LMFDB - Embedded newform 1960.1.bu.a.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,1,Mod(373,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 9, 4]))

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

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

chi := DirichletCharacter("1960.373");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1960.bu (of order $12$, 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:	$0.978167424761$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.21952000000.3

Embedding invariants

Embedding label			557.2
Root			$-0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1960.557
Dual form			1960.1.bu.a.373.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.258819 + 0.965926i) q^{2} +(1.78480 - 0.478235i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(0.130526 + 0.991445i) q^{5} +1.84776i q^{6} +(0.707107 - 0.707107i) q^{8} +(2.09077 - 1.20711i) q^{9} +O(q^{10})$	\(q+(-0.258819 + 0.965926i) q^{2} +(1.78480 - 0.478235i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(0.130526 + 0.991445i) q^{5} +1.84776i q^{6} +(0.707107 - 0.707107i) q^{8} +(2.09077 - 1.20711i) q^{9} +(-0.991445 - 0.130526i) q^{10} +(-1.78480 - 0.478235i) q^{12} +(0.541196 + 0.541196i) q^{13} +(0.707107 + 1.70711i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.624844 + 2.33195i) q^{18} +(-0.382683 - 0.662827i) q^{19} +(0.382683 - 0.923880i) q^{20} +(-1.36603 - 0.366025i) q^{23} +(0.923880 - 1.60021i) q^{24} +(-0.965926 + 0.258819i) q^{25} +(-0.662827 + 0.382683i) q^{26} +(1.84776 - 1.84776i) q^{27} +(-1.83195 + 0.241181i) q^{30} +(-0.965926 + 0.258819i) q^{32} -2.41421 q^{36} +(0.739288 - 0.198092i) q^{38} +(1.22474 + 0.707107i) q^{39} +(0.793353 + 0.608761i) q^{40} +(1.46968 + 1.91532i) q^{45} +(0.707107 - 1.22474i) q^{46} +(1.30656 + 1.30656i) q^{48} -1.00000i q^{50} +(-0.198092 - 0.739288i) q^{52} +(1.30656 + 2.26303i) q^{54} +(-1.00000 - 1.00000i) q^{57} +(-0.923880 + 1.60021i) q^{59} +(0.241181 - 1.83195i) q^{60} +(1.60021 - 0.923880i) q^{61} -1.00000i q^{64} +(-0.465926 + 0.607206i) q^{65} -2.61313 q^{69} -1.41421 q^{71} +(0.624844 - 2.33195i) q^{72} +(-1.60021 + 0.923880i) q^{75} +0.765367i q^{76} +(-1.00000 + 1.00000i) q^{78} +(-1.22474 + 0.707107i) q^{79} +(-0.793353 + 0.608761i) q^{80} +(1.20711 - 2.09077i) q^{81} +(-0.541196 - 0.541196i) q^{83} +(-2.23044 + 0.923880i) q^{90} +(1.00000 + 1.00000i) q^{92} +(0.607206 - 0.465926i) q^{95} +(-1.60021 + 0.923880i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 8 q^{16} - 8 q^{18} - 8 q^{23} - 16 q^{36} - 16 q^{57} + 8 q^{60} + 8 q^{65} - 8 q^{72} - 16 q^{78} + 8 q^{81} + 16 q^{92}+O(q^{100}) $ 16 * q + 8 * q^16 - 8 * q^18 - 8 * q^23 - 16 * q^36 - 16 * q^57 + 8 * q^60 + 8 * q^65 - 8 * q^72 - 16 * q^78 + 8 * q^81 + 16 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{4}\right)$	$1$

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 $

$2$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$2$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$3$	1.78480	−	0.478235i	1.78480	−	0.478235i	0.793353	−	0.608761i	$-0.208333\pi$
$3$	1.78480	−	0.478235i	1.78480	−	0.478235i	0.991445	+	0.130526i	$0.0416667\pi$
$4$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$5$	0.130526	+	0.991445i	0.130526	+	0.991445i
$5$	0.130526	+	0.991445i	0.130526	+	0.991445i
$6$			1.84776i			1.84776i
$7$		0			0
$7$		0			0
$8$	0.707107	−	0.707107i	0.707107	−	0.707107i
$9$	2.09077	−	1.20711i	2.09077	−	1.20711i
$10$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−1.78480	−	0.478235i	−1.78480	−	0.478235i
$13$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$13$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$14$		0			0
$15$	0.707107	+	1.70711i	0.707107	+	1.70711i
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$17$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$18$	0.624844	+	2.33195i	0.624844	+	2.33195i
$19$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$19$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$20$	0.382683	−	0.923880i	0.382683	−	0.923880i
$21$		0			0
$22$		0			0
$23$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.923880	−	1.60021i	0.923880	−	1.60021i
$25$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$26$	−0.662827	+	0.382683i	−0.662827	+	0.382683i
$27$	1.84776	−	1.84776i	1.84776	−	1.84776i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−1.83195	+	0.241181i	−1.83195	+	0.241181i
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.41421			−2.41421
$37$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$37$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$38$	0.739288	−	0.198092i	0.739288	−	0.198092i
$39$	1.22474	+	0.707107i	1.22474	+	0.707107i
$40$	0.793353	+	0.608761i	0.793353	+	0.608761i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	1.46968	+	1.91532i	1.46968	+	1.91532i
$46$	0.707107	−	1.22474i	0.707107	−	1.22474i
$47$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$47$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$48$	1.30656	+	1.30656i	1.30656	+	1.30656i
$49$		0			0
$50$		−	1.00000i		−	1.00000i
$51$		0			0
$52$	−0.198092	−	0.739288i	−0.198092	−	0.739288i
$53$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$53$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$54$	1.30656	+	2.26303i	1.30656	+	2.26303i
$55$		0			0
$56$		0			0
$57$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$58$		0			0
$59$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$59$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$60$	0.241181	−	1.83195i	0.241181	−	1.83195i
$61$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.608761	−	0.793353i	$-0.291667\pi$
$61$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.991445	−	0.130526i	$-0.0416667\pi$
$62$		0			0
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$	−0.465926	+	0.607206i	−0.465926	+	0.607206i
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$		0			0
$69$	−2.61313			−2.61313
$70$		0			0
$71$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$71$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$72$	0.624844	−	2.33195i	0.624844	−	2.33195i
$73$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$73$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$74$		0			0
$75$	−1.60021	+	0.923880i	−1.60021	+	0.923880i
$76$			0.765367i			0.765367i
$77$		0			0
$78$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$79$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$79$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$80$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$81$	1.20711	−	2.09077i	1.20711	−	2.09077i
$82$		0			0
$83$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$83$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	−2.23044	+	0.923880i	−2.23044	+	0.923880i
$91$		0			0
$92$	1.00000	+	1.00000i	1.00000	+	1.00000i
$93$		0			0
$94$		0			0
$95$	0.607206	−	0.465926i	0.607206	−	0.465926i
$96$	−1.60021	+	0.923880i	−1.60021	+	0.923880i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$	0.965926	+	0.258819i	0.965926	+	0.258819i
$101$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$101$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.608761	+	0.793353i	$0.291667\pi$
$102$		0			0
$103$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$103$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$104$	0.765367			0.765367
$105$		0			0
$106$		0			0
$107$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$107$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$108$	−2.52409	+	0.676327i	−2.52409	+	0.676327i
$109$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$109$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$	1.22474	−	0.707107i	1.22474	−	0.707107i
$115$	0.184592	−	1.40211i	0.184592	−	1.40211i
$116$		0			0
$117$	1.78480	+	0.478235i	1.78480	+	0.478235i
$118$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$119$		0			0
$120$	1.70711	+	0.707107i	1.70711	+	0.707107i
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	0.478235	+	1.78480i	0.478235	+	1.78480i
$123$		0			0
$124$		0			0
$125$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$126$		0			0
$127$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$127$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$128$	0.965926	+	0.258819i	0.965926	+	0.258819i
$129$		0			0
$130$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$131$	0.662827	−	0.382683i	0.662827	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$131$	0.662827	−	0.382683i	0.662827	−	0.382683i	0.793353	+	0.608761i	$0.208333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.07313	+	1.59077i	2.07313	+	1.59077i
$136$		0			0
$137$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0.676327	−	2.52409i	0.676327	−	2.52409i
$139$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$139$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$140$		0			0
$141$		0			0
$142$	0.366025	−	1.36603i	0.366025	−	1.36603i
$143$		0			0
$144$	2.09077	+	1.20711i	2.09077	+	1.20711i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	−0.478235	−	1.78480i	−0.478235	−	1.78480i
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$	−0.739288	−	0.198092i	−0.739288	−	0.198092i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$157$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	−0.608761	−	0.793353i	$-0.708333\pi$
$157$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	0.130526	−	0.991445i	$-0.458333\pi$
$158$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$159$		0			0
$160$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$161$		0			0
$162$	1.70711	+	1.70711i	1.70711	+	1.70711i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$		0			0
$165$		0			0
$166$	0.662827	−	0.382683i	0.662827	−	0.382683i
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		−	0.414214i		−	0.414214i
$170$		0			0
$171$	−1.60021	−	0.923880i	−1.60021	−	0.923880i
$172$		0			0
$173$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	0.130526	+	0.991445i	$0.458333\pi$
$173$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	−0.608761	+	0.793353i	$0.708333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.883663	+	3.29788i	−0.883663	+	3.29788i
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−0.315118	−	2.39356i	−0.315118	−	2.39356i
$181$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$181$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$182$		0			0
$183$	2.41421	−	2.41421i	2.41421	−	2.41421i
$184$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.292893	+	0.707107i	0.292893	+	0.707107i
$191$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$191$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$192$	−0.478235	−	1.78480i	−0.478235	−	1.78480i
$193$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$193$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$194$		0			0
$195$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$201$		0			0
$202$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.29788	+	0.883663i	−3.29788	+	0.883663i
$208$	−0.198092	+	0.739288i	−0.198092	+	0.739288i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−2.52409	+	0.676327i	−2.52409	+	0.676327i
$214$		0			0
$215$		0			0
$216$		−	2.61313i		−	2.61313i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$226$		0			0
$227$	0.198092	+	0.739288i	0.198092	+	0.739288i	0.991445	+	0.130526i	$0.0416667\pi$
$227$	0.198092	+	0.739288i	0.198092	+	0.739288i	−0.793353	+	0.608761i	$0.791667\pi$
$228$	0.366025	+	1.36603i	0.366025	+	1.36603i
$229$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$229$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$230$	1.30656	+	0.541196i	1.30656	+	0.541196i
$231$		0			0
$232$		0			0
$233$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$234$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$235$		0			0
$236$	1.60021	−	0.923880i	1.60021	−	0.923880i
$237$	−1.84776	+	1.84776i	−1.84776	+	1.84776i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−1.12484	+	1.46593i	−1.12484	+	1.46593i
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	0.965926	−	0.258819i	0.965926	−	0.258819i
$243$	0.478235	−	1.78480i	0.478235	−	1.78480i
$244$	−1.84776			−1.84776
$245$		0			0
$246$		0			0
$247$	0.151613	−	0.565826i	0.151613	−	0.565826i
$248$		0			0
$249$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$250$	0.991445	−	0.130526i	0.991445	−	0.130526i
$251$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$251$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$252$		0			0
$253$		0			0
$254$	1.22474	−	0.707107i	1.22474	−	0.707107i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$257$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$258$		0			0
$259$		0			0
$260$	0.707107	−	0.292893i	0.707107	−	0.292893i
$261$		0			0
$262$	0.198092	+	0.739288i	0.198092	+	0.739288i
$263$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$263$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$269$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$270$	−2.07313	+	1.59077i	−2.07313	+	1.59077i
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		−	1.41421i		−	1.41421i
$275$		0			0
$276$	2.26303	+	1.30656i	2.26303	+	1.30656i
$277$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$277$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$278$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.130526	−	0.991445i	$-0.458333\pi$
$283$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.608761	+	0.793353i	$0.291667\pi$
$284$	1.22474	+	0.707107i	1.22474	+	0.707107i
$285$	0.860919	−	1.12197i	0.860919	−	1.12197i
$286$		0			0
$287$		0			0
$288$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$289$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$293$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$294$		0			0
$295$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$300$	1.84776			1.84776
$301$		0			0
$302$		0			0
$303$	3.29788	+	0.883663i	3.29788	+	0.883663i
$304$	0.382683	−	0.662827i	0.382683	−	0.662827i
$305$	1.12484	+	1.46593i	1.12484	+	1.46593i
$306$		0			0
$307$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$307$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$	1.36603	−	0.366025i	1.36603	−	0.366025i
$313$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$313$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$314$	1.84776			1.84776
$315$		0			0
$316$	1.41421			1.41421
$317$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$317$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$318$		0			0
$319$		0			0
$320$	0.991445	−	0.130526i	0.991445	−	0.130526i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.09077	+	1.20711i	−2.09077	+	1.20711i
$325$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	0.198092	+	0.739288i	0.198092	+	0.739288i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$337$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$338$	0.400100	+	0.107206i	0.400100	+	0.107206i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	1.30656	−	1.30656i	1.30656	−	1.30656i
$343$		0			0
$344$		0			0
$345$	−0.341081	−	2.59077i	−0.341081	−	2.59077i
$346$	−1.60021	−	0.923880i	−1.60021	−	0.923880i
$347$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$347$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$349$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$	2.00000			2.00000
$352$		0			0
$353$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$353$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$354$	−2.95680	−	1.70711i	−2.95680	−	1.70711i
$355$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$359$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$360$	2.39356	+	0.315118i	2.39356	+	0.315118i
$361$	0.207107	−	0.358719i	0.207107	−	0.358719i
$362$	0.739288	+	0.198092i	0.739288	+	0.198092i
$363$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$364$		0			0
$365$		0			0
$366$	1.70711	+	2.95680i	1.70711	+	2.95680i
$367$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$367$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$368$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$373$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$374$		0			0
$375$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i
$381$	−2.26303	−	1.30656i	−2.26303	−	1.30656i
$382$	1.36603	−	0.366025i	1.36603	−	0.366025i
$383$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$383$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$384$	1.84776			1.84776
$385$		0			0
$386$	−2.00000			−2.00000
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$	−1.12197	−	0.860919i	−1.12197	−	0.860919i
$391$		0			0
$392$		0			0
$393$	1.00000	−	1.00000i	1.00000	−	1.00000i
$394$		0			0
$395$	−0.860919	−	1.12197i	−0.860919	−	1.12197i
$396$		0			0
$397$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$397$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.991445	+	0.130526i	$0.958333\pi$
$398$		0			0
$399$		0			0
$400$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$401$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$401$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$402$		0			0
$403$		0			0
$404$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$405$	2.23044	+	0.923880i	2.23044	+	0.923880i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	−2.26303	+	1.30656i	−2.26303	+	1.30656i
$412$		0			0
$413$		0			0
$414$		−	3.41421i		−	3.41421i
$415$	0.465926	−	0.607206i	0.465926	−	0.607206i
$416$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$417$	3.29788	−	0.883663i	3.29788	−	0.883663i
$418$		0			0
$419$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$419$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		−	2.61313i		−	2.61313i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$431$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$432$	2.52409	+	0.676327i	2.52409	+	0.676327i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.280144	+	1.04551i	0.280144	+	1.04551i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$443$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.20711	−	2.09077i	−1.20711	−	2.09077i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.765367			−0.765367
$455$		0			0
$456$	−1.41421			−1.41421
$457$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$457$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$458$	1.78480	−	0.478235i	1.78480	−	0.478235i
$459$		0			0
$460$	−0.860919	+	1.12197i	−0.860919	+	1.12197i
$461$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$461$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$462$		0			0
$463$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$463$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	0.707107	−	1.22474i	0.707107	−	1.22474i
$467$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.991445	−	0.130526i	$-0.0416667\pi$
$467$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.793353	+	0.608761i	$0.208333\pi$
$468$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$469$		0			0
$470$		0			0
$471$	−1.70711	−	2.95680i	−1.70711	−	2.95680i
$472$	0.478235	+	1.78480i	0.478235	+	1.78480i
$473$		0			0
$474$	−1.30656	−	2.26303i	−1.30656	−	2.26303i
$475$	0.541196	+	0.541196i	0.541196	+	0.541196i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$481$		0			0
$482$		0			0
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$	1.60021	+	0.923880i	1.60021	+	0.923880i
$487$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$487$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$488$	0.478235	−	1.78480i	0.478235	−	1.78480i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	0.507306	+	0.292893i	0.507306	+	0.292893i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	1.00000	−	1.00000i	1.00000	−	1.00000i
$499$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$499$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$500$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$501$		0			0
$502$	1.78480	+	0.478235i	1.78480	+	0.478235i
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$506$		0			0
$507$	−0.198092	−	0.739288i	−0.198092	−	0.739288i
$508$	0.366025	+	1.36603i	0.366025	+	1.36603i
$509$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$509$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$510$		0			0
$511$		0			0
$512$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$513$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			3.41421i			3.41421i
$520$	0.0999004	+	0.758819i	0.0999004	+	0.758819i
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	0.130526	+	0.991445i	$0.458333\pi$
$523$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	−0.608761	+	0.793353i	$0.708333\pi$
$524$	−0.765367			−0.765367
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	+	0.500000i	0.866025	+	0.500000i
$530$		0			0
$531$			4.46088i			4.46088i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$539$		0			0
$540$	−1.00000	−	2.41421i	−1.00000	−	2.41421i
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	1.36603	+	0.366025i	1.36603	+	0.366025i
$549$	2.23044	−	3.86324i	2.23044	−	3.86324i
$550$		0			0
$551$		0			0
$552$	−1.84776	+	1.84776i	−1.84776	+	1.84776i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.60021	−	0.923880i	−1.60021	−	0.923880i
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.991445	−	0.130526i	$-0.958333\pi$
$563$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.793353	+	0.608761i	$0.791667\pi$
$564$		0			0
$565$		0			0
$566$			0.765367i			0.765367i
$567$		0			0
$568$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$569$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$569$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$570$	0.860919	+	1.12197i	0.860919	+	1.12197i
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−1.84776	−	1.84776i	−1.84776	−	1.84776i
$574$		0			0
$575$	1.41421			1.41421
$576$	−1.20711	−	2.09077i	−1.20711	−	2.09077i
$577$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$577$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$578$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$579$	1.84776	+	3.20041i	1.84776	+	3.20041i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.241181	+	1.83195i	−0.241181	+	1.83195i
$586$	−1.60021	+	0.923880i	−1.60021	+	0.923880i
$587$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$587$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$588$		0			0
$589$		0			0
$590$	1.12484	−	1.46593i	1.12484	−	1.46593i
$591$		0			0
$592$		0			0
$593$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$593$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.04551	−	0.280144i	1.04551	−	0.280144i
$599$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$599$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$600$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.793353	−	0.608761i	0.793353	−	0.608761i
$606$	−1.70711	+	2.95680i	−1.70711	+	2.95680i
$607$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$607$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$608$	0.541196	+	0.541196i	0.541196	+	0.541196i
$609$		0			0
$610$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$611$		0			0
$612$		0			0
$613$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$613$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$614$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$619$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$620$		0			0
$621$	−3.20041	+	1.84776i	−3.20041	+	1.84776i
$622$		0			0
$623$		0			0
$624$			1.41421i			1.41421i
$625$	0.866025	−	0.500000i	0.866025	−	0.500000i
$626$		0			0
$627$		0			0
$628$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$629$		0			0
$630$		0			0
$631$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$631$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$632$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$633$		0			0
$634$		0			0
$635$	0.860919	−	1.12197i	0.860919	−	1.12197i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.95680	+	1.70711i	−2.95680	+	1.70711i
$640$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$641$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$641$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$642$		0			0
$643$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$643$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$647$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$648$	−0.624844	−	2.33195i	−0.624844	−	2.33195i
$649$		0			0
$650$	0.541196	−	0.541196i	0.541196	−	0.541196i
$651$		0			0
$652$		0			0
$653$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$653$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$654$		0			0
$655$	0.465926	+	0.607206i	0.465926	+	0.607206i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	0.662827	+	0.382683i	0.662827	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$661$	0.662827	+	0.382683i	0.662827	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$662$		0			0
$663$		0			0
$664$	−0.765367			−0.765367
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$674$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$675$	−1.30656	+	2.26303i	−1.30656	+	2.26303i
$676$	−0.207107	+	0.358719i	−0.207107	+	0.358719i
$677$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$677$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.991445	+	0.130526i	$0.958333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.707107	+	1.22474i	0.707107	+	1.22474i
$682$		0			0
$683$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$683$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$684$	0.923880	+	1.60021i	0.923880	+	1.60021i
$685$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$686$		0			0
$687$	−2.41421	−	2.41421i	−2.41421	−	2.41421i
$688$		0			0
$689$		0			0
$690$	2.59077	+	0.341081i	2.59077	+	0.341081i
$691$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	−0.793353	−	0.608761i	$-0.791667\pi$
$691$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	0.130526	+	0.991445i	$0.458333\pi$
$692$	1.30656	−	1.30656i	1.30656	−	1.30656i
$693$		0			0
$694$		0			0
$695$	0.241181	+	1.83195i	0.241181	+	1.83195i
$696$		0			0
$697$		0			0
$698$	−0.198092	+	0.739288i	−0.198092	+	0.739288i
$699$	−2.61313			−2.61313
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	2.41421	−	2.41421i	2.41421	−	2.41421i
$709$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$709$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$710$	1.40211	+	0.184592i	1.40211	+	0.184592i
$711$	−1.70711	+	2.95680i	−1.70711	+	2.95680i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.517638	+	1.93185i	0.517638	+	1.93185i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−0.923880	+	2.23044i	−0.923880	+	2.23044i
$721$		0			0
$722$	0.292893	+	0.292893i	0.292893	+	0.292893i
$723$		0			0
$724$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$725$		0			0
$726$	1.60021	−	0.923880i	1.60021	−	0.923880i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	−3.29788	+	0.883663i	−3.29788	+	0.883663i
$733$	0.478235	−	1.78480i	0.478235	−	1.78480i	−0.130526	−	0.991445i	$-0.541667\pi$
$733$	0.478235	−	1.78480i	0.478235	−	1.78480i	0.608761	−	0.793353i	$-0.291667\pi$
$734$		0			0
$735$		0			0
$736$	1.41421			1.41421
$737$		0			0
$738$		0			0
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$		−	1.08239i		−	1.08239i
$742$		0			0
$743$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$743$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.78480	−	0.478235i	−1.78480	−	0.478235i
$748$		0			0
$749$		0			0
$750$	1.70711	−	0.707107i	1.70711	−	0.707107i
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	−0.883663	−	3.29788i	−0.883663	−	3.29788i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$	0.0999004	−	0.758819i	0.0999004	−	0.758819i
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$	1.84776	−	1.84776i	1.84776	−	1.84776i
$763$		0			0
$764$			1.41421i			1.41421i
$765$		0			0
$766$		0			0
$767$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$768$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	0.517638	−	1.93185i	0.517638	−	1.93185i
$773$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.130526	−	0.991445i	$-0.458333\pi$
$773$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.608761	+	0.793353i	$0.291667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	1.12197	−	0.860919i	1.12197	−	0.860919i
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.70711	−	0.707107i	1.70711	−	0.707107i
$786$	0.707107	+	1.22474i	0.707107	+	1.22474i
$787$	0.478235	+	1.78480i	0.478235	+	1.78480i	0.608761	+	0.793353i	$0.291667\pi$
$787$	0.478235	+	1.78480i	0.478235	+	1.78480i	−0.130526	+	0.991445i	$0.541667\pi$
$788$		0			0
$789$		0			0
$790$	1.30656	−	0.541196i	1.30656	−	0.541196i
$791$		0			0
$792$		0			0
$793$	1.36603	+	0.366025i	1.36603	+	0.366025i
$794$	0.923880	−	1.60021i	0.923880	−	1.60021i
$795$		0			0
$796$		0			0
$797$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$797$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	−	0.500000i	0.866025	−	0.500000i
$801$		0			0
$802$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$808$	1.78480	−	0.478235i	1.78480	−	0.478235i
$809$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$809$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$810$	−1.46968	+	1.91532i	−1.46968	+	1.91532i
$811$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$811$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$812$		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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−0.676327	−	2.52409i	−0.676327	−	2.52409i
$823$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$823$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	3.29788	+	0.883663i	3.29788	+	0.883663i
$829$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$829$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$830$	0.465926	+	0.607206i	0.465926	+	0.607206i
$831$		0			0
$832$	0.541196	−	0.541196i	0.541196	−	0.541196i
$833$		0			0
$834$			3.41421i			3.41421i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.198092	+	0.739288i	−0.198092	+	0.739288i
$839$		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
$845$	0.410670	−	0.0540657i	0.410670	−	0.0540657i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.22474	−	0.707107i	1.22474	−	0.707107i
$850$		0			0
$851$		0			0
$852$	2.52409	+	0.676327i	2.52409	+	0.676327i
$853$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$853$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$854$		0			0
$855$	0.707107	−	1.70711i	0.707107	−	1.70711i
$856$		0			0
$857$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$857$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$858$		0			0
$859$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$859$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$860$		0			0
$861$		0			0
$862$	1.41421	+	1.41421i	1.41421	+	1.41421i
$863$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$863$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$864$	−1.30656	+	2.26303i	−1.30656	+	2.26303i
$865$	−1.83195	−	0.241181i	−1.83195	−	0.241181i
$866$		0			0
$867$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1.08239			−1.08239
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$		0			0
$879$	2.95680	+	1.70711i	2.95680	+	1.70711i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$	−3.38500	−	0.445644i	−3.38500	−	0.445644i
$886$		0			0
$887$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$887$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$898$		0			0
$899$		0			0
$900$	2.33195	−	0.624844i	2.33195	−	0.624844i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.758819	−	0.0999004i	0.758819	−	0.0999004i
$906$		0			0
$907$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$907$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$908$	0.198092	−	0.739288i	0.198092	−	0.739288i
$909$	4.46088			4.46088
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	0.366025	−	1.36603i	0.366025	−	1.36603i
$913$		0			0
$914$		0			0
$915$	2.70868	+	2.07844i	2.70868	+	2.07844i
$916$			1.84776i			1.84776i
$917$		0			0
$918$		0			0
$919$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$919$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$920$	−0.860919	−	1.12197i	−0.860919	−	1.12197i
$921$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$922$	−0.739288	−	0.198092i	−0.739288	−	0.198092i
$923$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$924$		0			0
$925$		0			0
$926$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	1.00000	+	1.00000i	1.00000	+	1.00000i
$933$		0			0
$934$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$935$		0			0
$936$	1.60021	−	0.923880i	1.60021	−	0.923880i
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.662827	+	0.382683i	0.662827	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$941$	0.662827	+	0.382683i	0.662827	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$942$	3.29788	−	0.883663i	3.29788	−	0.883663i
$943$		0			0
$944$	−1.84776			−1.84776
$945$		0			0
$946$		0			0
$947$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$947$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$948$	2.52409	−	0.676327i	2.52409	−	0.676327i
$949$		0			0
$950$	−0.662827	+	0.382683i	−0.662827	+	0.382683i
$951$		0			0
$952$		0			0
$953$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$953$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$954$		0			0
$955$	1.12197	−	0.860919i	1.12197	−	0.860919i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.70711	−	0.707107i	1.70711	−	0.707107i
$961$	0.500000	+	0.866025i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$966$		0			0
$967$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$967$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$968$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$969$		0			0
$970$		0			0
$971$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.608761	−	0.793353i	$-0.291667\pi$
$971$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.991445	−	0.130526i	$-0.0416667\pi$
$972$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$973$		0			0
$974$			1.41421i			1.41421i
$975$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$976$	1.60021	+	0.923880i	1.60021	+	0.923880i
$977$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$983$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$989$		0			0
$990$		0			0
$991$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$991$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	0.707107	+	1.22474i	0.707107	+	1.22474i
$997$	0.198092	+	0.739288i	0.198092	+	0.739288i	0.991445	+	0.130526i	$0.0416667\pi$
$997$	0.198092	+	0.739288i	0.198092	+	0.739288i	−0.793353	+	0.608761i	$0.791667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.1.bu.a.557.2		16
5.3	odd	4	inner	1960.1.bu.a.1733.4		16
7.2	even	3	inner	1960.1.bu.a.1157.3		16
7.3	odd	6		1960.1.u.a.197.2	yes	8
7.4	even	3		1960.1.u.a.197.1	✓	8
7.5	odd	6	inner	1960.1.bu.a.1157.4		16
7.6	odd	2	inner	1960.1.bu.a.557.1		16
8.5	even	2	inner	1960.1.bu.a.557.1		16
35.3	even	12		1960.1.u.a.1373.1	yes	8
35.13	even	4	inner	1960.1.bu.a.1733.3		16
35.18	odd	12		1960.1.u.a.1373.2	yes	8
35.23	odd	12	inner	1960.1.bu.a.373.1		16
35.33	even	12	inner	1960.1.bu.a.373.2		16
40.13	odd	4	inner	1960.1.bu.a.1733.3		16
56.5	odd	6	inner	1960.1.bu.a.1157.3		16
56.13	odd	2	CM	1960.1.bu.a.557.2		16
56.37	even	6	inner	1960.1.bu.a.1157.4		16
56.45	odd	6		1960.1.u.a.197.1	✓	8
56.53	even	6		1960.1.u.a.197.2	yes	8
280.13	even	4	inner	1960.1.bu.a.1733.4		16
280.53	odd	12		1960.1.u.a.1373.1	yes	8
280.93	odd	12	inner	1960.1.bu.a.373.2		16
280.173	even	12	inner	1960.1.bu.a.373.1		16
280.213	even	12		1960.1.u.a.1373.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.1.u.a.197.1	✓	8	7.4	even	3
1960.1.u.a.197.1	✓	8	56.45	odd	6
1960.1.u.a.197.2	yes	8	7.3	odd	6
1960.1.u.a.197.2	yes	8	56.53	even	6
1960.1.u.a.1373.1	yes	8	35.3	even	12
1960.1.u.a.1373.1	yes	8	280.53	odd	12
1960.1.u.a.1373.2	yes	8	35.18	odd	12
1960.1.u.a.1373.2	yes	8	280.213	even	12
1960.1.bu.a.373.1		16	35.23	odd	12	inner
1960.1.bu.a.373.1		16	280.173	even	12	inner
1960.1.bu.a.373.2		16	35.33	even	12	inner
1960.1.bu.a.373.2		16	280.93	odd	12	inner
1960.1.bu.a.557.1		16	7.6	odd	2	inner
1960.1.bu.a.557.1		16	8.5	even	2	inner
1960.1.bu.a.557.2		16	1.1	even	1	trivial
1960.1.bu.a.557.2		16	56.13	odd	2	CM
1960.1.bu.a.1157.3		16	7.2	even	3	inner
1960.1.bu.a.1157.3		16	56.5	odd	6	inner
1960.1.bu.a.1157.4		16	7.5	odd	6	inner
1960.1.bu.a.1157.4		16	56.37	even	6	inner
1960.1.bu.a.1733.3		16	35.13	even	4	inner
1960.1.bu.a.1733.3		16	40.13	odd	4	inner
1960.1.bu.a.1733.4		16	5.3	odd	4	inner
1960.1.bu.a.1733.4		16	280.13	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1960.bu (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.258819 + 0.965926i) q^{2} +(1.78480 - 0.478235i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(0.130526 + 0.991445i) q^{5} +1.84776i q^{6} +(0.707107 - 0.707107i) q^{8} +(2.09077 - 1.20711i) q^{9} +O(q^{10})\)	\(q+(-0.258819 + 0.965926i) q^{2} +(1.78480 - 0.478235i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(0.130526 + 0.991445i) q^{5} +1.84776i q^{6} +(0.707107 - 0.707107i) q^{8} +(2.09077 - 1.20711i) q^{9} +(-0.991445 - 0.130526i) q^{10} +(-1.78480 - 0.478235i) q^{12} +(0.541196 + 0.541196i) q^{13} +(0.707107 + 1.70711i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.624844 + 2.33195i) q^{18} +(-0.382683 - 0.662827i) q^{19} +(0.382683 - 0.923880i) q^{20} +(-1.36603 - 0.366025i) q^{23} +(0.923880 - 1.60021i) q^{24} +(-0.965926 + 0.258819i) q^{25} +(-0.662827 + 0.382683i) q^{26} +(1.84776 - 1.84776i) q^{27} +(-1.83195 + 0.241181i) q^{30} +(-0.965926 + 0.258819i) q^{32} -2.41421 q^{36} +(0.739288 - 0.198092i) q^{38} +(1.22474 + 0.707107i) q^{39} +(0.793353 + 0.608761i) q^{40} +(1.46968 + 1.91532i) q^{45} +(0.707107 - 1.22474i) q^{46} +(1.30656 + 1.30656i) q^{48} -1.00000i q^{50} +(-0.198092 - 0.739288i) q^{52} +(1.30656 + 2.26303i) q^{54} +(-1.00000 - 1.00000i) q^{57} +(-0.923880 + 1.60021i) q^{59} +(0.241181 - 1.83195i) q^{60} +(1.60021 - 0.923880i) q^{61} -1.00000i q^{64} +(-0.465926 + 0.607206i) q^{65} -2.61313 q^{69} -1.41421 q^{71} +(0.624844 - 2.33195i) q^{72} +(-1.60021 + 0.923880i) q^{75} +0.765367i q^{76} +(-1.00000 + 1.00000i) q^{78} +(-1.22474 + 0.707107i) q^{79} +(-0.793353 + 0.608761i) q^{80} +(1.20711 - 2.09077i) q^{81} +(-0.541196 - 0.541196i) q^{83} +(-2.23044 + 0.923880i) q^{90} +(1.00000 + 1.00000i) q^{92} +(0.607206 - 0.465926i) q^{95} +(-1.60021 + 0.923880i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 8 q^{16} - 8 q^{18} - 8 q^{23} - 16 q^{36} - 16 q^{57} + 8 q^{60} + 8 q^{65} - 8 q^{72} - 16 q^{78} + 8 q^{81} + 16 q^{92}+O(q^{100}) \) 16 * q + 8 * q^16 - 8 * q^18 - 8 * q^23 - 16 * q^36 - 16 * q^57 + 8 * q^60 + 8 * q^65 - 8 * q^72 - 16 * q^78 + 8 * q^81 + 16 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more