LMFDB - Embedded newform 3131.1.de.a.1766.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3131,1,Mod(30,3131)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3131, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 47]))

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

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

chi := DirichletCharacter("3131.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3131 = 31 \cdot 101 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3131.de (of order $50$, degree $20$, 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:	$1.56257255455$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

Embedding invariants

Embedding label			1766.1
Root			$0.992115 + 0.125333i$ of defining polynomial
Character	$\chi$	$=$	3131.1766
Dual form			3131.1.de.a.1952.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.211645 - 0.134314i) q^{2} +(-0.399026 - 0.847973i) q^{4} +(0.200808 + 0.316423i) q^{5} +(0.0922765 - 0.233064i) q^{7} +(-0.0608596 + 0.481754i) q^{8} +(-0.876307 + 0.481754i) q^{9} +O(q^{10})$	\(q+(-0.211645 - 0.134314i) q^{2} +(-0.399026 - 0.847973i) q^{4} +(0.200808 + 0.316423i) q^{5} +(0.0922765 - 0.233064i) q^{7} +(-0.0608596 + 0.481754i) q^{8} +(-0.876307 + 0.481754i) q^{9} -0.0939404i q^{10} +(-0.0508335 + 0.0369327i) q^{14} +(-0.519786 + 0.628313i) q^{16} +(0.250172 + 0.0157395i) q^{18} +(1.18532 + 1.43281i) q^{19} +(0.188190 - 0.296541i) q^{20} +(0.365980 - 0.777747i) q^{25} +(-0.234453 + 0.0147505i) q^{28} +(0.929776 + 0.368125i) q^{31} +(0.656217 - 0.213218i) q^{32} +(0.0922765 - 0.0176027i) q^{35} +(0.758183 + 0.550853i) q^{36} +(-0.0584213 - 0.462452i) q^{38} +(-0.164659 + 0.0774826i) q^{40} +(1.30209 + 0.423073i) q^{41} +(-0.328407 - 0.180543i) q^{45} +(-0.115808 - 0.607087i) q^{47} +(0.683165 + 0.641534i) q^{49} +(-0.181920 + 0.115450i) q^{50} +(0.106663 + 0.0586387i) q^{56} +(0.383238 + 0.317042i) q^{59} +(-0.147338 - 0.202793i) q^{62} +(0.0314168 + 0.248690i) q^{63} +(0.622305 + 0.159781i) q^{64} +(0.383238 - 1.49261i) q^{67} +(-0.0218941 - 0.00866849i) q^{70} +(-1.23480 + 0.317042i) q^{71} +(-0.178755 - 0.451483i) q^{72} +(0.742010 - 1.57685i) q^{76} +(-0.303189 - 0.0383017i) q^{80} +(0.535827 - 0.844328i) q^{81} +(-0.218755 - 0.264429i) q^{82} +(0.0452561 + 0.0823206i) q^{90} +(-0.0570300 + 0.144041i) q^{94} +(-0.215351 + 0.662783i) q^{95} +(0.263146 + 0.559214i) q^{97} +(-0.0584213 - 0.227536i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{8}+O(q^{10}) $ 20 * q + 20 * q^8	$ 20 q + 20 q^{8} - 10 q^{14} + 5 q^{16} - 5 q^{19} + 5 q^{20} + 5 q^{28} - 5 q^{36} + 5 q^{38} - 5 q^{40} - 5 q^{45} + 5 q^{50} - 20 q^{56} + 5 q^{59} + 20 q^{63} + 20 q^{64} + 5 q^{67} - 5 q^{70} - 5 q^{71} - 20 q^{72} - 5 q^{76} + 20 q^{82} + 5 q^{98}+O(q^{100}) $ 20 * q + 20 * q^8 - 10 * q^14 + 5 * q^16 - 5 * q^19 + 5 * q^20 + 5 * q^28 - 5 * q^36 + 5 * q^38 - 5 * q^40 - 5 * q^45 + 5 * q^50 - 20 * q^56 + 5 * q^59 + 20 * q^63 + 20 * q^64 + 5 * q^67 - 5 * q^70 - 5 * q^71 - 20 * q^72 - 5 * q^76 + 20 * q^82 + 5 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3131\mathbb{Z}\right)^\times$.

$n$	$809$	$2729$
$\chi(n)$	$-1$	$e\left(\frac{9}{50}\right)$

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 $

$2$	−0.211645	−	0.134314i	−0.211645	−	0.134314i	0.425779	−	0.904827i	$-0.360000\pi$
$2$	−0.211645	−	0.134314i	−0.211645	−	0.134314i	−0.637424	+	0.770513i	$0.720000\pi$
$3$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$3$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$4$	−0.399026	−	0.847973i	−0.399026	−	0.847973i
$5$	0.200808	+	0.316423i	0.200808	+	0.316423i	0.929776	−	0.368125i	$-0.120000\pi$
$5$	0.200808	+	0.316423i	0.200808	+	0.316423i	−0.728969	+	0.684547i	$0.760000\pi$
$6$		0			0
$7$	0.0922765	−	0.233064i	0.0922765	−	0.233064i	−0.876307	−	0.481754i	$-0.840000\pi$
$7$	0.0922765	−	0.233064i	0.0922765	−	0.233064i	0.968583	+	0.248690i	$0.0800000\pi$
$8$	−0.0608596	+	0.481754i	−0.0608596	+	0.481754i
$9$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$10$		−	0.0939404i		−	0.0939404i
$11$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$11$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$12$		0			0
$13$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$13$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$14$	−0.0508335	+	0.0369327i	−0.0508335	+	0.0369327i
$15$		0			0
$16$	−0.519786	+	0.628313i	−0.519786	+	0.628313i
$17$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$17$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$18$	0.250172	+	0.0157395i	0.250172	+	0.0157395i
$19$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.876307	+	0.481754i	$0.160000\pi$
$19$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.309017	+	0.951057i	$0.400000\pi$
$20$	0.188190	−	0.296541i	0.188190	−	0.296541i
$21$		0			0
$22$		0			0
$23$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$23$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$24$		0			0
$25$	0.365980	−	0.777747i	0.365980	−	0.777747i
$26$		0			0
$27$		0			0
$28$	−0.234453	+	0.0147505i	−0.234453	+	0.0147505i
$29$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$29$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$30$		0			0
$31$	0.929776	+	0.368125i	0.929776	+	0.368125i
$31$	0.929776	+	0.368125i	0.929776	+	0.368125i
$32$	0.656217	−	0.213218i	0.656217	−	0.213218i
$33$		0			0
$34$		0			0
$35$	0.0922765	−	0.0176027i	0.0922765	−	0.0176027i
$36$	0.758183	+	0.550853i	0.758183	+	0.550853i
$37$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$37$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$38$	−0.0584213	−	0.462452i	−0.0584213	−	0.462452i
$39$		0			0
$40$	−0.164659	+	0.0774826i	−0.164659	+	0.0774826i
$41$	1.30209	+	0.423073i	1.30209	+	0.423073i	0.876307	−	0.481754i	$-0.160000\pi$
$41$	1.30209	+	0.423073i	1.30209	+	0.423073i	0.425779	+	0.904827i	$0.360000\pi$
$42$		0			0
$43$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$43$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$44$		0			0
$45$	−0.328407	−	0.180543i	−0.328407	−	0.180543i
$46$		0			0
$47$	−0.115808	−	0.607087i	−0.115808	−	0.607087i	−0.992115	−	0.125333i	$-0.960000\pi$
$47$	−0.115808	−	0.607087i	−0.115808	−	0.607087i	0.876307	−	0.481754i	$-0.160000\pi$
$48$		0			0
$49$	0.683165	+	0.641534i	0.683165	+	0.641534i
$50$	−0.181920	+	0.115450i	−0.181920	+	0.115450i
$51$		0			0
$52$		0			0
$53$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$53$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$54$		0			0
$55$		0			0
$56$	0.106663	+	0.0586387i	0.106663	+	0.0586387i
$57$		0			0
$58$		0			0
$59$	0.383238	+	0.317042i	0.383238	+	0.317042i	0.809017	−	0.587785i	$-0.200000\pi$
$59$	0.383238	+	0.317042i	0.383238	+	0.317042i	−0.425779	+	0.904827i	$0.640000\pi$
$60$		0			0
$61$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$61$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$62$	−0.147338	−	0.202793i	−0.147338	−	0.202793i
$63$	0.0314168	+	0.248690i	0.0314168	+	0.248690i
$64$	0.622305	+	0.159781i	0.622305	+	0.159781i
$65$		0			0
$66$		0			0
$67$	0.383238	−	1.49261i	0.383238	−	1.49261i	−0.425779	−	0.904827i	$-0.640000\pi$
$67$	0.383238	−	1.49261i	0.383238	−	1.49261i	0.809017	−	0.587785i	$-0.200000\pi$
$68$		0			0
$69$		0			0
$70$	−0.0218941	−	0.00866849i	−0.0218941	−	0.00866849i
$71$	−1.23480	+	0.317042i	−1.23480	+	0.317042i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−1.23480	+	0.317042i	−1.23480	+	0.317042i	−0.425779	+	0.904827i	$0.640000\pi$
$72$	−0.178755	−	0.451483i	−0.178755	−	0.451483i
$73$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$73$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$74$		0			0
$75$		0			0
$76$	0.742010	−	1.57685i	0.742010	−	1.57685i
$77$		0			0
$78$		0			0
$79$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$79$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$80$	−0.303189	−	0.0383017i	−0.303189	−	0.0383017i
$81$	0.535827	−	0.844328i	0.535827	−	0.844328i
$82$	−0.218755	−	0.264429i	−0.218755	−	0.264429i
$83$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$83$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$89$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$90$	0.0452561	+	0.0823206i	0.0452561	+	0.0823206i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.0570300	+	0.144041i	−0.0570300	+	0.144041i
$95$	−0.215351	+	0.662783i	−0.215351	+	0.662783i
$96$		0			0
$97$	0.263146	+	0.559214i	0.263146	+	0.559214i	0.992115	−	0.125333i	$-0.0400000\pi$
$97$	0.263146	+	0.559214i	0.263146	+	0.559214i	−0.728969	+	0.684547i	$0.760000\pi$
$98$	−0.0584213	−	0.227536i	−0.0584213	−	0.227536i
$99$		0			0
$100$	−0.805544			−0.805544
$101$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$101$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$102$		0			0
$103$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$103$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$107$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$108$		0			0
$109$	−0.147338	+	1.16630i	−0.147338	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$109$	−0.147338	+	1.16630i	−0.147338	+	1.16630i	−0.876307	+	0.481754i	$0.840000\pi$
$110$		0			0
$111$		0			0
$112$	0.0984730	+	0.179122i	0.0984730	+	0.179122i
$113$	−0.383238	+	0.317042i	−0.383238	+	0.317042i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	−0.383238	+	0.317042i	−0.383238	+	0.317042i	0.425779	+	0.904827i	$0.360000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.0385271	−	0.118574i	−0.0385271	−	0.118574i
$119$		0			0
$120$		0			0
$121$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$122$		0			0
$123$		0			0
$124$	−0.0588452	−	0.935317i	−0.0588452	−	0.935317i
$125$	0.691396	−	0.0873436i	0.691396	−	0.0873436i
$126$	0.0267533	−	0.0568536i	0.0267533	−	0.0568536i
$127$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$127$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$128$	−0.582576	−	0.620380i	−0.582576	−	0.620380i
$129$		0			0
$130$		0			0
$131$	0.121636	−	0.0312307i	0.121636	−	0.0312307i	−0.187381	−	0.982287i	$-0.560000\pi$
$131$	0.121636	−	0.0312307i	0.121636	−	0.0312307i	0.309017	+	0.951057i	$0.400000\pi$
$132$		0			0
$133$	0.443314	−	0.144041i	0.443314	−	0.144041i
$134$	−0.281589	+	0.264429i	−0.281589	+	0.264429i
$135$		0			0
$136$		0			0
$137$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$137$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$138$		0			0
$139$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$139$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$140$	−0.0517473	−	0.0712241i	−0.0517473	−	0.0712241i
$141$		0			0
$142$	0.303921	+	0.0987500i	0.303921	+	0.0987500i
$143$		0			0
$144$	0.152800	−	0.801003i	0.152800	−	0.801003i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.23879	−	0.582932i	−1.23879	−	0.582932i	−0.309017	−	0.951057i	$-0.600000\pi$
$149$	−1.23879	−	0.582932i	−1.23879	−	0.582932i	−0.929776	+	0.368125i	$0.880000\pi$
$150$		0			0
$151$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$151$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$152$	−0.762400	+	0.483834i	−0.762400	+	0.483834i
$153$		0			0
$154$		0			0
$155$	0.0702235	+	0.368125i	0.0702235	+	0.368125i
$156$		0			0
$157$	0.328407	+	0.180543i	0.328407	+	0.180543i	0.637424	−	0.770513i	$-0.280000\pi$
$157$	0.328407	+	0.180543i	0.328407	+	0.180543i	−0.309017	+	0.951057i	$0.600000\pi$
$158$		0			0
$159$		0			0
$160$	0.199241	+	0.164826i	0.199241	+	0.164826i
$161$		0			0
$162$	−0.226810	+	0.106729i	−0.226810	+	0.106729i
$163$	0.992567	+	1.36615i	0.992567	+	1.36615i	0.929776	+	0.368125i	$0.120000\pi$
$163$	0.992567	+	1.36615i	0.992567	+	1.36615i	0.0627905	+	0.998027i	$0.480000\pi$
$164$	−0.160811	−	1.27295i	−0.160811	−	1.27295i
$165$		0			0
$166$		0			0
$167$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$167$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$168$		0			0
$169$	0.728969	−	0.684547i	0.728969	−	0.684547i
$170$		0			0
$171$	−1.72897	−	0.684547i	−1.72897	−	0.684547i
$172$		0			0
$173$	0.723208	+	1.82662i	0.723208	+	1.82662i	0.535827	+	0.844328i	$0.320000\pi$
$173$	0.723208	+	1.82662i	0.723208	+	1.82662i	0.187381	+	0.982287i	$0.440000\pi$
$174$		0			0
$175$	−0.147493	−	0.157064i	−0.147493	−	0.157064i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$179$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$180$	−0.0220530	+	0.350522i	−0.0220530	+	0.350522i
$181$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$181$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.468583	+	0.340446i	−0.468583	+	0.340446i
$189$		0			0
$190$	0.134599	−	0.111350i	0.134599	−	0.111350i
$191$	−0.354691	−	0.645180i	−0.354691	−	0.645180i	0.637424	−	0.770513i	$-0.280000\pi$
$191$	−0.354691	−	0.645180i	−0.354691	−	0.645180i	−0.992115	+	0.125333i	$0.960000\pi$
$192$		0			0
$193$	0.746226	−	0.410241i	0.746226	−	0.410241i	−0.0627905	−	0.998027i	$-0.520000\pi$
$193$	0.746226	−	0.410241i	0.746226	−	0.410241i	0.809017	+	0.587785i	$0.200000\pi$
$194$	0.0194167	−	0.153699i	0.0194167	−	0.153699i
$195$		0			0
$196$	0.271404	−	0.835295i	0.271404	−	0.835295i
$197$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$197$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$198$		0			0
$199$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$199$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$200$	0.352409	+	0.223646i	0.352409	+	0.223646i
$201$		0			0
$202$			0.250666i			0.250666i
$203$		0			0
$204$		0			0
$205$	0.127599	+	0.496966i	0.127599	+	0.496966i
$206$	−0.164472	−	0.349520i	−0.164472	−	0.349520i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.541587	+	0.297740i	−0.541587	+	0.297740i	−0.728969	−	0.684547i	$-0.760000\pi$
$211$	−0.541587	+	0.297740i	−0.541587	+	0.297740i	0.187381	+	0.982287i	$0.440000\pi$
$212$		0			0
$213$		0			0
$214$	−0.281589	+	0.232950i	−0.281589	+	0.232950i
$215$		0			0
$216$		0			0
$217$	0.171593	−	0.182728i	0.171593	−	0.182728i
$218$	0.187834	−	0.227052i	0.187834	−	0.227052i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$223$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$224$	0.0108600	−	0.172616i	0.0108600	−	0.172616i
$225$	0.0539718	+	0.857857i	0.0539718	+	0.857857i
$226$	0.123693	−	0.0156261i	0.123693	−	0.0156261i
$227$	0.824805	−	1.75280i	0.824805	−	1.75280i	0.187381	−	0.982287i	$-0.440000\pi$
$227$	0.824805	−	1.75280i	0.824805	−	1.75280i	0.637424	−	0.770513i	$-0.280000\pi$
$228$		0			0
$229$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$229$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	0.0627905	−	0.998027i	$-0.480000\pi$
$233$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	−0.637424	+	0.770513i	$0.720000\pi$
$234$		0			0
$235$	0.168841	−	0.158552i	0.168841	−	0.158552i
$236$	0.115921	−	0.451483i	0.115921	−	0.451483i
$237$		0			0
$238$		0			0
$239$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$239$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$240$		0			0
$241$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$241$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$242$	0.226810	−	0.106729i	0.226810	−	0.106729i
$243$		0			0
$244$		0			0
$245$	−0.0658111	+	0.344994i	−0.0658111	+	0.344994i
$246$		0			0
$247$		0			0
$248$	−0.233931	+	0.425519i	−0.233931	+	0.425519i
$249$		0			0
$250$	−0.158062	−	0.0743782i	−0.158062	−	0.0743782i
$251$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$251$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$252$	0.198346	−	0.125874i	0.198346	−	0.125874i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.0804172	−	0.421562i	−0.0804172	−	0.421562i
$257$	−0.916350	+	1.66683i	−0.916350	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$257$	−0.916350	+	1.66683i	−0.916350	+	1.66683i	−0.728969	+	0.684547i	$0.760000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.0299383	−	0.00972753i	−0.0299383	−	0.00972753i
$263$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$263$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$264$		0			0
$265$		0			0
$266$	−0.113172	−	0.0290576i	−0.113172	−	0.0290576i
$267$		0			0
$268$	−1.41862	+	0.270616i	−1.41862	+	0.270616i
$269$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$269$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$277$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$278$		0			0
$279$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$280$	0.00286424	+	0.0455258i	0.00286424	+	0.0455258i
$281$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i	0.876307	+	0.481754i	$0.160000\pi$
$281$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i	−0.929776	+	0.368125i	$0.880000\pi$
$282$		0			0
$283$	0.331159	−	0.521823i	0.331159	−	0.521823i	−0.637424	−	0.770513i	$-0.720000\pi$
$283$	0.331159	−	0.521823i	0.331159	−	0.521823i	0.968583	+	0.248690i	$0.0800000\pi$
$284$	0.761559	+	0.920567i	0.761559	+	0.920567i
$285$		0			0
$286$		0			0
$287$	0.218755	−	0.264429i	0.218755	−	0.264429i
$288$	−0.472329	+	0.502980i	−0.472329	+	0.502980i
$289$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$293$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$	−0.0233620	+	0.184930i	−0.0233620	+	0.184930i
$296$		0			0
$297$		0			0
$298$	0.183888	+	0.289762i	0.183888	+	0.289762i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.51637			−1.51637
$305$		0			0
$306$		0			0
$307$	−0.688925	−	1.46404i	−0.688925	−	1.46404i	−0.876307	−	0.481754i	$-0.840000\pi$
$307$	−0.688925	−	1.46404i	−0.688925	−	1.46404i	0.187381	−	0.982287i	$-0.440000\pi$
$308$		0			0
$309$		0			0
$310$	0.0345818	−	0.0873436i	0.0345818	−	0.0873436i
$311$	−0.147338	+	1.16630i	−0.147338	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$311$	−0.147338	+	1.16630i	−0.147338	+	1.16630i	−0.876307	+	0.481754i	$0.840000\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	−0.0452561	−	0.0823206i	−0.0452561	−	0.0823206i
$315$	−0.0723823	+	0.0598799i	−0.0723823	+	0.0598799i
$316$		0			0
$317$	−1.03137	+	0.749337i	−1.03137	+	0.749337i	−0.968583	−	0.248690i	$-0.920000\pi$
$317$	−1.03137	+	0.749337i	−1.03137	+	0.749337i	−0.0627905	+	0.998027i	$0.520000\pi$
$318$		0			0
$319$		0			0
$320$	0.0744055	+	0.228997i	0.0744055	+	0.228997i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.929776	−	0.117458i	−0.929776	−	0.117458i
$325$		0			0
$326$	−0.0265786	−	0.422454i	−0.0265786	−	0.422454i
$327$		0			0
$328$	−0.283062	+	0.601537i	−0.283062	+	0.601537i
$329$	−0.152176	−	0.0290292i	−0.152176	−	0.0290292i
$330$		0			0
$331$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$331$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.549253	−	0.178463i	0.549253	−	0.178463i
$336$		0			0
$337$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$337$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$338$	−0.246226	+	0.0469702i	−0.246226	+	0.0469702i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.273983	+	0.377105i	0.273983	+	0.377105i
$343$	0.439368	−	0.206751i	0.439368	−	0.206751i
$344$		0			0
$345$		0			0
$346$	0.0922765	−	0.483730i	0.0922765	−	0.483730i
$347$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$347$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$348$		0			0
$349$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$349$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$350$	0.0101203	+	0.0530522i	0.0101203	+	0.0530522i
$351$		0			0
$352$		0			0
$353$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$353$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$354$		0			0
$355$	−0.348276	−	0.327053i	−0.348276	−	0.327053i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.110048	−	0.0604991i	−0.110048	−	0.0604991i	0.425779	−	0.904827i	$-0.360000\pi$
$359$	−0.110048	−	0.0604991i	−0.110048	−	0.0604991i	−0.535827	+	0.844328i	$0.680000\pi$
$360$	0.106964	−	0.147223i	0.106964	−	0.147223i
$361$	−0.460572	+	2.41440i	−0.460572	+	2.41440i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$367$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$368$		0			0
$369$	−1.34484	+	0.256543i	−1.34484	+	0.256543i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.62954	−	0.645180i	−1.62954	−	0.645180i	−0.637424	−	0.770513i	$-0.720000\pi$
$373$	−1.62954	−	0.645180i	−1.62954	−	0.645180i	−0.992115	+	0.125333i	$0.960000\pi$
$374$		0			0
$375$		0			0
$376$	0.299514	−	0.0188438i	0.299514	−	0.0188438i
$377$		0			0
$378$		0			0
$379$	0.791759	−	1.68257i	0.791759	−	1.68257i	0.0627905	−	0.998027i	$-0.480000\pi$
$379$	0.791759	−	1.68257i	0.791759	−	1.68257i	0.728969	−	0.684547i	$-0.240000\pi$
$380$	0.647953	−	0.0818555i	0.647953	−	0.0818555i
$381$		0			0
$382$	−0.0115882	+	0.184189i	−0.0115882	+	0.184189i
$383$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$383$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$384$		0			0
$385$		0			0
$386$	−0.213036	−	0.0134031i	−0.213036	−	0.0134031i
$387$		0			0
$388$	0.369196	−	0.446282i	0.369196	−	0.446282i
$389$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$389$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$390$		0			0
$391$		0			0
$392$	−0.350639	+	0.290074i	−0.350639	+	0.290074i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.503997	−	1.27295i	0.503997	−	1.27295i	−0.425779	−	0.904827i	$-0.640000\pi$
$397$	0.503997	−	1.27295i	0.503997	−	1.27295i	0.929776	−	0.368125i	$-0.120000\pi$
$398$		0			0
$399$		0			0
$400$	0.298437	+	0.634211i	0.298437	+	0.634211i
$401$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$401$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$402$		0			0
$403$		0			0
$404$	−0.502159	+	0.791276i	−0.502159	+	0.791276i
$405$	0.374763			0.374763
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$409$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$410$	0.0397437	−	0.122319i	0.0397437	−	0.122319i
$411$		0			0
$412$	0.181006	−	1.43281i	0.181006	−	1.43281i
$413$	0.109255	−	0.0600633i	0.109255	−	0.0600633i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.30209	+	1.38658i	−1.30209	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$419$	−1.30209	+	1.38658i	−1.30209	+	1.38658i	−0.876307	+	0.481754i	$0.840000\pi$
$420$		0			0
$421$	−0.0388067	−	0.119435i	−0.0388067	−	0.119435i	0.929776	−	0.368125i	$-0.120000\pi$
$421$	−0.0388067	−	0.119435i	−0.0388067	−	0.119435i	−0.968583	+	0.248690i	$0.920000\pi$
$422$	0.154615	+	0.00972753i	0.154615	+	0.00972753i
$423$	0.393950	+	0.476203i	0.393950	+	0.476203i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.35556	+	0.171247i	−1.35556	+	0.171247i
$429$		0			0
$430$		0			0
$431$	−1.34484	−	1.43211i	−1.34484	−	1.43211i	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−1.34484	−	1.43211i	−1.34484	−	1.43211i	−0.535827	−	0.844328i	$-0.680000\pi$
$432$		0			0
$433$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$433$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$434$	−0.0608596	+	0.0156261i	−0.0608596	+	0.0156261i
$435$		0			0
$436$	1.04778	−	0.340446i	1.04778	−	0.340446i
$437$		0			0
$438$		0			0
$439$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$439$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$440$		0			0
$441$	−0.907724	−	0.233064i	−0.907724	−	0.233064i
$442$		0			0
$443$	−1.17325	−	1.61484i	−1.17325	−	1.61484i	−0.637424	−	0.770513i	$-0.720000\pi$
$443$	−1.17325	−	1.61484i	−1.17325	−	1.61484i	−0.535827	−	0.844328i	$-0.680000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0946633	−	0.130293i	0.0946633	−	0.130293i
$449$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$449$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$450$	0.103799	−	0.188810i	0.103799	−	0.188810i
$451$		0			0
$452$	0.421765	+	0.198467i	0.421765	+	0.198467i
$453$		0			0
$454$	−0.409991	+	0.260188i	−0.409991	+	0.260188i
$455$		0			0
$456$		0			0
$457$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$457$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$461$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$462$		0			0
$463$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$463$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$464$		0			0
$465$		0			0
$466$	0.0910599	+	0.125333i	0.0910599	+	0.125333i
$467$	−0.147338	−	1.16630i	−0.147338	−	1.16630i	−0.876307	−	0.481754i	$-0.840000\pi$
$467$	−0.147338	−	1.16630i	−0.147338	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$468$		0			0
$469$	−0.312510	−	0.227052i	−0.312510	−	0.227052i
$470$	−0.0570300	+	0.0108791i	−0.0570300	+	0.0108791i
$471$		0			0
$472$	−0.176060	+	0.165331i	−0.176060	+	0.165331i
$473$		0			0
$474$		0			0
$475$	1.54817	−	0.397502i	1.54817	−	0.397502i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.992115	−	0.125333i	$-0.960000\pi$
$479$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.968583	−	0.248690i	$-0.920000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.929776	+	0.117458i	0.929776	+	0.117458i
$485$	−0.124106	+	0.195560i	−0.124106	+	0.195560i
$486$		0			0
$487$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$487$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$488$		0			0
$489$		0			0
$490$	0.0602660	−	0.0641768i	0.0602660	−	0.0641768i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.714582	+	0.392845i	−0.714582	+	0.392845i
$497$	−0.0400517	+	0.317042i	−0.0400517	+	0.317042i
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$	−0.349950	−	0.551433i	−0.349950	−	0.551433i
$501$		0			0
$502$		0			0
$503$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$503$	1.30113	+	0.825723i	1.30113	+	0.825723i	0.309017	+	0.951057i	$0.400000\pi$
$504$	−0.121719			−0.121719
$505$	0.159566	−	0.339095i	0.159566	−	0.339095i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$509$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$510$		0			0
$511$		0			0
$512$	−0.352890	+	0.891298i	−0.352890	+	0.891298i
$513$		0			0
$514$	0.417819	−	0.229698i	0.417819	−	0.229698i
$515$			0.577519i			0.577519i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.542804	+	0.656137i	−0.542804	+	0.656137i	−0.968583	−	0.248690i	$-0.920000\pi$
$521$	−0.542804	+	0.656137i	−0.542804	+	0.656137i	0.425779	+	0.904827i	$0.360000\pi$
$522$		0			0
$523$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$523$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$524$	−0.0750186	−	0.0906820i	−0.0750186	−	0.0906820i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$530$		0			0
$531$	−0.488570	−	0.0931997i	−0.488570	−	0.0931997i
$532$	−0.299037	−	0.318442i	−0.299037	−	0.318442i
$533$		0			0
$534$		0			0
$535$	0.529215	−	0.135879i	0.529215	−	0.135879i
$536$	0.695748	+	0.275466i	0.695748	+	0.275466i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$541$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.398631	+	0.187581i	−0.398631	+	0.187581i
$546$		0			0
$547$	−1.39436	−	1.15352i	−1.39436	−	1.15352i	−0.968583	−	0.248690i	$-0.920000\pi$
$547$	−1.39436	−	1.15352i	−1.39436	−	1.15352i	−0.425779	−	0.904827i	$-0.640000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$557$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$558$	0.226810	+	0.106729i	0.226810	+	0.106729i
$559$		0			0
$560$	−0.0369040	+	0.0671281i	−0.0369040	+	0.0671281i
$561$		0			0
$562$	0.125467	−	0.172690i	0.125467	−	0.172690i
$563$	−0.348445	+	1.82662i	−0.348445	+	1.82662i	0.187381	+	0.982287i	$0.440000\pi$
$563$	−0.348445	+	1.82662i	−0.348445	+	1.82662i	−0.535827	+	0.844328i	$0.680000\pi$
$564$		0			0
$565$	−0.177276	−	0.0576006i	−0.177276	−	0.0576006i
$566$	−0.140176	+	0.0659619i	−0.140176	+	0.0659619i
$567$	−0.147338	−	0.202793i	−0.147338	−	0.202793i
$568$	−0.0775868	−	0.614163i	−0.0775868	−	0.614163i
$569$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$569$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$570$		0			0
$571$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$571$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$572$		0			0
$573$		0			0
$574$	−0.0818149	+	0.0265833i	−0.0818149	+	0.0265833i
$575$		0			0
$576$	−0.622305	+	0.159781i	−0.622305	+	0.159781i
$577$	−0.621636	−	1.57007i	−0.621636	−	1.57007i	−0.809017	−	0.587785i	$-0.800000\pi$
$577$	−0.621636	−	1.57007i	−0.621636	−	1.57007i	0.187381	−	0.982287i	$-0.440000\pi$
$578$	0.250172	−	0.0157395i	0.250172	−	0.0157395i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.255480	−	0.402572i	0.255480	−	0.402572i
$587$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$587$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$588$		0			0
$589$	0.574633	+	1.76854i	0.574633	+	1.76854i
$590$	0.0297830	−	0.0360015i	0.0297830	−	0.0360015i
$591$		0			0
$592$		0			0
$593$	1.50441	−	0.595638i	1.50441	−	0.595638i	0.535827	−	0.844328i	$-0.320000\pi$
$593$	1.50441	−	0.595638i	1.50441	−	0.595638i	0.968583	+	0.248690i	$0.0800000\pi$
$594$		0			0
$595$		0			0
$596$			1.28307i			1.28307i
$597$		0			0
$598$		0			0
$599$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$599$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$600$		0			0
$601$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$601$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$602$		0			0
$603$	0.383238	+	1.49261i	0.383238	+	1.49261i
$604$		0			0
$605$	−0.374763			−0.374763
$606$		0			0
$607$	0.851559			0.851559			0.425779	−	0.904827i	$-0.360000\pi$
$607$	0.851559			0.851559			0.425779	+	0.904827i	$0.360000\pi$
$608$	1.08333	+	0.687503i	1.08333	+	0.687503i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$613$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$614$	−0.0508335	+	0.402389i	−0.0508335	+	0.402389i
$615$		0			0
$616$		0			0
$617$	−0.659566	−	1.19975i	−0.659566	−	1.19975i	−0.968583	−	0.248690i	$-0.920000\pi$
$617$	−0.659566	−	1.19975i	−0.659566	−	1.19975i	0.309017	−	0.951057i	$-0.400000\pi$
$618$		0			0
$619$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$619$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$620$	0.284139	−	0.206439i	0.284139	−	0.206439i
$621$		0			0
$622$	0.187834	−	0.227052i	0.187834	−	0.227052i
$623$		0			0
$624$		0			0
$625$	−0.381424	−	0.461063i	−0.381424	−	0.461063i
$626$		0			0
$627$		0			0
$628$	0.0220530	−	0.350522i	0.0220530	−	0.350522i
$629$		0			0
$630$	0.0233620	−	0.00295131i	0.0233620	−	0.00295131i
$631$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$631$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$632$		0			0
$633$		0			0
$634$	0.318931	−	0.0200654i	0.318931	−	0.0200654i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.929324	−	0.872693i	0.929324	−	0.872693i
$640$	0.0793165	−	0.308917i	0.0793165	−	0.308917i
$641$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$641$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$642$		0			0
$643$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$643$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$647$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$648$	0.374148	+	0.309522i	0.374148	+	0.309522i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.762400	−	1.38680i	0.762400	−	1.38680i
$653$	0.200808	+	1.05267i	0.200808	+	1.05267i	0.929776	+	0.368125i	$0.120000\pi$
$653$	0.200808	+	1.05267i	0.200808	+	1.05267i	−0.728969	+	0.684547i	$0.760000\pi$
$654$		0			0
$655$	0.0343075	+	0.0322169i	0.0343075	+	0.0322169i
$656$	−0.942628	+	0.598210i	−0.942628	+	0.598210i
$657$		0			0
$658$	0.0283083	+	0.0265833i	0.0283083	+	0.0265833i
$659$	1.52794	+	0.718995i	1.52794	+	0.718995i	0.992115	−	0.125333i	$-0.0400000\pi$
$659$	1.52794	+	0.718995i	1.52794	+	0.718995i	0.535827	+	0.844328i	$0.320000\pi$
$660$		0			0
$661$	−0.742395	+	1.35041i	−0.742395	+	1.35041i	0.187381	+	0.982287i	$0.440000\pi$
$661$	−0.742395	+	1.35041i	−0.742395	+	1.35041i	−0.929776	+	0.368125i	$0.880000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.134599	+	0.111350i	0.134599	+	0.111350i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−0.140217	−	0.0360015i	−0.140217	−	0.0360015i
$671$		0			0
$672$		0			0
$673$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$673$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$674$		0			0
$675$		0			0
$676$	−0.871355	−	0.344994i	−0.871355	−	0.344994i
$677$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$677$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$678$		0			0
$679$	0.154615	−	0.00972753i	0.154615	−	0.00972753i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.06320	−	0.134314i	1.06320	−	0.134314i	0.425779	−	0.904827i	$-0.360000\pi$
$683$	1.06320	−	0.134314i	1.06320	−	0.134314i	0.637424	+	0.770513i	$0.280000\pi$
$684$	0.109426	+	1.73927i	0.109426	+	1.73927i
$685$		0			0
$686$	−0.120759	−	0.0152555i	−0.120759	−	0.0152555i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.683098	−	0.825723i	0.683098	−	0.825723i	−0.309017	−	0.951057i	$-0.600000\pi$
$691$	0.683098	−	0.825723i	0.683098	−	0.825723i	0.992115	+	0.125333i	$0.0400000\pi$
$692$	1.26034	−	1.34213i	1.26034	−	1.34213i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.0743328	+	0.187743i	−0.0743328	+	0.187743i
$701$	0.613161	−	1.88711i	0.613161	−	1.88711i	0.187381	−	0.982287i	$-0.440000\pi$
$701$	0.613161	−	1.88711i	0.613161	−	1.88711i	0.425779	−	0.904827i	$-0.360000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.246226	+	0.0469702i	−0.246226	+	0.0469702i
$708$		0			0
$709$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$709$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$710$	0.0297830	+	0.115997i	0.0297830	+	0.115997i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.0151651	+	0.0275852i	0.0151651	+	0.0275852i
$719$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$719$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$720$	0.284139	−	0.112499i	0.284139	−	0.112499i
$721$	0.312510	−	0.227052i	0.312510	−	0.227052i
$722$	0.421765	−	0.449134i	0.421765	−	0.449134i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.574221	−	0.904827i	0.574221	−	0.904827i	−0.425779	−	0.904827i	$-0.640000\pi$
$727$	0.574221	−	0.904827i	0.574221	−	0.904827i	1.00000			$0$
$728$		0			0
$729$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.92978	−	0.368125i	−1.92978	−	0.368125i	−0.929776	−	0.368125i	$-0.880000\pi$
$733$	−1.92978	−	0.368125i	−1.92978	−	0.368125i	−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.319086	+	0.126335i	0.319086	+	0.126335i
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$	−0.0643066	−	0.509040i	−0.0643066	−	0.509040i
$746$	0.258227	+	0.355418i	0.258227	+	0.355418i
$747$		0			0
$748$		0			0
$749$	−0.281589	−	0.232950i	−0.281589	−	0.232950i
$750$		0			0
$751$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$751$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$752$	0.441636	+	0.242791i	0.441636	+	0.242791i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$757$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$758$	−0.393565	+	0.249764i	−0.393565	+	0.249764i
$759$		0			0
$760$	−0.306192	−	0.144083i	−0.306192	−	0.144083i
$761$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$761$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$762$		0			0
$763$	0.258227	+	0.141961i	0.258227	+	0.141961i
$764$	−0.405565	+	0.558212i	−0.405565	+	0.558212i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$769$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$	−0.645638	−	0.469083i	−0.645638	−	0.469083i
$773$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$773$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$774$		0			0
$775$	0.626587	−	0.588404i	0.626587	−	0.588404i
$776$	−0.285418	+	0.0927380i	−0.285418	+	0.0927380i
$777$		0			0
$778$		0			0
$779$	0.937209	+	2.36712i	0.937209	+	2.36712i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.758183	+	0.0957808i	−0.758183	+	0.0957808i
$785$	0.00881874	+	0.140170i	0.00881874	+	0.140170i
$786$		0			0
$787$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$787$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.0385271	+	0.118574i	0.0385271	+	0.118574i
$792$		0			0
$793$		0			0
$794$	−0.277643	+	0.201720i	−0.277643	+	0.201720i
$795$		0			0
$796$		0			0
$797$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$797$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$798$		0			0
$799$		0			0
$800$	0.0743328	−	0.588404i	0.0743328	−	0.588404i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.439368	−	0.206751i	0.439368	−	0.206751i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.0793165	−	0.0503358i	−0.0793165	−	0.0503358i
$811$	−0.0623382	−	0.242791i	−0.0623382	−	0.242791i	0.929776	−	0.368125i	$-0.120000\pi$
$811$	−0.0623382	−	0.242791i	−0.0623382	−	0.242791i	−0.992115	+	0.125333i	$0.960000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.232966	+	0.588404i	−0.232966	+	0.588404i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.370498	−	0.306503i	0.370498	−	0.306503i
$821$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$821$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$822$		0			0
$823$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$823$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$824$	−0.476982	+	0.576572i	−0.476982	+	0.576572i
$825$		0			0
$826$	−0.0311905	−	0.00196234i	−0.0311905	−	0.00196234i
$827$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$827$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$828$		0			0
$829$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$829$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	0.461817	−	0.118574i	0.461817	−	0.118574i
$839$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.968583	+	0.248690i	$0.0800000\pi$
$839$	1.84489	+	0.730444i	1.84489	+	0.730444i	0.876307	+	0.481754i	$0.160000\pi$
$840$		0			0
$841$	−0.728969	+	0.684547i	−0.728969	+	0.684547i
$842$	−0.00782850	+	0.0304900i	−0.00782850	+	0.0304900i
$843$		0			0
$844$	0.468583	+	0.340446i	0.468583	+	0.340446i
$845$	0.362989	+	0.0931997i	0.362989	+	0.0931997i
$846$	−0.0194167	−	0.153699i	−0.0194167	−	0.153699i
$847$	0.147338	+	0.202793i	0.147338	+	0.202793i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.992115	−	0.125333i	$-0.960000\pi$
$853$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.637424	−	0.770513i	$-0.720000\pi$
$854$		0			0
$855$	−0.130584	−	0.684547i	−0.130584	−	0.684547i
$856$	0.640571	+	0.301430i	0.640571	+	0.301430i
$857$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.992115	−	0.125333i	$-0.960000\pi$
$857$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.187381	−	0.982287i	$-0.560000\pi$
$858$		0			0
$859$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$859$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$860$		0			0
$861$		0			0
$862$	0.0922765	+	0.483730i	0.0922765	+	0.483730i
$863$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$863$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$864$		0			0
$865$	−0.432756	+	0.595638i	−0.432756	+	0.595638i
$866$		0			0
$867$		0			0
$868$	−0.223419	−	0.0725931i	−0.223419	−	0.0725931i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.552903	−	0.141961i	−0.552903	−	0.141961i
$873$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$874$		0			0
$875$	0.0434429	−	0.169199i	0.0434429	−	0.169199i
$876$		0			0
$877$	−0.916350	+	0.297740i	−0.916350	+	0.297740i	−0.728969	−	0.684547i	$-0.760000\pi$
$877$	−0.916350	+	0.297740i	−0.916350	+	0.297740i	−0.187381	+	0.982287i	$0.560000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$881$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$882$	0.160811	+	0.171247i	0.160811	+	0.171247i
$883$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$883$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$884$		0			0
$885$		0			0
$886$	0.0314168	+	0.499356i	0.0314168	+	0.499356i
$887$	0.0915446	−	1.45506i	0.0915446	−	1.45506i	−0.637424	−	0.770513i	$-0.720000\pi$
$887$	0.0915446	−	1.45506i	0.0915446	−	1.45506i	0.728969	−	0.684547i	$-0.240000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.732570	−	0.885525i	0.732570	−	0.885525i
$894$		0			0
$895$		0			0
$896$	−0.198346	+	0.0785308i	−0.198346	+	0.0785308i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.705904	−	0.388074i	0.705904	−	0.388074i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.129412	−	0.203921i	−0.129412	−	0.203921i
$905$		0			0
$906$		0			0
$907$	1.15596	+	0.733597i	1.15596	+	0.733597i	0.968583	−	0.248690i	$-0.0800000\pi$
$907$	1.15596	+	0.733597i	1.15596	+	0.733597i	0.187381	+	0.982287i	$0.440000\pi$
$908$	−1.81545			−1.81545
$909$	0.876307	+	0.481754i	0.876307	+	0.481754i
$910$		0			0
$911$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$911$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.00394536	−	0.0312307i	0.00394536	−	0.0312307i
$918$		0			0
$919$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$919$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.53799	−	0.0967619i	−1.53799	−	0.0967619i
$928$		0			0
$929$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$929$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$930$		0			0
$931$	−0.109426	+	1.73927i	−0.109426	+	1.73927i
$932$	0.0363683	+	0.578058i	0.0363683	+	0.578058i
$933$		0			0
$934$	−0.125467	+	0.266631i	−0.125467	+	0.266631i
$935$		0			0
$936$		0			0
$937$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$937$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.992115	+	0.125333i	$0.0400000\pi$
$938$	0.0356449	+	0.0900287i	0.0356449	+	0.0900287i
$939$		0			0
$940$	−0.201820	−	0.0799061i	−0.201820	−	0.0799061i
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	−0.398403	+	0.0759994i	−0.398403	+	0.0759994i
$945$		0			0
$946$		0			0
$947$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$947$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$948$		0			0
$949$		0			0
$950$	−0.381052	−	0.123811i	−0.381052	−	0.123811i
$951$		0			0
$952$		0			0
$953$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$953$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$954$		0			0
$955$	0.132925	−	0.241789i	0.132925	−	0.241789i
$956$		0			0
$957$		0			0
$958$	0.364735	+	0.342509i	0.364735	+	0.342509i
$959$		0			0
$960$		0			0
$961$	0.728969	+	0.684547i	0.728969	+	0.684547i
$962$		0			0
$963$	0.273190	+	1.43211i	0.273190	+	1.43211i
$964$		0			0
$965$	0.279658	+	0.153743i	0.279658	+	0.153743i
$966$		0			0
$967$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$967$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$968$	−0.374148	−	0.309522i	−0.374148	−	0.309522i
$969$		0			0
$970$	0.0525328	−	0.0247201i	0.0525328	−	0.0247201i
$971$	−0.992567	−	1.36615i	−0.992567	−	1.36615i	−0.929776	−	0.368125i	$-0.880000\pi$
$971$	−0.992567	−	1.36615i	−0.992567	−	1.36615i	−0.0627905	−	0.998027i	$-0.520000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.929324	−	0.872693i	0.929324	−	0.872693i	−0.0627905	−	0.998027i	$-0.520000\pi$
$977$	0.929324	−	0.872693i	0.929324	−	0.872693i	0.992115	+	0.125333i	$0.0400000\pi$
$978$		0			0
$979$		0			0
$980$	0.318806	−	0.0818555i	0.318806	−	0.0818555i
$981$	−0.432756	−	1.09302i	−0.432756	−	1.09302i
$982$		0			0
$983$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$983$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$991$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$992$	0.688626	+	0.0433247i	0.688626	+	0.0433247i
$993$		0			0
$994$	0.0510598	−	0.0617207i	0.0510598	−	0.0617207i
$995$		0			0
$996$		0			0
$997$	0.574633	−	0.227513i	0.574633	−	0.227513i	−0.0627905	−	0.998027i	$-0.520000\pi$
$997$	0.574633	−	0.227513i	0.574633	−	0.227513i	0.637424	+	0.770513i	$0.280000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3131.1.de.a.1766.1	✓	20
31.30	odd	2	CM	3131.1.de.a.1766.1	✓	20
101.33	even	50	inner	3131.1.de.a.1952.1	yes	20
3131.1952	odd	50	inner	3131.1.de.a.1952.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3131.1.de.a.1766.1	✓	20	1.1	even	1	trivial
3131.1.de.a.1766.1	✓	20	31.30	odd	2	CM
3131.1.de.a.1952.1	yes	20	101.33	even	50	inner
3131.1.de.a.1952.1	yes	20	3131.1952	odd	50	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3131 = 31 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3131.de (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.211645 - 0.134314i) q^{2} +(-0.399026 - 0.847973i) q^{4} +(0.200808 + 0.316423i) q^{5} +(0.0922765 - 0.233064i) q^{7} +(-0.0608596 + 0.481754i) q^{8} +(-0.876307 + 0.481754i) q^{9} +O(q^{10})\)	\(q+(-0.211645 - 0.134314i) q^{2} +(-0.399026 - 0.847973i) q^{4} +(0.200808 + 0.316423i) q^{5} +(0.0922765 - 0.233064i) q^{7} +(-0.0608596 + 0.481754i) q^{8} +(-0.876307 + 0.481754i) q^{9} -0.0939404i q^{10} +(-0.0508335 + 0.0369327i) q^{14} +(-0.519786 + 0.628313i) q^{16} +(0.250172 + 0.0157395i) q^{18} +(1.18532 + 1.43281i) q^{19} +(0.188190 - 0.296541i) q^{20} +(0.365980 - 0.777747i) q^{25} +(-0.234453 + 0.0147505i) q^{28} +(0.929776 + 0.368125i) q^{31} +(0.656217 - 0.213218i) q^{32} +(0.0922765 - 0.0176027i) q^{35} +(0.758183 + 0.550853i) q^{36} +(-0.0584213 - 0.462452i) q^{38} +(-0.164659 + 0.0774826i) q^{40} +(1.30209 + 0.423073i) q^{41} +(-0.328407 - 0.180543i) q^{45} +(-0.115808 - 0.607087i) q^{47} +(0.683165 + 0.641534i) q^{49} +(-0.181920 + 0.115450i) q^{50} +(0.106663 + 0.0586387i) q^{56} +(0.383238 + 0.317042i) q^{59} +(-0.147338 - 0.202793i) q^{62} +(0.0314168 + 0.248690i) q^{63} +(0.622305 + 0.159781i) q^{64} +(0.383238 - 1.49261i) q^{67} +(-0.0218941 - 0.00866849i) q^{70} +(-1.23480 + 0.317042i) q^{71} +(-0.178755 - 0.451483i) q^{72} +(0.742010 - 1.57685i) q^{76} +(-0.303189 - 0.0383017i) q^{80} +(0.535827 - 0.844328i) q^{81} +(-0.218755 - 0.264429i) q^{82} +(0.0452561 + 0.0823206i) q^{90} +(-0.0570300 + 0.144041i) q^{94} +(-0.215351 + 0.662783i) q^{95} +(0.263146 + 0.559214i) q^{97} +(-0.0584213 - 0.227536i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{8}+O(q^{10}) \) 20 * q + 20 * q^8	\( 20 q + 20 q^{8} - 10 q^{14} + 5 q^{16} - 5 q^{19} + 5 q^{20} + 5 q^{28} - 5 q^{36} + 5 q^{38} - 5 q^{40} - 5 q^{45} + 5 q^{50} - 20 q^{56} + 5 q^{59} + 20 q^{63} + 20 q^{64} + 5 q^{67} - 5 q^{70} - 5 q^{71} - 20 q^{72} - 5 q^{76} + 20 q^{82} + 5 q^{98}+O(q^{100}) \) 20 * q + 20 * q^8 - 10 * q^14 + 5 * q^16 - 5 * q^19 + 5 * q^20 + 5 * q^28 - 5 * q^36 + 5 * q^38 - 5 * q^40 - 5 * q^45 + 5 * q^50 - 20 * q^56 + 5 * q^59 + 20 * q^63 + 20 * q^64 + 5 * q^67 - 5 * q^70 - 5 * q^71 - 20 * q^72 - 5 * q^76 + 20 * q^82 + 5 * q^98

Introduction

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