LMFDB - Embedded newform 5070.2.a.c.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:	$1$
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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} -1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -5.00000 q^{38} +1.00000 q^{40} -6.00000 q^{41} +3.00000 q^{42} -2.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -13.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.00000 q^{56} -5.00000 q^{57} -4.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +10.0000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +12.0000 q^{67} +4.00000 q^{69} +3.00000 q^{70} +2.00000 q^{71} -1.00000 q^{72} +16.0000 q^{73} -1.00000 q^{74} -1.00000 q^{75} +5.00000 q^{76} -3.00000 q^{77} -10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -3.00000 q^{84} +2.00000 q^{86} +1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{90} -4.00000 q^{92} +10.0000 q^{93} -9.00000 q^{94} -5.00000 q^{95} +1.00000 q^{96} -12.0000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	1.00000	0.213201
$23$	−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
$28$	3.00000	0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	3.00000	0.462910
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0	0
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−3.00000	−0.400892
$57$	−5.00000	−0.662266
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	1.00000	0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	10.0000	1.27000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	3.00000	0.358569
$71$	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
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	−1.00000	−0.116248
$75$	−1.00000	−0.115470
$76$	5.00000	0.573539
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	1.00000	0.106600
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	10.0000	1.03695
$94$	−9.00000	−0.928279
$95$	−5.00000	−0.512989
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	13.0000	1.26267
$107$	−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
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.00000	−0.0953463
$111$	−1.00000	−0.0949158
$112$	3.00000	0.283473
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	5.00000	0.468293
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	2.00000	0.181071
$123$	6.00000	0.541002
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	−3.00000	−0.267261
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	1.00000	0.0870388
$133$	15.0000	1.30066
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	−4.00000	−0.340503
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	−3.00000	−0.253546
$141$	−9.00000	−0.757937
$142$	−2.00000	−0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	−2.00000	−0.164957
$148$	1.00000	0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	3.00000	0.241747
$155$	10.0000	0.803219
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	10.0000	0.795557
$159$	13.0000	1.03097
$160$	1.00000	0.0790569
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−6.00000	−0.468521
$165$	−1.00000	−0.0778499
$166$	12.0000	0.931381
$167$	13.0000	1.00597	0.502985	−	0.864295i	$-0.332235\pi$
$167$	13.0000	1.00597	0.502985	+	0.864295i	$0.332235\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	0	0
$171$	5.00000	0.382360
$172$	−2.00000	−0.152499
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	1.00000	0.0749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.00000	−0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	4.00000	0.294884
$185$	−1.00000	−0.0735215
$186$	−10.0000	−0.733236
$187$	0	0
$188$	9.00000	0.656392
$189$	−3.00000	−0.218218
$190$	5.00000	0.362738
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−1.00000	−0.0721688
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	1.00000	0.0710669
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	−1.00000	−0.0707107
$201$	−12.0000	−0.846415
$202$	−4.00000	−0.281439
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	9.00000	0.627060
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−5.00000	−0.345857
$210$	−3.00000	−0.207020
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	−13.0000	−0.892844
$213$	−2.00000	−0.137038
$214$	6.00000	0.410152
$215$	2.00000	0.136399
$216$	1.00000	0.0680414
$217$	−30.0000	−2.03653
$218$	−10.0000	−0.677285
$219$	−16.0000	−1.08118
$220$	1.00000	0.0674200
$221$	0	0
$222$	1.00000	0.0671156
$223$	−11.0000	−0.736614	−0.368307	−	0.929704i	$-0.620063\pi$
$223$	−11.0000	−0.736614	−0.368307	+	0.929704i	$0.620063\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	−16.0000	−1.06430
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	−5.00000	−0.331133
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−4.00000	−0.263752
$231$	3.00000	0.197386
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	−4.00000	−0.260378
$237$	10.0000	0.649570
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	1.00000	0.0645497
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−2.00000	−0.127775
$246$	−6.00000	−0.382546
$247$	0	0
$248$	10.0000	0.635001
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	3.00000	0.188982
$253$	4.00000	0.251478
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	−2.00000	−0.124515
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	15.0000	0.926703
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	−1.00000	−0.0615457
$265$	13.0000	0.798584
$266$	−15.0000	−0.919709
$267$	1.00000	0.0611990
$268$	12.0000	0.733017
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	−1.00000	−0.0608581
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	16.0000	0.966595
$275$	−1.00000	−0.0603023
$276$	4.00000	0.240772
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	9.00000	0.539784
$279$	−10.0000	−0.598684
$280$	3.00000	0.179284
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	9.00000	0.535942
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	2.00000	0.118678
$285$	5.00000	0.296174
$286$	0	0
$287$	−18.0000	−1.06251
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	12.0000	0.703452
$292$	16.0000	0.936329
$293$	−13.0000	−0.759468	−0.379734	−	0.925096i	$-0.623985\pi$
$293$	−13.0000	−0.759468	−0.379734	+	0.925096i	$0.623985\pi$
$294$	2.00000	0.116642
$295$	4.00000	0.232889
$296$	−1.00000	−0.0581238
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−6.00000	−0.345834
$302$	−6.00000	−0.345261
$303$	−4.00000	−0.229794
$304$	5.00000	0.286770
$305$	2.00000	0.114520
$306$	0	0
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	−3.00000	−0.170941
$309$	9.00000	0.511992
$310$	−10.0000	−0.567962
$311$	−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
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	−1.00000	−0.0564333
$315$	−3.00000	−0.169031
$316$	−10.0000	−0.562544
$317$	−19.0000	−1.06715	−0.533573	−	0.845754i	$-0.679151\pi$
$317$	−19.0000	−1.06715	−0.533573	+	0.845754i	$0.679151\pi$
$318$	−13.0000	−0.729004
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	6.00000	0.334887
$322$	12.0000	0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−10.0000	−0.553001
$328$	6.00000	0.331295
$329$	27.0000	1.48856
$330$	1.00000	0.0550482
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−12.0000	−0.658586
$333$	1.00000	0.0547997
$334$	−13.0000	−0.711328
$335$	−12.0000	−0.655630
$336$	−3.00000	−0.163663
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	10.0000	0.541530
$342$	−5.00000	−0.270369
$343$	−15.0000	−0.809924
$344$	2.00000	0.107833
$345$	−4.00000	−0.215353
$346$	9.00000	0.483843
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	−36.0000	−1.92704	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000	−1.92704	−0.963518	+	0.267644i	$0.913755\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	1.00000	0.0533002
$353$	36.0000	1.91609	0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000	1.91609	0.958043	+	0.286623i	$0.0925328\pi$
$354$	−4.00000	−0.212598
$355$	−2.00000	−0.106149
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	1.00000	0.0527046
$361$	6.00000	0.315789
$362$	−6.00000	−0.315353
$363$	10.0000	0.524864
$364$	0	0
$365$	−16.0000	−0.837478
$366$	−2.00000	−0.104542
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	1.00000	0.0519875
$371$	−39.0000	−2.02478
$372$	10.0000	0.518476
$373$	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
$375$	1.00000	0.0516398
$376$	−9.00000	−0.464140
$377$	0	0
$378$	3.00000	0.154303
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	−5.00000	−0.256495
$381$	−5.00000	−0.256158
$382$	18.0000	0.920960
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	1.00000	0.0510310
$385$	3.00000	0.152894
$386$	16.0000	0.814379
$387$	−2.00000	−0.101666
$388$	−12.0000	−0.609208
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	0	0
$392$	−2.00000	−0.101015
$393$	15.0000	0.756650
$394$	−15.0000	−0.755689
$395$	10.0000	0.503155
$396$	−1.00000	−0.0502519
$397$	33.0000	1.65622	0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000	1.65622	0.828111	+	0.560564i	$0.189416\pi$
$398$	2.00000	0.100251
$399$	−15.0000	−0.750939
$400$	1.00000	0.0500000
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	4.00000	0.199007
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−1.00000	−0.0495682
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	−6.00000	−0.296319
$411$	16.0000	0.789222
$412$	−9.00000	−0.443398
$413$	−12.0000	−0.590481
$414$	4.00000	0.196589
$415$	12.0000	0.589057
$416$	0	0
$417$	9.00000	0.440732
$418$	5.00000	0.244558
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	3.00000	0.146385
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	15.0000	0.730189
$423$	9.00000	0.437595
$424$	13.0000	0.631336
$425$	0	0
$426$	2.00000	0.0969003
$427$	−6.00000	−0.290360
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	−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	$0.374389\pi$
$434$	30.0000	1.44005
$435$	0	0
$436$	10.0000	0.478913
$437$	−20.0000	−0.956730
$438$	16.0000	0.764510
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	−1.00000	−0.0476731
$441$	2.00000	0.0952381
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	−1.00000	−0.0474579
$445$	1.00000	0.0474045
$446$	11.0000	0.520865
$447$	−6.00000	−0.283790
$448$	3.00000	0.141737
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	16.0000	0.752577
$453$	−6.00000	−0.281905
$454$	−20.0000	−0.938647
$455$	0	0
$456$	5.00000	0.234146
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	4.00000	0.186501
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−3.00000	−0.139573
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−10.0000	−0.463739
$466$	−10.0000	−0.463241
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	9.00000	0.415139
$471$	−1.00000	−0.0460776
$472$	4.00000	0.184115
$473$	2.00000	0.0919601
$474$	−10.0000	−0.459315
$475$	5.00000	0.229416
$476$	0	0
$477$	−13.0000	−0.595229
$478$	−2.00000	−0.0914779
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	15.0000	0.683231
$483$	12.0000	0.546019
$484$	−10.0000	−0.454545
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	−35.0000	−1.58600	−0.793001	−	0.609221i	$-0.791482\pi$
$487$	−35.0000	−1.58600	−0.793001	+	0.609221i	$0.791482\pi$
$488$	2.00000	0.0905357
$489$	−20.0000	−0.904431
$490$	2.00000	0.0903508
$491$	25.0000	1.12823	0.564117	−	0.825695i	$-0.309217\pi$
$491$	25.0000	1.12823	0.564117	+	0.825695i	$0.309217\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	0	0
$495$	1.00000	0.0449467
$496$	−10.0000	−0.449013
$497$	6.00000	0.269137
$498$	−12.0000	−0.537733
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	−13.0000	−0.580797
$502$	−11.0000	−0.490954
$503$	−1.00000	−0.0445878	−0.0222939	−	0.999751i	$-0.507097\pi$
$503$	−1.00000	−0.0445878	−0.0222939	+	0.999751i	$0.507097\pi$
$504$	−3.00000	−0.133631
$505$	−4.00000	−0.177998
$506$	−4.00000	−0.177822
$507$	0	0
$508$	5.00000	0.221839
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	48.0000	2.12339
$512$	−1.00000	−0.0441942
$513$	−5.00000	−0.220755
$514$	24.0000	1.05859
$515$	9.00000	0.396587
$516$	2.00000	0.0880451
$517$	−9.00000	−0.395820
$518$	−3.00000	−0.131812
$519$	9.00000	0.395056
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−15.0000	−0.655278
$525$	−3.00000	−0.130931
$526$	−3.00000	−0.130806
$527$	0	0
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	−13.0000	−0.564684
$531$	−4.00000	−0.173585
$532$	15.0000	0.650332
$533$	0	0
$534$	−1.00000	−0.0432742
$535$	6.00000	0.259403
$536$	−12.0000	−0.518321
$537$	−12.0000	−0.517838
$538$	20.0000	0.862261
$539$	−2.00000	−0.0861461
$540$	1.00000	0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−24.0000	−1.03089
$543$	−6.00000	−0.257485
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	−16.0000	−0.683486
$549$	−2.00000	−0.0853579
$550$	1.00000	0.0426401
$551$	0	0
$552$	−4.00000	−0.170251
$553$	−30.0000	−1.27573
$554$	−23.0000	−0.977176
$555$	1.00000	0.0424476
$556$	−9.00000	−0.381685
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	10.0000	0.423334
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	10.0000	0.421825
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−9.00000	−0.378968
$565$	−16.0000	−0.673125
$566$	−2.00000	−0.0840663
$567$	3.00000	0.125988
$568$	−2.00000	−0.0839181
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	−5.00000	−0.209427
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	18.0000	0.751305
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	17.0000	0.707107
$579$	16.0000	0.664937
$580$	0	0
$581$	−36.0000	−1.49353
$582$	−12.0000	−0.497416
$583$	13.0000	0.538405
$584$	−16.0000	−0.662085
$585$	0	0
$586$	13.0000	0.537025
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	−0.0824786
$589$	−50.0000	−2.06021
$590$	−4.00000	−0.164677
$591$	−15.0000	−0.617018
$592$	1.00000	0.0410997
$593$	−28.0000	−1.14982	−0.574911	−	0.818216i	$-0.694963\pi$
$593$	−28.0000	−1.14982	−0.574911	+	0.818216i	$0.694963\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	2.00000	0.0818546
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	1.00000	0.0408248
$601$	−27.0000	−1.10135	−0.550676	−	0.834719i	$-0.685630\pi$
$601$	−27.0000	−1.10135	−0.550676	+	0.834719i	$0.685630\pi$
$602$	6.00000	0.244542
$603$	12.0000	0.488678
$604$	6.00000	0.244137
$605$	10.0000	0.406558
$606$	4.00000	0.162489
$607$	37.0000	1.50178	0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000	1.50178	0.750892	+	0.660425i	$0.229624\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	18.0000	0.726421
$615$	−6.00000	−0.241943
$616$	3.00000	0.120873
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−9.00000	−0.362033
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	10.0000	0.401610
$621$	4.00000	0.160514
$622$	12.0000	0.481156
$623$	−3.00000	−0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	34.0000	1.35891
$627$	5.00000	0.199681
$628$	1.00000	0.0399043
$629$	0	0
$630$	3.00000	0.119523
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	10.0000	0.397779
$633$	15.0000	0.596196
$634$	19.0000	0.754586
$635$	−5.00000	−0.198419
$636$	13.0000	0.515484
$637$	0	0
$638$	0	0
$639$	2.00000	0.0791188
$640$	1.00000	0.0395285
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	−6.00000	−0.236801
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−12.0000	−0.472866
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	−1.00000	−0.0392837
$649$	4.00000	0.157014
$650$	0	0
$651$	30.0000	1.17579
$652$	20.0000	0.783260
$653$	41.0000	1.60445	0.802227	−	0.597019i	$-0.203648\pi$
$653$	41.0000	1.60445	0.802227	+	0.597019i	$0.203648\pi$
$654$	10.0000	0.391031
$655$	15.0000	0.586098
$656$	−6.00000	−0.234261
$657$	16.0000	0.624219
$658$	−27.0000	−1.05257
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−1.00000	−0.0389249
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	12.0000	0.465690
$665$	−15.0000	−0.581675
$666$	−1.00000	−0.0387492
$667$	0	0
$668$	13.0000	0.502985
$669$	11.0000	0.425285
$670$	12.0000	0.463600
$671$	2.00000	0.0772091
$672$	3.00000	0.115728
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	16.0000	0.614476
$679$	−36.0000	−1.38155
$680$	0	0
$681$	−20.0000	−0.766402
$682$	−10.0000	−0.382920
$683$	26.0000	0.994862	0.497431	−	0.867503i	$-0.334277\pi$
$683$	26.0000	0.994862	0.497431	+	0.867503i	$0.334277\pi$
$684$	5.00000	0.191180
$685$	16.0000	0.611329
$686$	15.0000	0.572703
$687$	−10.0000	−0.381524
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	4.00000	0.152277
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	−9.00000	−0.342129
$693$	−3.00000	−0.113961
$694$	28.0000	1.06287
$695$	9.00000	0.341389
$696$	0	0
$697$	0	0
$698$	36.0000	1.36262
$699$	−10.0000	−0.378235
$700$	3.00000	0.113389
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	−1.00000	−0.0376889
$705$	9.00000	0.338960
$706$	−36.0000	−1.35488
$707$	12.0000	0.451306
$708$	4.00000	0.150329
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	2.00000	0.0750587
$711$	−10.0000	−0.375029
$712$	1.00000	0.0374766
$713$	40.0000	1.49801
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	−2.00000	−0.0746914
$718$	34.0000	1.26887
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	−27.0000	−1.00553
$722$	−6.00000	−0.223297
$723$	15.0000	0.557856
$724$	6.00000	0.222988
$725$	0	0
$726$	−10.0000	−0.371135
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.0000	0.592187
$731$	0	0
$732$	2.00000	0.0739221
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	4.00000	0.147442
$737$	−12.0000	−0.442026
$738$	6.00000	0.220863
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	39.0000	1.43174
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−10.0000	−0.366618
$745$	−6.00000	−0.219823
$746$	−10.0000	−0.366126
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−18.0000	−0.657706
$750$	−1.00000	−0.0365148
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	9.00000	0.328196
$753$	−11.0000	−0.400862
$754$	0	0
$755$	−6.00000	−0.218362
$756$	−3.00000	−0.109109
$757$	39.0000	1.41748	0.708740	−	0.705470i	$-0.249264\pi$
$757$	39.0000	1.41748	0.708740	+	0.705470i	$0.249264\pi$
$758$	−1.00000	−0.0363216
$759$	−4.00000	−0.145191
$760$	5.00000	0.181369
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	5.00000	0.181131
$763$	30.0000	1.08607
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−28.0000	−1.01168
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−3.00000	−0.108112
$771$	24.0000	0.864339
$772$	−16.0000	−0.575853
$773$	37.0000	1.33080	0.665399	−	0.746488i	$-0.268262\pi$
$773$	37.0000	1.33080	0.665399	+	0.746488i	$0.268262\pi$
$774$	2.00000	0.0718885
$775$	−10.0000	−0.359211
$776$	12.0000	0.430775
$777$	−3.00000	−0.107624
$778$	16.0000	0.573628
$779$	−30.0000	−1.07486
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	0	0
$784$	2.00000	0.0714286
$785$	−1.00000	−0.0356915
$786$	−15.0000	−0.535032
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	15.0000	0.534353
$789$	−3.00000	−0.106803
$790$	−10.0000	−0.355784
$791$	48.0000	1.70668
$792$	1.00000	0.0355335
$793$	0	0
$794$	−33.0000	−1.17113
$795$	−13.0000	−0.461062
$796$	−2.00000	−0.0708881
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	15.0000	0.530994
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−1.00000	−0.0353333
$802$	25.0000	0.882781
$803$	−16.0000	−0.564628
$804$	−12.0000	−0.423207
$805$	12.0000	0.422944
$806$	0	0
$807$	20.0000	0.704033
$808$	−4.00000	−0.140720
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	1.00000	0.0351364
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	1.00000	0.0350500
$815$	−20.0000	−0.700569
$816$	0	0
$817$	−10.0000	−0.349856
$818$	17.0000	0.594391
$819$	0	0
$820$	6.00000	0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−16.0000	−0.558064
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	9.00000	0.313530
$825$	1.00000	0.0348155
$826$	12.0000	0.417533
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	−4.00000	−0.139010
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	−12.0000	−0.416526
$831$	−23.0000	−0.797861
$832$	0	0
$833$	0	0
$834$	−9.00000	−0.311645
$835$	−13.0000	−0.449884
$836$	−5.00000	−0.172929
$837$	10.0000	0.345651
$838$	28.0000	0.967244
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	−3.00000	−0.103510
$841$	−29.0000	−1.00000
$842$	20.0000	0.689246
$843$	10.0000	0.344418
$844$	−15.0000	−0.516321
$845$	0	0
$846$	−9.00000	−0.309426
$847$	−30.0000	−1.03081
$848$	−13.0000	−0.446422
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	−4.00000	−0.137118
$852$	−2.00000	−0.0685189
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	6.00000	0.205316
$855$	−5.00000	−0.170996
$856$	6.00000	0.205076
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	2.00000	0.0681994
$861$	18.0000	0.613438
$862$	36.0000	1.22616
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	1.00000	0.0340207
$865$	9.00000	0.306009
$866$	−16.0000	−0.543702
$867$	17.0000	0.577350
$868$	−30.0000	−1.01827
$869$	10.0000	0.339227
$870$	0	0
$871$	0	0
$872$	−10.0000	−0.338643
$873$	−12.0000	−0.406138
$874$	20.0000	0.676510
$875$	−3.00000	−0.101419
$876$	−16.0000	−0.540590
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	10.0000	0.337484
$879$	13.0000	0.438479
$880$	1.00000	0.0337100
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	−2.00000	−0.0673435
$883$	42.0000	1.41341	0.706706	−	0.707507i	$-0.250180\pi$
$883$	42.0000	1.41341	0.706706	+	0.707507i	$0.250180\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	18.0000	0.604722
$887$	−13.0000	−0.436497	−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000	−0.436497	−0.218249	+	0.975893i	$0.570034\pi$
$888$	1.00000	0.0335578
$889$	15.0000	0.503084
$890$	−1.00000	−0.0335201
$891$	−1.00000	−0.0335013
$892$	−11.0000	−0.368307
$893$	45.0000	1.50587
$894$	6.00000	0.200670
$895$	−12.0000	−0.401116
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−15.0000	−0.500556
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	−6.00000	−0.199778
$903$	6.00000	0.199667
$904$	−16.0000	−0.532152
$905$	−6.00000	−0.199447
$906$	6.00000	0.199337
$907$	42.0000	1.39459	0.697294	−	0.716786i	$-0.254387\pi$
$907$	42.0000	1.39459	0.697294	+	0.716786i	$0.254387\pi$
$908$	20.0000	0.663723
$909$	4.00000	0.132672
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−5.00000	−0.165567
$913$	12.0000	0.397142
$914$	22.0000	0.727695
$915$	−2.00000	−0.0661180
$916$	10.0000	0.330409
$917$	−45.0000	−1.48603
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	−4.00000	−0.131876
$921$	18.0000	0.593120
$922$	−12.0000	−0.395199
$923$	0	0
$924$	3.00000	0.0986928
$925$	1.00000	0.0328798
$926$	−16.0000	−0.525793
$927$	−9.00000	−0.295599
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	10.0000	0.327913
$931$	10.0000	0.327737
$932$	10.0000	0.327561
$933$	12.0000	0.392862
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	−36.0000	−1.17544
$939$	34.0000	1.10955
$940$	−9.00000	−0.293548
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	1.00000	0.0325818
$943$	24.0000	0.781548
$944$	−4.00000	−0.130189
$945$	3.00000	0.0975900
$946$	−2.00000	−0.0650256
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	10.0000	0.324785
$949$	0	0
$950$	−5.00000	−0.162221
$951$	19.0000	0.616117
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	13.0000	0.420891
$955$	18.0000	0.582466
$956$	2.00000	0.0646846
$957$	0	0
$958$	−8.00000	−0.258468
$959$	−48.0000	−1.55000
$960$	1.00000	0.0322749
$961$	69.0000	2.22581
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−15.0000	−0.483117
$965$	16.0000	0.515058
$966$	−12.0000	−0.386094
$967$	−7.00000	−0.225105	−0.112552	−	0.993646i	$-0.535903\pi$
$967$	−7.00000	−0.225105	−0.112552	+	0.993646i	$0.535903\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	−12.0000	−0.385297
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	−1.00000	−0.0320750
$973$	−27.0000	−0.865580
$974$	35.0000	1.12147
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	20.0000	0.639529
$979$	1.00000	0.0319601
$980$	−2.00000	−0.0638877
$981$	10.0000	0.319275
$982$	−25.0000	−0.797782
$983$	−53.0000	−1.69044	−0.845219	−	0.534421i	$-0.820530\pi$
$983$	−53.0000	−1.69044	−0.845219	+	0.534421i	$0.820530\pi$
$984$	−6.00000	−0.191273
$985$	−15.0000	−0.477940
$986$	0	0
$987$	−27.0000	−0.859419
$988$	0	0
$989$	8.00000	0.254385
$990$	−1.00000	−0.0317821
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	10.0000	0.317500
$993$	28.0000	0.888553
$994$	−6.00000	−0.190308
$995$	2.00000	0.0634043
$996$	12.0000	0.380235
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	20.0000	0.633089
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.c.1.1		1
13.4	even	6		390.2.i.b.211.1	yes	2
13.5	odd	4		5070.2.b.a.1351.2		2
13.8	odd	4		5070.2.b.a.1351.1		2
13.10	even	6		390.2.i.b.61.1	✓	2
13.12	even	2		5070.2.a.q.1.1		1
39.17	odd	6		1170.2.i.j.991.1		2
39.23	odd	6		1170.2.i.j.451.1		2
65.4	even	6		1950.2.i.o.601.1		2
65.17	odd	12		1950.2.z.i.1849.2		4
65.23	odd	12		1950.2.z.i.1699.2		4
65.43	odd	12		1950.2.z.i.1849.1		4
65.49	even	6		1950.2.i.o.451.1		2
65.62	odd	12		1950.2.z.i.1699.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.b.61.1	✓	2	13.10	even	6
390.2.i.b.211.1	yes	2	13.4	even	6
1170.2.i.j.451.1		2	39.23	odd	6
1170.2.i.j.991.1		2	39.17	odd	6
1950.2.i.o.451.1		2	65.49	even	6
1950.2.i.o.601.1		2	65.4	even	6
1950.2.z.i.1699.1		4	65.62	odd	12
1950.2.z.i.1699.2		4	65.23	odd	12
1950.2.z.i.1849.1		4	65.43	odd	12
1950.2.z.i.1849.2		4	65.17	odd	12
5070.2.a.c.1.1		1	1.1	even	1	trivial
5070.2.a.q.1.1		1	13.12	even	2
5070.2.b.a.1351.1		2	13.8	odd	4
5070.2.b.a.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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