LMFDB - Embedded newform 117.5.j.b.73.4

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			73.4
Root			$-1.66937 - 1.66937i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.5.j.b.109.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.66937 - 1.66937i) q^{2} -10.4264i q^{4} +(28.4160 + 28.4160i) q^{5} +(-17.1341 + 17.1341i) q^{7} +(-44.1155 + 44.1155i) q^{8} -94.8736i q^{10} +(-54.2346 + 54.2346i) q^{11} +(-168.969 + 3.22849i) q^{13} +57.2062 q^{14} -19.5322 q^{16} +70.4411i q^{17} +(194.123 + 194.123i) q^{19} +(296.276 - 296.276i) q^{20} +181.075 q^{22} +716.329i q^{23} +989.935i q^{25} +(287.462 + 276.683i) q^{26} +(178.647 + 178.647i) q^{28} -725.866 q^{29} +(586.446 + 586.446i) q^{31} +(738.454 + 738.454i) q^{32} +(117.592 - 117.592i) q^{34} -973.762 q^{35} +(575.923 - 575.923i) q^{37} -648.126i q^{38} -2507.17 q^{40} +(1810.07 + 1810.07i) q^{41} -2879.48i q^{43} +(565.472 + 565.472i) q^{44} +(1195.82 - 1195.82i) q^{46} +(-1064.86 + 1064.86i) q^{47} +1813.85i q^{49} +(1652.57 - 1652.57i) q^{50} +(33.6615 + 1761.74i) q^{52} -2549.76 q^{53} -3082.26 q^{55} -1511.75i q^{56} +(1211.74 + 1211.74i) q^{58} +(2264.35 - 2264.35i) q^{59} +6434.68 q^{61} -1957.99i q^{62} -2152.99i q^{64} +(-4893.16 - 4709.68i) q^{65} +(-5472.27 - 5472.27i) q^{67} +734.447 q^{68} +(1625.57 + 1625.57i) q^{70} +(-4643.32 - 4643.32i) q^{71} +(-5605.57 + 5605.57i) q^{73} -1922.86 q^{74} +(2024.00 - 2024.00i) q^{76} -1858.52i q^{77} -6871.05 q^{79} +(-555.027 - 555.027i) q^{80} -6043.36i q^{82} +(2647.22 + 2647.22i) q^{83} +(-2001.65 + 2001.65i) q^{85} +(-4806.92 + 4806.92i) q^{86} -4785.17i q^{88} +(6804.23 - 6804.23i) q^{89} +(2839.81 - 2950.45i) q^{91} +7468.73 q^{92} +3555.29 q^{94} +11032.4i q^{95} +(9498.95 + 9498.95i) q^{97} +(3027.99 - 3027.99i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.66937	−	1.66937i	−0.417343	−	0.417343i	0.466944	−	0.884287i	$-0.345355\pi$
$2$	−1.66937	−	1.66937i	−0.417343	−	0.417343i	−0.884287	+	0.466944i	$0.845355\pi$
$3$		0			0
$3$		0			0
$4$		−	10.4264i		−	0.651650i
$5$	28.4160	+	28.4160i	1.13664	+	1.13664i	0.989049	+	0.147591i	$0.0471518\pi$
$5$	28.4160	+	28.4160i	1.13664	+	1.13664i	0.147591	+	0.989049i	$0.452848\pi$
$6$		0			0
$7$	−17.1341	+	17.1341i	−0.349675	+	0.349675i	−0.859988	−	0.510314i	$-0.829529\pi$
$7$	−17.1341	+	17.1341i	−0.349675	+	0.349675i	0.510314	+	0.859988i	$0.329529\pi$
$8$	−44.1155	+	44.1155i	−0.689304	+	0.689304i
$9$		0			0
$10$		−	94.8736i		−	0.948736i
$11$	−54.2346	+	54.2346i	−0.448220	+	0.448220i	−0.894762	−	0.446542i	$-0.852655\pi$
$11$	−54.2346	+	54.2346i	−0.448220	+	0.448220i	0.446542	+	0.894762i	$0.352655\pi$
$12$		0			0
$13$	−168.969	+	3.22849i	−0.999818	+	0.0191035i
$13$	−168.969	+	3.22849i	−0.999818	+	0.0191035i
$14$	57.2062			0.291868
$15$		0			0
$16$	−19.5322			−0.0762978
$17$			70.4411i			0.243741i	0.992546	+	0.121870i	$0.0388892\pi$
$17$			70.4411i			0.243741i	−0.992546	+	0.121870i	$0.961111\pi$
$18$		0			0
$19$	194.123	+	194.123i	0.537736	+	0.537736i	0.922863	−	0.385127i	$-0.125843\pi$
$19$	194.123	+	194.123i	0.537736	+	0.537736i	−0.385127	+	0.922863i	$0.625843\pi$
$20$	296.276	−	296.276i	0.740691	−	0.740691i
$21$		0			0
$22$	181.075			0.374123
$23$			716.329i			1.35412i	0.735928	+	0.677059i	$0.236746\pi$
$23$			716.329i			1.35412i	−0.735928	+	0.677059i	$0.763254\pi$
$24$		0			0
$25$			989.935i			1.58390i
$26$	287.462	+	276.683i	0.425239	+	0.409294i
$27$		0			0
$28$	178.647	+	178.647i	0.227866	+	0.227866i
$29$	−725.866			−0.863099			−0.431550	−	0.902089i	$-0.642033\pi$
$29$	−725.866			−0.863099			−0.431550	+	0.902089i	$0.642033\pi$
$30$		0			0
$31$	586.446	+	586.446i	0.610245	+	0.610245i	0.943010	−	0.332765i	$-0.107981\pi$
$31$	586.446	+	586.446i	0.610245	+	0.610245i	−0.332765	+	0.943010i	$0.607981\pi$
$32$	738.454	+	738.454i	0.721147	+	0.721147i
$33$		0			0
$34$	117.592	−	117.592i	0.101723	−	0.101723i
$35$	−973.762			−0.794908
$36$		0			0
$37$	575.923	−	575.923i	0.420689	−	0.420689i	−0.464752	−	0.885441i	$-0.653856\pi$
$37$	575.923	−	575.923i	0.420689	−	0.420689i	0.885441	+	0.464752i	$0.153856\pi$
$38$		−	648.126i		−	0.448840i
$39$		0			0
$40$	−2507.17			−1.56698
$41$	1810.07	+	1810.07i	1.07678	+	1.07678i	0.996796	+	0.0799854i	$0.0254874\pi$
$41$	1810.07	+	1810.07i	1.07678	+	1.07678i	0.0799854	+	0.996796i	$0.474513\pi$
$42$		0			0
$43$		−	2879.48i		−	1.55732i	−0.627448	−	0.778659i	$-0.715900\pi$
$43$		−	2879.48i		−	1.55732i	0.627448	−	0.778659i	$-0.284100\pi$
$44$	565.472	+	565.472i	0.292083	+	0.292083i
$45$		0			0
$46$	1195.82	−	1195.82i	0.565132	−	0.565132i
$47$	−1064.86	+	1064.86i	−0.482055	+	0.482055i	−0.905787	−	0.423733i	$-0.860720\pi$
$47$	−1064.86	+	1064.86i	−0.482055	+	0.482055i	0.423733	+	0.905787i	$0.360720\pi$
$48$		0			0
$49$			1813.85i			0.755455i
$50$	1652.57	−	1652.57i	0.661028	−	0.661028i
$51$		0			0
$52$	33.6615	+	1761.74i	0.0124488	+	0.651531i
$53$	−2549.76			−0.907710			−0.453855	−	0.891076i	$-0.649952\pi$
$53$	−2549.76			−0.907710			−0.453855	+	0.891076i	$0.649952\pi$
$54$		0			0
$55$	−3082.26			−1.01893
$56$		−	1511.75i		−	0.482065i
$57$		0			0
$58$	1211.74	+	1211.74i	0.360208	+	0.360208i
$59$	2264.35	−	2264.35i	0.650487	−	0.650487i	−0.302623	−	0.953110i	$-0.597862\pi$
$59$	2264.35	−	2264.35i	0.650487	−	0.650487i	0.953110	+	0.302623i	$0.0978624\pi$
$60$		0			0
$61$	6434.68			1.72929			0.864644	−	0.502386i	$-0.167544\pi$
$61$	6434.68			1.72929			0.864644	+	0.502386i	$0.167544\pi$
$62$		−	1957.99i		−	0.509363i
$63$		0			0
$64$		−	2152.99i		−	0.525633i
$65$	−4893.16	−	4709.68i	−1.15815	−	1.11472i
$66$		0			0
$67$	−5472.27	−	5472.27i	−1.21904	−	1.21904i	−0.967969	−	0.251070i	$-0.919217\pi$
$67$	−5472.27	−	5472.27i	−1.21904	−	1.21904i	−0.251070	−	0.967969i	$-0.580783\pi$
$68$	734.447			0.158834
$69$		0			0
$70$	1625.57	+	1625.57i	0.331749	+	0.331749i
$71$	−4643.32	−	4643.32i	−0.921111	−	0.921111i	0.0759967	−	0.997108i	$-0.475786\pi$
$71$	−4643.32	−	4643.32i	−0.921111	−	0.921111i	−0.997108	+	0.0759967i	$0.975786\pi$
$72$		0			0
$73$	−5605.57	+	5605.57i	−1.05190	+	1.05190i	−0.0533222	+	0.998577i	$0.516981\pi$
$73$	−5605.57	+	5605.57i	−1.05190	+	1.05190i	−0.998577	+	0.0533222i	$0.983019\pi$
$74$	−1922.86			−0.351143
$75$		0			0
$76$	2024.00	−	2024.00i	0.350416	−	0.350416i
$77$		−	1858.52i		−	0.313462i
$78$		0			0
$79$	−6871.05			−1.10095			−0.550477	−	0.834850i	$-0.685554\pi$
$79$	−6871.05			−1.10095			−0.550477	+	0.834850i	$0.685554\pi$
$80$	−555.027	−	555.027i	−0.0867230	−	0.0867230i
$81$		0			0
$82$		−	6043.36i		−	0.898774i
$83$	2647.22	+	2647.22i	0.384267	+	0.384267i	0.872637	−	0.488370i	$-0.162408\pi$
$83$	2647.22	+	2647.22i	0.384267	+	0.384267i	−0.488370	+	0.872637i	$0.662408\pi$
$84$		0			0
$85$	−2001.65	+	2001.65i	−0.277045	+	0.277045i
$86$	−4806.92	+	4806.92i	−0.649935	+	0.649935i
$87$		0			0
$88$		−	4785.17i		−	0.617920i
$89$	6804.23	−	6804.23i	0.859012	−	0.859012i	−0.132210	−	0.991222i	$-0.542207\pi$
$89$	6804.23	−	6804.23i	0.859012	−	0.859012i	0.991222	+	0.132210i	$0.0422073\pi$
$90$		0			0
$91$	2839.81	−	2950.45i	0.342931	−	0.356291i
$92$	7468.73			0.882412
$93$		0			0
$94$	3555.29			0.402364
$95$			11032.4i			1.22242i
$96$		0			0
$97$	9498.95	+	9498.95i	1.00956	+	1.00956i	0.999954	+	0.00960578i	$0.00305766\pi$
$97$	9498.95	+	9498.95i	1.00956	+	1.00956i	0.00960578	+	0.999954i	$0.496942\pi$
$98$	3027.99	−	3027.99i	0.315284	−	0.315284i
$99$		0			0
$100$	10321.5			1.03215
$101$			16541.4i			1.62155i	0.585361	+	0.810773i	$0.300953\pi$
$101$			16541.4i			1.62155i	−0.585361	+	0.810773i	$0.699047\pi$
$102$		0			0
$103$		−	11148.4i		−	1.05084i	−0.850842	−	0.525421i	$-0.823908\pi$
$103$		−	11148.4i		−	1.05084i	0.850842	−	0.525421i	$-0.176092\pi$
$104$	7311.73	−	7596.58i	0.676010	−	0.702347i
$105$		0			0
$106$	4256.49	+	4256.49i	0.378826	+	0.378826i
$107$	−9291.29			−0.811537			−0.405769	−	0.913976i	$-0.632996\pi$
$107$	−9291.29			−0.811537			−0.405769	+	0.913976i	$0.632996\pi$
$108$		0			0
$109$	1985.75	+	1985.75i	0.167136	+	0.167136i	0.785719	−	0.618583i	$-0.212293\pi$
$109$	1985.75	+	1985.75i	0.167136	+	0.167136i	−0.618583	+	0.785719i	$0.712293\pi$
$110$	5145.43	+	5145.43i	0.425243	+	0.425243i
$111$		0			0
$112$	334.666	−	334.666i	0.0266794	−	0.0266794i
$113$	1799.10			0.140896			0.0704478	−	0.997515i	$-0.477557\pi$
$113$	1799.10			0.140896			0.0704478	+	0.997515i	$0.477557\pi$
$114$		0			0
$115$	−20355.2	+	20355.2i	−1.53914	+	1.53914i
$116$			7568.17i			0.562439i
$117$		0			0
$118$	−7560.07			−0.542952
$119$	−1206.94	−	1206.94i	−0.0852300	−	0.0852300i
$120$		0			0
$121$			8758.21i			0.598198i
$122$	−10741.9	−	10741.9i	−0.721706	−	0.721706i
$123$		0			0
$124$	6114.52	−	6114.52i	0.397666	−	0.397666i
$125$	−10370.0	+	10370.0i	−0.663680	+	0.663680i
$126$		0			0
$127$		−	7557.96i		−	0.468594i	−0.972165	−	0.234297i	$-0.924721\pi$
$127$		−	7557.96i		−	0.468594i	0.972165	−	0.234297i	$-0.0752789\pi$
$128$	8221.12	−	8221.12i	0.501777	−	0.501777i
$129$		0			0
$130$	306.299	+	16030.7i	0.0181242	+	0.948563i
$131$	−9705.21			−0.565539			−0.282769	−	0.959188i	$-0.591253\pi$
$131$	−9705.21			−0.565539			−0.282769	+	0.959188i	$0.591253\pi$
$132$		0			0
$133$	−6652.22			−0.376065
$134$			18270.5i			1.01751i
$135$		0			0
$136$	−3107.54	−	3107.54i	−0.168012	−	0.168012i
$137$	7306.03	−	7306.03i	0.389261	−	0.389261i	−0.485163	−	0.874424i	$-0.661240\pi$
$137$	7306.03	−	7306.03i	0.389261	−	0.389261i	0.874424	+	0.485163i	$0.161240\pi$
$138$		0			0
$139$	14508.5			0.750921			0.375460	−	0.926838i	$-0.377485\pi$
$139$	14508.5			0.750921			0.375460	+	0.926838i	$0.377485\pi$
$140$			10152.8i			0.518002i
$141$		0			0
$142$			15502.9i			0.768838i
$143$	8988.88	−	9339.07i	0.439576	−	0.456701i
$144$		0			0
$145$	−20626.2	−	20626.2i	−0.981032	−	0.981032i
$146$	18715.6			0.878005
$147$		0			0
$148$	−6004.81	−	6004.81i	−0.274142	−	0.274142i
$149$	−6209.08	−	6209.08i	−0.279675	−	0.279675i	0.553304	−	0.832979i	$-0.313367\pi$
$149$	−6209.08	−	6209.08i	−0.279675	−	0.279675i	−0.832979	+	0.553304i	$0.813367\pi$
$150$		0			0
$151$	6322.22	−	6322.22i	0.277278	−	0.277278i	−0.554743	−	0.832021i	$-0.687184\pi$
$151$	6322.22	−	6322.22i	0.277278	−	0.277278i	0.832021	+	0.554743i	$0.187184\pi$
$152$	−17127.6			−0.741327
$153$		0			0
$154$	−3102.56	+	3102.56i	−0.130821	+	0.130821i
$155$			33328.9i			1.38726i
$156$		0			0
$157$	26942.7			1.09305			0.546526	−	0.837442i	$-0.315950\pi$
$157$	26942.7			1.09305			0.546526	+	0.837442i	$0.315950\pi$
$158$	11470.3	+	11470.3i	0.459475	+	0.459475i
$159$		0			0
$160$			41967.8i			1.63937i
$161$	−12273.6	−	12273.6i	−0.473501	−	0.473501i
$162$		0			0
$163$	22832.2	−	22832.2i	0.859355	−	0.859355i	−0.131907	−	0.991262i	$-0.542110\pi$
$163$	22832.2	−	22832.2i	0.859355	−	0.859355i	0.991262	+	0.131907i	$0.0421101\pi$
$164$	18872.5	−	18872.5i	0.701685	−	0.701685i
$165$		0			0
$166$		−	8838.37i		−	0.320742i
$167$	−7664.75	+	7664.75i	−0.274831	+	0.274831i	−0.831041	−	0.556211i	$-0.812255\pi$
$167$	−7664.75	+	7664.75i	−0.274831	+	0.274831i	0.556211	+	0.831041i	$0.312255\pi$
$168$		0			0
$169$	28540.2	−	1091.03i	0.999270	−	0.0382000i
$170$	6683.00			0.231246
$171$		0			0
$172$	−30022.6			−1.01483
$173$		−	9382.63i		−	0.313496i	−0.987639	−	0.156748i	$-0.949899\pi$
$173$		−	9382.63i		−	0.313496i	0.987639	−	0.156748i	$-0.0501011\pi$
$174$		0			0
$175$	−16961.6	−	16961.6i	−0.553849	−	0.553849i
$176$	1059.32	−	1059.32i	0.0341982	−	0.0341982i
$177$		0			0
$178$	−22717.6			−0.717005
$179$			322.023i			0.0100503i	0.999987	+	0.00502517i	$0.00159957\pi$
$179$			322.023i			0.0100503i	−0.999987	+	0.00502517i	$0.998400\pi$
$180$		0			0
$181$		−	28479.3i		−	0.869306i	−0.900598	−	0.434653i	$-0.856871\pi$
$181$		−	28479.3i		−	0.869306i	0.900598	−	0.434653i	$-0.143129\pi$
$182$	−9666.09	+	184.690i	−0.291815	+	0.00557571i
$183$		0			0
$184$	−31601.2	−	31601.2i	−0.933400	−	0.933400i
$185$	32730.8			0.956343
$186$		0			0
$187$	−3820.35	−	3820.35i	−0.109249	−	0.109249i
$188$	11102.6	+	11102.6i	0.314131	+	0.314131i
$189$		0			0
$190$	18417.1	−	18417.1i	0.510170	−	0.510170i
$191$	−39049.7			−1.07041			−0.535206	−	0.844722i	$-0.679766\pi$
$191$	−39049.7			−1.07041			−0.535206	+	0.844722i	$0.679766\pi$
$192$		0			0
$193$	−34849.0	+	34849.0i	−0.935570	+	0.935570i	−0.998046	−	0.0624768i	$-0.980100\pi$
$193$	−34849.0	+	34849.0i	−0.935570	+	0.935570i	0.0624768	+	0.998046i	$0.480100\pi$
$194$		−	31714.5i		−	0.842665i
$195$		0			0
$196$	18911.9			0.492292
$197$	−33755.4	−	33755.4i	−0.869784	−	0.869784i	0.122664	−	0.992448i	$-0.460856\pi$
$197$	−33755.4	−	33755.4i	−0.869784	−	0.869784i	−0.992448	+	0.122664i	$0.960856\pi$
$198$		0			0
$199$		−	12091.0i		−	0.305322i	−0.988279	−	0.152661i	$-0.951216\pi$
$199$		−	12091.0i		−	0.305322i	0.988279	−	0.152661i	$-0.0487842\pi$
$200$	−43671.5	−	43671.5i	−1.09179	−	1.09179i
$201$		0			0
$202$	27613.7	−	27613.7i	0.676740	−	0.676740i
$203$	12437.0	−	12437.0i	0.301804	−	0.301804i
$204$		0			0
$205$			102870.i			2.44782i
$206$	−18610.8	+	18610.8i	−0.438561	+	0.438561i
$207$		0			0
$208$	3300.34	−	63.0596i	0.0762838	−	0.00145755i
$209$	−21056.3			−0.482048
$210$		0			0
$211$	51063.6			1.14695			0.573477	−	0.819221i	$-0.305594\pi$
$211$	51063.6			1.14695			0.573477	+	0.819221i	$0.305594\pi$
$212$			26584.8i			0.591509i
$213$		0			0
$214$	15510.6	+	15510.6i	0.338689	+	0.338689i
$215$	81823.2	−	81823.2i	1.77011	−	1.77011i
$216$		0			0
$217$	−20096.4			−0.426775
$218$		−	6629.90i		−	0.139506i
$219$		0			0
$220$			32136.9i			0.663985i
$221$	−227.418	−	11902.4i	−0.00465630	−	0.243696i
$222$		0			0
$223$	7861.80	+	7861.80i	0.158093	+	0.158093i	0.781721	−	0.623628i	$-0.214342\pi$
$223$	7861.80	+	7861.80i	0.158093	+	0.158093i	−0.623628	+	0.781721i	$0.714342\pi$
$224$	−25305.4			−0.504333
$225$		0			0
$226$	−3003.36	−	3003.36i	−0.0588018	−	0.0588018i
$227$	9642.38	+	9642.38i	0.187125	+	0.187125i	0.794452	−	0.607327i	$-0.207758\pi$
$227$	9642.38	+	9642.38i	0.187125	+	0.187125i	−0.607327	+	0.794452i	$0.707758\pi$
$228$		0			0
$229$	36285.4	−	36285.4i	0.691928	−	0.691928i	−0.270728	−	0.962656i	$-0.587264\pi$
$229$	36285.4	−	36285.4i	0.691928	−	0.691928i	0.962656	+	0.270728i	$0.0872643\pi$
$230$	67960.7			1.28470
$231$		0			0
$232$	32021.9	−	32021.9i	0.594938	−	0.594938i
$233$		−	2365.03i		−	0.0435637i	−0.999763	−	0.0217819i	$-0.993066\pi$
$233$		−	2365.03i		−	0.0435637i	0.999763	−	0.0217819i	$-0.00693393\pi$
$234$		0			0
$235$	−60518.0			−1.09584
$236$	−23609.0	−	23609.0i	−0.423890	−	0.423890i
$237$		0			0
$238$			4029.67i			0.0711402i
$239$	18841.4	+	18841.4i	0.329850	+	0.329850i	0.852529	−	0.522679i	$-0.175067\pi$
$239$	18841.4	+	18841.4i	0.329850	+	0.329850i	−0.522679	+	0.852529i	$0.675067\pi$
$240$		0			0
$241$	−27569.6	+	27569.6i	−0.474676	+	0.474676i	−0.903424	−	0.428748i	$-0.858955\pi$
$241$	−27569.6	+	27569.6i	−0.474676	+	0.474676i	0.428748	+	0.903424i	$0.358955\pi$
$242$	14620.7	−	14620.7i	0.249653	−	0.249653i
$243$		0			0
$244$		−	67090.5i		−	1.12689i
$245$	−51542.3	+	51542.3i	−0.858680	+	0.858680i
$246$		0			0
$247$	−33427.5	−	32174.0i	−0.547910	−	0.527365i
$248$	−51742.7			−0.841289
$249$		0			0
$250$	34622.8			0.553964
$251$			52279.9i			0.829826i	0.909861	+	0.414913i	$0.136188\pi$
$251$			52279.9i			0.829826i	−0.909861	+	0.414913i	$0.863812\pi$
$252$		0			0
$253$	−38849.8	−	38849.8i	−0.606943	−	0.606943i
$254$	−12617.0	+	12617.0i	−0.195564	+	0.195564i
$255$		0			0
$256$	−61896.1			−0.944459
$257$			87118.8i			1.31900i	0.751703	+	0.659501i	$0.229232\pi$
$257$			87118.8i			1.31900i	−0.751703	+	0.659501i	$0.770768\pi$
$258$		0			0
$259$			19735.8i			0.294209i
$260$	−49105.0	+	51018.1i	−0.726406	+	0.754705i
$261$		0			0
$262$	16201.6	+	16201.6i	0.236023	+	0.236023i
$263$	92681.2			1.33992			0.669962	−	0.742395i	$-0.266310\pi$
$263$	92681.2			1.33992			0.669962	+	0.742395i	$0.266310\pi$
$264$		0			0
$265$	−72453.8	−	72453.8i	−1.03174	−	1.03174i
$266$	11105.0	+	11105.0i	0.156948	+	0.156948i
$267$		0			0
$268$	−57056.0	+	57056.0i	−0.794387	+	0.794387i
$269$	123465.			1.70624			0.853118	−	0.521717i	$-0.174708\pi$
$269$	123465.			1.70624			0.853118	+	0.521717i	$0.174708\pi$
$270$		0			0
$271$	60295.3	−	60295.3i	0.821003	−	0.821003i	−0.165249	−	0.986252i	$-0.552843\pi$
$271$	60295.3	−	60295.3i	0.821003	−	0.821003i	0.986252	+	0.165249i	$0.0528428\pi$
$272$		−	1375.87i		−	0.0185969i
$273$		0			0
$274$	−24393.0			−0.324910
$275$	−53688.8	−	53688.8i	−0.709934	−	0.709934i
$276$		0			0
$277$			62673.2i			0.816812i	0.912800	+	0.408406i	$0.133915\pi$
$277$			62673.2i			0.816812i	−0.912800	+	0.408406i	$0.866085\pi$
$278$	−24220.1	−	24220.1i	−0.313391	−	0.313391i
$279$		0			0
$280$	42958.0	−	42958.0i	0.547933	−	0.547933i
$281$	−17050.4	+	17050.4i	−0.215934	+	0.215934i	−0.806783	−	0.590848i	$-0.798793\pi$
$281$	−17050.4	+	17050.4i	−0.215934	+	0.215934i	0.590848	+	0.806783i	$0.298793\pi$
$282$		0			0
$283$			135558.i			1.69259i	0.532716	+	0.846294i	$0.321171\pi$
$283$			135558.i			1.69259i	−0.532716	+	0.846294i	$0.678829\pi$
$284$	−48413.1	+	48413.1i	−0.600242	+	0.600242i
$285$		0			0
$286$	−30596.2	+	584.600i	−0.374054	+	0.00714705i
$287$	−62027.7			−0.753047
$288$		0			0
$289$	78559.1			0.940590
$290$			68865.6i			0.818854i
$291$		0			0
$292$	58445.9	+	58445.9i	0.685470	+	0.685470i
$293$	33642.1	−	33642.1i	0.391876	−	0.391876i	−0.483480	−	0.875356i	$-0.660627\pi$
$293$	33642.1	−	33642.1i	0.391876	−	0.391876i	0.875356	+	0.483480i	$0.160627\pi$
$294$		0			0
$295$	128687.			1.47874
$296$			50814.3i			0.579966i
$297$		0			0
$298$			20730.5i			0.233441i
$299$	−2312.66	−	121037.i	−0.0258684	−	1.35387i
$300$		0			0
$301$	49337.2	+	49337.2i	0.544555	+	0.544555i
$302$	−21108.2			−0.231440
$303$		0			0
$304$	−3791.65	−	3791.65i	−0.0410281	−	0.0410281i
$305$	182848.	+	182848.i	1.96558	+	1.96558i
$306$		0			0
$307$	−121981.	+	121981.i	−1.29425	+	1.29425i	−0.362112	+	0.932135i	$0.617944\pi$
$307$	−121981.	+	121981.i	−1.29425	+	1.29425i	−0.932135	+	0.362112i	$0.882056\pi$
$308$	−19377.7			−0.204268
$309$		0			0
$310$	55638.2	−	55638.2i	0.578962	−	0.578962i
$311$		−	1370.53i		−	0.0141699i	−0.999975	−	0.00708495i	$-0.997745\pi$
$311$		−	1370.53i		−	0.0141699i	0.999975	−	0.00708495i	$-0.00225523\pi$
$312$		0			0
$313$	52462.0			0.535495			0.267748	−	0.963489i	$-0.413721\pi$
$313$	52462.0			0.535495			0.267748	+	0.963489i	$0.413721\pi$
$314$	−44977.3	−	44977.3i	−0.456178	−	0.456178i
$315$		0			0
$316$			71640.4i			0.717437i
$317$	101779.	+	101779.i	1.01284	+	1.01284i	0.999916	+	0.0129239i	$0.00411392\pi$
$317$	101779.	+	101779.i	1.01284	+	1.01284i	0.0129239	+	0.999916i	$0.495886\pi$
$318$		0			0
$319$	39367.1	−	39367.1i	0.386858	−	0.386858i
$320$	61179.4	−	61179.4i	0.597455	−	0.597455i
$321$		0			0
$322$			40978.5i			0.395225i
$323$	−13674.2	+	13674.2i	−0.131068	+	0.131068i
$324$		0			0
$325$	−3196.00	−	167269.i	−0.0302580	−	1.58361i
$326$	−76230.8			−0.717291
$327$		0			0
$328$	−159704.			−1.48446
$329$		−	36490.7i		−	0.337125i
$330$		0			0
$331$	90476.9	+	90476.9i	0.825813	+	0.825813i	0.986935	−	0.161122i	$-0.0515112\pi$
$331$	90476.9	+	90476.9i	0.825813	+	0.825813i	−0.161122	+	0.986935i	$0.551511\pi$
$332$	27600.9	−	27600.9i	0.250408	−	0.250408i
$333$		0			0
$334$	25590.6			0.229397
$335$		−	311000.i		−	2.77122i
$336$		0			0
$337$			48432.3i			0.426457i	0.977002	+	0.213228i	$0.0683979\pi$
$337$			48432.3i			0.426457i	−0.977002	+	0.213228i	$0.931602\pi$
$338$	−49465.4	−	45822.8i	−0.432981	−	0.401096i
$339$		0			0
$340$	20870.0	+	20870.0i	0.180537	+	0.180537i
$341$	−63611.3			−0.547048
$342$		0			0
$343$	−72217.5	−	72217.5i	−0.613838	−	0.613838i
$344$	127030.	+	127030.i	1.07347	+	1.07347i
$345$		0			0
$346$	−15663.1	+	15663.1i	−0.130835	+	0.130835i
$347$	−50065.9			−0.415799			−0.207899	−	0.978150i	$-0.566663\pi$
$347$	−50065.9			−0.415799			−0.207899	+	0.978150i	$0.566663\pi$
$348$		0			0
$349$	−36678.7	+	36678.7i	−0.301137	+	0.301137i	−0.841458	−	0.540322i	$-0.818302\pi$
$349$	−36678.7	+	36678.7i	−0.301137	+	0.301137i	0.540322	+	0.841458i	$0.318302\pi$
$350$			56630.5i			0.462290i
$351$		0			0
$352$	−80099.5			−0.646465
$353$	91471.6	+	91471.6i	0.734069	+	0.734069i	0.971423	−	0.237354i	$-0.0762803\pi$
$353$	91471.6	+	91471.6i	0.734069	+	0.734069i	−0.237354	+	0.971423i	$0.576280\pi$
$354$		0			0
$355$		−	263889.i		−	2.09394i
$356$	−70943.6	−	70943.6i	−0.559775	−	0.559775i
$357$		0			0
$358$	537.576	−	537.576i	0.00419444	−	0.00419444i
$359$	−19904.6	+	19904.6i	−0.154442	+	0.154442i	−0.780098	−	0.625657i	$-0.784831\pi$
$359$	−19904.6	+	19904.6i	−0.154442	+	0.154442i	0.625657	+	0.780098i	$0.284831\pi$
$360$		0			0
$361$		−	54953.8i		−	0.421680i
$362$	−47542.6	+	47542.6i	−0.362799	+	0.362799i
$363$		0			0
$364$	−30762.5	−	29609.0i	−0.232177	−	0.223471i
$365$	−318576.			−2.39126
$366$		0			0
$367$	13539.5			0.100524			0.0502620	−	0.998736i	$-0.483994\pi$
$367$	13539.5			0.100524			0.0502620	+	0.998736i	$0.483994\pi$
$368$		−	13991.5i		−	0.103316i
$369$		0			0
$370$	−54639.9	−	54639.9i	−0.399123	−	0.399123i
$371$	43687.7	−	43687.7i	0.317403	−	0.317403i
$372$		0			0
$373$	−76501.0			−0.549857			−0.274928	−	0.961465i	$-0.588654\pi$
$373$	−76501.0			−0.549857			−0.274928	+	0.961465i	$0.588654\pi$
$374$			12755.1i			0.0911890i
$375$		0			0
$376$		−	93953.5i		−	0.664565i
$377$	122649.	−	2343.45i	0.862942	−	0.0164882i
$378$		0			0
$379$	9873.34	+	9873.34i	0.0687362	+	0.0687362i	0.740639	−	0.671903i	$-0.234523\pi$
$379$	9873.34	+	9873.34i	0.0687362	+	0.0687362i	−0.671903	+	0.740639i	$0.734523\pi$
$380$	115028.			0.796592
$381$		0			0
$382$	65188.4	+	65188.4i	0.446729	+	0.446729i
$383$	−92064.1	−	92064.1i	−0.627614	−	0.627614i	0.319853	−	0.947467i	$-0.396366\pi$
$383$	−92064.1	−	92064.1i	−0.627614	−	0.627614i	−0.947467	+	0.319853i	$0.896366\pi$
$384$		0			0
$385$	52811.6	−	52811.6i	0.356294	−	0.356294i
$386$	116352.			0.780906
$387$		0			0
$388$	99039.8	−	99039.8i	0.657880	−	0.657880i
$389$			35489.2i			0.234529i	0.993101	+	0.117265i	$0.0374125\pi$
$389$			35489.2i			0.234529i	−0.993101	+	0.117265i	$0.962587\pi$
$390$		0			0
$391$	−50459.0			−0.330054
$392$	−80018.7	−	80018.7i	−0.520738	−	0.520738i
$393$		0			0
$394$			112701.i			0.725996i
$395$	−195248.	−	195248.i	−1.25139	−	1.25139i
$396$		0			0
$397$	45764.3	−	45764.3i	0.290366	−	0.290366i	−0.546859	−	0.837225i	$-0.684177\pi$
$397$	45764.3	−	45764.3i	0.290366	−	0.290366i	0.837225	+	0.546859i	$0.184177\pi$
$398$	−20184.4	+	20184.4i	−0.127424	+	0.127424i
$399$		0			0
$400$		−	19335.6i		−	0.120848i
$401$	−53143.4	+	53143.4i	−0.330492	+	0.330492i	−0.852773	−	0.522281i	$-0.825081\pi$
$401$	−53143.4	+	53143.4i	−0.330492	+	0.330492i	0.522281	+	0.852773i	$0.325081\pi$
$402$		0			0
$403$	−100985.	−	97197.9i	−0.621792	−	0.598476i
$404$	172467.			1.05668
$405$		0			0
$406$	−41524.1			−0.251911
$407$			62470.0i			0.377123i
$408$		0			0
$409$	75294.0	+	75294.0i	0.450105	+	0.450105i	0.895389	−	0.445284i	$-0.146897\pi$
$409$	75294.0	+	75294.0i	0.450105	+	0.450105i	−0.445284	+	0.895389i	$0.646897\pi$
$410$	171728.	−	171728.i	1.02158	−	1.02158i
$411$		0			0
$412$	−116237.			−0.684781
$413$			77594.9i			0.454918i
$414$		0			0
$415$			150447.i			0.873546i
$416$	−127160.	−	122392.i	−0.734791	−	0.707239i
$417$		0			0
$418$	35150.8	+	35150.8i	0.201179	+	0.201179i
$419$	264545.			1.50686			0.753429	−	0.657530i	$-0.228399\pi$
$419$	264545.			1.50686			0.753429	+	0.657530i	$0.228399\pi$
$420$		0			0
$421$	30687.4	+	30687.4i	0.173140	+	0.173140i	0.788357	−	0.615218i	$-0.210932\pi$
$421$	30687.4	+	30687.4i	0.173140	+	0.173140i	−0.615218	+	0.788357i	$0.710932\pi$
$422$	−85244.1	−	85244.1i	−0.478673	−	0.478673i
$423$		0			0
$424$	112484.	−	112484.i	0.625688	−	0.625688i
$425$	−69732.1			−0.386060
$426$		0			0
$427$	−110252.	+	110252.i	−0.604688	+	0.604688i
$428$			96874.7i			0.528838i
$429$		0			0
$430$	−273187.			−1.47748
$431$	−138755.	−	138755.i	−0.746956	−	0.746956i	0.226950	−	0.973906i	$-0.427124\pi$
$431$	−138755.	−	138755.i	−0.746956	−	0.746956i	−0.973906	+	0.226950i	$0.927124\pi$
$432$		0			0
$433$			192478.i			1.02661i	0.858206	+	0.513305i	$0.171579\pi$
$433$			192478.i			1.02661i	−0.858206	+	0.513305i	$0.828421\pi$
$434$	33548.3	+	33548.3i	0.178111	+	0.178111i
$435$		0			0
$436$	20704.2	−	20704.2i	0.108914	−	0.108914i
$437$	−139056.	+	139056.i	−0.728158	+	0.728158i
$438$		0			0
$439$		−	221370.i		−	1.14865i	−0.818626	−	0.574327i	$-0.805264\pi$
$439$		−	221370.i		−	1.14865i	0.818626	−	0.574327i	$-0.194736\pi$
$440$	135975.	−	135975.i	0.702352	−	0.702352i
$441$		0			0
$442$	−19489.8	+	20249.1i	−0.0997616	+	0.103648i
$443$	−354252.			−1.80511			−0.902557	−	0.430570i	$-0.858313\pi$
$443$	−354252.			−1.80511			−0.902557	+	0.430570i	$0.858313\pi$
$444$		0			0
$445$	386698.			1.95277
$446$		−	26248.5i		−	0.131958i
$447$		0			0
$448$	36889.5	+	36889.5i	0.183801	+	0.183801i
$449$	187721.	−	187721.i	0.931152	−	0.931152i	−0.0666259	−	0.997778i	$-0.521223\pi$
$449$	187721.	−	187721.i	0.931152	−	0.931152i	0.997778	+	0.0666259i	$0.0212234\pi$
$450$		0			0
$451$	−196337.			−0.965270
$452$		−	18758.1i		−	0.0918147i
$453$		0			0
$454$		−	32193.4i		−	0.156191i
$455$	164536.	−	3143.78i	0.794763	−	0.0151855i
$456$		0			0
$457$	−88792.4	−	88792.4i	−0.425151	−	0.425151i	0.461822	−	0.886973i	$-0.347196\pi$
$457$	−88792.4	−	88792.4i	−0.425151	−	0.425151i	−0.886973	+	0.461822i	$0.847196\pi$
$458$	−121148.			−0.577543
$459$		0			0
$460$	212231.	+	212231.i	1.00298	+	1.00298i
$461$	56596.4	+	56596.4i	0.266310	+	0.266310i	0.827611	−	0.561302i	$-0.189699\pi$
$461$	56596.4	+	56596.4i	0.266310	+	0.266310i	−0.561302	+	0.827611i	$0.689699\pi$
$462$		0			0
$463$	171067.	−	171067.i	0.798003	−	0.798003i	−0.184777	−	0.982780i	$-0.559156\pi$
$463$	171067.	−	171067.i	0.798003	−	0.798003i	0.982780	+	0.184777i	$0.0591563\pi$
$464$	14177.8			0.0658526
$465$		0			0
$466$	−3948.11	+	3948.11i	−0.0181810	+	0.0181810i
$467$			68140.5i			0.312443i	0.987722	+	0.156222i	$0.0499314\pi$
$467$			68140.5i			0.312443i	−0.987722	+	0.156222i	$0.950069\pi$
$468$		0			0
$469$	187524.			0.852534
$470$	101027.	+	101027.i	0.457343	+	0.457343i
$471$		0			0
$472$			199785.i			0.896767i
$473$	156168.	+	156168.i	0.698021	+	0.698021i
$474$		0			0
$475$	−192169.	+	192169.i	−0.851718	+	0.851718i
$476$	−12584.1	+	12584.1i	−0.0555401	+	0.0555401i
$477$		0			0
$478$		−	62906.5i		−	0.275321i
$479$	113713.	−	113713.i	0.495610	−	0.495610i	−0.414458	−	0.910068i	$-0.636029\pi$
$479$	113713.	−	113713.i	0.495610	−	0.495610i	0.910068	+	0.414458i	$0.136029\pi$
$480$		0			0
$481$	−95453.9	+	99172.6i	−0.412576	+	0.428649i
$482$	92047.9			0.396205
$483$		0			0
$484$	91316.6			0.389816
$485$			539844.i			2.29501i
$486$		0			0
$487$	30988.8	+	30988.8i	0.130661	+	0.130661i	0.769413	−	0.638752i	$-0.220549\pi$
$487$	30988.8	+	30988.8i	0.130661	+	0.130661i	−0.638752	+	0.769413i	$0.720549\pi$
$488$	−283869.	+	283869.i	−1.19201	+	1.19201i
$489$		0			0
$490$	172086.			0.716728
$491$			229476.i			0.951861i	0.879483	+	0.475931i	$0.157889\pi$
$491$			229476.i			0.951861i	−0.879483	+	0.475931i	$0.842111\pi$
$492$		0			0
$493$		−	51130.8i		−	0.210372i
$494$	2092.47	+	109513.i	0.00857442	+	0.448759i
$495$		0			0
$496$	−11454.6	−	11454.6i	−0.0465604	−	0.0465604i
$497$	159118.			0.644179
$498$		0			0
$499$	29259.2	+	29259.2i	0.117506	+	0.117506i	0.763415	−	0.645909i	$-0.223521\pi$
$499$	29259.2	+	29259.2i	0.117506	+	0.117506i	−0.645909	+	0.763415i	$0.723521\pi$
$500$	108122.	+	108122.i	0.432487	+	0.432487i
$501$		0			0
$502$	87274.5	−	87274.5i	0.346322	−	0.346322i
$503$	−432230.			−1.70836			−0.854180	−	0.519978i	$-0.825940\pi$
$503$	−432230.			−1.70836			−0.854180	+	0.519978i	$0.825940\pi$
$504$		0			0
$505$	−470040.	+	470040.i	−1.84311	+	1.84311i
$506$			129710.i			0.506607i
$507$		0			0
$508$	−78802.3			−0.305360
$509$	150630.	+	150630.i	0.581402	+	0.581402i	0.935289	−	0.353886i	$-0.115140\pi$
$509$	150630.	+	150630.i	0.581402	+	0.581402i	−0.353886	+	0.935289i	$0.615140\pi$
$510$		0			0
$511$		−	192092.i		−	0.735645i
$512$	−28210.4	−	28210.4i	−0.107614	−	0.107614i
$513$		0			0
$514$	145434.	−	145434.i	0.550476	−	0.550476i
$515$	316792.	−	316792.i	1.19443	−	1.19443i
$516$		0			0
$517$		−	115504.i		−	0.432133i
$518$	32946.4	−	32946.4i	0.122786	−	0.122786i
$519$		0			0
$520$	423634.	−	8094.37i	1.56669	−	0.0299348i
$521$	152305.			0.561096			0.280548	−	0.959840i	$-0.409484\pi$
$521$	152305.			0.561096			0.280548	+	0.959840i	$0.409484\pi$
$522$		0			0
$523$	78780.8			0.288016			0.144008	−	0.989577i	$-0.454001\pi$
$523$	78780.8			0.288016			0.144008	+	0.989577i	$0.454001\pi$
$524$			101190.i			0.368533i
$525$		0			0
$526$	−154719.	−	154719.i	−0.559208	−	0.559208i
$527$	−41309.9	+	41309.9i	−0.148742	+	0.148742i
$528$		0			0
$529$	−233286.			−0.833638
$530$			241905.i			0.861177i
$531$		0			0
$532$			69358.7i			0.245063i
$533$	−311690.	−	300002.i	−1.09716	−	1.05601i
$534$		0			0
$535$	−264021.	−	264021.i	−0.922425	−	0.922425i
$536$	482823.			1.68058
$537$		0			0
$538$	−206109.	−	206109.i	−0.712086	−	0.712086i
$539$	−98373.3	−	98373.3i	−0.338610	−	0.338610i
$540$		0			0
$541$	−35408.4	+	35408.4i	−0.120979	+	0.120979i	−0.765004	−	0.644025i	$-0.777263\pi$
$541$	−35408.4	+	35408.4i	−0.120979	+	0.120979i	0.644025	+	0.765004i	$0.277263\pi$
$542$	−201310.			−0.685279
$543$		0			0
$544$	−52017.5	+	52017.5i	−0.175773	+	0.175773i
$545$			112854.i			0.379947i
$546$		0			0
$547$	10939.9			0.0365628			0.0182814	−	0.999833i	$-0.494181\pi$
$547$	10939.9			0.0365628			0.0182814	+	0.999833i	$0.494181\pi$
$548$	−76175.6	−	76175.6i	−0.253662	−	0.253662i
$549$		0			0
$550$			179253.i			0.592572i
$551$	−140907.	−	140907.i	−0.464120	−	0.464120i
$552$		0			0
$553$	117729.	−	117729.i	0.384976	−	0.384976i
$554$	104625.	−	104625.i	0.340891	−	0.340891i
$555$		0			0
$556$		−	151272.i		−	0.489338i
$557$	−32717.3	+	32717.3i	−0.105455	+	0.105455i	−0.757866	−	0.652411i	$-0.773758\pi$
$557$	−32717.3	+	32717.3i	−0.105455	+	0.105455i	0.652411	+	0.757866i	$0.273758\pi$
$558$		0			0
$559$	9296.38	+	486543.i	0.0297502	+	1.55703i
$560$	19019.7			0.0606497
$561$		0			0
$562$	56926.8			0.180237
$563$			180795.i			0.570388i	0.958470	+	0.285194i	$0.0920580\pi$
$563$			180795.i			0.570388i	−0.958470	+	0.285194i	$0.907942\pi$
$564$		0			0
$565$	51123.1	+	51123.1i	0.160148	+	0.160148i
$566$	226296.	−	226296.i	0.706390	−	0.706390i
$567$		0			0
$568$	409685.			1.26985
$569$		−	500901.i		−	1.54713i	−0.633715	−	0.773566i	$-0.718471\pi$
$569$		−	500901.i		−	1.54713i	0.633715	−	0.773566i	$-0.281529\pi$
$570$		0			0
$571$			246396.i			0.755722i	0.925862	+	0.377861i	$0.123340\pi$
$571$			246396.i			0.755722i	−0.925862	+	0.377861i	$0.876660\pi$
$572$	−97372.9	−	93721.7i	−0.297609	−	0.286449i
$573$		0			0
$574$	103547.	+	103547.i	0.314279	+	0.314279i
$575$	−709119.			−2.14478
$576$		0			0
$577$	−184815.	−	184815.i	−0.555119	−	0.555119i	0.372794	−	0.927914i	$-0.378400\pi$
$577$	−184815.	−	184815.i	−0.555119	−	0.555119i	−0.927914	+	0.372794i	$0.878400\pi$
$578$	−131144.	−	131144.i	−0.392549	−	0.392549i
$579$		0			0
$580$	−215057.	+	215057.i	−0.639290	+	0.639290i
$581$	−90715.2			−0.268737
$582$		0			0
$583$	138285.	−	138285.i	0.406854	−	0.406854i
$584$		−	494585.i		−	1.45016i
$585$		0			0
$586$	−112322.			−0.327093
$587$	402989.	+	402989.i	1.16954	+	1.16954i	0.982317	+	0.187227i	$0.0599501\pi$
$587$	402989.	+	402989.i	1.16954	+	1.16954i	0.187227	+	0.982317i	$0.440050\pi$
$588$		0			0
$589$			227685.i			0.656302i
$590$	−214827.	−	214827.i	−0.617141	−	0.617141i
$591$		0			0
$592$	−11249.1	+	11249.1i	−0.0320976	+	0.0320976i
$593$	−180607.	+	180607.i	−0.513601	+	0.513601i	−0.915628	−	0.402027i	$-0.868306\pi$
$593$	−180607.	+	180607.i	−0.513601	+	0.513601i	0.402027	+	0.915628i	$0.368306\pi$
$594$		0			0
$595$		−	68592.9i		−	0.193751i
$596$	−64738.3	+	64738.3i	−0.182251	+	0.182251i
$597$		0			0
$598$	−198196.	+	205917.i	−0.554233	+	0.575825i
$599$	461474.			1.28615			0.643077	−	0.765801i	$-0.277657\pi$
$599$	461474.			1.28615			0.643077	+	0.765801i	$0.277657\pi$
$600$		0			0
$601$	−482801.			−1.33666			−0.668328	−	0.743867i	$-0.732990\pi$
$601$	−482801.			−1.33666			−0.668328	+	0.743867i	$0.732990\pi$
$602$		−	164724.i		−	0.454532i
$603$		0			0
$604$	−65918.0	−	65918.0i	−0.180688	−	0.180688i
$605$	−248873.	+	248873.i	−0.679935	+	0.679935i
$606$		0			0
$607$	219694.			0.596266			0.298133	−	0.954524i	$-0.403636\pi$
$607$	219694.			0.596266			0.298133	+	0.954524i	$0.403636\pi$
$608$			286701.i			0.775573i
$609$		0			0
$610$		−	610481.i		−	1.64064i
$611$	176490.	−	183366.i	0.472758	−	0.491176i
$612$		0			0
$613$	193600.	+	193600.i	0.515209	+	0.515209i	0.916118	−	0.400909i	$-0.131306\pi$
$613$	193600.	+	193600.i	0.515209	+	0.515209i	−0.400909	+	0.916118i	$0.631306\pi$
$614$	407265.			1.08029
$615$		0			0
$616$	81989.4	+	81989.4i	0.216071	+	0.216071i
$617$	−234942.	−	234942.i	−0.617149	−	0.617149i	0.327650	−	0.944799i	$-0.393743\pi$
$617$	−234942.	−	234942.i	−0.617149	−	0.617149i	−0.944799	+	0.327650i	$0.893743\pi$
$618$		0			0
$619$	518107.	−	518107.i	1.35219	−	1.35219i	0.468984	−	0.883206i	$-0.344620\pi$
$619$	518107.	−	518107.i	1.35219	−	1.35219i	0.883206	−	0.468984i	$-0.155380\pi$
$620$	347500.			0.904006
$621$		0			0
$622$	−2287.92	+	2287.92i	−0.00591371	+	0.00591371i
$623$			233168.i			0.600749i
$624$		0			0
$625$	29362.4			0.0751678
$626$	−87578.5	−	87578.5i	−0.223485	−	0.223485i
$627$		0			0
$628$		−	280915.i		−	0.712288i
$629$	40568.7	+	40568.7i	0.102539	+	0.102539i
$630$		0			0
$631$	−335596.	+	335596.i	−0.842866	+	0.842866i	−0.989231	−	0.146365i	$-0.953243\pi$
$631$	−335596.	+	335596.i	−0.842866	+	0.842866i	0.146365	+	0.989231i	$0.453243\pi$
$632$	303120.	−	303120.i	0.758892	−	0.758892i
$633$		0			0
$634$		−	339815.i		−	0.845403i
$635$	214767.	−	214767.i	0.532623	−	0.532623i
$636$		0			0
$637$	−5855.99	−	306484.i	−0.0144318	−	0.755317i
$638$	−131437.			−0.322905
$639$		0			0
$640$	467222.			1.14068
$641$		−	555098.i		−	1.35099i	−0.737362	−	0.675497i	$-0.763929\pi$
$641$		−	555098.i		−	1.35099i	0.737362	−	0.675497i	$-0.236071\pi$
$642$		0			0
$643$	263449.	+	263449.i	0.637197	+	0.637197i	0.949863	−	0.312666i	$-0.101222\pi$
$643$	263449.	+	263449.i	0.637197	+	0.637197i	−0.312666	+	0.949863i	$0.601222\pi$
$644$	−127970.	+	127970.i	−0.308557	+	0.308557i
$645$		0			0
$646$	45654.7			0.109401
$647$			396185.i			0.946433i	0.880946	+	0.473216i	$0.156907\pi$
$647$			396185.i			0.946433i	−0.880946	+	0.473216i	$0.843093\pi$
$648$		0			0
$649$			245612.i			0.583123i
$650$	−273898.	+	284569.i	−0.648279	+	0.673535i
$651$		0			0
$652$	−238058.	−	238058.i	−0.559999	−	0.559999i
$653$	−825533.			−1.93601			−0.968006	−	0.250929i	$-0.919264\pi$
$653$	−825533.			−1.93601			−0.968006	+	0.250929i	$0.919264\pi$
$654$		0			0
$655$	−275783.	−	275783.i	−0.642813	−	0.642813i
$656$	−35354.7	−	35354.7i	−0.0821560	−	0.0821560i
$657$		0			0
$658$	−60916.5	+	60916.5i	−0.140697	+	0.140697i
$659$	−123474.			−0.284317			−0.142159	−	0.989844i	$-0.545404\pi$
$659$	−123474.			−0.284317			−0.142159	+	0.989844i	$0.545404\pi$
$660$		0			0
$661$	121255.	−	121255.i	0.277522	−	0.277522i	−0.554597	−	0.832119i	$-0.687128\pi$
$661$	121255.	−	121255.i	0.277522	−	0.277522i	0.832119	+	0.554597i	$0.187128\pi$
$662$		−	302079.i		−	0.689294i
$663$		0			0
$664$	−233566.			−0.529754
$665$	−189029.	−	189029.i	−0.427451	−	0.427451i
$666$		0			0
$667$		−	519959.i		−	1.16874i
$668$	79915.8	+	79915.8i	0.179093	+	0.179093i
$669$		0			0
$670$	−519174.	+	519174.i	−1.15655	+	1.15655i
$671$	−348982.	+	348982.i	−0.775101	+	0.775101i
$672$		0			0
$673$			140601.i			0.310425i	0.987881	+	0.155213i	$0.0496062\pi$
$673$			140601.i			0.310425i	−0.987881	+	0.155213i	$0.950394\pi$
$674$	80851.5	−	80851.5i	0.177979	−	0.177979i
$675$		0			0
$676$	−11375.5	−	297571.i	−0.0248930	−	0.651174i
$677$	543307.			1.18541			0.592704	−	0.805420i	$-0.298060\pi$
$677$	543307.			1.18541			0.592704	+	0.805420i	$0.298060\pi$
$678$		0			0
$679$	−325511.			−0.706035
$680$		−	176608.i		−	0.381937i
$681$		0			0
$682$	106191.	+	106191.i	0.228307	+	0.228307i
$683$	475365.	−	475365.i	1.01903	−	1.01903i	0.0192116	−	0.999815i	$-0.493884\pi$
$683$	475365.	−	475365.i	1.01903	−	1.01903i	0.999815	−	0.0192116i	$-0.00611561\pi$
$684$		0			0
$685$	415216.			0.884898
$686$			241115.i			0.512362i
$687$		0			0
$688$			56242.7i			0.118820i
$689$	430830.	−	8231.87i	0.907544	−	0.0173404i
$690$		0			0
$691$	−212706.	−	212706.i	−0.445476	−	0.445476i	0.448371	−	0.893847i	$-0.352004\pi$
$691$	−212706.	−	212706.i	−0.445476	−	0.445476i	−0.893847	+	0.448371i	$0.852004\pi$
$692$	−97827.1			−0.204290
$693$		0			0
$694$	83578.6	+	83578.6i	0.173531	+	0.173531i
$695$	412274.	+	412274.i	0.853526	+	0.853526i
$696$		0			0
$697$	−127503.	+	127503.i	−0.262456	+	0.262456i
$698$	122461.			0.251354
$699$		0			0
$700$	−176849.	+	176849.i	−0.360916	+	0.360916i
$701$			56550.4i			0.115080i	0.998343	+	0.0575400i	$0.0183257\pi$
$701$			56550.4i			0.115080i	−0.998343	+	0.0575400i	$0.981674\pi$
$702$		0			0
$703$	223600.			0.452439
$704$	116767.	+	116767.i	0.235599	+	0.235599i
$705$		0			0
$706$		−	305400.i		−	0.612716i
$707$	−283421.	−	283421.i	−0.567013	−	0.567013i
$708$		0			0
$709$	284754.	−	284754.i	0.566470	−	0.566470i	−0.364668	−	0.931138i	$-0.618818\pi$
$709$	284754.	−	284754.i	0.566470	−	0.566470i	0.931138	+	0.364668i	$0.118818\pi$
$710$	−440529.	+	440529.i	−0.873892	+	0.873892i
$711$		0			0
$712$			600344.i			1.18424i
$713$	−420088.	+	420088.i	−0.826345	+	0.826345i
$714$		0			0
$715$	520807.	−	9951.05i	1.01874	−	0.0194651i
$716$	3357.54			0.00654931
$717$		0			0
$718$	66456.3			0.128910
$719$		−	513980.i		−	0.994234i	−0.867684	−	0.497117i	$-0.834392\pi$
$719$		−	513980.i		−	0.994234i	0.867684	−	0.497117i	$-0.165608\pi$
$720$		0			0
$721$	191017.	+	191017.i	0.367453	+	0.367453i
$722$	−91738.2	+	91738.2i	−0.175985	+	0.175985i
$723$		0			0
$724$	−296937.			−0.566483
$725$		−	718561.i		−	1.36706i
$726$		0			0
$727$		−	5325.23i		−	0.0100756i	−0.999987	−	0.00503779i	$-0.998396\pi$
$727$		−	5325.23i		−	0.0100756i	0.999987	−	0.00503779i	$-0.00160358\pi$
$728$	4880.69	+	255440.i	0.00920912	+	0.481977i
$729$		0			0
$730$	531821.	+	531821.i	0.997975	+	0.997975i
$731$	202834.			0.379582
$732$		0			0
$733$	471793.	+	471793.i	0.878100	+	0.878100i	0.993338	−	0.115238i	$-0.0367631\pi$
$733$	471793.	+	471793.i	0.878100	+	0.878100i	−0.115238	+	0.993338i	$0.536763\pi$
$734$	−22602.4	−	22602.4i	−0.0419530	−	0.0419530i
$735$		0			0
$736$	−528976.	+	528976.i	−0.976518	+	0.976518i
$737$	593573.			1.09280
$738$		0			0
$739$	383237.	−	383237.i	0.701745	−	0.701745i	−0.263040	−	0.964785i	$-0.584725\pi$
$739$	383237.	−	383237.i	0.701745	−	0.701745i	0.964785	+	0.263040i	$0.0847252\pi$
$740$		−	341265.i		−	0.623201i
$741$		0			0
$742$	−145862.			−0.264932
$743$	−77467.3	−	77467.3i	−0.140327	−	0.140327i	0.633454	−	0.773781i	$-0.281637\pi$
$743$	−77467.3	−	77467.3i	−0.140327	−	0.140327i	−0.773781	+	0.633454i	$0.781637\pi$
$744$		0			0
$745$		−	352874.i		−	0.635780i
$746$	127709.	+	127709.i	0.229479	+	0.229479i
$747$		0			0
$748$	−39832.4	+	39832.4i	−0.0711924	+	0.0711924i
$749$	159198.	−	159198.i	0.283774	−	0.283774i
$750$		0			0
$751$			287709.i			0.510121i	0.966925	+	0.255061i	$0.0820954\pi$
$751$			287709.i			0.510121i	−0.966925	+	0.255061i	$0.917905\pi$
$752$	20799.1	−	20799.1i	0.0367797	−	0.0367797i
$753$		0			0
$754$	−208659.	−	200835.i	−0.367024	−	0.353261i
$755$	359304.			0.630330
$756$		0			0
$757$	−962044.			−1.67882			−0.839408	−	0.543502i	$-0.817098\pi$
$757$	−962044.			−1.67882			−0.839408	+	0.543502i	$0.817098\pi$
$758$		−	32964.5i		−	0.0573731i
$759$		0			0
$760$	−486698.	−	486698.i	−0.842622	−	0.842622i
$761$	−496273.	+	496273.i	−0.856943	+	0.856943i	−0.990977	−	0.134034i	$-0.957207\pi$
$761$	−496273.	+	496273.i	−0.856943	+	0.856943i	0.134034	+	0.990977i	$0.457207\pi$
$762$		0			0
$763$	−68047.8			−0.116887
$764$			407148.i			0.697534i
$765$		0			0
$766$			307378.i			0.523860i
$767$	−375294.	+	389915.i	−0.637942	+	0.662795i
$768$		0			0
$769$	326544.	+	326544.i	0.552191	+	0.552191i	0.927073	−	0.374882i	$-0.122317\pi$
$769$	326544.	+	326544.i	0.552191	+	0.552191i	−0.374882	+	0.927073i	$0.622317\pi$
$770$	−176324.			−0.297393
$771$		0			0
$772$	363350.	+	363350.i	0.609664	+	0.609664i
$773$	−213799.	−	213799.i	−0.357804	−	0.357804i	0.505199	−	0.863003i	$-0.331419\pi$
$773$	−213799.	−	213799.i	−0.357804	−	0.357804i	−0.863003	+	0.505199i	$0.831419\pi$
$774$		0			0
$775$	−580544.	+	580544.i	−0.966566	+	0.966566i
$776$	−838101.			−1.39179
$777$		0			0
$778$	59244.6	−	59244.6i	0.0978790	−	0.0978790i
$779$			702751.i			1.15805i
$780$		0			0
$781$	503658.			0.825721
$782$	84234.8	+	84234.8i	0.137746	+	0.137746i
$783$		0			0
$784$		−	35428.5i		−	0.0576395i
$785$	765602.	+	765602.i	1.24241	+	1.24241i
$786$		0			0
$787$	642953.	−	642953.i	1.03808	−	1.03808i	0.0388320	−	0.999246i	$-0.487636\pi$
$787$	642953.	−	642953.i	1.03808	−	1.03808i	0.999246	−	0.0388320i	$-0.0123637\pi$
$788$	−351948.	+	351948.i	−0.566795	+	0.566795i
$789$		0			0
$790$			651882.i			1.04452i
$791$	−30825.8	+	30825.8i	−0.0492677	+	0.0492677i
$792$		0			0
$793$	−1.08726e6	+	20774.3i	−1.72897	+	0.0330354i
$794$	−152795.			−0.242364
$795$		0			0
$796$	−126066.			−0.198963
$797$		−	735073.i		−	1.15721i	−0.815606	−	0.578607i	$-0.803596\pi$
$797$		−	735073.i		−	1.15721i	0.815606	−	0.578607i	$-0.196404\pi$
$798$		0			0
$799$	−75009.8	−	75009.8i	−0.117496	−	0.117496i
$800$	−731022.	+	731022.i	−1.14222	+	1.14222i
$801$		0			0
$802$	177432.			0.275857
$803$		−	608032.i		−	0.942965i
$804$		0			0
$805$		−	697534.i		−	1.07640i
$806$	6321.36	+	330840.i	0.00973062	+	0.509270i
$807$		0			0
$808$	−729731.	−	729731.i	−1.11774	−	1.11774i
$809$	388192.			0.593130			0.296565	−	0.955013i	$-0.404159\pi$
$809$	388192.			0.593130			0.296565	+	0.955013i	$0.404159\pi$
$810$		0			0
$811$	−163285.	−	163285.i	−0.248259	−	0.248259i	0.571997	−	0.820256i	$-0.306169\pi$
$811$	−163285.	−	163285.i	−0.248259	−	0.248259i	−0.820256	+	0.571997i	$0.806169\pi$
$812$	−129674.	−	129674.i	−0.196671	−	0.196671i
$813$		0			0
$814$	104286.	−	104286.i	0.157389	−	0.157389i
$815$	1.29760e6			1.95355
$816$		0			0
$817$	558972.	−	558972.i	0.837426	−	0.837426i
$818$		−	251387.i		−	0.375696i
$819$		0			0
$820$	1.07256e6			1.59512
$821$	401933.	+	401933.i	0.596303	+	0.596303i	0.939327	−	0.343024i	$-0.111451\pi$
$821$	401933.	+	401933.i	0.596303	+	0.596303i	−0.343024	+	0.939327i	$0.611451\pi$
$822$		0			0
$823$		−	129326.i		−	0.190935i	−0.995433	−	0.0954674i	$-0.969565\pi$
$823$		−	129326.i		−	0.190935i	0.995433	−	0.0954674i	$-0.0304346\pi$
$824$	491816.	+	491816.i	0.724350	+	0.724350i
$825$		0			0
$826$	129535.	−	129535.i	0.189857	−	0.189857i
$827$	−51675.4	+	51675.4i	−0.0755567	+	0.0755567i	−0.743875	−	0.668319i	$-0.767014\pi$
$827$	−51675.4	+	51675.4i	−0.0755567	+	0.0755567i	0.668319	+	0.743875i	$0.267014\pi$
$828$		0			0
$829$			984088.i			1.43194i	0.698131	+	0.715970i	$0.254015\pi$
$829$			984088.i			1.43194i	−0.698131	+	0.715970i	$0.745985\pi$
$830$	251151.	−	251151.i	0.364568	−	0.364568i
$831$		0			0
$832$	6950.92	+	363789.i	0.0100414	+	0.525537i
$833$	−127769.			−0.184135
$834$		0			0
$835$	−435603.			−0.624767
$836$			219542.i			0.314127i
$837$		0			0
$838$	−441624.	−	441624.i	−0.628876	−	0.628876i
$839$	−378169.	+	378169.i	−0.537232	+	0.537232i	−0.922715	−	0.385483i	$-0.874035\pi$
$839$	−378169.	+	378169.i	−0.537232	+	0.537232i	0.385483	+	0.922715i	$0.374035\pi$
$840$		0			0
$841$	−180399.			−0.255060
$842$		−	102457.i		−	0.144517i
$843$		0			0
$844$		−	532409.i		−	0.747413i
$845$	841999.	+	779994.i	1.17923	+	1.09239i
$846$		0			0
$847$	−150064.	−	150064.i	−0.209175	−	0.209175i
$848$	49802.4			0.0692562
$849$		0			0
$850$	116409.	+	116409.i	0.161119	+	0.161119i
$851$	412551.	+	412551.i	0.569663	+	0.569663i
$852$		0			0
$853$	−296254.	+	296254.i	−0.407161	+	0.407161i	−0.880747	−	0.473586i	$-0.842959\pi$
$853$	−296254.	+	296254.i	−0.407161	+	0.407161i	0.473586	+	0.880747i	$0.342959\pi$
$854$	368104.			0.504724
$855$		0			0
$856$	409890.	−	409890.i	0.559396	−	0.559396i
$857$		−	1.04442e6i		−	1.42205i	−0.703168	−	0.711023i	$-0.748232\pi$
$857$		−	1.04442e6i		−	1.42205i	0.703168	−	0.711023i	$-0.251768\pi$
$858$		0			0
$859$	503818.			0.682790			0.341395	−	0.939920i	$-0.389101\pi$
$859$	503818.			0.682790			0.341395	+	0.939920i	$0.389101\pi$
$860$	−853122.	−	853122.i	−1.15349	−	1.15349i
$861$		0			0
$862$			463268.i			0.623473i
$863$	283034.	+	283034.i	0.380029	+	0.380029i	0.871113	−	0.491083i	$-0.163399\pi$
$863$	283034.	+	283034.i	0.380029	+	0.380029i	−0.491083	+	0.871113i	$0.663399\pi$
$864$		0			0
$865$	266617.	−	266617.i	0.356332	−	0.356332i
$866$	321318.	−	321318.i	0.428449	−	0.428449i
$867$		0			0
$868$			209533.i			0.278108i
$869$	372649.	−	372649.i	0.493470	−	0.493470i
$870$		0			0
$871$	942311.	+	906977.i	1.24210	+	1.19553i
$872$	−175204.			−0.230416
$873$		0			0
$874$	464271.			0.607783
$875$		−	355360.i		−	0.464144i
$876$		0			0
$877$	86669.2	+	86669.2i	0.112685	+	0.112685i	0.761201	−	0.648516i	$-0.224610\pi$
$877$	86669.2	+	86669.2i	0.112685	+	0.112685i	−0.648516	+	0.761201i	$0.724610\pi$
$878$	−369548.	+	369548.i	−0.479383	+	0.479383i
$879$		0			0
$880$	60203.4			0.0777420
$881$			346696.i			0.446681i	0.974741	+	0.223340i	$0.0716961\pi$
$881$			346696.i			0.446681i	−0.974741	+	0.223340i	$0.928304\pi$
$882$		0			0
$883$			570534.i			0.731746i	0.930665	+	0.365873i	$0.119230\pi$
$883$			570534.i			0.731746i	−0.930665	+	0.365873i	$0.880770\pi$
$884$	−124099.	+	2371.16i	−0.158805	+	0.00303428i
$885$		0			0
$886$	591378.	+	591378.i	0.753352	+	0.753352i
$887$	−988202.			−1.25603			−0.628013	−	0.778203i	$-0.716131\pi$
$887$	−988202.			−1.25603			−0.628013	+	0.778203i	$0.716131\pi$
$888$		0			0
$889$	129499.	+	129499.i	0.163856	+	0.163856i
$890$	−645542.	−	645542.i	−0.814976	−	0.814976i
$891$		0			0
$892$	81970.3	−	81970.3i	0.103021	−	0.103021i
$893$	−413427.			−0.518436
$894$		0			0
$895$	−9150.60	+	9150.60i	−0.0114236	+	0.0114236i
$896$			281722.i			0.350918i
$897$		0			0
$898$	−626753.			−0.777219
$899$	−425681.	−	425681.i	−0.526702	−	0.526702i
$900$		0			0
$901$		−	179608.i		−	0.221246i
$902$	327759.	+	327759.i	0.402848	+	0.402848i
$903$		0			0
$904$	−79368.0	+	79368.0i	−0.0971200	+	0.0971200i
$905$	809268.	−	809268.i	0.988088	−	0.988088i
$906$		0			0
$907$		−	720192.i		−	0.875455i	−0.899108	−	0.437728i	$-0.855783\pi$
$907$		−	720192.i		−	0.875455i	0.899108	−	0.437728i	$-0.144217\pi$
$908$	100535.	−	100535.i	0.121940	−	0.121940i
$909$		0			0
$910$	−279919.	−	269423.i	−0.338026	−	0.325351i
$911$	−963155.			−1.16054			−0.580269	−	0.814425i	$-0.697053\pi$
$911$	−963155.			−1.16054			−0.580269	+	0.814425i	$0.697053\pi$
$912$		0			0
$913$	−287142.			−0.344473
$914$			296455.i			0.354867i
$915$		0			0
$916$	−378326.	−	378326.i	−0.450895	−	0.450895i
$917$	166290.	−	166290.i	0.197755	−	0.197755i
$918$		0			0
$919$	−1.29069e6			−1.52823			−0.764117	−	0.645077i	$-0.776825\pi$
$919$	−1.29069e6			−1.52823			−0.764117	+	0.645077i	$0.776825\pi$
$920$		−	1.79596e6i		−	2.12188i
$921$		0			0
$922$		−	188961.i		−	0.222285i
$923$	799569.	+	769587.i	0.938540	+	0.903347i
$924$		0			0
$925$	570127.	+	570127.i	0.666328	+	0.666328i
$926$	−571149.			−0.666082
$927$		0			0
$928$	−536019.	−	536019.i	−0.622421	−	0.622421i
$929$	−620998.	−	620998.i	−0.719546	−	0.719546i	0.248966	−	0.968512i	$-0.419909\pi$
$929$	−620998.	−	620998.i	−0.719546	−	0.719546i	−0.968512	+	0.248966i	$0.919909\pi$
$930$		0			0
$931$	−352109.	+	352109.i	−0.406235	+	0.406235i
$932$	−24658.8			−0.0283883
$933$		0			0
$934$	113752.	−	113752.i	0.130396	−	0.130396i
$935$		−	217118.i		−	0.248354i
$936$		0			0
$937$	1.00082e6			1.13992			0.569960	−	0.821672i	$-0.306959\pi$
$937$	1.00082e6			1.13992			0.569960	+	0.821672i	$0.306959\pi$
$938$	−313048.	−	313048.i	−0.355799	−	0.355799i
$939$		0			0
$940$			630985.i			0.714107i
$941$	−902091.	−	902091.i	−1.01876	−	1.01876i	−0.999821	−	0.0189381i	$-0.993971\pi$
$941$	−902091.	−	902091.i	−1.01876	−	1.01876i	−0.0189381	−	0.999821i	$-0.506029\pi$
$942$		0			0
$943$	−1.29661e6	+	1.29661e6i	−1.45809	+	1.45809i
$944$	−44227.7	+	44227.7i	−0.0496307	+	0.0496307i
$945$		0			0
$946$		−	521403.i		−	0.582628i
$947$	64662.8	−	64662.8i	0.0721032	−	0.0721032i	−0.670136	−	0.742239i	$-0.733764\pi$
$947$	64662.8	−	64662.8i	0.0721032	−	0.0721032i	0.742239	+	0.670136i	$0.233764\pi$
$948$		0			0
$949$	929071.	−	965266.i	1.03161	−	1.07180i
$950$	641603.			0.710917
$951$		0			0
$952$	106490.			0.117499
$953$		−	1.57625e6i		−	1.73556i	−0.496948	−	0.867781i	$-0.665546\pi$
$953$		−	1.57625e6i		−	1.73556i	0.496948	−	0.867781i	$-0.334454\pi$
$954$		0			0
$955$	−1.10963e6	−	1.10963e6i	−1.21667	−	1.21667i
$956$	196448.	−	196448.i	0.214947	−	0.214947i
$957$		0			0
$958$	−379659.			−0.413679
$959$			250364.i			0.272229i
$960$		0			0
$961$		−	235683.i		−	0.255201i
$962$	324904.	−	6207.93i	0.351079	−	0.00670806i
$963$		0			0
$964$	287452.	+	287452.i	0.309322	+	0.309322i
$965$	−1.98054e6			−2.12681
$966$		0			0
$967$	−561644.	−	561644.i	−0.600632	−	0.600632i	0.339849	−	0.940480i	$-0.389624\pi$
$967$	−561644.	−	561644.i	−0.600632	−	0.600632i	−0.940480	+	0.339849i	$0.889624\pi$
$968$	−386373.	−	386373.i	−0.412340	−	0.412340i
$969$		0			0
$970$	901199.	−	901199.i	0.957806	−	0.957806i
$971$	70845.6			0.0751405			0.0375703	−	0.999294i	$-0.488038\pi$
$971$	70845.6			0.0751405			0.0375703	+	0.999294i	$0.488038\pi$
$972$		0			0
$973$	−248590.	+	248590.i	−0.262578	+	0.262578i
$974$		−	103464.i		−	0.109061i
$975$		0			0
$976$	−125684.			−0.131941
$977$	617175.	+	617175.i	0.646576	+	0.646576i	0.952164	−	0.305588i	$-0.0988531\pi$
$977$	617175.	+	617175.i	0.646576	+	0.646576i	−0.305588	+	0.952164i	$0.598853\pi$
$978$		0			0
$979$			738050.i			0.770052i
$980$	537400.	+	537400.i	0.559559	+	0.559559i
$981$		0			0
$982$	383080.	−	383080.i	0.397252	−	0.397252i
$983$	−665135.	+	665135.i	−0.688339	+	0.688339i	−0.961865	−	0.273526i	$-0.911810\pi$
$983$	−665135.	+	665135.i	−0.688339	+	0.688339i	0.273526	+	0.961865i	$0.411810\pi$
$984$		0			0
$985$		−	1.91839e6i		−	1.97726i
$986$	−85356.3	+	85356.3i	−0.0877974	+	0.0877974i
$987$		0			0
$988$	−335459.	+	348528.i	−0.343658	+	0.357046i
$989$	2.06265e6			2.10879
$990$		0			0
$991$	250003.			0.254565			0.127282	−	0.991867i	$-0.459375\pi$
$991$	250003.			0.254565			0.127282	+	0.991867i	$0.459375\pi$
$992$			866127.i			0.880153i
$993$		0			0
$994$	−265627.	−	265627.i	−0.268843	−	0.268843i
$995$	343579.	−	343579.i	0.347041	−	0.347041i
$996$		0			0
$997$	−1.47712e6			−1.48602			−0.743011	−	0.669279i	$-0.766603\pi$
$997$	−1.47712e6			−1.48602			−0.743011	+	0.669279i	$0.766603\pi$
$998$		−	97688.8i		−	0.0980808i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.73.4		20
3.2	odd	2		39.5.g.a.34.7	yes	20
13.5	odd	4	inner	117.5.j.b.109.4		20
39.5	even	4		39.5.g.a.31.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.7	✓	20	39.5	even	4
39.5.g.a.34.7	yes	20	3.2	odd	2
117.5.j.b.73.4		20	1.1	even	1	trivial
117.5.j.b.109.4		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)

\(f(q)\)	\(=\)	\(q+(-1.66937 - 1.66937i) q^{2} -10.4264i q^{4} +(28.4160 + 28.4160i) q^{5} +(-17.1341 + 17.1341i) q^{7} +(-44.1155 + 44.1155i) q^{8} -94.8736i q^{10} +(-54.2346 + 54.2346i) q^{11} +(-168.969 + 3.22849i) q^{13} +57.2062 q^{14} -19.5322 q^{16} +70.4411i q^{17} +(194.123 + 194.123i) q^{19} +(296.276 - 296.276i) q^{20} +181.075 q^{22} +716.329i q^{23} +989.935i q^{25} +(287.462 + 276.683i) q^{26} +(178.647 + 178.647i) q^{28} -725.866 q^{29} +(586.446 + 586.446i) q^{31} +(738.454 + 738.454i) q^{32} +(117.592 - 117.592i) q^{34} -973.762 q^{35} +(575.923 - 575.923i) q^{37} -648.126i q^{38} -2507.17 q^{40} +(1810.07 + 1810.07i) q^{41} -2879.48i q^{43} +(565.472 + 565.472i) q^{44} +(1195.82 - 1195.82i) q^{46} +(-1064.86 + 1064.86i) q^{47} +1813.85i q^{49} +(1652.57 - 1652.57i) q^{50} +(33.6615 + 1761.74i) q^{52} -2549.76 q^{53} -3082.26 q^{55} -1511.75i q^{56} +(1211.74 + 1211.74i) q^{58} +(2264.35 - 2264.35i) q^{59} +6434.68 q^{61} -1957.99i q^{62} -2152.99i q^{64} +(-4893.16 - 4709.68i) q^{65} +(-5472.27 - 5472.27i) q^{67} +734.447 q^{68} +(1625.57 + 1625.57i) q^{70} +(-4643.32 - 4643.32i) q^{71} +(-5605.57 + 5605.57i) q^{73} -1922.86 q^{74} +(2024.00 - 2024.00i) q^{76} -1858.52i q^{77} -6871.05 q^{79} +(-555.027 - 555.027i) q^{80} -6043.36i q^{82} +(2647.22 + 2647.22i) q^{83} +(-2001.65 + 2001.65i) q^{85} +(-4806.92 + 4806.92i) q^{86} -4785.17i q^{88} +(6804.23 - 6804.23i) q^{89} +(2839.81 - 2950.45i) q^{91} +7468.73 q^{92} +3555.29 q^{94} +11032.4i q^{95} +(9498.95 + 9498.95i) q^{97} +(3027.99 - 3027.99i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 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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