LMFDB - Embedded newform 5070.2.b.h.1351.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +5.00000 q^{29} -1.00000 q^{30} +11.0000i q^{31} +1.00000i q^{32} +5.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +3.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000 q^{43} +5.00000i q^{44} -1.00000i q^{45} +1.00000i q^{46} +9.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000i q^{54} -5.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} +5.00000i q^{58} -15.0000i q^{59} -1.00000i q^{60} +10.0000 q^{61} -11.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} -16.0000i q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -6.00000i q^{73} -3.00000 q^{74} +1.00000 q^{75} -2.00000i q^{76} -10.0000 q^{77} -11.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -2.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -2.00000i q^{85} +11.0000i q^{86} -5.00000 q^{87} -5.00000 q^{88} +2.00000i q^{89} +1.00000 q^{90} -1.00000 q^{92} -11.0000i q^{93} -9.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} -5.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{17} + 10 q^{22} + 2 q^{23} - 2 q^{25} - 2 q^{27} + 10 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 4 q^{38} - 2 q^{40} - 4 q^{42} + 22 q^{43} - 2 q^{48} + 6 q^{49} - 4 q^{51} + 12 q^{53} - 10 q^{55} - 4 q^{56} + 20 q^{61} - 22 q^{62} - 2 q^{64} - 10 q^{66} - 4 q^{68} - 2 q^{69} - 6 q^{74} + 2 q^{75} - 20 q^{77} - 22 q^{79} + 2 q^{81} - 4 q^{82} - 10 q^{87} - 10 q^{88} + 2 q^{90} - 2 q^{92} - 18 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 + 4 * q^17 + 10 * q^22 + 2 * q^23 - 2 * q^25 - 2 * q^27 + 10 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 4 * q^38 - 2 * q^40 - 4 * q^42 + 22 * q^43 - 2 * q^48 + 6 * q^49 - 4 * q^51 + 12 * q^53 - 10 * q^55 - 4 * q^56 + 20 * q^61 - 22 * q^62 - 2 * q^64 - 10 * q^66 - 4 * q^68 - 2 * q^69 - 6 * q^74 + 2 * q^75 - 20 * q^77 - 22 * q^79 + 2 * q^81 - 4 * q^82 - 10 * q^87 - 10 * q^88 + 2 * q^90 - 2 * q^92 - 18 * 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
$11$	− 5.00000i	− 1.50756i	−0.657129	−	0.753778i	$-0.728229\pi$
$11$	− 5.00000i	− 1.50756i	0.657129	−	0.753778i	$-0.271771\pi$
$12$	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
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$22$	5.00000	1.06600
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000i	0.377964i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−1.00000	−0.182574
$31$	11.0000i	1.97566i	0.155543	+	0.987829i	$0.450287\pi$
$31$	11.0000i	1.97566i	−0.155543	+	0.987829i	$0.549713\pi$
$32$	1.00000i	0.176777i
$33$	5.00000i	0.870388i
$34$	2.00000i	0.342997i
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	−2.00000	−0.308607
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	5.00000i	0.753778i
$45$	− 1.00000i	− 0.149071i
$46$	1.00000i	0.147442i
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	− 1.00000i	− 0.141421i
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−5.00000	−0.674200
$56$	−2.00000	−0.267261
$57$	− 2.00000i	− 0.264906i
$58$	5.00000i	0.656532i
$59$	− 15.0000i	− 1.95283i	−0.215894	−	0.976417i	$-0.569267\pi$
$59$	− 15.0000i	− 1.95283i	0.215894	−	0.976417i	$-0.430733\pi$
$60$	− 1.00000i	− 0.129099i
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−11.0000	−1.39700
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	− 16.0000i	− 1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	− 16.0000i	− 1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$	−2.00000	−0.242536
$69$	−1.00000	−0.120386
$70$	− 2.00000i	− 0.239046i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−3.00000	−0.348743
$75$	1.00000	0.115470
$76$	− 2.00000i	− 0.229416i
$77$	−10.0000	−1.13961
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	− 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
$85$	− 2.00000i	− 0.216930i
$86$	11.0000i	1.18616i
$87$	−5.00000	−0.536056
$88$	−5.00000	−0.533002
$89$	2.00000i	0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$	2.00000i	0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	− 11.0000i	− 1.14065i
$94$	−9.00000	−0.928279
$95$	2.00000	0.205196
$96$	− 1.00000i	− 0.102062i
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	3.00000i	0.303046i
$99$	− 5.00000i	− 0.502519i
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	− 2.00000i	− 0.198030i
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	6.00000i	0.582772i
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	1.00000	0.0962250
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	− 5.00000i	− 0.476731i
$111$	− 3.00000i	− 0.284747i
$112$	− 2.00000i	− 0.188982i
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	2.00000	0.187317
$115$	− 1.00000i	− 0.0932505i
$116$	−5.00000	−0.464238
$117$	0	0
$118$	15.0000	1.38086
$119$	− 4.00000i	− 0.366679i
$120$	1.00000	0.0912871
$121$	−14.0000	−1.27273
$122$	10.0000i	0.905357i
$123$	− 2.00000i	− 0.180334i
$124$	− 11.0000i	− 0.987829i
$125$	1.00000i	0.0894427i
$126$	2.00000	0.178174
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	− 5.00000i	− 0.435194i
$133$	4.00000	0.346844
$134$	16.0000	1.38219
$135$	1.00000i	0.0860663i
$136$	− 2.00000i	− 0.171499i
$137$	− 11.0000i	− 0.939793i	−0.882721	−	0.469897i	$-0.844291\pi$
$137$	− 11.0000i	− 0.939793i	0.882721	−	0.469897i	$-0.155709\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	2.00000	0.169031
$141$	− 9.00000i	− 0.757937i
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 5.00000i	− 0.415227i
$146$	6.00000	0.496564
$147$	−3.00000	−0.247436
$148$	− 3.00000i	− 0.246598i
$149$	− 17.0000i	− 1.39269i	−0.717705	−	0.696347i	$-0.754807\pi$
$149$	− 17.0000i	− 1.39269i	0.717705	−	0.696347i	$-0.245193\pi$
$150$	1.00000i	0.0816497i
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	2.00000	0.162221
$153$	2.00000	0.161690
$154$	− 10.0000i	− 0.805823i
$155$	11.0000	0.883541
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	− 11.0000i	− 0.875113i
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	− 2.00000i	− 0.157622i
$162$	1.00000i	0.0785674i
$163$	− 15.0000i	− 1.17489i	−0.809264	−	0.587445i	$-0.800134\pi$
$163$	− 15.0000i	− 1.17489i	0.809264	−	0.587445i	$-0.199866\pi$
$164$	− 2.00000i	− 0.156174i
$165$	5.00000	0.389249
$166$	6.00000	0.465690
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	2.00000	0.153393
$171$	2.00000i	0.152944i
$172$	−11.0000	−0.838742
$173$	−20.0000	−1.52057	−0.760286	−	0.649589i	$-0.774941\pi$
$173$	−20.0000	−1.52057	−0.760286	+	0.649589i	$0.774941\pi$
$174$	− 5.00000i	− 0.379049i
$175$	2.00000i	0.151186i
$176$	− 5.00000i	− 0.376889i
$177$	15.0000i	1.12747i
$178$	−2.00000	−0.149906
$179$	13.0000	0.971666	0.485833	−	0.874052i	$-0.338516\pi$
$179$	13.0000	0.971666	0.485833	+	0.874052i	$0.338516\pi$
$180$	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
$183$	−10.0000	−0.739221
$184$	− 1.00000i	− 0.0737210i
$185$	3.00000	0.220564
$186$	11.0000	0.806559
$187$	− 10.0000i	− 0.731272i
$188$	− 9.00000i	− 0.656392i
$189$	2.00000i	0.145479i
$190$	2.00000i	0.145095i
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000	0.0721688
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	5.00000	0.355335
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	1.00000i	0.0707107i
$201$	16.0000i	1.12855i
$202$	− 2.00000i	− 0.140720i
$203$	− 10.0000i	− 0.701862i
$204$	2.00000	0.140028
$205$	2.00000	0.139686
$206$	− 10.0000i	− 0.696733i
$207$	1.00000	0.0695048
$208$	0	0
$209$	10.0000	0.691714
$210$	2.00000i	0.138013i
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	10.0000i	0.683586i
$215$	− 11.0000i	− 0.750194i
$216$	1.00000i	0.0680414i
$217$	22.0000	1.49346
$218$	−2.00000	−0.135457
$219$	6.00000i	0.405442i
$220$	5.00000	0.337100
$221$	0	0
$222$	3.00000	0.201347
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	11.0000i	0.731709i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	2.00000i	0.132453i
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	1.00000	0.0659380
$231$	10.0000	0.657952
$232$	− 5.00000i	− 0.328266i
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	15.0000i	0.976417i
$237$	11.0000	0.714527
$238$	4.00000	0.259281
$239$	20.0000i	1.29369i	0.762620	+	0.646846i	$0.223912\pi$
$239$	20.0000i	1.29369i	−0.762620	+	0.646846i	$0.776088\pi$
$240$	1.00000i	0.0645497i
$241$	− 7.00000i	− 0.450910i	−0.974254	−	0.225455i	$-0.927613\pi$
$241$	− 7.00000i	− 0.450910i	0.974254	−	0.225455i	$-0.0723868\pi$
$242$	− 14.0000i	− 0.899954i
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	− 3.00000i	− 0.191663i
$246$	2.00000	0.127515
$247$	0	0
$248$	11.0000	0.698501
$249$	6.00000i	0.380235i
$250$	−1.00000	−0.0632456
$251$	−25.0000	−1.57799	−0.788993	−	0.614402i	$-0.789397\pi$
$251$	−25.0000	−1.57799	−0.788993	+	0.614402i	$0.789397\pi$
$252$	2.00000i	0.125988i
$253$	− 5.00000i	− 0.314347i
$254$	− 2.00000i	− 0.125491i
$255$	2.00000i	0.125245i
$256$	1.00000	0.0625000
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	− 11.0000i	− 0.684830i
$259$	6.00000	0.372822
$260$	0	0
$261$	5.00000	0.309492
$262$	− 1.00000i	− 0.0617802i
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	5.00000	0.307729
$265$	− 6.00000i	− 0.368577i
$266$	4.00000i	0.245256i
$267$	− 2.00000i	− 0.122398i
$268$	16.0000i	0.977356i
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−1.00000	−0.0608581
$271$	− 13.0000i	− 0.789694i	−0.918747	−	0.394847i	$-0.870798\pi$
$271$	− 13.0000i	− 0.789694i	0.918747	−	0.394847i	$-0.129202\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	11.0000	0.664534
$275$	5.00000i	0.301511i
$276$	1.00000	0.0601929
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	2.00000i	0.119952i
$279$	11.0000i	0.658553i
$280$	2.00000i	0.119523i
$281$	− 10.0000i	− 0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	− 10.0000i	− 0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	9.00000	0.535942
$283$	−19.0000	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$283$	−19.0000	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	4.00000	0.236113
$288$	1.00000i	0.0589256i
$289$	−13.0000	−0.764706
$290$	5.00000	0.293610
$291$	− 2.00000i	− 0.117242i
$292$	6.00000i	0.351123i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	− 3.00000i	− 0.174964i
$295$	−15.0000	−0.873334
$296$	3.00000	0.174371
$297$	5.00000i	0.290129i
$298$	17.0000	0.984784
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 22.0000i	− 1.26806i
$302$	−8.00000	−0.460348
$303$	2.00000	0.114897
$304$	2.00000i	0.114708i
$305$	− 10.0000i	− 0.572598i
$306$	2.00000i	0.114332i
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	10.0000	0.569803
$309$	10.0000	0.568880
$310$	11.0000i	0.624758i
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	− 7.00000i	− 0.395033i
$315$	−2.00000	−0.112687
$316$	11.0000	0.618798
$317$	− 16.0000i	− 0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$	− 16.0000i	− 0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$	− 6.00000i	− 0.336463i
$319$	− 25.0000i	− 1.39973i
$320$	1.00000i	0.0559017i
$321$	−10.0000	−0.558146
$322$	2.00000	0.111456
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	15.0000	0.830773
$327$	− 2.00000i	− 0.110600i
$328$	2.00000	0.110432
$329$	18.0000	0.992372
$330$	5.00000i	0.275241i
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	6.00000i	0.329293i
$333$	3.00000i	0.164399i
$334$	−3.00000	−0.164153
$335$	−16.0000	−0.874173
$336$	2.00000i	0.109109i
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	−11.0000	−0.597438
$340$	2.00000i	0.108465i
$341$	55.0000	2.97842
$342$	−2.00000	−0.108148
$343$	− 20.0000i	− 1.07990i
$344$	− 11.0000i	− 0.593080i
$345$	1.00000i	0.0538382i
$346$	− 20.0000i	− 1.07521i
$347$	34.0000	1.82522	0.912608	−	0.408836i	$-0.134065\pi$
$347$	34.0000	1.82522	0.912608	+	0.408836i	$0.134065\pi$
$348$	5.00000	0.268028
$349$	20.0000i	1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$	20.0000i	1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	5.00000	0.266501
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	− 2.00000i	− 0.106000i
$357$	4.00000i	0.211702i
$358$	13.0000i	0.687071i
$359$	12.0000i	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$	12.0000i	0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	−1.00000	−0.0527046
$361$	15.0000	0.789474
$362$	16.0000i	0.840941i
$363$	14.0000	0.734809
$364$	0	0
$365$	−6.00000	−0.314054
$366$	− 10.0000i	− 0.522708i
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.00000	0.0521286
$369$	2.00000i	0.104116i
$370$	3.00000i	0.155963i
$371$	− 12.0000i	− 0.623009i
$372$	11.0000i	0.570323i
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	10.0000	0.517088
$375$	− 1.00000i	− 0.0516398i
$376$	9.00000	0.464140
$377$	0	0
$378$	−2.00000	−0.102869
$379$	2.00000i	0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$	2.00000i	0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$	−2.00000	−0.102598
$381$	2.00000	0.102463
$382$	4.00000i	0.204658i
$383$	− 31.0000i	− 1.58403i	−0.610504	−	0.792013i	$-0.709033\pi$
$383$	− 31.0000i	− 1.58403i	0.610504	−	0.792013i	$-0.290967\pi$
$384$	1.00000i	0.0510310i
$385$	10.0000i	0.509647i
$386$	24.0000	1.22157
$387$	11.0000	0.559161
$388$	− 2.00000i	− 0.101535i
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	− 3.00000i	− 0.151523i
$393$	1.00000	0.0504433
$394$	−8.00000	−0.403034
$395$	11.0000i	0.553470i
$396$	5.00000i	0.251259i
$397$	3.00000i	0.150566i	0.997162	+	0.0752828i	$0.0239860\pi$
$397$	3.00000i	0.150566i	−0.997162	+	0.0752828i	$0.976014\pi$
$398$	− 24.0000i	− 1.20301i
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	36.0000i	1.79775i	0.438201	+	0.898877i	$0.355616\pi$
$401$	36.0000i	1.79775i	−0.438201	+	0.898877i	$0.644384\pi$
$402$	−16.0000	−0.798007
$403$	0	0
$404$	2.00000	0.0995037
$405$	− 1.00000i	− 0.0496904i
$406$	10.0000	0.496292
$407$	15.0000	0.743522
$408$	2.00000i	0.0990148i
$409$	− 30.0000i	− 1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$	− 30.0000i	− 1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$	2.00000i	0.0987730i
$411$	11.0000i	0.542590i
$412$	10.0000	0.492665
$413$	−30.0000	−1.47620
$414$	1.00000i	0.0491473i
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	10.0000i	0.489116i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−2.00000	−0.0975900
$421$	16.0000i	0.779792i	0.920859	+	0.389896i	$0.127489\pi$
$421$	16.0000i	0.779792i	−0.920859	+	0.389896i	$0.872511\pi$
$422$	− 4.00000i	− 0.194717i
$423$	9.00000i	0.437595i
$424$	− 6.00000i	− 0.291386i
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	−10.0000	−0.483368
$429$	0	0
$430$	11.0000	0.530467
$431$	− 40.0000i	− 1.92673i	−0.268190	−	0.963366i	$-0.586425\pi$
$431$	− 40.0000i	− 1.92673i	0.268190	−	0.963366i	$-0.413575\pi$
$432$	−1.00000	−0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	22.0000i	1.05603i
$435$	5.00000i	0.239732i
$436$	− 2.00000i	− 0.0957826i
$437$	2.00000i	0.0956730i
$438$	−6.00000	−0.286691
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	5.00000i	0.238366i
$441$	3.00000	0.142857
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	3.00000i	0.142374i
$445$	2.00000	0.0948091
$446$	26.0000	1.23114
$447$	17.0000i	0.804072i
$448$	2.00000i	0.0944911i
$449$	12.0000i	0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$	12.0000i	0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	10.0000	0.470882
$452$	−11.0000	−0.517396
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	−10.0000	−0.467269
$459$	−2.00000	−0.0933520
$460$	1.00000i	0.0466252i
$461$	39.0000i	1.81641i	0.418524	+	0.908206i	$0.362547\pi$
$461$	39.0000i	1.81641i	−0.418524	+	0.908206i	$0.637453\pi$
$462$	10.0000i	0.465242i
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	5.00000	0.232119
$465$	−11.0000	−0.510113
$466$	− 1.00000i	− 0.0463241i
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	9.00000i	0.415139i
$471$	7.00000	0.322543
$472$	−15.0000	−0.690431
$473$	− 55.0000i	− 2.52890i
$474$	11.0000i	0.505247i
$475$	− 2.00000i	− 0.0917663i
$476$	4.00000i	0.183340i
$477$	6.00000	0.274721
$478$	−20.0000	−0.914779
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	7.00000	0.318841
$483$	2.00000i	0.0910032i
$484$	14.0000	0.636364
$485$	2.00000	0.0908153
$486$	− 1.00000i	− 0.0453609i
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 10.0000i	− 0.452679i
$489$	15.0000i	0.678323i
$490$	3.00000	0.135526
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	2.00000i	0.0901670i
$493$	10.0000	0.450377
$494$	0	0
$495$	−5.00000	−0.224733
$496$	11.0000i	0.493915i
$497$	0	0
$498$	−6.00000	−0.268866
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 3.00000i	− 0.134030i
$502$	− 25.0000i	− 1.11580i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−2.00000	−0.0890871
$505$	2.00000i	0.0889988i
$506$	5.00000	0.222277
$507$	0	0
$508$	2.00000	0.0887357
$509$	− 29.0000i	− 1.28540i	−0.766117	−	0.642701i	$-0.777814\pi$
$509$	− 29.0000i	− 1.28540i	0.766117	−	0.642701i	$-0.222186\pi$
$510$	−2.00000	−0.0885615
$511$	−12.0000	−0.530849
$512$	1.00000i	0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	17.0000i	0.749838i
$515$	10.0000i	0.440653i
$516$	11.0000	0.484248
$517$	45.0000	1.97910
$518$	6.00000i	0.263625i
$519$	20.0000	0.877903
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	5.00000i	0.218844i
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	1.00000	0.0436852
$525$	− 2.00000i	− 0.0872872i
$526$	21.0000i	0.915644i
$527$	22.0000i	0.958335i
$528$	5.00000i	0.217597i
$529$	−22.0000	−0.956522
$530$	6.00000	0.260623
$531$	− 15.0000i	− 0.650945i
$532$	−4.00000	−0.173422
$533$	0	0
$534$	2.00000	0.0865485
$535$	− 10.0000i	− 0.432338i
$536$	−16.0000	−0.691095
$537$	−13.0000	−0.560991
$538$	14.0000i	0.603583i
$539$	− 15.0000i	− 0.646096i
$540$	− 1.00000i	− 0.0430331i
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	13.0000	0.558398
$543$	−16.0000	−0.686626
$544$	2.00000i	0.0857493i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	11.0000i	0.469897i
$549$	10.0000	0.426790
$550$	−5.00000	−0.213201
$551$	10.0000i	0.426014i
$552$	1.00000i	0.0425628i
$553$	22.0000i	0.935535i
$554$	11.0000i	0.467345i
$555$	−3.00000	−0.127343
$556$	−2.00000	−0.0848189
$557$	26.0000i	1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$	26.0000i	1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$	−11.0000	−0.465667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	10.0000i	0.422200i
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	9.00000i	0.378968i
$565$	− 11.0000i	− 0.462773i
$566$	− 19.0000i	− 0.798630i
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	− 2.00000i	− 0.0837708i
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	4.00000i	0.166957i
$575$	−1.00000	−0.0417029
$576$	−1.00000	−0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 13.0000i	− 0.540729i
$579$	24.0000i	0.997406i
$580$	5.00000i	0.207614i
$581$	−12.0000	−0.497844
$582$	2.00000	0.0829027
$583$	− 30.0000i	− 1.24247i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	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
$589$	−22.0000	−0.906494
$590$	− 15.0000i	− 0.617540i
$591$	− 8.00000i	− 0.329076i
$592$	3.00000i	0.123299i
$593$	31.0000i	1.27302i	0.771270	+	0.636509i	$0.219622\pi$
$593$	31.0000i	1.27302i	−0.771270	+	0.636509i	$0.780378\pi$
$594$	−5.00000	−0.205152
$595$	−4.00000	−0.163984
$596$	17.0000i	0.696347i
$597$	24.0000	0.982255
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	22.0000	0.896653
$603$	− 16.0000i	− 0.651570i
$604$	− 8.00000i	− 0.325515i
$605$	14.0000i	0.569181i
$606$	2.00000i	0.0812444i
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	−2.00000	−0.0811107
$609$	10.0000i	0.405220i
$610$	10.0000	0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	29.0000i	1.17130i	0.810564	+	0.585649i	$0.199160\pi$
$613$	29.0000i	1.17130i	−0.810564	+	0.585649i	$0.800840\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	10.0000i	0.402911i
$617$	− 41.0000i	− 1.65060i	−0.564696	−	0.825299i	$-0.691007\pi$
$617$	− 41.0000i	− 1.65060i	0.564696	−	0.825299i	$-0.308993\pi$
$618$	10.0000i	0.402259i
$619$	− 10.0000i	− 0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$	− 10.0000i	− 0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	−11.0000	−0.441771
$621$	−1.00000	−0.0401286
$622$	− 20.0000i	− 0.801927i
$623$	4.00000	0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.0000i	0.799361i
$627$	−10.0000	−0.399362
$628$	7.00000	0.279330
$629$	6.00000i	0.239236i
$630$	− 2.00000i	− 0.0796819i
$631$	40.0000i	1.59237i	0.605050	+	0.796187i	$0.293153\pi$
$631$	40.0000i	1.59237i	−0.605050	+	0.796187i	$0.706847\pi$
$632$	11.0000i	0.437557i
$633$	4.00000	0.158986
$634$	16.0000	0.635441
$635$	2.00000i	0.0793676i
$636$	6.00000	0.237915
$637$	0	0
$638$	25.0000	0.989759
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 10.0000i	− 0.394669i
$643$	− 8.00000i	− 0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$	− 8.00000i	− 0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$	2.00000i	0.0788110i
$645$	11.0000i	0.433125i
$646$	−4.00000	−0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−75.0000	−2.94401
$650$	0	0
$651$	−22.0000	−0.862248
$652$	15.0000i	0.587445i
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	2.00000	0.0782062
$655$	1.00000i	0.0390732i
$656$	2.00000i	0.0780869i
$657$	− 6.00000i	− 0.234082i
$658$	18.0000i	0.701713i
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	−5.00000	−0.194625
$661$	32.0000i	1.24466i	0.782757	+	0.622328i	$0.213813\pi$
$661$	32.0000i	1.24466i	−0.782757	+	0.622328i	$0.786187\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	− 4.00000i	− 0.155113i
$666$	−3.00000	−0.116248
$667$	5.00000	0.193601
$668$	− 3.00000i	− 0.116073i
$669$	26.0000i	1.00522i
$670$	− 16.0000i	− 0.618134i
$671$	− 50.0000i	− 1.93023i
$672$	−2.00000	−0.0771517
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	2.00000i	0.0770371i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	− 11.0000i	− 0.422452i
$679$	4.00000	0.153506
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	55.0000i	2.10606i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	− 2.00000i	− 0.0764719i
$685$	−11.0000	−0.420288
$686$	20.0000	0.763604
$687$	− 10.0000i	− 0.381524i
$688$	11.0000	0.419371
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	30.0000i	1.14125i	0.821209	+	0.570627i	$0.193300\pi$
$691$	30.0000i	1.14125i	−0.821209	+	0.570627i	$0.806700\pi$
$692$	20.0000	0.760286
$693$	−10.0000	−0.379869
$694$	34.0000i	1.29062i
$695$	− 2.00000i	− 0.0758643i
$696$	5.00000i	0.189525i
$697$	4.00000i	0.151511i
$698$	−20.0000	−0.757011
$699$	1.00000	0.0378235
$700$	− 2.00000i	− 0.0755929i
$701$	−13.0000	−0.491003	−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000	−0.491003	−0.245502	+	0.969396i	$0.578953\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	5.00000i	0.188445i
$705$	−9.00000	−0.338960
$706$	−18.0000	−0.677439
$707$	4.00000i	0.150435i
$708$	− 15.0000i	− 0.563735i
$709$	− 8.00000i	− 0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$	− 8.00000i	− 0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	2.00000	0.0749532
$713$	11.0000i	0.411953i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−13.0000	−0.485833
$717$	− 20.0000i	− 0.746914i
$718$	−12.0000	−0.447836
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	20.0000i	0.744839i
$722$	15.0000i	0.558242i
$723$	7.00000i	0.260333i
$724$	−16.0000	−0.594635
$725$	−5.00000	−0.185695
$726$	14.0000i	0.519589i
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.00000i	− 0.222070i
$731$	22.0000	0.813699
$732$	10.0000	0.369611
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	− 16.0000i	− 0.590571i
$735$	3.00000i	0.110657i
$736$	1.00000i	0.0368605i
$737$	−80.0000	−2.94684
$738$	−2.00000	−0.0736210
$739$	44.0000i	1.61857i	0.587419	+	0.809283i	$0.300144\pi$
$739$	44.0000i	1.61857i	−0.587419	+	0.809283i	$0.699856\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	12.0000	0.440534
$743$	− 51.0000i	− 1.87101i	−0.353315	−	0.935504i	$-0.614946\pi$
$743$	− 51.0000i	− 1.87101i	0.353315	−	0.935504i	$-0.385054\pi$
$744$	−11.0000	−0.403280
$745$	−17.0000	−0.622832
$746$	− 19.0000i	− 0.695639i
$747$	− 6.00000i	− 0.219529i
$748$	10.0000i	0.365636i
$749$	− 20.0000i	− 0.730784i
$750$	1.00000	0.0365148
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	9.00000i	0.328196i
$753$	25.0000	0.911051
$754$	0	0
$755$	8.00000	0.291150
$756$	− 2.00000i	− 0.0727393i
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−2.00000	−0.0726433
$759$	5.00000i	0.181489i
$760$	− 2.00000i	− 0.0725476i
$761$	20.0000i	0.724999i	0.931984	+	0.362500i	$0.118077\pi$
$761$	20.0000i	0.724999i	−0.931984	+	0.362500i	$0.881923\pi$
$762$	2.00000i	0.0724524i
$763$	4.00000	0.144810
$764$	−4.00000	−0.144715
$765$	− 2.00000i	− 0.0723102i
$766$	31.0000	1.12008
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 43.0000i	− 1.55062i	−0.631581	−	0.775310i	$-0.717594\pi$
$769$	− 43.0000i	− 1.55062i	0.631581	−	0.775310i	$-0.282406\pi$
$770$	−10.0000	−0.360375
$771$	−17.0000	−0.612240
$772$	24.0000i	0.863779i
$773$	− 32.0000i	− 1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$	− 32.0000i	− 1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$	11.0000i	0.395387i
$775$	− 11.0000i	− 0.395132i
$776$	2.00000	0.0717958
$777$	−6.00000	−0.215249
$778$	− 5.00000i	− 0.179259i
$779$	−4.00000	−0.143315
$780$	0	0
$781$	0	0
$782$	2.00000i	0.0715199i
$783$	−5.00000	−0.178685
$784$	3.00000	0.107143
$785$	7.00000i	0.249841i
$786$	1.00000i	0.0356688i
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	− 8.00000i	− 0.284988i
$789$	−21.0000	−0.747620
$790$	−11.0000	−0.391362
$791$	− 22.0000i	− 0.782230i
$792$	−5.00000	−0.177667
$793$	0	0
$794$	−3.00000	−0.106466
$795$	6.00000i	0.212798i
$796$	24.0000	0.850657
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	− 4.00000i	− 0.141598i
$799$	18.0000i	0.636794i
$800$	− 1.00000i	− 0.0353553i
$801$	2.00000i	0.0706665i
$802$	−36.0000	−1.27120
$803$	−30.0000	−1.05868
$804$	− 16.0000i	− 0.564276i
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	−14.0000	−0.492823
$808$	2.00000i	0.0703598i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	10.0000i	0.350931i
$813$	13.0000i	0.455930i
$814$	15.0000i	0.525750i
$815$	−15.0000	−0.525427
$816$	−2.00000	−0.0700140
$817$	22.0000i	0.769683i
$818$	30.0000	1.04893
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	51.0000i	1.77991i	0.456046	+	0.889956i	$0.349265\pi$
$821$	51.0000i	1.77991i	−0.456046	+	0.889956i	$0.650735\pi$
$822$	−11.0000	−0.383669
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	10.0000i	0.348367i
$825$	− 5.00000i	− 0.174078i
$826$	− 30.0000i	− 1.04383i
$827$	− 50.0000i	− 1.73867i	−0.494223	−	0.869335i	$-0.664547\pi$
$827$	− 50.0000i	− 1.73867i	0.494223	−	0.869335i	$-0.335453\pi$
$828$	−1.00000	−0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	− 6.00000i	− 0.208263i
$831$	−11.0000	−0.381586
$832$	0	0
$833$	6.00000	0.207888
$834$	− 2.00000i	− 0.0692543i
$835$	3.00000	0.103819
$836$	−10.0000	−0.345857
$837$	− 11.0000i	− 0.380216i
$838$	− 4.00000i	− 0.138178i
$839$	− 54.0000i	− 1.86429i	−0.362089	−	0.932144i	$-0.617936\pi$
$839$	− 54.0000i	− 1.86429i	0.362089	−	0.932144i	$-0.382064\pi$
$840$	− 2.00000i	− 0.0690066i
$841$	−4.00000	−0.137931
$842$	−16.0000	−0.551396
$843$	10.0000i	0.344418i
$844$	4.00000	0.137686
$845$	0	0
$846$	−9.00000	−0.309426
$847$	28.0000i	0.962091i
$848$	6.00000	0.206041
$849$	19.0000	0.652078
$850$	− 2.00000i	− 0.0685994i
$851$	3.00000i	0.102839i
$852$	0	0
$853$	− 49.0000i	− 1.67773i	−0.544341	−	0.838864i	$-0.683220\pi$
$853$	− 49.0000i	− 1.67773i	0.544341	−	0.838864i	$-0.316780\pi$
$854$	20.0000	0.684386
$855$	2.00000	0.0683986
$856$	− 10.0000i	− 0.341793i
$857$	25.0000	0.853984	0.426992	−	0.904255i	$-0.359573\pi$
$857$	25.0000	0.853984	0.426992	+	0.904255i	$0.359573\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	11.0000i	0.375097i
$861$	−4.00000	−0.136320
$862$	40.0000	1.36241
$863$	− 39.0000i	− 1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$	− 39.0000i	− 1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	20.0000i	0.680020i
$866$	− 28.0000i	− 0.951479i
$867$	13.0000	0.441503
$868$	−22.0000	−0.746729
$869$	55.0000i	1.86575i
$870$	−5.00000	−0.169516
$871$	0	0
$872$	2.00000	0.0677285
$873$	2.00000i	0.0676897i
$874$	−2.00000	−0.0676510
$875$	2.00000	0.0676123
$876$	− 6.00000i	− 0.202721i
$877$	19.0000i	0.641584i	0.947150	+	0.320792i	$0.103949\pi$
$877$	19.0000i	0.641584i	−0.947150	+	0.320792i	$0.896051\pi$
$878$	4.00000i	0.134993i
$879$	− 6.00000i	− 0.202375i
$880$	−5.00000	−0.168550
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	3.00000i	0.101015i
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	0	0
$885$	15.0000	0.504219
$886$	20.0000i	0.671913i
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	−3.00000	−0.100673
$889$	4.00000i	0.134156i
$890$	2.00000i	0.0670402i
$891$	− 5.00000i	− 0.167506i
$892$	26.0000i	0.870544i
$893$	−18.0000	−0.602347
$894$	−17.0000	−0.568565
$895$	− 13.0000i	− 0.434542i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−12.0000	−0.400445
$899$	55.0000i	1.83435i
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	10.0000i	0.332964i
$903$	22.0000i	0.732114i
$904$	− 11.0000i	− 0.365855i
$905$	− 16.0000i	− 0.531858i
$906$	8.00000	0.265782
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	−30.0000	−0.992855
$914$	38.0000	1.25693
$915$	10.0000i	0.330590i
$916$	− 10.0000i	− 0.330409i
$917$	2.00000i	0.0660458i
$918$	− 2.00000i	− 0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−39.0000	−1.28440
$923$	0	0
$924$	−10.0000	−0.328976
$925$	− 3.00000i	− 0.0986394i
$926$	−14.0000	−0.460069
$927$	−10.0000	−0.328443
$928$	5.00000i	0.164133i
$929$	16.0000i	0.524943i	0.964940	+	0.262471i	$0.0845376\pi$
$929$	16.0000i	0.524943i	−0.964940	+	0.262471i	$0.915462\pi$
$930$	− 11.0000i	− 0.360704i
$931$	6.00000i	0.196642i
$932$	1.00000	0.0327561
$933$	20.0000	0.654771
$934$	− 6.00000i	− 0.196326i
$935$	−10.0000	−0.327035
$936$	0	0
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	− 32.0000i	− 1.04484i
$939$	−20.0000	−0.652675
$940$	−9.00000	−0.293548
$941$	50.0000i	1.62995i	0.579494	+	0.814977i	$0.303250\pi$
$941$	50.0000i	1.62995i	−0.579494	+	0.814977i	$0.696750\pi$
$942$	7.00000i	0.228072i
$943$	2.00000i	0.0651290i
$944$	− 15.0000i	− 0.488208i
$945$	2.00000	0.0650600
$946$	55.0000	1.78820
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	−11.0000	−0.357263
$949$	0	0
$950$	2.00000	0.0648886
$951$	16.0000i	0.518836i
$952$	−4.00000	−0.129641
$953$	51.0000	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$954$	6.00000i	0.194257i
$955$	− 4.00000i	− 0.129437i
$956$	− 20.0000i	− 0.646846i
$957$	25.0000i	0.808135i
$958$	24.0000	0.775405
$959$	−22.0000	−0.710417
$960$	− 1.00000i	− 0.0322749i
$961$	−90.0000	−2.90323
$962$	0	0
$963$	10.0000	0.322245
$964$	7.00000i	0.225455i
$965$	−24.0000	−0.772587
$966$	−2.00000	−0.0643489
$967$	− 20.0000i	− 0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	− 20.0000i	− 0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$	14.0000i	0.449977i
$969$	− 4.00000i	− 0.128499i
$970$	2.00000i	0.0642161i
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	1.00000	0.0320750
$973$	− 4.00000i	− 0.128234i
$974$	2.00000	0.0640841
$975$	0	0
$976$	10.0000	0.320092
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	−15.0000	−0.479647
$979$	10.0000	0.319601
$980$	3.00000i	0.0958315i
$981$	2.00000i	0.0638551i
$982$	16.0000i	0.510581i
$983$	− 3.00000i	− 0.0956851i	−0.998855	−	0.0478426i	$-0.984765\pi$
$983$	− 3.00000i	− 0.0956851i	0.998855	−	0.0478426i	$-0.0152346\pi$
$984$	−2.00000	−0.0637577
$985$	8.00000	0.254901
$986$	10.0000i	0.318465i
$987$	−18.0000	−0.572946
$988$	0	0
$989$	11.0000	0.349780
$990$	− 5.00000i	− 0.158910i
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	−11.0000	−0.349250
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	24.0000i	0.760851i
$996$	− 6.00000i	− 0.190117i
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	28.0000	0.886325
$999$	− 3.00000i	− 0.0949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.h.1351.2		2
13.5	odd	4		5070.2.a.p.1.1		1
13.7	odd	12		390.2.i.f.211.1	yes	2
13.8	odd	4		5070.2.a.d.1.1		1
13.11	odd	12		390.2.i.f.61.1	✓	2
13.12	even	2	inner	5070.2.b.h.1351.1		2
39.11	even	12		1170.2.i.a.451.1		2
39.20	even	12		1170.2.i.a.991.1		2
65.7	even	12		1950.2.z.e.1849.1		4
65.24	odd	12		1950.2.i.d.451.1		2
65.33	even	12		1950.2.z.e.1849.2		4
65.37	even	12		1950.2.z.e.1699.2		4
65.59	odd	12		1950.2.i.d.601.1		2
65.63	even	12		1950.2.z.e.1699.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.f.61.1	✓	2	13.11	odd	12
390.2.i.f.211.1	yes	2	13.7	odd	12
1170.2.i.a.451.1		2	39.11	even	12
1170.2.i.a.991.1		2	39.20	even	12
1950.2.i.d.451.1		2	65.24	odd	12
1950.2.i.d.601.1		2	65.59	odd	12
1950.2.z.e.1699.1		4	65.63	even	12
1950.2.z.e.1699.2		4	65.37	even	12
1950.2.z.e.1849.1		4	65.7	even	12
1950.2.z.e.1849.2		4	65.33	even	12
5070.2.a.d.1.1		1	13.8	odd	4
5070.2.a.p.1.1		1	13.5	odd	4
5070.2.b.h.1351.1		2	13.12	even	2	inner
5070.2.b.h.1351.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +5.00000 q^{29} -1.00000 q^{30} +11.0000i q^{31} +1.00000i q^{32} +5.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +3.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000 q^{43} +5.00000i q^{44} -1.00000i q^{45} +1.00000i q^{46} +9.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000i q^{54} -5.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} +5.00000i q^{58} -15.0000i q^{59} -1.00000i q^{60} +10.0000 q^{61} -11.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} -16.0000i q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -6.00000i q^{73} -3.00000 q^{74} +1.00000 q^{75} -2.00000i q^{76} -10.0000 q^{77} -11.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -2.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -2.00000i q^{85} +11.0000i q^{86} -5.00000 q^{87} -5.00000 q^{88} +2.00000i q^{89} +1.00000 q^{90} -1.00000 q^{92} -11.0000i q^{93} -9.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 10 q^{22} + 2 q^{23} - 2 q^{25} - 2 q^{27} + 10 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 4 q^{38} - 2 q^{40} - 4 q^{42} + 22 q^{43} - 2 q^{48} + 6 q^{49} - 4 q^{51} + 12 q^{53} - 10 q^{55} - 4 q^{56} + 20 q^{61} - 22 q^{62} - 2 q^{64} - 10 q^{66} - 4 q^{68} - 2 q^{69} - 6 q^{74} + 2 q^{75} - 20 q^{77} - 22 q^{79} + 2 q^{81} - 4 q^{82} - 10 q^{87} - 10 q^{88} + 2 q^{90} - 2 q^{92} - 18 q^{94} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 10 * q^22 + 2 * q^23 - 2 * q^25 - 2 * q^27 + 10 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 4 * q^38 - 2 * q^40 - 4 * q^42 + 22 * q^43 - 2 * q^48 + 6 * q^49 - 4 * q^51 + 12 * q^53 - 10 * q^55 - 4 * q^56 + 20 * q^61 - 22 * q^62 - 2 * q^64 - 10 * q^66 - 4 * q^68 - 2 * q^69 - 6 * q^74 + 2 * q^75 - 20 * q^77 - 22 * q^79 + 2 * q^81 - 4 * q^82 - 10 * q^87 - 10 * q^88 + 2 * q^90 - 2 * q^92 - 18 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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