LMFDB - Embedded newform 288.7.b.c.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,7,Mod(271,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("288.271");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	288.b (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:	$66.2555760825$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 $ x^12 + 78x^10 + 3408x^8 + 73216x^6 + 13959168x^4 + 1308622848*x^2 + 68719476736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{66}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.1
Root			$-4.24448 + 6.78118i$ of defining polynomial
Character	$\chi$	$=$	288.271
Dual form			288.7.b.c.271.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-206.098i q^{5} -210.403i q^{7} +O(q^{10})$	$q-206.098i q^{5} -210.403i q^{7} -2261.83 q^{11} -3538.83i q^{13} +3485.68 q^{17} -9846.20 q^{19} -2096.31i q^{23} -26851.4 q^{25} +10106.1i q^{29} +1373.11i q^{31} -43363.6 q^{35} -55063.1i q^{37} +58478.9 q^{41} +106639. q^{43} +84028.3i q^{47} +73379.7 q^{49} +80372.5i q^{53} +466159. i q^{55} +84520.5 q^{59} -160512. i q^{61} -729346. q^{65} +229254. q^{67} +184976. i q^{71} -356258. q^{73} +475895. i q^{77} -443857. i q^{79} -356957. q^{83} -718392. i q^{85} +593976. q^{89} -744580. q^{91} +2.02928e6i q^{95} -783783. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 3936 q^{19} - 47796 q^{25} + 340704 q^{43} - 304644 q^{49} + 962112 q^{67} - 1069560 q^{73} - 775008 q^{91} - 86952 q^{97}+O(q^{100}) $ 12 * q - 3936 * q^19 - 47796 * q^25 + 340704 * q^43 - 304644 * q^49 + 962112 * q^67 - 1069560 * q^73 - 775008 * q^91 - 86952 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	− 206.098i	− 1.64878i	−0.566019	−	0.824392i	$-0.691517\pi$
$5$	− 206.098i	− 1.64878i	0.566019	−	0.824392i	$-0.308483\pi$
$6$	0	0
$7$	− 210.403i	− 0.613419i	−0.951803	−	0.306710i	$-0.900772\pi$
$7$	− 210.403i	− 0.613419i	0.951803	−	0.306710i	$-0.0992280\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2261.83	−1.69935	−0.849673	−	0.527310i	$-0.823201\pi$
$11$	−2261.83	−1.69935	−0.849673	+	0.527310i	$0.823201\pi$
$12$	0	0
$13$	− 3538.83i	− 1.61076i	−0.592762	−	0.805378i	$-0.701962\pi$
$13$	− 3538.83i	− 1.61076i	0.592762	−	0.805378i	$-0.298038\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3485.68	0.709481	0.354740	−	0.934965i	$-0.384569\pi$
$17$	3485.68	0.709481	0.354740	+	0.934965i	$0.384569\pi$
$18$	0	0
$19$	−9846.20	−1.43552	−0.717758	−	0.696293i	$-0.754832\pi$
$19$	−9846.20	−1.43552	−0.717758	+	0.696293i	$0.754832\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2096.31i	− 0.172294i	−0.996282	−	0.0861472i	$-0.972544\pi$
$23$	− 2096.31i	− 0.172294i	0.996282	−	0.0861472i	$-0.0274556\pi$
$24$	0	0
$25$	−26851.4	−1.71849
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10106.1i	0.414369i	0.978302	+	0.207185i	$0.0664301\pi$
$29$	10106.1i	0.414369i	−0.978302	+	0.207185i	$0.933570\pi$
$30$	0	0
$31$	1373.11i	0.0460914i	0.999734	+	0.0230457i	$0.00733632\pi$
$31$	1373.11i	0.0460914i	−0.999734	+	0.0230457i	$0.992664\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−43363.6	−1.01140
$36$	0	0
$37$	− 55063.1i	− 1.08706i	−0.839388	−	0.543532i	$-0.817087\pi$
$37$	− 55063.1i	− 1.08706i	0.839388	−	0.543532i	$-0.182913\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	58478.9	0.848491	0.424246	−	0.905547i	$-0.360539\pi$
$41$	58478.9	0.848491	0.424246	+	0.905547i	$0.360539\pi$
$42$	0	0
$43$	106639.	1.34126	0.670628	−	0.741794i	$-0.266025\pi$
$43$	106639.	1.34126	0.670628	+	0.741794i	$0.266025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	84028.3i	0.809342i	0.914462	+	0.404671i	$0.132614\pi$
$47$	84028.3i	0.809342i	−0.914462	+	0.404671i	$0.867386\pi$
$48$	0	0
$49$	73379.7	0.623717
$50$	0	0
$51$	0	0
$52$	0	0
$53$	80372.5i	0.539859i	0.962880	+	0.269929i	$0.0870003\pi$
$53$	80372.5i	0.539859i	−0.962880	+	0.269929i	$0.913000\pi$
$54$	0	0
$55$	466159.i	2.80186i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	84520.5	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$59$	84520.5	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$60$	0	0
$61$	− 160512.i	− 0.707159i	−0.935405	−	0.353579i	$-0.884964\pi$
$61$	− 160512.i	− 0.707159i	0.935405	−	0.353579i	$-0.115036\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−729346.	−2.65579
$66$	0	0
$67$	229254.	0.762240	0.381120	−	0.924525i	$-0.375538\pi$
$67$	229254.	0.762240	0.381120	+	0.924525i	$0.375538\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	184976.i	0.516821i	0.966035	+	0.258411i	$0.0831987\pi$
$71$	184976.i	0.516821i	−0.966035	+	0.258411i	$0.916801\pi$
$72$	0	0
$73$	−356258.	−0.915791	−0.457895	−	0.889006i	$-0.651397\pi$
$73$	−356258.	−0.915791	−0.457895	+	0.889006i	$0.651397\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	475895.i	1.04241i
$78$	0	0
$79$	− 443857.i	− 0.900247i	−0.892966	−	0.450124i	$-0.851380\pi$
$79$	− 443857.i	− 0.900247i	0.892966	−	0.450124i	$-0.148620\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−356957.	−0.624284	−0.312142	−	0.950035i	$-0.601046\pi$
$83$	−356957.	−0.624284	−0.312142	+	0.950035i	$0.601046\pi$
$84$	0	0
$85$	− 718392.i	− 1.16978i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	593976.	0.842556	0.421278	−	0.906932i	$-0.361582\pi$
$89$	593976.	0.842556	0.421278	+	0.906932i	$0.361582\pi$
$90$	0	0
$91$	−744580.	−0.988068
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.02928e6i	2.36686i
$96$	0	0
$97$	−783783.	−0.858778	−0.429389	−	0.903120i	$-0.641271\pi$
$97$	−783783.	−0.858778	−0.429389	+	0.903120i	$0.641271\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.57046e6i	1.52428i	0.647415	+	0.762138i	$0.275850\pi$
$101$	1.57046e6i	1.52428i	−0.647415	+	0.762138i	$0.724150\pi$
$102$	0	0
$103$	− 79752.8i	− 0.0729851i	−0.999334	−	0.0364925i	$-0.988381\pi$
$103$	− 79752.8i	− 0.0729851i	0.999334	−	0.0364925i	$-0.0116185\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.07013e6	−1.68984	−0.844922	−	0.534890i	$-0.820353\pi$
$107$	−2.07013e6	−1.68984	−0.844922	+	0.534890i	$0.820353\pi$
$108$	0	0
$109$	1.62532e6i	1.25505i	0.778598	+	0.627523i	$0.215931\pi$
$109$	1.62532e6i	1.25505i	−0.778598	+	0.627523i	$0.784069\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.61000e6	−1.11581	−0.557904	−	0.829905i	$-0.688394\pi$
$113$	−1.61000e6	−1.11581	−0.557904	+	0.829905i	$0.688394\pi$
$114$	0	0
$115$	−432045.	−0.284077
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 733397.i	− 0.435209i
$120$	0	0
$121$	3.34431e6	1.88778
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.31375e6i	1.18464i
$126$	0	0
$127$	− 499189.i	− 0.243699i	−0.992549	−	0.121849i	$-0.961117\pi$
$127$	− 499189.i	− 0.243699i	0.992549	−	0.121849i	$-0.0388825\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.65837e6	−0.737680	−0.368840	−	0.929493i	$-0.620245\pi$
$131$	−1.65837e6	−0.737680	−0.368840	+	0.929493i	$0.620245\pi$
$132$	0	0
$133$	2.07167e6i	0.880573i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.62443e6	−0.631742	−0.315871	−	0.948802i	$-0.602297\pi$
$137$	−1.62443e6	−0.631742	−0.315871	+	0.948802i	$0.602297\pi$
$138$	0	0
$139$	2.94535e6	1.09671	0.548355	−	0.836246i	$-0.315254\pi$
$139$	2.94535e6	1.09671	0.548355	+	0.836246i	$0.315254\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00423e6i	2.73723i
$144$	0	0
$145$	2.08284e6	0.683206
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.59888e6i	0.483346i	0.970358	+	0.241673i	$0.0776961\pi$
$149$	1.59888e6i	0.483346i	−0.970358	+	0.241673i	$0.922304\pi$
$150$	0	0
$151$	− 5.58054e6i	− 1.62086i	−0.585836	−	0.810429i	$-0.699234\pi$
$151$	− 5.58054e6i	− 1.62086i	0.585836	−	0.810429i	$-0.300766\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	282995.	0.0759948
$156$	0	0
$157$	4.58044e6i	1.18361i	0.806082	+	0.591804i	$0.201584\pi$
$157$	4.58044e6i	1.18361i	−0.806082	+	0.591804i	$0.798416\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−441069.	−0.105689
$162$	0	0
$163$	−4.50036e6	−1.03917	−0.519583	−	0.854420i	$-0.673913\pi$
$163$	−4.50036e6	−1.03917	−0.519583	+	0.854420i	$0.673913\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.07108e6i	− 1.30352i	−0.758427	−	0.651758i	$-0.774032\pi$
$167$	− 6.07108e6i	− 1.30352i	0.758427	−	0.651758i	$-0.225968\pi$
$168$	0	0
$169$	−7.69651e6	−1.59453
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.67216e6i	0.322954i	0.986877	+	0.161477i	$0.0516257\pi$
$173$	1.67216e6i	0.322954i	−0.986877	+	0.161477i	$0.948374\pi$
$174$	0	0
$175$	5.64962e6i	1.05416i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.71439e6	0.821990	0.410995	−	0.911638i	$-0.365181\pi$
$179$	4.71439e6	0.821990	0.410995	+	0.911638i	$0.365181\pi$
$180$	0	0
$181$	− 9.41548e6i	− 1.58784i	−0.608022	−	0.793920i	$-0.708037\pi$
$181$	− 9.41548e6i	− 1.58784i	0.608022	−	0.793920i	$-0.291963\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.13484e7	−1.79234
$186$	0	0
$187$	−7.88401e6	−1.20565
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.01454e6i	1.29373i	0.762604	+	0.646865i	$0.223920\pi$
$191$	9.01454e6i	1.29373i	−0.762604	+	0.646865i	$0.776080\pi$
$192$	0	0
$193$	803577.	0.111778	0.0558889	−	0.998437i	$-0.482201\pi$
$193$	803577.	0.111778	0.0558889	+	0.998437i	$0.482201\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.11135e6i	− 0.145362i	−0.997355	−	0.0726812i	$-0.976844\pi$
$197$	− 1.11135e6i	− 0.145362i	0.997355	−	0.0726812i	$-0.0231556\pi$
$198$	0	0
$199$	− 1.01000e6i	− 0.128163i	−0.997945	−	0.0640814i	$-0.979588\pi$
$199$	− 1.01000e6i	− 0.128163i	0.997945	−	0.0640814i	$-0.0204117\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.12634e6	0.254182
$204$	0	0
$205$	− 1.20524e7i	− 1.39898i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.22704e7	2.43944
$210$	0	0
$211$	1.52288e7	1.62113	0.810565	−	0.585648i	$-0.199160\pi$
$211$	1.52288e7	1.62113	0.810565	+	0.585648i	$0.199160\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 2.19781e7i	− 2.21144i
$216$	0	0
$217$	288906.	0.0282734
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.23352e7i	− 1.14280i
$222$	0	0
$223$	7.55064e6i	0.680877i	0.940267	+	0.340439i	$0.110576\pi$
$223$	7.55064e6i	0.680877i	−0.940267	+	0.340439i	$0.889424\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.93352e7	−1.65299	−0.826497	−	0.562941i	$-0.809670\pi$
$227$	−1.93352e7	−1.65299	−0.826497	+	0.562941i	$0.809670\pi$
$228$	0	0
$229$	1.03537e7i	0.862159i	0.902314	+	0.431079i	$0.141867\pi$
$229$	1.03537e7i	0.862159i	−0.902314	+	0.431079i	$0.858133\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.04567e7	0.826663	0.413331	−	0.910581i	$-0.364365\pi$
$233$	1.04567e7	0.826663	0.413331	+	0.910581i	$0.364365\pi$
$234$	0	0
$235$	1.73181e7	1.33443
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.58738e6i	0.189525i	0.995500	+	0.0947626i	$0.0302092\pi$
$239$	2.58738e6i	0.189525i	−0.995500	+	0.0947626i	$0.969791\pi$
$240$	0	0
$241$	−4.98127e6	−0.355868	−0.177934	−	0.984042i	$-0.556941\pi$
$241$	−4.98127e6	−0.355868	−0.177934	+	0.984042i	$0.556941\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.51234e7i	− 1.02838i
$246$	0	0
$247$	3.48440e7i	2.31227i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.36281e7	0.861816	0.430908	−	0.902396i	$-0.358193\pi$
$251$	1.36281e7	0.861816	0.430908	+	0.902396i	$0.358193\pi$
$252$	0	0
$253$	4.74149e6i	0.292788i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.80282e6	−0.341853	−0.170927	−	0.985284i	$-0.554676\pi$
$257$	−5.80282e6	−0.341853	−0.170927	+	0.985284i	$0.554676\pi$
$258$	0	0
$259$	−1.15854e7	−0.666826
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.82908e7i	− 1.55517i	−0.628778	−	0.777585i	$-0.716445\pi$
$263$	− 2.82908e7i	− 1.55517i	0.628778	−	0.777585i	$-0.283555\pi$
$264$	0	0
$265$	1.65646e7	0.890111
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 5.02831e6i	− 0.258324i	−0.991624	−	0.129162i	$-0.958771\pi$
$269$	− 5.02831e6i	− 0.258324i	0.991624	−	0.129162i	$-0.0412287\pi$
$270$	0	0
$271$	− 3.34374e7i	− 1.68006i	−0.542541	−	0.840029i	$-0.682538\pi$
$271$	− 3.34374e7i	− 1.68006i	0.542541	−	0.840029i	$-0.317462\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.07334e7	2.92031
$276$	0	0
$277$	− 1.94246e7i	− 0.913929i	−0.889485	−	0.456965i	$-0.848937\pi$
$277$	− 1.94246e7i	− 0.913929i	0.889485	−	0.456965i	$-0.151063\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.59526e7	1.16967	0.584833	−	0.811154i	$-0.301160\pi$
$281$	2.59526e7	1.16967	0.584833	+	0.811154i	$0.301160\pi$
$282$	0	0
$283$	336420.	0.0148430	0.00742152	−	0.999972i	$-0.497638\pi$
$283$	336420.	0.0148430	0.00742152	+	0.999972i	$0.497638\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.23041e7i	− 0.520481i
$288$	0	0
$289$	−1.19876e7	−0.496637
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.59063e7i	− 0.632364i	−0.948699	−	0.316182i	$-0.897599\pi$
$293$	− 1.59063e7i	− 0.632364i	0.948699	−	0.316182i	$-0.102401\pi$
$294$	0	0
$295$	− 1.74195e7i	− 0.678532i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.41847e6	−0.277524
$300$	0	0
$301$	− 2.24372e7i	− 0.822752i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.30811e7	−1.16595
$306$	0	0
$307$	−5.19952e7	−1.79700	−0.898500	−	0.438974i	$-0.855342\pi$
$307$	−5.19952e7	−1.79700	−0.898500	+	0.438974i	$0.855342\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.97946e6i	0.232028i	0.993248	+	0.116014i	$0.0370118\pi$
$311$	6.97946e6i	0.232028i	−0.993248	+	0.116014i	$0.962988\pi$
$312$	0	0
$313$	−1.81265e7	−0.591127	−0.295564	−	0.955323i	$-0.595507\pi$
$313$	−1.81265e7	−0.591127	−0.295564	+	0.955323i	$0.595507\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.82943e7i	− 1.20214i	−0.799195	−	0.601072i	$-0.794740\pi$
$317$	− 3.82943e7i	− 1.20214i	0.799195	−	0.601072i	$-0.205260\pi$
$318$	0	0
$319$	− 2.28582e7i	− 0.704157i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.43207e7	−1.01847
$324$	0	0
$325$	9.50227e7i	2.76807i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.76798e7	0.496466
$330$	0	0
$331$	−4.87879e6	−0.134533	−0.0672664	−	0.997735i	$-0.521428\pi$
$331$	−4.87879e6	−0.134533	−0.0672664	+	0.997735i	$0.521428\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 4.72488e7i	− 1.25677i
$336$	0	0
$337$	5.80289e6	0.151619	0.0758097	−	0.997122i	$-0.475846\pi$
$337$	5.80289e6	0.151619	0.0758097	+	0.997122i	$0.475846\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.10574e6i	− 0.0783253i
$342$	0	0
$343$	− 4.01930e7i	− 0.996019i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.11799e7	0.506916	0.253458	−	0.967346i	$-0.418432\pi$
$347$	2.11799e7	0.506916	0.253458	+	0.967346i	$0.418432\pi$
$348$	0	0
$349$	3.02066e7i	0.710600i	0.934752	+	0.355300i	$0.115621\pi$
$349$	3.02066e7i	0.710600i	−0.934752	+	0.355300i	$0.884379\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.47613e6	−0.124494	−0.0622472	−	0.998061i	$-0.519827\pi$
$353$	−5.47613e6	−0.124494	−0.0622472	+	0.998061i	$0.519827\pi$
$354$	0	0
$355$	3.81232e7	0.852127
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.76347e7i	1.46179i	0.682487	+	0.730897i	$0.260898\pi$
$359$	6.76347e7i	1.46179i	−0.682487	+	0.730897i	$0.739102\pi$
$360$	0	0
$361$	4.99019e7	1.06071
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.34241e7i	1.50994i
$366$	0	0
$367$	4.80835e7i	0.972743i	0.873752	+	0.486371i	$0.161680\pi$
$367$	4.80835e7i	0.972743i	−0.873752	+	0.486371i	$0.838320\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.69106e7	0.331160
$372$	0	0
$373$	− 5.09422e7i	− 0.981637i	−0.871262	−	0.490819i	$-0.836698\pi$
$373$	− 5.09422e7i	− 0.981637i	0.871262	−	0.490819i	$-0.163302\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.57636e7	0.667448
$378$	0	0
$379$	4.01401e7	0.737329	0.368665	−	0.929563i	$-0.379815\pi$
$379$	4.01401e7	0.737329	0.368665	+	0.929563i	$0.379815\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.67862e7i	− 0.476777i	−0.971170	−	0.238388i	$-0.923381\pi$
$383$	− 2.67862e7i	− 0.476777i	0.971170	−	0.238388i	$-0.0766191\pi$
$384$	0	0
$385$	9.80811e7	1.71871
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.82424e7i	0.989442i	0.869052	+	0.494721i	$0.164730\pi$
$389$	5.82424e7i	0.989442i	−0.869052	+	0.494721i	$0.835270\pi$
$390$	0	0
$391$	− 7.30705e6i	− 0.122240i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.14781e7	−1.48431
$396$	0	0
$397$	1.92002e7i	0.306856i	0.988160	+	0.153428i	$0.0490312\pi$
$397$	1.92002e7i	0.306856i	−0.988160	+	0.153428i	$0.950969\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.53461e6	0.0703245	0.0351623	−	0.999382i	$-0.488805\pi$
$401$	4.53461e6	0.0703245	0.0351623	+	0.999382i	$0.488805\pi$
$402$	0	0
$403$	4.85920e6	0.0742420
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.24543e8i	1.84730i
$408$	0	0
$409$	−1.02088e8	−1.49213	−0.746063	−	0.665876i	$-0.768058\pi$
$409$	−1.02088e8	−1.49213	−0.746063	+	0.665876i	$0.768058\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.77834e7i	− 0.252443i
$414$	0	0
$415$	7.35682e7i	1.02931i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.36648e8	−1.85764	−0.928820	−	0.370531i	$-0.879176\pi$
$419$	−1.36648e8	−1.85764	−0.928820	+	0.370531i	$0.879176\pi$
$420$	0	0
$421$	− 3.78060e7i	− 0.506657i	−0.967380	−	0.253329i	$-0.918475\pi$
$421$	− 3.78060e7i	− 0.506657i	0.967380	−	0.253329i	$-0.0815254\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.35955e7	−1.21924
$426$	0	0
$427$	−3.37721e7	−0.433785
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.82526e6i	− 0.0477781i	−0.999715	−	0.0238891i	$-0.992395\pi$
$431$	− 3.82526e6i	− 0.0477781i	0.999715	−	0.0238891i	$-0.00760485\pi$
$432$	0	0
$433$	−7.51604e7	−0.925818	−0.462909	−	0.886406i	$-0.653194\pi$
$433$	−7.51604e7	−0.925818	−0.462909	+	0.886406i	$0.653194\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.06407e7i	0.247332i
$438$	0	0
$439$	2.64865e7i	0.313063i	0.987673	+	0.156531i	$0.0500312\pi$
$439$	2.64865e7i	0.313063i	−0.987673	+	0.156531i	$0.949969\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.65074e7	0.995044	0.497522	−	0.867451i	$-0.334243\pi$
$443$	8.65074e7	0.995044	0.497522	+	0.867451i	$0.334243\pi$
$444$	0	0
$445$	− 1.22417e8i	− 1.38919i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.98909e7	−0.330217	−0.165109	−	0.986275i	$-0.552797\pi$
$449$	−2.98909e7	−0.330217	−0.165109	+	0.986275i	$0.552797\pi$
$450$	0	0
$451$	−1.32269e8	−1.44188
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.53456e8i	1.62911i
$456$	0	0
$457$	−7.17316e7	−0.751557	−0.375779	−	0.926709i	$-0.622625\pi$
$457$	−7.17316e7	−0.751557	−0.375779	+	0.926709i	$0.622625\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.44580e8i	1.47572i	0.674952	+	0.737861i	$0.264164\pi$
$461$	1.44580e8i	1.47572i	−0.674952	+	0.737861i	$0.735836\pi$
$462$	0	0
$463$	8.06977e6i	0.0813052i	0.999173	+	0.0406526i	$0.0129437\pi$
$463$	8.06977e6i	0.0813052i	−0.999173	+	0.0406526i	$0.987056\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.38502e7	0.823291	0.411646	−	0.911344i	$-0.364954\pi$
$467$	8.38502e7	0.823291	0.411646	+	0.911344i	$0.364954\pi$
$468$	0	0
$469$	− 4.82356e7i	− 0.467573i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.41200e8	−2.27926
$474$	0	0
$475$	2.64385e8	2.46692
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 9.09446e7i	− 0.827505i	−0.910389	−	0.413752i	$-0.864218\pi$
$479$	− 9.09446e7i	− 0.827505i	0.910389	−	0.413752i	$-0.135782\pi$
$480$	0	0
$481$	−1.94859e8	−1.75100
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.61536e8i	1.41594i
$486$	0	0
$487$	− 1.12400e8i	− 0.973151i	−0.873639	−	0.486575i	$-0.838246\pi$
$487$	− 1.12400e8i	− 0.973151i	0.873639	−	0.486575i	$-0.161754\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.04565e7	−0.341777	−0.170889	−	0.985290i	$-0.554664\pi$
$491$	−4.04565e7	−0.341777	−0.170889	+	0.985290i	$0.554664\pi$
$492$	0	0
$493$	3.52265e7i	0.293987i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.89194e7	0.317028
$498$	0	0
$499$	−1.36162e8	−1.09586	−0.547931	−	0.836524i	$-0.684584\pi$
$499$	−1.36162e8	−1.09586	−0.547931	+	0.836524i	$0.684584\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.62786e8i	1.27912i	0.768739	+	0.639562i	$0.220884\pi$
$503$	1.62786e8i	1.27912i	−0.768739	+	0.639562i	$0.779116\pi$
$504$	0	0
$505$	3.23669e8	2.51320
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.23236e8i	− 0.934511i	−0.884122	−	0.467255i	$-0.845243\pi$
$509$	− 1.23236e8i	− 0.934511i	0.884122	−	0.467255i	$-0.154757\pi$
$510$	0	0
$511$	7.49577e7i	0.561764i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.64369e7	−0.120337
$516$	0	0
$517$	− 1.90058e8i	− 1.37535i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.46388e7	−0.174223	−0.0871115	−	0.996199i	$-0.527764\pi$
$521$	−2.46388e7	−0.174223	−0.0871115	+	0.996199i	$0.527764\pi$
$522$	0	0
$523$	8.82039e7	0.616571	0.308285	−	0.951294i	$-0.400245\pi$
$523$	8.82039e7	0.616571	0.308285	+	0.951294i	$0.400245\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.78622e6i	0.0327010i
$528$	0	0
$529$	1.43641e8	0.970315
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.06947e8i	− 1.36671i
$534$	0	0
$535$	4.26650e8i	2.78619i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.65972e8	−1.05991
$540$	0	0
$541$	3.02964e8i	1.91337i	0.291128	+	0.956684i	$0.405969\pi$
$541$	3.02964e8i	1.91337i	−0.291128	+	0.956684i	$0.594031\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.34976e8	2.06930
$546$	0	0
$547$	−9.88974e6	−0.0604258	−0.0302129	−	0.999543i	$-0.509619\pi$
$547$	−9.88974e6	−0.0604258	−0.0302129	+	0.999543i	$0.509619\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 9.95063e7i	− 0.594834i
$552$	0	0
$553$	−9.33887e7	−0.552229
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.10741e8i	1.79818i	0.437763	+	0.899090i	$0.355771\pi$
$557$	3.10741e8i	1.79818i	−0.437763	+	0.899090i	$0.644229\pi$
$558$	0	0
$559$	− 3.77378e8i	− 2.16043i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.02595e8	−0.574913	−0.287457	−	0.957794i	$-0.592810\pi$
$563$	−1.02595e8	−0.574913	−0.287457	+	0.957794i	$0.592810\pi$
$564$	0	0
$565$	3.31817e8i	1.83973i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.40381e8	1.30486	0.652428	−	0.757851i	$-0.273751\pi$
$569$	2.40381e8	1.30486	0.652428	+	0.757851i	$0.273751\pi$
$570$	0	0
$571$	4.46256e7	0.239704	0.119852	−	0.992792i	$-0.461758\pi$
$571$	4.46256e7	0.239704	0.119852	+	0.992792i	$0.461758\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.62888e7i	0.296087i
$576$	0	0
$577$	1.24361e8	0.647377	0.323689	−	0.946164i	$-0.395077\pi$
$577$	1.24361e8	0.647377	0.323689	+	0.946164i	$0.395077\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.51048e7i	0.382948i
$582$	0	0
$583$	− 1.81789e8i	− 0.917407i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.65460e8	0.818046	0.409023	−	0.912524i	$-0.365870\pi$
$587$	1.65460e8	0.818046	0.409023	+	0.912524i	$0.365870\pi$
$588$	0	0
$589$	− 1.35199e7i	− 0.0661650i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.64689e8	−1.74887	−0.874437	−	0.485139i	$-0.838769\pi$
$593$	−3.64689e8	−1.74887	−0.874437	+	0.485139i	$0.838769\pi$
$594$	0	0
$595$	−1.51152e8	−0.717566
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 4.13919e8i	− 1.92590i	−0.269672	−	0.962952i	$-0.586915\pi$
$599$	− 4.13919e8i	− 1.92590i	0.269672	−	0.962952i	$-0.413085\pi$
$600$	0	0
$601$	−1.49949e8	−0.690747	−0.345373	−	0.938465i	$-0.612248\pi$
$601$	−1.49949e8	−0.690747	−0.345373	+	0.938465i	$0.612248\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.89257e8i	− 3.11254i
$606$	0	0
$607$	− 3.28292e8i	− 1.46789i	−0.679207	−	0.733946i	$-0.737676\pi$
$607$	− 3.28292e8i	− 1.46789i	0.679207	−	0.733946i	$-0.262324\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.97362e8	1.30365
$612$	0	0
$613$	− 9.28329e7i	− 0.403014i	−0.979487	−	0.201507i	$-0.935416\pi$
$613$	− 9.28329e7i	− 0.403014i	0.979487	−	0.201507i	$-0.0645839\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.94762e8	0.829178	0.414589	−	0.910009i	$-0.363925\pi$
$617$	1.94762e8	0.829178	0.414589	+	0.910009i	$0.363925\pi$
$618$	0	0
$619$	−5.23439e6	−0.0220696	−0.0110348	−	0.999939i	$-0.503513\pi$
$619$	−5.23439e6	−0.0220696	−0.0110348	+	0.999939i	$0.503513\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.24974e8i	− 0.516840i
$624$	0	0
$625$	5.73051e7	0.234722
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.91932e8i	− 0.771252i
$630$	0	0
$631$	− 2.99018e8i	− 1.19017i	−0.803662	−	0.595086i	$-0.797118\pi$
$631$	− 2.99018e8i	− 1.19017i	0.803662	−	0.595086i	$-0.202882\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.02882e8	−0.401807
$636$	0	0
$637$	− 2.59678e8i	− 1.00466i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.26180e8	−0.858777	−0.429389	−	0.903120i	$-0.641271\pi$
$641$	−2.26180e8	−0.858777	−0.429389	+	0.903120i	$0.641271\pi$
$642$	0	0
$643$	−4.23603e8	−1.59341	−0.796703	−	0.604371i	$-0.793425\pi$
$643$	−4.23603e8	−1.59341	−0.796703	+	0.604371i	$0.793425\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.69306e8i	− 0.994337i	−0.867654	−	0.497168i	$-0.834373\pi$
$647$	− 2.69306e8i	− 0.994337i	0.867654	−	0.497168i	$-0.165627\pi$
$648$	0	0
$649$	−1.91171e8	−0.699340
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.66960e8i	0.958754i	0.877609	+	0.479377i	$0.159137\pi$
$653$	2.66960e8i	0.958754i	−0.877609	+	0.479377i	$0.840863\pi$
$654$	0	0
$655$	3.41787e8i	1.21628i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.64400e7	0.267094	0.133547	−	0.991042i	$-0.457363\pi$
$659$	7.64400e7	0.267094	0.133547	+	0.991042i	$0.457363\pi$
$660$	0	0
$661$	1.85402e8i	0.641963i	0.947085	+	0.320981i	$0.104013\pi$
$661$	1.85402e8i	0.641963i	−0.947085	+	0.320981i	$0.895987\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.26967e8	1.45188
$666$	0	0
$667$	2.11854e7	0.0713936
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.63050e8i	1.20171i
$672$	0	0
$673$	−4.60309e7	−0.151009	−0.0755047	−	0.997145i	$-0.524057\pi$
$673$	−4.60309e7	−0.151009	−0.0755047	+	0.997145i	$0.524057\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.19932e8i	− 0.708797i	−0.935095	−	0.354398i	$-0.884686\pi$
$677$	− 2.19932e8i	− 0.708797i	0.935095	−	0.354398i	$-0.115314\pi$
$678$	0	0
$679$	1.64910e8i	0.526791i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.36350e8	1.68339	0.841697	−	0.539950i	$-0.181557\pi$
$683$	5.36350e8	1.68339	0.841697	+	0.539950i	$0.181557\pi$
$684$	0	0
$685$	3.34792e8i	1.04161i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.84425e8	0.869580
$690$	0	0
$691$	−4.81630e8	−1.45975	−0.729877	−	0.683579i	$-0.760423\pi$
$691$	−4.81630e8	−1.45975	−0.729877	+	0.683579i	$0.760423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 6.07030e8i	− 1.80824i
$696$	0	0
$697$	2.03839e8	0.601988
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.69995e8i	− 1.36439i	−0.731169	−	0.682196i	$-0.761025\pi$
$701$	− 4.69995e8i	− 1.36439i	0.731169	−	0.682196i	$-0.238975\pi$
$702$	0	0
$703$	5.42162e8i	1.56050i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.30430e8	0.935020
$708$	0	0
$709$	− 4.05087e8i	− 1.13660i	−0.822820	−	0.568302i	$-0.807600\pi$
$709$	− 4.05087e8i	− 1.13660i	0.822820	−	0.568302i	$-0.192400\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.87846e6	0.00794130
$714$	0	0
$715$	1.64966e9	4.51311
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.74151e8i	1.00661i	0.864109	+	0.503304i	$0.167882\pi$
$719$	3.74151e8i	1.00661i	−0.864109	+	0.503304i	$0.832118\pi$
$720$	0	0
$721$	−1.67802e7	−0.0447704
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.71362e8i	− 0.712090i
$726$	0	0
$727$	− 2.13805e8i	− 0.556435i	−0.960518	−	0.278218i	$-0.910256\pi$
$727$	− 2.13805e8i	− 0.556435i	0.960518	−	0.278218i	$-0.0897436\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.71710e8	0.951595
$732$	0	0
$733$	− 5.56686e8i	− 1.41351i	−0.707459	−	0.706754i	$-0.750159\pi$
$733$	− 5.56686e8i	− 1.41351i	0.707459	−	0.706754i	$-0.249841\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.18533e8	−1.29531
$738$	0	0
$739$	−2.10164e8	−0.520744	−0.260372	−	0.965508i	$-0.583845\pi$
$739$	−2.10164e8	−0.520744	−0.260372	+	0.965508i	$0.583845\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 5.71643e8i	− 1.39367i	−0.717234	−	0.696833i	$-0.754592\pi$
$743$	− 5.71643e8i	− 1.39367i	0.717234	−	0.696833i	$-0.245408\pi$
$744$	0	0
$745$	3.29527e8	0.796933
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.35561e8i	1.03658i
$750$	0	0
$751$	2.96917e8i	0.700996i	0.936564	+	0.350498i	$0.113988\pi$
$751$	2.96917e8i	0.700996i	−0.936564	+	0.350498i	$0.886012\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.15014e9	−2.67245
$756$	0	0
$757$	5.06152e8i	1.16679i	0.812188	+	0.583396i	$0.198276\pi$
$757$	5.06152e8i	1.16679i	−0.812188	+	0.583396i	$0.801724\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.88935e8	−0.882518	−0.441259	−	0.897380i	$-0.645468\pi$
$761$	−3.88935e8	−0.882518	−0.441259	+	0.897380i	$0.645468\pi$
$762$	0	0
$763$	3.41972e8	0.769869
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.99104e8i	− 0.662881i
$768$	0	0
$769$	3.96129e7	0.0871079	0.0435540	−	0.999051i	$-0.486132\pi$
$769$	3.96129e7	0.0871079	0.0435540	+	0.999051i	$0.486132\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.41418e8i	1.17218i	0.810246	+	0.586090i	$0.199334\pi$
$773$	5.41418e8i	1.17218i	−0.810246	+	0.586090i	$0.800666\pi$
$774$	0	0
$775$	− 3.68699e7i	− 0.0792077i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.75795e8	−1.21802
$780$	0	0
$781$	− 4.18384e8i	− 0.878258i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.44019e8	1.95151
$786$	0	0
$787$	−2.10149e8	−0.431126	−0.215563	−	0.976490i	$-0.569159\pi$
$787$	−2.10149e8	−0.431126	−0.215563	+	0.976490i	$0.569159\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.38748e8i	0.684459i
$792$	0	0
$793$	−5.68023e8	−1.13906
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.44520e8i	− 1.27310i	−0.771237	−	0.636548i	$-0.780362\pi$
$797$	− 6.44520e8i	− 1.27310i	0.771237	−	0.636548i	$-0.219638\pi$
$798$	0	0
$799$	2.92896e8i	0.574213i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.05795e8	1.55625
$804$	0	0
$805$	9.09034e7i	0.174258i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.12527e8	0.779125	0.389562	−	0.921000i	$-0.372626\pi$
$809$	4.12527e8	0.779125	0.389562	+	0.921000i	$0.372626\pi$
$810$	0	0
$811$	−1.60499e8	−0.300891	−0.150446	−	0.988618i	$-0.548071\pi$
$811$	−1.60499e8	−0.300891	−0.150446	+	0.988618i	$0.548071\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.27516e8i	1.71336i
$816$	0	0
$817$	−1.04999e9	−1.92539
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.60585e8i	− 0.651596i	−0.945439	−	0.325798i	$-0.894367\pi$
$821$	− 3.60585e8i	− 0.651596i	0.945439	−	0.325798i	$-0.105633\pi$
$822$	0	0
$823$	6.90554e8i	1.23879i	0.785079	+	0.619396i	$0.212622\pi$
$823$	6.90554e8i	1.23879i	−0.785079	+	0.619396i	$0.787378\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.02908e9	−1.81941	−0.909706	−	0.415252i	$-0.863693\pi$
$827$	−1.02908e9	−1.81941	−0.909706	+	0.415252i	$0.863693\pi$
$828$	0	0
$829$	− 3.71431e8i	− 0.651950i	−0.945378	−	0.325975i	$-0.894307\pi$
$829$	− 3.71431e8i	− 0.651950i	0.945378	−	0.325975i	$-0.105693\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.55778e8	0.442515
$834$	0	0
$835$	−1.25124e9	−2.14922
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.45962e7i	0.0247145i	0.999924	+	0.0123573i	$0.00393354\pi$
$839$	1.45962e7i	0.0247145i	−0.999924	+	0.0123573i	$0.996066\pi$
$840$	0	0
$841$	4.92691e8	0.828298
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.58624e9i	2.62904i
$846$	0	0
$847$	− 7.03653e8i	− 1.15800i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.15429e8	−0.187295
$852$	0	0
$853$	− 7.71644e8i	− 1.24328i	−0.783302	−	0.621641i	$-0.786466\pi$
$853$	− 7.71644e8i	− 1.24328i	0.783302	−	0.621641i	$-0.213534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.73106e8	−1.22828	−0.614139	−	0.789198i	$-0.710496\pi$
$857$	−7.73106e8	−1.22828	−0.614139	+	0.789198i	$0.710496\pi$
$858$	0	0
$859$	2.50724e8	0.395563	0.197782	−	0.980246i	$-0.436626\pi$
$859$	2.50724e8	0.395563	0.197782	+	0.980246i	$0.436626\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.52441e8i	1.17068i	0.810786	+	0.585342i	$0.199040\pi$
$863$	7.52441e8i	1.17068i	−0.810786	+	0.585342i	$0.800960\pi$
$864$	0	0
$865$	3.44630e8	0.532481
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.00393e9i	1.52983i
$870$	0	0
$871$	− 8.11290e8i	− 1.22778i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.86819e8	0.726680
$876$	0	0
$877$	6.37458e8i	0.945046i	0.881318	+	0.472523i	$0.156657\pi$
$877$	6.37458e8i	0.945046i	−0.881318	+	0.472523i	$0.843343\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.02174e9	−1.49421	−0.747107	−	0.664704i	$-0.768558\pi$
$881$	−1.02174e9	−1.49421	−0.747107	+	0.664704i	$0.768558\pi$
$882$	0	0
$883$	−2.63930e8	−0.383360	−0.191680	−	0.981457i	$-0.561394\pi$
$883$	−2.63930e8	−0.383360	−0.191680	+	0.981457i	$0.561394\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.72432e8i	0.533674i	0.963742	+	0.266837i	$0.0859785\pi$
$887$	3.72432e8i	0.533674i	−0.963742	+	0.266837i	$0.914022\pi$
$888$	0	0
$889$	−1.05031e8	−0.149490
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 8.27360e8i	− 1.16182i
$894$	0	0
$895$	− 9.71627e8i	− 1.35529i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.38767e7	−0.0190989
$900$	0	0
$901$	2.80153e8i	0.383019i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.94051e9	−2.61801
$906$	0	0
$907$	9.13146e8	1.22382	0.611911	−	0.790927i	$-0.290401\pi$
$907$	9.13146e8	1.22382	0.611911	+	0.790927i	$0.290401\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 9.98295e8i	− 1.32039i	−0.751092	−	0.660197i	$-0.770473\pi$
$911$	− 9.98295e8i	− 1.32039i	0.751092	−	0.660197i	$-0.229527\pi$
$912$	0	0
$913$	8.07377e8	1.06087
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.48926e8i	0.452507i
$918$	0	0
$919$	2.77108e8i	0.357029i	0.983937	+	0.178514i	$0.0571291\pi$
$919$	2.77108e8i	0.357029i	−0.983937	+	0.178514i	$0.942871\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.54598e8	0.832472
$924$	0	0
$925$	1.47852e9i	1.86811i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.71196e8	1.08660	0.543299	−	0.839539i	$-0.317175\pi$
$929$	8.71196e8	1.08660	0.543299	+	0.839539i	$0.317175\pi$
$930$	0	0
$931$	−7.22511e8	−0.895356
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.62488e9i	1.98786i
$936$	0	0
$937$	1.53726e7	0.0186865	0.00934324	−	0.999956i	$-0.497026\pi$
$937$	1.53726e7	0.0186865	0.00934324	+	0.999956i	$0.497026\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.32198e9i	1.58656i	0.608856	+	0.793281i	$0.291629\pi$
$941$	1.32198e9i	1.58656i	−0.608856	+	0.793281i	$0.708371\pi$
$942$	0	0
$943$	− 1.22590e8i	− 0.146190i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.85382e8	−0.689270	−0.344635	−	0.938737i	$-0.611997\pi$
$947$	−5.85382e8	−0.689270	−0.344635	+	0.938737i	$0.611997\pi$
$948$	0	0
$949$	1.26074e9i	1.47512i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.94470e8	−0.455759	−0.227880	−	0.973689i	$-0.573179\pi$
$953$	−3.94470e8	−0.455759	−0.227880	+	0.973689i	$0.573179\pi$
$954$	0	0
$955$	1.85788e9	2.13308
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.41785e8i	0.387523i
$960$	0	0
$961$	8.85618e8	0.997876
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.65616e8i	− 0.184297i
$966$	0	0
$967$	8.33074e8i	0.921306i	0.887580	+	0.460653i	$0.152385\pi$
$967$	8.33074e8i	0.921306i	−0.887580	+	0.460653i	$0.847615\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.01281e9	−1.10630	−0.553148	−	0.833083i	$-0.686574\pi$
$971$	−1.01281e9	−1.10630	−0.553148	+	0.833083i	$0.686574\pi$
$972$	0	0
$973$	− 6.19709e8i	− 0.672743i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.78748e6	0.00835051	0.00417526	−	0.999991i	$-0.498671\pi$
$977$	7.78748e6	0.00835051	0.00417526	+	0.999991i	$0.498671\pi$
$978$	0	0
$979$	−1.34347e9	−1.43179
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.06247e9i	1.11856i	0.828980	+	0.559278i	$0.188921\pi$
$983$	1.06247e9i	1.11856i	−0.828980	+	0.559278i	$0.811079\pi$
$984$	0	0
$985$	−2.29047e8	−0.239671
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.23549e8i	− 0.231091i
$990$	0	0
$991$	− 3.44802e8i	− 0.354281i	−0.984186	−	0.177141i	$-0.943315\pi$
$991$	− 3.44802e8i	− 0.354281i	0.984186	−	0.177141i	$-0.0566848\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.08159e8	−0.211313
$996$	0	0
$997$	1.24019e9i	1.25142i	0.780056	+	0.625709i	$0.215190\pi$
$997$	1.24019e9i	1.25142i	−0.780056	+	0.625709i	$0.784810\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.7.b.c.271.1		12
3.2	odd	2	inner	288.7.b.c.271.11		12
4.3	odd	2		72.7.b.d.19.10	yes	12
8.3	odd	2	inner	288.7.b.c.271.12		12
8.5	even	2		72.7.b.d.19.9	yes	12
12.11	even	2		72.7.b.d.19.3	✓	12
24.5	odd	2		72.7.b.d.19.4	yes	12
24.11	even	2	inner	288.7.b.c.271.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.b.d.19.3	✓	12	12.11	even	2
72.7.b.d.19.4	yes	12	24.5	odd	2
72.7.b.d.19.9	yes	12	8.5	even	2
72.7.b.d.19.10	yes	12	4.3	odd	2
288.7.b.c.271.1		12	1.1	even	1	trivial
288.7.b.c.271.2		12	24.11	even	2	inner
288.7.b.c.271.11		12	3.2	odd	2	inner
288.7.b.c.271.12		12	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	288.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-206.098i q^{5} -210.403i q^{7} +O(q^{10})\)	\(q-206.098i q^{5} -210.403i q^{7} -2261.83 q^{11} -3538.83i q^{13} +3485.68 q^{17} -9846.20 q^{19} -2096.31i q^{23} -26851.4 q^{25} +10106.1i q^{29} +1373.11i q^{31} -43363.6 q^{35} -55063.1i q^{37} +58478.9 q^{41} +106639. q^{43} +84028.3i q^{47} +73379.7 q^{49} +80372.5i q^{53} +466159. i q^{55} +84520.5 q^{59} -160512. i q^{61} -729346. q^{65} +229254. q^{67} +184976. i q^{71} -356258. q^{73} +475895. i q^{77} -443857. i q^{79} -356957. q^{83} -718392. i q^{85} +593976. q^{89} -744580. q^{91} +2.02928e6i q^{95} -783783. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 3936 q^{19} - 47796 q^{25} + 340704 q^{43} - 304644 q^{49} + 962112 q^{67} - 1069560 q^{73} - 775008 q^{91} - 86952 q^{97}+O(q^{100}) \) 12 * q - 3936 * q^19 - 47796 * q^25 + 340704 * q^43 - 304644 * q^49 + 962112 * q^67 - 1069560 * q^73 - 775008 * q^91 - 86952 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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