LMFDB - Embedded newform 3267.1.be.a.1703.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3267,1,Mod(245,3267)] 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("3267.245"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([5, 72])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3267.be (of order $90$, degree $24$, not 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:	$1.63044539627$
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:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			1703.1
Root			$-0.997564 - 0.0697565i$ of defining polynomial
Character	$\chi$	$=$	3267.1703
Dual form			3267.1.be.a.3173.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.719340 + 0.694658i) q^{3} +(0.559193 + 0.829038i) q^{4} +(-0.362486 + 0.580099i) q^{5} +(0.0348995 + 0.999391i) q^{9} +(-0.173648 + 0.984808i) q^{12} +(-0.663721 + 0.165484i) q^{15} +(-0.374607 + 0.927184i) q^{16} +(-0.683624 + 0.0238727i) q^{20} +(0.592396 - 1.62760i) q^{23} +(0.233252 + 0.478238i) q^{25} +(-0.669131 + 0.743145i) q^{27} +(0.213817 - 0.273673i) q^{31} +(-0.809017 + 0.587785i) q^{36} +(-0.196449 + 1.86909i) q^{37} +(-0.592396 - 0.342020i) q^{45} +(1.06579 + 0.718885i) q^{47} +(-0.913545 + 0.406737i) q^{48} +(-0.848048 - 0.529919i) q^{49} +(-1.87322 - 0.608645i) q^{53} +(-0.137393 - 1.96482i) q^{59} +(-0.508341 - 0.457712i) q^{60} +(-0.978148 + 0.207912i) q^{64} +(-0.0603074 - 0.342020i) q^{67} +(1.55676 - 0.759281i) q^{69} +(0.955369 + 0.860218i) q^{71} +(-0.164425 + 0.506047i) q^{75} +(-0.402069 - 0.553400i) q^{80} +(-0.997564 + 0.0697565i) q^{81} +(-1.50000 - 0.866025i) q^{89} +(1.68060 - 0.419021i) q^{92} +(0.343916 - 0.0483343i) q^{93} +(1.29929 - 0.811883i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) $ 24 * q + 3 * q^5 + 3 * q^15 + 6 * q^20 + 3 * q^25 - 3 * q^27 - 3 * q^31 - 6 * q^36 - 3 * q^47 - 3 * q^48 - 3 * q^59 + 3 * q^64 - 24 * q^67 - 6 * q^75 - 36 * q^89 - 6 * q^93 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$.

$n$	$244$	$3026$
$\chi(n)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{18}\right)$

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		−0.882948	−	0.469472i	$-0.844444\pi$
$2$		0			0		0.882948	+	0.469472i	$0.155556\pi$
$3$	0.719340	+	0.694658i	0.719340	+	0.694658i
$3$	0.719340	+	0.694658i	0.719340	+	0.694658i
$4$	0.559193	+	0.829038i	0.559193	+	0.829038i
$5$	−0.362486	+	0.580099i	−0.362486	+	0.580099i	−0.978148	−	0.207912i	$-0.933333\pi$
$5$	−0.362486	+	0.580099i	−0.362486	+	0.580099i	0.615661	+	0.788011i	$0.288889\pi$
$6$		0			0
$7$		0			0		0.275637	−	0.961262i	$-0.411111\pi$
$7$		0			0		−0.275637	+	0.961262i	$0.588889\pi$
$8$		0			0
$9$	0.0348995	+	0.999391i	0.0348995	+	0.999391i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$13$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$13$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$14$		0			0
$15$	−0.663721	+	0.165484i	−0.663721	+	0.165484i
$16$	−0.374607	+	0.927184i	−0.374607	+	0.927184i
$17$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$17$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$18$		0			0
$19$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$19$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$20$	−0.683624	+	0.0238727i	−0.683624	+	0.0238727i
$21$		0			0
$22$		0			0
$23$	0.592396	−	1.62760i	0.592396	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$23$	0.592396	−	1.62760i	0.592396	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$24$		0			0
$25$	0.233252	+	0.478238i	0.233252	+	0.478238i
$26$		0			0
$27$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$28$		0			0
$29$		0			0		0.719340	−	0.694658i	$-0.244444\pi$
$29$		0			0		−0.719340	+	0.694658i	$0.755556\pi$
$30$		0			0
$31$	0.213817	−	0.273673i	0.213817	−	0.273673i	−0.669131	−	0.743145i	$-0.733333\pi$
$31$	0.213817	−	0.273673i	0.213817	−	0.273673i	0.882948	+	0.469472i	$0.155556\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$37$	−0.196449	+	1.86909i	−0.196449	+	1.86909i	0.241922	+	0.970296i	$0.422222\pi$
$37$	−0.196449	+	1.86909i	−0.196449	+	1.86909i	−0.438371	+	0.898794i	$0.644444\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.719340	−	0.694658i	$-0.755556\pi$
$41$		0			0		0.719340	+	0.694658i	$0.244444\pi$
$42$		0			0
$43$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$43$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$44$		0			0
$45$	−0.592396	−	0.342020i	−0.592396	−	0.342020i
$46$		0			0
$47$	1.06579	+	0.718885i	1.06579	+	0.718885i	0.961262	−	0.275637i	$-0.0888889\pi$
$47$	1.06579	+	0.718885i	1.06579	+	0.718885i	0.104528	+	0.994522i	$0.466667\pi$
$48$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$49$	−0.848048	−	0.529919i	−0.848048	−	0.529919i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.87322	−	0.608645i	−1.87322	−	0.608645i	−0.990268	−	0.139173i	$-0.955556\pi$
$53$	−1.87322	−	0.608645i	−1.87322	−	0.608645i	−0.882948	−	0.469472i	$-0.844444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.137393	−	1.96482i	−0.137393	−	1.96482i	−0.241922	−	0.970296i	$-0.577778\pi$
$59$	−0.137393	−	1.96482i	−0.137393	−	1.96482i	0.104528	−	0.994522i	$-0.466667\pi$
$60$	−0.508341	−	0.457712i	−0.508341	−	0.457712i
$61$		0			0		0.788011	−	0.615661i	$-0.211111\pi$
$61$		0			0		−0.788011	+	0.615661i	$0.788889\pi$
$62$		0			0
$63$		0			0
$64$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$65$		0			0
$66$		0			0
$67$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	0.939693	−	0.342020i	$-0.111111\pi$
$67$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	−1.00000			$\pi$
$68$		0			0
$69$	1.55676	−	0.759281i	1.55676	−	0.759281i
$70$		0			0
$71$	0.955369	+	0.860218i	0.955369	+	0.860218i	0.990268	−	0.139173i	$-0.0444444\pi$
$71$	0.955369	+	0.860218i	0.955369	+	0.860218i	−0.0348995	+	0.999391i	$0.511111\pi$
$72$		0			0
$73$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$73$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$74$		0			0
$75$	−0.164425	+	0.506047i	−0.164425	+	0.506047i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.469472	−	0.882948i	$-0.344444\pi$
$79$		0			0		−0.469472	+	0.882948i	$0.655556\pi$
$80$	−0.402069	−	0.553400i	−0.402069	−	0.553400i
$81$	−0.997564	+	0.0697565i	−0.997564	+	0.0697565i
$82$		0			0
$83$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$83$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1.68060	−	0.419021i	1.68060	−	0.419021i
$93$	0.343916	−	0.0483343i	0.343916	−	0.0483343i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.29929	−	0.811883i	1.29929	−	0.811883i	0.309017	−	0.951057i	$-0.400000\pi$
$97$	1.29929	−	0.811883i	1.29929	−	0.811883i	0.990268	+	0.139173i	$0.0444444\pi$
$98$		0			0
$99$		0			0
$100$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$101$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$101$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$102$		0			0
$103$	1.87481	−	0.131099i	1.87481	−	0.131099i	0.913545	−	0.406737i	$-0.133333\pi$
$103$	1.87481	−	0.131099i	1.87481	−	0.131099i	0.961262	+	0.275637i	$0.0888889\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$107$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$108$	−0.990268	−	0.139173i	−0.990268	−	0.139173i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$112$		0			0
$113$	1.24739	+	0.311009i	1.24739	+	0.311009i	0.809017	−	0.587785i	$-0.200000\pi$
$113$	1.24739	+	0.311009i	1.24739	+	0.311009i	0.438371	+	0.898794i	$0.355556\pi$
$114$		0			0
$115$	0.729431	+	0.933629i	0.729431	+	0.933629i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	0.346450	+	0.0242262i	0.346450	+	0.0242262i
$125$	−1.04227	−	0.109547i	−1.04227	−	0.109547i
$126$		0			0
$127$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$127$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$131$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.188547	−	0.657542i	−0.188547	−	0.657542i
$136$		0			0
$137$	−0.274117	−	1.95045i	−0.274117	−	1.95045i	−0.309017	−	0.951057i	$-0.600000\pi$
$137$	−0.274117	−	1.95045i	−0.274117	−	1.95045i	0.0348995	−	0.999391i	$-0.488889\pi$
$138$		0			0
$139$		0			0		0.829038	−	0.559193i	$-0.188889\pi$
$139$		0			0		−0.829038	+	0.559193i	$0.811111\pi$
$140$		0			0
$141$	0.267286	+	1.25748i	0.267286	+	1.25748i
$142$		0			0
$143$		0			0
$144$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$145$		0			0
$146$		0			0
$147$	−0.241922	−	0.970296i	−0.241922	−	0.970296i
$148$	−1.65940	+	0.882318i	−1.65940	+	0.882318i
$149$		0			0		0.882948	−	0.469472i	$-0.155556\pi$
$149$		0			0		−0.882948	+	0.469472i	$0.844444\pi$
$150$		0			0
$151$		0			0		0.0697565	−	0.997564i	$-0.477778\pi$
$151$		0			0		−0.0697565	+	0.997564i	$0.522222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.0812519	+	0.223238i	0.0812519	+	0.223238i
$156$		0			0
$157$	0.0840186	−	0.336980i	0.0840186	−	0.336980i	−0.913545	−	0.406737i	$-0.866667\pi$
$157$	0.0840186	−	0.336980i	0.0840186	−	0.336980i	0.997564	+	0.0697565i	$0.0222222\pi$
$158$		0			0
$159$	−0.924678	−	1.73907i	−0.924678	−	1.73907i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.473442	−	1.45710i	0.473442	−	1.45710i	−0.374607	−	0.927184i	$-0.622222\pi$
$163$	0.473442	−	1.45710i	0.473442	−	1.45710i	0.848048	−	0.529919i	$-0.177778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.374607	−	0.927184i	$-0.622222\pi$
$167$		0			0		0.374607	+	0.927184i	$0.377778\pi$
$168$		0			0
$169$	−0.961262	−	0.275637i	−0.961262	−	0.275637i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.997564	−	0.0697565i	$-0.977778\pi$
$173$		0			0		0.997564	+	0.0697565i	$0.0222222\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.26604	−	1.50881i	1.26604	−	1.50881i
$178$		0			0
$179$	−0.801115	+	1.79933i	−0.801115	+	1.79933i	−0.241922	+	0.970296i	$0.577778\pi$
$179$	−0.801115	+	1.79933i	−0.801115	+	1.79933i	−0.559193	+	0.829038i	$0.688889\pi$
$180$	−0.0477162	−	0.682374i	−0.0477162	−	0.682374i
$181$	1.83832	+	0.390746i	1.83832	+	0.390746i	0.990268	−	0.139173i	$-0.0444444\pi$
$181$	1.83832	+	0.390746i	1.83832	+	0.390746i	0.848048	+	0.529919i	$0.177778\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.01305	−	0.791479i	−1.01305	−	0.791479i
$186$		0			0
$187$		0			0
$188$			1.28558i			1.28558i
$189$		0			0
$190$		0			0
$191$	0.893036	+	0.924765i	0.893036	+	0.924765i	0.997564	−	0.0697565i	$-0.0222222\pi$
$191$	0.893036	+	0.924765i	0.893036	+	0.924765i	−0.104528	+	0.994522i	$0.533333\pi$
$192$	−0.848048	−	0.529919i	−0.848048	−	0.529919i
$193$		0			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$193$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$194$		0			0
$195$		0			0
$196$	−0.0348995	−	0.999391i	−0.0348995	−	0.999391i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$199$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$200$		0			0
$201$	0.194206	−	0.287922i	0.194206	−	0.287922i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.64728	+	0.535233i	1.64728	+	0.535233i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.927184	−	0.374607i	$-0.122222\pi$
$211$		0			0		−0.927184	+	0.374607i	$0.877778\pi$
$212$	−0.542900	−	1.89332i	−0.542900	−	1.89332i
$213$	0.0896772	+	1.28244i	0.0896772	+	1.28244i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.559193	+	0.829038i	−0.559193	+	0.829038i	−0.997564	−	0.0697565i	$-0.977778\pi$
$223$	−0.559193	+	0.829038i	−0.559193	+	0.829038i	0.438371	+	0.898794i	$0.355556\pi$
$224$		0			0
$225$	−0.469807	+	0.249801i	−0.469807	+	0.249801i
$226$		0			0
$227$		0			0		−0.241922	−	0.970296i	$-0.577778\pi$
$227$		0			0		0.241922	+	0.970296i	$0.422222\pi$
$228$		0			0
$229$	0.990268	+	0.139173i	0.990268	+	0.139173i	0.615661	−	0.788011i	$-0.288889\pi$
$229$	0.990268	+	0.139173i	0.990268	+	0.139173i	0.374607	+	0.927184i	$0.377778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$233$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$234$		0			0
$235$	−0.803358	+	0.357678i	−0.803358	+	0.357678i
$236$	1.55208	−	1.21262i	1.55208	−	1.21262i
$237$		0			0
$238$		0			0
$239$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$239$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$240$	0.0952000	−	0.677383i	0.0952000	−	0.677383i
$241$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$241$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$242$		0			0
$243$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$244$		0			0
$245$	0.614811	−	0.299864i	0.614811	−	0.299864i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$251$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.719340	−	0.694658i	−0.719340	−	0.694658i
$257$	1.55676	+	0.759281i	1.55676	+	0.759281i	0.997564	−	0.0697565i	$-0.0222222\pi$
$257$	1.55676	+	0.759281i	1.55676	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$263$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$264$		0			0
$265$	1.03209	−	0.866025i	1.03209	−	0.866025i
$266$		0			0
$267$	−0.477418	−	1.66495i	−0.477418	−	1.66495i
$268$	0.249824	−	0.241252i	0.249824	−	0.241252i
$269$	1.22265	−	0.397265i	1.22265	−	0.397265i	0.374607	−	0.927184i	$-0.377778\pi$
$269$	1.22265	−	0.397265i	1.22265	−	0.397265i	0.848048	+	0.529919i	$0.177778\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.50000	+	0.866025i	1.50000	+	0.866025i
$277$		0			0		0.999391	−	0.0348995i	$-0.0111111\pi$
$277$		0			0		−0.999391	+	0.0348995i	$0.988889\pi$
$278$		0			0
$279$	0.280969	+	0.204136i	0.280969	+	0.204136i
$280$		0			0
$281$		0			0		0.374607	−	0.927184i	$-0.377778\pi$
$281$		0			0		−0.374607	+	0.927184i	$0.622222\pi$
$282$		0			0
$283$		0			0		0.694658	−	0.719340i	$-0.255556\pi$
$283$		0			0		−0.694658	+	0.719340i	$0.744444\pi$
$284$	−0.178917	+	1.27306i	−0.178917	+	1.27306i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$290$		0			0
$291$	1.49861	+	0.318539i	1.49861	+	0.318539i
$292$		0			0
$293$		0			0		−0.559193	−	0.829038i	$-0.688889\pi$
$293$		0			0		0.559193	+	0.829038i	$0.311111\pi$
$294$		0			0
$295$	1.18959	+	0.632517i	1.18959	+	0.632517i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.511477	+	0.146664i	−0.511477	+	0.146664i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$307$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$308$		0			0
$309$	1.43969	+	1.20805i	1.43969	+	1.20805i
$310$		0			0
$311$	−1.36821	+	1.41682i	−1.36821	+	1.41682i	−0.559193	+	0.829038i	$0.688889\pi$
$311$	−1.36821	+	1.41682i	−1.36821	+	1.41682i	−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−0.374607	+	0.927184i	−0.374607	+	0.927184i	0.615661	+	0.788011i	$0.288889\pi$
$313$	−0.374607	+	0.927184i	−0.374607	+	0.927184i	−0.990268	+	0.139173i	$0.955556\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$317$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$318$		0			0
$319$		0			0
$320$	0.233956	−	0.642788i	0.233956	−	0.642788i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.615661	−	0.788011i	−0.615661	−	0.788011i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$331$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−1.87481	−	0.131099i	−1.87481	−	0.131099i
$334$		0			0
$335$	0.220266	+	0.0889933i	0.220266	+	0.0889933i
$336$		0			0
$337$		0			0		−0.898794	−	0.438371i	$-0.855556\pi$
$337$		0			0		0.898794	+	0.438371i	$0.144444\pi$
$338$		0			0
$339$	0.681251	+	1.09023i	0.681251	+	1.09023i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.123845	+	1.17830i	−0.123845	+	1.17830i
$346$		0			0
$347$		0			0		0.0348995	−	0.999391i	$-0.488889\pi$
$347$		0			0		−0.0348995	+	0.999391i	$0.511111\pi$
$348$		0			0
$349$		0			0		0.898794	−	0.438371i	$-0.144444\pi$
$349$		0			0		−0.898794	+	0.438371i	$0.855556\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$353$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$354$		0			0
$355$	−0.845319	+	0.242391i	−0.845319	+	0.242391i
$356$	−0.120822	−	1.72783i	−0.120822	−	1.72783i
$357$		0			0
$358$		0			0
$359$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$359$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$360$		0			0
$361$	0.978148	−	0.207912i	0.978148	−	0.207912i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.370646	−	1.48658i	−0.370646	−	1.48658i	−0.809017	−	0.587785i	$-0.800000\pi$
$367$	−0.370646	−	1.48658i	−0.370646	−	1.48658i	0.438371	−	0.898794i	$-0.355556\pi$
$368$	1.28716	+	1.15897i	1.28716	+	1.15897i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.232387	+	0.258091i	0.232387	+	0.258091i
$373$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$373$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$374$		0			0
$375$	−0.673648	−	0.802823i	−0.673648	−	0.802823i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$379$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.82620	−	0.737831i	1.82620	−	0.737831i	0.848048	−	0.529919i	$-0.177778\pi$
$383$	1.82620	−	0.737831i	1.82620	−	0.737831i	0.978148	−	0.207912i	$-0.0666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.39963	+	0.623157i	1.39963	+	0.623157i
$389$	−1.24739	+	0.311009i	−1.24739	+	0.311009i	−0.809017	−	0.587785i	$-0.800000\pi$
$389$	−1.24739	+	0.311009i	−1.24739	+	0.311009i	−0.438371	+	0.898794i	$0.644444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$398$		0			0
$399$		0			0
$400$	−0.530793	+	0.0371166i	−0.530793	+	0.0371166i
$401$	−0.683624	−	0.0238727i	−0.683624	−	0.0238727i	−0.309017	−	0.951057i	$-0.600000\pi$
$401$	−0.683624	−	0.0238727i	−0.683624	−	0.0238727i	−0.374607	+	0.927184i	$0.622222\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.321137	−	0.603972i	0.321137	−	0.603972i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.788011	−	0.615661i	$-0.788889\pi$
$409$		0			0		0.788011	+	0.615661i	$0.211111\pi$
$410$		0			0
$411$	1.15771	−	1.59345i	1.15771	−	1.59345i
$412$	1.15707	+	1.48098i	1.15707	+	1.48098i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.673648	+	0.118782i	−0.673648	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−0.673648	+	0.118782i	−0.673648	+	0.118782i	−0.173648	+	0.984808i	$0.555556\pi$
$420$		0			0
$421$	−1.87481	−	0.131099i	−1.87481	−	0.131099i	−0.913545	−	0.406737i	$-0.866667\pi$
$421$	−1.87481	−	0.131099i	−1.87481	−	0.131099i	−0.961262	+	0.275637i	$0.911111\pi$
$422$		0			0
$423$	−0.681251	+	1.09023i	−0.681251	+	1.09023i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$431$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$432$	−0.438371	−	0.898794i	−0.438371	−	0.898794i
$433$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$433$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$439$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$440$		0			0
$441$	0.500000	−	0.866025i	0.500000	−	0.866025i
$442$		0			0
$443$	−0.0477162	+	0.682374i	−0.0477162	+	0.682374i	0.913545	+	0.406737i	$0.133333\pi$
$443$	−0.0477162	+	0.682374i	−0.0477162	+	0.682374i	−0.961262	+	0.275637i	$0.911111\pi$
$444$	−1.80658	−	0.518029i	−1.80658	−	0.518029i
$445$	1.04611	−	0.556226i	1.04611	−	0.556226i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.142220	+	0.669092i	0.142220	+	0.669092i	0.990268	+	0.139173i	$0.0444444\pi$
$449$	0.142220	+	0.669092i	0.142220	+	0.669092i	−0.848048	+	0.529919i	$0.822222\pi$
$450$		0			0
$451$		0			0
$452$	0.439693	+	1.20805i	0.439693	+	1.20805i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.139173	−	0.990268i	$-0.544444\pi$
$457$		0			0		0.139173	+	0.990268i	$0.455556\pi$
$458$		0			0
$459$		0			0
$460$	−0.366121	+	1.12680i	−0.366121	+	1.12680i
$461$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$461$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$463$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$464$		0			0
$465$	−0.0966263	+	0.217026i	−0.0966263	+	0.217026i
$466$		0			0
$467$	0.267286	−	1.25748i	0.267286	−	1.25748i	−0.615661	−	0.788011i	$-0.711111\pi$
$467$	0.267286	−	1.25748i	0.267286	−	1.25748i	0.882948	−	0.469472i	$-0.155556\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.294524	−	0.184039i	0.294524	−	0.184039i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.542900	−	1.89332i	0.542900	−	1.89332i
$478$		0			0
$479$		0			0		−0.615661	−	0.788011i	$-0.711111\pi$
$479$		0			0		0.615661	+	0.788011i	$0.288889\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1.04801i			1.04801i
$486$		0			0
$487$	−1.52045	+	1.10467i	−1.52045	+	1.10467i	−0.559193	+	0.829038i	$0.688889\pi$
$487$	−1.52045	+	1.10467i	−1.52045	+	1.10467i	−0.961262	+	0.275637i	$0.911111\pi$
$488$		0			0
$489$	1.35275	−	0.719272i	1.35275	−	0.719272i
$490$		0			0
$491$		0			0		0.997564	−	0.0697565i	$-0.0222222\pi$
$491$		0			0		−0.997564	+	0.0697565i	$0.977778\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.173648	+	0.300767i	0.173648	+	0.300767i
$497$		0			0
$498$		0			0
$499$	0.823868	−	1.68918i	0.823868	−	1.68918i	0.104528	−	0.994522i	$-0.466667\pi$
$499$	0.823868	−	1.68918i	0.823868	−	1.68918i	0.719340	−	0.694658i	$-0.244444\pi$
$500$	−0.492011	−	0.925338i	−0.492011	−	0.925338i
$501$		0			0
$502$		0			0
$503$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$503$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$508$		0			0
$509$	−0.477418	−	1.66495i	−0.477418	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$509$	−0.477418	−	1.66495i	−0.477418	−	1.66495i	0.241922	−	0.970296i	$-0.422222\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.603541	+	1.13510i	−0.603541	+	1.13510i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.278224	−	0.624902i	−0.278224	−	0.624902i	0.719340	−	0.694658i	$-0.244444\pi$
$521$	−0.278224	−	0.624902i	−0.278224	−	0.624902i	−0.997564	+	0.0697565i	$0.977778\pi$
$522$		0			0
$523$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$523$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$530$		0			0
$531$	1.95883	−	0.205881i	1.95883	−	0.205881i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.82620	+	0.737831i	−1.82620	+	0.737831i
$538$		0			0
$539$		0			0
$540$	0.439693	−	0.524005i	0.439693	−	0.524005i
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$	1.05094	+	1.55808i	1.05094	+	1.55808i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.829038	−	0.559193i	$-0.811111\pi$
$547$		0			0		0.829038	+	0.559193i	$0.188889\pi$
$548$	1.46371	−	1.31793i	1.46371	−	1.31793i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.178917	−	1.27306i	−0.178917	−	1.27306i
$556$		0			0
$557$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$557$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.615661	−	0.788011i	$-0.288889\pi$
$563$		0			0		−0.615661	+	0.788011i	$0.711111\pi$
$564$	−0.893036	+	0.924765i	−0.893036	+	0.924765i
$565$	−0.632577	+	0.610872i	−0.632577	+	0.610872i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.438371	−	0.898794i	$-0.644444\pi$
$569$		0			0		0.438371	+	0.898794i	$0.355556\pi$
$570$		0			0
$571$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$571$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$572$		0			0
$573$			1.28558i			1.28558i
$574$		0			0
$575$	0.916556	−	0.0963340i	0.916556	−	0.0963340i
$576$	−0.241922	−	0.970296i	−0.241922	−	0.970296i
$577$	1.25755	−	1.39666i	1.25755	−	1.39666i	0.374607	−	0.927184i	$-0.377778\pi$
$577$	1.25755	−	1.39666i	1.25755	−	1.39666i	0.882948	−	0.469472i	$-0.155556\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.188547	+	0.657542i	−0.188547	+	0.657542i	0.809017	+	0.587785i	$0.200000\pi$
$587$	−0.188547	+	0.657542i	−0.188547	+	0.657542i	−0.997564	+	0.0697565i	$0.977778\pi$
$588$	0.669131	−	0.743145i	0.669131	−	0.743145i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.65940	−	0.882318i	−1.65940	−	0.882318i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.370646	+	1.48658i	−0.370646	+	1.48658i
$598$		0			0
$599$	−0.917847	+	1.46886i	−0.917847	+	1.46886i	−0.0348995	+	0.999391i	$0.511111\pi$
$599$	−0.917847	+	1.46886i	−0.917847	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$600$		0			0
$601$		0			0		0.275637	−	0.961262i	$-0.411111\pi$
$601$		0			0		−0.275637	+	0.961262i	$0.588889\pi$
$602$		0			0
$603$	0.339707	−	0.0722070i	0.339707	−	0.0722070i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$607$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$613$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.673648	−	1.85083i	0.673648	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$617$	0.673648	−	1.85083i	0.673648	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$618$		0			0
$619$	−0.823868	−	1.68918i	−0.823868	−	1.68918i	−0.719340	−	0.694658i	$-0.755556\pi$
$619$	−0.823868	−	1.68918i	−0.823868	−	1.68918i	−0.104528	−	0.994522i	$-0.533333\pi$
$620$	−0.139637	+	0.192194i	−0.139637	+	0.192194i
$621$	0.813149	+	1.52931i	0.813149	+	1.52931i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.113770	−	0.145619i	0.113770	−	0.145619i
$626$		0			0
$627$		0			0
$628$	0.326352	−	0.118782i	0.326352	−	0.118782i
$629$		0			0
$630$		0			0
$631$	0.0363024	−	0.345394i	0.0363024	−	0.345394i	−0.961262	−	0.275637i	$-0.911111\pi$
$631$	0.0363024	−	0.345394i	0.0363024	−	0.345394i	0.997564	−	0.0697565i	$-0.0222222\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.924678	−	1.73907i	0.924678	−	1.73907i
$637$		0			0
$638$		0			0
$639$	−0.826352	+	0.984808i	−0.826352	+	0.984808i
$640$		0			0
$641$	−1.43594	−	0.968551i	−1.43594	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$641$	−1.43594	−	0.968551i	−1.43594	−	0.968551i	−0.438371	−	0.898794i	$-0.644444\pi$
$642$		0			0
$643$	−0.848048	−	0.529919i	−0.848048	−	0.529919i	0.0348995	−	0.999391i	$-0.488889\pi$
$643$	−0.848048	−	0.529919i	−0.848048	−	0.529919i	−0.882948	+	0.469472i	$0.844444\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.47274	−	0.422301i	1.47274	−	0.422301i
$653$	−0.0477162	−	0.682374i	−0.0477162	−	0.682374i	−0.961262	−	0.275637i	$-0.911111\pi$
$653$	−0.0477162	−	0.682374i	−0.0477162	−	0.682374i	0.913545	−	0.406737i	$-0.133333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$659$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$660$		0			0
$661$	−0.266044	−	1.50881i	−0.266044	−	1.50881i	−0.766044	−	0.642788i	$-0.777778\pi$
$661$	−0.266044	−	1.50881i	−0.266044	−	1.50881i	0.500000	−	0.866025i	$-0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.469472	−	0.882948i	$-0.344444\pi$
$673$		0			0		−0.469472	+	0.882948i	$0.655556\pi$
$674$		0			0
$675$	−0.511477	−	0.146664i	−0.511477	−	0.146664i
$676$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$677$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$677$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$683$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$684$		0			0
$685$	1.23082	+	0.547995i	1.23082	+	0.547995i
$686$		0			0
$687$	0.615661	+	0.788011i	0.615661	+	0.788011i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.59381	+	0.995922i	−1.59381	+	0.995922i	−0.615661	+	0.788011i	$0.711111\pi$
$691$	−1.59381	+	0.995922i	−1.59381	+	0.995922i	−0.978148	+	0.207912i	$0.933333\pi$
$692$		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			0
$701$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−0.826352	−	0.300767i	−0.826352	−	0.300767i
$706$		0			0
$707$		0			0
$708$	1.95883	+	0.205881i	1.95883	+	0.205881i
$709$	−1.15707	−	1.48098i	−1.15707	−	1.48098i	−0.848048	−	0.529919i	$-0.822222\pi$
$709$	−1.15707	−	1.48098i	−1.15707	−	1.48098i	−0.309017	−	0.951057i	$-0.600000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.318765	−	0.510131i	−0.318765	−	0.510131i
$714$		0			0
$715$		0			0
$716$	−1.93969	+	0.342020i	−1.93969	+	0.342020i
$717$		0			0
$718$		0			0
$719$	−1.27853	−	0.134379i	−1.27853	−	0.134379i	−0.559193	−	0.829038i	$-0.688889\pi$
$719$	−1.27853	−	0.134379i	−1.27853	−	0.134379i	−0.719340	+	0.694658i	$0.755556\pi$
$720$	0.539031	−	0.421137i	0.539031	−	0.421137i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.704030	+	1.74254i	0.704030	+	1.74254i
$725$		0			0
$726$		0			0
$727$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$727$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$728$		0			0
$729$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.829038	−	0.559193i	$-0.188889\pi$
$733$		0			0		−0.829038	+	0.559193i	$0.811111\pi$
$734$		0			0
$735$	0.650561	+	0.211380i	0.650561	+	0.211380i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$739$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$740$	0.0896772	−	1.28244i	0.0896772	−	1.28244i
$741$		0			0
$742$		0			0
$743$		0			0		0.882948	−	0.469472i	$-0.155556\pi$
$743$		0			0		−0.882948	+	0.469472i	$0.844444\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.370646	−	1.48658i	0.370646	−	1.48658i	−0.438371	−	0.898794i	$-0.644444\pi$
$751$	0.370646	−	1.48658i	0.370646	−	1.48658i	0.809017	−	0.587785i	$-0.200000\pi$
$752$	−1.06579	+	0.718885i	−1.06579	+	0.718885i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.107320	+	0.330298i	−0.107320	+	0.330298i	−0.990268	−	0.139173i	$-0.955556\pi$
$757$	−0.107320	+	0.330298i	−0.107320	+	0.330298i	0.882948	+	0.469472i	$0.155556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.374607	−	0.927184i	$-0.622222\pi$
$761$		0			0		0.374607	+	0.927184i	$0.377778\pi$
$762$		0			0
$763$		0			0
$764$	−0.267286	+	1.25748i	−0.267286	+	1.25748i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.0348995	−	0.999391i	−0.0348995	−	0.999391i
$769$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$769$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$770$		0			0
$771$	0.592396	+	1.62760i	0.592396	+	1.62760i
$772$		0			0
$773$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$773$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$774$		0			0
$775$	0.180754	+	0.0384205i	0.180754	+	0.0384205i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.809017	−	0.587785i	0.809017	−	0.587785i
$785$	0.165026	+	0.170890i	0.165026	+	0.170890i
$786$		0			0
$787$		0			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$787$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.34401	+	0.0939826i	1.34401	+	0.0939826i
$796$	−0.671624	+	1.37703i	−0.671624	+	1.37703i
$797$	−0.321137	−	0.603972i	−0.321137	−	0.603972i	0.669131	−	0.743145i	$-0.266667\pi$
$797$	−0.321137	−	0.603972i	−0.321137	−	0.603972i	−0.990268	+	0.139173i	$0.955556\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.813149	−	1.52931i	0.813149	−	1.52931i
$802$		0			0
$803$		0			0
$804$	0.347296			0.347296
$805$		0			0
$806$		0			0
$807$	1.15547	+	0.563559i	1.15547	+	0.563559i
$808$		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			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.673648	+	0.802823i	0.673648	+	0.802823i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.241922	−	0.970296i	$-0.577778\pi$
$821$		0			0		0.241922	+	0.970296i	$0.422222\pi$
$822$		0			0
$823$	−0.990268	−	0.139173i	−0.990268	−	0.139173i	−0.374607	−	0.927184i	$-0.622222\pi$
$823$	−0.990268	−	0.139173i	−0.990268	−	0.139173i	−0.615661	+	0.788011i	$0.711111\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$827$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$828$	0.477418	+	1.66495i	0.477418	+	1.66495i
$829$	0.317271	−	0.141258i	0.317271	−	0.141258i	−0.241922	−	0.970296i	$-0.577778\pi$
$829$	0.317271	−	0.141258i	0.317271	−	0.141258i	0.559193	+	0.829038i	$0.311111\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$838$		0			0
$839$	1.55676	−	0.759281i	1.55676	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$839$	1.55676	−	0.759281i	1.55676	−	0.759281i	0.997564	+	0.0697565i	$0.0222222\pi$
$840$		0			0
$841$	0.0348995	−	0.999391i	0.0348995	−	0.999391i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.508341	−	0.457712i	0.508341	−	0.457712i
$846$		0			0
$847$		0			0
$848$	1.26604	−	1.50881i	1.26604	−	1.50881i
$849$		0			0
$850$		0			0
$851$	2.92575	+	1.42698i	2.92575	+	1.42698i
$852$	−1.01305	+	0.791479i	−1.01305	+	0.791479i
$853$		0			0		−0.927184	−	0.374607i	$-0.877778\pi$
$853$		0			0		0.927184	+	0.374607i	$0.122222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$857$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$858$		0			0
$859$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$859$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.64728	+	0.535233i	−1.64728	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$863$	−1.64728	+	0.535233i	−1.64728	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.615661	−	0.788011i	0.615661	−	0.788011i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.856733	+	1.27016i	0.856733	+	1.27016i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.694658	−	0.719340i	$-0.255556\pi$
$877$		0			0		−0.694658	+	0.719340i	$0.744444\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$881$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$882$		0			0
$883$	0.160147	+	1.52370i	0.160147	+	1.52370i	0.719340	+	0.694658i	$0.244444\pi$
$883$	0.160147	+	1.52370i	0.160147	+	1.52370i	−0.559193	+	0.829038i	$0.688889\pi$
$884$		0			0
$885$	0.416337	+	1.28135i	0.416337	+	1.28135i
$886$		0			0
$887$		0			0		−0.559193	−	0.829038i	$-0.688889\pi$
$887$		0			0		0.559193	+	0.829038i	$0.311111\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.00000			−1.00000
$893$		0			0
$894$		0			0
$895$	−0.753399	−	1.11696i	−0.753399	−	1.11696i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.469807	−	0.249801i	−0.469807	−	0.249801i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.893036	+	0.924765i	−0.893036	+	0.924765i
$906$		0			0
$907$	0.374607	−	0.927184i	0.374607	−	0.927184i	−0.615661	−	0.788011i	$-0.711111\pi$
$907$	0.374607	−	0.927184i	0.374607	−	0.927184i	0.990268	−	0.139173i	$-0.0444444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.683624	−	0.0238727i	0.683624	−	0.0238727i	0.309017	−	0.951057i	$-0.400000\pi$
$911$	0.683624	−	0.0238727i	0.683624	−	0.0238727i	0.374607	+	0.927184i	$0.377778\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.438371	+	0.898794i	0.438371	+	0.898794i
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$926$		0			0
$927$	0.196449	+	1.86909i	0.196449	+	1.86909i
$928$		0			0
$929$	−1.82620	−	0.737831i	−1.82620	−	0.737831i	−0.978148	−	0.207912i	$-0.933333\pi$
$929$	−1.82620	−	0.737831i	−1.82620	−	0.737831i	−0.848048	−	0.529919i	$-0.822222\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.96842	+	0.0687386i	−1.96842	+	0.0687386i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$937$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$938$		0			0
$939$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$940$	−0.745761	−	0.466003i	−0.745761	−	0.466003i
$941$		0			0		0.0348995	−	0.999391i	$-0.488889\pi$
$941$		0			0		−0.0348995	+	0.999391i	$0.511111\pi$
$942$		0			0
$943$		0			0
$944$	1.87322	+	0.608645i	1.87322	+	0.608645i
$945$		0			0
$946$		0			0
$947$	1.93969	+	0.342020i	1.93969	+	0.342020i	1.00000			$0$
$947$	1.93969	+	0.342020i	1.93969	+	0.342020i	0.939693	+	0.342020i	$0.111111\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$953$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$954$		0			0
$955$	−0.860169	+	0.182834i	−0.860169	+	0.182834i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.614811	−	0.299864i	0.614811	−	0.299864i
$961$	0.212743	+	0.853264i	0.212743	+	0.853264i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$967$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$971$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$972$	0.104528	−	0.994522i	0.104528	−	0.994522i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.374607	−	0.927184i	$-0.622222\pi$
$977$		0			0		0.374607	+	0.927184i	$0.377778\pi$
$978$		0			0
$979$		0			0
$980$	0.592396	+	0.342020i	0.592396	+	0.342020i
$981$		0			0
$982$		0			0
$983$	−1.91111	+	0.476493i	−1.91111	+	0.476493i	−0.913545	+	0.406737i	$0.866667\pi$
$983$	−1.91111	+	0.476493i	−1.91111	+	0.476493i	−0.997564	+	0.0697565i	$0.977778\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$991$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$992$		0			0
$993$	−1.39963	−	0.623157i	−1.39963	−	0.623157i
$994$		0			0
$995$	−1.04737	−	0.0365750i	−1.04737	−	0.0365750i
$996$		0			0
$997$		0			0		−0.694658	−	0.719340i	$-0.744444\pi$
$997$		0			0		0.694658	+	0.719340i	$0.255556\pi$
$998$		0			0
$999$	−1.25755	−	1.39666i	−1.25755	−	1.39666i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3267.1.be.a.1703.1		24
11.2	odd	10	inner	3267.1.be.a.245.1		24
11.3	even	5	inner	3267.1.be.a.2864.1		24
11.4	even	5		3267.1.q.a.3026.1	yes	6
11.5	even	5	inner	3267.1.be.a.2810.1		24
11.6	odd	10	inner	3267.1.be.a.2810.1		24
11.7	odd	10		3267.1.q.a.3026.1	yes	6
11.8	odd	10	inner	3267.1.be.a.2864.1		24
11.9	even	5	inner	3267.1.be.a.245.1		24
11.10	odd	2	CM	3267.1.be.a.1703.1		24
27.14	odd	18	inner	3267.1.be.a.2066.1		24
297.14	odd	90	inner	3267.1.be.a.3227.1		24
297.41	even	90	inner	3267.1.be.a.3227.1		24
297.68	even	90	inner	3267.1.be.a.608.1		24
297.95	even	90		3267.1.q.a.122.1	✓	6
297.149	even	90	inner	3267.1.be.a.3173.1		24
297.203	odd	90	inner	3267.1.be.a.3173.1		24
297.230	even	18	inner	3267.1.be.a.2066.1		24
297.257	odd	90		3267.1.q.a.122.1	✓	6
297.284	odd	90	inner	3267.1.be.a.608.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3267.1.q.a.122.1	✓	6	297.95	even	90
3267.1.q.a.122.1	✓	6	297.257	odd	90
3267.1.q.a.3026.1	yes	6	11.4	even	5
3267.1.q.a.3026.1	yes	6	11.7	odd	10
3267.1.be.a.245.1		24	11.2	odd	10	inner
3267.1.be.a.245.1		24	11.9	even	5	inner
3267.1.be.a.608.1		24	297.68	even	90	inner
3267.1.be.a.608.1		24	297.284	odd	90	inner
3267.1.be.a.1703.1		24	1.1	even	1	trivial
3267.1.be.a.1703.1		24	11.10	odd	2	CM
3267.1.be.a.2066.1		24	27.14	odd	18	inner
3267.1.be.a.2066.1		24	297.230	even	18	inner
3267.1.be.a.2810.1		24	11.5	even	5	inner
3267.1.be.a.2810.1		24	11.6	odd	10	inner
3267.1.be.a.2864.1		24	11.3	even	5	inner
3267.1.be.a.2864.1		24	11.8	odd	10	inner
3267.1.be.a.3173.1		24	297.149	even	90	inner
3267.1.be.a.3173.1		24	297.203	odd	90	inner
3267.1.be.a.3227.1		24	297.14	odd	90	inner
3267.1.be.a.3227.1		24	297.41	even	90	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3267.be (of order \(90\), degree \(24\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.719340 + 0.694658i) q^{3} +(0.559193 + 0.829038i) q^{4} +(-0.362486 + 0.580099i) q^{5} +(0.0348995 + 0.999391i) q^{9} +(-0.173648 + 0.984808i) q^{12} +(-0.663721 + 0.165484i) q^{15} +(-0.374607 + 0.927184i) q^{16} +(-0.683624 + 0.0238727i) q^{20} +(0.592396 - 1.62760i) q^{23} +(0.233252 + 0.478238i) q^{25} +(-0.669131 + 0.743145i) q^{27} +(0.213817 - 0.273673i) q^{31} +(-0.809017 + 0.587785i) q^{36} +(-0.196449 + 1.86909i) q^{37} +(-0.592396 - 0.342020i) q^{45} +(1.06579 + 0.718885i) q^{47} +(-0.913545 + 0.406737i) q^{48} +(-0.848048 - 0.529919i) q^{49} +(-1.87322 - 0.608645i) q^{53} +(-0.137393 - 1.96482i) q^{59} +(-0.508341 - 0.457712i) q^{60} +(-0.978148 + 0.207912i) q^{64} +(-0.0603074 - 0.342020i) q^{67} +(1.55676 - 0.759281i) q^{69} +(0.955369 + 0.860218i) q^{71} +(-0.164425 + 0.506047i) q^{75} +(-0.402069 - 0.553400i) q^{80} +(-0.997564 + 0.0697565i) q^{81} +(-1.50000 - 0.866025i) q^{89} +(1.68060 - 0.419021i) q^{92} +(0.343916 - 0.0483343i) q^{93} +(1.29929 - 0.811883i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100}) \) 24 * q + 3 * q^5 + 3 * q^15 + 6 * q^20 + 3 * q^25 - 3 * q^27 - 3 * q^31 - 6 * q^36 - 3 * q^47 - 3 * q^48 - 3 * q^59 + 3 * q^64 - 24 * q^67 - 6 * q^75 - 36 * q^89 - 6 * q^93 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more