LMFDB - Embedded newform 200.12.c.c.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,12,Mod(49,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("200.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$153.668636112$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{109})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 55x^{2} + 729 $ x^4 + 55*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-5.72015i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.12.c.c.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-696.180i q^{3} +73591.5i q^{7} -307519. q^{9} -383508. q^{11} -1.15867e6i q^{13} -6.51693e6i q^{17} +1.39982e7 q^{19} +5.12329e7 q^{21} -1.37394e6i q^{23} +9.07624e7i q^{27} +7.46197e7 q^{29} +1.32297e7 q^{31} +2.66991e8i q^{33} -1.67200e7i q^{37} -8.06644e8 q^{39} +1.03298e9 q^{41} -1.93764e8i q^{43} -1.16005e9i q^{47} -3.43839e9 q^{49} -4.53696e9 q^{51} -4.44363e8i q^{53} -9.74529e9i q^{57} +1.28304e8 q^{59} +7.96097e9 q^{61} -2.26308e10i q^{63} -6.89243e9i q^{67} -9.56509e8 q^{69} -1.12698e10 q^{71} -3.34998e9i q^{73} -2.82230e10i q^{77} -5.36237e10 q^{79} +8.71083e9 q^{81} +6.31693e10i q^{83} -5.19487e10i q^{87} -9.62373e10 q^{89} +8.52685e10 q^{91} -9.21027e9i q^{93} -4.37466e10i q^{97} +1.17936e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1080404 q^{9} + 318160 q^{11} + 43733200 q^{19} + 80105088 q^{21} + 457655400 q^{29} + 129444224 q^{31} - 1751818144 q^{39} + 2402428392 q^{41} - 3532138148 q^{49} - 19249975840 q^{51} + 12025853168 q^{59}+ \cdots - 16629834512 q^{99}+O(q^{100}) $ 4 * q - 1080404 * q^9 + 318160 * q^11 + 43733200 * q^19 + 80105088 * q^21 + 457655400 * q^29 + 129444224 * q^31 - 1751818144 * q^39 + 2402428392 * q^41 - 3532138148 * q^49 - 19249975840 * q^51 + 12025853168 * q^59 + 19579585816 * q^61 + 42173328512 * q^69 + 8639982176 * q^71 - 103915246528 * q^79 - 19495920604 * q^81 - 142376583720 * q^89 + 166757785152 * q^91 - 16629834512 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 696.180i	− 1.65407i	−0.562149	−	0.827036i	$-0.690025\pi$
$3$	− 696.180i	− 1.65407i	0.562149	−	0.827036i	$-0.309975\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	73591.5i	1.65496i	0.561492	+	0.827482i	$0.310228\pi$
$7$	73591.5i	1.65496i	−0.561492	+	0.827482i	$0.689772\pi$
$8$	0	0
$9$	−307519.	−1.73595
$10$	0	0
$11$	−383508.	−0.717985	−0.358992	−	0.933340i	$-0.616880\pi$
$11$	−383508.	−0.717985	−0.358992	+	0.933340i	$0.616880\pi$
$12$	0	0
$13$	− 1.15867e6i	− 0.865510i	−0.901512	−	0.432755i	$-0.857541\pi$
$13$	− 1.15867e6i	− 0.865510i	0.901512	−	0.432755i	$-0.142459\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.51693e6i	− 1.11320i	−0.830780	−	0.556601i	$-0.812105\pi$
$17$	− 6.51693e6i	− 1.11320i	0.830780	−	0.556601i	$-0.187895\pi$
$18$	0	0
$19$	1.39982e7	1.29697	0.648483	−	0.761229i	$-0.275404\pi$
$19$	1.39982e7	1.29697	0.648483	+	0.761229i	$0.275404\pi$
$20$	0	0
$21$	5.12329e7	2.73743
$22$	0	0
$23$	− 1.37394e6i	− 0.0445107i	−0.999752	−	0.0222554i	$-0.992915\pi$
$23$	− 1.37394e6i	− 0.0445107i	0.999752	−	0.0222554i	$-0.00708469\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.07624e7i	1.21732i
$28$	0	0
$29$	7.46197e7	0.675561	0.337781	−	0.941225i	$-0.390324\pi$
$29$	7.46197e7	0.675561	0.337781	+	0.941225i	$0.390324\pi$
$30$	0	0
$31$	1.32297e7	0.0829969	0.0414985	−	0.999139i	$-0.486787\pi$
$31$	1.32297e7	0.0829969	0.0414985	+	0.999139i	$0.486787\pi$
$32$	0	0
$33$	2.66991e8i	1.18760i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.67200e7i	− 0.0396394i	−0.999804	−	0.0198197i	$-0.993691\pi$
$37$	− 1.67200e7i	− 0.0396394i	0.999804	−	0.0198197i	$-0.00630922\pi$
$38$	0	0
$39$	−8.06644e8	−1.43162
$40$	0	0
$41$	1.03298e9	1.39245	0.696225	−	0.717823i	$-0.254861\pi$
$41$	1.03298e9	1.39245	0.696225	+	0.717823i	$0.254861\pi$
$42$	0	0
$43$	− 1.93764e8i	− 0.201001i	−0.994937	−	0.100500i	$-0.967956\pi$
$43$	− 1.93764e8i	− 0.201001i	0.994937	−	0.100500i	$-0.0320443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.16005e9i	− 0.737803i	−0.929469	−	0.368901i	$-0.879734\pi$
$47$	− 1.16005e9i	− 0.737803i	0.929469	−	0.368901i	$-0.120266\pi$
$48$	0	0
$49$	−3.43839e9	−1.73891
$50$	0	0
$51$	−4.53696e9	−1.84132
$52$	0	0
$53$	− 4.44363e8i	− 0.145956i	−0.997334	−	0.0729778i	$-0.976750\pi$
$53$	− 4.44363e8i	− 0.145956i	0.997334	−	0.0729778i	$-0.0232502\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 9.74529e9i	− 2.14528i
$58$	0	0
$59$	1.28304e8	0.0233644	0.0116822	−	0.999932i	$-0.496281\pi$
$59$	1.28304e8	0.0233644	0.0116822	+	0.999932i	$0.496281\pi$
$60$	0	0
$61$	7.96097e9	1.20685	0.603423	−	0.797421i	$-0.293803\pi$
$61$	7.96097e9	1.20685	0.603423	+	0.797421i	$0.293803\pi$
$62$	0	0
$63$	− 2.26308e10i	− 2.87294i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.89243e9i	− 0.623678i	−0.950135	−	0.311839i	$-0.899055\pi$
$67$	− 6.89243e9i	− 0.623678i	0.950135	−	0.311839i	$-0.100945\pi$
$68$	0	0
$69$	−9.56509e8	−0.0736240
$70$	0	0
$71$	−1.12698e10	−0.741305	−0.370652	−	0.928772i	$-0.620866\pi$
$71$	−1.12698e10	−0.741305	−0.370652	+	0.928772i	$0.620866\pi$
$72$	0	0
$73$	− 3.34998e9i	− 0.189132i	−0.995519	−	0.0945662i	$-0.969854\pi$
$73$	− 3.34998e9i	− 0.189132i	0.995519	−	0.0945662i	$-0.0301464\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.82230e10i	− 1.18824i
$78$	0	0
$79$	−5.36237e10	−1.96068	−0.980342	−	0.197304i	$-0.936782\pi$
$79$	−5.36237e10	−1.96068	−0.980342	+	0.197304i	$0.936782\pi$
$80$	0	0
$81$	8.71083e9	0.277583
$82$	0	0
$83$	6.31693e10i	1.76026i	0.474733	+	0.880130i	$0.342545\pi$
$83$	6.31693e10i	1.76026i	−0.474733	+	0.880130i	$0.657455\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.19487e10i	− 1.11743i
$88$	0	0
$89$	−9.62373e10	−1.82683	−0.913416	−	0.407028i	$-0.866565\pi$
$89$	−9.62373e10	−1.82683	−0.913416	+	0.407028i	$0.866565\pi$
$90$	0	0
$91$	8.52685e10	1.43239
$92$	0	0
$93$	− 9.21027e9i	− 0.137283i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.37466e10i	− 0.517250i	−0.965978	−	0.258625i	$-0.916731\pi$
$97$	− 4.37466e10i	− 0.517250i	0.965978	−	0.258625i	$-0.0832693\pi$
$98$	0	0
$99$	1.17936e11	1.24639
$100$	0	0
$101$	−1.68061e11	−1.59111	−0.795554	−	0.605882i	$-0.792820\pi$
$101$	−1.68061e11	−1.59111	−0.795554	+	0.605882i	$0.792820\pi$
$102$	0	0
$103$	− 6.02325e10i	− 0.511948i	−0.966684	−	0.255974i	$-0.917604\pi$
$103$	− 6.02325e10i	− 0.511948i	0.966684	−	0.255974i	$-0.0823961\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.39380e11i	0.960701i	0.877077	+	0.480351i	$0.159491\pi$
$107$	1.39380e11i	0.960701i	−0.877077	+	0.480351i	$0.840509\pi$
$108$	0	0
$109$	−1.19601e11	−0.744540	−0.372270	−	0.928124i	$-0.621421\pi$
$109$	−1.19601e11	−0.744540	−0.372270	+	0.928124i	$0.621421\pi$
$110$	0	0
$111$	−1.16401e10	−0.0655664
$112$	0	0
$113$	− 2.69277e10i	− 0.137489i	−0.997634	−	0.0687445i	$-0.978101\pi$
$113$	− 2.69277e10i	− 0.137489i	0.997634	−	0.0687445i	$-0.0218993\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.56314e11i	1.50249i
$118$	0	0
$119$	4.79591e11	1.84231
$120$	0	0
$121$	−1.38233e11	−0.484498
$122$	0	0
$123$	− 7.19138e11i	− 2.30321i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.63631e11i	− 0.439486i	−0.975558	−	0.219743i	$-0.929478\pi$
$127$	− 1.63631e11i	− 0.439486i	0.975558	−	0.219743i	$-0.0705219\pi$
$128$	0	0
$129$	−1.34895e11	−0.332469
$130$	0	0
$131$	−2.54332e11	−0.575981	−0.287991	−	0.957633i	$-0.592987\pi$
$131$	−2.54332e11	−0.575981	−0.287991	+	0.957633i	$0.592987\pi$
$132$	0	0
$133$	1.03015e12i	2.14643i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.05270e12i	1.86355i	0.363032	+	0.931776i	$0.381741\pi$
$137$	1.05270e12i	1.86355i	−0.363032	+	0.931776i	$0.618259\pi$
$138$	0	0
$139$	9.02335e11	1.47498	0.737491	−	0.675357i	$-0.236011\pi$
$139$	9.02335e11	1.47498	0.737491	+	0.675357i	$0.236011\pi$
$140$	0	0
$141$	−8.07607e11	−1.22038
$142$	0	0
$143$	4.44361e11i	0.621423i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.39374e12i	2.87628i
$148$	0	0
$149$	−1.08288e12	−1.20797	−0.603986	−	0.796995i	$-0.706422\pi$
$149$	−1.08288e12	−1.20797	−0.603986	+	0.796995i	$0.706422\pi$
$150$	0	0
$151$	8.99725e11	0.932688	0.466344	−	0.884603i	$-0.345571\pi$
$151$	8.99725e11	0.932688	0.466344	+	0.884603i	$0.345571\pi$
$152$	0	0
$153$	2.00408e12i	1.93247i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.05620e11i	− 0.506701i	−0.967375	−	0.253351i	$-0.918467\pi$
$157$	− 6.05620e11i	− 0.506701i	0.967375	−	0.253351i	$-0.0815327\pi$
$158$	0	0
$159$	−3.09356e11	−0.241421
$160$	0	0
$161$	1.01110e11	0.0736637
$162$	0	0
$163$	− 5.92020e11i	− 0.402999i	−0.979489	−	0.201500i	$-0.935418\pi$
$163$	− 5.92020e11i	− 0.402999i	0.979489	−	0.201500i	$-0.0645815\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.67821e10i	− 0.0338276i	−0.999857	−	0.0169138i	$-0.994616\pi$
$167$	− 5.67821e10i	− 0.0338276i	0.999857	−	0.0169138i	$-0.00538409\pi$
$168$	0	0
$169$	4.49639e11	0.250892
$170$	0	0
$171$	−4.30473e12	−2.25147
$172$	0	0
$173$	− 2.27930e12i	− 1.11828i	−0.829075	−	0.559138i	$-0.811132\pi$
$173$	− 2.27930e12i	− 1.11828i	0.829075	−	0.559138i	$-0.188868\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.93228e10i	− 0.0386464i
$178$	0	0
$179$	1.91742e12	0.779876	0.389938	−	0.920841i	$-0.372497\pi$
$179$	1.91742e12	0.779876	0.389938	+	0.920841i	$0.372497\pi$
$180$	0	0
$181$	−3.61293e12	−1.38238	−0.691190	−	0.722673i	$-0.742913\pi$
$181$	−3.61293e12	−1.38238	−0.691190	+	0.722673i	$0.742913\pi$
$182$	0	0
$183$	− 5.54226e12i	− 1.99621i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.49930e12i	0.799263i
$188$	0	0
$189$	−6.67934e12	−2.01462
$190$	0	0
$191$	−6.07217e11	−0.172846	−0.0864232	−	0.996259i	$-0.527544\pi$
$191$	−6.07217e11	−0.172846	−0.0864232	+	0.996259i	$0.527544\pi$
$192$	0	0
$193$	− 4.64252e12i	− 1.24793i	−0.781454	−	0.623963i	$-0.785522\pi$
$193$	− 4.64252e12i	− 1.24793i	0.781454	−	0.623963i	$-0.214478\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.86139e12i	− 1.16734i	−0.811992	−	0.583668i	$-0.801617\pi$
$197$	− 4.86139e12i	− 1.16734i	0.811992	−	0.583668i	$-0.198383\pi$
$198$	0	0
$199$	−6.52349e12	−1.48180	−0.740898	−	0.671618i	$-0.765600\pi$
$199$	−6.52349e12	−1.48180	−0.740898	+	0.671618i	$0.765600\pi$
$200$	0	0
$201$	−4.79837e12	−1.03161
$202$	0	0
$203$	5.49138e12i	1.11803i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.22513e11i	0.0772686i
$208$	0	0
$209$	−5.36844e12	−0.931202
$210$	0	0
$211$	1.89227e12	0.311479	0.155740	−	0.987798i	$-0.450224\pi$
$211$	1.89227e12	0.311479	0.155740	+	0.987798i	$0.450224\pi$
$212$	0	0
$213$	7.84583e12i	1.22617i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.73597e11i	0.137357i
$218$	0	0
$219$	−2.33219e12	−0.312839
$220$	0	0
$221$	−7.55099e12	−0.963488
$222$	0	0
$223$	− 6.33591e12i	− 0.769365i	−0.923049	−	0.384682i	$-0.874311\pi$
$223$	− 6.33591e12i	− 0.769365i	0.923049	−	0.384682i	$-0.125689\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.95535e12i	0.545673i	0.962060	+	0.272837i	$0.0879618\pi$
$227$	4.95535e12i	0.545673i	−0.962060	+	0.272837i	$0.912038\pi$
$228$	0	0
$229$	−1.55285e13	−1.62943	−0.814714	−	0.579863i	$-0.803106\pi$
$229$	−1.55285e13	−1.62943	−0.814714	+	0.579863i	$0.803106\pi$
$230$	0	0
$231$	−1.96483e13	−1.96543
$232$	0	0
$233$	− 7.75664e12i	− 0.739973i	−0.929037	−	0.369986i	$-0.879362\pi$
$233$	− 7.75664e12i	− 0.739973i	0.929037	−	0.369986i	$-0.120638\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.73317e13i	3.24311i
$238$	0	0
$239$	−4.18729e12	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$239$	−4.18729e12	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$240$	0	0
$241$	8.28156e12	0.656173	0.328087	−	0.944648i	$-0.393596\pi$
$241$	8.28156e12	0.656173	0.328087	+	0.944648i	$0.393596\pi$
$242$	0	0
$243$	1.00140e13i	0.758180i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.62194e13i	− 1.12254i
$248$	0	0
$249$	4.39772e13	2.91160
$250$	0	0
$251$	−6.31791e11	−0.0400284	−0.0200142	−	0.999800i	$-0.506371\pi$
$251$	−6.31791e11	−0.0400284	−0.0200142	+	0.999800i	$0.506371\pi$
$252$	0	0
$253$	5.26918e11i	0.0319580i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.76085e13i	0.979691i	0.871809	+	0.489845i	$0.162947\pi$
$257$	1.76085e13i	0.979691i	−0.871809	+	0.489845i	$0.837053\pi$
$258$	0	0
$259$	1.23045e12	0.0656018
$260$	0	0
$261$	−2.29470e13	−1.17274
$262$	0	0
$263$	2.18141e13i	1.06901i	0.845165	+	0.534505i	$0.179502\pi$
$263$	2.18141e13i	1.06901i	−0.845165	+	0.534505i	$0.820498\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.69985e13i	3.02171i
$268$	0	0
$269$	1.09453e13	0.473795	0.236897	−	0.971535i	$-0.423869\pi$
$269$	1.09453e13	0.473795	0.236897	+	0.971535i	$0.423869\pi$
$270$	0	0
$271$	−4.38029e13	−1.82042	−0.910210	−	0.414146i	$-0.864080\pi$
$271$	−4.38029e13	−1.82042	−0.910210	+	0.414146i	$0.864080\pi$
$272$	0	0
$273$	− 5.93622e13i	− 2.36927i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3.83236e13i	− 1.41198i	−0.708224	−	0.705988i	$-0.750503\pi$
$277$	− 3.83236e13i	− 1.41198i	0.708224	−	0.705988i	$-0.249497\pi$
$278$	0	0
$279$	−4.06840e12	−0.144079
$280$	0	0
$281$	−1.88363e13	−0.641372	−0.320686	−	0.947186i	$-0.603913\pi$
$281$	−1.88363e13	−0.641372	−0.320686	+	0.947186i	$0.603913\pi$
$282$	0	0
$283$	3.59883e12i	0.117852i	0.998262	+	0.0589258i	$0.0187675\pi$
$283$	3.59883e12i	0.117852i	−0.998262	+	0.0589258i	$0.981232\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.60185e13i	2.30446i
$288$	0	0
$289$	−8.19853e12	−0.239220
$290$	0	0
$291$	−3.04555e13	−0.855568
$292$	0	0
$293$	1.89960e13i	0.513913i	0.966423	+	0.256957i	$0.0827198\pi$
$293$	1.89960e13i	0.513913i	−0.966423	+	0.256957i	$0.917280\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.48081e13i	− 0.874018i
$298$	0	0
$299$	−1.59195e12	−0.0385245
$300$	0	0
$301$	1.42594e13	0.332649
$302$	0	0
$303$	1.17001e14i	2.63181i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.07622e13i	− 1.06238i	−0.847253	−	0.531189i	$-0.821745\pi$
$307$	− 5.07622e13i	− 1.06238i	0.847253	−	0.531189i	$-0.178255\pi$
$308$	0	0
$309$	−4.19326e13	−0.846799
$310$	0	0
$311$	4.89908e13	0.954844	0.477422	−	0.878674i	$-0.341571\pi$
$311$	4.89908e13	0.954844	0.477422	+	0.878674i	$0.341571\pi$
$312$	0	0
$313$	− 2.02897e13i	− 0.381752i	−0.981614	−	0.190876i	$-0.938867\pi$
$313$	− 2.02897e13i	− 0.381752i	0.981614	−	0.190876i	$-0.0611329\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.80299e13i	1.01818i	0.860712	+	0.509091i	$0.170018\pi$
$317$	5.80299e13i	1.01818i	−0.860712	+	0.509091i	$0.829982\pi$
$318$	0	0
$319$	−2.86173e13	−0.485043
$320$	0	0
$321$	9.70332e13	1.58907
$322$	0	0
$323$	− 9.12256e13i	− 1.44379i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.32636e13i	1.23152i
$328$	0	0
$329$	8.53702e13	1.22104
$330$	0	0
$331$	−1.94810e13	−0.269500	−0.134750	−	0.990880i	$-0.543023\pi$
$331$	−1.94810e13	−0.269500	−0.134750	+	0.990880i	$0.543023\pi$
$332$	0	0
$333$	5.14172e12i	0.0688122i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.11994e14i	− 1.40356i	−0.712392	−	0.701782i	$-0.752388\pi$
$337$	− 1.11994e14i	− 1.40356i	0.712392	−	0.701782i	$-0.247612\pi$
$338$	0	0
$339$	−1.87465e13	−0.227417
$340$	0	0
$341$	−5.07372e12	−0.0595905
$342$	0	0
$343$	− 1.07522e14i	− 1.22287i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.43917e14i	− 1.53568i	−0.640640	−	0.767841i	$-0.721331\pi$
$347$	− 1.43917e14i	− 1.53568i	0.640640	−	0.767841i	$-0.278669\pi$
$348$	0	0
$349$	−1.17413e14	−1.21389	−0.606943	−	0.794745i	$-0.707604\pi$
$349$	−1.17413e14	−1.21389	−0.606943	+	0.794745i	$0.707604\pi$
$350$	0	0
$351$	1.05164e14	1.05360
$352$	0	0
$353$	3.14959e12i	0.0305839i	0.999883	+	0.0152919i	$0.00486777\pi$
$353$	3.14959e12i	0.0305839i	−0.999883	+	0.0152919i	$0.995132\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.33882e14i	− 3.04732i
$358$	0	0
$359$	2.16645e14	1.91748	0.958738	−	0.284291i	$-0.0917583\pi$
$359$	2.16645e14	1.91748	0.958738	+	0.284291i	$0.0917583\pi$
$360$	0	0
$361$	7.94605e13	0.682121
$362$	0	0
$363$	9.62349e13i	0.801394i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.32924e14i	− 1.04217i	−0.853505	−	0.521085i	$-0.825527\pi$
$367$	− 1.32924e14i	− 1.04217i	0.853505	−	0.521085i	$-0.174473\pi$
$368$	0	0
$369$	−3.17660e14	−2.41723
$370$	0	0
$371$	3.27014e13	0.241551
$372$	0	0
$373$	− 4.97449e13i	− 0.356738i	−0.983964	−	0.178369i	$-0.942918\pi$
$373$	− 4.97449e13i	− 0.356738i	0.983964	−	0.178369i	$-0.0570821\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.64598e13i	− 0.584705i
$378$	0	0
$379$	−6.41630e13	−0.421472	−0.210736	−	0.977543i	$-0.567586\pi$
$379$	−6.41630e13	−0.421472	−0.210736	+	0.977543i	$0.567586\pi$
$380$	0	0
$381$	−1.13917e14	−0.726942
$382$	0	0
$383$	− 1.53711e14i	− 0.953040i	−0.879164	−	0.476520i	$-0.841898\pi$
$383$	− 1.53711e14i	− 0.953040i	0.879164	−	0.476520i	$-0.158102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.95862e13i	0.348928i
$388$	0	0
$389$	−3.08115e14	−1.75384	−0.876921	−	0.480635i	$-0.840406\pi$
$389$	−3.08115e14	−1.75384	−0.876921	+	0.480635i	$0.840406\pi$
$390$	0	0
$391$	−8.95388e12	−0.0495495
$392$	0	0
$393$	1.77061e14i	0.952714i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.15252e14i	1.09547i	0.836653	+	0.547734i	$0.184509\pi$
$397$	2.15252e14i	1.09547i	−0.836653	+	0.547734i	$0.815491\pi$
$398$	0	0
$399$	7.17171e14	3.55035
$400$	0	0
$401$	−4.29414e13	−0.206815	−0.103407	−	0.994639i	$-0.532975\pi$
$401$	−4.29414e13	−0.206815	−0.103407	+	0.994639i	$0.532975\pi$
$402$	0	0
$403$	− 1.53289e13i	− 0.0718347i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.41227e12i	0.0284605i
$408$	0	0
$409$	1.78142e14	0.769639	0.384819	−	0.922992i	$-0.374264\pi$
$409$	1.78142e14	0.769639	0.384819	+	0.922992i	$0.374264\pi$
$410$	0	0
$411$	7.32869e14	3.08245
$412$	0	0
$413$	9.44211e12i	0.0386673i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.28187e14i	− 2.43972i
$418$	0	0
$419$	3.36601e14	1.27332	0.636662	−	0.771143i	$-0.280315\pi$
$419$	3.36601e14	1.27332	0.636662	+	0.771143i	$0.280315\pi$
$420$	0	0
$421$	6.36951e13	0.234722	0.117361	−	0.993089i	$-0.462557\pi$
$421$	6.36951e13	0.234722	0.117361	+	0.993089i	$0.462557\pi$
$422$	0	0
$423$	3.56739e14i	1.28079i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.85860e14i	1.99729i
$428$	0	0
$429$	3.09355e14	1.02788
$430$	0	0
$431$	1.38853e14	0.449707	0.224853	−	0.974393i	$-0.427810\pi$
$431$	1.38853e14	0.449707	0.224853	+	0.974393i	$0.427810\pi$
$432$	0	0
$433$	3.07253e14i	0.970093i	0.874488	+	0.485046i	$0.161197\pi$
$433$	3.07253e14i	0.970093i	−0.874488	+	0.485046i	$0.838803\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.92328e13i	− 0.0577289i
$438$	0	0
$439$	2.37747e14	0.695921	0.347961	−	0.937509i	$-0.386874\pi$
$439$	2.37747e14	0.695921	0.347961	+	0.937509i	$0.386874\pi$
$440$	0	0
$441$	1.05737e15	3.01866
$442$	0	0
$443$	6.82803e13i	0.190141i	0.995471	+	0.0950703i	$0.0303076\pi$
$443$	6.82803e13i	0.190141i	−0.995471	+	0.0950703i	$0.969692\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.53881e14i	1.99807i
$448$	0	0
$449$	−7.19557e14	−1.86084	−0.930422	−	0.366490i	$-0.880560\pi$
$449$	−7.19557e14	−1.86084	−0.930422	+	0.366490i	$0.880560\pi$
$450$	0	0
$451$	−3.96156e14	−0.999758
$452$	0	0
$453$	− 6.26370e14i	− 1.54273i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.95803e14i	− 1.39818i	−0.715033	−	0.699091i	$-0.753588\pi$
$457$	− 5.95803e14i	− 1.39818i	0.715033	−	0.699091i	$-0.246412\pi$
$458$	0	0
$459$	5.91492e14	1.35513
$460$	0	0
$461$	−2.48547e14	−0.555973	−0.277987	−	0.960585i	$-0.589667\pi$
$461$	−2.48547e14	−0.555973	−0.277987	+	0.960585i	$0.589667\pi$
$462$	0	0
$463$	5.55882e14i	1.21419i	0.794629	+	0.607096i	$0.207666\pi$
$463$	5.55882e14i	1.21419i	−0.794629	+	0.607096i	$0.792334\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.11081e13i	0.106475i	0.998582	+	0.0532374i	$0.0169540\pi$
$467$	5.11081e13i	0.106475i	−0.998582	+	0.0532374i	$0.983046\pi$
$468$	0	0
$469$	5.07224e14	1.03217
$470$	0	0
$471$	−4.21620e14	−0.838121
$472$	0	0
$473$	7.43103e13i	0.144315i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.36650e14i	0.253372i
$478$	0	0
$479$	−6.79608e14	−1.23144	−0.615720	−	0.787965i	$-0.711135\pi$
$479$	−6.79608e14	−1.23144	−0.615720	+	0.787965i	$0.711135\pi$
$480$	0	0
$481$	−1.93730e13	−0.0343083
$482$	0	0
$483$	− 7.03910e13i	− 0.121845i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.50601e14i	− 0.579967i	−0.957032	−	0.289984i	$-0.906350\pi$
$487$	− 3.50601e14i	− 0.579967i	0.957032	−	0.289984i	$-0.0936499\pi$
$488$	0	0
$489$	−4.12152e14	−0.666590
$490$	0	0
$491$	−5.60616e14	−0.886579	−0.443289	−	0.896379i	$-0.646189\pi$
$491$	−5.60616e14	−0.886579	−0.443289	+	0.896379i	$0.646189\pi$
$492$	0	0
$493$	− 4.86292e14i	− 0.752037i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.29365e14i	− 1.22683i
$498$	0	0
$499$	2.60119e14	0.376374	0.188187	−	0.982133i	$-0.439739\pi$
$499$	2.60119e14	0.376374	0.188187	+	0.982133i	$0.439739\pi$
$500$	0	0
$501$	−3.95306e13	−0.0559533
$502$	0	0
$503$	− 1.34386e15i	− 1.86093i	−0.366377	−	0.930466i	$-0.619402\pi$
$503$	− 1.34386e15i	− 1.86093i	0.366377	−	0.930466i	$-0.380598\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.13029e14i	− 0.414994i
$508$	0	0
$509$	−3.28205e14	−0.425792	−0.212896	−	0.977075i	$-0.568290\pi$
$509$	−3.28205e14	−0.425792	−0.212896	+	0.977075i	$0.568290\pi$
$510$	0	0
$511$	2.46530e14	0.313007
$512$	0	0
$513$	1.27051e15i	1.57882i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.44891e14i	0.529731i
$518$	0	0
$519$	−1.58681e15	−1.84971
$520$	0	0
$521$	3.73999e14	0.426838	0.213419	−	0.976961i	$-0.431540\pi$
$521$	3.73999e14	0.426838	0.213419	+	0.976961i	$0.431540\pi$
$522$	0	0
$523$	− 7.77988e13i	− 0.0869388i	−0.999055	−	0.0434694i	$-0.986159\pi$
$523$	− 7.77988e13i	− 0.0869388i	0.999055	−	0.0434694i	$-0.0138411\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.62173e13i	− 0.0923924i
$528$	0	0
$529$	9.50922e14	0.998019
$530$	0	0
$531$	−3.94560e13	−0.0405596
$532$	0	0
$533$	− 1.19688e15i	− 1.20518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.33487e15i	− 1.28997i
$538$	0	0
$539$	1.31865e15	1.24851
$540$	0	0
$541$	−1.17909e15	−1.09386	−0.546929	−	0.837179i	$-0.684203\pi$
$541$	−1.17909e15	−1.09386	−0.546929	+	0.837179i	$0.684203\pi$
$542$	0	0
$543$	2.51525e15i	2.28656i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.58961e14i	0.662658i	0.943515	+	0.331329i	$0.107497\pi$
$547$	7.58961e14i	0.662658i	−0.943515	+	0.331329i	$0.892503\pi$
$548$	0	0
$549$	−2.44815e15	−2.09503
$550$	0	0
$551$	1.04455e15	0.876180
$552$	0	0
$553$	− 3.94625e15i	− 3.24486i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.88953e14i	− 0.781578i	−0.920480	−	0.390789i	$-0.872202\pi$
$557$	− 9.88953e14i	− 0.781578i	0.920480	−	0.390789i	$-0.127798\pi$
$558$	0	0
$559$	−2.24509e14	−0.173968
$560$	0	0
$561$	1.73996e15	1.32204
$562$	0	0
$563$	1.70471e15i	1.27015i	0.772450	+	0.635075i	$0.219031\pi$
$563$	1.70471e15i	1.27015i	−0.772450	+	0.635075i	$0.780969\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.41044e14i	0.459389i
$568$	0	0
$569$	−5.26889e14	−0.370341	−0.185170	−	0.982706i	$-0.559284\pi$
$569$	−5.26889e14	−0.370341	−0.185170	+	0.982706i	$0.559284\pi$
$570$	0	0
$571$	−6.85464e14	−0.472592	−0.236296	−	0.971681i	$-0.575933\pi$
$571$	−6.85464e14	−0.472592	−0.236296	+	0.971681i	$0.575933\pi$
$572$	0	0
$573$	4.22732e14i	0.285900i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.25421e15i	0.816402i	0.912892	+	0.408201i	$0.133844\pi$
$577$	1.25421e15i	0.816402i	−0.912892	+	0.408201i	$0.866156\pi$
$578$	0	0
$579$	−3.23203e15	−2.06416
$580$	0	0
$581$	−4.64873e15	−2.91317
$582$	0	0
$583$	1.70417e14i	0.104794i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.56777e15i	1.52071i	0.649507	+	0.760356i	$0.274975\pi$
$587$	2.56777e15i	1.52071i	−0.649507	+	0.760356i	$0.725025\pi$
$588$	0	0
$589$	1.85193e14	0.107644
$590$	0	0
$591$	−3.38440e15	−1.93086
$592$	0	0
$593$	1.05019e15i	0.588123i	0.955786	+	0.294062i	$0.0950071\pi$
$593$	1.05019e15i	0.588123i	−0.955786	+	0.294062i	$0.904993\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.54152e15i	2.45100i
$598$	0	0
$599$	−1.16345e14	−0.0616455	−0.0308227	−	0.999525i	$-0.509813\pi$
$599$	−1.16345e14	−0.0616455	−0.0308227	+	0.999525i	$0.509813\pi$
$600$	0	0
$601$	−9.27714e14	−0.482619	−0.241310	−	0.970448i	$-0.577577\pi$
$601$	−9.27714e14	−0.482619	−0.241310	+	0.970448i	$0.577577\pi$
$602$	0	0
$603$	2.11955e15i	1.08268i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9.37322e14i	− 0.461691i	−0.972990	−	0.230845i	$-0.925851\pi$
$607$	− 9.37322e14i	− 0.461691i	0.972990	−	0.230845i	$-0.0741491\pi$
$608$	0	0
$609$	3.82299e15	1.84930
$610$	0	0
$611$	−1.34412e15	−0.638576
$612$	0	0
$613$	− 9.73439e13i	− 0.0454230i	−0.999742	−	0.0227115i	$-0.992770\pi$
$613$	− 9.73439e13i	− 0.0454230i	0.999742	−	0.0227115i	$-0.00722992\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.70501e15i	− 1.21787i	−0.793220	−	0.608935i	$-0.791597\pi$
$617$	− 2.70501e15i	− 1.21787i	0.793220	−	0.608935i	$-0.208403\pi$
$618$	0	0
$619$	−5.22797e14	−0.231225	−0.115612	−	0.993294i	$-0.536883\pi$
$619$	−5.22797e14	−0.231225	−0.115612	+	0.993294i	$0.536883\pi$
$620$	0	0
$621$	1.24702e14	0.0541839
$622$	0	0
$623$	− 7.08225e15i	− 3.02334i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.73740e15i	1.54027i
$628$	0	0
$629$	−1.08963e14	−0.0441267
$630$	0	0
$631$	−1.58461e15	−0.630611	−0.315305	−	0.948990i	$-0.602107\pi$
$631$	−1.58461e15	−0.630611	−0.315305	+	0.948990i	$0.602107\pi$
$632$	0	0
$633$	− 1.31736e15i	− 0.515209i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.98397e15i	1.50504i
$638$	0	0
$639$	3.46569e15	1.28687
$640$	0	0
$641$	3.67621e15	1.34178	0.670890	−	0.741557i	$-0.265912\pi$
$641$	3.67621e15	1.34178	0.670890	+	0.741557i	$0.265912\pi$
$642$	0	0
$643$	− 1.40268e15i	− 0.503268i	−0.967822	−	0.251634i	$-0.919032\pi$
$643$	− 1.40268e15i	− 0.503268i	0.967822	−	0.251634i	$-0.0809680\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.65400e14i	0.265409i	0.991156	+	0.132704i	$0.0423661\pi$
$647$	7.65400e14i	0.265409i	−0.991156	+	0.132704i	$0.957634\pi$
$648$	0	0
$649$	−4.92058e13	−0.0167753
$650$	0	0
$651$	6.77798e14	0.227198
$652$	0	0
$653$	3.06409e15i	1.00990i	0.863148	+	0.504951i	$0.168490\pi$
$653$	3.06409e15i	1.00990i	−0.863148	+	0.504951i	$0.831510\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.03018e15i	0.328325i
$658$	0	0
$659$	5.49550e15	1.72241	0.861207	−	0.508254i	$-0.169709\pi$
$659$	5.49550e15	1.72241	0.861207	+	0.508254i	$0.169709\pi$
$660$	0	0
$661$	−4.42413e15	−1.36370	−0.681851	−	0.731491i	$-0.738825\pi$
$661$	−4.42413e15	−1.36370	−0.681851	+	0.731491i	$0.738825\pi$
$662$	0	0
$663$	5.25685e15i	1.59368i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.02523e14i	− 0.0300697i
$668$	0	0
$669$	−4.41093e15	−1.27258
$670$	0	0
$671$	−3.05310e15	−0.866497
$672$	0	0
$673$	3.72652e15i	1.04045i	0.854029	+	0.520225i	$0.174152\pi$
$673$	3.72652e15i	1.04045i	−0.854029	+	0.520225i	$0.825848\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.80105e15i	1.02723i	0.858022	+	0.513613i	$0.171693\pi$
$677$	3.80105e15i	1.02723i	−0.858022	+	0.513613i	$0.828307\pi$
$678$	0	0
$679$	3.21938e15	0.856030
$680$	0	0
$681$	3.44982e15	0.902582
$682$	0	0
$683$	− 4.06733e15i	− 1.04712i	−0.851990	−	0.523558i	$-0.824604\pi$
$683$	− 4.06733e15i	− 1.04712i	0.851990	−	0.523558i	$-0.175396\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.08106e16i	2.69519i
$688$	0	0
$689$	−5.14871e14	−0.126326
$690$	0	0
$691$	−5.14416e15	−1.24218	−0.621091	−	0.783739i	$-0.713310\pi$
$691$	−5.14416e15	−1.24218	−0.621091	+	0.783739i	$0.713310\pi$
$692$	0	0
$693$	8.67910e15i	2.06273i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.73185e15i	− 1.55008i
$698$	0	0
$699$	−5.40001e15	−1.22397
$700$	0	0
$701$	−2.86572e15	−0.639417	−0.319708	−	0.947516i	$-0.603585\pi$
$701$	−2.86572e15	−0.639417	−0.319708	+	0.947516i	$0.603585\pi$
$702$	0	0
$703$	− 2.34051e14i	− 0.0514110i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.23679e16i	− 2.63323i
$708$	0	0
$709$	−5.54221e15	−1.16179	−0.580896	−	0.813978i	$-0.697298\pi$
$709$	−5.54221e15	−1.16179	−0.580896	+	0.813978i	$0.697298\pi$
$710$	0	0
$711$	1.64903e16	3.40366
$712$	0	0
$713$	− 1.81769e13i	− 0.00369425i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.91511e15i	0.574512i
$718$	0	0
$719$	−4.73660e15	−0.919301	−0.459651	−	0.888100i	$-0.652025\pi$
$719$	−4.73660e15	−0.919301	−0.459651	+	0.888100i	$0.652025\pi$
$720$	0	0
$721$	4.43260e15	0.847256
$722$	0	0
$723$	− 5.76545e15i	− 1.08536i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.01777e16i	− 1.85870i	−0.369205	−	0.929348i	$-0.620370\pi$
$727$	− 1.01777e16i	− 1.85870i	0.369205	−	0.929348i	$-0.379630\pi$
$728$	0	0
$729$	8.51462e15	1.53167
$730$	0	0
$731$	−1.26275e15	−0.223754
$732$	0	0
$733$	− 5.29486e15i	− 0.924235i	−0.886819	−	0.462117i	$-0.847090\pi$
$733$	− 5.29486e15i	− 0.924235i	0.886819	−	0.462117i	$-0.152910\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.64330e15i	0.447792i
$738$	0	0
$739$	−4.55855e15	−0.760821	−0.380411	−	0.924818i	$-0.624217\pi$
$739$	−4.55855e15	−0.760821	−0.380411	+	0.924818i	$0.624217\pi$
$740$	0	0
$741$	−1.12916e16	−1.85676
$742$	0	0
$743$	− 9.91668e14i	− 0.160667i	−0.996768	−	0.0803337i	$-0.974401\pi$
$743$	− 9.91668e14i	− 0.160667i	0.996768	−	0.0803337i	$-0.0255986\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.94258e16i	− 3.05573i
$748$	0	0
$749$	−1.02572e16	−1.58993
$750$	0	0
$751$	5.23831e15	0.800151	0.400075	−	0.916482i	$-0.368984\pi$
$751$	5.23831e15	0.800151	0.400075	+	0.916482i	$0.368984\pi$
$752$	0	0
$753$	4.39840e14i	0.0662098i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 7.31380e15i	− 1.06934i	−0.845061	−	0.534670i	$-0.820436\pi$
$757$	− 7.31380e15i	− 1.06934i	0.845061	−	0.534670i	$-0.179564\pi$
$758$	0	0
$759$	3.66829e14	0.0528609
$760$	0	0
$761$	−7.81344e15	−1.10975	−0.554877	−	0.831932i	$-0.687235\pi$
$761$	−7.81344e15	−1.10975	−0.554877	+	0.831932i	$0.687235\pi$
$762$	0	0
$763$	− 8.80161e15i	− 1.23219i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.48663e14i	− 0.0202222i
$768$	0	0
$769$	−2.19554e15	−0.294405	−0.147203	−	0.989106i	$-0.547027\pi$
$769$	−2.19554e15	−0.294405	−0.147203	+	0.989106i	$0.547027\pi$
$770$	0	0
$771$	1.22586e16	1.62048
$772$	0	0
$773$	− 1.32704e16i	− 1.72940i	−0.502286	−	0.864702i	$-0.667507\pi$
$773$	− 1.32704e16i	− 1.72940i	0.502286	−	0.864702i	$-0.332493\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.56615e14i	− 0.108510i
$778$	0	0
$779$	1.44599e16	1.80596
$780$	0	0
$781$	4.32208e15	0.532246
$782$	0	0
$783$	6.77266e15i	0.822375i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.40226e15i	0.165565i	0.996568	+	0.0827826i	$0.0263807\pi$
$787$	1.40226e15i	0.165565i	−0.996568	+	0.0827826i	$0.973619\pi$
$788$	0	0
$789$	1.51866e16	1.76822
$790$	0	0
$791$	1.98165e15	0.227539
$792$	0	0
$793$	− 9.22415e15i	− 1.04454i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.09627e15i	0.341049i	0.985353	+	0.170525i	$0.0545463\pi$
$797$	3.09627e15i	0.341049i	−0.985353	+	0.170525i	$0.945454\pi$
$798$	0	0
$799$	−7.56000e15	−0.821324
$800$	0	0
$801$	2.95948e16	3.17130
$802$	0	0
$803$	1.28474e15i	0.135794i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.61990e15i	− 0.783691i
$808$	0	0
$809$	1.62765e16	1.65137	0.825683	−	0.564135i	$-0.190790\pi$
$809$	1.62765e16	1.65137	0.825683	+	0.564135i	$0.190790\pi$
$810$	0	0
$811$	1.55106e16	1.55244	0.776219	−	0.630463i	$-0.217135\pi$
$811$	1.55106e16	1.55244	0.776219	+	0.630463i	$0.217135\pi$
$812$	0	0
$813$	3.04947e16i	3.01111i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.71236e15i	− 0.260691i
$818$	0	0
$819$	−2.62217e16	−2.48656
$820$	0	0
$821$	−5.57057e15	−0.521209	−0.260605	−	0.965446i	$-0.583922\pi$
$821$	−5.57057e15	−0.521209	−0.260605	+	0.965446i	$0.583922\pi$
$822$	0	0
$823$	− 4.05461e15i	− 0.374326i	−0.982329	−	0.187163i	$-0.940071\pi$
$823$	− 4.05461e15i	− 0.374326i	0.982329	−	0.187163i	$-0.0599292\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.59579e14i	0.0862582i	0.999070	+	0.0431291i	$0.0137327\pi$
$827$	9.59579e14i	0.0862582i	−0.999070	+	0.0431291i	$0.986267\pi$
$828$	0	0
$829$	1.05269e16	0.933791	0.466895	−	0.884313i	$-0.345373\pi$
$829$	1.05269e16	0.933791	0.466895	+	0.884313i	$0.345373\pi$
$830$	0	0
$831$	−2.66801e16	−2.33551
$832$	0	0
$833$	2.24078e16i	1.93576i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.20076e15i	0.101034i
$838$	0	0
$839$	−6.15316e15	−0.510984	−0.255492	−	0.966811i	$-0.582238\pi$
$839$	−6.15316e15	−0.510984	−0.255492	+	0.966811i	$0.582238\pi$
$840$	0	0
$841$	−6.63240e15	−0.543617
$842$	0	0
$843$	1.31134e16i	1.06087i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.01728e16i	− 0.801827i
$848$	0	0
$849$	2.50543e15	0.194935
$850$	0	0
$851$	−2.29723e13	−0.00176438
$852$	0	0
$853$	− 1.77560e16i	− 1.34625i	−0.739528	−	0.673126i	$-0.764951\pi$
$853$	− 1.77560e16i	− 1.34625i	0.739528	−	0.673126i	$-0.235049\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.26793e16i	1.67585i	0.545785	+	0.837925i	$0.316231\pi$
$857$	2.26793e16i	1.67585i	−0.545785	+	0.837925i	$0.683769\pi$
$858$	0	0
$859$	1.65429e15	0.120684	0.0603421	−	0.998178i	$-0.480781\pi$
$859$	1.65429e15	0.120684	0.0603421	+	0.998178i	$0.480781\pi$
$860$	0	0
$861$	5.29225e16	3.81174
$862$	0	0
$863$	2.31182e15i	0.164397i	0.996616	+	0.0821986i	$0.0261942\pi$
$863$	2.31182e15i	0.164397i	−0.996616	+	0.0821986i	$0.973806\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.70765e15i	0.395688i
$868$	0	0
$869$	2.05651e16	1.40774
$870$	0	0
$871$	−7.98606e15	−0.539800
$872$	0	0
$873$	1.34529e16i	0.897921i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.39330e16i	0.906874i	0.891288	+	0.453437i	$0.149802\pi$
$877$	1.39330e16i	0.906874i	−0.891288	+	0.453437i	$0.850198\pi$
$878$	0	0
$879$	1.32246e16	0.850050
$880$	0	0
$881$	4.04171e15	0.256565	0.128283	−	0.991738i	$-0.459054\pi$
$881$	4.04171e15	0.256565	0.128283	+	0.991738i	$0.459054\pi$
$882$	0	0
$883$	1.26327e16i	0.791975i	0.918256	+	0.395988i	$0.129598\pi$
$883$	1.26327e16i	0.791975i	−0.918256	+	0.395988i	$0.870402\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.96489e16i	− 1.20160i	−0.799401	−	0.600798i	$-0.794849\pi$
$887$	− 1.96489e16i	− 1.20160i	0.799401	−	0.600798i	$-0.205151\pi$
$888$	0	0
$889$	1.20419e16	0.727334
$890$	0	0
$891$	−3.34068e15	−0.199300
$892$	0	0
$893$	− 1.62387e16i	− 0.956905i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.10828e15i	0.0637223i
$898$	0	0
$899$	9.87200e14	0.0560695
$900$	0	0
$901$	−2.89588e15	−0.162478
$902$	0	0
$903$	− 9.92712e15i	− 0.550225i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.75609e15i	0.0949961i	0.998871	+	0.0474981i	$0.0151248\pi$
$907$	1.75609e15i	0.0949961i	−0.998871	+	0.0474981i	$0.984875\pi$
$908$	0	0
$909$	5.16820e16	2.76209
$910$	0	0
$911$	2.30636e16	1.21780	0.608899	−	0.793247i	$-0.291611\pi$
$911$	2.30636e16	1.21780	0.608899	+	0.793247i	$0.291611\pi$
$912$	0	0
$913$	− 2.42260e16i	− 1.26384i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.87167e16i	− 0.953229i
$918$	0	0
$919$	−9.41404e14	−0.0473741	−0.0236870	−	0.999719i	$-0.507541\pi$
$919$	−9.41404e14	−0.0473741	−0.0236870	+	0.999719i	$0.507541\pi$
$920$	0	0
$921$	−3.53396e16	−1.75725
$922$	0	0
$923$	1.30581e16i	0.641607i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.85226e16i	0.888718i
$928$	0	0
$929$	−1.77556e16	−0.841876	−0.420938	−	0.907089i	$-0.638299\pi$
$929$	−1.77556e16	−0.841876	−0.420938	+	0.907089i	$0.638299\pi$
$930$	0	0
$931$	−4.81314e16	−2.25530
$932$	0	0
$933$	− 3.41064e16i	− 1.57938i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.10107e16i	0.498020i	0.968501	+	0.249010i	$0.0801053\pi$
$937$	1.10107e16i	0.498020i	−0.968501	+	0.249010i	$0.919895\pi$
$938$	0	0
$939$	−1.41253e16	−0.631446
$940$	0	0
$941$	−2.50908e16	−1.10859	−0.554295	−	0.832320i	$-0.687012\pi$
$941$	−2.50908e16	−1.10859	−0.554295	+	0.832320i	$0.687012\pi$
$942$	0	0
$943$	− 1.41925e15i	− 0.0619790i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.97365e16i	0.842064i	0.907046	+	0.421032i	$0.138332\pi$
$947$	1.97365e16i	0.842064i	−0.907046	+	0.421032i	$0.861668\pi$
$948$	0	0
$949$	−3.88152e15	−0.163696
$950$	0	0
$951$	4.03992e16	1.68415
$952$	0	0
$953$	− 1.28343e16i	− 0.528884i	−0.964402	−	0.264442i	$-0.914812\pi$
$953$	− 1.28343e16i	− 0.528884i	0.964402	−	0.264442i	$-0.0851878\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.99228e16i	0.802296i
$958$	0	0
$959$	−7.74699e16	−3.08411
$960$	0	0
$961$	−2.52335e16	−0.993112
$962$	0	0
$963$	− 4.28619e16i	− 1.66773i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.73110e16i	− 1.79935i	−0.436558	−	0.899676i	$-0.643803\pi$
$967$	− 4.73110e16i	− 1.79935i	0.436558	−	0.899676i	$-0.356197\pi$
$968$	0	0
$969$	−6.35094e16	−2.38813
$970$	0	0
$971$	2.69052e16	1.00030	0.500149	−	0.865939i	$-0.333278\pi$
$971$	2.69052e16	1.00030	0.500149	+	0.865939i	$0.333278\pi$
$972$	0	0
$973$	6.64042e16i	2.44104i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.19615e16i	− 0.429899i	−0.976625	−	0.214950i	$-0.931041\pi$
$977$	− 1.19615e16i	− 0.429899i	0.976625	−	0.214950i	$-0.0689588\pi$
$978$	0	0
$979$	3.69078e16	1.31164
$980$	0	0
$981$	3.67795e16	1.29249
$982$	0	0
$983$	4.93179e16i	1.71380i	0.515483	+	0.856900i	$0.327613\pi$
$983$	4.93179e16i	1.71380i	−0.515483	+	0.856900i	$0.672387\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5.94330e16i	− 2.01968i
$988$	0	0
$989$	−2.66221e14	−0.00894669
$990$	0	0
$991$	4.42856e16	1.47183	0.735915	−	0.677074i	$-0.236753\pi$
$991$	4.42856e16	1.47183	0.735915	+	0.677074i	$0.236753\pi$
$992$	0	0
$993$	1.35623e16i	0.445772i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.10581e16i	0.677011i	0.940964	+	0.338506i	$0.109921\pi$
$997$	2.10581e16i	0.677011i	−0.940964	+	0.338506i	$0.890079\pi$
$998$	0	0
$999$	1.51755e15	0.0482539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.12.c.c.49.1		4
5.2	odd	4		200.12.a.d.1.1		2
5.3	odd	4		8.12.a.b.1.2	✓	2
5.4	even	2	inner	200.12.c.c.49.4		4
15.8	even	4		72.12.a.e.1.2		2
20.3	even	4		16.12.a.d.1.1		2
40.3	even	4		64.12.a.k.1.2		2
40.13	odd	4		64.12.a.h.1.1		2
60.23	odd	4		144.12.a.p.1.2		2
80.3	even	4		256.12.b.k.129.1		4
80.13	odd	4		256.12.b.h.129.4		4
80.43	even	4		256.12.b.k.129.4		4
80.53	odd	4		256.12.b.h.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.12.a.b.1.2	✓	2	5.3	odd	4
16.12.a.d.1.1		2	20.3	even	4
64.12.a.h.1.1		2	40.13	odd	4
64.12.a.k.1.2		2	40.3	even	4
72.12.a.e.1.2		2	15.8	even	4
144.12.a.p.1.2		2	60.23	odd	4
200.12.a.d.1.1		2	5.2	odd	4
200.12.c.c.49.1		4	1.1	even	1	trivial
200.12.c.c.49.4		4	5.4	even	2	inner
256.12.b.h.129.1		4	80.53	odd	4
256.12.b.h.129.4		4	80.13	odd	4
256.12.b.k.129.1		4	80.3	even	4
256.12.b.k.129.4		4	80.43	even	4

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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-696.180i q^{3} +73591.5i q^{7} -307519. q^{9} -383508. q^{11} -1.15867e6i q^{13} -6.51693e6i q^{17} +1.39982e7 q^{19} +5.12329e7 q^{21} -1.37394e6i q^{23} +9.07624e7i q^{27} +7.46197e7 q^{29} +1.32297e7 q^{31} +2.66991e8i q^{33} -1.67200e7i q^{37} -8.06644e8 q^{39} +1.03298e9 q^{41} -1.93764e8i q^{43} -1.16005e9i q^{47} -3.43839e9 q^{49} -4.53696e9 q^{51} -4.44363e8i q^{53} -9.74529e9i q^{57} +1.28304e8 q^{59} +7.96097e9 q^{61} -2.26308e10i q^{63} -6.89243e9i q^{67} -9.56509e8 q^{69} -1.12698e10 q^{71} -3.34998e9i q^{73} -2.82230e10i q^{77} -5.36237e10 q^{79} +8.71083e9 q^{81} +6.31693e10i q^{83} -5.19487e10i q^{87} -9.62373e10 q^{89} +8.52685e10 q^{91} -9.21027e9i q^{93} -4.37466e10i q^{97} +1.17936e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1080404 q^{9} + 318160 q^{11} + 43733200 q^{19} + 80105088 q^{21} + 457655400 q^{29} + 129444224 q^{31} - 1751818144 q^{39} + 2402428392 q^{41} - 3532138148 q^{49} - 19249975840 q^{51} + 12025853168 q^{59}+ \cdots - 16629834512 q^{99}+O(q^{100}) \) 4 * q - 1080404 * q^9 + 318160 * q^11 + 43733200 * q^19 + 80105088 * q^21 + 457655400 * q^29 + 129444224 * q^31 - 1751818144 * q^39 + 2402428392 * q^41 - 3532138148 * q^49 - 19249975840 * q^51 + 12025853168 * q^59 + 19579585816 * q^61 + 42173328512 * q^69 + 8639982176 * q^71 - 103915246528 * q^79 - 19495920604 * q^81 - 142376583720 * q^89 + 166757785152 * q^91 - 16629834512 * q^99

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