LMFDB - Embedded newform 3380.1.co.b.2799.1

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			2799.1
Root			$0.278217 + 0.960518i$ of defining polynomial
Character	$\chi$	$=$	3380.2799
Dual form			3380.1.co.b.2519.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.996757 + 0.0804666i) q^{2} +(0.987050 - 0.160411i) q^{4} +(0.919979 + 0.391967i) q^{5} +(-0.970942 + 0.239316i) q^{8} +(-0.428693 + 0.903450i) q^{9} +O(q^{10})$	\(q+(-0.996757 + 0.0804666i) q^{2} +(0.987050 - 0.160411i) q^{4} +(0.919979 + 0.391967i) q^{5} +(-0.970942 + 0.239316i) q^{8} +(-0.428693 + 0.903450i) q^{9} +(-0.948536 - 0.316668i) q^{10} +(-0.692724 + 0.721202i) q^{13} +(0.948536 - 0.316668i) q^{16} +(0.231378 - 0.222242i) q^{17} +(0.354605 - 0.935016i) q^{18} +(0.970942 + 0.239316i) q^{20} +(0.692724 + 0.721202i) q^{25} +(0.632445 - 0.774605i) q^{26} +(-0.0802707 + 0.00648012i) q^{29} +(-0.919979 + 0.391967i) q^{32} +(-0.212745 + 0.240139i) q^{34} +(-0.278217 + 0.960518i) q^{36} +(0.368039 + 0.156807i) q^{37} +(-0.987050 - 0.160411i) q^{40} +(-0.0860133 - 0.136019i) q^{41} +(-0.748511 + 0.663123i) q^{45} +(-0.0402659 + 0.999189i) q^{49} +(-0.748511 - 0.663123i) q^{50} +(-0.568065 + 0.822984i) q^{52} +(0.345190 + 1.40049i) q^{53} +(0.0794890 - 0.0129182i) q^{58} +(0.470293 + 1.62364i) q^{61} +(0.885456 - 0.464723i) q^{64} +(-0.919979 + 0.391967i) q^{65} +(0.192732 - 0.256479i) q^{68} +(0.200026 - 0.979791i) q^{72} +(-0.568065 + 0.822984i) q^{73} +(-0.379463 - 0.126683i) q^{74} +(0.996757 + 0.0804666i) q^{80} +(-0.632445 - 0.774605i) q^{81} +(0.0966793 + 0.128657i) q^{82} +(0.299974 - 0.113765i) q^{85} +(-1.14856 - 0.663123i) q^{89} +(0.692724 - 0.721202i) q^{90} +(0.0482209 - 0.236201i) q^{97} +(-0.0402659 - 0.999189i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - q^{10} - q^{13} + q^{16} + 3 q^{17} + 2 q^{18} + 2 q^{20} + q^{25} - q^{26} + q^{29} + q^{32} - q^{36} - q^{37} - q^{40} + 3 q^{41} - 2 q^{45} + q^{49} - 2 q^{50} + 2 q^{52} - 13 q^{53} + q^{58} + q^{61} - 2 q^{64} + q^{65} - 3 q^{68} - q^{72} + 2 q^{73} - 14 q^{74} - q^{80} + q^{81} - 3 q^{82} + 13 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 - q^10 - q^13 + q^16 + 3 * q^17 + 2 * q^18 + 2 * q^20 + q^25 - q^26 + q^29 + q^32 - q^36 - q^37 - q^40 + 3 * q^41 - 2 * q^45 + q^49 - 2 * q^50 + 2 * q^52 - 13 * q^53 + q^58 + q^61 - 2 * q^64 + q^65 - 3 * q^68 - q^72 + 2 * q^73 - 14 * q^74 - q^80 + q^81 - 3 * q^82 + 13 * 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 $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{37}{78}\right)$

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 $

$2$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$2$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$3$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$3$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$4$	0.987050	−	0.160411i	0.987050	−	0.160411i
$5$	0.919979	+	0.391967i	0.919979	+	0.391967i
$5$	0.919979	+	0.391967i	0.919979	+	0.391967i
$6$		0			0
$7$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$7$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$8$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$9$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$10$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$11$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$11$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$12$		0			0
$13$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$13$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$14$		0			0
$15$		0			0
$16$	0.948536	−	0.316668i	0.948536	−	0.316668i
$17$	0.231378	−	0.222242i	0.231378	−	0.222242i	−0.568065	−	0.822984i	$-0.692308\pi$
$17$	0.231378	−	0.222242i	0.231378	−	0.222242i	0.799443	+	0.600742i	$0.205128\pi$
$18$	0.354605	−	0.935016i	0.354605	−	0.935016i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	0.970942	+	0.239316i	0.970942	+	0.239316i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	0.692724	+	0.721202i	0.692724	+	0.721202i
$26$	0.632445	−	0.774605i	0.632445	−	0.774605i
$27$		0			0
$28$		0			0
$29$	−0.0802707	+	0.00648012i	−0.0802707	+	0.00648012i	−0.120537	−	0.992709i	$-0.538462\pi$
$29$	−0.0802707	+	0.00648012i	−0.0802707	+	0.00648012i	0.0402659	+	0.999189i	$0.487179\pi$
$30$		0			0
$31$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$31$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$32$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$33$		0			0
$34$	−0.212745	+	0.240139i	−0.212745	+	0.240139i
$35$		0			0
$36$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$37$	0.368039	+	0.156807i	0.368039	+	0.156807i	0.568065	−	0.822984i	$-0.307692\pi$
$37$	0.368039	+	0.156807i	0.368039	+	0.156807i	−0.200026	+	0.979791i	$0.564103\pi$
$38$		0			0
$39$		0			0
$40$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$41$	−0.0860133	−	0.136019i	−0.0860133	−	0.136019i	0.799443	−	0.600742i	$-0.205128\pi$
$41$	−0.0860133	−	0.136019i	−0.0860133	−	0.136019i	−0.885456	+	0.464723i	$0.846154\pi$
$42$		0			0
$43$		0			0		−0.391967	−	0.919979i	$-0.628205\pi$
$43$		0			0		0.391967	+	0.919979i	$0.371795\pi$
$44$		0			0
$45$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$46$		0			0
$47$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$47$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$48$		0			0
$49$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$50$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$51$		0			0
$52$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$53$	0.345190	+	1.40049i	0.345190	+	1.40049i	0.845190	+	0.534466i	$0.179487\pi$
$53$	0.345190	+	1.40049i	0.345190	+	1.40049i	−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.0794890	−	0.0129182i	0.0794890	−	0.0129182i
$59$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$59$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$60$		0			0
$61$	0.470293	+	1.62364i	0.470293	+	1.62364i	0.748511	+	0.663123i	$0.230769\pi$
$61$	0.470293	+	1.62364i	0.470293	+	1.62364i	−0.278217	+	0.960518i	$0.589744\pi$
$62$		0			0
$63$		0			0
$64$	0.885456	−	0.464723i	0.885456	−	0.464723i
$65$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$66$		0			0
$67$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$67$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$68$	0.192732	−	0.256479i	0.192732	−	0.256479i
$69$		0			0
$70$		0			0
$71$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$71$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$72$	0.200026	−	0.979791i	0.200026	−	0.979791i
$73$	−0.568065	+	0.822984i	−0.568065	+	0.822984i	−0.996757	−	0.0804666i	$-0.974359\pi$
$73$	−0.568065	+	0.822984i	−0.568065	+	0.822984i	0.428693	+	0.903450i	$0.358974\pi$
$74$	−0.379463	−	0.126683i	−0.379463	−	0.126683i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$79$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$80$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$81$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$82$	0.0966793	+	0.128657i	0.0966793	+	0.128657i
$83$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$83$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$84$		0			0
$85$	0.299974	−	0.113765i	0.299974	−	0.113765i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.14856	−	0.663123i	−1.14856	−	0.663123i	−0.200026	−	0.979791i	$-0.564103\pi$
$89$	−1.14856	−	0.663123i	−1.14856	−	0.663123i	−0.948536	+	0.316668i	$0.897436\pi$
$90$	0.692724	−	0.721202i	0.692724	−	0.721202i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0482209	−	0.236201i	0.0482209	−	0.236201i	−0.948536	−	0.316668i	$-0.897436\pi$
$97$	0.0482209	−	0.236201i	0.0482209	−	0.236201i	0.996757	+	0.0804666i	$0.0256410\pi$
$98$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$99$		0			0
$100$	0.799443	+	0.600742i	0.799443	+	0.600742i
$101$	−1.47094	−	1.10534i	−1.47094	−	1.10534i	−0.970942	−	0.239316i	$-0.923077\pi$
$101$	−1.47094	−	1.10534i	−1.47094	−	1.10534i	−0.500000	−	0.866025i	$-0.666667\pi$
$102$		0			0
$103$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$103$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$104$	0.500000	−	0.866025i	0.500000	−	0.866025i
$105$		0			0
$106$	−0.456763	−	1.36817i	−0.456763	−	1.36817i
$107$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$107$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$108$		0			0
$109$	1.85640	+	0.225408i	1.85640	+	0.225408i	0.970942	−	0.239316i	$-0.0769231\pi$
$109$	1.85640	+	0.225408i	1.85640	+	0.225408i	0.885456	+	0.464723i	$0.153846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.338496	+	0.535289i	−0.338496	+	0.535289i	−0.970942	−	0.239316i	$-0.923077\pi$
$113$	−0.338496	+	0.535289i	−0.338496	+	0.535289i	0.632445	+	0.774605i	$0.282051\pi$
$114$		0			0
$115$		0			0
$116$	−0.0781918	+	0.0192725i	−0.0781918	+	0.0192725i
$117$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.632445	−	0.774605i	0.632445	−	0.774605i
$122$	−0.599417	−	1.58053i	−0.599417	−	1.58053i
$123$		0			0
$124$		0			0
$125$	0.354605	+	0.935016i	0.354605	+	0.935016i
$126$		0			0
$127$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$127$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$128$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$129$		0			0
$130$	0.885456	−	0.464723i	0.885456	−	0.464723i
$131$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$131$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.171469	+	0.271156i	−0.171469	+	0.271156i
$137$	1.74527	−	0.743589i	1.74527	−	0.743589i	0.748511	−	0.663123i	$-0.230769\pi$
$137$	1.74527	−	0.743589i	1.74527	−	0.743589i	0.996757	−	0.0804666i	$-0.0256410\pi$
$138$		0			0
$139$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$139$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$145$	−0.0763874	−	0.0255019i	−0.0763874	−	0.0255019i
$146$	0.500000	−	0.866025i	0.500000	−	0.866025i
$147$		0			0
$148$	0.388427	+	0.0957386i	0.388427	+	0.0957386i
$149$	−0.930676	−	0.759873i	−0.930676	−	0.759873i	0.0402659	−	0.999189i	$-0.487179\pi$
$149$	−0.930676	−	0.759873i	−0.930676	−	0.759873i	−0.970942	+	0.239316i	$0.923077\pi$
$150$		0			0
$151$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$151$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$152$		0			0
$153$	0.101594	+	0.304312i	0.101594	+	0.304312i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.796732	+	0.899324i	0.796732	+	0.899324i	0.996757	−	0.0804666i	$-0.0256410\pi$
$157$	0.796732	+	0.899324i	0.796732	+	0.899324i	−0.200026	+	0.979791i	$0.564103\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000			−1.00000
$161$		0			0
$162$	0.692724	+	0.721202i	0.692724	+	0.721202i
$163$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$163$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$164$	−0.106718	−	0.120460i	−0.106718	−	0.120460i
$165$		0			0
$166$		0			0
$167$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$167$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$168$		0			0
$169$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$170$	−0.289847	+	0.137534i	−0.289847	+	0.137534i
$171$		0			0
$172$		0			0
$173$	0.149094	+	0.917410i	0.149094	+	0.917410i	0.948536	+	0.316668i	$0.102564\pi$
$173$	0.149094	+	0.917410i	0.149094	+	0.917410i	−0.799443	+	0.600742i	$0.794872\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.19820	+	0.568552i	1.19820	+	0.568552i
$179$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$179$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$180$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$181$	0.416498	−	0.368985i	0.416498	−	0.368985i	−0.428693	−	0.903450i	$-0.641026\pi$
$181$	0.416498	−	0.368985i	0.416498	−	0.368985i	0.845190	+	0.534466i	$0.179487\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.277125	+	0.288518i	0.277125	+	0.288518i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−1.10759	+	1.15312i	−1.10759	+	1.15312i	−0.120537	+	0.992709i	$0.538462\pi$
$193$	−1.10759	+	1.15312i	−1.10759	+	1.15312i	−0.987050	+	0.160411i	$0.948718\pi$
$194$	−0.0290582	+	0.239316i	−0.0290582	+	0.239316i
$195$		0			0
$196$	0.120537	+	0.992709i	0.120537	+	0.992709i
$197$	−1.41574	−	1.06386i	−1.41574	−	1.06386i	−0.987050	−	0.160411i	$-0.948718\pi$
$197$	−1.41574	−	1.06386i	−1.41574	−	1.06386i	−0.428693	−	0.903450i	$-0.641026\pi$
$198$		0			0
$199$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$199$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$200$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$201$		0			0
$202$	1.55512	+	0.983395i	1.55512	+	0.983395i
$203$		0			0
$204$		0			0
$205$	−0.0258155	−	0.158849i	−0.0258155	−	0.158849i
$206$		0			0
$207$		0			0
$208$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$209$		0			0
$210$		0			0
$211$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$211$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$212$	0.565375	+	1.32698i	0.565375	+	1.32698i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.86852	−	0.0752986i	−1.86852	−	0.0752986i
$219$		0			0
$220$		0			0
$221$			0.320823i			0.320823i
$222$		0			0
$223$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$223$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$224$		0			0
$225$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$226$	0.294326	−	0.560791i	0.294326	−	0.560791i
$227$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$227$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$228$		0			0
$229$	0.475142	+	0.0576926i	0.475142	+	0.0576926i	0.354605	−	0.935016i	$-0.384615\pi$
$229$	0.475142	+	0.0576926i	0.475142	+	0.0576926i	0.120537	+	0.992709i	$0.461538\pi$
$230$		0			0
$231$		0			0
$232$	0.0763874	−	0.0255019i	0.0763874	−	0.0255019i
$233$	0.922670	+	1.75800i	0.922670	+	1.75800i	0.568065	+	0.822984i	$0.307692\pi$
$233$	0.922670	+	1.75800i	0.922670	+	1.75800i	0.354605	+	0.935016i	$0.384615\pi$
$234$	0.428693	+	0.903450i	0.428693	+	0.903450i
$235$		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
$241$	−1.04733	−	0.213814i	−1.04733	−	0.213814i	−0.354605	−	0.935016i	$-0.615385\pi$
$241$	−1.04733	−	0.213814i	−1.04733	−	0.213814i	−0.692724	+	0.721202i	$0.743590\pi$
$242$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$243$		0			0
$244$	0.724653	+	1.52717i	0.724653	+	1.52717i
$245$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$251$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$251$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.799443	−	0.600742i	0.799443	−	0.600742i
$257$	1.48804	−	1.21495i	1.48804	−	1.21495i	0.568065	−	0.822984i	$-0.307692\pi$
$257$	1.48804	−	1.21495i	1.48804	−	1.21495i	0.919979	−	0.391967i	$-0.128205\pi$
$258$		0			0
$259$		0			0
$260$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$261$	0.0285570	−	0.0752986i	0.0285570	−	0.0752986i
$262$		0			0
$263$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$263$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$264$		0			0
$265$	−0.231378	+	1.42373i	−0.231378	+	1.42373i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.0602790	+	1.49581i	0.0602790	+	1.49581i	0.692724	+	0.721202i	$0.256410\pi$
$269$	0.0602790	+	1.49581i	0.0602790	+	1.49581i	−0.632445	+	0.774605i	$0.717949\pi$
$270$		0			0
$271$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$271$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$272$	0.149094	−	0.284074i	0.149094	−	0.284074i
$273$		0			0
$274$	−1.67977	+	0.881614i	−1.67977	+	0.881614i
$275$		0			0
$276$		0			0
$277$	1.66367	−	0.481887i	1.66367	−	0.481887i	0.692724	−	0.721202i	$-0.256410\pi$
$277$	1.66367	−	0.481887i	1.66367	−	0.481887i	0.970942	+	0.239316i	$0.0769231\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.670319	−	1.27719i	0.670319	−	1.27719i	−0.278217	−	0.960518i	$-0.589744\pi$
$281$	0.670319	−	1.27719i	0.670319	−	1.27719i	0.948536	−	0.316668i	$-0.102564\pi$
$282$		0			0
$283$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$283$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$289$	−0.0361215	+	0.896345i	−0.0361215	+	0.896345i
$290$	0.0781918	+	0.0192725i	0.0781918	+	0.0192725i
$291$		0			0
$292$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$293$	−0.511909	−	1.76731i	−0.511909	−	1.76731i	−0.632445	−	0.774605i	$-0.717949\pi$
$293$	−0.511909	−	1.76731i	−0.511909	−	1.76731i	0.120537	−	0.992709i	$-0.461538\pi$
$294$		0			0
$295$		0			0
$296$	−0.394871	−	0.0641728i	−0.394871	−	0.0641728i
$297$		0			0
$298$	0.988802	+	0.682521i	0.988802	+	0.682521i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.203753	+	1.67806i	−0.203753	+	1.67806i
$306$	−0.125752	−	0.295150i	−0.125752	−	0.295150i
$307$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$307$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$311$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$312$		0			0
$313$	1.63397	+	0.198399i	1.63397	+	0.198399i	0.885456	−	0.464723i	$-0.153846\pi$
$313$	1.63397	+	0.198399i	1.63397	+	0.198399i	0.748511	+	0.663123i	$0.230769\pi$
$314$	−0.866514	−	0.832298i	−0.866514	−	0.832298i
$315$		0			0
$316$		0			0
$317$	−1.34519	−	0.331560i	−1.34519	−	0.331560i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.34519	−	0.331560i	−1.34519	−	0.331560i	−0.845190	+	0.534466i	$0.820513\pi$
$318$		0			0
$319$		0			0
$320$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$325$	−1.00000			−1.00000
$326$		0			0
$327$		0			0
$328$	0.116065	+	0.111482i	0.116065	+	0.111482i
$329$		0			0
$330$		0			0
$331$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$331$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$332$		0			0
$333$	−0.299443	+	0.265283i	−0.299443	+	0.265283i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	0.783933i		−	0.783933i	−0.919979	−	0.391967i	$-0.871795\pi$
$337$		−	0.783933i		−	0.783933i	0.919979	−	0.391967i	$-0.128205\pi$
$338$	0.120537	+	0.992709i	0.120537	+	0.992709i
$339$		0			0
$340$	0.277840	−	0.160411i	0.277840	−	0.160411i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.222431	−	0.902438i	−0.222431	−	0.902438i
$347$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$347$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$348$		0			0
$349$	1.19820	−	0.568552i	1.19820	−	0.568552i	0.278217	−	0.960518i	$-0.410256\pi$
$349$	1.19820	−	0.568552i	1.19820	−	0.568552i	0.919979	+	0.391967i	$0.128205\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.154810	−	0.534466i	0.154810	−	0.534466i	−0.845190	−	0.534466i	$-0.820513\pi$
$353$	0.154810	−	0.534466i	0.154810	−	0.534466i	1.00000			$0$
$354$		0			0
$355$		0			0
$356$	−1.24006	−	0.470293i	−1.24006	−	0.470293i
$357$		0			0
$358$		0			0
$359$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$359$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$360$	0.568065	−	0.822984i	0.568065	−	0.822984i
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$	−0.385456	+	0.401302i	−0.385456	+	0.401302i
$363$		0			0
$364$		0			0
$365$	−0.845190	+	0.534466i	−0.845190	+	0.534466i
$366$		0			0
$367$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$367$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$368$		0			0
$369$	0.159760	−	0.0193983i	0.159760	−	0.0193983i
$370$	−0.299443	−	0.265283i	−0.299443	−	0.265283i
$371$		0			0
$372$		0			0
$373$	−0.0129497	+	0.160411i	−0.0129497	+	0.160411i	0.987050	+	0.160411i	$0.0512821\pi$
$373$	−0.0129497	+	0.160411i	−0.0129497	+	0.160411i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.0509320	−	0.0623804i	0.0509320	−	0.0623804i
$378$		0			0
$379$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$379$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$383$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$384$		0			0
$385$		0			0
$386$	1.01121	−	1.23850i	1.01121	−	1.23850i
$387$		0			0
$388$	0.00970705	−	0.240878i	0.00970705	−	0.240878i
$389$	0.388427	+	0.0957386i	0.388427	+	0.0957386i	0.428693	−	0.903450i	$-0.358974\pi$
$389$	0.388427	+	0.0957386i	0.388427	+	0.0957386i	−0.0402659	+	0.999189i	$0.512821\pi$
$390$		0			0
$391$		0			0
$392$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$393$		0			0
$394$	1.49676	+	0.946492i	1.49676	+	0.946492i
$395$		0			0
$396$		0			0
$397$	1.07766	+	0.359776i	1.07766	+	0.359776i	0.799443	−	0.600742i	$-0.205128\pi$
$397$	1.07766	+	0.359776i	1.07766	+	0.359776i	0.278217	+	0.960518i	$0.410256\pi$
$398$		0			0
$399$		0			0
$400$	0.885456	+	0.464723i	0.885456	+	0.464723i
$401$	−0.248247	−	0.743589i	−0.248247	−	0.743589i	−0.996757	−	0.0804666i	$-0.974359\pi$
$401$	−0.248247	−	0.743589i	−0.248247	−	0.743589i	0.748511	−	0.663123i	$-0.230769\pi$
$402$		0			0
$403$		0			0
$404$	−1.62920	−	0.855072i	−1.62920	−	0.855072i
$405$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.73065	+	0.0697427i	−1.73065	+	0.0697427i	−0.885456	−	0.464723i	$-0.846154\pi$
$409$	−1.73065	+	0.0697427i	−1.73065	+	0.0697427i	−0.845190	+	0.534466i	$0.820513\pi$
$410$	0.0385138	+	0.156257i	0.0385138	+	0.156257i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.354605	−	0.935016i	0.354605	−	0.935016i
$417$		0			0
$418$		0			0
$419$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$419$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$420$		0			0
$421$	0.496757	+	0.946492i	0.496757	+	0.946492i	0.996757	+	0.0804666i	$0.0256410\pi$
$421$	0.496757	+	0.946492i	0.496757	+	0.946492i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	−0.670319	−	1.27719i	−0.670319	−	1.27719i
$425$	0.320562	+	0.0129182i	0.320562	+	0.0129182i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$431$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$432$		0			0
$433$	0.620537	−	1.85873i	0.620537	−	1.85873i	0.120537	−	0.992709i	$-0.461538\pi$
$433$	0.620537	−	1.85873i	0.620537	−	1.85873i	0.500000	−	0.866025i	$-0.333333\pi$
$434$		0			0
$435$		0			0
$436$	1.86852	−	0.0752986i	1.86852	−	0.0752986i
$437$		0			0
$438$		0			0
$439$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$439$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$440$		0			0
$441$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$442$	−0.0258155	−	0.319782i	−0.0258155	−	0.319782i
$443$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$443$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$444$		0			0
$445$	−0.796732	−	1.06026i	−0.796732	−	1.06026i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.264032	−	1.62465i	−0.264032	−	1.62465i	−0.692724	−	0.721202i	$-0.743590\pi$
$449$	−0.264032	−	1.62465i	−0.264032	−	1.62465i	0.428693	−	0.903450i	$-0.358974\pi$
$450$	0.919979	−	0.391967i	0.919979	−	0.391967i
$451$		0			0
$452$	−0.248247	+	0.582656i	−0.248247	+	0.582656i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.0557864	−	1.38433i	0.0557864	−	1.38433i	−0.692724	−	0.721202i	$-0.743590\pi$
$457$	0.0557864	−	1.38433i	0.0557864	−	1.38433i	0.748511	−	0.663123i	$-0.230769\pi$
$458$	−0.478243	−	0.0192725i	−0.478243	−	0.0192725i
$459$		0			0
$460$		0			0
$461$	0.572188	+	0.271506i	0.572188	+	0.271506i	0.692724	−	0.721202i	$-0.256410\pi$
$461$	0.572188	+	0.271506i	0.572188	+	0.271506i	−0.120537	+	0.992709i	$0.538462\pi$
$462$		0			0
$463$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$463$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$464$	−0.0740877	+	0.0315658i	−0.0740877	+	0.0315658i
$465$		0			0
$466$	−1.06114	−	1.67806i	−1.06114	−	1.67806i
$467$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$467$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$468$	−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
$477$	−1.41325	−	0.288518i	−1.41325	−	0.288518i
$478$		0			0
$479$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$479$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$480$		0			0
$481$	−0.368039	+	0.156807i	−0.368039	+	0.156807i
$482$	1.06114	+	0.128845i	1.06114	+	0.128845i
$483$		0			0
$484$	0.500000	−	0.866025i	0.500000	−	0.866025i
$485$	0.136945	−	0.198399i	0.136945	−	0.198399i
$486$		0			0
$487$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$487$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$488$	−0.845190	−	1.46391i	−0.845190	−	1.46391i
$489$		0			0
$490$	0.354605	−	0.935016i	0.354605	−	0.935016i
$491$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$491$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$492$		0			0
$493$	−0.0171327	+	0.0193389i	−0.0171327	+	0.0193389i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$499$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$500$	0.500000	+	0.866025i	0.500000	+	0.866025i
$501$		0			0
$502$		0			0
$503$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$503$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$504$		0			0
$505$	−0.919979	−	1.59345i	−0.919979	−	1.59345i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.160803	+	1.99190i	0.160803	+	1.99190i	0.120537	+	0.992709i	$0.461538\pi$
$509$	0.160803	+	1.99190i	0.160803	+	1.99190i	0.0402659	+	0.999189i	$0.487179\pi$
$510$		0			0
$511$		0			0
$512$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$513$		0			0
$514$	−1.38546	+	1.33075i	−1.38546	+	1.33075i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.799443	−	0.600742i	0.799443	−	0.600742i
$521$	−1.49217	−	1.32194i	−1.49217	−	1.32194i	−0.799443	−	0.600742i	$-0.794872\pi$
$521$	−1.49217	−	1.32194i	−1.49217	−	1.32194i	−0.692724	−	0.721202i	$-0.743590\pi$
$522$	−0.0224054	+	0.0773523i	−0.0224054	+	0.0773523i
$523$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$523$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	+	0.866025i	0.500000	+	0.866025i
$530$	0.116065	−	1.43773i	0.116065	−	1.43773i
$531$		0			0
$532$		0			0
$533$	0.157681	+	0.0321908i	0.157681	+	0.0321908i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.180446	−	1.48611i	−0.180446	−	1.48611i
$539$		0			0
$540$		0			0
$541$	1.27388	−	1.43792i	1.27388	−	1.43792i	0.428693	−	0.903450i	$-0.358974\pi$
$541$	1.27388	−	1.43792i	1.27388	−	1.43792i	0.845190	−	0.534466i	$-0.179487\pi$
$542$		0			0
$543$		0			0
$544$	−0.125752	+	0.295150i	−0.125752	+	0.295150i
$545$	1.61950	+	0.935016i	1.61950	+	0.935016i
$546$		0			0
$547$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$547$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$548$	1.60339	−	1.01392i	1.60339	−	1.01392i
$549$	−1.66849	−	0.271156i	−1.66849	−	0.271156i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.61950	+	0.614194i	−1.61950	+	0.614194i
$555$		0			0
$556$		0			0
$557$	0.0402659	−	0.999189i	0.0402659	−	0.999189i	−0.845190	−	0.534466i	$-0.820513\pi$
$557$	0.0402659	−	0.999189i	0.0402659	−	0.999189i	0.885456	−	0.464723i	$-0.153846\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.565375	+	1.32698i	−0.565375	+	1.32698i
$563$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$563$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$564$		0			0
$565$	−0.521225	+	0.359776i	−0.521225	+	0.359776i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.141860	−	0.694877i	−0.141860	−	0.694877i	−0.987050	−	0.160411i	$-0.948718\pi$
$569$	−0.141860	−	0.694877i	−0.141860	−	0.694877i	0.845190	−	0.534466i	$-0.179487\pi$
$570$		0			0
$571$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$571$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.0402659	+	0.999189i	0.0402659	+	0.999189i
$577$	−1.26489			−1.26489			−0.632445	−	0.774605i	$-0.717949\pi$
$577$	−1.26489			−1.26489			−0.632445	+	0.774605i	$0.717949\pi$
$578$	−0.0361215	−	0.896345i	−0.0361215	−	0.896345i
$579$		0			0
$580$	−0.0794890	−	0.0129182i	−0.0794890	−	0.0129182i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.354605	−	0.935016i	0.354605	−	0.935016i
$585$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$586$	0.652458	+	1.72039i	0.652458	+	1.72039i
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.398754	+	0.0321908i	0.398754	+	0.0321908i
$593$	1.12001	−	0.587824i	1.12001	−	0.587824i	0.200026	−	0.979791i	$-0.435897\pi$
$593$	1.12001	−	0.587824i	1.12001	−	0.587824i	0.919979	+	0.391967i	$0.128205\pi$
$594$		0			0
$595$		0			0
$596$	−1.04052	−	0.600742i	−1.04052	−	0.600742i
$597$		0			0
$598$		0			0
$599$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$599$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$600$		0			0
$601$	−0.724653	−	1.52717i	−0.724653	−	1.52717i	−0.845190	−	0.534466i	$-0.820513\pi$
$601$	−0.724653	−	1.52717i	−0.724653	−	1.52717i	0.120537	−	0.992709i	$-0.461538\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.885456	−	0.464723i	0.885456	−	0.464723i
$606$		0			0
$607$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$607$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$608$		0			0
$609$		0			0
$610$	0.0680647	−	1.68901i	0.0680647	−	1.68901i
$611$		0			0
$612$	0.149094	+	0.284074i	0.149094	+	0.284074i
$613$	1.60339	−	0.535289i	1.60339	−	0.535289i	0.632445	−	0.774605i	$-0.282051\pi$
$613$	1.60339	−	0.535289i	1.60339	−	0.535289i	0.970942	+	0.239316i	$0.0769231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.0763874	+	0.0255019i	0.0763874	+	0.0255019i	0.354605	−	0.935016i	$-0.384615\pi$
$617$	0.0763874	+	0.0255019i	0.0763874	+	0.0255019i	−0.278217	+	0.960518i	$0.589744\pi$
$618$		0			0
$619$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$619$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$626$	−1.64463	−	0.0662764i	−1.64463	−	0.0662764i
$627$		0			0
$628$	0.930676	+	0.759873i	0.930676	+	0.759873i
$629$	0.120005	−	0.0455119i	0.120005	−	0.0455119i
$630$		0			0
$631$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$631$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$632$		0			0
$633$		0			0
$634$	1.36751	+	0.222242i	1.36751	+	0.222242i
$635$		0			0
$636$		0			0
$637$	−0.692724	−	0.721202i	−0.692724	−	0.721202i
$638$		0			0
$639$		0			0
$640$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$641$	0.996757	+	0.0804666i	0.996757	+	0.0804666i	0.568065	−	0.822984i	$-0.307692\pi$
$641$	0.996757	+	0.0804666i	0.996757	+	0.0804666i	0.428693	+	0.903450i	$0.358974\pi$
$642$		0			0
$643$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$643$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$647$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$648$	0.799443	+	0.600742i	0.799443	+	0.600742i
$649$		0			0
$650$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$651$		0			0
$652$		0			0
$653$	0.414507	+	0.239316i	0.414507	+	0.239316i	0.692724	−	0.721202i	$-0.256410\pi$
$653$	0.414507	+	0.239316i	0.414507	+	0.239316i	−0.278217	+	0.960518i	$0.589744\pi$
$654$		0			0
$655$		0			0
$656$	−0.124660	−	0.101781i	−0.124660	−	0.101781i
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$659$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$660$		0			0
$661$	1.48804	+	0.431017i	1.48804	+	0.431017i	0.919979	−	0.391967i	$-0.128205\pi$
$661$	1.48804	+	0.431017i	1.48804	+	0.431017i	0.568065	+	0.822984i	$0.307692\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.277125	−	0.288518i	0.277125	−	0.288518i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.02673	−	1.62364i	−1.02673	−	1.62364i	−0.748511	−	0.663123i	$-0.769231\pi$
$673$	−1.02673	−	1.62364i	−1.02673	−	1.62364i	−0.278217	−	0.960518i	$-0.589744\pi$
$674$	0.0630804	+	0.781391i	0.0630804	+	0.781391i
$675$		0			0
$676$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$677$		−	1.87003i		−	1.87003i	−0.354605	−	0.935016i	$-0.615385\pi$
$677$		−	1.87003i		−	1.87003i	0.354605	−	0.935016i	$-0.384615\pi$
$678$		0			0
$679$		0			0
$680$	−0.264032	+	0.182248i	−0.264032	+	0.182248i
$681$		0			0
$682$		0			0
$683$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$683$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$684$		0			0
$685$	1.89707			1.89707
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.24916	−	0.721202i	−1.24916	−	0.721202i
$690$		0			0
$691$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$691$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$692$	0.294326	+	0.881614i	0.294326	+	0.881614i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.0501307	−	0.0123561i	−0.0501307	−	0.0123561i
$698$	−1.14856	+	0.663123i	−1.14856	+	0.663123i
$699$		0			0
$700$		0			0
$701$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	0.120537	+	0.992709i	$0.461538\pi$
$701$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.111301	+	0.545190i	−0.111301	+	0.545190i
$707$		0			0
$708$		0			0
$709$	0.828000	−	1.30938i	0.828000	−	1.30938i	−0.120537	−	0.992709i	$-0.538462\pi$
$709$	0.828000	−	1.30938i	0.828000	−	1.30938i	0.948536	−	0.316668i	$-0.102564\pi$
$710$		0			0
$711$		0			0
$712$	1.27388	+	0.368985i	1.27388	+	0.368985i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$719$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$720$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$721$		0			0
$722$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$723$		0			0
$724$	0.351915	−	0.431017i	0.351915	−	0.431017i
$725$	−0.0602790	−	0.0534025i	−0.0602790	−	0.0534025i
$726$		0			0
$727$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$727$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$728$		0			0
$729$	0.970942	−	0.239316i	0.970942	−	0.239316i
$730$	0.799443	−	0.600742i	0.799443	−	0.600742i
$731$		0			0
$732$		0			0
$733$	−1.41574	−	0.743039i	−1.41574	−	0.743039i	−0.428693	−	0.903450i	$-0.641026\pi$
$733$	−1.41574	−	0.743039i	−1.41574	−	0.743039i	−0.987050	+	0.160411i	$0.948718\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.157681	+	0.0321908i	−0.157681	+	0.0321908i
$739$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$739$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$740$	0.319818	+	0.240328i	0.319818	+	0.240328i
$741$		0			0
$742$		0			0
$743$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$743$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$744$		0			0
$745$	−0.558358	−	1.06386i	−0.558358	−	1.06386i
$746$		−	0.160933i		−	0.160933i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$751$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$752$		0			0
$753$		0			0
$754$	−0.0457473	+	0.0662764i	−0.0457473	+	0.0662764i
$755$		0			0
$756$		0			0
$757$	−0.719954	+	0.587824i	−0.719954	+	0.587824i	−0.919979	−	0.391967i	$-0.871795\pi$
$757$	−0.719954	+	0.587824i	−0.719954	+	0.587824i	0.200026	+	0.979791i	$0.435897\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0385138	−	0.477079i	0.0385138	−	0.477079i	−0.948536	−	0.316668i	$-0.897436\pi$
$761$	0.0385138	−	0.477079i	0.0385138	−	0.477079i	0.987050	−	0.160411i	$-0.0512821\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.0258155	+	0.319782i	−0.0258155	+	0.319782i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.719954	+	0.587824i	−0.719954	+	0.587824i	−0.919979	−	0.391967i	$-0.871795\pi$
$769$	−0.719954	+	0.587824i	−0.719954	+	0.587824i	0.200026	+	0.979791i	$0.435897\pi$
$770$		0			0
$771$		0			0
$772$	−0.908271	+	1.31586i	−0.908271	+	1.31586i
$773$	0.141860	−	0.694877i	0.141860	−	0.694877i	−0.845190	−	0.534466i	$-0.820513\pi$
$773$	0.141860	−	0.694877i	0.141860	−	0.694877i	0.987050	−	0.160411i	$-0.0512821\pi$
$774$		0			0
$775$		0			0
$776$	0.00970705	+	0.240878i	0.00970705	+	0.240878i
$777$		0			0
$778$	−0.394871	−	0.0641728i	−0.394871	−	0.0641728i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.278217	+	0.960518i	0.278217	+	0.960518i
$785$	0.380472	+	1.13965i	0.380472	+	1.13965i
$786$		0			0
$787$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$787$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$788$	−1.56806	−	0.822984i	−1.56806	−	0.822984i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.49676	−	0.785559i	−1.49676	−	0.785559i
$794$	−1.10312	−	0.271894i	−1.10312	−	0.271894i
$795$		0			0
$796$		0			0
$797$	1.27497	+	1.04098i	1.27497	+	1.04098i	0.996757	+	0.0804666i	$0.0256410\pi$
$797$	1.27497	+	1.04098i	1.27497	+	1.04098i	0.278217	+	0.960518i	$0.410256\pi$
$798$		0			0
$799$		0			0
$800$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$801$	1.09148	−	0.753393i	1.09148	−	0.753393i
$802$	0.307276	+	0.721202i	0.307276	+	0.721202i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.69272	+	0.721202i	1.69272	+	0.721202i
$809$	−0.444838	+	1.53576i	−0.444838	+	1.53576i	0.354605	+	0.935016i	$0.384615\pi$
$809$	−0.444838	+	1.53576i	−0.444838	+	1.53576i	−0.799443	+	0.600742i	$0.794872\pi$
$810$	0.354605	+	0.935016i	0.354605	+	0.935016i
$811$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$811$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.71942	−	0.208776i	1.71942	−	0.208776i
$819$		0			0
$820$	−0.0509624	−	0.152651i	−0.0509624	−	0.152651i
$821$	1.43189	+	1.37535i	1.43189	+	1.37535i	0.799443	+	0.600742i	$0.205128\pi$
$821$	1.43189	+	1.37535i	1.43189	+	1.37535i	0.632445	+	0.774605i	$0.282051\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$827$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$828$		0			0
$829$	−1.31415	+	0.438727i	−1.31415	+	0.438727i	−0.885456	−	0.464723i	$-0.846154\pi$
$829$	−1.31415	+	0.438727i	−1.31415	+	0.438727i	−0.428693	+	0.903450i	$0.641026\pi$
$830$		0			0
$831$		0			0
$832$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$833$	0.212745	+	0.240139i	0.212745	+	0.240139i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.600742	−	0.799443i	$-0.705128\pi$
$839$		0			0		0.600742	+	0.799443i	$0.294872\pi$
$840$		0			0
$841$	−0.980649	+	0.159371i	−0.980649	+	0.159371i
$842$	−0.571307	−	0.903450i	−0.571307	−	0.903450i
$843$		0			0
$844$		0			0
$845$	0.354605	−	0.935016i	0.354605	−	0.935016i
$846$		0			0
$847$		0			0
$848$	0.770916	+	1.21911i	0.770916	+	1.21911i
$849$		0			0
$850$	−0.320562	+	0.0129182i	−0.320562	+	0.0129182i
$851$		0			0
$852$		0			0
$853$	−1.91674	+	0.472433i	−1.91674	+	0.472433i	−0.919979	+	0.391967i	$0.871795\pi$
$853$	−1.91674	+	0.472433i	−1.91674	+	0.472433i	−0.996757	+	0.0804666i	$0.974359\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.708833	−	0.800107i	−0.708833	−	0.800107i	0.278217	−	0.960518i	$-0.410256\pi$
$857$	−0.708833	−	0.800107i	−0.708833	−	0.800107i	−0.987050	+	0.160411i	$0.948718\pi$
$858$		0			0
$859$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$859$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$863$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$864$		0			0
$865$	−0.222431	+	0.902438i	−0.222431	+	0.902438i
$866$	−0.468959	+	1.90264i	−0.468959	+	1.90264i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.85640	+	0.225408i	−1.85640	+	0.225408i
$873$	0.192724	+	0.144823i	0.192724	+	0.144823i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.83399	+	0.781391i	−1.83399	+	0.781391i	−0.885456	+	0.464723i	$0.846154\pi$
$877$	−1.83399	+	0.781391i	−1.83399	+	0.781391i	−0.948536	+	0.316668i	$0.897436\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.554631	−	1.91481i	0.554631	−	1.91481i	0.200026	−	0.979791i	$-0.435897\pi$
$881$	0.554631	−	1.91481i	0.554631	−	1.91481i	0.354605	−	0.935016i	$-0.384615\pi$
$882$	0.919979	+	0.391967i	0.919979	+	0.391967i
$883$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$883$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$884$	0.0514636	+	0.316668i	0.0514636	+	0.316668i
$885$		0			0
$886$		0			0
$887$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$887$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$888$		0			0
$889$		0			0
$890$	0.879463	+	0.992709i	0.879463	+	0.992709i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.393906	+	1.59814i	0.393906	+	1.59814i
$899$		0			0
$900$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$901$	0.391117	+	0.247327i	0.391117	+	0.247327i
$902$		0			0
$903$		0			0
$904$	0.200557	−	0.600742i	0.200557	−	0.600742i
$905$	0.527799	−	0.176205i	0.527799	−	0.176205i
$906$		0			0
$907$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$907$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$908$		0			0
$909$	1.62920	−	0.855072i	1.62920	−	0.855072i
$910$		0			0
$911$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$911$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$912$		0			0
$913$		0			0
$914$	0.0557864	+	1.38433i	0.0557864	+	1.38433i
$915$		0			0
$916$	0.478243	−	0.0192725i	0.478243	−	0.0192725i
$917$		0			0
$918$		0			0
$919$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$919$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$920$		0			0
$921$		0			0
$922$	−0.592179	−	0.224584i	−0.592179	−	0.224584i
$923$		0			0
$924$		0			0
$925$	0.141860	+	0.374055i	0.141860	+	0.374055i
$926$		0			0
$927$		0			0
$928$	0.0713074	−	0.0374250i	0.0713074	−	0.0374250i
$929$	−0.0258155	+	0.319782i	−0.0258155	+	0.319782i	0.970942	+	0.239316i	$0.0769231\pi$
$929$	−0.0258155	+	0.319782i	−0.0258155	+	0.319782i	−0.996757	+	0.0804666i	$0.974359\pi$
$930$		0			0
$931$		0			0
$932$	1.19272	+	1.58723i	1.19272	+	1.58723i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.568065	+	0.822984i	0.568065	+	0.822984i
$937$	1.61950	+	0.614194i	1.61950	+	0.614194i	0.987050	−	0.160411i	$-0.0512821\pi$
$937$	1.61950	+	0.614194i	1.61950	+	0.614194i	0.632445	+	0.774605i	$0.282051\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.764919	+	0.527986i	0.764919	+	0.527986i	0.885456	−	0.464723i	$-0.153846\pi$
$941$	0.764919	+	0.527986i	0.764919	+	0.527986i	−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		−0.987050	−	0.160411i	$-0.948718\pi$
$947$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$948$		0			0
$949$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.892750	−	0.258588i	0.892750	−	0.258588i	0.200026	−	0.979791i	$-0.435897\pi$
$953$	0.892750	−	0.258588i	0.892750	−	0.258588i	0.692724	+	0.721202i	$0.256410\pi$
$954$	1.43189	+	0.173863i	1.43189	+	0.173863i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.970942	−	0.239316i	0.970942	−	0.239316i
$962$	0.354228	−	0.185913i	0.354228	−	0.185913i
$963$		0			0
$964$	−1.06806	−	0.0430415i	−1.06806	−	0.0430415i
$965$	−1.47094	+	0.626710i	−1.47094	+	0.626710i
$966$		0			0
$967$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$967$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$968$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$969$		0			0
$970$	−0.120537	+	0.208776i	−0.120537	+	0.208776i
$971$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$971$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.960245	+	1.39116i	0.960245	+	1.39116i
$977$	−1.47094	−	0.626710i	−1.47094	−	0.626710i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	−1.47094	−	0.626710i	−1.47094	−	0.626710i	−0.970942	+	0.239316i	$0.923077\pi$
$978$		0			0
$979$		0			0
$980$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$981$	−0.999468	+	1.58053i	−0.999468	+	1.58053i
$982$		0			0
$983$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$983$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$984$		0			0
$985$	−0.885456	−	1.53365i	−0.885456	−	1.53365i
$986$	0.0155210	−	0.0206548i	0.0155210	−	0.0206548i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.456763	−	0.438727i	0.456763	−	0.438727i	−0.428693	−	0.903450i	$-0.641026\pi$
$997$	0.456763	−	0.438727i	0.456763	−	0.438727i	0.885456	+	0.464723i	$0.153846\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.co.b.2799.1	yes	24
4.3	odd	2	CM	3380.1.co.b.2799.1	yes	24
5.4	even	2		3380.1.co.a.2799.1	yes	24
20.19	odd	2		3380.1.co.a.2799.1	yes	24
169.153	even	78		3380.1.co.a.2519.1	✓	24
676.491	odd	78		3380.1.co.a.2519.1	✓	24
845.829	even	78	inner	3380.1.co.b.2519.1	yes	24
3380.2519	odd	78	inner	3380.1.co.b.2519.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.co.a.2519.1	✓	24	169.153	even	78
3380.1.co.a.2519.1	✓	24	676.491	odd	78
3380.1.co.a.2799.1	yes	24	5.4	even	2
3380.1.co.a.2799.1	yes	24	20.19	odd	2
3380.1.co.b.2519.1	yes	24	845.829	even	78	inner
3380.1.co.b.2519.1	yes	24	3380.2519	odd	78	inner
3380.1.co.b.2799.1	yes	24	1.1	even	1	trivial
3380.1.co.b.2799.1	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.co (of order \(78\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.996757 + 0.0804666i) q^{2} +(0.987050 - 0.160411i) q^{4} +(0.919979 + 0.391967i) q^{5} +(-0.970942 + 0.239316i) q^{8} +(-0.428693 + 0.903450i) q^{9} +O(q^{10})\)	\(q+(-0.996757 + 0.0804666i) q^{2} +(0.987050 - 0.160411i) q^{4} +(0.919979 + 0.391967i) q^{5} +(-0.970942 + 0.239316i) q^{8} +(-0.428693 + 0.903450i) q^{9} +(-0.948536 - 0.316668i) q^{10} +(-0.692724 + 0.721202i) q^{13} +(0.948536 - 0.316668i) q^{16} +(0.231378 - 0.222242i) q^{17} +(0.354605 - 0.935016i) q^{18} +(0.970942 + 0.239316i) q^{20} +(0.692724 + 0.721202i) q^{25} +(0.632445 - 0.774605i) q^{26} +(-0.0802707 + 0.00648012i) q^{29} +(-0.919979 + 0.391967i) q^{32} +(-0.212745 + 0.240139i) q^{34} +(-0.278217 + 0.960518i) q^{36} +(0.368039 + 0.156807i) q^{37} +(-0.987050 - 0.160411i) q^{40} +(-0.0860133 - 0.136019i) q^{41} +(-0.748511 + 0.663123i) q^{45} +(-0.0402659 + 0.999189i) q^{49} +(-0.748511 - 0.663123i) q^{50} +(-0.568065 + 0.822984i) q^{52} +(0.345190 + 1.40049i) q^{53} +(0.0794890 - 0.0129182i) q^{58} +(0.470293 + 1.62364i) q^{61} +(0.885456 - 0.464723i) q^{64} +(-0.919979 + 0.391967i) q^{65} +(0.192732 - 0.256479i) q^{68} +(0.200026 - 0.979791i) q^{72} +(-0.568065 + 0.822984i) q^{73} +(-0.379463 - 0.126683i) q^{74} +(0.996757 + 0.0804666i) q^{80} +(-0.632445 - 0.774605i) q^{81} +(0.0966793 + 0.128657i) q^{82} +(0.299974 - 0.113765i) q^{85} +(-1.14856 - 0.663123i) q^{89} +(0.692724 - 0.721202i) q^{90} +(0.0482209 - 0.236201i) q^{97} +(-0.0402659 - 0.999189i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + q^{4} - q^{5} - 2 q^{8} - q^{9}+O(q^{10}) \) 24 * q + q^2 + q^4 - q^5 - 2 * q^8 - q^9	\( 24 q + q^{2} + q^{4} - q^{5} - 2 q^{8} - q^{9} - q^{10} - q^{13} + q^{16} + 3 q^{17} + 2 q^{18} + 2 q^{20} + q^{25} - q^{26} + q^{29} + q^{32} - q^{36} - q^{37} - q^{40} + 3 q^{41} - 2 q^{45} + q^{49} - 2 q^{50} + 2 q^{52} - 13 q^{53} + q^{58} + q^{61} - 2 q^{64} + q^{65} - 3 q^{68} - q^{72} + 2 q^{73} - 14 q^{74} - q^{80} + q^{81} - 3 q^{82} + 13 q^{85} + q^{90} - 2 q^{97} + q^{98}+O(q^{100}) \) 24 * q + q^2 + q^4 - q^5 - 2 * q^8 - q^9 - q^10 - q^13 + q^16 + 3 * q^17 + 2 * q^18 + 2 * q^20 + q^25 - q^26 + q^29 + q^32 - q^36 - q^37 - q^40 + 3 * q^41 - 2 * q^45 + q^49 - 2 * q^50 + 2 * q^52 - 13 * q^53 + q^58 + q^61 - 2 * q^64 + q^65 - 3 * q^68 - q^72 + 2 * q^73 - 14 * q^74 - q^80 + q^81 - 3 * q^82 + 13 * q^85 + q^90 - 2 * q^97 + q^98

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