LMFDB - Embedded newform 7440.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

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:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} -6.00000 q^{17} -2.00000 q^{21} +1.00000 q^{25} +1.00000 q^{27} +4.00000 q^{29} -1.00000 q^{31} +4.00000 q^{33} -2.00000 q^{35} +4.00000 q^{37} -4.00000 q^{39} -6.00000 q^{41} +8.00000 q^{43} +1.00000 q^{45} +12.0000 q^{47} -3.00000 q^{49} -6.00000 q^{51} -2.00000 q^{53} +4.00000 q^{55} -6.00000 q^{59} +10.0000 q^{61} -2.00000 q^{63} -4.00000 q^{65} +10.0000 q^{67} +14.0000 q^{71} +4.00000 q^{73} +1.00000 q^{75} -8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -6.00000 q^{85} +4.00000 q^{87} -8.00000 q^{89} +8.00000 q^{91} -1.00000 q^{93} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	14.0000	0.959264
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	16.0000	0.793091
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−20.0000	−0.967868
$428$	0	0
$429$	−16.0000	−0.772487
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	14.0000	0.662177
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−28.0000	−1.25597
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	48.0000	2.11104
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	18.0000	0.771035
$546$	0	0
$547$	−46.0000	−1.96682	−0.983409	−	0.181402i	$-0.941936\pi$
$547$	−46.0000	−1.96682	−0.983409	+	0.181402i	$0.941936\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	0	0
$655$	−14.0000	−0.547025
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	−20.0000	−0.746914
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	40.0000	1.47342
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	0	0
$780$	0	0
$781$	56.0000	2.00384
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	0	0
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	−14.0000	−0.480479
$850$	0	0
$851$	0	0
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	26.0000	0.876958
$880$	0	0
$881$	−56.0000	−1.88669	−0.943344	−	0.331816i	$-0.892339\pi$
$881$	−56.0000	−1.88669	−0.943344	+	0.331816i	$0.892339\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−58.0000	−1.92586	−0.962929	−	0.269754i	$-0.913058\pi$
$907$	−58.0000	−1.92586	−0.962929	+	0.269754i	$0.913058\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	10.0000	0.330590
$916$	0	0
$917$	28.0000	0.924641
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−56.0000	−1.84326
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.00000	−0.196431
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−4.00000	−0.130535
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.y.1.1		1
4.3	odd	2		930.2.a.d.1.1	✓	1
12.11	even	2		2790.2.a.t.1.1		1
20.3	even	4		4650.2.d.p.3349.2		2
20.7	even	4		4650.2.d.p.3349.1		2
20.19	odd	2		4650.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.d.1.1	✓	1	4.3	odd	2
2790.2.a.t.1.1		1	12.11	even	2
4650.2.a.bl.1.1		1	20.19	odd	2
4650.2.d.p.3349.1		2	20.7	even	4
4650.2.d.p.3349.2		2	20.3	even	4
7440.2.a.y.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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