LMFDB - Embedded newform 576.4.c.f.575.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(575,576)]

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([1, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("576.575");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.c (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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 161x^{4} + 4096 $ x^8 + 161*x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.8
Root			$-1.67746 - 1.67746i$ of defining polynomial
Character	$\chi$	$=$	576.575
Dual form			576.4.c.f.575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.6623i q^{5} +14.9783i q^{7} +O(q^{10})$	$q+17.6623i q^{5} +14.9783i q^{7} -55.1236 q^{11} +9.02175 q^{13} -14.1729i q^{17} +139.913i q^{19} -86.1748 q^{23} -186.957 q^{25} +3.64319i q^{29} -273.022i q^{31} -264.550 q^{35} +405.826 q^{37} -344.853i q^{41} -241.696i q^{43} +297.999 q^{47} +118.652 q^{49} +602.901i q^{53} -973.609i q^{55} -630.678 q^{59} -237.478 q^{61} +159.345i q^{65} -397.957i q^{67} -778.955 q^{71} -1155.04 q^{73} -825.655i q^{77} -127.326i q^{79} -301.627 q^{83} +250.326 q^{85} +338.059i q^{89} +135.130i q^{91} -2471.18 q^{95} +1617.35 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 256 q^{13} - 1128 q^{25} + 1776 q^{37} - 1992 q^{49} + 2512 q^{61} - 1152 q^{73} - 3696 q^{85} + 7424 q^{97}+O(q^{100}) $ 8 * q + 256 * q^13 - 1128 * q^25 + 1776 * q^37 - 1992 * q^49 + 2512 * q^61 - 1152 * q^73 - 3696 * q^85 + 7424 * 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
$5$	17.6623i	1.57976i	0.613259	+	0.789882i	$0.289858\pi$
$5$	17.6623i	1.57976i	−0.613259	+	0.789882i	$0.710142\pi$
$6$	0	0
$7$	14.9783i	0.808750i	0.914593	+	0.404375i	$0.132511\pi$
$7$	14.9783i	0.808750i	−0.914593	+	0.404375i	$0.867489\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−55.1236	−1.51094	−0.755472	−	0.655181i	$-0.772592\pi$
$11$	−55.1236	−1.51094	−0.755472	+	0.655181i	$0.772592\pi$
$12$	0	0
$13$	9.02175	0.192476	0.0962378	−	0.995358i	$-0.469319\pi$
$13$	9.02175	0.192476	0.0962378	+	0.995358i	$0.469319\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 14.1729i	− 0.202202i	−0.994876	−	0.101101i	$-0.967763\pi$
$17$	− 14.1729i	− 0.202202i	0.994876	−	0.101101i	$-0.0322365\pi$
$18$	0	0
$19$	139.913i	1.68938i	0.535255	+	0.844691i	$0.320216\pi$
$19$	139.913i	1.68938i	−0.535255	+	0.844691i	$0.679784\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−86.1748	−0.781247	−0.390623	−	0.920551i	$-0.627741\pi$
$23$	−86.1748	−0.781247	−0.390623	+	0.920551i	$0.627741\pi$
$24$	0	0
$25$	−186.957	−1.49565
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.64319i	0.0233284i	0.999932	+	0.0116642i	$0.00371291\pi$
$29$	3.64319i	0.0233284i	−0.999932	+	0.0116642i	$0.996287\pi$
$30$	0	0
$31$	− 273.022i	− 1.58181i	−0.611938	−	0.790906i	$-0.709610\pi$
$31$	− 273.022i	− 1.58181i	0.611938	−	0.790906i	$-0.290390\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−264.550	−1.27763
$36$	0	0
$37$	405.826	1.80317	0.901586	−	0.432599i	$-0.142404\pi$
$37$	405.826	1.80317	0.901586	+	0.432599i	$0.142404\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 344.853i	− 1.31358i	−0.754072	−	0.656792i	$-0.771913\pi$
$41$	− 344.853i	− 1.31358i	0.754072	−	0.656792i	$-0.228087\pi$
$42$	0	0
$43$	− 241.696i	− 0.857168i	−0.903502	−	0.428584i	$-0.859013\pi$
$43$	− 241.696i	− 0.857168i	0.903502	−	0.428584i	$-0.140987\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	297.999	0.924844	0.462422	−	0.886660i	$-0.346981\pi$
$47$	297.999	0.924844	0.462422	+	0.886660i	$0.346981\pi$
$48$	0	0
$49$	118.652	0.345924
$50$	0	0
$51$	0	0
$52$	0	0
$53$	602.901i	1.56254i	0.624191	+	0.781271i	$0.285429\pi$
$53$	602.901i	1.56254i	−0.624191	+	0.781271i	$0.714571\pi$
$54$	0	0
$55$	− 973.609i	− 2.38693i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−630.678	−1.39165	−0.695824	−	0.718212i	$-0.744961\pi$
$59$	−630.678	−1.39165	−0.695824	+	0.718212i	$0.744961\pi$
$60$	0	0
$61$	−237.478	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$61$	−237.478	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	159.345i	0.304066i
$66$	0	0
$67$	− 397.957i	− 0.725644i	−0.931859	−	0.362822i	$-0.881813\pi$
$67$	− 397.957i	− 0.725644i	0.931859	−	0.362822i	$-0.118187\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−778.955	−1.30204	−0.651021	−	0.759060i	$-0.725659\pi$
$71$	−778.955	−1.30204	−0.651021	+	0.759060i	$0.725659\pi$
$72$	0	0
$73$	−1155.04	−1.85188	−0.925942	−	0.377665i	$-0.876727\pi$
$73$	−1155.04	−1.85188	−0.925942	+	0.377665i	$0.876727\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 825.655i	− 1.22198i
$78$	0	0
$79$	− 127.326i	− 0.181333i	−0.995881	−	0.0906666i	$-0.971100\pi$
$79$	− 127.326i	− 0.181333i	0.995881	−	0.0906666i	$-0.0288998\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−301.627	−0.398890	−0.199445	−	0.979909i	$-0.563914\pi$
$83$	−301.627	−0.398890	−0.199445	+	0.979909i	$0.563914\pi$
$84$	0	0
$85$	250.326	0.319431
$86$	0	0
$87$	0	0
$88$	0	0
$89$	338.059i	0.402631i	0.979526	+	0.201315i	$0.0645216\pi$
$89$	338.059i	0.402631i	−0.979526	+	0.201315i	$0.935478\pi$
$90$	0	0
$91$	135.130i	0.155665i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2471.18	−2.66882
$96$	0	0
$97$	1617.35	1.69296	0.846478	−	0.532423i	$-0.178719\pi$
$97$	1617.35	1.69296	0.846478	+	0.532423i	$0.178719\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	451.672i	0.444981i	0.974935	+	0.222490i	$0.0714186\pi$
$101$	451.672i	0.444981i	−0.974935	+	0.222490i	$0.928581\pi$
$102$	0	0
$103$	1000.15i	0.956776i	0.878149	+	0.478388i	$0.158779\pi$
$103$	1000.15i	0.956776i	−0.878149	+	0.478388i	$0.841221\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1582.93	−1.43017	−0.715085	−	0.699038i	$-0.753612\pi$
$107$	−1582.93	−1.43017	−0.715085	+	0.699038i	$0.753612\pi$
$108$	0	0
$109$	−636.761	−0.559547	−0.279773	−	0.960066i	$-0.590259\pi$
$109$	−636.761	−0.559547	−0.279773	+	0.960066i	$0.590259\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	83.9922i	0.0699232i	0.999389	+	0.0349616i	$0.0111309\pi$
$113$	83.9922i	0.0699232i	−0.999389	+	0.0349616i	$0.988869\pi$
$114$	0	0
$115$	− 1522.04i	− 1.23419i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	212.285	0.163531
$120$	0	0
$121$	1707.61	1.28295
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1094.29i	− 0.783013i
$126$	0	0
$127$	923.239i	0.645073i	0.946557	+	0.322536i	$0.104535\pi$
$127$	923.239i	0.645073i	−0.946557	+	0.322536i	$0.895465\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−363.452	−0.242404	−0.121202	−	0.992628i	$-0.538675\pi$
$131$	−363.452	−0.242404	−0.121202	+	0.992628i	$0.538675\pi$
$132$	0	0
$133$	−2095.65	−1.36629
$134$	0	0
$135$	0	0
$136$	0	0
$137$	704.986i	0.439642i	0.975540	+	0.219821i	$0.0705474\pi$
$137$	704.986i	0.439642i	−0.975540	+	0.219821i	$0.929453\pi$
$138$	0	0
$139$	− 2065.35i	− 1.26029i	−0.776477	−	0.630146i	$-0.782995\pi$
$139$	− 2065.35i	− 1.26029i	0.776477	−	0.630146i	$-0.217005\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−497.311	−0.290820
$144$	0	0
$145$	−64.3470	−0.0368533
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1123.27i	− 0.617596i	−0.951128	−	0.308798i	$-0.900073\pi$
$149$	− 1123.27i	− 0.617596i	0.951128	−	0.308798i	$-0.0999267\pi$
$150$	0	0
$151$	− 748.239i	− 0.403250i	−0.979463	−	0.201625i	$-0.935378\pi$
$151$	− 748.239i	− 0.403250i	0.979463	−	0.201625i	$-0.0646223\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4822.19	2.49889
$156$	0	0
$157$	1164.78	0.592100	0.296050	−	0.955172i	$-0.404330\pi$
$157$	1164.78	0.592100	0.296050	+	0.955172i	$0.404330\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1290.75i	− 0.631833i
$162$	0	0
$163$	− 976.304i	− 0.469141i	−0.972099	−	0.234571i	$-0.924632\pi$
$163$	− 976.304i	− 0.469141i	0.972099	−	0.234571i	$-0.0753684\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1603.66	−0.743082	−0.371541	−	0.928417i	$-0.621170\pi$
$167$	−1603.66	−0.743082	−0.371541	+	0.928417i	$0.621170\pi$
$168$	0	0
$169$	−2115.61	−0.962953
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 266.534i	− 0.117134i	−0.998283	−	0.0585670i	$-0.981347\pi$
$173$	− 266.534i	− 0.117134i	0.998283	−	0.0585670i	$-0.0186531\pi$
$174$	0	0
$175$	− 2800.28i	− 1.20961i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−197.867	−0.0826216	−0.0413108	−	0.999146i	$-0.513153\pi$
$179$	−197.867	−0.0826216	−0.0413108	+	0.999146i	$0.513153\pi$
$180$	0	0
$181$	−2834.50	−1.16401	−0.582007	−	0.813184i	$-0.697732\pi$
$181$	−2834.50	−1.16401	−0.582007	+	0.813184i	$0.697732\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7167.82i	2.84859i
$186$	0	0
$187$	781.261i	0.305516i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5188.38	1.96554	0.982770	−	0.184835i	$-0.0591750\pi$
$191$	5188.38	1.96554	0.982770	+	0.184835i	$0.0591750\pi$
$192$	0	0
$193$	206.348	0.0769599	0.0384799	−	0.999259i	$-0.487748\pi$
$193$	206.348	0.0769599	0.0384799	+	0.999259i	$0.487748\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3306.60i	1.19587i	0.801546	+	0.597933i	$0.204011\pi$
$197$	3306.60i	1.19587i	−0.801546	+	0.597933i	$0.795989\pi$
$198$	0	0
$199$	3491.37i	1.24370i	0.783136	+	0.621851i	$0.213619\pi$
$199$	3491.37i	1.24370i	−0.783136	+	0.621851i	$0.786381\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−54.5686	−0.0188668
$204$	0	0
$205$	6090.89	2.07515
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7712.50i	− 2.55256i
$210$	0	0
$211$	4808.48i	1.56886i	0.620218	+	0.784429i	$0.287044\pi$
$211$	4808.48i	1.56886i	−0.620218	+	0.784429i	$0.712956\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4268.90	1.35412
$216$	0	0
$217$	4089.39	1.27929
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 127.864i	− 0.0389189i
$222$	0	0
$223$	1620.37i	0.486582i	0.969953	+	0.243291i	$0.0782271\pi$
$223$	1620.37i	0.486582i	−0.969953	+	0.243291i	$0.921773\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−761.369	−0.222616	−0.111308	−	0.993786i	$-0.535504\pi$
$227$	−761.369	−0.222616	−0.111308	+	0.993786i	$0.535504\pi$
$228$	0	0
$229$	−2397.76	−0.691915	−0.345957	−	0.938250i	$-0.612446\pi$
$229$	−2397.76	−0.691915	−0.345957	+	0.938250i	$0.612446\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4299.95i	− 1.20901i	−0.796602	−	0.604504i	$-0.793371\pi$
$233$	− 4299.95i	− 1.20901i	0.796602	−	0.604504i	$-0.206629\pi$
$234$	0	0
$235$	5263.35i	1.46103i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1287.79	−0.348538	−0.174269	−	0.984698i	$-0.555756\pi$
$239$	−1287.79	−0.348538	−0.174269	+	0.984698i	$0.555756\pi$
$240$	0	0
$241$	−4906.82	−1.31152	−0.655760	−	0.754969i	$-0.727652\pi$
$241$	−4906.82	−1.31152	−0.655760	+	0.754969i	$0.727652\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2095.67i	0.546478i
$246$	0	0
$247$	1262.26i	0.325165i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3501.68	−0.880575	−0.440288	−	0.897857i	$-0.645124\pi$
$251$	−3501.68	−0.880575	−0.440288	+	0.897857i	$0.645124\pi$
$252$	0	0
$253$	4750.26	1.18042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1295.48i	− 0.314435i	−0.987564	−	0.157218i	$-0.949748\pi$
$257$	− 1295.48i	− 0.314435i	0.987564	−	0.157218i	$-0.0502524\pi$
$258$	0	0
$259$	6078.56i	1.45831i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4117.27	0.965329	0.482665	−	0.875805i	$-0.339669\pi$
$263$	4117.27	0.965329	0.482665	+	0.875805i	$0.339669\pi$
$264$	0	0
$265$	−10648.6	−2.46845
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2974.78i	− 0.674259i	−0.941458	−	0.337129i	$-0.890544\pi$
$269$	− 2974.78i	− 0.674259i	0.941458	−	0.337129i	$-0.109456\pi$
$270$	0	0
$271$	5000.11i	1.12079i	0.828224	+	0.560396i	$0.189351\pi$
$271$	5000.11i	1.12079i	−0.828224	+	0.560396i	$0.810649\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10305.7	2.25985
$276$	0	0
$277$	213.586	0.0463291	0.0231645	−	0.999732i	$-0.492626\pi$
$277$	213.586	0.0463291	0.0231645	+	0.999732i	$0.492626\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7291.74i	1.54800i	0.633184	+	0.774002i	$0.281748\pi$
$281$	7291.74i	1.54800i	−0.633184	+	0.774002i	$0.718252\pi$
$282$	0	0
$283$	4761.83i	1.00022i	0.865963	+	0.500108i	$0.166706\pi$
$283$	4761.83i	1.00022i	−0.865963	+	0.500108i	$0.833294\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5165.29	1.06236
$288$	0	0
$289$	4712.13	0.959114
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1371.49i	0.273459i	0.990608	+	0.136730i	$0.0436592\pi$
$293$	1371.49i	0.273459i	−0.990608	+	0.136730i	$0.956341\pi$
$294$	0	0
$295$	− 11139.2i	− 2.19847i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−777.447	−0.150371
$300$	0	0
$301$	3620.18	0.693234
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4194.41i	− 0.787446i
$306$	0	0
$307$	511.696i	0.0951272i	0.998868	+	0.0475636i	$0.0151457\pi$
$307$	511.696i	0.0951272i	−0.998868	+	0.0475636i	$0.984854\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−518.033	−0.0944532	−0.0472266	−	0.998884i	$-0.515038\pi$
$311$	−518.033	−0.0944532	−0.0472266	+	0.998884i	$0.515038\pi$
$312$	0	0
$313$	2726.17	0.492308	0.246154	−	0.969231i	$-0.420833\pi$
$313$	2726.17	0.492308	0.246154	+	0.969231i	$0.420833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	566.346i	0.100344i	0.998741	+	0.0501722i	$0.0159770\pi$
$317$	566.346i	0.100344i	−0.998741	+	0.0501722i	$0.984023\pi$
$318$	0	0
$319$	− 200.826i	− 0.0352479i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1982.97	0.341596
$324$	0	0
$325$	−1686.67	−0.287877
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4463.51i	0.747967i
$330$	0	0
$331$	− 7531.08i	− 1.25059i	−0.780388	−	0.625296i	$-0.784978\pi$
$331$	− 7531.08i	− 1.25059i	0.780388	−	0.625296i	$-0.215022\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7028.82	1.14635
$336$	0	0
$337$	7241.22	1.17049	0.585244	−	0.810857i	$-0.300999\pi$
$337$	7241.22	1.17049	0.585244	+	0.810857i	$0.300999\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15049.9i	2.39003i
$342$	0	0
$343$	6914.74i	1.08852i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8937.55	1.38269	0.691344	−	0.722525i	$-0.257019\pi$
$347$	8937.55	1.38269	0.691344	+	0.722525i	$0.257019\pi$
$348$	0	0
$349$	−7003.04	−1.07411	−0.537055	−	0.843547i	$-0.680463\pi$
$349$	−7003.04	−1.07411	−0.537055	+	0.843547i	$0.680463\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3368.17i	− 0.507845i	−0.967225	−	0.253923i	$-0.918279\pi$
$353$	− 3368.17i	− 0.507845i	0.967225	−	0.253923i	$-0.0817208\pi$
$354$	0	0
$355$	− 13758.1i	− 2.05692i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6019.78	0.884992	0.442496	−	0.896771i	$-0.354093\pi$
$359$	6019.78	0.884992	0.442496	+	0.896771i	$0.354093\pi$
$360$	0	0
$361$	−12716.6	−1.85401
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 20400.7i	− 2.92554i
$366$	0	0
$367$	3759.59i	0.534738i	0.963594	+	0.267369i	$0.0861542\pi$
$367$	3759.59i	0.534738i	−0.963594	+	0.267369i	$0.913846\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9030.40	−1.26371
$372$	0	0
$373$	−5174.87	−0.718350	−0.359175	−	0.933270i	$-0.616942\pi$
$373$	−5174.87	−0.718350	−0.359175	+	0.933270i	$0.616942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.8679i	0.00449014i
$378$	0	0
$379$	3076.35i	0.416943i	0.978028	+	0.208471i	$0.0668488\pi$
$379$	3076.35i	0.416943i	−0.978028	+	0.208471i	$0.933151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10465.9	−1.39630	−0.698151	−	0.715951i	$-0.745993\pi$
$383$	−10465.9	−1.39630	−0.698151	+	0.715951i	$0.745993\pi$
$384$	0	0
$385$	14583.0	1.93043
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2556.64i	0.333231i	0.986022	+	0.166615i	$0.0532838\pi$
$389$	2556.64i	0.333231i	−0.986022	+	0.166615i	$0.946716\pi$
$390$	0	0
$391$	1221.35i	0.157970i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2248.87	0.286464
$396$	0	0
$397$	−1232.61	−0.155826	−0.0779130	−	0.996960i	$-0.524826\pi$
$397$	−1232.61	−0.155826	−0.0779130	+	0.996960i	$0.524826\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2278.88i	0.283795i	0.989881	+	0.141898i	$0.0453204\pi$
$401$	2278.88i	0.283795i	−0.989881	+	0.141898i	$0.954680\pi$
$402$	0	0
$403$	− 2463.13i	− 0.304460i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22370.6	−2.72449
$408$	0	0
$409$	950.130	0.114868	0.0574339	−	0.998349i	$-0.481708\pi$
$409$	950.130	0.114868	0.0574339	+	0.998349i	$0.481708\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9446.45i	− 1.12549i
$414$	0	0
$415$	− 5327.43i	− 0.630152i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6974.81	0.813226	0.406613	−	0.913600i	$-0.366710\pi$
$419$	6974.81	0.813226	0.406613	+	0.913600i	$0.366710\pi$
$420$	0	0
$421$	1713.41	0.198353	0.0991764	−	0.995070i	$-0.468379\pi$
$421$	1713.41	0.198353	0.0991764	+	0.995070i	$0.468379\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2649.71i	0.302424i
$426$	0	0
$427$	− 3557.01i	− 0.403128i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7234.34	−0.808506	−0.404253	−	0.914647i	$-0.632468\pi$
$431$	−7234.34	−0.808506	−0.404253	+	0.914647i	$0.632468\pi$
$432$	0	0
$433$	8283.91	0.919398	0.459699	−	0.888075i	$-0.347957\pi$
$433$	8283.91	0.919398	0.459699	+	0.888075i	$0.347957\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12057.0i	− 1.31982i
$438$	0	0
$439$	11500.7i	1.25034i	0.780489	+	0.625170i	$0.214970\pi$
$439$	11500.7i	1.25034i	−0.780489	+	0.625170i	$0.785030\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12976.7	−1.39174	−0.695869	−	0.718169i	$-0.744981\pi$
$443$	−12976.7	−1.39174	−0.695869	+	0.718169i	$0.744981\pi$
$444$	0	0
$445$	−5970.89	−0.636061
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8302.07i	0.872604i	0.899800	+	0.436302i	$0.143712\pi$
$449$	8302.07i	0.872604i	−0.899800	+	0.436302i	$0.856288\pi$
$450$	0	0
$451$	19009.5i	1.98475i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2386.71	−0.245913
$456$	0	0
$457$	−1977.43	−0.202408	−0.101204	−	0.994866i	$-0.532269\pi$
$457$	−1977.43	−0.202408	−0.101204	+	0.994866i	$0.532269\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1616.73i	0.163338i	0.996660	+	0.0816690i	$0.0260250\pi$
$461$	1616.73i	0.163338i	−0.996660	+	0.0816690i	$0.973975\pi$
$462$	0	0
$463$	− 1476.11i	− 0.148165i	−0.997252	−	0.0740827i	$-0.976397\pi$
$463$	− 1476.11i	− 0.148165i	0.997252	−	0.0740827i	$-0.0236029\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15942.8	−1.57975	−0.789875	−	0.613268i	$-0.789855\pi$
$467$	−15942.8	−1.57975	−0.789875	+	0.613268i	$0.789855\pi$
$468$	0	0
$469$	5960.69	0.586864
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13323.1i	1.29513i
$474$	0	0
$475$	− 26157.6i	− 2.52673i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14684.4	−1.40073	−0.700363	−	0.713786i	$-0.746979\pi$
$479$	−14684.4	−1.40073	−0.700363	+	0.713786i	$0.746979\pi$
$480$	0	0
$481$	3661.26	0.347067
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28566.1i	2.67447i
$486$	0	0
$487$	9871.54i	0.918526i	0.888300	+	0.459263i	$0.151886\pi$
$487$	9871.54i	0.918526i	−0.888300	+	0.459263i	$0.848114\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4356.26	0.400398	0.200199	−	0.979755i	$-0.435841\pi$
$491$	4356.26	0.400398	0.200199	+	0.979755i	$0.435841\pi$
$492$	0	0
$493$	51.6345	0.00471704
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11667.4i	− 1.05303i
$498$	0	0
$499$	− 15900.3i	− 1.42645i	−0.700937	−	0.713224i	$-0.747234\pi$
$499$	− 15900.3i	− 1.42645i	0.700937	−	0.713224i	$-0.252766\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9083.46	0.805192	0.402596	−	0.915378i	$-0.368108\pi$
$503$	9083.46	0.805192	0.402596	+	0.915378i	$0.368108\pi$
$504$	0	0
$505$	−7977.56	−0.702964
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4733.57i	0.412204i	0.978531	+	0.206102i	$0.0660778\pi$
$509$	4733.57i	0.412204i	−0.978531	+	0.206102i	$0.933922\pi$
$510$	0	0
$511$	− 17300.5i	− 1.49771i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17665.0	−1.51148
$516$	0	0
$517$	−16426.8	−1.39739
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 14468.8i	− 1.21668i	−0.793675	−	0.608341i	$-0.791835\pi$
$521$	− 14468.8i	− 1.21668i	0.793675	−	0.608341i	$-0.208165\pi$
$522$	0	0
$523$	12945.2i	1.08232i	0.840919	+	0.541161i	$0.182015\pi$
$523$	12945.2i	1.08232i	−0.840919	+	0.541161i	$0.817985\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3869.51	−0.319845
$528$	0	0
$529$	−4740.91	−0.389653
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3111.18i	− 0.252833i
$534$	0	0
$535$	− 27958.3i	− 2.25933i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6540.52	−0.522672
$540$	0	0
$541$	7859.59	0.624603	0.312301	−	0.949983i	$-0.398900\pi$
$541$	7859.59	0.624603	0.312301	+	0.949983i	$0.398900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 11246.7i	− 0.883952i
$546$	0	0
$547$	− 20633.2i	− 1.61282i	−0.591359	−	0.806408i	$-0.701408\pi$
$547$	− 20633.2i	− 1.61282i	0.591359	−	0.806408i	$-0.298592\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−509.729	−0.0394105
$552$	0	0
$553$	1907.12	0.146653
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22253.3i	− 1.69282i	−0.532531	−	0.846411i	$-0.678759\pi$
$557$	− 22253.3i	− 1.69282i	0.532531	−	0.846411i	$-0.321241\pi$
$558$	0	0
$559$	− 2180.52i	− 0.164984i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7722.80	0.578112	0.289056	−	0.957312i	$-0.406659\pi$
$563$	7722.80	0.578112	0.289056	+	0.957312i	$0.406659\pi$
$564$	0	0
$565$	−1483.50	−0.110462
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5085.66i	0.374696i	0.982294	+	0.187348i	$0.0599892\pi$
$569$	5085.66i	0.374696i	−0.982294	+	0.187348i	$0.940011\pi$
$570$	0	0
$571$	11021.0i	0.807728i	0.914819	+	0.403864i	$0.132333\pi$
$571$	11021.0i	0.807728i	−0.914819	+	0.403864i	$0.867667\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16110.9	1.16847
$576$	0	0
$577$	−16881.8	−1.21802	−0.609012	−	0.793161i	$-0.708434\pi$
$577$	−16881.8	−1.21802	−0.609012	+	0.793161i	$0.708434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4517.85i	− 0.322602i
$582$	0	0
$583$	− 33234.0i	− 2.36091i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3641.73	−0.256065	−0.128032	−	0.991770i	$-0.540866\pi$
$587$	−3641.73	−0.256065	−0.128032	+	0.991770i	$0.540866\pi$
$588$	0	0
$589$	38199.3	2.67228
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2475.73i	0.171444i	0.996319	+	0.0857219i	$0.0273197\pi$
$593$	2475.73i	0.171444i	−0.996319	+	0.0857219i	$0.972680\pi$
$594$	0	0
$595$	3749.44i	0.258340i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−306.543	−0.0209099	−0.0104549	−	0.999945i	$-0.503328\pi$
$599$	−306.543	−0.0209099	−0.0104549	+	0.999945i	$0.503328\pi$
$600$	0	0
$601$	−12411.0	−0.842358	−0.421179	−	0.906978i	$-0.638384\pi$
$601$	−12411.0	−0.842358	−0.421179	+	0.906978i	$0.638384\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30160.3i	2.02676i
$606$	0	0
$607$	1572.02i	0.105118i	0.998618	+	0.0525588i	$0.0167377\pi$
$607$	1572.02i	0.105118i	−0.998618	+	0.0525588i	$0.983262\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2688.47	0.178010
$612$	0	0
$613$	3597.65	0.237044	0.118522	−	0.992951i	$-0.462184\pi$
$613$	3597.65	0.237044	0.118522	+	0.992951i	$0.462184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26075.4i	1.70139i	0.525659	+	0.850695i	$0.323819\pi$
$617$	26075.4i	1.70139i	−0.525659	+	0.850695i	$0.676181\pi$
$618$	0	0
$619$	13844.9i	0.898986i	0.893284	+	0.449493i	$0.148395\pi$
$619$	13844.9i	0.898986i	−0.893284	+	0.449493i	$0.851605\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5063.53	−0.325627
$624$	0	0
$625$	−4041.83	−0.258677
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5751.73i	− 0.364605i
$630$	0	0
$631$	906.847i	0.0572124i	0.999591	+	0.0286062i	$0.00910687\pi$
$631$	906.847i	0.0572124i	−0.999591	+	0.0286062i	$0.990893\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16306.5	−1.01906
$636$	0	0
$637$	1070.45	0.0665820
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 28362.9i	− 1.74769i	−0.486209	−	0.873843i	$-0.661621\pi$
$641$	− 28362.9i	− 1.74769i	0.486209	−	0.873843i	$-0.338379\pi$
$642$	0	0
$643$	6606.82i	0.405206i	0.979261	+	0.202603i	$0.0649401\pi$
$643$	6606.82i	0.405206i	−0.979261	+	0.202603i	$0.935060\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18843.5	−1.14500	−0.572499	−	0.819905i	$-0.694026\pi$
$647$	−18843.5	−1.14500	−0.572499	+	0.819905i	$0.694026\pi$
$648$	0	0
$649$	34765.2	2.10270
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15382.1i	0.921819i	0.887447	+	0.460910i	$0.152477\pi$
$653$	15382.1i	0.921819i	−0.887447	+	0.460910i	$0.847523\pi$
$654$	0	0
$655$	− 6419.40i	− 0.382941i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6692.00	0.395574	0.197787	−	0.980245i	$-0.436625\pi$
$659$	6692.00	0.395574	0.197787	+	0.980245i	$0.436625\pi$
$660$	0	0
$661$	18708.8	1.10089	0.550444	−	0.834872i	$-0.314458\pi$
$661$	18708.8	1.10089	0.550444	+	0.834872i	$0.314458\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 37014.0i	− 2.15841i
$666$	0	0
$667$	− 313.951i	− 0.0182252i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13090.6	0.753142
$672$	0	0
$673$	−21815.7	−1.24953	−0.624766	−	0.780812i	$-0.714805\pi$
$673$	−21815.7	−1.24953	−0.624766	+	0.780812i	$0.714805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14764.7i	− 0.838191i	−0.907942	−	0.419095i	$-0.862347\pi$
$677$	− 14764.7i	− 0.838191i	0.907942	−	0.419095i	$-0.137653\pi$
$678$	0	0
$679$	24225.0i	1.36918i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10522.2	−0.589489	−0.294744	−	0.955576i	$-0.595235\pi$
$683$	−10522.2	−0.589489	−0.294744	+	0.955576i	$0.595235\pi$
$684$	0	0
$685$	−12451.7	−0.694531
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5439.22i	0.300751i
$690$	0	0
$691$	24495.2i	1.34854i	0.738486	+	0.674269i	$0.235541\pi$
$691$	24495.2i	1.34854i	−0.738486	+	0.674269i	$0.764459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	36478.8	1.99096
$696$	0	0
$697$	−4887.56	−0.265609
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22137.1i	1.19274i	0.802711	+	0.596368i	$0.203390\pi$
$701$	22137.1i	1.19274i	−0.802711	+	0.596368i	$0.796610\pi$
$702$	0	0
$703$	56780.3i	3.04625i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6765.26	−0.359878
$708$	0	0
$709$	−23366.2	−1.23771	−0.618854	−	0.785506i	$-0.712403\pi$
$709$	−23366.2	−1.23771	−0.618854	+	0.785506i	$0.712403\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23527.6i	1.23579i
$714$	0	0
$715$	− 8783.65i	− 0.459427i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3714.68	0.192676	0.0963380	−	0.995349i	$-0.469287\pi$
$719$	3714.68	0.192676	0.0963380	+	0.995349i	$0.469287\pi$
$720$	0	0
$721$	−14980.5	−0.773792
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 681.118i	− 0.0348911i
$726$	0	0
$727$	− 6646.80i	− 0.339087i	−0.985523	−	0.169543i	$-0.945771\pi$
$727$	− 6646.80i	− 0.339087i	0.985523	−	0.169543i	$-0.0542293\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3425.52	−0.173321
$732$	0	0
$733$	−22683.9	−1.14304	−0.571519	−	0.820589i	$-0.693646\pi$
$733$	−22683.9	−1.14304	−0.571519	+	0.820589i	$0.693646\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21936.8i	1.09641i
$738$	0	0
$739$	30377.3i	1.51211i	0.654510	+	0.756053i	$0.272875\pi$
$739$	30377.3i	1.51211i	−0.654510	+	0.756053i	$0.727125\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17871.8	−0.882439	−0.441219	−	0.897399i	$-0.645454\pi$
$743$	−17871.8	−0.882439	−0.441219	+	0.897399i	$0.645454\pi$
$744$	0	0
$745$	19839.5	0.975656
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 23709.6i	− 1.15665i
$750$	0	0
$751$	− 33183.5i	− 1.61236i	−0.591669	−	0.806181i	$-0.701531\pi$
$751$	− 33183.5i	− 1.61236i	0.591669	−	0.806181i	$-0.298469\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13215.6	0.637040
$756$	0	0
$757$	−30541.7	−1.46639	−0.733194	−	0.680020i	$-0.761971\pi$
$757$	−30541.7	−1.46639	−0.733194	+	0.680020i	$0.761971\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 14713.4i	− 0.700870i	−0.936587	−	0.350435i	$-0.886034\pi$
$761$	− 14713.4i	− 0.700870i	0.936587	−	0.350435i	$-0.113966\pi$
$762$	0	0
$763$	− 9537.56i	− 0.452533i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5689.82	−0.267858
$768$	0	0
$769$	1182.88	0.0554689	0.0277345	−	0.999615i	$-0.491171\pi$
$769$	1182.88	0.0554689	0.0277345	+	0.999615i	$0.491171\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 8341.02i	− 0.388106i	−0.980991	−	0.194053i	$-0.937837\pi$
$773$	− 8341.02i	− 0.388106i	0.980991	−	0.194053i	$-0.0621633\pi$
$774$	0	0
$775$	51043.2i	2.36584i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48249.4	2.21914
$780$	0	0
$781$	42938.8	1.96731
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20572.7i	0.935378i
$786$	0	0
$787$	− 9835.39i	− 0.445482i	−0.974878	−	0.222741i	$-0.928500\pi$
$787$	− 9835.39i	− 0.445482i	0.974878	−	0.222741i	$-0.0715003\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1258.06	−0.0565504
$792$	0	0
$793$	−2142.47	−0.0959410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20803.2i	0.924577i	0.886730	+	0.462288i	$0.152972\pi$
$797$	20803.2i	0.924577i	−0.886730	+	0.462288i	$0.847028\pi$
$798$	0	0
$799$	− 4223.51i	− 0.187005i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	63670.1	2.79809
$804$	0	0
$805$	22797.5	0.998147
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14439.4i	0.627518i	0.949503	+	0.313759i	$0.101588\pi$
$809$	14439.4i	0.627518i	−0.949503	+	0.313759i	$0.898412\pi$
$810$	0	0
$811$	6457.92i	0.279616i	0.990179	+	0.139808i	$0.0446485\pi$
$811$	6457.92i	0.279616i	−0.990179	+	0.139808i	$0.955352\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17243.8	0.741132
$816$	0	0
$817$	33816.3	1.44808
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 37203.3i	− 1.58149i	−0.612144	−	0.790747i	$-0.709693\pi$
$821$	− 37203.3i	− 1.58149i	0.612144	−	0.790747i	$-0.290307\pi$
$822$	0	0
$823$	1916.63i	0.0811781i	0.999176	+	0.0405890i	$0.0129234\pi$
$823$	1916.63i	0.0811781i	−0.999176	+	0.0405890i	$0.987077\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44003.2	1.85023	0.925117	−	0.379683i	$-0.123967\pi$
$827$	44003.2	1.85023	0.925117	+	0.379683i	$0.123967\pi$
$828$	0	0
$829$	−39853.8	−1.66970	−0.834849	−	0.550479i	$-0.814445\pi$
$829$	−39853.8	−1.66970	−0.834849	+	0.550479i	$0.814445\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1681.64i	− 0.0699465i
$834$	0	0
$835$	− 28324.2i	− 1.17389i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30695.6	1.26309	0.631543	−	0.775341i	$-0.282422\pi$
$839$	30695.6	1.26309	0.631543	+	0.775341i	$0.282422\pi$
$840$	0	0
$841$	24375.7	0.999456
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 37366.5i	− 1.52124i
$846$	0	0
$847$	25577.0i	1.03759i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−34972.0	−1.40872
$852$	0	0
$853$	36152.2	1.45115	0.725574	−	0.688144i	$-0.241574\pi$
$853$	36152.2	1.45115	0.725574	+	0.688144i	$0.241574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13536.1i	− 0.539540i	−0.962925	−	0.269770i	$-0.913052\pi$
$857$	− 13536.1i	− 0.539540i	0.962925	−	0.269770i	$-0.0869477\pi$
$858$	0	0
$859$	− 20932.8i	− 0.831454i	−0.909489	−	0.415727i	$-0.863527\pi$
$859$	− 20932.8i	− 0.831454i	0.909489	−	0.415727i	$-0.136473\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15068.3	0.594359	0.297179	−	0.954822i	$-0.403954\pi$
$863$	15068.3	0.594359	0.297179	+	0.954822i	$0.403954\pi$
$864$	0	0
$865$	4707.60	0.185044
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7018.68i	0.273984i
$870$	0	0
$871$	− 3590.26i	− 0.139669i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16390.6	0.633261
$876$	0	0
$877$	−40198.2	−1.54777	−0.773886	−	0.633325i	$-0.781689\pi$
$877$	−40198.2	−1.54777	−0.773886	+	0.633325i	$0.781689\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34387.5i	1.31503i	0.753441	+	0.657516i	$0.228393\pi$
$881$	34387.5i	1.31503i	−0.753441	+	0.657516i	$0.771607\pi$
$882$	0	0
$883$	− 9041.95i	− 0.344604i	−0.985044	−	0.172302i	$-0.944879\pi$
$883$	− 9041.95i	− 0.344604i	0.985044	−	0.172302i	$-0.0551206\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11974.5	−0.453286	−0.226643	−	0.973978i	$-0.572775\pi$
$887$	−11974.5	−0.453286	−0.226643	+	0.973978i	$0.572775\pi$
$888$	0	0
$889$	−13828.5	−0.521702
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41694.0i	1.56241i
$894$	0	0
$895$	− 3494.78i	− 0.130523i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	994.670	0.0369011
$900$	0	0
$901$	8544.85	0.315949
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 50063.7i	− 1.83887i
$906$	0	0
$907$	− 27657.0i	− 1.01250i	−0.862387	−	0.506249i	$-0.831032\pi$
$907$	− 27657.0i	− 1.01250i	0.862387	−	0.506249i	$-0.168968\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42952.4	1.56210	0.781052	−	0.624466i	$-0.214683\pi$
$911$	42952.4	1.56210	0.781052	+	0.624466i	$0.214683\pi$
$912$	0	0
$913$	16626.8	0.602701
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5443.88i	− 0.196044i
$918$	0	0
$919$	17076.0i	0.612934i	0.951881	+	0.306467i	$0.0991469\pi$
$919$	17076.0i	0.612934i	−0.951881	+	0.306467i	$0.900853\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7027.54	−0.250611
$924$	0	0
$925$	−75871.8	−2.69692
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26818.3i	0.947126i	0.880760	+	0.473563i	$0.157032\pi$
$929$	26818.3i	0.947126i	−0.880760	+	0.473563i	$0.842968\pi$
$930$	0	0
$931$	16601.0i	0.584398i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13798.9	−0.482642
$936$	0	0
$937$	38163.9	1.33059	0.665294	−	0.746582i	$-0.268306\pi$
$937$	38163.9	1.33059	0.665294	+	0.746582i	$0.268306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 10728.0i	− 0.371650i	−0.982583	−	0.185825i	$-0.940504\pi$
$941$	− 10728.0i	− 0.371650i	0.982583	−	0.185825i	$-0.0594957\pi$
$942$	0	0
$943$	29717.6i	1.02623i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28971.4	0.994133	0.497066	−	0.867713i	$-0.334411\pi$
$947$	28971.4	0.994133	0.497066	+	0.867713i	$0.334411\pi$
$948$	0	0
$949$	−10420.5	−0.356443
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 36897.2i	− 1.25416i	−0.778954	−	0.627081i	$-0.784250\pi$
$953$	− 36897.2i	− 1.25416i	0.778954	−	0.627081i	$-0.215750\pi$
$954$	0	0
$955$	91638.7i	3.10509i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10559.5	−0.355561
$960$	0	0
$961$	−44749.9	−1.50213
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3644.58i	0.121578i
$966$	0	0
$967$	2016.50i	0.0670591i	0.999438	+	0.0335295i	$0.0106748\pi$
$967$	2016.50i	0.0670591i	−0.999438	+	0.0335295i	$0.989325\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18482.0	−0.610830	−0.305415	−	0.952219i	$-0.598795\pi$
$971$	−18482.0	−0.610830	−0.305415	+	0.952219i	$0.598795\pi$
$972$	0	0
$973$	30935.3	1.01926
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22871.4i	0.748945i	0.927238	+	0.374473i	$0.122176\pi$
$977$	22871.4i	0.748945i	−0.927238	+	0.374473i	$0.877824\pi$
$978$	0	0
$979$	− 18635.0i	− 0.608352i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16630.8	0.539615	0.269808	−	0.962914i	$-0.413040\pi$
$983$	16630.8	0.539615	0.269808	+	0.962914i	$0.413040\pi$
$984$	0	0
$985$	−58402.1	−1.88918
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20828.1i	0.669660i
$990$	0	0
$991$	− 21036.2i	− 0.674307i	−0.941450	−	0.337153i	$-0.890536\pi$
$991$	− 21036.2i	− 0.674307i	0.941450	−	0.337153i	$-0.109464\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−61665.6	−1.96475
$996$	0	0
$997$	31241.8	0.992416	0.496208	−	0.868204i	$-0.334725\pi$
$997$	31241.8	0.992416	0.496208	+	0.868204i	$0.334725\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.c.f.575.8		8
3.2	odd	2	inner	576.4.c.f.575.2		8
4.3	odd	2	inner	576.4.c.f.575.7		8
8.3	odd	2		288.4.c.b.287.1	✓	8
8.5	even	2		288.4.c.b.287.2	yes	8
12.11	even	2	inner	576.4.c.f.575.1		8
16.3	odd	4		2304.4.f.h.1151.8		8
16.5	even	4		2304.4.f.h.1151.1		8
16.11	odd	4		2304.4.f.e.1151.2		8
16.13	even	4		2304.4.f.e.1151.7		8
24.5	odd	2		288.4.c.b.287.8	yes	8
24.11	even	2		288.4.c.b.287.7	yes	8
48.5	odd	4		2304.4.f.h.1151.7		8
48.11	even	4		2304.4.f.e.1151.8		8
48.29	odd	4		2304.4.f.e.1151.1		8
48.35	even	4		2304.4.f.h.1151.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.4.c.b.287.1	✓	8	8.3	odd	2
288.4.c.b.287.2	yes	8	8.5	even	2
288.4.c.b.287.7	yes	8	24.11	even	2
288.4.c.b.287.8	yes	8	24.5	odd	2
576.4.c.f.575.1		8	12.11	even	2	inner
576.4.c.f.575.2		8	3.2	odd	2	inner
576.4.c.f.575.7		8	4.3	odd	2	inner
576.4.c.f.575.8		8	1.1	even	1	trivial
2304.4.f.e.1151.1		8	48.29	odd	4
2304.4.f.e.1151.2		8	16.11	odd	4
2304.4.f.e.1151.7		8	16.13	even	4
2304.4.f.e.1151.8		8	48.11	even	4
2304.4.f.h.1151.1		8	16.5	even	4
2304.4.f.h.1151.2		8	48.35	even	4
2304.4.f.h.1151.7		8	48.5	odd	4
2304.4.f.h.1151.8		8	16.3	odd	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+17.6623i q^{5} +14.9783i q^{7} +O(q^{10})\)	\(q+17.6623i q^{5} +14.9783i q^{7} -55.1236 q^{11} +9.02175 q^{13} -14.1729i q^{17} +139.913i q^{19} -86.1748 q^{23} -186.957 q^{25} +3.64319i q^{29} -273.022i q^{31} -264.550 q^{35} +405.826 q^{37} -344.853i q^{41} -241.696i q^{43} +297.999 q^{47} +118.652 q^{49} +602.901i q^{53} -973.609i q^{55} -630.678 q^{59} -237.478 q^{61} +159.345i q^{65} -397.957i q^{67} -778.955 q^{71} -1155.04 q^{73} -825.655i q^{77} -127.326i q^{79} -301.627 q^{83} +250.326 q^{85} +338.059i q^{89} +135.130i q^{91} -2471.18 q^{95} +1617.35 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 256 q^{13} - 1128 q^{25} + 1776 q^{37} - 1992 q^{49} + 2512 q^{61} - 1152 q^{73} - 3696 q^{85} + 7424 q^{97}+O(q^{100}) \) 8 * q + 256 * q^13 - 1128 * q^25 + 1776 * q^37 - 1992 * q^49 + 2512 * q^61 - 1152 * q^73 - 3696 * q^85 + 7424 * q^97

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