LMFDB - Embedded newform 4140.2.s.b.737.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.10
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.818756 + 2.08078i) q^{5} +(0.623610 + 0.623610i) q^{7} +O(q^{10})$	$q+(-0.818756 + 2.08078i) q^{5} +(0.623610 + 0.623610i) q^{7} +1.13236i q^{11} +(1.31359 - 1.31359i) q^{13} +(4.45934 - 4.45934i) q^{17} -0.624713i q^{19} +(0.707107 + 0.707107i) q^{23} +(-3.65928 - 3.40730i) q^{25} -3.20702 q^{29} +9.14592 q^{31} +(-1.80818 + 0.787010i) q^{35} +(-0.268614 - 0.268614i) q^{37} +1.26380i q^{41} +(7.06538 - 7.06538i) q^{43} +(-1.93448 + 1.93448i) q^{47} -6.22222i q^{49} +(1.62940 + 1.62940i) q^{53} +(-2.35619 - 0.927126i) q^{55} +4.50670 q^{59} +0.362896 q^{61} +(1.65778 + 3.80880i) q^{65} +(-4.96236 - 4.96236i) q^{67} +2.82340i q^{71} +(7.41129 - 7.41129i) q^{73} +(-0.706151 + 0.706151i) q^{77} -9.11283i q^{79} +(6.52879 + 6.52879i) q^{83} +(5.62779 + 12.9300i) q^{85} -13.7034 q^{89} +1.63834 q^{91} +(1.29989 + 0.511487i) q^{95} +(1.67515 + 1.67515i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.818756	+	2.08078i	−0.366159	+	0.930552i
$5$	−0.818756	+	2.08078i	−0.366159	+	0.930552i
$6$		0			0
$7$	0.623610	+	0.623610i	0.235702	+	0.235702i	0.815068	−	0.579365i	$-0.196700\pi$
$7$	0.623610	+	0.623610i	0.235702	+	0.235702i	−0.579365	+	0.815068i	$0.696700\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.13236i			0.341419i	0.985321	+	0.170710i	$0.0546060\pi$
$11$			1.13236i			0.341419i	−0.985321	+	0.170710i	$0.945394\pi$
$12$		0			0
$13$	1.31359	−	1.31359i	0.364325	−	0.364325i	−0.501078	−	0.865402i	$-0.667063\pi$
$13$	1.31359	−	1.31359i	0.364325	−	0.364325i	0.865402	+	0.501078i	$0.167063\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.45934	−	4.45934i	1.08155	−	1.08155i	0.0851845	−	0.996365i	$-0.472852\pi$
$17$	4.45934	−	4.45934i	1.08155	−	1.08155i	0.996365	−	0.0851845i	$-0.0271480\pi$
$18$		0			0
$19$		−	0.624713i		−	0.143319i	−0.997429	−	0.0716595i	$-0.977171\pi$
$19$		−	0.624713i		−	0.143319i	0.997429	−	0.0716595i	$-0.0228295\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$24$		0			0
$25$	−3.65928	−	3.40730i	−0.731855	−	0.681460i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.20702			−0.595528			−0.297764	−	0.954639i	$-0.596241\pi$
$29$	−3.20702			−0.595528			−0.297764	+	0.954639i	$0.596241\pi$
$30$		0			0
$31$	9.14592			1.64266			0.821328	−	0.570456i	$-0.193234\pi$
$31$	9.14592			1.64266			0.821328	+	0.570456i	$0.193234\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.80818	+	0.787010i	−0.305638	+	0.133029i
$36$		0			0
$37$	−0.268614	−	0.268614i	−0.0441598	−	0.0441598i	0.684682	−	0.728842i	$-0.259941\pi$
$37$	−0.268614	−	0.268614i	−0.0441598	−	0.0441598i	−0.728842	+	0.684682i	$0.759941\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.26380i			0.197373i	0.995119	+	0.0986864i	$0.0314641\pi$
$41$			1.26380i			0.197373i	−0.995119	+	0.0986864i	$0.968536\pi$
$42$		0			0
$43$	7.06538	−	7.06538i	1.07746	−	1.07746i	0.0807237	−	0.996737i	$-0.474277\pi$
$43$	7.06538	−	7.06538i	1.07746	−	1.07746i	0.996737	−	0.0807237i	$-0.0257231\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.93448	+	1.93448i	−0.282173	+	0.282173i	−0.833975	−	0.551802i	$-0.813940\pi$
$47$	−1.93448	+	1.93448i	−0.282173	+	0.282173i	0.551802	+	0.833975i	$0.313940\pi$
$48$		0			0
$49$		−	6.22222i		−	0.888889i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.62940	+	1.62940i	0.223815	+	0.223815i	0.810103	−	0.586288i	$-0.199411\pi$
$53$	1.62940	+	1.62940i	0.223815	+	0.223815i	−0.586288	+	0.810103i	$0.699411\pi$
$54$		0			0
$55$	−2.35619	−	0.927126i	−0.317708	−	0.125014i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50670			0.586723			0.293361	−	0.956002i	$-0.405226\pi$
$59$	4.50670			0.586723			0.293361	+	0.956002i	$0.405226\pi$
$60$		0			0
$61$	0.362896			0.0464641			0.0232320	−	0.999730i	$-0.492604\pi$
$61$	0.362896			0.0464641			0.0232320	+	0.999730i	$0.492604\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.65778	+	3.80880i	0.205623	+	0.472424i
$66$		0			0
$67$	−4.96236	−	4.96236i	−0.606249	−	0.606249i	0.335715	−	0.941964i	$-0.391022\pi$
$67$	−4.96236	−	4.96236i	−0.606249	−	0.606249i	−0.941964	+	0.335715i	$0.891022\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.82340i			0.335076i	0.985866	+	0.167538i	$0.0535816\pi$
$71$			2.82340i			0.335076i	−0.985866	+	0.167538i	$0.946418\pi$
$72$		0			0
$73$	7.41129	−	7.41129i	0.867426	−	0.867426i	−0.124761	−	0.992187i	$-0.539816\pi$
$73$	7.41129	−	7.41129i	0.867426	−	0.867426i	0.992187	+	0.124761i	$0.0398163\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.706151	+	0.706151i	−0.0804733	+	0.0804733i
$78$		0			0
$79$		−	9.11283i		−	1.02527i	−0.858606	−	0.512637i	$-0.828669\pi$
$79$		−	9.11283i		−	1.02527i	0.858606	−	0.512637i	$-0.171331\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.52879	+	6.52879i	0.716628	+	0.716628i	0.967913	−	0.251285i	$-0.0808533\pi$
$83$	6.52879	+	6.52879i	0.716628	+	0.716628i	−0.251285	+	0.967913i	$0.580853\pi$
$84$		0			0
$85$	5.62779	+	12.9300i	0.610420	+	1.40246i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.7034			−1.45256			−0.726280	−	0.687399i	$-0.758752\pi$
$89$	−13.7034			−1.45256			−0.726280	+	0.687399i	$0.758752\pi$
$90$		0			0
$91$	1.63834			0.171744
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.29989	+	0.511487i	0.133366	+	0.0524775i
$96$		0			0
$97$	1.67515	+	1.67515i	0.170086	+	0.170086i	0.787017	−	0.616931i	$-0.211624\pi$
$97$	1.67515	+	1.67515i	0.170086	+	0.170086i	−0.616931	+	0.787017i	$0.711624\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			5.95418i			0.592463i	0.955116	+	0.296232i	$0.0957300\pi$
$101$			5.95418i			0.592463i	−0.955116	+	0.296232i	$0.904270\pi$
$102$		0			0
$103$	−10.9639	+	10.9639i	−1.08030	+	1.08030i	−0.0838225	+	0.996481i	$0.526713\pi$
$103$	−10.9639	+	10.9639i	−1.08030	+	1.08030i	−0.996481	+	0.0838225i	$0.973287\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.4478	−	14.4478i	1.39672	−	1.39672i	0.587491	−	0.809230i	$-0.300116\pi$
$107$	14.4478	−	14.4478i	1.39672	−	1.39672i	0.809230	−	0.587491i	$-0.199884\pi$
$108$		0			0
$109$			16.7352i			1.60294i	0.598037	+	0.801469i	$0.295948\pi$
$109$			16.7352i			1.60294i	−0.598037	+	0.801469i	$0.704052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.42170	+	6.42170i	0.604103	+	0.604103i	0.941399	−	0.337296i	$-0.109512\pi$
$113$	6.42170	+	6.42170i	0.604103	+	0.604103i	−0.337296	+	0.941399i	$0.609512\pi$
$114$		0			0
$115$	−2.05028	+	0.892384i	−0.191190	+	0.0832153i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.56178			0.509848
$120$		0			0
$121$	9.71776			0.883433
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.0859	−	4.82440i	0.902110	−	0.431507i
$126$		0			0
$127$	−3.96823	−	3.96823i	−0.352123	−	0.352123i	0.508776	−	0.860899i	$-0.330098\pi$
$127$	−3.96823	−	3.96823i	−0.352123	−	0.352123i	−0.860899	+	0.508776i	$0.830098\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.0578i			1.22823i	0.789216	+	0.614116i	$0.210487\pi$
$131$			14.0578i			1.22823i	−0.789216	+	0.614116i	$0.789513\pi$
$132$		0			0
$133$	0.389577	−	0.389577i	0.0337806	−	0.0337806i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.5029	+	14.5029i	−1.23907	+	1.23907i	−0.278688	+	0.960382i	$0.589899\pi$
$137$	−14.5029	+	14.5029i	−1.23907	+	1.23907i	−0.960382	+	0.278688i	$0.910101\pi$
$138$		0			0
$139$		−	2.17486i		−	0.184469i	−0.995737	−	0.0922344i	$-0.970599\pi$
$139$		−	2.17486i		−	0.184469i	0.995737	−	0.0922344i	$-0.0294009\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.48746	+	1.48746i	0.124387	+	0.124387i
$144$		0			0
$145$	2.62577	−	6.67309i	0.218058	−	0.554170i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.7775			1.37447			0.687233	−	0.726437i	$-0.258825\pi$
$149$	16.7775			1.37447			0.687233	+	0.726437i	$0.258825\pi$
$150$		0			0
$151$	3.73627			0.304054			0.152027	−	0.988376i	$-0.451420\pi$
$151$	3.73627			0.304054			0.152027	+	0.988376i	$0.451420\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.48828	+	19.0306i	−0.601473	+	1.52858i
$156$		0			0
$157$	10.9658	+	10.9658i	0.875164	+	0.875164i	0.993030	−	0.117865i	$-0.0376051\pi$
$157$	10.9658	+	10.9658i	0.875164	+	0.875164i	−0.117865	+	0.993030i	$0.537605\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.881918i			0.0695049i
$162$		0			0
$163$	6.83033	−	6.83033i	0.534993	−	0.534993i	−0.387061	−	0.922054i	$-0.626510\pi$
$163$	6.83033	−	6.83033i	0.534993	−	0.534993i	0.922054	+	0.387061i	$0.126510\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.57281	+	4.57281i	−0.353855	+	0.353855i	−0.861542	−	0.507687i	$-0.830501\pi$
$167$	−4.57281	+	4.57281i	−0.353855	+	0.353855i	0.507687	+	0.861542i	$0.330501\pi$
$168$		0			0
$169$			9.54895i			0.734535i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.8484	+	13.8484i	1.05288	+	1.05288i	0.998522	+	0.0543543i	$0.0173100\pi$
$173$	13.8484	+	13.8484i	1.05288	+	1.05288i	0.0543543	+	0.998522i	$0.482690\pi$
$174$		0			0
$175$	−0.157135	−	4.40679i	−0.0118783	−	0.333122i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.4034			1.82400			0.911998	−	0.410194i	$-0.134539\pi$
$179$	24.4034			1.82400			0.911998	+	0.410194i	$0.134539\pi$
$180$		0			0
$181$	−7.93663			−0.589925			−0.294962	−	0.955509i	$-0.595307\pi$
$181$	−7.93663			−0.589925			−0.294962	+	0.955509i	$0.595307\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.778854	−	0.338996i	0.0572625	−	0.0249235i
$186$		0			0
$187$	5.04958	+	5.04958i	0.369262	+	0.369262i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.8439i		−	0.856992i	−0.903544	−	0.428496i	$-0.859044\pi$
$191$		−	11.8439i		−	0.856992i	0.903544	−	0.428496i	$-0.140956\pi$
$192$		0			0
$193$	8.06350	−	8.06350i	0.580424	−	0.580424i	−0.354596	−	0.935020i	$-0.615382\pi$
$193$	8.06350	−	8.06350i	0.580424	−	0.580424i	0.935020	+	0.354596i	$0.115382\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.80885	+	2.80885i	−0.200122	+	0.200122i	−0.800052	−	0.599930i	$-0.795195\pi$
$197$	−2.80885	+	2.80885i	−0.200122	+	0.200122i	0.599930	+	0.800052i	$0.295195\pi$
$198$		0			0
$199$		−	21.7547i		−	1.54215i	−0.636743	−	0.771076i	$-0.719719\pi$
$199$		−	21.7547i		−	1.54215i	0.636743	−	0.771076i	$-0.280281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.99993	−	1.99993i	−0.140367	−	0.140367i
$204$		0			0
$205$	−2.62969	−	1.03475i	−0.183666	−	0.0722698i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.707399			0.0489318
$210$		0			0
$211$	−16.7588			−1.15372			−0.576862	−	0.816842i	$-0.695723\pi$
$211$	−16.7588			−1.15372			−0.576862	+	0.816842i	$0.695723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.91667	+	20.4863i	0.608112	+	1.39715i
$216$		0			0
$217$	5.70349	+	5.70349i	0.387178	+	0.387178i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	11.7155i		−	0.788071i
$222$		0			0
$223$	−2.29856	+	2.29856i	−0.153923	+	0.153923i	−0.779868	−	0.625945i	$-0.784714\pi$
$223$	−2.29856	+	2.29856i	−0.153923	+	0.153923i	0.625945	+	0.779868i	$0.284714\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.76892	+	9.76892i	−0.648386	+	0.648386i	−0.952603	−	0.304217i	$-0.901605\pi$
$227$	−9.76892	+	9.76892i	−0.648386	+	0.648386i	0.304217	+	0.952603i	$0.401605\pi$
$228$		0			0
$229$		−	17.3398i		−	1.14585i	−0.819609	−	0.572923i	$-0.805810\pi$
$229$		−	17.3398i		−	1.14585i	0.819609	−	0.572923i	$-0.194190\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.764139	−	0.764139i	−0.0500604	−	0.0500604i	0.681633	−	0.731694i	$-0.261270\pi$
$233$	−0.764139	−	0.764139i	−0.0500604	−	0.0500604i	−0.731694	+	0.681633i	$0.761270\pi$
$234$		0			0
$235$	−2.44135	−	5.60908i	−0.159256	−	0.365896i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.2937			−1.31269			−0.656344	−	0.754462i	$-0.727898\pi$
$239$	−20.2937			−1.31269			−0.656344	+	0.754462i	$0.727898\pi$
$240$		0			0
$241$	29.7221			1.91457			0.957286	−	0.289143i	$-0.0933705\pi$
$241$	29.7221			1.91457			0.957286	+	0.289143i	$0.0933705\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.9471	+	5.09448i	0.827158	+	0.325474i
$246$		0			0
$247$	−0.820617	−	0.820617i	−0.0522146	−	0.0522146i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			8.50308i			0.536710i	0.963320	+	0.268355i	$0.0864800\pi$
$251$			8.50308i			0.536710i	−0.963320	+	0.268355i	$0.913520\pi$
$252$		0			0
$253$	−0.800699	+	0.800699i	−0.0503395	+	0.0503395i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.49402	−	8.49402i	0.529843	−	0.529843i	−0.390683	−	0.920525i	$-0.627761\pi$
$257$	8.49402	−	8.49402i	0.529843	−	0.529843i	0.920525	+	0.390683i	$0.127761\pi$
$258$		0			0
$259$		−	0.335020i		−	0.0208171i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.84470	+	5.84470i	0.360400	+	0.360400i	0.863960	−	0.503560i	$-0.167977\pi$
$263$	5.84470	+	5.84470i	0.360400	+	0.360400i	−0.503560	+	0.863960i	$0.667977\pi$
$264$		0			0
$265$	−4.72450	+	2.05634i	−0.290224	+	0.126320i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.2120			−0.866520			−0.433260	−	0.901269i	$-0.642637\pi$
$269$	−14.2120			−0.866520			−0.433260	+	0.901269i	$0.642637\pi$
$270$		0			0
$271$	0.427968			0.0259972			0.0129986	−	0.999916i	$-0.495862\pi$
$271$	0.427968			0.0259972			0.0129986	+	0.999916i	$0.495862\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.85829	−	4.14362i	0.232664	−	0.249869i
$276$		0			0
$277$	15.8725	+	15.8725i	0.953686	+	0.953686i	0.998974	−	0.0452878i	$-0.0144205\pi$
$277$	15.8725	+	15.8725i	0.953686	+	0.953686i	−0.0452878	+	0.998974i	$0.514420\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	0.160395i		−	0.00956834i	−0.999989	−	0.00478417i	$-0.998477\pi$
$281$		−	0.160395i		−	0.00956834i	0.999989	−	0.00478417i	$-0.00152285\pi$
$282$		0			0
$283$	−8.51372	+	8.51372i	−0.506088	+	0.506088i	−0.913323	−	0.407235i	$-0.866493\pi$
$283$	−8.51372	+	8.51372i	−0.506088	+	0.506088i	0.407235	+	0.913323i	$0.366493\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.788120	+	0.788120i	−0.0465212	+	0.0465212i
$288$		0			0
$289$		−	22.7715i		−	1.33950i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.76506	+	5.76506i	0.336799	+	0.336799i	0.855161	−	0.518362i	$-0.173458\pi$
$293$	5.76506	+	5.76506i	0.336799	+	0.336799i	−0.518362	+	0.855161i	$0.673458\pi$
$294$		0			0
$295$	−3.68989	+	9.37745i	−0.214834	+	0.545976i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.85770			0.107433
$300$		0			0
$301$	8.81208			0.507920
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.297123	+	0.755106i	−0.0170132	+	0.0432372i
$306$		0			0
$307$	−0.367532	−	0.367532i	−0.0209762	−	0.0209762i	0.696541	−	0.717517i	$-0.254721\pi$
$307$	−0.367532	−	0.367532i	−0.0209762	−	0.0209762i	−0.717517	+	0.696541i	$0.754721\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.90890i		−	0.278358i	−0.990267	−	0.139179i	$-0.955554\pi$
$311$		−	4.90890i		−	0.278358i	0.990267	−	0.139179i	$-0.0444464\pi$
$312$		0			0
$313$	−7.08806	+	7.08806i	−0.400641	+	0.400641i	−0.878459	−	0.477818i	$-0.841428\pi$
$313$	−7.08806	+	7.08806i	−0.400641	+	0.400641i	0.477818	+	0.878459i	$0.341428\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.02285	−	1.02285i	0.0574490	−	0.0574490i	−0.677799	−	0.735248i	$-0.737066\pi$
$317$	1.02285	−	1.02285i	0.0574490	−	0.0574490i	0.735248	+	0.677799i	$0.237066\pi$
$318$		0			0
$319$		−	3.63150i		−	0.203325i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.78581	−	2.78581i	−0.155007	−	0.155007i
$324$		0			0
$325$	−9.28259	+	0.330995i	−0.514906	+	0.0183603i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.41272			−0.133018
$330$		0			0
$331$	10.8946			0.598822			0.299411	−	0.954124i	$-0.403210\pi$
$331$	10.8946			0.598822			0.299411	+	0.954124i	$0.403210\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14.3885	−	6.26261i	0.786130	−	0.342163i
$336$		0			0
$337$	−9.55236	−	9.55236i	−0.520350	−	0.520350i	0.397327	−	0.917677i	$-0.369938\pi$
$337$	−9.55236	−	9.55236i	−0.520350	−	0.520350i	−0.917677	+	0.397327i	$0.869938\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10.3565i			0.560834i
$342$		0			0
$343$	8.24551	−	8.24551i	0.445216	−	0.445216i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.2157	−	19.2157i	1.03155	−	1.03155i	0.0320662	−	0.999486i	$-0.489791\pi$
$347$	19.2157	−	19.2157i	1.03155	−	1.03155i	0.999486	−	0.0320662i	$-0.0102087\pi$
$348$		0			0
$349$		−	11.8597i		−	0.634835i	−0.948286	−	0.317418i	$-0.897184\pi$
$349$		−	11.8597i		−	0.634835i	0.948286	−	0.317418i	$-0.102816\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.61849	−	4.61849i	−0.245817	−	0.245817i	0.573434	−	0.819252i	$-0.305611\pi$
$353$	−4.61849	−	4.61849i	−0.245817	−	0.245817i	−0.819252	+	0.573434i	$0.805611\pi$
$354$		0			0
$355$	−5.87487	−	2.31167i	−0.311805	−	0.122691i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.7207			1.09359			0.546797	−	0.837265i	$-0.315847\pi$
$359$	20.7207			1.09359			0.546797	+	0.837265i	$0.315847\pi$
$360$		0			0
$361$	18.6097			0.979460
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.35322	+	21.4893i	0.489570	+	1.12480i
$366$		0			0
$367$	3.01265	+	3.01265i	0.157259	+	0.157259i	0.781351	−	0.624092i	$-0.214531\pi$
$367$	3.01265	+	3.01265i	0.157259	+	0.157259i	−0.624092	+	0.781351i	$0.714531\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.03222i			0.105508i
$372$		0			0
$373$	−22.4315	+	22.4315i	−1.16146	+	1.16146i	−0.177302	+	0.984156i	$0.556737\pi$
$373$	−22.4315	+	22.4315i	−1.16146	+	1.16146i	−0.984156	+	0.177302i	$0.943263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.21271	+	4.21271i	−0.216966	+	0.216966i
$378$		0			0
$379$		−	20.6977i		−	1.06317i	−0.847006	−	0.531584i	$-0.821597\pi$
$379$		−	20.6977i		−	1.06317i	0.847006	−	0.531584i	$-0.178403\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.32720	+	7.32720i	0.374403	+	0.374403i	0.869078	−	0.494675i	$-0.164713\pi$
$383$	7.32720	+	7.32720i	0.374403	+	0.374403i	−0.494675	+	0.869078i	$0.664713\pi$
$384$		0			0
$385$	−0.891178	−	2.04751i	−0.0454186	−	0.104351i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.20295			−0.415906			−0.207953	−	0.978139i	$-0.566680\pi$
$389$	−8.20295			−0.415906			−0.207953	+	0.978139i	$0.566680\pi$
$390$		0			0
$391$	6.30646			0.318932
$392$		0			0
$393$		0			0
$394$		0			0
$395$	18.9618	+	7.46118i	0.954070	+	0.375413i
$396$		0			0
$397$	16.5119	+	16.5119i	0.828710	+	0.828710i	0.987338	−	0.158629i	$-0.0507073\pi$
$397$	16.5119	+	16.5119i	0.828710	+	0.828710i	−0.158629	+	0.987338i	$0.550707\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.43717i		−	0.221582i	−0.993844	−	0.110791i	$-0.964662\pi$
$401$		−	4.43717i		−	0.221582i	0.993844	−	0.110791i	$-0.0353384\pi$
$402$		0			0
$403$	12.0140	−	12.0140i	0.598460	−	0.598460i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.304167	−	0.304167i	0.0150770	−	0.0150770i
$408$		0			0
$409$		−	7.34430i		−	0.363152i	−0.983377	−	0.181576i	$-0.941880\pi$
$409$		−	7.34430i		−	0.363152i	0.983377	−	0.181576i	$-0.0581198\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.81042	+	2.81042i	0.138292	+	0.138292i
$414$		0			0
$415$	−18.9304	+	8.23948i	−0.929259	+	0.404460i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.4582			−0.999447			−0.499723	−	0.866185i	$-0.666565\pi$
$419$	−20.4582			−0.999447			−0.499723	+	0.866185i	$0.666565\pi$
$420$		0			0
$421$	11.5128			0.561101			0.280550	−	0.959839i	$-0.409483\pi$
$421$	11.5128			0.561101			0.280550	+	0.959839i	$0.409483\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−31.5123	+	1.12365i	−1.52857	+	0.0545051i
$426$		0			0
$427$	0.226305	+	0.226305i	0.0109517	+	0.0109517i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	22.3755i		−	1.07779i	−0.842374	−	0.538894i	$-0.818842\pi$
$431$		−	22.3755i		−	1.07779i	0.842374	−	0.538894i	$-0.181158\pi$
$432$		0			0
$433$	−16.1481	+	16.1481i	−0.776029	+	0.776029i	−0.979153	−	0.203124i	$-0.934890\pi$
$433$	−16.1481	+	16.1481i	−0.776029	+	0.776029i	0.203124	+	0.979153i	$0.434890\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.441739	−	0.441739i	0.0211312	−	0.0211312i
$438$		0			0
$439$		−	24.2617i		−	1.15795i	−0.815345	−	0.578975i	$-0.803453\pi$
$439$		−	24.2617i		−	1.15795i	0.815345	−	0.578975i	$-0.196547\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.578048	−	0.578048i	−0.0274639	−	0.0274639i	0.693241	−	0.720705i	$-0.256182\pi$
$443$	−0.578048	−	0.578048i	−0.0274639	−	0.0274639i	−0.720705	+	0.693241i	$0.756182\pi$
$444$		0			0
$445$	11.2198	−	28.5138i	0.531868	−	1.35168i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.3971			−1.85927			−0.929633	−	0.368487i	$-0.879876\pi$
$449$	−39.3971			−1.85927			−0.929633	+	0.368487i	$0.879876\pi$
$450$		0			0
$451$	−1.43108			−0.0673868
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.34140	+	3.40902i	−0.0628857	+	0.159817i
$456$		0			0
$457$	22.9308	+	22.9308i	1.07266	+	1.07266i	0.997145	+	0.0755147i	$0.0240600\pi$
$457$	22.9308	+	22.9308i	1.07266	+	1.07266i	0.0755147	+	0.997145i	$0.475940\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			22.4406i			1.04516i	0.852589	+	0.522581i	$0.175031\pi$
$461$			22.4406i			1.04516i	−0.852589	+	0.522581i	$0.824969\pi$
$462$		0			0
$463$	16.4666	−	16.4666i	0.765270	−	0.765270i	−0.212000	−	0.977270i	$-0.567998\pi$
$463$	16.4666	−	16.4666i	0.765270	−	0.765270i	0.977270	+	0.212000i	$0.0679976\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.4948	+	13.4948i	−0.624466	+	0.624466i	−0.946670	−	0.322204i	$-0.895576\pi$
$467$	−13.4948	+	13.4948i	−0.624466	+	0.624466i	0.322204	+	0.946670i	$0.395576\pi$
$468$		0			0
$469$		−	6.18916i		−	0.285789i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00055	+	8.00055i	0.367866	+	0.367866i
$474$		0			0
$475$	−2.12858	+	2.28600i	−0.0976661	+	0.104889i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.14723			−0.189492			−0.0947459	−	0.995501i	$-0.530204\pi$
$479$	−4.14723			−0.189492			−0.0947459	+	0.995501i	$0.530204\pi$
$480$		0			0
$481$	−0.705697			−0.0321770
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.85717	+	2.11408i	−0.220553	+	0.0959955i
$486$		0			0
$487$	−5.79818	−	5.79818i	−0.262741	−	0.262741i	0.563426	−	0.826167i	$-0.309483\pi$
$487$	−5.79818	−	5.79818i	−0.262741	−	0.262741i	−0.826167	+	0.563426i	$0.809483\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.4345i		−	1.01245i	−0.862400	−	0.506227i	$-0.831040\pi$
$491$		−	22.4345i		−	1.01245i	0.862400	−	0.506227i	$-0.168960\pi$
$492$		0			0
$493$	−14.3012	+	14.3012i	−0.644093	+	0.644093i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.76070	+	1.76070i	−0.0789782	+	0.0789782i
$498$		0			0
$499$		−	8.30268i		−	0.371679i	−0.982580	−	0.185839i	$-0.940500\pi$
$499$		−	8.30268i		−	0.371679i	0.982580	−	0.185839i	$-0.0595004\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0203	−	15.0203i	−0.669722	−	0.669722i	0.287930	−	0.957652i	$-0.407033\pi$
$503$	−15.0203	−	15.0203i	−0.669722	−	0.669722i	−0.957652	+	0.287930i	$0.907033\pi$
$504$		0			0
$505$	−12.3893	−	4.87502i	−0.551318	−	0.216936i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.30097			−0.190637			−0.0953186	−	0.995447i	$-0.530387\pi$
$509$	−4.30097			−0.190637			−0.0953186	+	0.995447i	$0.530387\pi$
$510$		0			0
$511$	9.24351			0.408909
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.8367	−	31.7901i	−0.609716	−	1.40084i
$516$		0			0
$517$	−2.19052	−	2.19052i	−0.0963391	−	0.0963391i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	23.9287i		−	1.04833i	−0.851615	−	0.524167i	$-0.824377\pi$
$521$		−	23.9287i		−	1.04833i	0.851615	−	0.524167i	$-0.175623\pi$
$522$		0			0
$523$	−0.466545	+	0.466545i	−0.0204006	+	0.0204006i	−0.717234	−	0.696833i	$-0.754592\pi$
$523$	−0.466545	+	0.466545i	−0.0204006	+	0.0204006i	0.696833	+	0.717234i	$0.254592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	40.7848	−	40.7848i	1.77661	−	1.77661i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.66012	+	1.66012i	0.0719078	+	0.0719078i
$534$		0			0
$535$	18.2334	+	41.8919i	0.788301	+	1.81114i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.04579			0.303484
$540$		0			0
$541$	−4.26392			−0.183320			−0.0916602	−	0.995790i	$-0.529217\pi$
$541$	−4.26392			−0.183320			−0.0916602	+	0.995790i	$0.529217\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−34.8221	−	13.7020i	−1.49162	−	0.586930i
$546$		0			0
$547$	−6.79746	−	6.79746i	−0.290639	−	0.290639i	0.546694	−	0.837332i	$-0.315886\pi$
$547$	−6.79746	−	6.79746i	−0.290639	−	0.290639i	−0.837332	+	0.546694i	$0.815886\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			2.00347i			0.0853505i
$552$		0			0
$553$	5.68285	−	5.68285i	0.241659	−	0.241659i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.2685	+	25.2685i	−1.07066	+	1.07066i	−0.0733543	+	0.997306i	$0.523370\pi$
$557$	−25.2685	+	25.2685i	−1.07066	+	1.07066i	−0.997306	+	0.0733543i	$0.976630\pi$
$558$		0			0
$559$		−	18.5620i		−	0.785091i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.8024	−	12.8024i	−0.539559	−	0.539559i	0.383841	−	0.923399i	$-0.374601\pi$
$563$	−12.8024	−	12.8024i	−0.539559	−	0.539559i	−0.923399	+	0.383841i	$0.874601\pi$
$564$		0			0
$565$	−18.6199	+	8.10433i	−0.783347	+	0.340952i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−37.3676			−1.56653			−0.783266	−	0.621687i	$-0.786448\pi$
$569$	−37.3676			−1.56653			−0.783266	+	0.621687i	$0.786448\pi$
$570$		0			0
$571$	−0.473653			−0.0198217			−0.00991087	−	0.999951i	$-0.503155\pi$
$571$	−0.473653			−0.0198217			−0.00991087	+	0.999951i	$0.503155\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.178175	−	4.99682i	−0.00743040	−	0.208382i
$576$		0			0
$577$	−13.9004	−	13.9004i	−0.578680	−	0.578680i	0.355860	−	0.934539i	$-0.384188\pi$
$577$	−13.9004	−	13.9004i	−0.578680	−	0.578680i	−0.934539	+	0.355860i	$0.884188\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			8.14284i			0.337822i
$582$		0			0
$583$	−1.84507	+	1.84507i	−0.0764148	+	0.0764148i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0116	−	27.0116i	1.11489	−	1.11489i	0.122411	−	0.992480i	$-0.460938\pi$
$587$	27.0116	−	27.0116i	1.11489	−	1.11489i	0.992480	−	0.122411i	$-0.0390625\pi$
$588$		0			0
$589$		−	5.71357i		−	0.235424i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.2255	+	21.2255i	0.871628	+	0.871628i	0.992650	−	0.121022i	$-0.0386172\pi$
$593$	21.2255	+	21.2255i	0.871628	+	0.871628i	−0.121022	+	0.992650i	$0.538617\pi$
$594$		0			0
$595$	−4.55374	+	11.5728i	−0.186685	+	0.474440i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.8822			−0.648928			−0.324464	−	0.945898i	$-0.605184\pi$
$599$	−15.8822			−0.648928			−0.324464	+	0.945898i	$0.605184\pi$
$600$		0			0
$601$	0.462742			0.0188756			0.00943782	−	0.999955i	$-0.496996\pi$
$601$	0.462742			0.0188756			0.00943782	+	0.999955i	$0.496996\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.95648	+	20.2205i	−0.323477	+	0.822081i
$606$		0			0
$607$	7.35596	+	7.35596i	0.298569	+	0.298569i	0.840453	−	0.541884i	$-0.182289\pi$
$607$	7.35596	+	7.35596i	0.298569	+	0.298569i	−0.541884	+	0.840453i	$0.682289\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			5.08223i			0.205605i
$612$		0			0
$613$	3.88192	−	3.88192i	0.156789	−	0.156789i	−0.624353	−	0.781142i	$-0.714637\pi$
$613$	3.88192	−	3.88192i	0.156789	−	0.156789i	0.781142	+	0.624353i	$0.214637\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.05784	−	8.05784i	0.324396	−	0.324396i	−0.526055	−	0.850451i	$-0.676329\pi$
$617$	8.05784	−	8.05784i	0.324396	−	0.324396i	0.850451	+	0.526055i	$0.176329\pi$
$618$		0			0
$619$			13.0770i			0.525610i	0.964849	+	0.262805i	$0.0846475\pi$
$619$			13.0770i			0.525610i	−0.964849	+	0.262805i	$0.915352\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.54559	−	8.54559i	−0.342372	−	0.342372i
$624$		0			0
$625$	1.78062	+	24.9365i	0.0712246	+	0.997460i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.39568			−0.0955220
$630$		0			0
$631$	−5.82927			−0.232060			−0.116030	−	0.993246i	$-0.537017\pi$
$631$	−5.82927			−0.232060			−0.116030	+	0.993246i	$0.537017\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.5060	−	5.00799i	0.456602	−	0.198736i
$636$		0			0
$637$	−8.17346	−	8.17346i	−0.323844	−	0.323844i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			0.0988304i			0.00390357i	0.999998	+	0.00195178i	$0.000621272\pi$
$641$			0.0988304i			0.00390357i	−0.999998	+	0.00195178i	$0.999379\pi$
$642$		0			0
$643$	−1.75936	+	1.75936i	−0.0693822	+	0.0693822i	−0.740946	−	0.671564i	$-0.765623\pi$
$643$	−1.75936	+	1.75936i	−0.0693822	+	0.0693822i	0.671564	+	0.740946i	$0.265623\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.1455	−	14.1455i	0.556116	−	0.556116i	−0.372083	−	0.928199i	$-0.621356\pi$
$647$	14.1455	−	14.1455i	0.556116	−	0.556116i	0.928199	+	0.372083i	$0.121356\pi$
$648$		0			0
$649$			5.10321i			0.200318i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.66859	−	8.66859i	−0.339228	−	0.339228i	0.516849	−	0.856077i	$-0.327105\pi$
$653$	−8.66859	−	8.66859i	−0.339228	−	0.339228i	−0.856077	+	0.516849i	$0.827105\pi$
$654$		0			0
$655$	−29.2511	−	11.5099i	−1.14293	−	0.449728i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.4065			0.600152			0.300076	−	0.953915i	$-0.402988\pi$
$659$	15.4065			0.600152			0.300076	+	0.953915i	$0.402988\pi$
$660$		0			0
$661$	31.7123			1.23347			0.616733	−	0.787172i	$-0.288456\pi$
$661$	31.7123			1.23347			0.616733	+	0.787172i	$0.288456\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.491655	+	1.12959i	0.0190656	+	0.0438037i
$666$		0			0
$667$	−2.26770	−	2.26770i	−0.0878058	−	0.0878058i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			0.410929i			0.0158637i
$672$		0			0
$673$	−23.4907	+	23.4907i	−0.905502	+	0.905502i	−0.995905	−	0.0904035i	$-0.971184\pi$
$673$	−23.4907	+	23.4907i	−0.905502	+	0.905502i	0.0904035	+	0.995905i	$0.471184\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.06802	+	3.06802i	−0.117914	+	0.117914i	−0.763601	−	0.645688i	$-0.776571\pi$
$677$	−3.06802	+	3.06802i	−0.117914	+	0.117914i	0.645688	+	0.763601i	$0.276571\pi$
$678$		0			0
$679$			2.08929i			0.0801794i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5898	+	10.5898i	0.405209	+	0.405209i	0.880064	−	0.474855i	$-0.157499\pi$
$683$	10.5898	+	10.5898i	0.405209	+	0.405209i	−0.474855	+	0.880064i	$0.657499\pi$
$684$		0			0
$685$	−18.3030	−	42.0518i	−0.699323	−	1.60672i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.28073			0.163083
$690$		0			0
$691$	−28.9713			−1.10212			−0.551059	−	0.834466i	$-0.685776\pi$
$691$	−28.9713			−1.10212			−0.551059	+	0.834466i	$0.685776\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.52539	+	1.78068i	0.171658	+	0.0675449i
$696$		0			0
$697$	5.63573	+	5.63573i	0.213468	+	0.213468i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			10.0876i			0.381002i	0.981687	+	0.190501i	$0.0610112\pi$
$701$			10.0876i			0.381002i	−0.981687	+	0.190501i	$0.938989\pi$
$702$		0			0
$703$	−0.167806	+	0.167806i	−0.00632894	+	0.00632894i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.71309	+	3.71309i	−0.139645	+	0.139645i
$708$		0			0
$709$			42.5528i			1.59810i	0.601263	+	0.799051i	$0.294664\pi$
$709$			42.5528i			1.59810i	−0.601263	+	0.799051i	$0.705336\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.46714	+	6.46714i	0.242196	+	0.242196i
$714$		0			0
$715$	−4.31293	+	1.87720i	−0.161295	+	0.0702035i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.1969			0.454866			0.227433	−	0.973794i	$-0.426967\pi$
$719$	12.1969			0.454866			0.227433	+	0.973794i	$0.426967\pi$
$720$		0			0
$721$	−13.6744			−0.509260
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.7354	+	10.9273i	0.435841	+	0.405829i
$726$		0			0
$727$	22.7297	+	22.7297i	0.842997	+	0.842997i	0.989248	−	0.146251i	$-0.0467206\pi$
$727$	22.7297	+	22.7297i	0.842997	+	0.842997i	−0.146251	+	0.989248i	$0.546721\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	63.0139i		−	2.33065i
$732$		0			0
$733$	−29.0132	+	29.0132i	−1.07163	+	1.07163i	−0.0743995	+	0.997229i	$0.523704\pi$
$733$	−29.0132	+	29.0132i	−1.07163	+	1.07163i	−0.997229	+	0.0743995i	$0.976296\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.61918	−	5.61918i	0.206985	−	0.206985i
$738$		0			0
$739$			12.9343i			0.475795i	0.971290	+	0.237897i	$0.0764582\pi$
$739$			12.9343i			0.475795i	−0.971290	+	0.237897i	$0.923542\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.92156	−	2.92156i	−0.107182	−	0.107182i	0.651482	−	0.758664i	$-0.274148\pi$
$743$	−2.92156	−	2.92156i	−0.107182	−	0.107182i	−0.758664	+	0.651482i	$0.774148\pi$
$744$		0			0
$745$	−13.7367	+	34.9102i	−0.503273	+	1.27901i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0196			0.658421
$750$		0			0
$751$	20.1321			0.734629			0.367315	−	0.930097i	$-0.380277\pi$
$751$	20.1321			0.734629			0.367315	+	0.930097i	$0.380277\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.05910	+	7.77435i	−0.111332	+	0.282938i
$756$		0			0
$757$	2.26173	+	2.26173i	0.0822041	+	0.0822041i	0.747013	−	0.664809i	$-0.231487\pi$
$757$	2.26173	+	2.26173i	0.0822041	+	0.0822041i	−0.664809	+	0.747013i	$0.731487\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.02680i		−	0.0372214i	−0.999827	−	0.0186107i	$-0.994076\pi$
$761$		−	1.02680i		−	0.0372214i	0.999827	−	0.0186107i	$-0.00592431\pi$
$762$		0			0
$763$	−10.4362	+	10.4362i	−0.377816	+	0.377816i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.91997	−	5.91997i	0.213758	−	0.213758i
$768$		0			0
$769$			38.5758i			1.39108i	0.718489	+	0.695539i	$0.244834\pi$
$769$			38.5758i			1.39108i	−0.718489	+	0.695539i	$0.755166\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.8287	−	22.8287i	−0.821090	−	0.821090i	0.165174	−	0.986264i	$-0.447181\pi$
$773$	−22.8287	−	22.8287i	−0.821090	−	0.821090i	−0.986264	+	0.165174i	$0.947181\pi$
$774$		0			0
$775$	−33.4675	−	31.1629i	−1.20219	−	1.11940i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.789513			0.0282873
$780$		0			0
$781$	−3.19710			−0.114401
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.7957	+	13.8391i	−1.13484	+	0.493937i
$786$		0			0
$787$	19.3471	+	19.3471i	0.689649	+	0.689649i	0.962154	−	0.272505i	$-0.0878523\pi$
$787$	19.3471	+	19.3471i	0.689649	+	0.689649i	−0.272505	+	0.962154i	$0.587852\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			8.00927i			0.284777i
$792$		0			0
$793$	0.476697	−	0.476697i	0.0169280	−	0.0169280i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.51794	+	6.51794i	−0.230877	+	0.230877i	−0.813059	−	0.582181i	$-0.802199\pi$
$797$	−6.51794	+	6.51794i	−0.230877	+	0.230877i	0.582181	+	0.813059i	$0.302199\pi$
$798$		0			0
$799$			17.2530i			0.610367i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.39225	+	8.39225i	0.296156	+	0.296156i
$804$		0			0
$805$	−1.83508	−	0.722075i	−0.0646779	−	0.0254498i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.3554			−1.24303			−0.621514	−	0.783403i	$-0.713482\pi$
$809$	−35.3554			−1.24303			−0.621514	+	0.783403i	$0.713482\pi$
$810$		0			0
$811$	−56.6641			−1.98974			−0.994872	−	0.101139i	$-0.967751\pi$
$811$	−56.6641			−1.98974			−0.994872	+	0.101139i	$0.967751\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.62003	+	19.8048i	0.301946	+	0.693731i
$816$		0			0
$817$	−4.41383	−	4.41383i	−0.154420	−	0.154420i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.4359i		−	0.713217i	−0.934254	−	0.356608i	$-0.883933\pi$
$821$		−	20.4359i		−	0.713217i	0.934254	−	0.356608i	$-0.116067\pi$
$822$		0			0
$823$	−33.2544	+	33.2544i	−1.15917	+	1.15917i	−0.174520	+	0.984654i	$0.555837\pi$
$823$	−33.2544	+	33.2544i	−1.15917	+	1.15917i	−0.984654	+	0.174520i	$0.944163\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.4208	−	31.4208i	1.09261	−	1.09261i	0.0973572	−	0.995250i	$-0.468961\pi$
$827$	31.4208	−	31.4208i	1.09261	−	1.09261i	0.995250	−	0.0973572i	$-0.0310389\pi$
$828$		0			0
$829$		−	12.7478i		−	0.442748i	−0.975189	−	0.221374i	$-0.928946\pi$
$829$		−	12.7478i		−	0.442748i	0.975189	−	0.221374i	$-0.0710542\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−27.7470	−	27.7470i	−0.961377	−	0.961377i
$834$		0			0
$835$	−5.77099	−	13.2590i	−0.199713	−	0.458848i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−54.2611			−1.87330			−0.936650	−	0.350266i	$-0.886091\pi$
$839$	−54.2611			−1.87330			−0.936650	+	0.350266i	$0.886091\pi$
$840$		0			0
$841$	−18.7150			−0.645346
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.8693	−	7.81827i	−0.683523	−	0.268956i
$846$		0			0
$847$	6.06009	+	6.06009i	0.208227	+	0.208227i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	0.379877i		−	0.0130220i
$852$		0			0
$853$	23.8462	−	23.8462i	0.816478	−	0.816478i	−0.169118	−	0.985596i	$-0.554092\pi$
$853$	23.8462	−	23.8462i	0.816478	−	0.816478i	0.985596	+	0.169118i	$0.0540918\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.1524	−	34.1524i	1.16662	−	1.16662i	0.183627	−	0.982996i	$-0.441216\pi$
$857$	34.1524	−	34.1524i	1.16662	−	1.16662i	0.982996	−	0.183627i	$-0.0587837\pi$
$858$		0			0
$859$		−	46.8576i		−	1.59876i	−0.600825	−	0.799381i	$-0.705161\pi$
$859$		−	46.8576i		−	1.59876i	0.600825	−	0.799381i	$-0.294839\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.70291	+	7.70291i	0.262210	+	0.262210i	0.825951	−	0.563741i	$-0.190638\pi$
$863$	7.70291	+	7.70291i	0.262210	+	0.262210i	−0.563741	+	0.825951i	$0.690638\pi$
$864$		0			0
$865$	−40.1540	+	17.4770i	−1.36528	+	0.594236i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.3190			0.350048
$870$		0			0
$871$	−13.0370			−0.441743
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.29820	+	3.28112i	0.314337	+	0.110922i
$876$		0			0
$877$	22.1911	+	22.1911i	0.749339	+	0.749339i	0.974355	−	0.225016i	$-0.0722435\pi$
$877$	22.1911	+	22.1911i	0.749339	+	0.749339i	−0.225016	+	0.974355i	$0.572244\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	54.1051i		−	1.82285i	−0.411469	−	0.911424i	$-0.634984\pi$
$881$		−	54.1051i		−	1.82285i	0.411469	−	0.911424i	$-0.365016\pi$
$882$		0			0
$883$	−1.43068	+	1.43068i	−0.0481461	+	0.0481461i	−0.730770	−	0.682624i	$-0.760839\pi$
$883$	−1.43068	+	1.43068i	−0.0481461	+	0.0481461i	0.682624	+	0.730770i	$0.260839\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	33.7502	−	33.7502i	1.13322	−	1.13322i	0.143581	−	0.989639i	$-0.454138\pi$
$887$	33.7502	−	33.7502i	1.13322	−	1.13322i	0.989639	−	0.143581i	$-0.0458618\pi$
$888$		0			0
$889$		−	4.94925i		−	0.165993i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.20849	+	1.20849i	0.0404407	+	0.0404407i
$894$		0			0
$895$	−19.9804	+	50.7781i	−0.667873	+	1.69732i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−29.3311			−0.978248
$900$		0			0
$901$	14.5321			0.484135
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.49816	−	16.5144i	0.216006	−	0.548956i
$906$		0			0
$907$	35.8192	+	35.8192i	1.18936	+	1.18936i	0.977245	+	0.212112i	$0.0680343\pi$
$907$	35.8192	+	35.8192i	1.18936	+	1.18936i	0.212112	+	0.977245i	$0.431966\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			32.6416i			1.08147i	0.841194	+	0.540733i	$0.181853\pi$
$911$			32.6416i			1.08147i	−0.841194	+	0.540733i	$0.818147\pi$
$912$		0			0
$913$	−7.39294	+	7.39294i	−0.244670	+	0.244670i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.76656	+	8.76656i	−0.289497	+	0.289497i
$918$		0			0
$919$		−	47.9858i		−	1.58290i	−0.611231	−	0.791452i	$-0.709325\pi$
$919$		−	47.9858i		−	1.58290i	0.611231	−	0.791452i	$-0.290675\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.70879	+	3.70879i	0.122076	+	0.122076i
$924$		0			0
$925$	0.0676845	+	1.89818i	0.00222545	+	0.0624117i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29.4412			−0.965936			−0.482968	−	0.875638i	$-0.660441\pi$
$929$	−29.4412			−0.965936			−0.482968	+	0.875638i	$0.660441\pi$
$930$		0			0
$931$	−3.88710			−0.127395
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.6414	+	6.37268i	−0.478826	+	0.208409i
$936$		0			0
$937$	−13.7806	−	13.7806i	−0.450192	−	0.450192i	0.445226	−	0.895418i	$-0.353123\pi$
$937$	−13.7806	−	13.7806i	−0.450192	−	0.450192i	−0.895418	+	0.445226i	$0.853123\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			30.1956i			0.984349i	0.870497	+	0.492174i	$0.163798\pi$
$941$			30.1956i			0.984349i	−0.870497	+	0.492174i	$0.836202\pi$
$942$		0			0
$943$	−0.893643	+	0.893643i	−0.0291010	+	0.0291010i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.9730	+	11.9730i	−0.389071	+	0.389071i	−0.874356	−	0.485285i	$-0.838716\pi$
$947$	−11.9730	+	11.9730i	−0.389071	+	0.389071i	0.485285	+	0.874356i	$0.338716\pi$
$948$		0			0
$949$		−	19.4708i		−	0.632050i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.5550	−	15.5550i	−0.503876	−	0.503876i	0.408764	−	0.912640i	$-0.365960\pi$
$953$	−15.5550	−	15.5550i	−0.503876	−	0.503876i	−0.912640	+	0.408764i	$0.865960\pi$
$954$		0			0
$955$	24.6445	+	9.69723i	0.797476	+	0.313795i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0884			−0.584103
$960$		0			0
$961$	52.6479			1.69832
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.1763	+	23.3804i	0.327587	+	0.752642i
$966$		0			0
$967$	6.36177	+	6.36177i	0.204581	+	0.204581i	0.801959	−	0.597379i	$-0.203791\pi$
$967$	6.36177	+	6.36177i	0.204581	+	0.204581i	−0.597379	+	0.801959i	$0.703791\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	33.9274i		−	1.08878i	−0.838831	−	0.544391i	$-0.816761\pi$
$971$		−	33.9274i		−	1.08878i	0.838831	−	0.544391i	$-0.183239\pi$
$972$		0			0
$973$	1.35626	−	1.35626i	0.0434798	−	0.0434798i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−35.1738	+	35.1738i	−1.12531	+	1.12531i	−0.134378	+	0.990930i	$0.542904\pi$
$977$	−35.1738	+	35.1738i	−1.12531	+	1.12531i	−0.990930	+	0.134378i	$0.957096\pi$
$978$		0			0
$979$		−	15.5172i		−	0.495932i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.6369	+	28.6369i	0.913374	+	0.913374i	0.996536	−	0.0831623i	$-0.0265020\pi$
$983$	28.6369	+	28.6369i	0.913374	+	0.913374i	−0.0831623	+	0.996536i	$0.526502\pi$
$984$		0			0
$985$	−3.54483	−	8.14437i	−0.112948	−	0.259501i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.99195			0.317726
$990$		0			0
$991$	−36.4794			−1.15881			−0.579404	−	0.815041i	$-0.696715\pi$
$991$	−36.4794			−1.15881			−0.579404	+	0.815041i	$0.696715\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	45.2668	+	17.8118i	1.43505	+	0.564673i
$996$		0			0
$997$	−23.2787	−	23.2787i	−0.737243	−	0.737243i	0.234800	−	0.972044i	$-0.424556\pi$
$997$	−23.2787	−	23.2787i	−0.737243	−	0.737243i	−0.972044	+	0.234800i	$0.924556\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.10	✓	44
3.2	odd	2	inner	4140.2.s.b.737.13	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.13	yes	44
15.8	even	4	inner	4140.2.s.b.2393.10	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.10	✓	44	1.1	even	1	trivial
4140.2.s.b.737.13	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.10	yes	44	15.8	even	4	inner
4140.2.s.b.2393.13	yes	44	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.818756 + 2.08078i) q^{5} +(0.623610 + 0.623610i) q^{7} +O(q^{10})\)	\(q+(-0.818756 + 2.08078i) q^{5} +(0.623610 + 0.623610i) q^{7} +1.13236i q^{11} +(1.31359 - 1.31359i) q^{13} +(4.45934 - 4.45934i) q^{17} -0.624713i q^{19} +(0.707107 + 0.707107i) q^{23} +(-3.65928 - 3.40730i) q^{25} -3.20702 q^{29} +9.14592 q^{31} +(-1.80818 + 0.787010i) q^{35} +(-0.268614 - 0.268614i) q^{37} +1.26380i q^{41} +(7.06538 - 7.06538i) q^{43} +(-1.93448 + 1.93448i) q^{47} -6.22222i q^{49} +(1.62940 + 1.62940i) q^{53} +(-2.35619 - 0.927126i) q^{55} +4.50670 q^{59} +0.362896 q^{61} +(1.65778 + 3.80880i) q^{65} +(-4.96236 - 4.96236i) q^{67} +2.82340i q^{71} +(7.41129 - 7.41129i) q^{73} +(-0.706151 + 0.706151i) q^{77} -9.11283i q^{79} +(6.52879 + 6.52879i) q^{83} +(5.62779 + 12.9300i) q^{85} -13.7034 q^{89} +1.63834 q^{91} +(1.29989 + 0.511487i) q^{95} +(1.67515 + 1.67515i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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