LMFDB - Embedded newform 6930.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} -6.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +8.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +14.0000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} -10.0000 q^{73} -6.00000 q^{74} -8.00000 q^{76} -1.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} +8.00000 q^{92} -12.0000 q^{94} +8.00000 q^{95} +14.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	8.00000	1.17954
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	1.00000	0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−8.00000	−0.917663
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	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
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	8.00000	0.834058
$93$	0	0
$94$	−12.0000	−1.23771
$95$	8.00000	0.820783
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	14.0000	1.35980
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	8.00000	0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	−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
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−6.00000	−0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−8.00000	−0.648886
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	0	0
$172$	4.00000	0.304997
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	14.0000	1.04934
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	8.00000	0.589768
$185$	6.00000	0.441129
$186$	0	0
$187$	2.00000	0.146254
$188$	−12.0000	−0.875190
$189$	0	0
$190$	8.00000	0.580381
$191$	−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
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	10.0000	0.703598
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−2.00000	−0.139686
$206$	12.0000	0.836080
$207$	0	0
$208$	2.00000	0.138675
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	4.00000	0.269069
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−2.00000	−0.129641
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−10.0000	−0.640184
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	−2.00000	−0.124035
$261$	0	0
$262$	8.00000	0.494242
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	8.00000	0.490511
$267$	0	0
$268$	8.00000	0.488678
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−2.00000	−0.120824
$275$	1.00000	0.0603023
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	1.00000	0.0597614
$281$	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
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	2.00000	0.118262
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−6.00000	−0.348743
$297$	0	0
$298$	6.00000	0.347571
$299$	16.0000	0.925304
$300$	0	0
$301$	−4.00000	−0.230556
$302$	−12.0000	−0.690522
$303$	0	0
$304$	−8.00000	−0.458831
$305$	10.0000	0.572598
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	4.00000	0.225018
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−8.00000	−0.445823
$323$	−16.0000	−0.890264
$324$	0	0
$325$	2.00000	0.110940
$326$	−8.00000	−0.443079
$327$	0	0
$328$	2.00000	0.110432
$329$	12.0000	0.661581
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	20.0000	1.05703
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	22.0000	1.15629
$363$	0	0
$364$	−2.00000	−0.104828
$365$	10.0000	0.523424
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	6.00000	0.311925
$371$	−14.0000	−0.726844
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	12.0000	0.618031
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−24.0000	−1.22795
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	22.0000	1.11977
$387$	0	0
$388$	14.0000	0.710742
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	1.00000	0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−4.00000	−0.201262
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	12.0000	0.591198
$413$	4.00000	0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	2.00000	0.0980581
$417$	0	0
$418$	−8.00000	−0.391293
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−28.0000	−1.36302
$423$	0	0
$424$	14.0000	0.679900
$425$	2.00000	0.0970143
$426$	0	0
$427$	10.0000	0.483934
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−64.0000	−3.06154
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	4.00000	0.190261
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	6.00000	0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	2.00000	0.0937614
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−26.0000	−1.21490
$459$	0	0
$460$	−8.00000	−0.373002
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	12.0000	0.553519
$471$	0	0
$472$	−4.00000	−0.184115
$473$	4.00000	0.183920
$474$	0	0
$475$	−8.00000	−0.367065
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	20.0000	0.914779
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−10.0000	−0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	−16.0000	−0.719874
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	4.00000	0.178529
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	8.00000	0.355643
$507$	0	0
$508$	8.00000	0.354943
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.0000	0.441081
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−12.0000	−0.527759
$518$	6.00000	0.263625
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	−14.0000	−0.608121
$531$	0	0
$532$	8.00000	0.346844
$533$	4.00000	0.173259
$534$	0	0
$535$	12.0000	0.518805
$536$	8.00000	0.345547
$537$	0	0
$538$	−14.0000	−0.603583
$539$	1.00000	0.0430730
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	2.00000	0.0857493
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	1.00000	0.0426401
$551$	−48.0000	−2.04487
$552$	0	0
$553$	−4.00000	−0.170097
$554$	10.0000	0.424859
$555$	0	0
$556$	16.0000	0.678551
$557$	−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
$559$	8.00000	0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.00000	0.253095
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	8.00000	0.333623
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−4.00000	−0.165948
$582$	0	0
$583$	14.0000	0.579821
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−26.0000	−1.07405
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−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
$595$	2.00000	0.0819920
$596$	6.00000	0.245770
$597$	0	0
$598$	16.0000	0.654289
$599$	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
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−12.0000	−0.488273
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	10.0000	0.404888
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	16.0000	0.641542
$623$	−14.0000	−0.560898
$624$	0	0
$625$	1.00000	0.0400000
$626$	−18.0000	−0.719425
$627$	0	0
$628$	6.00000	0.239426
$629$	−12.0000	−0.478471
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	22.0000	0.873732
$635$	−8.00000	−0.317470
$636$	0	0
$637$	2.00000	0.0792429
$638$	6.00000	0.237542
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−16.0000	−0.629512
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	2.00000	0.0784465
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	2.00000	0.0780869
$657$	0	0
$658$	12.0000	0.467809
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	4.00000	0.155230
$665$	−8.00000	−0.310227
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−10.0000	−0.386046
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	0	0
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	28.0000	1.06672
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	−28.0000	−1.06287
$695$	−16.0000	−0.606915
$696$	0	0
$697$	4.00000	0.151511
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	1.00000	0.0376889
$705$	0	0
$706$	−22.0000	−0.827981
$707$	−10.0000	−0.376089
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	0	0
$712$	14.0000	0.524672
$713$	0	0
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	20.0000	0.747435
$717$	0	0
$718$	20.0000	0.746393
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	45.0000	1.67473
$723$	0	0
$724$	22.0000	0.817624
$725$	6.00000	0.222834
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	10.0000	0.370117
$731$	8.00000	0.295891
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	8.00000	0.294884
$737$	8.00000	0.294684
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	−14.0000	−0.513956
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	10.0000	0.366126
$747$	0	0
$748$	2.00000	0.0731272
$749$	12.0000	0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	12.0000	0.437014
$755$	12.0000	0.436725
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	8.00000	0.290191
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−24.0000	−0.868290
$765$	0	0
$766$	12.0000	0.433578
$767$	−8.00000	−0.288863
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	22.0000	0.791797
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	18.0000	0.645331
$779$	−16.0000	−0.573259
$780$	0	0
$781$	0	0
$782$	16.0000	0.572159
$783$	0	0
$784$	1.00000	0.0357143
$785$	−6.00000	−0.214149
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−4.00000	−0.142314
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−20.0000	−0.710221
$794$	22.0000	0.780751
$795$	0	0
$796$	16.0000	0.567105
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	14.0000	0.494357
$803$	−10.0000	−0.352892
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	0	0
$808$	10.0000	0.351799
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−6.00000	−0.210300
$815$	8.00000	0.280228
$816$	0	0
$817$	−32.0000	−1.11954
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	4.00000	0.139178
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	2.00000	0.0693375
$833$	2.00000	0.0692959
$834$	0	0
$835$	0	0
$836$	−8.00000	−0.276686
$837$	0	0
$838$	12.0000	0.414533
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	−28.0000	−0.963800
$845$	9.00000	0.309609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	14.0000	0.480762
$849$	0	0
$850$	2.00000	0.0685994
$851$	−48.0000	−1.64542
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	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
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−36.0000	−1.22616
$863$	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
$865$	−22.0000	−0.748022
$866$	−26.0000	−0.883516
$867$	0	0
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	16.0000	0.542139
$872$	2.00000	0.0677285
$873$	0	0
$874$	−64.0000	−2.16483
$875$	1.00000	0.0338062
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	−14.0000	−0.469281
$891$	0	0
$892$	−4.00000	−0.133930
$893$	96.0000	3.21252
$894$	0	0
$895$	−20.0000	−0.668526
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−18.0000	−0.600668
$899$	0	0
$900$	0	0
$901$	28.0000	0.932815
$902$	2.00000	0.0665927
$903$	0	0
$904$	6.00000	0.199557
$905$	−22.0000	−0.731305
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	2.00000	0.0662994
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	6.00000	0.198462
$915$	0	0
$916$	−26.0000	−0.859064
$917$	−8.00000	−0.264183
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	10.0000	0.329332
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−16.0000	−0.525793
$927$	0	0
$928$	6.00000	0.196960
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	10.0000	0.327561
$933$	0	0
$934$	−28.0000	−0.916188
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	12.0000	0.391397
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−4.00000	−0.130189
$945$	0	0
$946$	4.00000	0.130051
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	−8.00000	−0.259554
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	20.0000	0.646846
$957$	0	0
$958$	8.00000	0.258468
$959$	2.00000	0.0645834
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−22.0000	−0.708205
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−14.0000	−0.449513
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	14.0000	0.447442
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	28.0000	0.893516
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	12.0000	0.382158
$987$	0	0
$988$	−16.0000	−0.509028
$989$	32.0000	1.01754
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.w.1.1		1
3.2	odd	2		2310.2.a.k.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.k.1.1	✓	1	3.2	odd	2
6930.2.a.w.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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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