LMFDB - Embedded newform 2303.1.f.d.422.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2303,1,Mod(422,2303)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2303, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2303.422");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2303 = 7^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2303.f (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.14934672409$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 329)
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.143108492101942920287.1

Embedding invariants

Embedding label			422.1
Root			$0.669131 + 0.743145i$ of defining polynomial
Character	$\chi$	$=$	2303.422
Dual form			2303.1.f.d.704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.913545 + 1.58231i) q^{2} +(0.309017 + 0.535233i) q^{3} +(-1.16913 - 2.02499i) q^{4} -1.12920 q^{6} +2.44512 q^{8} +(0.309017 - 0.535233i) q^{9} +(0.722562 - 1.25151i) q^{12} +(-1.06460 + 1.84395i) q^{16} +(-0.978148 - 1.69420i) q^{17} +(0.564602 + 0.977920i) q^{18} +(0.755585 + 1.30871i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(-0.722562 - 1.25151i) q^{32} +3.57433 q^{34} -1.44512 q^{36} +(0.978148 - 1.69420i) q^{37} +(0.500000 - 0.866025i) q^{47} -1.31592 q^{48} +1.82709 q^{50} +(0.604528 - 1.04707i) q^{51} +(0.104528 + 0.181049i) q^{53} +(-0.913545 + 1.58231i) q^{54} +(0.913545 + 1.58231i) q^{59} +(0.913545 - 1.58231i) q^{61} +0.511170 q^{64} +(-2.28716 + 3.96149i) q^{68} -1.95630 q^{71} +(0.755585 - 1.30871i) q^{72} +(1.78716 + 3.09546i) q^{74} +(0.309017 - 0.535233i) q^{75} +(-0.309017 + 0.535233i) q^{79} +1.00000 q^{83} +(-0.809017 + 1.40126i) q^{89} +(0.913545 + 1.58231i) q^{94} +(0.446568 - 0.773479i) q^{96} +0.209057 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 2 q^{3} - 5 q^{4} - 4 q^{6} - 2 q^{8} - 2 q^{9} - 5 q^{12} - 6 q^{16} + q^{17} + 2 q^{18} + 8 q^{24} - 4 q^{25} + 8 q^{27} + 5 q^{32} + 2 q^{34} + 10 q^{36} - q^{37} + 4 q^{47} + 6 q^{48}+ \cdots - 2 q^{97}+O(q^{100}) $ 8 * q - q^2 - 2 * q^3 - 5 * q^4 - 4 * q^6 - 2 * q^8 - 2 * q^9 - 5 * q^12 - 6 * q^16 + q^17 + 2 * q^18 + 8 * q^24 - 4 * q^25 + 8 * q^27 + 5 * q^32 + 2 * q^34 + 10 * q^36 - q^37 + 4 * q^47 + 6 * q^48 + 2 * q^50 + 3 * q^51 - q^53 - q^54 + q^59 + q^61 + 8 * q^64 - 5 * q^68 + 2 * q^71 + 8 * q^72 + q^74 - 2 * q^75 + 2 * q^79 + 8 * q^83 - 2 * q^89 + q^94 + 10 * q^96 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2303\mathbb{Z}\right)^\times$.

$n$	$99$	$2257$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$2$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$3$	0.309017	+	0.535233i	0.309017	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$3$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$4$	−1.16913	−	2.02499i	−1.16913	−	2.02499i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−1.12920			−1.12920
$7$		0			0
$7$		0			0
$8$	2.44512			2.44512
$9$	0.309017	−	0.535233i	0.309017	−	0.535233i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	0.722562	−	1.25151i	0.722562	−	1.25151i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−1.06460	+	1.84395i	−1.06460	+	1.84395i
$17$	−0.978148	−	1.69420i	−0.978148	−	1.69420i	−0.669131	−	0.743145i	$-0.733333\pi$
$17$	−0.978148	−	1.69420i	−0.978148	−	1.69420i	−0.309017	−	0.951057i	$-0.600000\pi$
$18$	0.564602	+	0.977920i	0.564602	+	0.977920i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.755585	+	1.30871i	0.755585	+	1.30871i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.722562	−	1.25151i	−0.722562	−	1.25151i
$33$		0			0
$34$	3.57433			3.57433
$35$		0			0
$36$	−1.44512			−1.44512
$37$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		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
$48$	−1.31592			−1.31592
$49$		0			0
$50$	1.82709			1.82709
$51$	0.604528	−	1.04707i	0.604528	−	1.04707i
$52$		0			0
$53$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$53$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$54$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.809017	+	0.587785i	$0.200000\pi$
$59$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.104528	+	0.994522i	$0.466667\pi$
$60$		0			0
$61$	0.913545	−	1.58231i	0.913545	−	1.58231i	0.104528	−	0.994522i	$-0.466667\pi$
$61$	0.913545	−	1.58231i	0.913545	−	1.58231i	0.809017	−	0.587785i	$-0.200000\pi$
$62$		0			0
$63$		0			0
$64$	0.511170			0.511170
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−2.28716	+	3.96149i	−2.28716	+	3.96149i
$69$		0			0
$70$		0			0
$71$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$71$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$72$	0.755585	−	1.30871i	0.755585	−	1.30871i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	1.78716	+	3.09546i	1.78716	+	3.09546i
$75$	0.309017	−	0.535233i	0.309017	−	0.535233i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$79$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$83$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$89$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.913545	+	1.58231i	0.913545	+	1.58231i
$95$		0			0
$96$	0.446568	−	0.773479i	0.446568	−	0.773479i
$97$	0.209057			0.209057			0.104528	−	0.994522i	$-0.466667\pi$
$97$	0.209057			0.209057			0.104528	+	0.994522i	$0.466667\pi$
$98$		0			0
$99$		0			0
$100$	−1.16913	+	2.02499i	−1.16913	+	2.02499i
$101$	0.669131	+	1.15897i	0.669131	+	1.15897i	0.978148	+	0.207912i	$0.0666667\pi$
$101$	0.669131	+	1.15897i	0.669131	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$102$	1.10453	+	1.91310i	1.10453	+	1.91310i
$103$	0.669131	−	1.15897i	0.669131	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$103$	0.669131	−	1.15897i	0.669131	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$104$		0			0
$105$		0			0
$106$	−0.381966			−0.381966
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−1.16913	−	2.02499i	−1.16913	−	2.02499i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$	1.20906			1.20906
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−3.33826			−3.33826
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	1.66913	+	2.89102i	1.66913	+	2.89102i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.255585	−	0.442686i	0.255585	−	0.442686i
$129$		0			0
$130$		0			0
$131$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$131$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	0.809017	+	0.587785i	$0.200000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−2.39169	−	4.14253i	−2.39169	−	4.14253i
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.618034			0.618034
$142$	1.78716	−	3.09546i	1.78716	−	3.09546i
$143$		0			0
$144$	0.657960	+	1.13962i	0.657960	+	1.13962i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−4.57433			−4.57433
$149$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$149$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$150$	0.564602	+	0.977920i	0.564602	+	0.977920i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−1.20906			−1.20906
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$157$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$158$	−0.564602	−	0.977920i	−0.564602	−	0.977920i
$159$	−0.0646021	+	0.111894i	−0.0646021	+	0.111894i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$173$	−0.104528	+	0.181049i	−0.104528	+	0.181049i	0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.564602	+	0.977920i	−0.564602	+	0.977920i
$178$	−1.47815	−	2.56023i	−1.47815	−	2.56023i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.12920			1.12920
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.33826			−2.33826
$189$		0			0
$190$		0			0
$191$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$192$	0.157960	+	0.273595i	0.157960	+	0.273595i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$195$		0			0
$196$		0			0
$197$	−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	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$201$		0			0
$202$	−2.44512			−2.44512
$203$		0			0
$204$	−2.82709			−2.82709
$205$		0			0
$206$	1.22256	+	2.11754i	1.22256	+	2.11754i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.244415	−	0.423339i	0.244415	−	0.423339i
$213$	−0.604528	−	1.04707i	−0.604528	−	1.04707i
$214$		0			0
$215$		0			0
$216$	2.44512			2.44512
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−1.10453	+	1.91310i	−1.10453	+	1.91310i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−0.618034			−0.618034
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$	2.13611	−	3.69985i	2.13611	−	3.69985i
$237$	−0.381966			−0.381966
$238$		0			0
$239$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$239$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$240$		0			0
$241$	0.669131	+	1.15897i	0.669131	+	1.15897i	0.978148	+	0.207912i	$0.0666667\pi$
$241$	0.669131	+	1.15897i	0.669131	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$242$	−0.913545	−	1.58231i	−0.913545	−	1.58231i
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$	−4.27222			−4.27222
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.309017	+	0.535233i	0.309017	+	0.535233i
$250$		0			0
$251$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$251$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.722562	+	1.25151i	0.722562	+	1.25151i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$263$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$263$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$264$		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	$-0.333333\pi$
$269$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$270$		0			0
$271$	−0.978148	+	1.69420i	−0.978148	+	1.69420i	−0.309017	+	0.951057i	$0.600000\pi$
$271$	−0.978148	+	1.69420i	−0.978148	+	1.69420i	−0.669131	+	0.743145i	$0.733333\pi$
$272$	4.16535			4.16535
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$277$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−0.564602	+	0.977920i	−0.564602	+	0.977920i
$283$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	0.809017	−	0.587785i	$-0.200000\pi$
$283$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$284$	2.28716	+	3.96149i	2.28716	+	3.96149i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.893136			−0.893136
$289$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$290$		0			0
$291$	0.0646021	+	0.111894i	0.0646021	+	0.111894i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	2.39169	−	4.14253i	2.39169	−	4.14253i
$297$		0			0
$298$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$299$		0			0
$300$	−1.44512			−1.44512
$301$		0			0
$302$		0			0
$303$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$304$		0			0
$305$		0			0
$306$	1.10453	−	1.91310i	1.10453	−	1.91310i
$307$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$307$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$308$		0			0
$309$	0.827091			0.827091
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	2.95630			2.95630
$315$		0			0
$316$	1.44512			1.44512
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$	−0.118034	−	0.204441i	−0.118034	−	0.204441i
$319$		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			0
$330$		0			0
$331$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$331$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$332$	−1.16913	−	2.02499i	−1.16913	−	2.02499i
$333$	−0.604528	−	1.04707i	−0.604528	−	1.04707i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$337$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$338$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$347$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$347$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.809017	+	0.587785i	$0.200000\pi$
$353$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.104528	+	0.994522i	$0.466667\pi$
$354$	−1.03158	−	1.78675i	−1.03158	−	1.78675i
$355$		0			0
$356$	3.78339			3.78339
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	−0.618034			−0.618034
$364$		0			0
$365$		0			0
$366$	−1.03158	+	1.78675i	−1.03158	+	1.78675i
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	1.22256	−	2.11754i	1.22256	−	2.11754i
$377$		0			0
$378$		0			0
$379$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$379$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$380$		0			0
$381$		0			0
$382$	0.913545	+	1.58231i	0.913545	+	1.58231i
$383$	0.669131	−	1.15897i	0.669131	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$383$	0.669131	−	1.15897i	0.669131	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$384$	0.315921			0.315921
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.244415	−	0.423339i	−0.244415	−	0.423339i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−0.129204			−0.129204
$394$	0.913545	−	1.58231i	0.913545	−	1.58231i
$395$		0			0
$396$		0			0
$397$	0.669131	−	1.15897i	0.669131	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$397$	0.669131	−	1.15897i	0.669131	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$398$		0			0
$399$		0			0
$400$	2.12920			2.12920
$401$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$401$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$402$		0			0
$403$		0			0
$404$	1.56460	−	2.70997i	1.56460	−	2.70997i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.47815	−	2.56023i	1.47815	−	2.56023i
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	−3.12920			−3.12920
$413$		0			0
$414$		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
$423$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$424$	0.255585	+	0.442686i	0.255585	+	0.442686i
$425$	−0.978148	+	1.69420i	−0.978148	+	1.69420i
$426$	2.20906			2.20906
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$431$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$432$	−1.06460	+	1.84395i	−1.06460	+	1.84395i
$433$		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	0.500000	+	0.866025i	$0.333333\pi$
$439$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.41355	−	2.44833i	−1.41355	−	2.44833i
$445$		0			0
$446$		0			0
$447$	−0.827091			−0.827091
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.564602	−	0.977920i	0.564602	−	0.977920i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$457$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$458$		0			0
$459$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$460$		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	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.500000	−	0.866025i	0.500000	−	0.866025i
$472$	2.23373	+	3.86894i	2.23373	+	3.86894i
$473$		0			0
$474$	0.348943	−	0.604388i	0.348943	−	0.604388i
$475$		0			0
$476$		0			0
$477$	0.129204			0.129204
$478$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$479$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$479$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$480$		0			0
$481$		0			0
$482$	−2.44512			−2.44512
$483$		0			0
$484$	2.33826			2.33826
$485$		0			0
$486$	0.913545	+	1.58231i	0.913545	+	1.58231i
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$488$	2.23373	−	3.86894i	2.23373	−	3.86894i
$489$		0			0
$490$		0			0
$491$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.12920			−1.12920
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$	1.22256	−	2.11754i	1.22256	−	2.11754i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.309017	+	0.535233i	0.309017	+	0.535233i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−2.12920			−2.12920
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.129204			−0.129204
$520$		0			0
$521$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	0.809017	−	0.587785i	$-0.200000\pi$
$521$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$522$		0			0
$523$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$523$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$524$	0.488830			0.488830
$525$		0			0
$526$	−2.95630			−2.95630
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$	1.12920			1.12920
$532$		0			0
$533$		0			0
$534$	0.913545	−	1.58231i	0.913545	−	1.58231i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.82709			1.82709
$539$		0			0
$540$		0			0
$541$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$541$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$542$	−1.78716	−	3.09546i	−1.78716	−	3.09546i
$543$		0			0
$544$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−0.564602	−	0.977920i	−0.564602	−	0.977920i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.12920			1.12920
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$	−0.722562	−	1.25151i	−0.722562	−	1.25151i
$565$		0			0
$566$	0.381966			0.381966
$567$		0			0
$568$	−4.78339			−4.78339
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$571$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$572$		0			0
$573$	0.618034			0.618034
$574$		0			0
$575$		0			0
$576$	0.157960	−	0.273595i	0.157960	−	0.273595i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−2.58268	−	4.47333i	−2.58268	−	4.47333i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−0.236068			−0.236068
$583$		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			0
$589$		0			0
$590$		0			0
$591$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$592$	2.08268	+	3.60730i	2.08268	+	3.60730i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$	3.12920			3.12920
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.755585	−	1.30871i	0.755585	−	1.30871i
$601$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$601$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.755585	−	1.30871i	−0.755585	−	1.30871i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.41355	+	2.44833i	1.41355	+	2.44833i
$613$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$614$	−1.78716	+	3.09546i	−1.78716	+	3.09546i
$615$		0			0
$616$		0			0
$617$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$617$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$618$	−0.755585	+	1.30871i	−0.755585	+	1.30871i
$619$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$619$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−1.89169	+	3.27651i	−1.89169	+	3.27651i
$629$	−3.82709			−3.82709
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.755585	+	1.30871i	−0.755585	+	1.30871i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.302113			0.302113
$637$		0			0
$638$		0			0
$639$	−0.604528	+	1.04707i	−0.604528	+	1.04707i
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$643$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	0.809017	−	0.587785i	$-0.200000\pi$
$647$	−0.104528	−	0.181049i	−0.104528	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$653$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$659$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$660$		0			0
$661$	−0.978148	−	1.69420i	−0.978148	−	1.69420i	−0.669131	−	0.743145i	$-0.733333\pi$
$661$	−0.978148	−	1.69420i	−0.978148	−	1.69420i	−0.309017	−	0.951057i	$-0.600000\pi$
$662$	1.47815	+	2.56023i	1.47815	+	2.56023i
$663$		0			0
$664$	2.44512			2.44512
$665$		0			0
$666$	2.20906			2.20906
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.190983	−	0.330792i	0.190983	−	0.330792i
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	−1.16913	−	2.02499i	−1.16913	−	2.02499i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$683$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	0.488830			0.488830
$693$		0			0
$694$	2.44512			2.44512
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−3.33826			−3.33826
$707$		0			0
$708$	2.64037			2.64037
$709$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$710$		0			0
$711$	0.190983	+	0.330792i	0.190983	+	0.330792i
$712$	−1.97815	+	3.42625i	−1.97815	+	3.42625i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.413545	+	0.716282i	0.413545	+	0.716282i
$718$		0			0
$719$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$719$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$720$		0			0
$721$		0			0
$722$	1.82709			1.82709
$723$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$724$		0			0
$725$		0			0
$726$	0.564602	−	0.977920i	0.564602	−	0.977920i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.618034			0.618034
$730$		0			0
$731$		0			0
$732$	−1.32019	−	2.28663i	−1.32019	−	2.28663i
$733$	−0.978148	+	1.69420i	−0.978148	+	1.69420i	−0.309017	+	0.951057i	$0.600000\pi$
$733$	−0.978148	+	1.69420i	−0.978148	+	1.69420i	−0.669131	+	0.743145i	$0.733333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$739$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.309017	−	0.535233i	0.309017	−	0.535233i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	1.06460	+	1.84395i	1.06460	+	1.84395i
$753$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−1.66913	+	2.89102i	−1.66913	+	2.89102i
$759$		0			0
$760$		0			0
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−2.33826			−2.33826
$765$		0			0
$766$	1.22256	+	2.11754i	1.22256	+	2.11754i
$767$		0			0
$768$	−0.446568	+	0.773479i	−0.446568	+	0.773479i
$769$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$769$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.309017	+	0.535233i	0.309017	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$773$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$774$		0			0
$775$		0			0
$776$	0.511170			0.511170
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	0.118034	−	0.204441i	0.118034	−	0.204441i
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	1.16913	+	2.02499i	1.16913	+	2.02499i
$789$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.22256	+	2.11754i	1.22256	+	2.11754i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	−1.95630			−1.95630
$800$	−0.722562	+	1.25151i	−0.722562	+	1.25151i
$801$	0.500000	+	0.866025i	0.500000	+	0.866025i
$802$	1.78716	+	3.09546i	1.78716	+	3.09546i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.309017	−	0.535233i	0.309017	−	0.535233i
$808$	1.63611	+	2.83382i	1.63611	+	2.83382i
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$811$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$812$		0			0
$813$	−1.20906			−1.20906
$814$		0			0
$815$		0			0
$816$	1.28716	+	2.22943i	1.28716	+	2.22943i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$823$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$824$	1.63611	−	2.83382i	1.63611	−	2.83382i
$825$		0			0
$826$		0			0
$827$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$827$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$	0.190983	−	0.330792i	0.190983	−	0.330792i
$832$		0			0
$833$		0			0
$834$		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
$846$	1.12920			1.12920
$847$		0			0
$848$	−0.445125			−0.445125
$849$	0.0646021	−	0.111894i	0.0646021	−	0.111894i
$850$	−1.78716	−	3.09546i	−1.78716	−	3.09546i
$851$		0			0
$852$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$853$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$853$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$	−0.381966			−0.381966
$863$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$863$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$864$	−0.722562	−	1.25151i	−0.722562	−	1.25151i
$865$		0			0
$866$		0			0
$867$	−1.74724			−1.74724
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.0646021	−	0.111894i	0.0646021	−	0.111894i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	−0.913545	−	1.58231i	−0.913545	−	1.58231i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$883$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$	2.95630			2.95630
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.755585	−	1.30871i	0.755585	−	1.30871i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.722562	+	1.25151i	0.722562	+	1.25151i
$901$	0.204489	−	0.354185i	0.204489	−	0.354185i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	0.827091			0.827091
$910$		0			0
$911$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$911$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$912$		0			0
$913$		0			0
$914$	−0.564602	−	0.977920i	−0.564602	−	0.977920i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	3.57433			3.57433
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	0.604528	+	1.04707i	0.604528	+	1.04707i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.95630			−1.95630
$926$		0			0
$927$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$928$		0			0
$929$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$929$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$930$		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
$941$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$941$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$942$	0.913545	+	1.58231i	0.913545	+	1.58231i
$943$		0			0
$944$	−3.89025			−3.89025
$945$		0			0
$946$		0			0
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$948$	0.446568	+	0.773479i	0.446568	+	0.773479i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−0.118034	+	0.204441i	−0.118034	+	0.204441i
$955$		0			0
$956$	−1.56460	−	2.70997i	−1.56460	−	2.70997i
$957$		0			0
$958$	2.95630			2.95630
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$	1.56460	−	2.70997i	1.56460	−	2.70997i
$965$		0			0
$966$		0			0
$967$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$968$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−2.33826			−2.33826
$973$		0			0
$974$	−1.82709			−1.82709
$975$		0			0
$976$	1.94512	+	3.36906i	1.94512	+	3.36906i
$977$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$977$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.47815	−	2.56023i	1.47815	−	2.56023i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$991$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$992$		0			0
$993$	1.00000			1.00000
$994$		0			0
$995$		0			0
$996$	0.722562	−	1.25151i	0.722562	−	1.25151i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$	0.978148	−	1.69420i	0.978148	−	1.69420i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2303.1.f.d.422.1		8
7.2	even	3		2303.1.d.e.2255.4		4
7.3	odd	6		329.1.f.b.46.1	✓	8
7.4	even	3	inner	2303.1.f.d.704.1		8
7.5	odd	6		2303.1.d.d.2255.4		4
7.6	odd	2		329.1.f.b.93.1	yes	8
21.17	even	6		2961.1.x.d.46.4		8
21.20	even	2		2961.1.x.d.1738.4		8
47.46	odd	2	CM	2303.1.f.d.422.1		8
329.46	odd	6	inner	2303.1.f.d.704.1		8
329.93	odd	6		2303.1.d.e.2255.4		4
329.187	even	6		2303.1.d.d.2255.4		4
329.234	even	6		329.1.f.b.46.1	✓	8
329.328	even	2		329.1.f.b.93.1	yes	8
987.563	odd	6		2961.1.x.d.46.4		8
987.986	odd	2		2961.1.x.d.1738.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
329.1.f.b.46.1	✓	8	7.3	odd	6
329.1.f.b.46.1	✓	8	329.234	even	6
329.1.f.b.93.1	yes	8	7.6	odd	2
329.1.f.b.93.1	yes	8	329.328	even	2
2303.1.d.d.2255.4		4	7.5	odd	6
2303.1.d.d.2255.4		4	329.187	even	6
2303.1.d.e.2255.4		4	7.2	even	3
2303.1.d.e.2255.4		4	329.93	odd	6
2303.1.f.d.422.1		8	1.1	even	1	trivial
2303.1.f.d.422.1		8	47.46	odd	2	CM
2303.1.f.d.704.1		8	7.4	even	3	inner
2303.1.f.d.704.1		8	329.46	odd	6	inner
2961.1.x.d.46.4		8	21.17	even	6
2961.1.x.d.46.4		8	987.563	odd	6
2961.1.x.d.1738.4		8	21.20	even	2
2961.1.x.d.1738.4		8	987.986	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2303 = 7^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2303.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.913545 + 1.58231i) q^{2} +(0.309017 + 0.535233i) q^{3} +(-1.16913 - 2.02499i) q^{4} -1.12920 q^{6} +2.44512 q^{8} +(0.309017 - 0.535233i) q^{9} +(0.722562 - 1.25151i) q^{12} +(-1.06460 + 1.84395i) q^{16} +(-0.978148 - 1.69420i) q^{17} +(0.564602 + 0.977920i) q^{18} +(0.755585 + 1.30871i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(-0.722562 - 1.25151i) q^{32} +3.57433 q^{34} -1.44512 q^{36} +(0.978148 - 1.69420i) q^{37} +(0.500000 - 0.866025i) q^{47} -1.31592 q^{48} +1.82709 q^{50} +(0.604528 - 1.04707i) q^{51} +(0.104528 + 0.181049i) q^{53} +(-0.913545 + 1.58231i) q^{54} +(0.913545 + 1.58231i) q^{59} +(0.913545 - 1.58231i) q^{61} +0.511170 q^{64} +(-2.28716 + 3.96149i) q^{68} -1.95630 q^{71} +(0.755585 - 1.30871i) q^{72} +(1.78716 + 3.09546i) q^{74} +(0.309017 - 0.535233i) q^{75} +(-0.309017 + 0.535233i) q^{79} +1.00000 q^{83} +(-0.809017 + 1.40126i) q^{89} +(0.913545 + 1.58231i) q^{94} +(0.446568 - 0.773479i) q^{96} +0.209057 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 2 q^{3} - 5 q^{4} - 4 q^{6} - 2 q^{8} - 2 q^{9} - 5 q^{12} - 6 q^{16} + q^{17} + 2 q^{18} + 8 q^{24} - 4 q^{25} + 8 q^{27} + 5 q^{32} + 2 q^{34} + 10 q^{36} - q^{37} + 4 q^{47} + 6 q^{48}+ \cdots - 2 q^{97}+O(q^{100}) \) 8 * q - q^2 - 2 * q^3 - 5 * q^4 - 4 * q^6 - 2 * q^8 - 2 * q^9 - 5 * q^12 - 6 * q^16 + q^17 + 2 * q^18 + 8 * q^24 - 4 * q^25 + 8 * q^27 + 5 * q^32 + 2 * q^34 + 10 * q^36 - q^37 + 4 * q^47 + 6 * q^48 + 2 * q^50 + 3 * q^51 - q^53 - q^54 + q^59 + q^61 + 8 * q^64 - 5 * q^68 + 2 * q^71 + 8 * q^72 + q^74 - 2 * q^75 + 2 * q^79 + 8 * q^83 - 2 * q^89 + q^94 + 10 * q^96 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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