LMFDB - Embedded newform 252.9.c.a.197.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(197,252)]

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, 1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.197");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.c (of order $2$, degree $1$, 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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 $ x^16 + 4002260x^14 + 6534459751956x^12 + 5613923146579405376x^10 + 2733728904154246859079616x^8 + 757873148017661341349205888000x^6 + 113644318422397913452531577312640000x^4 + 8098650340007618970326973663348480000000*x^2 + 199066230990417435753898292645889849600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{37}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.10
Root			$348.567i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+348.567i q^{5} +907.493 q^{7} +O(q^{10})$	$q+348.567i q^{5} +907.493 q^{7} -11889.3i q^{11} -4095.93 q^{13} +67449.6i q^{17} -116874. q^{19} -68357.8i q^{23} +269126. q^{25} +774693. i q^{29} +807878. q^{31} +316322. i q^{35} +1.02479e6 q^{37} +850867. i q^{41} -3.69010e6 q^{43} -5.47190e6i q^{47} +823543. q^{49} -541263. i q^{53} +4.14422e6 q^{55} +8.95168e6i q^{59} +650282. q^{61} -1.42770e6i q^{65} -3.39158e7 q^{67} +1.63485e7i q^{71} +1.67469e7 q^{73} -1.07895e7i q^{77} -1.38232e7 q^{79} +1.98832e6i q^{83} -2.35107e7 q^{85} +6.05729e7i q^{89} -3.71702e6 q^{91} -4.07384e7i q^{95} -3.52027e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) $ 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	348.567i	0.557707i	0.960334	+	0.278853i	$0.0899544\pi$
$5$	348.567i	0.557707i	−0.960334	+	0.278853i	$0.910046\pi$
$6$	0	0
$7$	907.493	0.377964
$7$	907.493	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11889.3i	− 0.812056i	−0.913861	−	0.406028i	$-0.866913\pi$
$11$	− 11889.3i	− 0.812056i	0.913861	−	0.406028i	$-0.133087\pi$
$12$	0	0
$13$	−4095.93	−0.143410	−0.0717049	−	0.997426i	$-0.522844\pi$
$13$	−4095.93	−0.143410	−0.0717049	+	0.997426i	$0.522844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	67449.6i	0.807576i	0.914852	+	0.403788i	$0.132307\pi$
$17$	67449.6i	0.807576i	−0.914852	+	0.403788i	$0.867693\pi$
$18$	0	0
$19$	−116874.	−0.896816	−0.448408	−	0.893829i	$-0.648009\pi$
$19$	−116874.	−0.896816	−0.448408	+	0.893829i	$0.648009\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 68357.8i	− 0.244274i	−0.992513	−	0.122137i	$-0.961025\pi$
$23$	− 68357.8i	− 0.244274i	0.992513	−	0.122137i	$-0.0389747\pi$
$24$	0	0
$25$	269126.	0.688963
$26$	0	0
$27$	0	0
$28$	0	0
$29$	774693.i	1.09531i	0.836704	+	0.547656i	$0.184480\pi$
$29$	774693.i	1.09531i	−0.836704	+	0.547656i	$0.815520\pi$
$30$	0	0
$31$	807878.	0.874781	0.437390	−	0.899272i	$-0.355903\pi$
$31$	807878.	0.874781	0.437390	+	0.899272i	$0.355903\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	316322.i	0.210793i
$36$	0	0
$37$	1.02479e6	0.546799	0.273399	−	0.961901i	$-0.411852\pi$
$37$	1.02479e6	0.546799	0.273399	+	0.961901i	$0.411852\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	850867.i	0.301111i	0.988602	+	0.150555i	$0.0481062\pi$
$41$	850867.i	0.301111i	−0.988602	+	0.150555i	$0.951894\pi$
$42$	0	0
$43$	−3.69010e6	−1.07936	−0.539678	−	0.841872i	$-0.681454\pi$
$43$	−3.69010e6	−1.07936	−0.539678	+	0.841872i	$0.681454\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.47190e6i	− 1.12137i	−0.828031	−	0.560683i	$-0.810539\pi$
$47$	− 5.47190e6i	− 1.12137i	0.828031	−	0.560683i	$-0.189461\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 541263.i	− 0.0685969i	−0.999412	−	0.0342984i	$-0.989080\pi$
$53$	− 541263.i	− 0.0685969i	0.999412	−	0.0342984i	$-0.0109197\pi$
$54$	0	0
$55$	4.14422e6	0.452889
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.95168e6i	0.738748i	0.929281	+	0.369374i	$0.120428\pi$
$59$	8.95168e6i	0.738748i	−0.929281	+	0.369374i	$0.879572\pi$
$60$	0	0
$61$	650282.	0.0469659	0.0234829	−	0.999724i	$-0.492524\pi$
$61$	650282.	0.0469659	0.0234829	+	0.999724i	$0.492524\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.42770e6i	− 0.0799806i
$66$	0	0
$67$	−3.39158e7	−1.68307	−0.841536	−	0.540201i	$-0.818348\pi$
$67$	−3.39158e7	−1.68307	−0.841536	+	0.540201i	$0.818348\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.63485e7i	0.643346i	0.946851	+	0.321673i	$0.104245\pi$
$71$	1.63485e7i	0.643346i	−0.946851	+	0.321673i	$0.895755\pi$
$72$	0	0
$73$	1.67469e7	0.589716	0.294858	−	0.955541i	$-0.404728\pi$
$73$	1.67469e7	0.589716	0.294858	+	0.955541i	$0.404728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.07895e7i	− 0.306928i
$78$	0	0
$79$	−1.38232e7	−0.354895	−0.177448	−	0.984130i	$-0.556784\pi$
$79$	−1.38232e7	−0.354895	−0.177448	+	0.984130i	$0.556784\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.98832e6i	0.0418961i	0.999781	+	0.0209480i	$0.00666845\pi$
$83$	1.98832e6i	0.0418961i	−0.999781	+	0.0209480i	$0.993332\pi$
$84$	0	0
$85$	−2.35107e7	−0.450391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.05729e7i	0.965424i	0.875779	+	0.482712i	$0.160348\pi$
$89$	6.05729e7i	0.965424i	−0.875779	+	0.482712i	$0.839652\pi$
$90$	0	0
$91$	−3.71702e6	−0.0542038
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 4.07384e7i	− 0.500160i
$96$	0	0
$97$	−3.52027e7	−0.397639	−0.198819	−	0.980036i	$-0.563711\pi$
$97$	−3.52027e7	−0.397639	−0.198819	+	0.980036i	$0.563711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.19699e8i	1.15029i	0.818053	+	0.575143i	$0.195054\pi$
$101$	1.19699e8i	1.15029i	−0.818053	+	0.575143i	$0.804946\pi$
$102$	0	0
$103$	1.19816e7	0.106455	0.0532273	−	0.998582i	$-0.483049\pi$
$103$	1.19816e7	0.106455	0.0532273	+	0.998582i	$0.483049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.98152e8i	1.51169i	0.654749	+	0.755846i	$0.272774\pi$
$107$	1.98152e8i	1.51169i	−0.654749	+	0.755846i	$0.727226\pi$
$108$	0	0
$109$	−2.18524e7	−0.154808	−0.0774038	−	0.997000i	$-0.524663\pi$
$109$	−2.18524e7	−0.154808	−0.0774038	+	0.997000i	$0.524663\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.09321e8i	− 0.670485i	−0.942132	−	0.335243i	$-0.891182\pi$
$113$	− 1.09321e8i	− 0.670485i	0.942132	−	0.335243i	$-0.108818\pi$
$114$	0	0
$115$	2.38272e7	0.136233
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.12100e7i	0.305235i
$120$	0	0
$121$	7.30030e7	0.340564
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.29967e8i	0.941946i
$126$	0	0
$127$	−3.13988e8	−1.20698	−0.603488	−	0.797372i	$-0.706223\pi$
$127$	−3.13988e8	−1.20698	−0.603488	+	0.797372i	$0.706223\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.95386e8i	0.663451i	0.943376	+	0.331726i	$0.107631\pi$
$131$	1.95386e8i	0.663451i	−0.943376	+	0.331726i	$0.892369\pi$
$132$	0	0
$133$	−1.06062e8	−0.338964
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.32944e8i	1.22899i	0.788919	+	0.614497i	$0.210641\pi$
$137$	4.32944e8i	1.22899i	−0.788919	+	0.614497i	$0.789359\pi$
$138$	0	0
$139$	−5.25420e8	−1.40750	−0.703748	−	0.710450i	$-0.748492\pi$
$139$	−5.25420e8	−1.40750	−0.703748	+	0.710450i	$0.748492\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.86978e7i	0.116457i
$144$	0	0
$145$	−2.70032e8	−0.610863
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.49656e8i	0.506521i	0.967398	+	0.253261i	$0.0815030\pi$
$149$	2.49656e8i	0.506521i	−0.967398	+	0.253261i	$0.918497\pi$
$150$	0	0
$151$	−1.90108e8	−0.365672	−0.182836	−	0.983143i	$-0.558528\pi$
$151$	−1.90108e8	−0.365672	−0.182836	+	0.983143i	$0.558528\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.81599e8i	0.487871i
$156$	0	0
$157$	−4.70501e8	−0.774393	−0.387197	−	0.921997i	$-0.626557\pi$
$157$	−4.70501e8	−0.774393	−0.387197	+	0.921997i	$0.626557\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 6.20342e7i	− 0.0923267i
$162$	0	0
$163$	−4.63201e8	−0.656174	−0.328087	−	0.944648i	$-0.606404\pi$
$163$	−4.63201e8	−0.656174	−0.328087	+	0.944648i	$0.606404\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.36026e8i	0.560592i	0.959914	+	0.280296i	$0.0904325\pi$
$167$	4.36026e8i	0.560592i	−0.959914	+	0.280296i	$0.909567\pi$
$168$	0	0
$169$	−7.98954e8	−0.979434
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.28581e9i	1.43546i	0.696322	+	0.717729i	$0.254819\pi$
$173$	1.28581e9i	1.43546i	−0.696322	+	0.717729i	$0.745181\pi$
$174$	0	0
$175$	2.44230e8	0.260404
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.77990e8i	0.562999i	0.959561	+	0.281500i	$0.0908318\pi$
$179$	5.77990e8i	0.562999i	−0.959561	+	0.281500i	$0.909168\pi$
$180$	0	0
$181$	1.36134e9	1.26839	0.634195	−	0.773173i	$-0.281332\pi$
$181$	1.36134e9	1.26839	0.634195	+	0.773173i	$0.281332\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.57207e8i	0.304953i
$186$	0	0
$187$	8.01930e8	0.655798
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.97105e8i	0.298381i	0.988808	+	0.149191i	$0.0476668\pi$
$191$	3.97105e8i	0.298381i	−0.988808	+	0.149191i	$0.952333\pi$
$192$	0	0
$193$	−1.44780e9	−1.04347	−0.521734	−	0.853108i	$-0.674715\pi$
$193$	−1.44780e9	−1.04347	−0.521734	+	0.853108i	$0.674715\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.27633e9i	0.847420i	0.905798	+	0.423710i	$0.139272\pi$
$197$	1.27633e9i	0.847420i	−0.905798	+	0.423710i	$0.860728\pi$
$198$	0	0
$199$	−2.33138e9	−1.48663	−0.743313	−	0.668944i	$-0.766746\pi$
$199$	−2.33138e9	−1.48663	−0.743313	+	0.668944i	$0.766746\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.03029e8i	0.413989i
$204$	0	0
$205$	−2.96584e8	−0.167931
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.38955e9i	0.728265i
$210$	0	0
$211$	−5.69510e8	−0.287324	−0.143662	−	0.989627i	$-0.545888\pi$
$211$	−5.69510e8	−0.287324	−0.143662	+	0.989627i	$0.545888\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.28625e9i	− 0.601964i
$216$	0	0
$217$	7.33144e8	0.330636
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.76269e8i	− 0.115814i
$222$	0	0
$223$	−1.10060e8	−0.0445049	−0.0222525	−	0.999752i	$-0.507084\pi$
$223$	−1.10060e8	−0.0445049	−0.0222525	+	0.999752i	$0.507084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.62031e9i	0.610232i	0.952315	+	0.305116i	$0.0986953\pi$
$227$	1.62031e9i	0.610232i	−0.952315	+	0.305116i	$0.901305\pi$
$228$	0	0
$229$	−1.86805e9	−0.679278	−0.339639	−	0.940556i	$-0.610305\pi$
$229$	−1.86805e9	−0.679278	−0.339639	+	0.940556i	$0.610305\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.23197e9i	0.417999i	0.977916	+	0.209000i	$0.0670207\pi$
$233$	1.23197e9i	0.417999i	−0.977916	+	0.209000i	$0.932979\pi$
$234$	0	0
$235$	1.90732e9	0.625393
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1.64262e9i	− 0.503437i	−0.967800	−	0.251719i	$-0.919004\pi$
$239$	− 1.64262e9i	− 0.503437i	0.967800	−	0.251719i	$-0.0809958\pi$
$240$	0	0
$241$	−1.10426e9	−0.327342	−0.163671	−	0.986515i	$-0.552334\pi$
$241$	−1.10426e9	−0.327342	−0.163671	+	0.986515i	$0.552334\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.87060e8i	0.0796724i
$246$	0	0
$247$	4.78707e8	0.128612
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.57251e9i	0.396187i	0.980183	+	0.198093i	$0.0634749\pi$
$251$	1.57251e9i	0.396187i	−0.980183	+	0.198093i	$0.936525\pi$
$252$	0	0
$253$	−8.12727e8	−0.198364
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.88202e9i	0.431412i	0.976458	+	0.215706i	$0.0692053\pi$
$257$	1.88202e9i	0.431412i	−0.976458	+	0.215706i	$0.930795\pi$
$258$	0	0
$259$	9.29989e8	0.206671
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.28482e9i	− 0.268547i	−0.990944	−	0.134274i	$-0.957130\pi$
$263$	− 1.28482e9i	− 0.268547i	0.990944	−	0.134274i	$-0.0428701\pi$
$264$	0	0
$265$	1.88666e8	0.0382570
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 9.82675e9i	− 1.87673i	−0.345652	−	0.938363i	$-0.612342\pi$
$269$	− 9.82675e9i	− 1.87673i	0.345652	−	0.938363i	$-0.387658\pi$
$270$	0	0
$271$	2.37009e9	0.439428	0.219714	−	0.975564i	$-0.429488\pi$
$271$	2.37009e9	0.439428	0.219714	+	0.975564i	$0.429488\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.19973e9i	− 0.559477i
$276$	0	0
$277$	−3.04785e9	−0.517695	−0.258847	−	0.965918i	$-0.583343\pi$
$277$	−3.04785e9	−0.517695	−0.258847	+	0.965918i	$0.583343\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.73650e9i	− 1.24085i	−0.784266	−	0.620425i	$-0.786960\pi$
$281$	− 7.73650e9i	− 1.24085i	0.784266	−	0.620425i	$-0.213040\pi$
$282$	0	0
$283$	5.94321e9	0.926564	0.463282	−	0.886211i	$-0.346672\pi$
$283$	5.94321e9	0.926564	0.463282	+	0.886211i	$0.346672\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.72155e8i	0.113809i
$288$	0	0
$289$	2.42631e9	0.347820
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3.90739e9i	− 0.530172i	−0.964225	−	0.265086i	$-0.914600\pi$
$293$	− 3.90739e9i	− 0.530172i	0.964225	−	0.265086i	$-0.0854003\pi$
$294$	0	0
$295$	−3.12026e9	−0.412005
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.79988e8i	0.0350312i
$300$	0	0
$301$	−3.34874e9	−0.407958
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.26667e8i	0.0261932i
$306$	0	0
$307$	6.72279e9	0.756826	0.378413	−	0.925637i	$-0.376470\pi$
$307$	6.72279e9	0.756826	0.378413	+	0.925637i	$0.376470\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.74568e8i	0.0827977i	0.999143	+	0.0413988i	$0.0131814\pi$
$311$	7.74568e8i	0.0827977i	−0.999143	+	0.0413988i	$0.986819\pi$
$312$	0	0
$313$	1.68797e10	1.75868	0.879339	−	0.476197i	$-0.157985\pi$
$313$	1.68797e10	1.75868	0.879339	+	0.476197i	$0.157985\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.08895e9i	0.602983i	0.953469	+	0.301492i	$0.0974845\pi$
$317$	6.08895e9i	0.602983i	−0.953469	+	0.301492i	$0.902515\pi$
$318$	0	0
$319$	9.21058e9	0.889455
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7.88310e9i	− 0.724247i
$324$	0	0
$325$	−1.10232e9	−0.0988041
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.96571e9i	− 0.423836i
$330$	0	0
$331$	6.68142e9	0.556618	0.278309	−	0.960492i	$-0.410226\pi$
$331$	6.68142e9	0.556618	0.278309	+	0.960492i	$0.410226\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.18219e10i	− 0.938661i
$336$	0	0
$337$	8.18497e9	0.634596	0.317298	−	0.948326i	$-0.397224\pi$
$337$	8.18497e9	0.634596	0.317298	+	0.948326i	$0.397224\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.60512e9i	− 0.710371i
$342$	0	0
$343$	7.47359e8	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.41926e10i	− 0.978916i	−0.872027	−	0.489458i	$-0.837195\pi$
$347$	− 1.41926e10i	− 0.978916i	0.872027	−	0.489458i	$-0.162805\pi$
$348$	0	0
$349$	−4.87708e9	−0.328744	−0.164372	−	0.986398i	$-0.552560\pi$
$349$	−4.87708e9	−0.328744	−0.164372	+	0.986398i	$0.552560\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.09772e10i	0.706957i	0.935443	+	0.353479i	$0.115001\pi$
$353$	1.09772e10i	0.706957i	−0.935443	+	0.353479i	$0.884999\pi$
$354$	0	0
$355$	−5.69854e9	−0.358798
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.83944e10i	− 1.10741i	−0.832713	−	0.553704i	$-0.813214\pi$
$359$	− 1.83944e10i	− 1.10741i	0.832713	−	0.553704i	$-0.186786\pi$
$360$	0	0
$361$	−3.32406e9	−0.195722
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.83741e9i	0.328889i
$366$	0	0
$367$	1.73627e10	0.957093	0.478547	−	0.878062i	$-0.341164\pi$
$367$	1.73627e10	0.957093	0.478547	+	0.878062i	$0.341164\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.91192e8i	− 0.0259272i
$372$	0	0
$373$	8.25097e9	0.426255	0.213128	−	0.977024i	$-0.431635\pi$
$373$	8.25097e9	0.426255	0.213128	+	0.977024i	$0.431635\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.17309e9i	− 0.157078i
$378$	0	0
$379$	−1.09004e10	−0.528308	−0.264154	−	0.964481i	$-0.585093\pi$
$379$	−1.09004e10	−0.528308	−0.264154	+	0.964481i	$0.585093\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.93716e10i	− 1.36500i	−0.730886	−	0.682500i	$-0.760893\pi$
$383$	− 2.93716e10i	− 1.36500i	0.730886	−	0.682500i	$-0.239107\pi$
$384$	0	0
$385$	3.76085e9	0.171176
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8.08142e9i	− 0.352931i	−0.984307	−	0.176465i	$-0.943534\pi$
$389$	− 8.08142e9i	− 0.352931i	0.984307	−	0.176465i	$-0.0564663\pi$
$390$	0	0
$391$	4.61070e9	0.197270
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.81831e9i	− 0.197928i
$396$	0	0
$397$	−4.23249e9	−0.170386	−0.0851930	−	0.996364i	$-0.527151\pi$
$397$	−4.23249e9	−0.170386	−0.0851930	+	0.996364i	$0.527151\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.74199e9i	− 0.144719i	−0.997379	−	0.0723595i	$-0.976947\pi$
$401$	− 3.74199e9i	− 0.144719i	0.997379	−	0.0723595i	$-0.0230529\pi$
$402$	0	0
$403$	−3.30901e9	−0.125452
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.21840e10i	− 0.444032i
$408$	0	0
$409$	−9.46732e9	−0.338325	−0.169162	−	0.985588i	$-0.554106\pi$
$409$	−9.46732e9	−0.338325	−0.169162	+	0.985588i	$0.554106\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.12358e9i	0.279221i
$414$	0	0
$415$	−6.93061e8	−0.0233657
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.06975e10i	− 0.671525i	−0.941947	−	0.335762i	$-0.891006\pi$
$419$	− 2.06975e10i	− 0.671525i	0.941947	−	0.335762i	$-0.108994\pi$
$420$	0	0
$421$	2.83855e10	0.903582	0.451791	−	0.892124i	$-0.350785\pi$
$421$	2.83855e10	0.903582	0.451791	+	0.892124i	$0.350785\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.81525e10i	0.556390i
$426$	0	0
$427$	5.90126e8	0.0177514
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.33726e10i	0.967123i	0.875310	+	0.483561i	$0.160657\pi$
$431$	3.33726e10i	0.967123i	−0.875310	+	0.483561i	$0.839343\pi$
$432$	0	0
$433$	1.35508e10	0.385490	0.192745	−	0.981249i	$-0.438261\pi$
$433$	1.35508e10	0.385490	0.192745	+	0.981249i	$0.438261\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.98924e9i	0.219068i
$438$	0	0
$439$	−1.08521e10	−0.292183	−0.146091	−	0.989271i	$-0.546669\pi$
$439$	−1.08521e10	−0.292183	−0.146091	+	0.989271i	$0.546669\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.77764e10i	0.721209i	0.932719	+	0.360605i	$0.117430\pi$
$443$	2.77764e10i	0.721209i	−0.932719	+	0.360605i	$0.882570\pi$
$444$	0	0
$445$	−2.11137e10	−0.538424
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.44594e10i	1.09390i	0.837165	+	0.546950i	$0.184211\pi$
$449$	4.44594e10i	1.09390i	−0.837165	+	0.546950i	$0.815789\pi$
$450$	0	0
$451$	1.01162e10	0.244519
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1.29563e9i	− 0.0302298i
$456$	0	0
$457$	−8.26792e10	−1.89553	−0.947766	−	0.318966i	$-0.896665\pi$
$457$	−8.26792e10	−1.89553	−0.947766	+	0.318966i	$0.896665\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.06508e10i	1.12146i	0.828000	+	0.560728i	$0.189479\pi$
$461$	5.06508e10i	1.12146i	−0.828000	+	0.560728i	$0.810521\pi$
$462$	0	0
$463$	7.76023e10	1.68869	0.844346	−	0.535798i	$-0.179989\pi$
$463$	7.76023e10	1.68869	0.844346	+	0.535798i	$0.179989\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.58099e10i	− 0.542648i	−0.962488	−	0.271324i	$-0.912539\pi$
$467$	− 2.58099e10i	− 0.542648i	0.962488	−	0.271324i	$-0.0874615\pi$
$468$	0	0
$469$	−3.07783e10	−0.636142
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.38728e10i	0.876497i
$474$	0	0
$475$	−3.14538e10	−0.617873
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 3.61244e10i	− 0.686213i	−0.939296	−	0.343107i	$-0.888521\pi$
$479$	− 3.61244e10i	− 0.686213i	0.939296	−	0.343107i	$-0.111479\pi$
$480$	0	0
$481$	−4.19746e9	−0.0784163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.22705e10i	− 0.221766i
$486$	0	0
$487$	9.41764e9	0.167427	0.0837137	−	0.996490i	$-0.473322\pi$
$487$	9.41764e9	0.167427	0.0837137	+	0.996490i	$0.473322\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.29411e10i	0.394720i	0.980331	+	0.197360i	$0.0632368\pi$
$491$	2.29411e10i	0.394720i	−0.980331	+	0.197360i	$0.936763\pi$
$492$	0	0
$493$	−5.22528e10	−0.884548
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.48361e10i	0.243162i
$498$	0	0
$499$	1.95129e10	0.314717	0.157359	−	0.987542i	$-0.449702\pi$
$499$	1.95129e10	0.314717	0.157359	+	0.987542i	$0.449702\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.96172e10i	0.306454i	0.988191	+	0.153227i	$0.0489666\pi$
$503$	1.96172e10i	0.306454i	−0.988191	+	0.153227i	$0.951033\pi$
$504$	0	0
$505$	−4.17232e10	−0.641522
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.09459e10i	0.758993i	0.925193	+	0.379496i	$0.123903\pi$
$509$	5.09459e10i	0.758993i	−0.925193	+	0.379496i	$0.876097\pi$
$510$	0	0
$511$	1.51977e10	0.222892
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.17638e9i	0.0593705i
$516$	0	0
$517$	−6.50572e10	−0.910612
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.02686e11i	1.39368i	0.717229	+	0.696838i	$0.245410\pi$
$521$	1.02686e11i	1.39368i	−0.717229	+	0.696838i	$0.754590\pi$
$522$	0	0
$523$	−3.84556e10	−0.513987	−0.256994	−	0.966413i	$-0.582732\pi$
$523$	−3.84556e10	−0.513987	−0.256994	+	0.966413i	$0.582732\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.44911e10i	0.706452i
$528$	0	0
$529$	7.36382e10	0.940330
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.48509e9i	− 0.0431822i
$534$	0	0
$535$	−6.90692e10	−0.843081
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 9.79136e9i	− 0.116008i
$540$	0	0
$541$	−6.67196e10	−0.778869	−0.389434	−	0.921054i	$-0.627329\pi$
$541$	−6.67196e10	−0.778869	−0.389434	+	0.921054i	$0.627329\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7.61701e9i	− 0.0863373i
$546$	0	0
$547$	−1.55267e10	−0.173432	−0.0867158	−	0.996233i	$-0.527637\pi$
$547$	−1.55267e10	−0.173432	−0.0867158	+	0.996233i	$0.527637\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 9.05414e10i	− 0.982293i
$552$	0	0
$553$	−1.25445e10	−0.134138
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.18540e11i	1.23152i	0.787933	+	0.615762i	$0.211152\pi$
$557$	1.18540e11i	1.23152i	−0.787933	+	0.615762i	$0.788848\pi$
$558$	0	0
$559$	1.51144e10	0.154790
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.71879e10i	0.768273i	0.923276	+	0.384136i	$0.125501\pi$
$563$	7.71879e10i	0.768273i	−0.923276	+	0.384136i	$0.874499\pi$
$564$	0	0
$565$	3.81056e10	0.373934
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.17585e11i	1.12177i	0.827895	+	0.560883i	$0.189538\pi$
$569$	1.17585e11i	1.12177i	−0.827895	+	0.560883i	$0.810462\pi$
$570$	0	0
$571$	−1.04156e11	−0.979807	−0.489904	−	0.871777i	$-0.662968\pi$
$571$	−1.04156e11	−0.979807	−0.489904	+	0.871777i	$0.662968\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.83969e10i	− 0.168295i
$576$	0	0
$577$	−3.22534e10	−0.290986	−0.145493	−	0.989359i	$-0.546477\pi$
$577$	−3.22534e10	−0.290986	−0.145493	+	0.989359i	$0.546477\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.80438e9i	0.0158352i
$582$	0	0
$583$	−6.43524e9	−0.0557045
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.16532e11i	− 0.981509i	−0.871298	−	0.490754i	$-0.836721\pi$
$587$	− 1.16532e11i	− 0.981509i	0.871298	−	0.490754i	$-0.163279\pi$
$588$	0	0
$589$	−9.44199e10	−0.784517
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5.84175e10i	− 0.472415i	−0.971703	−	0.236208i	$-0.924095\pi$
$593$	− 5.84175e10i	− 0.472415i	0.971703	−	0.236208i	$-0.0759046\pi$
$594$	0	0
$595$	−2.13358e10	−0.170232
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.11787e10i	0.319864i	0.987128	+	0.159932i	$0.0511275\pi$
$599$	4.11787e10i	0.319864i	−0.987128	+	0.159932i	$0.948873\pi$
$600$	0	0
$601$	−7.25375e10	−0.555987	−0.277994	−	0.960583i	$-0.589669\pi$
$601$	−7.25375e10	−0.555987	−0.277994	+	0.960583i	$0.589669\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.54464e10i	0.189935i
$606$	0	0
$607$	8.61757e10	0.634790	0.317395	−	0.948293i	$-0.397192\pi$
$607$	8.61757e10	0.634790	0.317395	+	0.948293i	$0.397192\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.24125e10i	0.160815i
$612$	0	0
$613$	−1.89534e11	−1.34229	−0.671143	−	0.741328i	$-0.734196\pi$
$613$	−1.89534e11	−1.34229	−0.671143	+	0.741328i	$0.734196\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.36770e11i	0.943739i	0.881669	+	0.471869i	$0.156421\pi$
$617$	1.36770e11i	0.943739i	−0.881669	+	0.471869i	$0.843579\pi$
$618$	0	0
$619$	2.05400e11	1.39907	0.699534	−	0.714599i	$-0.253391\pi$
$619$	2.05400e11	1.39907	0.699534	+	0.714599i	$0.253391\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.49695e10i	0.364896i
$624$	0	0
$625$	2.49685e10	0.163633
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.91216e10i	0.441582i
$630$	0	0
$631$	1.44468e11	0.911288	0.455644	−	0.890162i	$-0.349409\pi$
$631$	1.44468e11	0.911288	0.455644	+	0.890162i	$0.349409\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.09446e11i	− 0.673138i
$636$	0	0
$637$	−3.37317e9	−0.0204871
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.48167e10i	0.0877648i	0.999037	+	0.0438824i	$0.0139727\pi$
$641$	1.48167e10i	0.0877648i	−0.999037	+	0.0438824i	$0.986027\pi$
$642$	0	0
$643$	3.20057e11	1.87234	0.936168	−	0.351553i	$-0.114346\pi$
$643$	3.20057e11	1.87234	0.936168	+	0.351553i	$0.114346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.36291e10i	0.134843i	0.997725	+	0.0674216i	$0.0214773\pi$
$647$	2.36291e10i	0.134843i	−0.997725	+	0.0674216i	$0.978523\pi$
$648$	0	0
$649$	1.06429e11	0.599905
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.18818e10i	0.0653473i	0.999466	+	0.0326737i	$0.0104022\pi$
$653$	1.18818e10i	0.0653473i	−0.999466	+	0.0326737i	$0.989598\pi$
$654$	0	0
$655$	−6.81052e10	−0.370011
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.02462e11i	− 1.60372i	−0.597510	−	0.801862i	$-0.703843\pi$
$659$	− 3.02462e11i	− 1.60372i	0.597510	−	0.801862i	$-0.296157\pi$
$660$	0	0
$661$	−1.94947e8	−0.00102120	−0.000510600	−	1.00000i	$-0.500163\pi$
$661$	−1.94947e8	−0.00102120	−0.000510600		1.00000i	$0.500163\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.69698e10i	− 0.189043i
$666$	0	0
$667$	5.29563e10	0.267556
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 7.73141e9i	− 0.0381389i
$672$	0	0
$673$	−5.04705e10	−0.246024	−0.123012	−	0.992405i	$-0.539255\pi$
$673$	−5.04705e10	−0.246024	−0.123012	+	0.992405i	$0.539255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.18944e10i	0.437456i	0.975786	+	0.218728i	$0.0701907\pi$
$677$	9.18944e10i	0.437456i	−0.975786	+	0.218728i	$0.929809\pi$
$678$	0	0
$679$	−3.19462e10	−0.150293
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.76743e11i	− 1.27173i	−0.771802	−	0.635863i	$-0.780644\pi$
$683$	− 2.76743e11i	− 1.27173i	0.771802	−	0.635863i	$-0.219356\pi$
$684$	0	0
$685$	−1.50910e11	−0.685418
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.21697e9i	0.00983747i
$690$	0	0
$691$	3.33822e11	1.46421	0.732104	−	0.681193i	$-0.238538\pi$
$691$	3.33822e11	1.46421	0.732104	+	0.681193i	$0.238538\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.83144e11i	− 0.784970i
$696$	0	0
$697$	−5.73906e10	−0.243170
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.07724e11i	− 1.68847i	−0.535971	−	0.844236i	$-0.680055\pi$
$701$	− 4.07724e11i	− 1.68847i	0.535971	−	0.844236i	$-0.319945\pi$
$702$	0	0
$703$	−1.19771e11	−0.490378
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.08626e11i	0.434767i
$708$	0	0
$709$	−4.90558e10	−0.194136	−0.0970678	−	0.995278i	$-0.530946\pi$
$709$	−4.90558e10	−0.194136	−0.0970678	+	0.995278i	$0.530946\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5.52248e10i	− 0.213686i
$714$	0	0
$715$	−1.69744e10	−0.0649488
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 4.69821e11i	− 1.75799i	−0.476830	−	0.878996i	$-0.658214\pi$
$719$	− 4.69821e11i	− 1.75799i	0.476830	−	0.878996i	$-0.341786\pi$
$720$	0	0
$721$	1.08732e10	0.0402361
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.08490e11i	0.754630i
$726$	0	0
$727$	1.92122e11	0.687764	0.343882	−	0.939013i	$-0.388258\pi$
$727$	1.92122e11	0.687764	0.343882	+	0.939013i	$0.388258\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.48896e11i	− 0.871662i
$732$	0	0
$733$	7.47278e10	0.258861	0.129430	−	0.991589i	$-0.458685\pi$
$733$	7.47278e10	0.258861	0.129430	+	0.991589i	$0.458685\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.03236e11i	1.36675i
$738$	0	0
$739$	1.49064e11	0.499799	0.249900	−	0.968272i	$-0.419602\pi$
$739$	1.49064e11	0.499799	0.249900	+	0.968272i	$0.419602\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.90655e11i	− 0.953723i	−0.878978	−	0.476862i	$-0.841774\pi$
$743$	− 2.90655e11i	− 0.953723i	0.878978	−	0.476862i	$-0.158226\pi$
$744$	0	0
$745$	−8.70219e10	−0.282490
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.79822e11i	0.571366i
$750$	0	0
$751$	−3.85693e10	−0.121250	−0.0606250	−	0.998161i	$-0.519309\pi$
$751$	−3.85693e10	−0.121250	−0.0606250	+	0.998161i	$0.519309\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.62652e10i	− 0.203938i
$756$	0	0
$757$	1.75336e11	0.533934	0.266967	−	0.963706i	$-0.413979\pi$
$757$	1.75336e11	0.533934	0.266967	+	0.963706i	$0.413979\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.42227e11i	− 1.61675i	−0.588668	−	0.808375i	$-0.700347\pi$
$761$	− 5.42227e11i	− 1.61675i	0.588668	−	0.808375i	$-0.299653\pi$
$762$	0	0
$763$	−1.98309e10	−0.0585118
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.66654e10i	− 0.105944i
$768$	0	0
$769$	2.67008e11	0.763517	0.381759	−	0.924262i	$-0.375319\pi$
$769$	2.67008e11	0.763517	0.381759	+	0.924262i	$0.375319\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.93424e11i	0.541743i	0.962616	+	0.270871i	$0.0873118\pi$
$773$	1.93424e11i	0.541743i	−0.962616	+	0.270871i	$0.912688\pi$
$774$	0	0
$775$	2.17421e11	0.602692
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.94441e10i	− 0.270041i
$780$	0	0
$781$	1.94372e11	0.522433
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.64001e11i	− 0.431884i
$786$	0	0
$787$	−8.52841e10	−0.222315	−0.111158	−	0.993803i	$-0.535456\pi$
$787$	−8.52841e10	−0.222315	−0.111158	+	0.993803i	$0.535456\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 9.92079e10i	− 0.253420i
$792$	0	0
$793$	−2.66351e9	−0.00673536
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.61423e11i	− 1.63925i	−0.572897	−	0.819627i	$-0.694181\pi$
$797$	− 6.61423e11i	− 1.63925i	0.572897	−	0.819627i	$-0.305819\pi$
$798$	0	0
$799$	3.69078e11	0.905588
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.99109e11i	− 0.478883i
$804$	0	0
$805$	2.16231e10	0.0514913
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.13264e11i	1.89862i	0.314344	+	0.949309i	$0.398215\pi$
$809$	8.13264e11i	1.89862i	−0.314344	+	0.949309i	$0.601785\pi$
$810$	0	0
$811$	4.22085e10	0.0975702	0.0487851	−	0.998809i	$-0.484465\pi$
$811$	4.22085e10	0.0975702	0.0487851	+	0.998809i	$0.484465\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.61456e11i	− 0.365953i
$816$	0	0
$817$	4.31277e11	0.967983
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.27883e11i	1.16189i	0.813943	+	0.580945i	$0.197317\pi$
$821$	5.27883e11i	1.16189i	−0.813943	+	0.580945i	$0.802683\pi$
$822$	0	0
$823$	−8.78060e10	−0.191393	−0.0956963	−	0.995411i	$-0.530508\pi$
$823$	−8.78060e10	−0.191393	−0.0956963	+	0.995411i	$0.530508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.95035e11i	− 0.416955i	−0.978027	−	0.208478i	$-0.933149\pi$
$827$	− 1.95035e11i	− 0.416955i	0.978027	−	0.208478i	$-0.0668509\pi$
$828$	0	0
$829$	−7.56585e11	−1.60192	−0.800958	−	0.598720i	$-0.795676\pi$
$829$	−7.56585e11	−1.60192	−0.800958	+	0.598720i	$0.795676\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.55476e10i	0.115368i
$834$	0	0
$835$	−1.51984e11	−0.312646
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 8.44106e10i	− 0.170353i	−0.996366	−	0.0851763i	$-0.972855\pi$
$839$	− 8.44106e10i	− 0.170353i	0.996366	−	0.0851763i	$-0.0271454\pi$
$840$	0	0
$841$	−9.99034e10	−0.199708
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.78489e11i	− 0.546237i
$846$	0	0
$847$	6.62497e10	0.128721
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.00523e10i	− 0.133569i
$852$	0	0
$853$	4.56707e11	0.862664	0.431332	−	0.902193i	$-0.358044\pi$
$853$	4.56707e11	0.862664	0.431332	+	0.902193i	$0.358044\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.28440e11i	− 0.608881i	−0.952531	−	0.304440i	$-0.901531\pi$
$857$	− 3.28440e11i	− 0.608881i	0.952531	−	0.304440i	$-0.0984694\pi$
$858$	0	0
$859$	1.00764e12	1.85069	0.925344	−	0.379129i	$-0.123776\pi$
$859$	1.00764e12	1.85069	0.925344	+	0.379129i	$0.123776\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.82398e11i	− 0.869685i	−0.900507	−	0.434842i	$-0.856804\pi$
$863$	− 4.82398e11i	− 0.869685i	0.900507	−	0.434842i	$-0.143196\pi$
$864$	0	0
$865$	−4.48189e11	−0.800565
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.64348e11i	0.288195i
$870$	0	0
$871$	1.38917e11	0.241369
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.08694e11i	0.356022i
$876$	0	0
$877$	−1.47431e10	−0.0249225	−0.0124613	−	0.999922i	$-0.503967\pi$
$877$	−1.47431e10	−0.0249225	−0.0124613	+	0.999922i	$0.503967\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.80542e11i	1.12967i	0.825204	+	0.564835i	$0.191060\pi$
$881$	6.80542e11i	1.12967i	−0.825204	+	0.564835i	$0.808940\pi$
$882$	0	0
$883$	−1.59710e11	−0.262717	−0.131358	−	0.991335i	$-0.541934\pi$
$883$	−1.59710e11	−0.262717	−0.131358	+	0.991335i	$0.541934\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.26548e11i	− 1.01218i	−0.862479	−	0.506092i	$-0.831089\pi$
$887$	− 6.26548e11i	− 1.01218i	0.862479	−	0.506092i	$-0.168911\pi$
$888$	0	0
$889$	−2.84942e11	−0.456194
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.39523e11i	1.00566i
$894$	0	0
$895$	−2.01468e11	−0.313989
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.25858e11i	0.958158i
$900$	0	0
$901$	3.65079e10	0.0553972
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.74519e11i	0.707390i
$906$	0	0
$907$	−9.67535e11	−1.42968	−0.714838	−	0.699290i	$-0.753500\pi$
$907$	−9.67535e11	−1.42968	−0.714838	+	0.699290i	$0.753500\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.30084e11i	0.769611i	0.922998	+	0.384805i	$0.125731\pi$
$911$	5.30084e11i	0.769611i	−0.922998	+	0.384805i	$0.874269\pi$
$912$	0	0
$913$	2.36397e10	0.0340220
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.77312e11i	0.250761i
$918$	0	0
$919$	−1.29433e12	−1.81461	−0.907304	−	0.420476i	$-0.861863\pi$
$919$	−1.29433e12	−1.81461	−0.907304	+	0.420476i	$0.861863\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 6.69623e10i	− 0.0922621i
$924$	0	0
$925$	2.75798e11	0.376724
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.17539e12i	− 1.57804i	−0.614369	−	0.789019i	$-0.710589\pi$
$929$	− 1.17539e12i	− 1.57804i	0.614369	−	0.789019i	$-0.289411\pi$
$930$	0	0
$931$	−9.62507e10	−0.128117
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.79526e11i	0.365743i
$936$	0	0
$937$	−7.22945e11	−0.937880	−0.468940	−	0.883230i	$-0.655364\pi$
$937$	−7.22945e11	−0.937880	−0.468940	+	0.883230i	$0.655364\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.19339e12i	1.52203i	0.648733	+	0.761016i	$0.275299\pi$
$941$	1.19339e12i	1.52203i	−0.648733	+	0.761016i	$0.724701\pi$
$942$	0	0
$943$	5.81633e10	0.0735534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.56063e12i	1.94044i	0.242226	+	0.970220i	$0.422123\pi$
$947$	1.56063e12i	1.94044i	−0.242226	+	0.970220i	$0.577877\pi$
$948$	0	0
$949$	−6.85941e10	−0.0845711
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8.68527e11i	− 1.05296i	−0.850188	−	0.526480i	$-0.823512\pi$
$953$	− 8.68527e11i	− 1.05296i	0.850188	−	0.526480i	$-0.176488\pi$
$954$	0	0
$955$	−1.38418e11	−0.166409
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.92894e11i	0.464516i
$960$	0	0
$961$	−2.00224e11	−0.234759
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.04655e11i	− 0.581949i
$966$	0	0
$967$	1.38012e12	1.57838	0.789189	−	0.614151i	$-0.210501\pi$
$967$	1.38012e12	1.57838	0.789189	+	0.614151i	$0.210501\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 8.75507e11i	− 0.984879i	−0.870347	−	0.492439i	$-0.836105\pi$
$971$	− 8.75507e11i	− 0.984879i	0.870347	−	0.492439i	$-0.163895\pi$
$972$	0	0
$973$	−4.76815e11	−0.531984
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.13074e11i	0.782630i	0.920257	+	0.391315i	$0.127980\pi$
$977$	7.13074e11i	0.782630i	−0.920257	+	0.391315i	$0.872020\pi$
$978$	0	0
$979$	7.20170e11	0.783979
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.62031e12i	1.73534i	0.497140	+	0.867670i	$0.334383\pi$
$983$	1.62031e12i	1.73534i	−0.497140	+	0.867670i	$0.665617\pi$
$984$	0	0
$985$	−4.44887e11	−0.472612
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.52247e11i	0.263658i
$990$	0	0
$991$	−1.45491e12	−1.50849	−0.754243	−	0.656595i	$-0.771996\pi$
$991$	−1.45491e12	−1.50849	−0.754243	+	0.656595i	$0.771996\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.12643e11i	− 0.829101i
$996$	0	0
$997$	−2.93410e11	−0.296958	−0.148479	−	0.988916i	$-0.547438\pi$
$997$	−2.93410e11	−0.296958	−0.148479	+	0.988916i	$0.547438\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.c.a.197.10	yes	16
3.2	odd	2	inner	252.9.c.a.197.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.7	✓	16	3.2	odd	2	inner
252.9.c.a.197.10	yes	16	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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+348.567i q^{5} +907.493 q^{7} +O(q^{10})\)	\(q+348.567i q^{5} +907.493 q^{7} -11889.3i q^{11} -4095.93 q^{13} +67449.6i q^{17} -116874. q^{19} -68357.8i q^{23} +269126. q^{25} +774693. i q^{29} +807878. q^{31} +316322. i q^{35} +1.02479e6 q^{37} +850867. i q^{41} -3.69010e6 q^{43} -5.47190e6i q^{47} +823543. q^{49} -541263. i q^{53} +4.14422e6 q^{55} +8.95168e6i q^{59} +650282. q^{61} -1.42770e6i q^{65} -3.39158e7 q^{67} +1.63485e7i q^{71} +1.67469e7 q^{73} -1.07895e7i q^{77} -1.38232e7 q^{79} +1.98832e6i q^{83} -2.35107e7 q^{85} +6.05729e7i q^{89} -3.71702e6 q^{91} -4.07384e7i q^{95} -3.52027e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) \) 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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