LMFDB - Embedded newform 8100.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1207701504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 14x^{4} + 43x^{2} - 36$ x^6 - 14x^4 + 43x^2 - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.17695$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.28835 q^{7} -5.09303 q^{11} -3.56951 q^{13} +0.895796 q^{17} -5.34015 q^{19} -5.06555 q^{23} +3.00000 q^{29} -6.59877 q^{31} +7.24970 q^{37} +7.84590 q^{41} -10.9299 q^{43} +2.96925 q^{47} -5.34015 q^{49} +4.78369 q^{53} +5.75287 q^{59} +4.34015 q^{61} +8.53805 q^{67} +5.34015 q^{71} +9.34600 q^{73} -6.56160 q^{77} -0.741379 q^{79} +9.21247 q^{83} +9.24713 q^{89} -4.59877 q^{91} +3.45882 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 2 q^{11} + 18 q^{29} - 6 q^{31} + 14 q^{41} + 34 q^{59} - 6 q^{61} - 6 q^{79} + 56 q^{89} + 6 q^{91}+O(q^{100})$ 6 * q + 2 * q^11 + 18 * q^29 - 6 * q^31 + 14 * q^41 + 34 * q^59 - 6 * q^61 - 6 * q^79 + 56 * q^89 + 6 * q^91

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.28835	0.486951	0.243475	−	0.969907i	$-0.421713\pi$
$7$	1.28835	0.486951	0.243475	+	0.969907i	$0.421713\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.09303	−1.53561	−0.767803	−	0.640686i	$-0.778650\pi$
$11$	−5.09303	−1.53561	−0.767803	+	0.640686i	$0.778650\pi$
$12$	0	0
$13$	−3.56951	−0.990003	−0.495002	−	0.868892i	$-0.664833\pi$
$13$	−3.56951	−0.990003	−0.495002	+	0.868892i	$0.664833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.895796	0.217262	0.108631	−	0.994082i	$-0.465353\pi$
$17$	0.895796	0.217262	0.108631	+	0.994082i	$0.465353\pi$
$18$	0	0
$19$	−5.34015	−1.22512	−0.612558	−	0.790426i	$-0.709859\pi$
$19$	−5.34015	−1.22512	−0.612558	+	0.790426i	$0.709859\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.06555	−1.05624	−0.528121	−	0.849169i	$-0.677103\pi$
$23$	−5.06555	−1.05624	−0.528121	+	0.849169i	$0.677103\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	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
$31$	−6.59877	−1.18517	−0.592587	−	0.805506i	$-0.701894\pi$
$31$	−6.59877	−1.18517	−0.592587	+	0.805506i	$0.701894\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.24970	1.19184	0.595922	−	0.803043i	$-0.296787\pi$
$37$	7.24970	1.19184	0.595922	+	0.803043i	$0.296787\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.84590	1.22532	0.612662	−	0.790345i	$-0.290099\pi$
$41$	7.84590	1.22532	0.612662	+	0.790345i	$0.290099\pi$
$42$	0	0
$43$	−10.9299	−1.66679	−0.833397	−	0.552675i	$-0.813607\pi$
$43$	−10.9299	−1.66679	−0.833397	+	0.552675i	$0.813607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.96925	0.433110	0.216555	−	0.976270i	$-0.430518\pi$
$47$	2.96925	0.433110	0.216555	+	0.976270i	$0.430518\pi$
$48$	0	0
$49$	−5.34015	−0.762879
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.78369	0.657090	0.328545	−	0.944488i	$-0.393442\pi$
$53$	4.78369	0.657090	0.328545	+	0.944488i	$0.393442\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.75287	0.748960	0.374480	−	0.927235i	$-0.377821\pi$
$59$	5.75287	0.748960	0.374480	+	0.927235i	$0.377821\pi$
$60$	0	0
$61$	4.34015	0.555700	0.277850	−	0.960625i	$-0.410378\pi$
$61$	4.34015	0.555700	0.277850	+	0.960625i	$0.410378\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.53805	1.04309	0.521544	−	0.853224i	$-0.325356\pi$
$67$	8.53805	1.04309	0.521544	+	0.853224i	$0.325356\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.34015	0.633760	0.316880	−	0.948466i	$-0.397365\pi$
$71$	5.34015	0.633760	0.316880	+	0.948466i	$0.397365\pi$
$72$	0	0
$73$	9.34600	1.09387	0.546933	−	0.837176i	$-0.315795\pi$
$73$	9.34600	1.09387	0.546933	+	0.837176i	$0.315795\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.56160	−0.747764
$78$	0	0
$79$	−0.741379	−0.0834116	−0.0417058	−	0.999130i	$-0.513279\pi$
$79$	−0.741379	−0.0834116	−0.0417058	+	0.999130i	$0.513279\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.21247	1.01120	0.505600	−	0.862768i	$-0.331271\pi$
$83$	9.21247	1.01120	0.505600	+	0.862768i	$0.331271\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.24713	0.980193	0.490097	−	0.871668i	$-0.336961\pi$
$89$	9.24713	0.980193	0.490097	+	0.871668i	$0.336961\pi$
$90$	0	0
$91$	−4.59877	−0.482083
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.45882	0.351190	0.175595	−	0.984462i	$-0.443815\pi$
$97$	3.45882	0.351190	0.175595	+	0.984462i	$0.443815\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.4332	1.63516	0.817581	−	0.575813i	$-0.195314\pi$
$101$	16.4332	1.63516	0.817581	+	0.575813i	$0.195314\pi$
$102$	0	0
$103$	−15.8619	−1.56292	−0.781461	−	0.623954i	$-0.785525\pi$
$103$	−15.8619	−1.56292	−0.781461	+	0.623954i	$0.785525\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.53086	0.921383	0.460691	−	0.887560i	$-0.347601\pi$
$107$	9.53086	0.921383	0.460691	+	0.887560i	$0.347601\pi$
$108$	0	0
$109$	16.2791	1.55925	0.779627	−	0.626245i	$-0.215409\pi$
$109$	16.2791	1.55925	0.779627	+	0.626245i	$0.215409\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.1796	−1.71019	−0.855096	−	0.518469i	$-0.826502\pi$
$113$	−18.1796	−1.71019	−0.855096	+	0.518469i	$0.826502\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.15410	0.105796
$120$	0	0
$121$	14.9389	1.35808
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.9969	1.06455	0.532275	−	0.846571i	$-0.321337\pi$
$127$	11.9969	1.06455	0.532275	+	0.846571i	$0.321337\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.93893	−0.868368	−0.434184	−	0.900824i	$-0.642963\pi$
$131$	−9.93893	−0.868368	−0.434184	+	0.900824i	$0.642963\pi$
$132$	0	0
$133$	−6.87999	−0.596571
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.66581	0.484063	0.242031	−	0.970268i	$-0.422186\pi$
$137$	5.66581	0.484063	0.242031	+	0.970268i	$0.422186\pi$
$138$	0	0
$139$	7.93893	0.673371	0.336686	−	0.941617i	$-0.390694\pi$
$139$	7.93893	0.673371	0.336686	+	0.941617i	$0.390694\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.1796	1.52025
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.8344	1.21528	0.607641	−	0.794212i	$-0.292116\pi$
$149$	14.8344	1.21528	0.607641	+	0.794212i	$0.292116\pi$
$150$	0	0
$151$	−0.598775	−0.0487276	−0.0243638	−	0.999703i	$-0.507756\pi$
$151$	−0.598775	−0.0487276	−0.0243638	+	0.999703i	$0.507756\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.9582	−1.43322	−0.716611	−	0.697473i	$-0.754308\pi$
$157$	−17.9582	−1.43322	−0.716611	+	0.697473i	$0.754308\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.52621	−0.514337
$162$	0	0
$163$	4.67300	0.366018	0.183009	−	0.983111i	$-0.441416\pi$
$163$	4.67300	0.366018	0.183009	+	0.983111i	$0.441416\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.96135	0.461303	0.230652	−	0.973036i	$-0.425914\pi$
$167$	5.96135	0.461303	0.230652	+	0.973036i	$0.425914\pi$
$168$	0	0
$169$	−0.258621	−0.0198939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.81272	0.746048	0.373024	−	0.927822i	$-0.378321\pi$
$173$	9.81272	0.746048	0.373024	+	0.927822i	$0.378321\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.3516	1.22218	0.611090	−	0.791561i	$-0.290731\pi$
$179$	16.3516	1.22218	0.611090	+	0.791561i	$0.290731\pi$
$180$	0	0
$181$	2.93893	0.218449	0.109224	−	0.994017i	$-0.465163\pi$
$181$	2.93893	0.218449	0.109224	+	0.994017i	$0.465163\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.56231	−0.333629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.59877	0.332756	0.166378	−	0.986062i	$-0.446793\pi$
$191$	4.59877	0.332756	0.166378	+	0.986062i	$0.446793\pi$
$192$	0	0
$193$	1.58389	0.114011	0.0570056	−	0.998374i	$-0.481845\pi$
$193$	1.58389	0.114011	0.0570056	+	0.998374i	$0.481845\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.0775	−0.931735	−0.465867	−	0.884855i	$-0.654258\pi$
$197$	−13.0775	−0.931735	−0.465867	+	0.884855i	$0.654258\pi$
$198$	0	0
$199$	−5.34015	−0.378553	−0.189277	−	0.981924i	$-0.560614\pi$
$199$	−5.34015	−0.378553	−0.189277	+	0.981924i	$0.560614\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.86505	0.271273
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.1975	1.88129
$210$	0	0
$211$	4.59877	0.316593	0.158296	−	0.987392i	$-0.449400\pi$
$211$	4.59877	0.316593	0.158296	+	0.987392i	$0.449400\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.50153	−0.577122
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.19755	−0.215090
$222$	0	0
$223$	−8.53805	−0.571750	−0.285875	−	0.958267i	$-0.592284\pi$
$223$	−8.53805	−0.571750	−0.285875	+	0.958267i	$0.592284\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.2838	−1.14717	−0.573583	−	0.819147i	$-0.694447\pi$
$227$	−17.2838	−1.14717	−0.573583	+	0.819147i	$0.694447\pi$
$228$	0	0
$229$	15.6803	1.03618	0.518092	−	0.855325i	$-0.326643\pi$
$229$	15.6803	1.03618	0.518092	+	0.855325i	$0.326643\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.7649	−1.03279	−0.516397	−	0.856349i	$-0.672727\pi$
$233$	−15.7649	−1.03279	−0.516397	+	0.856349i	$0.672727\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.56682	0.101349	0.0506745	−	0.998715i	$-0.483863\pi$
$239$	1.56682	0.101349	0.0506745	+	0.998715i	$0.483863\pi$
$240$	0	0
$241$	−21.0205	−1.35405	−0.677023	−	0.735961i	$-0.736730\pi$
$241$	−21.0205	−1.35405	−0.677023	+	0.735961i	$0.736730\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.0617	1.21287
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.85740	0.622193	0.311097	−	0.950378i	$-0.399304\pi$
$251$	9.85740	0.622193	0.311097	+	0.950378i	$0.399304\pi$
$252$	0	0
$253$	25.7990	1.62197
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.9097	−1.61620	−0.808101	−	0.589044i	$-0.799505\pi$
$257$	−25.9097	−1.61620	−0.808101	+	0.589044i	$0.799505\pi$
$258$	0	0
$259$	9.34015	0.580369
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.6391	−1.21100	−0.605500	−	0.795845i	$-0.707027\pi$
$263$	−19.6391	−1.21100	−0.605500	+	0.795845i	$0.707027\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.279082	−0.0170160	−0.00850798	−	0.999964i	$-0.502708\pi$
$269$	−0.279082	−0.0170160	−0.00850798	+	0.999964i	$0.502708\pi$
$270$	0	0
$271$	9.34015	0.567374	0.283687	−	0.958917i	$-0.408442\pi$
$271$	9.34015	0.567374	0.283687	+	0.958917i	$0.408442\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.0689	−1.08566	−0.542828	−	0.839844i	$-0.682646\pi$
$277$	−18.0689	−1.08566	−0.542828	+	0.839844i	$0.682646\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.02046	−0.418806	−0.209403	−	0.977829i	$-0.567152\pi$
$281$	−7.02046	−0.418806	−0.209403	+	0.977829i	$0.567152\pi$
$282$	0	0
$283$	18.3645	1.09165	0.545827	−	0.837898i	$-0.316216\pi$
$283$	18.3645	1.09165	0.545827	+	0.837898i	$0.316216\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.1083	0.596672
$288$	0	0
$289$	−16.1975	−0.952797
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.7085	−0.625598	−0.312799	−	0.949819i	$-0.601267\pi$
$293$	−10.7085	−0.625598	−0.312799	+	0.949819i	$0.601267\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.0815	1.04568
$300$	0	0
$301$	−14.0815	−0.811646
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.3145	−0.816974	−0.408487	−	0.912764i	$-0.633944\pi$
$307$	−14.3145	−0.816974	−0.408487	+	0.912764i	$0.633944\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.4766	1.38794	0.693971	−	0.720003i	$-0.255859\pi$
$311$	24.4766	1.38794	0.693971	+	0.720003i	$0.255859\pi$
$312$	0	0
$313$	31.9772	1.80746	0.903730	−	0.428103i	$-0.140818\pi$
$313$	31.9772	1.80746	0.903730	+	0.428103i	$0.140818\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.2144	1.47235	0.736174	−	0.676792i	$-0.236631\pi$
$317$	26.2144	1.47235	0.736174	+	0.676792i	$0.236631\pi$
$318$	0	0
$319$	−15.2791	−0.855464
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.78369	−0.266172
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.82544	0.210903
$330$	0	0
$331$	−2.74138	−0.150680	−0.0753399	−	0.997158i	$-0.524004\pi$
$331$	−2.74138	−0.150680	−0.0753399	+	0.997158i	$0.524004\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.6790	−1.78014	−0.890069	−	0.455825i	$-0.849344\pi$
$337$	−32.6790	−1.78014	−0.890069	+	0.455825i	$0.849344\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.6077	1.81996
$342$	0	0
$343$	−15.8984	−0.858435
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.1369	0.705225	0.352612	−	0.935769i	$-0.385293\pi$
$347$	13.1369	0.705225	0.352612	+	0.935769i	$0.385293\pi$
$348$	0	0
$349$	−14.1975	−0.759977	−0.379989	−	0.924991i	$-0.624072\pi$
$349$	−14.1975	−0.759977	−0.379989	+	0.924991i	$0.624072\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.10207	0.271556	0.135778	−	0.990739i	$-0.456647\pi$
$353$	5.10207	0.271556	0.135778	+	0.990739i	$0.456647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.8229	0.676767	0.338384	−	0.941008i	$-0.390120\pi$
$359$	12.8229	0.676767	0.338384	+	0.941008i	$0.390120\pi$
$360$	0	0
$361$	9.51724	0.500907
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.77649	0.301531	0.150765	−	0.988570i	$-0.451826\pi$
$367$	5.77649	0.301531	0.150765	+	0.988570i	$0.451826\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.16307	0.319970
$372$	0	0
$373$	12.8048	0.663008	0.331504	−	0.943454i	$-0.392444\pi$
$373$	12.8048	0.663008	0.331504	+	0.943454i	$0.392444\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.7085	−0.551517
$378$	0	0
$379$	−7.19755	−0.369713	−0.184857	−	0.982765i	$-0.559182\pi$
$379$	−7.19755	−0.369713	−0.184857	+	0.982765i	$0.559182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.41469	0.123385	0.0616923	−	0.998095i	$-0.480350\pi$
$383$	2.41469	0.123385	0.0616923	+	0.998095i	$0.480350\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.6484	0.539893	0.269946	−	0.962875i	$-0.412994\pi$
$389$	10.6484	0.539893	0.269946	+	0.962875i	$0.412994\pi$
$390$	0	0
$391$	−4.53770	−0.229482
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.3068	−0.517284	−0.258642	−	0.965973i	$-0.583275\pi$
$397$	−10.3068	−0.517284	−0.258642	+	0.965973i	$0.583275\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.7529	0.886536	0.443268	−	0.896389i	$-0.353819\pi$
$401$	17.7529	0.886536	0.443268	+	0.896389i	$0.353819\pi$
$402$	0	0
$403$	23.5544	1.17333
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−36.9229	−1.83020
$408$	0	0
$409$	−23.1365	−1.14403	−0.572013	−	0.820245i	$-0.693837\pi$
$409$	−23.1365	−1.14403	−0.572013	+	0.820245i	$0.693837\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.41172	0.364707
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28.2676	1.38096	0.690481	−	0.723351i	$-0.257399\pi$
$419$	28.2676	1.38096	0.690481	+	0.723351i	$0.257399\pi$
$420$	0	0
$421$	35.1365	1.71245	0.856224	−	0.516605i	$-0.172805\pi$
$421$	35.1365	1.71245	0.856224	+	0.516605i	$0.172805\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.59164	0.270598
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.8894	−1.77690	−0.888449	−	0.458976i	$-0.848216\pi$
$431$	−36.8894	−1.77690	−0.888449	+	0.458976i	$0.848216\pi$
$432$	0	0
$433$	−11.8120	−0.567649	−0.283825	−	0.958876i	$-0.591603\pi$
$433$	−11.8120	−0.567649	−0.283825	+	0.958876i	$0.591603\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.0508	1.29402
$438$	0	0
$439$	−0.741379	−0.0353841	−0.0176920	−	0.999843i	$-0.505632\pi$
$439$	−0.741379	−0.0353841	−0.0176920	+	0.999843i	$0.505632\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.37026	−0.255149	−0.127574	−	0.991829i	$-0.540719\pi$
$443$	−5.37026	−0.255149	−0.127574	+	0.991829i	$0.540719\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.1975	1.00037	0.500187	−	0.865917i	$-0.333265\pi$
$449$	21.1975	1.00037	0.500187	+	0.865917i	$0.333265\pi$
$450$	0	0
$451$	−39.9594	−1.88161
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.992806	0.0464415	0.0232208	−	0.999730i	$-0.492608\pi$
$457$	0.992806	0.0464415	0.0232208	+	0.999730i	$0.492608\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.3197	0.480636	0.240318	−	0.970694i	$-0.422748\pi$
$461$	10.3197	0.480636	0.240318	+	0.970694i	$0.422748\pi$
$462$	0	0
$463$	13.0638	0.607128	0.303564	−	0.952811i	$-0.401823\pi$
$463$	13.0638	0.607128	0.303564	+	0.952811i	$0.401823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.3938	−0.897437	−0.448718	−	0.893673i	$-0.648119\pi$
$467$	−19.3938	−0.897437	−0.448718	+	0.893673i	$0.648119\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	55.6663	2.55954
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.9159	−0.544453	−0.272227	−	0.962233i	$-0.587760\pi$
$479$	−11.9159	−0.544453	−0.272227	+	0.962233i	$0.587760\pi$
$480$	0	0
$481$	−25.8779	−1.17993
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.8027	−0.852031	−0.426016	−	0.904716i	$-0.640083\pi$
$487$	−18.8027	−0.852031	−0.426016	+	0.904716i	$0.640083\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.93893	−0.448538	−0.224269	−	0.974527i	$-0.571999\pi$
$491$	−9.93893	−0.448538	−0.224269	+	0.974527i	$0.571999\pi$
$492$	0	0
$493$	2.68739	0.121034
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.87999	0.308610
$498$	0	0
$499$	−9.93893	−0.444928	−0.222464	−	0.974941i	$-0.571410\pi$
$499$	−9.93893	−0.444928	−0.222464	+	0.974941i	$0.571410\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−36.0636	−1.60800	−0.803999	−	0.594630i	$-0.797298\pi$
$503$	−36.0636	−1.60800	−0.803999	+	0.594630i	$0.797298\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.8779	1.81188	0.905940	−	0.423407i	$-0.139166\pi$
$509$	40.8779	1.81188	0.905940	+	0.423407i	$0.139166\pi$
$510$	0	0
$511$	12.0409	0.532659
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15.1225	−0.665087
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.9504	0.786422	0.393211	−	0.919448i	$-0.371364\pi$
$521$	17.9504	0.786422	0.393211	+	0.919448i	$0.371364\pi$
$522$	0	0
$523$	−20.9732	−0.917092	−0.458546	−	0.888671i	$-0.651630\pi$
$523$	−20.9732	−0.917092	−0.458546	+	0.888671i	$0.651630\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.91116	−0.257494
$528$	0	0
$529$	2.65985	0.115645
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−28.0060	−1.21307
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	27.1975	1.17148
$540$	0	0
$541$	−13.6192	−0.585537	−0.292768	−	0.956183i	$-0.594576\pi$
$541$	−13.6192	−0.585537	−0.292768	+	0.956183i	$0.594576\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.79708	−0.376136	−0.188068	−	0.982156i	$-0.560223\pi$
$547$	−8.79708	−0.376136	−0.188068	+	0.982156i	$0.560223\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.0205	−0.682495
$552$	0	0
$553$	−0.955156	−0.0406174
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.9308	1.43770	0.718848	−	0.695168i	$-0.244670\pi$
$557$	33.9308	1.43770	0.718848	+	0.695168i	$0.244670\pi$
$558$	0	0
$559$	39.0143	1.65013
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.3626	1.40607	0.703033	−	0.711158i	$-0.251829\pi$
$563$	33.3626	1.40607	0.703033	+	0.711158i	$0.251829\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.63073	−0.319897	−0.159948	−	0.987125i	$-0.551133\pi$
$569$	−7.63073	−0.319897	−0.159948	+	0.987125i	$0.551133\pi$
$570$	0	0
$571$	26.4971	1.10887	0.554434	−	0.832227i	$-0.312935\pi$
$571$	26.4971	1.10887	0.554434	+	0.832227i	$0.312935\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.93709	0.413686	0.206843	−	0.978374i	$-0.433681\pi$
$577$	9.93709	0.413686	0.206843	+	0.978374i	$0.433681\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8689	0.492405
$582$	0	0
$583$	−24.3635	−1.00903
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.5179	0.970686	0.485343	−	0.874324i	$-0.338695\pi$
$587$	23.5179	0.970686	0.485343	+	0.874324i	$0.338695\pi$
$588$	0	0
$589$	35.2385	1.45198
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.6637	−0.478970	−0.239485	−	0.970900i	$-0.576979\pi$
$593$	−11.6637	−0.478970	−0.239485	+	0.970900i	$0.576979\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.06107	−0.0842131	−0.0421065	−	0.999113i	$-0.513407\pi$
$599$	−2.06107	−0.0842131	−0.0421065	+	0.999113i	$0.513407\pi$
$600$	0	0
$601$	−13.9389	−0.568581	−0.284290	−	0.958738i	$-0.591758\pi$
$601$	−13.9389	−0.568581	−0.284290	+	0.958738i	$0.591758\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	47.2526	1.91792	0.958961	−	0.283537i	$-0.0915079\pi$
$607$	47.2526	1.91792	0.958961	+	0.283537i	$0.0915079\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.5988	−0.428781
$612$	0	0
$613$	−31.4648	−1.27085	−0.635426	−	0.772162i	$-0.719175\pi$
$613$	−31.4648	−1.27085	−0.635426	+	0.772162i	$0.719175\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.0987	0.849400	0.424700	−	0.905334i	$-0.360380\pi$
$617$	21.0987	0.849400	0.424700	+	0.905334i	$0.360380\pi$
$618$	0	0
$619$	−3.91847	−0.157496	−0.0787482	−	0.996895i	$-0.525092\pi$
$619$	−3.91847	−0.157496	−0.0787482	+	0.996895i	$0.525092\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.9135	0.477306
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.49425	0.258943
$630$	0	0
$631$	44.5582	1.77383	0.886916	−	0.461930i	$-0.152843\pi$
$631$	44.5582	1.77383	0.886916	+	0.461930i	$0.152843\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.0617	0.755253
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.3287	0.605446	0.302723	−	0.953079i	$-0.402104\pi$
$641$	15.3287	0.605446	0.302723	+	0.953079i	$0.402104\pi$
$642$	0	0
$643$	5.37026	0.211783	0.105891	−	0.994378i	$-0.466230\pi$
$643$	5.37026	0.211783	0.105891	+	0.994378i	$0.466230\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.65087	0.104216	0.0521082	−	0.998641i	$-0.483406\pi$
$647$	2.65087	0.104216	0.0521082	+	0.998641i	$0.483406\pi$
$648$	0	0
$649$	−29.2995	−1.15011
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.71642	−0.301967	−0.150984	−	0.988536i	$-0.548244\pi$
$653$	−7.71642	−0.301967	−0.150984	+	0.988536i	$0.548244\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.7529	0.457827	0.228913	−	0.973447i	$-0.426483\pi$
$659$	11.7529	0.457827	0.228913	+	0.973447i	$0.426483\pi$
$660$	0	0
$661$	10.0611	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$661$	10.0611	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.1967	−0.588417
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.1045	−0.853336
$672$	0	0
$673$	32.1986	1.24116	0.620582	−	0.784141i	$-0.286896\pi$
$673$	32.1986	1.24116	0.620582	+	0.784141i	$0.286896\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.82360	−0.0700866	−0.0350433	−	0.999386i	$-0.511157\pi$
$677$	−1.82360	−0.0700866	−0.0350433	+	0.999386i	$0.511157\pi$
$678$	0	0
$679$	4.45617	0.171012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.5909	1.40011	0.700055	−	0.714089i	$-0.253159\pi$
$683$	36.5909	1.40011	0.700055	+	0.714089i	$0.253159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.0754	−0.650521
$690$	0	0
$691$	7.38076	0.280777	0.140389	−	0.990096i	$-0.455165\pi$
$691$	7.38076	0.280777	0.140389	+	0.990096i	$0.455165\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.02833	0.266217
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.1045	1.17480	0.587401	−	0.809296i	$-0.300151\pi$
$701$	31.1045	1.17480	0.587401	+	0.809296i	$0.300151\pi$
$702$	0	0
$703$	−38.7145	−1.46015
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.1717	0.796243
$708$	0	0
$709$	26.0754	0.979282	0.489641	−	0.871924i	$-0.337128\pi$
$709$	26.0754	0.979282	0.489641	+	0.871924i	$0.337128\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	33.4265	1.25183
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−43.0524	−1.60558	−0.802792	−	0.596259i	$-0.796653\pi$
$719$	−43.0524	−1.60558	−0.802792	+	0.596259i	$0.796653\pi$
$720$	0	0
$721$	−20.4357	−0.761066
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.2825	0.529710	0.264855	−	0.964288i	$-0.414676\pi$
$727$	14.2825	0.529710	0.264855	+	0.964288i	$0.414676\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.79096	−0.362132
$732$	0	0
$733$	3.53186	0.130452	0.0652260	−	0.997871i	$-0.479223\pi$
$733$	3.53186	0.130452	0.0652260	+	0.997871i	$0.479223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.4845	−1.60177
$738$	0	0
$739$	13.3606	0.491478	0.245739	−	0.969336i	$-0.420969\pi$
$739$	13.3606	0.491478	0.245739	+	0.969336i	$0.420969\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.1625	0.776378	0.388189	−	0.921580i	$-0.373101\pi$
$743$	21.1625	0.776378	0.388189	+	0.921580i	$0.373101\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.2791	0.448668
$750$	0	0
$751$	8.08153	0.294899	0.147450	−	0.989070i	$-0.452894\pi$
$751$	8.08153	0.294899	0.147450	+	0.989070i	$0.452894\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.3114	0.956305	0.478152	−	0.878277i	$-0.341307\pi$
$757$	26.3114	0.956305	0.478152	+	0.878277i	$0.341307\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.1631	0.984660	0.492330	−	0.870409i	$-0.336145\pi$
$761$	27.1631	0.984660	0.492330	+	0.870409i	$0.336145\pi$
$762$	0	0
$763$	20.9732	0.759279
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.5349	−0.741473
$768$	0	0
$769$	2.83693	0.102302	0.0511512	−	0.998691i	$-0.483711\pi$
$769$	2.83693	0.102302	0.0511512	+	0.998691i	$0.483711\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−41.8983	−1.50116
$780$	0	0
$781$	−27.1975	−0.973205
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.79088	−0.135130	−0.0675652	−	0.997715i	$-0.521523\pi$
$787$	−3.79088	−0.135130	−0.0675652	+	0.997715i	$0.521523\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.4217	−0.832779
$792$	0	0
$793$	−15.4922	−0.550144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.0756	−1.56124	−0.780619	−	0.625007i	$-0.785096\pi$
$797$	−44.0756	−1.56124	−0.780619	+	0.625007i	$0.785096\pi$
$798$	0	0
$799$	2.65985	0.0940986
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−47.5994	−1.67975
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.48276	0.0521310	0.0260655	−	0.999660i	$-0.491702\pi$
$809$	1.48276	0.0521310	0.0260655	+	0.999660i	$0.491702\pi$
$810$	0	0
$811$	51.2385	1.79923	0.899613	−	0.436688i	$-0.143849\pi$
$811$	51.2385	1.79923	0.899613	+	0.436688i	$0.143849\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	58.3673	2.04201
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.7123	0.792663	0.396332	−	0.918107i	$-0.370283\pi$
$821$	22.7123	0.792663	0.396332	+	0.918107i	$0.370283\pi$
$822$	0	0
$823$	−13.1004	−0.456650	−0.228325	−	0.973585i	$-0.573325\pi$
$823$	−13.1004	−0.456650	−0.228325	+	0.973585i	$0.573325\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.8631	1.21231	0.606155	−	0.795346i	$-0.292711\pi$
$827$	34.8631	1.21231	0.606155	+	0.795346i	$0.292711\pi$
$828$	0	0
$829$	−25.5988	−0.889082	−0.444541	−	0.895758i	$-0.646633\pi$
$829$	−25.5988	−0.889082	−0.444541	+	0.895758i	$0.646633\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.78369	−0.165745
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.3315	1.39240	0.696199	−	0.717849i	$-0.254873\pi$
$839$	40.3315	1.39240	0.696199	+	0.717849i	$0.254873\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.2466	0.661320
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−36.7238	−1.25887
$852$	0	0
$853$	−40.2984	−1.37979	−0.689896	−	0.723909i	$-0.742344\pi$
$853$	−40.2984	−1.37979	−0.689896	+	0.723909i	$0.742344\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.0401	0.752877	0.376438	−	0.926442i	$-0.377149\pi$
$857$	22.0401	0.752877	0.376438	+	0.926442i	$0.377149\pi$
$858$	0	0
$859$	29.2791	0.998989	0.499495	−	0.866317i	$-0.333519\pi$
$859$	29.2791	0.998989	0.499495	+	0.866317i	$0.333519\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.1491	−1.57093	−0.785466	−	0.618904i	$-0.787577\pi$
$863$	−46.1491	−1.57093	−0.785466	+	0.618904i	$0.787577\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.77586	0.128087
$870$	0	0
$871$	−30.4766	−1.03266
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.6363	0.933211	0.466606	−	0.884466i	$-0.345477\pi$
$877$	27.6363	0.933211	0.466606	+	0.884466i	$0.345477\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.79632	−0.161592	−0.0807961	−	0.996731i	$-0.525746\pi$
$881$	−4.79632	−0.161592	−0.0807961	+	0.996731i	$0.525746\pi$
$882$	0	0
$883$	21.7126	0.730687	0.365343	−	0.930873i	$-0.380952\pi$
$883$	21.7126	0.730687	0.365343	+	0.930873i	$0.380952\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.0401	0.740035	0.370018	−	0.929025i	$-0.379352\pi$
$887$	22.0401	0.740035	0.370018	+	0.929025i	$0.379352\pi$
$888$	0	0
$889$	15.4562	0.518383
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.8563	−0.530610
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.7963	−0.660244
$900$	0	0
$901$	4.28521	0.142761
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−39.3011	−1.30497	−0.652486	−	0.757801i	$-0.726274\pi$
$907$	−39.3011	−1.30497	−0.652486	+	0.757801i	$0.726274\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.9389	−0.528080	−0.264040	−	0.964512i	$-0.585055\pi$
$911$	−15.9389	−0.528080	−0.264040	+	0.964512i	$0.585055\pi$
$912$	0	0
$913$	−46.9194	−1.55280
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.8048	−0.422852
$918$	0	0
$919$	−6.16307	−0.203301	−0.101650	−	0.994820i	$-0.532412\pi$
$919$	−6.16307	−0.203301	−0.101650	+	0.994820i	$0.532412\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−19.0617	−0.627424
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.1020	0.331436	0.165718	−	0.986173i	$-0.447006\pi$
$929$	10.1020	0.331436	0.165718	+	0.986173i	$0.447006\pi$
$930$	0	0
$931$	28.5172	0.934615
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.00506	0.163508	0.0817541	−	0.996653i	$-0.473948\pi$
$937$	5.00506	0.163508	0.0817541	+	0.996653i	$0.473948\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.98851	0.325616	0.162808	−	0.986658i	$-0.447945\pi$
$941$	9.98851	0.325616	0.162808	+	0.986658i	$0.447945\pi$
$942$	0	0
$943$	−39.7438	−1.29424
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.6223	−1.54752	−0.773758	−	0.633481i	$-0.781625\pi$
$947$	−47.6223	−1.54752	−0.773758	+	0.633481i	$0.781625\pi$
$948$	0	0
$949$	−33.3606	−1.08293
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34.2355	−1.10900	−0.554499	−	0.832184i	$-0.687090\pi$
$953$	−34.2355	−1.10900	−0.554499	+	0.832184i	$0.687090\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.29954	0.235715
$960$	0	0
$961$	12.5438	0.404640
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.71671	−0.119521	−0.0597607	−	0.998213i	$-0.519034\pi$
$967$	−3.71671	−0.119521	−0.0597607	+	0.998213i	$0.519034\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.0959	1.44720	0.723598	−	0.690222i	$-0.242487\pi$
$971$	45.0959	1.44720	0.723598	+	0.690222i	$0.242487\pi$
$972$	0	0
$973$	10.2281	0.327898
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.58896	0.210799	0.105400	−	0.994430i	$-0.466388\pi$
$977$	6.58896	0.210799	0.105400	+	0.994430i	$0.466388\pi$
$978$	0	0
$979$	−47.0959	−1.50519
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.2319	−0.390138	−0.195069	−	0.980790i	$-0.562493\pi$
$983$	−12.2319	−0.390138	−0.195069	+	0.980790i	$0.562493\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	55.3660	1.76054
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.23744	0.102531	0.0512655	−	0.998685i	$-0.483675\pi$
$997$	3.23744	0.102531	0.0512655	+	0.998685i	$0.483675\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.bd.1.4		6
3.2	odd	2		8100.2.a.bc.1.4		6
5.2	odd	4		1620.2.d.d.649.3		6
5.3	odd	4		1620.2.d.d.649.4		6
5.4	even	2	inner	8100.2.a.bd.1.3		6
9.2	odd	6		900.2.i.f.301.6		12
9.4	even	3		2700.2.i.f.1801.3		12
9.5	odd	6		900.2.i.f.601.6		12
9.7	even	3		2700.2.i.f.901.3		12
15.2	even	4		1620.2.d.c.649.4		6
15.8	even	4		1620.2.d.c.649.3		6
15.14	odd	2		8100.2.a.bc.1.3		6
45.2	even	12		180.2.r.a.49.3	✓	12
45.4	even	6		2700.2.i.f.1801.4		12
45.7	odd	12		540.2.r.a.469.5		12
45.13	odd	12		540.2.r.a.289.5		12
45.14	odd	6		900.2.i.f.601.1		12
45.22	odd	12		540.2.r.a.289.1		12
45.23	even	12		180.2.r.a.169.3	yes	12
45.29	odd	6		900.2.i.f.301.1		12
45.32	even	12		180.2.r.a.169.4	yes	12
45.34	even	6		2700.2.i.f.901.4		12
45.38	even	12		180.2.r.a.49.4	yes	12
45.43	odd	12		540.2.r.a.469.1		12
180.7	even	12		2160.2.by.e.1009.5		12
180.23	odd	12		720.2.by.e.529.4		12
180.43	even	12		2160.2.by.e.1009.1		12
180.47	odd	12		720.2.by.e.49.4		12
180.67	even	12		2160.2.by.e.289.1		12
180.83	odd	12		720.2.by.e.49.3		12
180.103	even	12		2160.2.by.e.289.5		12
180.167	odd	12		720.2.by.e.529.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.3	✓	12	45.2	even	12
180.2.r.a.49.4	yes	12	45.38	even	12
180.2.r.a.169.3	yes	12	45.23	even	12
180.2.r.a.169.4	yes	12	45.32	even	12
540.2.r.a.289.1		12	45.22	odd	12
540.2.r.a.289.5		12	45.13	odd	12
540.2.r.a.469.1		12	45.43	odd	12
540.2.r.a.469.5		12	45.7	odd	12
720.2.by.e.49.3		12	180.83	odd	12
720.2.by.e.49.4		12	180.47	odd	12
720.2.by.e.529.3		12	180.167	odd	12
720.2.by.e.529.4		12	180.23	odd	12
900.2.i.f.301.1		12	45.29	odd	6
900.2.i.f.301.6		12	9.2	odd	6
900.2.i.f.601.1		12	45.14	odd	6
900.2.i.f.601.6		12	9.5	odd	6
1620.2.d.c.649.3		6	15.8	even	4
1620.2.d.c.649.4		6	15.2	even	4
1620.2.d.d.649.3		6	5.2	odd	4
1620.2.d.d.649.4		6	5.3	odd	4
2160.2.by.e.289.1		12	180.67	even	12
2160.2.by.e.289.5		12	180.103	even	12
2160.2.by.e.1009.1		12	180.43	even	12
2160.2.by.e.1009.5		12	180.7	even	12
2700.2.i.f.901.3		12	9.7	even	3
2700.2.i.f.901.4		12	45.34	even	6
2700.2.i.f.1801.3		12	9.4	even	3
2700.2.i.f.1801.4		12	45.4	even	6
8100.2.a.bc.1.3		6	15.14	odd	2
8100.2.a.bc.1.4		6	3.2	odd	2
8100.2.a.bd.1.3		6	5.4	even	2	inner
8100.2.a.bd.1.4		6	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8100.2.a.bd.1.4

Level	$8100$
Weight	$2$
Character	8100.1
Self dual	yes
Analytic conductor	$64.679$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$