LMFDB - Embedded newform 5070.2.b.b.1351.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5070,2,Mod(1351,5070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,-2,-2,0,0,0,0,2,-2,0,2,0,-6,0,2,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)

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.b.1351.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -3.00000i q^{19} -1.00000i q^{20} -3.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{30} -6.00000i q^{31} +1.00000i q^{32} +3.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} +9.00000i q^{37} +3.00000 q^{38} +1.00000 q^{40} +10.0000i q^{41} +3.00000 q^{42} +10.0000 q^{43} +3.00000i q^{44} +1.00000i q^{45} +4.00000i q^{46} -3.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} -1.00000i q^{50} +9.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} +3.00000 q^{56} +3.00000i q^{57} -4.00000i q^{58} +12.0000i q^{59} +1.00000i q^{60} -6.00000 q^{61} +6.00000 q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +8.00000i q^{67} -4.00000 q^{69} -3.00000i q^{70} +14.0000i q^{71} -1.00000i q^{72} -8.00000i q^{73} -9.00000 q^{74} +1.00000 q^{75} +3.00000i q^{76} +9.00000 q^{77} +6.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -16.0000i q^{83} +3.00000i q^{84} +10.0000i q^{86} +4.00000 q^{87} -3.00000 q^{88} -3.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +6.00000i q^{93} +3.00000 q^{94} +3.00000 q^{95} -1.00000i q^{96} -8.00000i q^{97} -2.00000i q^{98} -3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 6 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 8 q^{29} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 6 q^{38} + 2 q^{40} + 6 q^{42} + 20 q^{43}+ \cdots + 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 - 6 * q^14 + 2 * q^16 + 6 * q^22 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 8 * q^29 + 2 * q^30 - 6 * q^35 - 2 * q^36 + 6 * q^38 + 2 * q^40 + 6 * q^42 + 20 * q^43 - 2 * q^48 - 4 * q^49 + 18 * q^53 + 6 * q^55 + 6 * q^56 - 12 * q^61 + 12 * q^62 - 2 * q^64 - 6 * q^66 - 8 * q^69 - 18 * q^74 + 2 * q^75 + 18 * q^77 + 12 * q^79 + 2 * q^81 - 20 * q^82 + 8 * q^87 - 6 * q^88 - 2 * q^90 - 8 * q^92 + 6 * q^94 + 6 * q^95

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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000i	0.235702i
$19$	− 3.00000i	− 0.688247i	−0.938924	−	0.344124i	$-0.888176\pi$
$19$	− 3.00000i	− 0.688247i	0.938924	−	0.344124i	$-0.111824\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 3.00000i	− 0.654654i
$22$	3.00000	0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 3.00000i	− 0.566947i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	− 6.00000i	− 1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$	− 6.00000i	− 1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$	1.00000i	0.176777i
$33$	3.00000i	0.522233i
$34$	0	0
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	9.00000i	1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$	9.00000i	1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	1.00000	0.158114
$41$	10.0000i	1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$	10.0000i	1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	3.00000	0.462910
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	3.00000i	0.452267i
$45$	1.00000i	0.149071i
$46$	4.00000i	0.589768i
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	−1.00000	−0.144338
$49$	−2.00000	−0.285714
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	− 1.00000i	− 0.136083i
$55$	3.00000	0.404520
$56$	3.00000	0.400892
$57$	3.00000i	0.397360i
$58$	− 4.00000i	− 0.525226i
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	1.00000i	0.129099i
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	6.00000	0.762001
$63$	3.00000i	0.377964i
$64$	−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	$0.837479\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	− 3.00000i	− 0.358569i
$71$	14.0000i	1.66149i	0.556650	+	0.830747i	$0.312086\pi$
$71$	14.0000i	1.66149i	−0.556650	+	0.830747i	$0.687914\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	−9.00000	−1.04623
$75$	1.00000	0.115470
$76$	3.00000i	0.344124i
$77$	9.00000	1.02565
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	3.00000i	0.327327i
$85$	0	0
$86$	10.0000i	1.07833i
$87$	4.00000	0.428845
$88$	−3.00000	−0.319801
$89$	− 3.00000i	− 0.317999i	−0.987279	−	0.159000i	$-0.949173\pi$
$89$	− 3.00000i	− 0.317999i	0.987279	−	0.159000i	$-0.0508269\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	6.00000i	0.622171i
$94$	3.00000	0.309426
$95$	3.00000	0.307794
$96$	− 1.00000i	− 0.102062i
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	− 2.00000i	− 0.202031i
$99$	− 3.00000i	− 0.301511i
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	15.0000	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$103$	15.0000	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	9.00000i	0.874157i
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	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	$0.969465\pi$
$110$	3.00000i	0.286039i
$111$	− 9.00000i	− 0.854242i
$112$	3.00000i	0.283473i
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	−3.00000	−0.280976
$115$	4.00000i	0.373002i
$116$	4.00000	0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	2.00000	0.181818
$122$	− 6.00000i	− 0.543214i
$123$	− 10.0000i	− 0.901670i
$124$	6.00000i	0.538816i
$125$	− 1.00000i	− 0.0894427i
$126$	−3.00000	−0.267261
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	− 3.00000i	− 0.261116i
$133$	9.00000	0.780399
$134$	−8.00000	−0.691095
$135$	− 1.00000i	− 0.0860663i
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	3.00000	0.253546
$141$	3.00000i	0.252646i
$142$	−14.0000	−1.17485
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 4.00000i	− 0.332182i
$146$	8.00000	0.662085
$147$	2.00000	0.164957
$148$	− 9.00000i	− 0.739795i
$149$	2.00000i	0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$	2.00000i	0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$	1.00000i	0.0816497i
$151$	14.0000i	1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$	14.0000i	1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	9.00000i	0.725241i
$155$	6.00000	0.481932
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	6.00000i	0.477334i
$159$	−9.00000	−0.713746
$160$	−1.00000	−0.0790569
$161$	12.0000i	0.945732i
$162$	1.00000i	0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	− 10.0000i	− 0.780869i
$165$	−3.00000	−0.233550
$166$	16.0000	1.24184
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	− 3.00000i	− 0.229416i
$172$	−10.0000	−0.762493
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	4.00000i	0.303239i
$175$	− 3.00000i	− 0.226779i
$176$	− 3.00000i	− 0.226134i
$177$	− 12.0000i	− 0.901975i
$178$	3.00000	0.224860
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	− 4.00000i	− 0.294884i
$185$	−9.00000	−0.661693
$186$	−6.00000	−0.439941
$187$	0	0
$188$	3.00000i	0.218797i
$189$	− 3.00000i	− 0.218218i
$190$	3.00000i	0.217643i
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.00000	0.0721688
$193$	8.00000i	0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$	8.00000i	0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	2.00000	0.142857
$197$	− 11.0000i	− 0.783718i	−0.920025	−	0.391859i	$-0.871832\pi$
$197$	− 11.0000i	− 0.783718i	0.920025	−	0.391859i	$-0.128168\pi$
$198$	3.00000	0.213201
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	1.00000i	0.0707107i
$201$	− 8.00000i	− 0.564276i
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	−10.0000	−0.698430
$206$	15.0000i	1.04510i
$207$	4.00000	0.278019
$208$	0	0
$209$	−9.00000	−0.622543
$210$	3.00000i	0.207020i
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	−9.00000	−0.618123
$213$	− 14.0000i	− 0.959264i
$214$	2.00000i	0.136717i
$215$	10.0000i	0.681994i
$216$	1.00000i	0.0680414i
$217$	18.0000	1.22192
$218$	−2.00000	−0.135457
$219$	8.00000i	0.540590i
$220$	−3.00000	−0.202260
$221$	0	0
$222$	9.00000	0.604040
$223$	11.0000i	0.736614i	0.929704	+	0.368307i	$0.120063\pi$
$223$	11.0000i	0.736614i	−0.929704	+	0.368307i	$0.879937\pi$
$224$	−3.00000	−0.200446
$225$	−1.00000	−0.0666667
$226$	− 8.00000i	− 0.532152i
$227$	16.0000i	1.06196i	0.847385	+	0.530979i	$0.178176\pi$
$227$	16.0000i	1.06196i	−0.847385	+	0.530979i	$0.821824\pi$
$228$	− 3.00000i	− 0.198680i
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	−4.00000	−0.263752
$231$	−9.00000	−0.592157
$232$	4.00000i	0.262613i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	− 12.0000i	− 0.781133i
$237$	−6.00000	−0.389742
$238$	0	0
$239$	26.0000i	1.68180i	0.541190	+	0.840900i	$0.317974\pi$
$239$	26.0000i	1.68180i	−0.541190	+	0.840900i	$0.682026\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	7.00000i	0.450910i	0.974254	+	0.225455i	$0.0723868\pi$
$241$	7.00000i	0.450910i	−0.974254	+	0.225455i	$0.927613\pi$
$242$	2.00000i	0.128565i
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	− 2.00000i	− 0.127775i
$246$	10.0000	0.637577
$247$	0	0
$248$	−6.00000	−0.381000
$249$	16.0000i	1.01396i
$250$	1.00000	0.0632456
$251$	1.00000	0.0631194	0.0315597	−	0.999502i	$-0.489953\pi$
$251$	1.00000	0.0631194	0.0315597	+	0.999502i	$0.489953\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 12.0000i	− 0.754434i
$254$	− 3.00000i	− 0.188237i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	− 10.0000i	− 0.622573i
$259$	−27.0000	−1.67770
$260$	0	0
$261$	−4.00000	−0.247594
$262$	− 3.00000i	− 0.185341i
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	3.00000	0.184637
$265$	9.00000i	0.552866i
$266$	9.00000i	0.551825i
$267$	3.00000i	0.183597i
$268$	− 8.00000i	− 0.488678i
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	1.00000	0.0608581
$271$	− 12.0000i	− 0.728948i	−0.931214	−	0.364474i	$-0.881249\pi$
$271$	− 12.0000i	− 0.728948i	0.931214	−	0.364474i	$-0.118751\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	3.00000i	0.180907i
$276$	4.00000	0.240772
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	− 17.0000i	− 1.01959i
$279$	− 6.00000i	− 0.359211i
$280$	3.00000i	0.179284i
$281$	− 30.0000i	− 1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$	− 30.0000i	− 1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$	−3.00000	−0.178647
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	− 14.0000i	− 0.830747i
$285$	−3.00000	−0.177705
$286$	0	0
$287$	−30.0000	−1.77084
$288$	1.00000i	0.0589256i
$289$	−17.0000	−1.00000
$290$	4.00000	0.234888
$291$	8.00000i	0.468968i
$292$	8.00000i	0.468165i
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	2.00000i	0.116642i
$295$	−12.0000	−0.698667
$296$	9.00000	0.523114
$297$	3.00000i	0.174078i
$298$	−2.00000	−0.115857
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	30.0000i	1.72917i
$302$	−14.0000	−0.805609
$303$	0	0
$304$	− 3.00000i	− 0.172062i
$305$	− 6.00000i	− 0.343559i
$306$	0	0
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	−9.00000	−0.512823
$309$	−15.0000	−0.853320
$310$	6.00000i	0.340777i
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	− 17.0000i	− 0.959366i
$315$	−3.00000	−0.169031
$316$	−6.00000	−0.337526
$317$	− 17.0000i	− 0.954815i	−0.878682	−	0.477408i	$-0.841577\pi$
$317$	− 17.0000i	− 0.954815i	0.878682	−	0.477408i	$-0.158423\pi$
$318$	− 9.00000i	− 0.504695i
$319$	12.0000i	0.671871i
$320$	− 1.00000i	− 0.0559017i
$321$	−2.00000	−0.111629
$322$	−12.0000	−0.668734
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 2.00000i	− 0.110600i
$328$	10.0000	0.552158
$329$	9.00000	0.496186
$330$	− 3.00000i	− 0.165145i
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	16.0000i	0.878114i
$333$	9.00000i	0.493197i
$334$	−9.00000	−0.492458
$335$	−8.00000	−0.437087
$336$	− 3.00000i	− 0.163663i
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	−18.0000	−0.974755
$342$	3.00000	0.162221
$343$	15.0000i	0.809924i
$344$	− 10.0000i	− 0.539164i
$345$	− 4.00000i	− 0.215353i
$346$	− 13.0000i	− 0.698884i
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−4.00000	−0.214423
$349$	16.0000i	0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$	16.0000i	0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	3.00000	0.159901
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	12.0000	0.637793
$355$	−14.0000	−0.743043
$356$	3.00000i	0.159000i
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	30.0000i	1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$	30.0000i	1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$	1.00000	0.0527046
$361$	10.0000	0.526316
$362$	10.0000i	0.525588i
$363$	−2.00000	−0.104973
$364$	0	0
$365$	8.00000	0.418739
$366$	6.00000i	0.313625i
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000	0.208514
$369$	10.0000i	0.520579i
$370$	− 9.00000i	− 0.467888i
$371$	27.0000i	1.40177i
$372$	− 6.00000i	− 0.311086i
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	1.00000i	0.0516398i
$376$	−3.00000	−0.154713
$377$	0	0
$378$	3.00000	0.154303
$379$	33.0000i	1.69510i	0.530719	+	0.847548i	$0.321922\pi$
$379$	33.0000i	1.69510i	−0.530719	+	0.847548i	$0.678078\pi$
$380$	−3.00000	−0.153897
$381$	3.00000	0.153695
$382$	− 6.00000i	− 0.306987i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	1.00000i	0.0510310i
$385$	9.00000i	0.458682i
$386$	−8.00000	−0.407189
$387$	10.0000	0.508329
$388$	8.00000i	0.406138i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	2.00000i	0.101015i
$393$	3.00000	0.151330
$394$	11.0000	0.554172
$395$	6.00000i	0.301893i
$396$	3.00000i	0.150756i
$397$	− 23.0000i	− 1.15434i	−0.816625	−	0.577168i	$-0.804158\pi$
$397$	− 23.0000i	− 1.15434i	0.816625	−	0.577168i	$-0.195842\pi$
$398$	10.0000i	0.501255i
$399$	−9.00000	−0.450564
$400$	−1.00000	−0.0500000
$401$	− 27.0000i	− 1.34832i	−0.738587	−	0.674158i	$-0.764507\pi$
$401$	− 27.0000i	− 1.34832i	0.738587	−	0.674158i	$-0.235493\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	12.0000	0.595550
$407$	27.0000	1.33834
$408$	0	0
$409$	7.00000i	0.346128i	0.984911	+	0.173064i	$0.0553667\pi$
$409$	7.00000i	0.346128i	−0.984911	+	0.173064i	$0.944633\pi$
$410$	− 10.0000i	− 0.493865i
$411$	− 12.0000i	− 0.591916i
$412$	−15.0000	−0.738997
$413$	−36.0000	−1.77144
$414$	4.00000i	0.196589i
$415$	16.0000	0.785409
$416$	0	0
$417$	17.0000	0.832494
$418$	− 9.00000i	− 0.440204i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−3.00000	−0.146385
$421$	28.0000i	1.36464i	0.731055	+	0.682318i	$0.239028\pi$
$421$	28.0000i	1.36464i	−0.731055	+	0.682318i	$0.760972\pi$
$422$	9.00000i	0.438113i
$423$	− 3.00000i	− 0.145865i
$424$	− 9.00000i	− 0.437079i
$425$	0	0
$426$	14.0000	0.678302
$427$	− 18.0000i	− 0.871081i
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−10.0000	−0.482243
$431$	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
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	18.0000i	0.864028i
$435$	4.00000i	0.191785i
$436$	− 2.00000i	− 0.0957826i
$437$	− 12.0000i	− 0.574038i
$438$	−8.00000	−0.382255
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	− 3.00000i	− 0.143019i
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−10.0000	−0.475114	−0.237557	−	0.971374i	$-0.576347\pi$
$443$	−10.0000	−0.475114	−0.237557	+	0.971374i	$0.576347\pi$
$444$	9.00000i	0.427121i
$445$	3.00000	0.142214
$446$	−11.0000	−0.520865
$447$	− 2.00000i	− 0.0945968i
$448$	− 3.00000i	− 0.141737i
$449$	− 35.0000i	− 1.65175i	−0.563852	−	0.825876i	$-0.690681\pi$
$449$	− 35.0000i	− 1.65175i	0.563852	−	0.825876i	$-0.309319\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	30.0000	1.41264
$452$	8.00000	0.376288
$453$	− 14.0000i	− 0.657777i
$454$	−16.0000	−0.750917
$455$	0	0
$456$	3.00000	0.140488
$457$	42.0000i	1.96468i	0.187112	+	0.982339i	$0.440087\pi$
$457$	42.0000i	1.96468i	−0.187112	+	0.982339i	$0.559913\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	− 4.00000i	− 0.186501i
$461$	32.0000i	1.49039i	0.666847	+	0.745194i	$0.267643\pi$
$461$	32.0000i	1.49039i	−0.666847	+	0.745194i	$0.732357\pi$
$462$	− 9.00000i	− 0.418718i
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−4.00000	−0.185695
$465$	−6.00000	−0.278243
$466$	6.00000i	0.277945i
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	3.00000i	0.138380i
$471$	17.0000	0.783319
$472$	12.0000	0.552345
$473$	− 30.0000i	− 1.37940i
$474$	− 6.00000i	− 0.275589i
$475$	3.00000i	0.137649i
$476$	0	0
$477$	9.00000	0.412082
$478$	−26.0000	−1.18921
$479$	12.0000i	0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$	12.0000i	0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−7.00000	−0.318841
$483$	− 12.0000i	− 0.546019i
$484$	−2.00000	−0.0909091
$485$	8.00000	0.363261
$486$	− 1.00000i	− 0.0453609i
$487$	− 29.0000i	− 1.31412i	−0.753840	−	0.657058i	$-0.771801\pi$
$487$	− 29.0000i	− 1.31412i	0.753840	−	0.657058i	$-0.228199\pi$
$488$	6.00000i	0.271607i
$489$	− 20.0000i	− 0.904431i
$490$	2.00000	0.0903508
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	10.0000i	0.450835i
$493$	0	0
$494$	0	0
$495$	3.00000	0.134840
$496$	− 6.00000i	− 0.269408i
$497$	−42.0000	−1.88396
$498$	−16.0000	−0.716977
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	1.00000i	0.0447214i
$501$	− 9.00000i	− 0.402090i
$502$	1.00000i	0.0446322i
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	3.00000	0.133103
$509$	− 10.0000i	− 0.443242i	−0.975133	−	0.221621i	$-0.928865\pi$
$509$	− 10.0000i	− 0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	1.00000i	0.0441942i
$513$	3.00000i	0.132453i
$514$	− 12.0000i	− 0.529297i
$515$	15.0000i	0.660979i
$516$	10.0000	0.440225
$517$	−9.00000	−0.395820
$518$	− 27.0000i	− 1.18631i
$519$	13.0000	0.570637
$520$	0	0
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	− 4.00000i	− 0.175075i
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	3.00000	0.131056
$525$	3.00000i	0.130931i
$526$	− 31.0000i	− 1.35166i
$527$	0	0
$528$	3.00000i	0.130558i
$529$	−7.00000	−0.304348
$530$	−9.00000	−0.390935
$531$	12.0000i	0.520756i
$532$	−9.00000	−0.390199
$533$	0	0
$534$	−3.00000	−0.129823
$535$	2.00000i	0.0864675i
$536$	8.00000	0.345547
$537$	4.00000	0.172613
$538$	− 4.00000i	− 0.172452i
$539$	6.00000i	0.258438i
$540$	1.00000i	0.0430331i
$541$	− 22.0000i	− 0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$	− 22.0000i	− 0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$	12.0000	0.515444
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	− 12.0000i	− 0.512615i
$549$	−6.00000	−0.256074
$550$	−3.00000	−0.127920
$551$	12.0000i	0.511217i
$552$	4.00000i	0.170251i
$553$	18.0000i	0.765438i
$554$	31.0000i	1.31706i
$555$	9.00000	0.382029
$556$	17.0000	0.720961
$557$	− 9.00000i	− 0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$	− 9.00000i	− 0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$	6.00000	0.254000
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	30.0000	1.26547
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	− 3.00000i	− 0.126323i
$565$	− 8.00000i	− 0.336563i
$566$	− 6.00000i	− 0.252199i
$567$	3.00000i	0.125988i
$568$	14.0000	0.587427
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	− 3.00000i	− 0.125656i
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	− 30.0000i	− 1.25218i
$575$	−4.00000	−0.166812
$576$	−1.00000	−0.0416667
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	− 17.0000i	− 0.707107i
$579$	− 8.00000i	− 0.332469i
$580$	4.00000i	0.166091i
$581$	48.0000	1.99138
$582$	−8.00000	−0.331611
$583$	− 27.0000i	− 1.11823i
$584$	−8.00000	−0.331042
$585$	0	0
$586$	1.00000	0.0413096
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	−2.00000	−0.0824786
$589$	−18.0000	−0.741677
$590$	− 12.0000i	− 0.494032i
$591$	11.0000i	0.452480i
$592$	9.00000i	0.369898i
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	− 2.00000i	− 0.0819232i
$597$	−10.0000	−0.409273
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	−30.0000	−1.22271
$603$	8.00000i	0.325785i
$604$	− 14.0000i	− 0.569652i
$605$	2.00000i	0.0813116i
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	3.00000	0.121666
$609$	12.0000i	0.486265i
$610$	6.00000	0.242933
$611$	0	0
$612$	0	0
$613$	7.00000i	0.282727i	0.989958	+	0.141364i	$0.0451487\pi$
$613$	7.00000i	0.282727i	−0.989958	+	0.141364i	$0.954851\pi$
$614$	−26.0000	−1.04927
$615$	10.0000	0.403239
$616$	− 9.00000i	− 0.362620i
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 15.0000i	− 0.603388i
$619$	− 23.0000i	− 0.924448i	−0.886763	−	0.462224i	$-0.847052\pi$
$619$	− 23.0000i	− 0.924448i	0.886763	−	0.462224i	$-0.152948\pi$
$620$	−6.00000	−0.240966
$621$	−4.00000	−0.160514
$622$	4.00000i	0.160385i
$623$	9.00000	0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000i	1.03917i
$627$	9.00000	0.359425
$628$	17.0000	0.678374
$629$	0	0
$630$	− 3.00000i	− 0.119523i
$631$	12.0000i	0.477712i	0.971055	+	0.238856i	$0.0767725\pi$
$631$	12.0000i	0.477712i	−0.971055	+	0.238856i	$0.923228\pi$
$632$	− 6.00000i	− 0.238667i
$633$	−9.00000	−0.357718
$634$	17.0000	0.675156
$635$	− 3.00000i	− 0.119051i
$636$	9.00000	0.356873
$637$	0	0
$638$	−12.0000	−0.475085
$639$	14.0000i	0.553831i
$640$	1.00000	0.0395285
$641$	−35.0000	−1.38242	−0.691208	−	0.722655i	$-0.742921\pi$
$641$	−35.0000	−1.38242	−0.691208	+	0.722655i	$0.742921\pi$
$642$	− 2.00000i	− 0.0789337i
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	− 12.0000i	− 0.472866i
$645$	− 10.0000i	− 0.393750i
$646$	0	0
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	36.0000	1.41312
$650$	0	0
$651$	−18.0000	−0.705476
$652$	− 20.0000i	− 0.783260i
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	2.00000	0.0782062
$655$	− 3.00000i	− 0.117220i
$656$	10.0000i	0.390434i
$657$	− 8.00000i	− 0.312110i
$658$	9.00000i	0.350857i
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	3.00000	0.116775
$661$	30.0000i	1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$	30.0000i	1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−16.0000	−0.620920
$665$	9.00000i	0.349005i
$666$	−9.00000	−0.348743
$667$	−16.0000	−0.619522
$668$	− 9.00000i	− 0.348220i
$669$	− 11.0000i	− 0.425285i
$670$	− 8.00000i	− 0.309067i
$671$	18.0000i	0.694882i
$672$	3.00000	0.115728
$673$	−48.0000	−1.85026	−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000	−1.85026	−0.925132	+	0.379646i	$0.876046\pi$
$674$	− 6.00000i	− 0.231111i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	8.00000i	0.307238i
$679$	24.0000	0.921035
$680$	0	0
$681$	− 16.0000i	− 0.613121i
$682$	− 18.0000i	− 0.689256i
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	3.00000i	0.114708i
$685$	−12.0000	−0.458496
$686$	−15.0000	−0.572703
$687$	10.0000i	0.381524i
$688$	10.0000	0.381246
$689$	0	0
$690$	4.00000	0.152277
$691$	7.00000i	0.266293i	0.991096	+	0.133146i	$0.0425080\pi$
$691$	7.00000i	0.266293i	−0.991096	+	0.133146i	$0.957492\pi$
$692$	13.0000	0.494186
$693$	9.00000	0.341882
$694$	− 24.0000i	− 0.911028i
$695$	− 17.0000i	− 0.644847i
$696$	− 4.00000i	− 0.151620i
$697$	0	0
$698$	−16.0000	−0.605609
$699$	−6.00000	−0.226941
$700$	3.00000i	0.113389i
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	27.0000	1.01832
$704$	3.00000i	0.113067i
$705$	−3.00000	−0.112987
$706$	−8.00000	−0.301084
$707$	0	0
$708$	12.0000i	0.450988i
$709$	− 40.0000i	− 1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$	− 40.0000i	− 1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$	− 14.0000i	− 0.525411i
$711$	6.00000	0.225018
$712$	−3.00000	−0.112430
$713$	− 24.0000i	− 0.898807i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 26.0000i	− 0.970988i
$718$	−30.0000	−1.11959
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000i	0.0372678i
$721$	45.0000i	1.67589i
$722$	10.0000i	0.372161i
$723$	− 7.00000i	− 0.260333i
$724$	−10.0000	−0.371647
$725$	4.00000	0.148556
$726$	− 2.00000i	− 0.0742270i
$727$	3.00000	0.111264	0.0556319	−	0.998451i	$-0.482283\pi$
$727$	3.00000	0.111264	0.0556319	+	0.998451i	$0.482283\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.00000i	0.296093i
$731$	0	0
$732$	−6.00000	−0.221766
$733$	− 53.0000i	− 1.95760i	−0.204819	−	0.978800i	$-0.565661\pi$
$733$	− 53.0000i	− 1.95760i	0.204819	−	0.978800i	$-0.434339\pi$
$734$	8.00000i	0.295285i
$735$	2.00000i	0.0737711i
$736$	4.00000i	0.147442i
$737$	24.0000	0.884051
$738$	−10.0000	−0.368105
$739$	11.0000i	0.404642i	0.979319	+	0.202321i	$0.0648484\pi$
$739$	11.0000i	0.404642i	−0.979319	+	0.202321i	$0.935152\pi$
$740$	9.00000	0.330847
$741$	0	0
$742$	−27.0000	−0.991201
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	6.00000	0.219971
$745$	−2.00000	−0.0732743
$746$	− 26.0000i	− 0.951928i
$747$	− 16.0000i	− 0.585409i
$748$	0	0
$749$	6.00000i	0.219235i
$750$	−1.00000	−0.0365148
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	− 3.00000i	− 0.109399i
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−14.0000	−0.509512
$756$	3.00000i	0.109109i
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−33.0000	−1.19861
$759$	12.0000i	0.435572i
$760$	− 3.00000i	− 0.108821i
$761$	− 31.0000i	− 1.12375i	−0.827222	−	0.561875i	$-0.810080\pi$
$761$	− 31.0000i	− 1.12375i	0.827222	−	0.561875i	$-0.189920\pi$
$762$	3.00000i	0.108679i
$763$	−6.00000	−0.217215
$764$	6.00000	0.217072
$765$	0	0
$766$	4.00000	0.144526
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 46.0000i	− 1.65880i	−0.558653	−	0.829401i	$-0.688682\pi$
$769$	− 46.0000i	− 1.65880i	0.558653	−	0.829401i	$-0.311318\pi$
$770$	−9.00000	−0.324337
$771$	12.0000	0.432169
$772$	− 8.00000i	− 0.287926i
$773$	7.00000i	0.251773i	0.992045	+	0.125886i	$0.0401774\pi$
$773$	7.00000i	0.251773i	−0.992045	+	0.125886i	$0.959823\pi$
$774$	10.0000i	0.359443i
$775$	6.00000i	0.215526i
$776$	−8.00000	−0.287183
$777$	27.0000	0.968620
$778$	− 24.0000i	− 0.860442i
$779$	30.0000	1.07486
$780$	0	0
$781$	42.0000	1.50288
$782$	0	0
$783$	4.00000	0.142948
$784$	−2.00000	−0.0714286
$785$	− 17.0000i	− 0.606756i
$786$	3.00000i	0.107006i
$787$	− 36.0000i	− 1.28326i	−0.767014	−	0.641631i	$-0.778258\pi$
$787$	− 36.0000i	− 1.28326i	0.767014	−	0.641631i	$-0.221742\pi$
$788$	11.0000i	0.391859i
$789$	31.0000	1.10363
$790$	−6.00000	−0.213470
$791$	− 24.0000i	− 0.853342i
$792$	−3.00000	−0.106600
$793$	0	0
$794$	23.0000	0.816239
$795$	− 9.00000i	− 0.319197i
$796$	−10.0000	−0.354441
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	− 9.00000i	− 0.318597i
$799$	0	0
$800$	− 1.00000i	− 0.0353553i
$801$	− 3.00000i	− 0.106000i
$802$	27.0000	0.953403
$803$	−24.0000	−0.846942
$804$	8.00000i	0.282138i
$805$	−12.0000	−0.422944
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−1.00000	−0.0351364
$811$	31.0000i	1.08856i	0.838905	+	0.544279i	$0.183197\pi$
$811$	31.0000i	1.08856i	−0.838905	+	0.544279i	$0.816803\pi$
$812$	12.0000i	0.421117i
$813$	12.0000i	0.420858i
$814$	27.0000i	0.946350i
$815$	−20.0000	−0.700569
$816$	0	0
$817$	− 30.0000i	− 1.04957i
$818$	−7.00000	−0.244749
$819$	0	0
$820$	10.0000	0.349215
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	12.0000	0.418548
$823$	11.0000	0.383436	0.191718	−	0.981450i	$-0.438594\pi$
$823$	11.0000	0.383436	0.191718	+	0.981450i	$0.438594\pi$
$824$	− 15.0000i	− 0.522550i
$825$	− 3.00000i	− 0.104447i
$826$	− 36.0000i	− 1.25260i
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	−4.00000	−0.139010
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	16.0000i	0.555368i
$831$	−31.0000	−1.07538
$832$	0	0
$833$	0	0
$834$	17.0000i	0.588662i
$835$	−9.00000	−0.311458
$836$	9.00000	0.311272
$837$	6.00000i	0.207390i
$838$	− 12.0000i	− 0.414533i
$839$	− 6.00000i	− 0.207143i	−0.994622	−	0.103572i	$-0.966973\pi$
$839$	− 6.00000i	− 0.207143i	0.994622	−	0.103572i	$-0.0330271\pi$
$840$	− 3.00000i	− 0.103510i
$841$	−13.0000	−0.448276
$842$	−28.0000	−0.964944
$843$	30.0000i	1.03325i
$844$	−9.00000	−0.309793
$845$	0	0
$846$	3.00000	0.103142
$847$	6.00000i	0.206162i
$848$	9.00000	0.309061
$849$	6.00000	0.205919
$850$	0	0
$851$	36.0000i	1.23406i
$852$	14.0000i	0.479632i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	18.0000	0.615947
$855$	3.00000	0.102598
$856$	− 2.00000i	− 0.0683586i
$857$	50.0000	1.70797	0.853984	−	0.520300i	$-0.174180\pi$
$857$	50.0000	1.70797	0.853984	+	0.520300i	$0.174180\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	− 10.0000i	− 0.340997i
$861$	30.0000	1.02240
$862$	−12.0000	−0.408722
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 13.0000i	− 0.442013i
$866$	16.0000i	0.543702i
$867$	17.0000	0.577350
$868$	−18.0000	−0.610960
$869$	− 18.0000i	− 0.610608i
$870$	−4.00000	−0.135613
$871$	0	0
$872$	2.00000	0.0677285
$873$	− 8.00000i	− 0.270759i
$874$	12.0000	0.405906
$875$	3.00000	0.101419
$876$	− 8.00000i	− 0.270295i
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	30.0000i	1.01245i
$879$	1.00000i	0.0337292i
$880$	3.00000	0.101130
$881$	23.0000	0.774890	0.387445	−	0.921893i	$-0.373358\pi$
$881$	23.0000	0.774890	0.387445	+	0.921893i	$0.373358\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	− 10.0000i	− 0.335957i
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	−9.00000	−0.302020
$889$	− 9.00000i	− 0.301850i
$890$	3.00000i	0.100560i
$891$	− 3.00000i	− 0.100504i
$892$	− 11.0000i	− 0.368307i
$893$	−9.00000	−0.301174
$894$	2.00000	0.0668900
$895$	− 4.00000i	− 0.133705i
$896$	3.00000	0.100223
$897$	0	0
$898$	35.0000	1.16797
$899$	24.0000i	0.800445i
$900$	1.00000	0.0333333
$901$	0	0
$902$	30.0000i	0.998891i
$903$	− 30.0000i	− 0.998337i
$904$	8.00000i	0.266076i
$905$	10.0000i	0.332411i
$906$	14.0000	0.465119
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	− 16.0000i	− 0.530979i
$909$	0	0
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	3.00000i	0.0993399i
$913$	−48.0000	−1.58857
$914$	−42.0000	−1.38924
$915$	6.00000i	0.198354i
$916$	10.0000i	0.330409i
$917$	− 9.00000i	− 0.297206i
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	4.00000	0.131876
$921$	− 26.0000i	− 0.856729i
$922$	−32.0000	−1.05386
$923$	0	0
$924$	9.00000	0.296078
$925$	− 9.00000i	− 0.295918i
$926$	24.0000	0.788689
$927$	15.0000	0.492665
$928$	− 4.00000i	− 0.131306i
$929$	− 2.00000i	− 0.0656179i	−0.999462	−	0.0328089i	$-0.989555\pi$
$929$	− 2.00000i	− 0.0656179i	0.999462	−	0.0328089i	$-0.0104453\pi$
$930$	− 6.00000i	− 0.196748i
$931$	6.00000i	0.196642i
$932$	−6.00000	−0.196537
$933$	−4.00000	−0.130954
$934$	− 8.00000i	− 0.261768i
$935$	0	0
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	− 24.0000i	− 0.783628i
$939$	−26.0000	−0.848478
$940$	−3.00000	−0.0978492
$941$	32.0000i	1.04317i	0.853199	+	0.521585i	$0.174659\pi$
$941$	32.0000i	1.04317i	−0.853199	+	0.521585i	$0.825341\pi$
$942$	17.0000i	0.553890i
$943$	40.0000i	1.30258i
$944$	12.0000i	0.390567i
$945$	3.00000	0.0975900
$946$	30.0000	0.975384
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	6.00000	0.194871
$949$	0	0
$950$	−3.00000	−0.0973329
$951$	17.0000i	0.551263i
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	9.00000i	0.291386i
$955$	− 6.00000i	− 0.194155i
$956$	− 26.0000i	− 0.840900i
$957$	− 12.0000i	− 0.387905i
$958$	−12.0000	−0.387702
$959$	−36.0000	−1.16250
$960$	1.00000i	0.0322749i
$961$	−5.00000	−0.161290
$962$	0	0
$963$	2.00000	0.0644491
$964$	− 7.00000i	− 0.225455i
$965$	−8.00000	−0.257529
$966$	12.0000	0.386094
$967$	− 25.0000i	− 0.803946i	−0.915652	−	0.401973i	$-0.868325\pi$
$967$	− 25.0000i	− 0.803946i	0.915652	−	0.401973i	$-0.131675\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	8.00000i	0.256865i
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	1.00000	0.0320750
$973$	− 51.0000i	− 1.63498i
$974$	29.0000	0.929220
$975$	0	0
$976$	−6.00000	−0.192055
$977$	44.0000i	1.40768i	0.710356	+	0.703842i	$0.248534\pi$
$977$	44.0000i	1.40768i	−0.710356	+	0.703842i	$0.751466\pi$
$978$	20.0000	0.639529
$979$	−9.00000	−0.287641
$980$	2.00000i	0.0638877i
$981$	2.00000i	0.0638551i
$982$	− 5.00000i	− 0.159556i
$983$	55.0000i	1.75423i	0.480283	+	0.877114i	$0.340534\pi$
$983$	55.0000i	1.75423i	−0.480283	+	0.877114i	$0.659466\pi$
$984$	−10.0000	−0.318788
$985$	11.0000	0.350489
$986$	0	0
$987$	−9.00000	−0.286473
$988$	0	0
$989$	40.0000	1.27193
$990$	3.00000i	0.0953463i
$991$	58.0000	1.84243	0.921215	−	0.389053i	$-0.127198\pi$
$991$	58.0000	1.84243	0.921215	+	0.389053i	$0.127198\pi$
$992$	6.00000	0.190500
$993$	− 28.0000i	− 0.888553i
$994$	− 42.0000i	− 1.33216i
$995$	10.0000i	0.317021i
$996$	− 16.0000i	− 0.506979i
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	20.0000	0.633089
$999$	− 9.00000i	− 0.284747i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.e.61.1	✓	2	13.11	odd	12
390.2.i.e.211.1	yes	2	13.7	odd	12
1170.2.i.c.451.1		2	39.11	even	12
1170.2.i.c.991.1		2	39.20	even	12
1950.2.i.f.451.1		2	65.24	odd	12
1950.2.i.f.601.1		2	65.59	odd	12
1950.2.z.d.1699.1		4	65.63	even	12
1950.2.z.d.1699.2		4	65.37	even	12
1950.2.z.d.1849.1		4	65.7	even	12
1950.2.z.d.1849.2		4	65.33	even	12
5070.2.a.b.1.1		1	13.8	odd	4
5070.2.a.r.1.1		1	13.5	odd	4
5070.2.b.b.1351.1		2	13.12	even	2	inner
5070.2.b.b.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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +3.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -3.00000i q^{19} -1.00000i q^{20} -3.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{30} -6.00000i q^{31} +1.00000i q^{32} +3.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} +9.00000i q^{37} +3.00000 q^{38} +1.00000 q^{40} +10.0000i q^{41} +3.00000 q^{42} +10.0000 q^{43} +3.00000i q^{44} +1.00000i q^{45} +4.00000i q^{46} -3.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} -1.00000i q^{50} +9.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} +3.00000 q^{56} +3.00000i q^{57} -4.00000i q^{58} +12.0000i q^{59} +1.00000i q^{60} -6.00000 q^{61} +6.00000 q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +8.00000i q^{67} -4.00000 q^{69} -3.00000i q^{70} +14.0000i q^{71} -1.00000i q^{72} -8.00000i q^{73} -9.00000 q^{74} +1.00000 q^{75} +3.00000i q^{76} +9.00000 q^{77} +6.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -16.0000i q^{83} +3.00000i q^{84} +10.0000i q^{86} +4.00000 q^{87} -3.00000 q^{88} -3.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +6.00000i q^{93} +3.00000 q^{94} +3.00000 q^{95} -1.00000i q^{96} -8.00000i q^{97} -2.00000i q^{98} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 6 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 8 q^{29} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 6 q^{38} + 2 q^{40} + 6 q^{42} + 20 q^{43}+ \cdots + 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 - 6 * q^14 + 2 * q^16 + 6 * q^22 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 8 * q^29 + 2 * q^30 - 6 * q^35 - 2 * q^36 + 6 * q^38 + 2 * q^40 + 6 * q^42 + 20 * q^43 - 2 * q^48 - 4 * q^49 + 18 * q^53 + 6 * q^55 + 6 * q^56 - 12 * q^61 + 12 * q^62 - 2 * q^64 - 6 * q^66 - 8 * q^69 - 18 * q^74 + 2 * q^75 + 18 * q^77 + 12 * q^79 + 2 * q^81 - 20 * q^82 + 8 * q^87 - 6 * q^88 - 2 * q^90 - 8 * q^92 + 6 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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