LMFDB - Embedded newform 864.2.bh.a.335.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(47,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 7]))

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

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

chi := DirichletCharacter("864.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.bh (of order $18$, degree $6$, 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:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			335.1
Root			$1.39273 + 0.245576i$ of defining polynomial
Character	$\chi$	$=$	864.335
Dual form			864.2.bh.a.815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.42338 - 0.986906i) q^{3} +(1.05203 + 2.80949i) q^{9} +O(q^{10})$	$q+(-1.42338 - 0.986906i) q^{3} +(1.05203 + 2.80949i) q^{9} +(-1.85296 - 5.09097i) q^{11} +(-0.412814 + 0.238338i) q^{17} +(-2.39076 + 4.14092i) q^{19} +(-3.83022 - 3.21394i) q^{25} +(1.27526 - 5.03723i) q^{27} +(-2.38684 + 9.07509i) q^{33} +(-5.51466 - 6.57212i) q^{41} +(-12.3238 + 4.48548i) q^{43} +(-6.57785 - 2.39414i) q^{49} +(0.822809 + 0.0681621i) q^{51} +(7.48967 - 3.53466i) q^{57} +(-5.24706 + 14.4162i) q^{59} +(-8.26366 + 6.93404i) q^{67} +(-6.22028 + 10.7738i) q^{73} +(2.28002 + 8.35473i) q^{75} +(-6.78645 + 5.91135i) q^{81} +(9.11103 - 10.8581i) q^{83} +(-15.9495 - 9.20844i) q^{89} +(17.3366 - 6.31001i) q^{97} +(12.3536 - 10.5617i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) $ 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{13}{18}\right)$	$-1$

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
$3$	−1.42338	−	0.986906i	−0.821790	−	0.569790i
$3$	−1.42338	−	0.986906i	−0.821790	−	0.569790i
$4$		0			0
$5$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$5$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$6$		0			0
$7$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$7$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$8$		0			0
$9$	1.05203	+	2.80949i	0.350678	+	0.936496i
$10$		0			0
$11$	−1.85296	−	5.09097i	−0.558689	−	1.53498i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−1.85296	−	5.09097i	−0.558689	−	1.53498i	0.262853	−	0.964836i	$-0.415337\pi$
$12$		0			0
$13$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$13$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.412814	+	0.238338i	−0.100122	+	0.0578055i	−0.549225	−	0.835675i	$-0.685077\pi$
$17$	−0.412814	+	0.238338i	−0.100122	+	0.0578055i	0.449103	+	0.893480i	$0.351744\pi$
$18$		0			0
$19$	−2.39076	+	4.14092i	−0.548478	+	0.949992i	0.449901	+	0.893079i	$0.351459\pi$
$19$	−2.39076	+	4.14092i	−0.548478	+	0.949992i	−0.998379	+	0.0569137i	$0.981874\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$23$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$24$		0			0
$25$	−3.83022	−	3.21394i	−0.766044	−	0.642788i
$26$		0			0
$27$	1.27526	−	5.03723i	0.245423	−	0.969416i
$28$		0			0
$29$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$29$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$30$		0			0
$31$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$31$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$32$		0			0
$33$	−2.38684	+	9.07509i	−0.415495	+	1.57977i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.51466	−	6.57212i	−0.861245	−	1.02639i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−5.51466	−	6.57212i	−0.861245	−	1.02639i	0.138108	−	0.990417i	$-0.455898\pi$
$42$		0			0
$43$	−12.3238	+	4.48548i	−1.87936	+	0.684030i	−0.937795	+	0.347190i	$0.887136\pi$
$43$	−12.3238	+	4.48548i	−1.87936	+	0.684030i	−0.941562	+	0.336840i	$0.890642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$47$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$48$		0			0
$49$	−6.57785	−	2.39414i	−0.939693	−	0.342020i
$50$		0			0
$51$	0.822809	+	0.0681621i	0.115216	+	0.00954460i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.48967	−	3.53466i	0.992030	−	0.468176i
$58$		0			0
$59$	−5.24706	+	14.4162i	−0.683109	+	1.87683i	−0.292542	+	0.956253i	$0.594501\pi$
$59$	−5.24706	+	14.4162i	−0.683109	+	1.87683i	−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$61$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.26366	+	6.93404i	−1.00957	+	0.847127i	−0.988281	−	0.152646i	$-0.951221\pi$
$67$	−8.26366	+	6.93404i	−1.00957	+	0.847127i	−0.0212861	+	0.999773i	$0.506776\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	−6.22028	+	10.7738i	−0.728028	+	1.26098i	0.229687	+	0.973265i	$0.426230\pi$
$73$	−6.22028	+	10.7738i	−0.728028	+	1.26098i	−0.957715	+	0.287718i	$0.907104\pi$
$74$		0			0
$75$	2.28002	+	8.35473i	0.263274	+	0.964721i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	−6.78645	+	5.91135i	−0.754050	+	0.656817i
$82$		0			0
$83$	9.11103	−	10.8581i	1.00007	−	1.19183i	0.0186715	−	0.999826i	$-0.494056\pi$
$83$	9.11103	−	10.8581i	1.00007	−	1.19183i	0.981394	−	0.192006i	$-0.0614992\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.9495	−	9.20844i	−1.69064	−	0.976093i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−15.9495	−	9.20844i	−1.69064	−	0.976093i	−0.736644	−	0.676280i	$-0.763591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	17.3366	−	6.31001i	1.76026	−	0.640684i	0.760304	−	0.649567i	$-0.225050\pi$
$97$	17.3366	−	6.31001i	1.76026	−	0.640684i	0.999961	+	0.00888289i	$0.00282755\pi$
$98$		0			0
$99$	12.3536	−	10.5617i	1.24159	−	1.06149i
$100$		0			0
$101$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$101$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$102$		0			0
$103$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$103$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.0236i		−	1.83908i	−0.392992	−	0.919542i	$-0.628560\pi$
$107$		−	19.0236i		−	1.83908i	0.392992	−	0.919542i	$-0.371440\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.39681	−	9.33265i	0.319545	−	0.877942i	−0.671087	−	0.741379i	$-0.734172\pi$
$113$	3.39681	−	9.33265i	0.319545	−	0.877942i	0.990631	−	0.136563i	$-0.0436057\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−14.0580	+	11.7961i	−1.27800	+	1.07237i
$122$		0			0
$123$	1.36341	+	14.7971i	0.122934	+	1.33421i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$	21.9682	+	5.77784i	1.93419	+	0.508710i
$130$		0			0
$131$	22.3153	−	3.93479i	1.94969	−	0.343784i	0.950213	−	0.311602i	$-0.100866\pi$
$131$	22.3153	−	3.93479i	1.94969	−	0.343784i	0.999482	−	0.0321817i	$-0.0102455\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.32378	−	7.53639i	0.540277	−	0.643877i	−0.424973	−	0.905206i	$-0.639716\pi$
$137$	6.32378	−	7.53639i	0.540277	−	0.643877i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	−0.787687	−	4.46720i	−0.0668108	−	0.378903i	−0.999819	−	0.0190466i	$-0.993937\pi$
$139$	−0.787687	−	4.46720i	−0.0668108	−	0.378903i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	+	9.89949i	0.577350	+	0.816497i
$148$		0			0
$149$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$149$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$150$		0			0
$151$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$151$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$152$		0			0
$153$	−1.10390	−	0.909055i	−0.0892452	−	0.0734928i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$157$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	23.0454			1.80506			0.902528	−	0.430632i	$-0.141709\pi$
$163$	23.0454			1.80506			0.902528	+	0.430632i	$0.141709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$167$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$168$		0			0
$169$	2.25743	−	12.8025i	0.173648	−	0.984808i
$170$		0			0
$171$	−14.1490	−	2.36043i	−1.08200	−	0.180507i
$172$		0			0
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	21.6960	−	15.3414i	1.63077	−	1.15313i
$178$		0			0
$179$	−4.92679	+	2.84448i	−0.368245	+	0.212607i	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−4.92679	+	2.84448i	−0.368245	+	0.212607i	0.304446	+	0.952529i	$0.401529\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.97830	+	1.65999i	0.144668	+	0.121390i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$191$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$192$		0			0
$193$	4.59777	+	26.0753i	0.330955	+	1.87694i	0.464011	+	0.885830i	$0.346410\pi$
$193$	4.59777	+	26.0753i	0.330955	+	1.87694i	−0.133056	+	0.991109i	$0.542479\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$	18.6056	−	1.71433i	1.31234	−	0.120919i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	25.5113	+	4.49833i	1.76465	+	0.311156i
$210$		0			0
$211$	−27.2938	−	9.93411i	−1.87898	−	0.683893i	−0.946595	−	0.322424i	$-0.895502\pi$
$211$	−27.2938	−	9.93411i	−1.87898	−	0.683893i	−0.932384	−	0.361469i	$-0.882275\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	19.4866	−	9.19646i	1.31678	−	0.621439i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$223$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$224$		0			0
$225$	5.00000	−	14.1421i	0.333333	−	0.942809i
$226$		0			0
$227$	−5.85433	−	16.0847i	−0.388566	−	1.06758i	−0.967647	−	0.252306i	$-0.918811\pi$
$227$	−5.85433	−	16.0847i	−0.388566	−	1.06758i	0.579082	−	0.815270i	$-0.303411\pi$
$228$		0			0
$229$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$229$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.28241	+	2.47245i	−0.280550	+	0.161976i	−0.633672	−	0.773602i	$-0.718453\pi$
$233$	−4.28241	+	2.47245i	−0.280550	+	0.161976i	0.353122	+	0.935577i	$0.385120\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$239$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$240$		0			0
$241$	−9.34413	−	7.84065i	−0.601908	−	0.505061i	0.290150	−	0.956981i	$-0.406295\pi$
$241$	−9.34413	−	7.84065i	−0.601908	−	0.505061i	−0.892058	+	0.451920i	$0.850739\pi$
$242$		0			0
$243$	15.4937	−	1.71652i	0.993919	−	0.110115i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−23.6844	+	6.46350i	−1.50094	+	0.409608i
$250$		0			0
$251$	−17.3006	−	9.98849i	−1.09200	−	0.630468i	−0.157894	−	0.987456i	$-0.550470\pi$
$251$	−17.3006	−	9.98849i	−1.09200	−	0.630468i	−0.934109	+	0.356988i	$0.883804\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	20.2535	+	24.1372i	1.26338	+	1.50564i	0.773427	+	0.633885i	$0.218541\pi$
$257$	20.2535	+	24.1372i	1.26338	+	1.50564i	0.489951	+	0.871750i	$0.337015\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$263$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	13.6144	+	28.8478i	0.833185	+	1.76546i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.26480	+	25.4548i	−0.558689	+	1.53498i
$276$		0			0
$277$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$277$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.494671	−	1.35910i	−0.0295096	−	0.0810770i	0.924063	−	0.382240i	$-0.124847\pi$
$281$	−0.494671	−	1.35910i	−0.0295096	−	0.0810770i	−0.953573	+	0.301163i	$0.902625\pi$
$282$		0			0
$283$	8.46127	−	7.09985i	0.502970	−	0.422042i	−0.355677	−	0.934609i	$-0.615750\pi$
$283$	8.46127	−	7.09985i	0.502970	−	0.422042i	0.858647	+	0.512567i	$0.171305\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.38639	+	14.5257i	−0.493317	+	0.854450i
$290$		0			0
$291$	−30.9040	−	8.12804i	−1.81162	−	0.476474i
$292$		0			0
$293$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$293$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−28.0074	+	2.84151i	−1.62515	+	0.164881i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	14.5237	+	25.1558i	0.828912	+	1.43572i	0.898893	+	0.438169i	$0.144373\pi$
$307$	14.5237	+	25.1558i	0.828912	+	1.43572i	−0.0699810	+	0.997548i	$0.522294\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$311$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$312$		0			0
$313$	2.07825	−	0.756419i	0.117469	−	0.0427554i	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	2.07825	−	0.756419i	0.117469	−	0.0427554i	0.400086	+	0.916478i	$0.368980\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$317$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−18.7745	+	27.0779i	−1.04789	+	1.51134i
$322$		0			0
$323$		−	2.27924i		−	0.126820i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.08557	−	34.5130i	0.334493	−	1.89701i	−0.0976852	−	0.995217i	$-0.531144\pi$
$331$	6.08557	−	34.5130i	0.334493	−	1.89701i	0.432178	−	0.901788i	$-0.357745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.49720	+	7.13000i	−0.462872	+	0.388396i	−0.844187	−	0.536050i	$-0.819916\pi$
$337$	−8.49720	+	7.13000i	−0.462872	+	0.388396i	0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	−14.0454	+	9.93160i	−0.762842	+	0.539411i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−32.0062	+	5.64356i	−1.71818	+	0.302962i	−0.943987	−	0.329983i	$-0.892957\pi$
$347$	−32.0062	+	5.64356i	−1.71818	+	0.302962i	−0.774197	+	0.632945i	$0.781846\pi$
$348$		0			0
$349$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$349$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	23.5371	−	28.0504i	1.25275	−	1.49297i	0.454384	−	0.890806i	$-0.349859\pi$
$353$	23.5371	−	28.0504i	1.25275	−	1.49297i	0.798369	−	0.602168i	$-0.205696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−1.93148	−	3.34542i	−0.101657	−	0.176075i
$362$		0			0
$363$	31.6515	−	2.91638i	1.66127	−	0.153070i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$367$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$368$		0			0
$369$	12.6627	−	22.4075i	0.659193	−	1.16649i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$373$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	38.6105			1.98329			0.991643	−	0.129012i	$-0.0411807\pi$
$379$	38.6105			1.98329			0.991643	+	0.129012i	$0.0411807\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$383$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−25.5669	−	29.9046i	−1.29964	−	1.52014i
$388$		0			0
$389$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$389$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−35.6464	−	16.4224i	−1.79812	−	0.828399i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−38.6656	+	6.81778i	−1.93087	+	0.340464i	−0.999708	−	0.0241516i	$-0.992312\pi$
$401$	−38.6656	+	6.81778i	−1.93087	+	0.340464i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.71496	−	38.0824i	−0.332033	−	1.88305i	−0.454759	−	0.890614i	$-0.650275\pi$
$409$	−6.71496	−	38.0824i	−0.332033	−	1.88305i	0.122726	−	0.992441i	$-0.460836\pi$
$410$		0			0
$411$	−16.4389	+	4.48618i	−0.810869	+	0.221287i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.28752	+	7.13590i	−0.160991	+	0.349447i
$418$		0			0
$419$	1.79744	+	2.14210i	0.0878107	+	0.104649i	0.808161	−	0.588962i	$-0.200463\pi$
$419$	1.79744	+	2.14210i	0.0878107	+	0.104649i	−0.720350	+	0.693611i	$0.756019\pi$
$420$		0			0
$421$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$421$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.34717	+	0.413870i	0.113855	+	0.0200756i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	40.0182			1.92315			0.961575	−	0.274543i	$-0.0885264\pi$
$433$	40.0182			1.92315			0.961575	+	0.274543i	$0.0885264\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	−0.193805	−	20.9991i	−0.00922881	−	0.999957i
$442$		0			0
$443$	4.00403	+	11.0010i	0.190237	+	0.522672i	0.997740	−	0.0671913i	$-0.0214038\pi$
$443$	4.00403	+	11.0010i	0.190237	+	0.522672i	−0.807503	+	0.589863i	$0.799182\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−36.6712	+	21.1721i	−1.73062	+	0.999174i	−0.844990	+	0.534783i	$0.820394\pi$
$449$	−36.6712	+	21.1721i	−1.73062	+	0.999174i	−0.885630	+	0.464391i	$0.846273\pi$
$450$		0			0
$451$	−23.2400	+	40.2528i	−1.09433	+	1.89543i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−27.6087	−	23.1664i	−1.29148	−	1.08368i	−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−27.6087	−	23.1664i	−1.29148	−	1.08368i	−0.299928	−	0.953962i	$-0.596963\pi$
$458$		0			0
$459$	0.674122	+	2.38338i	0.0314653	+	0.111247i
$460$		0			0
$461$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$461$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$462$		0			0
$463$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$463$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.4335	+	9.48788i	0.760452	+	0.439047i	0.829458	−	0.558569i	$-0.188650\pi$
$467$	16.4335	+	9.48788i	0.760452	+	0.439047i	−0.0690063	+	0.997616i	$0.521983\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	45.6709	+	54.4285i	2.09995	+	2.50262i
$474$		0			0
$475$	22.4658	−	8.17688i	1.03080	−	0.375181i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$479$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−32.8024	−	22.7436i	−1.48338	−	1.02850i
$490$		0			0
$491$	−5.52827	+	15.1888i	−0.249487	+	0.685461i	0.750218	+	0.661190i	$0.229948\pi$
$491$	−5.52827	+	15.1888i	−0.249487	+	0.685461i	−0.999705	+	0.0242702i	$0.992274\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−29.1065	+	24.4232i	−1.30298	+	1.09333i	−0.313363	+	0.949633i	$0.601456\pi$
$499$	−29.1065	+	24.4232i	−1.30298	+	1.09333i	−0.989621	+	0.143700i	$0.954100\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.8480	+	15.9950i	−0.703836	+	0.710362i
$508$		0			0
$509$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$509$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	17.8100	+	17.3236i	0.786329	+	0.764854i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.3627	−	19.2620i	−1.46165	−	0.843882i	−0.462559	−	0.886588i	$-0.653069\pi$
$521$	−33.3627	−	19.2620i	−1.46165	−	0.843882i	−0.999088	+	0.0427062i	$0.986402\pi$
$522$		0			0
$523$	−20.5227	−	35.5464i	−0.897395	−	1.55433i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−20.5227	−	35.5464i	−0.897395	−	1.55433i	−0.0665832	−	0.997781i	$-0.521210\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.6129	−	7.86646i	0.939693	−	0.342020i
$530$		0			0
$531$	−46.0222	+	0.424748i	−1.99719	+	0.0184325i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	9.81993	+	0.813490i	0.423761	+	0.0351047i
$538$		0			0
$539$			37.9239i			1.63350i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.06614	−	40.0741i	0.302126	−	1.71344i	−0.334606	−	0.942358i	$-0.608603\pi$
$547$	7.06614	−	40.0741i	0.302126	−	1.71344i	0.636732	−	0.771085i	$-0.280286\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.17762	−	4.31519i	−0.0497192	−	0.182188i
$562$		0			0
$563$	35.3834	−	6.23904i	1.49123	−	0.262944i	0.632175	−	0.774826i	$-0.282163\pi$
$563$	35.3834	−	6.23904i	1.49123	−	0.262944i	0.859056	+	0.511882i	$0.171051\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.9010	+	27.2924i	−0.960061	+	1.14416i	0.0294311	+	0.999567i	$0.490630\pi$
$569$	−22.9010	+	27.2924i	−0.960061	+	1.14416i	−0.989492	+	0.144589i	$0.953814\pi$
$570$		0			0
$571$	−1.80910	−	10.2599i	−0.0757085	−	0.429364i	−0.998978	−	0.0452101i	$-0.985604\pi$
$571$	−1.80910	−	10.2599i	−0.0757085	−	0.429364i	0.923269	−	0.384154i	$-0.125507\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.7607	−	23.8343i	−0.572866	−	0.992234i	−0.996270	−	0.0862925i	$-0.972498\pi$
$577$	−13.7607	−	23.8343i	−0.572866	−	0.992234i	0.423403	−	0.905941i	$-0.360835\pi$
$578$		0			0
$579$	19.1894	−	41.6526i	0.797486	−	1.73102i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.0418	+	2.47595i	0.579567	+	0.102193i	0.455744	−	0.890111i	$-0.349373\pi$
$587$	14.0418	+	2.47595i	0.579567	+	0.102193i	0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	7.03896i		−	0.289055i	−0.989501	−	0.144528i	$-0.953834\pi$
$593$		−	7.03896i		−	0.289055i	0.989501	−	0.144528i	$-0.0461663\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$		0			0
$601$	−4.28921	+	24.3253i	−0.174960	+	0.992250i	0.763229	+	0.646128i	$0.223613\pi$
$601$	−4.28921	+	24.3253i	−0.174960	+	0.992250i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−28.1747	−	15.9218i	−1.14736	−	0.648387i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$607$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.3237	+	3.93628i	−0.898720	+	0.158469i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−22.3237	+	3.93628i	−0.898720	+	0.158469i	−0.294843	+	0.955546i	$0.595267\pi$
$618$		0			0
$619$	7.60017	+	6.37730i	0.305477	+	0.256325i	0.782620	−	0.622500i	$-0.213883\pi$
$619$	7.60017	+	6.37730i	0.305477	+	0.256325i	−0.477143	+	0.878826i	$0.658328\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	+	24.6202i	0.173648	+	0.984808i
$626$		0			0
$627$	−31.8729	−	31.5801i	−1.27288	−	1.26119i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	29.0454	+	41.0764i	1.15445	+	1.63264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0281	−	2.12088i	−0.475082	−	0.0837698i	−0.0690201	−	0.997615i	$-0.521987\pi$
$641$	−12.0281	−	2.12088i	−0.475082	−	0.0837698i	−0.406062	+	0.913845i	$0.633098\pi$
$642$		0			0
$643$	16.0112	+	5.82761i	0.631421	+	0.229818i	0.637850	−	0.770161i	$-0.279824\pi$
$643$	16.0112	+	5.82761i	0.631421	+	0.229818i	−0.00642884	+	0.999979i	$0.502046\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	83.1149			3.26255
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$653$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−36.8129	−	6.14136i	−1.43621	−	0.239597i
$658$		0			0
$659$	13.5543	+	37.2401i	0.528000	+	1.45067i	0.861422	+	0.507891i	$0.169575\pi$
$659$	13.5543	+	37.2401i	0.528000	+	1.45067i	−0.333422	+	0.942778i	$0.608203\pi$
$660$		0			0
$661$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$661$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	37.6057	+	31.5550i	1.44959	+	1.21635i	0.932875	+	0.360200i	$0.117292\pi$
$673$	37.6057	+	31.5550i	1.44959	+	1.21635i	0.516720	+	0.856154i	$0.327153\pi$
$674$		0			0
$675$	−21.0739	+	15.1951i	−0.811134	+	0.584861i
$676$		0			0
$677$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$677$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7.54108	+	28.6723i	−0.288975	+	1.09872i
$682$		0			0
$683$	−40.4993	−	23.3823i	−1.54966	−	0.894699i	−0.998167	−	0.0605215i	$-0.980724\pi$
$683$	−40.4993	−	23.3823i	−1.54966	−	0.894699i	−0.551497	−	0.834177i	$-0.685943\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.897023	−	0.326490i	0.0341244	−	0.0124203i	−0.324902	−	0.945748i	$-0.605331\pi$
$691$	0.897023	−	0.326490i	0.0341244	−	0.0124203i	0.359026	+	0.933328i	$0.383109\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.84291	+	1.39871i	0.145561	+	0.0529797i
$698$		0			0
$699$	8.53558	+	0.707094i	0.322845	+	0.0267447i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$709$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	5.56228	+	20.3820i	0.206863	+	0.758015i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$727$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	−23.7474	−	12.8475i	−0.879535	−	0.475834i
$730$		0			0
$731$	4.01836	−	4.78889i	0.148624	−	0.177124i
$732$		0			0
$733$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$733$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	50.6132	+	29.2215i	1.86436	+	1.07639i
$738$		0			0
$739$	−23.8429	−	41.2972i	−0.877077	−	1.51914i	−0.854534	−	0.519396i	$-0.826157\pi$
$739$	−23.8429	−	41.2972i	−0.877077	−	1.51914i	−0.0225433	−	0.999746i	$-0.507176\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$743$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	40.0908	+	14.1742i	1.46685	+	0.518608i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$751$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$752$		0			0
$753$	14.7676	+	31.2915i	0.538162	+	1.14032i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.86952	−	10.6314i	0.140270	−	0.385388i	−0.849589	−	0.527446i	$-0.823150\pi$
$761$	3.86952	−	10.6314i	0.140270	−	0.385388i	0.989858	+	0.142058i	$0.0453719\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	42.2020	−	35.4117i	1.52184	−	1.27698i	0.686709	−	0.726932i	$-0.259055\pi$
$769$	42.2020	−	35.4117i	1.52184	−	1.27698i	0.835134	−	0.550046i	$-0.185390\pi$
$770$		0			0
$771$	−5.00734	−	54.3447i	−0.180335	−	1.95718i
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	40.3988	−	7.12341i	1.44744	−	0.255222i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	8.16935	+	46.3307i	0.291206	+	1.65151i	0.682236	+	0.731132i	$0.261008\pi$
$787$	8.16935	+	46.3307i	0.291206	+	1.65151i	−0.391030	+	0.920378i	$0.627881\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$797$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	9.09162	−	54.4975i	0.321236	−	1.92557i
$802$		0			0
$803$	66.3752	+	11.7037i	2.34233	+	0.413016i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	39.4773i		−	1.38795i	−0.720000	−	0.693974i	$-0.755858\pi$
$809$		−	39.4773i		−	1.38795i	0.720000	−	0.693974i	$-0.244142\pi$
$810$		0			0
$811$	35.1832			1.23545			0.617725	−	0.786394i	$-0.288054\pi$
$811$	35.1832			1.23545			0.617725	+	0.786394i	$0.288054\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	10.8892	−	61.7554i	0.380963	−	2.16055i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$821$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$822$		0			0
$823$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$823$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$	38.3089	−	27.0885i	1.33374	−	0.943100i
$826$		0			0
$827$	−17.1464	+	9.89949i	−0.596240	+	0.344239i	−0.767561	−	0.640976i	$-0.778530\pi$
$827$	−17.1464	+	9.89949i	−0.596240	+	0.344239i	0.171321	+	0.985215i	$0.445196\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.28604	−	0.579418i	0.113855	−	0.0200756i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$839$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$840$		0			0
$841$	−5.03580	−	28.5594i	−0.173648	−	0.984808i
$842$		0			0
$843$	−0.637196	+	2.42271i	−0.0219462	+	0.0834425i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.0505	+	1.75532i	−0.653812	+	0.0602425i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$853$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.2837	−	3.92921i	−0.761195	−	0.134219i	−0.220441	−	0.975400i	$-0.570750\pi$
$857$	−22.2837	−	3.92921i	−0.761195	−	0.134219i	−0.540754	+	0.841181i	$0.681861\pi$
$858$		0			0
$859$	−1.61178	−	0.586641i	−0.0549933	−	0.0200159i	0.314377	−	0.949298i	$-0.398204\pi$
$859$	−1.61178	−	0.586641i	−0.0549933	−	0.0200159i	−0.369370	+	0.929282i	$0.620427\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	26.2725	−	12.3990i	0.892261	−	0.421091i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	35.9666	+	42.0686i	1.21728	+	1.42381i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$877$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.9949	−	21.9364i	1.28008	−	0.739055i	0.303218	−	0.952921i	$-0.401939\pi$
$881$	37.9949	−	21.9364i	1.28008	−	0.739055i	0.976863	+	0.213866i	$0.0686057\pi$
$882$		0			0
$883$	−18.3238	+	31.7377i	−0.616644	+	1.06806i	0.373450	+	0.927650i	$0.378175\pi$
$883$	−18.3238	+	31.7377i	−0.616644	+	1.06806i	−0.990094	+	0.140408i	$0.955158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	42.6695	+	23.5961i	1.42948	+	0.790499i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	53.3352	−	19.4124i	1.77097	−	0.644579i	0.770996	−	0.636841i	$-0.219759\pi$
$907$	53.3352	−	19.4124i	1.77097	−	0.644579i	0.999970	−	0.00773827i	$-0.00246319\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$911$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$912$		0			0
$913$	−72.1606	−	26.2643i	−2.38817	−	0.869222i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	4.15362	−	50.1398i	0.136866	−	1.65216i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.67379	−	26.5785i	−0.317387	−	0.872013i	−0.991112	−	0.133031i	$-0.957529\pi$
$929$	−9.67379	−	26.5785i	−0.317387	−	0.872013i	0.673725	−	0.738982i	$-0.264693\pi$
$930$		0			0
$931$	25.6400	−	21.5145i	0.840317	−	0.705110i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−30.5454	+	52.9062i	−0.997875	+	1.72837i	−0.442509	+	0.896764i	$0.645912\pi$
$937$	−30.5454	+	52.9062i	−0.997875	+	1.72837i	−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−3.70465	−	0.974359i	−0.120897	−	0.0317970i
$940$		0			0
$941$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$941$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.0666	−	16.7639i	0.457102	−	0.544753i	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	14.0666	−	16.7639i	0.457102	−	0.544753i	0.944536	+	0.328407i	$0.106512\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.0149	+	16.7518i	0.939884	+	0.542643i	0.889924	−	0.456109i	$-0.150757\pi$
$953$	29.0149	+	16.7518i	0.939884	+	0.542643i	0.0499603	+	0.998751i	$0.484091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.1305	+	10.6026i	−0.939693	+	0.342020i
$962$		0			0
$963$	53.4467	−	20.0135i	1.72230	−	0.644926i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$967$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$968$		0			0
$969$	−2.24939	+	3.24423i	−0.0722609	+	0.104220i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.2970	+	53.0180i	−0.617365	+	1.69620i	0.0959785	+	0.995383i	$0.469402\pi$
$977$	−19.2970	+	53.0180i	−0.617365	+	1.69620i	−0.713344	+	0.700814i	$0.752820\pi$
$978$		0			0
$979$	−17.3261	+	98.2612i	−0.553745	+	3.14044i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$983$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−42.7232	+	43.1193i	−1.35578	+	1.36835i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$997$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bh.a.335.1		12
4.3	odd	2		216.2.v.a.11.1	✓	12
8.3	odd	2	CM	864.2.bh.a.335.1		12
8.5	even	2		216.2.v.a.11.1	✓	12
12.11	even	2		648.2.v.a.35.2		12
24.5	odd	2		648.2.v.a.35.2		12
27.5	odd	18	inner	864.2.bh.a.815.1		12
108.59	even	18		216.2.v.a.59.1	yes	12
108.103	odd	18		648.2.v.a.611.2		12
216.5	odd	18		216.2.v.a.59.1	yes	12
216.59	even	18	inner	864.2.bh.a.815.1		12
216.157	even	18		648.2.v.a.611.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.11.1	✓	12	4.3	odd	2
216.2.v.a.11.1	✓	12	8.5	even	2
216.2.v.a.59.1	yes	12	108.59	even	18
216.2.v.a.59.1	yes	12	216.5	odd	18
648.2.v.a.35.2		12	12.11	even	2
648.2.v.a.35.2		12	24.5	odd	2
648.2.v.a.611.2		12	108.103	odd	18
648.2.v.a.611.2		12	216.157	even	18
864.2.bh.a.335.1		12	1.1	even	1	trivial
864.2.bh.a.335.1		12	8.3	odd	2	CM
864.2.bh.a.815.1		12	27.5	odd	18	inner
864.2.bh.a.815.1		12	216.59	even	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bh (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.42338 - 0.986906i) q^{3} +(1.05203 + 2.80949i) q^{9} +O(q^{10})\)	\(q+(-1.42338 - 0.986906i) q^{3} +(1.05203 + 2.80949i) q^{9} +(-1.85296 - 5.09097i) q^{11} +(-0.412814 + 0.238338i) q^{17} +(-2.39076 + 4.14092i) q^{19} +(-3.83022 - 3.21394i) q^{25} +(1.27526 - 5.03723i) q^{27} +(-2.38684 + 9.07509i) q^{33} +(-5.51466 - 6.57212i) q^{41} +(-12.3238 + 4.48548i) q^{43} +(-6.57785 - 2.39414i) q^{49} +(0.822809 + 0.0681621i) q^{51} +(7.48967 - 3.53466i) q^{57} +(-5.24706 + 14.4162i) q^{59} +(-8.26366 + 6.93404i) q^{67} +(-6.22028 + 10.7738i) q^{73} +(2.28002 + 8.35473i) q^{75} +(-6.78645 + 5.91135i) q^{81} +(9.11103 - 10.8581i) q^{83} +(-15.9495 - 9.20844i) q^{89} +(17.3366 - 6.31001i) q^{97} +(12.3536 - 10.5617i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) \) 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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