299.i		−	0.300084i	0.988680	−	0.150042i	$-0.0479409\pi$
$642$		0			0
$643$	−280732.	−	280732.i	−0.678999	−	0.678999i	0.280774	−	0.959774i	$-0.409409\pi$
$643$	−280732.	−	280732.i	−0.678999	−	0.678999i	−0.959774	+	0.280774i	$0.909409\pi$
$644$	−408343.	+	408343.i	−0.984586	+	0.984586i
$645$		0			0
$646$	810850.			1.94301
$647$		−	45435.6i		−	0.108539i	−0.998526	−	0.0542697i	$-0.982717\pi$
$647$		−	45435.6i		−	0.108539i	0.998526	−	0.0542697i	$-0.0172831\pi$
$648$		0			0
$649$			24258.9i			0.0575946i
$650$	−2994.45	+	312668.i	−0.00708746	+	0.740042i
$651$		0			0
$652$	−461375.	−	461375.i	−1.08532	−	1.08532i
$653$	378723.			0.888168			0.444084	−	0.895985i	$-0.353529\pi$
$653$	378723.			0.888168			0.444084	+	0.895985i	$0.353529\pi$
$654$		0			0
$655$	−21791.0	−	21791.0i	−0.0507920	−	0.0507920i
$656$	−80793.4	−	80793.4i	−0.187745	−	0.187745i
$657$		0			0
$658$	18597.9	−	18597.9i	0.0429548	−	0.0429548i
$659$	−289116.			−0.665735			−0.332867	−	0.942974i	$-0.608016\pi$
$659$	−289116.			−0.665735			−0.332867	+	0.942974i	$0.608016\pi$
$660$		0			0
$661$	−168774.	+	168774.i	−0.386281	+	0.386281i	−0.873359	−	0.487078i	$-0.838063\pi$
$661$	−168774.	+	168774.i	−0.386281	+	0.386281i	0.487078	+	0.873359i	$0.338063\pi$
$662$		−	220807.i		−	0.503845i
$663$		0			0
$664$	−227433.			−0.515842
$665$	267024.	+	267024.i	0.603819	+	0.603819i
$666$		0			0
$667$		−	375065.i		−	0.843054i
$668$	1.05317e6	+	1.05317e6i	2.36019	+	2.36019i
$669$		0			0
$670$	115567.	−	115567.i	0.257444	−	0.257444i
$671$	−411539.	+	411539.i	−0.914043	+	0.914043i
$672$		0			0
$673$		−	163719.i		−	0.361467i	−0.983532	−	0.180734i	$-0.942153\pi$
$673$		−	163719.i		−	0.361467i	0.983532	−	0.180734i	$-0.0578472\pi$
$674$	537482.	−	537482.i	1.18316	−	1.18316i
$675$		0			0
$676$	793630.	+	15202.7i	1.73670	+	0.0332681i
$677$	−462025.			−1.00806			−0.504032	−	0.863685i	$-0.668151\pi$
$677$	−462025.			−1.00806			−0.504032	+	0.863685i	$0.668151\pi$
$678$		0			0
$679$	−336931.			−0.730805
$680$		−	373753.i		−	0.808290i
$681$		0			0
$682$	−717693.	−	717693.i	−1.54301	−	1.54301i
$683$	213724.	−	213724.i	0.458155	−	0.458155i	−0.439895	−	0.898049i	$-0.644984\pi$
$683$	213724.	−	213724.i	0.458155	−	0.458155i	0.898049	+	0.439895i	$0.144984\pi$
$684$		0			0
$685$	−567838.			−1.21016
$686$			841544.i			1.78825i
$687$		0			0
$688$		−	138473.i		−	0.292541i
$689$	608054.	−	596517.i	1.28087	−	1.25656i
$690$		0			0
$691$	−344817.	−	344817.i	−0.722159	−	0.722159i	0.246886	−	0.969045i	$-0.420593\pi$
$691$	−344817.	−	344817.i	−0.722159	−	0.722159i	−0.969045	+	0.246886i	$0.920593\pi$
$692$	−47605.4			−0.0994132
$693$		0			0
$694$	437094.	+	437094.i	0.907520	+	0.907520i
$695$	61216.6	+	61216.6i	0.126736	+	0.126736i
$696$		0			0
$697$	−290248.	+	290248.i	−0.597452	+	0.597452i
$698$	135507.			0.278132
$699$		0			0
$700$	−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	$-0.0271050\pi$
$702$		0			0
$703$	708198.			1.43299
$704$	−368619.	−	368619.i	−0.743760	−	0.743760i
$705$		0			0
$706$			1.18928e6i			2.38602i
$707$	−440919.	−	440919.i	−0.882104	−	0.882104i
$708$		0			0
$709$	507737.	−	507737.i	1.01006	−	1.01006i	0.0101092	−	0.999949i	$-0.496782\pi$
$709$	507737.	−	507737.i	1.01006	−	1.01006i	0.999949	−	0.0101092i	$-0.00321793\pi$
$710$	−375037.	+	375037.i	−0.743973	+	0.743973i
$711$		0			0
$712$			665711.i			1.31318i
$713$	−634135.	+	634135.i	−1.24739	+	1.24739i
$714$		0			0
$715$	186450.	−	182912.i	0.364711	−	0.357792i
$716$	75051.0			0.146396
$717$		0			0
$718$	379640.			0.736415
$719$			50596.5i			0.0978729i	0.998802	+	0.0489364i	$0.0155832\pi$
$719$			50596.5i			0.0978729i	−0.998802	+	0.0489364i	$0.984417\pi$
$720$		0			0
$721$	394982.	+	394982.i	0.759814	+	0.759814i
$722$	448046.	−	448046.i	0.859505	−	0.859505i
$723$		0			0
$724$	870610.			1.66091
$725$		−	215666.i		−	0.410304i
$726$		0			0
$727$			891410.i			1.68659i	0.537453	+	0.843294i	$0.319387\pi$
$727$			891410.i			1.68659i	−0.537453	+	0.843294i	$0.680613\pi$
$728$	394678.	+	402310.i	0.744697	+	0.759099i
$729$		0			0
$730$	−103333.	−	103333.i	−0.193906	−	0.193906i
$731$	−497458.			−0.930939
$732$		0			0
$733$	−439182.	−	439182.i	−0.817404	−	0.817404i	0.168327	−	0.985731i	$-0.446164\pi$
$733$	−439182.	−	439182.i	−0.817404	−	0.817404i	−0.985731	+	0.168327i	$0.946164\pi$
$734$	190672.	+	190672.i	0.353912	+	0.353912i
$735$		0			0
$736$	−266071.	+	266071.i	−0.491182	+	0.491182i
$737$	110505.			0.203445
$738$		0			0
$739$	83855.7	−	83855.7i	0.153548	−	0.153548i	−0.626153	−	0.779701i	$-0.715371\pi$
$739$	83855.7	−	83855.7i	0.153548	−	0.153548i	0.779701	+	0.626153i	$0.215371\pi$
$740$		−	769353.i		−	1.40495i
$741$		0			0
$742$	1.42535e6			2.58890
$743$	153273.	+	153273.i	0.277643	+	0.277643i	0.832167	−	0.554524i	$-0.187100\pi$
$743$	153273.	+	153273.i	0.277643	+	0.277643i	−0.554524	+	0.832167i	$0.687100\pi$
$744$		0			0
$745$			10352.8i			0.0186529i
$746$	207543.	+	207543.i	0.372933	+	0.372933i
$747$		0			0
$748$	421143.	−	421143.i	0.752708	−	0.752708i
$749$	−430752.	+	430752.i	−0.767827	+	0.767827i
$750$		0			0
$751$			208551.i			0.369771i	0.982760	+	0.184886i	$0.0591915\pi$
$751$			208551.i			0.369771i	−0.982760	+	0.184886i	$0.940808\pi$
$752$	−4717.57	+	4717.57i	−0.00834225	+	0.00834225i
$753$		0			0
$754$	−862640.	−	8261.59i	−1.51735	−	0.0145319i
$755$	200304.			0.351395
$756$		0			0
$757$	−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
$759$		0			0
$760$	−487610.	−	487610.i	−0.844201	−	0.844201i
$761$	−790313.	+	790313.i	−1.36468	+	1.36468i	−0.496826	+	0.867850i	$0.665501\pi$
$761$	−790313.	+	790313.i	−1.36468	+	1.36468i	−0.867850	+	0.496826i	$0.834499\pi$
$762$		0			0
$763$	295326.			0.507287
$764$		−	1.46682e6i		−	2.51299i
$765$		0			0
$766$		−	1.55746e6i		−	2.65436i
$767$	472.140	−	49298.9i	0.000802565	−	0.0838005i
$768$		0			0
$769$	170652.	+	170652.i	0.288574	+	0.288574i	0.836516	−	0.547942i	$-0.184589\pi$
$769$	170652.	+	170652.i	0.288574	+	0.288574i	−0.547942	+	0.836516i	$0.684589\pi$
$770$	437061.			0.737158
$771$		0			0
$772$	39070.1	+	39070.1i	0.0655555	+	0.0655555i
$773$	−238774.	−	238774.i	−0.399602	−	0.399602i	0.478490	−	0.878093i	$-0.341184\pi$
$773$	−238774.	−	238774.i	−0.399602	−	0.399602i	−0.878093	+	0.478490i	$0.841184\pi$
$774$		0			0
$775$	−364634.	+	364634.i	−0.607090	+	0.607090i
$776$	615268.			1.02174
$777$		0			0
$778$	77091.7	−	77091.7i	0.127365	−	0.127365i
$779$			757333.i			1.24799i
$780$		0			0
$781$	−358610.			−0.587923
$782$	−586335.	−	586335.i	−0.958810	−	0.958810i
$783$		0			0
$784$		−	41233.3i		−	0.0670836i
$785$	−336263.	−	336263.i	−0.545682	−	0.545682i
$786$		0			0
$787$	−460701.	+	460701.i	−0.743823	+	0.743823i	−0.973311	−	0.229488i	$-0.926295\pi$
$787$	−460701.	+	460701.i	−0.743823	+	0.743823i	0.229488	+	0.973311i	$0.426295\pi$
$788$	282793.	−	282793.i	0.455424	−	0.455424i
$789$		0			0
$790$			497435.i			0.797044i
$791$	−103750.	+	103750.i	−0.165820	+	0.165820i
$792$		0			0
$793$	844339.	−	828320.i	1.34267	−	1.31720i
$794$	1.45872e6			2.31383
$795$		0			0
$796$	−206762.			−0.326320
$797$		−	940096.i		−	1.47998i	−0.672618	−	0.739989i	$-0.734830\pi$
$797$		−	940096.i		−	1.47998i	0.672618	−	0.739989i	$-0.265170\pi$
$798$		0			0
$799$	16947.7	+	16947.7i	0.0265471	+	0.0265471i
$800$	−152993.	+	152993.i	−0.239052	+	0.239052i
$801$		0			0
$802$	−1.33066e6			−2.06880
$803$		−	98806.5i		−	0.153234i
$804$		0			0
$805$		−	386176.i		−	0.595928i
$806$	1.44453e6	+	1.47246e6i	2.22359	+	2.26660i
$807$		0			0
$808$	805159.	+	805159.i	1.23327	+	1.23327i
$809$	−612686.			−0.936140			−0.468070	−	0.883691i	$-0.655050\pi$
$809$	−612686.			−0.936140			−0.468070	+	0.883691i	$0.655050\pi$
$810$		0			0
$811$	431732.	+	431732.i	0.656406	+	0.656406i	0.954528	−	0.298121i	$-0.0963601\pi$
$811$	431732.	+	431732.i	0.656406	+	0.656406i	−0.298121	+	0.954528i	$0.596360\pi$
$812$	−647809.	−	647809.i	−0.982505	−	0.982505i
$813$		0			0
$814$	579585.	−	579585.i	0.874719	−	0.874719i
$815$	436329.			0.656900
$816$		0			0
$817$	−648999.	+	648999.i	−0.972300	+	0.972300i
$818$		−	93111.7i		−	0.139155i
$819$		0			0
$820$	822730.			1.22357
$821$	193284.	+	193284.i	0.286754	+	0.286754i	0.835795	−	0.549041i	$-0.185007\pi$
$821$	193284.	+	193284.i	0.286754	+	0.286754i	−0.549041	+	0.835795i	$0.685007\pi$
$822$		0			0
$823$		−	547967.i		−	0.809011i	−0.914536	−	0.404506i	$-0.867444\pi$
$823$		−	547967.i		−	0.809011i	0.914536	−	0.404506i	$-0.132556\pi$
$824$	−721275.	−	721275.i	−1.06230	−	1.06230i
$825$		0			0
$826$	58334.8	−	58334.8i	0.0855003	−	0.0855003i
$827$	859953.	−	859953.i	1.25737	−	1.25737i	0.305029	−	0.952343i	$-0.401334\pi$
$827$	859953.	−	859953.i	1.25737	−	1.25737i	0.952343	−	0.305029i	$-0.0986660\pi$
$828$		0			0
$829$			726172.i			1.05665i	0.849043	+	0.528324i	$0.177179\pi$
$829$			726172.i			1.05665i	−0.849043	+	0.528324i	$0.822821\pi$
$830$	253460.	−	253460.i	0.367919	−	0.367919i
$831$		0			0
$832$	741933.	+	756282.i	1.07181	+	1.09254i
$833$	−148129.			−0.213477
$834$		0			0
$835$	−995999.			−1.42852
$836$		−	1.09887e6i		−	1.57230i
$837$		0			0
$838$	−83260.3	−	83260.3i	−0.118563	−	0.118563i
$839$	507389.	−	507389.i	0.720804	−	0.720804i	−0.247965	−	0.968769i	$-0.579762\pi$
$839$	507389.	−	507389.i	0.720804	−	0.720804i	0.968769	+	0.247965i	$0.0797618\pi$
$840$		0			0
$841$	−112265.			−0.158728
$842$			1.15338e6i			1.62685i
$843$		0			0
$844$		−	1.27501e6i		−	1.78990i
$845$	−382462.	+	368085.i	−0.535642	+	0.515507i
$846$		0			0
$847$	−233456.	−	233456.i	−0.325415	−	0.325415i
$848$	−361557.			−0.502789
$849$		0			0
$850$	−337148.	−	337148.i	−0.466641	−	0.466641i
$851$	−512107.	−	512107.i	−0.707133	−	0.707133i
$852$		0			0
$853$	192677.	−	192677.i	0.264808	−	0.264808i	−0.562196	−	0.827004i	$-0.690043\pi$
$853$	192677.	−	192677.i	0.264808	−	0.264808i	0.827004	+	0.562196i	$0.190043\pi$
$854$	1.97924e6			2.71383
$855$		0			0
$856$	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	$0.979001\pi$
$858$		0			0
$859$	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
$861$		0			0
$862$		−	763533.i		−	1.02757i
$863$	−1.02273e6	−	1.02273e6i	−1.37321	−	1.37321i	−0.855640	−	0.517571i	$-0.826836\pi$
$863$	−1.02273e6	−	1.02273e6i	−1.37321	−	1.37321i	−0.517571	−	0.855640i	$-0.673164\pi$
$864$		0			0
$865$	22510.5	−	22510.5i	0.0300853	−	0.0300853i
$866$	284499.	−	284499.i	0.379355	−	0.379355i
$867$		0			0
$868$			2.19054e6i			2.90745i
$869$	−237824.	+	237824.i	−0.314931	+	0.314931i
$870$		0			0
$871$	−224567.	−	2150.70i	−0.296013	−	0.00283494i
$872$	−539294.			−0.709239
$873$		0			0
$874$	−1.52990e6			−2.00282
$875$		−	718444.i		−	0.938376i
$876$		0			0
$877$	263404.	+	263404.i	0.342470	+	0.342470i	0.857295	−	0.514825i	$-0.172143\pi$
$877$	263404.	+	263404.i	0.342470	+	0.342470i	−0.514825	+	0.857295i	$0.672143\pi$
$878$	−1.28702e6	+	1.28702e6i	−1.66954	+	1.66954i
$879$		0			0
$880$	−110866.			−0.143163
$881$			580702.i			0.748171i	0.927394	+	0.374086i	$0.122043\pi$
$881$			580702.i			0.748171i	−0.927394	+	0.374086i	$0.877957\pi$
$882$		0			0
$883$		−	1.28367e6i		−	1.64638i	−0.567763	−	0.823192i	$-0.692191\pi$
$883$		−	1.28367e6i		−	1.64638i	0.567763	−	0.823192i	$-0.307809\pi$
$884$	−864042.	+	847649.i	−1.10568	+	1.08470i
$885$		0			0
$886$	−40787.4	−	40787.4i	−0.0519587	−	0.0519587i
$887$	−166700.			−0.211879			−0.105940	−	0.994373i	$-0.533785\pi$
$887$	−166700.			−0.211879			−0.105940	+	0.994373i	$0.533785\pi$
$888$		0			0
$889$	−663219.	−	663219.i	−0.839176	−	0.839176i
$890$	−741894.	−	741894.i	−0.936617	−	0.936617i
$891$		0			0
$892$	−727805.	+	727805.i	−0.914714	+	0.914714i
$893$	44221.1			0.0554532
$894$		0			0
$895$	−35488.4	+	35488.4i	−0.0443037	+	0.0443037i
$896$			1.24369e6i			1.54916i
$897$		0			0
$898$	2.30820e6			2.86234
$899$	−1.00601e6	−	1.00601e6i	−1.24475	−	1.24475i
$900$		0			0
$901$			1.29888e6i			1.60000i
$902$	619796.	+	619796.i	0.761791	+	0.761791i
$903$		0			0
$904$	189458.	−	189458.i	0.231833	−	0.231833i
$905$	−411674.	+	411674.i	−0.502639	+	0.502639i
$906$		0			0
$907$		−	1.06525e6i		−	1.29490i	−0.762108	−	0.647450i	$-0.775836\pi$
$907$		−	1.06525e6i		−	1.29490i	0.762108	−	0.647450i	$-0.224164\pi$
$908$	−567464.	+	567464.i	−0.688282	+	0.688282i
$909$		0			0
$910$	−888194.	−	8506.32i	−1.07257	−	0.0102721i
$911$	794671.			0.957526			0.478763	−	0.877944i	$-0.341085\pi$
$911$	794671.			0.957526			0.478763	+	0.877944i	$0.341085\pi$
$912$		0			0
$913$	242358.			0.290747
$914$		−	517031.i		−	0.618905i
$915$		0			0
$916$	−1.44823e6	−	1.44823e6i	−1.72603	−	1.72603i
$917$	−50105.1	+	50105.1i	−0.0595859	+	0.0595859i
$918$		0			0
$919$	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
$921$		0			0
$922$		−	294545.i		−	0.346489i
$923$	728766.	+	6979.46i	0.855431	+	0.00819254i
$924$		0			0
$925$	−294466.	−	294466.i	−0.344153	−	0.344153i
$926$	−1.26064e6			−1.47018
$927$		0			0
$928$	−422104.	−	422104.i	−0.490143	−	0.490143i
$929$	746377.	+	746377.i	0.864823	+	0.864823i	0.991894	−	0.127071i	$-0.0405576\pi$
$929$	746377.	+	746377.i	0.864823	+	0.864823i	−0.127071	+	0.991894i	$0.540558\pi$
$930$		0			0
$931$	−193254.	+	193254.i	−0.222961	+	0.222961i
$932$	1.27240e6			1.46484
$933$		0			0
$934$	499758.	−	499758.i	0.572884	−	0.572884i
$935$			398281.i			0.455582i
$936$		0			0
$937$	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
$939$		0			0
$940$		−	48039.7i		−	0.0543681i
$941$	906449.	+	906449.i	1.02368	+	1.02368i	0.999713	+	0.0239676i	$0.00762984\pi$
$941$	906449.	+	906449.i	1.02368	+	1.02368i	0.0239676	+	0.999713i	$0.492370\pi$
$942$		0			0
$943$	547636.	−	547636.i	0.615841	−	0.615841i
$944$	−14797.3	+	14797.3i	−0.0166050	+	0.0166050i
$945$		0			0
$946$			1.06227e6i			1.18701i
$947$	−290385.	+	290385.i	−0.323798	+	0.323798i	−0.850222	−	0.526424i	$-0.823532\pi$
$947$	−290385.	+	290385.i	−0.323798	+	0.323798i	0.526424	+	0.850222i	$0.323532\pi$
$948$		0			0
$949$	−1923.03	+	200794.i	−0.00213527	+	0.222956i
$950$	−879709.			−0.974747
$951$		0			0
$952$	−859388.			−0.948234
$953$		−	1.13841e6i		−	1.25347i	−0.779234	−	0.626733i	$-0.784392\pi$
$953$		−	1.13841e6i		−	1.25347i	0.779234	−	0.626733i	$-0.215608\pi$
$954$		0			0
$955$	693596.	+	693596.i	0.760501	+	0.760501i
$956$	−1.52731e6	+	1.52731e6i	−1.67114	+	1.67114i
$957$		0			0
$958$	648362.			0.706459
$959$			1.30566e6i			1.41968i
$960$		0			0
$961$			2.47827e6i			2.68350i
$962$	−1.18911e6	+	1.16655e6i	−1.28491	+	1.26053i
$963$		0			0
$964$	652395.	+	652395.i	0.702032	+	0.702032i
$965$	−36949.1			−0.0396780
$966$		0			0
$967$	19218.1	+	19218.1i	0.0205521	+	0.0205521i	0.717308	−	0.696756i	$-0.245374\pi$
$967$	19218.1	+	19218.1i	0.0205521	+	0.0205521i	−0.696756	+	0.717308i	$0.745374\pi$
$968$	426312.	+	426312.i	0.454964	+	0.454964i
$969$		0			0
$970$	−685678.	+	685678.i	−0.728747	+	0.728747i
$971$	−416812.			−0.442080			−0.221040	−	0.975265i	$-0.570945\pi$
$971$	−416812.			−0.442080			−0.221040	+	0.975265i	$0.570945\pi$
$972$		0			0
$973$	140758.	−	140758.i	0.148678	−	0.148678i
$974$		−	3.12758e6i		−	3.29679i
$975$		0			0
$976$	−502056.			−0.527051
$977$	−394416.	−	394416.i	−0.413205	−	0.413205i	0.469649	−	0.882853i	$-0.344381\pi$
$977$	−394416.	−	394416.i	−0.413205	−	0.413205i	−0.882853	+	0.469649i	$0.844381\pi$
$978$		0			0
$979$		−	709399.i		−	0.740159i
$980$	209942.	+	209942.i	0.218599	+	0.218599i
$981$		0			0
$982$	−956284.	+	956284.i	−0.991663	+	0.991663i
$983$	159479.	−	159479.i	0.165043	−	0.165043i	−0.619754	−	0.784796i	$-0.712767\pi$
$983$	159479.	−	159479.i	0.165043	−	0.165043i	0.784796	+	0.619754i	$0.212767\pi$
$984$		0			0
$985$			267441.i			0.275649i
$986$	930181.	−	930181.i	0.956783	−	0.956783i
$987$		0			0
$988$	−21386.9	+	2.23313e6i	−0.0219095	+	2.28770i
$989$	938598.			0.959593
$990$		0			0
$991$	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
$993$		0			0
$994$	862340.	+	862340.i	0.872782	+	0.872782i
$995$	97768.8	−	97768.8i	0.0987539	−	0.0987539i
$996$		0			0
$997$	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
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.9	✓	20	39.5	even	4
39.5.g.a.34.9	yes	20	3.2	odd	2
117.5.j.b.73.2		20	1.1	even	1	trivial
117.5.j.b.109.2		20	13.5	odd	4	inner

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.67933 - 4.67933i) q^{2} +27.7923i q^{4} +(-13.1418 - 13.1418i) q^{5} +(-30.2175 + 30.2175i) q^{7} +(55.1800 - 55.1800i) q^{8} +122.989i q^{10} +(-58.8012 + 58.8012i) q^{11} +(120.640 - 118.351i) q^{13} +282.795 q^{14} -71.7343 q^{16} +257.703i q^{17} +(336.208 + 336.208i) q^{19} +(365.240 - 365.240i) q^{20} +550.300 q^{22} -486.231i q^{23} -279.587i q^{25} +(-1118.32 - 10.7102i) q^{26} +(-839.813 - 839.813i) q^{28} +771.373 q^{29} +(-1304.18 - 1304.18i) q^{31} +(-547.211 - 547.211i) q^{32} +(1205.88 - 1205.88i) q^{34} +794.223 q^{35} +(1053.22 - 1053.22i) q^{37} -3146.45i q^{38} -1450.33 q^{40} +(1126.29 + 1126.29i) q^{41} +1930.35i q^{43} +(-1634.22 - 1634.22i) q^{44} +(-2275.24 + 2275.24i) q^{46} +(65.7645 - 65.7645i) q^{47} +574.806i q^{49} +(-1308.28 + 1308.28i) q^{50} +(3289.25 + 3352.86i) q^{52} +5040.23 q^{53} +1545.50 q^{55} +3334.80i q^{56} +(-3609.51 - 3609.51i) q^{58} +(206.279 - 206.279i) q^{59} +6998.83 q^{61} +12205.4i q^{62} +6268.91i q^{64} +(-3140.77 - 30.0794i) q^{65} +(-939.648 - 939.648i) q^{67} -7162.15 q^{68} +(-3716.43 - 3716.43i) q^{70} +(3049.34 + 3049.34i) q^{71} +(-840.175 + 840.175i) q^{73} -9856.69 q^{74} +(-9343.97 + 9343.97i) q^{76} -3553.65i q^{77} +4044.54 q^{79} +(942.716 + 942.716i) q^{80} -10540.5i q^{82} +(-2060.82 - 2060.82i) q^{83} +(3386.67 - 3386.67i) q^{85} +(9032.76 - 9032.76i) q^{86} +6489.30i q^{88} +(-6032.18 + 6032.18i) q^{89} +(-69.1630 + 7221.71i) q^{91} +13513.5 q^{92} -615.468 q^{94} -8836.73i q^{95} +(5575.10 + 5575.10i) q^{97} +(2689.71 - 2689.71i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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