LMFDB - Embedded newform 3380.1.cg.a.3167.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(47,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(52))

chi = DirichletCharacter(H, H._module([26, 13, 21]))

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

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

chi := DirichletCharacter("3380.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.cg (of order $52$, degree $24$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.68683974270$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{52})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{52}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{52} + \cdots)$

Embedding invariants

Embedding label			3167.1
Root			$0.822984 + 0.568065i$ of defining polynomial
Character	$\chi$	$=$	3380.3167
Dual form			3380.1.cg.a.603.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.239316 + 0.970942i) q^{2} +(-0.885456 + 0.464723i) q^{4} +(-0.748511 - 0.663123i) q^{5} +(-0.663123 - 0.748511i) q^{8} +(0.239316 + 0.970942i) q^{9} +O(q^{10})$	\(q+(0.239316 + 0.970942i) q^{2} +(-0.885456 + 0.464723i) q^{4} +(-0.748511 - 0.663123i) q^{5} +(-0.663123 - 0.748511i) q^{8} +(0.239316 + 0.970942i) q^{9} +(0.464723 - 0.885456i) q^{10} +(0.748511 + 0.663123i) q^{13} +(0.568065 - 0.822984i) q^{16} +(0.0359256 - 0.593921i) q^{17} +(-0.885456 + 0.464723i) q^{18} +(0.970942 + 0.239316i) q^{20} +(0.120537 + 0.992709i) q^{25} +(-0.464723 + 0.885456i) q^{26} +(-0.0576926 - 0.234068i) q^{29} +(0.935016 + 0.354605i) q^{32} +(0.585260 - 0.107253i) q^{34} +(-0.663123 - 0.748511i) q^{36} +(0.583668 + 1.53901i) q^{37} +1.00000i q^{40} +(-0.638104 + 0.814480i) q^{41} +(0.464723 - 0.885456i) q^{45} +(0.120537 - 0.992709i) q^{49} +(-0.935016 + 0.354605i) q^{50} +(-0.970942 - 0.239316i) q^{52} +(-0.120537 + 1.99271i) q^{53} +(0.213460 - 0.112032i) q^{58} +(1.48611 + 1.31658i) q^{61} +(-0.120537 + 0.992709i) q^{64} +(-0.120537 - 0.992709i) q^{65} +(0.244198 + 0.542586i) q^{68} +(0.568065 - 0.822984i) q^{72} +(-0.478631 + 1.94188i) q^{73} +(-1.35460 + 0.935016i) q^{74} +(-0.970942 + 0.239316i) q^{80} +(-0.885456 + 0.464723i) q^{81} +(-0.943521 - 0.424644i) q^{82} +(-0.420733 + 0.420733i) q^{85} +(-1.39105 - 1.39105i) q^{89} +(0.970942 + 0.239316i) q^{90} +(-1.53901 - 1.06230i) q^{97} +(0.992709 - 0.120537i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{4} - 2 q^{5}+O(q^{10}) $ 24 * q + 2 * q^4 - 2 * q^5	$ 24 q + 2 q^{4} - 2 q^{5} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 2 q^{18} + 2 q^{20} - 2 q^{25} - 2 q^{34} + 2 q^{41} - 2 q^{49} - 2 q^{52} + 2 q^{53} - 4 q^{58} + 2 q^{64} + 2 q^{65} - 2 q^{68} - 2 q^{72} - 26 q^{74} - 2 q^{80} + 2 q^{81} + 2 q^{82} + 2 q^{85} + 2 q^{89} + 2 q^{90}+O(q^{100}) $ 24 * q + 2 * q^4 - 2 * q^5 + 2 * q^13 - 2 * q^16 + 2 * q^17 + 2 * q^18 + 2 * q^20 - 2 * q^25 - 2 * q^34 + 2 * q^41 - 2 * q^49 - 2 * q^52 + 2 * q^53 - 4 * q^58 + 2 * q^64 + 2 * q^65 - 2 * q^68 - 2 * q^72 - 26 * q^74 - 2 * q^80 + 2 * q^81 + 2 * q^82 + 2 * q^85 + 2 * q^89 + 2 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{9}{52}\right)$

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 $

$2$	0.239316	+	0.970942i	0.239316	+	0.970942i
$2$	0.239316	+	0.970942i	0.239316	+	0.970942i
$3$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$3$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$4$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$5$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$5$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$6$		0			0
$7$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$7$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$8$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$9$	0.239316	+	0.970942i	0.239316	+	0.970942i
$10$	0.464723	−	0.885456i	0.464723	−	0.885456i
$11$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$11$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$12$		0			0
$13$	0.748511	+	0.663123i	0.748511	+	0.663123i
$13$	0.748511	+	0.663123i	0.748511	+	0.663123i
$14$		0			0
$15$		0			0
$16$	0.568065	−	0.822984i	0.568065	−	0.822984i
$17$	0.0359256	−	0.593921i	0.0359256	−	0.593921i	−0.935016	−	0.354605i	$-0.884615\pi$
$17$	0.0359256	−	0.593921i	0.0359256	−	0.593921i	0.970942	−	0.239316i	$-0.0769231\pi$
$18$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	0.970942	+	0.239316i	0.970942	+	0.239316i
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	0.120537	+	0.992709i	0.120537	+	0.992709i
$26$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$27$		0			0
$28$		0			0
$29$	−0.0576926	−	0.234068i	−0.0576926	−	0.234068i	0.935016	−	0.354605i	$-0.115385\pi$
$29$	−0.0576926	−	0.234068i	−0.0576926	−	0.234068i	−0.992709	+	0.120537i	$0.961538\pi$
$30$		0			0
$31$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$31$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$32$	0.935016	+	0.354605i	0.935016	+	0.354605i
$33$		0			0
$34$	0.585260	−	0.107253i	0.585260	−	0.107253i
$35$		0			0
$36$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$37$	0.583668	+	1.53901i	0.583668	+	1.53901i	0.822984	+	0.568065i	$0.192308\pi$
$37$	0.583668	+	1.53901i	0.583668	+	1.53901i	−0.239316	+	0.970942i	$0.576923\pi$
$38$		0			0
$39$		0			0
$40$			1.00000i			1.00000i
$41$	−0.638104	+	0.814480i	−0.638104	+	0.814480i	−0.992709	−	0.120537i	$-0.961538\pi$
$41$	−0.638104	+	0.814480i	−0.638104	+	0.814480i	0.354605	+	0.935016i	$0.384615\pi$
$42$		0			0
$43$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$43$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$44$		0			0
$45$	0.464723	−	0.885456i	0.464723	−	0.885456i
$46$		0			0
$47$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$47$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$48$		0			0
$49$	0.120537	−	0.992709i	0.120537	−	0.992709i
$50$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$51$		0			0
$52$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$53$	−0.120537	+	1.99271i	−0.120537	+	1.99271i			1.00000i	$0.5\pi$
$53$	−0.120537	+	1.99271i	−0.120537	+	1.99271i	−0.120537	+	0.992709i	$0.538462\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.213460	−	0.112032i	0.213460	−	0.112032i
$59$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$59$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$60$		0			0
$61$	1.48611	+	1.31658i	1.48611	+	1.31658i	0.822984	+	0.568065i	$0.192308\pi$
$61$	1.48611	+	1.31658i	1.48611	+	1.31658i	0.663123	+	0.748511i	$0.269231\pi$
$62$		0			0
$63$		0			0
$64$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$65$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$66$		0			0
$67$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$67$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$68$	0.244198	+	0.542586i	0.244198	+	0.542586i
$69$		0			0
$70$		0			0
$71$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$71$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$72$	0.568065	−	0.822984i	0.568065	−	0.822984i
$73$	−0.478631	+	1.94188i	−0.478631	+	1.94188i	−0.239316	+	0.970942i	$0.576923\pi$
$73$	−0.478631	+	1.94188i	−0.478631	+	1.94188i	−0.239316	+	0.970942i	$0.576923\pi$
$74$	−1.35460	+	0.935016i	−1.35460	+	0.935016i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$79$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$80$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$81$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$82$	−0.943521	−	0.424644i	−0.943521	−	0.424644i
$83$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$83$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$84$		0			0
$85$	−0.420733	+	0.420733i	−0.420733	+	0.420733i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.39105	−	1.39105i	−1.39105	−	1.39105i	−0.822984	−	0.568065i	$-0.807692\pi$
$89$	−1.39105	−	1.39105i	−1.39105	−	1.39105i	−0.568065	−	0.822984i	$-0.692308\pi$
$90$	0.970942	+	0.239316i	0.970942	+	0.239316i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.53901	−	1.06230i	−1.53901	−	1.06230i	−0.970942	−	0.239316i	$-0.923077\pi$
$97$	−1.53901	−	1.06230i	−1.53901	−	1.06230i	−0.568065	−	0.822984i	$-0.692308\pi$
$98$	0.992709	−	0.120537i	0.992709	−	0.120537i
$99$		0			0
$100$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$101$	1.74851	+	0.663123i	1.74851	+	0.663123i	1.00000			$0$
$101$	1.74851	+	0.663123i	1.74851	+	0.663123i	0.748511	+	0.663123i	$0.230769\pi$
$102$		0			0
$103$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$103$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$104$		−	1.00000i		−	1.00000i
$105$		0			0
$106$	−1.96365	+	0.359852i	−1.96365	+	0.359852i
$107$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$107$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$108$		0			0
$109$	0.783659	+	1.74122i	0.783659	+	1.74122i	0.663123	+	0.748511i	$0.269231\pi$
$109$	0.783659	+	1.74122i	0.783659	+	1.74122i	0.120537	+	0.992709i	$0.461538\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.222333	+	0.283788i	0.222333	+	0.283788i	0.885456	−	0.464723i	$-0.153846\pi$
$113$	0.222333	+	0.283788i	0.222333	+	0.283788i	−0.663123	+	0.748511i	$0.730769\pi$
$114$		0			0
$115$		0			0
$116$	0.159861	+	0.180446i	0.159861	+	0.180446i
$117$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.464723	−	0.885456i	0.464723	−	0.885456i
$122$	−0.922670	+	1.75800i	−0.922670	+	1.75800i
$123$		0			0
$124$		0			0
$125$	0.568065	−	0.822984i	0.568065	−	0.822984i
$126$		0			0
$127$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$127$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$128$	−0.992709	+	0.120537i	−0.992709	+	0.120537i
$129$		0			0
$130$	0.935016	−	0.354605i	0.935016	−	0.354605i
$131$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$131$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.468379	+	0.366951i	−0.468379	+	0.366951i
$137$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	0.568065	+	0.822984i	$0.307692\pi$
$137$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	−0.970942	+	0.239316i	$0.923077\pi$
$138$		0			0
$139$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$139$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.935016	+	0.354605i	0.935016	+	0.354605i
$145$	−0.112032	+	0.213460i	−0.112032	+	0.213460i
$146$	−2.00000			−2.00000
$147$		0			0
$148$	−1.23202	−	1.09148i	−1.23202	−	1.09148i
$149$	1.74122	−	0.542586i	1.74122	−	0.542586i	0.748511	−	0.663123i	$-0.230769\pi$
$149$	1.74122	−	0.542586i	1.74122	−	0.542586i	0.992709	+	0.120537i	$0.0384615\pi$
$150$		0			0
$151$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$151$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$152$		0			0
$153$	0.585260	−	0.107253i	0.585260	−	0.107253i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.147958	−	0.807380i	0.147958	−	0.807380i	−0.822984	−	0.568065i	$-0.807692\pi$
$157$	0.147958	−	0.807380i	0.147958	−	0.807380i	0.970942	−	0.239316i	$-0.0769231\pi$
$158$		0			0
$159$		0			0
$160$	−0.464723	−	0.885456i	−0.464723	−	0.885456i
$161$		0			0
$162$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$163$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$163$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$164$	0.186505	−	1.01773i	0.186505	−	1.01773i
$165$		0			0
$166$		0			0
$167$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$167$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$168$		0			0
$169$	0.120537	+	0.992709i	0.120537	+	0.992709i
$170$	−0.509195	−	0.307819i	−0.509195	−	0.307819i
$171$		0			0
$172$		0			0
$173$	−1.17759	−	0.366951i	−1.17759	−	0.366951i	−0.354605	−	0.935016i	$-0.615385\pi$
$173$	−1.17759	−	0.366951i	−1.17759	−	0.366951i	−0.822984	+	0.568065i	$0.807692\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.01773	−	1.68353i	1.01773	−	1.68353i
$179$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$179$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$180$			1.00000i			1.00000i
$181$	1.23202	−	0.850405i	1.23202	−	0.850405i	0.239316	−	0.970942i	$-0.423077\pi$
$181$	1.23202	−	0.850405i	1.23202	−	0.850405i	0.992709	+	0.120537i	$0.0384615\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.583668	−	1.53901i	0.583668	−	1.53901i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	1.24006	−	1.39974i	1.24006	−	1.39974i	0.354605	−	0.935016i	$-0.384615\pi$
$193$	1.24006	−	1.39974i	1.24006	−	1.39974i	0.885456	−	0.464723i	$-0.153846\pi$
$194$	0.663123	−	1.74851i	0.663123	−	1.74851i
$195$		0			0
$196$	0.354605	+	0.935016i	0.354605	+	0.935016i
$197$	−1.85640	−	0.704039i	−1.85640	−	0.704039i	−0.970942	−	0.239316i	$-0.923077\pi$
$197$	−1.85640	−	0.704039i	−1.85640	−	0.704039i	−0.885456	−	0.464723i	$-0.846154\pi$
$198$		0			0
$199$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$199$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$200$	0.663123	−	0.748511i	0.663123	−	0.748511i
$201$		0			0
$202$	−0.225408	+	1.85640i	−0.225408	+	1.85640i
$203$		0			0
$204$		0			0
$205$	1.01773	−	0.186505i	1.01773	−	0.186505i
$206$		0			0
$207$		0			0
$208$	0.970942	−	0.239316i	0.970942	−	0.239316i
$209$		0			0
$210$		0			0
$211$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$211$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$212$	−0.819328	−	1.82047i	−0.819328	−	1.82047i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.50308	+	1.17759i	−1.50308	+	1.17759i
$219$		0			0
$220$		0			0
$221$	0.420733	−	0.420733i	0.420733	−	0.420733i
$222$		0			0
$223$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$223$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$224$		0			0
$225$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$226$	−0.222333	+	0.283788i	−0.222333	+	0.283788i
$227$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$227$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$228$		0			0
$229$	−0.110118	+	0.0495602i	−0.110118	+	0.0495602i	−0.464723	−	0.885456i	$-0.653846\pi$
$229$	−0.110118	+	0.0495602i	−0.110118	+	0.0495602i	0.354605	+	0.935016i	$0.384615\pi$
$230$		0			0
$231$		0			0
$232$	−0.136945	+	0.198399i	−0.136945	+	0.198399i
$233$	0.646140	−	0.506219i	0.646140	−	0.506219i	−0.239316	−	0.970942i	$-0.576923\pi$
$233$	0.646140	−	0.506219i	0.646140	−	0.506219i	0.885456	+	0.464723i	$0.153846\pi$
$234$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−0.222333	−	1.21323i	−0.222333	−	1.21323i	−0.885456	−	0.464723i	$-0.846154\pi$
$241$	−0.222333	−	1.21323i	−0.222333	−	1.21323i	0.663123	−	0.748511i	$-0.269231\pi$
$242$	0.970942	+	0.239316i	0.970942	+	0.239316i
$243$		0			0
$244$	−1.92773	−	0.475142i	−1.92773	−	0.475142i
$245$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.935016	+	0.354605i	0.935016	+	0.354605i
$251$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$251$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$257$	0.593921	−	1.90596i	0.593921	−	1.90596i	0.239316	−	0.970942i	$-0.423077\pi$
$257$	0.593921	−	1.90596i	0.593921	−	1.90596i	0.354605	−	0.935016i	$-0.384615\pi$
$258$		0			0
$259$		0			0
$260$	0.568065	+	0.822984i	0.568065	+	0.822984i
$261$	0.213460	−	0.112032i	0.213460	−	0.112032i
$262$		0			0
$263$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$263$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$264$		0			0
$265$	1.41163	−	1.41163i	1.41163	−	1.41163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.63397	−	0.198399i	1.63397	−	0.198399i	0.748511	−	0.663123i	$-0.230769\pi$
$269$	1.63397	−	0.198399i	1.63397	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$270$		0			0
$271$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$271$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$272$	−0.468379	−	0.366951i	−0.468379	−	0.366951i
$273$		0			0
$274$	−1.12785	−	0.136945i	−1.12785	−	0.136945i
$275$		0			0
$276$		0			0
$277$	−1.41163	+	0.0853881i	−1.41163	+	0.0853881i	−0.748511	−	0.663123i	$-0.769231\pi$
$277$	−1.41163	+	0.0853881i	−1.41163	+	0.0853881i	−0.663123	+	0.748511i	$0.730769\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.0950579	+	0.0744731i	0.0950579	+	0.0744731i	0.663123	−	0.748511i	$-0.269231\pi$
$281$	0.0950579	+	0.0744731i	0.0950579	+	0.0744731i	−0.568065	+	0.822984i	$0.692308\pi$
$282$		0			0
$283$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$283$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$289$	0.641258	+	0.0778628i	0.641258	+	0.0778628i
$290$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i
$291$		0			0
$292$	−0.478631	−	1.94188i	−0.478631	−	1.94188i
$293$	0.470293	−	0.530851i	0.470293	−	0.530851i	−0.464723	−	0.885456i	$-0.653846\pi$
$293$	0.470293	−	0.530851i	0.470293	−	0.530851i	0.935016	+	0.354605i	$0.115385\pi$
$294$		0			0
$295$		0			0
$296$	0.764919	−	1.45743i	0.764919	−	1.45743i
$297$		0			0
$298$	0.943521	+	1.56077i	0.943521	+	1.56077i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.239316	−	1.97094i	−0.239316	−	1.97094i
$306$	0.244198	+	0.542586i	0.244198	+	0.542586i
$307$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$307$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$311$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$312$		0			0
$313$	0.424644	+	0.943521i	0.424644	+	0.943521i	0.992709	+	0.120537i	$0.0384615\pi$
$313$	0.424644	+	0.943521i	0.424644	+	0.943521i	−0.568065	+	0.822984i	$0.692308\pi$
$314$	0.819328	−	0.0495602i	0.819328	−	0.0495602i
$315$		0			0
$316$		0			0
$317$	0.992709	−	1.12054i	0.992709	−	1.12054i		−	1.00000i	$-0.5\pi$
$317$	0.992709	−	1.12054i	0.992709	−	1.12054i	0.992709	−	0.120537i	$-0.0384615\pi$
$318$		0			0
$319$		0			0
$320$	0.748511	−	0.663123i	0.748511	−	0.663123i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.568065	−	0.822984i	0.568065	−	0.822984i
$325$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$326$		0			0
$327$		0			0
$328$	1.03279	−	0.0624722i	1.03279	−	0.0624722i
$329$		0			0
$330$		0			0
$331$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$331$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$332$		0			0
$333$	−1.35460	+	0.935016i	−1.35460	+	0.935016i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.28962	+	1.28962i	−1.28962	+	1.28962i	−0.354605	+	0.935016i	$0.615385\pi$
$337$	−1.28962	+	1.28962i	−1.28962	+	1.28962i	−0.935016	+	0.354605i	$0.884615\pi$
$338$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$339$		0			0
$340$	0.177016	−	0.568065i	0.177016	−	0.568065i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.0744731	−	1.23119i	0.0744731	−	1.23119i
$347$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$347$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$348$		0			0
$349$	0.186505	+	0.308518i	0.186505	+	0.308518i	0.935016	−	0.354605i	$-0.115385\pi$
$349$	0.186505	+	0.308518i	0.186505	+	0.308518i	−0.748511	+	0.663123i	$0.769231\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.992709	+	0.879463i	−0.992709	+	0.879463i	−0.992709	−	0.120537i	$-0.961538\pi$
$353$	−0.992709	+	0.879463i	−0.992709	+	0.879463i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	1.87816	+	0.585260i	1.87816	+	0.585260i
$357$		0			0
$358$		0			0
$359$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$359$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$360$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$361$		−	1.00000i		−	1.00000i
$362$	1.12054	+	0.992709i	1.12054	+	0.992709i
$363$		0			0
$364$		0			0
$365$	1.64597	−	1.13613i	1.64597	−	1.13613i
$366$		0			0
$367$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$367$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$368$		0			0
$369$	−0.943521	−	0.424644i	−0.943521	−	0.424644i
$370$	1.63397	+	0.198399i	1.63397	+	0.198399i
$371$		0			0
$372$		0			0
$373$	−1.46472	+	0.885456i	−1.46472	+	0.885456i	−0.464723	+	0.885456i	$0.653846\pi$
$373$	−1.46472	+	0.885456i	−1.46472	+	0.885456i	−1.00000			$1.00000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.112032	−	0.213460i	0.112032	−	0.213460i
$378$		0			0
$379$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$379$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$383$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$384$		0			0
$385$		0			0
$386$	1.65583	+	0.869047i	1.65583	+	0.869047i
$387$		0			0
$388$	1.85640	+	0.225408i	1.85640	+	0.225408i
$389$	0.850405	+	0.753393i	0.850405	+	0.753393i	0.970942	−	0.239316i	$-0.0769231\pi$
$389$	0.850405	+	0.753393i	0.850405	+	0.753393i	−0.120537	+	0.992709i	$0.538462\pi$
$390$		0			0
$391$		0			0
$392$	−0.822984	+	0.568065i	−0.822984	+	0.568065i
$393$		0			0
$394$	0.239316	−	1.97094i	0.239316	−	1.97094i
$395$		0			0
$396$		0			0
$397$	0.271894	+	0.393906i	0.271894	+	0.393906i	0.935016	−	0.354605i	$-0.115385\pi$
$397$	0.271894	+	0.393906i	0.271894	+	0.393906i	−0.663123	+	0.748511i	$0.730769\pi$
$398$		0			0
$399$		0			0
$400$	0.885456	+	0.464723i	0.885456	+	0.464723i
$401$	0.147958	+	0.807380i	0.147958	+	0.807380i	0.970942	+	0.239316i	$0.0769231\pi$
$401$	0.147958	+	0.807380i	0.147958	+	0.807380i	−0.822984	+	0.568065i	$0.807692\pi$
$402$		0			0
$403$		0			0
$404$	−1.85640	+	0.225408i	−1.85640	+	0.225408i
$405$	0.970942	+	0.239316i	0.970942	+	0.239316i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.11325	−	0.872172i	−1.11325	−	0.872172i	−0.120537	−	0.992709i	$-0.538462\pi$
$409$	−1.11325	−	0.872172i	−1.11325	−	0.872172i	−0.992709	+	0.120537i	$0.961538\pi$
$410$	0.424644	+	0.943521i	0.424644	+	0.943521i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.464723	+	0.885456i	0.464723	+	0.885456i
$417$		0			0
$418$		0			0
$419$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$419$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$420$		0			0
$421$	1.23932	−	0.970942i	1.23932	−	0.970942i	0.239316	−	0.970942i	$-0.423077\pi$
$421$	1.23932	−	0.970942i	1.23932	−	0.970942i	1.00000			$0$
$422$		0			0
$423$		0			0
$424$	1.57149	−	1.23119i	1.57149	−	1.23119i
$425$	0.593921	−	0.0359256i	0.593921	−	0.0359256i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$431$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$432$		0			0
$433$	0.0649838	−	0.354605i	0.0649838	−	0.354605i	−0.935016	−	0.354605i	$-0.884615\pi$
$433$	0.0649838	−	0.354605i	0.0649838	−	0.354605i	1.00000			$0$
$434$		0			0
$435$		0			0
$436$	−1.50308	−	1.17759i	−1.50308	−	1.17759i
$437$		0			0
$438$		0			0
$439$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$439$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$440$		0			0
$441$	0.992709	−	0.120537i	0.992709	−	0.120537i
$442$	0.509195	+	0.307819i	0.509195	+	0.307819i
$443$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$443$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$444$		0			0
$445$	0.118779	+	1.96365i	0.118779	+	1.96365i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.63406	−	0.509195i	−1.63406	−	0.509195i	−0.663123	−	0.748511i	$-0.730769\pi$
$449$	−1.63406	−	0.509195i	−1.63406	−	0.509195i	−0.970942	+	0.239316i	$0.923077\pi$
$450$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$451$		0			0
$452$	−0.328749	−	0.147958i	−0.328749	−	0.147958i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.31658	−	0.159861i	−1.31658	−	0.159861i	−0.568065	−	0.822984i	$-0.692308\pi$
$457$	−1.31658	−	0.159861i	−1.31658	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$458$	−0.0744731	−	0.0950579i	−0.0744731	−	0.0950579i
$459$		0			0
$460$		0			0
$461$	−1.68353	−	1.01773i	−1.68353	−	1.01773i	−0.935016	−	0.354605i	$-0.884615\pi$
$461$	−1.68353	−	1.01773i	−1.68353	−	1.01773i	−0.748511	−	0.663123i	$-0.769231\pi$
$462$		0			0
$463$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$463$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$464$	−0.225408	−	0.0854858i	−0.225408	−	0.0854858i
$465$		0			0
$466$	0.646140	+	0.506219i	0.646140	+	0.506219i
$467$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$467$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$468$		−	1.00000i		−	1.00000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.96365	+	0.359852i	−1.96365	+	0.359852i
$478$		0			0
$479$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$479$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$480$		0			0
$481$	−0.583668	+	1.53901i	−0.583668	+	1.53901i
$482$	1.12477	−	0.506219i	1.12477	−	0.506219i
$483$		0			0
$484$			1.00000i			1.00000i
$485$	0.447528	+	1.81569i	0.447528	+	1.81569i
$486$		0			0
$487$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$487$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$488$		−	1.98542i		−	1.98542i
$489$		0			0
$490$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$491$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$491$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$492$		0			0
$493$	−0.141091	+	0.0258558i	−0.141091	+	0.0258558i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$499$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$500$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$501$		0			0
$502$		0			0
$503$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$503$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$504$		0			0
$505$	−0.869047	−	1.65583i	−0.869047	−	1.65583i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.638104	−	1.05555i	0.638104	−	1.05555i	−0.354605	−	0.935016i	$-0.615385\pi$
$509$	0.638104	−	1.05555i	0.638104	−	1.05555i	0.992709	−	0.120537i	$-0.0384615\pi$
$510$		0			0
$511$		0			0
$512$	0.822984	−	0.568065i	0.822984	−	0.568065i
$513$		0			0
$514$	1.99271	+	0.120537i	1.99271	+	0.120537i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$521$	0.271894	−	0.393906i	0.271894	−	0.393906i	−0.663123	−	0.748511i	$-0.730769\pi$
$521$	0.271894	−	0.393906i	0.271894	−	0.393906i	0.935016	+	0.354605i	$0.115385\pi$
$522$	0.159861	+	0.180446i	0.159861	+	0.180446i
$523$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$523$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			1.00000i			1.00000i
$530$	1.70844	+	1.03279i	1.70844	+	1.03279i
$531$		0			0
$532$		0			0
$533$	−1.01773	+	0.186505i	−1.01773	+	0.186505i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.583668	+	1.53901i	0.583668	+	1.53901i
$539$		0			0
$540$		0			0
$541$	−0.0217671	−	0.118779i	−0.0217671	−	0.118779i	0.970942	−	0.239316i	$-0.0769231\pi$
$541$	−0.0217671	−	0.118779i	−0.0217671	−	0.118779i	−0.992709	+	0.120537i	$0.961538\pi$
$542$		0			0
$543$		0			0
$544$	0.244198	−	0.542586i	0.244198	−	0.542586i
$545$	0.568065	−	1.82298i	0.568065	−	1.82298i
$546$		0			0
$547$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$547$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$548$	−0.136945	−	1.12785i	−0.136945	−	1.12785i
$549$	−0.922670	+	1.75800i	−0.922670	+	1.75800i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.420733	−	1.35018i	−0.420733	−	1.35018i
$555$		0			0
$556$		0			0
$557$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$557$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0495602	+	0.110118i	−0.0495602	+	0.110118i
$563$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$563$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$564$		0			0
$565$	0.0217671	−	0.359852i	0.0217671	−	0.359852i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.527986	+	0.764919i	0.527986	+	0.764919i	0.992709	−	0.120537i	$-0.0384615\pi$
$569$	0.527986	+	0.764919i	0.527986	+	0.764919i	−0.464723	+	0.885456i	$0.653846\pi$
$570$		0			0
$571$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$571$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.992709	+	0.120537i	−0.992709	+	0.120537i
$577$			1.77091i			1.77091i	0.464723	+	0.885456i	$0.346154\pi$
$577$			1.77091i			1.77091i	−0.464723	+	0.885456i	$0.653846\pi$
$578$	0.0778628	+	0.641258i	0.0778628	+	0.641258i
$579$		0			0
$580$		−	0.241073i		−	0.241073i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.77091	−	0.929446i	1.77091	−	0.929446i
$585$	0.935016	−	0.354605i	0.935016	−	0.354605i
$586$	0.627974	+	0.329586i	0.627974	+	0.329586i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.59814	+	0.393906i	1.59814	+	0.393906i
$593$	0.922670	+	0.112032i	0.922670	+	0.112032i	0.568065	−	0.822984i	$-0.307692\pi$
$593$	0.922670	+	0.112032i	0.922670	+	0.112032i	0.354605	+	0.935016i	$0.384615\pi$
$594$		0			0
$595$		0			0
$596$	−1.28962	+	1.28962i	−1.28962	+	1.28962i
$597$		0			0
$598$		0			0
$599$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$599$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$600$		0			0
$601$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	0.120537	−	0.992709i	$-0.461538\pi$
$601$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	−0.354605	+	0.935016i	$0.615385\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$606$		0			0
$607$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$607$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$608$		0			0
$609$		0			0
$610$	1.85640	−	0.704039i	1.85640	−	0.704039i
$611$		0			0
$612$	−0.468379	+	0.366951i	−0.468379	+	0.366951i
$613$	0.136945	−	0.198399i	0.136945	−	0.198399i	−0.748511	−	0.663123i	$-0.769231\pi$
$613$	0.136945	−	0.198399i	0.136945	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.63397	−	1.12785i	1.63397	−	1.12785i	0.748511	−	0.663123i	$-0.230769\pi$
$617$	1.63397	−	1.12785i	1.63397	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$618$		0			0
$619$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$619$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$626$	−0.814480	+	0.638104i	−0.814480	+	0.638104i
$627$		0			0
$628$	0.244198	+	0.783659i	0.244198	+	0.783659i
$629$	0.935016	−	0.291363i	0.935016	−	0.291363i
$630$		0			0
$631$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$631$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$632$		0			0
$633$		0			0
$634$	1.32555	+	0.695701i	1.32555	+	0.695701i
$635$		0			0
$636$		0			0
$637$	0.748511	−	0.663123i	0.748511	−	0.663123i
$638$		0			0
$639$		0			0
$640$	0.822984	+	0.568065i	0.822984	+	0.568065i
$641$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$641$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$642$		0			0
$643$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$643$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$647$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$648$	0.935016	+	0.354605i	0.935016	+	0.354605i
$649$		0			0
$650$	−0.935016	−	0.354605i	−0.935016	−	0.354605i
$651$		0			0
$652$		0			0
$653$	−1.41163	+	1.41163i	−1.41163	+	1.41163i	−0.663123	+	0.748511i	$0.730769\pi$
$653$	−1.41163	+	1.41163i	−1.41163	+	1.41163i	−0.748511	+	0.663123i	$0.769231\pi$
$654$		0			0
$655$		0			0
$656$	0.307819	+	0.987826i	0.307819	+	0.987826i
$657$	−2.00000			−2.00000
$658$		0			0
$659$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$659$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$660$		0			0
$661$	−0.0359256	+	0.593921i	−0.0359256	+	0.593921i	0.935016	+	0.354605i	$0.115385\pi$
$661$	−0.0359256	+	0.593921i	−0.0359256	+	0.593921i	−0.970942	+	0.239316i	$0.923077\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.23202	−	1.09148i	−1.23202	−	1.09148i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0744731	+	0.0950579i	−0.0744731	+	0.0950579i	−0.822984	−	0.568065i	$-0.807692\pi$
$673$	−0.0744731	+	0.0950579i	−0.0744731	+	0.0950579i	0.748511	+	0.663123i	$0.230769\pi$
$674$	−1.56077	−	0.943521i	−1.56077	−	0.943521i
$675$		0			0
$676$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$677$	−0.420733	−	0.420733i	−0.420733	−	0.420733i	0.464723	−	0.885456i	$-0.346154\pi$
$677$	−0.420733	−	0.420733i	−0.420733	−	0.420733i	−0.885456	+	0.464723i	$0.846154\pi$
$678$		0			0
$679$		0			0
$680$	0.593921	+	0.0359256i	0.593921	+	0.0359256i
$681$		0			0
$682$		0			0
$683$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$683$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$684$		0			0
$685$	1.00599	−	0.527986i	1.00599	−	0.527986i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.41163	+	1.41163i	−1.41163	+	1.41163i
$690$		0			0
$691$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$691$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$692$	1.21323	−	0.222333i	1.21323	−	0.222333i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.460812	+	0.408244i	0.460812	+	0.408244i
$698$	−0.254919	+	0.254919i	−0.254919	+	0.254919i
$699$		0			0
$700$		0			0
$701$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$701$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.885456	+	0.464723i	$0.846154\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.09148	−	0.753393i	−1.09148	−	0.753393i
$707$		0			0
$708$		0			0
$709$	−1.50308	+	1.17759i	−1.50308	+	1.17759i	−0.568065	+	0.822984i	$0.692308\pi$
$709$	−1.50308	+	1.17759i	−1.50308	+	1.17759i	−0.935016	+	0.354605i	$0.884615\pi$
$710$		0			0
$711$		0			0
$712$	−0.118779	+	1.96365i	−0.118779	+	1.96365i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$719$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$720$	−0.464723	−	0.885456i	−0.464723	−	0.885456i
$721$		0			0
$722$	0.970942	−	0.239316i	0.970942	−	0.239316i
$723$		0			0
$724$	−0.695701	+	1.32555i	−0.695701	+	1.32555i
$725$	0.225408	−	0.0854858i	0.225408	−	0.0854858i
$726$		0			0
$727$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$727$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$728$		0			0
$729$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$730$	1.49702	+	1.32625i	1.49702	+	1.32625i
$731$		0			0
$732$		0			0
$733$	0.0854858	+	0.704039i	0.0854858	+	0.704039i	0.970942	+	0.239316i	$0.0769231\pi$
$733$	0.0854858	+	0.704039i	0.0854858	+	0.704039i	−0.885456	+	0.464723i	$0.846154\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.186505	−	1.01773i	0.186505	−	1.01773i
$739$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$739$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$740$	0.198399	+	1.63397i	0.198399	+	1.63397i
$741$		0			0
$742$		0			0
$743$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$743$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$744$		0			0
$745$	−1.66312	−	0.748511i	−1.66312	−	0.748511i
$746$	−1.21026	−	1.21026i	−1.21026	−	1.21026i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$751$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$752$		0			0
$753$		0			0
$754$	0.234068	+	0.0576926i	0.234068	+	0.0576926i
$755$		0			0
$756$		0			0
$757$	1.50308	+	0.468379i	1.50308	+	0.468379i	0.935016	−	0.354605i	$-0.115385\pi$
$757$	1.50308	+	0.468379i	1.50308	+	0.468379i	0.568065	+	0.822984i	$0.307692\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.70844	−	1.03279i	1.70844	−	1.03279i	0.822984	−	0.568065i	$-0.192308\pi$
$761$	1.70844	−	1.03279i	1.70844	−	1.03279i	0.885456	−	0.464723i	$-0.153846\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.509195	−	0.307819i	−0.509195	−	0.307819i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.468379	−	1.50308i	0.468379	−	1.50308i	−0.354605	−	0.935016i	$-0.615385\pi$
$769$	0.468379	−	1.50308i	0.468379	−	1.50308i	0.822984	−	0.568065i	$-0.192308\pi$
$770$		0			0
$771$		0			0
$772$	−0.447528	+	1.81569i	−0.447528	+	1.81569i
$773$	0.527986	−	0.764919i	0.527986	−	0.764919i	−0.464723	−	0.885456i	$-0.653846\pi$
$773$	0.527986	−	0.764919i	0.527986	−	0.764919i	0.992709	+	0.120537i	$0.0384615\pi$
$774$		0			0
$775$		0			0
$776$	0.225408	+	1.85640i	0.225408	+	1.85640i
$777$		0			0
$778$	−0.527986	+	1.00599i	−0.527986	+	1.00599i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$785$	−0.646140	+	0.506219i	−0.646140	+	0.506219i
$786$		0			0
$787$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$787$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$788$	1.97094	−	0.239316i	1.97094	−	0.239316i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.239316	+	1.97094i	0.239316	+	1.97094i
$794$	−0.317391	+	0.358261i	−0.317391	+	0.358261i
$795$		0			0
$796$		0			0
$797$	1.63406	−	0.509195i	1.63406	−	0.509195i	0.663123	−	0.748511i	$-0.269231\pi$
$797$	1.63406	−	0.509195i	1.63406	−	0.509195i	0.970942	+	0.239316i	$0.0769231\pi$
$798$		0			0
$799$		0			0
$800$	−0.239316	+	0.970942i	−0.239316	+	0.970942i
$801$	1.01773	−	1.68353i	1.01773	−	1.68353i
$802$	−0.748511	+	0.336877i	−0.748511	+	0.336877i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.663123	−	1.74851i	−0.663123	−	1.74851i
$809$	−0.470293	−	0.530851i	−0.470293	−	0.530851i	0.464723	−	0.885456i	$-0.346154\pi$
$809$	−0.470293	−	0.530851i	−0.470293	−	0.530851i	−0.935016	+	0.354605i	$0.884615\pi$
$810$			1.00000i			1.00000i
$811$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$811$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.580411	−	1.28962i	0.580411	−	1.28962i
$819$		0			0
$820$	−0.814480	+	0.638104i	−0.814480	+	0.638104i
$821$	−0.110118	−	1.82047i	−0.110118	−	1.82047i	−0.464723	−	0.885456i	$-0.653846\pi$
$821$	−0.110118	−	1.82047i	−0.110118	−	1.82047i	0.354605	−	0.935016i	$-0.384615\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$827$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$828$		0			0
$829$	0.753393	−	1.09148i	0.753393	−	1.09148i	−0.239316	−	0.970942i	$-0.576923\pi$
$829$	0.753393	−	1.09148i	0.753393	−	1.09148i	0.992709	−	0.120537i	$-0.0384615\pi$
$830$		0			0
$831$		0			0
$832$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$833$	−0.585260	−	0.107253i	−0.585260	−	0.107253i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$839$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$840$		0			0
$841$	0.833997	−	0.437715i	0.833997	−	0.437715i
$842$	1.23932	+	0.970942i	1.23932	+	0.970942i
$843$		0			0
$844$		0			0
$845$	0.568065	−	0.822984i	0.568065	−	0.822984i
$846$		0			0
$847$		0			0
$848$	1.57149	+	1.23119i	1.57149	+	1.23119i
$849$		0			0
$850$	0.177016	+	0.568065i	0.177016	+	0.568065i
$851$		0			0
$852$		0			0
$853$	−1.17433	−	1.32555i	−1.17433	−	1.32555i	−0.935016	−	0.354605i	$-0.884615\pi$
$853$	−1.17433	−	1.32555i	−1.17433	−	1.32555i	−0.239316	−	0.970942i	$-0.576923\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.54858	−	0.283788i	−1.54858	−	0.283788i	−0.663123	−	0.748511i	$-0.730769\pi$
$857$	−1.54858	−	0.283788i	−1.54858	−	0.283788i	−0.885456	+	0.464723i	$0.846154\pi$
$858$		0			0
$859$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$859$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$863$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$864$		0			0
$865$	0.638104	+	1.05555i	0.638104	+	1.05555i
$866$	0.359852	−	0.0217671i	0.359852	−	0.0217671i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.783659	−	1.74122i	0.783659	−	1.74122i
$873$	0.663123	−	1.74851i	0.663123	−	1.74851i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.447528	+	0.169725i	0.447528	+	0.169725i	0.568065	−	0.822984i	$-0.307692\pi$
$877$	0.447528	+	0.169725i	0.447528	+	0.169725i	−0.120537	+	0.992709i	$0.538462\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.28771	−	1.45352i	−1.28771	−	1.45352i	−0.822984	−	0.568065i	$-0.807692\pi$
$881$	−1.28771	−	1.45352i	−1.28771	−	1.45352i	−0.464723	−	0.885456i	$-0.653846\pi$
$882$	0.354605	+	0.935016i	0.354605	+	0.935016i
$883$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$883$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$884$	−0.177016	+	0.568065i	−0.177016	+	0.568065i
$885$		0			0
$886$		0			0
$887$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$887$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$888$		0			0
$889$		0			0
$890$	−1.87816	+	0.585260i	−1.87816	+	0.585260i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.103342	−	1.70844i	0.103342	−	1.70844i
$899$		0			0
$900$	0.663123	−	0.748511i	0.663123	−	0.748511i
$901$	1.17918	+	0.143178i	1.17918	+	0.143178i
$902$		0			0
$903$		0			0
$904$	0.0649838	−	0.354605i	0.0649838	−	0.354605i
$905$	−1.48611	−	0.180446i	−1.48611	−	0.180446i
$906$		0			0
$907$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$907$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$908$		0			0
$909$	−0.225408	+	1.85640i	−0.225408	+	1.85640i
$910$		0			0
$911$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$911$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$912$		0			0
$913$		0			0
$914$	−0.159861	−	1.31658i	−0.159861	−	1.31658i
$915$		0			0
$916$	0.0744731	−	0.0950579i	0.0744731	−	0.0950579i
$917$		0			0
$918$		0			0
$919$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$919$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$920$		0			0
$921$		0			0
$922$	0.585260	−	1.87816i	0.585260	−	1.87816i
$923$		0			0
$924$		0			0
$925$	−1.45743	+	0.764919i	−1.45743	+	0.764919i
$926$		0			0
$927$		0			0
$928$	0.0290582	−	0.239316i	0.0290582	−	0.239316i
$929$	−1.63406	+	0.987826i	−1.63406	+	0.987826i	−0.663123	+	0.748511i	$0.730769\pi$
$929$	−1.63406	+	0.987826i	−1.63406	+	0.987826i	−0.970942	+	0.239316i	$0.923077\pi$
$930$		0			0
$931$		0			0
$932$	−0.336877	+	0.748511i	−0.336877	+	0.748511i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.970942	−	0.239316i	0.970942	−	0.239316i
$937$	1.35018	+	0.420733i	1.35018	+	0.420733i	0.885456	−	0.464723i	$-0.153846\pi$
$937$	1.35018	+	0.420733i	1.35018	+	0.420733i	0.464723	+	0.885456i	$0.346154\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.814480	+	1.34731i	0.814480	+	1.34731i	0.935016	+	0.354605i	$0.115385\pi$
$941$	0.814480	+	1.34731i	0.814480	+	1.34731i	−0.120537	+	0.992709i	$0.538462\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$947$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$948$		0			0
$949$	−1.64597	+	1.13613i	−1.64597	+	1.13613i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.0950579	+	1.57149i	0.0950579	+	1.57149i	0.663123	+	0.748511i	$0.269231\pi$
$953$	0.0950579	+	1.57149i	0.0950579	+	1.57149i	−0.568065	+	0.822984i	$0.692308\pi$
$954$	−0.819328	−	1.82047i	−0.819328	−	1.82047i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$962$	−1.63397	−	0.198399i	−1.63397	−	0.198399i
$963$		0			0
$964$	0.760684	+	0.970942i	0.760684	+	0.970942i
$965$	−1.85640	+	0.225408i	−1.85640	+	0.225408i
$966$		0			0
$967$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$967$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$968$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$969$		0			0
$970$	−1.65583	+	0.869047i	−1.65583	+	0.869047i
$971$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$971$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.92773	−	0.475142i	1.92773	−	0.475142i
$977$	−0.663123	+	0.251489i	−0.663123	+	0.251489i	−0.663123	−	0.748511i	$-0.730769\pi$
$977$	−0.663123	+	0.251489i	−0.663123	+	0.251489i			1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	0.354605	−	0.935016i	0.354605	−	0.935016i
$981$	−1.50308	+	1.17759i	−1.50308	+	1.17759i
$982$		0			0
$983$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$983$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$984$		0			0
$985$	0.922670	+	1.75800i	0.922670	+	1.75800i
$986$	−0.0588697	−	0.130803i	−0.0588697	−	0.130803i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.96365	+	0.118779i	1.96365	+	0.118779i	0.992709	−	0.120537i	$-0.0384615\pi$
$997$	1.96365	+	0.118779i	1.96365	+	0.118779i	0.970942	+	0.239316i	$0.0769231\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.cg.a.3167.1	yes	24
4.3	odd	2	CM	3380.1.cg.a.3167.1	yes	24
5.3	odd	4		3380.1.bz.a.463.1	✓	24
20.3	even	4		3380.1.bz.a.463.1	✓	24
169.96	odd	52		3380.1.bz.a.3307.1	yes	24
676.603	even	52		3380.1.bz.a.3307.1	yes	24
845.603	even	52	inner	3380.1.cg.a.603.1	yes	24
3380.603	odd	52	inner	3380.1.cg.a.603.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.bz.a.463.1	✓	24	5.3	odd	4
3380.1.bz.a.463.1	✓	24	20.3	even	4
3380.1.bz.a.3307.1	yes	24	169.96	odd	52
3380.1.bz.a.3307.1	yes	24	676.603	even	52
3380.1.cg.a.603.1	yes	24	845.603	even	52	inner
3380.1.cg.a.603.1	yes	24	3380.603	odd	52	inner
3380.1.cg.a.3167.1	yes	24	1.1	even	1	trivial
3380.1.cg.a.3167.1	yes	24	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.cg (of order \(52\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.239316 + 0.970942i) q^{2} +(-0.885456 + 0.464723i) q^{4} +(-0.748511 - 0.663123i) q^{5} +(-0.663123 - 0.748511i) q^{8} +(0.239316 + 0.970942i) q^{9} +O(q^{10})\)	\(q+(0.239316 + 0.970942i) q^{2} +(-0.885456 + 0.464723i) q^{4} +(-0.748511 - 0.663123i) q^{5} +(-0.663123 - 0.748511i) q^{8} +(0.239316 + 0.970942i) q^{9} +(0.464723 - 0.885456i) q^{10} +(0.748511 + 0.663123i) q^{13} +(0.568065 - 0.822984i) q^{16} +(0.0359256 - 0.593921i) q^{17} +(-0.885456 + 0.464723i) q^{18} +(0.970942 + 0.239316i) q^{20} +(0.120537 + 0.992709i) q^{25} +(-0.464723 + 0.885456i) q^{26} +(-0.0576926 - 0.234068i) q^{29} +(0.935016 + 0.354605i) q^{32} +(0.585260 - 0.107253i) q^{34} +(-0.663123 - 0.748511i) q^{36} +(0.583668 + 1.53901i) q^{37} +1.00000i q^{40} +(-0.638104 + 0.814480i) q^{41} +(0.464723 - 0.885456i) q^{45} +(0.120537 - 0.992709i) q^{49} +(-0.935016 + 0.354605i) q^{50} +(-0.970942 - 0.239316i) q^{52} +(-0.120537 + 1.99271i) q^{53} +(0.213460 - 0.112032i) q^{58} +(1.48611 + 1.31658i) q^{61} +(-0.120537 + 0.992709i) q^{64} +(-0.120537 - 0.992709i) q^{65} +(0.244198 + 0.542586i) q^{68} +(0.568065 - 0.822984i) q^{72} +(-0.478631 + 1.94188i) q^{73} +(-1.35460 + 0.935016i) q^{74} +(-0.970942 + 0.239316i) q^{80} +(-0.885456 + 0.464723i) q^{81} +(-0.943521 - 0.424644i) q^{82} +(-0.420733 + 0.420733i) q^{85} +(-1.39105 - 1.39105i) q^{89} +(0.970942 + 0.239316i) q^{90} +(-1.53901 - 1.06230i) q^{97} +(0.992709 - 0.120537i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) 24 * q + 2 * q^4 - 2 * q^5	\( 24 q + 2 q^{4} - 2 q^{5} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 2 q^{18} + 2 q^{20} - 2 q^{25} - 2 q^{34} + 2 q^{41} - 2 q^{49} - 2 q^{52} + 2 q^{53} - 4 q^{58} + 2 q^{64} + 2 q^{65} - 2 q^{68} - 2 q^{72} - 26 q^{74} - 2 q^{80} + 2 q^{81} + 2 q^{82} + 2 q^{85} + 2 q^{89} + 2 q^{90}+O(q^{100}) \) 24 * q + 2 * q^4 - 2 * q^5 + 2 * q^13 - 2 * q^16 + 2 * q^17 + 2 * q^18 + 2 * q^20 - 2 * q^25 - 2 * q^34 + 2 * q^41 - 2 * q^49 - 2 * q^52 + 2 * q^53 - 4 * q^58 + 2 * q^64 + 2 * q^65 - 2 * q^68 - 2 * q^72 - 26 * q^74 - 2 * q^80 + 2 * q^81 + 2 * q^82 + 2 * q^85 + 2 * q^89 + 2 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more