LMFDB - Embedded newform 288.8.d.d.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(145,288)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.d (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:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 $ x^14 - 6x^13 - 52x^12 + 300x^11 - 1005x^10 - 23250x^9 + 349930x^8 + 2867784x^7 - 20463993x^6 - 78987210x^5 + 94608296x^4 - 6477861300x^3 - 21982316987x^2 + 613270661346*x + 3813237677250
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.2
Root			$-6.99438 - 0.299706i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.8.d.d.145.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-455.347i q^{5} +743.502 q^{7} +O(q^{10})$	$q-455.347i q^{5} +743.502 q^{7} -5478.60i q^{11} -6218.86i q^{13} +26310.6 q^{17} -23342.9i q^{19} +50954.9 q^{23} -129216. q^{25} -185064. i q^{29} +184302. q^{31} -338551. i q^{35} +235455. i q^{37} +539438. q^{41} +304234. i q^{43} +923746. q^{47} -270747. q^{49} -1.21851e6i q^{53} -2.49466e6 q^{55} +2.07818e6i q^{59} +176051. i q^{61} -2.83174e6 q^{65} +2.56719e6i q^{67} -2.50521e6 q^{71} -3.68984e6 q^{73} -4.07335e6i q^{77} +393800. q^{79} +2.77326e6i q^{83} -1.19805e7i q^{85} +9.69741e6 q^{89} -4.62374e6i q^{91} -1.06291e7 q^{95} +9.83248e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 1372 q^{7}+O(q^{10}) $ 14 * q - 1372 * q^7	$ 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) $ 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * 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
$5$	− 455.347i	− 1.62910i	−0.580095	−	0.814549i	$-0.696984\pi$
$5$	− 455.347i	− 1.62910i	0.580095	−	0.814549i	$-0.303016\pi$
$6$	0	0
$7$	743.502	0.819293	0.409646	−	0.912244i	$-0.365652\pi$
$7$	743.502	0.819293	0.409646	+	0.912244i	$0.365652\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5478.60i	− 1.24107i	−0.784180	−	0.620533i	$-0.786916\pi$
$11$	− 5478.60i	− 1.24107i	0.784180	−	0.620533i	$-0.213084\pi$
$12$	0	0
$13$	− 6218.86i	− 0.785072i	−0.919737	−	0.392536i	$-0.871598\pi$
$13$	− 6218.86i	− 0.785072i	0.919737	−	0.392536i	$-0.128402\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26310.6	1.29885	0.649426	−	0.760425i	$-0.275009\pi$
$17$	26310.6	1.29885	0.649426	+	0.760425i	$0.275009\pi$
$18$	0	0
$19$	− 23342.9i	− 0.780759i	−0.920654	−	0.390380i	$-0.872344\pi$
$19$	− 23342.9i	− 0.780759i	0.920654	−	0.390380i	$-0.127656\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	50954.9	0.873249	0.436625	−	0.899644i	$-0.356174\pi$
$23$	50954.9	0.873249	0.436625	+	0.899644i	$0.356174\pi$
$24$	0	0
$25$	−129216.	−1.65396
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 185064.i	− 1.40905i	−0.709677	−	0.704527i	$-0.751159\pi$
$29$	− 185064.i	− 1.40905i	0.709677	−	0.704527i	$-0.248841\pi$
$30$	0	0
$31$	184302.	1.11113	0.555565	−	0.831473i	$-0.312502\pi$
$31$	184302.	1.11113	0.555565	+	0.831473i	$0.312502\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 338551.i	− 1.33471i
$36$	0	0
$37$	235455.i	0.764190i	0.924123	+	0.382095i	$0.124797\pi$
$37$	235455.i	0.764190i	−0.924123	+	0.382095i	$0.875203\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	539438.	1.22236	0.611178	−	0.791493i	$-0.290696\pi$
$41$	539438.	1.22236	0.611178	+	0.791493i	$0.290696\pi$
$42$	0	0
$43$	304234.i	0.583536i	0.956489	+	0.291768i	$0.0942435\pi$
$43$	304234.i	0.583536i	−0.956489	+	0.291768i	$0.905756\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	923746.	1.29781	0.648903	−	0.760871i	$-0.275228\pi$
$47$	923746.	1.29781	0.648903	+	0.760871i	$0.275228\pi$
$48$	0	0
$49$	−270747.	−0.328759
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.21851e6i	− 1.12425i	−0.827051	−	0.562126i	$-0.809983\pi$
$53$	− 1.21851e6i	− 1.12425i	0.827051	−	0.562126i	$-0.190017\pi$
$54$	0	0
$55$	−2.49466e6	−2.02182
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.07818e6i	1.31735i	0.752428	+	0.658674i	$0.228882\pi$
$59$	2.07818e6i	1.31735i	−0.752428	+	0.658674i	$0.771118\pi$
$60$	0	0
$61$	176051.i	0.0993082i	0.998766	+	0.0496541i	$0.0158119\pi$
$61$	176051.i	0.0993082i	−0.998766	+	0.0496541i	$0.984188\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.83174e6	−1.27896
$66$	0	0
$67$	2.56719e6i	1.04279i	0.853316	+	0.521394i	$0.174588\pi$
$67$	2.56719e6i	1.04279i	−0.853316	+	0.521394i	$0.825412\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.50521e6	−0.830691	−0.415345	−	0.909664i	$-0.636339\pi$
$71$	−2.50521e6	−0.830691	−0.415345	+	0.909664i	$0.636339\pi$
$72$	0	0
$73$	−3.68984e6	−1.11014	−0.555070	−	0.831804i	$-0.687309\pi$
$73$	−3.68984e6	−1.11014	−0.555070	+	0.831804i	$0.687309\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.07335e6i	− 1.01680i
$78$	0	0
$79$	393800.	0.0898631	0.0449315	−	0.998990i	$-0.485693\pi$
$79$	393800.	0.0898631	0.0449315	+	0.998990i	$0.485693\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.77326e6i	0.532374i	0.963921	+	0.266187i	$0.0857639\pi$
$83$	2.77326e6i	0.532374i	−0.963921	+	0.266187i	$0.914236\pi$
$84$	0	0
$85$	− 1.19805e7i	− 2.11596i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.69741e6	1.45811	0.729056	−	0.684454i	$-0.239960\pi$
$89$	9.69741e6	1.45811	0.729056	+	0.684454i	$0.239960\pi$
$90$	0	0
$91$	− 4.62374e6i	− 0.643204i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.06291e7	−1.27193
$96$	0	0
$97$	9.83248e6	1.09386	0.546930	−	0.837178i	$-0.315796\pi$
$97$	9.83248e6	1.09386	0.546930	+	0.837178i	$0.315796\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.54730e6i	− 0.922053i	−0.887386	−	0.461027i	$-0.847481\pi$
$101$	− 9.54730e6i	− 0.922053i	0.887386	−	0.461027i	$-0.152519\pi$
$102$	0	0
$103$	−1.97038e6	−0.177672	−0.0888360	−	0.996046i	$-0.528315\pi$
$103$	−1.97038e6	−0.177672	−0.0888360	+	0.996046i	$0.528315\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	534061.i	0.0421451i	0.999778	+	0.0210726i	$0.00670810\pi$
$107$	534061.i	0.0421451i	−0.999778	+	0.0210726i	$0.993292\pi$
$108$	0	0
$109$	− 7.44951e6i	− 0.550979i	−0.961304	−	0.275490i	$-0.911160\pi$
$109$	− 7.44951e6i	− 0.550979i	0.961304	−	0.275490i	$-0.0888400\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.32506e7	−1.51586	−0.757932	−	0.652334i	$-0.773790\pi$
$113$	−2.32506e7	−1.51586	−0.757932	+	0.652334i	$0.773790\pi$
$114$	0	0
$115$	− 2.32021e7i	− 1.42261i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.95620e7	1.06414
$120$	0	0
$121$	−1.05278e7	−0.540245
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.32640e7i	1.06537i
$126$	0	0
$127$	1.75615e6	0.0760764	0.0380382	−	0.999276i	$-0.487889\pi$
$127$	1.75615e6	0.0760764	0.0380382	+	0.999276i	$0.487889\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.17334e7i	− 0.456008i	−0.973660	−	0.228004i	$-0.926780\pi$
$131$	− 1.17334e7i	− 0.456008i	0.973660	−	0.228004i	$-0.0732200\pi$
$132$	0	0
$133$	− 1.73555e7i	− 0.639670i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.24223e7	−1.40952	−0.704762	−	0.709444i	$-0.748946\pi$
$137$	−4.24223e7	−1.40952	−0.704762	+	0.709444i	$0.748946\pi$
$138$	0	0
$139$	− 4.53967e7i	− 1.43375i	−0.697204	−	0.716873i	$-0.745573\pi$
$139$	− 4.53967e7i	− 1.43375i	0.697204	−	0.716873i	$-0.254427\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.40706e7	−0.974326
$144$	0	0
$145$	−8.42681e7	−2.29549
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.11277e7i	− 0.275583i	−0.990461	−	0.137791i	$-0.956000\pi$
$149$	− 1.11277e7i	− 0.275583i	0.990461	−	0.137791i	$-0.0440003\pi$
$150$	0	0
$151$	−5.69822e7	−1.34685	−0.673426	−	0.739255i	$-0.735178\pi$
$151$	−5.69822e7	−1.34685	−0.673426	+	0.739255i	$0.735178\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.39214e7i	− 1.81014i
$156$	0	0
$157$	2.09804e7i	0.432679i	0.976318	+	0.216340i	$0.0694118\pi$
$157$	2.09804e7i	0.432679i	−0.976318	+	0.216340i	$0.930588\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.78851e7	0.715447
$162$	0	0
$163$	3.99666e7i	0.722837i	0.932404	+	0.361419i	$0.117707\pi$
$163$	3.99666e7i	0.722837i	−0.932404	+	0.361419i	$0.882293\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.53815e7	−0.920147	−0.460073	−	0.887881i	$-0.652177\pi$
$167$	−5.53815e7	−0.920147	−0.460073	+	0.887881i	$0.652177\pi$
$168$	0	0
$169$	2.40743e7	0.383662
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.98071e7i	0.290844i	0.989370	+	0.145422i	$0.0464540\pi$
$173$	1.98071e7i	0.290844i	−0.989370	+	0.145422i	$0.953546\pi$
$174$	0	0
$175$	−9.60721e7	−1.35508
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 6.55079e6i	− 0.0853706i	−0.999089	−	0.0426853i	$-0.986409\pi$
$179$	− 6.55079e6i	− 0.0853706i	0.999089	−	0.0426853i	$-0.0135913\pi$
$180$	0	0
$181$	1.74311e7i	0.218499i	0.994014	+	0.109250i	$0.0348448\pi$
$181$	1.74311e7i	0.218499i	−0.994014	+	0.109250i	$0.965155\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.07214e8	1.24494
$186$	0	0
$187$	− 1.44145e8i	− 1.61196i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.12470e7	0.947549	0.473774	−	0.880646i	$-0.342891\pi$
$191$	9.12470e7	0.947549	0.473774	+	0.880646i	$0.342891\pi$
$192$	0	0
$193$	−1.32532e8	−1.32699	−0.663497	−	0.748179i	$-0.730928\pi$
$193$	−1.32532e8	−1.32699	−0.663497	+	0.748179i	$0.730928\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.09472e7i	− 0.661155i	−0.943779	−	0.330578i	$-0.892756\pi$
$197$	− 7.09472e7i	− 0.661155i	0.943779	−	0.330578i	$-0.107244\pi$
$198$	0	0
$199$	2.23837e7	0.201347	0.100674	−	0.994920i	$-0.467900\pi$
$199$	2.23837e7	0.201347	0.100674	+	0.994920i	$0.467900\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.37595e8i	− 1.15443i
$204$	0	0
$205$	− 2.45631e8i	− 1.99134i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.27886e8	−0.968974
$210$	0	0
$211$	2.42419e8i	1.77656i	0.459307	+	0.888278i	$0.348098\pi$
$211$	2.42419e8i	1.77656i	−0.459307	+	0.888278i	$0.651902\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.38532e8	0.950637
$216$	0	0
$217$	1.37029e8	0.910340
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.63622e8i	− 1.01969i
$222$	0	0
$223$	7.36062e7	0.444475	0.222237	−	0.974993i	$-0.428664\pi$
$223$	7.36062e7	0.444475	0.222237	+	0.974993i	$0.428664\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.60367e8i	0.909965i	0.890500	+	0.454983i	$0.150354\pi$
$227$	1.60367e8i	0.909965i	−0.890500	+	0.454983i	$0.849646\pi$
$228$	0	0
$229$	2.31854e8i	1.27583i	0.770109	+	0.637913i	$0.220202\pi$
$229$	2.31854e8i	1.27583i	−0.770109	+	0.637913i	$0.779798\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.11331e8	−0.576596	−0.288298	−	0.957541i	$-0.593089\pi$
$233$	−1.11331e8	−0.576596	−0.288298	+	0.957541i	$0.593089\pi$
$234$	0	0
$235$	− 4.20625e8i	− 2.11425i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.33267e8	−1.10525	−0.552625	−	0.833430i	$-0.686374\pi$
$239$	−2.33267e8	−1.10525	−0.552625	+	0.833430i	$0.686374\pi$
$240$	0	0
$241$	3.17291e8	1.46015	0.730077	−	0.683365i	$-0.239484\pi$
$241$	3.17291e8	1.46015	0.730077	+	0.683365i	$0.239484\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.23284e8i	0.535581i
$246$	0	0
$247$	−1.45166e8	−0.612952
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.66791e8i	1.06491i	0.846458	+	0.532456i	$0.178731\pi$
$251$	2.66791e8i	1.06491i	−0.846458	+	0.532456i	$0.821269\pi$
$252$	0	0
$253$	− 2.79161e8i	− 1.08376i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.64860e7	−0.281071	−0.140535	−	0.990076i	$-0.544882\pi$
$257$	−7.64860e7	−0.281071	−0.140535	+	0.990076i	$0.544882\pi$
$258$	0	0
$259$	1.75061e8i	0.626095i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.49906e8	−1.52502	−0.762511	−	0.646975i	$-0.776034\pi$
$263$	−4.49906e8	−1.52502	−0.762511	+	0.646975i	$0.776034\pi$
$264$	0	0
$265$	−5.54845e8	−1.83152
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.15668e8i	1.61524i	0.589703	+	0.807620i	$0.299245\pi$
$269$	5.15668e8i	1.61524i	−0.589703	+	0.807620i	$0.700755\pi$
$270$	0	0
$271$	4.64084e8	1.41646	0.708230	−	0.705982i	$-0.249494\pi$
$271$	4.64084e8	1.41646	0.708230	+	0.705982i	$0.249494\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.07920e8i	2.05267i
$276$	0	0
$277$	2.30171e8i	0.650685i	0.945596	+	0.325342i	$0.105480\pi$
$277$	2.30171e8i	0.650685i	−0.945596	+	0.325342i	$0.894520\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.80065e8	0.484124	0.242062	−	0.970261i	$-0.422176\pi$
$281$	1.80065e8	0.484124	0.242062	+	0.970261i	$0.422176\pi$
$282$	0	0
$283$	6.13365e8i	1.60867i	0.594177	+	0.804334i	$0.297478\pi$
$283$	6.13365e8i	1.60867i	−0.594177	+	0.804334i	$0.702522\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.01073e8	1.00147
$288$	0	0
$289$	2.81910e8	0.687017
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.55965e8i	− 1.52351i	−0.647868	−	0.761753i	$-0.724339\pi$
$293$	− 6.55965e8i	− 1.52351i	0.647868	−	0.761753i	$-0.275661\pi$
$294$	0	0
$295$	9.46292e8	2.14609
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.16881e8i	− 0.685563i
$300$	0	0
$301$	2.26198e8i	0.478087i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.01644e7	0.161783
$306$	0	0
$307$	3.62095e8i	0.714230i	0.934060	+	0.357115i	$0.116240\pi$
$307$	3.62095e8i	0.714230i	−0.934060	+	0.357115i	$0.883760\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.50306e8	0.660368	0.330184	−	0.943917i	$-0.392889\pi$
$311$	3.50306e8	0.660368	0.330184	+	0.943917i	$0.392889\pi$
$312$	0	0
$313$	6.44094e8	1.18726	0.593628	−	0.804740i	$-0.297695\pi$
$313$	6.44094e8	1.18726	0.593628	+	0.804740i	$0.297695\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.81017e8i	0.495480i	0.968827	+	0.247740i	$0.0796878\pi$
$317$	2.81017e8i	0.495480i	−0.968827	+	0.247740i	$0.920312\pi$
$318$	0	0
$319$	−1.01389e9	−1.74873
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.14166e8i	− 1.01409i
$324$	0	0
$325$	8.03575e8i	1.29848i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.86807e8	1.06328
$330$	0	0
$331$	− 2.68661e8i	− 0.407200i	−0.979054	−	0.203600i	$-0.934736\pi$
$331$	− 2.68661e8i	− 0.407200i	0.979054	−	0.203600i	$-0.0652641\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.16896e9	1.69880
$336$	0	0
$337$	6.54698e8	0.931829	0.465915	−	0.884830i	$-0.345725\pi$
$337$	6.54698e8	0.931829	0.465915	+	0.884830i	$0.345725\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.00972e9i	− 1.37898i
$342$	0	0
$343$	−8.13607e8	−1.08864
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.10925e8i	0.142521i	0.997458	+	0.0712603i	$0.0227021\pi$
$347$	1.10925e8i	0.142521i	−0.997458	+	0.0712603i	$0.977298\pi$
$348$	0	0
$349$	− 4.68487e8i	− 0.589941i	−0.955506	−	0.294971i	$-0.904690\pi$
$349$	− 4.68487e8i	− 0.589941i	0.955506	−	0.294971i	$-0.0953098\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.87951e8	1.07443	0.537214	−	0.843446i	$-0.319477\pi$
$353$	8.87951e8	1.07443	0.537214	+	0.843446i	$0.319477\pi$
$354$	0	0
$355$	1.14074e9i	1.35328i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.87179e8	−0.213514	−0.106757	−	0.994285i	$-0.534047\pi$
$359$	−1.87179e8	−0.213514	−0.106757	+	0.994285i	$0.534047\pi$
$360$	0	0
$361$	3.48981e8	0.390415
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.68016e9i	1.80853i
$366$	0	0
$367$	6.67452e8	0.704837	0.352419	−	0.935842i	$-0.385359\pi$
$367$	6.67452e8	0.704837	0.352419	+	0.935842i	$0.385359\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.05966e8i	− 0.921092i
$372$	0	0
$373$	9.07598e8i	0.905550i	0.891625	+	0.452775i	$0.149566\pi$
$373$	9.07598e8i	0.905550i	−0.891625	+	0.452775i	$0.850434\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.15088e9	−1.10621
$378$	0	0
$379$	− 2.75547e8i	− 0.259991i	−0.991515	−	0.129996i	$-0.958504\pi$
$379$	− 2.75547e8i	− 0.259991i	0.991515	−	0.129996i	$-0.0414963\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.22253e9	−1.11190	−0.555948	−	0.831217i	$-0.687645\pi$
$383$	−1.22253e9	−1.11190	−0.555948	+	0.831217i	$0.687645\pi$
$384$	0	0
$385$	−1.85479e9	−1.65646
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.16174e8i	0.789140i	0.918866	+	0.394570i	$0.129106\pi$
$389$	9.16174e8i	0.789140i	−0.918866	+	0.394570i	$0.870894\pi$
$390$	0	0
$391$	1.34065e9	1.13422
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.79316e8i	− 0.146396i
$396$	0	0
$397$	1.48443e9i	1.19067i	0.803477	+	0.595335i	$0.202981\pi$
$397$	1.48443e9i	1.19067i	−0.803477	+	0.595335i	$0.797019\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.09687e9	0.849472	0.424736	−	0.905317i	$-0.360367\pi$
$401$	1.09687e9	0.849472	0.424736	+	0.905317i	$0.360367\pi$
$402$	0	0
$403$	− 1.14615e9i	− 0.872316i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.28996e9	0.948410
$408$	0	0
$409$	−2.35351e8	−0.170092	−0.0850462	−	0.996377i	$-0.527104\pi$
$409$	−2.35351e8	−0.170092	−0.0850462	+	0.996377i	$0.527104\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.54513e9i	1.07929i
$414$	0	0
$415$	1.26279e9	0.867289
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.23181e9i	0.818076i	0.912517	+	0.409038i	$0.134136\pi$
$419$	1.23181e9i	0.818076i	−0.912517	+	0.409038i	$0.865864\pi$
$420$	0	0
$421$	2.59403e8i	0.169429i	0.996405	+	0.0847144i	$0.0269978\pi$
$421$	2.59403e8i	0.169429i	−0.996405	+	0.0847144i	$0.973002\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.39974e9	−2.14825
$426$	0	0
$427$	1.30895e8i	0.0813625i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.69789e9	−1.02150	−0.510750	−	0.859729i	$-0.670632\pi$
$431$	−1.69789e9	−1.02150	−0.510750	+	0.859729i	$0.670632\pi$
$432$	0	0
$433$	6.41532e8	0.379761	0.189881	−	0.981807i	$-0.439190\pi$
$433$	6.41532e8	0.379761	0.189881	+	0.981807i	$0.439190\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.18943e9i	− 0.681797i
$438$	0	0
$439$	−9.19486e8	−0.518704	−0.259352	−	0.965783i	$-0.583509\pi$
$439$	−9.19486e8	−0.518704	−0.259352	+	0.965783i	$0.583509\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.12482e9i	0.614709i	0.951595	+	0.307355i	$0.0994438\pi$
$443$	1.12482e9i	0.614709i	−0.951595	+	0.307355i	$0.900556\pi$
$444$	0	0
$445$	− 4.41568e9i	− 2.37541i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.46238e8	−0.128379	−0.0641893	−	0.997938i	$-0.520446\pi$
$449$	−2.46238e8	−0.128379	−0.0641893	+	0.997938i	$0.520446\pi$
$450$	0	0
$451$	− 2.95536e9i	− 1.51702i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.10540e9	−1.04784
$456$	0	0
$457$	1.01056e9	0.495286	0.247643	−	0.968851i	$-0.420344\pi$
$457$	1.01056e9	0.495286	0.247643	+	0.968851i	$0.420344\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.08713e8i	0.431990i	0.976395	+	0.215995i	$0.0692994\pi$
$461$	9.08713e8i	0.431990i	−0.976395	+	0.215995i	$0.930701\pi$
$462$	0	0
$463$	6.41118e8	0.300196	0.150098	−	0.988671i	$-0.452041\pi$
$463$	6.41118e8	0.300196	0.150098	+	0.988671i	$0.452041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.29050e9i	− 0.586338i	−0.956061	−	0.293169i	$-0.905290\pi$
$467$	− 1.29050e9i	− 0.586338i	0.956061	−	0.293169i	$-0.0947099\pi$
$468$	0	0
$469$	1.90871e9i	0.854349i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.66677e9	0.724207
$474$	0	0
$475$	3.01627e9i	1.29134i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.64358e9	0.683310	0.341655	−	0.939825i	$-0.389013\pi$
$479$	1.64358e9	0.683310	0.341655	+	0.939825i	$0.389013\pi$
$480$	0	0
$481$	1.46426e9	0.599944
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.47719e9i	− 1.78201i
$486$	0	0
$487$	−4.75452e9	−1.86533	−0.932664	−	0.360745i	$-0.882522\pi$
$487$	−4.75452e9	−1.86533	−0.932664	+	0.360745i	$0.882522\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.37630e8i	− 0.204974i	−0.994734	−	0.102487i	$-0.967320\pi$
$491$	− 5.37630e8i	− 0.204974i	0.994734	−	0.102487i	$-0.0326800\pi$
$492$	0	0
$493$	− 4.86913e9i	− 1.83015i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.86263e9	−0.680579
$498$	0	0
$499$	− 4.36235e9i	− 1.57170i	−0.618419	−	0.785849i	$-0.712227\pi$
$499$	− 4.36235e9i	− 1.57170i	0.618419	−	0.785849i	$-0.287773\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.75198e8	−0.0964180	−0.0482090	−	0.998837i	$-0.515351\pi$
$503$	−2.75198e8	−0.0964180	−0.0482090	+	0.998837i	$0.515351\pi$
$504$	0	0
$505$	−4.34733e9	−1.50211
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.51872e9i	− 1.18269i	−0.806417	−	0.591347i	$-0.798596\pi$
$509$	− 3.51872e9i	− 1.18269i	0.806417	−	0.591347i	$-0.201404\pi$
$510$	0	0
$511$	−2.74341e9	−0.909530
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.97205e8i	0.289445i
$516$	0	0
$517$	− 5.06083e9i	− 1.61066i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.74538e9	1.47007	0.735037	−	0.678027i	$-0.237165\pi$
$521$	4.74538e9	1.47007	0.735037	+	0.678027i	$0.237165\pi$
$522$	0	0
$523$	− 4.18246e8i	− 0.127843i	−0.997955	−	0.0639214i	$-0.979639\pi$
$523$	− 4.18246e8i	− 0.127843i	0.997955	−	0.0639214i	$-0.0203607\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.84910e9	1.44319
$528$	0	0
$529$	−8.08428e8	−0.237436
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.35469e9i	− 0.959637i
$534$	0	0
$535$	2.43183e8	0.0686586
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.48332e9i	0.408012i
$540$	0	0
$541$	5.81961e8i	0.158017i	0.996874	+	0.0790085i	$0.0251754\pi$
$541$	5.81961e8i	0.158017i	−0.996874	+	0.0790085i	$0.974825\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.39211e9	−0.897599
$546$	0	0
$547$	4.20858e9i	1.09946i	0.835342	+	0.549731i	$0.185270\pi$
$547$	4.20858e9i	1.09946i	−0.835342	+	0.549731i	$0.814730\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.31992e9	−1.10013
$552$	0	0
$553$	2.92791e8	0.0736242
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.75707e8i	− 0.0921205i	−0.998939	−	0.0460602i	$-0.985333\pi$
$557$	− 3.75707e8i	− 0.0921205i	0.998939	−	0.0460602i	$-0.0146666\pi$
$558$	0	0
$559$	1.89199e9	0.458118
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.85385e9i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$563$	− 2.85385e9i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$564$	0	0
$565$	1.05871e10i	2.46949i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.08256e9	0.929052	0.464526	−	0.885560i	$-0.346225\pi$
$569$	4.08256e9	0.929052	0.464526	+	0.885560i	$0.346225\pi$
$570$	0	0
$571$	6.86611e9i	1.54342i	0.635974	+	0.771710i	$0.280599\pi$
$571$	6.86611e9i	1.54342i	−0.635974	+	0.771710i	$0.719401\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.58417e9	−1.44432
$576$	0	0
$577$	−6.27974e9	−1.36090	−0.680450	−	0.732795i	$-0.738216\pi$
$577$	−6.27974e9	−1.36090	−0.680450	+	0.732795i	$0.738216\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.06192e9i	0.436170i
$582$	0	0
$583$	−6.67573e9	−1.39527
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.78860e9i	0.364989i	0.983207	+	0.182494i	$0.0584171\pi$
$587$	1.78860e9i	0.364989i	−0.983207	+	0.182494i	$0.941583\pi$
$588$	0	0
$589$	− 4.30215e9i	− 0.867524i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.82702e9	−1.34444	−0.672218	−	0.740353i	$-0.734658\pi$
$593$	−6.82702e9	−1.34444	−0.672218	+	0.740353i	$0.734658\pi$
$594$	0	0
$595$	− 8.90749e9i	− 1.73359i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.20473e9	−0.229031	−0.114516	−	0.993421i	$-0.536532\pi$
$599$	−1.20473e9	−0.229031	−0.114516	+	0.993421i	$0.536532\pi$
$600$	0	0
$601$	−6.66530e9	−1.25245	−0.626223	−	0.779644i	$-0.715400\pi$
$601$	−6.66530e9	−1.25245	−0.626223	+	0.779644i	$0.715400\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.79382e9i	0.880111i
$606$	0	0
$607$	−8.60606e8	−0.156187	−0.0780933	−	0.996946i	$-0.524883\pi$
$607$	−8.60606e8	−0.156187	−0.0780933	+	0.996946i	$0.524883\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.74465e9i	− 1.01887i
$612$	0	0
$613$	− 1.25607e9i	− 0.220244i	−0.993918	−	0.110122i	$-0.964876\pi$
$613$	− 1.25607e9i	− 0.220244i	0.993918	−	0.110122i	$-0.0351241\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.18040e9	−0.716506	−0.358253	−	0.933625i	$-0.616627\pi$
$617$	−4.18040e9	−0.716506	−0.358253	+	0.933625i	$0.616627\pi$
$618$	0	0
$619$	− 4.62152e9i	− 0.783191i	−0.920137	−	0.391596i	$-0.871923\pi$
$619$	− 4.62152e9i	− 0.783191i	0.920137	−	0.391596i	$-0.128077\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.21005e9	1.19462
$624$	0	0
$625$	4.98197e8	0.0816245
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.19496e9i	0.992570i
$630$	0	0
$631$	1.10201e10	1.74616	0.873079	−	0.487579i	$-0.162120\pi$
$631$	1.10201e10	1.74616	0.873079	+	0.487579i	$0.162120\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.99659e8i	− 0.123936i
$636$	0	0
$637$	1.68374e9i	0.258100i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.12548e10	1.68785	0.843923	−	0.536464i	$-0.180240\pi$
$641$	1.12548e10	1.68785	0.843923	+	0.536464i	$0.180240\pi$
$642$	0	0
$643$	− 2.79124e9i	− 0.414056i	−0.978335	−	0.207028i	$-0.933621\pi$
$643$	− 2.79124e9i	− 0.414056i	0.978335	−	0.207028i	$-0.0663792\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.67204e9	1.40395	0.701977	−	0.712200i	$-0.252301\pi$
$647$	9.67204e9	1.40395	0.701977	+	0.712200i	$0.252301\pi$
$648$	0	0
$649$	1.13855e10	1.63492
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.62597e9i	− 0.228515i	−0.993451	−	0.114258i	$-0.963551\pi$
$653$	− 1.62597e9i	− 0.228515i	0.993451	−	0.114258i	$-0.0364490\pi$
$654$	0	0
$655$	−5.34274e9	−0.742882
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.29858e7i	− 0.00721209i	−0.999993	−	0.00360604i	$-0.998852\pi$
$659$	− 5.29858e7i	− 0.00721209i	0.999993	−	0.00360604i	$-0.00114784\pi$
$660$	0	0
$661$	− 6.70820e9i	− 0.903444i	−0.892159	−	0.451722i	$-0.850810\pi$
$661$	− 6.70820e9i	− 0.903444i	0.892159	−	0.451722i	$-0.149190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.90277e9	−1.04209
$666$	0	0
$667$	− 9.42988e9i	− 1.23046i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.64514e8	0.123248
$672$	0	0
$673$	7.84241e9	0.991737	0.495869	−	0.868398i	$-0.334850\pi$
$673$	7.84241e9	0.991737	0.495869	+	0.868398i	$0.334850\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.82039e9i	− 0.225478i	−0.993625	−	0.112739i	$-0.964038\pi$
$677$	− 1.82039e9i	− 0.225478i	0.993625	−	0.112739i	$-0.0359624\pi$
$678$	0	0
$679$	7.31047e9	0.896192
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.99869e8i	− 0.0480226i	−0.999712	−	0.0240113i	$-0.992356\pi$
$683$	− 3.99869e8i	− 0.0480226i	0.999712	−	0.0240113i	$-0.00764376\pi$
$684$	0	0
$685$	1.93169e10i	2.29625i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.57775e9	−0.882619
$690$	0	0
$691$	8.62228e9i	0.994143i	0.867709	+	0.497072i	$0.165591\pi$
$691$	8.62228e9i	0.994143i	−0.867709	+	0.497072i	$0.834409\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.06712e10	−2.33571
$696$	0	0
$697$	1.41929e10	1.58766
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.99773e9i	0.219040i	0.993985	+	0.109520i	$0.0349314\pi$
$701$	1.99773e9i	0.219040i	−0.993985	+	0.109520i	$0.965069\pi$
$702$	0	0
$703$	5.49619e9	0.596648
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.09844e9i	− 0.755431i
$708$	0	0
$709$	− 1.25129e10i	− 1.31855i	−0.751904	−	0.659273i	$-0.770864\pi$
$709$	− 1.25129e10i	− 1.31855i	0.751904	−	0.659273i	$-0.229136\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.39109e9	0.970292
$714$	0	0
$715$	1.55140e10i	1.58727i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.15772e10	1.16159	0.580793	−	0.814051i	$-0.302742\pi$
$719$	1.15772e10	1.16159	0.580793	+	0.814051i	$0.302742\pi$
$720$	0	0
$721$	−1.46498e9	−0.145565
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.39131e10i	2.33052i
$726$	0	0
$727$	3.70175e9	0.357303	0.178651	−	0.983912i	$-0.442827\pi$
$727$	3.70175e9	0.357303	0.178651	+	0.983912i	$0.442827\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00457e9i	0.757927i
$732$	0	0
$733$	− 3.02987e9i	− 0.284158i	−0.989855	−	0.142079i	$-0.954621\pi$
$733$	− 3.02987e9i	− 0.284158i	0.989855	−	0.142079i	$-0.0453787\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.40646e10	1.29417
$738$	0	0
$739$	4.80537e9i	0.437998i	0.975725	+	0.218999i	$0.0702791\pi$
$739$	4.80537e9i	0.437998i	−0.975725	+	0.218999i	$0.929721\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.54835e9	−0.496253	−0.248127	−	0.968728i	$-0.579815\pi$
$743$	−5.54835e9	−0.496253	−0.248127	+	0.968728i	$0.579815\pi$
$744$	0	0
$745$	−5.06694e9	−0.448951
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.97075e8i	0.0345292i
$750$	0	0
$751$	1.33688e10	1.15173	0.575867	−	0.817543i	$-0.304664\pi$
$751$	1.33688e10	1.15173	0.575867	+	0.817543i	$0.304664\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.59466e10i	2.19415i
$756$	0	0
$757$	1.78827e10i	1.49829i	0.662405	+	0.749146i	$0.269536\pi$
$757$	1.78827e10i	1.49829i	−0.662405	+	0.749146i	$0.730464\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.42976e9	−0.282109	−0.141054	−	0.990002i	$-0.545049\pi$
$761$	−3.42976e9	−0.282109	−0.141054	+	0.990002i	$0.545049\pi$
$762$	0	0
$763$	− 5.53873e9i	− 0.451413i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.29239e10	1.03421
$768$	0	0
$769$	−4.79906e9	−0.380552	−0.190276	−	0.981731i	$-0.560938\pi$
$769$	−4.79906e9	−0.380552	−0.190276	+	0.981731i	$0.560938\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.01981e10i	− 1.57283i	−0.617697	−	0.786416i	$-0.711934\pi$
$773$	− 2.01981e10i	− 1.57283i	0.617697	−	0.786416i	$-0.288066\pi$
$774$	0	0
$775$	−2.38147e10	−1.83776
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.25920e10i	− 0.954366i
$780$	0	0
$781$	1.37250e10i	1.03094i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.55338e9	0.704877
$786$	0	0
$787$	2.43720e10i	1.78229i	0.453716	+	0.891146i	$0.350098\pi$
$787$	2.43720e10i	1.78229i	−0.453716	+	0.891146i	$0.649902\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.72869e10	−1.24194
$792$	0	0
$793$	1.09484e9	0.0779640
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.87528e10i	− 1.31208i	−0.754724	−	0.656042i	$-0.772229\pi$
$797$	− 1.87528e10i	− 1.31208i	0.754724	−	0.656042i	$-0.227771\pi$
$798$	0	0
$799$	2.43043e10	1.68566
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.02152e10i	1.37776i
$804$	0	0
$805$	− 1.72508e10i	− 1.16553i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.22263e10	−0.811850	−0.405925	−	0.913906i	$-0.633051\pi$
$809$	−1.22263e10	−0.811850	−0.405925	+	0.913906i	$0.633051\pi$
$810$	0	0
$811$	− 7.90707e9i	− 0.520526i	−0.965538	−	0.260263i	$-0.916191\pi$
$811$	− 7.90707e9i	− 0.520526i	0.965538	−	0.260263i	$-0.0838093\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.81987e10	1.17757
$816$	0	0
$817$	7.10169e9	0.455601
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.86830e10i	1.17827i	0.808033	+	0.589137i	$0.200532\pi$
$821$	1.86830e10i	1.17827i	−0.808033	+	0.589137i	$0.799468\pi$
$822$	0	0
$823$	−2.69016e10	−1.68221	−0.841103	−	0.540875i	$-0.818093\pi$
$823$	−2.69016e10	−1.68221	−0.841103	+	0.540875i	$0.818093\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.70970e10i	− 1.66591i	−0.553341	−	0.832955i	$-0.686647\pi$
$827$	− 2.70970e10i	− 1.66591i	0.553341	−	0.832955i	$-0.313353\pi$
$828$	0	0
$829$	2.34110e10i	1.42718i	0.700562	+	0.713591i	$0.252933\pi$
$829$	2.34110e10i	1.42718i	−0.700562	+	0.713591i	$0.747067\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.12353e9	−0.427010
$834$	0	0
$835$	2.52178e10i	1.49901i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.25141e10	0.731528	0.365764	−	0.930708i	$-0.380808\pi$
$839$	1.25141e10	0.731528	0.365764	+	0.930708i	$0.380808\pi$
$840$	0	0
$841$	−1.69986e10	−0.985435
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.09621e10i	− 0.625024i
$846$	0	0
$847$	−7.82747e9	−0.442619
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.19976e10i	0.667328i
$852$	0	0
$853$	− 1.23698e10i	− 0.682406i	−0.939990	−	0.341203i	$-0.889166\pi$
$853$	− 1.23698e10i	− 0.682406i	0.939990	−	0.341203i	$-0.110834\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.79673e10	0.975102	0.487551	−	0.873094i	$-0.337890\pi$
$857$	1.79673e10	0.975102	0.487551	+	0.873094i	$0.337890\pi$
$858$	0	0
$859$	− 1.52772e10i	− 0.822373i	−0.911551	−	0.411187i	$-0.865114\pi$
$859$	− 1.52772e10i	− 0.822373i	0.911551	−	0.411187i	$-0.134886\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.28852e9	−0.438974	−0.219487	−	0.975615i	$-0.570438\pi$
$863$	−8.28852e9	−0.438974	−0.219487	+	0.975615i	$0.570438\pi$
$864$	0	0
$865$	9.01912e9	0.473814
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.15747e9i	− 0.111526i
$870$	0	0
$871$	1.59650e10	0.818664
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.72968e10i	0.872846i
$876$	0	0
$877$	2.33208e10i	1.16747i	0.811946	+	0.583733i	$0.198408\pi$
$877$	2.33208e10i	1.16747i	−0.811946	+	0.583733i	$0.801592\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.84509e8	0.0238719	0.0119359	−	0.999929i	$-0.496201\pi$
$881$	4.84509e8	0.0238719	0.0119359	+	0.999929i	$0.496201\pi$
$882$	0	0
$883$	− 1.33164e9i	− 0.0650914i	−0.999470	−	0.0325457i	$-0.989639\pi$
$883$	− 1.33164e9i	− 0.0650914i	0.999470	−	0.0325457i	$-0.0103615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.50356e10	1.20455	0.602276	−	0.798288i	$-0.294261\pi$
$887$	2.50356e10	1.20455	0.602276	+	0.798288i	$0.294261\pi$
$888$	0	0
$889$	1.30570e9	0.0623288
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.15629e10i	− 1.01327i
$894$	0	0
$895$	−2.98288e9	−0.139077
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.41076e10i	− 1.56564i
$900$	0	0
$901$	− 3.20598e10i	− 1.46024i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.93720e9	0.355957
$906$	0	0
$907$	− 9.90907e9i	− 0.440969i	−0.975391	−	0.220484i	$-0.929236\pi$
$907$	− 9.90907e9i	− 0.440969i	0.975391	−	0.220484i	$-0.0707638\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.07733e10	1.34853	0.674263	−	0.738491i	$-0.264461\pi$
$911$	3.07733e10	1.34853	0.674263	+	0.738491i	$0.264461\pi$
$912$	0	0
$913$	1.51936e10	0.660711
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.72377e9i	− 0.373604i
$918$	0	0
$919$	3.38273e10	1.43768	0.718842	−	0.695173i	$-0.244672\pi$
$919$	3.38273e10	1.43768	0.718842	+	0.695173i	$0.244672\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.55795e10i	0.652152i
$924$	0	0
$925$	− 3.04244e10i	− 1.26394i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.57424e10	−1.05340	−0.526702	−	0.850050i	$-0.676572\pi$
$929$	−2.57424e10	−1.05340	−0.526702	+	0.850050i	$0.676572\pi$
$930$	0	0
$931$	6.32003e9i	0.256682i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.56360e10	−2.62604
$936$	0	0
$937$	−1.59703e10	−0.634198	−0.317099	−	0.948393i	$-0.602709\pi$
$937$	−1.59703e10	−0.634198	−0.317099	+	0.948393i	$0.602709\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.15360e10i	− 0.451329i	−0.974205	−	0.225665i	$-0.927545\pi$
$941$	− 1.15360e10i	− 0.451329i	0.974205	−	0.225665i	$-0.0724553\pi$
$942$	0	0
$943$	2.74870e10	1.06742
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.78852e9i	0.259747i	0.991531	+	0.129873i	$0.0414571\pi$
$947$	6.78852e9i	0.259747i	−0.991531	+	0.129873i	$0.958543\pi$
$948$	0	0
$949$	2.29466e10i	0.871540i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.37964e9	−0.126487	−0.0632434	−	0.997998i	$-0.520144\pi$
$953$	−3.37964e9	−0.126487	−0.0632434	+	0.997998i	$0.520144\pi$
$954$	0	0
$955$	− 4.15490e10i	− 1.54365i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.15411e10	−1.15481
$960$	0	0
$961$	6.45467e9	0.234608
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.03478e10i	2.16180i
$966$	0	0
$967$	−4.66583e10	−1.65934	−0.829672	−	0.558251i	$-0.811473\pi$
$967$	−4.66583e10	−1.65934	−0.829672	+	0.558251i	$0.811473\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.88076e10i	− 0.659273i	−0.944108	−	0.329637i	$-0.893074\pi$
$971$	− 1.88076e10i	− 0.659273i	0.944108	−	0.329637i	$-0.106926\pi$
$972$	0	0
$973$	− 3.37525e10i	− 1.17466i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.95802e10	1.35784	0.678918	−	0.734214i	$-0.262449\pi$
$977$	3.95802e10	1.35784	0.678918	+	0.734214i	$0.262449\pi$
$978$	0	0
$979$	− 5.31282e10i	− 1.80961i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.34757e10	−0.452494	−0.226247	−	0.974070i	$-0.572646\pi$
$983$	−1.34757e10	−0.452494	−0.226247	+	0.974070i	$0.572646\pi$
$984$	0	0
$985$	−3.23056e10	−1.07709
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.55022e10i	0.509572i
$990$	0	0
$991$	−4.14371e10	−1.35248	−0.676241	−	0.736680i	$-0.736392\pi$
$991$	−4.14371e10	−1.35248	−0.676241	+	0.736680i	$0.736392\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.01923e10i	− 0.328014i
$996$	0	0
$997$	− 3.75786e10i	− 1.20090i	−0.799661	−	0.600451i	$-0.794988\pi$
$997$	− 3.75786e10i	− 1.20090i	0.799661	−	0.600451i	$-0.205012\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.d.d.145.2		14
3.2	odd	2		96.8.d.a.49.7		14
4.3	odd	2		72.8.d.d.37.11		14
8.3	odd	2		72.8.d.d.37.12		14
8.5	even	2	inner	288.8.d.d.145.13		14
12.11	even	2		24.8.d.a.13.4	yes	14
24.5	odd	2		96.8.d.a.49.8		14
24.11	even	2		24.8.d.a.13.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.d.a.13.3	✓	14	24.11	even	2
24.8.d.a.13.4	yes	14	12.11	even	2
72.8.d.d.37.11		14	4.3	odd	2
72.8.d.d.37.12		14	8.3	odd	2
96.8.d.a.49.7		14	3.2	odd	2
96.8.d.a.49.8		14	24.5	odd	2
288.8.d.d.145.2		14	1.1	even	1	trivial
288.8.d.d.145.13		14	8.5	even	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-455.347i q^{5} +743.502 q^{7} +O(q^{10})\)	\(q-455.347i q^{5} +743.502 q^{7} -5478.60i q^{11} -6218.86i q^{13} +26310.6 q^{17} -23342.9i q^{19} +50954.9 q^{23} -129216. q^{25} -185064. i q^{29} +184302. q^{31} -338551. i q^{35} +235455. i q^{37} +539438. q^{41} +304234. i q^{43} +923746. q^{47} -270747. q^{49} -1.21851e6i q^{53} -2.49466e6 q^{55} +2.07818e6i q^{59} +176051. i q^{61} -2.83174e6 q^{65} +2.56719e6i q^{67} -2.50521e6 q^{71} -3.68984e6 q^{73} -4.07335e6i q^{77} +393800. q^{79} +2.77326e6i q^{83} -1.19805e7i q^{85} +9.69741e6 q^{89} -4.62374e6i q^{91} -1.06291e7 q^{95} +9.83248e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 1372 q^{7}+O(q^{10}) \) 14 * q - 1372 * q^7	\( 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) \) 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more