LMFDB - Embedded newform 5070.2.b.l.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} +5.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000i q^{18} -5.00000i q^{19} +1.00000i q^{20} +5.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -5.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} -2.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000i q^{34} +5.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} -5.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +5.00000 q^{42} -6.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} -18.0000 q^{49} +1.00000i q^{50} +8.00000 q^{51} +1.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} -5.00000 q^{56} -5.00000i q^{57} +4.00000i q^{58} -12.0000i q^{59} +1.00000i q^{60} +2.00000 q^{61} -2.00000 q^{62} +5.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} -8.00000 q^{68} +4.00000 q^{69} -5.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} +7.00000 q^{74} -1.00000 q^{75} +5.00000i q^{76} -15.0000 q^{77} -2.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000i q^{83} -5.00000i q^{84} -8.00000i q^{85} +6.00000i q^{86} -4.00000 q^{87} -3.00000 q^{88} +11.0000i q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000i q^{93} +3.00000 q^{94} -5.00000 q^{95} -1.00000i q^{96} +18.0000i q^{98} +3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 10 q^{14} + 2 q^{16} + 16 q^{17} + 6 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{27} - 8 q^{29} - 2 q^{30} + 10 q^{35} - 2 q^{36} - 10 q^{38} + 2 q^{40} + 10 q^{42} - 12 q^{43} + 2 q^{48} - 36 q^{49} + 16 q^{51} + 2 q^{53} + 6 q^{55} - 10 q^{56} + 4 q^{61} - 4 q^{62} - 2 q^{64} + 6 q^{66} - 16 q^{68} + 8 q^{69} + 14 q^{74} - 2 q^{75} - 30 q^{77} - 4 q^{79} + 2 q^{81} + 12 q^{82} - 8 q^{87} - 6 q^{88} - 2 q^{90} - 8 q^{92} + 6 q^{94} - 10 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 10 * q^14 + 2 * q^16 + 16 * q^17 + 6 * q^22 + 8 * q^23 - 2 * q^25 + 2 * q^27 - 8 * q^29 - 2 * q^30 + 10 * q^35 - 2 * q^36 - 10 * q^38 + 2 * q^40 + 10 * q^42 - 12 * q^43 + 2 * q^48 - 36 * q^49 + 16 * q^51 + 2 * q^53 + 6 * q^55 - 10 * q^56 + 4 * q^61 - 4 * q^62 - 2 * q^64 + 6 * q^66 - 16 * q^68 + 8 * q^69 + 14 * q^74 - 2 * q^75 - 30 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^82 - 8 * q^87 - 6 * q^88 - 2 * q^90 - 8 * q^92 + 6 * q^94 - 10 * 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
$7$	5.00000i	1.88982i	0.327327	+	0.944911i	$0.393852\pi$
$7$	5.00000i	1.88982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	5.00000	1.33631
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 5.00000i	− 1.14708i	−0.819178	−	0.573539i	$-0.805570\pi$
$19$	− 5.00000i	− 1.14708i	0.819178	−	0.573539i	$-0.194430\pi$
$20$	1.00000i	0.223607i
$21$	5.00000i	1.09109i
$22$	3.00000	0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 5.00000i	− 0.944911i
$29$	−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
$31$	− 2.00000i	− 0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$	− 2.00000i	− 0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.00000i	0.522233i
$34$	− 8.00000i	− 1.37199i
$35$	5.00000	0.845154
$36$	−1.00000	−0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	5.00000	0.771517
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	− 3.00000i	− 0.452267i
$45$	− 1.00000i	− 0.149071i
$46$	− 4.00000i	− 0.589768i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	1.00000	0.144338
$49$	−18.0000	−2.57143
$50$	1.00000i	0.141421i
$51$	8.00000	1.12022
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	− 1.00000i	− 0.136083i
$55$	3.00000	0.404520
$56$	−5.00000	−0.668153
$57$	− 5.00000i	− 0.662266i
$58$	4.00000i	0.525226i
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	1.00000i	0.129099i
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	5.00000i	0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	−8.00000	−0.970143
$69$	4.00000	0.481543
$70$	− 5.00000i	− 0.597614i
$71$	2.00000i	0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$	2.00000i	0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$	1.00000i	0.117851i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	7.00000	0.813733
$75$	−1.00000	−0.115470
$76$	5.00000i	0.573539i
$77$	−15.0000	−1.70941
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	− 5.00000i	− 0.545545i
$85$	− 8.00000i	− 0.867722i
$86$	6.00000i	0.646997i
$87$	−4.00000	−0.428845
$88$	−3.00000	−0.319801
$89$	11.0000i	1.16600i	0.812473	+	0.582999i	$0.198121\pi$
$89$	11.0000i	1.16600i	−0.812473	+	0.582999i	$0.801879\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	− 2.00000i	− 0.207390i
$94$	3.00000	0.309426
$95$	−5.00000	−0.512989
$96$	− 1.00000i	− 0.102062i
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	18.0000i	1.81827i
$99$	3.00000i	0.301511i
$100$	1.00000	0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	− 8.00000i	− 0.792118i
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	− 1.00000i	− 0.0971286i
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	− 3.00000i	− 0.286039i
$111$	7.00000i	0.664411i
$112$	5.00000i	0.472456i
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	−5.00000	−0.468293
$115$	− 4.00000i	− 0.373002i
$116$	4.00000	0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	40.0000i	3.66679i
$120$	1.00000	0.0912871
$121$	2.00000	0.181818
$122$	− 2.00000i	− 0.181071i
$123$	6.00000i	0.541002i
$124$	2.00000i	0.179605i
$125$	1.00000i	0.0894427i
$126$	5.00000	0.445435
$127$	21.0000	1.86345	0.931724	−	0.363166i	$-0.118304\pi$
$127$	21.0000	1.86345	0.931724	+	0.363166i	$0.118304\pi$
$128$	1.00000i	0.0883883i
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−19.0000	−1.66004	−0.830019	−	0.557735i	$-0.811670\pi$
$131$	−19.0000	−1.66004	−0.830019	+	0.557735i	$0.811670\pi$
$132$	− 3.00000i	− 0.261116i
$133$	25.0000	2.16777
$134$	8.00000	0.691095
$135$	− 1.00000i	− 0.0860663i
$136$	8.00000i	0.685994i
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 4.00000i	− 0.340503i
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	−5.00000	−0.422577
$141$	3.00000i	0.252646i
$142$	2.00000	0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000i	0.332182i
$146$	0	0
$147$	−18.0000	−1.48461
$148$	− 7.00000i	− 0.575396i
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	1.00000i	0.0816497i
$151$	− 22.0000i	− 1.79033i	−0.445730	−	0.895167i	$-0.647056\pi$
$151$	− 22.0000i	− 1.79033i	0.445730	−	0.895167i	$-0.352944\pi$
$152$	5.00000	0.405554
$153$	8.00000	0.646762
$154$	15.0000i	1.20873i
$155$	−2.00000	−0.160644
$156$	0	0
$157$	15.0000	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$157$	15.0000	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$158$	2.00000i	0.159111i
$159$	1.00000	0.0793052
$160$	−1.00000	−0.0790569
$161$	20.0000i	1.57622i
$162$	− 1.00000i	− 0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	− 6.00000i	− 0.468521i
$165$	3.00000	0.233550
$166$	8.00000	0.620920
$167$	23.0000i	1.77979i	0.456162	+	0.889897i	$0.349224\pi$
$167$	23.0000i	1.77979i	−0.456162	+	0.889897i	$0.650776\pi$
$168$	−5.00000	−0.385758
$169$	0	0
$170$	−8.00000	−0.613572
$171$	− 5.00000i	− 0.382360i
$172$	6.00000	0.457496
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	4.00000i	0.303239i
$175$	− 5.00000i	− 0.377964i
$176$	3.00000i	0.226134i
$177$	− 12.0000i	− 0.901975i
$178$	11.0000	0.824485
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	1.00000i	0.0745356i
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	4.00000i	0.294884i
$185$	7.00000	0.514650
$186$	−2.00000	−0.146647
$187$	24.0000i	1.75505i
$188$	− 3.00000i	− 0.218797i
$189$	5.00000i	0.363696i
$190$	5.00000i	0.362738i
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	−1.00000	−0.0721688
$193$	− 24.0000i	− 1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	− 24.0000i	− 1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	0	0
$195$	0	0
$196$	18.0000	1.28571
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	3.00000	0.213201
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	8.00000i	0.564276i
$202$	− 8.00000i	− 0.562878i
$203$	− 20.0000i	− 1.40372i
$204$	−8.00000	−0.560112
$205$	6.00000	0.419058
$206$	− 7.00000i	− 0.487713i
$207$	4.00000	0.278019
$208$	0	0
$209$	15.0000	1.03757
$210$	− 5.00000i	− 0.345033i
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	−1.00000	−0.0686803
$213$	2.00000i	0.137038i
$214$	6.00000i	0.410152i
$215$	6.00000i	0.409197i
$216$	1.00000i	0.0680414i
$217$	10.0000	0.678844
$218$	14.0000	0.948200
$219$	0	0
$220$	−3.00000	−0.202260
$221$	0	0
$222$	7.00000	0.469809
$223$	− 3.00000i	− 0.200895i	−0.994942	−	0.100447i	$-0.967973\pi$
$223$	− 3.00000i	− 0.200895i	0.994942	−	0.100447i	$-0.0320274\pi$
$224$	5.00000	0.334077
$225$	−1.00000	−0.0666667
$226$	− 8.00000i	− 0.532152i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	5.00000i	0.331133i
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	−4.00000	−0.263752
$231$	−15.0000	−0.986928
$232$	− 4.00000i	− 0.262613i
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	12.0000i	0.781133i
$237$	−2.00000	−0.129914
$238$	40.0000	2.59281
$239$	− 18.0000i	− 1.16432i	−0.813073	−	0.582162i	$-0.802207\pi$
$239$	− 18.0000i	− 1.16432i	0.813073	−	0.582162i	$-0.197793\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	25.0000i	1.61039i	0.593009	+	0.805196i	$0.297940\pi$
$241$	25.0000i	1.61039i	−0.593009	+	0.805196i	$0.702060\pi$
$242$	− 2.00000i	− 0.128565i
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	18.0000i	1.14998i
$246$	6.00000	0.382546
$247$	0	0
$248$	2.00000	0.127000
$249$	8.00000i	0.506979i
$250$	1.00000	0.0632456
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	− 5.00000i	− 0.314970i
$253$	12.0000i	0.754434i
$254$	− 21.0000i	− 1.31766i
$255$	− 8.00000i	− 0.500979i
$256$	1.00000	0.0625000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	6.00000i	0.373544i
$259$	−35.0000	−2.17479
$260$	0	0
$261$	−4.00000	−0.247594
$262$	19.0000i	1.17382i
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	−3.00000	−0.184637
$265$	− 1.00000i	− 0.0614295i
$266$	− 25.0000i	− 1.53285i
$267$	11.0000i	0.673189i
$268$	− 8.00000i	− 0.488678i
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	−1.00000	−0.0608581
$271$	− 4.00000i	− 0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$	− 4.00000i	− 0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	12.0000	0.724947
$275$	− 3.00000i	− 0.180907i
$276$	−4.00000	−0.240772
$277$	15.0000	0.901263	0.450631	−	0.892710i	$-0.351199\pi$
$277$	15.0000	0.901263	0.450631	+	0.892710i	$0.351199\pi$
$278$	− 7.00000i	− 0.419832i
$279$	− 2.00000i	− 0.119737i
$280$	5.00000i	0.298807i
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	3.00000	0.178647
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	− 2.00000i	− 0.118678i
$285$	−5.00000	−0.296174
$286$	0	0
$287$	−30.0000	−1.77084
$288$	− 1.00000i	− 0.0589256i
$289$	47.0000	2.76471
$290$	4.00000	0.234888
$291$	0	0
$292$	0	0
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	18.0000i	1.04978i
$295$	−12.0000	−0.698667
$296$	−7.00000	−0.406867
$297$	3.00000i	0.174078i
$298$	−2.00000	−0.115857
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 30.0000i	− 1.72917i
$302$	−22.0000	−1.26596
$303$	8.00000	0.459588
$304$	− 5.00000i	− 0.286770i
$305$	− 2.00000i	− 0.114520i
$306$	− 8.00000i	− 0.457330i
$307$	6.00000i	0.342438i	0.985233	+	0.171219i	$0.0547706\pi$
$307$	6.00000i	0.342438i	−0.985233	+	0.171219i	$0.945229\pi$
$308$	15.0000	0.854704
$309$	7.00000	0.398216
$310$	2.00000i	0.113592i
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	− 15.0000i	− 0.846499i
$315$	5.00000	0.281718
$316$	2.00000	0.112509
$317$	− 23.0000i	− 1.29181i	−0.763418	−	0.645904i	$-0.776480\pi$
$317$	− 23.0000i	− 1.29181i	0.763418	−	0.645904i	$-0.223520\pi$
$318$	− 1.00000i	− 0.0560772i
$319$	− 12.0000i	− 0.671871i
$320$	1.00000i	0.0559017i
$321$	−6.00000	−0.334887
$322$	20.0000	1.11456
$323$	− 40.0000i	− 2.22566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	14.0000i	0.774202i
$328$	−6.00000	−0.331295
$329$	−15.0000	−0.826977
$330$	− 3.00000i	− 0.165145i
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	− 8.00000i	− 0.439057i
$333$	7.00000i	0.383598i
$334$	23.0000	1.25850
$335$	8.00000	0.437087
$336$	5.00000i	0.272772i
$337$	−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
$339$	8.00000	0.434500
$340$	8.00000i	0.433861i
$341$	6.00000	0.324918
$342$	−5.00000	−0.270369
$343$	− 55.0000i	− 2.96972i
$344$	− 6.00000i	− 0.323498i
$345$	− 4.00000i	− 0.215353i
$346$	5.00000i	0.268802i
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	4.00000	0.214423
$349$	− 8.00000i	− 0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$	− 8.00000i	− 0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$	−5.00000	−0.267261
$351$	0	0
$352$	3.00000	0.159901
$353$	16.0000i	0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$	16.0000i	0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−12.0000	−0.637793
$355$	2.00000	0.106149
$356$	− 11.0000i	− 0.582999i
$357$	40.0000i	2.11702i
$358$	4.00000i	0.211407i
$359$	18.0000i	0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$	18.0000i	0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$	1.00000	0.0527046
$361$	−6.00000	−0.315789
$362$	− 2.00000i	− 0.105118i
$363$	2.00000	0.104973
$364$	0	0
$365$	0	0
$366$	− 2.00000i	− 0.104542i
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	4.00000	0.208514
$369$	6.00000i	0.312348i
$370$	− 7.00000i	− 0.363913i
$371$	5.00000i	0.259587i
$372$	2.00000i	0.103695i
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	24.0000	1.24101
$375$	1.00000i	0.0516398i
$376$	−3.00000	−0.154713
$377$	0	0
$378$	5.00000	0.257172
$379$	− 25.0000i	− 1.28416i	−0.766636	−	0.642082i	$-0.778071\pi$
$379$	− 25.0000i	− 1.28416i	0.766636	−	0.642082i	$-0.221929\pi$
$380$	5.00000	0.256495
$381$	21.0000	1.07586
$382$	− 2.00000i	− 0.102329i
$383$	− 28.0000i	− 1.43073i	−0.698749	−	0.715367i	$-0.746260\pi$
$383$	− 28.0000i	− 1.43073i	0.698749	−	0.715367i	$-0.253740\pi$
$384$	1.00000i	0.0510310i
$385$	15.0000i	0.764471i
$386$	−24.0000	−1.22157
$387$	−6.00000	−0.304997
$388$	0	0
$389$	32.0000	1.62246	0.811232	−	0.584724i	$-0.198797\pi$
$389$	32.0000	1.62246	0.811232	+	0.584724i	$0.198797\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	− 18.0000i	− 0.909137i
$393$	−19.0000	−0.958423
$394$	3.00000	0.151138
$395$	2.00000i	0.100631i
$396$	− 3.00000i	− 0.150756i
$397$	− 25.0000i	− 1.25471i	−0.778732	−	0.627357i	$-0.784137\pi$
$397$	− 25.0000i	− 1.25471i	0.778732	−	0.627357i	$-0.215863\pi$
$398$	22.0000i	1.10276i
$399$	25.0000	1.25157
$400$	−1.00000	−0.0500000
$401$	19.0000i	0.948815i	0.880305	+	0.474407i	$0.157338\pi$
$401$	19.0000i	0.948815i	−0.880305	+	0.474407i	$0.842662\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	−8.00000	−0.398015
$405$	− 1.00000i	− 0.0496904i
$406$	−20.0000	−0.992583
$407$	−21.0000	−1.04093
$408$	8.00000i	0.396059i
$409$	25.0000i	1.23617i	0.786111	+	0.618085i	$0.212091\pi$
$409$	25.0000i	1.23617i	−0.786111	+	0.618085i	$0.787909\pi$
$410$	− 6.00000i	− 0.296319i
$411$	12.0000i	0.591916i
$412$	−7.00000	−0.344865
$413$	60.0000	2.95241
$414$	− 4.00000i	− 0.196589i
$415$	8.00000	0.392705
$416$	0	0
$417$	7.00000	0.342791
$418$	− 15.0000i	− 0.733674i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−5.00000	−0.243975
$421$	12.0000i	0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$	12.0000i	0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$	15.0000i	0.730189i
$423$	3.00000i	0.145865i
$424$	1.00000i	0.0485643i
$425$	−8.00000	−0.388057
$426$	2.00000	0.0969003
$427$	10.0000i	0.483934i
$428$	6.00000	0.290021
$429$	0	0
$430$	6.00000	0.289346
$431$	12.0000i	0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$	12.0000i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	1.00000	0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	− 10.0000i	− 0.480015i
$435$	4.00000i	0.191785i
$436$	− 14.0000i	− 0.670478i
$437$	− 20.0000i	− 0.956730i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	3.00000i	0.143019i
$441$	−18.0000	−0.857143
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	− 7.00000i	− 0.332205i
$445$	11.0000	0.521450
$446$	−3.00000	−0.142054
$447$	− 2.00000i	− 0.0945968i
$448$	− 5.00000i	− 0.236228i
$449$	27.0000i	1.27421i	0.770778	+	0.637104i	$0.219868\pi$
$449$	27.0000i	1.27421i	−0.770778	+	0.637104i	$0.780132\pi$
$450$	1.00000i	0.0471405i
$451$	−18.0000	−0.847587
$452$	−8.00000	−0.376288
$453$	− 22.0000i	− 1.03365i
$454$	0	0
$455$	0	0
$456$	5.00000	0.234146
$457$	30.0000i	1.40334i	0.712502	+	0.701670i	$0.247562\pi$
$457$	30.0000i	1.40334i	−0.712502	+	0.701670i	$0.752438\pi$
$458$	−14.0000	−0.654177
$459$	8.00000	0.373408
$460$	4.00000i	0.186501i
$461$	8.00000i	0.372597i	0.982493	+	0.186299i	$0.0596492\pi$
$461$	8.00000i	0.372597i	−0.982493	+	0.186299i	$0.940351\pi$
$462$	15.0000i	0.697863i
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−4.00000	−0.185695
$465$	−2.00000	−0.0927478
$466$	− 14.0000i	− 0.648537i
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	− 3.00000i	− 0.138380i
$471$	15.0000	0.691164
$472$	12.0000	0.552345
$473$	− 18.0000i	− 0.827641i
$474$	2.00000i	0.0918630i
$475$	5.00000i	0.229416i
$476$	− 40.0000i	− 1.83340i
$477$	1.00000	0.0457869
$478$	−18.0000	−0.823301
$479$	28.0000i	1.27935i	0.768644	+	0.639676i	$0.220932\pi$
$479$	28.0000i	1.27935i	−0.768644	+	0.639676i	$0.779068\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	25.0000	1.13872
$483$	20.0000i	0.910032i
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	− 1.00000i	− 0.0453609i
$487$	37.0000i	1.67663i	0.545186	+	0.838315i	$0.316459\pi$
$487$	37.0000i	1.67663i	−0.545186	+	0.838315i	$0.683541\pi$
$488$	2.00000i	0.0905357i
$489$	20.0000i	0.904431i
$490$	18.0000	0.813157
$491$	−21.0000	−0.947717	−0.473858	−	0.880601i	$-0.657139\pi$
$491$	−21.0000	−0.947717	−0.473858	+	0.880601i	$0.657139\pi$
$492$	− 6.00000i	− 0.270501i
$493$	−32.0000	−1.44121
$494$	0	0
$495$	3.00000	0.134840
$496$	− 2.00000i	− 0.0898027i
$497$	−10.0000	−0.448561
$498$	8.00000	0.358489
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	23.0000i	1.02756i
$502$	15.0000i	0.669483i
$503$	−11.0000	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$504$	−5.00000	−0.222718
$505$	− 8.00000i	− 0.355995i
$506$	12.0000	0.533465
$507$	0	0
$508$	−21.0000	−0.931724
$509$	10.0000i	0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	10.0000i	0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$	−8.00000	−0.354246
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 5.00000i	− 0.220755i
$514$	28.0000i	1.23503i
$515$	− 7.00000i	− 0.308457i
$516$	6.00000	0.264135
$517$	−9.00000	−0.395820
$518$	35.0000i	1.53781i
$519$	−5.00000	−0.219476
$520$	0	0
$521$	5.00000	0.219054	0.109527	−	0.993984i	$-0.465066\pi$
$521$	5.00000	0.219054	0.109527	+	0.993984i	$0.465066\pi$
$522$	4.00000i	0.175075i
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	19.0000	0.830019
$525$	− 5.00000i	− 0.218218i
$526$	15.0000i	0.654031i
$527$	− 16.0000i	− 0.696971i
$528$	3.00000i	0.130558i
$529$	−7.00000	−0.304348
$530$	−1.00000	−0.0434372
$531$	− 12.0000i	− 0.520756i
$532$	−25.0000	−1.08389
$533$	0	0
$534$	11.0000	0.476017
$535$	6.00000i	0.259403i
$536$	−8.00000	−0.345547
$537$	−4.00000	−0.172613
$538$	− 4.00000i	− 0.172452i
$539$	− 54.0000i	− 2.32594i
$540$	1.00000i	0.0430331i
$541$	− 34.0000i	− 1.46177i	−0.682498	−	0.730887i	$-0.739107\pi$
$541$	− 34.0000i	− 1.46177i	0.682498	−	0.730887i	$-0.260893\pi$
$542$	−4.00000	−0.171815
$543$	2.00000	0.0858282
$544$	− 8.00000i	− 0.342997i
$545$	14.0000	0.599694
$546$	0	0
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	− 12.0000i	− 0.512615i
$549$	2.00000	0.0853579
$550$	−3.00000	−0.127920
$551$	20.0000i	0.852029i
$552$	4.00000i	0.170251i
$553$	− 10.0000i	− 0.425243i
$554$	− 15.0000i	− 0.637289i
$555$	7.00000	0.297133
$556$	−7.00000	−0.296866
$557$	− 15.0000i	− 0.635570i	−0.948163	−	0.317785i	$-0.897061\pi$
$557$	− 15.0000i	− 0.635570i	0.948163	−	0.317785i	$-0.102939\pi$
$558$	−2.00000	−0.0846668
$559$	0	0
$560$	5.00000	0.211289
$561$	24.0000i	1.01328i
$562$	−18.0000	−0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	− 3.00000i	− 0.126323i
$565$	− 8.00000i	− 0.336563i
$566$	− 10.0000i	− 0.420331i
$567$	5.00000i	0.209980i
$568$	−2.00000	−0.0839181
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	5.00000i	0.209427i
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	30.0000i	1.25218i
$575$	−4.00000	−0.166812
$576$	−1.00000	−0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 47.0000i	− 1.95494i
$579$	− 24.0000i	− 0.997406i
$580$	− 4.00000i	− 0.166091i
$581$	−40.0000	−1.65948
$582$	0	0
$583$	3.00000i	0.124247i
$584$	0	0
$585$	0	0
$586$	9.00000	0.371787
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	18.0000	0.742307
$589$	−10.0000	−0.412043
$590$	12.0000i	0.494032i
$591$	3.00000i	0.123404i
$592$	7.00000i	0.287698i
$593$	− 20.0000i	− 0.821302i	−0.911793	−	0.410651i	$-0.865302\pi$
$593$	− 20.0000i	− 0.821302i	0.911793	−	0.410651i	$-0.134698\pi$
$594$	3.00000	0.123091
$595$	40.0000	1.63984
$596$	2.00000i	0.0819232i
$597$	−22.0000	−0.900400
$598$	0	0
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	−30.0000	−1.22271
$603$	8.00000i	0.325785i
$604$	22.0000i	0.895167i
$605$	− 2.00000i	− 0.0813116i
$606$	− 8.00000i	− 0.324978i
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	−5.00000	−0.202777
$609$	− 20.0000i	− 0.810441i
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	−8.00000	−0.323381
$613$	25.0000i	1.00974i	0.863195	+	0.504870i	$0.168460\pi$
$613$	25.0000i	1.00974i	−0.863195	+	0.504870i	$0.831540\pi$
$614$	6.00000	0.242140
$615$	6.00000	0.241943
$616$	− 15.0000i	− 0.604367i
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	− 7.00000i	− 0.281581i
$619$	− 17.0000i	− 0.683288i	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	− 17.0000i	− 0.683288i	0.939829	−	0.341644i	$-0.110984\pi$
$620$	2.00000	0.0803219
$621$	4.00000	0.160514
$622$	− 12.0000i	− 0.481156i
$623$	−55.0000	−2.20353
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000i	0.239808i
$627$	15.0000	0.599042
$628$	−15.0000	−0.598565
$629$	56.0000i	2.23287i
$630$	− 5.00000i	− 0.199205i
$631$	− 12.0000i	− 0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$	− 12.0000i	− 0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$	− 2.00000i	− 0.0795557i
$633$	−15.0000	−0.596196
$634$	−23.0000	−0.913447
$635$	− 21.0000i	− 0.833360i
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	−12.0000	−0.475085
$639$	2.00000i	0.0791188i
$640$	1.00000	0.0395285
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	6.00000i	0.236801i
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	− 20.0000i	− 0.788110i
$645$	6.00000i	0.236250i
$646$	−40.0000	−1.57378
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	1.00000i	0.0392837i
$649$	36.0000	1.41312
$650$	0	0
$651$	10.0000	0.391931
$652$	− 20.0000i	− 0.783260i
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	14.0000	0.547443
$655$	19.0000i	0.742391i
$656$	6.00000i	0.234261i
$657$	0	0
$658$	15.0000i	0.584761i
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−3.00000	−0.116775
$661$	10.0000i	0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$	10.0000i	0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−8.00000	−0.310460
$665$	− 25.0000i	− 0.969458i
$666$	7.00000	0.271244
$667$	−16.0000	−0.619522
$668$	− 23.0000i	− 0.889897i
$669$	− 3.00000i	− 0.115987i
$670$	− 8.00000i	− 0.309067i
$671$	6.00000i	0.231627i
$672$	5.00000	0.192879
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	14.0000i	0.539260i
$675$	−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	$0.638940\pi$
$678$	− 8.00000i	− 0.307238i
$679$	0	0
$680$	8.00000	0.306786
$681$	0	0
$682$	− 6.00000i	− 0.229752i
$683$	− 30.0000i	− 1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$	− 30.0000i	− 1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$	5.00000i	0.191180i
$685$	12.0000	0.458496
$686$	−55.0000	−2.09991
$687$	− 14.0000i	− 0.534133i
$688$	−6.00000	−0.228748
$689$	0	0
$690$	−4.00000	−0.152277
$691$	17.0000i	0.646710i	0.946278	+	0.323355i	$0.104811\pi$
$691$	17.0000i	0.646710i	−0.946278	+	0.323355i	$0.895189\pi$
$692$	5.00000	0.190071
$693$	−15.0000	−0.569803
$694$	16.0000i	0.607352i
$695$	− 7.00000i	− 0.265525i
$696$	− 4.00000i	− 0.151620i
$697$	48.0000i	1.81813i
$698$	−8.00000	−0.302804
$699$	14.0000	0.529529
$700$	5.00000i	0.188982i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	35.0000	1.32005
$704$	− 3.00000i	− 0.113067i
$705$	3.00000	0.112987
$706$	16.0000	0.602168
$707$	40.0000i	1.50435i
$708$	12.0000i	0.450988i
$709$	− 32.0000i	− 1.20179i	−0.799330	−	0.600893i	$-0.794812\pi$
$709$	− 32.0000i	− 1.20179i	0.799330	−	0.600893i	$-0.205188\pi$
$710$	− 2.00000i	− 0.0750587i
$711$	−2.00000	−0.0750059
$712$	−11.0000	−0.412242
$713$	− 8.00000i	− 0.299602i
$714$	40.0000	1.49696
$715$	0	0
$716$	4.00000	0.149487
$717$	− 18.0000i	− 0.672222i
$718$	18.0000	0.671754
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	35.0000i	1.30347i
$722$	6.00000i	0.223297i
$723$	25.0000i	0.929760i
$724$	−2.00000	−0.0743294
$725$	4.00000	0.148556
$726$	− 2.00000i	− 0.0742270i
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	−2.00000	−0.0739221
$733$	− 43.0000i	− 1.58824i	−0.607760	−	0.794121i	$-0.707932\pi$
$733$	− 43.0000i	− 1.58824i	0.607760	−	0.794121i	$-0.292068\pi$
$734$	8.00000i	0.295285i
$735$	18.0000i	0.663940i
$736$	− 4.00000i	− 0.147442i
$737$	−24.0000	−0.884051
$738$	6.00000	0.220863
$739$	− 19.0000i	− 0.698926i	−0.936950	−	0.349463i	$-0.886364\pi$
$739$	− 19.0000i	− 0.698926i	0.936950	−	0.349463i	$-0.113636\pi$
$740$	−7.00000	−0.257325
$741$	0	0
$742$	5.00000	0.183556
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	2.00000	0.0733236
$745$	−2.00000	−0.0732743
$746$	− 38.0000i	− 1.39128i
$747$	8.00000i	0.292705i
$748$	− 24.0000i	− 0.877527i
$749$	− 30.0000i	− 1.09618i
$750$	1.00000	0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	3.00000i	0.109399i
$753$	−15.0000	−0.546630
$754$	0	0
$755$	−22.0000	−0.800662
$756$	− 5.00000i	− 0.181848i
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−25.0000	−0.908041
$759$	12.0000i	0.435572i
$760$	− 5.00000i	− 0.181369i
$761$	− 9.00000i	− 0.326250i	−0.986605	−	0.163125i	$-0.947843\pi$
$761$	− 9.00000i	− 0.326250i	0.986605	−	0.163125i	$-0.0521573\pi$
$762$	− 21.0000i	− 0.760750i
$763$	−70.0000	−2.53417
$764$	−2.00000	−0.0723575
$765$	− 8.00000i	− 0.289241i
$766$	−28.0000	−1.01168
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 34.0000i	− 1.22607i	−0.790055	−	0.613036i	$-0.789948\pi$
$769$	− 34.0000i	− 1.22607i	0.790055	−	0.613036i	$-0.210052\pi$
$770$	15.0000	0.540562
$771$	−28.0000	−1.00840
$772$	24.0000i	0.863779i
$773$	1.00000i	0.0359675i	0.999838	+	0.0179838i	$0.00572471\pi$
$773$	1.00000i	0.0359675i	−0.999838	+	0.0179838i	$0.994275\pi$
$774$	6.00000i	0.215666i
$775$	2.00000i	0.0718421i
$776$	0	0
$777$	−35.0000	−1.25562
$778$	− 32.0000i	− 1.14726i
$779$	30.0000	1.07486
$780$	0	0
$781$	−6.00000	−0.214697
$782$	− 32.0000i	− 1.14432i
$783$	−4.00000	−0.142948
$784$	−18.0000	−0.642857
$785$	− 15.0000i	− 0.535373i
$786$	19.0000i	0.677708i
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 3.00000i	− 0.106871i
$789$	−15.0000	−0.534014
$790$	2.00000	0.0711568
$791$	40.0000i	1.42224i
$792$	−3.00000	−0.106600
$793$	0	0
$794$	−25.0000	−0.887217
$795$	− 1.00000i	− 0.0354663i
$796$	22.0000	0.779769
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	− 25.0000i	− 0.884990i
$799$	24.0000i	0.849059i
$800$	1.00000i	0.0353553i
$801$	11.0000i	0.388666i
$802$	19.0000	0.670913
$803$	0	0
$804$	− 8.00000i	− 0.282138i
$805$	20.0000	0.704907
$806$	0	0
$807$	4.00000	0.140807
$808$	8.00000i	0.281439i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	−1.00000	−0.0351364
$811$	25.0000i	0.877869i	0.898519	+	0.438934i	$0.144644\pi$
$811$	25.0000i	0.877869i	−0.898519	+	0.438934i	$0.855356\pi$
$812$	20.0000i	0.701862i
$813$	− 4.00000i	− 0.140286i
$814$	21.0000i	0.736050i
$815$	20.0000	0.700569
$816$	8.00000	0.280056
$817$	30.0000i	1.04957i
$818$	25.0000	0.874105
$819$	0	0
$820$	−6.00000	−0.209529
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	12.0000	0.418548
$823$	35.0000	1.22002	0.610012	−	0.792392i	$-0.291165\pi$
$823$	35.0000	1.22002	0.610012	+	0.792392i	$0.291165\pi$
$824$	7.00000i	0.243857i
$825$	− 3.00000i	− 0.104447i
$826$	− 60.0000i	− 2.08767i
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	−4.00000	−0.139010
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	− 8.00000i	− 0.277684i
$831$	15.0000	0.520344
$832$	0	0
$833$	−144.000	−4.98930
$834$	− 7.00000i	− 0.242390i
$835$	23.0000	0.795948
$836$	−15.0000	−0.518786
$837$	− 2.00000i	− 0.0691301i
$838$	12.0000i	0.414533i
$839$	14.0000i	0.483334i	0.970359	+	0.241667i	$0.0776941\pi$
$839$	14.0000i	0.483334i	−0.970359	+	0.241667i	$0.922306\pi$
$840$	5.00000i	0.172516i
$841$	−13.0000	−0.448276
$842$	12.0000	0.413547
$843$	− 18.0000i	− 0.619953i
$844$	15.0000	0.516321
$845$	0	0
$846$	3.00000	0.103142
$847$	10.0000i	0.343604i
$848$	1.00000	0.0343401
$849$	10.0000	0.343199
$850$	8.00000i	0.274398i
$851$	28.0000i	0.959828i
$852$	− 2.00000i	− 0.0685189i
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	10.0000	0.342193
$855$	−5.00000	−0.170996
$856$	− 6.00000i	− 0.205076i
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	3.00000	0.102359	0.0511793	−	0.998689i	$-0.483702\pi$
$859$	3.00000	0.102359	0.0511793	+	0.998689i	$0.483702\pi$
$860$	− 6.00000i	− 0.204598i
$861$	−30.0000	−1.02240
$862$	12.0000	0.408722
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	5.00000i	0.170005i
$866$	16.0000i	0.543702i
$867$	47.0000	1.59620
$868$	−10.0000	−0.339422
$869$	− 6.00000i	− 0.203536i
$870$	4.00000	0.135613
$871$	0	0
$872$	−14.0000	−0.474100
$873$	0	0
$874$	−20.0000	−0.676510
$875$	−5.00000	−0.169031
$876$	0	0
$877$	− 54.0000i	− 1.82345i	−0.410801	−	0.911725i	$-0.634751\pi$
$877$	− 54.0000i	− 1.82345i	0.410801	−	0.911725i	$-0.365249\pi$
$878$	10.0000i	0.337484i
$879$	9.00000i	0.303562i
$880$	3.00000	0.101130
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	18.0000i	0.606092i
$883$	−30.0000	−1.00958	−0.504790	−	0.863242i	$-0.668430\pi$
$883$	−30.0000	−1.00958	−0.504790	+	0.863242i	$0.668430\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	− 6.00000i	− 0.201574i
$887$	41.0000	1.37665	0.688323	−	0.725405i	$-0.258347\pi$
$887$	41.0000	1.37665	0.688323	+	0.725405i	$0.258347\pi$
$888$	−7.00000	−0.234905
$889$	105.000i	3.52159i
$890$	− 11.0000i	− 0.368721i
$891$	3.00000i	0.100504i
$892$	3.00000i	0.100447i
$893$	15.0000	0.501956
$894$	−2.00000	−0.0668900
$895$	4.00000i	0.133705i
$896$	−5.00000	−0.167038
$897$	0	0
$898$	27.0000	0.901002
$899$	8.00000i	0.266815i
$900$	1.00000	0.0333333
$901$	8.00000	0.266519
$902$	18.0000i	0.599334i
$903$	− 30.0000i	− 0.998337i
$904$	8.00000i	0.266076i
$905$	− 2.00000i	− 0.0664822i
$906$	−22.0000	−0.730901
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	− 5.00000i	− 0.165567i
$913$	−24.0000	−0.794284
$914$	30.0000	0.992312
$915$	− 2.00000i	− 0.0661180i
$916$	14.0000i	0.462573i
$917$	− 95.0000i	− 3.13718i
$918$	− 8.00000i	− 0.264039i
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	4.00000	0.131876
$921$	6.00000i	0.197707i
$922$	8.00000	0.263466
$923$	0	0
$924$	15.0000	0.493464
$925$	− 7.00000i	− 0.230159i
$926$	−8.00000	−0.262896
$927$	7.00000	0.229910
$928$	4.00000i	0.131306i
$929$	2.00000i	0.0656179i	0.999462	+	0.0328089i	$0.0104453\pi$
$929$	2.00000i	0.0656179i	−0.999462	+	0.0328089i	$0.989555\pi$
$930$	2.00000i	0.0655826i
$931$	90.0000i	2.94963i
$932$	−14.0000	−0.458585
$933$	12.0000	0.392862
$934$	− 24.0000i	− 0.785304i
$935$	24.0000	0.784884
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	40.0000i	1.30605i
$939$	−6.00000	−0.195803
$940$	−3.00000	−0.0978492
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	− 15.0000i	− 0.488726i
$943$	24.0000i	0.781548i
$944$	− 12.0000i	− 0.390567i
$945$	5.00000	0.162650
$946$	−18.0000	−0.585230
$947$	− 58.0000i	− 1.88475i	−0.334563	−	0.942373i	$-0.608589\pi$
$947$	− 58.0000i	− 1.88475i	0.334563	−	0.942373i	$-0.391411\pi$
$948$	2.00000	0.0649570
$949$	0	0
$950$	5.00000	0.162221
$951$	− 23.0000i	− 0.745826i
$952$	−40.0000	−1.29641
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	− 1.00000i	− 0.0323762i
$955$	− 2.00000i	− 0.0647185i
$956$	18.0000i	0.582162i
$957$	− 12.0000i	− 0.387905i
$958$	28.0000	0.904639
$959$	−60.0000	−1.93750
$960$	1.00000i	0.0322749i
$961$	27.0000	0.870968
$962$	0	0
$963$	−6.00000	−0.193347
$964$	− 25.0000i	− 0.805196i
$965$	−24.0000	−0.772587
$966$	20.0000	0.643489
$967$	− 31.0000i	− 0.996893i	−0.866921	−	0.498446i	$-0.833904\pi$
$967$	− 31.0000i	− 0.996893i	0.866921	−	0.498446i	$-0.166096\pi$
$968$	2.00000i	0.0642824i
$969$	− 40.0000i	− 1.28499i
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	−1.00000	−0.0320750
$973$	35.0000i	1.12205i
$974$	37.0000	1.18556
$975$	0	0
$976$	2.00000	0.0640184
$977$	20.0000i	0.639857i	0.947442	+	0.319928i	$0.103659\pi$
$977$	20.0000i	0.639857i	−0.947442	+	0.319928i	$0.896341\pi$
$978$	20.0000	0.639529
$979$	−33.0000	−1.05468
$980$	− 18.0000i	− 0.574989i
$981$	14.0000i	0.446986i
$982$	21.0000i	0.670137i
$983$	− 23.0000i	− 0.733586i	−0.930303	−	0.366793i	$-0.880456\pi$
$983$	− 23.0000i	− 0.733586i	0.930303	−	0.366793i	$-0.119544\pi$
$984$	−6.00000	−0.191273
$985$	3.00000	0.0955879
$986$	32.0000i	1.01909i
$987$	−15.0000	−0.477455
$988$	0	0
$989$	−24.0000	−0.763156
$990$	− 3.00000i	− 0.0953463i
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	−2.00000	−0.0635001
$993$	4.00000i	0.126936i
$994$	10.0000i	0.317181i
$995$	22.0000i	0.697447i
$996$	− 8.00000i	− 0.253490i
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	4.00000	0.126618
$999$	7.00000i	0.221470i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +5.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} +5.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000i q^{18} -5.00000i q^{19} +1.00000i q^{20} +5.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -5.00000i q^{28} -4.00000 q^{29} -1.00000 q^{30} -2.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} -8.00000i q^{34} +5.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} -5.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +5.00000 q^{42} -6.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} -18.0000 q^{49} +1.00000i q^{50} +8.00000 q^{51} +1.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} -5.00000 q^{56} -5.00000i q^{57} +4.00000i q^{58} -12.0000i q^{59} +1.00000i q^{60} +2.00000 q^{61} -2.00000 q^{62} +5.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} -8.00000 q^{68} +4.00000 q^{69} -5.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} +7.00000 q^{74} -1.00000 q^{75} +5.00000i q^{76} -15.0000 q^{77} -2.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000i q^{83} -5.00000i q^{84} -8.00000i q^{85} +6.00000i q^{86} -4.00000 q^{87} -3.00000 q^{88} +11.0000i q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000i q^{93} +3.00000 q^{94} -5.00000 q^{95} -1.00000i q^{96} +18.0000i q^{98} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 10 q^{14} + 2 q^{16} + 16 q^{17} + 6 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{27} - 8 q^{29} - 2 q^{30} + 10 q^{35} - 2 q^{36} - 10 q^{38} + 2 q^{40} + 10 q^{42} - 12 q^{43} + 2 q^{48} - 36 q^{49} + 16 q^{51} + 2 q^{53} + 6 q^{55} - 10 q^{56} + 4 q^{61} - 4 q^{62} - 2 q^{64} + 6 q^{66} - 16 q^{68} + 8 q^{69} + 14 q^{74} - 2 q^{75} - 30 q^{77} - 4 q^{79} + 2 q^{81} + 12 q^{82} - 8 q^{87} - 6 q^{88} - 2 q^{90} - 8 q^{92} + 6 q^{94} - 10 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 10 * q^14 + 2 * q^16 + 16 * q^17 + 6 * q^22 + 8 * q^23 - 2 * q^25 + 2 * q^27 - 8 * q^29 - 2 * q^30 + 10 * q^35 - 2 * q^36 - 10 * q^38 + 2 * q^40 + 10 * q^42 - 12 * q^43 + 2 * q^48 - 36 * q^49 + 16 * q^51 + 2 * q^53 + 6 * q^55 - 10 * q^56 + 4 * q^61 - 4 * q^62 - 2 * q^64 + 6 * q^66 - 16 * q^68 + 8 * q^69 + 14 * q^74 - 2 * q^75 - 30 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^82 - 8 * q^87 - 6 * q^88 - 2 * q^90 - 8 * q^92 + 6 * q^94 - 10 * q^95

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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