LMFDB - Embedded newform 5070.2.b.e.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -8.00000 q^{17} -1.00000i q^{18} +6.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +4.00000 q^{22} -6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} -1.00000i q^{32} -4.00000i q^{33} +8.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +8.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} -4.00000 q^{55} -2.00000 q^{56} -6.00000i q^{57} +4.00000i q^{58} +4.00000i q^{59} +1.00000i q^{60} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000 q^{68} +6.00000 q^{69} +2.00000i q^{70} +8.00000i q^{71} +1.00000i q^{72} -8.00000i q^{73} -2.00000 q^{74} +1.00000 q^{75} -6.00000i q^{76} -8.00000 q^{77} +8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -8.00000i q^{85} -4.00000i q^{86} +4.00000 q^{87} -4.00000 q^{88} -14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} -6.00000 q^{95} +1.00000i q^{96} +16.0000i q^{97} -3.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 16 q^{17} + 8 q^{22} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 8 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 12 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} - 2 q^{48} + 6 q^{49} + 16 q^{51} - 20 q^{53} - 8 q^{55} - 4 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{68} + 12 q^{69} - 4 q^{74} + 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{82} + 8 q^{87} - 8 q^{88} + 2 q^{90} + 12 q^{92} - 12 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 - 16 * q^17 + 8 * q^22 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 8 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 + 12 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 - 2 * q^48 + 6 * q^49 + 16 * q^51 - 20 * q^53 - 8 * q^55 - 4 * q^56 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^68 + 12 * q^69 - 4 * q^74 + 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 4 * q^82 + 8 * q^87 - 8 * q^88 + 2 * q^90 + 12 * q^92 - 12 * 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	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	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
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	4.00000	0.852803
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 2.00000i	− 0.377964i
$29$	−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
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	8.00000i	1.37199i
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	6.00000	0.973329
$39$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	− 4.00000i	− 0.603023i
$45$	1.00000i	0.149071i
$46$	6.00000i	0.884652i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	8.00000	1.12022
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000i	0.136083i
$55$	−4.00000	−0.539360
$56$	−2.00000	−0.267261
$57$	− 6.00000i	− 0.794719i
$58$	4.00000i	0.525226i
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	1.00000i	0.129099i
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	8.00000	0.970143
$69$	6.00000	0.722315
$70$	2.00000i	0.239046i
$71$	8.00000i	0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$	8.00000i	0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$	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	$-0.155085\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	− 6.00000i	− 0.688247i
$77$	−8.00000	−0.911685
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	2.00000i	0.218218i
$85$	− 8.00000i	− 0.867722i
$86$	− 4.00000i	− 0.431331i
$87$	4.00000	0.428845
$88$	−4.00000	−0.426401
$89$	− 14.0000i	− 1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$	− 14.0000i	− 1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	1.00000i	0.102062i
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 3.00000i	− 0.303046i
$99$	4.00000i	0.402015i
$100$	1.00000	0.100000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	− 8.00000i	− 0.792118i
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	10.0000i	0.971286i
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	− 12.0000i	− 1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$	− 12.0000i	− 1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$	4.00000i	0.381385i
$111$	2.00000i	0.189832i
$112$	2.00000i	0.188982i
$113$	20.0000	1.88144	0.940721	−	0.339182i	$-0.110150\pi$
$113$	20.0000	1.88144	0.940721	+	0.339182i	$0.110150\pi$
$114$	−6.00000	−0.561951
$115$	− 6.00000i	− 0.559503i
$116$	4.00000	0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	− 16.0000i	− 1.46672i
$120$	1.00000	0.0912871
$121$	−5.00000	−0.454545
$122$	10.0000i	0.905357i
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	2.00000	0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	4.00000i	0.348155i
$133$	−12.0000	−1.04053
$134$	−12.0000	−1.03664
$135$	− 1.00000i	− 0.0860663i
$136$	− 8.00000i	− 0.685994i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 6.00000i	− 0.510754i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 4.00000i	− 0.332182i
$146$	−8.00000	−0.662085
$147$	−3.00000	−0.247436
$148$	2.00000i	0.164399i
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−6.00000	−0.486664
$153$	−8.00000	−0.646762
$154$	8.00000i	0.644658i
$155$	0	0
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	− 8.00000i	− 0.636446i
$159$	10.0000	0.793052
$160$	1.00000	0.0790569
$161$	− 12.0000i	− 0.945732i
$162$	− 1.00000i	− 0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	− 2.00000i	− 0.156174i
$165$	4.00000	0.311400
$166$	−12.0000	−0.931381
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	−8.00000	−0.613572
$171$	6.00000i	0.458831i
$172$	−4.00000	−0.304997
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	− 4.00000i	− 0.303239i
$175$	− 2.00000i	− 0.151186i
$176$	4.00000i	0.301511i
$177$	− 4.00000i	− 0.300658i
$178$	−14.0000	−1.04934
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	− 6.00000i	− 0.442326i
$185$	2.00000	0.147043
$186$	0	0
$187$	− 32.0000i	− 2.34007i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	6.00000i	0.435286i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	12.0000i	0.846415i
$202$	− 16.0000i	− 1.12576i
$203$	− 8.00000i	− 0.561490i
$204$	−8.00000	−0.560112
$205$	−2.00000	−0.139686
$206$	− 12.0000i	− 0.836080i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−24.0000	−1.66011
$210$	− 2.00000i	− 0.138013i
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000	0.686803
$213$	− 8.00000i	− 0.548151i
$214$	− 12.0000i	− 0.820303i
$215$	4.00000i	0.272798i
$216$	− 1.00000i	− 0.0680414i
$217$	0	0
$218$	−12.0000	−0.812743
$219$	8.00000i	0.540590i
$220$	4.00000	0.269680
$221$	0	0
$222$	2.00000	0.134231
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	− 20.0000i	− 1.33038i
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	6.00000i	0.397360i
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	−6.00000	−0.395628
$231$	8.00000	0.526361
$232$	− 4.00000i	− 0.262613i
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	− 4.00000i	− 0.260378i
$237$	−8.00000	−0.519656
$238$	−16.0000	−1.03713
$239$	− 16.0000i	− 1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$	− 16.0000i	− 1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	2.00000i	0.128831i	0.997923	+	0.0644157i	$0.0205183\pi$
$241$	2.00000i	0.128831i	−0.997923	+	0.0644157i	$0.979482\pi$
$242$	5.00000i	0.321412i
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	3.00000i	0.191663i
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	12.0000i	0.760469i
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 24.0000i	− 1.50887i
$254$	4.00000i	0.250982i
$255$	8.00000i	0.500979i
$256$	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	$0.622107\pi$
$258$	4.00000i	0.249029i
$259$	4.00000	0.248548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	− 10.0000i	− 0.617802i
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	4.00000	0.246183
$265$	− 10.0000i	− 0.614295i
$266$	12.0000i	0.735767i
$267$	14.0000i	0.856786i
$268$	12.0000i	0.733017i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−1.00000	−0.0608581
$271$	16.0000i	0.971931i	0.873978	+	0.485965i	$0.161532\pi$
$271$	16.0000i	0.971931i	−0.873978	+	0.485965i	$0.838468\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	−6.00000	−0.362473
$275$	− 4.00000i	− 0.241209i
$276$	−6.00000	−0.361158
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	− 2.00000i	− 0.119523i
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	− 8.00000i	− 0.474713i
$285$	6.00000	0.355409
$286$	0	0
$287$	−4.00000	−0.236113
$288$	− 1.00000i	− 0.0589256i
$289$	47.0000	2.76471
$290$	−4.00000	−0.234888
$291$	− 16.0000i	− 0.937937i
$292$	8.00000i	0.468165i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	3.00000i	0.174964i
$295$	−4.00000	−0.232889
$296$	2.00000	0.116248
$297$	− 4.00000i	− 0.232104i
$298$	10.0000	0.579284
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.00000i	0.461112i
$302$	0	0
$303$	−16.0000	−0.919176
$304$	6.00000i	0.344124i
$305$	− 10.0000i	− 0.572598i
$306$	8.00000i	0.457330i
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	8.00000	0.455842
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	− 22.0000i	− 1.24153i
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	− 10.0000i	− 0.560772i
$319$	− 16.0000i	− 0.895828i
$320$	− 1.00000i	− 0.0559017i
$321$	−12.0000	−0.669775
$322$	−12.0000	−0.668734
$323$	− 48.0000i	− 2.67079i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	12.0000i	0.663602i
$328$	−2.00000	−0.110432
$329$	0	0
$330$	− 4.00000i	− 0.220193i
$331$	− 10.0000i	− 0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$	− 10.0000i	− 0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	12.0000i	0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	−4.00000	−0.218870
$335$	12.0000	0.655630
$336$	− 2.00000i	− 0.109109i
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	8.00000i	0.433861i
$341$	0	0
$342$	6.00000	0.324443
$343$	20.0000i	1.07990i
$344$	4.00000i	0.215666i
$345$	6.00000i	0.323029i
$346$	22.0000i	1.18273i
$347$	−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
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−4.00000	−0.212598
$355$	−8.00000	−0.424596
$356$	14.0000i	0.741999i
$357$	16.0000i	0.846810i
$358$	− 10.0000i	− 0.528516i
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	−1.00000	−0.0527046
$361$	−17.0000	−0.894737
$362$	10.0000i	0.525588i
$363$	5.00000	0.262432
$364$	0	0
$365$	8.00000	0.418739
$366$	− 10.0000i	− 0.522708i
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	−6.00000	−0.312772
$369$	2.00000i	0.104116i
$370$	− 2.00000i	− 0.103975i
$371$	− 20.0000i	− 1.03835i
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	−32.0000	−1.65468
$375$	1.00000i	0.0516398i
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	18.0000i	0.924598i	0.886724	+	0.462299i	$0.152975\pi$
$379$	18.0000i	0.924598i	−0.886724	+	0.462299i	$0.847025\pi$
$380$	6.00000	0.307794
$381$	4.00000	0.204926
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 8.00000i	− 0.407718i
$386$	−4.00000	−0.203595
$387$	4.00000	0.203331
$388$	− 16.0000i	− 0.812277i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	3.00000i	0.151523i
$393$	−10.0000	−0.504433
$394$	18.0000	0.906827
$395$	8.00000i	0.402524i
$396$	− 4.00000i	− 0.201008i
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	−1.00000	−0.0500000
$401$	30.0000i	1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$	30.0000i	1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	−16.0000	−0.796030
$405$	1.00000i	0.0496904i
$406$	−8.00000	−0.397033
$407$	8.00000	0.396545
$408$	8.00000i	0.396059i
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	2.00000i	0.0987730i
$411$	6.00000i	0.295958i
$412$	−12.0000	−0.591198
$413$	−8.00000	−0.393654
$414$	6.00000i	0.294884i
$415$	12.0000	0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	24.0000i	1.17388i
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	−2.00000	−0.0975900
$421$	8.00000i	0.389896i	0.980814	+	0.194948i	$0.0624538\pi$
$421$	8.00000i	0.389896i	−0.980814	+	0.194948i	$0.937546\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	− 10.0000i	− 0.485643i
$425$	8.00000	0.388057
$426$	−8.00000	−0.387601
$427$	− 20.0000i	− 0.967868i
$428$	−12.0000	−0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	12.0000i	0.574696i
$437$	− 36.0000i	− 1.72211i
$438$	8.00000	0.382255
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	− 4.00000i	− 0.190693i
$441$	3.00000	0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	− 2.00000i	− 0.0949158i
$445$	14.0000	0.663664
$446$	2.00000	0.0947027
$447$	− 10.0000i	− 0.472984i
$448$	− 2.00000i	− 0.0944911i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	1.00000i	0.0471405i
$451$	−8.00000	−0.376705
$452$	−20.0000	−0.940721
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	6.00000	0.280976
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	4.00000	0.186908
$459$	8.00000	0.373408
$460$	6.00000i	0.279751i
$461$	6.00000i	0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$	6.00000i	0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$	− 8.00000i	− 0.372194i
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	24.0000i	1.11178i
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	−22.0000	−1.01371
$472$	−4.00000	−0.184115
$473$	16.0000i	0.735681i
$474$	8.00000i	0.367452i
$475$	− 6.00000i	− 0.275299i
$476$	16.0000i	0.733359i
$477$	−10.0000	−0.457869
$478$	−16.0000	−0.731823
$479$	32.0000i	1.46212i	0.682315	+	0.731059i	$0.260973\pi$
$479$	32.0000i	1.46212i	−0.682315	+	0.731059i	$0.739027\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	2.00000	0.0910975
$483$	12.0000i	0.546019i
$484$	5.00000	0.227273
$485$	−16.0000	−0.726523
$486$	1.00000i	0.0453609i
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	− 10.0000i	− 0.452679i
$489$	16.0000i	0.723545i
$490$	3.00000	0.135526
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	2.00000i	0.0901670i
$493$	32.0000	1.44121
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−16.0000	−0.717698
$498$	12.0000	0.537733
$499$	− 38.0000i	− 1.70111i	−0.525883	−	0.850557i	$-0.676265\pi$
$499$	− 38.0000i	− 1.70111i	0.525883	−	0.850557i	$-0.323735\pi$
$500$	1.00000i	0.0447214i
$501$	4.00000i	0.178707i
$502$	6.00000i	0.267793i
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	−2.00000	−0.0890871
$505$	16.0000i	0.711991i
$506$	−24.0000	−1.06693
$507$	0	0
$508$	4.00000	0.177471
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	8.00000	0.354246
$511$	16.0000	0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	− 6.00000i	− 0.264906i
$514$	12.0000i	0.529297i
$515$	12.0000i	0.528783i
$516$	4.00000	0.176090
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	22.0000	0.965693
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	4.00000i	0.175075i
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−10.0000	−0.436852
$525$	2.00000i	0.0872872i
$526$	− 2.00000i	− 0.0872041i
$527$	0	0
$528$	− 4.00000i	− 0.174078i
$529$	13.0000	0.565217
$530$	−10.0000	−0.434372
$531$	4.00000i	0.173585i
$532$	12.0000	0.520266
$533$	0	0
$534$	14.0000	0.605839
$535$	12.0000i	0.518805i
$536$	12.0000	0.518321
$537$	−10.0000	−0.431532
$538$	24.0000i	1.03471i
$539$	12.0000i	0.516877i
$540$	1.00000i	0.0430331i
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	16.0000	0.687259
$543$	10.0000	0.429141
$544$	8.00000i	0.342997i
$545$	12.0000	0.514024
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	6.00000i	0.256307i
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	− 24.0000i	− 1.02243i
$552$	6.00000i	0.255377i
$553$	16.0000i	0.680389i
$554$	− 10.0000i	− 0.424859i
$555$	−2.00000	−0.0848953
$556$	8.00000	0.339276
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	32.0000i	1.35104i
$562$	−6.00000	−0.253095
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	20.0000i	0.841406i
$566$	− 4.00000i	− 0.168133i
$567$	2.00000i	0.0839921i
$568$	−8.00000	−0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	− 6.00000i	− 0.251312i
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	4.00000i	0.166957i
$575$	6.00000	0.250217
$576$	−1.00000	−0.0416667
$577$	− 12.0000i	− 0.499567i	−0.968302	−	0.249783i	$-0.919641\pi$
$577$	− 12.0000i	− 0.499567i	0.968302	−	0.249783i	$-0.0803594\pi$
$578$	− 47.0000i	− 1.95494i
$579$	4.00000i	0.166234i
$580$	4.00000i	0.166091i
$581$	24.0000	0.995688
$582$	−16.0000	−0.663221
$583$	− 40.0000i	− 1.65663i
$584$	8.00000	0.331042
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	4.00000i	0.164677i
$591$	− 18.0000i	− 0.740421i
$592$	− 2.00000i	− 0.0821995i
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	−4.00000	−0.164122
$595$	16.0000	0.655936
$596$	− 10.0000i	− 0.409616i
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000i	0.0408248i
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	8.00000	0.326056
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	16.0000i	0.649956i
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	6.00000	0.243332
$609$	8.00000i	0.324176i
$610$	−10.0000	−0.404888
$611$	0	0
$612$	8.00000	0.323381
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	12.0000	0.484281
$615$	2.00000	0.0806478
$616$	− 8.00000i	− 0.322329i
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	12.0000i	0.482711i
$619$	− 10.0000i	− 0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$	− 10.0000i	− 0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	24.0000i	0.962312i
$623$	28.0000	1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 6.00000i	− 0.239808i
$627$	24.0000	0.958468
$628$	−22.0000	−0.877896
$629$	16.0000i	0.637962i
$630$	2.00000i	0.0796819i
$631$	12.0000i	0.477712i	0.971055	+	0.238856i	$0.0767725\pi$
$631$	12.0000i	0.477712i	−0.971055	+	0.238856i	$0.923228\pi$
$632$	8.00000i	0.318223i
$633$	20.0000	0.794929
$634$	30.0000	1.19145
$635$	− 4.00000i	− 0.158735i
$636$	−10.0000	−0.396526
$637$	0	0
$638$	−16.0000	−0.633446
$639$	8.00000i	0.316475i
$640$	−1.00000	−0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	12.0000i	0.473602i
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	12.0000i	0.472866i
$645$	− 4.00000i	− 0.157500i
$646$	−48.0000	−1.88853
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	1.00000i	0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	12.0000	0.469237
$655$	10.0000i	0.390732i
$656$	2.00000i	0.0780869i
$657$	− 8.00000i	− 0.312110i
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	−4.00000	−0.155700
$661$	48.0000i	1.86698i	0.358599	+	0.933492i	$0.383255\pi$
$661$	48.0000i	1.86698i	−0.358599	+	0.933492i	$0.616745\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	− 12.0000i	− 0.465340i
$666$	−2.00000	−0.0774984
$667$	24.0000	0.929284
$668$	4.00000i	0.154765i
$669$	− 2.00000i	− 0.0773245i
$670$	− 12.0000i	− 0.463600i
$671$	− 40.0000i	− 1.54418i
$672$	−2.00000	−0.0771517
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	14.0000i	0.539260i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	20.0000i	0.768095i
$679$	−32.0000	−1.22805
$680$	8.00000	0.306786
$681$	− 4.00000i	− 0.153280i
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	− 6.00000i	− 0.229416i
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	− 4.00000i	− 0.152610i
$688$	4.00000	0.152499
$689$	0	0
$690$	6.00000	0.228416
$691$	14.0000i	0.532585i	0.963892	+	0.266293i	$0.0857987\pi$
$691$	14.0000i	0.532585i	−0.963892	+	0.266293i	$0.914201\pi$
$692$	22.0000	0.836315
$693$	−8.00000	−0.303895
$694$	24.0000i	0.911028i
$695$	− 8.00000i	− 0.303457i
$696$	4.00000i	0.151620i
$697$	− 16.0000i	− 0.606043i
$698$	−28.0000	−1.05982
$699$	24.0000	0.907763
$700$	2.00000i	0.0755929i
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	− 4.00000i	− 0.150756i
$705$	0	0
$706$	14.0000	0.526897
$707$	32.0000i	1.20348i
$708$	4.00000i	0.150329i
$709$	− 4.00000i	− 0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$	− 4.00000i	− 0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$	8.00000i	0.300235i
$711$	8.00000	0.300023
$712$	14.0000	0.524672
$713$	0	0
$714$	16.0000	0.598785
$715$	0	0
$716$	−10.0000	−0.373718
$717$	16.0000i	0.597531i
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	1.00000i	0.0372678i
$721$	24.0000i	0.893807i
$722$	17.0000i	0.632674i
$723$	− 2.00000i	− 0.0743808i
$724$	10.0000	0.371647
$725$	4.00000	0.148556
$726$	− 5.00000i	− 0.185567i
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 8.00000i	− 0.296093i
$731$	−32.0000	−1.18356
$732$	−10.0000	−0.369611
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	36.0000i	1.32878i
$735$	− 3.00000i	− 0.110657i
$736$	6.00000i	0.221163i
$737$	48.0000	1.76810
$738$	2.00000	0.0736210
$739$	30.0000i	1.10357i	0.833987	+	0.551784i	$0.186053\pi$
$739$	30.0000i	1.10357i	−0.833987	+	0.551784i	$0.813947\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−20.0000	−0.734223
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	2.00000i	0.0732252i
$747$	− 12.0000i	− 0.439057i
$748$	32.0000i	1.17004i
$749$	24.0000i	0.876941i
$750$	1.00000	0.0365148
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	0	0
$756$	2.00000i	0.0727393i
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	18.0000	0.653789
$759$	24.0000i	0.871145i
$760$	− 6.00000i	− 0.217643i
$761$	22.0000i	0.797499i	0.917060	+	0.398750i	$0.130556\pi$
$761$	22.0000i	0.797499i	−0.917060	+	0.398750i	$0.869444\pi$
$762$	− 4.00000i	− 0.144905i
$763$	24.0000	0.868858
$764$	0	0
$765$	− 8.00000i	− 0.289241i
$766$	12.0000	0.433578
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 10.0000i	− 0.360609i	−0.983611	−	0.180305i	$-0.942292\pi$
$769$	− 10.0000i	− 0.360609i	0.983611	−	0.180305i	$-0.0577084\pi$
$770$	−8.00000	−0.288300
$771$	12.0000	0.432169
$772$	4.00000i	0.143963i
$773$	− 38.0000i	− 1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$	− 38.0000i	− 1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$	− 4.00000i	− 0.143777i
$775$	0	0
$776$	−16.0000	−0.574367
$777$	−4.00000	−0.143499
$778$	8.00000i	0.286814i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 48.0000i	− 1.71648i
$783$	4.00000	0.142948
$784$	3.00000	0.107143
$785$	22.0000i	0.785214i
$786$	10.0000i	0.356688i
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−2.00000	−0.0712019
$790$	8.00000	0.284627
$791$	40.0000i	1.42224i
$792$	−4.00000	−0.142134
$793$	0	0
$794$	14.0000	0.496841
$795$	10.0000i	0.354663i
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	− 12.0000i	− 0.424795i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	− 14.0000i	− 0.494666i
$802$	30.0000	1.05934
$803$	32.0000	1.12926
$804$	− 12.0000i	− 0.423207i
$805$	12.0000	0.422944
$806$	0	0
$807$	24.0000	0.844840
$808$	16.0000i	0.562878i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	1.00000	0.0351364
$811$	38.0000i	1.33436i	0.744896	+	0.667180i	$0.232499\pi$
$811$	38.0000i	1.33436i	−0.744896	+	0.667180i	$0.767501\pi$
$812$	8.00000i	0.280745i
$813$	− 16.0000i	− 0.561144i
$814$	− 8.00000i	− 0.280400i
$815$	16.0000	0.560456
$816$	8.00000	0.280056
$817$	24.0000i	0.839654i
$818$	26.0000	0.909069
$819$	0	0
$820$	2.00000	0.0698430
$821$	10.0000i	0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$	10.0000i	0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$	6.00000	0.209274
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	12.0000i	0.418040i
$825$	4.00000i	0.139262i
$826$	8.00000i	0.278356i
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	6.00000	0.208514
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	− 12.0000i	− 0.416526i
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−24.0000	−0.831551
$834$	− 8.00000i	− 0.277017i
$835$	4.00000	0.138426
$836$	24.0000	0.830057
$837$	0	0
$838$	− 10.0000i	− 0.345444i
$839$	− 40.0000i	− 1.38095i	−0.723355	−	0.690477i	$-0.757401\pi$
$839$	− 40.0000i	− 1.38095i	0.723355	−	0.690477i	$-0.242599\pi$
$840$	2.00000i	0.0690066i
$841$	−13.0000	−0.448276
$842$	8.00000	0.275698
$843$	6.00000i	0.206651i
$844$	20.0000	0.688428
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	−10.0000	−0.343401
$849$	−4.00000	−0.137280
$850$	− 8.00000i	− 0.274398i
$851$	12.0000i	0.411355i
$852$	8.00000i	0.274075i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−20.0000	−0.684386
$855$	−6.00000	−0.205196
$856$	12.0000i	0.410152i
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	− 4.00000i	− 0.136399i
$861$	4.00000	0.136320
$862$	0	0
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	1.00000i	0.0340207i
$865$	− 22.0000i	− 0.748022i
$866$	− 2.00000i	− 0.0679628i
$867$	−47.0000	−1.59620
$868$	0	0
$869$	32.0000i	1.08553i
$870$	4.00000	0.135613
$871$	0	0
$872$	12.0000	0.406371
$873$	16.0000i	0.541518i
$874$	−36.0000	−1.21772
$875$	2.00000	0.0676123
$876$	− 8.00000i	− 0.270295i
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 32.0000i	− 1.07995i
$879$	26.0000i	0.876958i
$880$	−4.00000	−0.134840
$881$	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
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	− 16.0000i	− 0.537531i
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	−2.00000	−0.0671156
$889$	− 8.00000i	− 0.268311i
$890$	− 14.0000i	− 0.469281i
$891$	4.00000i	0.134005i
$892$	− 2.00000i	− 0.0669650i
$893$	0	0
$894$	−10.0000	−0.334450
$895$	10.0000i	0.334263i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	1.00000	0.0333333
$901$	80.0000	2.66519
$902$	8.00000i	0.266371i
$903$	− 8.00000i	− 0.266223i
$904$	20.0000i	0.665190i
$905$	− 10.0000i	− 0.332411i
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	− 4.00000i	− 0.132745i
$909$	16.0000	0.530687
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	− 6.00000i	− 0.198680i
$913$	48.0000	1.58857
$914$	−8.00000	−0.264616
$915$	10.0000i	0.330590i
$916$	− 4.00000i	− 0.132164i
$917$	20.0000i	0.660458i
$918$	− 8.00000i	− 0.264039i
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	6.00000	0.197814
$921$	− 12.0000i	− 0.395413i
$922$	6.00000	0.197599
$923$	0	0
$924$	−8.00000	−0.263181
$925$	2.00000i	0.0657596i
$926$	−26.0000	−0.854413
$927$	12.0000	0.394132
$928$	4.00000i	0.131306i
$929$	30.0000i	0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$	30.0000i	0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$	0	0
$931$	18.0000i	0.589926i
$932$	24.0000	0.786146
$933$	24.0000	0.785725
$934$	28.0000i	0.916188i
$935$	32.0000	1.04651
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	− 24.0000i	− 0.783628i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	34.0000i	1.10837i	0.832394	+	0.554184i	$0.186970\pi$
$941$	34.0000i	1.10837i	−0.832394	+	0.554184i	$0.813030\pi$
$942$	22.0000i	0.716799i
$943$	− 12.0000i	− 0.390774i
$944$	4.00000i	0.130189i
$945$	2.00000	0.0650600
$946$	16.0000	0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	−6.00000	−0.194666
$951$	− 30.0000i	− 0.972817i
$952$	16.0000	0.518563
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	10.0000i	0.323762i
$955$	0	0
$956$	16.0000i	0.517477i
$957$	16.0000i	0.517207i
$958$	32.0000	1.03387
$959$	12.0000	0.387500
$960$	1.00000i	0.0322749i
$961$	31.0000	1.00000
$962$	0	0
$963$	12.0000	0.386695
$964$	− 2.00000i	− 0.0644157i
$965$	4.00000	0.128765
$966$	12.0000	0.386094
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	− 5.00000i	− 0.160706i
$969$	48.0000i	1.54198i
$970$	16.0000i	0.513729i
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	1.00000	0.0320750
$973$	− 16.0000i	− 0.512936i
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	16.0000	0.511624
$979$	56.0000	1.78977
$980$	− 3.00000i	− 0.0958315i
$981$	− 12.0000i	− 0.383131i
$982$	42.0000i	1.34027i
$983$	− 52.0000i	− 1.65854i	−0.558846	−	0.829271i	$-0.688756\pi$
$983$	− 52.0000i	− 1.65854i	0.558846	−	0.829271i	$-0.311244\pi$
$984$	2.00000	0.0637577
$985$	−18.0000	−0.573528
$986$	− 32.0000i	− 1.01909i
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	4.00000i	0.127128i
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	10.0000i	0.317340i
$994$	16.0000i	0.507489i
$995$	0	0
$996$	− 12.0000i	− 0.380235i
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−38.0000	−1.20287
$999$	2.00000i	0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.e.1351.1		2
13.5	odd	4		5070.2.a.e.1.1		1
13.8	odd	4		390.2.a.e.1.1	✓	1
13.12	even	2	inner	5070.2.b.e.1351.2		2
39.8	even	4		1170.2.a.e.1.1		1
52.47	even	4		3120.2.a.o.1.1		1
65.8	even	4		1950.2.e.f.1249.1		2
65.34	odd	4		1950.2.a.h.1.1		1
65.47	even	4		1950.2.e.f.1249.2		2
156.47	odd	4		9360.2.a.bh.1.1		1
195.8	odd	4		5850.2.e.i.5149.2		2
195.47	odd	4		5850.2.e.i.5149.1		2
195.164	even	4		5850.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.e.1.1	✓	1	13.8	odd	4
1170.2.a.e.1.1		1	39.8	even	4
1950.2.a.h.1.1		1	65.34	odd	4
1950.2.e.f.1249.1		2	65.8	even	4
1950.2.e.f.1249.2		2	65.47	even	4
3120.2.a.o.1.1		1	52.47	even	4
5070.2.a.e.1.1		1	13.5	odd	4
5070.2.b.e.1351.1		2	1.1	even	1	trivial
5070.2.b.e.1351.2		2	13.12	even	2	inner
5850.2.a.bi.1.1		1	195.164	even	4
5850.2.e.i.5149.1		2	195.47	odd	4
5850.2.e.i.5149.2		2	195.8	odd	4
9360.2.a.bh.1.1		1	156.47	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -8.00000 q^{17} -1.00000i q^{18} +6.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +4.00000 q^{22} -6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} -1.00000i q^{32} -4.00000i q^{33} +8.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +6.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000i q^{44} +1.00000i q^{45} +6.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +8.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} -4.00000 q^{55} -2.00000 q^{56} -6.00000i q^{57} +4.00000i q^{58} +4.00000i q^{59} +1.00000i q^{60} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000 q^{68} +6.00000 q^{69} +2.00000i q^{70} +8.00000i q^{71} +1.00000i q^{72} -8.00000i q^{73} -2.00000 q^{74} +1.00000 q^{75} -6.00000i q^{76} -8.00000 q^{77} +8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -8.00000i q^{85} -4.00000i q^{86} +4.00000 q^{87} -4.00000 q^{88} -14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} -6.00000 q^{95} +1.00000i q^{96} +16.0000i q^{97} -3.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 16 q^{17} + 8 q^{22} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 8 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 12 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} - 2 q^{48} + 6 q^{49} + 16 q^{51} - 20 q^{53} - 8 q^{55} - 4 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{68} + 12 q^{69} - 4 q^{74} + 2 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{82} + 8 q^{87} - 8 q^{88} + 2 q^{90} + 12 q^{92} - 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 16 * q^17 + 8 * q^22 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 8 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 + 12 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 - 2 * q^48 + 6 * q^49 + 16 * q^51 - 20 * q^53 - 8 * q^55 - 4 * q^56 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^68 + 12 * q^69 - 4 * q^74 + 2 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 + 4 * q^82 + 8 * q^87 - 8 * q^88 + 2 * q^90 + 12 * q^92 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more