LMFDB - Embedded newform 1008.4.h.a.575.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(575,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1008.575");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 $ x^12 - 144x^10 + 12024x^8 - 766296x^6 + 11751192x^4 + 565147728*x^2 + 9666232489
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.4
Root			$1.44334 + 3.80934i$ of defining polynomial
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.a.575.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.9196i q^{5} +7.00000i q^{7} +O(q^{10})$	$q-11.9196i q^{5} +7.00000i q^{7} -38.1517 q^{11} -24.4057 q^{13} -50.6466i q^{17} +92.9762i q^{19} +6.97792 q^{23} -17.0762 q^{25} +122.675i q^{29} +85.1533i q^{31} +83.4370 q^{35} +239.851 q^{37} -117.547i q^{41} +305.775i q^{43} -202.006 q^{47} -49.0000 q^{49} +100.531i q^{53} +454.751i q^{55} +646.008 q^{59} +101.435 q^{61} +290.906i q^{65} -513.673i q^{67} +346.765 q^{71} -743.214 q^{73} -267.062i q^{77} -343.366i q^{79} +985.795 q^{83} -603.685 q^{85} -77.9139i q^{89} -170.840i q^{91} +1108.24 q^{95} +208.826 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) $ 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	− 11.9196i	− 1.06612i	−0.846078	−	0.533059i	$-0.821042\pi$
$5$	− 11.9196i	− 1.06612i	0.846078	−	0.533059i	$-0.178958\pi$
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−38.1517	−1.04574	−0.522871	−	0.852412i	$-0.675139\pi$
$11$	−38.1517	−1.04574	−0.522871	+	0.852412i	$0.675139\pi$
$12$	0	0
$13$	−24.4057	−0.520686	−0.260343	−	0.965516i	$-0.583836\pi$
$13$	−24.4057	−0.520686	−0.260343	+	0.965516i	$0.583836\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 50.6466i	− 0.722565i	−0.932456	−	0.361282i	$-0.882339\pi$
$17$	− 50.6466i	− 0.722565i	0.932456	−	0.361282i	$-0.117661\pi$
$18$	0	0
$19$	92.9762i	1.12264i	0.827598	+	0.561321i	$0.189707\pi$
$19$	92.9762i	1.12264i	−0.827598	+	0.561321i	$0.810293\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.97792	0.0632608	0.0316304	−	0.999500i	$-0.489930\pi$
$23$	6.97792	0.0632608	0.0316304	+	0.999500i	$0.489930\pi$
$24$	0	0
$25$	−17.0762	−0.136609
$26$	0	0
$27$	0	0
$28$	0	0
$29$	122.675i	0.785524i	0.919640	+	0.392762i	$0.128480\pi$
$29$	122.675i	0.785524i	−0.919640	+	0.392762i	$0.871520\pi$
$30$	0	0
$31$	85.1533i	0.493354i	0.969098	+	0.246677i	$0.0793387\pi$
$31$	85.1533i	0.493354i	−0.969098	+	0.246677i	$0.920661\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	83.4370	0.402955
$36$	0	0
$37$	239.851	1.06571	0.532856	−	0.846206i	$-0.321119\pi$
$37$	239.851	1.06571	0.532856	+	0.846206i	$0.321119\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 117.547i	− 0.447750i	−0.974618	−	0.223875i	$-0.928129\pi$
$41$	− 117.547i	− 0.447750i	0.974618	−	0.223875i	$-0.0718707\pi$
$42$	0	0
$43$	305.775i	1.08443i	0.840241	+	0.542213i	$0.182413\pi$
$43$	305.775i	1.08443i	−0.840241	+	0.542213i	$0.817587\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−202.006	−0.626928	−0.313464	−	0.949600i	$-0.601490\pi$
$47$	−202.006	−0.626928	−0.313464	+	0.949600i	$0.601490\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	100.531i	0.260548i	0.991478	+	0.130274i	$0.0415856\pi$
$53$	100.531i	0.260548i	−0.991478	+	0.130274i	$0.958414\pi$
$54$	0	0
$55$	454.751i	1.11488i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	646.008	1.42548	0.712738	−	0.701430i	$-0.247455\pi$
$59$	646.008	1.42548	0.712738	+	0.701430i	$0.247455\pi$
$60$	0	0
$61$	101.435	0.212909	0.106454	−	0.994318i	$-0.466050\pi$
$61$	101.435	0.212909	0.106454	+	0.994318i	$0.466050\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	290.906i	0.555114i
$66$	0	0
$67$	− 513.673i	− 0.936644i	−0.883558	−	0.468322i	$-0.844859\pi$
$67$	− 513.673i	− 0.936644i	0.883558	−	0.468322i	$-0.155141\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	346.765	0.579626	0.289813	−	0.957083i	$-0.406407\pi$
$71$	346.765	0.579626	0.289813	+	0.957083i	$0.406407\pi$
$72$	0	0
$73$	−743.214	−1.19160	−0.595799	−	0.803134i	$-0.703164\pi$
$73$	−743.214	−1.19160	−0.595799	+	0.803134i	$0.703164\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 267.062i	− 0.395253i
$78$	0	0
$79$	− 343.366i	− 0.489009i	−0.969648	−	0.244505i	$-0.921375\pi$
$79$	− 343.366i	− 0.489009i	0.969648	−	0.244505i	$-0.0786254\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	985.795	1.30368	0.651838	−	0.758358i	$-0.273998\pi$
$83$	985.795	1.30368	0.651838	+	0.758358i	$0.273998\pi$
$84$	0	0
$85$	−603.685	−0.770340
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 77.9139i	− 0.0927961i	−0.998923	−	0.0463981i	$-0.985226\pi$
$89$	− 77.9139i	− 0.0927961i	0.998923	−	0.0463981i	$-0.0147743\pi$
$90$	0	0
$91$	− 170.840i	− 0.196801i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1108.24	1.19687
$96$	0	0
$97$	208.826	0.218588	0.109294	−	0.994009i	$-0.465141\pi$
$97$	208.826	0.218588	0.109294	+	0.994009i	$0.465141\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 80.3061i	− 0.0791164i	−0.999217	−	0.0395582i	$-0.987405\pi$
$101$	− 80.3061i	− 0.0791164i	0.999217	−	0.0395582i	$-0.0125950\pi$
$102$	0	0
$103$	1759.63i	1.68331i	0.540013	+	0.841656i	$0.318419\pi$
$103$	1759.63i	1.68331i	−0.540013	+	0.841656i	$0.681581\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2063.41	1.86428	0.932138	−	0.362103i	$-0.117941\pi$
$107$	2063.41	1.86428	0.932138	+	0.362103i	$0.117941\pi$
$108$	0	0
$109$	1751.07	1.53873	0.769366	−	0.638809i	$-0.220572\pi$
$109$	1751.07	1.53873	0.769366	+	0.638809i	$0.220572\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 124.420i	− 0.103579i	−0.998658	−	0.0517897i	$-0.983507\pi$
$113$	− 124.420i	− 0.103579i	0.998658	−	0.0517897i	$-0.0164926\pi$
$114$	0	0
$115$	− 83.1738i	− 0.0674435i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	354.526	0.273104
$120$	0	0
$121$	124.548	0.0935751
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1286.41i	− 0.920477i
$126$	0	0
$127$	1246.12i	0.870673i	0.900268	+	0.435337i	$0.143371\pi$
$127$	1246.12i	0.870673i	−0.900268	+	0.435337i	$0.856629\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	900.082	0.600309	0.300155	−	0.953891i	$-0.402962\pi$
$131$	900.082	0.600309	0.300155	+	0.953891i	$0.402962\pi$
$132$	0	0
$133$	−650.833	−0.424319
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2105.01i	1.31272i	0.754446	+	0.656362i	$0.227906\pi$
$137$	2105.01i	1.31272i	−0.754446	+	0.656362i	$0.772094\pi$
$138$	0	0
$139$	604.804i	0.369056i	0.982827	+	0.184528i	$0.0590757\pi$
$139$	604.804i	0.369056i	−0.982827	+	0.184528i	$0.940924\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	931.118	0.544503
$144$	0	0
$145$	1462.23	0.837462
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 284.632i	− 0.156496i	−0.996934	−	0.0782482i	$-0.975067\pi$
$149$	− 284.632i	− 0.156496i	0.996934	−	0.0782482i	$-0.0249327\pi$
$150$	0	0
$151$	2242.63i	1.20863i	0.796746	+	0.604314i	$0.206553\pi$
$151$	2242.63i	1.20863i	−0.796746	+	0.604314i	$0.793447\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1014.99	0.525974
$156$	0	0
$157$	86.8659	0.0441570	0.0220785	−	0.999756i	$-0.492972\pi$
$157$	86.8659	0.0441570	0.0220785	+	0.999756i	$0.492972\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	48.8455i	0.0239103i
$162$	0	0
$163$	2736.79i	1.31510i	0.753410	+	0.657551i	$0.228408\pi$
$163$	2736.79i	1.31510i	−0.753410	+	0.657551i	$0.771592\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1321.77	−0.612464	−0.306232	−	0.951957i	$-0.599068\pi$
$167$	−1321.77	−0.612464	−0.306232	+	0.951957i	$0.599068\pi$
$168$	0	0
$169$	−1601.36	−0.728886
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2142.36i	0.941505i	0.882265	+	0.470753i	$0.156018\pi$
$173$	2142.36i	0.941505i	−0.882265	+	0.470753i	$0.843982\pi$
$174$	0	0
$175$	− 119.533i	− 0.0516335i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3861.78	1.61253	0.806265	−	0.591555i	$-0.201486\pi$
$179$	3861.78	1.61253	0.806265	+	0.591555i	$0.201486\pi$
$180$	0	0
$181$	−4121.72	−1.69263	−0.846313	−	0.532686i	$-0.821183\pi$
$181$	−4121.72	−1.69263	−0.846313	+	0.532686i	$0.821183\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2858.93i	− 1.13618i
$186$	0	0
$187$	1932.25i	0.755616i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−437.994	−0.165927	−0.0829637	−	0.996553i	$-0.526439\pi$
$191$	−437.994	−0.165927	−0.0829637	+	0.996553i	$0.526439\pi$
$192$	0	0
$193$	4006.92	1.49443	0.747213	−	0.664585i	$-0.231392\pi$
$193$	4006.92	1.49443	0.747213	+	0.664585i	$0.231392\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 479.380i	− 0.173373i	−0.996236	−	0.0866863i	$-0.972372\pi$
$197$	− 479.380i	− 0.173373i	0.996236	−	0.0866863i	$-0.0276278\pi$
$198$	0	0
$199$	2470.01i	0.879870i	0.898030	+	0.439935i	$0.144999\pi$
$199$	2470.01i	0.879870i	−0.898030	+	0.439935i	$0.855001\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−858.726	−0.296900
$204$	0	0
$205$	−1401.11	−0.477355
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3547.19i	− 1.17399i
$210$	0	0
$211$	867.918i	0.283175i	0.989926	+	0.141588i	$0.0452207\pi$
$211$	867.918i	0.283175i	−0.989926	+	0.141588i	$0.954779\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3644.71	1.15613
$216$	0	0
$217$	−596.073	−0.186470
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1236.07i	0.376230i
$222$	0	0
$223$	979.908i	0.294258i	0.989117	+	0.147129i	$0.0470032\pi$
$223$	979.908i	0.294258i	−0.989117	+	0.147129i	$0.952997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	205.820	0.0601796	0.0300898	−	0.999547i	$-0.490421\pi$
$227$	205.820	0.0601796	0.0300898	+	0.999547i	$0.490421\pi$
$228$	0	0
$229$	2021.61	0.583371	0.291685	−	0.956514i	$-0.405784\pi$
$229$	2021.61	0.583371	0.291685	+	0.956514i	$0.405784\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3771.02i	1.06029i	0.847907	+	0.530145i	$0.177863\pi$
$233$	3771.02i	1.06029i	−0.847907	+	0.530145i	$0.822137\pi$
$234$	0	0
$235$	2407.83i	0.668380i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	502.357	0.135962	0.0679808	−	0.997687i	$-0.478344\pi$
$239$	502.357	0.135962	0.0679808	+	0.997687i	$0.478344\pi$
$240$	0	0
$241$	−3130.11	−0.836632	−0.418316	−	0.908301i	$-0.637380\pi$
$241$	−3130.11	−0.836632	−0.418316	+	0.908301i	$0.637380\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	584.059i	0.152303i
$246$	0	0
$247$	− 2269.15i	− 0.584544i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6547.91	1.64662	0.823308	−	0.567595i	$-0.192126\pi$
$251$	6547.91	1.64662	0.823308	+	0.567595i	$0.192126\pi$
$252$	0	0
$253$	−266.219	−0.0661544
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5282.60i	1.28218i	0.767467	+	0.641088i	$0.221517\pi$
$257$	5282.60i	1.28218i	−0.767467	+	0.641088i	$0.778483\pi$
$258$	0	0
$259$	1678.96i	0.402801i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4826.44	−1.13160	−0.565800	−	0.824542i	$-0.691433\pi$
$263$	−4826.44	−1.13160	−0.565800	+	0.824542i	$0.691433\pi$
$264$	0	0
$265$	1198.29	0.277775
$266$	0	0
$267$	0	0
$268$	0	0
$269$	205.516i	0.0465818i	0.999729	+	0.0232909i	$0.00741440\pi$
$269$	205.516i	0.0465818i	−0.999729	+	0.0232909i	$0.992586\pi$
$270$	0	0
$271$	− 8386.70i	− 1.87991i	−0.341296	−	0.939956i	$-0.610866\pi$
$271$	− 8386.70i	− 1.87991i	0.341296	−	0.939956i	$-0.389134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	651.485	0.142858
$276$	0	0
$277$	7078.99	1.53551	0.767754	−	0.640745i	$-0.221374\pi$
$277$	7078.99	1.53551	0.767754	+	0.640745i	$0.221374\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3673.31i	− 0.779826i	−0.920852	−	0.389913i	$-0.872505\pi$
$281$	− 3673.31i	− 0.779826i	0.920852	−	0.389913i	$-0.127495\pi$
$282$	0	0
$283$	− 345.797i	− 0.0726344i	−0.999340	−	0.0363172i	$-0.988437\pi$
$283$	− 345.797i	− 0.0726344i	0.999340	−	0.0363172i	$-0.0115627\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	822.828	0.169234
$288$	0	0
$289$	2347.92	0.477900
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4444.57i	− 0.886194i	−0.896474	−	0.443097i	$-0.853880\pi$
$293$	− 4444.57i	− 0.886194i	0.896474	−	0.443097i	$-0.146120\pi$
$294$	0	0
$295$	− 7700.14i	− 1.51973i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−170.301	−0.0329390
$300$	0	0
$301$	−2140.43	−0.409874
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1209.06i	− 0.226986i
$306$	0	0
$307$	− 3398.68i	− 0.631833i	−0.948787	−	0.315917i	$-0.897688\pi$
$307$	− 3398.68i	− 0.631833i	0.948787	−	0.315917i	$-0.102312\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.5307	−0.00356105	−0.00178052	−	0.999998i	$-0.500567\pi$
$311$	−19.5307	−0.00356105	−0.00178052	+	0.999998i	$0.500567\pi$
$312$	0	0
$313$	−4011.40	−0.724402	−0.362201	−	0.932100i	$-0.617975\pi$
$313$	−4011.40	−0.724402	−0.362201	+	0.932100i	$0.617975\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2479.02i	0.439230i	0.975587	+	0.219615i	$0.0704800\pi$
$317$	2479.02i	0.439230i	−0.975587	+	0.219615i	$0.929520\pi$
$318$	0	0
$319$	− 4680.26i	− 0.821455i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4708.92	0.811181
$324$	0	0
$325$	416.756	0.0711307
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1414.04i	− 0.236957i
$330$	0	0
$331$	6917.21i	1.14865i	0.818626	+	0.574326i	$0.194736\pi$
$331$	6917.21i	1.14865i	−0.818626	+	0.574326i	$0.805264\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6122.76	−0.998574
$336$	0	0
$337$	−3561.67	−0.575716	−0.287858	−	0.957673i	$-0.592943\pi$
$337$	−3561.67	−0.575716	−0.287858	+	0.957673i	$0.592943\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3248.74i	− 0.515921i
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8527.31	−1.31922	−0.659611	−	0.751607i	$-0.729279\pi$
$347$	−8527.31	−1.31922	−0.659611	+	0.751607i	$0.729279\pi$
$348$	0	0
$349$	5428.70	0.832640	0.416320	−	0.909218i	$-0.363320\pi$
$349$	5428.70	0.832640	0.416320	+	0.909218i	$0.363320\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6845.71i	− 1.03218i	−0.856534	−	0.516091i	$-0.827387\pi$
$353$	− 6845.71i	− 1.03218i	0.856534	−	0.516091i	$-0.172613\pi$
$354$	0	0
$355$	− 4133.29i	− 0.617950i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7131.98	−1.04850	−0.524250	−	0.851564i	$-0.675654\pi$
$359$	−7131.98	−1.04850	−0.524250	+	0.851564i	$0.675654\pi$
$360$	0	0
$361$	−1785.57	−0.260325
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8858.79i	1.27038i
$366$	0	0
$367$	− 2435.03i	− 0.346343i	−0.984892	−	0.173171i	$-0.944599\pi$
$367$	− 2435.03i	− 0.346343i	0.984892	−	0.173171i	$-0.0554014\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−703.719	−0.0984778
$372$	0	0
$373$	−6213.44	−0.862519	−0.431260	−	0.902228i	$-0.641931\pi$
$373$	−6213.44	−0.862519	−0.431260	+	0.902228i	$0.641931\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2993.97i	− 0.409012i
$378$	0	0
$379$	9818.83i	1.33076i	0.746503	+	0.665382i	$0.231731\pi$
$379$	9818.83i	1.33076i	−0.746503	+	0.665382i	$0.768269\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6503.67	−0.867681	−0.433841	−	0.900990i	$-0.642842\pi$
$383$	−6503.67	−0.867681	−0.433841	+	0.900990i	$0.642842\pi$
$384$	0	0
$385$	−3183.26	−0.421387
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 695.295i	− 0.0906242i	−0.998973	−	0.0453121i	$-0.985572\pi$
$389$	− 695.295i	− 0.0906242i	0.998973	−	0.0453121i	$-0.0144282\pi$
$390$	0	0
$391$	− 353.408i	− 0.0457100i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4092.78	−0.521342
$396$	0	0
$397$	−531.823	−0.0672328	−0.0336164	−	0.999435i	$-0.510702\pi$
$397$	−531.823	−0.0672328	−0.0336164	+	0.999435i	$0.510702\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7978.13i	0.993539i	0.867883	+	0.496769i	$0.165480\pi$
$401$	7978.13i	0.993539i	−0.867883	+	0.496769i	$0.834520\pi$
$402$	0	0
$403$	− 2078.23i	− 0.256883i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9150.73	−1.11446
$408$	0	0
$409$	1186.75	0.143475	0.0717375	−	0.997424i	$-0.477146\pi$
$409$	1186.75	0.143475	0.0717375	+	0.997424i	$0.477146\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4522.06i	0.538779i
$414$	0	0
$415$	− 11750.3i	− 1.38987i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2184.10	−0.254654	−0.127327	−	0.991861i	$-0.540640\pi$
$419$	−2184.10	−0.254654	−0.127327	+	0.991861i	$0.540640\pi$
$420$	0	0
$421$	−4437.46	−0.513702	−0.256851	−	0.966451i	$-0.582685\pi$
$421$	−4437.46	−0.513702	−0.256851	+	0.966451i	$0.582685\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	864.850i	0.0987092i
$426$	0	0
$427$	710.046i	0.0804720i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4890.43	−0.546551	−0.273276	−	0.961936i	$-0.588107\pi$
$431$	−4890.43	−0.546551	−0.273276	+	0.961936i	$0.588107\pi$
$432$	0	0
$433$	12968.2	1.43928	0.719642	−	0.694345i	$-0.244306\pi$
$433$	12968.2	1.43928	0.719642	+	0.694345i	$0.244306\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	648.780i	0.0710192i
$438$	0	0
$439$	2019.93i	0.219604i	0.993953	+	0.109802i	$0.0350217\pi$
$439$	2019.93i	0.219604i	−0.993953	+	0.109802i	$0.964978\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6620.34	−0.710027	−0.355014	−	0.934861i	$-0.615524\pi$
$443$	−6620.34	−0.710027	−0.355014	+	0.934861i	$0.615524\pi$
$444$	0	0
$445$	−928.700	−0.0989317
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13372.0i	1.40549i	0.711441	+	0.702746i	$0.248043\pi$
$449$	13372.0i	1.40549i	−0.711441	+	0.702746i	$0.751957\pi$
$450$	0	0
$451$	4484.61i	0.468231i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2036.34	−0.209813
$456$	0	0
$457$	6437.61	0.658947	0.329473	−	0.944165i	$-0.393129\pi$
$457$	6437.61	0.658947	0.329473	+	0.944165i	$0.393129\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12766.0i	− 1.28974i	−0.764291	−	0.644872i	$-0.776911\pi$
$461$	− 12766.0i	− 1.28974i	0.764291	−	0.644872i	$-0.223089\pi$
$462$	0	0
$463$	− 16030.1i	− 1.60903i	−0.593931	−	0.804516i	$-0.702425\pi$
$463$	− 16030.1i	− 1.60903i	0.593931	−	0.804516i	$-0.297575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6062.97	−0.600772	−0.300386	−	0.953818i	$-0.597116\pi$
$467$	−6062.97	−0.600772	−0.300386	+	0.953818i	$0.597116\pi$
$468$	0	0
$469$	3595.71	0.354018
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11665.8i	− 1.13403i
$474$	0	0
$475$	− 1587.68i	− 0.153364i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17909.2	−1.70833	−0.854166	−	0.520001i	$-0.825932\pi$
$479$	−17909.2	−1.70833	−0.854166	+	0.520001i	$0.825932\pi$
$480$	0	0
$481$	−5853.74	−0.554901
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 2489.11i	− 0.233041i
$486$	0	0
$487$	534.130i	0.0496997i	0.999691	+	0.0248498i	$0.00791076\pi$
$487$	534.130i	0.0496997i	−0.999691	+	0.0248498i	$0.992089\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17092.0	−1.57098	−0.785492	−	0.618872i	$-0.787590\pi$
$491$	−17092.0	−1.57098	−0.785492	+	0.618872i	$0.787590\pi$
$492$	0	0
$493$	6213.07	0.567592
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2427.35i	0.219078i
$498$	0	0
$499$	− 5684.58i	− 0.509974i	−0.966945	−	0.254987i	$-0.917929\pi$
$499$	− 5684.58i	− 0.509974i	0.966945	−	0.254987i	$-0.0820711\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14874.6	−1.31854	−0.659271	−	0.751906i	$-0.729135\pi$
$503$	−14874.6	−1.31854	−0.659271	+	0.751906i	$0.729135\pi$
$504$	0	0
$505$	−957.214	−0.0843474
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6291.35i	0.547857i	0.961750	+	0.273928i	$0.0883231\pi$
$509$	6291.35i	0.547857i	−0.961750	+	0.273928i	$0.911677\pi$
$510$	0	0
$511$	− 5202.50i	− 0.450382i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20974.0	1.79461
$516$	0	0
$517$	7706.87	0.655605
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9708.29i	0.816368i	0.912900	+	0.408184i	$0.133838\pi$
$521$	9708.29i	0.816368i	−0.912900	+	0.408184i	$0.866162\pi$
$522$	0	0
$523$	15098.5i	1.26236i	0.775638	+	0.631178i	$0.217428\pi$
$523$	15098.5i	1.26236i	−0.775638	+	0.631178i	$0.782572\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4312.72	0.356480
$528$	0	0
$529$	−12118.3	−0.995998
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2868.81i	0.233137i
$534$	0	0
$535$	− 24595.0i	− 1.98754i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1869.43	0.149392
$540$	0	0
$541$	−24342.5	−1.93450	−0.967251	−	0.253823i	$-0.918312\pi$
$541$	−24342.5	−1.93450	−0.967251	+	0.253823i	$0.918312\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 20872.0i	− 1.64047i
$546$	0	0
$547$	− 162.453i	− 0.0126983i	−0.999980	−	0.00634915i	$-0.997979\pi$
$547$	− 162.453i	− 0.0126983i	0.999980	−	0.00634915i	$-0.00202101\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11405.9	−0.881862
$552$	0	0
$553$	2403.56	0.184828
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22776.8i	− 1.73264i	−0.499487	−	0.866321i	$-0.666478\pi$
$557$	− 22776.8i	− 1.73264i	0.499487	−	0.866321i	$-0.333522\pi$
$558$	0	0
$559$	− 7462.66i	− 0.564645i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23184.9	1.73557	0.867786	−	0.496937i	$-0.165542\pi$
$563$	23184.9	1.73557	0.867786	+	0.496937i	$0.165542\pi$
$564$	0	0
$565$	−1483.04	−0.110428
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19030.0i	1.40207i	0.713125	+	0.701037i	$0.247279\pi$
$569$	19030.0i	1.40207i	−0.713125	+	0.701037i	$0.752721\pi$
$570$	0	0
$571$	− 9761.23i	− 0.715402i	−0.933836	−	0.357701i	$-0.883561\pi$
$571$	− 9761.23i	− 0.715402i	0.933836	−	0.357701i	$-0.116439\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−119.156	−0.00864202
$576$	0	0
$577$	7577.90	0.546746	0.273373	−	0.961908i	$-0.411861\pi$
$577$	7577.90	0.546746	0.273373	+	0.961908i	$0.411861\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6900.57i	0.492743i
$582$	0	0
$583$	− 3835.43i	− 0.272465i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	827.187	0.0581629	0.0290815	−	0.999577i	$-0.490742\pi$
$587$	827.187	0.0581629	0.0290815	+	0.999577i	$0.490742\pi$
$588$	0	0
$589$	−7917.23	−0.553860
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5405.17i	− 0.374306i	−0.982331	−	0.187153i	$-0.940074\pi$
$593$	− 5405.17i	− 0.374306i	0.982331	−	0.187153i	$-0.0599261\pi$
$594$	0	0
$595$	− 4225.80i	− 0.291161i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10448.1	−0.712684	−0.356342	−	0.934356i	$-0.615976\pi$
$599$	−10448.1	−0.712684	−0.356342	+	0.934356i	$0.615976\pi$
$600$	0	0
$601$	27029.6	1.83454	0.917272	−	0.398260i	$-0.130386\pi$
$601$	27029.6	1.83454	0.917272	+	0.398260i	$0.130386\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1484.56i	− 0.0997622i
$606$	0	0
$607$	− 10759.4i	− 0.719460i	−0.933056	−	0.359730i	$-0.882869\pi$
$607$	− 10759.4i	− 0.719460i	0.933056	−	0.359730i	$-0.117131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4930.10	0.326433
$612$	0	0
$613$	4616.83	0.304196	0.152098	−	0.988365i	$-0.451397\pi$
$613$	4616.83	0.304196	0.152098	+	0.988365i	$0.451397\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18550.7i	1.21041i	0.796070	+	0.605204i	$0.206909\pi$
$617$	18550.7i	1.21041i	−0.796070	+	0.605204i	$0.793091\pi$
$618$	0	0
$619$	− 5673.63i	− 0.368405i	−0.982888	−	0.184202i	$-0.941030\pi$
$619$	− 5673.63i	− 0.368405i	0.982888	−	0.184202i	$-0.0589702\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	545.397	0.0350736
$624$	0	0
$625$	−17467.9	−1.11795
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12147.6i	− 0.770045i
$630$	0	0
$631$	20960.0i	1.32235i	0.750230	+	0.661177i	$0.229943\pi$
$631$	20960.0i	1.32235i	−0.750230	+	0.661177i	$0.770057\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14853.2	0.928241
$636$	0	0
$637$	1195.88	0.0743838
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 18418.6i	− 1.13493i	−0.823398	−	0.567465i	$-0.807924\pi$
$641$	− 18418.6i	− 1.13493i	0.823398	−	0.567465i	$-0.192076\pi$
$642$	0	0
$643$	− 9383.51i	− 0.575505i	−0.957705	−	0.287752i	$-0.907092\pi$
$643$	− 9383.51i	− 0.575505i	0.957705	−	0.287752i	$-0.0929080\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21208.4	−1.28870	−0.644350	−	0.764731i	$-0.722872\pi$
$647$	−21208.4	−1.28870	−0.644350	+	0.764731i	$0.722872\pi$
$648$	0	0
$649$	−24646.3	−1.49068
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8873.57i	− 0.531776i	−0.964004	−	0.265888i	$-0.914335\pi$
$653$	− 8873.57i	− 0.531776i	0.964004	−	0.265888i	$-0.0856651\pi$
$654$	0	0
$655$	− 10728.6i	− 0.640001i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21529.7	1.27265	0.636327	−	0.771419i	$-0.280453\pi$
$659$	21529.7	1.27265	0.636327	+	0.771419i	$0.280453\pi$
$660$	0	0
$661$	31145.4	1.83270	0.916352	−	0.400374i	$-0.131120\pi$
$661$	31145.4	1.83270	0.916352	+	0.400374i	$0.131120\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7757.65i	0.452374i
$666$	0	0
$667$	856.017i	0.0496928i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3869.92	−0.222648
$672$	0	0
$673$	12945.1	0.741451	0.370726	−	0.928742i	$-0.379109\pi$
$673$	12945.1	0.741451	0.370726	+	0.928742i	$0.379109\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26590.7i	1.50955i	0.655985	+	0.754774i	$0.272254\pi$
$677$	26590.7i	1.50955i	−0.655985	+	0.754774i	$0.727746\pi$
$678$	0	0
$679$	1461.78i	0.0826185i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7769.98	0.435300	0.217650	−	0.976027i	$-0.430161\pi$
$683$	7769.98	0.435300	0.217650	+	0.976027i	$0.430161\pi$
$684$	0	0
$685$	25090.8	1.39952
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2453.54i	− 0.135664i
$690$	0	0
$691$	17860.1i	0.983257i	0.870805	+	0.491628i	$0.163598\pi$
$691$	17860.1i	0.983257i	−0.870805	+	0.491628i	$0.836402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7209.00	0.393458
$696$	0	0
$697$	−5953.35	−0.323528
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19269.6i	1.03824i	0.854702	+	0.519119i	$0.173740\pi$
$701$	19269.6i	1.03824i	−0.854702	+	0.519119i	$0.826260\pi$
$702$	0	0
$703$	22300.5i	1.19641i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	562.142	0.0299032
$708$	0	0
$709$	−16254.8	−0.861017	−0.430508	−	0.902587i	$-0.641666\pi$
$709$	−16254.8	−0.861017	−0.430508	+	0.902587i	$0.641666\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	594.193i	0.0312100i
$714$	0	0
$715$	− 11098.5i	− 0.580505i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24587.2	−1.27531	−0.637656	−	0.770322i	$-0.720096\pi$
$719$	−24587.2	−1.27531	−0.637656	+	0.770322i	$0.720096\pi$
$720$	0	0
$721$	−12317.4	−0.636232
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2094.82i	− 0.107310i
$726$	0	0
$727$	30281.8i	1.54483i	0.635118	+	0.772415i	$0.280951\pi$
$727$	30281.8i	1.54483i	−0.635118	+	0.772415i	$0.719049\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15486.5	0.783567
$732$	0	0
$733$	−6269.51	−0.315920	−0.157960	−	0.987445i	$-0.550492\pi$
$733$	−6269.51	−0.315920	−0.157960	+	0.987445i	$0.550492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19597.5i	0.979487i
$738$	0	0
$739$	20348.7i	1.01291i	0.862267	+	0.506455i	$0.169044\pi$
$739$	20348.7i	1.01291i	−0.862267	+	0.506455i	$0.830956\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14185.4	−0.700419	−0.350210	−	0.936671i	$-0.613890\pi$
$743$	−14185.4	−0.700419	−0.350210	+	0.936671i	$0.613890\pi$
$744$	0	0
$745$	−3392.69	−0.166844
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14443.9i	0.704630i
$750$	0	0
$751$	7279.09i	0.353685i	0.984239	+	0.176843i	$0.0565884\pi$
$751$	7279.09i	0.353685i	−0.984239	+	0.176843i	$0.943412\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26731.2	1.28854
$756$	0	0
$757$	−31142.3	−1.49522	−0.747612	−	0.664136i	$-0.768800\pi$
$757$	−31142.3	−1.49522	−0.747612	+	0.664136i	$0.768800\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17392.2i	− 0.828473i	−0.910169	−	0.414237i	$-0.864049\pi$
$761$	− 17392.2i	− 0.828473i	0.910169	−	0.414237i	$-0.135951\pi$
$762$	0	0
$763$	12257.5i	0.581586i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15766.3	−0.742226
$768$	0	0
$769$	30069.8	1.41007	0.705034	−	0.709173i	$-0.250932\pi$
$769$	30069.8	1.41007	0.705034	+	0.709173i	$0.250932\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 11469.9i	− 0.533693i	−0.963739	−	0.266847i	$-0.914018\pi$
$773$	− 11469.9i	− 0.533693i	0.963739	−	0.266847i	$-0.0859817\pi$
$774$	0	0
$775$	− 1454.09i	− 0.0673969i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10929.1	0.502663
$780$	0	0
$781$	−13229.7	−0.606139
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1035.40i	− 0.0470766i
$786$	0	0
$787$	21283.7i	0.964020i	0.876166	+	0.482010i	$0.160093\pi$
$787$	21283.7i	0.964020i	−0.876166	+	0.482010i	$0.839907\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	870.942	0.0391493
$792$	0	0
$793$	−2475.60	−0.110859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36139.8i	− 1.60619i	−0.595849	−	0.803096i	$-0.703184\pi$
$797$	− 36139.8i	− 1.60619i	0.595849	−	0.803096i	$-0.296816\pi$
$798$	0	0
$799$	10230.9i	0.452996i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28354.8	1.24610
$804$	0	0
$805$	582.217	0.0254912
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 43514.3i	− 1.89108i	−0.325510	−	0.945539i	$-0.605536\pi$
$809$	− 43514.3i	− 1.89108i	0.325510	−	0.945539i	$-0.394464\pi$
$810$	0	0
$811$	33974.1i	1.47101i	0.677518	+	0.735506i	$0.263056\pi$
$811$	33974.1i	1.47101i	−0.677518	+	0.735506i	$0.736944\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32621.3	1.40206
$816$	0	0
$817$	−28429.8	−1.21742
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36811.8i	1.56485i	0.622746	+	0.782424i	$0.286017\pi$
$821$	36811.8i	1.56485i	−0.622746	+	0.782424i	$0.713983\pi$
$822$	0	0
$823$	− 507.257i	− 0.0214847i	−0.999942	−	0.0107423i	$-0.996581\pi$
$823$	− 507.257i	− 0.0214847i	0.999942	−	0.0107423i	$-0.00341946\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17169.1	0.721922	0.360961	−	0.932581i	$-0.382449\pi$
$827$	17169.1	0.721922	0.360961	+	0.932581i	$0.382449\pi$
$828$	0	0
$829$	25323.6	1.06095	0.530474	−	0.847701i	$-0.322014\pi$
$829$	25323.6	1.06095	0.530474	+	0.847701i	$0.322014\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2481.68i	0.103224i
$834$	0	0
$835$	15754.9i	0.652959i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34081.3	1.40240	0.701201	−	0.712963i	$-0.252647\pi$
$839$	34081.3	1.40240	0.701201	+	0.712963i	$0.252647\pi$
$840$	0	0
$841$	9339.82	0.382952
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19087.5i	0.777079i
$846$	0	0
$847$	871.839i	0.0353681i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1673.66	0.0674177
$852$	0	0
$853$	19929.4	0.799965	0.399983	−	0.916523i	$-0.369016\pi$
$853$	19929.4	0.799965	0.399983	+	0.916523i	$0.369016\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 30514.0i	− 1.21626i	−0.793836	−	0.608132i	$-0.791919\pi$
$857$	− 30514.0i	− 1.21626i	0.793836	−	0.608132i	$-0.208081\pi$
$858$	0	0
$859$	− 27265.6i	− 1.08299i	−0.840703	−	0.541496i	$-0.817858\pi$
$859$	− 27265.6i	− 1.08299i	0.840703	−	0.541496i	$-0.182142\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36505.6	−1.43994	−0.719968	−	0.694007i	$-0.755844\pi$
$863$	−36505.6	−1.43994	−0.719968	+	0.694007i	$0.755844\pi$
$864$	0	0
$865$	25536.0	1.00376
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13100.0i	0.511377i
$870$	0	0
$871$	12536.6i	0.487698i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9004.84	0.347908
$876$	0	0
$877$	−18786.5	−0.723345	−0.361673	−	0.932305i	$-0.617794\pi$
$877$	−18786.5	−0.723345	−0.361673	+	0.932305i	$0.617794\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43324.4i	− 1.65680i	−0.560140	−	0.828398i	$-0.689253\pi$
$881$	− 43324.4i	− 1.65680i	0.560140	−	0.828398i	$-0.310747\pi$
$882$	0	0
$883$	17469.4i	0.665788i	0.942964	+	0.332894i	$0.108025\pi$
$883$	17469.4i	0.665788i	−0.942964	+	0.332894i	$0.891975\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35461.8	1.34238	0.671189	−	0.741286i	$-0.265784\pi$
$887$	35461.8	1.34238	0.671189	+	0.741286i	$0.265784\pi$
$888$	0	0
$889$	−8722.86	−0.329084
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 18781.8i	− 0.703816i
$894$	0	0
$895$	− 46030.7i	− 1.71915i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10446.2	−0.387542
$900$	0	0
$901$	5091.56	0.188263
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49129.2i	1.80454i
$906$	0	0
$907$	32380.1i	1.18541i	0.805421	+	0.592703i	$0.201939\pi$
$907$	32380.1i	1.18541i	−0.805421	+	0.592703i	$0.798061\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37179.4	1.35215	0.676075	−	0.736833i	$-0.263680\pi$
$911$	37179.4	1.35215	0.676075	+	0.736833i	$0.263680\pi$
$912$	0	0
$913$	−37609.7	−1.36331
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6300.57i	0.226896i
$918$	0	0
$919$	40076.1i	1.43851i	0.694748	+	0.719254i	$0.255516\pi$
$919$	40076.1i	1.43851i	−0.694748	+	0.719254i	$0.744484\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8463.04	−0.301803
$924$	0	0
$925$	−4095.75	−0.145586
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36577.8i	1.29180i	0.763423	+	0.645899i	$0.223517\pi$
$929$	36577.8i	1.29180i	−0.763423	+	0.645899i	$0.776483\pi$
$930$	0	0
$931$	− 4555.83i	− 0.160377i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	23031.6	0.805576
$936$	0	0
$937$	32565.3	1.13539	0.567696	−	0.823238i	$-0.307835\pi$
$937$	32565.3	1.13539	0.567696	+	0.823238i	$0.307835\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11866.6i	0.411095i	0.978647	+	0.205547i	$0.0658974\pi$
$941$	11866.6i	0.411095i	−0.978647	+	0.205547i	$0.934103\pi$
$942$	0	0
$943$	− 820.233i	− 0.0283250i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20683.1	0.709725	0.354862	−	0.934919i	$-0.384528\pi$
$947$	20683.1	0.709725	0.354862	+	0.934919i	$0.384528\pi$
$948$	0	0
$949$	18138.7	0.620449
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44835.4i	1.52399i	0.647584	+	0.761994i	$0.275780\pi$
$953$	44835.4i	1.52399i	−0.647584	+	0.761994i	$0.724220\pi$
$954$	0	0
$955$	5220.70i	0.176898i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14735.1	−0.496163
$960$	0	0
$961$	22539.9	0.756601
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 47760.7i	− 1.59323i
$966$	0	0
$967$	− 31756.4i	− 1.05607i	−0.849224	−	0.528033i	$-0.822930\pi$
$967$	− 31756.4i	− 1.05607i	0.849224	−	0.528033i	$-0.177070\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27608.5	0.912459	0.456230	−	0.889862i	$-0.349200\pi$
$971$	27608.5	0.912459	0.456230	+	0.889862i	$0.349200\pi$
$972$	0	0
$973$	−4233.63	−0.139490
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10084.8i	− 0.330238i	−0.986274	−	0.165119i	$-0.947199\pi$
$977$	− 10084.8i	− 0.330238i	0.986274	−	0.165119i	$-0.0528009\pi$
$978$	0	0
$979$	2972.54i	0.0970408i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14097.1	0.457405	0.228703	−	0.973496i	$-0.426552\pi$
$983$	14097.1	0.457405	0.228703	+	0.973496i	$0.426552\pi$
$984$	0	0
$985$	−5714.00	−0.184836
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2133.68i	0.0686015i
$990$	0	0
$991$	14222.4i	0.455893i	0.973674	+	0.227946i	$0.0732011\pi$
$991$	14222.4i	0.455893i	−0.973674	+	0.227946i	$0.926799\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29441.4	0.938046
$996$	0	0
$997$	−111.143	−0.00353052	−0.00176526	−	0.999998i	$-0.500562\pi$
$997$	−111.143	−0.00353052	−0.00176526	+	0.999998i	$0.500562\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.a.575.4	yes	12
3.2	odd	2	inner	1008.4.h.a.575.10	yes	12
4.3	odd	2	inner	1008.4.h.a.575.3	✓	12
12.11	even	2	inner	1008.4.h.a.575.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.a.575.3	✓	12	4.3	odd	2	inner
1008.4.h.a.575.4	yes	12	1.1	even	1	trivial
1008.4.h.a.575.9	yes	12	12.11	even	2	inner
1008.4.h.a.575.10	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.9196i q^{5} +7.00000i q^{7} +O(q^{10})\)	\(q-11.9196i q^{5} +7.00000i q^{7} -38.1517 q^{11} -24.4057 q^{13} -50.6466i q^{17} +92.9762i q^{19} +6.97792 q^{23} -17.0762 q^{25} +122.675i q^{29} +85.1533i q^{31} +83.4370 q^{35} +239.851 q^{37} -117.547i q^{41} +305.775i q^{43} -202.006 q^{47} -49.0000 q^{49} +100.531i q^{53} +454.751i q^{55} +646.008 q^{59} +101.435 q^{61} +290.906i q^{65} -513.673i q^{67} +346.765 q^{71} -743.214 q^{73} -267.062i q^{77} -343.366i q^{79} +985.795 q^{83} -603.685 q^{85} -77.9139i q^{89} -170.840i q^{91} +1108.24 q^{95} +208.826 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) \) 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

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