LMFDB - Embedded newform 5520.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5520,2,Mod(1,5520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5520.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,-1,0,-3,0,1,0,2,0,2,0,-1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} +3.00000 q^{31} +2.00000 q^{33} +3.00000 q^{35} +3.00000 q^{37} +2.00000 q^{39} -7.00000 q^{41} -1.00000 q^{45} +2.00000 q^{49} -1.00000 q^{51} -5.00000 q^{53} -2.00000 q^{55} -6.00000 q^{57} -9.00000 q^{59} +12.0000 q^{61} -3.00000 q^{63} -2.00000 q^{65} -3.00000 q^{67} +1.00000 q^{69} -3.00000 q^{71} +2.00000 q^{73} +1.00000 q^{75} -6.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -7.00000 q^{83} +1.00000 q^{85} +3.00000 q^{87} -16.0000 q^{89} -6.00000 q^{91} +3.00000 q^{93} +6.00000 q^{95} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})$

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
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−7.00000	−0.631169
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	18.0000	1.56080
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−9.00000	−0.631676
$204$	0	0
$205$	7.00000	0.488901
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−3.00000	−0.205557
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.00000	−0.610960
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	−7.00000	−0.443607
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	0	0
$267$	−16.0000	−0.979184
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−29.0000	−1.72387	−0.861936	−	0.507018i	$-0.830748\pi$
$283$	−29.0000	−1.72387	−0.861936	+	0.507018i	$0.830748\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	21.0000	1.23959
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.00000	0.402139
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	3.00000	0.163908
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	19.0000	1.03194
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	3.00000	0.159223
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	0	0
$369$	−7.00000	−0.364405
$370$	0	0
$371$	15.0000	0.778761
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	18.0000	0.901127
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	27.0000	1.32858
$414$	0	0
$415$	7.00000	0.343616
$416$	0	0
$417$	−15.0000	−0.734553
$418$	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
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−36.0000	−1.74216
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	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
$445$	16.0000	0.758473
$446$	0	0
$447$	20.0000	0.945968
$448$	0	0
$449$	11.0000	0.519122	0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000	0.519122	0.259561	+	0.965727i	$0.416422\pi$
$450$	0	0
$451$	−14.0000	−0.659234
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	−7.00000	−0.323921	−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000	−0.323921	−0.161961	+	0.986797i	$0.551782\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	−7.00000	−0.322543
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−5.00000	−0.228934
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	13.0000	0.586682	0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000	0.586682	0.293341	+	0.956008i	$0.405233\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	9.00000	0.403705
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	−7.00000	−0.311496
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.00000	−0.390567
$532$	0	0
$533$	−14.0000	−0.606407
$534$	0	0
$535$	15.0000	0.648507
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	12.0000	0.512148
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	0	0
$565$	−19.0000	−0.799336
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	21.0000	0.871227
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	10.0000	0.412744	0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000	0.412744	0.206372	+	0.978474i	$0.433834\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−10.0000	−0.409273
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	3.00000	0.122373	0.0611863	−	0.998126i	$-0.480512\pi$
$601$	3.00000	0.122373	0.0611863	+	0.998126i	$0.480512\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	7.00000	0.282267
$616$	0	0
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	48.0000	1.92308
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	0	0
$651$	−9.00000	−0.352738
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−18.0000	−0.698010
$666$	0	0
$667$	3.00000	0.116160
$668$	0	0
$669$	−22.0000	−0.850569
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	18.0000	0.690777
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	15.0000	0.568982
$696$	0	0
$697$	7.00000	0.265144
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.0000	−0.789786
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−5.00000	−0.186728
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	31.0000	1.14973	0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000	1.14973	0.574863	+	0.818250i	$0.305055\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	19.0000	0.701781	0.350891	−	0.936416i	$-0.385879\pi$
$733$	19.0000	0.701781	0.350891	+	0.936416i	$0.385879\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	−7.00000	−0.256117
$748$	0	0
$749$	45.0000	1.64426
$750$	0	0
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	−19.0000	−0.690567	−0.345283	−	0.938498i	$-0.612217\pi$
$757$	−19.0000	−0.690567	−0.345283	+	0.938498i	$0.612217\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	42.0000	1.52050
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−18.0000	−0.649942
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	−9.00000	−0.322873
$778$	0	0
$779$	42.0000	1.50481
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	0	0
$789$	−1.00000	−0.0356009
$790$	0	0
$791$	−57.0000	−2.02669
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	5.00000	0.177332
$796$	0	0
$797$	−35.0000	−1.23976	−0.619882	−	0.784695i	$-0.712819\pi$
$797$	−35.0000	−1.23976	−0.619882	+	0.784695i	$0.712819\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	4.00000	0.141157
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	31.0000	1.08990	0.544951	−	0.838468i	$-0.316548\pi$
$809$	31.0000	1.08990	0.544951	+	0.838468i	$0.316548\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	0	0
$813$	5.00000	0.175358
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	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
$825$	2.00000	0.0696311
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−30.0000	−1.03325
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	−29.0000	−0.995277
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−25.0000	−0.852989	−0.426494	−	0.904490i	$-0.640252\pi$
$859$	−25.0000	−0.852989	−0.426494	+	0.904490i	$0.640252\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	0	0
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−6.00000	−0.203302
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	1.00000	0.0337292
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	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
$885$	9.00000	0.302532
$886$	0	0
$887$	34.0000	1.14161	0.570804	−	0.821086i	$-0.306632\pi$
$887$	34.0000	1.14161	0.570804	+	0.821086i	$0.306632\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	5.00000	0.166574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	7.00000	0.232175
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	0	0
$915$	−12.0000	−0.396708
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	−6.00000	−0.197492
$924$	0	0
$925$	3.00000	0.0986394
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	51.0000	1.67326	0.836628	−	0.547772i	$-0.184524\pi$
$929$	51.0000	1.67326	0.836628	+	0.547772i	$0.184524\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	−7.00000	−0.227951
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	0	0
$957$	6.00000	0.193952
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−15.0000	−0.483368
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	45.0000	1.44263
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	−51.0000	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−51.0000	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	31.0000	0.984747	0.492374	−	0.870384i	$-0.336129\pi$
$991$	31.0000	0.984747	0.492374	+	0.870384i	$0.336129\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.t.1.1		1
4.3	odd	2		2760.2.a.b.1.1	✓	1
12.11	even	2		8280.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.b.1.1	✓	1	4.3	odd	2
5520.2.a.t.1.1		1	1.1	even	1	trivial
8280.2.a.t.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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