LMFDB - Embedded newform 704.4.e.g.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(703,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("704.703");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.e (of order $2$, degree $1$, not 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:	$41.5373446440$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.2
Character	$\chi$	$=$	704.703
Dual form			704.4.e.g.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.41697i q^{3} +17.7991 q^{5} +30.3804 q^{7} -61.6793 q^{9} +(36.4823 - 0.203999i) q^{11} -73.3735i q^{13} +167.613i q^{15} +33.5998i q^{17} +3.79105 q^{19} +286.091i q^{21} -39.3421i q^{23} +191.807 q^{25} -326.574i q^{27} +80.3580i q^{29} +132.048i q^{31} +(1.92105 + 343.553i) q^{33} +540.743 q^{35} +177.408 q^{37} +690.957 q^{39} +155.637i q^{41} +49.0555 q^{43} -1097.84 q^{45} -363.000i q^{47} +579.967 q^{49} -316.409 q^{51} -313.897 q^{53} +(649.351 - 3.63099i) q^{55} +35.7002i q^{57} -31.5479i q^{59} -204.455i q^{61} -1873.84 q^{63} -1305.98i q^{65} -638.489i q^{67} +370.484 q^{69} +712.511i q^{71} +747.515i q^{73} +1806.24i q^{75} +(1108.35 - 6.19755i) q^{77} +722.055 q^{79} +1410.00 q^{81} +6.82976 q^{83} +598.046i q^{85} -756.729 q^{87} -1160.53 q^{89} -2229.12i q^{91} -1243.49 q^{93} +67.4771 q^{95} -855.560 q^{97} +(-2250.20 + 12.5825i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) $ 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			9.41697i			1.81230i	0.422960	+	0.906148i	$0.360991\pi$
$3$			9.41697i			1.81230i	−0.422960	+	0.906148i	$0.639009\pi$
$4$		0			0
$5$	17.7991			1.59200			0.795999	−	0.605298i	$-0.206946\pi$
$5$	17.7991			1.59200			0.795999	+	0.605298i	$0.206946\pi$
$6$		0			0
$7$	30.3804			1.64039			0.820193	−	0.572087i	$-0.193866\pi$
$7$	30.3804			1.64039			0.820193	+	0.572087i	$0.193866\pi$
$8$		0			0
$9$	−61.6793			−2.28442
$10$		0			0
$11$	36.4823	−	0.203999i	0.999984	−	0.00559163i
$11$	36.4823	−	0.203999i	0.999984	−	0.00559163i
$12$		0			0
$13$		−	73.3735i		−	1.56540i	−0.622401	−	0.782698i	$-0.713843\pi$
$13$		−	73.3735i		−	1.56540i	0.622401	−	0.782698i	$-0.286157\pi$
$14$		0			0
$15$			167.613i			2.88517i
$16$		0			0
$17$			33.5998i			0.479362i	0.970852	+	0.239681i	$0.0770429\pi$
$17$			33.5998i			0.479362i	−0.970852	+	0.239681i	$0.922957\pi$
$18$		0			0
$19$	3.79105			0.0457750			0.0228875	−	0.999738i	$-0.492714\pi$
$19$	3.79105			0.0457750			0.0228875	+	0.999738i	$0.492714\pi$
$20$		0			0
$21$			286.091i			2.97287i
$22$		0			0
$23$		−	39.3421i		−	0.356670i	−0.983970	−	0.178335i	$-0.942929\pi$
$23$		−	39.3421i		−	0.356670i	0.983970	−	0.178335i	$-0.0570710\pi$
$24$		0			0
$25$	191.807			1.53446
$26$		0			0
$27$		−	326.574i		−	2.32775i
$28$		0			0
$29$			80.3580i			0.514555i	0.966338	+	0.257278i	$0.0828255\pi$
$29$			80.3580i			0.514555i	−0.966338	+	0.257278i	$0.917175\pi$
$30$		0			0
$31$			132.048i			0.765047i	0.923946	+	0.382523i	$0.124945\pi$
$31$			132.048i			0.765047i	−0.923946	+	0.382523i	$0.875055\pi$
$32$		0			0
$33$	1.92105	+	343.553i	0.0101337	+	1.81227i
$34$		0			0
$35$	540.743			2.61149
$36$		0			0
$37$	177.408			0.788260			0.394130	−	0.919055i	$-0.371046\pi$
$37$	177.408			0.788260			0.394130	+	0.919055i	$0.371046\pi$
$38$		0			0
$39$	690.957			2.83696
$40$		0			0
$41$			155.637i			0.592841i	0.955058	+	0.296420i	$0.0957929\pi$
$41$			155.637i			0.592841i	−0.955058	+	0.296420i	$0.904207\pi$
$42$		0			0
$43$	49.0555			0.173974			0.0869871	−	0.996209i	$-0.472276\pi$
$43$	49.0555			0.173974			0.0869871	+	0.996209i	$0.472276\pi$
$44$		0			0
$45$	−1097.84			−3.63679
$46$		0			0
$47$		−	363.000i		−	1.12658i	−0.826261	−	0.563288i	$-0.809536\pi$
$47$		−	363.000i		−	1.12658i	0.826261	−	0.563288i	$-0.190464\pi$
$48$		0			0
$49$	579.967			1.69087
$50$		0			0
$51$	−316.409			−0.868747
$52$		0			0
$53$	−313.897			−0.813530			−0.406765	−	0.913533i	$-0.633343\pi$
$53$	−313.897			−0.813530			−0.406765	+	0.913533i	$0.633343\pi$
$54$		0			0
$55$	649.351	−	3.63099i	1.59197	−	0.00890186i
$56$		0			0
$57$			35.7002i			0.0829580i
$58$		0			0
$59$		−	31.5479i		−	0.0696134i	−0.999394	−	0.0348067i	$-0.988918\pi$
$59$		−	31.5479i		−	0.0696134i	0.999394	−	0.0348067i	$-0.0110815\pi$
$60$		0			0
$61$		−	204.455i		−	0.429144i	−0.976708	−	0.214572i	$-0.931164\pi$
$61$		−	204.455i		−	0.429144i	0.976708	−	0.214572i	$-0.0688356\pi$
$62$		0			0
$63$	−1873.84			−3.74733
$64$		0			0
$65$		−	1305.98i		−	2.49211i
$66$		0			0
$67$		−	638.489i		−	1.16424i	−0.813104	−	0.582118i	$-0.802224\pi$
$67$		−	638.489i		−	1.16424i	0.813104	−	0.582118i	$-0.197776\pi$
$68$		0			0
$69$	370.484			0.646391
$70$		0			0
$71$			712.511i			1.19098i	0.803363	+	0.595489i	$0.203042\pi$
$71$			712.511i			1.19098i	−0.803363	+	0.595489i	$0.796958\pi$
$72$		0			0
$73$			747.515i			1.19849i	0.800564	+	0.599247i	$0.204533\pi$
$73$			747.515i			1.19849i	−0.800564	+	0.599247i	$0.795467\pi$
$74$		0			0
$75$			1806.24i			2.78089i
$76$		0			0
$77$	1108.35	−	6.19755i	1.64036	−	0.00917243i
$78$		0			0
$79$	722.055			1.02832			0.514162	−	0.857693i	$-0.328103\pi$
$79$	722.055			1.02832			0.514162	+	0.857693i	$0.328103\pi$
$80$		0			0
$81$	1410.00			1.93416
$82$		0			0
$83$	6.82976			0.00903209			0.00451605	−	0.999990i	$-0.498562\pi$
$83$	6.82976			0.00903209			0.00451605	+	0.999990i	$0.498562\pi$
$84$		0			0
$85$			598.046i			0.763144i
$86$		0			0
$87$	−756.729			−0.932527
$88$		0			0
$89$	−1160.53			−1.38220			−0.691099	−	0.722760i	$-0.742873\pi$
$89$	−1160.53			−1.38220			−0.691099	+	0.722760i	$0.742873\pi$
$90$		0			0
$91$		−	2229.12i		−	2.56786i
$92$		0			0
$93$	−1243.49			−1.38649
$94$		0			0
$95$	67.4771			0.0728738
$96$		0			0
$97$	−855.560			−0.895556			−0.447778	−	0.894145i	$-0.647784\pi$
$97$	−855.560			−0.895556			−0.447778	+	0.894145i	$0.647784\pi$
$98$		0			0
$99$	−2250.20	+	12.5825i	−2.28438	+	0.0127736i
$100$		0			0
$101$			532.344i			0.524457i	0.965006	+	0.262229i	$0.0844574\pi$
$101$			532.344i			0.524457i	−0.965006	+	0.262229i	$0.915543\pi$
$102$		0			0
$103$			1207.14i			1.15479i	0.816465	+	0.577395i	$0.195931\pi$
$103$			1207.14i			1.15479i	−0.816465	+	0.577395i	$0.804069\pi$
$104$		0			0
$105$			5092.16i			4.73280i
$106$		0			0
$107$	649.928			0.587205			0.293603	−	0.955928i	$-0.405146\pi$
$107$	649.928			0.587205			0.293603	+	0.955928i	$0.405146\pi$
$108$		0			0
$109$			660.536i			0.580439i	0.956960	+	0.290220i	$0.0937284\pi$
$109$			660.536i			0.580439i	−0.956960	+	0.290220i	$0.906272\pi$
$110$		0			0
$111$			1670.64i			1.42856i
$112$		0			0
$113$	−1731.78			−1.44170			−0.720851	−	0.693090i	$-0.756249\pi$
$113$	−1731.78			−1.44170			−0.720851	+	0.693090i	$0.756249\pi$
$114$		0			0
$115$		−	700.254i		−	0.567817i
$116$		0			0
$117$			4525.63i			3.57602i
$118$		0			0
$119$			1020.78i			0.786339i
$120$		0			0
$121$	1330.92	−	14.8847i	0.999937	−	0.0111831i
$122$		0			0
$123$	−1465.63			−1.07440
$124$		0			0
$125$	1189.11			0.850856
$126$		0			0
$127$	−2756.86			−1.92623			−0.963116	−	0.269088i	$-0.913278\pi$
$127$	−2756.86			−1.92623			−0.963116	+	0.269088i	$0.913278\pi$
$128$		0			0
$129$			461.954i			0.315293i
$130$		0			0
$131$	−1322.04			−0.881734			−0.440867	−	0.897573i	$-0.645329\pi$
$131$	−1322.04			−0.881734			−0.440867	+	0.897573i	$0.645329\pi$
$132$		0			0
$133$	115.173			0.0750887
$134$		0			0
$135$		−	5812.72i		−	3.70578i
$136$		0			0
$137$	−1272.26			−0.793408			−0.396704	−	0.917947i	$-0.629846\pi$
$137$	−1272.26			−0.793408			−0.396704	+	0.917947i	$0.629846\pi$
$138$		0			0
$139$	2210.02			1.34857			0.674287	−	0.738469i	$-0.264451\pi$
$139$	2210.02			1.34857			0.674287	+	0.738469i	$0.264451\pi$
$140$		0			0
$141$	3418.36			2.04169
$142$		0			0
$143$	−14.9681	−	2676.84i	−0.00875311	−	1.56537i
$144$		0			0
$145$			1430.30i			0.819171i
$146$		0			0
$147$			5461.54i			3.06435i
$148$		0			0
$149$		−	3036.72i		−	1.66965i	−0.550515	−	0.834825i	$-0.685569\pi$
$149$		−	3036.72i		−	1.66965i	0.550515	−	0.834825i	$-0.314431\pi$
$150$		0			0
$151$	−2031.12			−1.09464			−0.547319	−	0.836924i	$-0.684352\pi$
$151$	−2031.12			−1.09464			−0.547319	+	0.836924i	$0.684352\pi$
$152$		0			0
$153$		−	2072.42i		−	1.09506i
$154$		0			0
$155$			2350.33i			1.21795i
$156$		0			0
$157$	−1074.14			−0.546026			−0.273013	−	0.962010i	$-0.588020\pi$
$157$	−1074.14			−0.546026			−0.273013	+	0.962010i	$0.588020\pi$
$158$		0			0
$159$		−	2955.96i		−	1.47436i
$160$		0			0
$161$		−	1195.23i		−	0.585076i
$162$		0			0
$163$		−	876.432i		−	0.421150i	−0.977578	−	0.210575i	$-0.932466\pi$
$163$		−	876.432i		−	0.421150i	0.977578	−	0.210575i	$-0.0675336\pi$
$164$		0			0
$165$	34.1929	+	6114.92i	0.0161328	+	2.88513i
$166$		0			0
$167$	2312.96			1.07175			0.535874	−	0.844298i	$-0.319982\pi$
$167$	2312.96			1.07175			0.535874	+	0.844298i	$0.319982\pi$
$168$		0			0
$169$	−3186.68			−1.45047
$170$		0			0
$171$	−233.829			−0.104569
$172$		0			0
$173$		−	2571.97i		−	1.13031i	−0.824985	−	0.565155i	$-0.808816\pi$
$173$		−	2571.97i		−	1.13031i	0.824985	−	0.565155i	$-0.191184\pi$
$174$		0			0
$175$	5827.18			2.51710
$176$		0			0
$177$	297.086			0.126160
$178$		0			0
$179$		−	2882.88i		−	1.20378i	−0.798580	−	0.601889i	$-0.794415\pi$
$179$		−	2882.88i		−	1.20378i	0.798580	−	0.601889i	$-0.205585\pi$
$180$		0			0
$181$	−1451.53			−0.596084			−0.298042	−	0.954553i	$-0.596334\pi$
$181$	−1451.53			−0.596084			−0.298042	+	0.954553i	$0.596334\pi$
$182$		0			0
$183$	1925.35			0.777736
$184$		0			0
$185$	3157.69			1.25491
$186$		0			0
$187$	6.85432	+	1225.80i	0.00268041	+	0.479355i
$188$		0			0
$189$		−	9921.46i		−	3.81841i
$190$		0			0
$191$			2702.75i			1.02389i	0.859017	+	0.511947i	$0.171076\pi$
$191$			2702.75i			1.02389i	−0.859017	+	0.511947i	$0.828924\pi$
$192$		0			0
$193$		−	3196.49i		−	1.19217i	−0.802923	−	0.596083i	$-0.796723\pi$
$193$		−	3196.49i		−	1.19217i	0.802923	−	0.596083i	$-0.203277\pi$
$194$		0			0
$195$	12298.4			4.51644
$196$		0			0
$197$		−	407.301i		−	0.147305i	−0.997284	−	0.0736523i	$-0.976534\pi$
$197$		−	407.301i		−	0.147305i	0.997284	−	0.0736523i	$-0.0234655\pi$
$198$		0			0
$199$		−	1740.63i		−	0.620051i	−0.950728	−	0.310025i	$-0.899662\pi$
$199$		−	1740.63i		−	0.620051i	0.950728	−	0.310025i	$-0.100338\pi$
$200$		0			0
$201$	6012.63			2.10994
$202$		0			0
$203$			2441.31i			0.844069i
$204$		0			0
$205$			2770.20i			0.943801i
$206$		0			0
$207$			2426.60i			0.814783i
$208$		0			0
$209$	138.306	−	0.773368i	0.0457743	−	0.000255957i
$210$		0			0
$211$	−2418.44			−0.789062			−0.394531	−	0.918883i	$-0.629093\pi$
$211$	−2418.44			−0.789062			−0.394531	+	0.918883i	$0.629093\pi$
$212$		0			0
$213$	−6709.70			−2.15841
$214$		0			0
$215$	873.142			0.276967
$216$		0			0
$217$			4011.66i			1.25497i
$218$		0			0
$219$	−7039.33			−2.17203
$220$		0			0
$221$	2465.34			0.750392
$222$		0			0
$223$		−	1252.23i		−	0.376034i	−0.982166	−	0.188017i	$-0.939794\pi$
$223$		−	1252.23i		−	0.376034i	0.982166	−	0.188017i	$-0.0602059\pi$
$224$		0			0
$225$	−11830.5			−3.50535
$226$		0			0
$227$	888.191			0.259697			0.129849	−	0.991534i	$-0.458551\pi$
$227$	888.191			0.259697			0.129849	+	0.991534i	$0.458551\pi$
$228$		0			0
$229$	3316.99			0.957174			0.478587	−	0.878040i	$-0.341149\pi$
$229$	3316.99			0.957174			0.478587	+	0.878040i	$0.341149\pi$
$230$		0			0
$231$	58.3622	+	10437.3i	0.0166232	+	2.97282i
$232$		0			0
$233$		−	266.045i		−	0.0748035i	−0.999300	−	0.0374017i	$-0.988092\pi$
$233$		−	266.045i		−	0.0748035i	0.999300	−	0.0374017i	$-0.0119081\pi$
$234$		0			0
$235$		−	6461.07i		−	1.79351i
$236$		0			0
$237$			6799.57i			1.86363i
$238$		0			0
$239$	−1852.14			−0.501276			−0.250638	−	0.968081i	$-0.580640\pi$
$239$	−1852.14			−0.501276			−0.250638	+	0.968081i	$0.580640\pi$
$240$		0			0
$241$		−	7072.56i		−	1.89039i	−0.326510	−	0.945194i	$-0.605873\pi$
$241$		−	7072.56i		−	1.89039i	0.326510	−	0.945194i	$-0.394127\pi$
$242$		0			0
$243$			4460.42i			1.17751i
$244$		0			0
$245$	10322.9			2.69186
$246$		0			0
$247$		−	278.162i		−	0.0716561i
$248$		0			0
$249$			64.3157i			0.0163688i
$250$		0			0
$251$			3720.78i			0.935672i	0.883815	+	0.467836i	$0.154966\pi$
$251$			3720.78i			0.935672i	−0.883815	+	0.467836i	$0.845034\pi$
$252$		0			0
$253$	−8.02574	−	1435.29i	−0.00199436	−	0.356664i
$254$		0			0
$255$	−5631.78			−1.38304
$256$		0			0
$257$	−6472.20			−1.57091			−0.785457	−	0.618916i	$-0.787572\pi$
$257$	−6472.20			−1.57091			−0.785457	+	0.618916i	$0.787572\pi$
$258$		0			0
$259$	5389.71			1.29305
$260$		0			0
$261$		−	4956.43i		−	1.17546i
$262$		0			0
$263$	6331.30			1.48443			0.742214	−	0.670163i	$-0.233776\pi$
$263$	6331.30			1.48443			0.742214	+	0.670163i	$0.233776\pi$
$264$		0			0
$265$	−5587.08			−1.29514
$266$		0			0
$267$		−	10928.7i		−	2.50495i
$268$		0			0
$269$	−5551.45			−1.25828			−0.629141	−	0.777291i	$-0.716593\pi$
$269$	−5551.45			−1.25828			−0.629141	+	0.777291i	$0.716593\pi$
$270$		0			0
$271$	2092.42			0.469023			0.234512	−	0.972113i	$-0.424651\pi$
$271$	2092.42			0.469023			0.234512	+	0.972113i	$0.424651\pi$
$272$		0			0
$273$	20991.5			4.65372
$274$		0			0
$275$	6997.57	−	39.1284i	1.53443	−	0.00858011i
$276$		0			0
$277$		−	1204.35i		−	0.261236i	−0.991433	−	0.130618i	$-0.958304\pi$
$277$		−	1204.35i		−	0.261236i	0.991433	−	0.130618i	$-0.0416961\pi$
$278$		0			0
$279$		−	8144.61i		−	1.74769i
$280$		0			0
$281$			5812.55i			1.23398i	0.786972	+	0.616988i	$0.211647\pi$
$281$			5812.55i			1.23398i	−0.786972	+	0.616988i	$0.788353\pi$
$282$		0			0
$283$	−2834.78			−0.595443			−0.297721	−	0.954653i	$-0.596227\pi$
$283$	−2834.78			−0.595443			−0.297721	+	0.954653i	$0.596227\pi$
$284$		0			0
$285$			635.430i			0.132069i
$286$		0			0
$287$			4728.32i			0.972488i
$288$		0			0
$289$	3784.05			0.770212
$290$		0			0
$291$		−	8056.78i		−	1.62301i
$292$		0			0
$293$		−	7014.62i		−	1.39863i	−0.714814	−	0.699314i	$-0.753489\pi$
$293$		−	7014.62i		−	1.39863i	0.714814	−	0.699314i	$-0.246511\pi$
$294$		0			0
$295$		−	561.524i		−	0.110824i
$296$		0			0
$297$	−66.6207	−	11914.2i	−0.0130159	−	2.32771i
$298$		0			0
$299$	−2886.67			−0.558330
$300$		0			0
$301$	1490.32			0.285385
$302$		0			0
$303$	−5013.06			−0.950472
$304$		0			0
$305$		−	3639.11i		−	0.683196i
$306$		0			0
$307$	6706.63			1.24680			0.623400	−	0.781903i	$-0.285751\pi$
$307$	6706.63			1.24680			0.623400	+	0.781903i	$0.285751\pi$
$308$		0			0
$309$	−11367.6			−2.09282
$310$		0			0
$311$			2375.27i			0.433084i	0.976273	+	0.216542i	$0.0694778\pi$
$311$			2375.27i			0.433084i	−0.976273	+	0.216542i	$0.930522\pi$
$312$		0			0
$313$	188.963			0.0341240			0.0170620	−	0.999854i	$-0.494569\pi$
$313$	188.963			0.0341240			0.0170620	+	0.999854i	$0.494569\pi$
$314$		0			0
$315$	−33352.7			−5.96574
$316$		0			0
$317$	−1511.86			−0.267870			−0.133935	−	0.990990i	$-0.542761\pi$
$317$	−1511.86			−0.267870			−0.133935	+	0.990990i	$0.542761\pi$
$318$		0			0
$319$	16.3929	+	2931.64i	0.00287720	+	0.514547i
$320$		0			0
$321$			6120.36i			1.06419i
$322$		0			0
$323$			127.379i			0.0219428i
$324$		0			0
$325$		−	14073.6i		−	2.40204i
$326$		0			0
$327$	−6220.25			−1.05193
$328$		0			0
$329$		−	11028.1i		−	1.84802i
$330$		0			0
$331$			4462.63i			0.741052i	0.928822	+	0.370526i	$0.120823\pi$
$331$			4462.63i			0.741052i	−0.928822	+	0.370526i	$0.879177\pi$
$332$		0			0
$333$	−10942.4			−1.80072
$334$		0			0
$335$		−	11364.5i		−	1.85346i
$336$		0			0
$337$			10849.8i			1.75378i	0.480689	+	0.876891i	$0.340387\pi$
$337$			10849.8i			1.75378i	−0.480689	+	0.876891i	$0.659613\pi$
$338$		0			0
$339$		−	16308.1i		−	2.61279i
$340$		0			0
$341$	26.9375	+	4817.40i	0.00427786	+	0.765035i
$342$		0			0
$343$	7199.16			1.13329
$344$		0			0
$345$	6594.27			1.02905
$346$		0			0
$347$	8318.75			1.28696			0.643478	−	0.765464i	$-0.277491\pi$
$347$	8318.75			1.28696			0.643478	+	0.765464i	$0.277491\pi$
$348$		0			0
$349$			4475.92i			0.686506i	0.939243	+	0.343253i	$0.111529\pi$
$349$			4475.92i			0.686506i	−0.939243	+	0.343253i	$0.888471\pi$
$350$		0			0
$351$	−23961.9			−3.64385
$352$		0			0
$353$	4020.70			0.606233			0.303117	−	0.952954i	$-0.401973\pi$
$353$	4020.70			0.606233			0.303117	+	0.952954i	$0.401973\pi$
$354$		0			0
$355$			12682.0i			1.89604i
$356$		0			0
$357$	−9612.62			−1.42508
$358$		0			0
$359$	−1198.45			−0.176189			−0.0880944	−	0.996112i	$-0.528078\pi$
$359$	−1198.45			−0.176189			−0.0880944	+	0.996112i	$0.528078\pi$
$360$		0			0
$361$	−6844.63			−0.997905
$362$		0			0
$363$	140.169	+	12533.2i	0.0202671	+	1.81218i
$364$		0			0
$365$			13305.1i			1.90800i
$366$		0			0
$367$			12127.9i			1.72498i	0.506070	+	0.862492i	$0.331098\pi$
$367$			12127.9i			1.72498i	−0.506070	+	0.862492i	$0.668902\pi$
$368$		0			0
$369$		−	9599.61i		−	1.35430i
$370$		0			0
$371$	−9536.31			−1.33450
$372$		0			0
$373$			4567.00i			0.633968i	0.948431	+	0.316984i	$0.102670\pi$
$373$			4567.00i			0.633968i	−0.948431	+	0.316984i	$0.897330\pi$
$374$		0			0
$375$			11197.8i			1.54200i
$376$		0			0
$377$	5896.15			0.805483
$378$		0			0
$379$			5811.56i			0.787652i	0.919185	+	0.393826i	$0.128849\pi$
$379$			5811.56i			0.787652i	−0.919185	+	0.393826i	$0.871151\pi$
$380$		0			0
$381$		−	25961.2i		−	3.49090i
$382$		0			0
$383$		−	9585.96i		−	1.27890i	−0.768832	−	0.639451i	$-0.779162\pi$
$383$		−	9585.96i		−	1.27890i	0.768832	−	0.639451i	$-0.220838\pi$
$384$		0			0
$385$	19727.5	−	110.311i	2.61145	−	0.0146025i
$386$		0			0
$387$	−3025.71			−0.397430
$388$		0			0
$389$	417.280			0.0543880			0.0271940	−	0.999630i	$-0.491343\pi$
$389$	417.280			0.0543880			0.0271940	+	0.999630i	$0.491343\pi$
$390$		0			0
$391$	1321.89			0.170974
$392$		0			0
$393$		−	12449.6i		−	1.59796i
$394$		0			0
$395$	12851.9			1.63709
$396$		0			0
$397$	−4548.41			−0.575007			−0.287504	−	0.957780i	$-0.592825\pi$
$397$	−4548.41			−0.575007			−0.287504	+	0.957780i	$0.592825\pi$
$398$		0			0
$399$			1084.58i			0.136083i
$400$		0			0
$401$	−15243.5			−1.89832			−0.949159	−	0.314796i	$-0.898064\pi$
$401$	−15243.5			−1.89832			−0.949159	+	0.314796i	$0.898064\pi$
$402$		0			0
$403$	9688.80			1.19760
$404$		0			0
$405$	25096.7			3.07917
$406$		0			0
$407$	6472.24	−	36.1909i	0.788248	−	0.00440766i
$408$		0			0
$409$		−	15182.5i		−	1.83552i	−0.397141	−	0.917758i	$-0.629998\pi$
$409$		−	15182.5i		−	1.83552i	0.397141	−	0.917758i	$-0.370002\pi$
$410$		0			0
$411$		−	11980.9i		−	1.43789i
$412$		0			0
$413$		−	958.437i		−	0.114193i
$414$		0			0
$415$	121.563			0.0143791
$416$		0			0
$417$			20811.7i			2.44402i
$418$		0			0
$419$			4620.31i			0.538704i	0.963042	+	0.269352i	$0.0868095\pi$
$419$			4620.31i			0.538704i	−0.963042	+	0.269352i	$0.913191\pi$
$420$		0			0
$421$	7886.52			0.912983			0.456491	−	0.889728i	$-0.349106\pi$
$421$	7886.52			0.912983			0.456491	+	0.889728i	$0.349106\pi$
$422$		0			0
$423$			22389.6i			2.57357i
$424$		0			0
$425$			6444.69i			0.735561i
$426$		0			0
$427$		−	6211.42i		−	0.703961i
$428$		0			0
$429$	25207.7	−	140.954i	2.83692	−	0.0158632i
$430$		0			0
$431$	4609.02			0.515101			0.257551	−	0.966265i	$-0.417085\pi$
$431$	4609.02			0.515101			0.257551	+	0.966265i	$0.417085\pi$
$432$		0			0
$433$	10466.4			1.16162			0.580811	−	0.814039i	$-0.302736\pi$
$433$	10466.4			1.16162			0.580811	+	0.814039i	$0.302736\pi$
$434$		0			0
$435$	−13469.1			−1.48458
$436$		0			0
$437$		−	149.148i		−	0.0163266i
$438$		0			0
$439$	5328.49			0.579305			0.289652	−	0.957132i	$-0.406460\pi$
$439$	5328.49			0.579305			0.289652	+	0.957132i	$0.406460\pi$
$440$		0			0
$441$	−35772.0			−3.86265
$442$		0			0
$443$			2967.26i			0.318237i	0.987260	+	0.159118i	$0.0508651\pi$
$443$			2967.26i			0.318237i	−0.987260	+	0.159118i	$0.949135\pi$
$444$		0			0
$445$	−20656.3			−2.20046
$446$		0			0
$447$	28596.7			3.02590
$448$		0			0
$449$	−15406.0			−1.61927			−0.809637	−	0.586932i	$-0.800336\pi$
$449$	−15406.0			−1.61927			−0.809637	+	0.586932i	$0.800336\pi$
$450$		0			0
$451$	31.7498	+	5678.01i	0.00331494	+	0.592831i
$452$		0			0
$453$		−	19127.0i		−	1.98381i
$454$		0			0
$455$		−	39676.2i		−	4.08802i
$456$		0			0
$457$		−	14676.6i		−	1.50228i	−0.660141	−	0.751142i	$-0.729503\pi$
$457$		−	14676.6i		−	1.50228i	0.660141	−	0.751142i	$-0.270497\pi$
$458$		0			0
$459$	10972.8			1.11584
$460$		0			0
$461$			12794.0i			1.29258i	0.763094	+	0.646288i	$0.223679\pi$
$461$			12794.0i			1.29258i	−0.763094	+	0.646288i	$0.776321\pi$
$462$		0			0
$463$		−	4138.32i		−	0.415387i	−0.978194	−	0.207694i	$-0.933404\pi$
$463$		−	4138.32i		−	0.415387i	0.978194	−	0.207694i	$-0.0665957\pi$
$464$		0			0
$465$	−22133.0			−2.20729
$466$		0			0
$467$			2466.57i			0.244410i	0.992505	+	0.122205i	$0.0389965\pi$
$467$			2466.57i			0.244410i	−0.992505	+	0.122205i	$0.961004\pi$
$468$		0			0
$469$		−	19397.5i		−	1.90980i
$470$		0			0
$471$		−	10115.2i		−	0.989561i
$472$		0			0
$473$	1789.66	−	10.0072i	0.173971	−	0.000972799i
$474$		0			0
$475$	727.150			0.0702399
$476$		0			0
$477$	19361.0			1.85844
$478$		0			0
$479$	−2084.12			−0.198802			−0.0994009	−	0.995047i	$-0.531693\pi$
$479$	−2084.12			−0.198802			−0.0994009	+	0.995047i	$0.531693\pi$
$480$		0			0
$481$		−	13017.0i		−	1.23394i
$482$		0			0
$483$	11255.4			1.06033
$484$		0			0
$485$	−15228.2			−1.42572
$486$		0			0
$487$		−	4131.19i		−	0.384399i	−0.981356	−	0.192199i	$-0.938438\pi$
$487$		−	4131.19i		−	0.384399i	0.981356	−	0.192199i	$-0.0615620\pi$
$488$		0			0
$489$	8253.33			0.763249
$490$		0			0
$491$	10166.3			0.934413			0.467207	−	0.884148i	$-0.345260\pi$
$491$	10166.3			0.934413			0.467207	+	0.884148i	$0.345260\pi$
$492$		0			0
$493$	−2700.01			−0.246658
$494$		0			0
$495$	−40051.6	+	223.957i	−3.63674	+	0.0203356i
$496$		0			0
$497$			21646.4i			1.95367i
$498$		0			0
$499$		−	13074.0i		−	1.17289i	−0.809989	−	0.586446i	$-0.800527\pi$
$499$		−	13074.0i		−	1.17289i	0.809989	−	0.586446i	$-0.199473\pi$
$500$		0			0
$501$			21781.0i			1.94233i
$502$		0			0
$503$	4251.49			0.376868			0.188434	−	0.982086i	$-0.439659\pi$
$503$	4251.49			0.376868			0.188434	+	0.982086i	$0.439659\pi$
$504$		0			0
$505$			9475.22i			0.834935i
$506$		0			0
$507$		−	30008.8i		−	2.62868i
$508$		0			0
$509$	14717.7			1.28163			0.640817	−	0.767694i	$-0.278596\pi$
$509$	14717.7			1.28163			0.640817	+	0.767694i	$0.278596\pi$
$510$		0			0
$511$			22709.8i			1.96599i
$512$		0			0
$513$		−	1238.06i		−	0.106553i
$514$		0			0
$515$			21486.0i			1.83842i
$516$		0			0
$517$	−74.0516	−	13243.1i	−0.00629939	−	1.12656i
$518$		0			0
$519$	24220.2			2.04846
$520$		0			0
$521$	−406.171			−0.0341548			−0.0170774	−	0.999854i	$-0.505436\pi$
$521$	−406.171			−0.0341548			−0.0170774	+	0.999854i	$0.505436\pi$
$522$		0			0
$523$	−13155.5			−1.09990			−0.549951	−	0.835197i	$-0.685353\pi$
$523$	−13155.5			−1.09990			−0.549951	+	0.835197i	$0.685353\pi$
$524$		0			0
$525$			54874.4i			4.56174i
$526$		0			0
$527$	−4436.78			−0.366735
$528$		0			0
$529$	10619.2			0.872787
$530$		0			0
$531$			1945.85i			0.159026i
$532$		0			0
$533$	11419.7			0.928031
$534$		0			0
$535$	11568.1			0.934829
$536$		0			0
$537$	27148.0			2.18160
$538$		0			0
$539$	21158.5	−	118.313i	1.69084	−	0.00945470i
$540$		0			0
$541$		−	3349.50i		−	0.266185i	−0.991104	−	0.133092i	$-0.957509\pi$
$541$		−	3349.50i		−	0.266185i	0.991104	−	0.133092i	$-0.0424907\pi$
$542$		0			0
$543$		−	13669.0i		−	1.08028i
$544$		0			0
$545$			11756.9i			0.924058i
$546$		0			0
$547$	12489.8			0.976280			0.488140	−	0.872765i	$-0.337675\pi$
$547$	12489.8			0.976280			0.488140	+	0.872765i	$0.337675\pi$
$548$		0			0
$549$			12610.6i			0.980345i
$550$		0			0
$551$			304.641i			0.0235538i
$552$		0			0
$553$	21936.3			1.68685
$554$		0			0
$555$			29735.9i			2.27427i
$556$		0			0
$557$			19262.8i			1.46533i	0.680589	+	0.732665i	$0.261724\pi$
$557$			19262.8i			1.46533i	−0.680589	+	0.732665i	$0.738276\pi$
$558$		0			0
$559$		−	3599.37i		−	0.272339i
$560$		0			0
$561$	−11543.3	+	64.5469i	−0.868733	+	0.00485771i
$562$		0			0
$563$	−8441.34			−0.631900			−0.315950	−	0.948776i	$-0.602323\pi$
$563$	−8441.34			−0.631900			−0.315950	+	0.948776i	$0.602323\pi$
$564$		0			0
$565$	−30824.1			−2.29519
$566$		0			0
$567$	42836.3			3.17276
$568$		0			0
$569$			10765.4i			0.793164i	0.917999	+	0.396582i	$0.129804\pi$
$569$			10765.4i			0.793164i	−0.917999	+	0.396582i	$0.870196\pi$
$570$		0			0
$571$	−12604.1			−0.923757			−0.461879	−	0.886943i	$-0.652824\pi$
$571$	−12604.1			−0.923757			−0.461879	+	0.886943i	$0.652824\pi$
$572$		0			0
$573$	−25451.7			−1.85560
$574$		0			0
$575$		−	7546.11i		−	0.547295i
$576$		0			0
$577$	−9182.15			−0.662492			−0.331246	−	0.943544i	$-0.607469\pi$
$577$	−9182.15			−0.662492			−0.331246	+	0.943544i	$0.607469\pi$
$578$		0			0
$579$	30101.2			2.16056
$580$		0			0
$581$	207.491			0.0148161
$582$		0			0
$583$	−11451.7	+	64.0345i	−0.813517	+	0.00454895i
$584$		0			0
$585$			80552.1i			5.69302i
$586$		0			0
$587$			14346.4i			1.00875i	0.863484	+	0.504377i	$0.168278\pi$
$587$			14346.4i			1.00875i	−0.863484	+	0.504377i	$0.831722\pi$
$588$		0			0
$589$			500.599i			0.0350200i
$590$		0			0
$591$	3835.54			0.266960
$592$		0			0
$593$			19737.0i			1.36678i	0.730053	+	0.683391i	$0.239495\pi$
$593$			19737.0i			1.36678i	−0.730053	+	0.683391i	$0.760505\pi$
$594$		0			0
$595$			18168.9i			1.25185i
$596$		0			0
$597$	16391.5			1.12372
$598$		0			0
$599$		−	13732.3i		−	0.936706i	−0.883542	−	0.468353i	$-0.844848\pi$
$599$		−	13732.3i		−	0.936706i	0.883542	−	0.468353i	$-0.155152\pi$
$600$		0			0
$601$		−	23513.7i		−	1.59592i	−0.602713	−	0.797958i	$-0.705914\pi$
$601$		−	23513.7i		−	1.59592i	0.602713	−	0.797958i	$-0.294086\pi$
$602$		0			0
$603$			39381.6i			2.65960i
$604$		0			0
$605$	23689.1	−	264.934i	1.59190	−	0.0178034i
$606$		0			0
$607$	1069.45			0.0715121			0.0357560	−	0.999361i	$-0.488616\pi$
$607$	1069.45			0.0715121			0.0357560	+	0.999361i	$0.488616\pi$
$608$		0			0
$609$	−22989.7			−1.52970
$610$		0			0
$611$	−26634.6			−1.76354
$612$		0			0
$613$			17502.7i			1.15322i	0.817018	+	0.576612i	$0.195625\pi$
$613$			17502.7i			1.15322i	−0.817018	+	0.576612i	$0.804375\pi$
$614$		0			0
$615$	−26086.9			−1.71045
$616$		0			0
$617$	−18013.9			−1.17538			−0.587692	−	0.809085i	$-0.699963\pi$
$617$	−18013.9			−1.17538			−0.587692	+	0.809085i	$0.699963\pi$
$618$		0			0
$619$		−	9108.77i		−	0.591458i	−0.955272	−	0.295729i	$-0.904437\pi$
$619$		−	9108.77i		−	0.591458i	0.955272	−	0.295729i	$-0.0955625\pi$
$620$		0			0
$621$	−12848.1			−0.830238
$622$		0			0
$623$	−35257.3			−2.26734
$624$		0			0
$625$	−2810.89			−0.179897
$626$		0			0
$627$	7.28278	+	1302.42i	0.000463870	+	0.0829567i
$628$		0			0
$629$			5960.87i			0.377862i
$630$		0			0
$631$			27228.8i			1.71785i	0.512105	+	0.858923i	$0.328866\pi$
$631$			27228.8i			1.71785i	−0.512105	+	0.858923i	$0.671134\pi$
$632$		0			0
$633$		−	22774.3i		−	1.43001i
$634$		0			0
$635$	−49069.5			−3.06656
$636$		0			0
$637$		−	42554.3i		−	2.64688i
$638$		0			0
$639$		−	43947.2i		−	2.72070i
$640$		0			0
$641$	14904.7			0.918409			0.459204	−	0.888331i	$-0.348135\pi$
$641$	14904.7			0.918409			0.459204	+	0.888331i	$0.348135\pi$
$642$		0			0
$643$		−	12264.4i		−	0.752194i	−0.926580	−	0.376097i	$-0.877266\pi$
$643$		−	12264.4i		−	0.752194i	0.926580	−	0.376097i	$-0.122734\pi$
$644$		0			0
$645$			8222.36i			0.501946i
$646$		0			0
$647$			26840.7i			1.63094i	0.578801	+	0.815469i	$0.303521\pi$
$647$			26840.7i			1.63094i	−0.578801	+	0.815469i	$0.696479\pi$
$648$		0			0
$649$	−6.43573	−	1150.94i	−0.000389252	−	0.0696123i
$650$		0			0
$651$	−37777.7			−2.27438
$652$		0			0
$653$	12511.4			0.749785			0.374892	−	0.927068i	$-0.377680\pi$
$653$	12511.4			0.749785			0.374892	+	0.927068i	$0.377680\pi$
$654$		0			0
$655$	−23531.1			−1.40372
$656$		0			0
$657$		−	46106.3i		−	2.73786i
$658$		0			0
$659$	17122.7			1.01215			0.506076	−	0.862489i	$-0.331096\pi$
$659$	17122.7			1.01215			0.506076	+	0.862489i	$0.331096\pi$
$660$		0			0
$661$	27540.4			1.62057			0.810284	−	0.586037i	$-0.199313\pi$
$661$	27540.4			1.62057			0.810284	+	0.586037i	$0.199313\pi$
$662$		0			0
$663$			23216.0i			1.35993i
$664$		0			0
$665$	2049.98			0.119541
$666$		0			0
$667$	3161.45			0.183526
$668$		0			0
$669$	11792.2			0.681485
$670$		0			0
$671$	−41.7085	−	7458.98i	−0.00239961	−	0.429137i
$672$		0			0
$673$		−	26046.9i		−	1.49188i	−0.666012	−	0.745941i	$-0.732000\pi$
$673$		−	26046.9i		−	1.49188i	0.666012	−	0.745941i	$-0.268000\pi$
$674$		0			0
$675$		−	62639.3i		−	3.57184i
$676$		0			0
$677$			10233.3i			0.580942i	0.956884	+	0.290471i	$0.0938121\pi$
$677$			10233.3i			0.580942i	−0.956884	+	0.290471i	$0.906188\pi$
$678$		0			0
$679$	−25992.2			−1.46906
$680$		0			0
$681$			8364.07i			0.470649i
$682$		0			0
$683$		−	22601.7i		−	1.26622i	−0.774060	−	0.633112i	$-0.781777\pi$
$683$		−	22601.7i		−	1.26622i	0.774060	−	0.633112i	$-0.218223\pi$
$684$		0			0
$685$	−22645.1			−1.26310
$686$		0			0
$687$			31236.0i			1.73468i
$688$		0			0
$689$			23031.7i			1.27350i
$690$		0			0
$691$		−	17886.9i		−	0.984731i	−0.870389	−	0.492365i	$-0.836132\pi$
$691$		−	17886.9i		−	0.984731i	0.870389	−	0.492365i	$-0.163868\pi$
$692$		0			0
$693$	−68362.1	+	382.261i	−3.74727	+	0.0209537i
$694$		0			0
$695$	39336.4			2.14693
$696$		0			0
$697$	−5229.39			−0.284185
$698$		0			0
$699$	2505.34			0.135566
$700$		0			0
$701$		−	8089.76i		−	0.435872i	−0.975963	−	0.217936i	$-0.930068\pi$
$701$		−	8089.76i		−	0.435872i	0.975963	−	0.217936i	$-0.0699323\pi$
$702$		0			0
$703$	672.560			0.0360826
$704$		0			0
$705$	60843.7			3.25037
$706$		0			0
$707$			16172.8i			0.860312i
$708$		0			0
$709$	274.224			0.0145257			0.00726283	−	0.999974i	$-0.497688\pi$
$709$	274.224			0.0145257			0.00726283	+	0.999974i	$0.497688\pi$
$710$		0			0
$711$	−44535.9			−2.34912
$712$		0			0
$713$	5195.04			0.272869
$714$		0			0
$715$	−266.418	−	47645.2i	−0.0139349	−	2.49207i
$716$		0			0
$717$		−	17441.6i		−	0.908461i
$718$		0			0
$719$			949.067i			0.0492270i	0.999697	+	0.0246135i	$0.00783551\pi$
$719$			949.067i			0.0492270i	−0.999697	+	0.0246135i	$0.992164\pi$
$720$		0			0
$721$			36673.5i			1.89430i
$722$		0			0
$723$	66602.1			3.42594
$724$		0			0
$725$			15413.2i			0.789563i
$726$		0			0
$727$		−	23767.7i		−	1.21251i	−0.795270	−	0.606256i	$-0.792671\pi$
$727$		−	23767.7i		−	1.21251i	0.795270	−	0.606256i	$-0.207329\pi$
$728$		0			0
$729$	−3933.63			−0.199849
$730$		0			0
$731$			1648.26i			0.0833966i
$732$		0			0
$733$		−	17509.7i		−	0.882312i	−0.897430	−	0.441156i	$-0.854569\pi$
$733$		−	17509.7i		−	0.882312i	0.897430	−	0.441156i	$-0.145431\pi$
$734$		0			0
$735$			97210.3i			4.87844i
$736$		0			0
$737$	−130.251	−	23293.5i	−0.00650997	−	1.16422i
$738$		0			0
$739$	−37975.9			−1.89035			−0.945174	−	0.326567i	$-0.894108\pi$
$739$	−37975.9			−1.89035			−0.945174	+	0.326567i	$0.894108\pi$
$740$		0			0
$741$	2619.45			0.129862
$742$		0			0
$743$	27755.4			1.37045			0.685226	−	0.728330i	$-0.259703\pi$
$743$	27755.4			1.37045			0.685226	+	0.728330i	$0.259703\pi$
$744$		0			0
$745$		−	54050.9i		−	2.65808i
$746$		0			0
$747$	−421.255			−0.0206331
$748$		0			0
$749$	19745.1			0.963243
$750$		0			0
$751$		−	34722.2i		−	1.68713i	−0.537031	−	0.843563i	$-0.680454\pi$
$751$		−	34722.2i		−	1.68713i	0.537031	−	0.843563i	$-0.319546\pi$
$752$		0			0
$753$	−35038.5			−1.69571
$754$		0			0
$755$	−36152.1			−1.74266
$756$		0			0
$757$	14197.0			0.681637			0.340818	−	0.940129i	$-0.389296\pi$
$757$	14197.0			0.681637			0.340818	+	0.940129i	$0.389296\pi$
$758$		0			0
$759$	13516.1	−	75.5782i	0.646381	−	0.00361438i
$760$		0			0
$761$		−	8653.44i		−	0.412204i	−0.978531	−	0.206102i	$-0.933922\pi$
$761$		−	8653.44i		−	0.412204i	0.978531	−	0.206102i	$-0.0660778\pi$
$762$		0			0
$763$			20067.3i			0.952145i
$764$		0			0
$765$		−	36887.1i		−	1.74334i
$766$		0			0
$767$	−2314.78			−0.108973
$768$		0			0
$769$			18768.1i			0.880099i	0.897974	+	0.440049i	$0.145039\pi$
$769$			18768.1i			0.880099i	−0.897974	+	0.440049i	$0.854961\pi$
$770$		0			0
$771$		−	60948.5i		−	2.84696i
$772$		0			0
$773$	−14263.3			−0.663666			−0.331833	−	0.943338i	$-0.607667\pi$
$773$	−14263.3			−0.663666			−0.331833	+	0.943338i	$0.607667\pi$
$774$		0			0
$775$			25327.7i			1.17393i
$776$		0			0
$777$			50754.8i			2.34339i
$778$		0			0
$779$			590.028i			0.0271373i
$780$		0			0
$781$	145.351	+	25994.0i	0.00665951	+	1.19096i
$782$		0			0
$783$	26242.9			1.19776
$784$		0			0
$785$	−19118.8			−0.869272
$786$		0			0
$787$	13427.9			0.608201			0.304100	−	0.952640i	$-0.401644\pi$
$787$	13427.9			0.608201			0.304100	+	0.952640i	$0.401644\pi$
$788$		0			0
$789$			59621.6i			2.69022i
$790$		0			0
$791$	−52612.2			−2.36495
$792$		0			0
$793$	−15001.6			−0.671780
$794$		0			0
$795$		−	52613.4i		−	2.34717i
$796$		0			0
$797$	8225.98			0.365595			0.182797	−	0.983151i	$-0.441485\pi$
$797$	8225.98			0.365595			0.182797	+	0.983151i	$0.441485\pi$
$798$		0			0
$799$	12196.8			0.540038
$800$		0			0
$801$	71580.6			3.15752
$802$		0			0
$803$	152.492	+	27271.1i	0.00670153	+	1.19848i
$804$		0			0
$805$		−	21274.0i		−	0.931440i
$806$		0			0
$807$		−	52277.9i		−	2.28038i
$808$		0			0
$809$			31308.8i			1.36064i	0.732914	+	0.680321i	$0.238160\pi$
$809$			31308.8i			1.36064i	−0.732914	+	0.680321i	$0.761840\pi$
$810$		0			0
$811$	−14023.8			−0.607203			−0.303601	−	0.952799i	$-0.598189\pi$
$811$	−14023.8			−0.607203			−0.303601	+	0.952799i	$0.598189\pi$
$812$		0			0
$813$			19704.2i			0.850010i
$814$		0			0
$815$		−	15599.7i		−	0.670470i
$816$		0			0
$817$	185.972			0.00796367
$818$		0			0
$819$			137490.i			5.86606i
$820$		0			0
$821$			30198.1i			1.28370i	0.766828	+	0.641852i	$0.221834\pi$
$821$			30198.1i			1.28370i	−0.766828	+	0.641852i	$0.778166\pi$
$822$		0			0
$823$		−	35007.9i		−	1.48275i	−0.671093	−	0.741373i	$-0.734175\pi$
$823$		−	35007.9i		−	1.48275i	0.671093	−	0.741373i	$-0.265825\pi$
$824$		0			0
$825$	368.471	+	65895.9i	0.0155497	+	2.78085i
$826$		0			0
$827$	19445.1			0.817622			0.408811	−	0.912619i	$-0.365943\pi$
$827$	19445.1			0.817622			0.408811	+	0.912619i	$0.365943\pi$
$828$		0			0
$829$	−33474.0			−1.40241			−0.701206	−	0.712959i	$-0.747355\pi$
$829$	−33474.0			−1.40241			−0.701206	+	0.712959i	$0.747355\pi$
$830$		0			0
$831$	11341.3			0.473437
$832$		0			0
$833$			19486.8i			0.810538i
$834$		0			0
$835$	41168.5			1.70622
$836$		0			0
$837$	43123.4			1.78084
$838$		0			0
$839$			18202.2i			0.749000i	0.927227	+	0.374500i	$0.122186\pi$
$839$			18202.2i			0.749000i	−0.927227	+	0.374500i	$0.877814\pi$
$840$		0			0
$841$	17931.6			0.735233
$842$		0			0
$843$	−54736.6			−2.23633
$844$		0			0
$845$	−56719.9			−2.30914
$846$		0			0
$847$	40433.8	−	452.202i	1.64028	−	0.0183446i
$848$		0			0
$849$		−	26695.1i		−	1.07912i
$850$		0			0
$851$		−	6979.60i		−	0.281149i
$852$		0			0
$853$		−	3788.44i		−	0.152068i	−0.997105	−	0.0760338i	$-0.975774\pi$
$853$		−	3788.44i		−	0.152068i	0.997105	−	0.0760338i	$-0.0242257\pi$
$854$		0			0
$855$	−4161.95			−0.166474
$856$		0			0
$857$		−	31127.7i		−	1.24073i	−0.784314	−	0.620364i	$-0.786985\pi$
$857$		−	31127.7i		−	1.24073i	0.784314	−	0.620364i	$-0.213015\pi$
$858$		0			0
$859$			17355.0i			0.689342i	0.938723	+	0.344671i	$0.112010\pi$
$859$			17355.0i			0.689342i	−0.938723	+	0.344671i	$0.887990\pi$
$860$		0			0
$861$	−44526.5			−1.76244
$862$		0			0
$863$			40088.8i			1.58127i	0.612285	+	0.790637i	$0.290251\pi$
$863$			40088.8i			1.58127i	−0.612285	+	0.790637i	$0.709749\pi$
$864$		0			0
$865$		−	45778.8i		−	1.79945i
$866$		0			0
$867$			35634.3i			1.39585i
$868$		0			0
$869$	26342.2	−	147.298i	1.02831	−	0.00575000i
$870$		0			0
$871$	−46848.2			−1.82249
$872$		0			0
$873$	52770.4			2.04583
$874$		0			0
$875$	36125.5			1.39573
$876$		0			0
$877$		−	14573.4i		−	0.561127i	−0.959835	−	0.280563i	$-0.909479\pi$
$877$		−	14573.4i		−	0.561127i	0.959835	−	0.280563i	$-0.0905213\pi$
$878$		0			0
$879$	66056.4			2.53473
$880$		0			0
$881$	16920.2			0.647055			0.323528	−	0.946219i	$-0.395131\pi$
$881$	16920.2			0.647055			0.323528	+	0.946219i	$0.395131\pi$
$882$		0			0
$883$		−	52003.4i		−	1.98194i	−0.134077	−	0.990971i	$-0.542807\pi$
$883$		−	52003.4i		−	1.98194i	0.134077	−	0.990971i	$-0.457193\pi$
$884$		0			0
$885$	5287.85			0.200847
$886$		0			0
$887$	4547.91			0.172158			0.0860789	−	0.996288i	$-0.472566\pi$
$887$	4547.91			0.172158			0.0860789	+	0.996288i	$0.472566\pi$
$888$		0			0
$889$	−83754.3			−3.15976
$890$		0			0
$891$	51440.0	−	287.638i	1.93413	−	0.0108151i
$892$		0			0
$893$		−	1376.15i		−	0.0515690i
$894$		0			0
$895$		−	51312.5i		−	1.91641i
$896$		0			0
$897$		−	27183.7i		−	1.01186i
$898$		0			0
$899$	−10611.1			−0.393659
$900$		0			0
$901$		−	10546.9i		−	0.389975i
$902$		0			0
$903$			14034.3i			0.517202i
$904$		0			0
$905$	−25835.9			−0.948965
$906$		0			0
$907$			9470.21i			0.346696i	0.984861	+	0.173348i	$0.0554585\pi$
$907$			9470.21i			0.346696i	−0.984861	+	0.173348i	$0.944541\pi$
$908$		0			0
$909$		−	32834.6i		−	1.19808i
$910$		0			0
$911$			47456.6i			1.72592i	0.505276	+	0.862958i	$0.331391\pi$
$911$			47456.6i			1.72592i	−0.505276	+	0.862958i	$0.668609\pi$
$912$		0			0
$913$	249.165	−	1.39326i	0.00903195	−	5.05041e-5i
$914$		0			0
$915$	34269.4			1.23815
$916$		0			0
$917$	−40164.0			−1.44638
$918$		0			0
$919$	1873.51			0.0672484			0.0336242	−	0.999435i	$-0.489295\pi$
$919$	1873.51			0.0672484			0.0336242	+	0.999435i	$0.489295\pi$
$920$		0			0
$921$			63156.1i			2.25957i
$922$		0			0
$923$	52279.5			1.86435
$924$		0			0
$925$	34028.1			1.20955
$926$		0			0
$927$		−	74455.8i		−	2.63803i
$928$		0			0
$929$	3698.04			0.130601			0.0653007	−	0.997866i	$-0.479199\pi$
$929$	3698.04			0.130601			0.0653007	+	0.997866i	$0.479199\pi$
$930$		0			0
$931$	2198.68			0.0773995
$932$		0			0
$933$	−22367.8			−0.784877
$934$		0			0
$935$	122.001	+	21818.1i	0.00426721	+	0.763132i
$936$		0			0
$937$		−	11195.9i		−	0.390344i	−0.980769	−	0.195172i	$-0.937473\pi$
$937$		−	11195.9i		−	0.390344i	0.980769	−	0.195172i	$-0.0625266\pi$
$938$		0			0
$939$			1779.46i			0.0618428i
$940$		0			0
$941$		−	19628.8i		−	0.680000i	−0.940425	−	0.340000i	$-0.889573\pi$
$941$		−	19628.8i		−	0.680000i	0.940425	−	0.340000i	$-0.110427\pi$
$942$		0			0
$943$	6123.10			0.211448
$944$		0			0
$945$		−	176593.i		−	6.07890i
$946$		0			0
$947$			827.647i			0.0284001i	0.999899	+	0.0142001i	$0.00452017\pi$
$947$			827.647i			0.0284001i	−0.999899	+	0.0142001i	$0.995480\pi$
$948$		0			0
$949$	54847.8			1.87612
$950$		0			0
$951$		−	14237.2i		−	0.485459i
$952$		0			0
$953$			21364.6i			0.726199i	0.931750	+	0.363100i	$0.118281\pi$
$953$			21364.6i			0.726199i	−0.931750	+	0.363100i	$0.881719\pi$
$954$		0			0
$955$			48106.4i			1.63004i
$956$		0			0
$957$	−27607.2	+	154.372i	−0.932512	+	0.00521434i
$958$		0			0
$959$	−38651.9			−1.30150
$960$		0			0
$961$	12354.4			0.414703
$962$		0			0
$963$	−40087.2			−1.34142
$964$		0			0
$965$		−	56894.5i		−	1.89793i
$966$		0			0
$967$	14549.5			0.483846			0.241923	−	0.970295i	$-0.422222\pi$
$967$	14549.5			0.483846			0.241923	+	0.970295i	$0.422222\pi$
$968$		0			0
$969$	−1199.52			−0.0397669
$970$		0			0
$971$			14973.2i			0.494864i	0.968905	+	0.247432i	$0.0795867\pi$
$971$			14973.2i			0.494864i	−0.968905	+	0.247432i	$0.920413\pi$
$972$		0			0
$973$	67141.4			2.21218
$974$		0			0
$975$	132530.			4.35320
$976$		0			0
$977$	−31130.0			−1.01938			−0.509691	−	0.860358i	$-0.670240\pi$
$977$	−31130.0			−1.01938			−0.509691	+	0.860358i	$0.670240\pi$
$978$		0			0
$979$	−42338.7	+	236.746i	−1.38218	+	0.00772874i
$980$		0			0
$981$		−	40741.4i		−	1.32597i
$982$		0			0
$983$			28874.9i			0.936894i	0.883492	+	0.468447i	$0.155186\pi$
$983$			28874.9i			0.936894i	−0.883492	+	0.468447i	$0.844814\pi$
$984$		0			0
$985$		−	7249.59i		−	0.234509i
$986$		0			0
$987$	103851.			3.34916
$988$		0			0
$989$		−	1929.95i		−	0.0620513i
$990$		0			0
$991$			11280.6i			0.361594i	0.983520	+	0.180797i	$0.0578678\pi$
$991$			11280.6i			0.361594i	−0.983520	+	0.180797i	$0.942132\pi$
$992$		0			0
$993$	−42024.4			−1.34301
$994$		0			0
$995$		−	30981.6i		−	0.987119i
$996$		0			0
$997$		−	50409.4i		−	1.60128i	−0.599143	−	0.800642i	$-0.704492\pi$
$997$		−	50409.4i		−	1.60128i	0.599143	−	0.800642i	$-0.295508\pi$
$998$		0			0
$999$		−	57936.8i		−	1.83487i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.e.g.703.2		36
4.3	odd	2	inner	704.4.e.g.703.11		36
8.3	odd	2		352.4.e.a.351.26	yes	36
8.5	even	2		352.4.e.a.351.35	yes	36
11.10	odd	2	inner	704.4.e.g.703.12		36
44.43	even	2	inner	704.4.e.g.703.1		36
88.21	odd	2		352.4.e.a.351.25	✓	36
88.43	even	2		352.4.e.a.351.36	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.4.e.a.351.25	✓	36	88.21	odd	2
352.4.e.a.351.26	yes	36	8.3	odd	2
352.4.e.a.351.35	yes	36	8.5	even	2
352.4.e.a.351.36	yes	36	88.43	even	2
704.4.e.g.703.1		36	44.43	even	2	inner
704.4.e.g.703.2		36	1.1	even	1	trivial
704.4.e.g.703.11		36	4.3	odd	2	inner
704.4.e.g.703.12		36	11.10	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.41697i q^{3} +17.7991 q^{5} +30.3804 q^{7} -61.6793 q^{9} +(36.4823 - 0.203999i) q^{11} -73.3735i q^{13} +167.613i q^{15} +33.5998i q^{17} +3.79105 q^{19} +286.091i q^{21} -39.3421i q^{23} +191.807 q^{25} -326.574i q^{27} +80.3580i q^{29} +132.048i q^{31} +(1.92105 + 343.553i) q^{33} +540.743 q^{35} +177.408 q^{37} +690.957 q^{39} +155.637i q^{41} +49.0555 q^{43} -1097.84 q^{45} -363.000i q^{47} +579.967 q^{49} -316.409 q^{51} -313.897 q^{53} +(649.351 - 3.63099i) q^{55} +35.7002i q^{57} -31.5479i q^{59} -204.455i q^{61} -1873.84 q^{63} -1305.98i q^{65} -638.489i q^{67} +370.484 q^{69} +712.511i q^{71} +747.515i q^{73} +1806.24i q^{75} +(1108.35 - 6.19755i) q^{77} +722.055 q^{79} +1410.00 q^{81} +6.82976 q^{83} +598.046i q^{85} -756.729 q^{87} -1160.53 q^{89} -2229.12i q^{91} -1243.49 q^{93} +67.4771 q^{95} -855.560 q^{97} +(-2250.20 + 12.5825i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) \) 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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