LMFDB - Embedded newform 5070.2.b.c.1351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5070.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,-2,-2,0,0,0,0,2,-2,0,2,0,0,0,2,12,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)

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

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

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000i q^{18} +1.00000i q^{20} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +1.00000 q^{30} -1.00000i q^{32} -6.00000i q^{34} -1.00000 q^{36} +6.00000i q^{37} +1.00000 q^{40} +2.00000i q^{41} +4.00000 q^{43} -1.00000i q^{45} -4.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} +10.0000i q^{58} -1.00000i q^{60} +6.00000 q^{61} -1.00000 q^{64} +4.00000i q^{67} -6.00000 q^{68} -4.00000 q^{69} +16.0000i q^{71} +1.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +1.00000 q^{75} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000i q^{83} -6.00000i q^{85} -4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +1.00000i q^{96} +14.0000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 20 q^{29} + 2 q^{30} - 2 q^{36} + 2 q^{40} + 8 q^{43} - 2 q^{48} + 14 q^{49} - 12 q^{51}+ \cdots - 8 q^{92}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^16 + 12 * q^17 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 20 * q^29 + 2 * q^30 - 2 * q^36 + 2 * q^40 + 8 * q^43 - 2 * q^48 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 12 * q^61 - 2 * q^64 - 12 * q^68 - 8 * q^69 + 12 * q^74 + 2 * q^75 + 2 * q^81 + 4 * q^82 + 20 * q^87 - 2 * q^90 - 8 * q^92

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	0	0
$23$	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
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	1.00000	0.182574
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 6.00000i	− 1.02899i
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	− 4.00000i	− 0.589768i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.00000	−0.144338
$49$	7.00000	1.00000
$50$	1.00000i	0.141421i
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000i	0.136083i
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000i	1.31306i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	− 1.00000i	− 0.129099i
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	−6.00000	−0.727607
$69$	−4.00000	−0.481543
$70$	0	0
$71$	16.0000i	1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$	16.0000i	1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$	1.00000i	0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	− 4.00000i	− 0.431331i
$87$	10.0000	1.07211
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00000i	0.102062i
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000i	0.594089i
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	6.00000i	0.582772i
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	− 6.00000i	− 0.569495i
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	10.0000	0.928477
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	11.0000	1.00000
$122$	− 6.00000i	− 0.543214i
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	1.00000i	0.0860663i
$136$	6.00000i	0.514496i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	4.00000i	0.340503i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	16.0000	1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	10.0000i	0.830455i
$146$	2.00000	0.165521
$147$	−7.00000	−0.577350
$148$	− 6.00000i	− 0.493197i
$149$	− 14.0000i	− 1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	− 14.0000i	− 1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	16.0000i	1.30206i	0.759051	+	0.651031i	$0.225663\pi$
$151$	16.0000i	1.30206i	−0.759051	+	0.651031i	$0.774337\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	− 2.00000i	− 0.156174i
$165$	0	0
$166$	4.00000	0.310460
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	0	0
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−4.00000	−0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	− 10.0000i	− 0.758098i
$175$	0	0
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000i	0.0745356i
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	4.00000i	0.294884i
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 4.00000i	− 0.282138i
$202$	6.00000i	0.422159i
$203$	0	0
$204$	6.00000	0.420084
$205$	2.00000	0.139686
$206$	12.0000i	0.836080i
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	− 16.0000i	− 1.09630i
$214$	4.00000i	0.273434i
$215$	− 4.00000i	− 0.272798i
$216$	− 1.00000i	− 0.0680414i
$217$	0	0
$218$	−14.0000	−0.948200
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	0	0
$222$	−6.00000	−0.402694
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	− 10.0000i	− 0.665190i
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	− 2.00000i	− 0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$	− 2.00000i	− 0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	− 10.0000i	− 0.656532i
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000i	1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$	16.0000i	1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$	1.00000i	0.0645497i
$241$	14.0000i	0.901819i	0.892570	+	0.450910i	$0.148900\pi$
$241$	14.0000i	0.901819i	−0.892570	+	0.450910i	$0.851100\pi$
$242$	− 11.0000i	− 0.707107i
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	− 7.00000i	− 0.447214i
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	− 4.00000i	− 0.253490i
$250$	1.00000	0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	0	0
$254$	− 12.0000i	− 0.752947i
$255$	6.00000i	0.375735i
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	− 12.0000i	− 0.741362i
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	− 4.00000i	− 0.244339i
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	1.00000	0.0608581
$271$	24.0000i	1.45790i	0.684569	+	0.728948i	$0.259990\pi$
$271$	24.0000i	1.45790i	−0.684569	+	0.728948i	$0.740010\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	6.00000i	0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$	6.00000i	0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	− 16.0000i	− 0.949425i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	19.0000	1.11765
$290$	10.0000	0.587220
$291$	− 14.0000i	− 0.820695i
$292$	− 2.00000i	− 0.117041i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	7.00000i	0.408248i
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−14.0000	−0.810998
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	0	0
$305$	− 6.00000i	− 0.343559i
$306$	− 6.00000i	− 0.342997i
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	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
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	14.0000i	0.790066i
$315$	0	0
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	14.0000i	0.774202i
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	− 4.00000i	− 0.219529i
$333$	6.00000i	0.328798i
$334$	−24.0000	−1.31322
$335$	4.00000	0.218543
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	6.00000i	0.325396i
$341$	0	0
$342$	0	0
$343$	0	0
$344$	4.00000i	0.215666i
$345$	4.00000i	0.215353i
$346$	− 14.0000i	− 0.752645i
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−10.0000	−0.536056
$349$	− 34.0000i	− 1.81998i	−0.414632	−	0.909989i	$-0.636090\pi$
$349$	− 34.0000i	− 1.81998i	0.414632	−	0.909989i	$-0.363910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	− 6.00000i	− 0.317999i
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	− 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
$361$	19.0000	1.00000
$362$	− 10.0000i	− 0.525588i
$363$	−11.0000	−0.577350
$364$	0	0
$365$	2.00000	0.104685
$366$	6.00000i	0.313625i
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	4.00000	0.208514
$369$	2.00000i	0.104116i
$370$	− 6.00000i	− 0.311925i
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	− 1.00000i	− 0.0516398i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.0000i	1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$	24.0000i	1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	− 24.0000i	− 1.22795i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000	0.203331
$388$	− 14.0000i	− 0.710742i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	7.00000i	0.353553i
$393$	−12.0000	−0.605320
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	− 18.0000i	− 0.898877i	−0.893311	−	0.449439i	$-0.851624\pi$
$401$	− 18.0000i	− 0.898877i	0.893311	−	0.449439i	$-0.148376\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	2.00000i	0.0988936i	0.998777	+	0.0494468i	$0.0157458\pi$
$409$	2.00000i	0.0988936i	−0.998777	+	0.0494468i	$0.984254\pi$
$410$	− 2.00000i	− 0.0987730i
$411$	6.00000i	0.295958i
$412$	12.0000	0.591198
$413$	0	0
$414$	− 4.00000i	− 0.196589i
$415$	4.00000	0.196352
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	− 6.00000i	− 0.291386i
$425$	−6.00000	−0.291043
$426$	−16.0000	−0.775203
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	−4.00000	−0.192897
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	− 10.0000i	− 0.479463i
$436$	14.0000i	0.670478i
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	6.00000i	0.284747i
$445$	6.00000	0.284427
$446$	16.0000	0.757622
$447$	14.0000i	0.662177i
$448$	0	0
$449$	14.0000i	0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$	14.0000i	0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	1.00000i	0.0471405i
$451$	0	0
$452$	−10.0000	−0.470360
$453$	− 16.0000i	− 0.751746i
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	−2.00000	−0.0934539
$459$	−6.00000	−0.280056
$460$	4.00000i	0.186501i
$461$	− 14.0000i	− 0.652045i	−0.945362	−	0.326023i	$-0.894291\pi$
$461$	− 14.0000i	− 0.652045i	0.945362	−	0.326023i	$-0.105709\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	18.0000i	0.833834i
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	16.0000	0.731823
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	14.0000	0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	14.0000	0.635707
$486$	1.00000i	0.0453609i
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	6.00000i	0.271607i
$489$	− 4.00000i	− 0.180886i
$490$	−7.00000	−0.316228
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	2.00000i	0.0901670i
$493$	−60.0000	−2.70226
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−4.00000	−0.179244
$499$	− 40.0000i	− 1.79065i	−0.445418	−	0.895323i	$-0.646945\pi$
$499$	− 40.0000i	− 1.79065i	0.445418	−	0.895323i	$-0.353055\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	24.0000i	1.07224i
$502$	− 4.00000i	− 0.178529i
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	6.00000i	0.266996i
$506$	0	0
$507$	0	0
$508$	−12.0000	−0.532414
$509$	− 38.0000i	− 1.68432i	−0.539227	−	0.842160i	$-0.681284\pi$
$509$	− 38.0000i	− 1.68432i	0.539227	−	0.842160i	$-0.318716\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 6.00000i	− 0.264649i
$515$	12.0000i	0.528783i
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	10.0000i	0.437688i
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	28.0000i	1.22086i
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	4.00000i	0.172935i
$536$	−4.00000	−0.172774
$537$	−20.0000	−0.863064
$538$	− 14.0000i	− 0.603583i
$539$	0	0
$540$	− 1.00000i	− 0.0430331i
$541$	22.0000i	0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$	22.0000i	0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$	24.0000	1.03089
$543$	−10.0000	−0.429141
$544$	− 6.00000i	− 0.257248i
$545$	−14.0000	−0.599694
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000i	0.256307i
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	2.00000i	0.0849719i
$555$	−6.00000	−0.254686
$556$	−4.00000	−0.169638
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	− 10.0000i	− 0.420703i
$566$	− 12.0000i	− 0.504398i
$567$	0	0
$568$	−16.0000	−0.671345
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	−4.00000	−0.166812
$576$	−1.00000	−0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 19.0000i	− 0.790296i
$579$	14.0000i	0.581820i
$580$	− 10.0000i	− 0.415227i
$581$	0	0
$582$	−14.0000	−0.580319
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	0	0
$591$	22.0000i	0.904959i
$592$	6.00000i	0.246598i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	14.0000i	0.573462i
$597$	−16.0000	−0.654836
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000i	0.0408248i
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	− 16.0000i	− 0.651031i
$605$	− 11.0000i	− 0.447214i
$606$	− 6.00000i	− 0.243733i
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	0	0
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−6.00000	−0.242536
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	4.00000	0.161427
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 12.0000i	− 0.482711i
$619$	− 40.0000i	− 1.60774i	−0.594808	−	0.803868i	$-0.702772\pi$
$619$	− 40.0000i	− 1.60774i	0.594808	−	0.803868i	$-0.297228\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 16.0000i	− 0.641542i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 26.0000i	− 1.03917i
$627$	0	0
$628$	14.0000	0.558661
$629$	36.0000i	1.43541i
$630$	0	0
$631$	48.0000i	1.91085i	0.295234	+	0.955425i	$0.404602\pi$
$631$	48.0000i	1.91085i	−0.295234	+	0.955425i	$0.595398\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	2.00000	0.0794301
$635$	− 12.0000i	− 0.476205i
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	16.0000i	0.632950i
$640$	1.00000	0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	− 4.00000i	− 0.157867i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	14.0000	0.547443
$655$	− 12.0000i	− 0.468879i
$656$	2.00000i	0.0780869i
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	30.0000i	1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$	30.0000i	1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	−40.0000	−1.54881
$668$	24.0000i	0.928588i
$669$	− 16.0000i	− 0.618596i
$670$	− 4.00000i	− 0.154533i
$671$	0	0
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	18.0000i	0.693334i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	10.0000i	0.384048i
$679$	0	0
$680$	6.00000	0.230089
$681$	− 20.0000i	− 0.766402i
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	4.00000	0.152499
$689$	0	0
$690$	4.00000	0.152277
$691$	32.0000i	1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$	32.0000i	1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	− 12.0000i	− 0.455514i
$695$	− 4.00000i	− 0.151729i
$696$	10.0000i	0.379049i
$697$	12.0000i	0.454532i
$698$	−34.0000	−1.28692
$699$	18.0000	0.680823
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−26.0000	−0.978523
$707$	0	0
$708$	0	0
$709$	− 2.00000i	− 0.0751116i	−0.999295	−	0.0375558i	$-0.988043\pi$
$709$	− 2.00000i	− 0.0751116i	0.999295	−	0.0375558i	$-0.0119572\pi$
$710$	− 16.0000i	− 0.600469i
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 16.0000i	− 0.597531i
$718$	−24.0000	−0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	0	0
$722$	− 19.0000i	− 0.707107i
$723$	− 14.0000i	− 0.520666i
$724$	−10.0000	−0.371647
$725$	10.0000	0.371391
$726$	11.0000i	0.408248i
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.00000i	− 0.0740233i
$731$	24.0000	0.887672
$732$	6.00000	0.221766
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	− 20.0000i	− 0.738213i
$735$	7.00000i	0.258199i
$736$	− 4.00000i	− 0.147442i
$737$	0	0
$738$	2.00000	0.0736210
$739$	40.0000i	1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$	40.0000i	1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	− 10.0000i	− 0.366126i
$747$	4.00000i	0.146352i
$748$	0	0
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	0	0
$761$	22.0000i	0.797499i	0.917060	+	0.398750i	$0.130556\pi$
$761$	22.0000i	0.797499i	−0.917060	+	0.398750i	$0.869444\pi$
$762$	12.0000i	0.434714i
$763$	0	0
$764$	−24.0000	−0.868290
$765$	− 6.00000i	− 0.216930i
$766$	−8.00000	−0.289052
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	34.0000i	1.22607i	0.790055	+	0.613036i	$0.210052\pi$
$769$	34.0000i	1.22607i	−0.790055	+	0.613036i	$0.789948\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	14.0000i	0.503871i
$773$	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
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 24.0000i	− 0.858238i
$783$	10.0000	0.357371
$784$	7.00000	0.250000
$785$	14.0000i	0.499681i
$786$	12.0000i	0.428026i
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	22.0000i	0.783718i
$789$	28.0000	0.996826
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	14.0000	0.496841
$795$	− 6.00000i	− 0.212798i
$796$	−16.0000	−0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	6.00000i	0.212000i
$802$	−18.0000	−0.635602
$803$	0	0
$804$	4.00000i	0.141069i
$805$	0	0
$806$	0	0
$807$	−14.0000	−0.492823
$808$	− 6.00000i	− 0.211079i
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−1.00000	−0.0351364
$811$	32.0000i	1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$	32.0000i	1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	0	0
$815$	4.00000	0.140114
$816$	−6.00000	−0.210042
$817$	0	0
$818$	2.00000	0.0699284
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	6.00000	0.209274
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	− 12.0000i	− 0.418040i
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	−4.00000	−0.139010
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	− 4.00000i	− 0.138842i
$831$	2.00000	0.0693792
$832$	0	0
$833$	42.0000	1.45521
$834$	4.00000i	0.138509i
$835$	−24.0000	−0.830554
$836$	0	0
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−22.0000	−0.758170
$843$	− 6.00000i	− 0.206651i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−12.0000	−0.411839
$850$	6.00000i	0.205798i
$851$	24.0000i	0.822709i
$852$	16.0000i	0.548151i
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	0	0
$855$	0	0
$856$	− 4.00000i	− 0.136717i
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	4.00000i	0.136399i
$861$	0	0
$862$	−24.0000	−0.817443
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	1.00000i	0.0340207i
$865$	− 14.0000i	− 0.476014i
$866$	2.00000i	0.0679628i
$867$	−19.0000	−0.645274
$868$	0	0
$869$	0	0
$870$	−10.0000	−0.339032
$871$	0	0
$872$	14.0000	0.474100
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	2.00000i	0.0675737i
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 24.0000i	− 0.809961i
$879$	26.0000i	0.876958i
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	− 7.00000i	− 0.235702i
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	4.00000i	0.134383i
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	− 6.00000i	− 0.201120i
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	14.0000	0.468230
$895$	− 20.0000i	− 0.668526i
$896$	0	0
$897$	0	0
$898$	14.0000	0.467186
$899$	0	0
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	10.0000i	0.332595i
$905$	− 10.0000i	− 0.332411i
$906$	−16.0000	−0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	0	0
$914$	6.00000	0.198462
$915$	6.00000i	0.198354i
$916$	2.00000i	0.0660819i
$917$	0	0
$918$	6.00000i	0.198030i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	4.00000	0.131876
$921$	− 4.00000i	− 0.131804i
$922$	−14.0000	−0.461065
$923$	0	0
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	24.0000	0.788689
$927$	−12.0000	−0.394132
$928$	10.0000i	0.328266i
$929$	− 46.0000i	− 1.50921i	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	− 46.0000i	− 1.50921i	0.656179	−	0.754606i	$-0.272172\pi$
$930$	0	0
$931$	0	0
$932$	18.0000	0.589610
$933$	−16.0000	−0.523816
$934$	− 28.0000i	− 0.916188i
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	10.0000i	0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$	10.0000i	0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$	− 14.0000i	− 0.456145i
$943$	8.00000i	0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 2.00000i	− 0.0648544i
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	6.00000i	0.194257i
$955$	− 24.0000i	− 0.776622i
$956$	− 16.0000i	− 0.517477i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	− 1.00000i	− 0.0322749i
$961$	31.0000	1.00000
$962$	0	0
$963$	−4.00000	−0.128898
$964$	− 14.0000i	− 0.450910i
$965$	−14.0000	−0.450676
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	− 14.0000i	− 0.449513i
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	− 2.00000i	− 0.0639857i	−0.999488	−	0.0319928i	$-0.989815\pi$
$977$	− 2.00000i	− 0.0639857i	0.999488	−	0.0319928i	$-0.0101854\pi$
$978$	−4.00000	−0.127906
$979$	0	0
$980$	7.00000i	0.223607i
$981$	− 14.0000i	− 0.446986i
$982$	20.0000i	0.638226i
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	2.00000	0.0637577
$985$	−22.0000	−0.700978
$986$	60.0000i	1.91079i
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	− 16.0000i	− 0.507234i
$996$	4.00000i	0.126745i
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−40.0000	−1.26618
$999$	− 6.00000i	− 0.189832i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.a.1.1	✓	1	13.5	odd	4
1170.2.a.m.1.1		1	39.5	even	4
1950.2.a.y.1.1		1	65.44	odd	4
1950.2.e.l.1249.1		2	65.57	even	4
1950.2.e.l.1249.2		2	65.18	even	4
3120.2.a.q.1.1		1	52.31	even	4
5070.2.a.s.1.1		1	13.8	odd	4
5070.2.b.c.1351.1		2	1.1	even	1	trivial
5070.2.b.c.1351.2		2	13.12	even	2	inner
5850.2.a.m.1.1		1	195.44	even	4
5850.2.e.p.5149.1		2	195.83	odd	4
5850.2.e.p.5149.2		2	195.122	odd	4
9360.2.a.bn.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000i q^{18} +1.00000i q^{20} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +1.00000 q^{30} -1.00000i q^{32} -6.00000i q^{34} -1.00000 q^{36} +6.00000i q^{37} +1.00000 q^{40} +2.00000i q^{41} +4.00000 q^{43} -1.00000i q^{45} -4.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} +10.0000i q^{58} -1.00000i q^{60} +6.00000 q^{61} -1.00000 q^{64} +4.00000i q^{67} -6.00000 q^{68} -4.00000 q^{69} +16.0000i q^{71} +1.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +1.00000 q^{75} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000i q^{83} -6.00000i q^{85} -4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +1.00000i q^{96} +14.0000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 20 q^{29} + 2 q^{30} - 2 q^{36} + 2 q^{40} + 8 q^{43} - 2 q^{48} + 14 q^{49} - 12 q^{51}+ \cdots - 8 q^{92}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^16 + 12 * q^17 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 20 * q^29 + 2 * q^30 - 2 * q^36 + 2 * q^40 + 8 * q^43 - 2 * q^48 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 12 * q^61 - 2 * q^64 - 12 * q^68 - 8 * q^69 + 12 * q^74 + 2 * q^75 + 2 * q^81 + 4 * q^82 + 20 * q^87 - 2 * q^90 - 8 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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