LMFDB - Embedded newform 3520.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(1,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, 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("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.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:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} -7.00000 q^{19} -5.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +7.00000 q^{31} +1.00000 q^{33} -5.00000 q^{35} +7.00000 q^{37} -2.00000 q^{39} +6.00000 q^{41} +8.00000 q^{43} -2.00000 q^{45} -6.00000 q^{47} +18.0000 q^{49} +3.00000 q^{51} +3.00000 q^{53} +1.00000 q^{55} -7.00000 q^{57} -6.00000 q^{59} +1.00000 q^{61} +10.0000 q^{63} -2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{69} -3.00000 q^{71} +2.00000 q^{73} +1.00000 q^{75} -5.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +3.00000 q^{85} +3.00000 q^{87} +9.00000 q^{89} +10.0000 q^{91} +7.00000 q^{93} -7.00000 q^{95} -4.00000 q^{97} -2.00000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−5.00000	−0.845154
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	10.0000	1.25988
$64$	0	0
$65$	−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	$0.337479\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$85$	3.00000	0.325396
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	0	0
$93$	7.00000	0.725866
$94$	0	0
$95$	−7.00000	−0.718185
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	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
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−15.0000	−1.37505
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	35.0000	3.03488
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	18.0000	1.48461
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	7.00000	0.562254
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−30.0000	−2.36433
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	14.0000	1.07061
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	7.00000	0.514650
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	25.0000	1.81848
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	−15.0000	−1.05279
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	−3.00000	−0.205557
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−35.0000	−2.37595
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	18.0000	1.14998
$246$	0	0
$247$	14.0000	0.890799
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−35.0000	−2.17479
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	9.00000	0.550791
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	10.0000	0.605228
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	−14.0000	−0.838158
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	0	0
$285$	−7.00000	−0.414644
$286$	0	0
$287$	−30.0000	−1.77084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−40.0000	−2.30556
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	10.0000	0.563436
$316$	0	0
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−21.0000	−1.16847
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	30.0000	1.65395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	7.00000	0.379071
$342$	0	0
$343$	−55.0000	−2.96972
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−15.0000	−0.778761
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	−16.0000	−0.813326
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	35.0000	1.75219
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	30.0000	1.47620
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−5.00000	−0.241967
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	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
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	−42.0000	−2.00913
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	9.00000	0.426641
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	10.0000	0.468807
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	7.00000	0.324617
$466$	0	0
$467$	−39.0000	−1.80470	−0.902352	−	0.430999i	$-0.858161\pi$
$467$	−39.0000	−1.80470	−0.902352	+	0.430999i	$0.858161\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	−30.0000	−1.36505
$484$	0	0
$485$	−4.00000	−0.181631
$486$	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
$489$	5.00000	0.226108
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	35.0000	1.54529
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−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
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	21.0000	0.914774
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−41.0000	−1.76273	−0.881364	−	0.472438i	$-0.843374\pi$
$541$	−41.0000	−1.76273	−0.881364	+	0.472438i	$0.843374\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−21.0000	−0.894630
$552$	0	0
$553$	−50.0000	−2.12622
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	23.0000	0.955847
$580$	0	0
$581$	30.0000	1.24461
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−49.0000	−2.01901
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	49.0000	1.98885	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	49.0000	1.98885	0.994424	+	0.105453i	$0.0336291\pi$
$608$	0	0
$609$	−15.0000	−0.607831
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−48.0000	−1.93241	−0.966204	−	0.257780i	$-0.917009\pi$
$617$	−48.0000	−1.93241	−0.966204	+	0.257780i	$0.917009\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	−45.0000	−1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.00000	−0.279553
$628$	0	0
$629$	21.0000	0.837325
$630$	0	0
$631$	−35.0000	−1.39333	−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000	−1.39333	−0.696664	+	0.717398i	$0.745333\pi$
$632$	0	0
$633$	5.00000	0.198732
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−36.0000	−1.42637
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−35.0000	−1.37176
$652$	0	0
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	35.0000	1.35724
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	1.00000	0.0386046
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	27.0000	1.02123
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	0	0
$703$	−49.0000	−1.84807
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	−30.0000	−1.12827
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	42.0000	1.57291
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	51.0000	1.90198	0.950990	−	0.309223i	$-0.100069\pi$
$719$	51.0000	1.90198	0.950990	+	0.309223i	$0.100069\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	−22.0000	−0.818189
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	14.0000	0.514303
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	70.0000	2.53417
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	0	0
$777$	−35.0000	−1.25562
$778$	0	0
$779$	−42.0000	−1.50481
$780$	0	0
$781$	−3.00000	−0.107348
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	−60.0000	−2.13335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	3.00000	0.106399
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	−30.0000	−1.05736
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	5.00000	0.175142
$816$	0	0
$817$	−56.0000	−1.95919
$818$	0	0
$819$	−20.0000	−0.698857
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	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
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	54.0000	1.87099
$834$	0	0
$835$	−21.0000	−0.726735
$836$	0	0
$837$	−35.0000	−1.20978
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	2.00000	0.0686398
$850$	0	0
$851$	42.0000	1.43974
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	14.0000	0.478790
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−30.0000	−1.02240
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−5.00000	−0.169031
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−80.0000	−2.68311
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	21.0000	0.700389
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−40.0000	−1.33112
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	1.00000	0.0330590
$916$	0	0
$917$	15.0000	0.495344
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	−126.000	−4.12948
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	25.0000	0.813250
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	21.0000	0.680972
$952$	0	0
$953$	−27.0000	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	3.00000	0.0969762
$958$	0	0
$959$	−60.0000	−1.93750
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23.0000	0.740396
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	−21.0000	−0.674617
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	28.0000	0.893971
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.0000	0.954911
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	−11.0000	−0.348723
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	0	0
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.a.z.1.1		1
4.3	odd	2		3520.2.a.l.1.1		1
8.3	odd	2		110.2.a.a.1.1	✓	1
8.5	even	2		880.2.a.c.1.1		1
24.5	odd	2		7920.2.a.s.1.1		1
24.11	even	2		990.2.a.l.1.1		1
40.3	even	4		550.2.b.b.199.2		2
40.13	odd	4		4400.2.b.g.4049.1		2
40.19	odd	2		550.2.a.i.1.1		1
40.27	even	4		550.2.b.b.199.1		2
40.29	even	2		4400.2.a.w.1.1		1
40.37	odd	4		4400.2.b.g.4049.2		2
56.27	even	2		5390.2.a.h.1.1		1
88.21	odd	2		9680.2.a.j.1.1		1
88.43	even	2		1210.2.a.k.1.1		1
120.59	even	2		4950.2.a.a.1.1		1
120.83	odd	4		4950.2.c.a.199.1		2
120.107	odd	4		4950.2.c.a.199.2		2
440.219	even	2		6050.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.a.1.1	✓	1	8.3	odd	2
550.2.a.i.1.1		1	40.19	odd	2
550.2.b.b.199.1		2	40.27	even	4
550.2.b.b.199.2		2	40.3	even	4
880.2.a.c.1.1		1	8.5	even	2
990.2.a.l.1.1		1	24.11	even	2
1210.2.a.k.1.1		1	88.43	even	2
3520.2.a.l.1.1		1	4.3	odd	2
3520.2.a.z.1.1		1	1.1	even	1	trivial
4400.2.a.w.1.1		1	40.29	even	2
4400.2.b.g.4049.1		2	40.13	odd	4
4400.2.b.g.4049.2		2	40.37	odd	4
4950.2.a.a.1.1		1	120.59	even	2
4950.2.c.a.199.1		2	120.83	odd	4
4950.2.c.a.199.2		2	120.107	odd	4
5390.2.a.h.1.1		1	56.27	even	2
6050.2.a.i.1.1		1	440.219	even	2
7920.2.a.s.1.1		1	24.5	odd	2
9680.2.a.j.1.1		1	88.21	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:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.a (trivial)

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