LMFDB - Embedded newform 3380.1.cs.a.267.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(7,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 39, 107]))

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

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

chi := DirichletCharacter("3380.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.cs (of order $156$, degree $48$, 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:	$48$
Coefficient field:	$\Q(\zeta_{156})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1 $ x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{156}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{156} - \cdots)$

Embedding invariants

Embedding label			267.1
Root			$-0.979791 - 0.200026i$ of defining polynomial
Character	$\chi$	$=$	3380.267
Dual form			3380.1.cs.a.2823.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.391967 + 0.919979i) q^{2} +(-0.692724 - 0.721202i) q^{4} +(0.987050 - 0.160411i) q^{5} +(0.935016 - 0.354605i) q^{8} +(-0.600742 - 0.799443i) q^{9} +O(q^{10})$	\(q+(-0.391967 + 0.919979i) q^{2} +(-0.692724 - 0.721202i) q^{4} +(0.987050 - 0.160411i) q^{5} +(0.935016 - 0.354605i) q^{8} +(-0.600742 - 0.799443i) q^{9} +(-0.239316 + 0.970942i) q^{10} +(0.632445 + 0.774605i) q^{13} +(-0.0402659 + 0.999189i) q^{16} +(-0.201003 - 1.98947i) q^{17} +(0.970942 - 0.239316i) q^{18} +(-0.799443 - 0.600742i) q^{20} +(0.948536 - 0.316668i) q^{25} +(-0.960518 + 0.278217i) q^{26} +(-0.156807 + 0.368039i) q^{29} +(-0.903450 - 0.428693i) q^{32} +(1.90905 + 0.594885i) q^{34} +(-0.160411 + 0.987050i) q^{36} +(-0.458243 - 0.965727i) q^{37} +(0.866025 - 0.500000i) q^{40} +(1.65988 + 0.828977i) q^{41} +(-0.721202 - 0.692724i) q^{45} +(-0.200026 - 0.979791i) q^{49} +(-0.0804666 + 0.996757i) q^{50} +(0.120537 - 0.992709i) q^{52} +(-1.81456 - 0.816668i) q^{53} +(-0.277125 - 0.288518i) q^{58} +(0.625134 - 0.101594i) q^{61} +(0.748511 - 0.663123i) q^{64} +(0.748511 + 0.663123i) q^{65} +(-1.29557 + 1.52312i) q^{68} +(-0.845190 - 0.534466i) q^{72} +(0.992709 + 0.120537i) q^{73} +(1.06806 - 0.0430415i) q^{74} +(0.120537 + 0.992709i) q^{80} +(-0.278217 + 0.960518i) q^{81} +(-1.41326 + 1.20212i) q^{82} +(-0.517533 - 1.93146i) q^{85} +(1.84438 - 0.494200i) q^{89} +(0.919979 - 0.391967i) q^{90} +(-0.879714 + 1.39116i) q^{97} +(0.979791 + 0.200026i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 48 * q - 2 * q^4 + 2 * q^5	$ 48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) $ 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Display 100 coefficients 10 coefficients

Character values

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

$n$	$677$	$1691$	$1861$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{59}{156}\right)$

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 $

$2$	−0.391967	+	0.919979i	−0.391967	+	0.919979i
$2$	−0.391967	+	0.919979i	−0.391967	+	0.919979i
$3$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$3$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$4$	−0.692724	−	0.721202i	−0.692724	−	0.721202i
$5$	0.987050	−	0.160411i	0.987050	−	0.160411i
$5$	0.987050	−	0.160411i	0.987050	−	0.160411i
$6$		0			0
$7$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$7$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$8$	0.935016	−	0.354605i	0.935016	−	0.354605i
$9$	−0.600742	−	0.799443i	−0.600742	−	0.799443i
$10$	−0.239316	+	0.970942i	−0.239316	+	0.970942i
$11$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$11$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$12$		0			0
$13$	0.632445	+	0.774605i	0.632445	+	0.774605i
$13$	0.632445	+	0.774605i	0.632445	+	0.774605i
$14$		0			0
$15$		0			0
$16$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$17$	−0.201003	−	1.98947i	−0.201003	−	1.98947i	−0.120537	−	0.992709i	$-0.538462\pi$
$17$	−0.201003	−	1.98947i	−0.201003	−	1.98947i	−0.0804666	−	0.996757i	$-0.525641\pi$
$18$	0.970942	−	0.239316i	0.970942	−	0.239316i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	−0.799443	−	0.600742i	−0.799443	−	0.600742i
$21$		0			0
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	0.948536	−	0.316668i	0.948536	−	0.316668i
$26$	−0.960518	+	0.278217i	−0.960518	+	0.278217i
$27$		0			0
$28$		0			0
$29$	−0.156807	+	0.368039i	−0.156807	+	0.368039i	−0.979791	−	0.200026i	$-0.935897\pi$
$29$	−0.156807	+	0.368039i	−0.156807	+	0.368039i	0.822984	+	0.568065i	$0.192308\pi$
$30$		0			0
$31$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$31$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$32$	−0.903450	−	0.428693i	−0.903450	−	0.428693i
$33$		0			0
$34$	1.90905	+	0.594885i	1.90905	+	0.594885i
$35$		0			0
$36$	−0.160411	+	0.987050i	−0.160411	+	0.987050i
$37$	−0.458243	−	0.965727i	−0.458243	−	0.965727i	−0.992709	−	0.120537i	$-0.961538\pi$
$37$	−0.458243	−	0.965727i	−0.458243	−	0.965727i	0.534466	−	0.845190i	$-0.320513\pi$
$38$		0			0
$39$		0			0
$40$	0.866025	−	0.500000i	0.866025	−	0.500000i
$41$	1.65988	+	0.828977i	1.65988	+	0.828977i	0.996757	+	0.0804666i	$0.0256410\pi$
$41$	1.65988	+	0.828977i	1.65988	+	0.828977i	0.663123	+	0.748511i	$0.269231\pi$
$42$		0			0
$43$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$43$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$44$		0			0
$45$	−0.721202	−	0.692724i	−0.721202	−	0.692724i
$46$		0			0
$47$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$47$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$48$		0			0
$49$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$50$	−0.0804666	+	0.996757i	−0.0804666	+	0.996757i
$51$		0			0
$52$	0.120537	−	0.992709i	0.120537	−	0.992709i
$53$	−1.81456	−	0.816668i	−1.81456	−	0.816668i	−0.948536	−	0.316668i	$-0.897436\pi$
$53$	−1.81456	−	0.816668i	−1.81456	−	0.816668i	−0.866025	−	0.500000i	$-0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.277125	−	0.288518i	−0.277125	−	0.288518i
$59$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$59$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$60$		0			0
$61$	0.625134	−	0.101594i	0.625134	−	0.101594i	0.160411	−	0.987050i	$-0.448718\pi$
$61$	0.625134	−	0.101594i	0.625134	−	0.101594i	0.464723	+	0.885456i	$0.346154\pi$
$62$		0			0
$63$		0			0
$64$	0.748511	−	0.663123i	0.748511	−	0.663123i
$65$	0.748511	+	0.663123i	0.748511	+	0.663123i
$66$		0			0
$67$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$67$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$68$	−1.29557	+	1.52312i	−1.29557	+	1.52312i
$69$		0			0
$70$		0			0
$71$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$71$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$72$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$73$	0.992709	+	0.120537i	0.992709	+	0.120537i	0.600742	−	0.799443i	$-0.294872\pi$
$73$	0.992709	+	0.120537i	0.992709	+	0.120537i	0.391967	+	0.919979i	$0.371795\pi$
$74$	1.06806	−	0.0430415i	1.06806	−	0.0430415i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$79$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$80$	0.120537	+	0.992709i	0.120537	+	0.992709i
$81$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$82$	−1.41326	+	1.20212i	−1.41326	+	1.20212i
$83$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$83$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$84$		0			0
$85$	−0.517533	−	1.93146i	−0.517533	−	1.93146i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.84438	−	0.494200i	1.84438	−	0.494200i	0.845190	−	0.534466i	$-0.179487\pi$
$89$	1.84438	−	0.494200i	1.84438	−	0.494200i	0.999189	+	0.0402659i	$0.0128205\pi$
$90$	0.919979	−	0.391967i	0.919979	−	0.391967i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	0.0402659	+	0.999189i	$0.487179\pi$
$97$	−0.879714	+	1.39116i	−0.879714	+	1.39116i	−0.919979	+	0.391967i	$0.871795\pi$
$98$	0.979791	+	0.200026i	0.979791	+	0.200026i
$99$		0			0
$100$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$101$	−0.145395	−	1.80104i	−0.145395	−	1.80104i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−0.145395	−	1.80104i	−0.145395	−	1.80104i	0.354605	−	0.935016i	$-0.384615\pi$
$102$		0			0
$103$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$103$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$104$	0.866025	+	0.500000i	0.866025	+	0.500000i
$105$		0			0
$106$	1.46257	−	1.34925i	1.46257	−	1.34925i
$107$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$107$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$108$		0			0
$109$	−1.68353	−	0.308518i	−1.68353	−	0.308518i	−0.748511	−	0.663123i	$-0.769231\pi$
$109$	−1.68353	−	0.308518i	−1.68353	−	0.308518i	−0.935016	+	0.354605i	$0.884615\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.21323	−	0.605913i	1.21323	−	0.605913i	0.278217	−	0.960518i	$-0.410256\pi$
$113$	1.21323	−	0.605913i	1.21323	−	0.605913i	0.935016	+	0.354605i	$0.115385\pi$
$114$		0			0
$115$		0			0
$116$	0.374055	−	0.141860i	0.374055	−	0.141860i
$117$	0.239316	−	0.970942i	0.239316	−	0.970942i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.960518	−	0.278217i	0.960518	−	0.278217i
$122$	−0.151567	+	0.614932i	−0.151567	+	0.614932i
$123$		0			0
$124$		0			0
$125$	0.885456	−	0.464723i	0.885456	−	0.464723i
$126$		0			0
$127$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$127$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$128$	0.316668	+	0.948536i	0.316668	+	0.948536i
$129$		0			0
$130$	−0.903450	+	0.428693i	−0.903450	+	0.428693i
$131$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$131$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.893416	−	1.78891i	−0.893416	−	1.78891i
$137$	−0.0345234	+	0.0727566i	−0.0345234	+	0.0727566i	−0.919979	−	0.391967i	$-0.871795\pi$
$137$	−0.0345234	+	0.0727566i	−0.0345234	+	0.0727566i	0.885456	+	0.464723i	$0.153846\pi$
$138$		0			0
$139$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$139$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.822984	−	0.568065i	0.822984	−	0.568065i
$145$	−0.0957386	+	0.388427i	−0.0957386	+	0.388427i
$146$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$147$		0			0
$148$	−0.379048	+	0.999468i	−0.379048	+	0.999468i
$149$	1.33440	+	0.734991i	1.33440	+	0.734991i	0.979791	−	0.200026i	$-0.0641026\pi$
$149$	1.33440	+	0.734991i	1.33440	+	0.734991i	0.354605	+	0.935016i	$0.384615\pi$
$150$		0			0
$151$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$151$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$152$		0			0
$153$	−1.46971	+	1.35585i	−1.46971	+	1.35585i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.385514	+	1.23716i	0.385514	+	1.23716i	0.919979	+	0.391967i	$0.128205\pi$
$157$	0.385514	+	1.23716i	0.385514	+	1.23716i	−0.534466	+	0.845190i	$0.679487\pi$
$158$		0			0
$159$		0			0
$160$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$161$		0			0
$162$	−0.774605	−	0.632445i	−0.774605	−	0.632445i
$163$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$163$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$164$	−0.551979	−	1.77136i	−0.551979	−	1.77136i
$165$		0			0
$166$		0			0
$167$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$167$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$168$		0			0
$169$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$170$	1.97976	+	0.280948i	1.97976	+	0.280948i
$171$		0			0
$172$		0			0
$173$	0.00243169	−	0.120733i	0.00243169	−	0.120733i	−0.996757	−	0.0804666i	$-0.974359\pi$
$173$	0.00243169	−	0.120733i	0.00243169	−	0.120733i	0.999189	−	0.0402659i	$-0.0128205\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.268281	+	1.89050i	−0.268281	+	1.89050i
$179$		0			0		−0.632445	−	0.774605i	$-0.717949\pi$
$179$		0			0		0.632445	+	0.774605i	$0.282051\pi$
$180$			1.00000i			1.00000i
$181$	−0.917410	+	1.74798i	−0.917410	+	1.74798i	−0.316668	+	0.948536i	$0.602564\pi$
$181$	−0.917410	+	1.74798i	−0.917410	+	1.74798i	−0.600742	+	0.799443i	$0.705128\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.607222	−	0.879714i	−0.607222	−	0.879714i
$186$		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
$193$	0.124660	−	0.101781i	0.124660	−	0.101781i	−0.568065	−	0.822984i	$-0.692308\pi$
$193$	0.124660	−	0.101781i	0.124660	−	0.101781i	0.692724	+	0.721202i	$0.256410\pi$
$194$	−0.935016	−	1.35460i	−0.935016	−	1.35460i
$195$		0			0
$196$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$197$	0.106718	+	1.32194i	0.106718	+	1.32194i	0.799443	+	0.600742i	$0.205128\pi$
$197$	0.106718	+	1.32194i	0.106718	+	1.32194i	−0.692724	+	0.721202i	$0.743590\pi$
$198$		0			0
$199$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$199$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$200$	0.774605	−	0.632445i	0.774605	−	0.632445i
$201$		0			0
$202$	1.71391	+	0.572188i	1.71391	+	0.572188i
$203$		0			0
$204$		0			0
$205$	1.77136	+	0.551979i	1.77136	+	0.551979i
$206$		0			0
$207$		0			0
$208$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$209$		0			0
$210$		0			0
$211$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$211$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$212$	0.668008	+	1.87439i	0.668008	+	1.87439i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.943716	−	1.42788i	0.943716	−	1.42788i
$219$		0			0
$220$		0			0
$221$	1.41393	−	1.41393i	1.41393	−	1.41393i
$222$		0			0
$223$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$223$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$224$		0			0
$225$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$226$	0.0818806	+	1.35365i	0.0818806	+	1.35365i
$227$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$227$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$228$		0			0
$229$	−0.328749	+	1.79393i	−0.328749	+	1.79393i	0.239316	+	0.970942i	$0.423077\pi$
$229$	−0.328749	+	1.79393i	−0.328749	+	1.79393i	−0.568065	+	0.822984i	$0.692308\pi$
$230$		0			0
$231$		0			0
$232$	−0.0161084	+	0.399727i	−0.0161084	+	0.399727i
$233$	−1.96365	−	0.118779i	−1.96365	−	0.118779i	−0.970942	−	0.239316i	$-0.923077\pi$
$233$	−1.96365	−	0.118779i	−1.96365	−	0.118779i	−0.992709	+	0.120537i	$0.961538\pi$
$234$	0.799443	+	0.600742i	0.799443	+	0.600742i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	1.74555	−	0.393130i	1.74555	−	0.393130i	0.774605	−	0.632445i	$-0.217949\pi$
$241$	1.74555	−	0.393130i	1.74555	−	0.393130i	0.970942	+	0.239316i	$0.0769231\pi$
$242$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$243$		0			0
$244$	−0.506316	−	0.380472i	−0.506316	−	0.380472i
$245$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.0804666	+	0.996757i	0.0804666	+	0.996757i
$251$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$251$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.996757	−	0.0804666i	−0.996757	−	0.0804666i
$257$	0.564016	+	1.02399i	0.564016	+	1.02399i	0.992709	+	0.120537i	$0.0384615\pi$
$257$	0.564016	+	1.02399i	0.564016	+	1.02399i	−0.428693	+	0.903450i	$0.641026\pi$
$258$		0			0
$259$		0			0
$260$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$261$	0.388427	−	0.0957386i	0.388427	−	0.0957386i
$262$		0			0
$263$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$263$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$264$		0			0
$265$	−1.92207	−	0.515016i	−1.92207	−	0.515016i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.910663	+	0.185913i	0.910663	+	0.185913i	0.632445	−	0.774605i	$-0.282051\pi$
$269$	0.910663	+	0.185913i	0.910663	+	0.185913i	0.278217	+	0.960518i	$0.410256\pi$
$270$		0			0
$271$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$271$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$272$	1.99595	−	0.120733i	1.99595	−	0.120733i
$273$		0			0
$274$	−0.0534025	−	0.0602790i	−0.0534025	−	0.0602790i
$275$		0			0
$276$		0			0
$277$	−1.12921	+	1.56746i	−1.12921	+	1.56746i	−0.354605	+	0.935016i	$0.615385\pi$
$277$	−1.12921	+	1.56746i	−1.12921	+	1.56746i	−0.774605	+	0.632445i	$0.782051\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.200677	−	0.0121387i	0.200677	−	0.0121387i	0.0402659	−	0.999189i	$-0.487179\pi$
$281$	0.200677	−	0.0121387i	0.200677	−	0.0121387i	0.160411	+	0.987050i	$0.448718\pi$
$282$		0			0
$283$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$283$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.200026	+	0.979791i	0.200026	+	0.979791i
$289$	−2.93778	+	0.599753i	−2.93778	+	0.599753i
$290$	−0.319818	−	0.240328i	−0.319818	−	0.240328i
$291$		0			0
$292$	−0.600742	−	0.799443i	−0.600742	−	0.799443i
$293$	−0.137534	−	0.846282i	−0.137534	−	0.846282i	−0.960518	−	0.278217i	$-0.910256\pi$
$293$	−0.137534	−	0.846282i	−0.137534	−	0.846282i	0.822984	−	0.568065i	$-0.192308\pi$
$294$		0			0
$295$		0			0
$296$	−0.770916	−	0.740475i	−0.770916	−	0.740475i
$297$		0			0
$298$	−1.19921	+	0.939525i	−1.19921	+	0.939525i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.600742	−	0.200557i	0.600742	−	0.200557i
$306$	−0.671273	−	1.88355i	−0.671273	−	1.88355i
$307$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$307$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$311$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$312$		0			0
$313$	−1.54858	−	0.283788i	−1.54858	−	0.283788i	−0.663123	−	0.748511i	$-0.730769\pi$
$313$	−1.54858	−	0.283788i	−1.54858	−	0.283788i	−0.885456	+	0.464723i	$0.846154\pi$
$314$	−1.28927	−	0.130260i	−1.28927	−	0.130260i
$315$		0			0
$316$		0			0
$317$	−1.18269	−	0.448536i	−1.18269	−	0.448536i	−0.316668	−	0.948536i	$-0.602564\pi$
$317$	−1.18269	−	0.448536i	−1.18269	−	0.448536i	−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$	0.632445	−	0.774605i	0.632445	−	0.774605i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.885456	−	0.464723i	0.885456	−	0.464723i
$325$	0.845190	+	0.534466i	0.845190	+	0.534466i
$326$		0			0
$327$		0			0
$328$	1.84597	+	0.186506i	1.84597	+	0.186506i
$329$		0			0
$330$		0			0
$331$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$331$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$332$		0			0
$333$	−0.496757	+	0.946492i	−0.496757	+	0.946492i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.33214	−	1.33214i	1.33214	−	1.33214i	0.428693	−	0.903450i	$-0.358974\pi$
$337$	1.33214	−	1.33214i	1.33214	−	1.33214i	0.903450	−	0.428693i	$-0.141026\pi$
$338$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$339$		0			0
$340$	−1.03447	+	1.71122i	−1.03447	+	1.71122i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.110118	+	0.0495602i	0.110118	+	0.0495602i
$347$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$347$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$348$		0			0
$349$	0.0835998	+	0.589104i	0.0835998	+	0.589104i	0.987050	+	0.160411i	$0.0512821\pi$
$349$	0.0835998	+	0.589104i	0.0835998	+	0.589104i	−0.903450	+	0.428693i	$0.858974\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.316668	+	0.0514636i	0.316668	+	0.0514636i	0.316668	−	0.948536i	$-0.397436\pi$
$353$	0.316668	+	0.0514636i	0.316668	+	0.0514636i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	−1.63406	−	0.987826i	−1.63406	−	0.987826i
$357$		0			0
$358$		0			0
$359$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$359$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$360$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$361$	0.866025	+	0.500000i	0.866025	+	0.500000i
$362$	−1.24851	−	1.52915i	−1.24851	−	1.52915i
$363$		0			0
$364$		0			0
$365$	0.999189	−	0.0402659i	0.999189	−	0.0402659i
$366$		0			0
$367$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$367$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$368$		0			0
$369$	−0.334440	−	1.82498i	−0.334440	−	1.82498i
$370$	1.04733	−	0.213814i	1.04733	−	0.213814i
$371$		0			0
$372$		0			0
$373$	−0.278798	+	0.692724i	−0.278798	+	0.692724i	0.721202	+	0.692724i	$0.243590\pi$
$373$	−0.278798	+	0.692724i	−0.278798	+	0.692724i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.384257	+	0.111301i	−0.384257	+	0.111301i
$378$		0			0
$379$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$379$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$383$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$384$		0			0
$385$		0			0
$386$	0.0447744	+	0.154579i	0.0447744	+	0.154579i
$387$		0			0
$388$	1.61270	−	0.329236i	1.61270	−	0.329236i
$389$	−0.599417	+	1.58053i	−0.599417	+	1.58053i	0.200026	+	0.979791i	$0.435897\pi$
$389$	−0.599417	+	1.58053i	−0.599417	+	1.58053i	−0.799443	+	0.600742i	$0.794872\pi$
$390$		0			0
$391$		0			0
$392$	−0.534466	−	0.845190i	−0.534466	−	0.845190i
$393$		0			0
$394$	−1.25799	−	0.419979i	−1.25799	−	0.419979i
$395$		0			0
$396$		0			0
$397$	−0.0799447	−	1.98381i	−0.0799447	−	1.98381i	−0.160411	−	0.987050i	$-0.551282\pi$
$397$	−0.0799447	−	1.98381i	−0.0799447	−	1.98381i	0.0804666	−	0.996757i	$-0.474359\pi$
$398$		0			0
$399$		0			0
$400$	0.278217	+	0.960518i	0.278217	+	0.960518i
$401$	0.455256	+	0.493489i	0.455256	+	0.493489i	0.919979	−	0.391967i	$-0.128205\pi$
$401$	0.455256	+	0.493489i	0.455256	+	0.493489i	−0.464723	+	0.885456i	$0.653846\pi$
$402$		0			0
$403$		0			0
$404$	−1.19820	+	1.35248i	−1.19820	+	1.35248i
$405$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.285414	−	0.431843i	−0.285414	−	0.431843i	0.663123	−	0.748511i	$-0.269231\pi$
$409$	−0.285414	−	0.431843i	−0.285414	−	0.431843i	−0.948536	+	0.316668i	$0.897436\pi$
$410$	−1.20212	+	1.41326i	−1.20212	+	1.41326i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.239316	−	0.970942i	−0.239316	−	0.970942i
$417$		0			0
$418$		0			0
$419$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$419$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$420$		0			0
$421$	−0.891967	−	0.0539540i	−0.891967	−	0.0539540i	−0.391967	−	0.919979i	$-0.628205\pi$
$421$	−0.891967	−	0.0539540i	−0.891967	−	0.0539540i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	−1.98624	−	0.120145i	−1.98624	−	0.120145i
$425$	−0.820659	−	1.82343i	−0.820659	−	1.82343i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$431$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$432$		0			0
$433$	−1.32298	+	1.43409i	−1.32298	+	1.43409i	−0.500000	+	0.866025i	$0.666667\pi$
$433$	−1.32298	+	1.43409i	−1.32298	+	1.43409i	−0.822984	+	0.568065i	$0.807692\pi$
$434$		0			0
$435$		0			0
$436$	0.943716	+	1.42788i	0.943716	+	1.42788i
$437$		0			0
$438$		0			0
$439$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$439$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$440$		0			0
$441$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$442$	0.746571	+	1.85500i	0.746571	+	1.85500i
$443$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$443$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$444$		0			0
$445$	1.74122	−	0.783659i	1.74122	−	0.783659i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.0248378	−	1.23319i	0.0248378	−	1.23319i	−0.774605	−	0.632445i	$-0.782051\pi$
$449$	0.0248378	−	1.23319i	0.0248378	−	1.23319i	0.799443	−	0.600742i	$-0.205128\pi$
$450$	0.845190	−	0.534466i	0.845190	−	0.534466i
$451$		0			0
$452$	−1.27742	−	0.455256i	−1.27742	−	0.455256i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.51790	+	0.309882i	−1.51790	+	0.309882i	−0.885456	−	0.464723i	$-0.846154\pi$
$457$	−1.51790	+	0.309882i	−1.51790	+	0.309882i	−0.632445	+	0.774605i	$0.717949\pi$
$458$	−1.52152	−	1.00560i	−1.52152	−	1.00560i
$459$		0			0
$460$		0			0
$461$	−1.45543	−	0.206540i	−1.45543	−	0.206540i	−0.632445	−	0.774605i	$-0.717949\pi$
$461$	−1.45543	−	0.206540i	−1.45543	−	0.206540i	−0.822984	+	0.568065i	$0.807692\pi$
$462$		0			0
$463$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$463$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$464$	−0.361427	−	0.171499i	−0.361427	−	0.171499i
$465$		0			0
$466$	0.878960	−	1.75996i	0.878960	−	1.75996i
$467$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$467$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$468$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.437205	+	1.94125i	0.437205	+	1.94125i
$478$		0			0
$479$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$479$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$480$		0			0
$481$	0.458243	−	0.965727i	0.458243	−	0.965727i
$482$	−0.322525	+	1.75996i	−0.322525	+	1.75996i
$483$		0			0
$484$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$485$	−0.645164	+	1.51426i	−0.645164	+	1.51426i
$486$		0			0
$487$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$487$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$488$	0.548485	−	0.316668i	0.548485	−	0.316668i
$489$		0			0
$490$	0.999189	+	0.0402659i	0.999189	+	0.0402659i
$491$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$491$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$492$		0			0
$493$	0.763720	+	0.237985i	0.763720	+	0.237985i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$499$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$500$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$501$		0			0
$502$		0			0
$503$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$503$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$504$		0			0
$505$	−0.432420	−	1.75440i	−0.432420	−	1.75440i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.54786	−	0.622958i	1.54786	−	0.622958i	0.568065	−	0.822984i	$-0.307692\pi$
$509$	1.54786	−	0.622958i	1.54786	−	0.622958i	0.979791	+	0.200026i	$0.0641026\pi$
$510$		0			0
$511$		0			0
$512$	0.464723	−	0.885456i	0.464723	−	0.885456i
$513$		0			0
$514$	−1.16312	+	0.117515i	−1.16312	+	0.117515i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.935016	+	0.354605i	0.935016	+	0.354605i
$521$	−0.694138	+	0.364312i	−0.694138	+	0.364312i	−0.774605	−	0.632445i	$-0.782051\pi$
$521$	−0.694138	+	0.364312i	−0.694138	+	0.364312i	0.0804666	+	0.996757i	$0.474359\pi$
$522$	−0.0641728	+	0.394871i	−0.0641728	+	0.394871i
$523$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$523$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$	1.22719	−	1.56639i	1.22719	−	1.56639i
$531$		0			0
$532$		0			0
$533$	0.407653	+	1.81003i	0.407653	+	1.81003i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.527986	+	0.764919i	−0.527986	+	0.764919i
$539$		0			0
$540$		0			0
$541$	−0.482775	+	1.54928i	−0.482775	+	1.54928i	0.316668	+	0.948536i	$0.397436\pi$
$541$	−0.482775	+	1.54928i	−0.482775	+	1.54928i	−0.799443	+	0.600742i	$0.794872\pi$
$542$		0			0
$543$		0			0
$544$	−0.671273	+	1.88355i	−0.671273	+	1.88355i
$545$	−1.71122	−	0.0344658i	−1.71122	−	0.0344658i
$546$		0			0
$547$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$547$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$548$	0.0763874	−	0.0255019i	0.0763874	−	0.0255019i
$549$	−0.456763	−	0.438727i	−0.456763	−	0.438727i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.999420	−	1.65324i	−0.999420	−	1.65324i
$555$		0			0
$556$		0			0
$557$	−0.346455	−	1.69705i	−0.346455	−	1.69705i	−0.663123	−	0.748511i	$-0.730769\pi$
$557$	−0.346455	−	1.69705i	−0.346455	−	1.69705i	0.316668	−	0.948536i	$-0.397436\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0674914	+	0.189377i	−0.0674914	+	0.189377i
$563$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$563$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$564$		0			0
$565$	1.10033	−	0.792683i	1.10033	−	0.792683i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.404534	−	0.255812i	0.404534	−	0.255812i	−0.316668	−	0.948536i	$-0.602564\pi$
$569$	0.404534	−	0.255812i	0.404534	−	0.255812i	0.721202	+	0.692724i	$0.243590\pi$
$570$		0			0
$571$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$571$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.979791	−	0.200026i	−0.979791	−	0.200026i
$577$			0.556435i			0.556435i	0.960518	+	0.278217i	$0.0897436\pi$
$577$			0.556435i			0.556435i	−0.960518	+	0.278217i	$0.910256\pi$
$578$	0.599753	−	2.93778i	0.599753	−	2.93778i
$579$		0			0
$580$	0.346455	−	0.200026i	0.346455	−	0.200026i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.970942	−	0.239316i	0.970942	−	0.239316i
$585$	0.0804666	−	0.996757i	0.0804666	−	0.996757i
$586$	0.832471	+	0.205186i	0.832471	+	0.205186i
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.983395	−	0.418986i	0.983395	−	0.418986i
$593$	−1.27388	−	1.43792i	−1.27388	−	1.43792i	−0.845190	−	0.534466i	$-0.820513\pi$
$593$	−1.27388	−	1.43792i	−1.27388	−	1.43792i	−0.428693	−	0.903450i	$-0.641026\pi$
$594$		0			0
$595$		0			0
$596$	−0.394291	−	1.47152i	−0.394291	−	1.47152i
$597$		0			0
$598$		0			0
$599$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$599$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$600$		0			0
$601$	1.51660	+	1.13965i	1.51660	+	1.13965i	0.948536	+	0.316668i	$0.102564\pi$
$601$	1.51660	+	1.13965i	1.51660	+	1.13965i	0.568065	+	0.822984i	$0.307692\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.903450	−	0.428693i	0.903450	−	0.428693i
$606$		0			0
$607$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$607$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$608$		0			0
$609$		0			0
$610$	−0.0509624	+	0.631282i	−0.0509624	+	0.631282i
$611$		0			0
$612$	1.99595	+	0.120733i	1.99595	+	0.120733i
$613$	−0.0763874	+	1.89553i	−0.0763874	+	1.89553i	0.278217	+	0.960518i	$0.410256\pi$
$613$	−0.0763874	+	1.89553i	−0.0763874	+	1.89553i	−0.354605	+	0.935016i	$0.615385\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.95799	+	0.0789044i	−1.95799	+	0.0789044i	−0.987050	−	0.160411i	$-0.948718\pi$
$617$	−1.95799	+	0.0789044i	−1.95799	+	0.0789044i	−0.970942	+	0.239316i	$0.923077\pi$
$618$		0			0
$619$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$619$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.799443	−	0.600742i	0.799443	−	0.600742i
$626$	0.868070	−	1.31343i	0.868070	−	1.31343i
$627$		0			0
$628$	0.625186	−	1.13504i	0.625186	−	1.13504i
$629$	−1.82917	+	1.10577i	−1.82917	+	1.10577i
$630$		0			0
$631$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$631$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$632$		0			0
$633$		0			0
$634$	0.876221	−	0.912242i	0.876221	−	0.912242i
$635$		0			0
$636$		0			0
$637$	0.632445	−	0.774605i	0.632445	−	0.774605i
$638$		0			0
$639$		0			0
$640$	0.464723	+	0.885456i	0.464723	+	0.885456i
$641$	−0.678906	−	1.59345i	−0.678906	−	1.59345i	−0.799443	−	0.600742i	$-0.794872\pi$
$641$	−0.678906	−	1.59345i	−0.678906	−	1.59345i	0.120537	−	0.992709i	$-0.461538\pi$
$642$		0			0
$643$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$643$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.927686	−	0.373361i	$-0.121795\pi$
$647$		0			0		−0.927686	+	0.373361i	$0.878205\pi$
$648$	0.0804666	+	0.996757i	0.0804666	+	0.996757i
$649$		0			0
$650$	−0.822984	+	0.568065i	−0.822984	+	0.568065i
$651$		0			0
$652$		0			0
$653$	0.212445	+	0.792857i	0.212445	+	0.792857i	0.987050	+	0.160411i	$0.0512821\pi$
$653$	0.212445	+	0.792857i	0.212445	+	0.792857i	−0.774605	+	0.632445i	$0.782051\pi$
$654$		0			0
$655$		0			0
$656$	−0.895142	+	1.62515i	−0.895142	+	1.62515i
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$659$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$660$		0			0
$661$	−0.782914	+	0.564016i	−0.782914	+	0.564016i	−0.903450	−	0.428693i	$-0.858974\pi$
$661$	−0.782914	+	0.564016i	−0.782914	+	0.564016i	0.120537	+	0.992709i	$0.461538\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.676041	−	0.828000i	−0.676041	−	0.828000i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.45177	−	0.725045i	−1.45177	−	0.725045i	−0.464723	−	0.885456i	$-0.653846\pi$
$673$	−1.45177	−	0.725045i	−1.45177	−	0.725045i	−0.987050	+	0.160411i	$0.948718\pi$
$674$	0.703389	+	1.74770i	0.703389	+	1.74770i
$675$		0			0
$676$	0.845190	−	0.534466i	0.845190	−	0.534466i
$677$	0.731626	+	0.731626i	0.731626	+	0.731626i	0.970942	−	0.239316i	$-0.0769231\pi$
$677$	0.731626	+	0.731626i	0.731626	+	0.731626i	−0.239316	+	0.970942i	$0.576923\pi$
$678$		0			0
$679$		0			0
$680$	−1.16881	−	1.62243i	−1.16881	−	1.62243i
$681$		0			0
$682$		0			0
$683$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$683$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$684$		0			0
$685$	−0.0224054	+	0.0773523i	−0.0224054	+	0.0773523i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.515016	−	1.92207i	−0.515016	−	1.92207i
$690$		0			0
$691$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$691$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$692$	−0.0887571	+	0.0818806i	−0.0887571	+	0.0818806i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.31558	−	3.46890i	1.31558	−	3.46890i
$698$	−0.574732	−	0.153999i	−0.574732	−	0.153999i
$699$		0			0
$700$		0			0
$701$	1.53901	−	1.06230i	1.53901	−	1.06230i	0.568065	−	0.822984i	$-0.307692\pi$
$701$	1.53901	−	1.06230i	1.53901	−	1.06230i	0.970942	−	0.239316i	$-0.0769231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.171469	+	0.271156i	−0.171469	+	0.271156i
$707$		0			0
$708$		0			0
$709$	−0.782718	−	1.56725i	−0.782718	−	1.56725i	−0.822984	−	0.568065i	$-0.807692\pi$
$709$	−0.782718	−	1.56725i	−0.782718	−	1.56725i	0.0402659	−	0.999189i	$-0.487179\pi$
$710$		0			0
$711$		0			0
$712$	1.54928	−	1.11611i	1.54928	−	1.11611i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$719$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$720$	0.721202	−	0.692724i	0.721202	−	0.692724i
$721$		0			0
$722$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$723$		0			0
$724$	1.89616	−	0.549229i	1.89616	−	0.549229i
$725$	−0.0321908	+	0.398754i	−0.0321908	+	0.398754i
$726$		0			0
$727$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$727$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$728$		0			0
$729$	0.935016	−	0.354605i	0.935016	−	0.354605i
$730$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$731$		0			0
$732$		0			0
$733$	−1.49217	−	1.32194i	−1.49217	−	1.32194i	−0.799443	−	0.600742i	$-0.794872\pi$
$733$	−1.49217	−	1.32194i	−1.49217	−	1.32194i	−0.692724	−	0.721202i	$-0.743590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.81003	+	0.407653i	1.81003	+	0.407653i
$739$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$739$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$740$	−0.213814	+	1.04733i	−0.213814	+	1.04733i
$741$		0			0
$742$		0			0
$743$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$743$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$744$		0			0
$745$	1.43502	+	0.511421i	1.43502	+	0.511421i
$746$	−0.528013	−	0.528013i	−0.528013	−	0.528013i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$751$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$752$		0			0
$753$		0			0
$754$	0.0482209	−	0.397135i	0.0482209	−	0.397135i
$755$		0			0
$756$		0			0
$757$	−1.74864	+	0.963158i	−1.74864	+	0.963158i	−0.845190	+	0.534466i	$0.820513\pi$
$757$	−1.74864	+	0.963158i	−1.74864	+	0.963158i	−0.903450	+	0.428693i	$0.858974\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.306465	+	0.761468i	−0.306465	+	0.761468i	0.692724	+	0.721202i	$0.256410\pi$
$761$	−0.306465	+	0.761468i	−0.306465	+	0.761468i	−0.999189	+	0.0402659i	$0.987179\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.23319	+	1.57405i	−1.23319	+	1.57405i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.963158	+	1.74864i	0.963158	+	1.74864i	0.534466	+	0.845190i	$0.320513\pi$
$769$	0.963158	+	1.74864i	0.963158	+	1.74864i	0.428693	+	0.903450i	$0.358974\pi$
$770$		0			0
$771$		0			0
$772$	−0.159760	−	0.0193983i	−0.159760	−	0.0193983i
$773$	0.404534	+	0.255812i	0.404534	+	0.255812i	0.721202	−	0.692724i	$-0.243590\pi$
$773$	0.404534	+	0.255812i	0.404534	+	0.255812i	−0.316668	+	0.948536i	$0.602564\pi$
$774$		0			0
$775$		0			0
$776$	−0.329236	+	1.61270i	−0.329236	+	1.61270i
$777$		0			0
$778$	−1.21911	−	1.17097i	−1.21911	−	1.17097i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.987050	−	0.160411i	0.987050	−	0.160411i
$785$	0.578975	+	1.15930i	0.578975	+	1.15930i
$786$		0			0
$787$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$787$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$788$	0.879463	−	0.992709i	0.879463	−	0.992709i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.474059	+	0.419979i	0.474059	+	0.419979i
$794$	1.85640	+	0.704039i	1.85640	+	0.704039i
$795$		0			0
$796$		0			0
$797$	1.08039	+	0.595084i	1.08039	+	0.595084i	0.919979	−	0.391967i	$-0.128205\pi$
$797$	1.08039	+	0.595084i	1.08039	+	0.595084i	0.160411	+	0.987050i	$0.448718\pi$
$798$		0			0
$799$		0			0
$800$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$801$	−1.50308	−	1.17759i	−1.50308	−	1.17759i
$802$	−0.632445	+	0.225395i	−0.632445	+	0.225395i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.774605	−	1.63245i	−0.774605	−	1.63245i
$809$	−0.319782	+	1.96770i	−0.319782	+	1.96770i	−0.0804666	+	0.996757i	$0.525641\pi$
$809$	−0.319782	+	1.96770i	−0.319782	+	1.96770i	−0.239316	+	0.970942i	$0.576923\pi$
$810$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$811$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$811$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.509159	−	0.0933069i	0.509159	−	0.0933069i
$819$		0			0
$820$	−0.828977	−	1.65988i	−0.828977	−	1.65988i
$821$	0.0362392	−	0.358684i	0.0362392	−	0.358684i	−0.960518	−	0.278217i	$-0.910256\pi$
$821$	0.0362392	−	0.358684i	0.0362392	−	0.358684i	0.996757	−	0.0804666i	$-0.0256410\pi$
$822$		0			0
$823$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$823$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$827$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$828$		0			0
$829$	−0.0623804	+	1.54795i	−0.0623804	+	1.54795i	0.600742	+	0.799443i	$0.294872\pi$
$829$	−0.0623804	+	1.54795i	−0.0623804	+	1.54795i	−0.663123	+	0.748511i	$0.730769\pi$
$830$		0			0
$831$		0			0
$832$	0.987050	+	0.160411i	0.987050	+	0.160411i
$833$	−1.90905	+	0.594885i	−1.90905	+	0.594885i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$839$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$840$		0			0
$841$	0.581860	+	0.605780i	0.581860	+	0.605780i
$842$	0.399258	−	0.799443i	0.399258	−	0.799443i
$843$		0			0
$844$		0			0
$845$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$846$		0			0
$847$		0			0
$848$	0.889071	−	1.78021i	0.889071	−	1.78021i
$849$		0			0
$850$	1.99919	−	0.0402659i	1.99919	−	0.0402659i
$851$		0			0
$852$		0			0
$853$	1.29542	−	0.491287i	1.29542	−	0.491287i	0.391967	−	0.919979i	$-0.371795\pi$
$853$	1.29542	−	0.491287i	1.29542	−	0.491287i	0.903450	+	0.428693i	$0.141026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.853136	+	0.265848i	−0.853136	+	0.265848i	−0.692724	−	0.721202i	$-0.743590\pi$
$857$	−0.853136	+	0.265848i	−0.853136	+	0.265848i	−0.160411	+	0.987050i	$0.551282\pi$
$858$		0			0
$859$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$859$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$863$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$864$		0			0
$865$	−0.0169667	−	0.119559i	−0.0169667	−	0.119559i
$866$	−0.800768	−	1.77923i	−0.800768	−	1.77923i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.68353	+	0.308518i	−1.68353	+	0.308518i
$873$	1.64063	−	0.132445i	1.64063	−	0.132445i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.708245	+	0.336066i	0.708245	+	0.336066i	0.748511	−	0.663123i	$-0.230769\pi$
$877$	0.708245	+	0.336066i	0.708245	+	0.336066i	−0.0402659	+	0.999189i	$0.512821\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.295150	+	1.81613i	−0.295150	+	1.81613i	0.239316	+	0.970942i	$0.423077\pi$
$881$	−0.295150	+	1.81613i	−0.295150	+	1.81613i	−0.534466	+	0.845190i	$0.679487\pi$
$882$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$883$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$883$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$884$	−1.99919	−	0.0402659i	−1.99919	−	0.0402659i
$885$		0			0
$886$		0			0
$887$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$887$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$888$		0			0
$889$		0			0
$890$	0.0384505	+	1.90905i	0.0384505	+	1.90905i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.12477	+	0.506219i	1.12477	+	0.506219i
$899$		0			0
$900$	0.160411	+	0.987050i	0.160411	+	0.987050i
$901$	−1.26000	+	3.77416i	−1.26000	+	3.77416i
$902$		0			0
$903$		0			0
$904$	0.919533	−	0.996757i	0.919533	−	0.996757i
$905$	−0.625134	+	1.87251i	−0.625134	+	1.87251i
$906$		0			0
$907$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$907$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$908$		0			0
$909$	−1.35248	+	1.19820i	−1.35248	+	1.19820i
$910$		0			0
$911$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$911$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$912$		0			0
$913$		0			0
$914$	0.309882	−	1.51790i	0.309882	−	1.51790i
$915$		0			0
$916$	1.52152	−	1.00560i	1.52152	−	1.00560i
$917$		0			0
$918$		0			0
$919$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$919$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$920$		0			0
$921$		0			0
$922$	0.760492	−	1.25801i	0.760492	−	1.25801i
$923$		0			0
$924$		0			0
$925$	−0.740475	−	0.770916i	−0.740475	−	0.770916i
$926$		0			0
$927$		0			0
$928$	0.299443	−	0.265283i	0.299443	−	0.265283i
$929$	0.0150368	−	0.0373617i	0.0150368	−	0.0373617i	−0.919979	−	0.391967i	$-0.871795\pi$
$929$	0.0150368	−	0.0373617i	0.0150368	−	0.0373617i	0.935016	+	0.354605i	$0.115385\pi$
$930$		0			0
$931$		0			0
$932$	1.27460	+	1.49847i	1.27460	+	1.49847i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$937$	−0.442985	−	0.267794i	−0.442985	−	0.267794i	0.278217	−	0.960518i	$-0.410256\pi$
$937$	−0.442985	−	0.267794i	−0.442985	−	0.267794i	−0.721202	+	0.692724i	$0.756410\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.57149	−	1.23119i	1.57149	−	1.23119i	0.748511	−	0.663123i	$-0.230769\pi$
$941$	1.57149	−	1.23119i	1.57149	−	1.23119i	0.822984	−	0.568065i	$-0.192308\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$947$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$948$		0			0
$949$	0.534466	+	0.845190i	0.534466	+	0.845190i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.61980	+	1.16691i	1.61980	+	1.16691i	0.845190	+	0.534466i	$0.179487\pi$
$953$	1.61980	+	1.16691i	1.61980	+	1.16691i	0.774605	+	0.632445i	$0.217949\pi$
$954$	−1.95728	−	0.358684i	−1.95728	−	0.358684i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.935016	−	0.354605i	0.935016	−	0.354605i
$962$	0.708833	+	0.800107i	0.708833	+	0.800107i
$963$		0			0
$964$	−1.49271	−	0.986562i	−1.49271	−	0.986562i
$965$	0.106718	−	0.120460i	0.106718	−	0.120460i
$966$		0			0
$967$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$967$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$968$	0.799443	−	0.600742i	0.799443	−	0.600742i
$969$		0			0
$970$	−1.14020	−	1.18708i	−1.14020	−	1.18708i
$971$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$971$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.0763402	+	0.628718i	0.0763402	+	0.628718i
$977$	1.80104	−	0.854605i	1.80104	−	0.854605i	0.866025	−	0.500000i	$-0.166667\pi$
$977$	1.80104	−	0.854605i	1.80104	−	0.854605i	0.935016	−	0.354605i	$-0.115385\pi$
$978$		0			0
$979$		0			0
$980$	−0.428693	+	0.903450i	−0.428693	+	0.903450i
$981$	0.764724	+	1.53122i	0.764724	+	1.53122i
$982$		0			0
$983$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$983$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$984$		0			0
$985$	0.317391	+	1.28771i	0.317391	+	1.28771i
$986$	−0.518294	+	0.609325i	−0.518294	+	0.609325i
$987$		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
$997$	−1.46257	+	0.147768i	−1.46257	+	0.147768i	−0.799443	−	0.600742i	$-0.794872\pi$
$997$	−1.46257	+	0.147768i	−1.46257	+	0.147768i	−0.663123	+	0.748511i	$0.730769\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.cs.a.267.1	✓	48
4.3	odd	2	CM	3380.1.cs.a.267.1	✓	48
5.3	odd	4		3380.1.cz.a.943.1	yes	48
20.3	even	4		3380.1.cz.a.943.1	yes	48
169.119	odd	156		3380.1.cz.a.2147.1	yes	48
676.119	even	156		3380.1.cz.a.2147.1	yes	48
845.288	even	156	inner	3380.1.cs.a.2823.1	yes	48
3380.2823	odd	156	inner	3380.1.cs.a.2823.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.cs.a.267.1	✓	48	1.1	even	1	trivial
3380.1.cs.a.267.1	✓	48	4.3	odd	2	CM
3380.1.cs.a.2823.1	yes	48	845.288	even	156	inner
3380.1.cs.a.2823.1	yes	48	3380.2823	odd	156	inner
3380.1.cz.a.943.1	yes	48	5.3	odd	4
3380.1.cz.a.943.1	yes	48	20.3	even	4
3380.1.cz.a.2147.1	yes	48	169.119	odd	156
3380.1.cz.a.2147.1	yes	48	676.119	even	156

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.cs (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.391967 + 0.919979i) q^{2} +(-0.692724 - 0.721202i) q^{4} +(0.987050 - 0.160411i) q^{5} +(0.935016 - 0.354605i) q^{8} +(-0.600742 - 0.799443i) q^{9} +O(q^{10})\)	\(q+(-0.391967 + 0.919979i) q^{2} +(-0.692724 - 0.721202i) q^{4} +(0.987050 - 0.160411i) q^{5} +(0.935016 - 0.354605i) q^{8} +(-0.600742 - 0.799443i) q^{9} +(-0.239316 + 0.970942i) q^{10} +(0.632445 + 0.774605i) q^{13} +(-0.0402659 + 0.999189i) q^{16} +(-0.201003 - 1.98947i) q^{17} +(0.970942 - 0.239316i) q^{18} +(-0.799443 - 0.600742i) q^{20} +(0.948536 - 0.316668i) q^{25} +(-0.960518 + 0.278217i) q^{26} +(-0.156807 + 0.368039i) q^{29} +(-0.903450 - 0.428693i) q^{32} +(1.90905 + 0.594885i) q^{34} +(-0.160411 + 0.987050i) q^{36} +(-0.458243 - 0.965727i) q^{37} +(0.866025 - 0.500000i) q^{40} +(1.65988 + 0.828977i) q^{41} +(-0.721202 - 0.692724i) q^{45} +(-0.200026 - 0.979791i) q^{49} +(-0.0804666 + 0.996757i) q^{50} +(0.120537 - 0.992709i) q^{52} +(-1.81456 - 0.816668i) q^{53} +(-0.277125 - 0.288518i) q^{58} +(0.625134 - 0.101594i) q^{61} +(0.748511 - 0.663123i) q^{64} +(0.748511 + 0.663123i) q^{65} +(-1.29557 + 1.52312i) q^{68} +(-0.845190 - 0.534466i) q^{72} +(0.992709 + 0.120537i) q^{73} +(1.06806 - 0.0430415i) q^{74} +(0.120537 + 0.992709i) q^{80} +(-0.278217 + 0.960518i) q^{81} +(-1.41326 + 1.20212i) q^{82} +(-0.517533 - 1.93146i) q^{85} +(1.84438 - 0.494200i) q^{89} +(0.919979 - 0.391967i) q^{90} +(-0.879714 + 1.39116i) q^{97} +(0.979791 + 0.200026i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 48 * q - 2 * q^4 + 2 * q^5	\( 48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) \) 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more