LMFDB - Embedded newform 968.1.j.a.233.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,1,Mod(161,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 7]))

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

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

chi := DirichletCharacter("968.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	968.j (of order $10$, degree $4$, 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:	$0.483094932229$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.21296.1

Embedding invariants

Embedding label			233.2
Root			$1.34500 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	968.233
Dual form			968.1.j.a.457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.809017 - 0.587785i) q^{3} +(0.309017 + 0.951057i) q^{5} +(0.831254 - 1.14412i) q^{7} +(0.809017 + 0.587785i) q^{15} +(-1.34500 + 0.437016i) q^{17} -1.41421i q^{21} -1.00000 q^{23} +(0.309017 + 0.951057i) q^{27} +(0.309017 - 0.951057i) q^{31} +(1.34500 + 0.437016i) q^{35} +(-0.809017 - 0.587785i) q^{37} +1.41421i q^{43} +(-0.309017 - 0.951057i) q^{49} +(-0.831254 + 1.14412i) q^{51} +(-0.809017 - 0.587785i) q^{59} +1.00000 q^{67} +(-0.809017 + 0.587785i) q^{69} +(-0.309017 - 0.951057i) q^{71} +(-1.34500 - 0.437016i) q^{79} +(0.809017 + 0.587785i) q^{81} +(1.34500 - 0.437016i) q^{83} +(-0.831254 - 1.14412i) q^{85} -1.00000 q^{89} +(-0.309017 - 0.951057i) q^{93} +(-0.309017 + 0.951057i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 2 q^{5} + 2 q^{15} - 8 q^{23} - 2 q^{27} - 2 q^{31} - 2 q^{37} + 2 q^{49} - 2 q^{59} + 8 q^{67} - 2 q^{69} + 2 q^{71} + 2 q^{81} - 8 q^{89} + 2 q^{93} + 2 q^{97}+O(q^{100}) $ 8 * q + 2 * q^3 - 2 * q^5 + 2 * q^15 - 8 * q^23 - 2 * q^27 - 2 * q^31 - 2 * q^37 + 2 * q^49 - 2 * q^59 + 8 * q^67 - 2 * q^69 + 2 * q^71 + 2 * q^81 - 8 * q^89 + 2 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{10}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$4$		0			0
$5$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$5$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$6$		0			0
$7$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$7$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$13$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$	0.809017	+	0.587785i	0.809017	+	0.587785i
$16$		0			0
$17$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$17$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$18$		0			0
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$		0			0
$21$		−	1.41421i		−	1.41421i
$22$		0			0
$23$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.309017	+	0.951057i	0.309017	+	0.951057i
$28$		0			0
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$31$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.34500	+	0.437016i	1.34500	+	0.437016i
$36$		0			0
$37$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$37$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$		0			0
$43$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$43$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$47$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$48$		0			0
$49$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$50$		0			0
$51$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$52$		0			0
$53$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$53$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$59$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$60$		0			0
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$70$		0			0
$71$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$71$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$72$		0			0
$73$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$73$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$79$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$80$		0			0
$81$	0.809017	+	0.587785i	0.809017	+	0.587785i
$82$		0			0
$83$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$83$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$84$		0			0
$85$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$97$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$101$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$102$		0			0
$103$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$103$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$104$		0			0
$105$	1.34500	−	0.437016i	1.34500	−	0.437016i
$106$		0			0
$107$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$107$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$108$		0			0
$109$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$109$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$	−1.00000			−1.00000
$112$		0			0
$113$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$113$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$114$		0			0
$115$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.809017	+	0.587785i	0.809017	+	0.587785i
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$		0			0
$129$	0.831254	+	1.14412i	0.831254	+	1.14412i
$130$		0			0
$131$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$131$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$136$		0			0
$137$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$137$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$138$		0			0
$139$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$139$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$148$		0			0
$149$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$149$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$150$		0			0
$151$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$151$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.00000			1.00000
$156$		0			0
$157$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$157$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$162$		0			0
$163$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$163$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$173$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.00000			−1.00000
$178$		0			0
$179$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$179$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$180$		0			0
$181$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$181$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.309017	−	0.951057i	0.309017	−	0.951057i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.34500	+	0.437016i	1.34500	+	0.437016i
$190$		0			0
$191$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$191$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$192$		0			0
$193$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$193$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$197$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	0.809017	−	0.587785i	0.809017	−	0.587785i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$		0			0
$213$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$214$		0			0
$215$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$216$		0			0
$217$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$223$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$227$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$228$		0			0
$229$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$229$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$238$		0			0
$239$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$239$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.809017	−	0.587785i	0.809017	−	0.587785i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.831254	−	1.14412i	0.831254	−	1.14412i
$250$		0			0
$251$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$251$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$256$		0			0
$257$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$257$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$258$		0			0
$259$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$263$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$268$		0			0
$269$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$269$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$277$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$281$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$282$		0			0
$283$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$283$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.809017	−	0.587785i	0.809017	−	0.587785i
$290$		0			0
$291$	0.309017	+	0.951057i	0.309017	+	0.951057i
$292$		0			0
$293$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$293$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$294$		0			0
$295$	0.309017	−	0.951057i	0.309017	−	0.951057i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.61803	+	1.17557i	1.61803	+	1.17557i
$302$		0			0
$303$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$311$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$312$		0			0
$313$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$313$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$317$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.34500	+	0.437016i	1.34500	+	0.437016i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.831254	+	1.14412i	0.831254	+	1.14412i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$331$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.309017	+	0.951057i	0.309017	+	0.951057i
$336$		0			0
$337$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$337$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$338$		0			0
$339$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$346$		0			0
$347$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$347$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$349$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	0.809017	−	0.587785i	0.809017	−	0.587785i
$356$		0			0
$357$	0.618034	+	1.90211i	0.618034	+	1.90211i
$358$		0			0
$359$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$359$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$360$		0			0
$361$	0.309017	−	0.951057i	0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$367$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$373$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$374$		0			0
$375$	1.00000			1.00000
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$379$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$383$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$389$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$390$		0			0
$391$	1.34500	−	0.437016i	1.34500	−	0.437016i
$392$		0			0
$393$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$394$		0			0
$395$		−	1.41421i		−	1.41421i
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$401$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$409$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$410$		0			0
$411$	0.809017	+	0.587785i	0.809017	+	0.587785i
$412$		0			0
$413$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$414$		0			0
$415$	0.831254	+	1.14412i	0.831254	+	1.14412i
$416$		0			0
$417$			1.41421i			1.41421i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$421$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$431$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$432$		0			0
$433$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$433$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$443$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$444$		0			0
$445$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$446$		0			0
$447$	0.831254	−	1.14412i	0.831254	−	1.14412i
$448$		0			0
$449$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$449$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$457$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$458$		0			0
$459$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$	0.809017	−	0.587785i	0.809017	−	0.587785i
$466$		0			0
$467$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$467$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$468$		0			0
$469$	0.831254	−	1.14412i	0.831254	−	1.14412i
$470$		0			0
$471$	0.309017	−	0.951057i	0.309017	−	0.951057i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			1.41421i			1.41421i
$484$		0			0
$485$	−1.00000			−1.00000
$486$		0			0
$487$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$487$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$491$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$498$		0			0
$499$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$499$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$503$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$504$		0			0
$505$		−	1.41421i		−	1.41421i
$506$		0			0
$507$	−1.00000			−1.00000
$508$		0			0
$509$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$509$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.34500	+	0.437016i	1.34500	+	0.437016i
$520$		0			0
$521$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$521$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$522$		0			0
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.41421i			1.41421i
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$536$		0			0
$537$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$		0			0
$543$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$544$		0			0
$545$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$554$		0			0
$555$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$556$		0			0
$557$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$557$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$566$		0			0
$567$	1.34500	−	0.437016i	1.34500	−	0.437016i
$568$		0			0
$569$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$569$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	1.00000			1.00000
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$577$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.618034	−	1.90211i	0.618034	−	1.90211i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$587$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	0.831254	+	1.14412i	0.831254	+	1.14412i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−2.00000			−2.00000
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$599$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$600$		0			0
$601$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$601$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$607$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$613$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$619$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$620$		0			0
$621$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$622$		0			0
$623$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$624$		0			0
$625$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.34500	+	0.437016i	1.34500	+	0.437016i
$630$		0			0
$631$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$631$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$641$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$642$		0			0
$643$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$643$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$644$		0			0
$645$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$646$		0			0
$647$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$647$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$652$		0			0
$653$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$653$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$654$		0			0
$655$	1.34500	−	0.437016i	1.34500	−	0.437016i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$659$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$673$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$677$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$678$		0			0
$679$	0.831254	+	1.14412i	0.831254	+	1.14412i
$680$		0			0
$681$		−	1.41421i		−	1.41421i
$682$		0			0
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$684$		0			0
$685$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$686$		0			0
$687$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$691$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$701$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$708$		0			0
$709$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$709$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$718$		0			0
$719$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$719$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$732$		0			0
$733$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$733$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$734$		0			0
$735$	0.309017	−	0.951057i	0.309017	−	0.951057i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$739$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$	0.831254	+	1.14412i	0.831254	+	1.14412i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.00000			2.00000
$750$		0			0
$751$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$751$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$752$		0			0
$753$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$754$		0			0
$755$	0.831254	−	1.14412i	0.831254	−	1.14412i
$756$		0			0
$757$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$757$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$761$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$762$		0			0
$763$	1.61803	+	1.17557i	1.61803	+	1.17557i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$773$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.809017	+	0.587785i	0.809017	+	0.587785i
$786$		0			0
$787$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$787$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$788$		0			0
$789$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$790$		0			0
$791$			1.41421i			1.41421i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$797$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$809$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$810$		0			0
$811$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$811$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$821$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$822$		0			0
$823$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$823$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$827$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$828$		0			0
$829$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$829$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$830$		0			0
$831$	1.34500	−	0.437016i	1.34500	−	0.437016i
$832$		0			0
$833$	0.831254	+	1.14412i	0.831254	+	1.14412i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.00000			1.00000
$838$		0			0
$839$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$839$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$		0			0
$843$	0.831254	−	1.14412i	0.831254	−	1.14412i
$844$		0			0
$845$	0.309017	−	0.951057i	0.309017	−	0.951057i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.809017	+	0.587785i	0.809017	+	0.587785i
$852$		0			0
$853$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$853$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$863$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$864$		0			0
$865$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$866$		0			0
$867$	0.309017	−	0.951057i	0.309017	−	0.951057i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.34500	−	0.437016i	1.34500	−	0.437016i
$876$		0			0
$877$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$877$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$881$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$883$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$884$		0			0
$885$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$886$		0			0
$887$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$887$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	2.00000			2.00000
$904$		0			0
$905$	0.809017	−	0.587785i	0.809017	−	0.587785i
$906$		0			0
$907$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$907$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$911$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$918$		0			0
$919$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$919$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$929$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.618034	−	1.90211i	0.618034	−	1.90211i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$937$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$	0.809017	+	0.587785i	0.809017	+	0.587785i
$940$		0			0
$941$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$941$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$			1.41421i			1.41421i
$946$		0			0
$947$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.309017	+	0.951057i	0.309017	+	0.951057i
$952$		0			0
$953$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$953$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$954$		0			0
$955$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.34500	+	0.437016i	1.34500	+	0.437016i
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$971$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$972$		0			0
$973$	0.618034	+	1.90211i	0.618034	+	1.90211i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$977$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$983$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$984$		0			0
$985$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.41421i		−	1.41421i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	0.809017	−	0.587785i	0.809017	−	0.587785i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$997$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$998$		0			0
$999$	0.309017	−	0.951057i	0.309017	−	0.951057i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.1.j.a.233.2		8
4.3	odd	2		1936.1.n.b.1201.1		8
11.2	odd	10	inner	968.1.j.a.481.1		8
11.3	even	5	inner	968.1.j.a.161.1		8
11.4	even	5		968.1.h.a.241.2	yes	2
11.5	even	5	inner	968.1.j.a.457.1		8
11.6	odd	10	inner	968.1.j.a.457.2		8
11.7	odd	10		968.1.h.a.241.1	✓	2
11.8	odd	10	inner	968.1.j.a.161.2		8
11.9	even	5	inner	968.1.j.a.481.2		8
11.10	odd	2	inner	968.1.j.a.233.1		8
44.3	odd	10		1936.1.n.b.161.2		8
44.7	even	10		1936.1.h.a.241.2		2
44.15	odd	10		1936.1.h.a.241.1		2
44.19	even	10		1936.1.n.b.161.1		8
44.27	odd	10		1936.1.n.b.1425.2		8
44.31	odd	10		1936.1.n.b.481.1		8
44.35	even	10		1936.1.n.b.481.2		8
44.39	even	10		1936.1.n.b.1425.1		8
44.43	even	2		1936.1.n.b.1201.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.1.h.a.241.1	✓	2	11.7	odd	10
968.1.h.a.241.2	yes	2	11.4	even	5
968.1.j.a.161.1		8	11.3	even	5	inner
968.1.j.a.161.2		8	11.8	odd	10	inner
968.1.j.a.233.1		8	11.10	odd	2	inner
968.1.j.a.233.2		8	1.1	even	1	trivial
968.1.j.a.457.1		8	11.5	even	5	inner
968.1.j.a.457.2		8	11.6	odd	10	inner
968.1.j.a.481.1		8	11.2	odd	10	inner
968.1.j.a.481.2		8	11.9	even	5	inner
1936.1.h.a.241.1		2	44.15	odd	10
1936.1.h.a.241.2		2	44.7	even	10
1936.1.n.b.161.1		8	44.19	even	10
1936.1.n.b.161.2		8	44.3	odd	10
1936.1.n.b.481.1		8	44.31	odd	10
1936.1.n.b.481.2		8	44.35	even	10
1936.1.n.b.1201.1		8	4.3	odd	2
1936.1.n.b.1201.2		8	44.43	even	2
1936.1.n.b.1425.1		8	44.39	even	10
1936.1.n.b.1425.2		8	44.27	odd	10

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	968.j (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 - 0.587785i) q^{3} +(0.309017 + 0.951057i) q^{5} +(0.831254 - 1.14412i) q^{7} +(0.809017 + 0.587785i) q^{15} +(-1.34500 + 0.437016i) q^{17} -1.41421i q^{21} -1.00000 q^{23} +(0.309017 + 0.951057i) q^{27} +(0.309017 - 0.951057i) q^{31} +(1.34500 + 0.437016i) q^{35} +(-0.809017 - 0.587785i) q^{37} +1.41421i q^{43} +(-0.309017 - 0.951057i) q^{49} +(-0.831254 + 1.14412i) q^{51} +(-0.809017 - 0.587785i) q^{59} +1.00000 q^{67} +(-0.809017 + 0.587785i) q^{69} +(-0.309017 - 0.951057i) q^{71} +(-1.34500 - 0.437016i) q^{79} +(0.809017 + 0.587785i) q^{81} +(1.34500 - 0.437016i) q^{83} +(-0.831254 - 1.14412i) q^{85} -1.00000 q^{89} +(-0.309017 - 0.951057i) q^{93} +(-0.309017 + 0.951057i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 2 q^{5} + 2 q^{15} - 8 q^{23} - 2 q^{27} - 2 q^{31} - 2 q^{37} + 2 q^{49} - 2 q^{59} + 8 q^{67} - 2 q^{69} + 2 q^{71} + 2 q^{81} - 8 q^{89} + 2 q^{93} + 2 q^{97}+O(q^{100}) \) 8 * q + 2 * q^3 - 2 * q^5 + 2 * q^15 - 8 * q^23 - 2 * q^27 - 2 * q^31 - 2 * q^37 + 2 * q^49 - 2 * q^59 + 8 * q^67 - 2 * q^69 + 2 * q^71 + 2 * q^81 - 8 * q^89 + 2 * q^93 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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