LMFDB - Embedded newform 3380.1.co.a.3299.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(179,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(78))

chi = DirichletCharacter(H, H._module([39, 39, 5]))

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

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

chi := DirichletCharacter("3380.179");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.co (of order $78$, 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:	$\Q(\zeta_{39})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 $ x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{78}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{78} - \cdots)$

Embedding invariants

Embedding label			3299.1
Root			$-0.919979 - 0.391967i$ of defining polynomial
Character	$\chi$	$=$	3380.3299
Dual form			3380.1.co.a.459.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.845190 + 0.534466i) q^{2} +(0.428693 + 0.903450i) q^{4} +(-0.919979 - 0.391967i) q^{5} +(-0.120537 + 0.992709i) q^{8} +(0.0402659 - 0.999189i) q^{9} +O(q^{10})$	\(q+(0.845190 + 0.534466i) q^{2} +(0.428693 + 0.903450i) q^{4} +(-0.919979 - 0.391967i) q^{5} +(-0.120537 + 0.992709i) q^{8} +(0.0402659 - 0.999189i) q^{9} +(-0.568065 - 0.822984i) q^{10} +(0.799443 - 0.600742i) q^{13} +(-0.632445 + 0.774605i) q^{16} +(1.08548 - 1.44451i) q^{17} +(0.568065 - 0.822984i) q^{18} +(-0.0402659 - 0.999189i) q^{20} +(0.692724 + 0.721202i) q^{25} +(0.996757 - 0.0804666i) q^{26} +(0.470293 + 0.297395i) q^{29} +(-0.948536 + 0.316668i) q^{32} +(1.68948 - 0.640736i) q^{34} +(0.919979 - 0.391967i) q^{36} +(-1.87251 - 0.625134i) q^{37} +(0.500000 - 0.866025i) q^{40} +(0.770916 - 0.740475i) q^{41} +(-0.428693 + 0.903450i) q^{45} +(0.278217 + 0.960518i) q^{49} +(0.200026 + 0.979791i) q^{50} +(0.885456 + 0.464723i) q^{52} +(1.19272 + 0.144823i) q^{53} +(0.238540 + 0.502711i) q^{58} +(1.27458 + 0.543050i) q^{61} +(-0.970942 - 0.239316i) q^{64} +(-0.970942 + 0.239316i) q^{65} +(1.77038 + 0.361427i) q^{68} +(0.987050 + 0.160411i) q^{72} +(0.885456 + 0.464723i) q^{73} +(-1.24851 - 1.52915i) q^{74} +(0.885456 - 0.464723i) q^{80} +(-0.996757 - 0.0804666i) q^{81} +(1.04733 - 0.213814i) q^{82} +(-1.56482 + 0.903450i) q^{85} +(1.61950 - 0.935016i) q^{89} +(-0.845190 + 0.534466i) q^{90} +(-1.47764 - 0.240139i) q^{97} +(-0.278217 + 0.960518i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} + q^{4} + q^{5} + 2 q^{8} - q^{9}+O(q^{10}) $ 24 * q - q^2 + q^4 + q^5 + 2 * q^8 - q^9	$ 24 q - q^{2} + q^{4} + q^{5} + 2 q^{8} - q^{9} + 2 q^{10} + q^{13} + q^{16} - 3 q^{17} - 2 q^{18} + q^{20} + q^{25} - q^{26} + q^{29} - q^{32} - q^{36} + q^{37} + 12 q^{40} + 3 q^{41} - q^{45} + q^{49} - q^{50} - 2 q^{52} + 13 q^{53} - q^{58} + q^{61} - 2 q^{64} - 2 q^{65} + 3 q^{68} + q^{72} - 2 q^{73} - 14 q^{74} - 2 q^{80} + q^{81} + 3 q^{82} + 3 q^{85} + q^{90} + 2 q^{97} - q^{98}+O(q^{100}) $ 24 * q - q^2 + q^4 + q^5 + 2 * q^8 - q^9 + 2 * q^10 + q^13 + q^16 - 3 * q^17 - 2 * q^18 + q^20 + q^25 - q^26 + q^29 - q^32 - q^36 + q^37 + 12 * q^40 + 3 * q^41 - q^45 + q^49 - q^50 - 2 * q^52 + 13 * q^53 - q^58 + q^61 - 2 * q^64 - 2 * q^65 + 3 * q^68 + q^72 - 2 * q^73 - 14 * q^74 - 2 * q^80 + q^81 + 3 * q^82 + 3 * q^85 + q^90 + 2 * q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{53}{78}\right)$

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 $

$2$	0.845190	+	0.534466i	0.845190	+	0.534466i
$2$	0.845190	+	0.534466i	0.845190	+	0.534466i
$3$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$3$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$4$	0.428693	+	0.903450i	0.428693	+	0.903450i
$5$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$5$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$6$		0			0
$7$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$7$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$8$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$9$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$10$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$11$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$11$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$12$		0			0
$13$	0.799443	−	0.600742i	0.799443	−	0.600742i
$13$	0.799443	−	0.600742i	0.799443	−	0.600742i
$14$		0			0
$15$		0			0
$16$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$17$	1.08548	−	1.44451i	1.08548	−	1.44451i	0.200026	−	0.979791i	$-0.435897\pi$
$17$	1.08548	−	1.44451i	1.08548	−	1.44451i	0.885456	−	0.464723i	$-0.153846\pi$
$18$	0.568065	−	0.822984i	0.568065	−	0.822984i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	0.692724	+	0.721202i	0.692724	+	0.721202i
$26$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$27$		0			0
$28$		0			0
$29$	0.470293	+	0.297395i	0.470293	+	0.297395i	0.748511	−	0.663123i	$-0.230769\pi$
$29$	0.470293	+	0.297395i	0.470293	+	0.297395i	−0.278217	+	0.960518i	$0.589744\pi$
$30$		0			0
$31$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$31$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$32$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$33$		0			0
$34$	1.68948	−	0.640736i	1.68948	−	0.640736i
$35$		0			0
$36$	0.919979	−	0.391967i	0.919979	−	0.391967i
$37$	−1.87251	−	0.625134i	−1.87251	−	0.625134i	−0.987050	−	0.160411i	$-0.948718\pi$
$37$	−1.87251	−	0.625134i	−1.87251	−	0.625134i	−0.885456	−	0.464723i	$-0.846154\pi$
$38$		0			0
$39$		0			0
$40$	0.500000	−	0.866025i	0.500000	−	0.866025i
$41$	0.770916	−	0.740475i	0.770916	−	0.740475i	−0.200026	−	0.979791i	$-0.564103\pi$
$41$	0.770916	−	0.740475i	0.770916	−	0.740475i	0.970942	+	0.239316i	$0.0769231\pi$
$42$		0			0
$43$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$43$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$44$		0			0
$45$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$46$		0			0
$47$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$47$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$48$		0			0
$49$	0.278217	+	0.960518i	0.278217	+	0.960518i
$50$	0.200026	+	0.979791i	0.200026	+	0.979791i
$51$		0			0
$52$	0.885456	+	0.464723i	0.885456	+	0.464723i
$53$	1.19272	+	0.144823i	1.19272	+	0.144823i	0.692724	−	0.721202i	$-0.256410\pi$
$53$	1.19272	+	0.144823i	1.19272	+	0.144823i	0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.238540	+	0.502711i	0.238540	+	0.502711i
$59$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$59$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$60$		0			0
$61$	1.27458	+	0.543050i	1.27458	+	0.543050i	0.919979	−	0.391967i	$-0.128205\pi$
$61$	1.27458	+	0.543050i	1.27458	+	0.543050i	0.354605	+	0.935016i	$0.384615\pi$
$62$		0			0
$63$		0			0
$64$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$65$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$66$		0			0
$67$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$67$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$68$	1.77038	+	0.361427i	1.77038	+	0.361427i
$69$		0			0
$70$		0			0
$71$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$71$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$72$	0.987050	+	0.160411i	0.987050	+	0.160411i
$73$	0.885456	+	0.464723i	0.885456	+	0.464723i	0.845190	−	0.534466i	$-0.179487\pi$
$73$	0.885456	+	0.464723i	0.885456	+	0.464723i	0.0402659	+	0.999189i	$0.487179\pi$
$74$	−1.24851	−	1.52915i	−1.24851	−	1.52915i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$79$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$80$	0.885456	−	0.464723i	0.885456	−	0.464723i
$81$	−0.996757	−	0.0804666i	−0.996757	−	0.0804666i
$82$	1.04733	−	0.213814i	1.04733	−	0.213814i
$83$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$83$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$84$		0			0
$85$	−1.56482	+	0.903450i	−1.56482	+	0.903450i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.61950	−	0.935016i	1.61950	−	0.935016i	0.632445	−	0.774605i	$-0.282051\pi$
$89$	1.61950	−	0.935016i	1.61950	−	0.935016i	0.987050	−	0.160411i	$-0.0512821\pi$
$90$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.47764	−	0.240139i	−1.47764	−	0.240139i	−0.632445	−	0.774605i	$-0.717949\pi$
$97$	−1.47764	−	0.240139i	−1.47764	−	0.240139i	−0.845190	+	0.534466i	$0.820513\pi$
$98$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$99$		0			0
$100$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$101$	−0.379463	+	1.85873i	−0.379463	+	1.85873i	0.120537	+	0.992709i	$0.461538\pi$
$101$	−0.379463	+	1.85873i	−0.379463	+	1.85873i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$103$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$104$	0.500000	+	0.866025i	0.500000	+	0.866025i
$105$		0			0
$106$	0.930676	+	0.759873i	0.930676	+	0.759873i
$107$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$107$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$108$		0			0
$109$	−1.09148	+	1.23202i	−1.09148	+	1.23202i	−0.120537	+	0.992709i	$0.538462\pi$
$109$	−1.09148	+	1.23202i	−1.09148	+	1.23202i	−0.970942	+	0.239316i	$0.923077\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.11729	−	1.07318i	−1.11729	−	1.07318i	−0.996757	−	0.0804666i	$-0.974359\pi$
$113$	−1.11729	−	1.07318i	−1.11729	−	1.07318i	−0.120537	−	0.992709i	$-0.538462\pi$
$114$		0			0
$115$		0			0
$116$	−0.0670708	+	0.552378i	−0.0670708	+	0.552378i
$117$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$122$	0.787025	+	1.14020i	0.787025	+	1.14020i
$123$		0			0
$124$		0			0
$125$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$126$		0			0
$127$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$127$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$128$	−0.692724	−	0.721202i	−0.692724	−	0.721202i
$129$		0			0
$130$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$131$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$131$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.30314	+	1.25168i	1.30314	+	1.25168i
$137$	−1.19979	+	0.400550i	−1.19979	+	0.400550i	−0.845190	−	0.534466i	$-0.820513\pi$
$137$	−1.19979	+	0.400550i	−1.19979	+	0.400550i	−0.354605	+	0.935016i	$0.615385\pi$
$138$		0			0
$139$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$139$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.748511	+	0.663123i	0.748511	+	0.663123i
$145$	−0.316091	−	0.457937i	−0.316091	−	0.457937i
$146$	0.500000	+	0.866025i	0.500000	+	0.866025i
$147$		0			0
$148$	−0.237952	−	1.95971i	−0.237952	−	1.95971i
$149$	−0.157681	−	1.95323i	−0.157681	−	1.95323i	−0.278217	−	0.960518i	$-0.589744\pi$
$149$	−0.157681	−	1.95323i	−0.157681	−	1.95323i	0.120537	−	0.992709i	$-0.461538\pi$
$150$		0			0
$151$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$151$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$152$		0			0
$153$	−1.39963	−	1.14277i	−1.39963	−	1.14277i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.83224	−	0.694877i	−1.83224	−	0.694877i	−0.987050	−	0.160411i	$-0.948718\pi$
$157$	−1.83224	−	0.694877i	−1.83224	−	0.694877i	−0.845190	−	0.534466i	$-0.820513\pi$
$158$		0			0
$159$		0			0
$160$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$161$		0			0
$162$	−0.799443	−	0.600742i	−0.799443	−	0.600742i
$163$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$163$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$164$	0.999468	+	0.379048i	0.999468	+	0.379048i
$165$		0			0
$166$		0			0
$167$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$167$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$168$		0			0
$169$	0.278217	−	0.960518i	0.278217	−	0.960518i
$170$	−1.80544	−	0.0727566i	−1.80544	−	0.0727566i
$171$		0			0
$172$		0			0
$173$	0.432420	−	0.205186i	0.432420	−	0.205186i	−0.200026	−	0.979791i	$-0.564103\pi$
$173$	0.432420	−	0.205186i	0.432420	−	0.205186i	0.632445	+	0.774605i	$0.282051\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.86852	+	0.0752986i	1.86852	+	0.0752986i
$179$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$179$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$180$	−1.00000			−1.00000
$181$	−0.652458	+	1.72039i	−0.652458	+	1.72039i	0.0402659	+	0.999189i	$0.487179\pi$
$181$	−0.652458	+	1.72039i	−0.652458	+	1.72039i	−0.692724	+	0.721202i	$0.743590\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.47764	+	1.30907i	1.47764	+	1.30907i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−0.319818	+	0.240328i	−0.319818	+	0.240328i	−0.748511	−	0.663123i	$-0.769231\pi$
$193$	−0.319818	+	0.240328i	−0.319818	+	0.240328i	0.428693	+	0.903450i	$0.358974\pi$
$194$	−1.12054	−	0.992709i	−1.12054	−	0.992709i
$195$		0			0
$196$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$197$	0.388427	−	1.90264i	0.388427	−	1.90264i	−0.0402659	−	0.999189i	$-0.512821\pi$
$197$	0.388427	−	1.90264i	0.388427	−	1.90264i	0.428693	−	0.903450i	$-0.358974\pi$
$198$		0			0
$199$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$199$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$200$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$201$		0			0
$202$	−1.31415	+	1.36817i	−1.31415	+	1.36817i
$203$		0			0
$204$		0			0
$205$	−0.999468	+	0.379048i	−0.999468	+	0.379048i
$206$		0			0
$207$		0			0
$208$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$209$		0			0
$210$		0			0
$211$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$211$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$212$	0.380472	+	1.13965i	0.380472	+	1.13965i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.58098	+	0.457937i	−1.58098	+	0.457937i
$219$		0			0
$220$		0			0
$221$		−	1.80690i		−	1.80690i
$222$		0			0
$223$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$223$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$224$		0			0
$225$	0.748511	−	0.663123i	0.748511	−	0.663123i
$226$	−0.370750	−	1.50419i	−0.370750	−	1.50419i
$227$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$227$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$228$		0			0
$229$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.568065	+	0.822984i	$0.692308\pi$
$229$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.748511	+	0.663123i	$0.769231\pi$
$230$		0			0
$231$		0			0
$232$	−0.351915	+	0.431017i	−0.351915	+	0.431017i
$233$	−0.317391	+	1.28771i	−0.317391	+	1.28771i	0.568065	+	0.822984i	$0.307692\pi$
$233$	−0.317391	+	1.28771i	−0.317391	+	1.28771i	−0.885456	+	0.464723i	$0.846154\pi$
$234$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−0.231378	+	1.42373i	−0.231378	+	1.42373i	0.568065	+	0.822984i	$0.307692\pi$
$241$	−0.231378	+	1.42373i	−0.231378	+	1.42373i	−0.799443	+	0.600742i	$0.794872\pi$
$242$	0.885456	+	0.464723i	0.885456	+	0.464723i
$243$		0			0
$244$	0.0557864	+	1.38433i	0.0557864	+	1.38433i
$245$	0.120537	−	0.992709i	0.120537	−	0.992709i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.200026	−	0.979791i	0.200026	−	0.979791i
$251$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$251$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$257$	0.0630804	−	0.781391i	0.0630804	−	0.781391i	−0.885456	−	0.464723i	$-0.846154\pi$
$257$	0.0630804	−	0.781391i	0.0630804	−	0.781391i	0.948536	−	0.316668i	$-0.102564\pi$
$258$		0			0
$259$		0			0
$260$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$261$	0.316091	−	0.457937i	0.316091	−	0.457937i
$262$		0			0
$263$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$263$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$264$		0			0
$265$	−1.04052	−	0.600742i	−1.04052	−	0.600742i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	0.799443	+	0.600742i	$0.205128\pi$
$269$	−0.197315	+	0.681209i	−0.197315	+	0.681209i	−0.996757	+	0.0804666i	$0.974359\pi$
$270$		0			0
$271$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$271$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$272$	0.432420	+	1.75440i	0.432420	+	1.75440i
$273$		0			0
$274$	−1.22814	−	0.302708i	−1.22814	−	0.302708i
$275$		0			0
$276$		0			0
$277$	−0.678906	+	1.59345i	−0.678906	+	1.59345i	0.120537	+	0.992709i	$0.461538\pi$
$277$	−0.678906	+	1.59345i	−0.678906	+	1.59345i	−0.799443	+	0.600742i	$0.794872\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.287534	+	1.16657i	0.287534	+	1.16657i	0.919979	+	0.391967i	$0.128205\pi$
$281$	0.287534	+	1.16657i	0.287534	+	1.16657i	−0.632445	+	0.774605i	$0.717949\pi$
$282$		0			0
$283$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$283$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.278217	+	0.960518i	0.278217	+	0.960518i
$289$	−0.630132	−	2.17547i	−0.630132	−	2.17547i
$290$	−0.0224054	−	0.555984i	−0.0224054	−	0.555984i
$291$		0			0
$292$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$293$	1.74527	+	0.743589i	1.74527	+	0.743589i	0.996757	+	0.0804666i	$0.0256410\pi$
$293$	1.74527	+	0.743589i	1.74527	+	0.743589i	0.748511	+	0.663123i	$0.230769\pi$
$294$		0			0
$295$		0			0
$296$	0.846282	−	1.78350i	0.846282	−	1.78350i
$297$		0			0
$298$	0.910663	−	1.73512i	0.910663	−	1.73512i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.959734	−	0.999189i	−0.959734	−	0.999189i
$306$	−0.572188	−	1.71391i	−0.572188	−	1.71391i
$307$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$307$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$311$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$312$		0			0
$313$	0.616337	−	0.695701i	0.616337	−	0.695701i	−0.354605	−	0.935016i	$-0.615385\pi$
$313$	0.616337	−	0.695701i	0.616337	−	0.695701i	0.970942	+	0.239316i	$0.0769231\pi$
$314$	−1.17720	−	1.56657i	−1.17720	−	1.56657i
$315$		0			0
$316$		0			0
$317$	−0.192724	−	1.58723i	−0.192724	−	1.58723i	−0.692724	−	0.721202i	$-0.743590\pi$
$317$	−0.192724	−	1.58723i	−0.192724	−	1.58723i	0.500000	−	0.866025i	$-0.333333\pi$
$318$		0			0
$319$		0			0
$320$	0.799443	+	0.600742i	0.799443	+	0.600742i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$325$	0.987050	+	0.160411i	0.987050	+	0.160411i
$326$		0			0
$327$		0			0
$328$	0.642152	+	0.854550i	0.642152	+	0.854550i
$329$		0			0
$330$		0			0
$331$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$331$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$332$		0			0
$333$	−0.700026	+	1.84582i	−0.700026	+	1.84582i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	0.633336i		−	0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$337$		−	0.633336i		−	0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$338$	0.748511	−	0.663123i	0.748511	−	0.663123i
$339$		0			0
$340$	−1.48705	−	1.02644i	−1.48705	−	1.02644i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.475142	+	0.0576926i	0.475142	+	0.0576926i
$347$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$347$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$348$		0			0
$349$	−1.86852	+	0.0752986i	−1.86852	+	0.0752986i	−0.948536	−	0.316668i	$-0.897436\pi$
$349$	−1.86852	+	0.0752986i	−1.86852	+	0.0752986i	−0.919979	+	0.391967i	$0.871795\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.69272	+	0.721202i	−1.69272	+	0.721202i	−0.692724	+	0.721202i	$0.743590\pi$
$353$	−1.69272	+	0.721202i	−1.69272	+	0.721202i	−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	1.53901	+	1.06230i	1.53901	+	1.06230i
$357$		0			0
$358$		0			0
$359$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$359$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$360$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$	−1.47094	+	1.10534i	−1.47094	+	1.10534i
$363$		0			0
$364$		0			0
$365$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$366$		0			0
$367$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$367$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$368$		0			0
$369$	−0.708833	−	0.800107i	−0.708833	−	0.800107i
$370$	0.549229	+	1.89616i	0.549229	+	1.89616i
$371$		0			0
$372$		0			0
$373$	0.571307	+	0.903450i	0.571307	+	0.903450i	1.00000			$0$
$373$	0.571307	+	0.903450i	0.571307	+	0.903450i	−0.428693	+	0.903450i	$0.641026\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.554631	−	0.0447744i	0.554631	−	0.0447744i
$378$		0			0
$379$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$379$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$383$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$384$		0			0
$385$		0			0
$386$	−0.398754	+	0.0321908i	−0.398754	+	0.0321908i
$387$		0			0
$388$	−0.416498	−	1.43792i	−0.416498	−	1.43792i
$389$	0.237952	+	1.95971i	0.237952	+	1.95971i	0.278217	+	0.960518i	$0.410256\pi$
$389$	0.237952	+	1.95971i	0.237952	+	1.95971i	−0.0402659	+	0.999189i	$0.512821\pi$
$390$		0			0
$391$		0			0
$392$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$393$		0			0
$394$	1.34519	−	1.40049i	1.34519	−	1.40049i
$395$		0			0
$396$		0			0
$397$	1.12001	+	1.37176i	1.12001	+	1.37176i	0.919979	+	0.391967i	$0.128205\pi$
$397$	1.12001	+	1.37176i	1.12001	+	1.37176i	0.200026	+	0.979791i	$0.435897\pi$
$398$		0			0
$399$		0			0
$400$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$401$	−0.490585	−	0.400550i	−0.490585	−	0.400550i	0.354605	−	0.935016i	$-0.384615\pi$
$401$	−0.490585	−	0.400550i	−0.490585	−	0.400550i	−0.845190	+	0.534466i	$0.820513\pi$
$402$		0			0
$403$		0			0
$404$	−1.84195	+	0.453999i	−1.84195	+	0.453999i
$405$	0.885456	+	0.464723i	0.885456	+	0.464723i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.66367	+	0.481887i	1.66367	+	0.481887i	0.970942	−	0.239316i	$-0.0769231\pi$
$409$	1.66367	+	0.481887i	1.66367	+	0.481887i	0.692724	+	0.721202i	$0.256410\pi$
$410$	−1.04733	−	0.213814i	−1.04733	−	0.213814i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$417$		0			0
$418$		0			0
$419$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$419$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$420$		0			0
$421$	0.345190	−	1.40049i	0.345190	−	1.40049i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.345190	−	1.40049i	0.345190	−	1.40049i	0.845190	−	0.534466i	$-0.179487\pi$
$422$		0			0
$423$		0			0
$424$	−0.287534	+	1.16657i	−0.287534	+	1.16657i
$425$	1.79373	−	0.217798i	1.79373	−	0.217798i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$431$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$432$		0			0
$433$	0.248511	−	0.202903i	0.248511	−	0.202903i	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	0.248511	−	0.202903i	0.248511	−	0.202903i	0.748511	+	0.663123i	$0.230769\pi$
$434$		0			0
$435$		0			0
$436$	−1.58098	−	0.457937i	−1.58098	−	0.457937i
$437$		0			0
$438$		0			0
$439$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$439$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$440$		0			0
$441$	0.970942	−	0.239316i	0.970942	−	0.239316i
$442$	0.965727	−	1.52717i	0.965727	−	1.52717i
$443$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$443$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$444$		0			0
$445$	−1.85640	+	0.225408i	−1.85640	+	0.225408i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.839709	+	0.398447i	−0.839709	+	0.398447i	−0.799443	−	0.600742i	$-0.794872\pi$
$449$	−0.839709	+	0.398447i	−0.839709	+	0.398447i	−0.0402659	+	0.999189i	$0.512821\pi$
$450$	0.987050	−	0.160411i	0.987050	−	0.160411i
$451$		0			0
$452$	0.490585	−	1.46948i	0.490585	−	1.46948i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.444838	+	1.53576i	0.444838	+	1.53576i	0.799443	+	0.600742i	$0.205128\pi$
$457$	0.444838	+	1.53576i	0.444838	+	1.53576i	−0.354605	+	0.935016i	$0.615385\pi$
$458$	−1.90703	+	0.552378i	−1.90703	+	0.552378i
$459$		0			0
$460$		0			0
$461$	1.54795	+	0.0623804i	1.54795	+	0.0623804i	0.799443	−	0.600742i	$-0.205128\pi$
$461$	1.54795	+	0.0623804i	1.54795	+	0.0623804i	0.748511	+	0.663123i	$0.230769\pi$
$462$		0			0
$463$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$463$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$464$	−0.527799	+	0.176205i	−0.527799	+	0.176205i
$465$		0			0
$466$	−0.956491	+	0.918722i	−0.956491	+	0.918722i
$467$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$467$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$468$	0.500000	−	0.866025i	0.500000	−	0.866025i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.192732	−	1.18593i	0.192732	−	1.18593i
$478$		0			0
$479$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$479$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$480$		0			0
$481$	−1.87251	+	0.625134i	−1.87251	+	0.625134i
$482$	−0.956491	+	1.07966i	−0.956491	+	1.07966i
$483$		0			0
$484$	0.500000	+	0.866025i	0.500000	+	0.866025i
$485$	1.26527	+	0.800107i	1.26527	+	0.800107i
$486$		0			0
$487$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$487$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$488$	−0.692724	+	1.19983i	−0.692724	+	1.19983i
$489$		0			0
$490$	0.632445	−	0.774605i	0.632445	−	0.774605i
$491$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$491$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$492$		0			0
$493$	0.940087	−	0.356528i	0.940087	−	0.356528i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$499$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$500$	0.692724	−	0.721202i	0.692724	−	0.721202i
$501$		0			0
$502$		0			0
$503$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$503$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$504$		0			0
$505$	1.07766	−	1.56126i	1.07766	−	1.56126i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.02673	+	1.62364i	−1.02673	+	1.62364i	−0.278217	+	0.960518i	$0.589744\pi$
$509$	−1.02673	+	1.62364i	−1.02673	+	1.62364i	−0.748511	+	0.663123i	$0.769231\pi$
$510$		0			0
$511$		0			0
$512$	0.354605	−	0.935016i	0.354605	−	0.935016i
$513$		0			0
$514$	0.470942	−	0.626710i	0.470942	−	0.626710i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$521$	−0.599417	−	1.58053i	−0.599417	−	1.58053i	−0.799443	−	0.600742i	$-0.794872\pi$
$521$	−0.599417	−	1.58053i	−0.599417	−	1.58053i	0.200026	−	0.979791i	$-0.435897\pi$
$522$	0.511909	−	0.218104i	0.511909	−	0.218104i
$523$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$523$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$	−0.558358	−	1.06386i	−0.558358	−	1.06386i
$531$		0			0
$532$		0			0
$533$	0.171469	−	1.05509i	0.171469	−	1.05509i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.530851	+	0.470293i	−0.530851	+	0.470293i
$539$		0			0
$540$		0			0
$541$	−0.732990	+	0.277987i	−0.732990	+	0.277987i	−0.692724	−	0.721202i	$-0.743590\pi$
$541$	−0.732990	+	0.277987i	−0.732990	+	0.277987i	−0.0402659	+	0.999189i	$0.512821\pi$
$542$		0			0
$543$		0			0
$544$	−0.572188	+	1.71391i	−0.572188	+	1.71391i
$545$	1.48705	−	0.705614i	1.48705	−	0.705614i
$546$		0			0
$547$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$547$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$548$	−0.876221	−	0.912242i	−0.876221	−	0.912242i
$549$	0.593932	−	1.25168i	0.593932	−	1.25168i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.42545	+	0.983917i	−1.42545	+	0.983917i
$555$		0			0
$556$		0			0
$557$	0.278217	+	0.960518i	0.278217	+	0.960518i	0.970942	+	0.239316i	$0.0769231\pi$
$557$	0.278217	+	0.960518i	0.278217	+	0.960518i	−0.692724	+	0.721202i	$0.743590\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.380472	+	1.13965i	−0.380472	+	1.13965i
$563$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$563$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$564$		0			0
$565$	0.607239	+	1.42524i	0.607239	+	1.42524i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.12142	+	0.182248i	−1.12142	+	0.182248i	−0.692724	−	0.721202i	$-0.743590\pi$
$569$	−1.12142	+	0.182248i	−1.12142	+	0.182248i	−0.428693	+	0.903450i	$0.641026\pi$
$570$		0			0
$571$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$571$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$577$	1.99351			1.99351			0.996757	−	0.0804666i	$-0.0256410\pi$
$577$	1.99351			1.99351			0.996757	+	0.0804666i	$0.0256410\pi$
$578$	0.630132	−	2.17547i	0.630132	−	2.17547i
$579$		0			0
$580$	0.278217	−	0.481887i	0.278217	−	0.481887i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$585$	0.200026	+	0.979791i	0.200026	+	0.979791i
$586$	1.07766	+	1.56126i	1.07766	+	1.56126i
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.66849	−	1.05509i	1.66849	−	1.05509i
$593$	1.93559	+	0.477079i	1.93559	+	0.477079i	0.987050	+	0.160411i	$0.0512821\pi$
$593$	1.93559	+	0.477079i	1.93559	+	0.477079i	0.948536	+	0.316668i	$0.102564\pi$
$594$		0			0
$595$		0			0
$596$	1.69705	−	0.979791i	1.69705	−	0.979791i
$597$		0			0
$598$		0			0
$599$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$599$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$600$		0			0
$601$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	−0.748511	−	0.663123i	$-0.769231\pi$
$601$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	0.692724	−	0.721202i	$-0.256410\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$606$		0			0
$607$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$607$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$608$		0			0
$609$		0			0
$610$	−0.277125	−	1.35745i	−0.277125	−	1.35745i
$611$		0			0
$612$	0.432420	−	1.75440i	0.432420	−	1.75440i
$613$	−0.876221	+	1.07318i	−0.876221	+	1.07318i	0.120537	+	0.992709i	$0.461538\pi$
$613$	−0.876221	+	1.07318i	−0.876221	+	1.07318i	−0.996757	+	0.0804666i	$0.974359\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.351915	−	0.431017i	−0.351915	−	0.431017i	0.568065	−	0.822984i	$-0.307692\pi$
$617$	−0.351915	−	0.431017i	−0.351915	−	0.431017i	−0.919979	+	0.391967i	$0.871795\pi$
$618$		0			0
$619$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$619$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$626$	0.892750	−	0.258588i	0.892750	−	0.258588i
$627$		0			0
$628$	−0.157681	−	1.95323i	−0.157681	−	1.95323i
$629$	−2.93559	+	2.02629i	−2.93559	+	2.02629i
$630$		0			0
$631$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$631$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$632$		0			0
$633$		0			0
$634$	0.685430	−	1.44451i	0.685430	−	1.44451i
$635$		0			0
$636$		0			0
$637$	0.799443	+	0.600742i	0.799443	+	0.600742i
$638$		0			0
$639$		0			0
$640$	0.354605	+	0.935016i	0.354605	+	0.935016i
$641$	0.845190	−	0.534466i	0.845190	−	0.534466i	−0.0402659	−	0.999189i	$-0.512821\pi$
$641$	0.845190	−	0.534466i	0.845190	−	0.534466i	0.885456	+	0.464723i	$0.153846\pi$
$642$		0			0
$643$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$643$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$647$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$648$	0.200026	−	0.979791i	0.200026	−	0.979791i
$649$		0			0
$650$	0.748511	+	0.663123i	0.748511	+	0.663123i
$651$		0			0
$652$		0			0
$653$	−1.71942	+	0.992709i	−1.71942	+	0.992709i	−0.799443	+	0.600742i	$0.794872\pi$
$653$	−1.71942	+	0.992709i	−1.71942	+	0.992709i	−0.919979	+	0.391967i	$0.871795\pi$
$654$		0			0
$655$		0			0
$656$	0.0860133	+	1.06547i	0.0860133	+	1.06547i
$657$	0.500000	−	0.866025i	0.500000	−	0.866025i
$658$		0			0
$659$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$659$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$660$		0			0
$661$	−0.0630804	−	0.148055i	−0.0630804	−	0.148055i	0.885456	−	0.464723i	$-0.153846\pi$
$661$	−0.0630804	−	0.148055i	−0.0630804	−	0.148055i	−0.948536	+	0.316668i	$0.897436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.57818	+	1.18593i	−1.57818	+	1.18593i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.565375	+	0.543050i	−0.565375	+	0.543050i	−0.919979	−	0.391967i	$-0.871795\pi$
$673$	−0.565375	+	0.543050i	−0.565375	+	0.543050i	0.354605	+	0.935016i	$0.384615\pi$
$674$	0.338496	−	0.535289i	0.338496	−	0.535289i
$675$		0			0
$676$	0.987050	−	0.160411i	0.987050	−	0.160411i
$677$		−	1.64597i		−	1.64597i	−0.568065	−	0.822984i	$-0.692308\pi$
$677$		−	1.64597i		−	1.64597i	0.568065	−	0.822984i	$-0.307692\pi$
$678$		0			0
$679$		0			0
$680$	−0.708245	−	1.66231i	−0.708245	−	1.66231i
$681$		0			0
$682$		0			0
$683$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$683$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$684$		0			0
$685$	1.26079	+	0.101781i	1.26079	+	0.101781i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.04052	−	0.600742i	1.04052	−	0.600742i
$690$		0			0
$691$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$691$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$692$	0.370750	+	0.302708i	0.370750	+	0.302708i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.232811	−	1.91737i	−0.232811	−	1.91737i
$698$	−1.61950	−	0.935016i	−1.61950	−	0.935016i
$699$		0			0
$700$		0			0
$701$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$701$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.81613	−	0.295150i	−1.81613	−	0.295150i
$707$		0			0
$708$		0			0
$709$	0.116065	+	0.111482i	0.116065	+	0.111482i	0.748511	−	0.663123i	$-0.230769\pi$
$709$	0.116065	+	0.111482i	0.116065	+	0.111482i	−0.632445	+	0.774605i	$0.717949\pi$
$710$		0			0
$711$		0			0
$712$	0.732990	+	1.72039i	0.732990	+	1.72039i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$719$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$720$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$721$		0			0
$722$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$723$		0			0
$724$	−1.83399	+	0.148055i	−1.83399	+	0.148055i
$725$	0.111301	+	0.545190i	0.111301	+	0.545190i
$726$		0			0
$727$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$727$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$728$		0			0
$729$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$730$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$731$		0			0
$732$		0			0
$733$	0.388427	−	0.0957386i	0.388427	−	0.0957386i	−0.0402659	−	0.999189i	$-0.512821\pi$
$733$	0.388427	−	0.0957386i	0.388427	−	0.0957386i	0.428693	+	0.903450i	$0.358974\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.171469	−	1.05509i	−0.171469	−	1.05509i
$739$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$739$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$740$	−0.549229	+	1.89616i	−0.549229	+	1.89616i
$741$		0			0
$742$		0			0
$743$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$743$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$744$		0			0
$745$	−0.620537	+	1.85873i	−0.620537	+	1.85873i
$746$			1.06893i			1.06893i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$751$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$752$		0			0
$753$		0			0
$754$	0.492699	+	0.258588i	0.492699	+	0.258588i
$755$		0			0
$756$		0			0
$757$	0.0385138	−	0.477079i	0.0385138	−	0.477079i	−0.948536	−	0.316668i	$-0.897436\pi$
$757$	0.0385138	−	0.477079i	0.0385138	−	0.477079i	0.987050	−	0.160411i	$-0.0512821\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.06114	+	1.67806i	1.06114	+	1.67806i	0.632445	+	0.774605i	$0.282051\pi$
$761$	1.06114	+	1.67806i	1.06114	+	1.67806i	0.428693	+	0.903450i	$0.358974\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.839709	+	1.59993i	0.839709	+	1.59993i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.0385138	+	0.477079i	−0.0385138	+	0.477079i	0.948536	+	0.316668i	$0.102564\pi$
$769$	−0.0385138	+	0.477079i	−0.0385138	+	0.477079i	−0.987050	+	0.160411i	$0.948718\pi$
$770$		0			0
$771$		0			0
$772$	−0.354228	−	0.185913i	−0.354228	−	0.185913i
$773$	−1.12142	−	0.182248i	−1.12142	−	0.182248i	−0.428693	−	0.903450i	$-0.641026\pi$
$773$	−1.12142	−	0.182248i	−1.12142	−	0.182248i	−0.692724	+	0.721202i	$0.743590\pi$
$774$		0			0
$775$		0			0
$776$	0.416498	−	1.43792i	0.416498	−	1.43792i
$777$		0			0
$778$	−0.846282	+	1.78350i	−0.846282	+	1.78350i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$785$	1.41325	+	1.35745i	1.41325	+	1.35745i
$786$		0			0
$787$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$787$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$788$	1.88546	−	0.464723i	1.88546	−	0.464723i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.34519	−	0.331560i	1.34519	−	0.331560i
$794$	0.213460	+	1.75800i	0.213460	+	1.75800i
$795$		0			0
$796$		0			0
$797$	0.0747894	+	0.926432i	0.0747894	+	0.926432i	0.919979	+	0.391967i	$0.128205\pi$
$797$	0.0747894	+	0.926432i	0.0747894	+	0.926432i	−0.845190	+	0.534466i	$0.820513\pi$
$798$		0			0
$799$		0			0
$800$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$801$	−0.869047	−	1.65583i	−0.869047	−	1.65583i
$802$	−0.200557	−	0.600742i	−0.200557	−	0.600742i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.79944	−	0.600742i	−1.79944	−	0.600742i
$809$	−0.368039	+	0.156807i	−0.368039	+	0.156807i	−0.568065	−	0.822984i	$-0.692308\pi$
$809$	−0.368039	+	0.156807i	−0.368039	+	0.156807i	0.200026	+	0.979791i	$0.435897\pi$
$810$	0.500000	+	0.866025i	0.500000	+	0.866025i
$811$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$811$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.14856	+	1.29646i	1.14856	+	1.29646i
$819$		0			0
$820$	−0.770916	−	0.740475i	−0.770916	−	0.740475i
$821$	0.796732	+	1.06026i	0.796732	+	1.06026i	0.996757	+	0.0804666i	$0.0256410\pi$
$821$	0.796732	+	1.06026i	0.796732	+	1.06026i	−0.200026	+	0.979791i	$0.564103\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$827$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$828$		0			0
$829$	1.01121	−	1.23850i	1.01121	−	1.23850i	0.0402659	−	0.999189i	$-0.487179\pi$
$829$	1.01121	−	1.23850i	1.01121	−	1.23850i	0.970942	−	0.239316i	$-0.0769231\pi$
$830$		0			0
$831$		0			0
$832$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$833$	1.68948	+	0.640736i	1.68948	+	0.640736i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$839$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$840$		0			0
$841$	−0.295961	−	0.623724i	−0.295961	−	0.623724i
$842$	1.04027	−	0.999189i	1.04027	−	0.999189i
$843$		0			0
$844$		0			0
$845$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$846$		0			0
$847$		0			0
$848$	−0.866514	+	0.832298i	−0.866514	+	0.832298i
$849$		0			0
$850$	1.63245	+	0.774605i	1.63245	+	0.774605i
$851$		0			0
$852$		0			0
$853$	−0.103346	+	0.851134i	−0.103346	+	0.851134i	0.845190	+	0.534466i	$0.179487\pi$
$853$	−0.103346	+	0.851134i	−0.103346	+	0.851134i	−0.948536	+	0.316668i	$0.897436\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.34867	+	0.511484i	1.34867	+	0.511484i	0.919979	−	0.391967i	$-0.128205\pi$
$857$	1.34867	+	0.511484i	1.34867	+	0.511484i	0.428693	+	0.903450i	$0.358974\pi$
$858$		0			0
$859$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$859$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$863$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$864$		0			0
$865$	−0.478243	+	0.0192725i	−0.478243	+	0.0192725i
$866$	0.318483	−	0.0386709i	0.318483	−	0.0386709i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.09148	−	1.23202i	−1.09148	−	1.23202i
$873$	−0.299443	+	1.46677i	−0.299443	+	1.46677i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.60339	+	0.535289i	−1.60339	+	0.535289i	−0.970942	−	0.239316i	$-0.923077\pi$
$877$	−1.60339	+	0.535289i	−1.60339	+	0.535289i	−0.632445	+	0.774605i	$0.717949\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.55512	+	0.662573i	−1.55512	+	0.662573i	−0.987050	−	0.160411i	$-0.948718\pi$
$881$	−1.55512	+	0.662573i	−1.55512	+	0.662573i	−0.568065	+	0.822984i	$0.692308\pi$
$882$	0.948536	+	0.316668i	0.948536	+	0.316668i
$883$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$883$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$884$	1.63245	−	0.774605i	1.63245	−	0.774605i
$885$		0			0
$886$		0			0
$887$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$887$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$888$		0			0
$889$		0			0
$890$	−1.68948	−	0.801669i	−1.68948	−	0.801669i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.922670	−	0.112032i	−0.922670	−	0.112032i
$899$		0			0
$900$	0.919979	+	0.391967i	0.919979	+	0.391967i
$901$	1.50388	−	1.56570i	1.50388	−	1.56570i
$902$		0			0
$903$		0			0
$904$	1.20003	−	0.979791i	1.20003	−	0.979791i
$905$	1.27458	−	1.32698i	1.27458	−	1.32698i
$906$		0			0
$907$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$907$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$908$		0			0
$909$	1.84195	+	0.453999i	1.84195	+	0.453999i
$910$		0			0
$911$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$911$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$912$		0			0
$913$		0			0
$914$	−0.444838	+	1.53576i	−0.444838	+	1.53576i
$915$		0			0
$916$	−1.90703	−	0.552378i	−1.90703	−	0.552378i
$917$		0			0
$918$		0			0
$919$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$919$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$920$		0			0
$921$		0			0
$922$	1.27497	+	0.880052i	1.27497	+	0.880052i
$923$		0			0
$924$		0			0
$925$	−0.846282	−	1.78350i	−0.846282	−	1.78350i
$926$		0			0
$927$		0			0
$928$	−0.540266	−	0.133164i	−0.540266	−	0.133164i
$929$	−0.965727	−	1.52717i	−0.965727	−	1.52717i	−0.845190	−	0.534466i	$-0.820513\pi$
$929$	−0.965727	−	1.52717i	−0.965727	−	1.52717i	−0.120537	−	0.992709i	$-0.538462\pi$
$930$		0			0
$931$		0			0
$932$	−1.29944	+	0.265283i	−1.29944	+	0.265283i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.885456	−	0.464723i	0.885456	−	0.464723i
$937$	−1.42545	−	0.983917i	−1.42545	−	0.983917i	−0.996757	−	0.0804666i	$-0.974359\pi$
$937$	−1.42545	−	0.983917i	−1.42545	−	0.983917i	−0.428693	−	0.903450i	$-0.641026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.222431	+	0.423807i	−0.222431	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$941$	−0.222431	+	0.423807i	−0.222431	+	0.423807i	0.748511	+	0.663123i	$0.230769\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$947$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$948$		0			0
$949$	0.987050	−	0.160411i	0.987050	−	0.160411i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.187607	−	0.440331i	0.187607	−	0.440331i	−0.799443	−	0.600742i	$-0.794872\pi$
$953$	0.187607	−	0.440331i	0.187607	−	0.440331i	0.987050	+	0.160411i	$0.0512821\pi$
$954$	0.796732	−	0.899324i	0.796732	−	0.899324i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$962$	−1.91674	−	0.472433i	−1.91674	−	0.472433i
$963$		0			0
$964$	−1.38546	+	0.401302i	−1.38546	+	0.401302i
$965$	0.388427	−	0.0957386i	0.388427	−	0.0957386i
$966$		0			0
$967$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$967$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$968$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$969$		0			0
$970$	0.641762	+	1.35248i	0.641762	+	1.35248i
$971$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$971$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.22675	+	0.643850i	−1.22675	+	0.643850i
$977$	0.379463	+	0.126683i	0.379463	+	0.126683i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	0.379463	+	0.126683i	0.379463	+	0.126683i	−0.120537	+	0.992709i	$0.538462\pi$
$978$		0			0
$979$		0			0
$980$	0.948536	−	0.316668i	0.948536	−	0.316668i
$981$	1.18708	+	1.14020i	1.18708	+	1.14020i
$982$		0			0
$983$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$983$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$984$		0			0
$985$	−1.10312	+	1.59814i	−1.10312	+	1.59814i
$986$	0.985104	+	0.201110i	0.985104	+	0.201110i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.930676	−	1.23850i	0.930676	−	1.23850i	−0.0402659	−	0.999189i	$-0.512821\pi$
$997$	0.930676	−	1.23850i	0.930676	−	1.23850i	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.co.a.3299.1	yes	24
4.3	odd	2	CM	3380.1.co.a.3299.1	yes	24
5.4	even	2		3380.1.co.b.3299.1	yes	24
20.19	odd	2		3380.1.co.b.3299.1	yes	24
169.121	even	78		3380.1.co.b.459.1	yes	24
676.459	odd	78		3380.1.co.b.459.1	yes	24
845.459	even	78	inner	3380.1.co.a.459.1	✓	24
3380.459	odd	78	inner	3380.1.co.a.459.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.co.a.459.1	✓	24	845.459	even	78	inner
3380.1.co.a.459.1	✓	24	3380.459	odd	78	inner
3380.1.co.a.3299.1	yes	24	1.1	even	1	trivial
3380.1.co.a.3299.1	yes	24	4.3	odd	2	CM
3380.1.co.b.459.1	yes	24	169.121	even	78
3380.1.co.b.459.1	yes	24	676.459	odd	78
3380.1.co.b.3299.1	yes	24	5.4	even	2
3380.1.co.b.3299.1	yes	24	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.co (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.845190 + 0.534466i) q^{2} +(0.428693 + 0.903450i) q^{4} +(-0.919979 - 0.391967i) q^{5} +(-0.120537 + 0.992709i) q^{8} +(0.0402659 - 0.999189i) q^{9} +O(q^{10})\)	\(q+(0.845190 + 0.534466i) q^{2} +(0.428693 + 0.903450i) q^{4} +(-0.919979 - 0.391967i) q^{5} +(-0.120537 + 0.992709i) q^{8} +(0.0402659 - 0.999189i) q^{9} +(-0.568065 - 0.822984i) q^{10} +(0.799443 - 0.600742i) q^{13} +(-0.632445 + 0.774605i) q^{16} +(1.08548 - 1.44451i) q^{17} +(0.568065 - 0.822984i) q^{18} +(-0.0402659 - 0.999189i) q^{20} +(0.692724 + 0.721202i) q^{25} +(0.996757 - 0.0804666i) q^{26} +(0.470293 + 0.297395i) q^{29} +(-0.948536 + 0.316668i) q^{32} +(1.68948 - 0.640736i) q^{34} +(0.919979 - 0.391967i) q^{36} +(-1.87251 - 0.625134i) q^{37} +(0.500000 - 0.866025i) q^{40} +(0.770916 - 0.740475i) q^{41} +(-0.428693 + 0.903450i) q^{45} +(0.278217 + 0.960518i) q^{49} +(0.200026 + 0.979791i) q^{50} +(0.885456 + 0.464723i) q^{52} +(1.19272 + 0.144823i) q^{53} +(0.238540 + 0.502711i) q^{58} +(1.27458 + 0.543050i) q^{61} +(-0.970942 - 0.239316i) q^{64} +(-0.970942 + 0.239316i) q^{65} +(1.77038 + 0.361427i) q^{68} +(0.987050 + 0.160411i) q^{72} +(0.885456 + 0.464723i) q^{73} +(-1.24851 - 1.52915i) q^{74} +(0.885456 - 0.464723i) q^{80} +(-0.996757 - 0.0804666i) q^{81} +(1.04733 - 0.213814i) q^{82} +(-1.56482 + 0.903450i) q^{85} +(1.61950 - 0.935016i) q^{89} +(-0.845190 + 0.534466i) q^{90} +(-1.47764 - 0.240139i) q^{97} +(-0.278217 + 0.960518i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} + q^{4} + q^{5} + 2 q^{8} - q^{9}+O(q^{10}) \) 24 * q - q^2 + q^4 + q^5 + 2 * q^8 - q^9	\( 24 q - q^{2} + q^{4} + q^{5} + 2 q^{8} - q^{9} + 2 q^{10} + q^{13} + q^{16} - 3 q^{17} - 2 q^{18} + q^{20} + q^{25} - q^{26} + q^{29} - q^{32} - q^{36} + q^{37} + 12 q^{40} + 3 q^{41} - q^{45} + q^{49} - q^{50} - 2 q^{52} + 13 q^{53} - q^{58} + q^{61} - 2 q^{64} - 2 q^{65} + 3 q^{68} + q^{72} - 2 q^{73} - 14 q^{74} - 2 q^{80} + q^{81} + 3 q^{82} + 3 q^{85} + q^{90} + 2 q^{97} - q^{98}+O(q^{100}) \) 24 * q - q^2 + q^4 + q^5 + 2 * q^8 - q^9 + 2 * q^10 + q^13 + q^16 - 3 * q^17 - 2 * q^18 + q^20 + q^25 - q^26 + q^29 - q^32 - q^36 + q^37 + 12 * q^40 + 3 * q^41 - q^45 + q^49 - q^50 - 2 * q^52 + 13 * q^53 - q^58 + q^61 - 2 * q^64 - 2 * q^65 + 3 * q^68 + q^72 - 2 * q^73 - 14 * q^74 - 2 * q^80 + q^81 + 3 * q^82 + 3 * q^85 + q^90 + 2 * q^97 - q^98

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