LMFDB - Embedded newform 1521.4.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 82x^{10} + 2137x^{8} - 22488x^{6} + 89784x^{4} - 67392x^{2} + 11664 $ x^12 - 82x^10 + 2137x^8 - 22488x^6 + 89784x^4 - 67392*x^2 + 11664
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.27062$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53857 q^{2} -1.55568 q^{4} -8.18941 q^{5} -10.7837 q^{7} +24.2577 q^{8} +O(q^{10})$	$q-2.53857 q^{2} -1.55568 q^{4} -8.18941 q^{5} -10.7837 q^{7} +24.2577 q^{8} +20.7894 q^{10} -10.9912 q^{11} +27.3753 q^{14} -49.1344 q^{16} +4.85282 q^{17} +65.2735 q^{19} +12.7401 q^{20} +27.9019 q^{22} -166.613 q^{23} -57.9336 q^{25} +16.7761 q^{28} +92.4849 q^{29} -3.63236 q^{31} -69.3309 q^{32} -12.3192 q^{34} +88.3125 q^{35} +40.6031 q^{37} -165.701 q^{38} -198.656 q^{40} +242.842 q^{41} +161.438 q^{43} +17.0988 q^{44} +422.958 q^{46} +296.631 q^{47} -226.711 q^{49} +147.068 q^{50} +662.165 q^{53} +90.0113 q^{55} -261.589 q^{56} -234.779 q^{58} +391.583 q^{59} -323.232 q^{61} +9.22099 q^{62} +569.076 q^{64} -558.555 q^{67} -7.54945 q^{68} -224.187 q^{70} -106.813 q^{71} +79.5194 q^{73} -103.074 q^{74} -101.545 q^{76} +118.526 q^{77} +480.970 q^{79} +402.382 q^{80} -616.469 q^{82} +1251.53 q^{83} -39.7418 q^{85} -409.822 q^{86} -266.621 q^{88} +1446.47 q^{89} +259.197 q^{92} -753.016 q^{94} -534.551 q^{95} +1394.73 q^{97} +575.520 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 32 q^{4}+O(q^{10}) $ 12 * q + 32 * q^4	$ 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) $ 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

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 $

$2$	−2.53857	−0.897519	−0.448759	−	0.893653i	$-0.648134\pi$
$2$	−2.53857	−0.897519	−0.448759	+	0.893653i	$0.648134\pi$
$3$	0	0
$3$	0	0
$4$	−1.55568	−0.194460
$5$	−8.18941	−0.732483	−0.366242	−	0.930520i	$-0.619356\pi$
$5$	−8.18941	−0.732483	−0.366242	+	0.930520i	$0.619356\pi$
$6$	0	0
$7$	−10.7837	−0.582268	−0.291134	−	0.956682i	$-0.594032\pi$
$7$	−10.7837	−0.582268	−0.291134	+	0.956682i	$0.594032\pi$
$8$	24.2577	1.07205
$9$	0	0
$10$	20.7894	0.657417
$11$	−10.9912	−0.301270	−0.150635	−	0.988589i	$-0.548132\pi$
$11$	−10.9912	−0.301270	−0.150635	+	0.988589i	$0.548132\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	27.3753	0.522596
$15$	0	0
$16$	−49.1344	−0.767725
$17$	4.85282	0.0692343	0.0346171	−	0.999401i	$-0.488979\pi$
$17$	4.85282	0.0692343	0.0346171	+	0.999401i	$0.488979\pi$
$18$	0	0
$19$	65.2735	0.788146	0.394073	−	0.919079i	$-0.371066\pi$
$19$	65.2735	0.788146	0.394073	+	0.919079i	$0.371066\pi$
$20$	12.7401	0.142439
$21$	0	0
$22$	27.9019	0.270395
$23$	−166.613	−1.51049	−0.755243	−	0.655445i	$-0.772481\pi$
$23$	−166.613	−1.51049	−0.755243	+	0.655445i	$0.772481\pi$
$24$	0	0
$25$	−57.9336	−0.463469
$26$	0	0
$27$	0	0
$28$	16.7761	0.113228
$29$	92.4849	0.592207	0.296104	−	0.955156i	$-0.404313\pi$
$29$	92.4849	0.592207	0.296104	+	0.955156i	$0.404313\pi$
$30$	0	0
$31$	−3.63236	−0.0210449	−0.0105224	−	0.999945i	$-0.503349\pi$
$31$	−3.63236	−0.0210449	−0.0105224	+	0.999945i	$0.503349\pi$
$32$	−69.3309	−0.383003
$33$	0	0
$34$	−12.3192	−0.0621390
$35$	88.3125	0.426501
$36$	0	0
$37$	40.6031	0.180408	0.0902041	−	0.995923i	$-0.471248\pi$
$37$	40.6031	0.180408	0.0902041	+	0.995923i	$0.471248\pi$
$38$	−165.701	−0.707375
$39$	0	0
$40$	−198.656	−0.785259
$41$	242.842	0.925012	0.462506	−	0.886616i	$-0.346950\pi$
$41$	242.842	0.925012	0.462506	+	0.886616i	$0.346950\pi$
$42$	0	0
$43$	161.438	0.572537	0.286269	−	0.958149i	$-0.407585\pi$
$43$	161.438	0.572537	0.286269	+	0.958149i	$0.407585\pi$
$44$	17.0988	0.0585850
$45$	0	0
$46$	422.958	1.35569
$47$	296.631	0.920596	0.460298	−	0.887764i	$-0.347743\pi$
$47$	296.631	0.920596	0.460298	+	0.887764i	$0.347743\pi$
$48$	0	0
$49$	−226.711	−0.660965
$50$	147.068	0.415972
$51$	0	0
$52$	0	0
$53$	662.165	1.71614	0.858070	−	0.513533i	$-0.171664\pi$
$53$	662.165	1.71614	0.858070	+	0.513533i	$0.171664\pi$
$54$	0	0
$55$	90.0113	0.220675
$56$	−261.589	−0.624220
$57$	0	0
$58$	−234.779	−0.531517
$59$	391.583	0.864063	0.432031	−	0.901859i	$-0.357797\pi$
$59$	391.583	0.864063	0.432031	+	0.901859i	$0.357797\pi$
$60$	0	0
$61$	−323.232	−0.678452	−0.339226	−	0.940705i	$-0.610165\pi$
$61$	−323.232	−0.678452	−0.339226	+	0.940705i	$0.610165\pi$
$62$	9.22099	0.0188882
$63$	0	0
$64$	569.076	1.11148
$65$	0	0
$66$	0	0
$67$	−558.555	−1.01848	−0.509241	−	0.860624i	$-0.670074\pi$
$67$	−558.555	−1.01848	−0.509241	+	0.860624i	$0.670074\pi$
$68$	−7.54945	−0.0134633
$69$	0	0
$70$	−224.187	−0.382793
$71$	−106.813	−0.178541	−0.0892705	−	0.996007i	$-0.528454\pi$
$71$	−106.813	−0.178541	−0.0892705	+	0.996007i	$0.528454\pi$
$72$	0	0
$73$	79.5194	0.127494	0.0637468	−	0.997966i	$-0.479695\pi$
$73$	79.5194	0.127494	0.0637468	+	0.997966i	$0.479695\pi$
$74$	−103.074	−0.161920
$75$	0	0
$76$	−101.545	−0.153263
$77$	118.526	0.175420
$78$	0	0
$79$	480.970	0.684980	0.342490	−	0.939522i	$-0.388730\pi$
$79$	480.970	0.684980	0.342490	+	0.939522i	$0.388730\pi$
$80$	402.382	0.562346
$81$	0	0
$82$	−616.469	−0.830215
$83$	1251.53	1.65510	0.827550	−	0.561392i	$-0.189734\pi$
$83$	1251.53	1.65510	0.827550	+	0.561392i	$0.189734\pi$
$84$	0	0
$85$	−39.7418	−0.0507129
$86$	−409.822	−0.513863
$87$	0	0
$88$	−266.621	−0.322976
$89$	1446.47	1.72276	0.861378	−	0.507965i	$-0.169602\pi$
$89$	1446.47	1.72276	0.861378	+	0.507965i	$0.169602\pi$
$90$	0	0
$91$	0	0
$92$	259.197	0.293730
$93$	0	0
$94$	−753.016	−0.826252
$95$	−534.551	−0.577303
$96$	0	0
$97$	1394.73	1.45993	0.729967	−	0.683482i	$-0.239535\pi$
$97$	1394.73	1.45993	0.729967	+	0.683482i	$0.239535\pi$
$98$	575.520	0.593228
$99$	0	0
$100$	90.1262	0.0901262
$101$	−672.744	−0.662778	−0.331389	−	0.943494i	$-0.607517\pi$
$101$	−672.744	−0.662778	−0.331389	+	0.943494i	$0.607517\pi$
$102$	0	0
$103$	131.830	0.126113	0.0630563	−	0.998010i	$-0.479915\pi$
$103$	131.830	0.126113	0.0630563	+	0.998010i	$0.479915\pi$
$104$	0	0
$105$	0	0
$106$	−1680.95	−1.54027
$107$	−2046.60	−1.84909	−0.924544	−	0.381076i	$-0.875554\pi$
$107$	−2046.60	−1.84909	−0.924544	+	0.381076i	$0.875554\pi$
$108$	0	0
$109$	−1380.13	−1.21277	−0.606386	−	0.795170i	$-0.707381\pi$
$109$	−1380.13	−1.21277	−0.606386	+	0.795170i	$0.707381\pi$
$110$	−228.500	−0.198060
$111$	0	0
$112$	529.853	0.447021
$113$	1038.82	0.864816	0.432408	−	0.901678i	$-0.357664\pi$
$113$	1038.82	0.864816	0.432408	+	0.901678i	$0.357664\pi$
$114$	0	0
$115$	1364.46	1.10641
$116$	−143.877	−0.115161
$117$	0	0
$118$	−994.058	−0.775513
$119$	−52.3316	−0.0403129
$120$	0	0
$121$	−1210.19	−0.909236
$122$	820.545	0.608924
$123$	0	0
$124$	5.65080	0.00409239
$125$	1498.12	1.07197
$126$	0	0
$127$	−2167.47	−1.51442	−0.757211	−	0.653170i	$-0.773439\pi$
$127$	−2167.47	−1.51442	−0.757211	+	0.653170i	$0.773439\pi$
$128$	−889.991	−0.614569
$129$	0	0
$130$	0	0
$131$	2034.60	1.35698	0.678489	−	0.734611i	$-0.262635\pi$
$131$	2034.60	1.35698	0.678489	+	0.734611i	$0.262635\pi$
$132$	0	0
$133$	−703.893	−0.458912
$134$	1417.93	0.914107
$135$	0	0
$136$	117.718	0.0742226
$137$	−828.385	−0.516596	−0.258298	−	0.966065i	$-0.583162\pi$
$137$	−828.385	−0.516596	−0.258298	+	0.966065i	$0.583162\pi$
$138$	0	0
$139$	−1479.76	−0.902963	−0.451481	−	0.892281i	$-0.649104\pi$
$139$	−1479.76	−0.902963	−0.451481	+	0.892281i	$0.649104\pi$
$140$	−137.386	−0.0829375
$141$	0	0
$142$	271.153	0.160244
$143$	0	0
$144$	0	0
$145$	−757.396	−0.433782
$146$	−201.865	−0.114428
$147$	0	0
$148$	−63.1654	−0.0350822
$149$	−2366.48	−1.30114	−0.650569	−	0.759447i	$-0.725470\pi$
$149$	−2366.48	−1.30114	−0.650569	+	0.759447i	$0.725470\pi$
$150$	0	0
$151$	−878.334	−0.473363	−0.236681	−	0.971587i	$-0.576060\pi$
$151$	−878.334	−0.473363	−0.236681	+	0.971587i	$0.576060\pi$
$152$	1583.39	0.844932
$153$	0	0
$154$	−300.887	−0.157442
$155$	29.7469	0.0154150
$156$	0	0
$157$	−2780.63	−1.41349	−0.706746	−	0.707467i	$-0.749838\pi$
$157$	−2780.63	−1.41349	−0.706746	+	0.707467i	$0.749838\pi$
$158$	−1220.97	−0.614782
$159$	0	0
$160$	567.779	0.280543
$161$	1796.71	0.879507
$162$	0	0
$163$	3899.94	1.87403	0.937014	−	0.349292i	$-0.113578\pi$
$163$	3899.94	1.87403	0.937014	+	0.349292i	$0.113578\pi$
$164$	−377.784	−0.179878
$165$	0	0
$166$	−3177.09	−1.48548
$167$	−206.066	−0.0954843	−0.0477422	−	0.998860i	$-0.515203\pi$
$167$	−206.066	−0.0954843	−0.0477422	+	0.998860i	$0.515203\pi$
$168$	0	0
$169$	0	0
$170$	100.887	0.0455158
$171$	0	0
$172$	−251.147	−0.111336
$173$	253.029	0.111199	0.0555995	−	0.998453i	$-0.482293\pi$
$173$	253.029	0.111199	0.0555995	+	0.998453i	$0.482293\pi$
$174$	0	0
$175$	624.741	0.269863
$176$	540.045	0.231292
$177$	0	0
$178$	−3671.95	−1.54621
$179$	−3989.22	−1.66574	−0.832872	−	0.553466i	$-0.813305\pi$
$179$	−3989.22	−1.66574	−0.832872	+	0.553466i	$0.813305\pi$
$180$	0	0
$181$	3392.65	1.39323	0.696613	−	0.717447i	$-0.254690\pi$
$181$	3392.65	1.39323	0.696613	+	0.717447i	$0.254690\pi$
$182$	0	0
$183$	0	0
$184$	−4041.65	−1.61932
$185$	−332.515	−0.132146
$186$	0	0
$187$	−53.3383	−0.0208582
$188$	−461.463	−0.179019
$189$	0	0
$190$	1356.99	0.518141
$191$	1267.95	0.480342	0.240171	−	0.970731i	$-0.422796\pi$
$191$	1267.95	0.480342	0.240171	+	0.970731i	$0.422796\pi$
$192$	0	0
$193$	−1335.26	−0.498002	−0.249001	−	0.968503i	$-0.580102\pi$
$193$	−1335.26	−0.498002	−0.249001	+	0.968503i	$0.580102\pi$
$194$	−3540.62	−1.31032
$195$	0	0
$196$	352.690	0.128531
$197$	1475.15	0.533504	0.266752	−	0.963765i	$-0.414050\pi$
$197$	1475.15	0.533504	0.266752	+	0.963765i	$0.414050\pi$
$198$	0	0
$199$	−1473.30	−0.524821	−0.262411	−	0.964956i	$-0.584517\pi$
$199$	−1473.30	−0.524821	−0.262411	+	0.964956i	$0.584517\pi$
$200$	−1405.34	−0.496862
$201$	0	0
$202$	1707.81	0.594855
$203$	−997.333	−0.344823
$204$	0	0
$205$	−1988.73	−0.677555
$206$	−334.659	−0.113188
$207$	0	0
$208$	0	0
$209$	−717.433	−0.237445
$210$	0	0
$211$	−737.421	−0.240598	−0.120299	−	0.992738i	$-0.538385\pi$
$211$	−737.421	−0.240598	−0.120299	+	0.992738i	$0.538385\pi$
$212$	−1030.12	−0.333721
$213$	0	0
$214$	5195.43	1.65959
$215$	−1322.08	−0.419374
$216$	0	0
$217$	39.1705	0.0122537
$218$	3503.54	1.08849
$219$	0	0
$220$	−140.029	−0.0429125
$221$	0	0
$222$	0	0
$223$	−4810.24	−1.44447	−0.722236	−	0.691646i	$-0.756886\pi$
$223$	−4810.24	−1.44447	−0.722236	+	0.691646i	$0.756886\pi$
$224$	747.647	0.223010
$225$	0	0
$226$	−2637.12	−0.776188
$227$	3404.30	0.995381	0.497690	−	0.867355i	$-0.334182\pi$
$227$	3404.30	0.995381	0.497690	+	0.867355i	$0.334182\pi$
$228$	0	0
$229$	4254.11	1.22760	0.613798	−	0.789463i	$-0.289641\pi$
$229$	4254.11	1.22760	0.613798	+	0.789463i	$0.289641\pi$
$230$	−3463.78	−0.993020
$231$	0	0
$232$	2243.47	0.634876
$233$	5134.37	1.44362	0.721810	−	0.692091i	$-0.243310\pi$
$233$	5134.37	1.44362	0.721810	+	0.692091i	$0.243310\pi$
$234$	0	0
$235$	−2429.23	−0.674321
$236$	−609.178	−0.168026
$237$	0	0
$238$	132.847	0.0361815
$239$	−4079.91	−1.10422	−0.552108	−	0.833773i	$-0.686176\pi$
$239$	−4079.91	−1.10422	−0.552108	+	0.833773i	$0.686176\pi$
$240$	0	0
$241$	1367.03	0.365387	0.182693	−	0.983170i	$-0.441518\pi$
$241$	1367.03	0.365387	0.182693	+	0.983170i	$0.441518\pi$
$242$	3072.16	0.816057
$243$	0	0
$244$	502.846	0.131932
$245$	1856.63	0.484145
$246$	0	0
$247$	0	0
$248$	−88.1128	−0.0225612
$249$	0	0
$250$	−3803.07	−0.962109
$251$	−2667.42	−0.670782	−0.335391	−	0.942079i	$-0.608868\pi$
$251$	−2667.42	−0.670782	−0.335391	+	0.942079i	$0.608868\pi$
$252$	0	0
$253$	1831.27	0.455064
$254$	5502.26	1.35922
$255$	0	0
$256$	−2293.31	−0.559890
$257$	3443.50	0.835797	0.417898	−	0.908494i	$-0.362767\pi$
$257$	3443.50	0.835797	0.417898	+	0.908494i	$0.362767\pi$
$258$	0	0
$259$	−437.853	−0.105046
$260$	0	0
$261$	0	0
$262$	−5164.97	−1.21791
$263$	5069.18	1.18851	0.594257	−	0.804275i	$-0.297446\pi$
$263$	5069.18	1.18851	0.594257	+	0.804275i	$0.297446\pi$
$264$	0	0
$265$	−5422.74	−1.25704
$266$	1786.88	0.411882
$267$	0	0
$268$	868.933	0.198054
$269$	−5418.00	−1.22803	−0.614017	−	0.789293i	$-0.710447\pi$
$269$	−5418.00	−1.22803	−0.614017	+	0.789293i	$0.710447\pi$
$270$	0	0
$271$	3159.73	0.708266	0.354133	−	0.935195i	$-0.384776\pi$
$271$	3159.73	0.708266	0.354133	+	0.935195i	$0.384776\pi$
$272$	−238.441	−0.0531529
$273$	0	0
$274$	2102.91	0.463655
$275$	636.759	0.139629
$276$	0	0
$277$	−5587.69	−1.21203	−0.606014	−	0.795454i	$-0.707233\pi$
$277$	−5587.69	−1.21203	−0.606014	+	0.795454i	$0.707233\pi$
$278$	3756.47	0.810426
$279$	0	0
$280$	2142.26	0.457231
$281$	−3911.43	−0.830379	−0.415189	−	0.909735i	$-0.636285\pi$
$281$	−3911.43	−0.830379	−0.415189	+	0.909735i	$0.636285\pi$
$282$	0	0
$283$	−8515.78	−1.78873	−0.894365	−	0.447337i	$-0.852372\pi$
$283$	−8515.78	−1.78873	−0.894365	+	0.447337i	$0.852372\pi$
$284$	166.168	0.0347191
$285$	0	0
$286$	0	0
$287$	−2618.74	−0.538604
$288$	0	0
$289$	−4889.45	−0.995207
$290$	1922.70	0.389327
$291$	0	0
$292$	−123.707	−0.0247925
$293$	−4744.17	−0.945930	−0.472965	−	0.881081i	$-0.656816\pi$
$293$	−4744.17	−0.945930	−0.472965	+	0.881081i	$0.656816\pi$
$294$	0	0
$295$	−3206.83	−0.632911
$296$	984.938	0.193407
$297$	0	0
$298$	6007.47	1.16780
$299$	0	0
$300$	0	0
$301$	−1740.91	−0.333370
$302$	2229.71	0.424852
$303$	0	0
$304$	−3207.17	−0.605079
$305$	2647.08	0.496955
$306$	0	0
$307$	−7553.03	−1.40415	−0.702075	−	0.712103i	$-0.747743\pi$
$307$	−7553.03	−1.40415	−0.702075	+	0.712103i	$0.747743\pi$
$308$	−184.389	−0.0341121
$309$	0	0
$310$	−75.5145	−0.0138353
$311$	1432.07	0.261111	0.130555	−	0.991441i	$-0.458324\pi$
$311$	1432.07	0.261111	0.130555	+	0.991441i	$0.458324\pi$
$312$	0	0
$313$	−6510.68	−1.17574	−0.587868	−	0.808957i	$-0.700033\pi$
$313$	−6510.68	−1.17574	−0.587868	+	0.808957i	$0.700033\pi$
$314$	7058.81	1.26864
$315$	0	0
$316$	−748.237	−0.133201
$317$	−9640.66	−1.70812	−0.854059	−	0.520177i	$-0.825866\pi$
$317$	−9640.66	−1.70812	−0.854059	+	0.520177i	$0.825866\pi$
$318$	0	0
$319$	−1016.52	−0.178414
$320$	−4660.40	−0.814138
$321$	0	0
$322$	−4561.07	−0.789374
$323$	316.761	0.0545667
$324$	0	0
$325$	0	0
$326$	−9900.25	−1.68198
$327$	0	0
$328$	5890.78	0.991659
$329$	−3198.79	−0.536033
$330$	0	0
$331$	−2246.85	−0.373105	−0.186553	−	0.982445i	$-0.559731\pi$
$331$	−2246.85	−0.373105	−0.186553	+	0.982445i	$0.559731\pi$
$332$	−1946.98	−0.321851
$333$	0	0
$334$	523.113	0.0856990
$335$	4574.23	0.746021
$336$	0	0
$337$	1589.67	0.256958	0.128479	−	0.991712i	$-0.458991\pi$
$337$	1589.67	0.256958	0.128479	+	0.991712i	$0.458991\pi$
$338$	0	0
$339$	0	0
$340$	61.8255	0.00986165
$341$	39.9240	0.00634019
$342$	0	0
$343$	6143.62	0.967126
$344$	3916.13	0.613789
$345$	0	0
$346$	−642.331	−0.0998032
$347$	−979.741	−0.151571	−0.0757856	−	0.997124i	$-0.524146\pi$
$347$	−979.741	−0.151571	−0.0757856	+	0.997124i	$0.524146\pi$
$348$	0	0
$349$	1972.93	0.302604	0.151302	−	0.988488i	$-0.451653\pi$
$349$	1972.93	0.302604	0.151302	+	0.988488i	$0.451653\pi$
$350$	−1585.95	−0.242207
$351$	0	0
$352$	762.029	0.115387
$353$	−10035.9	−1.51319	−0.756596	−	0.653883i	$-0.773139\pi$
$353$	−10035.9	−1.51319	−0.756596	+	0.653883i	$0.773139\pi$
$354$	0	0
$355$	874.738	0.130778
$356$	−2250.24	−0.335008
$357$	0	0
$358$	10126.9	1.49504
$359$	2749.28	0.404183	0.202091	−	0.979367i	$-0.435226\pi$
$359$	2749.28	0.404183	0.202091	+	0.979367i	$0.435226\pi$
$360$	0	0
$361$	−2598.37	−0.378826
$362$	−8612.47	−1.25045
$363$	0	0
$364$	0	0
$365$	−651.217	−0.0933870
$366$	0	0
$367$	−842.338	−0.119808	−0.0599042	−	0.998204i	$-0.519080\pi$
$367$	−842.338	−0.119808	−0.0599042	+	0.998204i	$0.519080\pi$
$368$	8186.42	1.15964
$369$	0	0
$370$	844.111	0.118603
$371$	−7140.62	−0.999252
$372$	0	0
$373$	2785.32	0.386644	0.193322	−	0.981135i	$-0.438074\pi$
$373$	2785.32	0.386644	0.193322	+	0.981135i	$0.438074\pi$
$374$	135.403	0.0187206
$375$	0	0
$376$	7195.58	0.986925
$377$	0	0
$378$	0	0
$379$	−7950.32	−1.07752	−0.538761	−	0.842459i	$-0.681107\pi$
$379$	−7950.32	−1.07752	−0.538761	+	0.842459i	$0.681107\pi$
$380$	831.592	0.112263
$381$	0	0
$382$	−3218.76	−0.431116
$383$	−7220.60	−0.963331	−0.481665	−	0.876355i	$-0.659968\pi$
$383$	−7220.60	−0.963331	−0.481665	+	0.876355i	$0.659968\pi$
$384$	0	0
$385$	−970.659	−0.128492
$386$	3389.65	0.446966
$387$	0	0
$388$	−2169.76	−0.283899
$389$	−781.125	−0.101811	−0.0509057	−	0.998703i	$-0.516211\pi$
$389$	−781.125	−0.101811	−0.0509057	+	0.998703i	$0.516211\pi$
$390$	0	0
$391$	−808.543	−0.104577
$392$	−5499.49	−0.708587
$393$	0	0
$394$	−3744.77	−0.478830
$395$	−3938.86	−0.501736
$396$	0	0
$397$	−6287.11	−0.794813	−0.397407	−	0.917643i	$-0.630090\pi$
$397$	−6287.11	−0.794813	−0.397407	+	0.917643i	$0.630090\pi$
$398$	3740.07	0.471037
$399$	0	0
$400$	2846.53	0.355816
$401$	−11933.0	−1.48605	−0.743025	−	0.669264i	$-0.766610\pi$
$401$	−11933.0	−1.48605	−0.743025	+	0.669264i	$0.766610\pi$
$402$	0	0
$403$	0	0
$404$	1046.58	0.128884
$405$	0	0
$406$	2531.80	0.309485
$407$	−446.276	−0.0543515
$408$	0	0
$409$	7864.96	0.950849	0.475425	−	0.879756i	$-0.342294\pi$
$409$	7864.96	0.950849	0.475425	+	0.879756i	$0.342294\pi$
$410$	5048.52	0.608119
$411$	0	0
$412$	−205.086	−0.0245239
$413$	−4222.73	−0.503116
$414$	0	0
$415$	−10249.3	−1.21233
$416$	0	0
$417$	0	0
$418$	1821.25	0.213111
$419$	3997.33	0.466068	0.233034	−	0.972469i	$-0.425135\pi$
$419$	3997.33	0.466068	0.233034	+	0.972469i	$0.425135\pi$
$420$	0	0
$421$	849.972	0.0983969	0.0491984	−	0.998789i	$-0.484333\pi$
$421$	849.972	0.0983969	0.0491984	+	0.998789i	$0.484333\pi$
$422$	1871.99	0.215941
$423$	0	0
$424$	16062.6	1.83979
$425$	−281.141	−0.0320879
$426$	0	0
$427$	3485.65	0.395041
$428$	3183.86	0.359574
$429$	0	0
$430$	3356.20	0.376396
$431$	−13367.7	−1.49396	−0.746982	−	0.664845i	$-0.768498\pi$
$431$	−13367.7	−1.49396	−0.746982	+	0.664845i	$0.768498\pi$
$432$	0	0
$433$	6594.88	0.731940	0.365970	−	0.930627i	$-0.380737\pi$
$433$	6594.88	0.731940	0.365970	+	0.930627i	$0.380737\pi$
$434$	−99.4368	−0.0109980
$435$	0	0
$436$	2147.04	0.235836
$437$	−10875.4	−1.19048
$438$	0	0
$439$	10805.0	1.17471	0.587354	−	0.809330i	$-0.300170\pi$
$439$	10805.0	1.17471	0.587354	+	0.809330i	$0.300170\pi$
$440$	2183.47	0.236575
$441$	0	0
$442$	0	0
$443$	−11448.1	−1.22780	−0.613901	−	0.789383i	$-0.710401\pi$
$443$	−11448.1	−1.22780	−0.613901	+	0.789383i	$0.710401\pi$
$444$	0	0
$445$	−11845.7	−1.26189
$446$	12211.1	1.29644
$447$	0	0
$448$	−6136.77	−0.647177
$449$	−3583.71	−0.376672	−0.188336	−	0.982105i	$-0.560309\pi$
$449$	−3583.71	−0.376672	−0.188336	+	0.982105i	$0.560309\pi$
$450$	0	0
$451$	−2669.12	−0.278678
$452$	−1616.08	−0.168172
$453$	0	0
$454$	−8642.04	−0.893373
$455$	0	0
$456$	0	0
$457$	12377.1	1.26691	0.633453	−	0.773781i	$-0.281637\pi$
$457$	12377.1	1.26691	0.633453	+	0.773781i	$0.281637\pi$
$458$	−10799.3	−1.10179
$459$	0	0
$460$	−2122.67	−0.215152
$461$	−437.626	−0.0442132	−0.0221066	−	0.999756i	$-0.507037\pi$
$461$	−437.626	−0.0442132	−0.0221066	+	0.999756i	$0.507037\pi$
$462$	0	0
$463$	−6568.13	−0.659281	−0.329640	−	0.944107i	$-0.606927\pi$
$463$	−6568.13	−0.659281	−0.329640	+	0.944107i	$0.606927\pi$
$464$	−4544.19	−0.454652
$465$	0	0
$466$	−13033.9	−1.29568
$467$	−897.583	−0.0889404	−0.0444702	−	0.999011i	$-0.514160\pi$
$467$	−897.583	−0.0889404	−0.0444702	+	0.999011i	$0.514160\pi$
$468$	0	0
$469$	6023.31	0.593029
$470$	6166.76	0.605216
$471$	0	0
$472$	9498.90	0.926319
$473$	−1774.40	−0.172488
$474$	0	0
$475$	−3781.53	−0.365281
$476$	81.4113	0.00783925
$477$	0	0
$478$	10357.1	0.991055
$479$	7078.48	0.675207	0.337603	−	0.941288i	$-0.390384\pi$
$479$	7078.48	0.675207	0.337603	+	0.941288i	$0.390384\pi$
$480$	0	0
$481$	0	0
$482$	−3470.30	−0.327942
$483$	0	0
$484$	1882.68	0.176810
$485$	−11422.0	−1.06938
$486$	0	0
$487$	3533.38	0.328773	0.164387	−	0.986396i	$-0.447436\pi$
$487$	3533.38	0.328773	0.164387	+	0.986396i	$0.447436\pi$
$488$	−7840.87	−0.727335
$489$	0	0
$490$	−4713.17	−0.434529
$491$	2025.52	0.186172	0.0930860	−	0.995658i	$-0.470327\pi$
$491$	2025.52	0.186172	0.0930860	+	0.995658i	$0.470327\pi$
$492$	0	0
$493$	448.813	0.0410010
$494$	0	0
$495$	0	0
$496$	178.474	0.0161567
$497$	1151.85	0.103959
$498$	0	0
$499$	−1397.04	−0.125331	−0.0626653	−	0.998035i	$-0.519960\pi$
$499$	−1397.04	−0.125331	−0.0626653	+	0.998035i	$0.519960\pi$
$500$	−2330.59	−0.208455
$501$	0	0
$502$	6771.43	0.602040
$503$	15884.5	1.40806	0.704031	−	0.710169i	$-0.251381\pi$
$503$	15884.5	1.40806	0.704031	+	0.710169i	$0.251381\pi$
$504$	0	0
$505$	5509.38	0.485473
$506$	−4648.81	−0.408428
$507$	0	0
$508$	3371.89	0.294495
$509$	14732.2	1.28289	0.641447	−	0.767167i	$-0.278334\pi$
$509$	14732.2	1.28289	0.641447	+	0.767167i	$0.278334\pi$
$510$	0	0
$511$	−857.517	−0.0742354
$512$	12941.6	1.11708
$513$	0	0
$514$	−8741.56	−0.750143
$515$	−1079.61	−0.0923754
$516$	0	0
$517$	−3260.32	−0.277348
$518$	1111.52	0.0942806
$519$	0	0
$520$	0	0
$521$	18090.5	1.52123	0.760613	−	0.649205i	$-0.224898\pi$
$521$	18090.5	1.52123	0.760613	+	0.649205i	$0.224898\pi$
$522$	0	0
$523$	13506.2	1.12922	0.564612	−	0.825356i	$-0.309026\pi$
$523$	13506.2	1.12922	0.564612	+	0.825356i	$0.309026\pi$
$524$	−3165.19	−0.263878
$525$	0	0
$526$	−12868.5	−1.06671
$527$	−17.6272	−0.00145703
$528$	0	0
$529$	15592.9	1.28157
$530$	13766.0	1.12822
$531$	0	0
$532$	1095.03	0.0892401
$533$	0	0
$534$	0	0
$535$	16760.5	1.35443
$536$	−13549.3	−1.09186
$537$	0	0
$538$	13753.9	1.10218
$539$	2491.82	0.199129
$540$	0	0
$541$	−18868.5	−1.49949	−0.749743	−	0.661730i	$-0.769823\pi$
$541$	−18868.5	−1.49949	−0.749743	+	0.661730i	$0.769823\pi$
$542$	−8021.19	−0.635682
$543$	0	0
$544$	−336.451	−0.0265169
$545$	11302.4	0.888335
$546$	0	0
$547$	−13405.6	−1.04786	−0.523931	−	0.851761i	$-0.675535\pi$
$547$	−13405.6	−1.04786	−0.523931	+	0.851761i	$0.675535\pi$
$548$	1288.70	0.100457
$549$	0	0
$550$	−1616.45	−0.125320
$551$	6036.81	0.466746
$552$	0	0
$553$	−5186.66	−0.398841
$554$	14184.7	1.08782
$555$	0	0
$556$	2302.04	0.175590
$557$	11863.2	0.902443	0.451222	−	0.892412i	$-0.350988\pi$
$557$	11863.2	0.902443	0.451222	+	0.892412i	$0.350988\pi$
$558$	0	0
$559$	0	0
$560$	−4339.18	−0.327436
$561$	0	0
$562$	9929.43	0.745280
$563$	−4252.00	−0.318296	−0.159148	−	0.987255i	$-0.550875\pi$
$563$	−4252.00	−0.318296	−0.159148	+	0.987255i	$0.550875\pi$
$564$	0	0
$565$	−8507.34	−0.633463
$566$	21617.9	1.60542
$567$	0	0
$568$	−2591.05	−0.191405
$569$	−4832.72	−0.356060	−0.178030	−	0.984025i	$-0.556972\pi$
$569$	−4832.72	−0.356060	−0.178030	+	0.984025i	$0.556972\pi$
$570$	0	0
$571$	−3819.95	−0.279965	−0.139982	−	0.990154i	$-0.544705\pi$
$571$	−3819.95	−0.279965	−0.139982	+	0.990154i	$0.544705\pi$
$572$	0	0
$573$	0	0
$574$	6647.85	0.483407
$575$	9652.48	0.700063
$576$	0	0
$577$	24973.7	1.80185	0.900926	−	0.433972i	$-0.142888\pi$
$577$	24973.7	1.80185	0.900926	+	0.433972i	$0.142888\pi$
$578$	12412.2	0.893217
$579$	0	0
$580$	1178.27	0.0843533
$581$	−13496.2	−0.963711
$582$	0	0
$583$	−7277.98	−0.517021
$584$	1928.96	0.136680
$585$	0	0
$586$	12043.4	0.848990
$587$	−1467.88	−0.103213	−0.0516064	−	0.998668i	$-0.516434\pi$
$587$	−1467.88	−0.103213	−0.0516064	+	0.998668i	$0.516434\pi$
$588$	0	0
$589$	−237.097	−0.0165864
$590$	8140.75	0.568050
$591$	0	0
$592$	−1995.01	−0.138504
$593$	−6559.32	−0.454231	−0.227115	−	0.973868i	$-0.572929\pi$
$593$	−6559.32	−0.454231	−0.227115	+	0.973868i	$0.572929\pi$
$594$	0	0
$595$	428.565	0.0295285
$596$	3681.49	0.253020
$597$	0	0
$598$	0	0
$599$	27069.0	1.84642	0.923212	−	0.384290i	$-0.125554\pi$
$599$	27069.0	1.84642	0.923212	+	0.384290i	$0.125554\pi$
$600$	0	0
$601$	8817.65	0.598468	0.299234	−	0.954180i	$-0.403269\pi$
$601$	8817.65	0.598468	0.299234	+	0.954180i	$0.403269\pi$
$602$	4419.41	0.299206
$603$	0	0
$604$	1366.41	0.0920502
$605$	9910.77	0.666000
$606$	0	0
$607$	23750.0	1.58811	0.794054	−	0.607847i	$-0.207966\pi$
$607$	23750.0	1.58811	0.794054	+	0.607847i	$0.207966\pi$
$608$	−4525.47	−0.301862
$609$	0	0
$610$	−6719.78	−0.446026
$611$	0	0
$612$	0	0
$613$	23474.2	1.54668	0.773338	−	0.633994i	$-0.218586\pi$
$613$	23474.2	1.54668	0.773338	+	0.633994i	$0.218586\pi$
$614$	19173.9	1.26025
$615$	0	0
$616$	2875.18	0.188059
$617$	−19145.1	−1.24919	−0.624596	−	0.780948i	$-0.714736\pi$
$617$	−19145.1	−1.24919	−0.624596	+	0.780948i	$0.714736\pi$
$618$	0	0
$619$	−15484.0	−1.00542	−0.502710	−	0.864455i	$-0.667663\pi$
$619$	−15484.0	−1.00542	−0.502710	+	0.864455i	$0.667663\pi$
$620$	−46.2767	−0.00299761
$621$	0	0
$622$	−3635.42	−0.234352
$623$	−15598.3	−1.00310
$624$	0	0
$625$	−5027.01	−0.321728
$626$	16527.8	1.05525
$627$	0	0
$628$	4325.77	0.274868
$629$	197.039	0.0124904
$630$	0	0
$631$	−20312.0	−1.28147	−0.640736	−	0.767761i	$-0.721371\pi$
$631$	−20312.0	−1.28147	−0.640736	+	0.767761i	$0.721371\pi$
$632$	11667.2	0.734333
$633$	0	0
$634$	24473.4	1.53307
$635$	17750.3	1.10929
$636$	0	0
$637$	0	0
$638$	2580.50	0.160130
$639$	0	0
$640$	7288.50	0.450161
$641$	−14247.2	−0.877896	−0.438948	−	0.898512i	$-0.644649\pi$
$641$	−14247.2	−0.877896	−0.438948	+	0.898512i	$0.644649\pi$
$642$	0	0
$643$	−19342.6	−1.18631	−0.593155	−	0.805088i	$-0.702118\pi$
$643$	−19342.6	−1.18631	−0.593155	+	0.805088i	$0.702118\pi$
$644$	−2795.11	−0.171029
$645$	0	0
$646$	−804.118	−0.0489746
$647$	−23481.4	−1.42681	−0.713407	−	0.700750i	$-0.752849\pi$
$647$	−23481.4	−1.42681	−0.713407	+	0.700750i	$0.752849\pi$
$648$	0	0
$649$	−4303.96	−0.260316
$650$	0	0
$651$	0	0
$652$	−6067.06	−0.364424
$653$	−24294.7	−1.45594	−0.727969	−	0.685611i	$-0.759535\pi$
$653$	−24294.7	−1.45594	−0.727969	+	0.685611i	$0.759535\pi$
$654$	0	0
$655$	−16662.2	−0.993963
$656$	−11931.9	−0.710155
$657$	0	0
$658$	8120.34	0.481100
$659$	27441.7	1.62212	0.811060	−	0.584963i	$-0.198891\pi$
$659$	27441.7	1.62212	0.811060	+	0.584963i	$0.198891\pi$
$660$	0	0
$661$	9184.52	0.540449	0.270224	−	0.962797i	$-0.412902\pi$
$661$	9184.52	0.540449	0.270224	+	0.962797i	$0.412902\pi$
$662$	5703.77	0.334869
$663$	0	0
$664$	30359.3	1.77435
$665$	5764.47	0.336145
$666$	0	0
$667$	−15409.2	−0.894521
$668$	320.574	0.0185679
$669$	0	0
$670$	−11612.0	−0.669568
$671$	3552.70	0.204397
$672$	0	0
$673$	−1968.42	−0.112744	−0.0563722	−	0.998410i	$-0.517953\pi$
$673$	−1968.42	−0.112744	−0.0563722	+	0.998410i	$0.517953\pi$
$674$	−4035.48	−0.230625
$675$	0	0
$676$	0	0
$677$	14505.5	0.823472	0.411736	−	0.911303i	$-0.364923\pi$
$677$	14505.5	0.823472	0.411736	+	0.911303i	$0.364923\pi$
$678$	0	0
$679$	−15040.4	−0.850072
$680$	−964.045	−0.0543668
$681$	0	0
$682$	−101.350	−0.00569044
$683$	5153.57	0.288720	0.144360	−	0.989525i	$-0.453888\pi$
$683$	5153.57	0.288720	0.144360	+	0.989525i	$0.453888\pi$
$684$	0	0
$685$	6783.98	0.378398
$686$	−15596.0	−0.868013
$687$	0	0
$688$	−7932.17	−0.439551
$689$	0	0
$690$	0	0
$691$	32224.4	1.77406	0.887029	−	0.461713i	$-0.152765\pi$
$691$	32224.4	1.77406	0.887029	+	0.461713i	$0.152765\pi$
$692$	−393.633	−0.0216238
$693$	0	0
$694$	2487.14	0.136038
$695$	12118.4	0.661405
$696$	0	0
$697$	1178.47	0.0640425
$698$	−5008.42	−0.271592
$699$	0	0
$700$	−971.898	−0.0524776
$701$	−25310.1	−1.36370	−0.681848	−	0.731494i	$-0.738823\pi$
$701$	−25310.1	−1.36370	−0.681848	+	0.731494i	$0.738823\pi$
$702$	0	0
$703$	2650.30	0.142188
$704$	−6254.83	−0.334855
$705$	0	0
$706$	25476.8	1.35812
$707$	7254.70	0.385914
$708$	0	0
$709$	−21069.6	−1.11606	−0.558028	−	0.829822i	$-0.688442\pi$
$709$	−21069.6	−1.11606	−0.558028	+	0.829822i	$0.688442\pi$
$710$	−2220.58	−0.117376
$711$	0	0
$712$	35088.0	1.84688
$713$	605.198	0.0317880
$714$	0	0
$715$	0	0
$716$	6205.95	0.323921
$717$	0	0
$718$	−6979.23	−0.362761
$719$	−18499.0	−0.959523	−0.479762	−	0.877399i	$-0.659277\pi$
$719$	−18499.0	−0.959523	−0.479762	+	0.877399i	$0.659277\pi$
$720$	0	0
$721$	−1421.62	−0.0734313
$722$	6596.13	0.340004
$723$	0	0
$724$	−5277.89	−0.270927
$725$	−5357.98	−0.274469
$726$	0	0
$727$	−24659.7	−1.25801	−0.629007	−	0.777400i	$-0.716538\pi$
$727$	−24659.7	−1.25801	−0.629007	+	0.777400i	$0.716538\pi$
$728$	0	0
$729$	0	0
$730$	1653.16	0.0838165
$731$	783.432	0.0396392
$732$	0	0
$733$	−1210.80	−0.0610119	−0.0305060	−	0.999535i	$-0.509712\pi$
$733$	−1210.80	−0.0610119	−0.0305060	+	0.999535i	$0.509712\pi$
$734$	2138.33	0.107530
$735$	0	0
$736$	11551.4	0.578521
$737$	6139.18	0.306838
$738$	0	0
$739$	−17111.2	−0.851753	−0.425877	−	0.904781i	$-0.640034\pi$
$739$	−17111.2	−0.851753	−0.425877	+	0.904781i	$0.640034\pi$
$740$	517.288	0.0256971
$741$	0	0
$742$	18126.9	0.896848
$743$	1521.57	0.0751294	0.0375647	−	0.999294i	$-0.488040\pi$
$743$	1521.57	0.0751294	0.0375647	+	0.999294i	$0.488040\pi$
$744$	0	0
$745$	19380.1	0.953062
$746$	−7070.71	−0.347020
$747$	0	0
$748$	82.9774	0.00405609
$749$	22070.0	1.07666
$750$	0	0
$751$	25923.6	1.25961	0.629803	−	0.776755i	$-0.283136\pi$
$751$	25923.6	1.25961	0.629803	+	0.776755i	$0.283136\pi$
$752$	−14574.8	−0.706765
$753$	0	0
$754$	0	0
$755$	7193.03	0.346730
$756$	0	0
$757$	25495.4	1.22410	0.612051	−	0.790818i	$-0.290345\pi$
$757$	25495.4	1.22410	0.612051	+	0.790818i	$0.290345\pi$
$758$	20182.4	0.967096
$759$	0	0
$760$	−12967.0	−0.618898
$761$	−17675.8	−0.841981	−0.420990	−	0.907065i	$-0.638317\pi$
$761$	−17675.8	−0.841981	−0.420990	+	0.907065i	$0.638317\pi$
$762$	0	0
$763$	14882.9	0.706158
$764$	−1972.52	−0.0934074
$765$	0	0
$766$	18330.0	0.864607
$767$	0	0
$768$	0	0
$769$	28854.8	1.35309	0.676547	−	0.736399i	$-0.263476\pi$
$769$	28854.8	1.35309	0.676547	+	0.736399i	$0.263476\pi$
$770$	2464.08	0.115324
$771$	0	0
$772$	2077.24	0.0968415
$773$	−35496.4	−1.65164	−0.825820	−	0.563933i	$-0.809287\pi$
$773$	−35496.4	−1.65164	−0.825820	+	0.563933i	$0.809287\pi$
$774$	0	0
$775$	210.436	0.00975364
$776$	33833.0	1.56512
$777$	0	0
$778$	1982.94	0.0913776
$779$	15851.1	0.729044
$780$	0	0
$781$	1174.01	0.0537890
$782$	2052.54	0.0938602
$783$	0	0
$784$	11139.3	0.507439
$785$	22771.7	1.03536
$786$	0	0
$787$	13249.0	0.600096	0.300048	−	0.953924i	$-0.402997\pi$
$787$	13249.0	0.600096	0.300048	+	0.953924i	$0.402997\pi$
$788$	−2294.87	−0.103745
$789$	0	0
$790$	9999.06	0.450317
$791$	−11202.4	−0.503554
$792$	0	0
$793$	0	0
$794$	15960.2	0.713360
$795$	0	0
$796$	2291.99	0.102057
$797$	32231.8	1.43251	0.716254	−	0.697840i	$-0.245855\pi$
$797$	32231.8	1.43251	0.716254	+	0.697840i	$0.245855\pi$
$798$	0	0
$799$	1439.50	0.0637368
$800$	4016.59	0.177510
$801$	0	0
$802$	30292.7	1.33376
$803$	−874.012	−0.0384100
$804$	0	0
$805$	−14714.0	−0.644224
$806$	0	0
$807$	0	0
$808$	−16319.2	−0.710531
$809$	12637.6	0.549216	0.274608	−	0.961556i	$-0.411452\pi$
$809$	12637.6	0.549216	0.274608	+	0.961556i	$0.411452\pi$
$810$	0	0
$811$	41102.4	1.77966	0.889828	−	0.456296i	$-0.150824\pi$
$811$	41102.4	1.77966	0.889828	+	0.456296i	$0.150824\pi$
$812$	1551.53	0.0670544
$813$	0	0
$814$	1132.90	0.0487815
$815$	−31938.2	−1.37269
$816$	0	0
$817$	10537.6	0.451243
$818$	−19965.7	−0.853405
$819$	0	0
$820$	3093.83	0.131758
$821$	−13071.1	−0.555644	−0.277822	−	0.960633i	$-0.589613\pi$
$821$	−13071.1	−0.555644	−0.277822	+	0.960633i	$0.589613\pi$
$822$	0	0
$823$	−29316.1	−1.24167	−0.620836	−	0.783941i	$-0.713207\pi$
$823$	−29316.1	−1.24167	−0.620836	+	0.783941i	$0.713207\pi$
$824$	3197.90	0.135199
$825$	0	0
$826$	10719.7	0.451556
$827$	4300.87	0.180842	0.0904208	−	0.995904i	$-0.471179\pi$
$827$	4300.87	0.180842	0.0904208	+	0.995904i	$0.471179\pi$
$828$	0	0
$829$	−15590.5	−0.653174	−0.326587	−	0.945167i	$-0.605899\pi$
$829$	−15590.5	−0.653174	−0.326587	+	0.945167i	$0.605899\pi$
$830$	26018.5	1.08809
$831$	0	0
$832$	0	0
$833$	−1100.19	−0.0457614
$834$	0	0
$835$	1687.56	0.0699407
$836$	1116.10	0.0461735
$837$	0	0
$838$	−10147.5	−0.418305
$839$	18957.4	0.780073	0.390037	−	0.920799i	$-0.372462\pi$
$839$	18957.4	0.780073	0.390037	+	0.920799i	$0.372462\pi$
$840$	0	0
$841$	−15835.6	−0.649291
$842$	−2157.71	−0.0883130
$843$	0	0
$844$	1147.19	0.0467868
$845$	0	0
$846$	0	0
$847$	13050.4	0.529419
$848$	−32535.1	−1.31752
$849$	0	0
$850$	713.696	0.0287995
$851$	−6764.99	−0.272504
$852$	0	0
$853$	−36949.2	−1.48314	−0.741568	−	0.670878i	$-0.765918\pi$
$853$	−36949.2	−1.48314	−0.741568	+	0.670878i	$0.765918\pi$
$854$	−8848.55	−0.354556
$855$	0	0
$856$	−49645.9	−1.98231
$857$	26643.0	1.06197	0.530985	−	0.847381i	$-0.321822\pi$
$857$	26643.0	1.06197	0.530985	+	0.847381i	$0.321822\pi$
$858$	0	0
$859$	−39165.5	−1.55566	−0.777829	−	0.628477i	$-0.783679\pi$
$859$	−39165.5	−1.55566	−0.777829	+	0.628477i	$0.783679\pi$
$860$	2056.74	0.0815516
$861$	0	0
$862$	33934.7	1.34086
$863$	2483.04	0.0979418	0.0489709	−	0.998800i	$-0.484406\pi$
$863$	2483.04	0.0979418	0.0489709	+	0.998800i	$0.484406\pi$
$864$	0	0
$865$	−2072.16	−0.0814514
$866$	−16741.5	−0.656929
$867$	0	0
$868$	−60.9368	−0.00238287
$869$	−5286.44	−0.206364
$870$	0	0
$871$	0	0
$872$	−33478.8	−1.30015
$873$	0	0
$874$	27607.9	1.06848
$875$	−16155.3	−0.624171
$876$	0	0
$877$	3191.16	0.122871	0.0614355	−	0.998111i	$-0.480432\pi$
$877$	3191.16	0.122871	0.0614355	+	0.998111i	$0.480432\pi$
$878$	−27429.3	−1.05432
$879$	0	0
$880$	−4422.65	−0.169418
$881$	24314.3	0.929817	0.464908	−	0.885359i	$-0.346087\pi$
$881$	24314.3	0.929817	0.464908	+	0.885359i	$0.346087\pi$
$882$	0	0
$883$	7703.20	0.293583	0.146791	−	0.989167i	$-0.453105\pi$
$883$	7703.20	0.293583	0.146791	+	0.989167i	$0.453105\pi$
$884$	0	0
$885$	0	0
$886$	29061.8	1.10198
$887$	−34386.1	−1.30166	−0.650829	−	0.759224i	$-0.725579\pi$
$887$	−34386.1	−1.30166	−0.650829	+	0.759224i	$0.725579\pi$
$888$	0	0
$889$	23373.4	0.881799
$890$	30071.1	1.13257
$891$	0	0
$892$	7483.20	0.280892
$893$	19362.1	0.725564
$894$	0	0
$895$	32669.3	1.22013
$896$	9597.43	0.357843
$897$	0	0
$898$	9097.47	0.338070
$899$	−335.938	−0.0124629
$900$	0	0
$901$	3213.37	0.118816
$902$	6775.73	0.250119
$903$	0	0
$904$	25199.5	0.927126
$905$	−27783.8	−1.02051
$906$	0	0
$907$	1088.24	0.0398395	0.0199197	−	0.999802i	$-0.493659\pi$
$907$	1088.24	0.0398395	0.0199197	+	0.999802i	$0.493659\pi$
$908$	−5296.01	−0.193562
$909$	0	0
$910$	0	0
$911$	−38964.1	−1.41705	−0.708527	−	0.705683i	$-0.750640\pi$
$911$	−38964.1	−1.41705	−0.708527	+	0.705683i	$0.750640\pi$
$912$	0	0
$913$	−13755.8	−0.498632
$914$	−31420.1	−1.13707
$915$	0	0
$916$	−6618.04	−0.238719
$917$	−21940.6	−0.790124
$918$	0	0
$919$	23304.0	0.836483	0.418242	−	0.908336i	$-0.362646\pi$
$919$	23304.0	0.836483	0.418242	+	0.908336i	$0.362646\pi$
$920$	33098.7	1.18612
$921$	0	0
$922$	1110.94	0.0396822
$923$	0	0
$924$	0	0
$925$	−2352.28	−0.0836135
$926$	16673.6	0.591717
$927$	0	0
$928$	−6412.06	−0.226817
$929$	34155.1	1.20624	0.603118	−	0.797652i	$-0.293925\pi$
$929$	34155.1	1.20624	0.603118	+	0.797652i	$0.293925\pi$
$930$	0	0
$931$	−14798.2	−0.520936
$932$	−7987.44	−0.280727
$933$	0	0
$934$	2278.57	0.0798257
$935$	436.809	0.0152783
$936$	0	0
$937$	−5453.10	−0.190123	−0.0950614	−	0.995471i	$-0.530305\pi$
$937$	−5453.10	−0.190123	−0.0950614	+	0.995471i	$0.530305\pi$
$938$	−15290.6	−0.532255
$939$	0	0
$940$	3779.11	0.131129
$941$	−32998.0	−1.14315	−0.571574	−	0.820550i	$-0.693667\pi$
$941$	−32998.0	−1.14315	−0.571574	+	0.820550i	$0.693667\pi$
$942$	0	0
$943$	−40460.5	−1.39722
$944$	−19240.2	−0.663363
$945$	0	0
$946$	4504.43	0.154811
$947$	7589.27	0.260420	0.130210	−	0.991486i	$-0.458435\pi$
$947$	7589.27	0.260420	0.130210	+	0.991486i	$0.458435\pi$
$948$	0	0
$949$	0	0
$950$	9599.66	0.327846
$951$	0	0
$952$	−1269.45	−0.0432174
$953$	19839.2	0.674349	0.337175	−	0.941442i	$-0.390529\pi$
$953$	19839.2	0.674349	0.337175	+	0.941442i	$0.390529\pi$
$954$	0	0
$955$	−10383.7	−0.351842
$956$	6347.05	0.214726
$957$	0	0
$958$	−17969.2	−0.606011
$959$	8933.09	0.300797
$960$	0	0
$961$	−29777.8	−0.999557
$962$	0	0
$963$	0	0
$964$	−2126.67	−0.0710532
$965$	10935.0	0.364778
$966$	0	0
$967$	−33260.8	−1.10610	−0.553049	−	0.833149i	$-0.686536\pi$
$967$	−33260.8	−1.10610	−0.553049	+	0.833149i	$0.686536\pi$
$968$	−29356.6	−0.974747
$969$	0	0
$970$	28995.6	0.959786
$971$	−6643.15	−0.219556	−0.109778	−	0.993956i	$-0.535014\pi$
$971$	−6643.15	−0.219556	−0.109778	+	0.993956i	$0.535014\pi$
$972$	0	0
$973$	15957.4	0.525766
$974$	−8969.71	−0.295080
$975$	0	0
$976$	15881.8	0.520865
$977$	10622.3	0.347839	0.173920	−	0.984760i	$-0.444357\pi$
$977$	10622.3	0.347839	0.173920	+	0.984760i	$0.444357\pi$
$978$	0	0
$979$	−15898.4	−0.519014
$980$	−2888.32	−0.0941470
$981$	0	0
$982$	−5141.92	−0.167093
$983$	9298.71	0.301712	0.150856	−	0.988556i	$-0.451797\pi$
$983$	9298.71	0.301712	0.150856	+	0.988556i	$0.451797\pi$
$984$	0	0
$985$	−12080.6	−0.390783
$986$	−1139.34	−0.0367992
$987$	0	0
$988$	0	0
$989$	−26897.7	−0.864810
$990$	0	0
$991$	−43517.4	−1.39493	−0.697465	−	0.716619i	$-0.745689\pi$
$991$	−43517.4	−1.39493	−0.697465	+	0.716619i	$0.745689\pi$
$992$	251.835	0.00806025
$993$	0	0
$994$	−2924.04	−0.0933048
$995$	12065.5	0.384423
$996$	0	0
$997$	−12099.0	−0.384332	−0.192166	−	0.981362i	$-0.561551\pi$
$997$	−12099.0	−0.384332	−0.192166	+	0.981362i	$0.561551\pi$
$998$	3546.47	0.112487
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bl.1.3		12
3.2	odd	2	inner	1521.4.a.bl.1.9		12
13.6	odd	12		117.4.q.f.10.5	yes	12
13.11	odd	12		117.4.q.f.82.5	yes	12
13.12	even	2	inner	1521.4.a.bl.1.10		12
39.11	even	12		117.4.q.f.82.2	yes	12
39.32	even	12		117.4.q.f.10.2	✓	12
39.38	odd	2	inner	1521.4.a.bl.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.q.f.10.2	✓	12	39.32	even	12
117.4.q.f.10.5	yes	12	13.6	odd	12
117.4.q.f.82.2	yes	12	39.11	even	12
117.4.q.f.82.5	yes	12	13.11	odd	12
1521.4.a.bl.1.3		12	1.1	even	1	trivial
1521.4.a.bl.1.4		12	39.38	odd	2	inner
1521.4.a.bl.1.9		12	3.2	odd	2	inner
1521.4.a.bl.1.10		12	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53857 q^{2} -1.55568 q^{4} -8.18941 q^{5} -10.7837 q^{7} +24.2577 q^{8} +O(q^{10})\)	\(q-2.53857 q^{2} -1.55568 q^{4} -8.18941 q^{5} -10.7837 q^{7} +24.2577 q^{8} +20.7894 q^{10} -10.9912 q^{11} +27.3753 q^{14} -49.1344 q^{16} +4.85282 q^{17} +65.2735 q^{19} +12.7401 q^{20} +27.9019 q^{22} -166.613 q^{23} -57.9336 q^{25} +16.7761 q^{28} +92.4849 q^{29} -3.63236 q^{31} -69.3309 q^{32} -12.3192 q^{34} +88.3125 q^{35} +40.6031 q^{37} -165.701 q^{38} -198.656 q^{40} +242.842 q^{41} +161.438 q^{43} +17.0988 q^{44} +422.958 q^{46} +296.631 q^{47} -226.711 q^{49} +147.068 q^{50} +662.165 q^{53} +90.0113 q^{55} -261.589 q^{56} -234.779 q^{58} +391.583 q^{59} -323.232 q^{61} +9.22099 q^{62} +569.076 q^{64} -558.555 q^{67} -7.54945 q^{68} -224.187 q^{70} -106.813 q^{71} +79.5194 q^{73} -103.074 q^{74} -101.545 q^{76} +118.526 q^{77} +480.970 q^{79} +402.382 q^{80} -616.469 q^{82} +1251.53 q^{83} -39.7418 q^{85} -409.822 q^{86} -266.621 q^{88} +1446.47 q^{89} +259.197 q^{92} -753.016 q^{94} -534.551 q^{95} +1394.73 q^{97} +575.520 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 32 q^{4}+O(q^{10}) \) 12 * q + 32 * q^4	\( 12 q + 32 q^{4} - 112 q^{10} + 184 q^{16} - 584 q^{22} + 92 q^{25} - 448 q^{40} - 1620 q^{43} - 2136 q^{49} - 920 q^{55} - 2588 q^{61} + 184 q^{64} - 6380 q^{79} - 2536 q^{82} - 10280 q^{88} - 3320 q^{94}+O(q^{100}) \) 12 * q + 32 * q^4 - 112 * q^10 + 184 * q^16 - 584 * q^22 + 92 * q^25 - 448 * q^40 - 1620 * q^43 - 2136 * q^49 - 920 * q^55 - 2588 * q^61 + 184 * q^64 - 6380 * q^79 - 2536 * q^82 - 10280 * q^88 - 3320 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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