LMFDB - Embedded newform 432.8.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,Mod(1,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("432.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	432.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:	$134.950331009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 257x^{2} - 702x - 63 $ x^4 - 2x^3 - 257x^2 - 702*x - 63
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-13.3438$ of defining polynomial
Character	$\chi$	$=$	432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+522.364 q^{5} +265.581 q^{7} +4309.87 q^{11} +5224.48 q^{13} +9351.88 q^{17} -50864.5 q^{19} +54812.3 q^{23} +194739. q^{25} +178588. q^{29} +92496.4 q^{31} +138730. q^{35} -450966. q^{37} +401145. q^{41} -187132. q^{43} +387436. q^{47} -753010. q^{49} +1.05304e6 q^{53} +2.25132e6 q^{55} +703560. q^{59} +2.53537e6 q^{61} +2.72908e6 q^{65} -1.10770e6 q^{67} -3.82539e6 q^{71} -1.95280e6 q^{73} +1.14462e6 q^{77} -8.46423e6 q^{79} -1.93900e6 q^{83} +4.88508e6 q^{85} -2.68227e6 q^{89} +1.38752e6 q^{91} -2.65698e7 q^{95} +4.35880e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 104 q^{5} + 492 q^{7} + 2104 q^{11} + 2356 q^{13} - 4136 q^{17} - 5516 q^{19} + 17848 q^{23} + 66476 q^{25} + 150720 q^{29} + 78256 q^{31} - 195432 q^{35} - 42324 q^{37} + 280704 q^{41} + 51200 q^{43}+ \cdots + 9596660 q^{97}+O(q^{100}) $ 4 * q + 104 * q^5 + 492 * q^7 + 2104 * q^11 + 2356 * q^13 - 4136 * q^17 - 5516 * q^19 + 17848 * q^23 + 66476 * q^25 + 150720 * q^29 + 78256 * q^31 - 195432 * q^35 - 42324 * q^37 + 280704 * q^41 + 51200 * q^43 - 829608 * q^47 + 13256 * q^49 + 1763120 * q^53 + 519968 * q^55 - 1881992 * q^59 + 869116 * q^61 + 5534696 * q^65 + 249700 * q^67 - 9344848 * q^71 - 461660 * q^73 + 11536584 * q^77 + 936116 * q^79 - 15826384 * q^83 + 5871776 * q^85 + 22823736 * q^89 + 2375484 * q^91 - 32094136 * q^95 + 9596660 * q^97

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
$5$	522.364	1.86887	0.934433	−	0.356140i	$-0.115907\pi$
$5$	522.364	1.86887	0.934433	+	0.356140i	$0.115907\pi$
$6$	0	0
$7$	265.581	0.292653	0.146327	−	0.989236i	$-0.453255\pi$
$7$	265.581	0.292653	0.146327	+	0.989236i	$0.453255\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4309.87	0.976315	0.488158	−	0.872755i	$-0.337669\pi$
$11$	4309.87	0.976315	0.488158	+	0.872755i	$0.337669\pi$
$12$	0	0
$13$	5224.48	0.659540	0.329770	−	0.944061i	$-0.393029\pi$
$13$	5224.48	0.659540	0.329770	+	0.944061i	$0.393029\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9351.88	0.461666	0.230833	−	0.972993i	$-0.425855\pi$
$17$	9351.88	0.461666	0.230833	+	0.972993i	$0.425855\pi$
$18$	0	0
$19$	−50864.5	−1.70129	−0.850643	−	0.525743i	$-0.823787\pi$
$19$	−50864.5	−1.70129	−0.850643	+	0.525743i	$0.823787\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	54812.3	0.939357	0.469679	−	0.882837i	$-0.344370\pi$
$23$	54812.3	0.939357	0.469679	+	0.882837i	$0.344370\pi$
$24$	0	0
$25$	194739.	2.49266
$26$	0	0
$27$	0	0
$28$	0	0
$29$	178588.	1.35975	0.679875	−	0.733328i	$-0.262034\pi$
$29$	178588.	1.35975	0.679875	+	0.733328i	$0.262034\pi$
$30$	0	0
$31$	92496.4	0.557646	0.278823	−	0.960342i	$-0.410056\pi$
$31$	92496.4	0.557646	0.278823	+	0.960342i	$0.410056\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	138730.	0.546930
$36$	0	0
$37$	−450966.	−1.46365	−0.731827	−	0.681491i	$-0.761332\pi$
$37$	−450966.	−1.46365	−0.731827	+	0.681491i	$0.761332\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	401145.	0.908986	0.454493	−	0.890750i	$-0.349820\pi$
$41$	401145.	0.908986	0.454493	+	0.890750i	$0.349820\pi$
$42$	0	0
$43$	−187132.	−0.358929	−0.179465	−	0.983764i	$-0.557437\pi$
$43$	−187132.	−0.358929	−0.179465	+	0.983764i	$0.557437\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	387436.	0.544324	0.272162	−	0.962251i	$-0.412261\pi$
$47$	387436.	0.544324	0.272162	+	0.962251i	$0.412261\pi$
$48$	0	0
$49$	−753010.	−0.914354
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.05304e6	0.971584	0.485792	−	0.874074i	$-0.338531\pi$
$53$	1.05304e6	0.971584	0.485792	+	0.874074i	$0.338531\pi$
$54$	0	0
$55$	2.25132e6	1.82460
$56$	0	0
$57$	0	0
$58$	0	0
$59$	703560.	0.445984	0.222992	−	0.974820i	$-0.428418\pi$
$59$	703560.	0.445984	0.222992	+	0.974820i	$0.428418\pi$
$60$	0	0
$61$	2.53537e6	1.43017	0.715083	−	0.699040i	$-0.246389\pi$
$61$	2.53537e6	1.43017	0.715083	+	0.699040i	$0.246389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.72908e6	1.23259
$66$	0	0
$67$	−1.10770e6	−0.449945	−0.224972	−	0.974365i	$-0.572229\pi$
$67$	−1.10770e6	−0.449945	−0.224972	+	0.974365i	$0.572229\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.82539e6	−1.26844	−0.634222	−	0.773151i	$-0.718679\pi$
$71$	−3.82539e6	−1.26844	−0.634222	+	0.773151i	$0.718679\pi$
$72$	0	0
$73$	−1.95280e6	−0.587526	−0.293763	−	0.955878i	$-0.594908\pi$
$73$	−1.95280e6	−0.587526	−0.293763	+	0.955878i	$0.594908\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.14462e6	0.285722
$78$	0	0
$79$	−8.46423e6	−1.93149	−0.965746	−	0.259490i	$-0.916445\pi$
$79$	−8.46423e6	−1.93149	−0.965746	+	0.259490i	$0.916445\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.93900e6	−0.372225	−0.186113	−	0.982528i	$-0.559589\pi$
$83$	−1.93900e6	−0.372225	−0.186113	+	0.982528i	$0.559589\pi$
$84$	0	0
$85$	4.88508e6	0.862791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.68227e6	−0.403309	−0.201654	−	0.979457i	$-0.564632\pi$
$89$	−2.68227e6	−0.403309	−0.201654	+	0.979457i	$0.564632\pi$
$90$	0	0
$91$	1.38752e6	0.193017
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.65698e7	−3.17947
$96$	0	0
$97$	4.35880e6	0.484915	0.242458	−	0.970162i	$-0.422046\pi$
$97$	4.35880e6	0.484915	0.242458	+	0.970162i	$0.422046\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.33427e7	1.28860	0.644302	−	0.764771i	$-0.277148\pi$
$101$	1.33427e7	1.28860	0.644302	+	0.764771i	$0.277148\pi$
$102$	0	0
$103$	2.03211e7	1.83238	0.916191	−	0.400742i	$-0.131248\pi$
$103$	2.03211e7	1.83238	0.916191	+	0.400742i	$0.131248\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.35494e7	−1.06924	−0.534621	−	0.845092i	$-0.679546\pi$
$107$	−1.35494e7	−1.06924	−0.534621	+	0.845092i	$0.679546\pi$
$108$	0	0
$109$	−6.25636e6	−0.462732	−0.231366	−	0.972867i	$-0.574319\pi$
$109$	−6.25636e6	−0.462732	−0.231366	+	0.972867i	$0.574319\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.06466e7	−1.99806	−0.999028	−	0.0440883i	$-0.985962\pi$
$113$	−3.06466e7	−1.99806	−0.999028	+	0.0440883i	$0.985962\pi$
$114$	0	0
$115$	2.86320e7	1.75553
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.48368e6	0.135108
$120$	0	0
$121$	−912159.	−0.0468082
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.09148e7	2.78957
$126$	0	0
$127$	−1.04187e7	−0.451337	−0.225669	−	0.974204i	$-0.572457\pi$
$127$	−1.04187e7	−0.451337	−0.225669	+	0.974204i	$0.572457\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.55899e7	−1.38318	−0.691588	−	0.722292i	$-0.743089\pi$
$131$	−3.55899e7	−1.38318	−0.691588	+	0.722292i	$0.743089\pi$
$132$	0	0
$133$	−1.35086e7	−0.497887
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.88948e7	1.29232	0.646159	−	0.763203i	$-0.276374\pi$
$137$	3.88948e7	1.29232	0.646159	+	0.763203i	$0.276374\pi$
$138$	0	0
$139$	−3.25439e7	−1.02782	−0.513911	−	0.857843i	$-0.671804\pi$
$139$	−3.25439e7	−1.02782	−0.513911	+	0.857843i	$0.671804\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.25168e7	0.643919
$144$	0	0
$145$	9.32878e7	2.54119
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.38219e7	0.837619	0.418810	−	0.908074i	$-0.362447\pi$
$149$	3.38219e7	0.837619	0.418810	+	0.908074i	$0.362447\pi$
$150$	0	0
$151$	4.46014e7	1.05421	0.527107	−	0.849799i	$-0.323277\pi$
$151$	4.46014e7	1.05421	0.527107	+	0.849799i	$0.323277\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.83168e7	1.04217
$156$	0	0
$157$	6.65287e7	1.37202	0.686010	−	0.727593i	$-0.259361\pi$
$157$	6.65287e7	1.37202	0.686010	+	0.727593i	$0.259361\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.45571e7	0.274906
$162$	0	0
$163$	−5.88647e6	−0.106463	−0.0532314	−	0.998582i	$-0.516952\pi$
$163$	−5.88647e6	−0.106463	−0.0532314	+	0.998582i	$0.516952\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.45734e7	0.408279	0.204139	−	0.978942i	$-0.434560\pi$
$167$	2.45734e7	0.408279	0.204139	+	0.978942i	$0.434560\pi$
$168$	0	0
$169$	−3.54533e7	−0.565007
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.86691e7	1.44884	0.724419	−	0.689360i	$-0.242108\pi$
$173$	9.86691e7	1.44884	0.724419	+	0.689360i	$0.242108\pi$
$174$	0	0
$175$	5.17189e7	0.729484
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.97181e7	−0.778252	−0.389126	−	0.921185i	$-0.627223\pi$
$179$	−5.97181e7	−0.778252	−0.389126	+	0.921185i	$0.627223\pi$
$180$	0	0
$181$	−2.39240e6	−0.0299887	−0.0149944	−	0.999888i	$-0.504773\pi$
$181$	−2.39240e6	−0.0299887	−0.0149944	+	0.999888i	$0.504773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.35569e8	−2.73537
$186$	0	0
$187$	4.03054e7	0.450731
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.61514e7	0.375412	0.187706	−	0.982225i	$-0.439895\pi$
$191$	3.61514e7	0.375412	0.187706	+	0.982225i	$0.439895\pi$
$192$	0	0
$193$	−1.76829e6	−0.0177053	−0.00885263	−	0.999961i	$-0.502818\pi$
$193$	−1.76829e6	−0.0177053	−0.00885263	+	0.999961i	$0.502818\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.13163e7	0.385025	0.192513	−	0.981294i	$-0.438336\pi$
$197$	4.13163e7	0.385025	0.192513	+	0.981294i	$0.438336\pi$
$198$	0	0
$199$	2.09634e8	1.88571	0.942857	−	0.333197i	$-0.108128\pi$
$199$	2.09634e8	1.88571	0.942857	+	0.333197i	$0.108128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.74295e7	0.397935
$204$	0	0
$205$	2.09543e8	1.69877
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.19220e8	−1.66099
$210$	0	0
$211$	2.36862e8	1.73583	0.867913	−	0.496717i	$-0.165461\pi$
$211$	2.36862e8	1.73583	0.867913	+	0.496717i	$0.165461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.77511e7	−0.670790
$216$	0	0
$217$	2.45653e7	0.163197
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.88587e7	0.304487
$222$	0	0
$223$	−9.04978e7	−0.546476	−0.273238	−	0.961946i	$-0.588095\pi$
$223$	−9.04978e7	−0.546476	−0.273238	+	0.961946i	$0.588095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.17972e8	0.669404	0.334702	−	0.942324i	$-0.391364\pi$
$227$	1.17972e8	0.669404	0.334702	+	0.942324i	$0.391364\pi$
$228$	0	0
$229$	−4.94105e7	−0.271891	−0.135946	−	0.990716i	$-0.543407\pi$
$229$	−4.94105e7	−0.271891	−0.135946	+	0.990716i	$0.543407\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.05321e8	−1.06338	−0.531688	−	0.846940i	$-0.678442\pi$
$233$	−2.05321e8	−1.06338	−0.531688	+	0.846940i	$0.678442\pi$
$234$	0	0
$235$	2.02382e8	1.01727
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.75221e8	1.77785	0.888924	−	0.458055i	$-0.151454\pi$
$239$	3.75221e8	1.77785	0.888924	+	0.458055i	$0.151454\pi$
$240$	0	0
$241$	3.63529e8	1.67294	0.836469	−	0.548014i	$-0.184616\pi$
$241$	3.63529e8	1.67294	0.836469	+	0.548014i	$0.184616\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.93345e8	−1.70880
$246$	0	0
$247$	−2.65741e8	−1.12207
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.64675e8	−1.05647	−0.528233	−	0.849099i	$-0.677145\pi$
$251$	−2.64675e8	−1.05647	−0.528233	+	0.849099i	$0.677145\pi$
$252$	0	0
$253$	2.36234e8	0.917109
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.45349e8	−0.901608	−0.450804	−	0.892623i	$-0.648863\pi$
$257$	−2.45349e8	−0.901608	−0.450804	+	0.892623i	$0.648863\pi$
$258$	0	0
$259$	−1.19768e8	−0.428343
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.98499e8	−1.01181	−0.505903	−	0.862590i	$-0.668841\pi$
$263$	−2.98499e8	−1.01181	−0.505903	+	0.862590i	$0.668841\pi$
$264$	0	0
$265$	5.50071e8	1.81576
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.66264e8	0.520792	0.260396	−	0.965502i	$-0.416147\pi$
$269$	1.66264e8	0.520792	0.260396	+	0.965502i	$0.416147\pi$
$270$	0	0
$271$	2.66040e8	0.811996	0.405998	−	0.913874i	$-0.366924\pi$
$271$	2.66040e8	0.811996	0.405998	+	0.913874i	$0.366924\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.39300e8	2.43362
$276$	0	0
$277$	−1.71731e8	−0.485478	−0.242739	−	0.970092i	$-0.578046\pi$
$277$	−1.71731e8	−0.485478	−0.242739	+	0.970092i	$0.578046\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.33022e8	−0.626507	−0.313253	−	0.949670i	$-0.601419\pi$
$281$	−2.33022e8	−0.626507	−0.313253	+	0.949670i	$0.601419\pi$
$282$	0	0
$283$	5.33684e8	1.39969	0.699844	−	0.714296i	$-0.253253\pi$
$283$	5.33684e8	1.39969	0.699844	+	0.714296i	$0.253253\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.06536e8	0.266018
$288$	0	0
$289$	−3.22881e8	−0.786865
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.00418e7	0.162675	0.0813374	−	0.996687i	$-0.474081\pi$
$293$	7.00418e7	0.162675	0.0813374	+	0.996687i	$0.474081\pi$
$294$	0	0
$295$	3.67514e8	0.833484
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.86366e8	0.619544
$300$	0	0
$301$	−4.96987e7	−0.105042
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.32438e9	2.67279
$306$	0	0
$307$	−1.85895e8	−0.366676	−0.183338	−	0.983050i	$-0.558690\pi$
$307$	−1.85895e8	−0.366676	−0.183338	+	0.983050i	$0.558690\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.31517e8	1.19048	0.595242	−	0.803546i	$-0.297056\pi$
$311$	6.31517e8	1.19048	0.595242	+	0.803546i	$0.297056\pi$
$312$	0	0
$313$	−7.46422e8	−1.37588	−0.687938	−	0.725769i	$-0.741484\pi$
$313$	−7.46422e8	−1.37588	−0.687938	+	0.725769i	$0.741484\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.45994e8	−0.433727	−0.216864	−	0.976202i	$-0.569583\pi$
$317$	−2.45994e8	−0.433727	−0.216864	+	0.976202i	$0.569583\pi$
$318$	0	0
$319$	7.69691e8	1.32754
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.75679e8	−0.785425
$324$	0	0
$325$	1.01741e9	1.64401
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.02896e8	0.159298
$330$	0	0
$331$	4.12731e8	0.625560	0.312780	−	0.949826i	$-0.398740\pi$
$331$	4.12731e8	0.625560	0.312780	+	0.949826i	$0.398740\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.78621e8	−0.840886
$336$	0	0
$337$	1.29110e8	0.183762	0.0918808	−	0.995770i	$-0.470712\pi$
$337$	1.29110e8	0.183762	0.0918808	+	0.995770i	$0.470712\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.98648e8	0.544439
$342$	0	0
$343$	−4.18702e8	−0.560242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.72908e8	0.222158	0.111079	−	0.993812i	$-0.464569\pi$
$347$	1.72908e8	0.222158	0.111079	+	0.993812i	$0.464569\pi$
$348$	0	0
$349$	−6.82165e8	−0.859014	−0.429507	−	0.903064i	$-0.641313\pi$
$349$	−6.82165e8	−0.859014	−0.429507	+	0.903064i	$0.641313\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.12159e9	1.35714	0.678569	−	0.734537i	$-0.262601\pi$
$353$	1.12159e9	1.35714	0.678569	+	0.734537i	$0.262601\pi$
$354$	0	0
$355$	−1.99824e9	−2.37055
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.13392e9	−1.29345	−0.646726	−	0.762723i	$-0.723862\pi$
$359$	−1.13392e9	−1.29345	−0.646726	+	0.762723i	$0.723862\pi$
$360$	0	0
$361$	1.69333e9	1.89438
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.02007e9	−1.09801
$366$	0	0
$367$	−1.01893e9	−1.07601	−0.538003	−	0.842943i	$-0.680821\pi$
$367$	−1.01893e9	−1.07601	−0.538003	+	0.842943i	$0.680821\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.79668e8	0.284337
$372$	0	0
$373$	−8.04353e8	−0.802538	−0.401269	−	0.915960i	$-0.631431\pi$
$373$	−8.04353e8	−0.802538	−0.401269	+	0.915960i	$0.631431\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.33029e8	0.896810
$378$	0	0
$379$	7.19650e8	0.679023	0.339511	−	0.940602i	$-0.389738\pi$
$379$	7.19650e8	0.679023	0.339511	+	0.940602i	$0.389738\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.42256e8	0.129382	0.0646910	−	0.997905i	$-0.479394\pi$
$383$	1.42256e8	0.129382	0.0646910	+	0.997905i	$0.479394\pi$
$384$	0	0
$385$	5.97908e8	0.533976
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.66902e9	−1.43760	−0.718799	−	0.695218i	$-0.755308\pi$
$389$	−1.66902e9	−1.43760	−0.718799	+	0.695218i	$0.755308\pi$
$390$	0	0
$391$	5.12598e8	0.433669
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.42141e9	−3.60970
$396$	0	0
$397$	−1.76066e9	−1.41224	−0.706119	−	0.708093i	$-0.749556\pi$
$397$	−1.76066e9	−1.41224	−0.706119	+	0.708093i	$0.749556\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.68377e8	0.440181	0.220090	−	0.975479i	$-0.429365\pi$
$401$	5.68377e8	0.440181	0.220090	+	0.975479i	$0.429365\pi$
$402$	0	0
$403$	4.83246e8	0.367790
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.94361e9	−1.42899
$408$	0	0
$409$	1.08355e9	0.783099	0.391549	−	0.920157i	$-0.371939\pi$
$409$	1.08355e9	0.783099	0.391549	+	0.920157i	$0.371939\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.86852e8	0.130519
$414$	0	0
$415$	−1.01287e9	−0.695638
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.89720e9	−1.25998	−0.629991	−	0.776602i	$-0.716942\pi$
$419$	−1.89720e9	−1.25998	−0.629991	+	0.776602i	$0.716942\pi$
$420$	0	0
$421$	−1.40976e9	−0.920781	−0.460391	−	0.887716i	$-0.652291\pi$
$421$	−1.40976e9	−0.920781	−0.460391	+	0.887716i	$0.652291\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.82117e9	1.15077
$426$	0	0
$427$	6.73344e8	0.418543
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.54622e9	0.930253	0.465127	−	0.885244i	$-0.346009\pi$
$431$	1.54622e9	0.930253	0.465127	+	0.885244i	$0.346009\pi$
$432$	0	0
$433$	−2.49764e9	−1.47850	−0.739251	−	0.673430i	$-0.764820\pi$
$433$	−2.49764e9	−1.47850	−0.739251	+	0.673430i	$0.764820\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.78800e9	−1.59812
$438$	0	0
$439$	−2.35187e9	−1.32674	−0.663372	−	0.748290i	$-0.730875\pi$
$439$	−2.35187e9	−1.32674	−0.663372	+	0.748290i	$0.730875\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.71779e9	1.48526	0.742631	−	0.669700i	$-0.233577\pi$
$443$	2.71779e9	1.48526	0.742631	+	0.669700i	$0.233577\pi$
$444$	0	0
$445$	−1.40112e9	−0.753730
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.96174e9	1.02277	0.511386	−	0.859351i	$-0.329132\pi$
$449$	1.96174e9	1.02277	0.511386	+	0.859351i	$0.329132\pi$
$450$	0	0
$451$	1.72888e9	0.887457
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.24791e8	0.360722
$456$	0	0
$457$	−3.64205e9	−1.78500	−0.892502	−	0.451043i	$-0.851052\pi$
$457$	−3.64205e9	−1.78500	−0.892502	+	0.451043i	$0.851052\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.08990e9	1.46890	0.734448	−	0.678665i	$-0.237441\pi$
$461$	3.08990e9	1.46890	0.734448	+	0.678665i	$0.237441\pi$
$462$	0	0
$463$	−1.10959e9	−0.519552	−0.259776	−	0.965669i	$-0.583649\pi$
$463$	−1.10959e9	−0.519552	−0.259776	+	0.965669i	$0.583649\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.21871e8	0.237112	0.118556	−	0.992947i	$-0.462173\pi$
$467$	5.21871e8	0.237112	0.118556	+	0.992947i	$0.462173\pi$
$468$	0	0
$469$	−2.94183e8	−0.131678
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.06516e8	−0.350428
$474$	0	0
$475$	−9.90530e9	−4.24072
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.43894e8	0.226121	0.113060	−	0.993588i	$-0.463935\pi$
$479$	5.43894e8	0.226121	0.113060	+	0.993588i	$0.463935\pi$
$480$	0	0
$481$	−2.35607e9	−0.965338
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.27688e9	0.906241
$486$	0	0
$487$	4.73331e9	1.85701	0.928503	−	0.371325i	$-0.121097\pi$
$487$	4.73331e9	1.85701	0.928503	+	0.371325i	$0.121097\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.79481e9	−0.684281	−0.342140	−	0.939649i	$-0.611152\pi$
$491$	−1.79481e9	−0.684281	−0.342140	+	0.939649i	$0.611152\pi$
$492$	0	0
$493$	1.67013e9	0.627750
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.01595e9	−0.371214
$498$	0	0
$499$	1.63272e9	0.588246	0.294123	−	0.955768i	$-0.404972\pi$
$499$	1.63272e9	0.588246	0.294123	+	0.955768i	$0.404972\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.10548e9	1.08803	0.544015	−	0.839075i	$-0.316903\pi$
$503$	3.10548e9	1.08803	0.544015	+	0.839075i	$0.316903\pi$
$504$	0	0
$505$	6.96975e9	2.40823
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.82949e9	−0.614920	−0.307460	−	0.951561i	$-0.599479\pi$
$509$	−1.82949e9	−0.614920	−0.307460	+	0.951561i	$0.599479\pi$
$510$	0	0
$511$	−5.18625e8	−0.171941
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.06150e10	3.42447
$516$	0	0
$517$	1.66980e9	0.531432
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.61188e9	−0.499345	−0.249673	−	0.968330i	$-0.580323\pi$
$521$	−1.61188e9	−0.499345	−0.249673	+	0.968330i	$0.580323\pi$
$522$	0	0
$523$	−2.70753e9	−0.827593	−0.413797	−	0.910369i	$-0.635798\pi$
$523$	−2.70753e9	−0.827593	−0.413797	+	0.910369i	$0.635798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.65015e8	0.257446
$528$	0	0
$529$	−4.00434e8	−0.117608
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.09577e9	0.599513
$534$	0	0
$535$	−7.07770e9	−1.99827
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.24538e9	−0.892698
$540$	0	0
$541$	−3.86803e9	−1.05027	−0.525133	−	0.851020i	$-0.675984\pi$
$541$	−3.86803e9	−1.05027	−0.525133	+	0.851020i	$0.675984\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.26810e9	−0.864783
$546$	0	0
$547$	2.49705e8	0.0652337	0.0326168	−	0.999468i	$-0.489616\pi$
$547$	2.49705e8	0.0652337	0.0326168	+	0.999468i	$0.489616\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.08379e9	−2.31332
$552$	0	0
$553$	−2.24794e9	−0.565257
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.06873e9	−0.507236	−0.253618	−	0.967304i	$-0.581621\pi$
$557$	−2.06873e9	−0.507236	−0.253618	+	0.967304i	$0.581621\pi$
$558$	0	0
$559$	−9.77668e8	−0.236728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.36904e9	1.26799	0.633997	−	0.773336i	$-0.281413\pi$
$563$	5.36904e9	1.26799	0.633997	+	0.773336i	$0.281413\pi$
$564$	0	0
$565$	−1.60087e10	−3.73410
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.59458e9	−1.50070	−0.750351	−	0.661040i	$-0.770116\pi$
$569$	−6.59458e9	−1.50070	−0.750351	+	0.661040i	$0.770116\pi$
$570$	0	0
$571$	1.42340e9	0.319964	0.159982	−	0.987120i	$-0.448856\pi$
$571$	1.42340e9	0.319964	0.159982	+	0.987120i	$0.448856\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.06741e10	2.34150
$576$	0	0
$577$	4.95933e9	1.07475	0.537375	−	0.843344i	$-0.319416\pi$
$577$	4.95933e9	1.07475	0.537375	+	0.843344i	$0.319416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.14962e8	−0.108933
$582$	0	0
$583$	4.53848e9	0.948573
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.12391e9	−0.229349	−0.114675	−	0.993403i	$-0.536583\pi$
$587$	−1.12391e9	−0.229349	−0.114675	+	0.993403i	$0.536583\pi$
$588$	0	0
$589$	−4.70479e9	−0.948716
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.05432e9	−1.78305	−0.891527	−	0.452968i	$-0.850365\pi$
$593$	−9.05432e9	−1.78305	−0.891527	+	0.452968i	$0.850365\pi$
$594$	0	0
$595$	1.29738e9	0.252499
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.96819e9	−1.13462	−0.567308	−	0.823506i	$-0.692015\pi$
$599$	−5.96819e9	−1.13462	−0.567308	+	0.823506i	$0.692015\pi$
$600$	0	0
$601$	−2.58084e8	−0.0484954	−0.0242477	−	0.999706i	$-0.507719\pi$
$601$	−2.58084e8	−0.0484954	−0.0242477	+	0.999706i	$0.507719\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.76479e8	−0.0874782
$606$	0	0
$607$	6.56803e9	1.19200	0.595998	−	0.802986i	$-0.296757\pi$
$607$	6.56803e9	1.19200	0.595998	+	0.802986i	$0.296757\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.02415e9	0.359004
$612$	0	0
$613$	2.05883e9	0.361001	0.180501	−	0.983575i	$-0.442228\pi$
$613$	2.05883e9	0.361001	0.180501	+	0.983575i	$0.442228\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.59084e9	0.958251	0.479125	−	0.877746i	$-0.340954\pi$
$617$	5.59084e9	0.958251	0.479125	+	0.877746i	$0.340954\pi$
$618$	0	0
$619$	4.44511e9	0.753296	0.376648	−	0.926357i	$-0.377077\pi$
$619$	4.44511e9	0.753296	0.376648	+	0.926357i	$0.377077\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.12360e8	−0.118030
$624$	0	0
$625$	1.66057e10	2.72068
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.21738e9	−0.675718
$630$	0	0
$631$	−5.69707e9	−0.902711	−0.451356	−	0.892344i	$-0.649059\pi$
$631$	−5.69707e9	−0.902711	−0.451356	+	0.892344i	$0.649059\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.44236e9	−0.843488
$636$	0	0
$637$	−3.93408e9	−0.603053
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.27140e10	−1.90669	−0.953343	−	0.301888i	$-0.902383\pi$
$641$	−1.27140e10	−1.90669	−0.953343	+	0.301888i	$0.902383\pi$
$642$	0	0
$643$	−4.35960e9	−0.646709	−0.323354	−	0.946278i	$-0.604811\pi$
$643$	−4.35960e9	−0.646709	−0.323354	+	0.946278i	$0.604811\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.19424e9	0.318507	0.159254	−	0.987238i	$-0.449091\pi$
$647$	2.19424e9	0.318507	0.159254	+	0.987238i	$0.449091\pi$
$648$	0	0
$649$	3.03226e9	0.435421
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.13375e10	1.59339	0.796695	−	0.604382i	$-0.206580\pi$
$653$	1.13375e10	1.59339	0.796695	+	0.604382i	$0.206580\pi$
$654$	0	0
$655$	−1.85909e10	−2.58497
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.11947e8	0.110517	0.0552584	−	0.998472i	$-0.482402\pi$
$659$	8.11947e8	0.110517	0.0552584	+	0.998472i	$0.482402\pi$
$660$	0	0
$661$	−4.62556e9	−0.622958	−0.311479	−	0.950253i	$-0.600824\pi$
$661$	−4.62556e9	−0.622958	−0.311479	+	0.950253i	$0.600824\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.05642e9	−0.930484
$666$	0	0
$667$	9.78882e9	1.27729
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.09271e10	1.39629
$672$	0	0
$673$	−3.43380e9	−0.434232	−0.217116	−	0.976146i	$-0.569665\pi$
$673$	−3.43380e9	−0.434232	−0.217116	+	0.976146i	$0.569665\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.42805e9	0.920057	0.460028	−	0.887904i	$-0.347839\pi$
$677$	7.42805e9	0.920057	0.460028	+	0.887904i	$0.347839\pi$
$678$	0	0
$679$	1.15761e9	0.141912
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.63189e10	−1.95983	−0.979916	−	0.199411i	$-0.936097\pi$
$683$	−1.63189e10	−1.95983	−0.979916	+	0.199411i	$0.936097\pi$
$684$	0	0
$685$	2.03172e10	2.41517
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.50160e9	0.640799
$690$	0	0
$691$	−7.36245e9	−0.848886	−0.424443	−	0.905455i	$-0.639530\pi$
$691$	−7.36245e9	−0.848886	−0.424443	+	0.905455i	$0.639530\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.69998e10	−1.92086
$696$	0	0
$697$	3.75145e9	0.419648
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.47242e8	−0.0709666	−0.0354833	−	0.999370i	$-0.511297\pi$
$701$	−6.47242e8	−0.0709666	−0.0354833	+	0.999370i	$0.511297\pi$
$702$	0	0
$703$	2.29382e10	2.49009
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.54357e9	0.377114
$708$	0	0
$709$	−1.83467e9	−0.193329	−0.0966644	−	0.995317i	$-0.530817\pi$
$709$	−1.83467e9	−0.193329	−0.0966644	+	0.995317i	$0.530817\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.06994e9	0.523829
$714$	0	0
$715$	1.17620e10	1.20340
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.60916e10	1.61454	0.807268	−	0.590185i	$-0.200945\pi$
$719$	1.60916e10	1.61454	0.807268	+	0.590185i	$0.200945\pi$
$720$	0	0
$721$	5.39688e9	0.536253
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.47780e10	3.38939
$726$	0	0
$727$	−1.09234e10	−1.05435	−0.527176	−	0.849756i	$-0.676749\pi$
$727$	−1.09234e10	−1.05435	−0.527176	+	0.849756i	$0.676749\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.75004e9	−0.165705
$732$	0	0
$733$	−4.29507e9	−0.402816	−0.201408	−	0.979507i	$-0.564552\pi$
$733$	−4.29507e9	−0.402816	−0.201408	+	0.979507i	$0.564552\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.77403e9	−0.439288
$738$	0	0
$739$	1.17116e10	1.06748	0.533739	−	0.845649i	$-0.320786\pi$
$739$	1.17116e10	1.06748	0.533739	+	0.845649i	$0.320786\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.42164e9	−0.842686	−0.421343	−	0.906901i	$-0.638441\pi$
$743$	−9.42164e9	−0.842686	−0.421343	+	0.906901i	$0.638441\pi$
$744$	0	0
$745$	1.76674e10	1.56540
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.59845e9	−0.312917
$750$	0	0
$751$	2.23111e10	1.92212	0.961062	−	0.276334i	$-0.0891195\pi$
$751$	2.23111e10	1.92212	0.961062	+	0.276334i	$0.0891195\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.32981e10	1.97018
$756$	0	0
$757$	−2.22897e10	−1.86753	−0.933767	−	0.357882i	$-0.883499\pi$
$757$	−2.22897e10	−1.86753	−0.933767	+	0.357882i	$0.883499\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.17891e10	0.969696	0.484848	−	0.874599i	$-0.338875\pi$
$761$	1.17891e10	0.969696	0.484848	+	0.874599i	$0.338875\pi$
$762$	0	0
$763$	−1.66157e9	−0.135420
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.67574e9	0.294144
$768$	0	0
$769$	1.32094e10	1.04747	0.523735	−	0.851881i	$-0.324538\pi$
$769$	1.32094e10	1.04747	0.523735	+	0.851881i	$0.324538\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.97934e9	0.154132	0.0770660	−	0.997026i	$-0.475445\pi$
$773$	1.97934e9	0.154132	0.0770660	+	0.997026i	$0.475445\pi$
$774$	0	0
$775$	1.80126e10	1.39002
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.04040e10	−1.54645
$780$	0	0
$781$	−1.64869e10	−1.23840
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.47522e10	2.56412
$786$	0	0
$787$	−1.07245e10	−0.784271	−0.392135	−	0.919907i	$-0.628264\pi$
$787$	−1.07245e10	−0.784271	−0.392135	+	0.919907i	$0.628264\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.13914e9	−0.584737
$792$	0	0
$793$	1.32460e10	0.943252
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.10393e10	−0.772389	−0.386195	−	0.922417i	$-0.626211\pi$
$797$	−1.10393e10	−0.772389	−0.386195	+	0.922417i	$0.626211\pi$
$798$	0	0
$799$	3.62325e9	0.251296
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.41631e9	−0.573611
$804$	0	0
$805$	7.60410e9	0.513762
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.16871e10	0.776047	0.388023	−	0.921650i	$-0.373158\pi$
$809$	1.16871e10	0.776047	0.388023	+	0.921650i	$0.373158\pi$
$810$	0	0
$811$	−5.63002e9	−0.370627	−0.185313	−	0.982679i	$-0.559330\pi$
$811$	−5.63002e9	−0.370627	−0.185313	+	0.982679i	$0.559330\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.07488e9	−0.198965
$816$	0	0
$817$	9.51839e9	0.610642
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.26349e10	−0.796838	−0.398419	−	0.917204i	$-0.630441\pi$
$821$	−1.26349e10	−0.796838	−0.398419	+	0.917204i	$0.630441\pi$
$822$	0	0
$823$	−2.37659e9	−0.148612	−0.0743061	−	0.997235i	$-0.523674\pi$
$823$	−2.37659e9	−0.148612	−0.0743061	+	0.997235i	$0.523674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.86630e9	−0.299178	−0.149589	−	0.988748i	$-0.547795\pi$
$827$	−4.86630e9	−0.299178	−0.149589	+	0.988748i	$0.547795\pi$
$828$	0	0
$829$	2.85668e9	0.174149	0.0870744	−	0.996202i	$-0.472248\pi$
$829$	2.85668e9	0.174149	0.0870744	+	0.996202i	$0.472248\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.04205e9	−0.422126
$834$	0	0
$835$	1.28362e10	0.763018
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.73854e10	−1.01629	−0.508145	−	0.861271i	$-0.669669\pi$
$839$	−1.73854e10	−1.01629	−0.508145	+	0.861271i	$0.669669\pi$
$840$	0	0
$841$	1.46438e10	0.848920
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.85195e10	−1.05592
$846$	0	0
$847$	−2.42252e8	−0.0136986
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.47185e10	−1.37489
$852$	0	0
$853$	7.39690e9	0.408064	0.204032	−	0.978964i	$-0.434595\pi$
$853$	7.39690e9	0.408064	0.204032	+	0.978964i	$0.434595\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.73904e9	0.0943792	0.0471896	−	0.998886i	$-0.484974\pi$
$857$	1.73904e9	0.0943792	0.0471896	+	0.998886i	$0.484974\pi$
$858$	0	0
$859$	6.72568e9	0.362043	0.181021	−	0.983479i	$-0.442060\pi$
$859$	6.72568e9	0.362043	0.181021	+	0.983479i	$0.442060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.71700e9	−0.355744	−0.177872	−	0.984054i	$-0.556921\pi$
$863$	−6.71700e9	−0.355744	−0.177872	+	0.984054i	$0.556921\pi$
$864$	0	0
$865$	5.15412e10	2.70768
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.64798e10	−1.88574
$870$	0	0
$871$	−5.78714e9	−0.296757
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.61778e10	0.816378
$876$	0	0
$877$	−1.42410e10	−0.712920	−0.356460	−	0.934311i	$-0.616016\pi$
$877$	−1.42410e10	−0.712920	−0.356460	+	0.934311i	$0.616016\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.95272e8	0.00962110	0.00481055	−	0.999988i	$-0.498469\pi$
$881$	1.95272e8	0.00962110	0.00481055	+	0.999988i	$0.498469\pi$
$882$	0	0
$883$	−8.89022e8	−0.0434560	−0.0217280	−	0.999764i	$-0.506917\pi$
$883$	−8.89022e8	−0.0434560	−0.0217280	+	0.999764i	$0.506917\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.50425e10	−1.20488	−0.602442	−	0.798163i	$-0.705806\pi$
$887$	−2.50425e10	−1.20488	−0.602442	+	0.798163i	$0.705806\pi$
$888$	0	0
$889$	−2.76701e9	−0.132085
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.97067e10	−0.926051
$894$	0	0
$895$	−3.11946e10	−1.45445
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.65187e10	0.758260
$900$	0	0
$901$	9.84793e9	0.448547
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.24970e9	−0.0560449
$906$	0	0
$907$	−1.75963e10	−0.783062	−0.391531	−	0.920165i	$-0.628054\pi$
$907$	−1.75963e10	−0.783062	−0.391531	+	0.920165i	$0.628054\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.00803e10	1.31816	0.659079	−	0.752073i	$-0.270946\pi$
$911$	3.00803e10	1.31816	0.659079	+	0.752073i	$0.270946\pi$
$912$	0	0
$913$	−8.35687e9	−0.363409
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.45200e9	−0.404791
$918$	0	0
$919$	2.22921e10	0.947428	0.473714	−	0.880679i	$-0.342913\pi$
$919$	2.22921e10	0.947428	0.473714	+	0.880679i	$0.342913\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.99857e10	−0.836589
$924$	0	0
$925$	−8.78207e10	−3.64839
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.57367e10	1.05317	0.526584	−	0.850123i	$-0.323473\pi$
$929$	2.57367e10	1.05317	0.526584	+	0.850123i	$0.323473\pi$
$930$	0	0
$931$	3.83015e10	1.55558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.10541e10	0.842356
$936$	0	0
$937$	2.71461e10	1.07800	0.539000	−	0.842306i	$-0.318802\pi$
$937$	2.71461e10	1.07800	0.539000	+	0.842306i	$0.318802\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.39586e9	0.211104	0.105552	−	0.994414i	$-0.466339\pi$
$941$	5.39586e9	0.211104	0.105552	+	0.994414i	$0.466339\pi$
$942$	0	0
$943$	2.19877e10	0.853863
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.29682e10	0.878825	0.439413	−	0.898285i	$-0.355187\pi$
$947$	2.29682e10	0.878825	0.439413	+	0.898285i	$0.355187\pi$
$948$	0	0
$949$	−1.02023e10	−0.387497
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.48516e9	−0.317567	−0.158783	−	0.987313i	$-0.550757\pi$
$953$	−8.48516e9	−0.317567	−0.158783	+	0.987313i	$0.550757\pi$
$954$	0	0
$955$	1.88842e10	0.701594
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.03297e10	0.378201
$960$	0	0
$961$	−1.89570e10	−0.689030
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.23689e8	−0.0330887
$966$	0	0
$967$	1.23655e10	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$967$	1.23655e10	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.57190e10	0.901545	0.450772	−	0.892639i	$-0.351149\pi$
$971$	2.57190e10	0.901545	0.450772	+	0.892639i	$0.351149\pi$
$972$	0	0
$973$	−8.64304e9	−0.300796
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.39299e10	−1.16400	−0.581998	−	0.813191i	$-0.697729\pi$
$977$	−3.39299e10	−1.16400	−0.581998	+	0.813191i	$0.697729\pi$
$978$	0	0
$979$	−1.15603e10	−0.393757
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.17172e10	1.06502	0.532509	−	0.846424i	$-0.321249\pi$
$983$	3.17172e10	1.06502	0.532509	+	0.846424i	$0.321249\pi$
$984$	0	0
$985$	2.15821e10	0.719560
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.02572e10	−0.337163
$990$	0	0
$991$	−3.97025e10	−1.29587	−0.647933	−	0.761697i	$-0.724366\pi$
$991$	−3.97025e10	−1.29587	−0.647933	+	0.761697i	$0.724366\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.09505e11	3.52415
$996$	0	0
$997$	1.77486e10	0.567193	0.283597	−	0.958944i	$-0.408472\pi$
$997$	1.77486e10	0.567193	0.283597	+	0.958944i	$0.408472\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.a.y.1.4		4
3.2	odd	2		432.8.a.x.1.1		4
4.3	odd	2		216.8.a.g.1.4	yes	4
12.11	even	2		216.8.a.f.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.8.a.f.1.1	✓	4	12.11	even	2
216.8.a.g.1.4	yes	4	4.3	odd	2
432.8.a.x.1.1		4	3.2	odd	2
432.8.a.y.1.4		4	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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

\(f(q)\)	\(=\)	\(q+522.364 q^{5} +265.581 q^{7} +4309.87 q^{11} +5224.48 q^{13} +9351.88 q^{17} -50864.5 q^{19} +54812.3 q^{23} +194739. q^{25} +178588. q^{29} +92496.4 q^{31} +138730. q^{35} -450966. q^{37} +401145. q^{41} -187132. q^{43} +387436. q^{47} -753010. q^{49} +1.05304e6 q^{53} +2.25132e6 q^{55} +703560. q^{59} +2.53537e6 q^{61} +2.72908e6 q^{65} -1.10770e6 q^{67} -3.82539e6 q^{71} -1.95280e6 q^{73} +1.14462e6 q^{77} -8.46423e6 q^{79} -1.93900e6 q^{83} +4.88508e6 q^{85} -2.68227e6 q^{89} +1.38752e6 q^{91} -2.65698e7 q^{95} +4.35880e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 104 q^{5} + 492 q^{7} + 2104 q^{11} + 2356 q^{13} - 4136 q^{17} - 5516 q^{19} + 17848 q^{23} + 66476 q^{25} + 150720 q^{29} + 78256 q^{31} - 195432 q^{35} - 42324 q^{37} + 280704 q^{41} + 51200 q^{43}+ \cdots + 9596660 q^{97}+O(q^{100}) \) 4 * q + 104 * q^5 + 492 * q^7 + 2104 * q^11 + 2356 * q^13 - 4136 * q^17 - 5516 * q^19 + 17848 * q^23 + 66476 * q^25 + 150720 * q^29 + 78256 * q^31 - 195432 * q^35 - 42324 * q^37 + 280704 * q^41 + 51200 * q^43 - 829608 * q^47 + 13256 * q^49 + 1763120 * q^53 + 519968 * q^55 - 1881992 * q^59 + 869116 * q^61 + 5534696 * q^65 + 249700 * q^67 - 9344848 * q^71 - 461660 * q^73 + 11536584 * q^77 + 936116 * q^79 - 15826384 * q^83 + 5871776 * q^85 + 22823736 * q^89 + 2375484 * q^91 - 32094136 * q^95 + 9596660 * q^97

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