LMFDB - Embedded newform 5070.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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, 0]))

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

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

chi := DirichletCharacter("5070.1");

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.a (trivial)

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:	yes
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

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

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	5.00000	1.33631
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	−1.00000	−0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−1.00000	−0.223607
$21$	−5.00000	−1.09109
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−5.00000	−0.944911
$29$	−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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	8.00000	1.37199
$35$	5.00000	0.845154
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	5.00000	0.771517
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−3.00000	−0.452267
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	18.0000	2.57143
$50$	−1.00000	−0.141421
$51$	−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	$0.478121\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	5.00000	0.668153
$57$	−5.00000	−0.662266
$58$	4.00000	0.525226
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−1.00000	−0.129099
$61$	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
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−8.00000	−0.970143
$69$	−4.00000	−0.481543
$70$	−5.00000	−0.597614
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	7.00000	0.813733
$75$	1.00000	0.115470
$76$	−5.00000	−0.573539
$77$	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	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−5.00000	−0.545545
$85$	8.00000	0.867722
$86$	−6.00000	−0.646997
$87$	−4.00000	−0.428845
$88$	3.00000	0.319801
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−2.00000	−0.207390
$94$	3.00000	0.309426
$95$	5.00000	0.512989
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−18.0000	−1.81827
$99$	−3.00000	−0.301511
$100$	1.00000	0.100000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	8.00000	0.792118
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	−1.00000	−0.0971286
$107$	−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
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−3.00000	−0.286039
$111$	−7.00000	−0.664411
$112$	−5.00000	−0.472456
$113$	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
$115$	4.00000	0.373002
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	40.0000	3.66679
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	5.00000	0.445435
$127$	−21.0000	−1.86345	−0.931724	−	0.363166i	$-0.881696\pi$
$127$	−21.0000	−1.86345	−0.931724	+	0.363166i	$0.881696\pi$
$128$	−1.00000	−0.0883883
$129$	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	$0.811670\pi$
$132$	−3.00000	−0.261116
$133$	25.0000	2.16777
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	8.00000	0.685994
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	4.00000	0.340503
$139$	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
$141$	−3.00000	−0.252646
$142$	−2.00000	−0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	0	0
$147$	18.0000	1.48461
$148$	−7.00000	−0.575396
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	−1.00000	−0.0816497
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	5.00000	0.405554
$153$	−8.00000	−0.646762
$154$	−15.0000	−1.20873
$155$	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	$0.295738\pi$
$158$	2.00000	0.159111
$159$	1.00000	0.0793052
$160$	1.00000	0.0790569
$161$	20.0000	1.57622
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	3.00000	0.233550
$166$	−8.00000	−0.620920
$167$	−23.0000	−1.77979	−0.889897	−	0.456162i	$-0.849224\pi$
$167$	−23.0000	−1.77979	−0.889897	+	0.456162i	$0.849224\pi$
$168$	5.00000	0.385758
$169$	0	0
$170$	−8.00000	−0.613572
$171$	−5.00000	−0.382360
$172$	6.00000	0.457496
$173$	5.00000	0.380143	0.190071	−	0.981770i	$-0.439128\pi$
$173$	5.00000	0.380143	0.190071	+	0.981770i	$0.439128\pi$
$174$	4.00000	0.303239
$175$	−5.00000	−0.377964
$176$	−3.00000	−0.226134
$177$	12.0000	0.901975
$178$	11.0000	0.824485
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	4.00000	0.294884
$185$	7.00000	0.514650
$186$	2.00000	0.146647
$187$	24.0000	1.75505
$188$	−3.00000	−0.218797
$189$	−5.00000	−0.363696
$190$	−5.00000	−0.362738
$191$	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
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	18.0000	1.28571
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	3.00000	0.213201
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	8.00000	0.562878
$203$	20.0000	1.40372
$204$	−8.00000	−0.560112
$205$	−6.00000	−0.419058
$206$	7.00000	0.487713
$207$	−4.00000	−0.278019
$208$	0	0
$209$	15.0000	1.03757
$210$	−5.00000	−0.345033
$211$	−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
$213$	2.00000	0.137038
$214$	6.00000	0.410152
$215$	−6.00000	−0.409197
$216$	−1.00000	−0.0680414
$217$	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
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	5.00000	0.334077
$225$	1.00000	0.0666667
$226$	−8.00000	−0.532152
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−5.00000	−0.331133
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−4.00000	−0.263752
$231$	15.0000	0.986928
$232$	4.00000	0.262613
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	12.0000	0.781133
$237$	−2.00000	−0.129914
$238$	−40.0000	−2.59281
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	−1.00000	−0.0645497
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−18.0000	−1.14998
$246$	−6.00000	−0.382546
$247$	0	0
$248$	2.00000	0.127000
$249$	8.00000	0.506979
$250$	1.00000	0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	−5.00000	−0.314970
$253$	12.0000	0.754434
$254$	21.0000	1.31766
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	−6.00000	−0.373544
$259$	35.0000	2.17479
$260$	0	0
$261$	−4.00000	−0.247594
$262$	19.0000	1.17382
$263$	−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
$265$	−1.00000	−0.0614295
$266$	−25.0000	−1.53285
$267$	−11.0000	−0.673189
$268$	8.00000	0.488678
$269$	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
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	12.0000	0.724947
$275$	−3.00000	−0.180907
$276$	−4.00000	−0.240772
$277$	−15.0000	−0.901263	−0.450631	−	0.892710i	$-0.648801\pi$
$277$	−15.0000	−0.901263	−0.450631	+	0.892710i	$0.648801\pi$
$278$	−7.00000	−0.419832
$279$	−2.00000	−0.119737
$280$	−5.00000	−0.298807
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	3.00000	0.178647
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	2.00000	0.118678
$285$	5.00000	0.296174
$286$	0	0
$287$	−30.0000	−1.77084
$288$	−1.00000	−0.0589256
$289$	47.0000	2.76471
$290$	−4.00000	−0.234888
$291$	0	0
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−18.0000	−1.04978
$295$	−12.0000	−0.698667
$296$	7.00000	0.406867
$297$	−3.00000	−0.174078
$298$	2.00000	0.115857
$299$	0	0
$300$	1.00000	0.0577350
$301$	−30.0000	−1.72917
$302$	−22.0000	−1.26596
$303$	−8.00000	−0.459588
$304$	−5.00000	−0.286770
$305$	−2.00000	−0.114520
$306$	8.00000	0.457330
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	15.0000	0.854704
$309$	−7.00000	−0.398216
$310$	−2.00000	−0.113592
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−15.0000	−0.846499
$315$	5.00000	0.281718
$316$	−2.00000	−0.112509
$317$	−23.0000	−1.29181	−0.645904	−	0.763418i	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	−1.00000	−0.0560772
$319$	12.0000	0.671871
$320$	−1.00000	−0.0559017
$321$	−6.00000	−0.334887
$322$	−20.0000	−1.11456
$323$	40.0000	2.22566
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	14.0000	0.774202
$328$	−6.00000	−0.331295
$329$	15.0000	0.826977
$330$	−3.00000	−0.165145
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	8.00000	0.439057
$333$	−7.00000	−0.383598
$334$	23.0000	1.25850
$335$	−8.00000	−0.437087
$336$	−5.00000	−0.272772
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	8.00000	0.433861
$341$	6.00000	0.324918
$342$	5.00000	0.270369
$343$	−55.0000	−2.96972
$344$	−6.00000	−0.323498
$345$	4.00000	0.215353
$346$	−5.00000	−0.268802
$347$	−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
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	5.00000	0.267261
$351$	0	0
$352$	3.00000	0.159901
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	−12.0000	−0.637793
$355$	−2.00000	−0.106149
$356$	−11.0000	−0.582999
$357$	40.0000	2.11702
$358$	−4.00000	−0.211407
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	1.00000	0.0527046
$361$	6.00000	0.315789
$362$	2.00000	0.105118
$363$	−2.00000	−0.104973
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	−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
$369$	6.00000	0.312348
$370$	−7.00000	−0.363913
$371$	−5.00000	−0.259587
$372$	−2.00000	−0.103695
$373$	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
$375$	−1.00000	−0.0516398
$376$	3.00000	0.154713
$377$	0	0
$378$	5.00000	0.257172
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	5.00000	0.256495
$381$	−21.0000	−1.07586
$382$	−2.00000	−0.102329
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	−1.00000	−0.0510310
$385$	−15.0000	−0.764471
$386$	−24.0000	−1.22157
$387$	6.00000	0.304997
$388$	0	0
$389$	−32.0000	−1.62246	−0.811232	−	0.584724i	$-0.801203\pi$
$389$	−32.0000	−1.62246	−0.811232	+	0.584724i	$0.801203\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	−18.0000	−0.909137
$393$	−19.0000	−0.958423
$394$	−3.00000	−0.151138
$395$	2.00000	0.100631
$396$	−3.00000	−0.150756
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−22.0000	−1.10276
$399$	25.0000	1.25157
$400$	1.00000	0.0500000
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	−8.00000	−0.398015
$405$	−1.00000	−0.0496904
$406$	−20.0000	−0.992583
$407$	21.0000	1.04093
$408$	8.00000	0.396059
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	6.00000	0.296319
$411$	−12.0000	−0.591916
$412$	−7.00000	−0.344865
$413$	−60.0000	−2.95241
$414$	4.00000	0.196589
$415$	−8.00000	−0.392705
$416$	0	0
$417$	7.00000	0.342791
$418$	−15.0000	−0.733674
$419$	−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
$421$	12.0000	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$422$	15.0000	0.730189
$423$	−3.00000	−0.145865
$424$	−1.00000	−0.0485643
$425$	−8.00000	−0.388057
$426$	−2.00000	−0.0969003
$427$	−10.0000	−0.483934
$428$	−6.00000	−0.290021
$429$	0	0
$430$	6.00000	0.289346
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−10.0000	−0.480015
$435$	4.00000	0.191785
$436$	14.0000	0.670478
$437$	20.0000	0.956730
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	−3.00000	−0.143019
$441$	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	$0.454475\pi$
$444$	−7.00000	−0.332205
$445$	11.0000	0.521450
$446$	3.00000	0.142054
$447$	−2.00000	−0.0945968
$448$	−5.00000	−0.236228
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	−1.00000	−0.0471405
$451$	−18.0000	−0.847587
$452$	8.00000	0.376288
$453$	22.0000	1.03365
$454$	0	0
$455$	0	0
$456$	5.00000	0.234146
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	−14.0000	−0.654177
$459$	−8.00000	−0.373408
$460$	4.00000	0.186501
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	−15.0000	−0.697863
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−4.00000	−0.185695
$465$	2.00000	0.0927478
$466$	14.0000	0.648537
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	−3.00000	−0.138380
$471$	15.0000	0.691164
$472$	−12.0000	−0.552345
$473$	−18.0000	−0.827641
$474$	2.00000	0.0918630
$475$	−5.00000	−0.229416
$476$	40.0000	1.83340
$477$	1.00000	0.0457869
$478$	18.0000	0.823301
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	25.0000	1.13872
$483$	20.0000	0.910032
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	37.0000	1.67663	0.838315	−	0.545186i	$-0.183541\pi$
$487$	37.0000	1.67663	0.838315	+	0.545186i	$0.183541\pi$
$488$	−2.00000	−0.0905357
$489$	−20.0000	−0.904431
$490$	18.0000	0.813157
$491$	21.0000	0.947717	0.473858	−	0.880601i	$-0.342861\pi$
$491$	21.0000	0.947717	0.473858	+	0.880601i	$0.342861\pi$
$492$	6.00000	0.270501
$493$	32.0000	1.44121
$494$	0	0
$495$	3.00000	0.134840
$496$	−2.00000	−0.0898027
$497$	−10.0000	−0.448561
$498$	−8.00000	−0.358489
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	−23.0000	−1.02756
$502$	−15.0000	−0.669483
$503$	−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
$505$	8.00000	0.355995
$506$	−12.0000	−0.533465
$507$	0	0
$508$	−21.0000	−0.931724
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−8.00000	−0.354246
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−5.00000	−0.220755
$514$	−28.0000	−1.23503
$515$	7.00000	0.308457
$516$	6.00000	0.264135
$517$	9.00000	0.395820
$518$	−35.0000	−1.53781
$519$	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	$0.465066\pi$
$522$	4.00000	0.175075
$523$	−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
$525$	−5.00000	−0.218218
$526$	15.0000	0.654031
$527$	16.0000	0.696971
$528$	−3.00000	−0.130558
$529$	−7.00000	−0.304348
$530$	1.00000	0.0434372
$531$	12.0000	0.520756
$532$	25.0000	1.08389
$533$	0	0
$534$	11.0000	0.476017
$535$	6.00000	0.259403
$536$	−8.00000	−0.345547
$537$	4.00000	0.172613
$538$	−4.00000	−0.172452
$539$	−54.0000	−2.32594
$540$	−1.00000	−0.0430331
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−4.00000	−0.171815
$543$	−2.00000	−0.0858282
$544$	8.00000	0.342997
$545$	−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	$0.459057\pi$
$548$	−12.0000	−0.512615
$549$	2.00000	0.0853579
$550$	3.00000	0.127920
$551$	20.0000	0.852029
$552$	4.00000	0.170251
$553$	10.0000	0.425243
$554$	15.0000	0.637289
$555$	7.00000	0.297133
$556$	7.00000	0.296866
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	2.00000	0.0846668
$559$	0	0
$560$	5.00000	0.211289
$561$	24.0000	1.01328
$562$	−18.0000	−0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−3.00000	−0.126323
$565$	−8.00000	−0.336563
$566$	10.0000	0.420331
$567$	−5.00000	−0.209980
$568$	−2.00000	−0.0839181
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	−5.00000	−0.209427
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	30.0000	1.25218
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−47.0000	−1.95494
$579$	24.0000	0.997406
$580$	4.00000	0.166091
$581$	−40.0000	−1.65948
$582$	0	0
$583$	−3.00000	−0.124247
$584$	0	0
$585$	0	0
$586$	9.00000	0.371787
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	18.0000	0.742307
$589$	10.0000	0.412043
$590$	12.0000	0.494032
$591$	3.00000	0.123404
$592$	−7.00000	−0.287698
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	3.00000	0.123091
$595$	−40.0000	−1.63984
$596$	−2.00000	−0.0819232
$597$	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	$0.255581\pi$
$600$	−1.00000	−0.0408248
$601$	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
$603$	8.00000	0.325785
$604$	22.0000	0.895167
$605$	2.00000	0.0813116
$606$	8.00000	0.324978
$607$	−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
$609$	20.0000	0.810441
$610$	2.00000	0.0809776
$611$	0	0
$612$	−8.00000	−0.323381
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	6.00000	0.242140
$615$	−6.00000	−0.241943
$616$	−15.0000	−0.604367
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	7.00000	0.281581
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	2.00000	0.0803219
$621$	−4.00000	−0.160514
$622$	12.0000	0.481156
$623$	55.0000	2.20353
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	15.0000	0.599042
$628$	15.0000	0.598565
$629$	56.0000	2.23287
$630$	−5.00000	−0.199205
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	2.00000	0.0795557
$633$	−15.0000	−0.596196
$634$	23.0000	0.913447
$635$	21.0000	0.833360
$636$	1.00000	0.0396526
$637$	0	0
$638$	−12.0000	−0.475085
$639$	2.00000	0.0791188
$640$	1.00000	0.0395285
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	6.00000	0.236801
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	20.0000	0.788110
$645$	−6.00000	−0.236250
$646$	−40.0000	−1.57378
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	−1.00000	−0.0392837
$649$	−36.0000	−1.41312
$650$	0	0
$651$	10.0000	0.391931
$652$	−20.0000	−0.783260
$653$	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
$655$	19.0000	0.742391
$656$	6.00000	0.234261
$657$	0	0
$658$	−15.0000	−0.584761
$659$	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
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−8.00000	−0.310460
$665$	−25.0000	−0.969458
$666$	7.00000	0.271244
$667$	16.0000	0.619522
$668$	−23.0000	−0.889897
$669$	−3.00000	−0.115987
$670$	8.00000	0.309067
$671$	−6.00000	−0.231627
$672$	5.00000	0.192879
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	−14.0000	−0.539260
$675$	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	$0.638940\pi$
$678$	−8.00000	−0.307238
$679$	0	0
$680$	−8.00000	−0.306786
$681$	0	0
$682$	−6.00000	−0.229752
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	−5.00000	−0.191180
$685$	12.0000	0.458496
$686$	55.0000	2.09991
$687$	14.0000	0.534133
$688$	6.00000	0.228748
$689$	0	0
$690$	−4.00000	−0.152277
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	5.00000	0.190071
$693$	15.0000	0.569803
$694$	16.0000	0.607352
$695$	−7.00000	−0.265525
$696$	4.00000	0.151620
$697$	−48.0000	−1.81813
$698$	−8.00000	−0.302804
$699$	−14.0000	−0.529529
$700$	−5.00000	−0.188982
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	35.0000	1.32005
$704$	−3.00000	−0.113067
$705$	3.00000	0.112987
$706$	−16.0000	−0.602168
$707$	40.0000	1.50435
$708$	12.0000	0.450988
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	2.00000	0.0750587
$711$	−2.00000	−0.0750059
$712$	11.0000	0.412242
$713$	8.00000	0.299602
$714$	−40.0000	−1.49696
$715$	0	0
$716$	4.00000	0.149487
$717$	−18.0000	−0.672222
$718$	18.0000	0.671754
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	−1.00000	−0.0372678
$721$	35.0000	1.30347
$722$	−6.00000	−0.223297
$723$	−25.0000	−0.929760
$724$	−2.00000	−0.0743294
$725$	−4.00000	−0.148556
$726$	2.00000	0.0742270
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	2.00000	0.0739221
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	8.00000	0.295285
$735$	−18.0000	−0.663940
$736$	4.00000	0.147442
$737$	−24.0000	−0.884051
$738$	−6.00000	−0.220863
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	7.00000	0.257325
$741$	0	0
$742$	5.00000	0.183556
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	2.00000	0.0733236
$745$	2.00000	0.0732743
$746$	−38.0000	−1.39128
$747$	8.00000	0.292705
$748$	24.0000	0.877527
$749$	30.0000	1.09618
$750$	1.00000	0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−3.00000	−0.109399
$753$	15.0000	0.546630
$754$	0	0
$755$	−22.0000	−0.800662
$756$	−5.00000	−0.181848
$757$	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
$759$	12.0000	0.435572
$760$	−5.00000	−0.181369
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	21.0000	0.760750
$763$	−70.0000	−2.53417
$764$	2.00000	0.0723575
$765$	8.00000	0.289241
$766$	28.0000	1.01168
$767$	0	0
$768$	1.00000	0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	15.0000	0.540562
$771$	28.0000	1.00840
$772$	24.0000	0.863779
$773$	1.00000	0.0359675	0.0179838	−	0.999838i	$-0.494275\pi$
$773$	1.00000	0.0359675	0.0179838	+	0.999838i	$0.494275\pi$
$774$	−6.00000	−0.215666
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	35.0000	1.25562
$778$	32.0000	1.14726
$779$	−30.0000	−1.07486
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−32.0000	−1.14432
$783$	−4.00000	−0.142948
$784$	18.0000	0.642857
$785$	−15.0000	−0.535373
$786$	19.0000	0.677708
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	3.00000	0.106871
$789$	−15.0000	−0.534014
$790$	−2.00000	−0.0711568
$791$	−40.0000	−1.42224
$792$	3.00000	0.106600
$793$	0	0
$794$	−25.0000	−0.887217
$795$	−1.00000	−0.0354663
$796$	22.0000	0.779769
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−25.0000	−0.884990
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	−11.0000	−0.388666
$802$	19.0000	0.670913
$803$	0	0
$804$	8.00000	0.282138
$805$	−20.0000	−0.704907
$806$	0	0
$807$	4.00000	0.140807
$808$	8.00000	0.281439
$809$	−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
$811$	25.0000	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$811$	25.0000	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$812$	20.0000	0.701862
$813$	4.00000	0.140286
$814$	−21.0000	−0.736050
$815$	20.0000	0.700569
$816$	−8.00000	−0.280056
$817$	−30.0000	−1.04957
$818$	−25.0000	−0.874105
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	12.0000	0.418548
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	7.00000	0.243857
$825$	−3.00000	−0.104447
$826$	60.0000	2.08767
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	−4.00000	−0.139010
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	8.00000	0.277684
$831$	−15.0000	−0.520344
$832$	0	0
$833$	−144.000	−4.98930
$834$	−7.00000	−0.242390
$835$	23.0000	0.795948
$836$	15.0000	0.518786
$837$	−2.00000	−0.0691301
$838$	12.0000	0.414533
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	−5.00000	−0.172516
$841$	−13.0000	−0.448276
$842$	−12.0000	−0.413547
$843$	18.0000	0.619953
$844$	−15.0000	−0.516321
$845$	0	0
$846$	3.00000	0.103142
$847$	10.0000	0.343604
$848$	1.00000	0.0343401
$849$	−10.0000	−0.343199
$850$	8.00000	0.274398
$851$	28.0000	0.959828
$852$	2.00000	0.0685189
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	10.0000	0.342193
$855$	5.00000	0.170996
$856$	6.00000	0.205076
$857$	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
$859$	3.00000	0.102359	0.0511793	−	0.998689i	$-0.483702\pi$
$859$	3.00000	0.102359	0.0511793	+	0.998689i	$0.483702\pi$
$860$	−6.00000	−0.204598
$861$	−30.0000	−1.02240
$862$	−12.0000	−0.408722
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	−5.00000	−0.170005
$866$	−16.0000	−0.543702
$867$	47.0000	1.59620
$868$	10.0000	0.339422
$869$	6.00000	0.203536
$870$	−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
$877$	−54.0000	−1.82345	−0.911725	−	0.410801i	$-0.865249\pi$
$877$	−54.0000	−1.82345	−0.911725	+	0.410801i	$0.865249\pi$
$878$	−10.0000	−0.337484
$879$	−9.00000	−0.303562
$880$	3.00000	0.101130
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	−18.0000	−0.606092
$883$	30.0000	1.00958	0.504790	−	0.863242i	$-0.331570\pi$
$883$	30.0000	1.00958	0.504790	+	0.863242i	$0.331570\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	−6.00000	−0.201574
$887$	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
$889$	105.000	3.52159
$890$	−11.0000	−0.368721
$891$	−3.00000	−0.100504
$892$	−3.00000	−0.100447
$893$	15.0000	0.501956
$894$	2.00000	0.0668900
$895$	−4.00000	−0.133705
$896$	5.00000	0.167038
$897$	0	0
$898$	27.0000	0.901002
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	−8.00000	−0.266519
$902$	18.0000	0.599334
$903$	−30.0000	−0.998337
$904$	−8.00000	−0.266076
$905$	2.00000	0.0664822
$906$	−22.0000	−0.730901
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	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	$0.607492\pi$
$912$	−5.00000	−0.165567
$913$	−24.0000	−0.794284
$914$	−30.0000	−0.992312
$915$	−2.00000	−0.0661180
$916$	14.0000	0.462573
$917$	95.0000	3.13718
$918$	8.00000	0.264039
$919$	−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
$921$	−6.00000	−0.197707
$922$	−8.00000	−0.263466
$923$	0	0
$924$	15.0000	0.493464
$925$	−7.00000	−0.230159
$926$	−8.00000	−0.262896
$927$	−7.00000	−0.229910
$928$	4.00000	0.131306
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	−2.00000	−0.0655826
$931$	−90.0000	−2.94963
$932$	−14.0000	−0.458585
$933$	−12.0000	−0.392862
$934$	24.0000	0.785304
$935$	−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	$0.336986\pi$
$938$	40.0000	1.30605
$939$	−6.00000	−0.195803
$940$	3.00000	0.0978492
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−15.0000	−0.488726
$943$	−24.0000	−0.781548
$944$	12.0000	0.390567
$945$	5.00000	0.162650
$946$	18.0000	0.585230
$947$	58.0000	1.88475	0.942373	−	0.334563i	$-0.108589\pi$
$947$	58.0000	1.88475	0.942373	+	0.334563i	$0.108589\pi$
$948$	−2.00000	−0.0649570
$949$	0	0
$950$	5.00000	0.162221
$951$	−23.0000	−0.745826
$952$	−40.0000	−1.29641
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−1.00000	−0.0323762
$955$	−2.00000	−0.0647185
$956$	−18.0000	−0.582162
$957$	12.0000	0.387905
$958$	28.0000	0.904639
$959$	60.0000	1.93750
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−25.0000	−0.805196
$965$	−24.0000	−0.772587
$966$	−20.0000	−0.643489
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	2.00000	0.0642824
$969$	40.0000	1.28499
$970$	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
$973$	−35.0000	−1.12205
$974$	−37.0000	−1.18556
$975$	0	0
$976$	2.00000	0.0640184
$977$	20.0000	0.639857	0.319928	−	0.947442i	$-0.396341\pi$
$977$	20.0000	0.639857	0.319928	+	0.947442i	$0.396341\pi$
$978$	20.0000	0.639529
$979$	33.0000	1.05468
$980$	−18.0000	−0.574989
$981$	14.0000	0.446986
$982$	−21.0000	−0.670137
$983$	23.0000	0.733586	0.366793	−	0.930303i	$-0.380456\pi$
$983$	23.0000	0.733586	0.366793	+	0.930303i	$0.380456\pi$
$984$	−6.00000	−0.191273
$985$	−3.00000	−0.0955879
$986$	−32.0000	−1.01909
$987$	15.0000	0.477455
$988$	0	0
$989$	−24.0000	−0.763156
$990$	−3.00000	−0.0953463
$991$	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
$993$	4.00000	0.126936
$994$	10.0000	0.317181
$995$	−22.0000	−0.697447
$996$	8.00000	0.253490
$997$	−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
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.i.1.1		1
13.3	even	3		390.2.i.d.61.1	✓	2
13.5	odd	4		5070.2.b.l.1351.2		2
13.8	odd	4		5070.2.b.l.1351.1		2
13.9	even	3		390.2.i.d.211.1	yes	2
13.12	even	2		5070.2.a.x.1.1		1
39.29	odd	6		1170.2.i.g.451.1		2
39.35	odd	6		1170.2.i.g.991.1		2
65.3	odd	12		1950.2.z.j.1699.1		4
65.9	even	6		1950.2.i.i.601.1		2
65.22	odd	12		1950.2.z.j.1849.1		4
65.29	even	6		1950.2.i.i.451.1		2
65.42	odd	12		1950.2.z.j.1699.2		4
65.48	odd	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	13.3	even	3
390.2.i.d.211.1	yes	2	13.9	even	3
1170.2.i.g.451.1		2	39.29	odd	6
1170.2.i.g.991.1		2	39.35	odd	6
1950.2.i.i.451.1		2	65.29	even	6
1950.2.i.i.601.1		2	65.9	even	6
1950.2.z.j.1699.1		4	65.3	odd	12
1950.2.z.j.1699.2		4	65.42	odd	12
1950.2.z.j.1849.1		4	65.22	odd	12
1950.2.z.j.1849.2		4	65.48	odd	12
5070.2.a.i.1.1		1	1.1	even	1	trivial
5070.2.a.x.1.1		1	13.12	even	2
5070.2.b.l.1351.1		2	13.8	odd	4
5070.2.b.l.1351.2		2	13.5	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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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