LMFDB - Embedded newform 864.2.d.c.433.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(433,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("864.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.629407744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 2x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.5
Root			$-0.767178 + 1.18804i$ of defining polynomial
Character	$\chi$	$=$	864.433
Dual form			864.2.d.c.433.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.841723i q^{5} -2.64575 q^{7} +O(q^{10})$	$q+0.841723i q^{5} -2.64575 q^{7} -3.91044i q^{11} -0.645751i q^{13} -5.59388 q^{17} +4.29150i q^{19} -8.66259 q^{23} +4.29150 q^{25} -7.82087i q^{29} -2.00000 q^{31} -2.22699i q^{35} -6.64575i q^{37} -6.13742 q^{41} -7.29150i q^{43} -2.52517 q^{47} -7.82087i q^{53} +3.29150 q^{55} +7.27733i q^{59} +13.9373i q^{61} +0.543544 q^{65} +3.00000i q^{67} -9.58301 q^{73} +10.3460i q^{77} +3.35425 q^{79} -9.50432i q^{83} -4.70850i q^{85} +5.59388 q^{89} +1.70850i q^{91} -3.61226 q^{95} -8.29150 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{25} - 16 q^{31} - 16 q^{55} + 8 q^{73} + 48 q^{79} - 24 q^{97}+O(q^{100}) $ 8 * q - 8 * q^25 - 16 * q^31 - 16 * q^55 + 8 * q^73 + 48 * q^79 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	0.841723i	0.376430i	0.982128	+	0.188215i	$0.0602702\pi$
$5$	0.841723i	0.376430i	−0.982128	+	0.188215i	$0.939730\pi$
$6$	0	0
$7$	−2.64575	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$7$	−2.64575	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.91044i	− 1.17904i	−0.807754	−	0.589520i	$-0.799317\pi$
$11$	− 3.91044i	− 1.17904i	0.807754	−	0.589520i	$-0.200683\pi$
$12$	0	0
$13$	− 0.645751i	− 0.179099i	−0.995982	−	0.0895496i	$-0.971457\pi$
$13$	− 0.645751i	− 0.179099i	0.995982	−	0.0895496i	$-0.0285428\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.59388	−1.35672	−0.678358	−	0.734732i	$-0.737308\pi$
$17$	−5.59388	−1.35672	−0.678358	+	0.734732i	$0.737308\pi$
$18$	0	0
$19$	4.29150i	0.984538i	0.870443	+	0.492269i	$0.163832\pi$
$19$	4.29150i	0.984538i	−0.870443	+	0.492269i	$0.836168\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.66259	−1.80628	−0.903138	−	0.429351i	$-0.858742\pi$
$23$	−8.66259	−1.80628	−0.903138	+	0.429351i	$0.858742\pi$
$24$	0	0
$25$	4.29150	0.858301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.82087i	− 1.45230i	−0.687537	−	0.726150i	$-0.741308\pi$
$29$	− 7.82087i	− 1.45230i	0.687537	−	0.726150i	$-0.258692\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.22699i	− 0.376430i
$36$	0	0
$37$	− 6.64575i	− 1.09255i	−0.837604	−	0.546277i	$-0.816044\pi$
$37$	− 6.64575i	− 1.09255i	0.837604	−	0.546277i	$-0.183956\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.13742	−0.958505	−0.479252	−	0.877677i	$-0.659092\pi$
$41$	−6.13742	−0.958505	−0.479252	+	0.877677i	$0.659092\pi$
$42$	0	0
$43$	− 7.29150i	− 1.11194i	−0.831201	−	0.555972i	$-0.812346\pi$
$43$	− 7.29150i	− 1.11194i	0.831201	−	0.555972i	$-0.187654\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.52517	−0.368334	−0.184167	−	0.982895i	$-0.558959\pi$
$47$	−2.52517	−0.368334	−0.184167	+	0.982895i	$0.558959\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.82087i	− 1.07428i	−0.843493	−	0.537140i	$-0.819505\pi$
$53$	− 7.82087i	− 1.07428i	0.843493	−	0.537140i	$-0.180495\pi$
$54$	0	0
$55$	3.29150	0.443826
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.27733i	0.947427i	0.880679	+	0.473714i	$0.157087\pi$
$59$	7.27733i	0.947427i	−0.880679	+	0.473714i	$0.842913\pi$
$60$	0	0
$61$	13.9373i	1.78448i	0.451559	+	0.892241i	$0.350868\pi$
$61$	13.9373i	1.78448i	−0.451559	+	0.892241i	$0.649132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.543544	0.0674183
$66$	0	0
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−9.58301	−1.12161	−0.560803	−	0.827949i	$-0.689507\pi$
$73$	−9.58301	−1.12161	−0.560803	+	0.827949i	$0.689507\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3460i	1.17904i
$78$	0	0
$79$	3.35425	0.377382	0.188691	−	0.982036i	$-0.439575\pi$
$79$	3.35425	0.377382	0.188691	+	0.982036i	$0.439575\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.50432i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$83$	− 9.50432i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$84$	0	0
$85$	− 4.70850i	− 0.510708i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.59388	0.592950	0.296475	−	0.955041i	$-0.404189\pi$
$89$	5.59388	0.592950	0.296475	+	0.955041i	$0.404189\pi$
$90$	0	0
$91$	1.70850i	0.179099i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.61226	−0.370610
$96$	0	0
$97$	−8.29150	−0.841875	−0.420937	−	0.907090i	$-0.638299\pi$
$97$	−8.29150	−0.841875	−0.420937	+	0.907090i	$0.638299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.68345i	− 0.167509i	−0.996486	−	0.0837546i	$-0.973309\pi$
$101$	− 1.68345i	− 0.167509i	0.996486	−	0.0837546i	$-0.0266912\pi$
$102$	0	0
$103$	10.6458	1.04896	0.524479	−	0.851424i	$-0.324260\pi$
$103$	10.6458	1.04896	0.524479	+	0.851424i	$0.324260\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.96077i	− 0.866271i	−0.901329	−	0.433135i	$-0.857407\pi$
$107$	− 8.96077i	− 0.866271i	0.901329	−	0.433135i	$-0.142593\pi$
$108$	0	0
$109$	4.70850i	0.450992i	0.974244	+	0.225496i	$0.0724003\pi$
$109$	4.70850i	0.450992i	−0.974244	+	0.225496i	$0.927600\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.7313	−1.10359	−0.551794	−	0.833980i	$-0.686057\pi$
$113$	−11.7313	−1.10359	−0.551794	+	0.833980i	$0.686057\pi$
$114$	0	0
$115$	− 7.29150i	− 0.679936i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.8000	1.35672
$120$	0	0
$121$	−4.29150	−0.390137
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.82087i	0.699520i
$126$	0	0
$127$	5.29150	0.469545	0.234772	−	0.972050i	$-0.424565\pi$
$127$	5.29150	0.469545	0.234772	+	0.972050i	$0.424565\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.9583i	1.21954i	0.792578	+	0.609771i	$0.208739\pi$
$131$	13.9583i	1.21954i	−0.792578	+	0.609771i	$0.791261\pi$
$132$	0	0
$133$	− 11.3542i	− 0.984538i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7313	1.00227	0.501137	−	0.865368i	$-0.332915\pi$
$137$	11.7313	1.00227	0.501137	+	0.865368i	$0.332915\pi$
$138$	0	0
$139$	− 1.70850i	− 0.144913i	−0.997372	−	0.0724564i	$-0.976916\pi$
$139$	− 1.70850i	− 0.144913i	0.997372	−	0.0724564i	$-0.0230838\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.52517	−0.211165
$144$	0	0
$145$	6.58301	0.546689
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.6417i	1.28142i	0.767782	+	0.640711i	$0.221360\pi$
$149$	15.6417i	1.28142i	−0.767782	+	0.640711i	$0.778640\pi$
$150$	0	0
$151$	−7.35425	−0.598480	−0.299240	−	0.954178i	$-0.596733\pi$
$151$	−7.35425	−0.598480	−0.299240	+	0.954178i	$0.596733\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.68345i	− 0.135218i
$156$	0	0
$157$	19.2915i	1.53963i	0.638267	+	0.769815i	$0.279651\pi$
$157$	19.2915i	1.53963i	−0.638267	+	0.769815i	$0.720349\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.9191	1.80628
$162$	0	0
$163$	− 4.29150i	− 0.336136i	−0.985775	−	0.168068i	$-0.946247\pi$
$163$	− 4.29150i	− 0.336136i	0.985775	−	0.168068i	$-0.0537529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.7129	1.06114	0.530569	−	0.847642i	$-0.321978\pi$
$167$	13.7129	1.06114	0.530569	+	0.847642i	$0.321978\pi$
$168$	0	0
$169$	12.5830	0.967923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.8712i	− 0.978580i	−0.872121	−	0.489290i	$-0.837256\pi$
$173$	− 12.8712i	− 0.978580i	0.872121	−	0.489290i	$-0.162744\pi$
$174$	0	0
$175$	−11.3542	−0.858301
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.0086i	1.42077i	0.703812	+	0.710386i	$0.251480\pi$
$179$	19.0086i	1.42077i	−0.703812	+	0.710386i	$0.748520\pi$
$180$	0	0
$181$	− 11.3542i	− 0.843955i	−0.906606	−	0.421977i	$-0.861336\pi$
$181$	− 11.3542i	− 0.843955i	0.906606	−	0.421977i	$-0.138664\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.59388	0.411270
$186$	0	0
$187$	21.8745i	1.59962i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.8504	1.43632	0.718161	−	0.695877i	$-0.244984\pi$
$191$	19.8504	1.43632	0.718161	+	0.695877i	$0.244984\pi$
$192$	0	0
$193$	6.29150	0.452872	0.226436	−	0.974026i	$-0.427293\pi$
$193$	6.29150	0.452872	0.226436	+	0.974026i	$0.427293\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.97915i	0.497244i	0.968601	+	0.248622i	$0.0799776\pi$
$197$	6.97915i	0.497244i	−0.968601	+	0.248622i	$0.920022\pi$
$198$	0	0
$199$	−18.5203	−1.31287	−0.656433	−	0.754384i	$-0.727936\pi$
$199$	−18.5203	−1.31287	−0.656433	+	0.754384i	$0.727936\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.6921i	1.45230i
$204$	0	0
$205$	− 5.16601i	− 0.360810i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.7816	1.16081
$210$	0	0
$211$	− 10.2915i	− 0.708496i	−0.935151	−	0.354248i	$-0.884737\pi$
$211$	− 10.2915i	− 0.708496i	0.935151	−	0.354248i	$-0.115263\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.13742	0.418569
$216$	0	0
$217$	5.29150	0.359211
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.61226i	0.242987i
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.36689i	− 0.223468i	−0.993738	−	0.111734i	$-0.964359\pi$
$227$	− 3.36689i	− 0.223468i	0.993738	−	0.111734i	$-0.0356405\pi$
$228$	0	0
$229$	− 29.1660i	− 1.92734i	−0.267088	−	0.963672i	$-0.586061\pi$
$229$	− 29.1660i	− 1.92734i	0.267088	−	0.963672i	$-0.413939\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.1878	−0.732935	−0.366467	−	0.930431i	$-0.619433\pi$
$233$	−11.1878	−0.732935	−0.366467	+	0.930431i	$0.619433\pi$
$234$	0	0
$235$	− 2.12549i	− 0.138652i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.05034	−0.326679	−0.163340	−	0.986570i	$-0.552227\pi$
$239$	−5.05034	−0.326679	−0.163340	+	0.986570i	$0.552227\pi$
$240$	0	0
$241$	−4.87451	−0.313995	−0.156997	−	0.987599i	$-0.550181\pi$
$241$	−4.87451	−0.313995	−0.156997	+	0.987599i	$0.550181\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.77124	0.176330
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.77053i	0.174874i	0.996170	+	0.0874372i	$0.0278677\pi$
$251$	2.77053i	0.174874i	−0.996170	+	0.0874372i	$0.972132\pi$
$252$	0	0
$253$	33.8745i	2.12967i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.05034	0.315031	0.157516	−	0.987516i	$-0.449652\pi$
$257$	5.05034	0.315031	0.157516	+	0.987516i	$0.449652\pi$
$258$	0	0
$259$	17.5830i	1.09255i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.3252	−1.06832	−0.534158	−	0.845384i	$-0.679371\pi$
$263$	−17.3252	−1.06832	−0.534158	+	0.845384i	$0.679371\pi$
$264$	0	0
$265$	6.58301	0.404391
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.3460i	− 0.630809i	−0.948957	−	0.315405i	$-0.897860\pi$
$269$	− 10.3460i	− 0.630809i	0.948957	−	0.315405i	$-0.102140\pi$
$270$	0	0
$271$	−5.22876	−0.317624	−0.158812	−	0.987309i	$-0.550766\pi$
$271$	−5.22876	−0.317624	−0.158812	+	0.987309i	$0.550766\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 16.7816i	− 1.01197i
$276$	0	0
$277$	− 19.2915i	− 1.15911i	−0.814932	−	0.579557i	$-0.803226\pi$
$277$	− 19.2915i	− 1.15911i	0.814932	−	0.579557i	$-0.196774\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.05034	−0.301278	−0.150639	−	0.988589i	$-0.548133\pi$
$281$	−5.05034	−0.301278	−0.150639	+	0.988589i	$0.548133\pi$
$282$	0	0
$283$	16.7085i	0.993217i	0.867975	+	0.496609i	$0.165422\pi$
$283$	16.7085i	0.993217i	−0.867975	+	0.496609i	$0.834578\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.2381	0.958505
$288$	0	0
$289$	14.2915	0.840677
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.29570i	− 0.309378i	−0.987963	−	0.154689i	$-0.950562\pi$
$293$	− 5.29570i	− 0.309378i	0.987963	−	0.154689i	$-0.0494376\pi$
$294$	0	0
$295$	−6.12549	−0.356640
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.59388i	0.323502i
$300$	0	0
$301$	19.2915i	1.11194i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.7313	−0.671733
$306$	0	0
$307$	− 21.8745i	− 1.24844i	−0.781247	−	0.624222i	$-0.785416\pi$
$307$	− 21.8745i	− 1.24844i	0.781247	−	0.624222i	$-0.214584\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.66259	−0.491211	−0.245605	−	0.969370i	$-0.578987\pi$
$311$	−8.66259	−0.491211	−0.245605	+	0.969370i	$0.578987\pi$
$312$	0	0
$313$	−16.8745	−0.953804	−0.476902	−	0.878957i	$-0.658240\pi$
$313$	−16.8745	−0.953804	−0.476902	+	0.878957i	$0.658240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.8798i	1.79055i	0.445514	+	0.895275i	$0.353021\pi$
$317$	31.8798i	1.79055i	−0.445514	+	0.895275i	$0.646979\pi$
$318$	0	0
$319$	−30.5830	−1.71232
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0062i	− 1.33574i
$324$	0	0
$325$	− 2.77124i	− 0.153721i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.68097	0.368334
$330$	0	0
$331$	− 16.2915i	− 0.895462i	−0.894168	−	0.447731i	$-0.852232\pi$
$331$	− 16.2915i	− 0.895462i	0.894168	−	0.447731i	$-0.147768\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.52517	−0.137965
$336$	0	0
$337$	−8.29150	−0.451667	−0.225833	−	0.974166i	$-0.572511\pi$
$337$	−8.29150	−0.451667	−0.225833	+	0.974166i	$0.572511\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.82087i	0.423524i
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.8295i	− 1.44028i	−0.693827	−	0.720142i	$-0.744077\pi$
$347$	− 26.8295i	− 1.44028i	0.693827	−	0.720142i	$-0.255923\pi$
$348$	0	0
$349$	5.35425i	0.286606i	0.989679	+	0.143303i	$0.0457724\pi$
$349$	5.35425i	0.286606i	−0.989679	+	0.143303i	$0.954228\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.4626	−1.24879	−0.624394	−	0.781109i	$-0.714654\pi$
$353$	−23.4626	−1.24879	−0.624394	+	0.781109i	$0.714654\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.61226	−0.190648	−0.0953238	−	0.995446i	$-0.530389\pi$
$359$	−3.61226	−0.190648	−0.0953238	+	0.995446i	$0.530389\pi$
$360$	0	0
$361$	0.583005	0.0306845
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.06623i	− 0.422206i
$366$	0	0
$367$	32.5203	1.69754	0.848772	−	0.528759i	$-0.177343\pi$
$367$	32.5203	1.69754	0.848772	+	0.528759i	$0.177343\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.6921i	1.07428i
$372$	0	0
$373$	− 1.47974i	− 0.0766181i	−0.999266	−	0.0383090i	$-0.987803\pi$
$373$	− 1.47974i	− 0.0766181i	0.999266	−	0.0383090i	$-0.0121971\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.05034	−0.260106
$378$	0	0
$379$	0.416995i	0.0214196i	0.999943	+	0.0107098i	$0.00340910\pi$
$379$	0.416995i	0.0214196i	−0.999943	+	0.0107098i	$0.996591\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.05034	−0.258060	−0.129030	−	0.991641i	$-0.541186\pi$
$383$	−5.05034	−0.258060	−0.129030	+	0.991641i	$0.541186\pi$
$384$	0	0
$385$	−8.70850	−0.443826
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.4835i	− 0.835745i	−0.908506	−	0.417872i	$-0.862776\pi$
$389$	− 16.4835i	− 0.835745i	0.908506	−	0.417872i	$-0.137224\pi$
$390$	0	0
$391$	48.4575	2.45060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.82335i	0.142058i
$396$	0	0
$397$	4.70850i	0.236313i	0.992995	+	0.118156i	$0.0376984\pi$
$397$	4.70850i	0.236313i	−0.992995	+	0.118156i	$0.962302\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.4626	1.17167	0.585833	−	0.810431i	$-0.300767\pi$
$401$	23.4626	1.17167	0.585833	+	0.810431i	$0.300767\pi$
$402$	0	0
$403$	1.29150i	0.0643343i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.9878	−1.28817
$408$	0	0
$409$	−2.29150	−0.113308	−0.0566538	−	0.998394i	$-0.518043\pi$
$409$	−2.29150	−0.113308	−0.0566538	+	0.998394i	$0.518043\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 19.2540i	− 0.947427i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 3.91044i	− 0.191037i	−0.995428	−	0.0955186i	$-0.969549\pi$
$419$	− 3.91044i	− 0.191037i	0.995428	−	0.0955186i	$-0.0304509\pi$
$420$	0	0
$421$	− 17.3542i	− 0.845794i	−0.906178	−	0.422897i	$-0.861013\pi$
$421$	− 17.3542i	− 0.845794i	0.906178	−	0.422897i	$-0.138987\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−24.0062	−1.16447
$426$	0	0
$427$	− 36.8745i	− 1.78448i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.57551	−0.364899	−0.182450	−	0.983215i	$-0.558403\pi$
$431$	−7.57551	−0.364899	−0.182450	+	0.983215i	$0.558403\pi$
$432$	0	0
$433$	−31.8745	−1.53179	−0.765896	−	0.642965i	$-0.777704\pi$
$433$	−31.8745	−1.53179	−0.765896	+	0.642965i	$0.777704\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 37.1755i	− 1.77835i
$438$	0	0
$439$	−31.1660	−1.48747	−0.743736	−	0.668473i	$-0.766948\pi$
$439$	−31.1660	−1.48747	−0.743736	+	0.668473i	$0.766948\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10.5914i	− 0.503213i	−0.967830	−	0.251606i	$-0.919041\pi$
$443$	− 10.5914i	− 0.503213i	0.967830	−	0.251606i	$-0.0809588\pi$
$444$	0	0
$445$	4.70850i	0.223204i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.1939	−1.66090	−0.830452	−	0.557091i	$-0.811918\pi$
$449$	−35.1939	−1.66090	−0.830452	+	0.557091i	$0.811918\pi$
$450$	0	0
$451$	24.0000i	1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.43808	−0.0674183
$456$	0	0
$457$	−7.87451	−0.368354	−0.184177	−	0.982893i	$-0.558962\pi$
$457$	−7.87451	−0.368354	−0.184177	+	0.982893i	$0.558962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.1166i	0.610900i	0.952208	+	0.305450i	$0.0988068\pi$
$461$	13.1166i	0.610900i	−0.952208	+	0.305450i	$0.901193\pi$
$462$	0	0
$463$	−4.77124	−0.221738	−0.110869	−	0.993835i	$-0.535363\pi$
$463$	−4.77124	−0.221738	−0.110869	+	0.993835i	$0.535363\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.4148i	0.620761i	0.950612	+	0.310380i	$0.100456\pi$
$467$	13.4148i	0.620761i	−0.950612	+	0.310380i	$0.899544\pi$
$468$	0	0
$469$	− 7.93725i	− 0.366508i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.5129	−1.31103
$474$	0	0
$475$	18.4170i	0.845030i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.7375	−1.63289	−0.816443	−	0.577426i	$-0.804057\pi$
$479$	−35.7375	−1.63289	−0.816443	+	0.577426i	$0.804057\pi$
$480$	0	0
$481$	−4.29150	−0.195676
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 6.97915i	− 0.316907i
$486$	0	0
$487$	−21.9373	−0.994072	−0.497036	−	0.867730i	$-0.665578\pi$
$487$	−21.9373	−0.994072	−0.497036	+	0.867730i	$0.665578\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21.2356i	− 0.958350i	−0.877719	−	0.479175i	$-0.840936\pi$
$491$	− 21.2356i	− 0.958350i	0.877719	−	0.479175i	$-0.159064\pi$
$492$	0	0
$493$	43.7490i	1.97036i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.7490i	0.884087i	0.896994	+	0.442044i	$0.145746\pi$
$499$	19.7490i	0.884087i	−0.896994	+	0.442044i	$0.854254\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.9878	−1.15874	−0.579369	−	0.815065i	$-0.696701\pi$
$503$	−25.9878	−1.15874	−0.579369	+	0.815065i	$0.696701\pi$
$504$	0	0
$505$	1.41699	0.0630554
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 43.9093i	− 1.94625i	−0.230285	−	0.973123i	$-0.573966\pi$
$509$	− 43.9093i	− 1.94625i	0.230285	−	0.973123i	$-0.426034\pi$
$510$	0	0
$511$	25.3542	1.12161
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.96077i	0.394859i
$516$	0	0
$517$	9.87451i	0.434280i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.7816	0.735217	0.367609	−	0.929981i	$-0.380177\pi$
$521$	16.7816	0.735217	0.367609	+	0.929981i	$0.380177\pi$
$522$	0	0
$523$	− 30.8745i	− 1.35005i	−0.737796	−	0.675024i	$-0.764133\pi$
$523$	− 30.8745i	− 1.35005i	0.737796	−	0.675024i	$-0.235867\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.1878	0.487347
$528$	0	0
$529$	52.0405	2.26263
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.96325i	0.171667i
$534$	0	0
$535$	7.54249	0.326090
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 10.5203i	− 0.452301i	−0.974092	−	0.226151i	$-0.927386\pi$
$541$	− 10.5203i	− 0.452301i	0.974092	−	0.226151i	$-0.0726142\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.96325	−0.169767
$546$	0	0
$547$	− 11.5830i	− 0.495254i	−0.968856	−	0.247627i	$-0.920349\pi$
$547$	− 11.5830i	− 0.495254i	0.968856	−	0.247627i	$-0.0796507\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	33.5633	1.42984
$552$	0	0
$553$	−8.87451	−0.377382
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.1669i	0.769757i	0.922967	+	0.384878i	$0.125757\pi$
$557$	18.1669i	0.769757i	−0.922967	+	0.384878i	$0.874243\pi$
$558$	0	0
$559$	−4.70850	−0.199148
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 34.0540i	− 1.43521i	−0.696453	−	0.717603i	$-0.745239\pi$
$563$	− 34.0540i	− 1.43521i	0.696453	−	0.717603i	$-0.254761\pi$
$564$	0	0
$565$	− 9.87451i	− 0.415424i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.68097	−0.280081	−0.140040	−	0.990146i	$-0.544723\pi$
$569$	−6.68097	−0.280081	−0.140040	+	0.990146i	$0.544723\pi$
$570$	0	0
$571$	16.2915i	0.681778i	0.940104	+	0.340889i	$0.110728\pi$
$571$	16.2915i	0.681778i	−0.940104	+	0.340889i	$0.889272\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−37.1755	−1.55033
$576$	0	0
$577$	2.41699	0.100621	0.0503104	−	0.998734i	$-0.483979\pi$
$577$	2.41699	0.100621	0.0503104	+	0.998734i	$0.483979\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25.1461i	1.04323i
$582$	0	0
$583$	−30.5830	−1.26662
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.6896i	1.06032i	0.847897	+	0.530162i	$0.177869\pi$
$587$	25.6896i	1.06032i	−0.847897	+	0.530162i	$0.822131\pi$
$588$	0	0
$589$	− 8.58301i	− 0.353657i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.4123	0.756101	0.378051	−	0.925785i	$-0.376594\pi$
$593$	18.4123	0.756101	0.378051	+	0.925785i	$0.376594\pi$
$594$	0	0
$595$	12.4575i	0.510708i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.7375	1.46019	0.730097	−	0.683344i	$-0.239475\pi$
$599$	35.7375	1.46019	0.730097	+	0.683344i	$0.239475\pi$
$600$	0	0
$601$	−0.125492	−0.00511893	−0.00255947	−	0.999997i	$-0.500815\pi$
$601$	−0.125492	−0.00511893	−0.00255947	+	0.999997i	$0.500815\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.61226i	− 0.146859i
$606$	0	0
$607$	−38.6458	−1.56858	−0.784291	−	0.620393i	$-0.786973\pi$
$607$	−38.6458	−1.56858	−0.784291	+	0.620393i	$0.786973\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.63063i	0.0659683i
$612$	0	0
$613$	− 37.1033i	− 1.49859i	−0.662238	−	0.749293i	$-0.730393\pi$
$613$	− 37.1033i	− 1.49859i	0.662238	−	0.749293i	$-0.269607\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.8687	0.719368	0.359684	−	0.933074i	$-0.382885\pi$
$617$	17.8687	0.719368	0.359684	+	0.933074i	$0.382885\pi$
$618$	0	0
$619$	− 39.0000i	− 1.56754i	−0.621050	−	0.783771i	$-0.713294\pi$
$619$	− 39.0000i	− 1.56754i	0.621050	−	0.783771i	$-0.286706\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.8000	−0.592950
$624$	0	0
$625$	14.8745	0.594980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.1755i	1.48229i
$630$	0	0
$631$	7.22876	0.287772	0.143886	−	0.989594i	$-0.454040\pi$
$631$	7.22876	0.287772	0.143886	+	0.989594i	$0.454040\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.45398i	0.176751i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.2748	−0.484827	−0.242414	−	0.970173i	$-0.577939\pi$
$641$	−12.2748	−0.484827	−0.242414	+	0.970173i	$0.577939\pi$
$642$	0	0
$643$	16.7085i	0.658919i	0.944170	+	0.329459i	$0.106866\pi$
$643$	16.7085i	0.658919i	−0.944170	+	0.329459i	$0.893134\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.4123	0.723861	0.361931	−	0.932205i	$-0.382118\pi$
$647$	18.4123	0.723861	0.361931	+	0.932205i	$0.382118\pi$
$648$	0	0
$649$	28.4575	1.11706
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.77053i	− 0.108419i	−0.998530	−	0.0542097i	$-0.982736\pi$
$653$	− 2.77053i	− 0.108419i	0.998530	−	0.0542097i	$-0.0172639\pi$
$654$	0	0
$655$	−11.7490	−0.459072
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20.6921i	− 0.806049i	−0.915189	−	0.403024i	$-0.867959\pi$
$659$	− 20.6921i	− 0.806049i	0.915189	−	0.403024i	$-0.132041\pi$
$660$	0	0
$661$	− 22.5203i	− 0.875937i	−0.898990	−	0.437968i	$-0.855698\pi$
$661$	− 22.5203i	− 0.875937i	0.898990	−	0.437968i	$-0.144302\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.55713	0.370610
$666$	0	0
$667$	67.7490i	2.62325i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	54.5007	2.10398
$672$	0	0
$673$	−9.12549	−0.351762	−0.175881	−	0.984411i	$-0.556277\pi$
$673$	−9.12549	−0.351762	−0.175881	+	0.984411i	$0.556277\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.5792i	1.40585i	0.711263	+	0.702926i	$0.248124\pi$
$677$	36.5792i	1.40585i	−0.711263	+	0.702926i	$0.751876\pi$
$678$	0	0
$679$	21.9373	0.841875
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 4.45398i	− 0.170427i	−0.996363	−	0.0852134i	$-0.972843\pi$
$683$	− 4.45398i	− 0.170427i	0.996363	−	0.0852134i	$-0.0271572\pi$
$684$	0	0
$685$	9.87451i	0.377286i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.05034	−0.192403
$690$	0	0
$691$	− 16.7085i	− 0.635621i	−0.948154	−	0.317811i	$-0.897052\pi$
$691$	− 16.7085i	− 0.635621i	0.948154	−	0.317811i	$-0.102948\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.43808	0.0545495
$696$	0	0
$697$	34.3320	1.30042
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.20861i	− 0.158957i	−0.996837	−	0.0794786i	$-0.974674\pi$
$701$	− 4.20861i	− 0.158957i	0.996837	−	0.0794786i	$-0.0253255\pi$
$702$	0	0
$703$	28.5203	1.07566
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.45398i	0.167509i
$708$	0	0
$709$	40.5203i	1.52177i	0.648887	+	0.760885i	$0.275235\pi$
$709$	40.5203i	1.52177i	−0.648887	+	0.760885i	$0.724765\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.3252	0.648833
$714$	0	0
$715$	− 2.12549i	− 0.0794889i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.96325	0.147804	0.0739021	−	0.997265i	$-0.476455\pi$
$719$	3.96325	0.147804	0.0739021	+	0.997265i	$0.476455\pi$
$720$	0	0
$721$	−28.1660	−1.04896
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 33.5633i	− 1.24651i
$726$	0	0
$727$	−23.4170	−0.868488	−0.434244	−	0.900795i	$-0.642984\pi$
$727$	−23.4170	−0.868488	−0.434244	+	0.900795i	$0.642984\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	40.7878i	1.50859i
$732$	0	0
$733$	− 24.4575i	− 0.903359i	−0.892180	−	0.451679i	$-0.850825\pi$
$733$	− 24.4575i	− 0.903359i	0.892180	−	0.451679i	$-0.149175\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.7313	0.432128
$738$	0	0
$739$	12.4575i	0.458257i	0.973396	+	0.229129i	$0.0735876\pi$
$739$	12.4575i	0.458257i	−0.973396	+	0.229129i	$0.926412\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.2626	−1.40372	−0.701860	−	0.712315i	$-0.747647\pi$
$743$	−38.2626	−1.40372	−0.701860	+	0.712315i	$0.747647\pi$
$744$	0	0
$745$	−13.1660	−0.482365
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.7080i	0.866271i
$750$	0	0
$751$	−39.9373	−1.45733	−0.728666	−	0.684870i	$-0.759859\pi$
$751$	−39.9373	−1.45733	−0.728666	+	0.684870i	$0.759859\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.19024i	− 0.225286i
$756$	0	0
$757$	1.93725i	0.0704107i	0.999380	+	0.0352053i	$0.0112085\pi$
$757$	1.93725i	0.0704107i	−0.999380	+	0.0352053i	$0.988791\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	51.4320	1.86441	0.932205	−	0.361932i	$-0.117883\pi$
$761$	51.4320	1.86441	0.932205	+	0.361932i	$0.117883\pi$
$762$	0	0
$763$	− 12.4575i	− 0.450992i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.69934	0.169683
$768$	0	0
$769$	−6.16601	−0.222352	−0.111176	−	0.993801i	$-0.535462\pi$
$769$	−6.16601	−0.222352	−0.111176	+	0.993801i	$0.535462\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 25.1461i	− 0.904441i	−0.891906	−	0.452220i	$-0.850632\pi$
$773$	− 25.1461i	− 0.904441i	0.891906	−	0.452220i	$-0.149368\pi$
$774$	0	0
$775$	−8.58301	−0.308311
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 26.3388i	− 0.943684i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.2381	−0.579563
$786$	0	0
$787$	− 44.6235i	− 1.59066i	−0.606179	−	0.795328i	$-0.707298\pi$
$787$	− 44.6235i	− 1.59066i	0.606179	−	0.795328i	$-0.292702\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	31.0381	1.10359
$792$	0	0
$793$	9.00000	0.319599
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.8663i	0.809964i	0.914325	+	0.404982i	$0.132722\pi$
$797$	22.8663i	0.809964i	−0.914325	+	0.404982i	$0.867278\pi$
$798$	0	0
$799$	14.1255	0.499724
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.4737i	1.32242i
$804$	0	0
$805$	19.2915i	0.679936i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	− 31.2915i	− 1.09879i	−0.835562	−	0.549397i	$-0.814858\pi$
$811$	− 31.2915i	− 1.09879i	0.835562	−	0.549397i	$-0.185142\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.61226	0.126532
$816$	0	0
$817$	31.2915	1.09475
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.2037i	0.495711i	0.968797	+	0.247856i	$0.0797258\pi$
$821$	14.2037i	0.495711i	−0.968797	+	0.247856i	$0.920274\pi$
$822$	0	0
$823$	35.9373	1.25269	0.626347	−	0.779544i	$-0.284549\pi$
$823$	35.9373	1.25269	0.626347	+	0.779544i	$0.284549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.7032i	1.20675i	0.797458	+	0.603374i	$0.206177\pi$
$827$	34.7032i	1.20675i	−0.797458	+	0.603374i	$0.793823\pi$
$828$	0	0
$829$	− 33.2288i	− 1.15408i	−0.816715	−	0.577041i	$-0.804207\pi$
$829$	− 33.2288i	− 1.15408i	0.816715	−	0.577041i	$-0.195793\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.5425i	0.399444i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.08709	0.0375304	0.0187652	−	0.999824i	$-0.494026\pi$
$839$	1.08709	0.0375304	0.0187652	+	0.999824i	$0.494026\pi$
$840$	0	0
$841$	−32.1660	−1.10917
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.5914i	0.364355i
$846$	0	0
$847$	11.3542	0.390137
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.5694i	1.97345i
$852$	0	0
$853$	17.8118i	0.609863i	0.952374	+	0.304932i	$0.0986336\pi$
$853$	17.8118i	0.609863i	−0.952374	+	0.304932i	$0.901366\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.1007	0.345032	0.172516	−	0.985007i	$-0.444810\pi$
$857$	10.1007	0.345032	0.172516	+	0.985007i	$0.444810\pi$
$858$	0	0
$859$	47.5830i	1.62351i	0.583997	+	0.811756i	$0.301488\pi$
$859$	47.5830i	1.62351i	−0.583997	+	0.811756i	$0.698512\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.4001	1.51140	0.755698	−	0.654921i	$-0.227298\pi$
$863$	44.4001	1.51140	0.755698	+	0.654921i	$0.227298\pi$
$864$	0	0
$865$	10.8340	0.368367
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 13.1166i	− 0.444949i
$870$	0	0
$871$	1.93725	0.0656413
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 20.6921i	− 0.699520i
$876$	0	0
$877$	20.3948i	0.688682i	0.938845	+	0.344341i	$0.111898\pi$
$877$	20.3948i	0.688682i	−0.938845	+	0.344341i	$0.888102\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.1572	1.31924	0.659619	−	0.751600i	$-0.270718\pi$
$881$	39.1572	1.31924	0.659619	+	0.751600i	$0.270718\pi$
$882$	0	0
$883$	− 26.1660i	− 0.880556i	−0.897861	−	0.440278i	$-0.854880\pi$
$883$	− 26.1660i	− 0.880556i	0.897861	−	0.440278i	$-0.145120\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.5129	0.957371	0.478686	−	0.877986i	$-0.341113\pi$
$887$	28.5129	0.957371	0.478686	+	0.877986i	$0.341113\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.8368i	− 0.362639i
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.6417i	0.521681i
$900$	0	0
$901$	43.7490i	1.45749i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.55713	0.317690
$906$	0	0
$907$	18.0405i	0.599026i	0.954092	+	0.299513i	$0.0968241\pi$
$907$	18.0405i	0.599026i	−0.954092	+	0.299513i	$0.903176\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38.6136	1.27933	0.639663	−	0.768655i	$-0.279074\pi$
$911$	38.6136	1.27933	0.639663	+	0.768655i	$0.279074\pi$
$912$	0	0
$913$	−37.1660	−1.23002
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 36.9302i	− 1.21954i
$918$	0	0
$919$	0.583005	0.0192316	0.00961578	−	0.999954i	$-0.496939\pi$
$919$	0.583005	0.0192316	0.00961578	+	0.999954i	$0.496939\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 28.5203i	− 0.937740i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.1130	1.90663	0.953313	−	0.301985i	$-0.0976494\pi$
$929$	58.1130	1.90663	0.953313	+	0.301985i	$0.0976494\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.4123	−0.602146
$936$	0	0
$937$	−40.8745	−1.33531	−0.667656	−	0.744470i	$-0.732702\pi$
$937$	−40.8745	−1.33531	−0.667656	+	0.744470i	$0.732702\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23.2172i	0.756861i	0.925630	+	0.378430i	$0.123536\pi$
$941$	23.2172i	0.756861i	−0.925630	+	0.378430i	$0.876464\pi$
$942$	0	0
$943$	53.1660	1.73132
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.5018i	0.471246i	0.971844	+	0.235623i	$0.0757131\pi$
$947$	14.5018i	0.471246i	−0.971844	+	0.235623i	$0.924287\pi$
$948$	0	0
$949$	6.18824i	0.200879i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.3449	−1.63083	−0.815416	−	0.578875i	$-0.803492\pi$
$953$	−50.3449	−1.63083	−0.815416	+	0.578875i	$0.803492\pi$
$954$	0	0
$955$	16.7085i	0.540674i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.0381	−1.00227
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.29570i	0.170475i
$966$	0	0
$967$	53.9373	1.73451	0.867253	−	0.497868i	$-0.165884\pi$
$967$	53.9373	1.73451	0.867253	+	0.497868i	$0.165884\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 8.96077i	− 0.287565i	−0.989609	−	0.143782i	$-0.954073\pi$
$971$	− 8.96077i	− 0.287565i	0.989609	−	0.143782i	$-0.0459265\pi$
$972$	0	0
$973$	4.52026i	0.144913i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.2381	0.519503	0.259751	−	0.965676i	$-0.416359\pi$
$977$	16.2381	0.519503	0.259751	+	0.965676i	$0.416359\pi$
$978$	0	0
$979$	− 21.8745i	− 0.699112i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.9007	−0.794209	−0.397104	−	0.917773i	$-0.629985\pi$
$983$	−24.9007	−0.794209	−0.397104	+	0.917773i	$0.629985\pi$
$984$	0	0
$985$	−5.87451	−0.187177
$986$	0	0
$987$	0	0
$988$	0	0
$989$	63.1633i	2.00848i
$990$	0	0
$991$	−6.06275	−0.192589	−0.0962947	−	0.995353i	$-0.530699\pi$
$991$	−6.06275	−0.192589	−0.0962947	+	0.995353i	$0.530699\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 15.5889i	− 0.494202i
$996$	0	0
$997$	− 9.87451i	− 0.312729i	−0.987699	−	0.156364i	$-0.950023\pi$
$997$	− 9.87451i	− 0.312729i	0.987699	−	0.156364i	$-0.0499774\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.d.c.433.5		8
3.2	odd	2	inner	864.2.d.c.433.3		8
4.3	odd	2		216.2.d.c.109.6	yes	8
8.3	odd	2		216.2.d.c.109.5	yes	8
8.5	even	2	inner	864.2.d.c.433.4		8
9.2	odd	6		2592.2.r.q.433.6		16
9.4	even	3		2592.2.r.q.2161.5		16
9.5	odd	6		2592.2.r.q.2161.3		16
9.7	even	3		2592.2.r.q.433.4		16
12.11	even	2		216.2.d.c.109.3	✓	8
16.3	odd	4		6912.2.a.cc.1.3		4
16.5	even	4		6912.2.a.ci.1.2		4
16.11	odd	4		6912.2.a.cj.1.2		4
16.13	even	4		6912.2.a.cd.1.3		4
24.5	odd	2	inner	864.2.d.c.433.6		8
24.11	even	2		216.2.d.c.109.4	yes	8
36.7	odd	6		648.2.n.q.109.6		16
36.11	even	6		648.2.n.q.109.3		16
36.23	even	6		648.2.n.q.541.8		16
36.31	odd	6		648.2.n.q.541.1		16
48.5	odd	4		6912.2.a.ci.1.3		4
48.11	even	4		6912.2.a.cj.1.3		4
48.29	odd	4		6912.2.a.cd.1.2		4
48.35	even	4		6912.2.a.cc.1.2		4
72.5	odd	6		2592.2.r.q.2161.6		16
72.11	even	6		648.2.n.q.109.8		16
72.13	even	6		2592.2.r.q.2161.4		16
72.29	odd	6		2592.2.r.q.433.3		16
72.43	odd	6		648.2.n.q.109.1		16
72.59	even	6		648.2.n.q.541.3		16
72.61	even	6		2592.2.r.q.433.5		16
72.67	odd	6		648.2.n.q.541.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.c.109.3	✓	8	12.11	even	2
216.2.d.c.109.4	yes	8	24.11	even	2
216.2.d.c.109.5	yes	8	8.3	odd	2
216.2.d.c.109.6	yes	8	4.3	odd	2
648.2.n.q.109.1		16	72.43	odd	6
648.2.n.q.109.3		16	36.11	even	6
648.2.n.q.109.6		16	36.7	odd	6
648.2.n.q.109.8		16	72.11	even	6
648.2.n.q.541.1		16	36.31	odd	6
648.2.n.q.541.3		16	72.59	even	6
648.2.n.q.541.6		16	72.67	odd	6
648.2.n.q.541.8		16	36.23	even	6
864.2.d.c.433.3		8	3.2	odd	2	inner
864.2.d.c.433.4		8	8.5	even	2	inner
864.2.d.c.433.5		8	1.1	even	1	trivial
864.2.d.c.433.6		8	24.5	odd	2	inner
2592.2.r.q.433.3		16	72.29	odd	6
2592.2.r.q.433.4		16	9.7	even	3
2592.2.r.q.433.5		16	72.61	even	6
2592.2.r.q.433.6		16	9.2	odd	6
2592.2.r.q.2161.3		16	9.5	odd	6
2592.2.r.q.2161.4		16	72.13	even	6
2592.2.r.q.2161.5		16	9.4	even	3
2592.2.r.q.2161.6		16	72.5	odd	6
6912.2.a.cc.1.2		4	48.35	even	4
6912.2.a.cc.1.3		4	16.3	odd	4
6912.2.a.cd.1.2		4	48.29	odd	4
6912.2.a.cd.1.3		4	16.13	even	4
6912.2.a.ci.1.2		4	16.5	even	4
6912.2.a.ci.1.3		4	48.5	odd	4
6912.2.a.cj.1.2		4	16.11	odd	4
6912.2.a.cj.1.3		4	48.11	even	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.841723i q^{5} -2.64575 q^{7} +O(q^{10})\)	\(q+0.841723i q^{5} -2.64575 q^{7} -3.91044i q^{11} -0.645751i q^{13} -5.59388 q^{17} +4.29150i q^{19} -8.66259 q^{23} +4.29150 q^{25} -7.82087i q^{29} -2.00000 q^{31} -2.22699i q^{35} -6.64575i q^{37} -6.13742 q^{41} -7.29150i q^{43} -2.52517 q^{47} -7.82087i q^{53} +3.29150 q^{55} +7.27733i q^{59} +13.9373i q^{61} +0.543544 q^{65} +3.00000i q^{67} -9.58301 q^{73} +10.3460i q^{77} +3.35425 q^{79} -9.50432i q^{83} -4.70850i q^{85} +5.59388 q^{89} +1.70850i q^{91} -3.61226 q^{95} -8.29150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{25} - 16 q^{31} - 16 q^{55} + 8 q^{73} + 48 q^{79} - 24 q^{97}+O(q^{100}) \) 8 * q - 8 * q^25 - 16 * q^31 - 16 * q^55 + 8 * q^73 + 48 * q^79 - 24 * q^97

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