LMFDB - Embedded newform 8046.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
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	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +4.00000 q^{19} -2.00000 q^{20} +5.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{40} +9.00000 q^{41} -4.00000 q^{43} +5.00000 q^{46} +6.00000 q^{47} -6.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{56} -1.00000 q^{58} -2.00000 q^{59} -6.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{68} -2.00000 q^{70} +3.00000 q^{71} -6.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} -2.00000 q^{80} +9.00000 q^{82} -6.00000 q^{83} +8.00000 q^{85} -4.00000 q^{86} -1.00000 q^{89} -2.00000 q^{91} +5.00000 q^{92} +6.00000 q^{94} -8.00000 q^{95} +10.0000 q^{97} -6.00000 q^{98} +O(q^{100})$

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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	$0.661206\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−2.00000	−0.316228
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	5.00000	0.737210
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	−2.00000	−0.239046
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	9.00000	0.993884
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	5.00000	0.521286
$93$	0	0
$94$	6.00000	0.618853
$95$	−8.00000	−0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−10.0000	−0.932505
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−2.00000	−0.184115
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	0	0
$124$	−5.00000	−0.449013
$125$	12.0000	1.07331
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	3.00000	0.251754
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	−6.00000	−0.496564
$147$	0	0
$148$	−6.00000	−0.493197
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	−21.0000	−1.67598	−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000	−1.67598	−0.837991	+	0.545684i	$0.816270\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−2.00000	−0.158114
$161$	5.00000	0.394055
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−6.00000	−0.465690
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	8.00000	0.613572
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	5.00000	0.368605
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−14.0000	−0.985037
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	−18.0000	−1.25717
$206$	7.00000	0.487713
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	−1.00000	−0.0686803
$213$	0	0
$214$	16.0000	1.09374
$215$	8.00000	0.545595
$216$	0	0
$217$	−5.00000	−0.339422
$218$	−17.0000	−1.15139
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	4.00000	0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−10.0000	−0.659380
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−2.00000	−0.130189
$237$	0	0
$238$	−4.00000	−0.259281
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−6.00000	−0.384111
$245$	12.0000	0.766652
$246$	0	0
$247$	−8.00000	−0.509028
$248$	−5.00000	−0.317500
$249$	0	0
$250$	12.0000	0.758947
$251$	25.0000	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$251$	25.0000	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.00000	−0.311891	−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000	−0.311891	−0.155946	+	0.987766i	$0.549842\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	4.00000	0.248069
$261$	0	0
$262$	0	0
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	4.00000	0.245256
$267$	0	0
$268$	0	0
$269$	23.0000	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$269$	23.0000	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−21.0000	−1.26866
$275$	0	0
$276$	0	0
$277$	−27.0000	−1.62227	−0.811136	−	0.584857i	$-0.801151\pi$
$277$	−27.0000	−1.62227	−0.811136	+	0.584857i	$0.801151\pi$
$278$	−9.00000	−0.539784
$279$	0	0
$280$	−2.00000	−0.119523
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−3.00000	−0.178331	−0.0891657	−	0.996017i	$-0.528420\pi$
$283$	−3.00000	−0.178331	−0.0891657	+	0.996017i	$0.528420\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	2.00000	0.117444
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−6.00000	−0.348743
$297$	0	0
$298$	1.00000	0.0579284
$299$	−10.0000	−0.578315
$300$	0	0
$301$	−4.00000	−0.230556
$302$	20.0000	1.15087
$303$	0	0
$304$	4.00000	0.229416
$305$	12.0000	0.687118
$306$	0	0
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	0	0
$309$	0	0
$310$	10.0000	0.567962
$311$	−31.0000	−1.75785	−0.878924	−	0.476961i	$-0.841738\pi$
$311$	−31.0000	−1.75785	−0.878924	+	0.476961i	$0.841738\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−21.0000	−1.18510
$315$	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
$319$	0	0
$320$	−2.00000	−0.111803
$321$	0	0
$322$	5.00000	0.278639
$323$	−16.0000	−0.890264
$324$	0	0
$325$	2.00000	0.110940
$326$	−11.0000	−0.609234
$327$	0	0
$328$	9.00000	0.496942
$329$	6.00000	0.330791
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	8.00000	0.433861
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−13.0000	−0.698884
$347$	5.00000	0.268414	0.134207	−	0.990953i	$-0.457151\pi$
$347$	5.00000	0.268414	0.134207	+	0.990953i	$0.457151\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	3.00000	0.158555
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−5.00000	−0.262794
$363$	0	0
$364$	−2.00000	−0.104828
$365$	12.0000	0.628109
$366$	0	0
$367$	29.0000	1.51379	0.756894	−	0.653538i	$-0.226716\pi$
$367$	29.0000	1.51379	0.756894	+	0.653538i	$0.226716\pi$
$368$	5.00000	0.260643
$369$	0	0
$370$	12.0000	0.623850
$371$	−1.00000	−0.0519174
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	2.00000	0.103005
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−10.0000	−0.511645
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	10.0000	0.507673
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−24.0000	−1.20910
$395$	16.0000	0.805047
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	0	0
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	−18.0000	−0.888957
$411$	0	0
$412$	7.00000	0.344865
$413$	−2.00000	−0.0984136
$414$	0	0
$415$	12.0000	0.589057
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	19.0000	0.924906
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	4.00000	0.194029
$426$	0	0
$427$	−6.00000	−0.290360
$428$	16.0000	0.773389
$429$	0	0
$430$	8.00000	0.385794
$431$	−26.0000	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$431$	−26.0000	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	−17.0000	−0.814152
$437$	20.0000	0.956730
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	−14.0000	−0.662919
$447$	0	0
$448$	1.00000	0.0472456
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	0	0
$452$	4.00000	0.188144
$453$	0	0
$454$	−18.0000	−0.844782
$455$	4.00000	0.187523
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	−10.0000	−0.466252
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−17.0000	−0.787510
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	0	0
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−2.00000	−0.0920575
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−4.00000	−0.183340
$477$	0	0
$478$	9.00000	0.411650
$479$	−13.0000	−0.593985	−0.296993	−	0.954880i	$-0.595984\pi$
$479$	−13.0000	−0.593985	−0.296993	+	0.954880i	$0.595984\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−20.0000	−0.908153
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	12.0000	0.542105
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−5.00000	−0.224507
$497$	3.00000	0.134568
$498$	0	0
$499$	−9.00000	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$499$	−9.00000	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	25.0000	1.11580
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.00000	−0.220541
$515$	−14.0000	−0.616914
$516$	0	0
$517$	0	0
$518$	−6.00000	−0.263625
$519$	0	0
$520$	4.00000	0.175412
$521$	−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
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	−30.0000	−1.30806
$527$	20.0000	0.871214
$528$	0	0
$529$	2.00000	0.0869565
$530$	2.00000	0.0868744
$531$	0	0
$532$	4.00000	0.173422
$533$	−18.0000	−0.779667
$534$	0	0
$535$	−32.0000	−1.38348
$536$	0	0
$537$	0	0
$538$	23.0000	0.991600
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	−4.00000	−0.171499
$545$	34.0000	1.45640
$546$	0	0
$547$	−18.0000	−0.769624	−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000	−0.769624	−0.384812	+	0.922995i	$0.625734\pi$
$548$	−21.0000	−0.897076
$549$	0	0
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−27.0000	−1.14712
$555$	0	0
$556$	−9.00000	−0.381685
$557$	13.0000	0.550828	0.275414	−	0.961326i	$-0.411185\pi$
$557$	13.0000	0.550828	0.275414	+	0.961326i	$0.411185\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	2.00000	0.0843649
$563$	13.0000	0.547885	0.273942	−	0.961746i	$-0.411672\pi$
$563$	13.0000	0.547885	0.273942	+	0.961746i	$0.411672\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	−3.00000	−0.126099
$567$	0	0
$568$	3.00000	0.125877
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	9.00000	0.375653
$575$	−5.00000	−0.208514
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	2.00000	0.0830455
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−7.00000	−0.289167
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	4.00000	0.164677
$591$	0	0
$592$	−6.00000	−0.246598
$593$	17.0000	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$593$	17.0000	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	1.00000	0.0409616
$597$	0	0
$598$	−10.0000	−0.408930
$599$	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
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	20.0000	0.813788
$605$	22.0000	0.894427
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	12.0000	0.485866
$611$	−12.0000	−0.485468
$612$	0	0
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	6.00000	0.242140
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	−31.0000	−1.24299
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	−19.0000	−0.760000
$626$	20.0000	0.799361
$627$	0	0
$628$	−21.0000	−0.837991
$629$	24.0000	0.956943
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	2.00000	0.0794301
$635$	16.0000	0.634941
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−31.0000	−1.21874	−0.609368	−	0.792888i	$-0.708577\pi$
$647$	−31.0000	−1.21874	−0.609368	+	0.792888i	$0.708577\pi$
$648$	0	0
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	−11.0000	−0.430793
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	0	0
$656$	9.00000	0.351391
$657$	0	0
$658$	6.00000	0.233904
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−8.00000	−0.310227
$666$	0	0
$667$	−5.00000	−0.193601
$668$	16.0000	0.619059
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	31.0000	1.19408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	8.00000	0.306786
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	42.0000	1.60474
$686$	−13.0000	−0.496342
$687$	0	0
$688$	−4.00000	−0.152499
$689$	2.00000	0.0761939
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	−13.0000	−0.494186
$693$	0	0
$694$	5.00000	0.189797
$695$	18.0000	0.682779
$696$	0	0
$697$	−36.0000	−1.36360
$698$	19.0000	0.719161
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	10.0000	0.376355
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	−25.0000	−0.936257
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	0	0
$718$	−16.0000	−0.597115
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−5.00000	−0.185824
$725$	1.00000	0.0371391
$726$	0	0
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	12.0000	0.444140
$731$	16.0000	0.591781
$732$	0	0
$733$	−17.0000	−0.627909	−0.313955	−	0.949438i	$-0.601654\pi$
$733$	−17.0000	−0.627909	−0.313955	+	0.949438i	$0.601654\pi$
$734$	29.0000	1.07041
$735$	0	0
$736$	5.00000	0.184302
$737$	0	0
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	−1.00000	−0.0367112
$743$	33.0000	1.21065	0.605326	−	0.795977i	$-0.293043\pi$
$743$	33.0000	1.21065	0.605326	+	0.795977i	$0.293043\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	23.0000	0.842090
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	29.0000	1.05823	0.529113	−	0.848552i	$-0.322525\pi$
$751$	29.0000	1.05823	0.529113	+	0.848552i	$0.322525\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	2.00000	0.0728357
$755$	−40.0000	−1.45575
$756$	0	0
$757$	25.0000	0.908640	0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000	0.908640	0.454320	+	0.890838i	$0.349882\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	−8.00000	−0.290191
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−17.0000	−0.615441
$764$	−10.0000	−0.361787
$765$	0	0
$766$	24.0000	0.867155
$767$	4.00000	0.144432
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	19.0000	0.683383	0.341691	−	0.939812i	$-0.389000\pi$
$773$	19.0000	0.683383	0.341691	+	0.939812i	$0.389000\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	10.0000	0.358979
$777$	0	0
$778$	−26.0000	−0.932145
$779$	36.0000	1.28983
$780$	0	0
$781$	0	0
$782$	−20.0000	−0.715199
$783$	0	0
$784$	−6.00000	−0.214286
$785$	42.0000	1.49904
$786$	0	0
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	16.0000	0.569254
$791$	4.00000	0.142224
$792$	0	0
$793$	12.0000	0.426132
$794$	2.00000	0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−24.0000	−0.847469
$803$	0	0
$804$	0	0
$805$	−10.0000	−0.352454
$806$	10.0000	0.352235
$807$	0	0
$808$	−14.0000	−0.492518
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	0	0
$815$	22.0000	0.770626
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−38.0000	−1.32864
$819$	0	0
$820$	−18.0000	−0.628587
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−2.00000	−0.0695889
$827$	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
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	24.0000	0.831551
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	0	0
$838$	23.0000	0.794522
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−26.0000	−0.896019
$843$	0	0
$844$	19.0000	0.654007
$845$	18.0000	0.619219
$846$	0	0
$847$	−11.0000	−0.377964
$848$	−1.00000	−0.0343401
$849$	0	0
$850$	4.00000	0.137199
$851$	−30.0000	−1.02839
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	16.0000	0.546869
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−26.0000	−0.885564
$863$	−34.0000	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$864$	0	0
$865$	26.0000	0.884027
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	−5.00000	−0.169711
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−17.0000	−0.575693
$873$	0	0
$874$	20.0000	0.676510
$875$	12.0000	0.405674
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	27.0000	0.907083
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	2.00000	0.0670402
$891$	0	0
$892$	−14.0000	−0.468755
$893$	24.0000	0.803129
$894$	0	0
$895$	−6.00000	−0.200558
$896$	1.00000	0.0334077
$897$	0	0
$898$	27.0000	0.901002
$899$	5.00000	0.166759
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	4.00000	0.133038
$905$	10.0000	0.332411
$906$	0	0
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	−18.0000	−0.597351
$909$	0	0
$910$	4.00000	0.132599
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	0	0
$913$	0	0
$914$	−34.0000	−1.12462
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	7.00000	0.230909	0.115454	−	0.993313i	$-0.463168\pi$
$919$	7.00000	0.230909	0.115454	+	0.993313i	$0.463168\pi$
$920$	−10.0000	−0.329690
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−6.00000	−0.197492
$924$	0	0
$925$	6.00000	0.197279
$926$	13.0000	0.427207
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	−17.0000	−0.556854
$933$	0	0
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	0	0
$940$	−12.0000	−0.391397
$941$	25.0000	0.814977	0.407488	−	0.913210i	$-0.366405\pi$
$941$	25.0000	0.814977	0.407488	+	0.913210i	$0.366405\pi$
$942$	0	0
$943$	45.0000	1.46540
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	0	0
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−4.00000	−0.129641
$953$	5.00000	0.161966	0.0809829	−	0.996715i	$-0.474194\pi$
$953$	5.00000	0.161966	0.0809829	+	0.996715i	$0.474194\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	9.00000	0.291081
$957$	0	0
$958$	−13.0000	−0.420011
$959$	−21.0000	−0.678125
$960$	0	0
$961$	−6.00000	−0.193548
$962$	12.0000	0.386896
$963$	0	0
$964$	−22.0000	−0.708572
$965$	8.00000	0.257529
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−20.0000	−0.642161
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−9.00000	−0.288527
$974$	32.0000	1.02535
$975$	0	0
$976$	−6.00000	−0.192055
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	0	0
$980$	12.0000	0.383326
$981$	0	0
$982$	−24.0000	−0.765871
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	4.00000	0.127386
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−20.0000	−0.635963
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	3.00000	0.0951542
$995$	−40.0000	−1.26809
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−9.00000	−0.284890
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.b.1.1	yes	1
3.2	odd	2		8046.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.a.1.1	✓	1	3.2	odd	2
8046.2.a.b.1.1	yes	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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