LMFDB - Embedded newform 288.3.h.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,3,Mod(17,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("288.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	288.h (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:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.33808912384.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 11x^{6} - 18x^{5} + 47x^{4} - 28x^{3} - 44x^{2} + 48x + 36 $ x^8 - 2x^7 + 11x^6 - 18x^5 + 47x^4 - 28x^3 - 44x^2 + 48*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$1.15139 - 0.593052i$ of defining polynomial
Character	$\chi$	$=$	288.17
Dual form			288.3.h.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.67101 q^{5} +7.21110 q^{7} +O(q^{10})$	$q-7.67101 q^{5} +7.21110 q^{7} +6.05164 q^{11} +2.29014i q^{13} +21.8103i q^{17} +34.8355i q^{19} +21.5117i q^{23} +33.8444 q^{25} -10.9098 q^{29} +37.6333 q^{31} -55.3164 q^{35} +34.8355i q^{37} -13.3250i q^{41} -60.5104i q^{43} -3.34701i q^{47} +3.00000 q^{49} +35.1163 q^{53} -46.4222 q^{55} +37.1615 q^{59} +25.6749i q^{61} -17.5677i q^{65} +25.6749i q^{67} +37.2881i q^{71} -77.6888 q^{73} +43.6390 q^{77} -31.3221 q^{79} +55.3164 q^{83} -167.307i q^{85} +5.43682i q^{89} +16.5144i q^{91} -267.223i q^{95} -52.8444 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} + 128 q^{31} + 24 q^{49} - 256 q^{55} - 160 q^{73} + 384 q^{79} - 192 q^{97}+O(q^{100}) $ 8 * q + 40 * q^25 + 128 * q^31 + 24 * q^49 - 256 * q^55 - 160 * q^73 + 384 * q^79 - 192 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−7.67101	−1.53420	−0.767101	−	0.641526i	$-0.778302\pi$
$5$	−7.67101	−1.53420	−0.767101	+	0.641526i	$0.778302\pi$
$6$	0	0
$7$	7.21110	1.03016	0.515079	−	0.857143i	$-0.327763\pi$
$7$	7.21110	1.03016	0.515079	+	0.857143i	$0.327763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.05164	0.550149	0.275075	−	0.961423i	$-0.411297\pi$
$11$	6.05164	0.550149	0.275075	+	0.961423i	$0.411297\pi$
$12$	0	0
$13$	2.29014i	0.176164i	0.996113	+	0.0880821i	$0.0280738\pi$
$13$	2.29014i	0.176164i	−0.996113	+	0.0880821i	$0.971926\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	21.8103i	1.28296i	0.767140	+	0.641479i	$0.221679\pi$
$17$	21.8103i	1.28296i	−0.767140	+	0.641479i	$0.778321\pi$
$18$	0	0
$19$	34.8355i	1.83345i	0.399523	+	0.916723i	$0.369176\pi$
$19$	34.8355i	1.83345i	−0.399523	+	0.916723i	$0.630824\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	21.5117i	0.935293i	0.883915	+	0.467647i	$0.154898\pi$
$23$	21.5117i	0.935293i	−0.883915	+	0.467647i	$0.845102\pi$
$24$	0	0
$25$	33.8444	1.35378
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.9098	−0.376198	−0.188099	−	0.982150i	$-0.560233\pi$
$29$	−10.9098	−0.376198	−0.188099	+	0.982150i	$0.560233\pi$
$30$	0	0
$31$	37.6333	1.21398	0.606989	−	0.794710i	$-0.292377\pi$
$31$	37.6333	1.21398	0.606989	+	0.794710i	$0.292377\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−55.3164	−1.58047
$36$	0	0
$37$	34.8355i	0.941499i	0.882267	+	0.470750i	$0.156017\pi$
$37$	34.8355i	0.941499i	−0.882267	+	0.470750i	$0.843983\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 13.3250i	− 0.325000i	−0.986709	−	0.162500i	$-0.948044\pi$
$41$	− 13.3250i	− 0.325000i	0.986709	−	0.162500i	$-0.0519558\pi$
$42$	0	0
$43$	− 60.5104i	− 1.40722i	−0.710587	−	0.703609i	$-0.751570\pi$
$43$	− 60.5104i	− 1.40722i	0.710587	−	0.703609i	$-0.248430\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.34701i	− 0.0712129i	−0.999366	−	0.0356065i	$-0.988664\pi$
$47$	− 3.34701i	− 0.0712129i	0.999366	−	0.0356065i	$-0.0113363\pi$
$48$	0	0
$49$	3.00000	0.0612245
$50$	0	0
$51$	0	0
$52$	0	0
$53$	35.1163	0.662572	0.331286	−	0.943530i	$-0.392518\pi$
$53$	35.1163	0.662572	0.331286	+	0.943530i	$0.392518\pi$
$54$	0	0
$55$	−46.4222	−0.844040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	37.1615	0.629856	0.314928	−	0.949116i	$-0.398020\pi$
$59$	37.1615	0.629856	0.314928	+	0.949116i	$0.398020\pi$
$60$	0	0
$61$	25.6749i	0.420901i	0.977605	+	0.210450i	$0.0674930\pi$
$61$	25.6749i	0.420901i	−0.977605	+	0.210450i	$0.932507\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 17.5677i	− 0.270272i
$66$	0	0
$67$	25.6749i	0.383208i	0.981472	+	0.191604i	$0.0613689\pi$
$67$	25.6749i	0.383208i	−0.981472	+	0.191604i	$0.938631\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	37.2881i	0.525185i	0.964907	+	0.262592i	$0.0845775\pi$
$71$	37.2881i	0.525185i	−0.964907	+	0.262592i	$0.915423\pi$
$72$	0	0
$73$	−77.6888	−1.06423	−0.532115	−	0.846672i	$-0.678603\pi$
$73$	−77.6888	−1.06423	−0.532115	+	0.846672i	$0.678603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	43.6390	0.566740
$78$	0	0
$79$	−31.3221	−0.396483	−0.198241	−	0.980153i	$-0.563523\pi$
$79$	−31.3221	−0.396483	−0.198241	+	0.980153i	$0.563523\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	55.3164	0.666463	0.333232	−	0.942845i	$-0.391861\pi$
$83$	55.3164	0.666463	0.333232	+	0.942845i	$0.391861\pi$
$84$	0	0
$85$	− 167.307i	− 1.96832i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.43682i	0.0610878i	0.999533	+	0.0305439i	$0.00972394\pi$
$89$	5.43682i	0.0610878i	−0.999533	+	0.0305439i	$0.990276\pi$
$90$	0	0
$91$	16.5144i	0.181477i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 267.223i	− 2.81288i
$96$	0	0
$97$	−52.8444	−0.544788	−0.272394	−	0.962186i	$-0.587815\pi$
$97$	−52.8444	−0.544788	−0.272394	+	0.962186i	$0.587815\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−90.0069	−0.891158	−0.445579	−	0.895243i	$-0.647002\pi$
$101$	−90.0069	−0.891158	−0.445579	+	0.895243i	$0.647002\pi$
$102$	0	0
$103$	−119.211	−1.15739	−0.578695	−	0.815544i	$-0.696438\pi$
$103$	−119.211	−1.15739	−0.578695	+	0.815544i	$0.696438\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	24.2066	0.226230	0.113115	−	0.993582i	$-0.463917\pi$
$107$	24.2066	0.226230	0.113115	+	0.993582i	$0.463917\pi$
$108$	0	0
$109$	− 132.472i	− 1.21533i	−0.794192	−	0.607667i	$-0.792105\pi$
$109$	− 132.472i	− 1.21533i	0.794192	−	0.607667i	$-0.207895\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 160.560i	− 1.42089i	−0.703754	−	0.710444i	$-0.748494\pi$
$113$	− 160.560i	− 1.42089i	0.703754	−	0.710444i	$-0.251506\pi$
$114$	0	0
$115$	− 165.017i	− 1.43493i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	157.276i	1.32165i
$120$	0	0
$121$	−84.3776	−0.697336
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−67.8456	−0.542765
$126$	0	0
$127$	146.478	1.15337	0.576684	−	0.816967i	$-0.304346\pi$
$127$	146.478	1.15337	0.576684	+	0.816967i	$0.304346\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	184.956	1.41188	0.705939	−	0.708273i	$-0.250525\pi$
$131$	184.956	1.41188	0.705939	+	0.708273i	$0.250525\pi$
$132$	0	0
$133$	251.202i	1.88874i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	133.313i	0.973089i	0.873656	+	0.486544i	$0.161743\pi$
$137$	133.313i	0.973089i	−0.873656	+	0.486544i	$0.838257\pi$
$138$	0	0
$139$	165.017i	1.18717i	0.804771	+	0.593586i	$0.202288\pi$
$139$	165.017i	1.18717i	−0.804771	+	0.593586i	$0.797712\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8591i	0.0969166i
$144$	0	0
$145$	83.6888	0.577164
$146$	0	0
$147$	0	0
$148$	0	0
$149$	228.937	1.53649	0.768244	−	0.640157i	$-0.221131\pi$
$149$	228.937	1.53649	0.768244	+	0.640157i	$0.221131\pi$
$150$	0	0
$151$	−162.478	−1.07601	−0.538006	−	0.842941i	$-0.680822\pi$
$151$	−162.478	−1.07601	−0.538006	+	0.842941i	$0.680822\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−288.686	−1.86249
$156$	0	0
$157$	113.667i	0.723993i	0.932179	+	0.361997i	$0.117905\pi$
$157$	113.667i	0.723993i	−0.932179	+	0.361997i	$0.882095\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	155.123i	0.963499i
$162$	0	0
$163$	− 111.860i	− 0.686259i	−0.939288	−	0.343130i	$-0.888513\pi$
$163$	− 111.860i	− 0.686259i	0.939288	−	0.343130i	$-0.111487\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	151.541i	0.907430i	0.891147	+	0.453715i	$0.149902\pi$
$167$	151.541i	0.907430i	−0.891147	+	0.453715i	$0.850098\pi$
$168$	0	0
$169$	163.755	0.968966
$170$	0	0
$171$	0	0
$172$	0	0
$173$	98.0198	0.566588	0.283294	−	0.959033i	$-0.408573\pi$
$173$	98.0198	0.566588	0.283294	+	0.959033i	$0.408573\pi$
$174$	0	0
$175$	244.056	1.39460
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−222.117	−1.24088	−0.620440	−	0.784254i	$-0.713046\pi$
$179$	−222.117	−1.24088	−0.620440	+	0.784254i	$0.713046\pi$
$180$	0	0
$181$	− 118.731i	− 0.655971i	−0.944683	−	0.327985i	$-0.893630\pi$
$181$	− 118.731i	− 0.655971i	0.944683	−	0.327985i	$-0.106370\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 267.223i	− 1.44445i
$186$	0	0
$187$	131.988i	0.705818i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 151.541i	− 0.793408i	−0.917947	−	0.396704i	$-0.870154\pi$
$191$	− 151.541i	− 0.793408i	0.917947	−	0.396704i	$-0.129846\pi$
$192$	0	0
$193$	113.378	0.587449	0.293724	−	0.955890i	$-0.405105\pi$
$193$	113.378	0.587449	0.293724	+	0.955890i	$0.405105\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−319.454	−1.62159	−0.810796	−	0.585329i	$-0.800965\pi$
$197$	−319.454	−1.62159	−0.810796	+	0.585329i	$0.800965\pi$
$198$	0	0
$199$	41.0109	0.206085	0.103043	−	0.994677i	$-0.467142\pi$
$199$	41.0109	0.206085	0.103043	+	0.994677i	$0.467142\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−78.6713	−0.387544
$204$	0	0
$205$	102.216i	0.498616i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	210.812i	1.00867i
$210$	0	0
$211$	95.3459i	0.451876i	0.974142	+	0.225938i	$0.0725447\pi$
$211$	95.3459i	0.451876i	−0.974142	+	0.225938i	$0.927455\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	464.176i	2.15896i
$216$	0	0
$217$	271.378	1.25059
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−49.9485	−0.226011
$222$	0	0
$223$	268.167	1.20254	0.601270	−	0.799046i	$-0.294662\pi$
$223$	268.167	1.20254	0.601270	+	0.799046i	$0.294662\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	65.7164	0.289499	0.144750	−	0.989468i	$-0.453762\pi$
$227$	65.7164	0.289499	0.144750	+	0.989468i	$0.453762\pi$
$228$	0	0
$229$	155.373i	0.678484i	0.940699	+	0.339242i	$0.110171\pi$
$229$	155.373i	0.678484i	−0.940699	+	0.339242i	$0.889829\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	285.451i	1.22511i	0.790427	+	0.612556i	$0.209859\pi$
$233$	285.451i	1.22511i	−0.790427	+	0.612556i	$0.790141\pi$
$234$	0	0
$235$	25.6749i	0.109255i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 421.153i	− 1.76214i	−0.472982	−	0.881072i	$-0.656822\pi$
$239$	− 421.153i	− 1.76214i	0.472982	−	0.881072i	$-0.343178\pi$
$240$	0	0
$241$	−191.600	−0.795019	−0.397510	−	0.917598i	$-0.630126\pi$
$241$	−191.600	−0.795019	−0.397510	+	0.917598i	$0.630126\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−23.0130	−0.0939307
$246$	0	0
$247$	−79.7779	−0.322988
$248$	0	0
$249$	0	0
$250$	0	0
$251$	483.190	1.92506	0.962529	−	0.271178i	$-0.0874131\pi$
$251$	483.190	1.92506	0.962529	+	0.271178i	$0.0874131\pi$
$252$	0	0
$253$	130.181i	0.514551i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2132i	0.0825416i	0.999148	+	0.0412708i	$0.0131406\pi$
$257$	21.2132i	0.0825416i	−0.999148	+	0.0412708i	$0.986859\pi$
$258$	0	0
$259$	251.202i	0.969893i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	246.670i	0.937910i	0.883222	+	0.468955i	$0.155369\pi$
$263$	246.670i	0.937910i	−0.883222	+	0.468955i	$0.844631\pi$
$264$	0	0
$265$	−269.378	−1.01652
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−241.040	−0.896060	−0.448030	−	0.894019i	$-0.647874\pi$
$269$	−241.040	−0.896060	−0.448030	+	0.894019i	$0.647874\pi$
$270$	0	0
$271$	−61.7443	−0.227839	−0.113919	−	0.993490i	$-0.536341\pi$
$271$	−61.7443	−0.227839	−0.113919	+	0.993490i	$0.536341\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	204.814	0.744779
$276$	0	0
$277$	− 379.093i	− 1.36857i	−0.729215	−	0.684284i	$-0.760115\pi$
$277$	− 379.093i	− 1.36857i	0.729215	−	0.684284i	$-0.239885\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 301.227i	− 1.07198i	−0.844223	−	0.535992i	$-0.819938\pi$
$281$	− 301.227i	− 1.07198i	0.844223	−	0.535992i	$-0.180062\pi$
$282$	0	0
$283$	− 399.705i	− 1.41238i	−0.708020	−	0.706192i	$-0.750412\pi$
$283$	− 399.705i	− 1.41238i	0.708020	−	0.706192i	$-0.249588\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 96.0880i	− 0.334801i
$288$	0	0
$289$	−186.689	−0.645982
$290$	0	0
$291$	0	0
$292$	0	0
$293$	258.085	0.880838	0.440419	−	0.897792i	$-0.354830\pi$
$293$	258.085	0.880838	0.440419	+	0.897792i	$0.354830\pi$
$294$	0	0
$295$	−285.066	−0.966327
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−49.2648	−0.164765
$300$	0	0
$301$	− 436.347i	− 1.44966i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 196.953i	− 0.645747i
$306$	0	0
$307$	286.038i	0.931719i	0.884859	+	0.465859i	$0.154255\pi$
$307$	286.038i	0.931719i	−0.884859	+	0.465859i	$0.845745\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 464.647i	− 1.49404i	−0.664800	−	0.747021i	$-0.731483\pi$
$311$	− 464.647i	− 1.49404i	0.664800	−	0.747021i	$-0.268517\pi$
$312$	0	0
$313$	158.000	0.504792	0.252396	−	0.967624i	$-0.418781\pi$
$313$	158.000	0.504792	0.252396	+	0.967624i	$0.418781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.6388	0.0903433	0.0451717	−	0.998979i	$-0.485617\pi$
$317$	28.6388	0.0903433	0.0451717	+	0.998979i	$0.485617\pi$
$318$	0	0
$319$	−66.0219	−0.206965
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−759.772	−2.35224
$324$	0	0
$325$	77.5083i	0.238487i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 24.1356i	− 0.0733605i
$330$	0	0
$331$	− 428.993i	− 1.29605i	−0.761619	−	0.648026i	$-0.775595\pi$
$331$	− 428.993i	− 1.29605i	0.761619	−	0.648026i	$-0.224405\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 196.953i	− 0.587919i
$336$	0	0
$337$	290.755	0.862775	0.431388	−	0.902167i	$-0.358024\pi$
$337$	290.755	0.862775	0.431388	+	0.902167i	$0.358024\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	227.743	0.667869
$342$	0	0
$343$	−331.711	−0.967087
$344$	0	0
$345$	0	0
$346$	0	0
$347$	179.756	0.518029	0.259014	−	0.965873i	$-0.416602\pi$
$347$	179.756	0.518029	0.259014	+	0.965873i	$0.416602\pi$
$348$	0	0
$349$	− 225.527i	− 0.646210i	−0.946363	−	0.323105i	$-0.895273\pi$
$349$	− 225.527i	− 0.646210i	0.946363	−	0.323105i	$-0.104727\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 133.784i	− 0.378992i	−0.981881	−	0.189496i	$-0.939315\pi$
$353$	− 133.784i	− 0.378992i	0.981881	−	0.189496i	$-0.0606854\pi$
$354$	0	0
$355$	− 286.038i	− 0.805740i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	239.018i	0.665787i	0.942964	+	0.332894i	$0.108025\pi$
$359$	239.018i	0.665787i	−0.942964	+	0.332894i	$0.891975\pi$
$360$	0	0
$361$	−852.511	−2.36153
$362$	0	0
$363$	0	0
$364$	0	0
$365$	595.952	1.63274
$366$	0	0
$367$	188.389	0.513320	0.256660	−	0.966502i	$-0.417378\pi$
$367$	188.389	0.513320	0.256660	+	0.966502i	$0.417378\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	253.227	0.682554
$372$	0	0
$373$	561.108i	1.50431i	0.658985	+	0.752156i	$0.270986\pi$
$373$	561.108i	1.50431i	−0.658985	+	0.752156i	$0.729014\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.9848i	− 0.0662727i
$378$	0	0
$379$	− 328.227i	− 0.866034i	−0.901386	−	0.433017i	$-0.857449\pi$
$379$	− 328.227i	− 0.866034i	0.901386	−	0.433017i	$-0.142551\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	204.118i	0.532945i	0.963843	+	0.266472i	$0.0858581\pi$
$383$	204.118i	0.532945i	−0.963843	+	0.266472i	$0.914142\pi$
$384$	0	0
$385$	−334.755	−0.869494
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−60.1746	−0.154690	−0.0773452	−	0.997004i	$-0.524644\pi$
$389$	−60.1746	−0.154690	−0.0773452	+	0.997004i	$0.524644\pi$
$390$	0	0
$391$	−469.177	−1.19994
$392$	0	0
$393$	0	0
$394$	0	0
$395$	240.272	0.608285
$396$	0	0
$397$	592.203i	1.49170i	0.666116	+	0.745848i	$0.267955\pi$
$397$	592.203i	1.49170i	−0.666116	+	0.745848i	$0.732045\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 653.178i	− 1.62887i	−0.580254	−	0.814436i	$-0.697047\pi$
$401$	− 653.178i	− 1.62887i	0.580254	−	0.814436i	$-0.302953\pi$
$402$	0	0
$403$	86.1854i	0.213859i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	210.812i	0.517965i
$408$	0	0
$409$	355.156	0.868351	0.434176	−	0.900828i	$-0.357040\pi$
$409$	355.156	0.868351	0.434176	+	0.900828i	$0.357040\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	267.976	0.648851
$414$	0	0
$415$	−424.333	−1.02249
$416$	0	0
$417$	0	0
$418$	0	0
$419$	299.085	0.713808	0.356904	−	0.934141i	$-0.383832\pi$
$419$	299.085	0.713808	0.356904	+	0.934141i	$0.383832\pi$
$420$	0	0
$421$	− 769.638i	− 1.82812i	−0.405582	−	0.914059i	$-0.632931\pi$
$421$	− 769.638i	− 1.82812i	0.405582	−	0.914059i	$-0.367069\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	738.156i	1.73684i
$426$	0	0
$427$	185.145i	0.433594i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 411.583i	− 0.954948i	−0.878646	−	0.477474i	$-0.841552\pi$
$431$	− 411.583i	− 0.954948i	0.878646	−	0.477474i	$-0.158448\pi$
$432$	0	0
$433$	516.133	1.19199	0.595996	−	0.802987i	$-0.296757\pi$
$433$	516.133	1.19199	0.595996	+	0.802987i	$0.296757\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−749.372	−1.71481
$438$	0	0
$439$	865.788	1.97218	0.986091	−	0.166205i	$-0.0531513\pi$
$439$	865.788	1.97218	0.986091	+	0.166205i	$0.0531513\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−546.261	−1.23310	−0.616548	−	0.787318i	$-0.711469\pi$
$443$	−546.261	−1.23310	−0.616548	+	0.787318i	$0.711469\pi$
$444$	0	0
$445$	− 41.7059i	− 0.0937211i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	428.978i	0.955407i	0.878521	+	0.477704i	$0.158531\pi$
$449$	428.978i	0.955407i	−0.878521	+	0.477704i	$0.841469\pi$
$450$	0	0
$451$	− 80.6382i	− 0.178799i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 126.682i	− 0.278422i
$456$	0	0
$457$	75.5997	0.165426	0.0827130	−	0.996573i	$-0.473642\pi$
$457$	75.5997	0.165426	0.0827130	+	0.996573i	$0.473642\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−771.365	−1.67324	−0.836622	−	0.547781i	$-0.815473\pi$
$461$	−771.365	−1.67324	−0.836622	+	0.547781i	$0.815473\pi$
$462$	0	0
$463$	468.278	1.01140	0.505699	−	0.862710i	$-0.331235\pi$
$463$	468.278	1.01140	0.505699	+	0.862710i	$0.331235\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−452.842	−0.969682	−0.484841	−	0.874602i	$-0.661123\pi$
$467$	−452.842	−0.969682	−0.484841	+	0.874602i	$0.661123\pi$
$468$	0	0
$469$	185.145i	0.394765i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 366.187i	− 0.774180i
$474$	0	0
$475$	1178.99i	2.48208i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	467.523i	0.976040i	0.872832	+	0.488020i	$0.162281\pi$
$479$	467.523i	0.976040i	−0.872832	+	0.488020i	$0.837719\pi$
$480$	0	0
$481$	−79.7779	−0.165859
$482$	0	0
$483$	0	0
$484$	0	0
$485$	405.370	0.835815
$486$	0	0
$487$	863.766	1.77365	0.886824	−	0.462108i	$-0.152907\pi$
$487$	863.766	1.77365	0.886824	+	0.462108i	$0.152907\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−149.498	−0.304476	−0.152238	−	0.988344i	$-0.548648\pi$
$491$	−149.498	−0.304476	−0.152238	+	0.988344i	$0.548648\pi$
$492$	0	0
$493$	− 237.945i	− 0.482647i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	268.889i	0.541023i
$498$	0	0
$499$	− 205.400i	− 0.411622i	−0.978592	−	0.205811i	$-0.934017\pi$
$499$	− 205.400i	− 0.411622i	0.978592	−	0.205811i	$-0.0659833\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 349.923i	− 0.695673i	−0.937555	−	0.347836i	$-0.886916\pi$
$503$	− 349.923i	− 0.695673i	0.937555	−	0.347836i	$-0.113084\pi$
$504$	0	0
$505$	690.444	1.36722
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−529.468	−1.04021	−0.520106	−	0.854102i	$-0.674108\pi$
$509$	−529.468	−1.04021	−0.520106	+	0.854102i	$0.674108\pi$
$510$	0	0
$511$	−560.222	−1.09632
$512$	0	0
$513$	0	0
$514$	0	0
$515$	914.470	1.77567
$516$	0	0
$517$	− 20.2549i	− 0.0391777i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	367.318i	0.705026i	0.935807	+	0.352513i	$0.114673\pi$
$521$	367.318i	0.705026i	−0.935807	+	0.352513i	$0.885327\pi$
$522$	0	0
$523$	577.495i	1.10420i	0.833779	+	0.552099i	$0.186173\pi$
$523$	577.495i	1.10420i	−0.833779	+	0.552099i	$0.813827\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	820.793i	1.55748i
$528$	0	0
$529$	66.2447	0.125226
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.5161	0.0572534
$534$	0	0
$535$	−185.689	−0.347082
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.1549	0.0336826
$540$	0	0
$541$	− 326.777i	− 0.604023i	−0.953304	−	0.302012i	$-0.902342\pi$
$541$	− 326.777i	− 0.604023i	0.953304	−	0.302012i	$-0.0976582\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1016.19i	1.86457i
$546$	0	0
$547$	273.137i	0.499336i	0.968332	+	0.249668i	$0.0803214\pi$
$547$	273.137i	0.499336i	−0.968332	+	0.249668i	$0.919679\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 380.046i	− 0.689739i
$552$	0	0
$553$	−225.867	−0.408440
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−93.2457	−0.167407	−0.0837035	−	0.996491i	$-0.526675\pi$
$557$	−93.2457	−0.167407	−0.0837035	+	0.996491i	$0.526675\pi$
$558$	0	0
$559$	138.577	0.247902
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.3615	0.0752424	0.0376212	−	0.999292i	$-0.488022\pi$
$563$	42.3615	0.0752424	0.0376212	+	0.999292i	$0.488022\pi$
$564$	0	0
$565$	1231.66i	2.17993i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1055.57i	− 1.85513i	−0.373663	−	0.927564i	$-0.621898\pi$
$569$	− 1055.57i	− 1.85513i	0.373663	−	0.927564i	$-0.378102\pi$
$570$	0	0
$571$	− 21.9344i	− 0.0384141i	−0.999816	−	0.0192070i	$-0.993886\pi$
$571$	− 21.9344i	− 0.0384141i	0.999816	−	0.0192070i	$-0.00611417\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	728.052i	1.26618i
$576$	0	0
$577$	161.378	0.279684	0.139842	−	0.990174i	$-0.455341\pi$
$577$	161.378	0.279684	0.139842	+	0.990174i	$0.455341\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	398.893	0.686562
$582$	0	0
$583$	212.511	0.364513
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−481.396	−0.820096	−0.410048	−	0.912064i	$-0.634488\pi$
$587$	−481.396	−0.820096	−0.410048	+	0.912064i	$0.634488\pi$
$588$	0	0
$589$	1310.97i	2.22576i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 321.781i	− 0.542632i	−0.962490	−	0.271316i	$-0.912541\pi$
$593$	− 321.781i	− 0.542632i	0.962490	−	0.271316i	$-0.0874588\pi$
$594$	0	0
$595$	− 1206.47i	− 2.02768i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.8419i	0.0598362i	0.999552	+	0.0299181i	$0.00952464\pi$
$599$	35.8419i	0.0598362i	−0.999552	+	0.0299181i	$0.990475\pi$
$600$	0	0
$601$	147.867	0.246035	0.123018	−	0.992404i	$-0.460743\pi$
$601$	147.867	0.246035	0.123018	+	0.992404i	$0.460743\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	647.262	1.06985
$606$	0	0
$607$	636.611	1.04878	0.524391	−	0.851478i	$-0.324293\pi$
$607$	636.611	1.04878	0.524391	+	0.851478i	$0.324293\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.66510	0.0125452
$612$	0	0
$613$	− 870.887i	− 1.42070i	−0.703850	−	0.710348i	$-0.748537\pi$
$613$	− 870.887i	− 1.42070i	0.703850	−	0.710348i	$-0.251463\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1126.69i	1.82607i	0.407876	+	0.913037i	$0.366270\pi$
$617$	1126.69i	1.82607i	−0.407876	+	0.913037i	$0.633730\pi$
$618$	0	0
$619$	− 311.713i	− 0.503575i	−0.967783	−	0.251787i	$-0.918982\pi$
$619$	− 311.713i	− 0.503575i	0.967783	−	0.251787i	$-0.0810183\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	39.2054i	0.0629301i
$624$	0	0
$625$	−325.666	−0.521066
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−759.772	−1.20790
$630$	0	0
$631$	203.278	0.322153	0.161076	−	0.986942i	$-0.448503\pi$
$631$	203.278	0.322153	0.161076	+	0.986942i	$0.448503\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1123.63	−1.76950
$636$	0	0
$637$	6.87041i	0.0107856i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 586.490i	− 0.914960i	−0.889220	−	0.457480i	$-0.848752\pi$
$641$	− 586.490i	− 0.914960i	0.889220	−	0.457480i	$-0.151248\pi$
$642$	0	0
$643$	− 9.16054i	− 0.0142466i	−0.999975	−	0.00712328i	$-0.997733\pi$
$643$	− 9.16054i	− 0.0142466i	0.999975	−	0.00712328i	$-0.00226743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 859.040i	− 1.32773i	−0.747853	−	0.663864i	$-0.768915\pi$
$647$	− 859.040i	− 1.32773i	0.747853	−	0.663864i	$-0.231085\pi$
$648$	0	0
$649$	224.888	0.346515
$650$	0	0
$651$	0	0
$652$	0	0
$653$	131.775	0.201799	0.100899	−	0.994897i	$-0.467828\pi$
$653$	131.775	0.201799	0.100899	+	0.994897i	$0.467828\pi$
$654$	0	0
$655$	−1418.80	−2.16611
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−305.137	−0.463030	−0.231515	−	0.972831i	$-0.574368\pi$
$659$	−305.137	−0.463030	−0.231515	+	0.972831i	$0.574368\pi$
$660$	0	0
$661$	− 623.298i	− 0.942962i	−0.881876	−	0.471481i	$-0.843720\pi$
$661$	− 623.298i	− 0.942962i	0.881876	−	0.471481i	$-0.156280\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1926.97i	− 2.89771i
$666$	0	0
$667$	− 234.688i	− 0.351856i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	155.376i	0.231558i
$672$	0	0
$673$	−263.867	−0.392076	−0.196038	−	0.980596i	$-0.562808\pi$
$673$	−263.867	−0.392076	−0.196038	+	0.980596i	$0.562808\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	788.411	1.16457	0.582283	−	0.812986i	$-0.302160\pi$
$677$	788.411	1.16457	0.582283	+	0.812986i	$0.302160\pi$
$678$	0	0
$679$	−381.066	−0.561217
$680$	0	0
$681$	0	0
$682$	0	0
$683$	905.773	1.32617	0.663084	−	0.748545i	$-0.269247\pi$
$683$	905.773	1.32617	0.663084	+	0.748545i	$0.269247\pi$
$684$	0	0
$685$	− 1022.65i	− 1.49291i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	80.4211i	0.116721i
$690$	0	0
$691$	− 45.8027i	− 0.0662847i	−0.999451	−	0.0331423i	$-0.989449\pi$
$691$	− 45.8027i	− 0.0662847i	0.999451	−	0.0331423i	$-0.0105515\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1265.85i	− 1.82136i
$696$	0	0
$697$	290.622	0.416962
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1019.73	1.45468	0.727339	−	0.686279i	$-0.240757\pi$
$701$	1019.73	1.45468	0.727339	+	0.686279i	$0.240757\pi$
$702$	0	0
$703$	−1213.51	−1.72619
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−649.049	−0.918033
$708$	0	0
$709$	− 84.7350i	− 0.119513i	−0.998213	−	0.0597567i	$-0.980968\pi$
$709$	− 84.7350i	− 0.119513i	0.998213	−	0.0597567i	$-0.0190325\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	809.558i	1.13543i
$714$	0	0
$715$	− 106.313i	− 0.148690i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	646.311i	0.898903i	0.893305	+	0.449451i	$0.148381\pi$
$719$	646.311i	0.898903i	−0.893305	+	0.449451i	$0.851619\pi$
$720$	0	0
$721$	−859.643	−1.19229
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−369.234	−0.509288
$726$	0	0
$727$	648.789	0.892419	0.446210	−	0.894928i	$-0.352774\pi$
$727$	648.789	0.892419	0.446210	+	0.894928i	$0.352774\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1319.75	1.80540
$732$	0	0
$733$	− 696.353i	− 0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$733$	− 696.353i	− 0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	155.376i	0.210822i
$738$	0	0
$739$	740.833i	1.00248i	0.865308	+	0.501240i	$0.167123\pi$
$739$	740.833i	1.00248i	−0.865308	+	0.501240i	$0.832877\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	111.377i	0.149901i	0.997187	+	0.0749507i	$0.0238800\pi$
$743$	111.377i	0.149901i	−0.997187	+	0.0749507i	$0.976120\pi$
$744$	0	0
$745$	−1756.18	−2.35728
$746$	0	0
$747$	0	0
$748$	0	0
$749$	174.556	0.233052
$750$	0	0
$751$	−923.699	−1.22996	−0.614979	−	0.788543i	$-0.710836\pi$
$751$	−923.699	−1.22996	−0.614979	+	0.788543i	$0.710836\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1246.37	1.65082
$756$	0	0
$757$	1233.47i	1.62941i	0.579873	+	0.814707i	$0.303102\pi$
$757$	1233.47i	1.62941i	−0.579873	+	0.814707i	$0.696898\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	179.070i	0.235309i	0.993055	+	0.117654i	$0.0375375\pi$
$761$	179.070i	0.235309i	−0.993055	+	0.117654i	$0.962463\pi$
$762$	0	0
$763$	− 955.266i	− 1.25199i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	85.1049i	0.110958i
$768$	0	0
$769$	−251.511	−0.327062	−0.163531	−	0.986538i	$-0.552288\pi$
$769$	−251.511	−0.327062	−0.163531	+	0.986538i	$0.552288\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−184.110	−0.238176	−0.119088	−	0.992884i	$-0.537997\pi$
$773$	−184.110	−0.238176	−0.119088	+	0.992884i	$0.537997\pi$
$774$	0	0
$775$	1273.68	1.64345
$776$	0	0
$777$	0	0
$778$	0	0
$779$	464.183	0.595870
$780$	0	0
$781$	225.654i	0.288930i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 871.941i	− 1.11075i
$786$	0	0
$787$	1351.36i	1.71710i	0.512731	+	0.858550i	$0.328634\pi$
$787$	1351.36i	1.71710i	−0.512731	+	0.858550i	$0.671366\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1157.82i	− 1.46374i
$792$	0	0
$793$	−58.7991	−0.0741476
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1407.20	−1.76562	−0.882812	−	0.469727i	$-0.844352\pi$
$797$	−1407.20	−1.76562	−0.882812	+	0.469727i	$0.844352\pi$
$798$	0	0
$799$	72.9992	0.0913632
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−470.145	−0.585485
$804$	0	0
$805$	− 1189.95i	− 1.47820i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	741.991i	0.917171i	0.888650	+	0.458585i	$0.151644\pi$
$809$	741.991i	0.917171i	−0.888650	+	0.458585i	$0.848356\pi$
$810$	0	0
$811$	− 716.837i	− 0.883893i	−0.897041	−	0.441947i	$-0.854288\pi$
$811$	− 716.837i	− 0.883893i	0.897041	−	0.441947i	$-0.145712\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	858.082i	1.05286i
$816$	0	0
$817$	2107.91	2.58006
$818$	0	0
$819$	0	0
$820$	0	0
$821$	244.447	0.297743	0.148871	−	0.988857i	$-0.452436\pi$
$821$	244.447	0.297743	0.148871	+	0.988857i	$0.452436\pi$
$822$	0	0
$823$	−331.500	−0.402795	−0.201398	−	0.979510i	$-0.564548\pi$
$823$	−331.500	−0.402795	−0.201398	+	0.979510i	$0.564548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	724.314	0.875833	0.437916	−	0.899016i	$-0.355717\pi$
$827$	724.314	0.875833	0.437916	+	0.899016i	$0.355717\pi$
$828$	0	0
$829$	− 77.5083i	− 0.0934961i	−0.998907	−	0.0467481i	$-0.985114\pi$
$829$	− 77.5083i	− 0.0934961i	0.998907	−	0.0467481i	$-0.0148858\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	65.4309i	0.0785485i
$834$	0	0
$835$	− 1162.47i	− 1.39218i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 952.252i	− 1.13498i	−0.823379	−	0.567492i	$-0.807914\pi$
$839$	− 952.252i	− 1.13498i	0.823379	−	0.567492i	$-0.192086\pi$
$840$	0	0
$841$	−721.977	−0.858475
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1256.17	−1.48659
$846$	0	0
$847$	−608.456	−0.718366
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−749.372	−0.880578
$852$	0	0
$853$	− 282.424i	− 0.331095i	−0.986202	−	0.165548i	$-0.947061\pi$
$853$	− 282.424i	− 0.331095i	0.986202	−	0.165548i	$-0.0529392\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 539.665i	− 0.629714i	−0.949139	−	0.314857i	$-0.898044\pi$
$857$	− 539.665i	− 0.629714i	0.949139	−	0.314857i	$-0.101956\pi$
$858$	0	0
$859$	− 53.2837i	− 0.0620299i	−0.999519	−	0.0310150i	$-0.990126\pi$
$859$	− 53.2837i	− 0.0620299i	0.999519	−	0.0310150i	$-0.00987395\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 632.435i	− 0.732834i	−0.930451	−	0.366417i	$-0.880584\pi$
$863$	− 632.435i	− 0.732834i	0.930451	−	0.366417i	$-0.119416\pi$
$864$	0	0
$865$	−751.911	−0.869261
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−189.550	−0.218125
$870$	0	0
$871$	−58.7991	−0.0675075
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−489.241	−0.559133
$876$	0	0
$877$	− 346.548i	− 0.395152i	−0.980288	−	0.197576i	$-0.936693\pi$
$877$	− 346.548i	− 0.395152i	0.980288	−	0.197576i	$-0.0633069\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1394.26i	− 1.58258i	−0.611438	−	0.791292i	$-0.709409\pi$
$881$	− 1394.26i	− 1.58258i	0.611438	−	0.791292i	$-0.290591\pi$
$882$	0	0
$883$	1714.29i	1.94144i	0.240212	+	0.970720i	$0.422783\pi$
$883$	1714.29i	1.94144i	−0.240212	+	0.970720i	$0.577217\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1149.19i	1.29559i	0.761815	+	0.647795i	$0.224309\pi$
$887$	1149.19i	1.29559i	−0.761815	+	0.647795i	$0.775691\pi$
$888$	0	0
$889$	1056.27	1.18815
$890$	0	0
$891$	0	0
$892$	0	0
$893$	116.595	0.130565
$894$	0	0
$895$	1703.87	1.90376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−410.570	−0.456696
$900$	0	0
$901$	765.897i	0.850052i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	910.785i	1.00639i
$906$	0	0
$907$	− 507.952i	− 0.560035i	−0.959995	−	0.280017i	$-0.909660\pi$
$907$	− 507.952i	− 0.560035i	0.959995	−	0.280017i	$-0.0903402\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1488.13i	1.63351i	0.576984	+	0.816756i	$0.304230\pi$
$911$	1488.13i	1.63351i	−0.576984	+	0.816756i	$0.695770\pi$
$912$	0	0
$913$	334.755	0.366654
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1333.74	1.45446
$918$	0	0
$919$	335.790	0.365386	0.182693	−	0.983170i	$-0.441519\pi$
$919$	335.790	0.365386	0.182693	+	0.983170i	$0.441519\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−85.3949	−0.0925188
$924$	0	0
$925$	1178.99i	1.27458i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1685.87i	1.81471i	0.420363	+	0.907356i	$0.361903\pi$
$929$	1685.87i	1.81471i	−0.420363	+	0.907356i	$0.638097\pi$
$930$	0	0
$931$	104.506i	0.112252i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1012.48i	− 1.08287i
$936$	0	0
$937$	−49.7342	−0.0530781	−0.0265390	−	0.999648i	$-0.508449\pi$
$937$	−49.7342	−0.0530781	−0.0265390	+	0.999648i	$0.508449\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1689.50	1.79543	0.897715	−	0.440577i	$-0.145226\pi$
$941$	1689.50	1.79543	0.897715	+	0.440577i	$0.145226\pi$
$942$	0	0
$943$	286.644	0.303971
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1036.44	−1.09445	−0.547225	−	0.836986i	$-0.684316\pi$
$947$	−1036.44	−1.09445	−0.547225	+	0.836986i	$0.684316\pi$
$948$	0	0
$949$	− 177.918i	− 0.187479i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	388.814i	0.407989i	0.978972	+	0.203995i	$0.0653925\pi$
$953$	388.814i	0.407989i	−0.978972	+	0.203995i	$0.934608\pi$
$954$	0	0
$955$	1162.47i	1.21725i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	961.335i	1.00243i
$960$	0	0
$961$	455.266	0.473742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−869.721	−0.901265
$966$	0	0
$967$	201.899	0.208789	0.104395	−	0.994536i	$-0.466710\pi$
$967$	201.899	0.208789	0.104395	+	0.994536i	$0.466710\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1500.54	−1.54535	−0.772676	−	0.634800i	$-0.781082\pi$
$971$	−1500.54	−1.54535	−0.772676	+	0.634800i	$0.781082\pi$
$972$	0	0
$973$	1189.95i	1.22297i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	866.378i	0.886774i	0.896330	+	0.443387i	$0.146223\pi$
$977$	866.378i	0.886774i	−0.896330	+	0.443387i	$0.853777\pi$
$978$	0	0
$979$	32.9017i	0.0336074i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	613.800i	0.624415i	0.950014	+	0.312207i	$0.101068\pi$
$983$	613.800i	0.624415i	−0.950014	+	0.312207i	$0.898932\pi$
$984$	0	0
$985$	2450.53	2.48785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1301.68	1.31616
$990$	0	0
$991$	−1460.50	−1.47376	−0.736882	−	0.676022i	$-0.763703\pi$
$991$	−1460.50	−1.47376	−0.736882	+	0.676022i	$0.763703\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−314.595	−0.316176
$996$	0	0
$997$	1378.84i	1.38299i	0.722382	+	0.691494i	$0.243047\pi$
$997$	1378.84i	1.38299i	−0.722382	+	0.691494i	$0.756953\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.3.h.a.17.2		8
3.2	odd	2	inner	288.3.h.a.17.8		8
4.3	odd	2		72.3.h.a.53.3	✓	8
8.3	odd	2		72.3.h.a.53.5	yes	8
8.5	even	2	inner	288.3.h.a.17.7		8
12.11	even	2		72.3.h.a.53.6	yes	8
16.3	odd	4		2304.3.e.n.1025.7		8
16.5	even	4		2304.3.e.o.1025.2		8
16.11	odd	4		2304.3.e.n.1025.2		8
16.13	even	4		2304.3.e.o.1025.7		8
24.5	odd	2	inner	288.3.h.a.17.1		8
24.11	even	2		72.3.h.a.53.4	yes	8
48.5	odd	4		2304.3.e.o.1025.8		8
48.11	even	4		2304.3.e.n.1025.8		8
48.29	odd	4		2304.3.e.o.1025.1		8
48.35	even	4		2304.3.e.n.1025.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.h.a.53.3	✓	8	4.3	odd	2
72.3.h.a.53.4	yes	8	24.11	even	2
72.3.h.a.53.5	yes	8	8.3	odd	2
72.3.h.a.53.6	yes	8	12.11	even	2
288.3.h.a.17.1		8	24.5	odd	2	inner
288.3.h.a.17.2		8	1.1	even	1	trivial
288.3.h.a.17.7		8	8.5	even	2	inner
288.3.h.a.17.8		8	3.2	odd	2	inner
2304.3.e.n.1025.1		8	48.35	even	4
2304.3.e.n.1025.2		8	16.11	odd	4
2304.3.e.n.1025.7		8	16.3	odd	4
2304.3.e.n.1025.8		8	48.11	even	4
2304.3.e.o.1025.1		8	48.29	odd	4
2304.3.e.o.1025.2		8	16.5	even	4
2304.3.e.o.1025.7		8	16.13	even	4
2304.3.e.o.1025.8		8	48.5	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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.67101 q^{5} +7.21110 q^{7} +O(q^{10})\)	\(q-7.67101 q^{5} +7.21110 q^{7} +6.05164 q^{11} +2.29014i q^{13} +21.8103i q^{17} +34.8355i q^{19} +21.5117i q^{23} +33.8444 q^{25} -10.9098 q^{29} +37.6333 q^{31} -55.3164 q^{35} +34.8355i q^{37} -13.3250i q^{41} -60.5104i q^{43} -3.34701i q^{47} +3.00000 q^{49} +35.1163 q^{53} -46.4222 q^{55} +37.1615 q^{59} +25.6749i q^{61} -17.5677i q^{65} +25.6749i q^{67} +37.2881i q^{71} -77.6888 q^{73} +43.6390 q^{77} -31.3221 q^{79} +55.3164 q^{83} -167.307i q^{85} +5.43682i q^{89} +16.5144i q^{91} -267.223i q^{95} -52.8444 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} + 128 q^{31} + 24 q^{49} - 256 q^{55} - 160 q^{73} + 384 q^{79} - 192 q^{97}+O(q^{100}) \) 8 * q + 40 * q^25 + 128 * q^31 + 24 * q^49 - 256 * q^55 - 160 * q^73 + 384 * q^79 - 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more