LMFDB - Embedded newform 329.1.f.b.46.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [329,1,Mod(46,329)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(329, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("329.46"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 329 = 7 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	329.f (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.164192389156$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{15})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.143108492101942920287.1

Embedding invariants

Embedding label			46.4
Root			$-0.104528 + 0.994522i$ of defining polynomial
Character	$\chi$	$=$	329.46
Dual form			329.1.f.b.93.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.978148 + 1.69420i) q^{2} +(0.809017 - 1.40126i) q^{3} +(-1.41355 + 2.44833i) q^{4} +3.16535 q^{6} +(-0.978148 - 0.207912i) q^{7} -3.57433 q^{8} +(-0.809017 - 1.40126i) q^{9} +(2.28716 + 3.96149i) q^{12} +(-0.604528 - 1.86055i) q^{14} +(-2.08268 - 3.60730i) q^{16} +(0.104528 - 0.181049i) q^{17} +(1.58268 - 2.74128i) q^{18} +(-1.08268 + 1.20243i) q^{21} +(-2.89169 + 5.00856i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(1.89169 - 2.10094i) q^{28} +(2.28716 - 3.96149i) q^{32} +0.408977 q^{34} +4.57433 q^{36} +(0.104528 + 0.181049i) q^{37} +(-3.09618 - 0.658114i) q^{42} +(-0.500000 - 0.866025i) q^{47} -6.73968 q^{48} +(0.913545 + 0.406737i) q^{49} -1.95630 q^{50} +(-0.169131 - 0.292943i) q^{51} +(-0.669131 + 1.15897i) q^{53} +(-0.978148 - 1.69420i) q^{54} +(3.49622 + 0.743145i) q^{56} +(0.978148 - 1.69420i) q^{59} +(0.978148 + 1.69420i) q^{61} +(0.500000 + 1.53884i) q^{63} +4.78339 q^{64} +(0.295511 + 0.511841i) q^{68} -0.209057 q^{71} +(2.89169 + 5.00856i) q^{72} +(-0.204489 + 0.354185i) q^{74} +(0.809017 + 1.40126i) q^{75} +(0.809017 + 1.40126i) q^{79} -1.00000 q^{83} +(-1.41355 - 4.35045i) q^{84} +(-0.309017 - 0.535233i) q^{89} +(0.978148 - 1.69420i) q^{94} +(-3.70071 - 6.40982i) q^{96} +1.33826 q^{97} +(0.204489 + 1.94558i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 2 q^{3} - 5 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 2 q^{9} + 5 q^{12} - 3 q^{14} - 6 q^{16} - q^{17} + 2 q^{18} + 2 q^{21} - 8 q^{24} - 4 q^{25} - 8 q^{27} + 5 q^{32} - 2 q^{34} + 10 q^{36}+ \cdots - q^{98}+O(q^{100}) $ 8 * q - q^2 + 2 * q^3 - 5 * q^4 + 4 * q^6 + q^7 - 2 * q^8 - 2 * q^9 + 5 * q^12 - 3 * q^14 - 6 * q^16 - q^17 + 2 * q^18 + 2 * q^21 - 8 * q^24 - 4 * q^25 - 8 * q^27 + 5 * q^32 - 2 * q^34 + 10 * q^36 - q^37 - 7 * q^42 - 4 * q^47 - 6 * q^48 + q^49 + 2 * q^50 + 3 * q^51 - q^53 + q^54 + 11 * q^56 - q^59 - q^61 + 4 * q^63 + 8 * q^64 + 5 * q^68 + 2 * q^71 + 8 * q^72 + q^74 + 2 * q^75 + 2 * q^79 - 8 * q^83 - 5 * q^84 + 2 * q^89 - q^94 - 10 * q^96 + 2 * q^97 - q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/329\mathbb{Z}\right)^\times$.

$n$	$99$	$283$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$2$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$3$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$3$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$4$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	3.16535			3.16535
$7$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$7$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$8$	−3.57433			−3.57433
$9$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	2.28716	+	3.96149i	2.28716	+	3.96149i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.604528	−	1.86055i	−0.604528	−	1.86055i
$15$		0			0
$16$	−2.08268	−	3.60730i	−2.08268	−	3.60730i
$17$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$17$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$18$	1.58268	−	2.74128i	1.58268	−	2.74128i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$	−1.08268	+	1.20243i	−1.08268	+	1.20243i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−2.89169	+	5.00856i	−2.89169	+	5.00856i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−1.00000			−1.00000
$28$	1.89169	−	2.10094i	1.89169	−	2.10094i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	2.28716	−	3.96149i	2.28716	−	3.96149i
$33$		0			0
$34$	0.408977			0.408977
$35$		0			0
$36$	4.57433			4.57433
$37$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$37$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−3.09618	−	0.658114i	−3.09618	−	0.658114i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$47$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$48$	−6.73968			−6.73968
$49$	0.913545	+	0.406737i	0.913545	+	0.406737i
$50$	−1.95630			−1.95630
$51$	−0.169131	−	0.292943i	−0.169131	−	0.292943i
$52$		0			0
$53$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$53$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$54$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$55$		0			0
$56$	3.49622	+	0.743145i	3.49622	+	0.743145i
$57$		0			0
$58$		0			0
$59$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$59$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$60$		0			0
$61$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$61$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$	0.500000	+	1.53884i	0.500000	+	1.53884i
$64$	4.78339			4.78339
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	0.295511	+	0.511841i	0.295511	+	0.511841i
$69$		0			0
$70$		0			0
$71$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$71$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$72$	2.89169	+	5.00856i	2.89169	+	5.00856i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$75$	0.809017	+	1.40126i	0.809017	+	1.40126i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$79$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$84$	−1.41355	−	4.35045i	−1.41355	−	4.35045i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$89$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.978148	−	1.69420i	0.978148	−	1.69420i
$95$		0			0
$96$	−3.70071	−	6.40982i	−3.70071	−	6.40982i
$97$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$97$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$98$	0.204489	+	1.94558i	0.204489	+	1.94558i
$99$		0			0
$100$	−1.41355	−	2.44833i	−1.41355	−	2.44833i
$101$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$101$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$102$	0.330869	−	0.573083i	0.330869	−	0.573083i
$103$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$103$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$104$		0			0
$105$		0			0
$106$	−2.61803			−2.61803
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	1.41355	−	2.44833i	1.41355	−	2.44833i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$	0.338261			0.338261
$112$	1.28716	+	3.96149i	1.28716	+	3.96149i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	3.82709			3.82709
$119$	−0.139886	+	0.155360i	−0.139886	+	0.155360i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	−1.91355	+	3.31436i	−1.91355	+	3.31436i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−2.11803	+	2.35232i	−2.11803	+	2.35232i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	2.39169	+	4.14253i	2.39169	+	4.14253i
$129$		0			0
$130$		0			0
$131$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$131$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.373619	+	0.647127i	−0.373619	+	0.647127i
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−1.61803			−1.61803
$142$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$143$		0			0
$144$	−3.36984	+	5.83674i	−3.36984	+	5.83674i
$145$		0			0
$146$		0			0
$147$	1.30902	−	0.951057i	1.30902	−	0.951057i
$148$	−0.591023			−0.591023
$149$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$149$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$150$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	−0.338261			−0.338261
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$157$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$158$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$159$	1.08268	+	1.87525i	1.08268	+	1.87525i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	3.86984	−	4.29789i	3.86984	−	4.29789i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$173$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$174$		0			0
$175$	0.669131	−	0.743145i	0.669131	−	0.743145i
$176$		0			0
$177$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$178$	0.604528	−	1.04707i	0.604528	−	1.04707i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	3.16535			3.16535
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	2.82709			2.82709
$189$	0.978148	+	0.207912i	0.978148	+	0.207912i
$190$		0			0
$191$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$191$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$192$	3.86984	−	6.70276i	3.86984	−	6.70276i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	1.30902	+	2.26728i	1.30902	+	2.26728i
$195$		0			0
$196$	−2.28716	+	1.66172i	−2.28716	+	1.66172i
$197$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	1.78716	−	3.09546i	1.78716	−	3.09546i
$201$		0			0
$202$	−3.57433			−3.57433
$203$		0			0
$204$	0.956295			0.956295
$205$		0			0
$206$	1.78716	−	3.09546i	1.78716	−	3.09546i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$213$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$214$		0			0
$215$		0			0
$216$	3.57433			3.57433
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	0.330869	+	0.573083i	0.330869	+	0.573083i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−3.06082	+	3.39939i	−3.06082	+	3.39939i
$225$	1.61803			1.61803
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$	2.76531	+	4.78966i	2.76531	+	4.78966i
$237$	2.61803			2.61803
$238$	−0.400040	−	0.0850311i	−0.400040	−	0.0850311i
$239$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$239$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$240$		0			0
$241$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$241$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$242$	0.978148	−	1.69420i	0.978148	−	1.69420i
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	−5.53062			−5.53062
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$250$		0			0
$251$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$251$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$252$	−4.47437	−	0.951057i	−4.47437	−	0.951057i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.28716	+	3.96149i	−2.28716	+	3.96149i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i
$260$		0			0
$261$		0			0
$262$	1.30902	−	2.26728i	1.30902	−	2.26728i
$263$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$263$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.00000			−1.00000
$268$		0			0
$269$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$270$		0			0
$271$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$271$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$272$	−0.870796			−0.870796
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$277$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$283$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$283$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$284$	0.295511	−	0.511841i	0.295511	−	0.511841i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−7.40142			−7.40142
$289$	0.478148	+	0.828176i	0.478148	+	0.828176i
$290$		0			0
$291$	1.08268	−	1.87525i	1.08268	−	1.87525i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	2.89169	+	1.28746i	2.89169	+	1.28746i
$295$		0			0
$296$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$297$		0			0
$298$	1.78716	−	3.09546i	1.78716	−	3.09546i
$299$		0			0
$300$	−4.57433			−4.57433
$301$		0			0
$302$		0			0
$303$	1.47815	+	2.56023i	1.47815	+	2.56023i
$304$		0			0
$305$		0			0
$306$	−0.330869	−	0.573083i	−0.330869	−	0.573083i
$307$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$307$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$308$		0			0
$309$	−2.95630			−2.95630
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	−1.20906			−1.20906
$315$		0			0
$316$	−4.57433			−4.57433
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$	−2.11803	+	3.66854i	−2.11803	+	3.66854i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.309017	+	0.951057i	0.309017	+	0.951057i
$330$		0			0
$331$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$331$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$332$	1.41355	−	2.44833i	1.41355	−	2.44833i
$333$	0.169131	−	0.292943i	0.169131	−	0.292943i
$334$		0			0
$335$		0			0
$336$	6.59240	+	1.40126i	6.59240	+	1.40126i
$337$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$337$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$338$	0.978148	+	1.69420i	0.978148	+	1.69420i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$344$		0			0
$345$		0			0
$346$	1.30902	−	2.26728i	1.30902	−	2.26728i
$347$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$347$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	1.91355	+	0.406737i	1.91355	+	0.406737i
$351$		0			0
$352$		0			0
$353$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$353$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$354$	3.09618	−	5.36274i	3.09618	−	5.36274i
$355$		0			0
$356$	1.74724			1.74724
$357$	0.104528	+	0.321706i	0.104528	+	0.321706i
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$	−1.61803			−1.61803
$364$		0			0
$365$		0			0
$366$	3.09618	+	5.36274i	3.09618	+	5.36274i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.895472	−	0.994522i	0.895472	−	0.994522i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	1.78716	+	3.09546i	1.78716	+	3.09546i
$377$		0			0
$378$	0.604528	+	1.86055i	0.604528	+	1.86055i
$379$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$379$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$380$		0			0
$381$		0			0
$382$	−0.978148	+	1.69420i	−0.978148	+	1.69420i
$383$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$384$	7.73968			7.73968
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.89169	+	3.27651i	−1.89169	+	3.27651i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−3.26531	−	1.45381i	−3.26531	−	1.45381i
$393$	−2.16535			−2.16535
$394$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$395$		0			0
$396$		0			0
$397$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$397$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$398$		0			0
$399$		0			0
$400$	4.16535			4.16535
$401$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$401$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$		0			0
$404$	−2.58268	−	4.47333i	−2.58268	−	4.47333i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.604528	+	1.04707i	0.604528	+	1.04707i
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$	5.16535			5.16535
$413$	−1.30902	+	1.45381i	−1.30902	+	1.45381i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$424$	2.39169	−	4.14253i	2.39169	−	4.14253i
$425$	0.104528	+	0.181049i	0.104528	+	0.181049i
$426$	−0.661739			−0.661739
$427$	−0.604528	−	1.86055i	−0.604528	−	1.86055i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$431$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$432$	2.08268	+	3.60730i	2.08268	+	3.60730i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.169131	−	1.60917i	−0.169131	−	1.60917i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	−0.478148	+	0.828176i	−0.478148	+	0.828176i
$445$		0			0
$446$		0			0
$447$	−2.95630			−2.95630
$448$	−4.67886	−	0.994522i	−4.67886	−	0.994522i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	1.58268	+	2.74128i	1.58268	+	2.74128i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$458$		0			0
$459$	−0.104528	+	0.181049i	−0.104528	+	0.181049i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.500000	+	0.866025i	0.500000	+	0.866025i
$472$	−3.49622	+	6.05563i	−3.49622	+	6.05563i
$473$		0			0
$474$	2.56082	+	4.43548i	2.56082	+	4.43548i
$475$		0			0
$476$	−0.182636	−	0.562096i	−0.182636	−	0.562096i
$477$	2.16535			2.16535
$478$	1.78716	+	3.09546i	1.78716	+	3.09546i
$479$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$479$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$480$		0			0
$481$		0			0
$482$	−3.57433			−3.57433
$483$		0			0
$484$	2.82709			2.82709
$485$		0			0
$486$	0.978148	−	1.69420i	0.978148	−	1.69420i
$487$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$488$	−3.49622	−	6.05563i	−3.49622	−	6.05563i
$489$		0			0
$490$		0			0
$491$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$491$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.204489	+	0.0434654i	0.204489	+	0.0434654i
$498$	−3.16535			−3.16535
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$		0			0
$502$	1.78716	+	3.09546i	1.78716	+	3.09546i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−1.78716	−	5.50033i	−1.78716	−	5.50033i
$505$		0			0
$506$		0			0
$507$	0.809017	−	1.40126i	0.809017	−	1.40126i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−4.16535			−4.16535
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.273659	−	0.303929i	0.273659	−	0.303929i
$519$	−2.16535			−2.16535
$520$		0			0
$521$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$521$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$522$		0			0
$523$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$523$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$524$	3.78339			3.78339
$525$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$526$	−1.20906			−1.20906
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$	−3.16535			−3.16535
$532$		0			0
$533$		0			0
$534$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.95630			1.95630
$539$		0			0
$540$		0			0
$541$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$541$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$542$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$543$		0			0
$544$	−0.478148	−	0.828176i	−0.478148	−	0.828176i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	1.58268	−	2.74128i	1.58268	−	2.74128i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$554$	3.16535			3.16535
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$	2.28716	−	3.96149i	2.28716	−	3.96149i
$565$		0			0
$566$	−2.61803			−2.61803
$567$		0			0
$568$	0.747238			0.747238
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$571$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$572$		0			0
$573$	1.61803			1.61803
$574$		0			0
$575$		0			0
$576$	−3.86984	−	6.70276i	−3.86984	−	6.70276i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.935398	+	1.62016i	−0.935398	+	1.62016i
$579$		0			0
$580$		0			0
$581$	0.978148	+	0.207912i	0.978148	+	0.207912i
$582$	4.23607			4.23607
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.478148	+	4.54927i	0.478148	+	4.54927i
$589$		0			0
$590$		0			0
$591$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$592$	0.435398	−	0.754131i	0.435398	−	0.754131i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	5.16535			5.16535
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−2.89169	−	5.00856i	−2.89169	−	5.00856i
$601$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$601$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−2.89169	+	5.00856i	−2.89169	+	5.00856i
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.478148	−	0.828176i	0.478148	−	0.828176i
$613$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$613$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$614$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$615$		0			0
$616$		0			0
$617$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$617$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$618$	−2.89169	−	5.00856i	−2.89169	−	5.00856i
$619$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$619$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.190983	+	0.587785i	0.190983	+	0.587785i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−0.873619	−	1.51315i	−0.873619	−	1.51315i
$629$	0.0437048			0.0437048
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−2.89169	−	5.00856i	−2.89169	−	5.00856i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−6.12165			−6.12165
$637$		0			0
$638$		0			0
$639$	0.169131	+	0.292943i	0.169131	+	0.292943i
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$643$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$647$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$653$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−1.30902	+	1.45381i	−1.30902	+	1.45381i
$659$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$661$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$662$	0.604528	−	1.04707i	0.604528	−	1.04707i
$663$		0			0
$664$	3.57433			3.57433
$665$		0			0
$666$	0.661739			0.661739
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	2.28716	+	7.03917i	2.28716	+	7.03917i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	1.30902	+	2.26728i	1.30902	+	2.26728i
$675$	0.500000	−	0.866025i	0.500000	−	0.866025i
$676$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−1.30902	−	0.278240i	−1.30902	−	0.278240i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$683$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$684$		0			0
$685$		0			0
$686$	0.204489	−	1.94558i	0.204489	−	1.94558i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	3.78339			3.78339
$693$		0			0
$694$	−3.57433			−3.57433
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.873619	+	2.68872i	0.873619	+	2.68872i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	3.82709			3.82709
$707$	1.22256	−	1.35779i	1.22256	−	1.35779i
$708$	8.94874			8.94874
$709$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$709$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$710$		0			0
$711$	1.30902	−	2.26728i	1.30902	−	2.26728i
$712$	1.10453	+	1.91310i	1.10453	+	1.91310i
$713$		0			0
$714$	−0.442790	+	0.491768i	−0.442790	+	0.491768i
$715$		0			0
$716$		0			0
$717$	1.47815	−	2.56023i	1.47815	−	2.56023i
$718$		0			0
$719$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$719$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$720$		0			0
$721$	0.564602	+	1.73767i	0.564602	+	1.73767i
$722$	−1.95630			−1.95630
$723$	1.47815	+	2.56023i	1.47815	+	2.56023i
$724$		0			0
$725$		0			0
$726$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1.61803			−1.61803
$730$		0			0
$731$		0			0
$732$	−4.47437	+	7.74983i	−4.47437	+	7.74983i
$733$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$733$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$739$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$740$		0			0
$741$		0			0
$742$	2.56082	+	0.544320i	2.56082	+	0.544320i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.809017	+	1.40126i	0.809017	+	1.40126i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−2.08268	+	3.60730i	−2.08268	+	3.60730i
$753$	1.47815	−	2.56023i	1.47815	−	2.56023i
$754$		0			0
$755$		0			0
$756$	−1.89169	+	2.10094i	−1.89169	+	2.10094i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−1.91355	−	3.31436i	−1.91355	−	3.31436i
$759$		0			0
$760$		0			0
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	−2.82709			−2.82709
$765$		0			0
$766$	1.78716	−	3.09546i	1.78716	−	3.09546i
$767$		0			0
$768$	3.70071	+	6.40982i	3.70071	+	6.40982i
$769$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$773$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$774$		0			0
$775$		0			0
$776$	−4.78339			−4.78339
$777$	−0.330869	−	0.0703285i	−0.330869	−	0.0703285i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.435398	−	4.14253i	−0.435398	−	4.14253i
$785$		0			0
$786$	−2.11803	−	3.66854i	−2.11803	−	3.66854i
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	1.41355	−	2.44833i	1.41355	−	2.44833i
$789$	0.500000	+	0.866025i	0.500000	+	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.78716	−	3.09546i	1.78716	−	3.09546i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	−0.209057			−0.209057
$800$	2.28716	+	3.96149i	2.28716	+	3.96149i
$801$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$802$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$808$	3.26531	−	5.65569i	3.26531	−	5.65569i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$811$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$812$		0			0
$813$	0.338261			0.338261
$814$		0			0
$815$		0			0
$816$	−0.704489	+	1.22021i	−0.704489	+	1.22021i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$824$	3.26531	+	5.65569i	3.26531	+	5.65569i
$825$		0			0
$826$	−3.74346	−	0.795697i	−3.74346	−	0.795697i
$827$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$827$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$832$		0			0
$833$	0.169131	−	0.122881i	0.169131	−	0.122881i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	−3.16535			−3.16535
$847$	0.309017	+	0.951057i	0.309017	+	0.951057i
$848$	5.57433			5.57433
$849$	1.08268	+	1.87525i	1.08268	+	1.87525i
$850$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$851$		0			0
$852$	−0.478148	−	0.828176i	−0.478148	−	0.828176i
$853$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$853$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$854$	2.56082	−	2.84408i	2.56082	−	2.84408i
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−2.61803			−2.61803
$863$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$863$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$864$	−2.28716	+	3.96149i	−2.28716	+	3.96149i
$865$		0			0
$866$		0			0
$867$	1.54732			1.54732
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.08268	−	1.87525i	−1.08268	−	1.87525i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	−0.978148	+	1.69420i	−0.978148	+	1.69420i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	2.56082	−	1.86055i	2.56082	−	1.86055i
$883$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$883$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$	−1.20906			−1.20906
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−2.89169	−	5.00856i	−2.89169	−	5.00856i
$895$		0			0
$896$	−1.47815	−	4.54927i	−1.47815	−	4.54927i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.28716	+	3.96149i	−2.28716	+	3.96149i
$901$	0.139886	+	0.242290i	0.139886	+	0.242290i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$908$		0			0
$909$	2.95630			2.95630
$910$		0			0
$911$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$911$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$912$		0			0
$913$		0			0
$914$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$915$		0			0
$916$		0			0
$917$	0.413545	+	1.27276i	0.413545	+	1.27276i
$918$	−0.408977			−0.408977
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.209057			−0.209057
$926$		0			0
$927$	−1.47815	+	2.56023i	−1.47815	+	2.56023i
$928$		0			0
$929$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$929$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$941$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$942$	−0.978148	+	1.69420i	−0.978148	+	1.69420i
$943$		0			0
$944$	−8.14866			−8.14866
$945$		0			0
$946$		0			0
$947$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$947$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$948$	−3.70071	+	6.40982i	−3.70071	+	6.40982i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.500000	−	0.555306i	0.500000	−	0.555306i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	2.11803	+	3.66854i	2.11803	+	3.66854i
$955$		0			0
$956$	−2.58268	+	4.47333i	−2.58268	+	4.47333i
$957$		0			0
$958$	−1.20906			−1.20906
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$	−2.58268	−	4.47333i	−2.58268	−	4.47333i
$965$		0			0
$966$		0			0
$967$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$968$	1.78716	+	3.09546i	1.78716	+	3.09546i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	2.82709			2.82709
$973$		0			0
$974$	1.95630			1.95630
$975$		0			0
$976$	4.07433	−	7.05695i	4.07433	−	7.05695i
$977$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.604528	+	1.04707i	0.604528	+	1.04707i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.58268	+	0.336408i	1.58268	+	0.336408i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$991$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$992$		0			0
$993$	−1.00000			−1.00000
$994$	0.126381	+	0.388960i	0.126381	+	0.388960i
$995$		0			0
$996$	−2.28716	−	3.96149i	−2.28716	−	3.96149i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	−0.104528	−	0.181049i	−0.104528	−	0.181049i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	329.1.f.b.46.4	✓	8
3.2	odd	2		2961.1.x.d.46.1		8
7.2	even	3	inner	329.1.f.b.93.4	yes	8
7.3	odd	6		2303.1.d.e.2255.1		4
7.4	even	3		2303.1.d.d.2255.1		4
7.5	odd	6		2303.1.f.d.422.4		8
7.6	odd	2		2303.1.f.d.704.4		8
21.2	odd	6		2961.1.x.d.1738.1		8
47.46	odd	2	CM	329.1.f.b.46.4	✓	8
141.140	even	2		2961.1.x.d.46.1		8
329.46	odd	6		2303.1.d.d.2255.1		4
329.93	odd	6	inner	329.1.f.b.93.4	yes	8
329.187	even	6		2303.1.f.d.422.4		8
329.234	even	6		2303.1.d.e.2255.1		4
329.328	even	2		2303.1.f.d.704.4		8
987.422	even	6		2961.1.x.d.1738.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
329.1.f.b.46.4	✓	8	1.1	even	1	trivial
329.1.f.b.46.4	✓	8	47.46	odd	2	CM
329.1.f.b.93.4	yes	8	7.2	even	3	inner
329.1.f.b.93.4	yes	8	329.93	odd	6	inner
2303.1.d.d.2255.1		4	7.4	even	3
2303.1.d.d.2255.1		4	329.46	odd	6
2303.1.d.e.2255.1		4	7.3	odd	6
2303.1.d.e.2255.1		4	329.234	even	6
2303.1.f.d.422.4		8	7.5	odd	6
2303.1.f.d.422.4		8	329.187	even	6
2303.1.f.d.704.4		8	7.6	odd	2
2303.1.f.d.704.4		8	329.328	even	2
2961.1.x.d.46.1		8	3.2	odd	2
2961.1.x.d.46.1		8	141.140	even	2
2961.1.x.d.1738.1		8	21.2	odd	6
2961.1.x.d.1738.1		8	987.422	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 329 = 7 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	329.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.978148 + 1.69420i) q^{2} +(0.809017 - 1.40126i) q^{3} +(-1.41355 + 2.44833i) q^{4} +3.16535 q^{6} +(-0.978148 - 0.207912i) q^{7} -3.57433 q^{8} +(-0.809017 - 1.40126i) q^{9} +(2.28716 + 3.96149i) q^{12} +(-0.604528 - 1.86055i) q^{14} +(-2.08268 - 3.60730i) q^{16} +(0.104528 - 0.181049i) q^{17} +(1.58268 - 2.74128i) q^{18} +(-1.08268 + 1.20243i) q^{21} +(-2.89169 + 5.00856i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(1.89169 - 2.10094i) q^{28} +(2.28716 - 3.96149i) q^{32} +0.408977 q^{34} +4.57433 q^{36} +(0.104528 + 0.181049i) q^{37} +(-3.09618 - 0.658114i) q^{42} +(-0.500000 - 0.866025i) q^{47} -6.73968 q^{48} +(0.913545 + 0.406737i) q^{49} -1.95630 q^{50} +(-0.169131 - 0.292943i) q^{51} +(-0.669131 + 1.15897i) q^{53} +(-0.978148 - 1.69420i) q^{54} +(3.49622 + 0.743145i) q^{56} +(0.978148 - 1.69420i) q^{59} +(0.978148 + 1.69420i) q^{61} +(0.500000 + 1.53884i) q^{63} +4.78339 q^{64} +(0.295511 + 0.511841i) q^{68} -0.209057 q^{71} +(2.89169 + 5.00856i) q^{72} +(-0.204489 + 0.354185i) q^{74} +(0.809017 + 1.40126i) q^{75} +(0.809017 + 1.40126i) q^{79} -1.00000 q^{83} +(-1.41355 - 4.35045i) q^{84} +(-0.309017 - 0.535233i) q^{89} +(0.978148 - 1.69420i) q^{94} +(-3.70071 - 6.40982i) q^{96} +1.33826 q^{97} +(0.204489 + 1.94558i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 2 q^{3} - 5 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 2 q^{9} + 5 q^{12} - 3 q^{14} - 6 q^{16} - q^{17} + 2 q^{18} + 2 q^{21} - 8 q^{24} - 4 q^{25} - 8 q^{27} + 5 q^{32} - 2 q^{34} + 10 q^{36}+ \cdots - q^{98}+O(q^{100}) \) 8 * q - q^2 + 2 * q^3 - 5 * q^4 + 4 * q^6 + q^7 - 2 * q^8 - 2 * q^9 + 5 * q^12 - 3 * q^14 - 6 * q^16 - q^17 + 2 * q^18 + 2 * q^21 - 8 * q^24 - 4 * q^25 - 8 * q^27 + 5 * q^32 - 2 * q^34 + 10 * q^36 - q^37 - 7 * q^42 - 4 * q^47 - 6 * q^48 + q^49 + 2 * q^50 + 3 * q^51 - q^53 + q^54 + 11 * q^56 - q^59 - q^61 + 4 * q^63 + 8 * q^64 + 5 * q^68 + 2 * q^71 + 8 * q^72 + q^74 + 2 * q^75 + 2 * q^79 - 8 * q^83 - 5 * q^84 + 2 * q^89 - q^94 - 10 * q^96 + 2 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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