LMFDB - Embedded newform 252.12.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.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:	$193.622481501$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 522x + 2520 $ x^3 - x^2 - 522*x + 2520
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-24.5020$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1711.58 q^{5} -16807.0 q^{7} +O(q^{10})$	$q+1711.58 q^{5} -16807.0 q^{7} +1.01255e6 q^{11} +1.21443e6 q^{13} +6.36074e6 q^{17} +1.67527e7 q^{19} +3.73496e7 q^{23} -4.58986e7 q^{25} +6.45735e7 q^{29} -4.71053e7 q^{31} -2.87665e7 q^{35} -2.34605e7 q^{37} +4.92161e8 q^{41} -8.70671e8 q^{43} -2.40360e9 q^{47} +2.82475e8 q^{49} -4.63788e7 q^{53} +1.73306e9 q^{55} -5.70675e9 q^{59} +6.61158e9 q^{61} +2.07860e9 q^{65} +1.35215e10 q^{67} +2.43620e10 q^{71} +1.59448e10 q^{73} -1.70180e10 q^{77} -4.58936e10 q^{79} +1.11643e10 q^{83} +1.08869e10 q^{85} -6.45889e10 q^{89} -2.04110e10 q^{91} +2.86735e10 q^{95} -4.01929e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) $ 3 * q - 4762 * q^5 - 50421 * q^7	$ 3 q - 4762 q^{5} - 50421 q^{7} - 80468 q^{11} + 996546 q^{13} + 7924090 q^{17} - 15261732 q^{19} + 61428352 q^{23} - 109205331 q^{25} + 160359182 q^{29} - 342148464 q^{31} + 80034934 q^{35} - 722039190 q^{37} - 285288606 q^{41} - 476523612 q^{43} - 1225105680 q^{47} + 847425747 q^{49} - 4030571514 q^{53} + 4822100040 q^{55} - 5222175892 q^{59} + 5684837106 q^{61} - 8153014116 q^{65} + 2837154348 q^{67} + 16928678200 q^{71} - 14560325442 q^{73} + 1352425676 q^{77} - 32832340032 q^{79} + 115451875348 q^{83} - 36782181276 q^{85} + 15661530882 q^{89} - 16748948622 q^{91} + 117828203728 q^{95} + 40693412742 q^{97}+O(q^{100}) $ 3 * q - 4762 * q^5 - 50421 * q^7 - 80468 * q^11 + 996546 * q^13 + 7924090 * q^17 - 15261732 * q^19 + 61428352 * q^23 - 109205331 * q^25 + 160359182 * q^29 - 342148464 * q^31 + 80034934 * q^35 - 722039190 * q^37 - 285288606 * q^41 - 476523612 * q^43 - 1225105680 * q^47 + 847425747 * q^49 - 4030571514 * q^53 + 4822100040 * q^55 - 5222175892 * q^59 + 5684837106 * q^61 - 8153014116 * q^65 + 2837154348 * q^67 + 16928678200 * q^71 - 14560325442 * q^73 + 1352425676 * q^77 - 32832340032 * q^79 + 115451875348 * q^83 - 36782181276 * q^85 + 15661530882 * q^89 - 16748948622 * q^91 + 117828203728 * q^95 + 40693412742 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1711.58	0.244941	0.122471	−	0.992472i	$-0.460918\pi$
$5$	1711.58	0.244941	0.122471	+	0.992472i	$0.460918\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.01255e6	1.89565	0.947825	−	0.318791i	$-0.103277\pi$
$11$	1.01255e6	1.89565	0.947825	+	0.318791i	$0.103277\pi$
$12$	0	0
$13$	1.21443e6	0.907164	0.453582	−	0.891215i	$-0.350146\pi$
$13$	1.21443e6	0.907164	0.453582	+	0.891215i	$0.350146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.36074e6	1.08652	0.543261	−	0.839564i	$-0.317189\pi$
$17$	6.36074e6	1.08652	0.543261	+	0.839564i	$0.317189\pi$
$18$	0	0
$19$	1.67527e7	1.55217	0.776085	−	0.630628i	$-0.217203\pi$
$19$	1.67527e7	1.55217	0.776085	+	0.630628i	$0.217203\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.73496e7	1.20999	0.604997	−	0.796228i	$-0.293174\pi$
$23$	3.73496e7	1.20999	0.604997	+	0.796228i	$0.293174\pi$
$24$	0	0
$25$	−4.58986e7	−0.940004
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.45735e7	0.584609	0.292305	−	0.956325i	$-0.405578\pi$
$29$	6.45735e7	0.584609	0.292305	+	0.956325i	$0.405578\pi$
$30$	0	0
$31$	−4.71053e7	−0.295515	−0.147758	−	0.989024i	$-0.547206\pi$
$31$	−4.71053e7	−0.295515	−0.147758	+	0.989024i	$0.547206\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.87665e7	−0.0925790
$36$	0	0
$37$	−2.34605e7	−0.0556195	−0.0278097	−	0.999613i	$-0.508853\pi$
$37$	−2.34605e7	−0.0556195	−0.0278097	+	0.999613i	$0.508853\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.92161e8	0.663431	0.331715	−	0.943380i	$-0.392373\pi$
$41$	4.92161e8	0.663431	0.331715	+	0.943380i	$0.392373\pi$
$42$	0	0
$43$	−8.70671e8	−0.903187	−0.451593	−	0.892224i	$-0.649144\pi$
$43$	−8.70671e8	−0.903187	−0.451593	+	0.892224i	$0.649144\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.40360e9	−1.52871	−0.764354	−	0.644797i	$-0.776942\pi$
$47$	−2.40360e9	−1.52871	−0.764354	+	0.644797i	$0.776942\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.63788e7	−0.0152336	−0.00761680	−	0.999971i	$-0.502425\pi$
$53$	−4.63788e7	−0.0152336	−0.00761680	+	0.999971i	$0.502425\pi$
$54$	0	0
$55$	1.73306e9	0.464323
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.70675e9	−1.03921	−0.519604	−	0.854407i	$-0.673921\pi$
$59$	−5.70675e9	−1.03921	−0.519604	+	0.854407i	$0.673921\pi$
$60$	0	0
$61$	6.61158e9	1.00229	0.501143	−	0.865365i	$-0.332913\pi$
$61$	6.61158e9	1.00229	0.501143	+	0.865365i	$0.332913\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.07860e9	0.222202
$66$	0	0
$67$	1.35215e10	1.22352	0.611762	−	0.791042i	$-0.290461\pi$
$67$	1.35215e10	1.22352	0.611762	+	0.791042i	$0.290461\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.43620e10	1.60248	0.801238	−	0.598346i	$-0.204175\pi$
$71$	2.43620e10	1.60248	0.801238	+	0.598346i	$0.204175\pi$
$72$	0	0
$73$	1.59448e10	0.900211	0.450106	−	0.892975i	$-0.351386\pi$
$73$	1.59448e10	0.900211	0.450106	+	0.892975i	$0.351386\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.70180e10	−0.716488
$78$	0	0
$79$	−4.58936e10	−1.67804	−0.839021	−	0.544100i	$-0.816871\pi$
$79$	−4.58936e10	−1.67804	−0.839021	+	0.544100i	$0.816871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.11643e10	0.311100	0.155550	−	0.987828i	$-0.450285\pi$
$83$	1.11643e10	0.311100	0.155550	+	0.987828i	$0.450285\pi$
$84$	0	0
$85$	1.08869e10	0.266134
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.45889e10	−1.22606	−0.613032	−	0.790058i	$-0.710050\pi$
$89$	−6.45889e10	−1.22606	−0.613032	+	0.790058i	$0.710050\pi$
$90$	0	0
$91$	−2.04110e10	−0.342876
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.86735e10	0.380190
$96$	0	0
$97$	−4.01929e10	−0.475231	−0.237615	−	0.971359i	$-0.576366\pi$
$97$	−4.01929e10	−0.475231	−0.237615	+	0.971359i	$0.576366\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.90909e10	−0.180742	−0.0903708	−	0.995908i	$-0.528805\pi$
$101$	−1.90909e10	−0.180742	−0.0903708	+	0.995908i	$0.528805\pi$
$102$	0	0
$103$	−1.31109e10	−0.111437	−0.0557183	−	0.998447i	$-0.517745\pi$
$103$	−1.31109e10	−0.111437	−0.0557183	+	0.998447i	$0.517745\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.26145e10	−0.224802	−0.112401	−	0.993663i	$-0.535854\pi$
$107$	−3.26145e10	−0.224802	−0.112401	+	0.993663i	$0.535854\pi$
$108$	0	0
$109$	1.49510e11	0.930730	0.465365	−	0.885119i	$-0.345923\pi$
$109$	1.49510e11	0.930730	0.465365	+	0.885119i	$0.345923\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.56886e10	−0.437514	−0.218757	−	0.975779i	$-0.570200\pi$
$113$	−8.56886e10	−0.437514	−0.218757	+	0.975779i	$0.570200\pi$
$114$	0	0
$115$	6.39268e10	0.296377
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.06905e11	−0.410667
$120$	0	0
$121$	7.39953e11	2.59349
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.62132e11	−0.475187
$126$	0	0
$127$	−7.32613e10	−0.196768	−0.0983839	−	0.995149i	$-0.531367\pi$
$127$	−7.32613e10	−0.196768	−0.0983839	+	0.995149i	$0.531367\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.42953e10	0.145609	0.0728043	−	0.997346i	$-0.476805\pi$
$131$	6.42953e10	0.145609	0.0728043	+	0.997346i	$0.476805\pi$
$132$	0	0
$133$	−2.81562e11	−0.586665
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.48141e11	0.439275	0.219637	−	0.975582i	$-0.429513\pi$
$137$	2.48141e11	0.439275	0.219637	+	0.975582i	$0.429513\pi$
$138$	0	0
$139$	−9.94986e11	−1.62643	−0.813216	−	0.581962i	$-0.802285\pi$
$139$	−9.94986e11	−1.62643	−0.813216	+	0.581962i	$0.802285\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.22968e12	1.71967
$144$	0	0
$145$	1.10523e11	0.143195
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.14091e11	0.685028	0.342514	−	0.939513i	$-0.388722\pi$
$149$	6.14091e11	0.685028	0.342514	+	0.939513i	$0.388722\pi$
$150$	0	0
$151$	−1.87982e12	−1.94869	−0.974344	−	0.225065i	$-0.927741\pi$
$151$	−1.87982e12	−1.94869	−0.974344	+	0.225065i	$0.927741\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.06243e10	−0.0723839
$156$	0	0
$157$	1.02673e12	0.859028	0.429514	−	0.903060i	$-0.358685\pi$
$157$	1.02673e12	0.859028	0.429514	+	0.903060i	$0.358685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.27735e11	−0.457335
$162$	0	0
$163$	2.63690e12	1.79499	0.897493	−	0.441029i	$-0.145386\pi$
$163$	2.63690e12	1.79499	0.897493	+	0.441029i	$0.145386\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.62389e12	1.56316	0.781582	−	0.623803i	$-0.214413\pi$
$167$	2.62389e12	1.56316	0.781582	+	0.623803i	$0.214413\pi$
$168$	0	0
$169$	−3.17309e11	−0.177054
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.26755e11	0.258437	0.129219	−	0.991616i	$-0.458753\pi$
$173$	5.26755e11	0.258437	0.129219	+	0.991616i	$0.458753\pi$
$174$	0	0
$175$	7.71418e11	0.355288
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.48559e12	−0.604236	−0.302118	−	0.953270i	$-0.597694\pi$
$179$	−1.48559e12	−0.604236	−0.302118	+	0.953270i	$0.597694\pi$
$180$	0	0
$181$	−3.66034e12	−1.40052	−0.700260	−	0.713887i	$-0.746933\pi$
$181$	−3.66034e12	−1.40052	−0.700260	+	0.713887i	$0.746933\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.01544e10	−0.0136235
$186$	0	0
$187$	6.44059e12	2.05967
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.90442e12	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$191$	3.90442e12	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$192$	0	0
$193$	−4.63719e12	−1.24649	−0.623246	−	0.782026i	$-0.714187\pi$
$193$	−4.63719e12	−1.24649	−0.623246	+	0.782026i	$0.714187\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.58785e12	−1.58190	−0.790951	−	0.611879i	$-0.790414\pi$
$197$	−6.58785e12	−1.58190	−0.790951	+	0.611879i	$0.790414\pi$
$198$	0	0
$199$	5.33064e12	1.21084	0.605421	−	0.795906i	$-0.293005\pi$
$199$	5.33064e12	1.21084	0.605421	+	0.795906i	$0.293005\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.08529e12	−0.220961
$204$	0	0
$205$	8.42371e11	0.162501
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.69630e13	2.94237
$210$	0	0
$211$	−2.06713e12	−0.340263	−0.170132	−	0.985421i	$-0.554419\pi$
$211$	−2.06713e12	−0.340263	−0.170132	+	0.985421i	$0.554419\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.49022e12	−0.221227
$216$	0	0
$217$	7.91698e11	0.111694
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.72471e12	0.985654
$222$	0	0
$223$	−5.00875e12	−0.608209	−0.304105	−	0.952639i	$-0.598357\pi$
$223$	−5.00875e12	−0.608209	−0.304105	+	0.952639i	$0.598357\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.11038e13	1.22273	0.611363	−	0.791350i	$-0.290621\pi$
$227$	1.11038e13	1.22273	0.611363	+	0.791350i	$0.290621\pi$
$228$	0	0
$229$	−8.13563e12	−0.853682	−0.426841	−	0.904327i	$-0.640374\pi$
$229$	−8.13563e12	−0.853682	−0.426841	+	0.904327i	$0.640374\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.06769e12	−0.769647	−0.384823	−	0.922990i	$-0.625738\pi$
$233$	−8.06769e12	−0.769647	−0.384823	+	0.922990i	$0.625738\pi$
$234$	0	0
$235$	−4.11395e12	−0.374443
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.46057e12	0.452949	0.226475	−	0.974017i	$-0.427280\pi$
$239$	5.46057e12	0.452949	0.226475	+	0.974017i	$0.427280\pi$
$240$	0	0
$241$	1.16419e13	0.922422	0.461211	−	0.887291i	$-0.347415\pi$
$241$	1.16419e13	0.922422	0.461211	+	0.887291i	$0.347415\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.83478e11	0.0349916
$246$	0	0
$247$	2.03450e13	1.40807
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.59718e11	0.0354620	0.0177310	−	0.999843i	$-0.494356\pi$
$251$	5.59718e11	0.0354620	0.0177310	+	0.999843i	$0.494356\pi$
$252$	0	0
$253$	3.78185e13	2.29372
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.68101e11	−0.0427352	−0.0213676	−	0.999772i	$-0.506802\pi$
$257$	−7.68101e11	−0.0427352	−0.0213676	+	0.999772i	$0.506802\pi$
$258$	0	0
$259$	3.94300e11	0.0210222
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.33756e13	1.63558	0.817790	−	0.575516i	$-0.195199\pi$
$263$	3.33756e13	1.63558	0.817790	+	0.575516i	$0.195199\pi$
$264$	0	0
$265$	−7.93810e10	−0.00373133
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.60879e13	1.56215	0.781077	−	0.624435i	$-0.214671\pi$
$269$	3.60879e13	1.56215	0.781077	+	0.624435i	$0.214671\pi$
$270$	0	0
$271$	3.51521e13	1.46090	0.730449	−	0.682968i	$-0.239311\pi$
$271$	3.51521e13	1.46090	0.730449	+	0.682968i	$0.239311\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.64748e13	−1.78192
$276$	0	0
$277$	2.21948e13	0.817736	0.408868	−	0.912594i	$-0.365924\pi$
$277$	2.21948e13	0.817736	0.408868	+	0.912594i	$0.365924\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.31593e13	0.448074	0.224037	−	0.974581i	$-0.428076\pi$
$281$	1.31593e13	0.448074	0.224037	+	0.974581i	$0.428076\pi$
$282$	0	0
$283$	−3.39132e13	−1.11056	−0.555282	−	0.831662i	$-0.687390\pi$
$283$	−3.39132e13	−1.11056	−0.555282	+	0.831662i	$0.687390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.27174e12	−0.250753
$288$	0	0
$289$	6.18718e12	0.180532
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.72820e12	0.127916	0.0639579	−	0.997953i	$-0.479628\pi$
$293$	4.72820e12	0.127916	0.0639579	+	0.997953i	$0.479628\pi$
$294$	0	0
$295$	−9.76754e12	−0.254545
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.53587e13	1.09766
$300$	0	0
$301$	1.46334e13	0.341372
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.13162e13	0.245501
$306$	0	0
$307$	−4.64898e13	−0.972964	−0.486482	−	0.873691i	$-0.661720\pi$
$307$	−4.64898e13	−0.972964	−0.486482	+	0.873691i	$0.661720\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.47839e13	0.483044	0.241522	−	0.970395i	$-0.422353\pi$
$311$	2.47839e13	0.483044	0.241522	+	0.970395i	$0.422353\pi$
$312$	0	0
$313$	−3.04738e13	−0.573368	−0.286684	−	0.958025i	$-0.592553\pi$
$313$	−3.04738e13	−0.573368	−0.286684	+	0.958025i	$0.592553\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.26908e13	1.45088	0.725439	−	0.688286i	$-0.241637\pi$
$317$	8.26908e13	1.45088	0.725439	+	0.688286i	$0.241637\pi$
$318$	0	0
$319$	6.53842e13	1.10821
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.06560e14	1.68647
$324$	0	0
$325$	−5.57409e13	−0.852737
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.03973e13	0.577797
$330$	0	0
$331$	−9.64392e13	−1.33413	−0.667067	−	0.744998i	$-0.732451\pi$
$331$	−9.64392e13	−1.33413	−0.667067	+	0.744998i	$0.732451\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.31430e13	0.299691
$336$	0	0
$337$	1.25678e14	1.57506	0.787528	−	0.616278i	$-0.211360\pi$
$337$	1.25678e14	1.57506	0.787528	+	0.616278i	$0.211360\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.76966e13	−0.560194
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.59006e13	0.596492	0.298246	−	0.954489i	$-0.403598\pi$
$347$	5.59006e13	0.596492	0.298246	+	0.954489i	$0.403598\pi$
$348$	0	0
$349$	1.01479e14	1.04915	0.524573	−	0.851365i	$-0.324225\pi$
$349$	1.01479e14	1.04915	0.524573	+	0.851365i	$0.324225\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.99801e14	−1.94016	−0.970079	−	0.242790i	$-0.921937\pi$
$353$	−1.99801e14	−1.94016	−0.970079	+	0.242790i	$0.921937\pi$
$354$	0	0
$355$	4.16974e13	0.392512
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.39511e14	1.23478	0.617388	−	0.786659i	$-0.288191\pi$
$359$	1.39511e14	1.23478	0.617388	+	0.786659i	$0.288191\pi$
$360$	0	0
$361$	1.64162e14	1.40923
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.72908e13	0.220499
$366$	0	0
$367$	1.96641e13	0.154174	0.0770871	−	0.997024i	$-0.475438\pi$
$367$	1.96641e13	0.154174	0.0770871	+	0.997024i	$0.475438\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.79489e11	0.00575776
$372$	0	0
$373$	−8.74269e13	−0.626969	−0.313485	−	0.949593i	$-0.601496\pi$
$373$	−8.74269e13	−0.626969	−0.313485	+	0.949593i	$0.601496\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.84203e13	0.530336
$378$	0	0
$379$	−1.57214e14	−1.03270	−0.516350	−	0.856377i	$-0.672710\pi$
$379$	−1.57214e14	−1.03270	−0.516350	+	0.856377i	$0.672710\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.11638e14	1.31220	0.656100	−	0.754674i	$-0.272205\pi$
$383$	2.11638e14	1.31220	0.656100	+	0.754674i	$0.272205\pi$
$384$	0	0
$385$	−2.91276e13	−0.175497
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.40928e14	−1.37140	−0.685700	−	0.727884i	$-0.740504\pi$
$389$	−2.40928e14	−1.37140	−0.685700	+	0.727884i	$0.740504\pi$
$390$	0	0
$391$	2.37571e14	1.31469
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.85504e13	−0.411021
$396$	0	0
$397$	−3.05279e14	−1.55364	−0.776818	−	0.629725i	$-0.783168\pi$
$397$	−3.05279e14	−1.55364	−0.776818	+	0.629725i	$0.783168\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.84391e13	−0.0888069	−0.0444034	−	0.999014i	$-0.514139\pi$
$401$	−1.84391e13	−0.0888069	−0.0444034	+	0.999014i	$0.514139\pi$
$402$	0	0
$403$	−5.72063e13	−0.268081
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.37550e13	−0.105435
$408$	0	0
$409$	−4.59934e13	−0.198709	−0.0993544	−	0.995052i	$-0.531678\pi$
$409$	−4.59934e13	−0.198709	−0.0993544	+	0.995052i	$0.531678\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.59133e13	0.392784
$414$	0	0
$415$	1.91085e13	0.0762012
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.22347e14	−1.21940	−0.609701	−	0.792632i	$-0.708710\pi$
$419$	−3.22347e14	−1.21940	−0.609701	+	0.792632i	$0.708710\pi$
$420$	0	0
$421$	−2.51212e14	−0.925739	−0.462870	−	0.886426i	$-0.653180\pi$
$421$	−2.51212e14	−0.925739	−0.462870	+	0.886426i	$0.653180\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.91949e14	−1.02134
$426$	0	0
$427$	−1.11121e14	−0.378828
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.55171e14	−0.502559	−0.251279	−	0.967915i	$-0.580851\pi$
$431$	−1.55171e14	−0.502559	−0.251279	+	0.967915i	$0.580851\pi$
$432$	0	0
$433$	4.58103e14	1.44637	0.723186	−	0.690653i	$-0.242677\pi$
$433$	4.58103e14	1.44637	0.723186	+	0.690653i	$0.242677\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.25706e14	1.87812
$438$	0	0
$439$	4.57796e13	0.134004	0.0670019	−	0.997753i	$-0.478657\pi$
$439$	4.57796e13	0.134004	0.0670019	+	0.997753i	$0.478657\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.43125e13	−0.123397	−0.0616986	−	0.998095i	$-0.519652\pi$
$443$	−4.43125e13	−0.123397	−0.0616986	+	0.998095i	$0.519652\pi$
$444$	0	0
$445$	−1.10549e14	−0.300313
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.04826e14	0.788311	0.394155	−	0.919044i	$-0.371037\pi$
$449$	3.04826e14	0.788311	0.394155	+	0.919044i	$0.371037\pi$
$450$	0	0
$451$	4.98339e14	1.25763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.49350e13	−0.0839843
$456$	0	0
$457$	4.43161e14	1.03997	0.519986	−	0.854174i	$-0.325937\pi$
$457$	4.43161e14	1.03997	0.519986	+	0.854174i	$0.325937\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.97542e14	−1.33664	−0.668318	−	0.743876i	$-0.732986\pi$
$461$	−5.97542e14	−1.33664	−0.668318	+	0.743876i	$0.732986\pi$
$462$	0	0
$463$	7.15527e13	0.156290	0.0781449	−	0.996942i	$-0.475100\pi$
$463$	7.15527e13	0.156290	0.0781449	+	0.996942i	$0.475100\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.77146e14	−0.369053	−0.184526	−	0.982828i	$-0.559075\pi$
$467$	−1.77146e14	−0.369053	−0.184526	+	0.982828i	$0.559075\pi$
$468$	0	0
$469$	−2.27255e14	−0.462448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.81601e14	−1.71213
$474$	0	0
$475$	−7.68925e14	−1.45905
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.12813e14	−0.748010	−0.374005	−	0.927427i	$-0.622016\pi$
$479$	−4.12813e14	−0.748010	−0.374005	+	0.927427i	$0.622016\pi$
$480$	0	0
$481$	−2.84912e13	−0.0504560
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.87932e13	−0.116404
$486$	0	0
$487$	−4.02966e14	−0.666591	−0.333295	−	0.942822i	$-0.608161\pi$
$487$	−4.02966e14	−0.666591	−0.333295	+	0.942822i	$0.608161\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.33629e14	−1.16019	−0.580094	−	0.814550i	$-0.696984\pi$
$491$	−7.33629e14	−1.16019	−0.580094	+	0.814550i	$0.696984\pi$
$492$	0	0
$493$	4.10736e14	0.635191
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.09452e14	−0.605679
$498$	0	0
$499$	−7.31362e14	−1.05823	−0.529114	−	0.848551i	$-0.677476\pi$
$499$	−7.31362e14	−1.05823	−0.529114	+	0.848551i	$0.677476\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.24178e13	0.114129	0.0570646	−	0.998370i	$-0.481826\pi$
$503$	8.24178e13	0.114129	0.0570646	+	0.998370i	$0.481826\pi$
$504$	0	0
$505$	−3.26755e13	−0.0442710
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.95798e14	−1.16215	−0.581075	−	0.813850i	$-0.697368\pi$
$509$	−8.95798e14	−1.16215	−0.581075	+	0.813850i	$0.697368\pi$
$510$	0	0
$511$	−2.67985e14	−0.340248
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.24403e13	−0.0272954
$516$	0	0
$517$	−2.43378e15	−2.89790
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.97211e14	0.681586	0.340793	−	0.940138i	$-0.389305\pi$
$521$	5.97211e14	0.681586	0.340793	+	0.940138i	$0.389305\pi$
$522$	0	0
$523$	1.01136e15	1.13017	0.565086	−	0.825032i	$-0.308843\pi$
$523$	1.01136e15	1.13017	0.565086	+	0.825032i	$0.308843\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.99625e14	−0.321084
$528$	0	0
$529$	4.42184e14	0.464084
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.97697e14	0.601840
$534$	0	0
$535$	−5.58222e13	−0.0550632
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.86021e14	0.270807
$540$	0	0
$541$	5.36928e14	0.498117	0.249059	−	0.968488i	$-0.419879\pi$
$541$	5.36928e14	0.498117	0.249059	+	0.968488i	$0.419879\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.55897e14	0.227974
$546$	0	0
$547$	−7.37403e14	−0.643835	−0.321918	−	0.946768i	$-0.604327\pi$
$547$	−7.37403e14	−0.643835	−0.321918	+	0.946768i	$0.604327\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.08178e15	0.907413
$552$	0	0
$553$	7.71333e14	0.634240
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.65880e14	0.368188	0.184094	−	0.982909i	$-0.441065\pi$
$557$	4.65880e14	0.368188	0.184094	+	0.982909i	$0.441065\pi$
$558$	0	0
$559$	−1.05737e15	−0.819338
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.93664e14	−0.740361	−0.370180	−	0.928960i	$-0.620704\pi$
$563$	−9.93664e14	−0.740361	−0.370180	+	0.928960i	$0.620704\pi$
$564$	0	0
$565$	−1.46663e14	−0.107165
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.06262e15	−1.44978	−0.724891	−	0.688864i	$-0.758110\pi$
$569$	−2.06262e15	−1.44978	−0.724891	+	0.688864i	$0.758110\pi$
$570$	0	0
$571$	8.49236e14	0.585504	0.292752	−	0.956188i	$-0.405429\pi$
$571$	8.49236e14	0.585504	0.292752	+	0.956188i	$0.405429\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.71430e15	−1.13740
$576$	0	0
$577$	−2.56261e15	−1.66807	−0.834036	−	0.551710i	$-0.813976\pi$
$577$	−2.56261e15	−1.66807	−0.834036	+	0.551710i	$0.813976\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.87638e14	−0.117585
$582$	0	0
$583$	−4.69611e13	−0.0288776
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.13285e15	1.85537	0.927684	−	0.373365i	$-0.121796\pi$
$587$	3.13285e15	1.85537	0.927684	+	0.373365i	$0.121796\pi$
$588$	0	0
$589$	−7.89140e14	−0.458690
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.22770e15	−1.24754	−0.623772	−	0.781607i	$-0.714401\pi$
$593$	−2.22770e15	−1.24754	−0.623772	+	0.781607i	$0.714401\pi$
$594$	0	0
$595$	−1.82976e14	−0.100589
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.15302e14	0.114078	0.0570389	−	0.998372i	$-0.481834\pi$
$599$	2.15302e14	0.114078	0.0570389	+	0.998372i	$0.481834\pi$
$600$	0	0
$601$	1.73955e15	0.904954	0.452477	−	0.891776i	$-0.350541\pi$
$601$	1.73955e15	0.904954	0.452477	+	0.891776i	$0.350541\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.26649e15	0.635252
$606$	0	0
$607$	2.37679e15	1.17072	0.585360	−	0.810774i	$-0.300953\pi$
$607$	2.37679e15	1.17072	0.585360	+	0.810774i	$0.300953\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.91902e15	−1.38679
$612$	0	0
$613$	5.05220e14	0.235748	0.117874	−	0.993029i	$-0.462392\pi$
$613$	5.05220e14	0.235748	0.117874	+	0.993029i	$0.462392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.82836e15	−1.27340	−0.636701	−	0.771110i	$-0.719702\pi$
$617$	−2.82836e15	−1.27340	−0.636701	+	0.771110i	$0.719702\pi$
$618$	0	0
$619$	−3.14424e15	−1.39065	−0.695324	−	0.718696i	$-0.744739\pi$
$619$	−3.14424e15	−1.39065	−0.695324	+	0.718696i	$0.744739\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.08555e15	0.463408
$624$	0	0
$625$	1.96364e15	0.823611
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.49226e14	−0.0604318
$630$	0	0
$631$	2.88224e15	1.14701	0.573507	−	0.819201i	$-0.305583\pi$
$631$	2.88224e15	1.14701	0.573507	+	0.819201i	$0.305583\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.25392e14	−0.0481965
$636$	0	0
$637$	3.43048e14	0.129595
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.25647e14	−0.301353	−0.150676	−	0.988583i	$-0.548145\pi$
$641$	−8.25647e14	−0.301353	−0.150676	+	0.988583i	$0.548145\pi$
$642$	0	0
$643$	−6.47800e14	−0.232424	−0.116212	−	0.993224i	$-0.537075\pi$
$643$	−6.47800e14	−0.232424	−0.116212	+	0.993224i	$0.537075\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.96109e14	0.102678	0.0513390	−	0.998681i	$-0.483651\pi$
$647$	2.96109e14	0.102678	0.0513390	+	0.998681i	$0.483651\pi$
$648$	0	0
$649$	−5.77839e15	−1.96998
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.68770e15	0.556254	0.278127	−	0.960544i	$-0.410286\pi$
$653$	1.68770e15	0.556254	0.278127	+	0.960544i	$0.410286\pi$
$654$	0	0
$655$	1.10046e14	0.0356655
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.02991e15	1.26307	0.631533	−	0.775349i	$-0.282426\pi$
$659$	4.02991e15	1.26307	0.631533	+	0.775349i	$0.282426\pi$
$660$	0	0
$661$	−8.80347e14	−0.271360	−0.135680	−	0.990753i	$-0.543322\pi$
$661$	−8.80347e14	−0.271360	−0.135680	+	0.990753i	$0.543322\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.81916e14	−0.143698
$666$	0	0
$667$	2.41180e15	0.707373
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.69458e15	1.89998
$672$	0	0
$673$	1.28570e15	0.358970	0.179485	−	0.983761i	$-0.442557\pi$
$673$	1.28570e15	0.358970	0.179485	+	0.983761i	$0.442557\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.27788e15	0.885840	0.442920	−	0.896561i	$-0.353942\pi$
$677$	3.27788e15	0.885840	0.442920	+	0.896561i	$0.353942\pi$
$678$	0	0
$679$	6.75522e14	0.179620
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.01456e15	−0.518640	−0.259320	−	0.965792i	$-0.583498\pi$
$683$	−2.01456e15	−0.518640	−0.259320	+	0.965792i	$0.583498\pi$
$684$	0	0
$685$	4.24713e14	0.107596
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.63241e13	−0.0138194
$690$	0	0
$691$	6.51077e15	1.57218	0.786091	−	0.618110i	$-0.212101\pi$
$691$	6.51077e15	1.57218	0.786091	+	0.618110i	$0.212101\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.70300e15	−0.398380
$696$	0	0
$697$	3.13051e15	0.720833
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.12910e15	0.475058	0.237529	−	0.971381i	$-0.423663\pi$
$701$	2.12910e15	0.475058	0.237529	+	0.971381i	$0.423663\pi$
$702$	0	0
$703$	−3.93025e14	−0.0863309
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.20860e14	0.0683139
$708$	0	0
$709$	−7.00228e15	−1.46786	−0.733931	−	0.679224i	$-0.762316\pi$
$709$	−7.00228e15	−1.46786	−0.733931	+	0.679224i	$0.762316\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.75936e15	−0.357572
$714$	0	0
$715$	2.10469e15	0.421217
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.89802e15	−0.950630	−0.475315	−	0.879816i	$-0.657666\pi$
$719$	−4.89802e15	−0.950630	−0.475315	+	0.879816i	$0.657666\pi$
$720$	0	0
$721$	2.20355e14	0.0421190
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.96384e15	−0.549535
$726$	0	0
$727$	5.74389e15	1.04898	0.524490	−	0.851417i	$-0.324256\pi$
$727$	5.74389e15	1.04898	0.524490	+	0.851417i	$0.324256\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.53811e15	−0.981333
$732$	0	0
$733$	2.90437e15	0.506968	0.253484	−	0.967340i	$-0.418423\pi$
$733$	2.90437e15	0.506968	0.253484	+	0.967340i	$0.418423\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.36912e16	2.31937
$738$	0	0
$739$	−3.60011e15	−0.600858	−0.300429	−	0.953804i	$-0.597130\pi$
$739$	−3.60011e15	−0.600858	−0.300429	+	0.953804i	$0.597130\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.11121e16	1.80036	0.900179	−	0.435521i	$-0.143436\pi$
$743$	1.11121e16	1.80036	0.900179	+	0.435521i	$0.143436\pi$
$744$	0	0
$745$	1.05106e15	0.167791
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.48152e14	0.0849671
$750$	0	0
$751$	−3.98036e14	−0.0607999	−0.0303999	−	0.999538i	$-0.509678\pi$
$751$	−3.98036e14	−0.0607999	−0.0303999	+	0.999538i	$0.509678\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.21745e15	−0.477313
$756$	0	0
$757$	−1.55801e15	−0.227795	−0.113897	−	0.993493i	$-0.536333\pi$
$757$	−1.55801e15	−0.227795	−0.113897	+	0.993493i	$0.536333\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.47483e15	−0.209472	−0.104736	−	0.994500i	$-0.533400\pi$
$761$	−1.47483e15	−0.209472	−0.104736	+	0.994500i	$0.533400\pi$
$762$	0	0
$763$	−2.51281e15	−0.351783
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.93047e15	−0.942732
$768$	0	0
$769$	−9.29764e15	−1.24675	−0.623373	−	0.781925i	$-0.714238\pi$
$769$	−9.29764e15	−1.24675	−0.623373	+	0.781925i	$0.714238\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.22041e16	1.59045	0.795224	−	0.606315i	$-0.207353\pi$
$773$	1.22041e16	1.59045	0.795224	+	0.606315i	$0.207353\pi$
$774$	0	0
$775$	2.16207e15	0.277786
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.24501e15	1.02976
$780$	0	0
$781$	2.46678e16	3.03773
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.75732e15	0.210411
$786$	0	0
$787$	1.02319e16	1.20808	0.604039	−	0.796955i	$-0.293557\pi$
$787$	1.02319e16	1.20808	0.604039	+	0.796955i	$0.293557\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.44017e15	0.165365
$792$	0	0
$793$	8.02933e15	0.909237
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.74682e15	−0.412707	−0.206354	−	0.978478i	$-0.566160\pi$
$797$	−3.74682e15	−0.412707	−0.206354	+	0.978478i	$0.566160\pi$
$798$	0	0
$799$	−1.52887e16	−1.66098
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.61450e16	1.70649
$804$	0	0
$805$	−1.07442e15	−0.112020
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.52925e16	1.55154	0.775768	−	0.631018i	$-0.217363\pi$
$809$	1.52925e16	1.55154	0.775768	+	0.631018i	$0.217363\pi$
$810$	0	0
$811$	1.62525e16	1.62669	0.813347	−	0.581778i	$-0.197643\pi$
$811$	1.62525e16	1.62669	0.813347	+	0.581778i	$0.197643\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.51325e15	0.439666
$816$	0	0
$817$	−1.45861e16	−1.40190
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.81713e15	−0.544279	−0.272140	−	0.962258i	$-0.587731\pi$
$821$	−5.81713e15	−0.544279	−0.272140	+	0.962258i	$0.587731\pi$
$822$	0	0
$823$	−1.54173e16	−1.42334	−0.711670	−	0.702514i	$-0.752061\pi$
$823$	−1.54173e16	−1.42334	−0.711670	+	0.702514i	$0.752061\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.09031e16	1.87901	0.939507	−	0.342529i	$-0.111283\pi$
$827$	2.09031e16	1.87901	0.939507	+	0.342529i	$0.111283\pi$
$828$	0	0
$829$	−1.01919e16	−0.904080	−0.452040	−	0.891998i	$-0.649304\pi$
$829$	−1.01919e16	−0.904080	−0.452040	+	0.891998i	$0.649304\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.79675e15	0.155218
$834$	0	0
$835$	4.49098e15	0.382883
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.77545e15	0.147441	0.0737203	−	0.997279i	$-0.476513\pi$
$839$	1.77545e15	0.147441	0.0737203	+	0.997279i	$0.476513\pi$
$840$	0	0
$841$	−8.03077e15	−0.658232
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.43099e14	−0.0433678
$846$	0	0
$847$	−1.24364e16	−0.980247
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.76239e14	−0.0672992
$852$	0	0
$853$	1.27871e15	0.0969508	0.0484754	−	0.998824i	$-0.484564\pi$
$853$	1.27871e15	0.0969508	0.0484754	+	0.998824i	$0.484564\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.60708e15	0.192646	0.0963230	−	0.995350i	$-0.469292\pi$
$857$	2.60708e15	0.192646	0.0963230	+	0.995350i	$0.469292\pi$
$858$	0	0
$859$	2.14061e16	1.56162	0.780808	−	0.624771i	$-0.214808\pi$
$859$	2.14061e16	1.56162	0.780808	+	0.624771i	$0.214808\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.19381e16	−0.848934	−0.424467	−	0.905443i	$-0.639539\pi$
$863$	−1.19381e16	−0.848934	−0.424467	+	0.905443i	$0.639539\pi$
$864$	0	0
$865$	9.01582e14	0.0633019
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.64697e16	−3.18098
$870$	0	0
$871$	1.64209e16	1.10994
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.72496e15	0.179604
$876$	0	0
$877$	−1.65246e16	−1.07555	−0.537777	−	0.843087i	$-0.680736\pi$
$877$	−1.65246e16	−1.07555	−0.537777	+	0.843087i	$0.680736\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.01498e16	−1.27909	−0.639547	−	0.768752i	$-0.720878\pi$
$881$	−2.01498e16	−1.27909	−0.639547	+	0.768752i	$0.720878\pi$
$882$	0	0
$883$	−1.09493e16	−0.686443	−0.343221	−	0.939255i	$-0.611518\pi$
$883$	−1.09493e16	−0.686443	−0.343221	+	0.939255i	$0.611518\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.62641e16	1.60614	0.803069	−	0.595887i	$-0.203199\pi$
$887$	2.62641e16	1.60614	0.803069	+	0.595887i	$0.203199\pi$
$888$	0	0
$889$	1.23130e15	0.0743712
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.02668e16	−2.37282
$894$	0	0
$895$	−2.54270e15	−0.148002
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.04175e15	−0.172761
$900$	0	0
$901$	−2.95004e14	−0.0165517
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.26496e15	−0.343045
$906$	0	0
$907$	−1.10077e16	−0.595464	−0.297732	−	0.954650i	$-0.596230\pi$
$907$	−1.10077e16	−0.595464	−0.297732	+	0.954650i	$0.596230\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.88649e15	−0.258016	−0.129008	−	0.991644i	$-0.541179\pi$
$911$	−4.88649e15	−0.258016	−0.129008	+	0.991644i	$0.541179\pi$
$912$	0	0
$913$	1.13044e16	0.589737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.08061e15	−0.0550349
$918$	0	0
$919$	2.49773e16	1.25693	0.628465	−	0.777838i	$-0.283684\pi$
$919$	2.49773e16	1.25693	0.628465	+	0.777838i	$0.283684\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.95860e16	1.45371
$924$	0	0
$925$	1.07680e15	0.0522825
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.57265e15	0.406470	0.203235	−	0.979130i	$-0.434854\pi$
$929$	8.57265e15	0.406470	0.203235	+	0.979130i	$0.434854\pi$
$930$	0	0
$931$	4.73222e15	0.221739
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.10236e16	0.504497
$936$	0	0
$937$	9.41746e15	0.425957	0.212979	−	0.977057i	$-0.431683\pi$
$937$	9.41746e15	0.425957	0.212979	+	0.977057i	$0.431683\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.69233e15	0.295689	0.147844	−	0.989011i	$-0.452767\pi$
$941$	6.69233e15	0.295689	0.147844	+	0.989011i	$0.452767\pi$
$942$	0	0
$943$	1.83820e16	0.802747
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.92898e16	0.823005	0.411503	−	0.911409i	$-0.365004\pi$
$947$	1.92898e16	0.823005	0.411503	+	0.911409i	$0.365004\pi$
$948$	0	0
$949$	1.93640e16	0.816639
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.29671e15	0.0534358	0.0267179	−	0.999643i	$-0.491494\pi$
$953$	1.29671e15	0.0534358	0.0267179	+	0.999643i	$0.491494\pi$
$954$	0	0
$955$	6.68271e15	0.272229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.17051e15	−0.166030
$960$	0	0
$961$	−2.31896e16	−0.912671
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.93691e15	−0.305317
$966$	0	0
$967$	2.04409e16	0.777419	0.388709	−	0.921360i	$-0.372921\pi$
$967$	2.04409e16	0.777419	0.388709	+	0.921360i	$0.372921\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.25004e16	−1.58011	−0.790054	−	0.613037i	$-0.789948\pi$
$971$	−4.25004e16	−1.58011	−0.790054	+	0.613037i	$0.789948\pi$
$972$	0	0
$973$	1.67227e16	0.614733
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.80920e15	0.0650228	0.0325114	−	0.999471i	$-0.489649\pi$
$977$	1.80920e15	0.0650228	0.0325114	+	0.999471i	$0.489649\pi$
$978$	0	0
$979$	−6.53997e16	−2.32419
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.77082e16	−0.962861	−0.481430	−	0.876484i	$-0.659883\pi$
$983$	−2.77082e16	−0.962861	−0.481430	+	0.876484i	$0.659883\pi$
$984$	0	0
$985$	−1.12756e16	−0.387473
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.25192e16	−1.09285
$990$	0	0
$991$	−3.50876e16	−1.16613	−0.583067	−	0.812424i	$-0.698148\pi$
$991$	−3.50876e16	−1.16613	−0.583067	+	0.812424i	$0.698148\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.12380e15	0.296585
$996$	0	0
$997$	−5.52194e16	−1.77529	−0.887644	−	0.460531i	$-0.847659\pi$
$997$	−5.52194e16	−1.77529	−0.887644	+	0.460531i	$0.847659\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.f.1.3		3
3.2	odd	2		28.12.a.a.1.3	✓	3
12.11	even	2		112.12.a.g.1.1		3
21.2	odd	6		196.12.e.e.165.1		6
21.5	even	6		196.12.e.d.165.3		6
21.11	odd	6		196.12.e.e.177.1		6
21.17	even	6		196.12.e.d.177.3		6
21.20	even	2		196.12.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.12.a.a.1.3	✓	3	3.2	odd	2
112.12.a.g.1.1		3	12.11	even	2
196.12.a.c.1.1		3	21.20	even	2
196.12.e.d.165.3		6	21.5	even	6
196.12.e.d.177.3		6	21.17	even	6
196.12.e.e.165.1		6	21.2	odd	6
196.12.e.e.177.1		6	21.11	odd	6
252.12.a.f.1.3		3	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q+1711.58 q^{5} -16807.0 q^{7} +O(q^{10})\)	\(q+1711.58 q^{5} -16807.0 q^{7} +1.01255e6 q^{11} +1.21443e6 q^{13} +6.36074e6 q^{17} +1.67527e7 q^{19} +3.73496e7 q^{23} -4.58986e7 q^{25} +6.45735e7 q^{29} -4.71053e7 q^{31} -2.87665e7 q^{35} -2.34605e7 q^{37} +4.92161e8 q^{41} -8.70671e8 q^{43} -2.40360e9 q^{47} +2.82475e8 q^{49} -4.63788e7 q^{53} +1.73306e9 q^{55} -5.70675e9 q^{59} +6.61158e9 q^{61} +2.07860e9 q^{65} +1.35215e10 q^{67} +2.43620e10 q^{71} +1.59448e10 q^{73} -1.70180e10 q^{77} -4.58936e10 q^{79} +1.11643e10 q^{83} +1.08869e10 q^{85} -6.45889e10 q^{89} -2.04110e10 q^{91} +2.86735e10 q^{95} -4.01929e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) \) 3 * q - 4762 * q^5 - 50421 * q^7	\( 3 q - 4762 q^{5} - 50421 q^{7} - 80468 q^{11} + 996546 q^{13} + 7924090 q^{17} - 15261732 q^{19} + 61428352 q^{23} - 109205331 q^{25} + 160359182 q^{29} - 342148464 q^{31} + 80034934 q^{35} - 722039190 q^{37} - 285288606 q^{41} - 476523612 q^{43} - 1225105680 q^{47} + 847425747 q^{49} - 4030571514 q^{53} + 4822100040 q^{55} - 5222175892 q^{59} + 5684837106 q^{61} - 8153014116 q^{65} + 2837154348 q^{67} + 16928678200 q^{71} - 14560325442 q^{73} + 1352425676 q^{77} - 32832340032 q^{79} + 115451875348 q^{83} - 36782181276 q^{85} + 15661530882 q^{89} - 16748948622 q^{91} + 117828203728 q^{95} + 40693412742 q^{97}+O(q^{100}) \) 3 * q - 4762 * q^5 - 50421 * q^7 - 80468 * q^11 + 996546 * q^13 + 7924090 * q^17 - 15261732 * q^19 + 61428352 * q^23 - 109205331 * q^25 + 160359182 * q^29 - 342148464 * q^31 + 80034934 * q^35 - 722039190 * q^37 - 285288606 * q^41 - 476523612 * q^43 - 1225105680 * q^47 + 847425747 * q^49 - 4030571514 * q^53 + 4822100040 * q^55 - 5222175892 * q^59 + 5684837106 * q^61 - 8153014116 * q^65 + 2837154348 * q^67 + 16928678200 * q^71 - 14560325442 * q^73 + 1352425676 * q^77 - 32832340032 * q^79 + 115451875348 * q^83 - 36782181276 * q^85 + 15661530882 * q^89 - 16748948622 * q^91 + 117828203728 * q^95 + 40693412742 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more