LMFDB - Embedded newform 864.2.f.a.431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(431,864)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.f (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:	8.0.23123460096.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 $ x^8 + 2x^6 + 6x^4 + 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.5
Root			$-1.08766 - 0.903873i$ of defining polynomial
Character	$\chi$	$=$	864.431
Dual form			864.2.f.a.431.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59245 q^{5} -1.43937i q^{7} +O(q^{10})$	$q+1.59245 q^{5} -1.43937i q^{7} +4.93886i q^{11} +5.37182i q^{13} +1.32336i q^{17} -0.267949 q^{19} +5.94311 q^{23} -2.46410 q^{25} +8.70131 q^{29} -7.86488i q^{31} -2.29213i q^{35} -2.49307i q^{37} +9.87771i q^{41} -2.00000 q^{43} +9.12801 q^{47} +4.92820 q^{49} -5.51641 q^{53} +7.86488i q^{55} +4.93886i q^{59} -5.37182i q^{61} +8.55435i q^{65} -1.19615 q^{67} -5.51641 q^{71} -7.92820 q^{73} +7.10886 q^{77} +12.1830i q^{79} +7.23099i q^{83} +2.10739i q^{85} -15.7853i q^{89} +7.73205 q^{91} -0.426696 q^{95} +9.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{19} + 8 q^{25} - 16 q^{43} - 16 q^{49} + 32 q^{67} - 8 q^{73} + 48 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^19 + 8 * q^25 - 16 * q^43 - 16 * q^49 + 32 * q^67 - 8 * q^73 + 48 * q^91 - 8 * 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
$5$	1.59245	0.712165	0.356083	−	0.934454i	$-0.384112\pi$
$5$	1.59245	0.712165	0.356083	+	0.934454i	$0.384112\pi$
$6$	0	0
$7$	− 1.43937i	− 0.544032i	−0.962293	−	0.272016i	$-0.912310\pi$
$7$	− 1.43937i	− 0.544032i	0.962293	−	0.272016i	$-0.0876904\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.93886i	1.48912i	0.667555	+	0.744561i	$0.267341\pi$
$11$	4.93886i	1.48912i	−0.667555	+	0.744561i	$0.732659\pi$
$12$	0	0
$13$	5.37182i	1.48987i	0.667135	+	0.744937i	$0.267520\pi$
$13$	5.37182i	1.48987i	−0.667135	+	0.744937i	$0.732480\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.32336i	0.320963i	0.987039	+	0.160481i	$0.0513046\pi$
$17$	1.32336i	0.320963i	−0.987039	+	0.160481i	$0.948695\pi$
$18$	0	0
$19$	−0.267949	−0.0614718	−0.0307359	−	0.999528i	$-0.509785\pi$
$19$	−0.267949	−0.0614718	−0.0307359	+	0.999528i	$0.509785\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.94311	1.23922	0.619612	−	0.784909i	$-0.287290\pi$
$23$	5.94311	1.23922	0.619612	+	0.784909i	$0.287290\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.70131	1.61579	0.807896	−	0.589324i	$-0.200606\pi$
$29$	8.70131	1.61579	0.807896	+	0.589324i	$0.200606\pi$
$30$	0	0
$31$	− 7.86488i	− 1.41257i	−0.707925	−	0.706287i	$-0.750369\pi$
$31$	− 7.86488i	− 1.41257i	0.707925	−	0.706287i	$-0.249631\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.29213i	− 0.387441i
$36$	0	0
$37$	− 2.49307i	− 0.409858i	−0.978777	−	0.204929i	$-0.934304\pi$
$37$	− 2.49307i	− 0.409858i	0.978777	−	0.204929i	$-0.0656963\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.87771i	1.54264i	0.636448	+	0.771320i	$0.280403\pi$
$41$	9.87771i	1.54264i	−0.636448	+	0.771320i	$0.719597\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.12801	1.33146	0.665728	−	0.746194i	$-0.268121\pi$
$47$	9.12801	1.33146	0.665728	+	0.746194i	$0.268121\pi$
$48$	0	0
$49$	4.92820	0.704029
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.51641	−0.757737	−0.378869	−	0.925450i	$-0.623687\pi$
$53$	−5.51641	−0.757737	−0.378869	+	0.925450i	$0.623687\pi$
$54$	0	0
$55$	7.86488i	1.06050i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.93886i	0.642984i	0.946912	+	0.321492i	$0.104184\pi$
$59$	4.93886i	0.642984i	−0.946912	+	0.321492i	$0.895816\pi$
$60$	0	0
$61$	− 5.37182i	− 0.687791i	−0.939008	−	0.343895i	$-0.888253\pi$
$61$	− 5.37182i	− 0.687791i	0.939008	−	0.343895i	$-0.111747\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.55435i	1.06104i
$66$	0	0
$67$	−1.19615	−0.146133	−0.0730666	−	0.997327i	$-0.523279\pi$
$67$	−1.19615	−0.146133	−0.0730666	+	0.997327i	$0.523279\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.51641	−0.654677	−0.327339	−	0.944907i	$-0.606152\pi$
$71$	−5.51641	−0.654677	−0.327339	+	0.944907i	$0.606152\pi$
$72$	0	0
$73$	−7.92820	−0.927926	−0.463963	−	0.885855i	$-0.653573\pi$
$73$	−7.92820	−0.927926	−0.463963	+	0.885855i	$0.653573\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.10886	0.810130
$78$	0	0
$79$	12.1830i	1.37070i	0.728216	+	0.685348i	$0.240350\pi$
$79$	12.1830i	1.37070i	−0.728216	+	0.685348i	$0.759650\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.23099i	0.793704i	0.917883	+	0.396852i	$0.129897\pi$
$83$	7.23099i	0.793704i	−0.917883	+	0.396852i	$0.870103\pi$
$84$	0	0
$85$	2.10739i	0.228578i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 15.7853i	− 1.67324i	−0.547782	−	0.836621i	$-0.684528\pi$
$89$	− 15.7853i	− 1.67324i	0.547782	−	0.836621i	$-0.315472\pi$
$90$	0	0
$91$	7.73205	0.810539
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.426696	−0.0437781
$96$	0	0
$97$	9.39230	0.953644	0.476822	−	0.879000i	$-0.341789\pi$
$97$	9.39230	0.953644	0.476822	+	0.879000i	$0.341789\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.18490	0.316909	0.158455	−	0.987366i	$-0.449349\pi$
$101$	3.18490	0.316909	0.158455	+	0.987366i	$0.449349\pi$
$102$	0	0
$103$	1.43937i	0.141826i	0.997483	+	0.0709129i	$0.0225912\pi$
$103$	1.43937i	0.141826i	−0.997483	+	0.0709129i	$0.977409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.1698i	− 1.17650i	−0.808678	−	0.588252i	$-0.799816\pi$
$107$	− 12.1698i	− 1.17650i	0.808678	−	0.588252i	$-0.200184\pi$
$108$	0	0
$109$	− 13.6224i	− 1.30479i	−0.757880	−	0.652394i	$-0.773765\pi$
$109$	− 13.6224i	− 1.30479i	0.757880	−	0.652394i	$-0.226235\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 11.2011i	− 1.05371i	−0.849956	−	0.526854i	$-0.823371\pi$
$113$	− 11.2011i	− 1.05371i	0.849956	−	0.526854i	$-0.176629\pi$
$114$	0	0
$115$	9.46410	0.882532
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.90481	0.174614
$120$	0	0
$121$	−13.3923	−1.21748
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8862	−1.06314
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.23099i	0.631774i	0.948797	+	0.315887i	$0.102302\pi$
$131$	7.23099i	0.631774i	−0.948797	+	0.315887i	$0.897698\pi$
$132$	0	0
$133$	0.385679i	0.0334426i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.90763i	0.504722i	0.967633	+	0.252361i	$0.0812071\pi$
$137$	5.90763i	0.504722i	−0.967633	+	0.252361i	$0.918793\pi$
$138$	0	0
$139$	−7.19615	−0.610370	−0.305185	−	0.952293i	$-0.598718\pi$
$139$	−7.19615	−0.610370	−0.305185	+	0.952293i	$0.598718\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.5306	−2.21860
$144$	0	0
$145$	13.8564	1.15071
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.18490	−0.260917	−0.130459	−	0.991454i	$-0.541645\pi$
$149$	−3.18490	−0.260917	−0.130459	+	0.991454i	$0.541645\pi$
$150$	0	0
$151$	− 6.42551i	− 0.522901i	−0.965217	−	0.261450i	$-0.915799\pi$
$151$	− 6.42551i	− 0.522901i	0.965217	−	0.261450i	$-0.0842008\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 12.5244i	− 1.00599i
$156$	0	0
$157$	− 2.10739i	− 0.168188i	−0.996458	−	0.0840940i	$-0.973200\pi$
$157$	− 2.10739i	− 0.168188i	0.996458	−	0.0840940i	$-0.0267996\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.55435i	− 0.674177i
$162$	0	0
$163$	−19.1962	−1.50356	−0.751779	−	0.659415i	$-0.770804\pi$
$163$	−19.1962	−1.50356	−0.751779	+	0.659415i	$0.770804\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.08971	−0.393854	−0.196927	−	0.980418i	$-0.563096\pi$
$167$	−5.08971	−0.393854	−0.196927	+	0.980418i	$0.563096\pi$
$168$	0	0
$169$	−15.8564	−1.21972
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.8862	−0.903692	−0.451846	−	0.892096i	$-0.649234\pi$
$173$	−11.8862	−0.903692	−0.451846	+	0.892096i	$0.649234\pi$
$174$	0	0
$175$	3.54676i	0.268110i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.87771i	− 0.738295i	−0.929371	−	0.369147i	$-0.879650\pi$
$179$	− 9.87771i	− 0.738295i	0.929371	−	0.369147i	$-0.120350\pi$
$180$	0	0
$181$	16.1154i	1.19785i	0.800804	+	0.598926i	$0.204406\pi$
$181$	16.1154i	1.19785i	−0.800804	+	0.598926i	$0.795594\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.97009i	− 0.291887i
$186$	0	0
$187$	−6.53590	−0.477952
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.12801	0.660479	0.330240	−	0.943897i	$-0.392870\pi$
$191$	9.12801	0.660479	0.330240	+	0.943897i	$0.392870\pi$
$192$	0	0
$193$	15.3923	1.10796	0.553981	−	0.832529i	$-0.313108\pi$
$193$	15.3923	1.10796	0.553981	+	0.832529i	$0.313108\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.8485	−1.41414	−0.707072	−	0.707141i	$-0.749984\pi$
$197$	−19.8485	−1.41414	−0.707072	+	0.707141i	$0.749984\pi$
$198$	0	0
$199$	− 14.2904i	− 1.01302i	−0.862234	−	0.506510i	$-0.830935\pi$
$199$	− 14.2904i	− 1.01302i	0.862234	−	0.506510i	$-0.169065\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.5244i	− 0.879043i
$204$	0	0
$205$	15.7298i	1.09861i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.32336i	− 0.0915389i
$210$	0	0
$211$	17.0526	1.17395	0.586973	−	0.809606i	$-0.300319\pi$
$211$	17.0526	1.17395	0.586973	+	0.809606i	$0.300319\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.18490	−0.217208
$216$	0	0
$217$	−11.3205	−0.768486
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.10886	−0.478194
$222$	0	0
$223$	− 19.3799i	− 1.29777i	−0.760885	−	0.648886i	$-0.775235\pi$
$223$	− 19.3799i	− 1.29777i	0.760885	−	0.648886i	$-0.224765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.87771i	− 0.655607i	−0.944746	−	0.327803i	$-0.893692\pi$
$227$	− 9.87771i	− 0.655607i	0.944746	−	0.327803i	$-0.106308\pi$
$228$	0	0
$229$	− 15.7298i	− 1.03945i	−0.854333	−	0.519726i	$-0.826034\pi$
$229$	− 15.7298i	− 1.03945i	0.854333	−	0.519726i	$-0.173966\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.64673i	− 0.173393i	−0.996235	−	0.0866964i	$-0.972369\pi$
$233$	− 2.64673i	− 0.173393i	0.996235	−	0.0866964i	$-0.0276310\pi$
$234$	0	0
$235$	14.5359	0.948217
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.36980	−0.412028	−0.206014	−	0.978549i	$-0.566049\pi$
$239$	−6.36980	−0.412028	−0.206014	+	0.978549i	$0.566049\pi$
$240$	0	0
$241$	15.3923	0.991506	0.495753	−	0.868464i	$-0.334892\pi$
$241$	15.3923	0.991506	0.495753	+	0.868464i	$0.334892\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.84792	0.501385
$246$	0	0
$247$	− 1.43937i	− 0.0915852i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.64673i	0.167060i	0.996505	+	0.0835299i	$0.0266194\pi$
$251$	2.64673i	0.167060i	−0.996505	+	0.0835299i	$0.973381\pi$
$252$	0	0
$253$	29.3521i	1.84535i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.87771i	0.616155i	0.951361	+	0.308077i	$0.0996856\pi$
$257$	9.87771i	0.616155i	−0.951361	+	0.308077i	$0.900314\pi$
$258$	0	0
$259$	−3.58846	−0.222976
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.7724	−1.46587	−0.732935	−	0.680298i	$-0.761850\pi$
$263$	−23.7724	−1.46587	−0.732935	+	0.680298i	$0.761850\pi$
$264$	0	0
$265$	−8.78461	−0.539634
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.739059	−0.0450612	−0.0225306	−	0.999746i	$-0.507172\pi$
$269$	−0.739059	−0.0450612	−0.0225306	+	0.999746i	$0.507172\pi$
$270$	0	0
$271$	− 4.31812i	− 0.262307i	−0.991362	−	0.131154i	$-0.958132\pi$
$271$	− 4.31812i	− 0.262307i	0.991362	−	0.131154i	$-0.0418681\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.1698i	− 0.733869i
$276$	0	0
$277$	2.10739i	0.126621i	0.997994	+	0.0633104i	$0.0201658\pi$
$277$	2.10739i	0.126621i	−0.997994	+	0.0633104i	$0.979834\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.23099i	− 0.431365i	−0.976464	−	0.215682i	$-0.930802\pi$
$281$	− 7.23099i	− 0.431365i	0.976464	−	0.215682i	$-0.0691975\pi$
$282$	0	0
$283$	−0.143594	−0.00853575	−0.00426787	−	0.999991i	$-0.501359\pi$
$283$	−0.143594	−0.00853575	−0.00426787	+	0.999991i	$0.501359\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.2177	0.839246
$288$	0	0
$289$	15.2487	0.896983
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.10886	0.415304	0.207652	−	0.978203i	$-0.433418\pi$
$293$	7.10886	0.415304	0.207652	+	0.978203i	$0.433418\pi$
$294$	0	0
$295$	7.86488i	0.457911i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	31.9253i	1.84629i
$300$	0	0
$301$	2.87875i	0.165928i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.55435i	− 0.489821i
$306$	0	0
$307$	−7.07180	−0.403609	−0.201804	−	0.979426i	$-0.564681\pi$
$307$	−7.07180	−0.403609	−0.201804	+	0.979426i	$0.564681\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.9759	0.962616	0.481308	−	0.876551i	$-0.340162\pi$
$311$	16.9759	0.962616	0.481308	+	0.876551i	$0.340162\pi$
$312$	0	0
$313$	−11.3923	−0.643931	−0.321966	−	0.946751i	$-0.604344\pi$
$313$	−11.3923	−0.643931	−0.321966	+	0.946751i	$0.604344\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.33151	−0.130951	−0.0654753	−	0.997854i	$-0.520856\pi$
$317$	−2.33151	−0.130951	−0.0654753	+	0.997854i	$0.520856\pi$
$318$	0	0
$319$	42.9745i	2.40611i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.354594i	− 0.0197301i
$324$	0	0
$325$	− 13.2367i	− 0.734240i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 13.1386i	− 0.724355i
$330$	0	0
$331$	−26.1244	−1.43592	−0.717962	−	0.696082i	$-0.754925\pi$
$331$	−26.1244	−1.43592	−0.717962	+	0.696082i	$0.754925\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.90481	−0.104071
$336$	0	0
$337$	9.39230	0.511631	0.255816	−	0.966726i	$-0.417656\pi$
$337$	9.39230	0.511631	0.255816	+	0.966726i	$0.417656\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	38.8435	2.10350
$342$	0	0
$343$	− 17.1691i	− 0.927047i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.7554i	1.06053i	0.847833	+	0.530263i	$0.177907\pi$
$347$	19.7554i	1.06053i	−0.847833	+	0.530263i	$0.822093\pi$
$348$	0	0
$349$	− 23.9803i	− 1.28364i	−0.766856	−	0.641819i	$-0.778180\pi$
$349$	− 23.9803i	− 1.28364i	0.766856	−	0.641819i	$-0.221820\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.7554i	− 1.05148i	−0.850647	−	0.525738i	$-0.823789\pi$
$353$	− 19.7554i	− 1.05148i	0.850647	−	0.525738i	$-0.176211\pi$
$354$	0	0
$355$	−8.78461	−0.466239
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.79650	0.358705	0.179353	−	0.983785i	$-0.442600\pi$
$359$	6.79650	0.358705	0.179353	+	0.983785i	$0.442600\pi$
$360$	0	0
$361$	−18.9282	−0.996221
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.6253	−0.660837
$366$	0	0
$367$	− 16.3978i	− 0.855957i	−0.903789	−	0.427979i	$-0.859226\pi$
$367$	− 16.3978i	− 0.855957i	0.903789	−	0.427979i	$-0.140774\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.94018i	0.412233i
$372$	0	0
$373$	− 18.2228i	− 0.943543i	−0.881721	−	0.471771i	$-0.843615\pi$
$373$	− 18.2228i	− 0.943543i	0.881721	−	0.471771i	$-0.156385\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46.7418i	2.40733i
$378$	0	0
$379$	29.7321	1.52723	0.763616	−	0.645670i	$-0.223422\pi$
$379$	29.7321	1.52723	0.763616	+	0.645670i	$0.223422\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.8862	−0.607357	−0.303679	−	0.952775i	$-0.598215\pi$
$383$	−11.8862	−0.607357	−0.303679	+	0.952775i	$0.598215\pi$
$384$	0	0
$385$	11.3205	0.576947
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.96225	0.403702	0.201851	−	0.979416i	$-0.435304\pi$
$389$	7.96225	0.403702	0.201851	+	0.979416i	$0.435304\pi$
$390$	0	0
$391$	7.86488i	0.397744i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.4008i	0.976162i
$396$	0	0
$397$	− 13.6224i	− 0.683688i	−0.939757	−	0.341844i	$-0.888949\pi$
$397$	− 13.6224i	− 0.683688i	0.939757	−	0.341844i	$-0.111051\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.5109i	1.97308i	0.163527	+	0.986539i	$0.447713\pi$
$401$	39.5109i	1.97308i	−0.163527	+	0.986539i	$0.552287\pi$
$402$	0	0
$403$	42.2487	2.10456
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.3129	0.610328
$408$	0	0
$409$	−29.3923	−1.45336	−0.726678	−	0.686978i	$-0.758937\pi$
$409$	−29.3923	−1.45336	−0.726678	+	0.686978i	$0.758937\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.10886	0.349804
$414$	0	0
$415$	11.5150i	0.565249i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.354594i	0.0173230i	0.999962	+	0.00866152i	$0.00275708\pi$
$419$	0.354594i	0.0173230i	−0.999962	+	0.00866152i	$0.997243\pi$
$420$	0	0
$421$	13.2367i	0.645117i	0.946549	+	0.322559i	$0.104543\pi$
$421$	13.2367i	0.645117i	−0.946549	+	0.322559i	$0.895457\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.26090i	− 0.158177i
$426$	0	0
$427$	−7.73205	−0.374180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.4168	1.85047	0.925237	−	0.379390i	$-0.123866\pi$
$431$	38.4168	1.85047	0.925237	+	0.379390i	$0.123866\pi$
$432$	0	0
$433$	11.4641	0.550930	0.275465	−	0.961311i	$-0.411168\pi$
$433$	11.4641	0.550930	0.275465	+	0.961311i	$0.411168\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.59245	−0.0761772
$438$	0	0
$439$	35.1096i	1.67569i	0.545907	+	0.837846i	$0.316185\pi$
$439$	35.1096i	1.67569i	−0.545907	+	0.837846i	$0.683815\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 31.5707i	− 1.49997i	−0.661456	−	0.749984i	$-0.730061\pi$
$443$	− 31.5707i	− 1.49997i	0.661456	−	0.749984i	$-0.269939\pi$
$444$	0	0
$445$	− 25.1374i	− 1.19163i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.0163i	1.08621i	0.839666	+	0.543104i	$0.182751\pi$
$449$	23.0163i	1.08621i	−0.839666	+	0.543104i	$0.817249\pi$
$450$	0	0
$451$	−48.7846	−2.29718
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.3129	0.577238
$456$	0	0
$457$	−14.3923	−0.673244	−0.336622	−	0.941640i	$-0.609284\pi$
$457$	−14.3923	−0.673244	−0.336622	+	0.941640i	$0.609284\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.9951	0.884689	0.442344	−	0.896845i	$-0.354147\pi$
$461$	18.9951	0.884689	0.442344	+	0.896845i	$0.354147\pi$
$462$	0	0
$463$	− 30.7915i	− 1.43100i	−0.698611	−	0.715502i	$-0.746198\pi$
$463$	− 30.7915i	− 1.43100i	0.698611	−	0.715502i	$-0.253802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.4633i	0.808105i	0.914736	+	0.404052i	$0.132399\pi$
$467$	17.4633i	0.808105i	−0.914736	+	0.404052i	$0.867601\pi$
$468$	0	0
$469$	1.72171i	0.0795012i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 9.87771i	− 0.454178i
$474$	0	0
$475$	0.660254	0.0302945
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.18490	−0.145522	−0.0727609	−	0.997349i	$-0.523181\pi$
$479$	−3.18490	−0.145522	−0.0727609	+	0.997349i	$0.523181\pi$
$480$	0	0
$481$	13.3923	0.610637
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.9568	0.679152
$486$	0	0
$487$	37.8850i	1.71674i	0.513035	+	0.858368i	$0.328521\pi$
$487$	37.8850i	1.71674i	−0.513035	+	0.858368i	$0.671479\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 16.7541i	− 0.756102i	−0.925785	−	0.378051i	$-0.876594\pi$
$491$	− 16.7541i	− 0.756102i	0.925785	−	0.378051i	$-0.123406\pi$
$492$	0	0
$493$	11.5150i	0.518609i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.94018i	0.356166i
$498$	0	0
$499$	26.3923	1.18148	0.590741	−	0.806861i	$-0.298836\pi$
$499$	26.3923	1.18148	0.590741	+	0.806861i	$0.298836\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.3785	1.53286	0.766432	−	0.642326i	$-0.222030\pi$
$503$	34.3785	1.53286	0.766432	+	0.642326i	$0.222030\pi$
$504$	0	0
$505$	5.07180	0.225692
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.1800	0.983110	0.491555	−	0.870847i	$-0.336429\pi$
$509$	22.1800	0.983110	0.491555	+	0.870847i	$0.336429\pi$
$510$	0	0
$511$	11.4116i	0.504822i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.29213i	0.101003i
$516$	0	0
$517$	45.0819i	1.98270i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.32336i	0.0579776i	0.999580	+	0.0289888i	$0.00922871\pi$
$521$	1.32336i	0.0579776i	−0.999580	+	0.0289888i	$0.990771\pi$
$522$	0	0
$523$	6.41154	0.280357	0.140179	−	0.990126i	$-0.455232\pi$
$523$	6.41154	0.280357	0.140179	+	0.990126i	$0.455232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.4081	0.453384
$528$	0	0
$529$	12.3205	0.535674
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−53.0613	−2.29834
$534$	0	0
$535$	− 19.3799i	− 0.837865i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.3397i	1.04838i
$540$	0	0
$541$	− 12.4653i	− 0.535927i	−0.963429	−	0.267963i	$-0.913649\pi$
$541$	− 12.4653i	− 0.535927i	0.963429	−	0.267963i	$-0.0863506\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 21.6930i	− 0.929224i
$546$	0	0
$547$	0.411543	0.0175963	0.00879815	−	0.999961i	$-0.497199\pi$
$547$	0.411543	0.0175963	0.00879815	+	0.999961i	$0.497199\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.33151	−0.0993256
$552$	0	0
$553$	17.5359	0.745702
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.3977	1.54222	0.771110	−	0.636702i	$-0.219702\pi$
$557$	36.3977	1.54222	0.771110	+	0.636702i	$0.219702\pi$
$558$	0	0
$559$	− 10.7436i	− 0.454407i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.8153i	0.497953i	0.968509	+	0.248977i	$0.0800943\pi$
$563$	11.8153i	0.497953i	−0.968509	+	0.248977i	$0.919906\pi$
$564$	0	0
$565$	− 17.8372i	− 0.750415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.5408i	1.48995i	0.667094	+	0.744973i	$0.267538\pi$
$569$	35.5408i	1.48995i	−0.667094	+	0.744973i	$0.732462\pi$
$570$	0	0
$571$	−7.19615	−0.301150	−0.150575	−	0.988599i	$-0.548112\pi$
$571$	−7.19615	−0.301150	−0.150575	+	0.988599i	$0.548112\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.6444	−0.610714
$576$	0	0
$577$	−1.92820	−0.0802722	−0.0401361	−	0.999194i	$-0.512779\pi$
$577$	−1.92820	−0.0802722	−0.0401361	+	0.999194i	$0.512779\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.4081	0.431801
$582$	0	0
$583$	− 27.2448i	− 1.12836i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.7541i	− 0.691516i	−0.938324	−	0.345758i	$-0.887622\pi$
$587$	− 16.7541i	− 0.691516i	0.938324	−	0.345758i	$-0.112378\pi$
$588$	0	0
$589$	2.10739i	0.0868335i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.93754i	0.0795651i	0.999208	+	0.0397826i	$0.0126665\pi$
$593$	1.93754i	0.0795651i	−0.999208	+	0.0397826i	$0.987333\pi$
$594$	0	0
$595$	3.03332	0.124354
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.5066	−1.77763	−0.888815	−	0.458267i	$-0.848471\pi$
$599$	−43.5066	−1.77763	−0.888815	+	0.458267i	$0.848471\pi$
$600$	0	0
$601$	9.60770	0.391906	0.195953	−	0.980613i	$-0.437220\pi$
$601$	9.60770	0.391906	0.195953	+	0.980613i	$0.437220\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.3266	−0.867049
$606$	0	0
$607$	− 27.1414i	− 1.10164i	−0.834625	−	0.550818i	$-0.814316\pi$
$607$	− 27.1414i	− 1.10164i	0.834625	−	0.550818i	$-0.185684\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	49.0340i	1.98370i
$612$	0	0
$613$	2.49307i	0.100694i	0.998732	+	0.0503470i	$0.0160327\pi$
$613$	2.49307i	0.100694i	−0.998732	+	0.0503470i	$0.983967\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23.7255i	− 0.955153i	−0.878590	−	0.477577i	$-0.841515\pi$
$617$	− 23.7255i	− 0.955153i	0.878590	−	0.477577i	$-0.158485\pi$
$618$	0	0
$619$	−26.8038	−1.07734	−0.538669	−	0.842518i	$-0.681073\pi$
$619$	−26.8038	−1.07734	−0.538669	+	0.842518i	$0.681073\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22.7210	−0.910298
$624$	0	0
$625$	−6.60770	−0.264308
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.29923	0.131549
$630$	0	0
$631$	− 17.9405i	− 0.714200i	−0.934066	−	0.357100i	$-0.883766\pi$
$631$	− 17.9405i	− 0.714200i	0.934066	−	0.357100i	$-0.116234\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.4734i	1.04891i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 11.8153i	− 0.466674i	−0.972396	−	0.233337i	$-0.925035\pi$
$641$	− 11.8153i	− 0.466674i	0.972396	−	0.233337i	$-0.0749646\pi$
$642$	0	0
$643$	−22.7846	−0.898537	−0.449269	−	0.893397i	$-0.648315\pi$
$643$	−22.7846	−0.898537	−0.449269	+	0.893397i	$0.648315\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.47812	0.0581108	0.0290554	−	0.999578i	$-0.490750\pi$
$647$	1.47812	0.0581108	0.0290554	+	0.999578i	$0.490750\pi$
$648$	0	0
$649$	−24.3923	−0.957482
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.8107	−1.08832	−0.544159	−	0.838982i	$-0.683151\pi$
$653$	−27.8107	−1.08832	−0.544159	+	0.838982i	$0.683151\pi$
$654$	0	0
$655$	11.5150i	0.449928i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 22.4022i	− 0.872664i	−0.899786	−	0.436332i	$-0.856277\pi$
$659$	− 22.4022i	− 0.872664i	0.899786	−	0.436332i	$-0.143723\pi$
$660$	0	0
$661$	41.2528i	1.60455i	0.596956	+	0.802274i	$0.296377\pi$
$661$	41.2528i	1.60455i	−0.596956	+	0.802274i	$0.703623\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.614175i	0.0238167i
$666$	0	0
$667$	51.7128	2.00233
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.5306	1.02420
$672$	0	0
$673$	−13.2487	−0.510700	−0.255350	−	0.966849i	$-0.582191\pi$
$673$	−13.2487	−0.510700	−0.255350	+	0.966849i	$0.582191\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.0005	0.461218	0.230609	−	0.973046i	$-0.425928\pi$
$677$	12.0005	0.461218	0.230609	+	0.973046i	$0.425928\pi$
$678$	0	0
$679$	− 13.5190i	− 0.518813i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 34.9266i	− 1.33643i	−0.743969	−	0.668214i	$-0.767059\pi$
$683$	− 34.9266i	− 1.33643i	0.743969	−	0.668214i	$-0.232941\pi$
$684$	0	0
$685$	9.40760i	0.359446i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 29.6331i	− 1.12893i
$690$	0	0
$691$	23.8564	0.907540	0.453770	−	0.891119i	$-0.350079\pi$
$691$	23.8564	0.907540	0.453770	+	0.891119i	$0.350079\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.4595	−0.434684
$696$	0	0
$697$	−13.0718	−0.495130
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.2883	0.652970	0.326485	−	0.945202i	$-0.394136\pi$
$701$	17.2883	0.652970	0.326485	+	0.945202i	$0.394136\pi$
$702$	0	0
$703$	0.668016i	0.0251947i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.58426i	− 0.172409i
$708$	0	0
$709$	− 27.6304i	− 1.03768i	−0.854870	−	0.518841i	$-0.826364\pi$
$709$	− 27.6304i	− 1.03768i	0.854870	−	0.518841i	$-0.173636\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 46.7418i	− 1.75050i
$714$	0	0
$715$	−42.2487	−1.58001
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.0711	0.562058	0.281029	−	0.959699i	$-0.409324\pi$
$719$	15.0711	0.562058	0.281029	+	0.959699i	$0.409324\pi$
$720$	0	0
$721$	2.07180	0.0771577
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.4409	−0.796296
$726$	0	0
$727$	− 19.3799i	− 0.718760i	−0.933191	−	0.359380i	$-0.882988\pi$
$727$	− 19.3799i	− 0.718760i	0.933191	−	0.359380i	$-0.117012\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.64673i	− 0.0978927i
$732$	0	0
$733$	45.0819i	1.66514i	0.553921	+	0.832569i	$0.313131\pi$
$733$	45.0819i	1.66514i	−0.553921	+	0.832569i	$0.686869\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.90763i	− 0.217610i
$738$	0	0
$739$	−48.6410	−1.78929	−0.894644	−	0.446779i	$-0.852571\pi$
$739$	−48.6410	−1.78929	−0.894644	+	0.446779i	$0.852571\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.3085	−1.58884	−0.794418	−	0.607372i	$-0.792224\pi$
$743$	−43.3085	−1.58884	−0.794418	+	0.607372i	$0.792224\pi$
$744$	0	0
$745$	−5.07180	−0.185816
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.5170	−0.640056
$750$	0	0
$751$	− 7.19687i	− 0.262617i	−0.991342	−	0.131309i	$-0.958082\pi$
$751$	− 7.19687i	− 0.262617i	0.991342	−	0.131309i	$-0.0419179\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10.2323i	− 0.372392i
$756$	0	0
$757$	− 36.0600i	− 1.31062i	−0.755359	−	0.655311i	$-0.772537\pi$
$757$	− 36.0600i	− 1.31062i	0.755359	−	0.655311i	$-0.227463\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.3722i	0.955993i	0.878362	+	0.477996i	$0.158637\pi$
$761$	26.3722i	0.955993i	−0.878362	+	0.477996i	$0.841363\pi$
$762$	0	0
$763$	−19.6077	−0.709846
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.5306	−0.957965
$768$	0	0
$769$	17.9282	0.646508	0.323254	−	0.946312i	$-0.395223\pi$
$769$	17.9282	0.646508	0.323254	+	0.946312i	$0.395223\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.1805	−1.22939	−0.614694	−	0.788766i	$-0.710720\pi$
$773$	−34.1805	−1.22939	−0.614694	+	0.788766i	$0.710720\pi$
$774$	0	0
$775$	19.3799i	0.696146i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.64673i	− 0.0948288i
$780$	0	0
$781$	− 27.2448i	− 0.974894i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3.35591i	− 0.119778i
$786$	0	0
$787$	51.4449	1.83381	0.916906	−	0.399104i	$-0.130679\pi$
$787$	51.4449	1.83381	0.916906	+	0.399104i	$0.130679\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.1225	−0.573251
$792$	0	0
$793$	28.8564	1.02472
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.84792	0.277988	0.138994	−	0.990293i	$-0.455613\pi$
$797$	7.84792	0.277988	0.138994	+	0.990293i	$0.455613\pi$
$798$	0	0
$799$	12.0797i	0.427348i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 39.1563i	− 1.38179i
$804$	0	0
$805$	− 13.6224i	− 0.480126i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 2.64673i	− 0.0930539i	−0.998917	−	0.0465270i	$-0.985185\pi$
$809$	− 2.64673i	− 0.0930539i	0.998917	−	0.0465270i	$-0.0148153\pi$
$810$	0	0
$811$	40.9282	1.43718	0.718592	−	0.695432i	$-0.244787\pi$
$811$	40.9282	1.43718	0.718592	+	0.695432i	$0.244787\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−30.5689	−1.07078
$816$	0	0
$817$	0.535898	0.0187487
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.6636	−0.581562	−0.290781	−	0.956790i	$-0.593915\pi$
$821$	−16.6636	−0.581562	−0.290781	+	0.956790i	$0.593915\pi$
$822$	0	0
$823$	28.6841i	0.999866i	0.866064	+	0.499933i	$0.166642\pi$
$823$	28.6841i	0.999866i	−0.866064	+	0.499933i	$0.833358\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.00132i	− 0.104366i	−0.998638	−	0.0521830i	$-0.983382\pi$
$827$	− 3.00132i	− 0.104366i	0.998638	−	0.0521830i	$-0.0166179\pi$
$828$	0	0
$829$	12.4653i	0.432939i	0.976289	+	0.216470i	$0.0694542\pi$
$829$	12.4653i	0.432939i	−0.976289	+	0.216470i	$0.930546\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.52180i	0.225967i
$834$	0	0
$835$	−8.10512	−0.280489
$836$	0	0
$837$	0	0
$838$	0	0
$839$	57.0996	1.97130	0.985648	−	0.168815i	$-0.0539942\pi$
$839$	57.0996	1.97130	0.985648	+	0.168815i	$0.0539942\pi$
$840$	0	0
$841$	46.7128	1.61079
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−25.2505	−0.868645
$846$	0	0
$847$	19.2765i	0.662349i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 14.8166i	− 0.507905i
$852$	0	0
$853$	− 0.385679i	− 0.0132054i	−0.999978	−	0.00660270i	$-0.997898\pi$
$853$	− 0.385679i	− 0.0132054i	0.999978	−	0.00660270i	$-0.00210172\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 36.8641i	− 1.25925i	−0.776897	−	0.629627i	$-0.783208\pi$
$857$	− 36.8641i	− 1.25925i	0.776897	−	0.629627i	$-0.216792\pi$
$858$	0	0
$859$	19.5885	0.668350	0.334175	−	0.942511i	$-0.391542\pi$
$859$	19.5885	0.668350	0.334175	+	0.942511i	$0.391542\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.27462	−0.281671	−0.140836	−	0.990033i	$-0.544979\pi$
$863$	−8.27462	−0.281671	−0.140836	+	0.990033i	$0.544979\pi$
$864$	0	0
$865$	−18.9282	−0.643578
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.1701	−2.04113
$870$	0	0
$871$	− 6.42551i	− 0.217720i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17.1087i	0.578380i
$876$	0	0
$877$	− 21.8729i	− 0.738597i	−0.929311	−	0.369298i	$-0.879598\pi$
$877$	− 21.8729i	− 0.738597i	0.929311	−	0.369298i	$-0.120402\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.5408i	1.19740i	0.800974	+	0.598699i	$0.204316\pi$
$881$	35.5408i	1.19740i	−0.800974	+	0.598699i	$0.795684\pi$
$882$	0	0
$883$	−19.8756	−0.668869	−0.334434	−	0.942419i	$-0.608545\pi$
$883$	−19.8756	−0.668869	−0.334434	+	0.942419i	$0.608545\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.7341	0.662607	0.331304	−	0.943524i	$-0.392512\pi$
$887$	19.7341	0.662607	0.331304	+	0.943524i	$0.392512\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.44584	−0.0818470
$894$	0	0
$895$	− 15.7298i	− 0.525788i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 68.4348i	− 2.28243i
$900$	0	0
$901$	− 7.30021i	− 0.243205i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.6631i	0.853069i
$906$	0	0
$907$	−26.3731	−0.875703	−0.437852	−	0.899047i	$-0.644260\pi$
$907$	−26.3731	−0.875703	−0.437852	+	0.899047i	$0.644260\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.3599	−1.46971	−0.734855	−	0.678224i	$-0.762750\pi$
$911$	−44.3599	−1.46971	−0.734855	+	0.678224i	$0.762750\pi$
$912$	0	0
$913$	−35.7128	−1.18192
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.4081	0.343706
$918$	0	0
$919$	30.8949i	1.01913i	0.860433	+	0.509564i	$0.170193\pi$
$919$	30.8949i	1.01913i	−0.860433	+	0.509564i	$0.829807\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 29.6331i	− 0.975387i
$924$	0	0
$925$	6.14317i	0.201986i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.4620i	0.474482i	0.971451	+	0.237241i	$0.0762431\pi$
$929$	14.4620i	0.474482i	−0.971451	+	0.237241i	$0.923757\pi$
$930$	0	0
$931$	−1.32051	−0.0432779
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.4081	−0.340381
$936$	0	0
$937$	−2.60770	−0.0851897	−0.0425948	−	0.999092i	$-0.513562\pi$
$937$	−2.60770	−0.0851897	−0.0425948	+	0.999092i	$0.513562\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.7347	1.03452	0.517260	−	0.855828i	$-0.326952\pi$
$941$	31.7347	1.03452	0.517260	+	0.855828i	$0.326952\pi$
$942$	0	0
$943$	58.7043i	1.91167i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.6318i	0.865418i	0.901534	+	0.432709i	$0.142442\pi$
$947$	26.6318i	0.865418i	−0.901534	+	0.432709i	$0.857558\pi$
$948$	0	0
$949$	− 42.5888i	− 1.38249i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 57.9429i	− 1.87696i	−0.345340	−	0.938478i	$-0.612236\pi$
$953$	− 57.9429i	− 1.87696i	0.345340	−	0.938478i	$-0.387764\pi$
$954$	0	0
$955$	14.5359	0.470371
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.50328	0.274585
$960$	0	0
$961$	−30.8564	−0.995368
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.5115	0.789053
$966$	0	0
$967$	6.42551i	0.206630i	0.994649	+	0.103315i	$0.0329451\pi$
$967$	6.42551i	0.206630i	−0.994649	+	0.103315i	$0.967055\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.5122i	1.36428i	0.731221	+	0.682140i	$0.238951\pi$
$971$	42.5122i	1.36428i	−0.731221	+	0.682140i	$0.761049\pi$
$972$	0	0
$973$	10.3580i	0.332061i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 41.4484i	− 1.32605i	−0.748597	−	0.663026i	$-0.769272\pi$
$977$	− 41.4484i	− 1.32605i	0.748597	−	0.663026i	$-0.230728\pi$
$978$	0	0
$979$	77.9615	2.49166
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.1127	−1.72593	−0.862963	−	0.505267i	$-0.831394\pi$
$983$	−54.1127	−1.72593	−0.862963	+	0.505267i	$0.831394\pi$
$984$	0	0
$985$	−31.6077	−1.00710
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.8862	−0.377960
$990$	0	0
$991$	− 0.668016i	− 0.0212202i	−0.999944	−	0.0106101i	$-0.996623\pi$
$991$	− 0.668016i	− 0.0212202i	0.999944	−	0.0106101i	$-0.00337737\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 22.7567i	− 0.721437i
$996$	0	0
$997$	29.3521i	0.929592i	0.885418	+	0.464796i	$0.153872\pi$
$997$	29.3521i	0.929592i	−0.885418	+	0.464796i	$0.846128\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.f.a.431.5		8
3.2	odd	2	inner	864.2.f.a.431.3		8
4.3	odd	2		216.2.f.a.107.7	yes	8
8.3	odd	2	inner	864.2.f.a.431.4		8
8.5	even	2		216.2.f.a.107.1	✓	8
9.2	odd	6		2592.2.p.f.2159.5		16
9.4	even	3		2592.2.p.f.431.4		16
9.5	odd	6		2592.2.p.f.431.6		16
9.7	even	3		2592.2.p.f.2159.3		16
12.11	even	2		216.2.f.a.107.2	yes	8
24.5	odd	2		216.2.f.a.107.8	yes	8
24.11	even	2	inner	864.2.f.a.431.6		8
36.7	odd	6		648.2.l.f.539.2		16
36.11	even	6		648.2.l.f.539.7		16
36.23	even	6		648.2.l.f.107.4		16
36.31	odd	6		648.2.l.f.107.5		16
72.5	odd	6		648.2.l.f.107.2		16
72.11	even	6		2592.2.p.f.2159.4		16
72.13	even	6		648.2.l.f.107.7		16
72.29	odd	6		648.2.l.f.539.5		16
72.43	odd	6		2592.2.p.f.2159.6		16
72.59	even	6		2592.2.p.f.431.3		16
72.61	even	6		648.2.l.f.539.4		16
72.67	odd	6		2592.2.p.f.431.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.f.a.107.1	✓	8	8.5	even	2
216.2.f.a.107.2	yes	8	12.11	even	2
216.2.f.a.107.7	yes	8	4.3	odd	2
216.2.f.a.107.8	yes	8	24.5	odd	2
648.2.l.f.107.2		16	72.5	odd	6
648.2.l.f.107.4		16	36.23	even	6
648.2.l.f.107.5		16	36.31	odd	6
648.2.l.f.107.7		16	72.13	even	6
648.2.l.f.539.2		16	36.7	odd	6
648.2.l.f.539.4		16	72.61	even	6
648.2.l.f.539.5		16	72.29	odd	6
648.2.l.f.539.7		16	36.11	even	6
864.2.f.a.431.3		8	3.2	odd	2	inner
864.2.f.a.431.4		8	8.3	odd	2	inner
864.2.f.a.431.5		8	1.1	even	1	trivial
864.2.f.a.431.6		8	24.11	even	2	inner
2592.2.p.f.431.3		16	72.59	even	6
2592.2.p.f.431.4		16	9.4	even	3
2592.2.p.f.431.5		16	72.67	odd	6
2592.2.p.f.431.6		16	9.5	odd	6
2592.2.p.f.2159.3		16	9.7	even	3
2592.2.p.f.2159.4		16	72.11	even	6
2592.2.p.f.2159.5		16	9.2	odd	6
2592.2.p.f.2159.6		16	72.43	odd	6

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.59245 q^{5} -1.43937i q^{7} +O(q^{10})\)	\(q+1.59245 q^{5} -1.43937i q^{7} +4.93886i q^{11} +5.37182i q^{13} +1.32336i q^{17} -0.267949 q^{19} +5.94311 q^{23} -2.46410 q^{25} +8.70131 q^{29} -7.86488i q^{31} -2.29213i q^{35} -2.49307i q^{37} +9.87771i q^{41} -2.00000 q^{43} +9.12801 q^{47} +4.92820 q^{49} -5.51641 q^{53} +7.86488i q^{55} +4.93886i q^{59} -5.37182i q^{61} +8.55435i q^{65} -1.19615 q^{67} -5.51641 q^{71} -7.92820 q^{73} +7.10886 q^{77} +12.1830i q^{79} +7.23099i q^{83} +2.10739i q^{85} -15.7853i q^{89} +7.73205 q^{91} -0.426696 q^{95} +9.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{19} + 8 q^{25} - 16 q^{43} - 16 q^{49} + 32 q^{67} - 8 q^{73} + 48 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^19 + 8 * q^25 - 16 * q^43 - 16 * q^49 + 32 * q^67 - 8 * q^73 + 48 * q^91 - 8 * q^97

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