LMFDB - Embedded newform 5070.2.b.f.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.f.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -2.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} -4.00000 q^{22} -2.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.00000i q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -6.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} +4.00000i q^{44} +1.00000i q^{45} +2.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +6.00000 q^{53} +1.00000i q^{54} +4.00000 q^{55} -2.00000 q^{56} +2.00000i q^{57} -8.00000i q^{58} +12.0000i q^{59} +1.00000i q^{60} -2.00000 q^{61} +4.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} +4.00000 q^{68} +2.00000 q^{69} +2.00000i q^{70} +1.00000i q^{72} -6.00000 q^{74} +1.00000 q^{75} +2.00000i q^{76} +8.00000 q^{77} -8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -8.00000 q^{87} +4.00000 q^{88} +10.0000i q^{89} +1.00000 q^{90} +2.00000 q^{92} -4.00000i q^{93} +2.00000 q^{95} +1.00000i q^{96} -8.00000i q^{97} -3.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{17} - 8 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{27} + 16 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 4 q^{38} - 2 q^{40} - 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} + 8 q^{51} + 12 q^{53} + 8 q^{55} - 4 q^{56} - 4 q^{61} + 8 q^{62} - 2 q^{64} + 8 q^{66} + 8 q^{68} + 4 q^{69} - 12 q^{74} + 2 q^{75} + 16 q^{77} - 16 q^{79} + 2 q^{81} + 20 q^{82} - 16 q^{87} + 8 q^{88} + 2 q^{90} + 4 q^{92} + 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 8 * q^17 - 8 * q^22 - 4 * q^23 - 2 * q^25 - 2 * q^27 + 16 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 4 * q^38 - 2 * q^40 - 4 * q^42 - 8 * q^43 - 2 * q^48 + 6 * q^49 + 8 * q^51 + 12 * q^53 + 8 * q^55 - 4 * q^56 - 4 * q^61 + 8 * q^62 - 2 * q^64 + 8 * q^66 + 8 * q^68 + 4 * q^69 - 12 * q^74 + 2 * q^75 + 16 * q^77 - 16 * q^79 + 2 * q^81 + 20 * q^82 - 16 * q^87 + 8 * q^88 + 2 * q^90 + 4 * q^92 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.00000i	0.408248i
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	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	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	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
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	−4.00000	−0.852803
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 2.00000i	− 0.377964i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	−1.00000	−0.182574
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	4.00000i	0.685994i
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	−2.00000	−0.324443
$39$	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	$0.714777\pi$
$42$	−2.00000	−0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000i	0.603023i
$45$	1.00000i	0.149071i
$46$	2.00000i	0.294884i
$47$	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
$51$	4.00000	0.560112
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000i	0.136083i
$55$	4.00000	0.539360
$56$	−2.00000	−0.267261
$57$	2.00000i	0.264906i
$58$	− 8.00000i	− 1.05045i
$59$	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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	4.00000	0.485071
$69$	2.00000	0.240772
$70$	2.00000i	0.239046i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000i	0.117851i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	2.00000i	0.229416i
$77$	8.00000	0.911685
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	− 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
$85$	− 4.00000i	− 0.433861i
$86$	4.00000i	0.431331i
$87$	−8.00000	−0.857690
$88$	4.00000	0.426401
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	2.00000	0.208514
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	2.00000	0.205196
$96$	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	$-0.133125\pi$
$98$	− 3.00000i	− 0.303046i
$99$	− 4.00000i	− 0.402015i
$100$	1.00000	0.100000
$101$	−20.0000	−1.99007	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−20.0000	−1.99007	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	− 4.00000i	− 0.396059i
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	− 6.00000i	− 0.582772i
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	− 4.00000i	− 0.381385i
$111$	6.00000i	0.569495i
$112$	2.00000i	0.188982i
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	2.00000	0.187317
$115$	− 2.00000i	− 0.186501i
$116$	−8.00000	−0.742781
$117$	0	0
$118$	12.0000	1.10469
$119$	− 8.00000i	− 0.733359i
$120$	1.00000	0.0912871
$121$	−5.00000	−0.454545
$122$	2.00000i	0.181071i
$123$	− 10.0000i	− 0.901670i
$124$	− 4.00000i	− 0.359211i
$125$	− 1.00000i	− 0.0894427i
$126$	2.00000	0.178174
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	− 4.00000i	− 0.348155i
$133$	4.00000	0.346844
$134$	−8.00000	−0.691095
$135$	− 1.00000i	− 0.0860663i
$136$	− 4.00000i	− 0.342997i
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	− 2.00000i	− 0.170251i
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	8.00000i	0.664364i
$146$	0	0
$147$	−3.00000	−0.247436
$148$	6.00000i	0.493197i
$149$	− 22.0000i	− 1.80231i	−0.433497	−	0.901155i	$-0.642720\pi$
$149$	− 22.0000i	− 1.80231i	0.433497	−	0.901155i	$-0.357280\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	4.00000i	0.325515i	0.986666	+	0.162758i	$0.0520389\pi$
$151$	4.00000i	0.325515i	−0.986666	+	0.162758i	$0.947961\pi$
$152$	2.00000	0.162221
$153$	−4.00000	−0.323381
$154$	− 8.00000i	− 0.644658i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000i	0.636446i
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	− 4.00000i	− 0.315244i
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	− 10.0000i	− 0.780869i
$165$	−4.00000	−0.311400
$166$	−12.0000	−0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	−4.00000	−0.306786
$171$	− 2.00000i	− 0.152944i
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	8.00000i	0.606478i
$175$	− 2.00000i	− 0.151186i
$176$	− 4.00000i	− 0.301511i
$177$	− 12.0000i	− 0.901975i
$178$	10.0000	0.749532
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	− 2.00000i	− 0.147442i
$185$	6.00000	0.441129
$186$	−4.00000	−0.293294
$187$	16.0000i	1.17004i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	− 2.00000i	− 0.145095i
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	− 12.0000i	− 0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$	− 12.0000i	− 0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	8.00000i	0.564276i
$202$	20.0000i	1.40720i
$203$	16.0000i	1.12298i
$204$	−4.00000	−0.280056
$205$	−10.0000	−0.698430
$206$	4.00000i	0.278693i
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−8.00000	−0.553372
$210$	− 2.00000i	− 0.138013i
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	− 4.00000i	− 0.273434i
$215$	− 4.00000i	− 0.272798i
$216$	− 1.00000i	− 0.0680414i
$217$	−8.00000	−0.543075
$218$	4.00000	0.270914
$219$	0	0
$220$	−4.00000	−0.269680
$221$	0	0
$222$	6.00000	0.402694
$223$	− 22.0000i	− 1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$	− 22.0000i	− 1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	16.0000i	1.06430i
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 2.00000i	− 0.132453i
$229$	− 28.0000i	− 1.85029i	−0.379611	−	0.925146i	$-0.623942\pi$
$229$	− 28.0000i	− 1.85029i	0.379611	−	0.925146i	$-0.376058\pi$
$230$	−2.00000	−0.131876
$231$	−8.00000	−0.526361
$232$	8.00000i	0.525226i
$233$	−28.0000	−1.83434	−0.917170	−	0.398495i	$-0.869533\pi$
$233$	−28.0000	−1.83434	−0.917170	+	0.398495i	$0.869533\pi$
$234$	0	0
$235$	0	0
$236$	− 12.0000i	− 0.781133i
$237$	8.00000	0.519656
$238$	−8.00000	−0.518563
$239$	− 8.00000i	− 0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$	− 8.00000i	− 0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	22.0000i	1.41714i	0.705638	+	0.708572i	$0.250660\pi$
$241$	22.0000i	1.41714i	−0.705638	+	0.708572i	$0.749340\pi$
$242$	5.00000i	0.321412i
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	3.00000i	0.191663i
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−4.00000	−0.254000
$249$	12.0000i	0.760469i
$250$	−1.00000	−0.0632456
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	− 2.00000i	− 0.125988i
$253$	8.00000i	0.502956i
$254$	20.0000i	1.25491i
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	− 4.00000i	− 0.249029i
$259$	12.0000	0.745644
$260$	0	0
$261$	8.00000	0.495188
$262$	− 14.0000i	− 0.864923i
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	−4.00000	−0.246183
$265$	6.00000i	0.368577i
$266$	− 4.00000i	− 0.245256i
$267$	− 10.0000i	− 0.611990i
$268$	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
$271$	28.0000i	1.70088i	0.526073	+	0.850439i	$0.323664\pi$
$271$	28.0000i	1.70088i	−0.526073	+	0.850439i	$0.676336\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	2.00000	0.120824
$275$	4.00000i	0.241209i
$276$	−2.00000	−0.120386
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	16.0000i	0.959616i
$279$	4.00000i	0.239474i
$280$	− 2.00000i	− 0.119523i
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−20.0000	−1.18056
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	8.00000	0.469776
$291$	8.00000i	0.468968i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	3.00000i	0.174964i
$295$	−12.0000	−0.698667
$296$	6.00000	0.348743
$297$	4.00000i	0.232104i
$298$	−22.0000	−1.27443
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 8.00000i	− 0.461112i
$302$	4.00000	0.230174
$303$	20.0000	1.14897
$304$	− 2.00000i	− 0.114708i
$305$	− 2.00000i	− 0.114520i
$306$	4.00000i	0.228665i
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−8.00000	−0.455842
$309$	4.00000	0.227552
$310$	4.00000i	0.227185i
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	10.0000i	0.564333i
$315$	−2.00000	−0.112687
$316$	8.00000	0.450035
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	6.00000i	0.336463i
$319$	− 32.0000i	− 1.79166i
$320$	− 1.00000i	− 0.0559017i
$321$	−4.00000	−0.223258
$322$	−4.00000	−0.222911
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	− 4.00000i	− 0.221201i
$328$	−10.0000	−0.552158
$329$	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
$333$	− 6.00000i	− 0.328798i
$334$	12.0000	0.656611
$335$	8.00000	0.437087
$336$	− 2.00000i	− 0.109109i
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	4.00000i	0.216930i
$341$	16.0000	0.866449
$342$	−2.00000	−0.108148
$343$	20.0000i	1.07990i
$344$	− 4.00000i	− 0.215666i
$345$	2.00000i	0.107676i
$346$	14.0000i	0.752645i
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	8.00000	0.428845
$349$	28.0000i	1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$	28.0000i	1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	−4.00000	−0.213201
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	− 10.0000i	− 0.529999i
$357$	8.00000i	0.423405i
$358$	2.00000i	0.105703i
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	−1.00000	−0.0527046
$361$	15.0000	0.789474
$362$	− 22.0000i	− 1.15629i
$363$	5.00000	0.262432
$364$	0	0
$365$	0	0
$366$	− 2.00000i	− 0.104542i
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−2.00000	−0.104257
$369$	10.0000i	0.520579i
$370$	− 6.00000i	− 0.311925i
$371$	12.0000i	0.623009i
$372$	4.00000i	0.207390i
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	16.0000	0.827340
$375$	1.00000i	0.0516398i
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	− 14.0000i	− 0.719132i	−0.933120	−	0.359566i	$-0.882925\pi$
$379$	− 14.0000i	− 0.719132i	0.933120	−	0.359566i	$-0.117075\pi$
$380$	−2.00000	−0.102598
$381$	20.0000	1.02463
$382$	8.00000i	0.409316i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	8.00000i	0.407718i
$386$	−12.0000	−0.610784
$387$	−4.00000	−0.203331
$388$	8.00000i	0.406138i
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	3.00000i	0.151523i
$393$	−14.0000	−0.706207
$394$	10.0000	0.503793
$395$	− 8.00000i	− 0.402524i
$396$	4.00000i	0.201008i
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	24.0000i	1.20301i
$399$	−4.00000	−0.200250
$400$	−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	$0.730503\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	20.0000	0.995037
$405$	1.00000i	0.0496904i
$406$	16.0000	0.794067
$407$	−24.0000	−1.18964
$408$	4.00000i	0.198030i
$409$	− 18.0000i	− 0.890043i	−0.895520	−	0.445021i	$-0.853196\pi$
$409$	− 18.0000i	− 0.890043i	0.895520	−	0.445021i	$-0.146804\pi$
$410$	10.0000i	0.493865i
$411$	− 2.00000i	− 0.0986527i
$412$	4.00000	0.197066
$413$	−24.0000	−1.18096
$414$	2.00000i	0.0982946i
$415$	12.0000	0.589057
$416$	0	0
$417$	16.0000	0.783523
$418$	8.00000i	0.391293i
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	−2.00000	−0.0975900
$421$	− 16.0000i	− 0.779792i	−0.920859	−	0.389896i	$-0.872511\pi$
$421$	− 16.0000i	− 0.779792i	0.920859	−	0.389896i	$-0.127489\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	6.00000i	0.291386i
$425$	4.00000	0.194029
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−4.00000	−0.192897
$431$	16.0000i	0.770693i	0.922772	+	0.385346i	$0.125918\pi$
$431$	16.0000i	0.770693i	−0.922772	+	0.385346i	$0.874082\pi$
$432$	−1.00000	−0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	8.00000i	0.384012i
$435$	− 8.00000i	− 0.383571i
$436$	− 4.00000i	− 0.191565i
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	4.00000i	0.190693i
$441$	3.00000	0.142857
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	− 6.00000i	− 0.284747i
$445$	−10.0000	−0.474045
$446$	−22.0000	−1.04173
$447$	22.0000i	1.04056i
$448$	− 2.00000i	− 0.0944911i
$449$	18.0000i	0.849473i	0.905317	+	0.424736i	$0.139633\pi$
$449$	18.0000i	0.849473i	−0.905317	+	0.424736i	$0.860367\pi$
$450$	1.00000i	0.0471405i
$451$	40.0000	1.88353
$452$	16.0000	0.752577
$453$	− 4.00000i	− 0.187936i
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 16.0000i	− 0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$	− 16.0000i	− 0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$	−28.0000	−1.30835
$459$	4.00000	0.186704
$460$	2.00000i	0.0932505i
$461$	30.0000i	1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$	30.0000i	1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	8.00000i	0.372194i
$463$	− 2.00000i	− 0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$	− 2.00000i	− 0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$	8.00000	0.371391
$465$	4.00000	0.185496
$466$	28.0000i	1.29707i
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	16.0000i	0.735681i
$474$	− 8.00000i	− 0.367452i
$475$	2.00000i	0.0917663i
$476$	8.00000i	0.366679i
$477$	6.00000	0.274721
$478$	−8.00000	−0.365911
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	22.0000	1.00207
$483$	4.00000i	0.182006i
$484$	5.00000	0.227273
$485$	8.00000	0.363261
$486$	1.00000i	0.0453609i
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	12.0000i	0.542659i
$490$	3.00000	0.135526
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	10.0000i	0.450835i
$493$	−32.0000	−1.44121
$494$	0	0
$495$	4.00000	0.179787
$496$	4.00000i	0.179605i
$497$	0	0
$498$	12.0000	0.537733
$499$	− 14.0000i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	− 14.0000i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$	1.00000i	0.0447214i
$501$	− 12.0000i	− 0.536120i
$502$	10.0000i	0.446322i
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	−2.00000	−0.0890871
$505$	− 20.0000i	− 0.889988i
$506$	8.00000	0.355643
$507$	0	0
$508$	20.0000	0.887357
$509$	38.0000i	1.68432i	0.539227	+	0.842160i	$0.318716\pi$
$509$	38.0000i	1.68432i	−0.539227	+	0.842160i	$0.681284\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	2.00000i	0.0883022i
$514$	− 8.00000i	− 0.352865i
$515$	− 4.00000i	− 0.176261i
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 12.0000i	− 0.527250i
$519$	14.0000	0.614532
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	− 8.00000i	− 0.350150i
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−14.0000	−0.611593
$525$	2.00000i	0.0872872i
$526$	18.0000i	0.784837i
$527$	− 16.0000i	− 0.696971i
$528$	4.00000i	0.174078i
$529$	−19.0000	−0.826087
$530$	6.00000	0.260623
$531$	12.0000i	0.520756i
$532$	−4.00000	−0.173422
$533$	0	0
$534$	−10.0000	−0.432742
$535$	4.00000i	0.172935i
$536$	8.00000	0.345547
$537$	2.00000	0.0863064
$538$	4.00000i	0.172452i
$539$	− 12.0000i	− 0.516877i
$540$	1.00000i	0.0430331i
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	28.0000	1.20270
$543$	−22.0000	−0.944110
$544$	4.00000i	0.171499i
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	−2.00000	−0.0853579
$550$	4.00000	0.170561
$551$	− 16.0000i	− 0.681623i
$552$	2.00000i	0.0851257i
$553$	− 16.0000i	− 0.680389i
$554$	22.0000i	0.934690i
$555$	−6.00000	−0.254686
$556$	16.0000	0.678551
$557$	10.0000i	0.423714i	0.977301	+	0.211857i	$0.0679510\pi$
$557$	10.0000i	0.423714i	−0.977301	+	0.211857i	$0.932049\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	− 16.0000i	− 0.675521i
$562$	10.0000	0.421825
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	− 16.0000i	− 0.673125i
$566$	28.0000i	1.17693i
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	2.00000i	0.0837708i
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	20.0000i	0.834784i
$575$	2.00000	0.0834058
$576$	−1.00000	−0.0416667
$577$	− 4.00000i	− 0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$	− 4.00000i	− 0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$	1.00000i	0.0415945i
$579$	12.0000i	0.498703i
$580$	− 8.00000i	− 0.332182i
$581$	24.0000	0.995688
$582$	8.00000	0.331611
$583$	− 24.0000i	− 0.993978i
$584$	0	0
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	3.00000	0.123718
$589$	8.00000	0.329634
$590$	12.0000i	0.494032i
$591$	− 10.0000i	− 0.411345i
$592$	− 6.00000i	− 0.246598i
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	4.00000	0.164122
$595$	8.00000	0.327968
$596$	22.0000i	0.901155i
$597$	24.0000	0.982255
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000i	0.0408248i
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	−8.00000	−0.326056
$603$	− 8.00000i	− 0.325785i
$604$	− 4.00000i	− 0.162758i
$605$	− 5.00000i	− 0.203279i
$606$	− 20.0000i	− 0.812444i
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	−2.00000	−0.0811107
$609$	− 16.0000i	− 0.648353i
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	4.00000	0.161690
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	10.0000	0.403239
$616$	8.00000i	0.322329i
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	− 4.00000i	− 0.160904i
$619$	− 26.0000i	− 1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	− 26.0000i	− 1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	4.00000	0.160644
$621$	2.00000	0.0802572
$622$	− 16.0000i	− 0.641542i
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000i	0.399680i
$627$	8.00000	0.319489
$628$	10.0000	0.399043
$629$	24.0000i	0.956943i
$630$	2.00000i	0.0796819i
$631$	− 40.0000i	− 1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$	− 40.0000i	− 1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$	− 8.00000i	− 0.318223i
$633$	4.00000	0.158986
$634$	−2.00000	−0.0794301
$635$	− 20.0000i	− 0.793676i
$636$	6.00000	0.237915
$637$	0	0
$638$	−32.0000	−1.26689
$639$	0	0
$640$	−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	$0.615682\pi$
$642$	4.00000i	0.157867i
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	4.00000i	0.157622i
$645$	4.00000i	0.157500i
$646$	8.00000	0.314756
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000i	0.0392837i
$649$	48.0000	1.88416
$650$	0	0
$651$	8.00000	0.313545
$652$	12.0000i	0.469956i
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	−4.00000	−0.156412
$655$	14.0000i	0.547025i
$656$	10.0000i	0.390434i
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	4.00000	0.155700
$661$	16.0000i	0.622328i	0.950356	+	0.311164i	$0.100719\pi$
$661$	16.0000i	0.622328i	−0.950356	+	0.311164i	$0.899281\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	4.00000i	0.155113i
$666$	−6.00000	−0.232495
$667$	−16.0000	−0.619522
$668$	− 12.0000i	− 0.464294i
$669$	22.0000i	0.850569i
$670$	− 8.00000i	− 0.309067i
$671$	8.00000i	0.308837i
$672$	−2.00000	−0.0771517
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	− 26.0000i	− 1.00148i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	− 16.0000i	− 0.614476i
$679$	16.0000	0.614024
$680$	4.00000	0.153393
$681$	− 12.0000i	− 0.459841i
$682$	− 16.0000i	− 0.612672i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	2.00000i	0.0764719i
$685$	−2.00000	−0.0764161
$686$	20.0000	0.763604
$687$	28.0000i	1.06827i
$688$	−4.00000	−0.152499
$689$	0	0
$690$	2.00000	0.0761387
$691$	− 18.0000i	− 0.684752i	−0.939563	−	0.342376i	$-0.888768\pi$
$691$	− 18.0000i	− 0.684752i	0.939563	−	0.342376i	$-0.111232\pi$
$692$	14.0000	0.532200
$693$	8.00000	0.303895
$694$	− 16.0000i	− 0.607352i
$695$	− 16.0000i	− 0.606915i
$696$	− 8.00000i	− 0.303239i
$697$	− 40.0000i	− 1.51511i
$698$	28.0000	1.05982
$699$	28.0000	1.05906
$700$	2.00000i	0.0755929i
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	4.00000i	0.150756i
$705$	0	0
$706$	6.00000	0.225813
$707$	− 40.0000i	− 1.50435i
$708$	12.0000i	0.450988i
$709$	− 4.00000i	− 0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$	− 4.00000i	− 0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−10.0000	−0.374766
$713$	− 8.00000i	− 0.299602i
$714$	8.00000	0.299392
$715$	0	0
$716$	2.00000	0.0747435
$717$	8.00000i	0.298765i
$718$	−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
$721$	− 8.00000i	− 0.297936i
$722$	− 15.0000i	− 0.558242i
$723$	− 22.0000i	− 0.818189i
$724$	−22.0000	−0.817624
$725$	−8.00000	−0.297113
$726$	− 5.00000i	− 0.185567i
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	−2.00000	−0.0739221
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	4.00000i	0.147643i
$735$	− 3.00000i	− 0.110657i
$736$	2.00000i	0.0737210i
$737$	−32.0000	−1.17874
$738$	10.0000	0.368105
$739$	− 50.0000i	− 1.83928i	−0.392763	−	0.919640i	$-0.628481\pi$
$739$	− 50.0000i	− 1.83928i	0.392763	−	0.919640i	$-0.371519\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	12.0000	0.440534
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	4.00000	0.146647
$745$	22.0000	0.806018
$746$	− 14.0000i	− 0.512576i
$747$	− 12.0000i	− 0.439057i
$748$	− 16.0000i	− 0.585018i
$749$	8.00000i	0.292314i
$750$	1.00000	0.0365148
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	−4.00000	−0.145575
$756$	2.00000i	0.0727393i
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−14.0000	−0.508503
$759$	− 8.00000i	− 0.290382i
$760$	2.00000i	0.0725476i
$761$	46.0000i	1.66750i	0.552143	+	0.833749i	$0.313810\pi$
$761$	46.0000i	1.66750i	−0.552143	+	0.833749i	$0.686190\pi$
$762$	− 20.0000i	− 0.724524i
$763$	−8.00000	−0.289619
$764$	8.00000	0.289430
$765$	− 4.00000i	− 0.144620i
$766$	−20.0000	−0.722629
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	10.0000i	0.360609i	0.983611	+	0.180305i	$0.0577084\pi$
$769$	10.0000i	0.360609i	−0.983611	+	0.180305i	$0.942292\pi$
$770$	8.00000	0.288300
$771$	−8.00000	−0.288113
$772$	12.0000i	0.431889i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	4.00000i	0.143777i
$775$	− 4.00000i	− 0.143684i
$776$	8.00000	0.287183
$777$	−12.0000	−0.430498
$778$	20.0000i	0.717035i
$779$	20.0000	0.716574
$780$	0	0
$781$	0	0
$782$	− 8.00000i	− 0.286079i
$783$	−8.00000	−0.285897
$784$	3.00000	0.107143
$785$	− 10.0000i	− 0.356915i
$786$	14.0000i	0.499363i
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 10.0000i	− 0.356235i
$789$	18.0000	0.640817
$790$	−8.00000	−0.284627
$791$	− 32.0000i	− 1.13779i
$792$	4.00000	0.142134
$793$	0	0
$794$	−6.00000	−0.212932
$795$	− 6.00000i	− 0.212798i
$796$	24.0000	0.850657
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	10.0000i	0.353333i
$802$	30.0000	1.05934
$803$	0	0
$804$	− 8.00000i	− 0.282138i
$805$	4.00000	0.140981
$806$	0	0
$807$	4.00000	0.140807
$808$	− 20.0000i	− 0.703598i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	− 34.0000i	− 1.19390i	−0.802278	−	0.596951i	$-0.796379\pi$
$811$	− 34.0000i	− 1.19390i	0.802278	−	0.596951i	$-0.203621\pi$
$812$	− 16.0000i	− 0.561490i
$813$	− 28.0000i	− 0.982003i
$814$	24.0000i	0.841200i
$815$	12.0000	0.420342
$816$	4.00000	0.140028
$817$	8.00000i	0.279885i
$818$	−18.0000	−0.629355
$819$	0	0
$820$	10.0000	0.349215
$821$	42.0000i	1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$	42.0000i	1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$	−2.00000	−0.0697580
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	− 4.00000i	− 0.139347i
$825$	− 4.00000i	− 0.139262i
$826$	24.0000i	0.835067i
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	2.00000	0.0695048
$829$	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
$831$	22.0000	0.763172
$832$	0	0
$833$	−12.0000	−0.415775
$834$	− 16.0000i	− 0.554035i
$835$	−12.0000	−0.415277
$836$	8.00000	0.276686
$837$	− 4.00000i	− 0.138260i
$838$	− 14.0000i	− 0.483622i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	2.00000i	0.0690066i
$841$	35.0000	1.20690
$842$	−16.0000	−0.551396
$843$	− 10.0000i	− 0.344418i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	6.00000	0.206041
$849$	28.0000	0.960958
$850$	− 4.00000i	− 0.137199i
$851$	12.0000i	0.411355i
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−4.00000	−0.136877
$855$	2.00000	0.0683986
$856$	4.00000i	0.136717i
$857$	16.0000	0.546550	0.273275	−	0.961936i	$-0.411893\pi$
$857$	16.0000	0.546550	0.273275	+	0.961936i	$0.411893\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	4.00000i	0.136399i
$861$	20.0000	0.681598
$862$	16.0000	0.544962
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	1.00000i	0.0340207i
$865$	− 14.0000i	− 0.476014i
$866$	− 26.0000i	− 0.883516i
$867$	1.00000	0.0339618
$868$	8.00000	0.271538
$869$	32.0000i	1.08553i
$870$	−8.00000	−0.271225
$871$	0	0
$872$	−4.00000	−0.135457
$873$	− 8.00000i	− 0.270759i
$874$	4.00000	0.135302
$875$	2.00000	0.0676123
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	8.00000i	0.269987i
$879$	− 6.00000i	− 0.202375i
$880$	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
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	16.0000i	0.537531i
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−6.00000	−0.201347
$889$	− 40.0000i	− 1.34156i
$890$	10.0000i	0.335201i
$891$	− 4.00000i	− 0.134005i
$892$	22.0000i	0.736614i
$893$	0	0
$894$	22.0000	0.735790
$895$	− 2.00000i	− 0.0668526i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	18.0000	0.600668
$899$	32.0000i	1.06726i
$900$	1.00000	0.0333333
$901$	−24.0000	−0.799556
$902$	− 40.0000i	− 1.33185i
$903$	8.00000i	0.266223i
$904$	− 16.0000i	− 0.532152i
$905$	22.0000i	0.731305i
$906$	−4.00000	−0.132891
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−20.0000	−0.663358
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	2.00000i	0.0662266i
$913$	−48.0000	−1.58857
$914$	−16.0000	−0.529233
$915$	2.00000i	0.0661180i
$916$	28.0000i	0.925146i
$917$	28.0000i	0.924641i
$918$	− 4.00000i	− 0.132020i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	2.00000	0.0659380
$921$	0	0
$922$	30.0000	0.987997
$923$	0	0
$924$	8.00000	0.263181
$925$	6.00000i	0.197279i
$926$	−2.00000	−0.0657241
$927$	−4.00000	−0.131377
$928$	− 8.00000i	− 0.262613i
$929$	− 34.0000i	− 1.11550i	−0.830008	−	0.557752i	$-0.811664\pi$
$929$	− 34.0000i	− 1.11550i	0.830008	−	0.557752i	$-0.188336\pi$
$930$	− 4.00000i	− 0.131165i
$931$	− 6.00000i	− 0.196642i
$932$	28.0000	0.917170
$933$	−16.0000	−0.523816
$934$	12.0000i	0.392652i
$935$	−16.0000	−0.523256
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	− 16.0000i	− 0.522419i
$939$	10.0000	0.326338
$940$	0	0
$941$	− 38.0000i	− 1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$	− 38.0000i	− 1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$	− 10.0000i	− 0.325818i
$943$	− 20.0000i	− 0.651290i
$944$	12.0000i	0.390567i
$945$	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
$950$	2.00000	0.0648886
$951$	2.00000i	0.0648544i
$952$	8.00000	0.259281
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	− 6.00000i	− 0.194257i
$955$	− 8.00000i	− 0.258874i
$956$	8.00000i	0.258738i
$957$	32.0000i	1.03441i
$958$	−24.0000	−0.775405
$959$	−4.00000	−0.129167
$960$	1.00000i	0.0322749i
$961$	15.0000	0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	− 22.0000i	− 0.708572i
$965$	12.0000	0.386294
$966$	4.00000	0.128698
$967$	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
$969$	− 8.00000i	− 0.256997i
$970$	− 8.00000i	− 0.256865i
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	1.00000	0.0320750
$973$	− 32.0000i	− 1.02587i
$974$	38.0000	1.21760
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	12.0000	0.383718
$979$	40.0000	1.27841
$980$	− 3.00000i	− 0.0958315i
$981$	4.00000i	0.127710i
$982$	38.0000i	1.21263i
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	10.0000	0.318788
$985$	−10.0000	−0.318626
$986$	32.0000i	1.01909i
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	− 4.00000i	− 0.127128i
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000	0.127000
$993$	10.0000i	0.317340i
$994$	0	0
$995$	− 24.0000i	− 0.760851i
$996$	− 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	$0.635070\pi$
$998$	−14.0000	−0.443162
$999$	6.00000i	0.189832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.f.1351.1		2
13.5	odd	4		390.2.a.b.1.1	✓	1
13.8	odd	4		5070.2.a.n.1.1		1
13.12	even	2	inner	5070.2.b.f.1351.2		2
39.5	even	4		1170.2.a.j.1.1		1
52.31	even	4		3120.2.a.y.1.1		1
65.18	even	4		1950.2.e.m.1249.2		2
65.44	odd	4		1950.2.a.ba.1.1		1
65.57	even	4		1950.2.e.m.1249.1		2
156.83	odd	4		9360.2.a.v.1.1		1
195.44	even	4		5850.2.a.s.1.1		1
195.83	odd	4		5850.2.e.h.5149.1		2
195.122	odd	4		5850.2.e.h.5149.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.b.1.1	✓	1	13.5	odd	4
1170.2.a.j.1.1		1	39.5	even	4
1950.2.a.ba.1.1		1	65.44	odd	4
1950.2.e.m.1249.1		2	65.57	even	4
1950.2.e.m.1249.2		2	65.18	even	4
3120.2.a.y.1.1		1	52.31	even	4
5070.2.a.n.1.1		1	13.8	odd	4
5070.2.b.f.1351.1		2	1.1	even	1	trivial
5070.2.b.f.1351.2		2	13.12	even	2	inner
5850.2.a.s.1.1		1	195.44	even	4
5850.2.e.h.5149.1		2	195.83	odd	4
5850.2.e.h.5149.2		2	195.122	odd	4
9360.2.a.v.1.1		1	156.83	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -2.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} -4.00000 q^{22} -2.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.00000i q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -6.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} +4.00000i q^{44} +1.00000i q^{45} +2.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +6.00000 q^{53} +1.00000i q^{54} +4.00000 q^{55} -2.00000 q^{56} +2.00000i q^{57} -8.00000i q^{58} +12.0000i q^{59} +1.00000i q^{60} -2.00000 q^{61} +4.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} +4.00000 q^{68} +2.00000 q^{69} +2.00000i q^{70} +1.00000i q^{72} -6.00000 q^{74} +1.00000 q^{75} +2.00000i q^{76} +8.00000 q^{77} -8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -8.00000 q^{87} +4.00000 q^{88} +10.0000i q^{89} +1.00000 q^{90} +2.00000 q^{92} -4.00000i q^{93} +2.00000 q^{95} +1.00000i q^{96} -8.00000i q^{97} -3.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 8 q^{17} - 8 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{27} + 16 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 4 q^{38} - 2 q^{40} - 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} + 8 q^{51} + 12 q^{53} + 8 q^{55} - 4 q^{56} - 4 q^{61} + 8 q^{62} - 2 q^{64} + 8 q^{66} + 8 q^{68} + 4 q^{69} - 12 q^{74} + 2 q^{75} + 16 q^{77} - 16 q^{79} + 2 q^{81} + 20 q^{82} - 16 q^{87} + 8 q^{88} + 2 q^{90} + 4 q^{92} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 8 * q^17 - 8 * q^22 - 4 * q^23 - 2 * q^25 - 2 * q^27 + 16 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 4 * q^38 - 2 * q^40 - 4 * q^42 - 8 * q^43 - 2 * q^48 + 6 * q^49 + 8 * q^51 + 12 * q^53 + 8 * q^55 - 4 * q^56 - 4 * q^61 + 8 * q^62 - 2 * q^64 + 8 * q^66 + 8 * q^68 + 4 * q^69 - 12 * q^74 + 2 * q^75 + 16 * q^77 - 16 * q^79 + 2 * q^81 + 20 * q^82 - 16 * q^87 + 8 * q^88 + 2 * q^90 + 4 * q^92 + 4 * q^95

Introduction

L-functions

Modular forms

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Database

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