LMFDB - Embedded newform 7406.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +3.00000 q^{18} -4.00000 q^{19} -4.00000 q^{22} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} -3.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +4.00000 q^{47} +1.00000 q^{49} +5.00000 q^{50} -2.00000 q^{52} -12.0000 q^{53} +1.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} -4.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -4.00000 q^{68} +8.00000 q^{71} +3.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} -4.00000 q^{76} -4.00000 q^{77} +8.00000 q^{79} +9.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{86} -4.00000 q^{88} +4.00000 q^{89} +2.00000 q^{91} -4.00000 q^{94} -12.0000 q^{97} -1.00000 q^{98} -12.0000 q^{99} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	3.00000	0.707107
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−5.00000	−1.00000
$26$	2.00000	0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	−3.00000	−0.500000
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.00000	0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−4.00000	−0.508001
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	3.00000	0.353553
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	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
$81$	9.00000	1.00000
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−4.00000	−0.426401
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−1.00000	−0.101015
$99$	−12.0000	−1.20605
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	6.00000	0.554700
$118$	−8.00000	−0.736460
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	−3.00000	−0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	4.00000	0.342997
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	−3.00000	−0.250000
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	−4.00000	−0.328798
$149$	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
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000	0.324443
$153$	12.0000	0.970143
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	24.0000	1.91541	0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−9.00000	−0.707107
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	12.0000	0.917663
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	4.00000	0.301511
$177$	0	0
$178$	−4.00000	−0.299813
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−16.0000	−1.17004
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	12.0000	0.852803
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	5.00000	0.353553
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−16.0000	−1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−4.00000	−0.270914
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	1.00000	0.0668153
$225$	15.0000	1.00000
$226$	16.0000	1.06430
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	−4.00000	−0.259281
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	−4.00000	−0.254000
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	3.00000	0.188982
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	0	0
$275$	−20.0000	−1.20605
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	8.00000	0.479808
$279$	−12.0000	−0.718421
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	8.00000	0.473050
$287$	6.00000	0.354169
$288$	3.00000	0.176777
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−10.0000	−0.585206
$293$	32.0000	1.86946	0.934730	−	0.355359i	$-0.115641\pi$
$293$	32.0000	1.86946	0.934730	+	0.355359i	$0.115641\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	−20.0000	−1.15857
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−12.0000	−0.685994
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	−24.0000	−1.35440
$315$	0	0
$316$	8.00000	0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	9.00000	0.500000
$325$	10.0000	0.554700
$326$	−12.0000	−0.664619
$327$	0	0
$328$	6.00000	0.331295
$329$	−4.00000	−0.220527
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	12.0000	0.658586
$333$	12.0000	0.657596
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	16.0000	0.866449
$342$	−12.0000	−0.648886
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−5.00000	−0.267261
$351$	0	0
$352$	−4.00000	−0.213201
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	0	0
$356$	4.00000	0.212000
$357$	0	0
$358$	12.0000	0.634220
$359$	−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
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−4.00000	−0.206284
$377$	−4.00000	−0.206010
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	8.00000	0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	−12.0000	−0.609994
$388$	−12.0000	−0.609208
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	−12.0000	−0.603023
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−5.00000	−0.250000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	2.00000	0.0992583
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	16.0000	0.788263
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	16.0000	0.782586
$419$	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
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−12.0000	−0.584151
$423$	−12.0000	−0.583460
$424$	12.0000	0.582772
$425$	20.0000	0.970143
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	4.00000	0.191565
$437$	0	0
$438$	0	0
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−8.00000	−0.380521
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	0	0
$446$	12.0000	0.568216
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−15.0000	−0.707107
$451$	−24.0000	−1.13012
$452$	−16.0000	−0.752577
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−26.0000	−1.20443
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	6.00000	0.277350
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	−8.00000	−0.368230
$473$	16.0000	0.735681
$474$	0	0
$475$	20.0000	0.917663
$476$	4.00000	0.183340
$477$	36.0000	1.64833
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−28.0000	−1.27537
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	−8.00000	−0.359937
$495$	0	0
$496$	4.00000	0.179605
$497$	−8.00000	−0.358849
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	20.0000	0.892644
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	16.0000	0.703679
$518$	−4.00000	−0.175750
$519$	0	0
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	6.00000	0.262613
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−24.0000	−1.04151
$532$	4.00000	0.173422
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	−18.0000	−0.776035
$539$	4.00000	0.172292
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	0	0
$550$	20.0000	0.852803
$551$	−8.00000	−0.340811
$552$	0	0
$553$	−8.00000	−0.340195
$554$	22.0000	0.934690
$555$	0	0
$556$	−8.00000	−0.339276
$557$	−20.0000	−0.847427	−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000	−0.847427	−0.423714	+	0.905796i	$0.639274\pi$
$558$	12.0000	0.508001
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	8.00000	0.337460
$563$	−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
$565$	0	0
$566$	−4.00000	−0.168133
$567$	−9.00000	−0.377964
$568$	−8.00000	−0.335673
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	−3.00000	−0.125000
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−48.0000	−1.98796
$584$	10.0000	0.413803
$585$	0	0
$586$	−32.0000	−1.32191
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	4.00000	0.163028
$603$	12.0000	0.488678
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	12.0000	0.485071
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	4.00000	0.161165
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	−28.0000	−1.12270
$623$	−4.00000	−0.160257
$624$	0	0
$625$	25.0000	1.00000
$626$	4.00000	0.159872
$627$	0	0
$628$	24.0000	0.957704
$629$	16.0000	0.637962
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	−8.00000	−0.316723
$639$	−24.0000	−0.949425
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	−9.00000	−0.353553
$649$	32.0000	1.25611
$650$	−10.0000	−0.392232
$651$	0	0
$652$	12.0000	0.469956
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	30.0000	1.17041
$658$	4.00000	0.155936
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−12.0000	−0.464991
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	0	0
$682$	−16.0000	−0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	24.0000	0.914327
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−14.0000	−0.532200
$693$	12.0000	0.455842
$694$	36.0000	1.36654
$695$	0	0
$696$	0	0
$697$	24.0000	0.909065
$698$	30.0000	1.13552
$699$	0	0
$700$	5.00000	0.188982
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	4.00000	0.150756
$705$	0	0
$706$	2.00000	0.0752710
$707$	−6.00000	−0.225653
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	−4.00000	−0.149906
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	8.00000	0.298557
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	3.00000	0.111648
$723$	0	0
$724$	8.00000	0.297318
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−2.00000	−0.0741249
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	−18.0000	−0.662589
$739$	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
$742$	−12.0000	−0.440534
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−36.0000	−1.31717
$748$	−16.0000	−0.585018
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	0	0
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	12.0000	0.431331
$775$	−20.0000	−0.718421
$776$	12.0000	0.430775
$777$	0	0
$778$	12.0000	0.430221
$779$	24.0000	0.859889
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	16.0000	0.568895
$792$	12.0000	0.426401
$793$	0	0
$794$	2.00000	0.0709773
$795$	0	0
$796$	24.0000	0.850657
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	5.00000	0.176777
$801$	−12.0000	−0.423999
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	16.0000	0.560800
$815$	0	0
$816$	0	0
$817$	−16.0000	−0.559769
$818$	26.0000	0.909069
$819$	−6.00000	−0.209657
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	8.00000	0.278356
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	−4.00000	−0.138592
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	28.0000	0.964944
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	12.0000	0.412568
$847$	−5.00000	−0.171802
$848$	−12.0000	−0.412082
$849$	0	0
$850$	−20.0000	−0.685994
$851$	0	0
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	0	0
$865$	0	0
$866$	−20.0000	−0.679628
$867$	0	0
$868$	−4.00000	−0.135769
$869$	32.0000	1.08553
$870$	0	0
$871$	8.00000	0.271070
$872$	−4.00000	−0.135457
$873$	36.0000	1.21842
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−36.0000	−1.21494
$879$	0	0
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	3.00000	0.101015
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	36.0000	1.20605
$892$	−12.0000	−0.401790
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	14.0000	0.467186
$899$	8.00000	0.266815
$900$	15.0000	0.500000
$901$	48.0000	1.59911
$902$	24.0000	0.799113
$903$	0	0
$904$	16.0000	0.532152
$905$	0	0
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	20.0000	0.663723
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−8.00000	−0.264327
$917$	0	0
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−16.0000	−0.526646
$924$	0	0
$925$	20.0000	0.657596
$926$	8.00000	0.262896
$927$	−48.0000	−1.57653
$928$	−2.00000	−0.0656532
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	26.0000	0.851658
$933$	0	0
$934$	20.0000	0.654420
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	−20.0000	−0.648886
$951$	0	0
$952$	−4.00000	−0.129641
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	−36.0000	−1.16554
$955$	0	0
$956$	0	0
$957$	0	0
$958$	16.0000	0.516937
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−8.00000	−0.257930
$963$	−36.0000	−1.16008
$964$	28.0000	0.901819
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	8.00000	0.256337
$975$	0	0
$976$	0	0
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−4.00000	−0.127645
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	8.00000	0.254772
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.b.1.1	✓	1
23.22	odd	2		7406.2.a.c.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.b.1.1	✓	1	1.1	even	1	trivial
7406.2.a.c.1.1	yes	1	23.22	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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