LMFDB - Embedded newform 5070.2.b.j.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not 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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.j.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} -2.00000i q^{21} +3.00000 q^{22} -3.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +3.00000 q^{29} -1.00000 q^{30} +5.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} +6.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} +2.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} -2.00000 q^{42} +1.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} +3.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} +2.00000 q^{56} +2.00000i q^{57} -3.00000i q^{58} +9.00000i q^{59} +1.00000i q^{60} +2.00000 q^{61} +5.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} +2.00000i q^{70} -12.0000i q^{71} +1.00000i q^{72} -14.0000i q^{73} +7.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} +6.00000 q^{77} +5.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -6.00000i q^{83} +2.00000i q^{84} +6.00000i q^{85} -1.00000i q^{86} +3.00000 q^{87} -3.00000 q^{88} +18.0000i q^{89} -1.00000 q^{90} +3.00000 q^{92} +5.00000i q^{93} +3.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +14.0000i q^{97} -3.00000i q^{98} +3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} - 12 q^{17} + 6 q^{22} - 6 q^{23} - 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 4 q^{38} + 2 q^{40} - 4 q^{42} + 2 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{51} - 12 q^{53} + 6 q^{55} + 4 q^{56} + 4 q^{61} + 10 q^{62} - 2 q^{64} + 6 q^{66} + 12 q^{68} - 6 q^{69} + 14 q^{74} - 2 q^{75} + 12 q^{77} + 10 q^{79} + 2 q^{81} + 12 q^{82} + 6 q^{87} - 6 q^{88} - 2 q^{90} + 6 q^{92} + 6 q^{94} + 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 - 12 * q^17 + 6 * q^22 - 6 * q^23 - 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 + 4 * q^38 + 2 * q^40 - 4 * q^42 + 2 * q^43 + 2 * q^48 + 6 * q^49 - 12 * q^51 - 12 * q^53 + 6 * q^55 + 4 * q^56 + 4 * q^61 + 10 * q^62 - 2 * q^64 + 6 * q^66 + 12 * q^68 - 6 * q^69 + 14 * q^74 - 2 * q^75 + 12 * q^77 + 10 * q^79 + 2 * q^81 + 12 * q^82 + 6 * q^87 - 6 * q^88 - 2 * q^90 + 6 * q^92 + 6 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	− 1.00000i	− 0.235702i
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	1.00000i	0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	3.00000	0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.00000i	0.377964i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−1.00000	−0.182574
$31$	5.00000i	0.898027i	0.893525	+	0.449013i	$0.148224\pi$
$31$	5.00000i	0.898027i	−0.893525	+	0.449013i	$0.851776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.00000i	0.522233i
$34$	6.00000i	1.02899i
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	−2.00000	−0.308607
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	− 3.00000i	− 0.452267i
$45$	− 1.00000i	− 0.149071i
$46$	3.00000i	0.442326i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	1.00000	0.144338
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	− 1.00000i	− 0.136083i
$55$	3.00000	0.404520
$56$	2.00000	0.267261
$57$	2.00000i	0.264906i
$58$	− 3.00000i	− 0.393919i
$59$	9.00000i	1.17170i	0.810419	+	0.585850i	$0.199239\pi$
$59$	9.00000i	1.17170i	−0.810419	+	0.585850i	$0.800761\pi$
$60$	1.00000i	0.129099i
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	5.00000	0.635001
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	6.00000	0.727607
$69$	−3.00000	−0.361158
$70$	2.00000i	0.239046i
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	1.00000i	0.117851i
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	7.00000	0.813733
$75$	−1.00000	−0.115470
$76$	− 2.00000i	− 0.229416i
$77$	6.00000	0.683763
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	2.00000i	0.218218i
$85$	6.00000i	0.650791i
$86$	− 1.00000i	− 0.107833i
$87$	3.00000	0.321634
$88$	−3.00000	−0.319801
$89$	18.0000i	1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$	18.0000i	1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	3.00000	0.312772
$93$	5.00000i	0.518476i
$94$	3.00000	0.309426
$95$	2.00000	0.205196
$96$	− 1.00000i	− 0.102062i
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 3.00000i	− 0.303046i
$99$	3.00000i	0.301511i
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000i	0.594089i
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	6.00000i	0.582772i
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	− 3.00000i	− 0.286039i
$111$	7.00000i	0.664411i
$112$	− 2.00000i	− 0.188982i
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	2.00000	0.187317
$115$	3.00000i	0.279751i
$116$	−3.00000	−0.278543
$117$	0	0
$118$	9.00000	0.828517
$119$	12.0000i	1.10004i
$120$	1.00000	0.0912871
$121$	2.00000	0.181818
$122$	− 2.00000i	− 0.181071i
$123$	6.00000i	0.541002i
$124$	− 5.00000i	− 0.449013i
$125$	1.00000i	0.0894427i
$126$	−2.00000	−0.178174
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000i	0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	− 3.00000i	− 0.261116i
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	− 1.00000i	− 0.0860663i
$136$	− 6.00000i	− 0.514496i
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	3.00000i	0.255377i
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	2.00000	0.169031
$141$	3.00000i	0.252646i
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 3.00000i	− 0.249136i
$146$	−14.0000	−1.15865
$147$	3.00000	0.247436
$148$	− 7.00000i	− 0.575396i
$149$	− 9.00000i	− 0.737309i	−0.929567	−	0.368654i	$-0.879819\pi$
$149$	− 9.00000i	− 0.737309i	0.929567	−	0.368654i	$-0.120181\pi$
$150$	1.00000i	0.0816497i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	−2.00000	−0.162221
$153$	−6.00000	−0.485071
$154$	− 6.00000i	− 0.483494i
$155$	5.00000	0.401610
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	− 5.00000i	− 0.397779i
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	6.00000i	0.472866i
$162$	− 1.00000i	− 0.0785674i
$163$	13.0000i	1.01824i	0.860696	+	0.509119i	$0.170029\pi$
$163$	13.0000i	1.01824i	−0.860696	+	0.509119i	$0.829971\pi$
$164$	− 6.00000i	− 0.468521i
$165$	3.00000	0.233550
$166$	−6.00000	−0.465690
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	6.00000	0.460179
$171$	2.00000i	0.152944i
$172$	−1.00000	−0.0762493
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	− 3.00000i	− 0.227429i
$175$	2.00000i	0.151186i
$176$	3.00000i	0.226134i
$177$	9.00000i	0.676481i
$178$	18.0000	1.34916
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	1.00000i	0.0745356i
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	− 3.00000i	− 0.221163i
$185$	7.00000	0.514650
$186$	5.00000	0.366618
$187$	− 18.0000i	− 1.31629i
$188$	− 3.00000i	− 0.218797i
$189$	− 2.00000i	− 0.145479i
$190$	− 2.00000i	− 0.145095i
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	3.00000	0.213201
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	8.00000i	0.564276i
$202$	6.00000i	0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	6.00000	0.420084
$205$	6.00000	0.419058
$206$	14.0000i	0.975426i
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−6.00000	−0.415029
$210$	2.00000i	0.138013i
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	− 12.0000i	− 0.822226i
$214$	6.00000i	0.410152i
$215$	− 1.00000i	− 0.0681994i
$216$	1.00000i	0.0680414i
$217$	10.0000	0.678844
$218$	14.0000	0.948200
$219$	− 14.0000i	− 0.946032i
$220$	−3.00000	−0.202260
$221$	0	0
$222$	7.00000	0.469809
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−2.00000	−0.133631
$225$	−1.00000	−0.0666667
$226$	− 15.0000i	− 0.997785i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	− 2.00000i	− 0.132453i
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	3.00000	0.197814
$231$	6.00000	0.394771
$232$	3.00000i	0.196960i
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	− 9.00000i	− 0.585850i
$237$	5.00000	0.324785
$238$	12.0000	0.777844
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 17.0000i	− 1.09507i	−0.836784	−	0.547533i	$-0.815567\pi$
$241$	− 17.0000i	− 1.09507i	0.836784	−	0.547533i	$-0.184433\pi$
$242$	− 2.00000i	− 0.128565i
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	− 3.00000i	− 0.191663i
$246$	6.00000	0.382546
$247$	0	0
$248$	−5.00000	−0.317500
$249$	− 6.00000i	− 0.380235i
$250$	1.00000	0.0632456
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	2.00000i	0.125988i
$253$	− 9.00000i	− 0.565825i
$254$	14.0000i	0.878438i
$255$	6.00000i	0.375735i
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	− 1.00000i	− 0.0622573i
$259$	14.0000	0.869918
$260$	0	0
$261$	3.00000	0.185695
$262$	− 9.00000i	− 0.556022i
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	−3.00000	−0.184637
$265$	6.00000i	0.368577i
$266$	− 4.00000i	− 0.245256i
$267$	18.0000i	1.10158i
$268$	− 8.00000i	− 0.488678i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	− 11.0000i	− 0.668202i	−0.942537	−	0.334101i	$-0.891567\pi$
$271$	− 11.0000i	− 0.668202i	0.942537	−	0.334101i	$-0.108433\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−9.00000	−0.543710
$275$	− 3.00000i	− 0.180907i
$276$	3.00000	0.180579
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	− 14.0000i	− 0.839664i
$279$	5.00000i	0.299342i
$280$	− 2.00000i	− 0.119523i
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	3.00000	0.178647
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	12.0000i	0.712069i
$285$	2.00000	0.118470
$286$	0	0
$287$	12.0000	0.708338
$288$	− 1.00000i	− 0.0589256i
$289$	19.0000	1.11765
$290$	−3.00000	−0.176166
$291$	14.0000i	0.820695i
$292$	14.0000i	0.819288i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	− 3.00000i	− 0.174964i
$295$	9.00000	0.524000
$296$	−7.00000	−0.406867
$297$	3.00000i	0.174078i
$298$	−9.00000	−0.521356
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 2.00000i	− 0.115278i
$302$	−8.00000	−0.460348
$303$	−6.00000	−0.344691
$304$	2.00000i	0.114708i
$305$	− 2.00000i	− 0.114520i
$306$	6.00000i	0.342997i
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	−6.00000	−0.341882
$309$	−14.0000	−0.796432
$310$	− 5.00000i	− 0.283981i
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	13.0000i	0.733632i
$315$	−2.00000	−0.112687
$316$	−5.00000	−0.281272
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	6.00000i	0.336463i
$319$	9.00000i	0.503903i
$320$	1.00000i	0.0559017i
$321$	−6.00000	−0.334887
$322$	6.00000	0.334367
$323$	− 12.0000i	− 0.667698i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	13.0000	0.720003
$327$	14.0000i	0.774202i
$328$	−6.00000	−0.331295
$329$	6.00000	0.330791
$330$	− 3.00000i	− 0.165145i
$331$	32.0000i	1.75888i	0.476011	+	0.879440i	$0.342082\pi$
$331$	32.0000i	1.75888i	−0.476011	+	0.879440i	$0.657918\pi$
$332$	6.00000i	0.329293i
$333$	7.00000i	0.383598i
$334$	9.00000	0.492458
$335$	8.00000	0.437087
$336$	− 2.00000i	− 0.109109i
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	− 6.00000i	− 0.325396i
$341$	−15.0000	−0.812296
$342$	2.00000	0.108148
$343$	− 20.0000i	− 1.07990i
$344$	1.00000i	0.0539164i
$345$	3.00000i	0.161515i
$346$	12.0000i	0.645124i
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	−3.00000	−0.160817
$349$	− 8.00000i	− 0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$	− 8.00000i	− 0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	3.00000	0.159901
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	9.00000	0.478345
$355$	−12.0000	−0.636894
$356$	− 18.0000i	− 0.953998i
$357$	12.0000i	0.635107i
$358$	− 3.00000i	− 0.158555i
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	1.00000	0.0527046
$361$	15.0000	0.789474
$362$	− 16.0000i	− 0.840941i
$363$	2.00000	0.104973
$364$	0	0
$365$	−14.0000	−0.732793
$366$	− 2.00000i	− 0.104542i
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−3.00000	−0.156386
$369$	6.00000i	0.312348i
$370$	− 7.00000i	− 0.363913i
$371$	12.0000i	0.623009i
$372$	− 5.00000i	− 0.259238i
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	−18.0000	−0.930758
$375$	1.00000i	0.0516398i
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−2.00000	−0.102869
$379$	38.0000i	1.95193i	0.217930	+	0.975964i	$0.430070\pi$
$379$	38.0000i	1.95193i	−0.217930	+	0.975964i	$0.569930\pi$
$380$	−2.00000	−0.102598
$381$	−14.0000	−0.717242
$382$	12.0000i	0.613973i
$383$	− 21.0000i	− 1.07305i	−0.843884	−	0.536525i	$-0.819737\pi$
$383$	− 21.0000i	− 1.07305i	0.843884	−	0.536525i	$-0.180263\pi$
$384$	1.00000i	0.0510310i
$385$	− 6.00000i	− 0.305788i
$386$	4.00000	0.203595
$387$	1.00000	0.0508329
$388$	− 14.0000i	− 0.710742i
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000i	0.151523i
$393$	9.00000	0.453990
$394$	24.0000	1.20910
$395$	− 5.00000i	− 0.251577i
$396$	− 3.00000i	− 0.150756i
$397$	31.0000i	1.55585i	0.628360	+	0.777923i	$0.283727\pi$
$397$	31.0000i	1.55585i	−0.628360	+	0.777923i	$0.716273\pi$
$398$	8.00000i	0.401004i
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	12.0000i	0.599251i	0.954057	+	0.299626i	$0.0968618\pi$
$401$	12.0000i	0.599251i	−0.954057	+	0.299626i	$0.903138\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	6.00000	0.298511
$405$	− 1.00000i	− 0.0496904i
$406$	−6.00000	−0.297775
$407$	−21.0000	−1.04093
$408$	− 6.00000i	− 0.297044i
$409$	− 10.0000i	− 0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$	− 10.0000i	− 0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$	− 6.00000i	− 0.296319i
$411$	− 9.00000i	− 0.443937i
$412$	14.0000	0.689730
$413$	18.0000	0.885722
$414$	3.00000i	0.147442i
$415$	−6.00000	−0.294528
$416$	0	0
$417$	14.0000	0.685583
$418$	6.00000i	0.293470i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	2.00000	0.0975900
$421$	− 16.0000i	− 0.779792i	−0.920859	−	0.389896i	$-0.872511\pi$
$421$	− 16.0000i	− 0.779792i	0.920859	−	0.389896i	$-0.127489\pi$
$422$	− 20.0000i	− 0.973585i
$423$	3.00000i	0.145865i
$424$	− 6.00000i	− 0.291386i
$425$	6.00000	0.291043
$426$	−12.0000	−0.581402
$427$	− 4.00000i	− 0.193574i
$428$	6.00000	0.290021
$429$	0	0
$430$	−1.00000	−0.0482243
$431$	12.0000i	0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$	12.0000i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	1.00000	0.0481125
$433$	40.0000	1.92228	0.961139	−	0.276066i	$-0.0890309\pi$
$433$	40.0000	1.92228	0.961139	+	0.276066i	$0.0890309\pi$
$434$	− 10.0000i	− 0.480015i
$435$	− 3.00000i	− 0.143839i
$436$	− 14.0000i	− 0.670478i
$437$	− 6.00000i	− 0.287019i
$438$	−14.0000	−0.668946
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	3.00000i	0.143019i
$441$	3.00000	0.142857
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	− 7.00000i	− 0.332205i
$445$	18.0000	0.853282
$446$	−10.0000	−0.473514
$447$	− 9.00000i	− 0.425685i
$448$	2.00000i	0.0944911i
$449$	− 36.0000i	− 1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	− 36.0000i	− 1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$	1.00000i	0.0471405i
$451$	−18.0000	−0.847587
$452$	−15.0000	−0.705541
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	−14.0000	−0.654177
$459$	−6.00000	−0.280056
$460$	− 3.00000i	− 0.139876i
$461$	15.0000i	0.698620i	0.937007	+	0.349310i	$0.113584\pi$
$461$	15.0000i	0.698620i	−0.937007	+	0.349310i	$0.886416\pi$
$462$	− 6.00000i	− 0.279145i
$463$	34.0000i	1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$	34.0000i	1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	3.00000	0.139272
$465$	5.00000	0.231869
$466$	21.0000i	0.972806i
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	− 3.00000i	− 0.138380i
$471$	−13.0000	−0.599008
$472$	−9.00000	−0.414259
$473$	3.00000i	0.137940i
$474$	− 5.00000i	− 0.229658i
$475$	− 2.00000i	− 0.0917663i
$476$	− 12.0000i	− 0.550019i
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−17.0000	−0.774329
$483$	6.00000i	0.273009i
$484$	−2.00000	−0.0909091
$485$	14.0000	0.635707
$486$	− 1.00000i	− 0.0453609i
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	2.00000i	0.0905357i
$489$	13.0000i	0.587880i
$490$	−3.00000	−0.135526
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 6.00000i	− 0.270501i
$493$	−18.0000	−0.810679
$494$	0	0
$495$	3.00000	0.134840
$496$	5.00000i	0.224507i
$497$	−24.0000	−1.07655
$498$	−6.00000	−0.268866
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	9.00000i	0.402090i
$502$	15.0000i	0.669483i
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	2.00000	0.0890871
$505$	6.00000i	0.266996i
$506$	−9.00000	−0.400099
$507$	0	0
$508$	14.0000	0.621150
$509$	3.00000i	0.132973i	0.997787	+	0.0664863i	$0.0211789\pi$
$509$	3.00000i	0.132973i	−0.997787	+	0.0664863i	$0.978821\pi$
$510$	6.00000	0.265684
$511$	−28.0000	−1.23865
$512$	− 1.00000i	− 0.0441942i
$513$	2.00000i	0.0883022i
$514$	− 21.0000i	− 0.926270i
$515$	14.0000i	0.616914i
$516$	−1.00000	−0.0440225
$517$	−9.00000	−0.395820
$518$	− 14.0000i	− 0.615125i
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 3.00000i	− 0.131306i
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	−9.00000	−0.393167
$525$	2.00000i	0.0872872i
$526$	15.0000i	0.654031i
$527$	− 30.0000i	− 1.30682i
$528$	3.00000i	0.130558i
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	9.00000i	0.390567i
$532$	−4.00000	−0.173422
$533$	0	0
$534$	18.0000	0.778936
$535$	6.00000i	0.259403i
$536$	−8.00000	−0.345547
$537$	3.00000	0.129460
$538$	− 18.0000i	− 0.776035i
$539$	9.00000i	0.387657i
$540$	1.00000i	0.0430331i
$541$	− 20.0000i	− 0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	− 20.0000i	− 0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−11.0000	−0.472490
$543$	16.0000	0.686626
$544$	6.00000i	0.257248i
$545$	14.0000	0.599694
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	9.00000i	0.384461i
$549$	2.00000	0.0853579
$550$	−3.00000	−0.127920
$551$	6.00000i	0.255609i
$552$	− 3.00000i	− 0.127688i
$553$	− 10.0000i	− 0.425243i
$554$	− 1.00000i	− 0.0424859i
$555$	7.00000	0.297133
$556$	−14.0000	−0.593732
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	5.00000	0.211667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	− 18.0000i	− 0.759961i
$562$	−18.0000	−0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	− 3.00000i	− 0.126323i
$565$	− 15.0000i	− 0.631055i
$566$	− 31.0000i	− 1.30303i
$567$	− 2.00000i	− 0.0839921i
$568$	12.0000	0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	− 2.00000i	− 0.0837708i
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	− 12.0000i	− 0.500870i
$575$	3.00000	0.125109
$576$	−1.00000	−0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 19.0000i	− 0.790296i
$579$	4.00000i	0.166234i
$580$	3.00000i	0.124568i
$581$	−12.0000	−0.497844
$582$	14.0000	0.580319
$583$	− 18.0000i	− 0.745484i
$584$	14.0000	0.579324
$585$	0	0
$586$	30.0000	1.23929
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	−3.00000	−0.123718
$589$	−10.0000	−0.412043
$590$	− 9.00000i	− 0.370524i
$591$	24.0000i	0.987228i
$592$	7.00000i	0.287698i
$593$	− 27.0000i	− 1.10876i	−0.832265	−	0.554379i	$-0.812956\pi$
$593$	− 27.0000i	− 1.10876i	0.832265	−	0.554379i	$-0.187044\pi$
$594$	3.00000	0.123091
$595$	12.0000	0.491952
$596$	9.00000i	0.368654i
$597$	−8.00000	−0.327418
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−2.00000	−0.0815139
$603$	8.00000i	0.325785i
$604$	8.00000i	0.325515i
$605$	− 2.00000i	− 0.0813116i
$606$	6.00000i	0.243733i
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	2.00000	0.0811107
$609$	− 6.00000i	− 0.243132i
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	6.00000	0.242536
$613$	− 31.0000i	− 1.25208i	−0.779792	−	0.626039i	$-0.784675\pi$
$613$	− 31.0000i	− 1.25208i	0.779792	−	0.626039i	$-0.215325\pi$
$614$	−8.00000	−0.322854
$615$	6.00000	0.241943
$616$	6.00000i	0.241747i
$617$	21.0000i	0.845428i	0.906263	+	0.422714i	$0.138923\pi$
$617$	21.0000i	0.845428i	−0.906263	+	0.422714i	$0.861077\pi$
$618$	14.0000i	0.563163i
$619$	46.0000i	1.84890i	0.381308	+	0.924448i	$0.375474\pi$
$619$	46.0000i	1.84890i	−0.381308	+	0.924448i	$0.624526\pi$
$620$	−5.00000	−0.200805
$621$	−3.00000	−0.120386
$622$	− 12.0000i	− 0.481156i
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 8.00000i	− 0.319744i
$627$	−6.00000	−0.239617
$628$	13.0000	0.518756
$629$	− 42.0000i	− 1.67465i
$630$	2.00000i	0.0796819i
$631$	16.0000i	0.636950i	0.947931	+	0.318475i	$0.103171\pi$
$631$	16.0000i	0.636950i	−0.947931	+	0.318475i	$0.896829\pi$
$632$	5.00000i	0.198889i
$633$	20.0000	0.794929
$634$	12.0000	0.476581
$635$	14.0000i	0.555573i
$636$	6.00000	0.237915
$637$	0	0
$638$	9.00000	0.356313
$639$	− 12.0000i	− 0.474713i
$640$	1.00000	0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	6.00000i	0.236801i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	− 6.00000i	− 0.236433i
$645$	− 1.00000i	− 0.0393750i
$646$	−12.0000	−0.472134
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000i	0.0392837i
$649$	−27.0000	−1.05984
$650$	0	0
$651$	10.0000	0.391931
$652$	− 13.0000i	− 0.509119i
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	14.0000	0.547443
$655$	− 9.00000i	− 0.351659i
$656$	6.00000i	0.234261i
$657$	− 14.0000i	− 0.546192i
$658$	− 6.00000i	− 0.233904i
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	−3.00000	−0.116775
$661$	− 32.0000i	− 1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	− 32.0000i	− 1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	32.0000	1.24372
$663$	0	0
$664$	6.00000	0.232845
$665$	− 4.00000i	− 0.155113i
$666$	7.00000	0.271244
$667$	−9.00000	−0.348481
$668$	− 9.00000i	− 0.348220i
$669$	− 10.0000i	− 0.386622i
$670$	− 8.00000i	− 0.309067i
$671$	6.00000i	0.231627i
$672$	−2.00000	−0.0771517
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	14.0000i	0.539260i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	− 15.0000i	− 0.576072i
$679$	28.0000	1.07454
$680$	−6.00000	−0.230089
$681$	0	0
$682$	15.0000i	0.574380i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	− 2.00000i	− 0.0764719i
$685$	−9.00000	−0.343872
$686$	−20.0000	−0.763604
$687$	− 14.0000i	− 0.534133i
$688$	1.00000	0.0381246
$689$	0	0
$690$	3.00000	0.114208
$691$	− 46.0000i	− 1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$	− 46.0000i	− 1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$	12.0000	0.456172
$693$	6.00000	0.227921
$694$	30.0000i	1.13878i
$695$	− 14.0000i	− 0.531050i
$696$	3.00000i	0.113715i
$697$	− 36.0000i	− 1.36360i
$698$	−8.00000	−0.302804
$699$	−21.0000	−0.794293
$700$	− 2.00000i	− 0.0755929i
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	− 3.00000i	− 0.113067i
$705$	3.00000	0.112987
$706$	30.0000	1.12906
$707$	12.0000i	0.451306i
$708$	− 9.00000i	− 0.338241i
$709$	− 32.0000i	− 1.20179i	−0.799330	−	0.600893i	$-0.794812\pi$
$709$	− 32.0000i	− 1.20179i	0.799330	−	0.600893i	$-0.205188\pi$
$710$	12.0000i	0.450352i
$711$	5.00000	0.187515
$712$	−18.0000	−0.674579
$713$	− 15.0000i	− 0.561754i
$714$	12.0000	0.449089
$715$	0	0
$716$	−3.00000	−0.112115
$717$	24.0000i	0.896296i
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	28.0000i	1.04277i
$722$	− 15.0000i	− 0.558242i
$723$	− 17.0000i	− 0.632237i
$724$	−16.0000	−0.594635
$725$	−3.00000	−0.111417
$726$	− 2.00000i	− 0.0742270i
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.0000i	0.518163i
$731$	−6.00000	−0.221918
$732$	−2.00000	−0.0739221
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	− 20.0000i	− 0.738213i
$735$	− 3.00000i	− 0.110657i
$736$	3.00000i	0.110581i
$737$	−24.0000	−0.884051
$738$	6.00000	0.220863
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	−7.00000	−0.257325
$741$	0	0
$742$	12.0000	0.440534
$743$	− 9.00000i	− 0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$	− 9.00000i	− 0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$	−5.00000	−0.183309
$745$	−9.00000	−0.329734
$746$	25.0000i	0.915315i
$747$	− 6.00000i	− 0.219529i
$748$	18.0000i	0.658145i
$749$	12.0000i	0.438470i
$750$	1.00000	0.0365148
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	3.00000i	0.109399i
$753$	−15.0000	−0.546630
$754$	0	0
$755$	−8.00000	−0.291150
$756$	2.00000i	0.0727393i
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	38.0000	1.38022
$759$	− 9.00000i	− 0.326679i
$760$	2.00000i	0.0725476i
$761$	12.0000i	0.435000i	0.976060	+	0.217500i	$0.0697902\pi$
$761$	12.0000i	0.435000i	−0.976060	+	0.217500i	$0.930210\pi$
$762$	14.0000i	0.507166i
$763$	28.0000	1.01367
$764$	12.0000	0.434145
$765$	6.00000i	0.216930i
$766$	−21.0000	−0.758761
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 13.0000i	− 0.468792i	−0.972141	−	0.234396i	$-0.924689\pi$
$769$	− 13.0000i	− 0.468792i	0.972141	−	0.234396i	$-0.0753112\pi$
$770$	−6.00000	−0.216225
$771$	21.0000	0.756297
$772$	− 4.00000i	− 0.143963i
$773$	− 48.0000i	− 1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$	− 48.0000i	− 1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$	− 1.00000i	− 0.0359443i
$775$	− 5.00000i	− 0.179605i
$776$	−14.0000	−0.502571
$777$	14.0000	0.502247
$778$	3.00000i	0.107555i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	36.0000	1.28818
$782$	− 18.0000i	− 0.643679i
$783$	3.00000	0.107211
$784$	3.00000	0.107143
$785$	13.0000i	0.463990i
$786$	− 9.00000i	− 0.321019i
$787$	− 35.0000i	− 1.24762i	−0.781578	−	0.623808i	$-0.785585\pi$
$787$	− 35.0000i	− 1.24762i	0.781578	−	0.623808i	$-0.214415\pi$
$788$	− 24.0000i	− 0.854965i
$789$	−15.0000	−0.534014
$790$	−5.00000	−0.177892
$791$	− 30.0000i	− 1.06668i
$792$	−3.00000	−0.106600
$793$	0	0
$794$	31.0000	1.10015
$795$	6.00000i	0.212798i
$796$	8.00000	0.283552
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	− 4.00000i	− 0.141598i
$799$	− 18.0000i	− 0.636794i
$800$	1.00000i	0.0353553i
$801$	18.0000i	0.635999i
$802$	12.0000	0.423735
$803$	42.0000	1.48215
$804$	− 8.00000i	− 0.282138i
$805$	6.00000	0.211472
$806$	0	0
$807$	18.0000	0.633630
$808$	− 6.00000i	− 0.211079i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	−1.00000	−0.0351364
$811$	32.0000i	1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$	32.0000i	1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$	6.00000i	0.210559i
$813$	− 11.0000i	− 0.385787i
$814$	21.0000i	0.736050i
$815$	13.0000	0.455370
$816$	−6.00000	−0.210042
$817$	2.00000i	0.0699711i
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−6.00000	−0.209529
$821$	27.0000i	0.942306i	0.882051	+	0.471153i	$0.156162\pi$
$821$	27.0000i	0.942306i	−0.882051	+	0.471153i	$0.843838\pi$
$822$	−9.00000	−0.313911
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	− 14.0000i	− 0.487713i
$825$	− 3.00000i	− 0.104447i
$826$	− 18.0000i	− 0.626300i
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	3.00000	0.104257
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	6.00000i	0.208263i
$831$	1.00000	0.0346896
$832$	0	0
$833$	−18.0000	−0.623663
$834$	− 14.0000i	− 0.484780i
$835$	9.00000	0.311458
$836$	6.00000	0.207514
$837$	5.00000i	0.172825i
$838$	12.0000i	0.414533i
$839$	− 42.0000i	− 1.45000i	−0.688748	−	0.725001i	$-0.741839\pi$
$839$	− 42.0000i	− 1.45000i	0.688748	−	0.725001i	$-0.258161\pi$
$840$	− 2.00000i	− 0.0690066i
$841$	−20.0000	−0.689655
$842$	−16.0000	−0.551396
$843$	− 18.0000i	− 0.619953i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	3.00000	0.103142
$847$	− 4.00000i	− 0.137442i
$848$	−6.00000	−0.206041
$849$	31.0000	1.06392
$850$	− 6.00000i	− 0.205798i
$851$	− 21.0000i	− 0.719871i
$852$	12.0000i	0.411113i
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	−4.00000	−0.136877
$855$	2.00000	0.0683986
$856$	− 6.00000i	− 0.205076i
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−46.0000	−1.56950	−0.784750	−	0.619813i	$-0.787209\pi$
$859$	−46.0000	−1.56950	−0.784750	+	0.619813i	$0.787209\pi$
$860$	1.00000i	0.0340997i
$861$	12.0000	0.408959
$862$	12.0000	0.408722
$863$	− 45.0000i	− 1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$	− 45.0000i	− 1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	12.0000i	0.408012i
$866$	− 40.0000i	− 1.35926i
$867$	19.0000	0.645274
$868$	−10.0000	−0.339422
$869$	15.0000i	0.508840i
$870$	−3.00000	−0.101710
$871$	0	0
$872$	−14.0000	−0.474100
$873$	14.0000i	0.473828i
$874$	−6.00000	−0.202953
$875$	2.00000	0.0676123
$876$	14.0000i	0.473016i
$877$	23.0000i	0.776655i	0.921521	+	0.388327i	$0.126947\pi$
$877$	23.0000i	0.776655i	−0.921521	+	0.388327i	$0.873053\pi$
$878$	− 4.00000i	− 0.134993i
$879$	30.0000i	1.01187i
$880$	3.00000	0.101130
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	− 3.00000i	− 0.101015i
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	36.0000i	1.20944i
$887$	−57.0000	−1.91387	−0.956936	−	0.290298i	$-0.906246\pi$
$887$	−57.0000	−1.91387	−0.956936	+	0.290298i	$0.906246\pi$
$888$	−7.00000	−0.234905
$889$	28.0000i	0.939090i
$890$	− 18.0000i	− 0.603361i
$891$	3.00000i	0.100504i
$892$	10.0000i	0.334825i
$893$	−6.00000	−0.200782
$894$	−9.00000	−0.301005
$895$	− 3.00000i	− 0.100279i
$896$	2.00000	0.0668153
$897$	0	0
$898$	−36.0000	−1.20134
$899$	15.0000i	0.500278i
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	18.0000i	0.599334i
$903$	− 2.00000i	− 0.0665558i
$904$	15.0000i	0.498893i
$905$	− 16.0000i	− 0.531858i
$906$	−8.00000	−0.265782
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	2.00000i	0.0662266i
$913$	18.0000	0.595713
$914$	2.00000	0.0661541
$915$	− 2.00000i	− 0.0661180i
$916$	14.0000i	0.462573i
$917$	− 18.0000i	− 0.594412i
$918$	6.00000i	0.198030i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−3.00000	−0.0989071
$921$	− 8.00000i	− 0.263609i
$922$	15.0000	0.493999
$923$	0	0
$924$	−6.00000	−0.197386
$925$	− 7.00000i	− 0.230159i
$926$	34.0000	1.11731
$927$	−14.0000	−0.459820
$928$	− 3.00000i	− 0.0984798i
$929$	− 12.0000i	− 0.393707i	−0.980433	−	0.196854i	$-0.936928\pi$
$929$	− 12.0000i	− 0.393707i	0.980433	−	0.196854i	$-0.0630724\pi$
$930$	− 5.00000i	− 0.163956i
$931$	6.00000i	0.196642i
$932$	21.0000	0.687878
$933$	12.0000	0.392862
$934$	18.0000i	0.588978i
$935$	−18.0000	−0.588663
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	− 16.0000i	− 0.522419i
$939$	8.00000	0.261070
$940$	−3.00000	−0.0978492
$941$	42.0000i	1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$	42.0000i	1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$	13.0000i	0.423563i
$943$	− 18.0000i	− 0.586161i
$944$	9.00000i	0.292925i
$945$	−2.00000	−0.0650600
$946$	3.00000	0.0975384
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	−5.00000	−0.162392
$949$	0	0
$950$	−2.00000	−0.0648886
$951$	12.0000i	0.389127i
$952$	−12.0000	−0.388922
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	6.00000i	0.194257i
$955$	12.0000i	0.388311i
$956$	− 24.0000i	− 0.776215i
$957$	9.00000i	0.290929i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	1.00000i	0.0322749i
$961$	6.00000	0.193548
$962$	0	0
$963$	−6.00000	−0.193347
$964$	17.0000i	0.547533i
$965$	4.00000	0.128765
$966$	6.00000	0.193047
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	2.00000i	0.0642824i
$969$	− 12.0000i	− 0.385496i
$970$	− 14.0000i	− 0.449513i
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−1.00000	−0.0320750
$973$	− 28.0000i	− 0.897639i
$974$	2.00000	0.0640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 57.0000i	− 1.82359i	−0.410644	−	0.911796i	$-0.634696\pi$
$977$	− 57.0000i	− 1.82359i	0.410644	−	0.911796i	$-0.365304\pi$
$978$	13.0000	0.415694
$979$	−54.0000	−1.72585
$980$	3.00000i	0.0958315i
$981$	14.0000i	0.446986i
$982$	0	0
$983$	− 9.00000i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$	− 9.00000i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$	−6.00000	−0.191273
$985$	24.0000	0.764704
$986$	18.0000i	0.573237i
$987$	6.00000	0.190982
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	− 3.00000i	− 0.0953463i
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	5.00000	0.158750
$993$	32.0000i	1.01549i
$994$	24.0000i	0.761234i
$995$	8.00000i	0.253617i
$996$	6.00000i	0.190117i
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	32.0000	1.01294
$999$	7.00000i	0.221470i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.j.1351.1		2
13.2	odd	12		390.2.i.c.61.1	✓	2
13.5	odd	4		5070.2.a.j.1.1		1
13.6	odd	12		390.2.i.c.211.1	yes	2
13.8	odd	4		5070.2.a.v.1.1		1
13.12	even	2	inner	5070.2.b.j.1351.2		2
39.2	even	12		1170.2.i.d.451.1		2
39.32	even	12		1170.2.i.d.991.1		2
65.2	even	12		1950.2.z.k.1699.2		4
65.19	odd	12		1950.2.i.n.601.1		2
65.28	even	12		1950.2.z.k.1699.1		4
65.32	even	12		1950.2.z.k.1849.1		4
65.54	odd	12		1950.2.i.n.451.1		2
65.58	even	12		1950.2.z.k.1849.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.c.61.1	✓	2	13.2	odd	12
390.2.i.c.211.1	yes	2	13.6	odd	12
1170.2.i.d.451.1		2	39.2	even	12
1170.2.i.d.991.1		2	39.32	even	12
1950.2.i.n.451.1		2	65.54	odd	12
1950.2.i.n.601.1		2	65.19	odd	12
1950.2.z.k.1699.1		4	65.28	even	12
1950.2.z.k.1699.2		4	65.2	even	12
1950.2.z.k.1849.1		4	65.32	even	12
1950.2.z.k.1849.2		4	65.58	even	12
5070.2.a.j.1.1		1	13.5	odd	4
5070.2.a.v.1.1		1	13.8	odd	4
5070.2.b.j.1351.1		2	1.1	even	1	trivial
5070.2.b.j.1351.2		2	13.12	even	2	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} -2.00000i q^{21} +3.00000 q^{22} -3.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +3.00000 q^{29} -1.00000 q^{30} +5.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} +6.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} +2.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} -2.00000 q^{42} +1.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} +3.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} +2.00000 q^{56} +2.00000i q^{57} -3.00000i q^{58} +9.00000i q^{59} +1.00000i q^{60} +2.00000 q^{61} +5.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} +2.00000i q^{70} -12.0000i q^{71} +1.00000i q^{72} -14.0000i q^{73} +7.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} +6.00000 q^{77} +5.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -6.00000i q^{83} +2.00000i q^{84} +6.00000i q^{85} -1.00000i q^{86} +3.00000 q^{87} -3.00000 q^{88} +18.0000i q^{89} -1.00000 q^{90} +3.00000 q^{92} +5.00000i q^{93} +3.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +14.0000i q^{97} -3.00000i q^{98} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} - 12 q^{17} + 6 q^{22} - 6 q^{23} - 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 4 q^{38} + 2 q^{40} - 4 q^{42} + 2 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{51} - 12 q^{53} + 6 q^{55} + 4 q^{56} + 4 q^{61} + 10 q^{62} - 2 q^{64} + 6 q^{66} + 12 q^{68} - 6 q^{69} + 14 q^{74} - 2 q^{75} + 12 q^{77} + 10 q^{79} + 2 q^{81} + 12 q^{82} + 6 q^{87} - 6 q^{88} - 2 q^{90} + 6 q^{92} + 6 q^{94} + 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 - 12 * q^17 + 6 * q^22 - 6 * q^23 - 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 + 4 * q^38 + 2 * q^40 - 4 * q^42 + 2 * q^43 + 2 * q^48 + 6 * q^49 - 12 * q^51 - 12 * q^53 + 6 * q^55 + 4 * q^56 + 4 * q^61 + 10 * q^62 - 2 * q^64 + 6 * q^66 + 12 * q^68 - 6 * q^69 + 14 * q^74 - 2 * q^75 + 12 * q^77 + 10 * q^79 + 2 * q^81 + 12 * q^82 + 6 * q^87 - 6 * q^88 - 2 * q^90 + 6 * q^92 + 6 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more