LMFDB - Embedded newform 329.1.f.b.46.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [329,1,Mod(46,329)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			46.2
Root			$0.913545 + 0.406737i$ of defining polynomial
Character	$\chi$	$=$	329.46
Dual form			329.1.f.b.93.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.669131 - 1.15897i) q^{2} +(0.809017 - 1.40126i) q^{3} +(-0.395472 + 0.684977i) q^{4} -2.16535 q^{6} +(0.669131 + 0.743145i) q^{7} -0.279773 q^{8} +(-0.809017 - 1.40126i) q^{9} +(0.639886 + 1.10832i) q^{12} +(0.413545 - 1.27276i) q^{14} +(0.582676 + 1.00922i) q^{16} +(-0.913545 + 1.58231i) q^{17} +(-1.08268 + 1.87525i) q^{18} +(1.58268 - 0.336408i) q^{21} +(-0.226341 + 0.392034i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(-0.773659 + 0.164446i) q^{28} +(0.639886 - 1.10832i) q^{32} +2.44512 q^{34} +1.27977 q^{36} +(-0.913545 - 1.58231i) q^{37} +(-1.44890 - 1.60917i) q^{42} +(-0.500000 - 0.866025i) q^{47} +1.88558 q^{48} +(-0.104528 + 0.994522i) q^{49} +1.33826 q^{50} +(1.47815 + 2.56023i) q^{51} +(0.978148 - 1.69420i) q^{53} +(0.669131 + 1.15897i) q^{54} +(-0.187205 - 0.207912i) q^{56} +(-0.669131 + 1.15897i) q^{59} +(-0.669131 - 1.15897i) q^{61} +(0.500000 - 1.53884i) q^{63} -0.547318 q^{64} +(-0.722562 - 1.25151i) q^{68} +1.82709 q^{71} +(0.226341 + 0.392034i) q^{72} +(-1.22256 + 2.11754i) q^{74} +(0.809017 + 1.40126i) q^{75} +(0.809017 + 1.40126i) q^{79} -1.00000 q^{83} +(-0.395472 + 1.21714i) q^{84} +(-0.309017 - 0.535233i) q^{89} +(-0.669131 + 1.15897i) q^{94} +(-1.03536 - 1.79329i) q^{96} -1.95630 q^{97} +(1.22256 - 0.544320i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - q^{2} + 2 q^{3} - 5 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 2 q^{9} + 5 q^{12} - 3 q^{14} - 6 q^{16} - q^{17} + 2 q^{18} + 2 q^{21} - 8 q^{24} - 4 q^{25} - 8 q^{27} + 5 q^{32} - 2 q^{34} + 10 q^{36}+ \cdots - q^{98}+O(q^{100})$ 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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$2$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	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	$-0.133333\pi$
$4$	−0.395472	+	0.684977i	−0.395472	+	0.684977i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−2.16535			−2.16535
$7$	0.669131	+	0.743145i	0.669131	+	0.743145i
$7$	0.669131	+	0.743145i	0.669131	+	0.743145i
$8$	−0.279773			−0.279773
$9$	−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	$0.666667\pi$
$12$	0.639886	+	1.10832i	0.639886	+	1.10832i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.413545	−	1.27276i	0.413545	−	1.27276i
$15$		0			0
$16$	0.582676	+	1.00922i	0.582676	+	1.00922i
$17$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$17$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$18$	−1.08268	+	1.87525i	−1.08268	+	1.87525i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$	1.58268	−	0.336408i	1.58268	−	0.336408i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.226341	+	0.392034i	−0.226341	+	0.392034i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−1.00000			−1.00000
$28$	−0.773659	+	0.164446i	−0.773659	+	0.164446i
$29$		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	$0.666667\pi$
$32$	0.639886	−	1.10832i	0.639886	−	1.10832i
$33$		0			0
$34$	2.44512			2.44512
$35$		0			0
$36$	1.27977			1.27977
$37$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$37$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−1.44890	−	1.60917i	−1.44890	−	1.60917i
$43$		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
$48$	1.88558			1.88558
$49$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$50$	1.33826			1.33826
$51$	1.47815	+	2.56023i	1.47815	+	2.56023i
$52$		0			0
$53$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$53$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$54$	0.669131	+	1.15897i	0.669131	+	1.15897i
$55$		0			0
$56$	−0.187205	−	0.207912i	−0.187205	−	0.207912i
$57$		0			0
$58$		0			0
$59$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$59$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$60$		0			0
$61$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$61$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$62$		0			0
$63$	0.500000	−	1.53884i	0.500000	−	1.53884i
$64$	−0.547318			−0.547318
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−0.722562	−	1.25151i	−0.722562	−	1.25151i
$69$		0			0
$70$		0			0
$71$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$71$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$72$	0.226341	+	0.392034i	0.226341	+	0.392034i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$75$	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	$0.666667\pi$
$84$	−0.395472	+	1.21714i	−0.395472	+	1.21714i
$85$		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
$94$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$95$		0			0
$96$	−1.03536	−	1.79329i	−1.03536	−	1.79329i
$97$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$97$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$98$	1.22256	−	0.544320i	1.22256	−	0.544320i
$99$		0			0
$100$	−0.395472	−	0.684977i	−0.395472	−	0.684977i
$101$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$101$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$102$	1.97815	−	3.42625i	1.97815	−	3.42625i
$103$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$103$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$104$		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	$0.333333\pi$
$108$	0.395472	−	0.684977i	0.395472	−	0.684977i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$	−2.95630			−2.95630
$112$	−0.360114	+	1.10832i	−0.360114	+	1.10832i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.79094			1.79094
$119$	−1.78716	+	0.379874i	−1.78716	+	0.379874i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	−0.895472	+	1.55100i	−0.895472	+	1.55100i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−2.11803	+	0.450202i	−2.11803	+	0.450202i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$129$		0			0
$130$		0			0
$131$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$131$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.255585	−	0.442686i	0.255585	−	0.442686i
$137$		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
$142$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$143$		0			0
$144$	0.942790	−	1.63296i	0.942790	−	1.63296i
$145$		0			0
$146$		0			0
$147$	1.30902	+	0.951057i	1.30902	+	0.951057i
$148$	1.44512			1.44512
$149$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$149$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$150$	1.08268	−	1.87525i	1.08268	−	1.87525i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	2.95630			2.95630
$154$		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	$0.266667\pi$
$158$	1.08268	−	1.87525i	1.08268	−	1.87525i
$159$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$160$		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
$166$	0.669131	+	1.15897i	0.669131	+	1.15897i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$	−0.442790	+	0.0941179i	−0.442790	+	0.0941179i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$173$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$176$		0			0
$177$	1.08268	+	1.87525i	1.08268	+	1.87525i
$178$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$179$		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
$183$	−2.16535			−2.16535
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.790943			0.790943
$189$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$190$		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	$0.666667\pi$
$192$	−0.442790	+	0.766934i	−0.442790	+	0.766934i
$193$		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
$196$	−0.639886	−	0.464905i	−0.639886	−	0.464905i
$197$	−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	$0.666667\pi$
$200$	0.139886	−	0.242290i	0.139886	−	0.242290i
$201$		0			0
$202$	−0.279773			−0.279773
$203$		0			0
$204$	−2.33826			−2.33826
$205$		0			0
$206$	0.139886	−	0.242290i	0.139886	−	0.242290i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.773659	+	1.34002i	0.773659	+	1.34002i
$213$	1.47815	−	2.56023i	1.47815	−	2.56023i
$214$		0			0
$215$		0			0
$216$	0.279773			0.279773
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	1.97815	+	3.42625i	1.97815	+	3.42625i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	1.25181	−	0.266080i	1.25181	−	0.266080i
$225$	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
$236$	−0.529244	−	0.916678i	−0.529244	−	0.916678i
$237$	2.61803			2.61803
$238$	1.63611	+	1.81708i	1.63611	+	1.81708i
$239$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$239$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$240$		0			0
$241$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$242$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	1.05849			1.05849
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$250$		0			0
$251$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$251$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$252$	0.856335	+	0.951057i	0.856335	+	0.951057i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.639886	+	1.10832i	−0.639886	+	1.10832i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	0.564602	−	1.73767i	0.564602	−	1.73767i
$260$		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
$271$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$271$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$272$	−2.12920			−2.12920
$273$		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			$\pi$
$282$	1.08268	+	1.87525i	1.08268	+	1.87525i
$283$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$283$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$284$	−0.722562	+	1.25151i	−0.722562	+	1.25151i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.07072			−2.07072
$289$	−1.16913	−	2.02499i	−1.16913	−	2.02499i
$290$		0			0
$291$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	0.226341	−	2.15349i	0.226341	−	2.15349i
$295$		0			0
$296$	0.255585	+	0.442686i	0.255585	+	0.442686i
$297$		0			0
$298$	0.139886	−	0.242290i	0.139886	−	0.242290i
$299$		0			0
$300$	−1.27977			−1.27977
$301$		0			0
$302$		0			0
$303$	−0.169131	−	0.292943i	−0.169131	−	0.292943i
$304$		0			0
$305$		0			0
$306$	−1.97815	−	3.42625i	−1.97815	−	3.42625i
$307$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$307$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$308$		0			0
$309$	0.338261			0.338261
$310$		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	$0.333333\pi$
$314$	0.827091			0.827091
$315$		0			0
$316$	−1.27977			−1.27977
$317$		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	$0.933333\pi$
$332$	0.395472	−	0.684977i	0.395472	−	0.684977i
$333$	−1.47815	+	2.56023i	−1.47815	+	2.56023i
$334$		0			0
$335$		0			0
$336$	1.26170	+	1.40126i	1.26170	+	1.40126i
$337$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$337$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$338$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$339$		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
$347$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$347$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	0.895472	+	0.994522i	0.895472	+	0.994522i
$351$		0			0
$352$		0			0
$353$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$353$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$354$	1.44890	−	2.50957i	1.44890	−	2.50957i
$355$		0			0
$356$	0.488830			0.488830
$357$	−0.913545	+	2.81160i	−0.913545	+	2.81160i
$358$		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
$366$	1.44890	+	2.50957i	1.44890	+	2.50957i
$367$		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
$371$	1.91355	−	0.406737i	1.91355	−	0.406737i
$372$		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
$376$	0.139886	+	0.242290i	0.139886	+	0.242290i
$377$		0			0
$378$	−0.413545	+	1.27276i	−0.413545	+	1.27276i
$379$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$379$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$380$		0			0
$381$		0			0
$382$	0.669131	−	1.15897i	0.669131	−	1.15897i
$383$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$383$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$384$	−0.885579			−0.885579
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.773659	−	1.34002i	0.773659	−	1.34002i
$389$		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
$392$	0.0292442	−	0.278240i	0.0292442	−	0.278240i
$393$	3.16535			3.16535
$394$	0.669131	+	1.15897i	0.669131	+	1.15897i
$395$		0			0
$396$		0			0
$397$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$397$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$398$		0			0
$399$		0			0
$400$	−1.16535			−1.16535
$401$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$401$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$402$		0			0
$403$		0			0
$404$	0.0826761	+	0.143199i	0.0826761	+	0.143199i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$409$		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
$412$	−0.165352			−0.165352
$413$	−1.30902	+	0.278240i	−1.30902	+	0.278240i
$414$		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
$424$	−0.273659	+	0.473991i	−0.273659	+	0.473991i
$425$	−0.913545	−	1.58231i	−0.913545	−	1.58231i
$426$	−3.95630			−3.95630
$427$	0.413545	−	1.27276i	0.413545	−	1.27276i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$431$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$432$	−0.582676	−	1.00922i	−0.582676	−	1.00922i
$433$		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
$441$	1.47815	−	0.658114i	1.47815	−	0.658114i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	1.16913	−	2.02499i	1.16913	−	2.02499i
$445$		0			0
$446$		0			0
$447$	0.338261			0.338261
$448$	−0.366227	−	0.406737i	−0.366227	−	0.406737i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.08268	−	1.87525i	−1.08268	−	1.87525i
$451$		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
$459$	0.913545	−	1.58231i	0.913545	−	1.58231i
$460$		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
$472$	0.187205	−	0.324248i	0.187205	−	0.324248i
$473$		0			0
$474$	−1.75181	−	3.03422i	−1.75181	−	3.03422i
$475$		0			0
$476$	0.446568	−	1.37440i	0.446568	−	1.37440i
$477$	−3.16535			−3.16535
$478$	0.139886	+	0.242290i	0.139886	+	0.242290i
$479$	−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
$482$	−0.279773			−0.279773
$483$		0			0
$484$	0.790943			0.790943
$485$		0			0
$486$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$487$	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			$0$
$488$	0.187205	+	0.324248i	0.187205	+	0.324248i
$489$		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
$497$	1.22256	+	1.35779i	1.22256	+	1.35779i
$498$	2.16535			2.16535
$499$		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
$502$	0.139886	+	0.242290i	0.139886	+	0.242290i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−0.139886	+	0.430526i	−0.139886	+	0.430526i
$505$		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
$512$	1.16535			1.16535
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−2.39169	+	0.508370i	−2.39169	+	0.508370i
$519$	3.16535			3.16535
$520$		0			0
$521$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$521$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$522$		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	$0.666667\pi$
$524$	−1.54732			−1.54732
$525$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$526$	0.827091			0.827091
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$	2.16535			2.16535
$532$		0			0
$533$		0			0
$534$	0.669131	+	1.15897i	0.669131	+	1.15897i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.33826			−1.33826
$539$		0			0
$540$		0			0
$541$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$541$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$542$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$543$		0			0
$544$	1.16913	+	2.02499i	1.16913	+	2.02499i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−1.08268	+	1.87525i	−1.08268	+	1.87525i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$554$	−2.16535			−2.16535
$555$		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	$0.666667\pi$
$564$	0.639886	−	1.10832i	0.639886	−	1.10832i
$565$		0			0
$566$	−2.61803			−2.61803
$567$		0			0
$568$	−0.511170			−0.511170
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$571$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$	1.61803			1.61803
$574$		0			0
$575$		0			0
$576$	0.442790	+	0.766934i	0.442790	+	0.766934i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−1.56460	+	2.70997i	−1.56460	+	2.70997i
$579$		0			0
$580$		0			0
$581$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$582$	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			$\pi$
$588$	−1.16913	+	0.520530i	−1.16913	+	0.520530i
$589$		0			0
$590$		0			0
$591$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$592$	1.06460	−	1.84395i	1.06460	−	1.84395i
$593$		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
$596$	−0.165352			−0.165352
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−0.226341	−	0.392034i	−0.226341	−	0.392034i
$601$	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
$606$	−0.226341	+	0.392034i	−0.226341	+	0.392034i
$607$		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
$612$	−1.16913	+	2.02499i	−1.16913	+	2.02499i
$613$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$613$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$614$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$615$		0			0
$616$		0			0
$617$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$617$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$618$	−0.226341	−	0.392034i	−0.226341	−	0.392034i
$619$	−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
$628$	−0.244415	−	0.423339i	−0.244415	−	0.423339i
$629$	3.33826			3.33826
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.226341	−	0.392034i	−0.226341	−	0.392034i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	2.50361			2.50361
$637$		0			0
$638$		0			0
$639$	−1.47815	−	2.56023i	−1.47815	−	2.56023i
$640$		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
$643$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$643$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$647$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$648$		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	+	0.278240i	−1.30902	+	0.278240i
$659$	−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
$661$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$661$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$662$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$663$		0			0
$664$	0.279773			0.279773
$665$		0			0
$666$	3.95630			3.95630
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.639886	−	1.96937i	0.639886	−	1.96937i
$673$		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
$676$	−0.395472	+	0.684977i	−0.395472	+	0.684977i
$677$		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	−	1.45381i	−1.30902	−	1.45381i
$680$		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
$686$	1.22256	+	0.544320i	1.22256	+	0.544320i
$687$		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	$0.333333\pi$
$692$	−1.54732			−1.54732
$693$		0			0
$694$	−0.279773			−0.279773
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.244415	−	0.752232i	0.244415	−	0.752232i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.79094			1.79094
$707$	0.204489	−	0.0434654i	0.204489	−	0.0434654i
$708$	−1.71267			−1.71267
$709$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$709$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$710$		0			0
$711$	1.30902	−	2.26728i	1.30902	−	2.26728i
$712$	0.0864545	+	0.149744i	0.0864545	+	0.149744i
$713$		0			0
$714$	3.86984	−	0.822560i	3.86984	−	0.822560i
$715$		0			0
$716$		0			0
$717$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$718$		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
$721$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$722$	1.33826			1.33826
$723$	−0.169131	−	0.292943i	−0.169131	−	0.292943i
$724$		0			0
$725$		0			0
$726$	1.08268	+	1.87525i	1.08268	+	1.87525i
$727$		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
$732$	0.856335	−	1.48322i	0.856335	−	1.48322i
$733$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$733$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$739$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$740$		0			0
$741$		0			0
$742$	−1.75181	−	1.94558i	−1.75181	−	1.94558i
$743$		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	$0.333333\pi$
$752$	0.582676	−	1.00922i	0.582676	−	1.00922i
$753$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$754$		0			0
$755$		0			0
$756$	0.773659	−	0.164446i	0.773659	−	0.164446i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.895472	−	1.55100i	−0.895472	−	1.55100i
$759$		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
$764$	−0.790943			−0.790943
$765$		0			0
$766$	0.139886	−	0.242290i	0.139886	−	0.242290i
$767$		0			0
$768$	1.03536	+	1.79329i	1.03536	+	1.79329i
$769$	−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
$776$	0.547318			0.547318
$777$	−1.97815	−	2.19696i	−1.97815	−	2.19696i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.06460	+	0.473991i	−1.06460	+	0.473991i
$785$		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	$0.666667\pi$
$788$	0.395472	−	0.684977i	0.395472	−	0.684977i
$789$	0.500000	+	0.866025i	0.500000	+	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.139886	−	0.242290i	0.139886	−	0.242290i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	1.82709			1.82709
$800$	0.639886	+	1.10832i	0.639886	+	1.10832i
$801$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$802$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$808$	−0.0292442	+	0.0506525i	−0.0292442	+	0.0506525i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$811$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$812$		0			0
$813$	−2.95630			−2.95630
$814$		0			0
$815$		0			0
$816$	−1.72256	+	2.98357i	−1.72256	+	2.98357i
$817$		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			$0$
$824$	−0.0292442	−	0.0506525i	−0.0292442	−	0.0506525i
$825$		0			0
$826$	1.19837	+	1.33093i	1.19837	+	1.33093i
$827$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$827$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$828$		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
$833$	−1.47815	−	1.07394i	−1.47815	−	1.07394i
$834$		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
$846$	2.16535			2.16535
$847$	0.309017	−	0.951057i	0.309017	−	0.951057i
$848$	2.27977			2.27977
$849$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$850$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$851$		0			0
$852$	1.16913	+	2.02499i	1.16913	+	2.02499i
$853$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$853$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$854$	−1.75181	+	0.372358i	−1.75181	+	0.372358i
$855$		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	$0.533333\pi$
$864$	−0.639886	+	1.10832i	−0.639886	+	1.10832i
$865$		0			0
$866$		0			0
$867$	−3.78339			−3.78339
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.58268	+	2.74128i	1.58268	+	2.74128i
$874$		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	$0.333333\pi$
$878$	0.669131	−	1.15897i	0.669131	−	1.15897i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.75181	−	1.27276i	−1.75181	−	1.27276i
$883$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$883$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$884$		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	$0.333333\pi$
$888$	0.827091			0.827091
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.226341	−	0.392034i	−0.226341	−	0.392034i
$895$		0			0
$896$	0.169131	−	0.520530i	0.169131	−	0.520530i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.639886	+	1.10832i	−0.639886	+	1.10832i
$901$	1.78716	+	3.09546i	1.78716	+	3.09546i
$902$		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
$909$	−0.338261			−0.338261
$910$		0			0
$911$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$911$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$912$		0			0
$913$		0			0
$914$	1.08268	−	1.87525i	1.08268	−	1.87525i
$915$		0			0
$916$		0			0
$917$	−0.604528	+	1.86055i	−0.604528	+	1.86055i
$918$	−2.44512			−2.44512
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$	1.47815	−	2.56023i	1.47815	−	2.56023i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.82709			1.82709
$926$		0			0
$927$	0.169131	−	0.292943i	0.169131	−	0.292943i
$928$		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	$0.266667\pi$
$942$	0.669131	−	1.15897i	0.669131	−	1.15897i
$943$		0			0
$944$	−1.55955			−1.55955
$945$		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	$0.666667\pi$
$948$	−1.03536	+	1.79329i	−1.03536	+	1.79329i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.500000	−	0.106278i	0.500000	−	0.106278i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	2.11803	+	3.66854i	2.11803	+	3.66854i
$955$		0			0
$956$	0.0826761	−	0.143199i	0.0826761	−	0.143199i
$957$		0			0
$958$	0.827091			0.827091
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$	0.0826761	+	0.143199i	0.0826761	+	0.143199i
$965$		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	$0.400000\pi$
$968$	0.139886	+	0.242290i	0.139886	+	0.242290i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	0.790943			0.790943
$973$		0			0
$974$	−1.33826			−1.33826
$975$		0			0
$976$	0.779773	−	1.35061i	0.779773	−	1.35061i
$977$	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
$982$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$983$		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
$987$	−1.08268	−	1.20243i	−1.08268	−	1.20243i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$991$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$992$		0			0
$993$	−1.00000			−1.00000
$994$	0.755585	−	2.32545i	0.755585	−	2.32545i
$995$		0			0
$996$	−0.639886	−	1.10832i	−0.639886	−	1.10832i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	0.913545	+	1.58231i	0.913545	+	1.58231i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	329.1.f.b.46.2	✓	8
3.2	odd	2		2961.1.x.d.46.3		8
7.2	even	3	inner	329.1.f.b.93.2	yes	8
7.3	odd	6		2303.1.d.e.2255.3		4
7.4	even	3		2303.1.d.d.2255.3		4
7.5	odd	6		2303.1.f.d.422.2		8
7.6	odd	2		2303.1.f.d.704.2		8
21.2	odd	6		2961.1.x.d.1738.3		8
47.46	odd	2	CM	329.1.f.b.46.2	✓	8
141.140	even	2		2961.1.x.d.46.3		8
329.46	odd	6		2303.1.d.d.2255.3		4
329.93	odd	6	inner	329.1.f.b.93.2	yes	8
329.187	even	6		2303.1.f.d.422.2		8
329.234	even	6		2303.1.d.e.2255.3		4
329.328	even	2		2303.1.f.d.704.2		8
987.422	even	6		2961.1.x.d.1738.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
329.1.f.b.46.2	✓	8	1.1	even	1	trivial
329.1.f.b.46.2	✓	8	47.46	odd	2	CM
329.1.f.b.93.2	yes	8	7.2	even	3	inner
329.1.f.b.93.2	yes	8	329.93	odd	6	inner
2303.1.d.d.2255.3		4	7.4	even	3
2303.1.d.d.2255.3		4	329.46	odd	6
2303.1.d.e.2255.3		4	7.3	odd	6
2303.1.d.e.2255.3		4	329.234	even	6
2303.1.f.d.422.2		8	7.5	odd	6
2303.1.f.d.422.2		8	329.187	even	6
2303.1.f.d.704.2		8	7.6	odd	2
2303.1.f.d.704.2		8	329.328	even	2
2961.1.x.d.46.3		8	3.2	odd	2
2961.1.x.d.46.3		8	141.140	even	2
2961.1.x.d.1738.3		8	21.2	odd	6
2961.1.x.d.1738.3		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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	329.1.f.b.46.2

Level	$329$
Weight	$1$
Character	329.46
Analytic conductor	$0.164$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{15}$
CM discriminant	-47
Inner twists	$4$