LMFDB - Embedded newform 3477.1.gk.a.1544.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3477,1,Mod(161,3477)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3477.161"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3477, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 40, 69])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3477 = 3 \cdot 19 \cdot 61 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3477.gk (of order $90$, degree $24$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.73524904892$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{45})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{90}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{90} - \cdots)$

Embedding invariants

Embedding label			1544.1
Root			$-0.882948 + 0.469472i$ of defining polynomial
Character	$\chi$	$=$	3477.1544
Dual form			3477.1.gk.a.1448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.882948 - 0.469472i) q^{3} +(-0.438371 + 0.898794i) q^{4} +(0.195217 - 0.918425i) q^{7} +(0.559193 + 0.829038i) q^{9} +(0.809017 - 0.587785i) q^{12} +(-0.0534691 + 0.0448659i) q^{13} +(-0.615661 - 0.788011i) q^{16} +(0.104528 + 0.994522i) q^{19} +(-0.603541 + 0.719272i) q^{21} +(0.0348995 - 0.999391i) q^{25} +(-0.104528 - 0.994522i) q^{27} +(0.739897 + 0.578071i) q^{28} +(0.524293 - 0.170353i) q^{31} +(-0.990268 + 0.139173i) q^{36} +(1.70961 - 0.555485i) q^{37} +(0.0682737 - 0.0145120i) q^{39} +(-1.77028 + 0.863423i) q^{43} +(0.173648 + 0.984808i) q^{48} +(0.108151 + 0.0481518i) q^{49} +(-0.0168859 - 0.0677257i) q^{52} +(0.374607 - 0.927184i) q^{57} +(-0.500000 - 0.866025i) q^{61} +(0.870573 - 0.351734i) q^{63} +(0.978148 - 0.207912i) q^{64} +(1.77028 + 0.863423i) q^{67} +(1.76159 + 0.123183i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-0.939693 - 0.342020i) q^{76} +(-1.37461 - 0.927184i) q^{79} +(-0.374607 + 0.927184i) q^{81} +(-0.381903 - 0.857767i) q^{84} +(0.0307679 + 0.0578660i) q^{91} +(-0.542900 - 0.0957278i) q^{93} +(1.93726 - 0.272264i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 6 q^{12} - 3 q^{13} - 3 q^{19} + 3 q^{21} + 3 q^{27} + 3 q^{28} - 6 q^{43} - 3 q^{49} + 3 q^{52} - 12 q^{61} - 3 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 12 q^{75} - 24 q^{79} + 6 q^{91} + 3 q^{93}+O(q^{100}) $ 24 * q + 6 * q^12 - 3 * q^13 - 3 * q^19 + 3 * q^21 + 3 * q^27 + 3 * q^28 - 6 * q^43 - 3 * q^49 + 3 * q^52 - 12 * q^61 - 3 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 12 * q^75 - 24 * q^79 + 6 * q^91 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3477\mathbb{Z}\right)^\times$.

$n$	$856$	$1160$	$1465$
$\chi(n)$	$e\left(\frac{13}{30}\right)$	$-1$	$e\left(\frac{8}{9}\right)$

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		−0.529919	−	0.848048i	$-0.677778\pi$
$2$		0			0		0.529919	+	0.848048i	$0.322222\pi$
$3$	−0.882948	−	0.469472i	−0.882948	−	0.469472i
$3$	−0.882948	−	0.469472i	−0.882948	−	0.469472i
$4$	−0.438371	+	0.898794i	−0.438371	+	0.898794i
$5$		0			0		0.719340	−	0.694658i	$-0.244444\pi$
$5$		0			0		−0.719340	+	0.694658i	$0.755556\pi$
$6$		0			0
$7$	0.195217	−	0.918425i	0.195217	−	0.918425i	−0.766044	−	0.642788i	$-0.777778\pi$
$7$	0.195217	−	0.918425i	0.195217	−	0.918425i	0.961262	−	0.275637i	$-0.0888889\pi$
$8$		0			0
$9$	0.559193	+	0.829038i	0.559193	+	0.829038i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$	0.809017	−	0.587785i	0.809017	−	0.587785i
$13$	−0.0534691	+	0.0448659i	−0.0534691	+	0.0448659i	−0.669131	−	0.743145i	$-0.733333\pi$
$13$	−0.0534691	+	0.0448659i	−0.0534691	+	0.0448659i	0.615661	+	0.788011i	$0.288889\pi$
$14$		0			0
$15$		0			0
$16$	−0.615661	−	0.788011i	−0.615661	−	0.788011i
$17$		0			0		−0.694658	−	0.719340i	$-0.744444\pi$
$17$		0			0		0.694658	+	0.719340i	$0.255556\pi$
$18$		0			0
$19$	0.104528	+	0.994522i	0.104528	+	0.994522i
$19$	0.104528	+	0.994522i	0.104528	+	0.994522i
$20$		0			0
$21$	−0.603541	+	0.719272i	−0.603541	+	0.719272i
$22$		0			0
$23$		0			0		−0.0697565	−	0.997564i	$-0.522222\pi$
$23$		0			0		0.0697565	+	0.997564i	$0.477778\pi$
$24$		0			0
$25$	0.0348995	−	0.999391i	0.0348995	−	0.999391i
$26$		0			0
$27$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$28$	0.739897	+	0.578071i	0.739897	+	0.578071i
$29$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$	0.524293	−	0.170353i	0.524293	−	0.170353i	−0.0348995	−	0.999391i	$-0.511111\pi$
$31$	0.524293	−	0.170353i	0.524293	−	0.170353i	0.559193	+	0.829038i	$0.311111\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.990268	+	0.139173i	−0.990268	+	0.139173i
$37$	1.70961	−	0.555485i	1.70961	−	0.555485i	0.719340	−	0.694658i	$-0.244444\pi$
$37$	1.70961	−	0.555485i	1.70961	−	0.555485i	0.990268	+	0.139173i	$0.0444444\pi$
$38$		0			0
$39$	0.0682737	−	0.0145120i	0.0682737	−	0.0145120i
$40$		0			0
$41$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$41$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$42$		0			0
$43$	−1.77028	+	0.863423i	−1.77028	+	0.863423i	−0.809017	+	0.587785i	$0.800000\pi$
$43$	−1.77028	+	0.863423i	−1.77028	+	0.863423i	−0.961262	+	0.275637i	$0.911111\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$47$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$48$	0.173648	+	0.984808i	0.173648	+	0.984808i
$49$	0.108151	+	0.0481518i	0.108151	+	0.0481518i
$50$		0			0
$51$		0			0
$52$	−0.0168859	−	0.0677257i	−0.0168859	−	0.0677257i
$53$		0			0		−0.970296	−	0.241922i	$-0.922222\pi$
$53$		0			0		0.970296	+	0.241922i	$0.0777778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.374607	−	0.927184i	0.374607	−	0.927184i
$58$		0			0
$59$		0			0		−0.529919	−	0.848048i	$-0.677778\pi$
$59$		0			0		0.529919	+	0.848048i	$0.322222\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$62$		0			0
$63$	0.870573	−	0.351734i	0.870573	−	0.351734i
$64$	0.978148	−	0.207912i	0.978148	−	0.207912i
$65$		0			0
$66$		0			0
$67$	1.77028	+	0.863423i	1.77028	+	0.863423i	0.961262	+	0.275637i	$0.0888889\pi$
$67$	1.77028	+	0.863423i	1.77028	+	0.863423i	0.809017	+	0.587785i	$0.200000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.829038	−	0.559193i	$-0.811111\pi$
$71$		0			0		0.829038	+	0.559193i	$0.188889\pi$
$72$		0			0
$73$	1.76159	+	0.123183i	1.76159	+	0.123183i	0.913545	−	0.406737i	$-0.133333\pi$
$73$	1.76159	+	0.123183i	1.76159	+	0.123183i	0.848048	+	0.529919i	$0.177778\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$76$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$77$		0			0
$78$		0			0
$79$	−1.37461	−	0.927184i	−1.37461	−	0.927184i	−0.374607	−	0.927184i	$-0.622222\pi$
$79$	−1.37461	−	0.927184i	−1.37461	−	0.927184i	−1.00000			$\pi$
$80$		0			0
$81$	−0.374607	+	0.927184i	−0.374607	+	0.927184i
$82$		0			0
$83$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$83$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$84$	−0.381903	−	0.857767i	−0.381903	−	0.857767i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$89$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$90$		0			0
$91$	0.0307679	+	0.0578660i	0.0307679	+	0.0578660i
$92$		0			0
$93$	−0.542900	−	0.0957278i	−0.542900	−	0.0957278i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.93726	−	0.272264i	1.93726	−	0.272264i	0.939693	−	0.342020i	$-0.111111\pi$
$97$	1.93726	−	0.272264i	1.93726	−	0.272264i	0.997564	+	0.0697565i	$0.0222222\pi$
$98$		0			0
$99$		0			0
$100$	0.882948	+	0.469472i	0.882948	+	0.469472i
$101$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$101$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$102$		0			0
$103$	1.09395	−	0.232525i	1.09395	−	0.232525i	0.374607	−	0.927184i	$-0.377778\pi$
$103$	1.09395	−	0.232525i	1.09395	−	0.232525i	0.719340	+	0.694658i	$0.244444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$107$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$108$	0.939693	+	0.342020i	0.939693	+	0.342020i
$109$	0.317271	−	1.79933i	0.317271	−	1.79933i	−0.241922	−	0.970296i	$-0.577778\pi$
$109$	0.317271	−	1.79933i	0.317271	−	1.79933i	0.559193	−	0.829038i	$-0.311111\pi$
$110$		0			0
$111$	−1.77028	−	0.312148i	−1.77028	−	0.312148i
$112$	−0.843916	+	0.411606i	−0.843916	+	0.411606i
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.0670951	−	0.0192392i	−0.0670951	−	0.0192392i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−0.0767226	+	0.545910i	−0.0767226	+	0.545910i
$125$		0			0
$126$		0			0
$127$	1.33500	−	0.0933524i	1.33500	−	0.0933524i	0.615661	−	0.788011i	$-0.288889\pi$
$127$	1.33500	−	0.0933524i	1.33500	−	0.0933524i	0.719340	+	0.694658i	$0.244444\pi$
$128$		0			0
$129$	1.96842	+	0.0687386i	1.96842	+	0.0687386i
$130$		0			0
$131$		0			0		−0.961262	−	0.275637i	$-0.911111\pi$
$131$		0			0		0.961262	+	0.275637i	$0.0888889\pi$
$132$		0			0
$133$	0.933799	+	0.0981463i	0.933799	+	0.0981463i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$137$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$138$		0			0
$139$	1.60882	−	1.08516i	1.60882	−	1.08516i	0.669131	−	0.743145i	$-0.266667\pi$
$139$	1.60882	−	1.08516i	1.60882	−	1.08516i	0.939693	−	0.342020i	$-0.111111\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.309017	−	0.951057i	0.309017	−	0.951057i
$145$		0			0
$146$		0			0
$147$	−0.0728856	−	0.0932893i	−0.0728856	−	0.0932893i
$148$	−0.250176	+	1.78009i	−0.250176	+	1.78009i
$149$		0			0		−0.438371	−	0.898794i	$-0.644444\pi$
$149$		0			0		0.438371	+	0.898794i	$0.355556\pi$
$150$		0			0
$151$	1.28716	+	0.743145i	1.28716	+	0.743145i	0.978148	−	0.207912i	$-0.0666667\pi$
$151$	1.28716	+	0.743145i	1.28716	+	0.743145i	0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.0168859	+	0.0677257i	−0.0168859	+	0.0677257i
$157$	1.99756	−	0.0697565i	1.99756	−	0.0697565i	0.997564	−	0.0697565i	$-0.0222222\pi$
$157$	1.99756	−	0.0697565i	1.99756	−	0.0697565i	1.00000			$0$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.40724	−	0.299118i	−1.40724	−	0.299118i	−0.559193	−	0.829038i	$-0.688889\pi$
$163$	−1.40724	−	0.299118i	−1.40724	−	0.299118i	−0.848048	+	0.529919i	$0.822222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.0348995	−	0.999391i	$-0.488889\pi$
$167$		0			0		−0.0348995	+	0.999391i	$0.511111\pi$
$168$		0			0
$169$	−0.172802	+	0.980010i	−0.172802	+	0.980010i
$170$		0			0
$171$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$172$		−	1.96962i		−	1.96962i
$173$		0			0		0.898794	−	0.438371i	$-0.144444\pi$
$173$		0			0		−0.898794	+	0.438371i	$0.855556\pi$
$174$		0			0
$175$	−0.911053	−	0.227151i	−0.911053	−	0.227151i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$179$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$180$		0			0
$181$	0.163608	−	1.16413i	0.163608	−	1.16413i	−0.719340	−	0.694658i	$-0.755556\pi$
$181$	0.163608	−	1.16413i	0.163608	−	1.16413i	0.882948	−	0.469472i	$-0.155556\pi$
$182$		0			0
$183$	0.0348995	+	0.999391i	0.0348995	+	0.999391i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.933799	−	0.0981463i	−0.933799	−	0.0981463i
$190$		0			0
$191$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$191$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$192$	−0.961262	−	0.275637i	−0.961262	−	0.275637i
$193$	−0.00973193	+	0.139173i	−0.00973193	+	0.139173i	0.990268	+	0.139173i	$0.0444444\pi$
$193$	−0.00973193	+	0.139173i	−0.00973193	+	0.139173i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−0.0906888	+	0.0760969i	−0.0906888	+	0.0760969i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.74871	+	0.929805i	−1.74871	+	0.929805i	−0.809017	+	0.587785i	$0.800000\pi$
$199$	−1.74871	+	0.929805i	−1.74871	+	0.929805i	−0.939693	+	0.342020i	$0.888889\pi$
$200$		0			0
$201$	−1.15771	−	1.59345i	−1.15771	−	1.59345i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.0682737	+	0.0145120i	0.0682737	+	0.0145120i
$209$		0			0
$210$		0			0
$211$	−1.85324	+	0.0647165i	−1.85324	+	0.0647165i	−0.939693	−	0.342020i	$-0.888889\pi$
$211$	−1.85324	+	0.0647165i	−1.85324	+	0.0647165i	−0.913545	+	0.406737i	$0.866667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0541056	−	0.514780i	−0.0541056	−	0.514780i
$218$		0			0
$219$	−1.49756	−	0.935782i	−1.49756	−	0.935782i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.55208	+	0.273673i	1.55208	+	0.273673i	0.882948	−	0.469472i	$-0.155556\pi$
$223$	1.55208	+	0.273673i	1.55208	+	0.273673i	0.669131	+	0.743145i	$0.266667\pi$
$224$		0			0
$225$	0.848048	−	0.529919i	0.848048	−	0.529919i
$226$		0			0
$227$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$227$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$228$	0.669131	+	0.743145i	0.669131	+	0.743145i
$229$	−1.75631	−	0.781961i	−1.75631	−	0.781961i	−0.990268	−	0.139173i	$-0.955556\pi$
$229$	−1.75631	−	0.781961i	−1.75631	−	0.781961i	−0.766044	−	0.642788i	$-0.777778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$233$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.778419	+	1.46399i	0.778419	+	1.46399i
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$		0			0
$241$	0.741922	+	0.104270i	0.741922	+	0.104270i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	0.741922	+	0.104270i	0.741922	+	0.104270i	0.241922	+	0.970296i	$0.422222\pi$
$242$		0			0
$243$	0.766044	−	0.642788i	0.766044	−	0.642788i
$244$	0.997564	−	0.0697565i	0.997564	−	0.0697565i
$245$		0			0
$246$		0			0
$247$	−0.0502092	−	0.0484865i	−0.0502092	−	0.0484865i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$251$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$252$	−0.0654974	+	0.936656i	−0.0654974	+	0.936656i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.241922	+	0.970296i	−0.241922	+	0.970296i
$257$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$257$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$258$		0			0
$259$	−0.176427	−	1.67859i	−0.176427	−	1.67859i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.848048	−	0.529919i	$-0.177778\pi$
$263$		0			0		−0.848048	+	0.529919i	$0.822222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.55208	+	1.21262i	−1.55208	+	1.21262i
$269$		0			0		0.374607	−	0.927184i	$-0.377778\pi$
$269$		0			0		−0.374607	+	0.927184i	$0.622222\pi$
$270$		0			0
$271$	−1.82264	−	0.127451i	−1.82264	−	0.127451i	−0.882948	−	0.469472i	$-0.844444\pi$
$271$	−1.82264	−	0.127451i	−1.82264	−	0.127451i	−0.939693	+	0.342020i	$0.888889\pi$
$272$		0			0
$273$		−	0.0655373i		−	0.0655373i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$277$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$278$		0			0
$279$	0.434410	+	0.339399i	0.434410	+	0.339399i
$280$		0			0
$281$		0			0		0.529919	−	0.848048i	$-0.322222\pi$
$281$		0			0		−0.529919	+	0.848048i	$0.677778\pi$
$282$		0			0
$283$	−0.177290	+	0.110783i	−0.177290	+	0.110783i	−0.615661	−	0.788011i	$-0.711111\pi$
$283$	−0.177290	+	0.110783i	−0.177290	+	0.110783i	0.438371	+	0.898794i	$0.355556\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.0348995	+	0.999391i	−0.0348995	+	0.999391i
$290$		0			0
$291$	−1.83832	−	0.669092i	−1.83832	−	0.669092i
$292$	−0.882948	+	1.52931i	−0.882948	+	1.52931i
$293$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$293$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.559193	−	0.829038i	−0.559193	−	0.829038i
$301$	0.447400	+	1.79442i	0.447400	+	1.79442i
$302$		0			0
$303$		0			0
$304$	0.719340	−	0.694658i	0.719340	−	0.694658i
$305$		0			0
$306$		0			0
$307$	−0.812978	+	0.0283898i	−0.812978	+	0.0283898i	−0.438371	−	0.898794i	$-0.644444\pi$
$307$	−0.812978	+	0.0283898i	−0.812978	+	0.0283898i	−0.374607	+	0.927184i	$0.622222\pi$
$308$		0			0
$309$	−1.07506	−	0.308269i	−1.07506	−	0.308269i
$310$		0			0
$311$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$311$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$312$		0			0
$313$	−0.409677	−	1.42871i	−0.409677	−	1.42871i	−0.848048	−	0.529919i	$-0.822222\pi$
$313$	−0.409677	−	1.42871i	−0.409677	−	1.42871i	0.438371	−	0.898794i	$-0.355556\pi$
$314$		0			0
$315$		0			0
$316$	1.43594	−	0.829038i	1.43594	−	0.829038i
$317$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$317$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$325$	0.0429726	+	0.0550024i	0.0429726	+	0.0550024i
$326$		0			0
$327$	−1.12487	+	1.43977i	−1.12487	+	1.43977i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.220353	+	1.03668i	0.220353	+	1.03668i	0.939693	+	0.342020i	$0.111111\pi$
$331$	0.220353	+	1.03668i	0.220353	+	1.03668i	−0.719340	+	0.694658i	$0.755556\pi$
$332$		0			0
$333$	1.41652	+	1.10671i	1.41652	+	1.10671i
$334$		0			0
$335$		0			0
$336$	0.938371	+	0.0327686i	0.938371	+	0.0327686i
$337$	−0.475174	−	1.30553i	−0.475174	−	1.30553i	−0.913545	−	0.406737i	$-0.866667\pi$
$337$	−0.475174	−	1.30553i	−0.475174	−	1.30553i	0.438371	−	0.898794i	$-0.355556\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.617234	−	0.849549i	0.617234	−	0.849549i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$347$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$348$		0			0
$349$	−0.755642	+	1.04005i	−0.755642	+	1.04005i	0.241922	+	0.970296i	$0.422222\pi$
$349$	−0.755642	+	1.04005i	−0.755642	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$350$		0			0
$351$	0.0502092	+	0.0484865i	0.0502092	+	0.0484865i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.927184	−	0.374607i	$-0.877778\pi$
$359$		0			0		0.927184	+	0.374607i	$0.122222\pi$
$360$		0			0
$361$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$362$		0			0
$363$	−0.0348995	−	0.999391i	−0.0348995	−	0.999391i
$364$	−0.0654974	+	0.00228722i	−0.0654974	+	0.00228722i
$365$		0			0
$366$		0			0
$367$	−0.343916	−	1.95045i	−0.343916	−	1.95045i	−0.309017	−	0.951057i	$-0.600000\pi$
$367$	−0.343916	−	1.95045i	−0.343916	−	1.95045i	−0.0348995	−	0.999391i	$-0.511111\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.324031	−	0.445991i	0.324031	−	0.445991i
$373$	−0.244415	+	1.14988i	−0.244415	+	1.14988i	0.669131	+	0.743145i	$0.266667\pi$
$373$	−0.244415	+	1.14988i	−0.244415	+	1.14988i	−0.913545	+	0.406737i	$0.866667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$379$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$380$		0			0
$381$	−1.22256	−	0.544320i	−1.22256	−	0.544320i
$382$		0			0
$383$		0			0		0.970296	−	0.241922i	$-0.0777778\pi$
$383$		0			0		−0.970296	+	0.241922i	$0.922222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$388$	−0.604528	+	1.86055i	−0.604528	+	1.86055i
$389$		0			0		0.829038	−	0.559193i	$-0.188889\pi$
$389$		0			0		−0.829038	+	0.559193i	$0.811111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.0952000	+	0.677383i	0.0952000	+	0.677383i	0.978148	+	0.207912i	$0.0666667\pi$
$397$	0.0952000	+	0.677383i	0.0952000	+	0.677383i	−0.882948	+	0.469472i	$0.844444\pi$
$398$		0			0
$399$	−0.778419	−	0.525050i	−0.778419	−	0.525050i
$400$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$401$		0			0		0.469472	−	0.882948i	$-0.344444\pi$
$401$		0			0		−0.469472	+	0.882948i	$0.655556\pi$
$402$		0			0
$403$	−0.0203905	+	0.0326316i	−0.0203905	+	0.0326316i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.288855	−	0.299118i	0.288855	−	0.299118i	−0.559193	−	0.829038i	$-0.688889\pi$
$409$	0.288855	−	0.299118i	0.288855	−	0.299118i	0.848048	+	0.529919i	$0.177778\pi$
$410$		0			0
$411$		0			0
$412$	−0.270562	+	1.08516i	−0.270562	+	1.08516i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.92996	+	0.202847i	−1.92996	+	0.202847i
$418$		0			0
$419$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	−0.524293	+	1.82843i	−0.524293	+	1.82843i	0.0348995	+	0.999391i	$0.488889\pi$
$421$	−0.524293	+	1.82843i	−0.524293	+	1.82843i	−0.559193	+	0.829038i	$0.688889\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.892988	+	0.290149i	−0.892988	+	0.290149i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.559193	−	0.829038i	$-0.688889\pi$
$431$		0			0		0.559193	+	0.829038i	$0.311111\pi$
$432$	−0.719340	+	0.694658i	−0.719340	+	0.694658i
$433$	0.878646	−	0.592654i	0.878646	−	0.592654i	−0.0348995	−	0.999391i	$-0.511111\pi$
$433$	0.878646	−	0.592654i	0.878646	−	0.592654i	0.913545	+	0.406737i	$0.133333\pi$
$434$		0			0
$435$		0			0
$436$	1.47815	+	1.07394i	1.47815	+	1.07394i
$437$		0			0
$438$		0			0
$439$	−0.874607	−	1.79321i	−0.874607	−	1.79321i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−0.874607	−	1.79321i	−0.874607	−	1.79321i	−0.374607	−	0.927184i	$-0.622222\pi$
$440$		0			0
$441$	0.0205575	+	0.116587i	0.0205575	+	0.116587i
$442$		0			0
$443$		0			0		−0.848048	−	0.529919i	$-0.822222\pi$
$443$		0			0		0.848048	+	0.529919i	$0.177778\pi$
$444$	1.05660	−	1.45428i	1.05660	−	1.45428i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	0.938943i		−	0.938943i
$449$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$449$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.787614	−	1.26045i	−0.787614	−	1.26045i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.327673	−	1.54158i	0.327673	−	1.54158i	−0.438371	−	0.898794i	$-0.644444\pi$
$457$	0.327673	−	1.54158i	0.327673	−	1.54158i	0.766044	−	0.642788i	$-0.222222\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.997564	−	0.0697565i	$-0.977778\pi$
$461$		0			0		0.997564	+	0.0697565i	$0.0222222\pi$
$462$		0			0
$463$	0.149516	+	0.460163i	0.149516	+	0.460163i	0.997564	−	0.0697565i	$-0.0222222\pi$
$463$	0.149516	+	0.460163i	0.149516	+	0.460163i	−0.848048	+	0.529919i	$0.822222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i
$469$	1.13858	−	1.45731i	1.13858	−	1.45731i
$470$		0			0
$471$	−1.79649	−	0.876208i	−1.79649	−	0.876208i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.997564	−	0.0697565i	0.997564	−	0.0697565i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.997564	−	0.0697565i	$-0.0222222\pi$
$479$		0			0		−0.997564	+	0.0697565i	$0.977778\pi$
$480$		0			0
$481$	−0.0664889	+	0.106404i	−0.0664889	+	0.106404i
$482$		0			0
$483$		0			0
$484$	−0.997564	+	0.0697565i	−0.997564	+	0.0697565i
$485$		0			0
$486$		0			0
$487$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$487$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$488$		0			0
$489$	1.10209	+	0.924765i	1.10209	+	0.924765i
$490$		0			0
$491$		0			0		−0.882948	−	0.469472i	$-0.844444\pi$
$491$		0			0		0.882948	+	0.469472i	$0.155556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.457027	−	0.308269i	−0.457027	−	0.308269i
$497$		0			0
$498$		0			0
$499$	−1.28479	+	1.53116i	−1.28479	+	1.53116i	−0.615661	+	0.788011i	$0.711111\pi$
$499$	−1.28479	+	1.53116i	−1.28479	+	1.53116i	−0.669131	+	0.743145i	$0.733333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.719340	−	0.694658i	$-0.755556\pi$
$503$		0			0		0.719340	+	0.694658i	$0.244444\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.612662	−	0.784172i	0.612662	−	0.784172i
$508$	−0.501321	+	1.24081i	−0.501321	+	1.24081i
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	0.457027	−	1.59384i	0.457027	−	1.59384i
$512$		0			0
$513$	0.978148	−	0.207912i	0.978148	−	0.207912i
$514$		0			0
$515$		0			0
$516$	−0.924678	+	1.73907i	−0.924678	+	1.73907i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$521$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$522$		0			0
$523$	−1.71934	+	0.694658i	−1.71934	+	0.694658i	−0.719340	+	0.694658i	$0.755556\pi$
$523$	−1.71934	+	0.694658i	−1.71934	+	0.694658i	−1.00000			$\pi$
$524$		0			0
$525$	0.697771	+	0.628276i	0.697771	+	0.628276i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.990268	+	0.139173i	−0.990268	+	0.139173i
$530$		0			0
$531$		0			0
$532$	−0.497564	+	0.796269i	−0.497564	+	0.796269i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.354353	+	1.23577i	0.354353	+	1.23577i	0.913545	+	0.406737i	$0.133333\pi$
$541$	0.354353	+	1.23577i	0.354353	+	1.23577i	−0.559193	+	0.829038i	$0.688889\pi$
$542$		0			0
$543$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.219340	+	0.171367i	0.219340	+	0.171367i	0.719340	−	0.694658i	$-0.244444\pi$
$547$	0.219340	+	0.171367i	0.219340	+	0.171367i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	0.438371	−	0.898794i	0.438371	−	0.898794i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.11990	+	1.08147i	−1.11990	+	1.08147i
$554$		0			0
$555$		0			0
$556$	0.270078	+	1.92171i	0.270078	+	1.92171i
$557$		0			0		0.694658	−	0.719340i	$-0.255556\pi$
$557$		0			0		−0.694658	+	0.719340i	$0.744444\pi$
$558$		0			0
$559$	0.0559170	−	0.125592i	0.0559170	−	0.125592i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.778419	+	0.525050i	0.778419	+	0.525050i
$568$		0			0
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	0.857583	−	0.182285i	0.857583	−	0.182285i	0.241922	−	0.970296i	$-0.422222\pi$
$571$	0.857583	−	0.182285i	0.857583	−	0.182285i	0.615661	+	0.788011i	$0.288889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.719340	+	0.694658i	0.719340	+	0.694658i
$577$	0.413545	+	1.94558i	0.413545	+	1.94558i	0.309017	+	0.951057i	$0.400000\pi$
$577$	0.413545	+	1.94558i	0.413545	+	1.94558i	0.104528	+	0.994522i	$0.466667\pi$
$578$		0			0
$579$	0.0739306	−	0.118314i	0.0739306	−	0.118314i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.275637	−	0.961262i	$-0.411111\pi$
$587$		0			0		−0.275637	+	0.961262i	$0.588889\pi$
$588$	0.115799	−	0.0246138i	0.115799	−	0.0246138i
$589$	0.224224	+	0.503615i	0.224224	+	0.503615i
$590$		0			0
$591$		0			0
$592$	−1.49027	−	1.00520i	−1.49027	−	1.00520i
$593$		0			0		−0.898794	−	0.438371i	$-0.855556\pi$
$593$		0			0		0.898794	+	0.438371i	$0.144444\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.98054			1.98054
$598$		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
$601$	0.0916445	−	0.871939i	0.0916445	−	0.871939i	−0.848048	−	0.529919i	$-0.822222\pi$
$601$	0.0916445	−	0.871939i	0.0916445	−	0.871939i	0.939693	−	0.342020i	$-0.111111\pi$
$602$		0			0
$603$	0.274117	+	1.95045i	0.274117	+	1.95045i
$604$	−1.23219	+	0.831123i	−1.23219	+	0.831123i
$605$		0			0
$606$		0			0
$607$	0.594092	−	1.82843i	0.594092	−	1.82843i	0.0348995	−	0.999391i	$-0.488889\pi$
$607$	0.594092	−	1.82843i	0.594092	−	1.82843i	0.559193	−	0.829038i	$-0.311111\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.177290	−	0.110783i	0.177290	−	0.110783i	−0.438371	−	0.898794i	$-0.644444\pi$
$613$	0.177290	−	0.110783i	0.177290	−	0.110783i	0.615661	+	0.788011i	$0.288889\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.469472	−	0.882948i	$-0.655556\pi$
$617$		0			0		0.469472	+	0.882948i	$0.344444\pi$
$618$		0			0
$619$	1.61323	−	0.718254i	1.61323	−	0.718254i	0.615661	−	0.788011i	$-0.288889\pi$
$619$	1.61323	−	0.718254i	1.61323	−	0.718254i	0.997564	+	0.0697565i	$0.0222222\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.0534691	−	0.0448659i	−0.0534691	−	0.0448659i
$625$	−0.997564	−	0.0697565i	−0.997564	−	0.0697565i
$626$		0			0
$627$		0			0
$628$	−0.812978	+	1.82598i	−0.812978	+	1.82598i
$629$		0			0
$630$		0			0
$631$	−0.475174	+	1.30553i	−0.475174	+	1.30553i	0.438371	+	0.898794i	$0.355556\pi$
$631$	−0.475174	+	1.30553i	−0.475174	+	1.30553i	−0.913545	+	0.406737i	$0.866667\pi$
$632$		0			0
$633$	1.66669	+	0.812901i	1.66669	+	0.812901i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.00794311	+	0.00227765i	−0.00794311	+	0.00227765i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.788011	−	0.615661i	$-0.211111\pi$
$641$		0			0		−0.788011	+	0.615661i	$0.788889\pi$
$642$		0			0
$643$	−0.870573	+	1.63731i	−0.870573	+	1.63731i	−0.104528	+	0.994522i	$0.533333\pi$
$643$	−0.870573	+	1.63731i	−0.870573	+	1.63731i	−0.766044	+	0.642788i	$0.777778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$647$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.193902	+	0.479925i	−0.193902	+	0.479925i
$652$	0.885740	−	1.13369i	0.885740	−	1.13369i
$653$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$653$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.882948	+	1.52931i	0.882948	+	1.52931i
$658$		0			0
$659$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$659$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$660$		0			0
$661$		0			0		−0.719340	−	0.694658i	$-0.755556\pi$
$661$		0			0		0.719340	+	0.694658i	$0.244444\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.24192	−	0.970296i	−1.24192	−	0.970296i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.90095	+	0.617657i	1.90095	+	0.617657i	0.961262	+	0.275637i	$0.0888889\pi$
$673$	1.90095	+	0.617657i	1.90095	+	0.617657i	0.939693	+	0.342020i	$0.111111\pi$
$674$		0			0
$675$	−0.997564	+	0.0697565i	−0.997564	+	0.0697565i
$676$	−0.805076	−	0.584922i	−0.805076	−	0.584922i
$677$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$677$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$678$		0			0
$679$	0.128132	−	1.83238i	0.128132	−	1.83238i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$683$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$684$	−0.241922	−	0.970296i	−0.241922	−	0.970296i
$685$		0			0
$686$		0			0
$687$	1.18362	+	1.51497i	1.18362	+	1.51497i
$688$	1.77028	+	0.863423i	1.77028	+	0.863423i
$689$		0			0
$690$		0			0
$691$	1.08268	−	1.20243i	1.08268	−	1.20243i	0.104528	−	0.994522i	$-0.466667\pi$
$691$	1.08268	−	1.20243i	1.08268	−	1.20243i	0.978148	−	0.207912i	$-0.0666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.603541	−	0.719272i	0.603541	−	0.719272i
$701$		0			0		−0.139173	−	0.990268i	$-0.544444\pi$
$701$		0			0		0.139173	+	0.990268i	$0.455556\pi$
$702$		0			0
$703$	0.731145	+	1.64218i	0.731145	+	1.64218i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.125393	−	0.0611585i	−0.125393	−	0.0611585i	0.374607	−	0.927184i	$-0.377778\pi$
$709$	−0.125393	−	0.0611585i	−0.125393	−	0.0611585i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		−	1.65808i		−	1.65808i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$719$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$720$		0			0
$721$		−	1.05010i		−	1.05010i
$722$		0			0
$723$	−0.606126	−	0.440376i	−0.606126	−	0.440376i
$724$	0.974592	+	0.657371i	0.974592	+	0.657371i
$725$		0			0
$726$		0			0
$727$	1.38295	−	1.33550i	1.38295	−	1.33550i	0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.38295	−	1.33550i	1.38295	−	1.33550i	0.882948	−	0.469472i	$-0.155556\pi$
$728$		0			0
$729$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$730$		0			0
$731$		0			0
$732$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$733$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$733$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.258078	−	0.104270i	−0.258078	−	0.104270i	0.241922	−	0.970296i	$-0.422222\pi$
$739$	−0.258078	−	0.104270i	−0.258078	−	0.104270i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$	0.0215691	+	0.0663828i	0.0215691	+	0.0663828i
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.635369	−	1.57259i	−0.635369	−	1.57259i	−0.809017	−	0.587785i	$-0.800000\pi$
$751$	−0.635369	−	1.57259i	−0.635369	−	1.57259i	0.173648	−	0.984808i	$-0.444444\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.497564	−	0.796269i	0.497564	−	0.796269i
$757$	−0.635369	+	1.57259i	−0.635369	+	1.57259i	0.173648	+	0.984808i	$0.444444\pi$
$757$	−0.635369	+	1.57259i	−0.635369	+	1.57259i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−1.59062	−	0.642651i	−1.59062	−	0.642651i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.669131	−	0.743145i	0.669131	−	0.743145i
$769$	−0.683624	−	0.0238727i	−0.683624	−	0.0238727i	−0.309017	−	0.951057i	$-0.600000\pi$
$769$	−0.683624	−	0.0238727i	−0.683624	−	0.0238727i	−0.374607	+	0.927184i	$0.622222\pi$
$770$		0			0
$771$		0			0
$772$	−0.120822	−	0.0697565i	−0.120822	−	0.0697565i
$773$		0			0		0.882948	−	0.469472i	$-0.155556\pi$
$773$		0			0		−0.882948	+	0.469472i	$0.844444\pi$
$774$		0			0
$775$	−0.151952	−	0.529919i	−0.151952	−	0.529919i
$776$		0			0
$777$	−0.632273	+	1.56493i	−0.632273	+	1.56493i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.0286401	−	0.114869i	−0.0286401	−	0.114869i
$785$		0			0
$786$		0			0
$787$	−0.816620	+	1.12398i	−0.816620	+	1.12398i	0.173648	+	0.984808i	$0.444444\pi$
$787$	−0.816620	+	1.12398i	−0.816620	+	1.12398i	−0.990268	+	0.139173i	$0.955556\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0655896	+	0.0238727i	0.0655896	+	0.0238727i
$794$		0			0
$795$		0			0
$796$	−0.0691197	−	1.97933i	−0.0691197	−	1.97933i
$797$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$797$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.93969	−	0.342020i	1.93969	−	0.342020i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$809$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$810$		0			0
$811$	0.789310	+	0.196797i	0.789310	+	0.196797i	0.615661	−	0.788011i	$-0.288889\pi$
$811$	0.789310	+	0.196797i	0.789310	+	0.196797i	0.173648	+	0.984808i	$0.444444\pi$
$812$		0			0
$813$	1.54946	+	0.968211i	1.54946	+	0.968211i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.04374	−	1.67033i	−1.04374	−	1.67033i
$818$		0			0
$819$	−0.0307679	+	0.0578660i	−0.0307679	+	0.0578660i
$820$		0			0
$821$		0			0		0.927184	−	0.374607i	$-0.122222\pi$
$821$		0			0		−0.927184	+	0.374607i	$0.877778\pi$
$822$		0			0
$823$	−0.415570	−	0.0145120i	−0.415570	−	0.0145120i	−0.173648	−	0.984808i	$-0.555556\pi$
$823$	−0.415570	−	0.0145120i	−0.415570	−	0.0145120i	−0.241922	+	0.970296i	$0.577778\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$827$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$828$		0			0
$829$	−0.149516	−	0.460163i	−0.149516	−	0.460163i	0.848048	−	0.529919i	$-0.177778\pi$
$829$	−0.149516	−	0.460163i	−0.149516	−	0.460163i	−0.997564	+	0.0697565i	$0.977778\pi$
$830$		0			0
$831$	1.68060	−	0.419021i	1.68060	−	0.419021i
$832$	−0.0429726	+	0.0550024i	−0.0429726	+	0.0550024i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.224224	−	0.503615i	−0.224224	−	0.503615i
$838$		0			0
$839$		0			0		0.559193	−	0.829038i	$-0.311111\pi$
$839$		0			0		−0.559193	+	0.829038i	$0.688889\pi$
$840$		0			0
$841$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	0.754239	−	1.69405i	0.754239	−	1.69405i
$845$		0			0
$846$		0			0
$847$	0.892988	−	0.290149i	0.892988	−	0.290149i
$848$		0			0
$849$	0.208548	−	0.0145831i	0.208548	−	0.0145831i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.87481	−	0.682374i	1.87481	−	0.682374i	0.913545	−	0.406737i	$-0.133333\pi$
$853$	1.87481	−	0.682374i	1.87481	−	0.682374i	0.961262	−	0.275637i	$-0.0888889\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$857$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$858$		0			0
$859$	−0.688547	−	1.02081i	−0.688547	−	1.02081i	−0.997564	−	0.0697565i	$-0.977778\pi$
$859$	−0.688547	−	1.02081i	−0.688547	−	1.02081i	0.309017	−	0.951057i	$-0.400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$863$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.500000	−	0.866025i	0.500000	−	0.866025i
$868$	0.486400	+	0.177035i	0.486400	+	0.177035i
$869$		0			0
$870$		0			0
$871$	−0.133394	+	0.0332587i	−0.133394	+	0.0332587i
$872$		0			0
$873$	1.30902	+	1.45381i	1.30902	+	1.45381i
$874$		0			0
$875$		0			0
$876$	1.49756	−	0.935782i	1.49756	−	0.935782i
$877$	−1.15180	−	1.19272i	−1.15180	−	1.19272i	−0.978148	−	0.207912i	$-0.933333\pi$
$877$	−1.15180	−	1.19272i	−1.15180	−	1.19272i	−0.173648	−	0.984808i	$-0.555556\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$881$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$882$		0			0
$883$	−0.354353	−	0.422301i	−0.354353	−	0.422301i	0.559193	−	0.829038i	$-0.311111\pi$
$883$	−0.354353	−	0.422301i	−0.354353	−	0.422301i	−0.913545	+	0.406737i	$0.866667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$887$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$888$		0			0
$889$	0.174878	−	1.24432i	0.174878	−	1.24432i
$890$		0			0
$891$		0			0
$892$	−0.926362	+	1.27503i	−0.926362	+	1.27503i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.104528	+	0.994522i	0.104528	+	0.994522i
$901$		0			0
$902$		0			0
$903$	0.447400	−	1.79442i	0.447400	−	1.79442i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.132685	+	1.89748i	−0.132685	+	1.89748i	0.241922	+	0.970296i	$0.422222\pi$
$907$	−0.132685	+	1.89748i	−0.132685	+	1.89748i	−0.374607	+	0.927184i	$0.622222\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$911$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$912$	−0.961262	+	0.275637i	−0.961262	+	0.275637i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.47274	−	1.23577i	1.47274	−	1.23577i
$917$		0			0
$918$		0			0
$919$	0.606126	+	0.440376i	0.606126	+	0.440376i	0.848048	−	0.529919i	$-0.177778\pi$
$919$	0.606126	+	0.440376i	0.606126	+	0.440376i	−0.241922	+	0.970296i	$0.577778\pi$
$920$		0			0
$921$	0.731145	+	0.356603i	0.731145	+	0.356603i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.495482	−	1.72795i	−0.495482	−	1.72795i
$926$		0			0
$927$	0.804499	+	0.776896i	0.804499	+	0.776896i
$928$		0			0
$929$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$929$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$	−0.0365832	+	0.112592i	−0.0365832	+	0.112592i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.704030	−	1.74254i	−0.704030	−	1.74254i	−0.669131	−	0.743145i	$-0.733333\pi$
$937$	−0.704030	−	1.74254i	−0.704030	−	1.74254i	−0.0348995	−	0.999391i	$-0.511111\pi$
$938$		0			0
$939$	−0.309017	+	1.45381i	−0.309017	+	1.45381i
$940$		0			0
$941$		0			0		0.927184	−	0.374607i	$-0.122222\pi$
$941$		0			0		−0.927184	+	0.374607i	$0.877778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$947$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$948$	−1.65707	+	0.0578660i	−1.65707	+	0.0578660i
$949$	−0.0997176	+	0.0724491i	−0.0997176	+	0.0724491i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.694658	−	0.719340i	$-0.744444\pi$
$953$		0			0		0.694658	+	0.719340i	$0.255556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.563154	+	0.409155i	−0.563154	+	0.409155i
$962$		0			0
$963$		0			0
$964$	−0.418955	+	0.621126i	−0.418955	+	0.621126i
$965$		0			0
$966$		0			0
$967$	1.47274	+	0.422301i	1.47274	+	0.422301i	0.913545	−	0.406737i	$-0.133333\pi$
$967$	1.47274	+	0.422301i	1.47274	+	0.422301i	0.559193	+	0.829038i	$0.311111\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.241922	−	0.970296i	$-0.577778\pi$
$971$		0			0		0.241922	+	0.970296i	$0.422222\pi$
$972$	0.241922	+	0.970296i	0.241922	+	0.970296i
$973$	−0.682573	−	1.68943i	−0.682573	−	1.68943i
$974$		0			0
$975$	−0.0121205	−	0.0687386i	−0.0121205	−	0.0687386i
$976$	−0.374607	+	0.927184i	−0.374607	+	0.927184i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.66913	−	0.743145i	1.66913	−	0.743145i
$982$		0			0
$983$		0			0		−0.139173	−	0.990268i	$-0.544444\pi$
$983$		0			0		0.139173	+	0.990268i	$0.455556\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.0655896	−	0.0238727i	0.0655896	−	0.0238727i
$989$		0			0
$990$		0			0
$991$	−0.194206	+	0.287922i	−0.194206	+	0.287922i	−0.913545	−	0.406737i	$-0.866667\pi$
$991$	−0.194206	+	0.287922i	−0.194206	+	0.287922i	0.719340	+	0.694658i	$0.244444\pi$
$992$		0			0
$993$	0.292131	−	1.01878i	0.292131	−	1.01878i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.36821	+	0.241252i	−1.36821	+	0.241252i	−0.809017	−	0.587785i	$-0.800000\pi$
$997$	−1.36821	+	0.241252i	−1.36821	+	0.241252i	−0.559193	+	0.829038i	$0.688889\pi$
$998$		0			0
$999$	−0.731145	−	1.64218i	−0.731145	−	1.64218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3477.1.gk.a.1544.1	yes	24
3.2	odd	2	CM	3477.1.gk.a.1544.1	yes	24
19.4	even	9		3477.1.gw.a.80.1	yes	24
57.23	odd	18		3477.1.gw.a.80.1	yes	24
61.45	even	30		3477.1.gw.a.2912.1	yes	24
183.167	odd	30		3477.1.gw.a.2912.1	yes	24
1159.289	even	90	inner	3477.1.gk.a.1448.1	✓	24
3477.1448	odd	90	inner	3477.1.gk.a.1448.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3477.1.gk.a.1448.1	✓	24	1159.289	even	90	inner
3477.1.gk.a.1448.1	✓	24	3477.1448	odd	90	inner
3477.1.gk.a.1544.1	yes	24	1.1	even	1	trivial
3477.1.gk.a.1544.1	yes	24	3.2	odd	2	CM
3477.1.gw.a.80.1	yes	24	19.4	even	9
3477.1.gw.a.80.1	yes	24	57.23	odd	18
3477.1.gw.a.2912.1	yes	24	61.45	even	30
3477.1.gw.a.2912.1	yes	24	183.167	odd	30

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3477 = 3 \cdot 19 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3477.gk (of order \(90\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.882948 - 0.469472i) q^{3} +(-0.438371 + 0.898794i) q^{4} +(0.195217 - 0.918425i) q^{7} +(0.559193 + 0.829038i) q^{9} +(0.809017 - 0.587785i) q^{12} +(-0.0534691 + 0.0448659i) q^{13} +(-0.615661 - 0.788011i) q^{16} +(0.104528 + 0.994522i) q^{19} +(-0.603541 + 0.719272i) q^{21} +(0.0348995 - 0.999391i) q^{25} +(-0.104528 - 0.994522i) q^{27} +(0.739897 + 0.578071i) q^{28} +(0.524293 - 0.170353i) q^{31} +(-0.990268 + 0.139173i) q^{36} +(1.70961 - 0.555485i) q^{37} +(0.0682737 - 0.0145120i) q^{39} +(-1.77028 + 0.863423i) q^{43} +(0.173648 + 0.984808i) q^{48} +(0.108151 + 0.0481518i) q^{49} +(-0.0168859 - 0.0677257i) q^{52} +(0.374607 - 0.927184i) q^{57} +(-0.500000 - 0.866025i) q^{61} +(0.870573 - 0.351734i) q^{63} +(0.978148 - 0.207912i) q^{64} +(1.77028 + 0.863423i) q^{67} +(1.76159 + 0.123183i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-0.939693 - 0.342020i) q^{76} +(-1.37461 - 0.927184i) q^{79} +(-0.374607 + 0.927184i) q^{81} +(-0.381903 - 0.857767i) q^{84} +(0.0307679 + 0.0578660i) q^{91} +(-0.542900 - 0.0957278i) q^{93} +(1.93726 - 0.272264i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{12} - 3 q^{13} - 3 q^{19} + 3 q^{21} + 3 q^{27} + 3 q^{28} - 6 q^{43} - 3 q^{49} + 3 q^{52} - 12 q^{61} - 3 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 12 q^{75} - 24 q^{79} + 6 q^{91} + 3 q^{93}+O(q^{100}) \) 24 * q + 6 * q^12 - 3 * q^13 - 3 * q^19 + 3 * q^21 + 3 * q^27 + 3 * q^28 - 6 * q^43 - 3 * q^49 + 3 * q^52 - 12 * q^61 - 3 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 12 * q^75 - 24 * q^79 + 6 * q^91 + 3 * q^93

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