LMFDB - Embedded newform 576.7.h.b.161.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$132.511152165$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.16
Character	$\chi$	$=$	576.161
Dual form			576.7.h.b.161.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-83.0572 q^{5} -37.1638 q^{7} +O(q^{10})$	$q-83.0572 q^{5} -37.1638 q^{7} -1959.80 q^{11} +4306.41i q^{13} -7855.72i q^{17} -6114.84i q^{19} -15556.2i q^{23} -8726.51 q^{25} -1668.36 q^{29} -27871.2 q^{31} +3086.72 q^{35} -52355.1i q^{37} +51807.9i q^{41} +135857. i q^{43} +93610.2i q^{47} -116268. q^{49} +104925. q^{53} +162776. q^{55} -176394. q^{59} -926.866i q^{61} -357678. i q^{65} -163928. i q^{67} -610911. i q^{71} +333300. q^{73} +72833.8 q^{77} -605169. q^{79} +486745. q^{83} +652474. i q^{85} +826800. i q^{89} -160043. i q^{91} +507882. i q^{95} -1.11632e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) $ 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	−83.0572	−0.664457	−0.332229	−	0.943199i	$-0.607801\pi$
$5$	−83.0572	−0.664457	−0.332229	+	0.943199i	$0.607801\pi$
$6$	0	0
$7$	−37.1638	−0.108349	−0.0541747	−	0.998531i	$-0.517253\pi$
$7$	−37.1638	−0.108349	−0.0541747	+	0.998531i	$0.517253\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1959.80	−1.47243	−0.736214	−	0.676748i	$-0.763388\pi$
$11$	−1959.80	−1.47243	−0.736214	+	0.676748i	$0.763388\pi$
$12$	0	0
$13$	4306.41i	1.96013i	0.198675	+	0.980065i	$0.436336\pi$
$13$	4306.41i	1.96013i	−0.198675	+	0.980065i	$0.563664\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7855.72i	− 1.59897i	−0.600688	−	0.799483i	$-0.705107\pi$
$17$	− 7855.72i	− 1.59897i	0.600688	−	0.799483i	$-0.294893\pi$
$18$	0	0
$19$	− 6114.84i	− 0.891507i	−0.895156	−	0.445753i	$-0.852936\pi$
$19$	− 6114.84i	− 0.891507i	0.895156	−	0.445753i	$-0.147064\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 15556.2i	− 1.27856i	−0.768975	−	0.639279i	$-0.779233\pi$
$23$	− 15556.2i	− 1.27856i	0.768975	−	0.639279i	$-0.220767\pi$
$24$	0	0
$25$	−8726.51	−0.558497
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1668.36	−0.0684061	−0.0342030	−	0.999415i	$-0.510889\pi$
$29$	−1668.36	−0.0684061	−0.0342030	+	0.999415i	$0.510889\pi$
$30$	0	0
$31$	−27871.2	−0.935558	−0.467779	−	0.883846i	$-0.654946\pi$
$31$	−27871.2	−0.935558	−0.467779	+	0.883846i	$0.654946\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3086.72	0.0719935
$36$	0	0
$37$	− 52355.1i	− 1.03360i	−0.856105	−	0.516802i	$-0.827122\pi$
$37$	− 52355.1i	− 1.03360i	0.856105	−	0.516802i	$-0.172878\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	51807.9i	0.751699i	0.926681	+	0.375850i	$0.122649\pi$
$41$	51807.9i	0.751699i	−0.926681	+	0.375850i	$0.877351\pi$
$42$	0	0
$43$	135857.i	1.70874i	0.519664	+	0.854371i	$0.326057\pi$
$43$	135857.i	1.70874i	−0.519664	+	0.854371i	$0.673943\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	93610.2i	0.901633i	0.892617	+	0.450816i	$0.148867\pi$
$47$	93610.2i	0.901633i	−0.892617	+	0.450816i	$0.851133\pi$
$48$	0	0
$49$	−116268.	−0.988260
$50$	0	0
$51$	0	0
$52$	0	0
$53$	104925.	0.704773	0.352387	−	0.935854i	$-0.385370\pi$
$53$	104925.	0.704773	0.352387	+	0.935854i	$0.385370\pi$
$54$	0	0
$55$	162776.	0.978366
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−176394.	−0.858870	−0.429435	−	0.903098i	$-0.641287\pi$
$59$	−176394.	−0.858870	−0.429435	+	0.903098i	$0.641287\pi$
$60$	0	0
$61$	− 926.866i	− 0.00408345i	−0.999998	−	0.00204173i	$-0.999350\pi$
$61$	− 926.866i	− 0.00408345i	0.999998	−	0.00204173i	$-0.000649902\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 357678.i	− 1.30242i
$66$	0	0
$67$	− 163928.i	− 0.545039i	−0.962150	−	0.272520i	$-0.912143\pi$
$67$	− 163928.i	− 0.545039i	0.962150	−	0.272520i	$-0.0878570\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 610911.i	− 1.70688i	−0.521190	−	0.853441i	$-0.674512\pi$
$71$	− 610911.i	− 1.70688i	0.521190	−	0.853441i	$-0.325488\pi$
$72$	0	0
$73$	333300.	0.856774	0.428387	−	0.903595i	$-0.359082\pi$
$73$	333300.	0.856774	0.428387	+	0.903595i	$0.359082\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	72833.8	0.159537
$78$	0	0
$79$	−605169.	−1.22743	−0.613713	−	0.789529i	$-0.710325\pi$
$79$	−605169.	−1.22743	−0.613713	+	0.789529i	$0.710325\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	486745.	0.851270	0.425635	−	0.904895i	$-0.360051\pi$
$83$	486745.	0.851270	0.425635	+	0.904895i	$0.360051\pi$
$84$	0	0
$85$	652474.i	1.06245i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	826800.i	1.17282i	0.810015	+	0.586409i	$0.199459\pi$
$89$	826800.i	1.17282i	−0.810015	+	0.586409i	$0.800541\pi$
$90$	0	0
$91$	− 160043.i	− 0.212379i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	507882.i	0.592368i
$96$	0	0
$97$	−1.11632e6	−1.22313	−0.611565	−	0.791194i	$-0.709460\pi$
$97$	−1.11632e6	−1.22313	−0.611565	+	0.791194i	$0.709460\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−153614.	−0.149096	−0.0745481	−	0.997217i	$-0.523751\pi$
$101$	−153614.	−0.149096	−0.0745481	+	0.997217i	$0.523751\pi$
$102$	0	0
$103$	1.73563e6	1.58835	0.794173	−	0.607692i	$-0.207904\pi$
$103$	1.73563e6	1.58835	0.794173	+	0.607692i	$0.207904\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.41074e6	1.15159	0.575793	−	0.817596i	$-0.304694\pi$
$107$	1.41074e6	1.15159	0.575793	+	0.817596i	$0.304694\pi$
$108$	0	0
$109$	121880.i	0.0941137i	0.998892	+	0.0470568i	$0.0149842\pi$
$109$	121880.i	0.0941137i	−0.998892	+	0.0470568i	$0.985016\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.38095e6i	0.957069i	0.878069	+	0.478534i	$0.158832\pi$
$113$	1.38095e6i	0.957069i	−0.878069	+	0.478534i	$0.841168\pi$
$114$	0	0
$115$	1.29205e6i	0.849547i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	291949.i	0.173247i
$120$	0	0
$121$	2.06926e6	1.16805
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.02257e6	1.03555
$126$	0	0
$127$	2.04249e6	0.997124	0.498562	−	0.866854i	$-0.333862\pi$
$127$	2.04249e6	0.997124	0.498562	+	0.866854i	$0.333862\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.55809e6	−0.693072	−0.346536	−	0.938037i	$-0.612642\pi$
$131$	−1.55809e6	−0.693072	−0.346536	+	0.938037i	$0.612642\pi$
$132$	0	0
$133$	227251.i	0.0965942i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 227489.i	− 0.0884707i	−0.999021	−	0.0442354i	$-0.985915\pi$
$137$	− 227489.i	− 0.0884707i	0.999021	−	0.0442354i	$-0.0140851\pi$
$138$	0	0
$139$	− 1.83311e6i	− 0.682565i	−0.939961	−	0.341282i	$-0.889139\pi$
$139$	− 1.83311e6i	− 0.682565i	0.939961	−	0.341282i	$-0.110861\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.43971e6i	− 2.88615i
$144$	0	0
$145$	138569.	0.0454529
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.77808e6	1.74673	0.873363	−	0.487070i	$-0.161935\pi$
$149$	5.77808e6	1.74673	0.873363	+	0.487070i	$0.161935\pi$
$150$	0	0
$151$	−479414.	−0.139245	−0.0696226	−	0.997573i	$-0.522179\pi$
$151$	−479414.	−0.139245	−0.0696226	+	0.997573i	$0.522179\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.31490e6	0.621638
$156$	0	0
$157$	5.54477e6i	1.43280i	0.697692	+	0.716398i	$0.254211\pi$
$157$	5.54477e6i	1.43280i	−0.697692	+	0.716398i	$0.745789\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	578128.i	0.138531i
$162$	0	0
$163$	1.96511e6i	0.453759i	0.973923	+	0.226879i	$0.0728523\pi$
$163$	1.96511e6i	0.453759i	−0.973923	+	0.226879i	$0.927148\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.09517e6i	0.449852i	0.974376	+	0.224926i	$0.0722140\pi$
$167$	2.09517e6i	0.449852i	−0.974376	+	0.224926i	$0.927786\pi$
$168$	0	0
$169$	−1.37183e7	−2.84211
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−964175.	−0.186216	−0.0931081	−	0.995656i	$-0.529680\pi$
$173$	−964175.	−0.186216	−0.0931081	+	0.995656i	$0.529680\pi$
$174$	0	0
$175$	324311.	0.0605128
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.77781e6	1.35612	0.678060	−	0.735007i	$-0.262821\pi$
$179$	7.77781e6	1.35612	0.678060	+	0.735007i	$0.262821\pi$
$180$	0	0
$181$	5.24935e6i	0.885258i	0.896705	+	0.442629i	$0.145954\pi$
$181$	5.24935e6i	0.885258i	−0.896705	+	0.442629i	$0.854046\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.34847e6i	0.686786i
$186$	0	0
$187$	1.53957e7i	2.35436i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.21395e7i	− 1.74221i	−0.491100	−	0.871103i	$-0.663405\pi$
$191$	− 1.21395e7i	− 1.74221i	0.491100	−	0.871103i	$-0.336595\pi$
$192$	0	0
$193$	2.06934e6	0.287846	0.143923	−	0.989589i	$-0.454028\pi$
$193$	2.06934e6	0.287846	0.143923	+	0.989589i	$0.454028\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.58221e6	0.860940	0.430470	−	0.902605i	$-0.358348\pi$
$197$	6.58221e6	0.860940	0.430470	+	0.902605i	$0.358348\pi$
$198$	0	0
$199$	1.08503e7	1.37683	0.688416	−	0.725316i	$-0.258306\pi$
$199$	1.08503e7	1.37683	0.688416	+	0.725316i	$0.258306\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	62002.5	0.00741176
$204$	0	0
$205$	− 4.30301e6i	− 0.499472i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.19839e7i	1.31268i
$210$	0	0
$211$	− 1.29771e6i	− 0.138143i	−0.997612	−	0.0690715i	$-0.977996\pi$
$211$	− 1.29771e6i	− 0.138143i	0.997612	−	0.0690715i	$-0.0220036\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.12839e7i	− 1.13539i
$216$	0	0
$217$	1.03580e6	0.101367
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.38299e7	3.13418
$222$	0	0
$223$	7.77428e6	0.701044	0.350522	−	0.936554i	$-0.386004\pi$
$223$	7.77428e6	0.701044	0.350522	+	0.936554i	$0.386004\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.27506e6	0.365481	0.182740	−	0.983161i	$-0.441503\pi$
$227$	4.27506e6	0.365481	0.182740	+	0.983161i	$0.441503\pi$
$228$	0	0
$229$	8.38344e6i	0.698097i	0.937105	+	0.349049i	$0.113495\pi$
$229$	8.38344e6i	0.698097i	−0.937105	+	0.349049i	$0.886505\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.76991e6i	− 0.377087i	−0.982065	−	0.188544i	$-0.939623\pi$
$233$	− 4.76991e6i	− 0.377087i	0.982065	−	0.188544i	$-0.0603767\pi$
$234$	0	0
$235$	− 7.77500e6i	− 0.599096i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6.88451e6i	− 0.504289i	−0.967690	−	0.252144i	$-0.918864\pi$
$239$	− 6.88451e6i	− 0.504289i	0.967690	−	0.252144i	$-0.0811358\pi$
$240$	0	0
$241$	1.90446e6	0.136057	0.0680284	−	0.997683i	$-0.478329\pi$
$241$	1.90446e6	0.136057	0.0680284	+	0.997683i	$0.478329\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.65688e6	0.656657
$246$	0	0
$247$	2.63330e7	1.74747
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.61930e6	0.481830	0.240915	−	0.970546i	$-0.422553\pi$
$251$	7.61930e6	0.481830	0.240915	+	0.970546i	$0.422553\pi$
$252$	0	0
$253$	3.04871e7i	1.88258i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.21558e7i	0.716115i	0.933700	+	0.358058i	$0.116561\pi$
$257$	1.21558e7i	0.716115i	−0.933700	+	0.358058i	$0.883439\pi$
$258$	0	0
$259$	1.94572e6i	0.111990i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.70043e7i	− 1.48445i	−0.670149	−	0.742226i	$-0.733770\pi$
$263$	− 2.70043e7i	− 1.48445i	0.670149	−	0.742226i	$-0.266230\pi$
$264$	0	0
$265$	−8.71473e6	−0.468292
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.21804e7	−1.13949	−0.569747	−	0.821820i	$-0.692959\pi$
$269$	−2.21804e7	−1.13949	−0.569747	+	0.821820i	$0.692959\pi$
$270$	0	0
$271$	−8.95677e6	−0.450032	−0.225016	−	0.974355i	$-0.572243\pi$
$271$	−8.95677e6	−0.450032	−0.225016	+	0.974355i	$0.572243\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.71022e7	0.822346
$276$	0	0
$277$	2.73042e7i	1.28466i	0.766427	+	0.642332i	$0.222033\pi$
$277$	2.73042e7i	1.28466i	−0.766427	+	0.642332i	$0.777967\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.84445e7i	1.73267i	0.499466	+	0.866334i	$0.333530\pi$
$281$	3.84445e7i	1.73267i	−0.499466	+	0.866334i	$0.666470\pi$
$282$	0	0
$283$	2.02032e7i	0.891374i	0.895189	+	0.445687i	$0.147041\pi$
$283$	2.02032e7i	0.891374i	−0.895189	+	0.445687i	$0.852959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.92538e6i	− 0.0814462i
$288$	0	0
$289$	−3.75748e7	−1.55669
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.25402e7	0.498541	0.249270	−	0.968434i	$-0.419809\pi$
$293$	1.25402e7	0.498541	0.249270	+	0.968434i	$0.419809\pi$
$294$	0	0
$295$	1.46508e7	0.570682
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.69914e7	2.50614
$300$	0	0
$301$	− 5.04896e6i	− 0.185141i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	76982.8i	0.00271328i
$306$	0	0
$307$	8.56216e6i	0.295916i	0.988994	+	0.147958i	$0.0472700\pi$
$307$	8.56216e6i	0.295916i	−0.988994	+	0.147958i	$0.952730\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.46955e7i	− 1.15343i	−0.816944	−	0.576716i	$-0.804334\pi$
$311$	− 3.46955e7i	− 1.15343i	0.816944	−	0.576716i	$-0.195666\pi$
$312$	0	0
$313$	3.49394e7	1.13942	0.569708	−	0.821847i	$-0.307056\pi$
$313$	3.49394e7	1.13942	0.569708	+	0.821847i	$0.307056\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.33835e7	1.67583	0.837913	−	0.545804i	$-0.183776\pi$
$317$	5.33835e7	1.67583	0.837913	+	0.545804i	$0.183776\pi$
$318$	0	0
$319$	3.26965e6	0.100723
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.80365e7	−1.42549
$324$	0	0
$325$	− 3.75799e7i	− 1.09473i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.47891e6i	− 0.0976913i
$330$	0	0
$331$	− 3.32801e7i	− 0.917699i	−0.888514	−	0.458850i	$-0.848262\pi$
$331$	− 3.32801e7i	− 0.917699i	0.888514	−	0.458850i	$-0.151738\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.36154e7i	0.362155i
$336$	0	0
$337$	−1.44209e7	−0.376792	−0.188396	−	0.982093i	$-0.560329\pi$
$337$	−1.44209e7	−0.376792	−0.188396	+	0.982093i	$0.560329\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.46220e7	1.37754
$342$	0	0
$343$	8.69325e6	0.215427
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.05006e7	−1.44801	−0.724005	−	0.689795i	$-0.757701\pi$
$347$	−6.05006e7	−1.44801	−0.724005	+	0.689795i	$0.757701\pi$
$348$	0	0
$349$	2.78673e7i	0.655569i	0.944753	+	0.327785i	$0.106302\pi$
$349$	2.78673e7i	0.655569i	−0.944753	+	0.327785i	$0.893698\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.20925e7i	− 0.729591i	−0.931088	−	0.364796i	$-0.881139\pi$
$353$	− 3.20925e7i	− 0.729591i	0.931088	−	0.364796i	$-0.118861\pi$
$354$	0	0
$355$	5.07406e7i	1.13415i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.17808e6i	0.198367i	0.995069	+	0.0991833i	$0.0316230\pi$
$359$	9.17808e6i	0.198367i	−0.995069	+	0.0991833i	$0.968377\pi$
$360$	0	0
$361$	9.65456e6	0.205216
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.76829e7	−0.569290
$366$	0	0
$367$	8.02971e7	1.62443	0.812216	−	0.583357i	$-0.198261\pi$
$367$	8.02971e7	1.62443	0.812216	+	0.583357i	$0.198261\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.89940e6	−0.0763618
$372$	0	0
$373$	2.61467e7i	0.503837i	0.967748	+	0.251918i	$0.0810615\pi$
$373$	2.61467e7i	0.503837i	−0.967748	+	0.251918i	$0.918939\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.18462e6i	− 0.134085i
$378$	0	0
$379$	8.11681e7i	1.49097i	0.666524	+	0.745483i	$0.267781\pi$
$379$	8.11681e7i	1.49097i	−0.666524	+	0.745483i	$0.732219\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 7.20532e7i	− 1.28250i	−0.767332	−	0.641250i	$-0.778416\pi$
$383$	− 7.20532e7i	− 1.28250i	0.767332	−	0.641250i	$-0.221584\pi$
$384$	0	0
$385$	−6.04937e6	−0.106005
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.30952e7	−1.41165	−0.705825	−	0.708386i	$-0.749423\pi$
$389$	−8.30952e7	−1.41165	−0.705825	+	0.708386i	$0.749423\pi$
$390$	0	0
$391$	−1.22205e8	−2.04437
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.02636e7	0.815572
$396$	0	0
$397$	− 4.67473e7i	− 0.747111i	−0.927608	−	0.373555i	$-0.878139\pi$
$397$	− 4.67473e7i	− 0.747111i	0.927608	−	0.373555i	$-0.121861\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.29945e7i	− 0.666775i	−0.942790	−	0.333388i	$-0.891808\pi$
$401$	− 4.29945e7i	− 0.666775i	0.942790	−	0.333388i	$-0.108192\pi$
$402$	0	0
$403$	− 1.20025e8i	− 1.83382i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.02606e8i	1.52191i
$408$	0	0
$409$	−1.10094e8	−1.60914	−0.804571	−	0.593857i	$-0.797604\pi$
$409$	−1.10094e8	−1.60914	−0.804571	+	0.593857i	$0.797604\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.55547e6	0.0930580
$414$	0	0
$415$	−4.04277e7	−0.565633
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.89084e7	0.664877	0.332439	−	0.943125i	$-0.392129\pi$
$419$	4.89084e7	0.664877	0.332439	+	0.943125i	$0.392129\pi$
$420$	0	0
$421$	− 1.81777e7i	− 0.243608i	−0.992554	−	0.121804i	$-0.961132\pi$
$421$	− 1.81777e7i	− 0.243608i	0.992554	−	0.121804i	$-0.0388680\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.85530e7i	0.893017i
$426$	0	0
$427$	34445.9i	0 0.000442439i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.09271e7i	0.386285i	0.981171	+	0.193143i	$0.0618680\pi$
$431$	3.09271e7i	0.386285i	−0.981171	+	0.193143i	$0.938132\pi$
$432$	0	0
$433$	5.51429e7	0.679244	0.339622	−	0.940562i	$-0.389701\pi$
$433$	5.51429e7	0.679244	0.339622	+	0.940562i	$0.389701\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.51238e7	−1.13984
$438$	0	0
$439$	−1.34146e8	−1.58557	−0.792784	−	0.609503i	$-0.791369\pi$
$439$	−1.34146e8	−1.58557	−0.792784	+	0.609503i	$0.791369\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.59880e7	−0.183900	−0.0919501	−	0.995764i	$-0.529310\pi$
$443$	−1.59880e7	−0.183900	−0.0919501	+	0.995764i	$0.529310\pi$
$444$	0	0
$445$	− 6.86717e7i	− 0.779287i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.72078e7i	0.852947i	0.904500	+	0.426473i	$0.140244\pi$
$449$	7.72078e7i	0.852947i	−0.904500	+	0.426473i	$0.859756\pi$
$450$	0	0
$451$	− 1.01533e8i	− 1.10682i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.32927e7i	0.141117i
$456$	0	0
$457$	1.09149e8	1.14359	0.571796	−	0.820396i	$-0.306247\pi$
$457$	1.09149e8	1.14359	0.571796	+	0.820396i	$0.306247\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.95186e7	−0.811645	−0.405822	−	0.913952i	$-0.633015\pi$
$461$	−7.95186e7	−0.811645	−0.405822	+	0.913952i	$0.633015\pi$
$462$	0	0
$463$	7.28884e7	0.734371	0.367185	−	0.930148i	$-0.380321\pi$
$463$	7.28884e7	0.734371	0.367185	+	0.930148i	$0.380321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.96554e8	−1.92988	−0.964940	−	0.262470i	$-0.915463\pi$
$467$	−1.96554e8	−1.92988	−0.964940	+	0.262470i	$0.915463\pi$
$468$	0	0
$469$	6.09218e6i	0.0590547i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.66253e8i	− 2.51600i
$474$	0	0
$475$	5.33612e7i	0.497903i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.05958e8i	1.87401i	0.349320	+	0.937004i	$0.386413\pi$
$479$	2.05958e8i	1.87401i	−0.349320	+	0.937004i	$0.613587\pi$
$480$	0	0
$481$	2.25463e8	2.02600
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.27182e7	0.812718
$486$	0	0
$487$	1.47902e8	1.28052	0.640262	−	0.768157i	$-0.278826\pi$
$487$	1.47902e8	1.28052	0.640262	+	0.768157i	$0.278826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.18634e7	−0.269183	−0.134592	−	0.990901i	$-0.542972\pi$
$491$	−3.18634e7	−0.269183	−0.134592	+	0.990901i	$0.542972\pi$
$492$	0	0
$493$	1.31061e7i	0.109379i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.27038e7i	0.184940i
$498$	0	0
$499$	− 1.10011e8i	− 0.885386i	−0.896673	−	0.442693i	$-0.854023\pi$
$499$	− 1.10011e8i	− 0.885386i	0.896673	−	0.442693i	$-0.145977\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.58001e7i	0.281307i	0.990059	+	0.140653i	$0.0449203\pi$
$503$	3.58001e7i	0.281307i	−0.990059	+	0.140653i	$0.955080\pi$
$504$	0	0
$505$	1.27587e7	0.0990680
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.16724e8	−1.64344	−0.821720	−	0.569892i	$-0.806985\pi$
$509$	−2.16724e8	−1.64344	−0.821720	+	0.569892i	$0.806985\pi$
$510$	0	0
$511$	−1.23867e7	−0.0928309
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.44156e8	−1.05539
$516$	0	0
$517$	− 1.83457e8i	− 1.32759i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.01755e8i	− 0.719518i	−0.933045	−	0.359759i	$-0.882859\pi$
$521$	− 1.01755e8i	− 0.719518i	0.933045	−	0.359759i	$-0.117141\pi$
$522$	0	0
$523$	− 1.10857e8i	− 0.774920i	−0.921887	−	0.387460i	$-0.873353\pi$
$523$	− 1.10857e8i	− 0.774920i	0.921887	−	0.387460i	$-0.126647\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.18948e8i	1.49593i
$528$	0	0
$529$	−9.39596e7	−0.634709
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.23106e8	−1.47343
$534$	0	0
$535$	−1.17172e8	−0.765179
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.27862e8	1.45514
$540$	0	0
$541$	− 1.47693e8i	− 0.932756i	−0.884586	−	0.466378i	$-0.845559\pi$
$541$	− 1.47693e8i	− 0.932756i	0.884586	−	0.466378i	$-0.154441\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.01230e7i	− 0.0625345i
$546$	0	0
$547$	8.41332e7i	0.514050i	0.966405	+	0.257025i	$0.0827423\pi$
$547$	8.41332e7i	0.514050i	−0.966405	+	0.257025i	$0.917258\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.02017e7i	0.0609845i
$552$	0	0
$553$	2.24904e7	0.132991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.09820e7	−0.295020	−0.147510	−	0.989061i	$-0.547126\pi$
$557$	−5.09820e7	−0.295020	−0.147510	+	0.989061i	$0.547126\pi$
$558$	0	0
$559$	−5.85055e8	−3.34936
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.51243e8	−0.847521	−0.423761	−	0.905774i	$-0.639290\pi$
$563$	−1.51243e8	−0.847521	−0.423761	+	0.905774i	$0.639290\pi$
$564$	0	0
$565$	− 1.14698e8i	− 0.635931i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.34052e8i	0.727671i	0.931463	+	0.363836i	$0.118533\pi$
$569$	1.34052e8i	0.727671i	−0.931463	+	0.363836i	$0.881467\pi$
$570$	0	0
$571$	− 8.15488e7i	− 0.438035i	−0.975721	−	0.219018i	$-0.929715\pi$
$571$	− 8.15488e7i	− 0.438035i	0.975721	−	0.219018i	$-0.0702852\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.35751e8i	0.714070i
$576$	0	0
$577$	1.66709e6	0.00867826	0.00433913	−	0.999991i	$-0.498619\pi$
$577$	1.66709e6	0.00867826	0.00433913	+	0.999991i	$0.498619\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.80893e7	−0.0922346
$582$	0	0
$583$	−2.05631e8	−1.03773
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.33311e8	0.659098	0.329549	−	0.944138i	$-0.393103\pi$
$587$	1.33311e8	0.659098	0.329549	+	0.944138i	$0.393103\pi$
$588$	0	0
$589$	1.70428e8i	0.834056i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.62556e8i	− 1.25910i	−0.776962	−	0.629548i	$-0.783240\pi$
$593$	− 2.62556e8i	− 1.25910i	0.776962	−	0.629548i	$-0.216760\pi$
$594$	0	0
$595$	− 2.42484e7i	− 0.115115i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.70406e7i	− 0.125816i	−0.998019	−	0.0629081i	$-0.979963\pi$
$599$	− 2.70406e7i	− 0.125816i	0.998019	−	0.0629081i	$-0.0200375\pi$
$600$	0	0
$601$	−1.33675e8	−0.615782	−0.307891	−	0.951422i	$-0.599623\pi$
$601$	−1.33675e8	−0.615782	−0.307891	+	0.951422i	$0.599623\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.71867e8	−0.776117
$606$	0	0
$607$	1.25106e8	0.559387	0.279693	−	0.960089i	$-0.409767\pi$
$607$	1.25106e8	0.559387	0.279693	+	0.960089i	$0.409767\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.03124e8	−1.76732
$612$	0	0
$613$	1.23831e6i	0.00537588i	0.999996	+	0.00268794i	$0.000855599\pi$
$613$	1.23831e6i	0.00537588i	−0.999996	+	0.00268794i	$0.999144\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.35967e7i	− 0.185608i	−0.995684	−	0.0928042i	$-0.970417\pi$
$617$	− 4.35967e7i	− 0.185608i	0.995684	−	0.0928042i	$-0.0295831\pi$
$618$	0	0
$619$	3.54527e8i	1.49478i	0.664386	+	0.747390i	$0.268693\pi$
$619$	3.54527e8i	1.49478i	−0.664386	+	0.747390i	$0.731307\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.07271e7i	− 0.127074i
$624$	0	0
$625$	−3.16370e7	−0.129585
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.11288e8	−1.65270
$630$	0	0
$631$	3.25768e8	1.29664	0.648322	−	0.761367i	$-0.275471\pi$
$631$	3.25768e8	1.29664	0.648322	+	0.761367i	$0.275471\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.69644e8	−0.662547
$636$	0	0
$637$	− 5.00697e8i	− 1.93712i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 5.13127e8i	− 1.94828i	−0.225954	−	0.974138i	$-0.572550\pi$
$641$	− 5.13127e8i	− 1.94828i	0.225954	−	0.974138i	$-0.427450\pi$
$642$	0	0
$643$	− 4.59513e8i	− 1.72848i	−0.503077	−	0.864242i	$-0.667799\pi$
$643$	− 4.59513e8i	− 1.72848i	0.503077	−	0.864242i	$-0.332201\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.56031e8i	0.945323i	0.881244	+	0.472662i	$0.156707\pi$
$647$	2.56031e8i	0.945323i	−0.881244	+	0.472662i	$0.843293\pi$
$648$	0	0
$649$	3.45697e8	1.26462
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.52625e8	1.62555	0.812773	−	0.582580i	$-0.197957\pi$
$653$	4.52625e8	1.62555	0.812773	+	0.582580i	$0.197957\pi$
$654$	0	0
$655$	1.29410e8	0.460516
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.27984e8	−0.796613	−0.398306	−	0.917252i	$-0.630402\pi$
$659$	−2.27984e8	−0.796613	−0.398306	+	0.917252i	$0.630402\pi$
$660$	0	0
$661$	1.28280e8i	0.444177i	0.975027	+	0.222089i	$0.0712874\pi$
$661$	1.28280e8i	0.444177i	−0.975027	+	0.222089i	$0.928713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.88748e7i	− 0.0641827i
$666$	0	0
$667$	2.59533e7i	0.0874611i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.81647e6i	0.00601259i
$672$	0	0
$673$	−1.51573e8	−0.497253	−0.248627	−	0.968599i	$-0.579979\pi$
$673$	−1.51573e8	−0.497253	−0.248627	+	0.968599i	$0.579979\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.60675e8	1.48467	0.742333	−	0.670031i	$-0.233719\pi$
$677$	4.60675e8	1.48467	0.742333	+	0.670031i	$0.233719\pi$
$678$	0	0
$679$	4.14867e7	0.132525
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.03627e8	0.325245	0.162623	−	0.986688i	$-0.448005\pi$
$683$	1.03627e8	0.325245	0.162623	+	0.986688i	$0.448005\pi$
$684$	0	0
$685$	1.88946e7i	0.0587850i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.51848e8i	1.38145i
$690$	0	0
$691$	− 5.59603e7i	− 0.169608i	−0.996398	−	0.0848039i	$-0.972974\pi$
$691$	− 5.59603e7i	− 0.169608i	0.996398	−	0.0848039i	$-0.0270264\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.52253e8i	0.453535i
$696$	0	0
$697$	4.06988e8	1.20194
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.95564e8	−1.14832	−0.574159	−	0.818744i	$-0.694671\pi$
$701$	−3.95564e8	−1.14832	−0.574159	+	0.818744i	$0.694671\pi$
$702$	0	0
$703$	−3.20144e8	−0.921465
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.70888e6	0.0161545
$708$	0	0
$709$	− 7.41424e7i	− 0.208031i	−0.994576	−	0.104015i	$-0.966831\pi$
$709$	− 7.41424e7i	− 0.208031i	0.994576	−	0.104015i	$-0.0331691\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.33570e8i	1.19616i
$714$	0	0
$715$	7.00978e8i	1.91772i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 5.26454e8i	− 1.41636i	−0.706032	−	0.708180i	$-0.749517\pi$
$719$	− 5.26454e8i	− 1.41636i	0.706032	−	0.708180i	$-0.250483\pi$
$720$	0	0
$721$	−6.45026e7	−0.172096
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.45589e7	0.0382046
$726$	0	0
$727$	−2.60552e8	−0.678096	−0.339048	−	0.940769i	$-0.610105\pi$
$727$	−2.60552e8	−0.678096	−0.339048	+	0.940769i	$0.610105\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.06725e9	2.73222
$732$	0	0
$733$	1.52419e8i	0.387014i	0.981099	+	0.193507i	$0.0619861\pi$
$733$	1.52419e8i	0.387014i	−0.981099	+	0.193507i	$0.938014\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.21266e8i	0.802531i
$738$	0	0
$739$	− 7.15919e8i	− 1.77391i	−0.461860	−	0.886953i	$-0.652818\pi$
$739$	− 7.15919e8i	− 1.77391i	0.461860	−	0.886953i	$-0.347182\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.70029e8i	1.14593i	0.819580	+	0.572965i	$0.194207\pi$
$743$	4.70029e8i	1.14593i	−0.819580	+	0.572965i	$0.805793\pi$
$744$	0	0
$745$	−4.79911e8	−1.16062
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.24286e7	−0.124774
$750$	0	0
$751$	2.67972e8	0.632659	0.316329	−	0.948649i	$-0.397550\pi$
$751$	2.67972e8	0.632659	0.316329	+	0.948649i	$0.397550\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.98188e7	0.0925225
$756$	0	0
$757$	1.90775e8i	0.439778i	0.975525	+	0.219889i	$0.0705695\pi$
$757$	1.90775e8i	0.439778i	−0.975525	+	0.219889i	$0.929431\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.63151e8i	0.370200i	0.982720	+	0.185100i	$0.0592608\pi$
$761$	1.63151e8i	0.370200i	−0.982720	+	0.185100i	$0.940739\pi$
$762$	0	0
$763$	− 4.52953e6i	− 0.0101972i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.59623e8i	− 1.68350i
$768$	0	0
$769$	−3.52840e8	−0.775888	−0.387944	−	0.921683i	$-0.626815\pi$
$769$	−3.52840e8	−0.775888	−0.387944	+	0.921683i	$0.626815\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.92662e8	−0.417116	−0.208558	−	0.978010i	$-0.566877\pi$
$773$	−1.92662e8	−0.417116	−0.208558	+	0.978010i	$0.566877\pi$
$774$	0	0
$775$	2.43218e8	0.522506
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.16797e8	0.670145
$780$	0	0
$781$	1.19727e9i	2.51326i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.60533e8i	− 0.952032i
$786$	0	0
$787$	− 1.39654e8i	− 0.286503i	−0.989686	−	0.143252i	$-0.954244\pi$
$787$	− 1.39654e8i	− 0.286503i	0.989686	−	0.143252i	$-0.0457558\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 5.13215e7i	− 0.103698i
$792$	0	0
$793$	3.99146e6	0.00800410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.74248e8	0.739239	0.369619	−	0.929183i	$-0.379488\pi$
$797$	3.74248e8	0.739239	0.369619	+	0.929183i	$0.379488\pi$
$798$	0	0
$799$	7.35376e8	1.44168
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.53201e8	−1.26154
$804$	0	0
$805$	− 4.80177e7i	− 0.0920479i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.07376e8i	− 1.52486i	−0.647071	−	0.762430i	$-0.724006\pi$
$809$	− 8.07376e8i	− 1.52486i	0.647071	−	0.762430i	$-0.275994\pi$
$810$	0	0
$811$	6.75232e8i	1.26587i	0.774204	+	0.632937i	$0.218151\pi$
$811$	6.75232e8i	1.26587i	−0.774204	+	0.632937i	$0.781849\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.63217e8i	− 0.301503i
$816$	0	0
$817$	8.30744e8	1.52335
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.61450e8	0.833865	0.416932	−	0.908938i	$-0.363105\pi$
$821$	4.61450e8	0.833865	0.416932	+	0.908938i	$0.363105\pi$
$822$	0	0
$823$	3.57228e8	0.640835	0.320418	−	0.947276i	$-0.396177\pi$
$823$	3.57228e8	0.640835	0.320418	+	0.947276i	$0.396177\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.57478e8	1.33922	0.669612	−	0.742711i	$-0.266460\pi$
$827$	7.57478e8	1.33922	0.669612	+	0.742711i	$0.266460\pi$
$828$	0	0
$829$	− 1.96526e8i	− 0.344950i	−0.985014	−	0.172475i	$-0.944824\pi$
$829$	− 1.96526e8i	− 0.344950i	0.985014	−	0.172475i	$-0.0551764\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.13368e8i	1.58020i
$834$	0	0
$835$	− 1.74019e8i	− 0.298907i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.04024e9i	− 1.76136i	−0.473711	−	0.880680i	$-0.657086\pi$
$839$	− 1.04024e9i	− 1.76136i	0.473711	−	0.880680i	$-0.342914\pi$
$840$	0	0
$841$	−5.92040e8	−0.995321
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.13941e9	1.88846
$846$	0	0
$847$	−7.69018e7	−0.126557
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.14447e8	−1.32152
$852$	0	0
$853$	6.97415e8i	1.12368i	0.827244	+	0.561842i	$0.189907\pi$
$853$	6.97415e8i	1.12368i	−0.827244	+	0.561842i	$0.810093\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.24118e8i	0.832696i	0.909205	+	0.416348i	$0.136690\pi$
$857$	5.24118e8i	0.832696i	−0.909205	+	0.416348i	$0.863310\pi$
$858$	0	0
$859$	1.07319e8i	0.169315i	0.996410	+	0.0846575i	$0.0269796\pi$
$859$	1.07319e8i	0.169315i	−0.996410	+	0.0846575i	$0.973020\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 2.28624e8i	− 0.355705i	−0.984057	−	0.177852i	$-0.943085\pi$
$863$	− 2.28624e8i	− 0.355705i	0.984057	−	0.177852i	$-0.0569150\pi$
$864$	0	0
$865$	8.00816e7	0.123733
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.18601e9	1.80730
$870$	0	0
$871$	7.05939e8	1.06835
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.51664e7	−0.112202
$876$	0	0
$877$	− 5.54337e8i	− 0.821818i	−0.911676	−	0.410909i	$-0.865211\pi$
$877$	− 5.54337e8i	− 0.821818i	0.911676	−	0.410909i	$-0.134789\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.17732e9i	1.72174i	0.508824	+	0.860870i	$0.330080\pi$
$881$	1.17732e9i	1.72174i	−0.508824	+	0.860870i	$0.669920\pi$
$882$	0	0
$883$	7.67777e8i	1.11520i	0.830109	+	0.557600i	$0.188278\pi$
$883$	7.67777e8i	1.11520i	−0.830109	+	0.557600i	$0.811722\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.89440e8i	− 0.987929i	−0.869482	−	0.493964i	$-0.835547\pi$
$887$	− 6.89440e8i	− 0.987929i	0.869482	−	0.493964i	$-0.164453\pi$
$888$	0	0
$889$	−7.59069e7	−0.108038
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.72412e8	0.803811
$894$	0	0
$895$	−6.46003e8	−0.901084
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.64991e7	0.0639978
$900$	0	0
$901$	− 8.24258e8i	− 1.12691i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 4.35996e8i	− 0.588216i
$906$	0	0
$907$	3.13153e8i	0.419696i	0.977734	+	0.209848i	$0.0672969\pi$
$907$	3.13153e8i	0.419696i	−0.977734	+	0.209848i	$0.932703\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.81208e8i	1.03326i	0.856207	+	0.516632i	$0.172815\pi$
$911$	7.81208e8i	1.03326i	−0.856207	+	0.516632i	$0.827185\pi$
$912$	0	0
$913$	−9.53924e8	−1.25343
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.79045e7	0.0750939
$918$	0	0
$919$	5.24800e8	0.676157	0.338078	−	0.941118i	$-0.390223\pi$
$919$	5.24800e8	0.676157	0.338078	+	0.941118i	$0.390223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.63083e9	3.34571
$924$	0	0
$925$	4.56878e8i	0.577264i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6.73499e8i	− 0.840021i	−0.907519	−	0.420010i	$-0.862026\pi$
$929$	− 6.73499e8i	− 0.840021i	0.907519	−	0.420010i	$-0.137974\pi$
$930$	0	0
$931$	7.10960e8i	0.881041i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.27872e9i	− 1.56437i
$936$	0	0
$937$	−1.46278e9	−1.77812	−0.889060	−	0.457790i	$-0.848641\pi$
$937$	−1.46278e9	−1.77812	−0.889060	+	0.457790i	$0.848641\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.46136e9	−1.75384	−0.876920	−	0.480637i	$-0.840405\pi$
$941$	−1.46136e9	−1.75384	−0.876920	+	0.480637i	$0.840405\pi$
$942$	0	0
$943$	8.05934e8	0.961091
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.23223e8	−1.08707	−0.543534	−	0.839387i	$-0.682914\pi$
$947$	−9.23223e8	−1.08707	−0.543534	+	0.839387i	$0.682914\pi$
$948$	0	0
$949$	1.43532e9i	1.67939i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.21757e9i	1.40675i	0.710820	+	0.703374i	$0.248324\pi$
$953$	1.21757e9i	1.40675i	−0.710820	+	0.703374i	$0.751676\pi$
$954$	0	0
$955$	1.00827e9i	1.15762i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.45438e6i	0.00958575i
$960$	0	0
$961$	−1.10700e8	−0.124732
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.71874e8	−0.191261
$966$	0	0
$967$	−1.22852e8	−0.135863	−0.0679317	−	0.997690i	$-0.521640\pi$
$967$	−1.22852e8	−0.135863	−0.0679317	+	0.997690i	$0.521640\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.16709e9	1.27481	0.637405	−	0.770529i	$-0.280008\pi$
$971$	1.16709e9	1.27481	0.637405	+	0.770529i	$0.280008\pi$
$972$	0	0
$973$	6.81253e7i	0.0739554i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.31100e8i	− 0.355038i	−0.984117	−	0.177519i	$-0.943193\pi$
$977$	− 3.31100e8i	− 0.355038i	0.984117	−	0.177519i	$-0.0568071\pi$
$978$	0	0
$979$	− 1.62037e9i	− 1.72689i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 9.27161e8i	− 0.976100i	−0.872816	−	0.488050i	$-0.837708\pi$
$983$	− 9.27161e8i	− 0.976100i	0.872816	−	0.488050i	$-0.162292\pi$
$984$	0	0
$985$	−5.46700e8	−0.572058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.11342e9	2.18472
$990$	0	0
$991$	−5.96968e8	−0.613381	−0.306691	−	0.951809i	$-0.599222\pi$
$991$	−5.96968e8	−0.613381	−0.306691	+	0.951809i	$0.599222\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.01192e8	−0.914846
$996$	0	0
$997$	1.50998e9i	1.52365i	0.647782	+	0.761826i	$0.275697\pi$
$997$	1.50998e9i	1.52365i	−0.647782	+	0.761826i	$0.724303\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.h.b.161.16	yes	32
3.2	odd	2	inner	576.7.h.b.161.18	yes	32
4.3	odd	2	inner	576.7.h.b.161.14	yes	32
8.3	odd	2	inner	576.7.h.b.161.17	yes	32
8.5	even	2	inner	576.7.h.b.161.19	yes	32
12.11	even	2	inner	576.7.h.b.161.20	yes	32
24.5	odd	2	inner	576.7.h.b.161.13	✓	32
24.11	even	2	inner	576.7.h.b.161.15	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.7.h.b.161.13	✓	32	24.5	odd	2	inner
576.7.h.b.161.14	yes	32	4.3	odd	2	inner
576.7.h.b.161.15	yes	32	24.11	even	2	inner
576.7.h.b.161.16	yes	32	1.1	even	1	trivial
576.7.h.b.161.17	yes	32	8.3	odd	2	inner
576.7.h.b.161.18	yes	32	3.2	odd	2	inner
576.7.h.b.161.19	yes	32	8.5	even	2	inner
576.7.h.b.161.20	yes	32	12.11	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-83.0572 q^{5} -37.1638 q^{7} +O(q^{10})\)	\(q-83.0572 q^{5} -37.1638 q^{7} -1959.80 q^{11} +4306.41i q^{13} -7855.72i q^{17} -6114.84i q^{19} -15556.2i q^{23} -8726.51 q^{25} -1668.36 q^{29} -27871.2 q^{31} +3086.72 q^{35} -52355.1i q^{37} +51807.9i q^{41} +135857. i q^{43} +93610.2i q^{47} -116268. q^{49} +104925. q^{53} +162776. q^{55} -176394. q^{59} -926.866i q^{61} -357678. i q^{65} -163928. i q^{67} -610911. i q^{71} +333300. q^{73} +72833.8 q^{77} -605169. q^{79} +486745. q^{83} +652474. i q^{85} +826800. i q^{89} -160043. i q^{91} +507882. i q^{95} -1.11632e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) \) 32 * q + 169888 * q^25 + 829792 * q^49 - 1493888 * q^73 - 15893248 * q^97

Introduction

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