LMFDB - Embedded newform 2151.4.a.g.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55261 q^{2} +4.62104 q^{4} -5.70782 q^{5} +19.6278 q^{7} +12.0041 q^{8} +O(q^{10})$	\(q-3.55261 q^{2} +4.62104 q^{4} -5.70782 q^{5} +19.6278 q^{7} +12.0041 q^{8} +20.2777 q^{10} +53.6545 q^{11} -46.6698 q^{13} -69.7299 q^{14} -79.6143 q^{16} -16.3957 q^{17} +139.195 q^{19} -26.3761 q^{20} -190.614 q^{22} -72.3809 q^{23} -92.4208 q^{25} +165.800 q^{26} +90.7008 q^{28} +102.408 q^{29} +256.252 q^{31} +186.806 q^{32} +58.2474 q^{34} -112.032 q^{35} -166.515 q^{37} -494.506 q^{38} -68.5174 q^{40} -487.233 q^{41} +422.930 q^{43} +247.940 q^{44} +257.141 q^{46} -439.594 q^{47} +42.2497 q^{49} +328.335 q^{50} -215.663 q^{52} -764.941 q^{53} -306.250 q^{55} +235.614 q^{56} -363.815 q^{58} -498.354 q^{59} -109.988 q^{61} -910.364 q^{62} -26.7331 q^{64} +266.383 q^{65} -175.561 q^{67} -75.7651 q^{68} +398.006 q^{70} -855.221 q^{71} -60.8046 q^{73} +591.565 q^{74} +643.226 q^{76} +1053.12 q^{77} -752.201 q^{79} +454.424 q^{80} +1730.95 q^{82} +802.845 q^{83} +93.5836 q^{85} -1502.51 q^{86} +644.075 q^{88} -76.5782 q^{89} -916.025 q^{91} -334.475 q^{92} +1561.71 q^{94} -794.500 q^{95} +607.054 q^{97} -150.097 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

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 $

$2$	−3.55261	−1.25604	−0.628019	−	0.778198i	$-0.716134\pi$
$2$	−3.55261	−1.25604	−0.628019	+	0.778198i	$0.716134\pi$
$3$	0	0
$3$	0	0
$4$	4.62104	0.577630
$5$	−5.70782	−0.510523	−0.255262	−	0.966872i	$-0.582162\pi$
$5$	−5.70782	−0.510523	−0.255262	+	0.966872i	$0.582162\pi$
$6$	0	0
$7$	19.6278	1.05980	0.529900	−	0.848060i	$-0.322229\pi$
$7$	19.6278	1.05980	0.529900	+	0.848060i	$0.322229\pi$
$8$	12.0041	0.530512
$9$	0	0
$10$	20.2777	0.641236
$11$	53.6545	1.47068	0.735338	−	0.677700i	$-0.237023\pi$
$11$	53.6545	1.47068	0.735338	+	0.677700i	$0.237023\pi$
$12$	0	0
$13$	−46.6698	−0.995683	−0.497841	−	0.867268i	$-0.665874\pi$
$13$	−46.6698	−0.995683	−0.497841	+	0.867268i	$0.665874\pi$
$14$	−69.7299	−1.33115
$15$	0	0
$16$	−79.6143	−1.24397
$17$	−16.3957	−0.233914	−0.116957	−	0.993137i	$-0.537314\pi$
$17$	−16.3957	−0.233914	−0.116957	+	0.993137i	$0.537314\pi$
$18$	0	0
$19$	139.195	1.68071	0.840356	−	0.542035i	$-0.182346\pi$
$19$	139.195	1.68071	0.840356	+	0.542035i	$0.182346\pi$
$20$	−26.3761	−0.294894
$21$	0	0
$22$	−190.614	−1.84722
$23$	−72.3809	−0.656194	−0.328097	−	0.944644i	$-0.606407\pi$
$23$	−72.3809	−0.656194	−0.328097	+	0.944644i	$0.606407\pi$
$24$	0	0
$25$	−92.4208	−0.739366
$26$	165.800	1.25061
$27$	0	0
$28$	90.7008	0.612173
$29$	102.408	0.655747	0.327874	−	0.944722i	$-0.393668\pi$
$29$	102.408	0.655747	0.327874	+	0.944722i	$0.393668\pi$
$30$	0	0
$31$	256.252	1.48465	0.742326	−	0.670039i	$-0.233722\pi$
$31$	256.252	1.48465	0.742326	+	0.670039i	$0.233722\pi$
$32$	186.806	1.03197
$33$	0	0
$34$	58.2474	0.293804
$35$	−112.032	−0.541053
$36$	0	0
$37$	−166.515	−0.739864	−0.369932	−	0.929059i	$-0.620619\pi$
$37$	−166.515	−0.739864	−0.369932	+	0.929059i	$0.620619\pi$
$38$	−494.506	−2.11104
$39$	0	0
$40$	−68.5174	−0.270839
$41$	−487.233	−1.85593	−0.927963	−	0.372672i	$-0.878442\pi$
$41$	−487.233	−1.85593	−0.927963	+	0.372672i	$0.878442\pi$
$42$	0	0
$43$	422.930	1.49991	0.749957	−	0.661487i	$-0.230074\pi$
$43$	422.930	1.49991	0.749957	+	0.661487i	$0.230074\pi$
$44$	247.940	0.849507
$45$	0	0
$46$	257.141	0.824204
$47$	−439.594	−1.36429	−0.682143	−	0.731219i	$-0.738952\pi$
$47$	−439.594	−1.36429	−0.682143	+	0.731219i	$0.738952\pi$
$48$	0	0
$49$	42.2497	0.123177
$50$	328.335	0.928672
$51$	0	0
$52$	−215.663	−0.575136
$53$	−764.941	−1.98250	−0.991252	−	0.131981i	$-0.957866\pi$
$53$	−764.941	−1.98250	−0.991252	+	0.131981i	$0.957866\pi$
$54$	0	0
$55$	−306.250	−0.750814
$56$	235.614	0.562237
$57$	0	0
$58$	−363.815	−0.823643
$59$	−498.354	−1.09966	−0.549832	−	0.835275i	$-0.685308\pi$
$59$	−498.354	−1.09966	−0.549832	+	0.835275i	$0.685308\pi$
$60$	0	0
$61$	−109.988	−0.230862	−0.115431	−	0.993316i	$-0.536825\pi$
$61$	−109.988	−0.230862	−0.115431	+	0.993316i	$0.536825\pi$
$62$	−910.364	−1.86478
$63$	0	0
$64$	−26.7331	−0.0522132
$65$	266.383	0.508319
$66$	0	0
$67$	−175.561	−0.320121	−0.160061	−	0.987107i	$-0.551169\pi$
$67$	−175.561	−0.320121	−0.160061	+	0.987107i	$0.551169\pi$
$68$	−75.7651	−0.135116
$69$	0	0
$70$	398.006	0.679582
$71$	−855.221	−1.42952	−0.714761	−	0.699369i	$-0.753464\pi$
$71$	−855.221	−1.42952	−0.714761	+	0.699369i	$0.753464\pi$
$72$	0	0
$73$	−60.8046	−0.0974882	−0.0487441	−	0.998811i	$-0.515522\pi$
$73$	−60.8046	−0.0974882	−0.0487441	+	0.998811i	$0.515522\pi$
$74$	591.565	0.929297
$75$	0	0
$76$	643.226	0.970830
$77$	1053.12	1.55862
$78$	0	0
$79$	−752.201	−1.07126	−0.535628	−	0.844454i	$-0.679925\pi$
$79$	−752.201	−1.07126	−0.535628	+	0.844454i	$0.679925\pi$
$80$	454.424	0.635077
$81$	0	0
$82$	1730.95	2.33111
$83$	802.845	1.06173	0.530866	−	0.847456i	$-0.321867\pi$
$83$	802.845	1.06173	0.530866	+	0.847456i	$0.321867\pi$
$84$	0	0
$85$	93.5836	0.119418
$86$	−1502.51	−1.88395
$87$	0	0
$88$	644.075	0.780212
$89$	−76.5782	−0.0912053	−0.0456026	−	0.998960i	$-0.514521\pi$
$89$	−76.5782	−0.0912053	−0.0456026	+	0.998960i	$0.514521\pi$
$90$	0	0
$91$	−916.025	−1.05522
$92$	−334.475	−0.379037
$93$	0	0
$94$	1561.71	1.71359
$95$	−794.500	−0.858042
$96$	0	0
$97$	607.054	0.635433	0.317716	−	0.948186i	$-0.397084\pi$
$97$	607.054	0.635433	0.317716	+	0.948186i	$0.397084\pi$
$98$	−150.097	−0.154715
$99$	0	0
$100$	−427.080	−0.427080
$101$	1175.60	1.15819	0.579094	−	0.815261i	$-0.303406\pi$
$101$	1175.60	1.15819	0.579094	+	0.815261i	$0.303406\pi$
$102$	0	0
$103$	1921.32	1.83799	0.918995	−	0.394268i	$-0.129002\pi$
$103$	1921.32	1.83799	0.918995	+	0.394268i	$0.129002\pi$
$104$	−560.230	−0.528222
$105$	0	0
$106$	2717.54	2.49010
$107$	−912.413	−0.824358	−0.412179	−	0.911103i	$-0.635232\pi$
$107$	−912.413	−0.824358	−0.412179	+	0.911103i	$0.635232\pi$
$108$	0	0
$109$	−1109.78	−0.975206	−0.487603	−	0.873066i	$-0.662129\pi$
$109$	−1109.78	−0.975206	−0.487603	+	0.873066i	$0.662129\pi$
$110$	1087.99	0.943051
$111$	0	0
$112$	−1562.65	−1.31836
$113$	−797.111	−0.663592	−0.331796	−	0.943351i	$-0.607655\pi$
$113$	−797.111	−0.663592	−0.331796	+	0.943351i	$0.607655\pi$
$114$	0	0
$115$	413.137	0.335002
$116$	473.231	0.378779
$117$	0	0
$118$	1770.46	1.38122
$119$	−321.811	−0.247902
$120$	0	0
$121$	1547.81	1.16289
$122$	390.746	0.289971
$123$	0	0
$124$	1184.15	0.857580
$125$	1241.00	0.887987
$126$	0	0
$127$	1902.88	1.32955	0.664777	−	0.747042i	$-0.268527\pi$
$127$	1902.88	1.32955	0.664777	+	0.747042i	$0.268527\pi$
$128$	−1399.47	−0.966383
$129$	0	0
$130$	−946.355	−0.638468
$131$	185.394	0.123649	0.0618244	−	0.998087i	$-0.480308\pi$
$131$	185.394	0.123649	0.0618244	+	0.998087i	$0.480308\pi$
$132$	0	0
$133$	2732.09	1.78122
$134$	623.698	0.402085
$135$	0	0
$136$	−196.816	−0.124094
$137$	2711.65	1.69104	0.845518	−	0.533947i	$-0.179292\pi$
$137$	2711.65	1.69104	0.845518	+	0.533947i	$0.179292\pi$
$138$	0	0
$139$	−1594.19	−0.972787	−0.486394	−	0.873740i	$-0.661688\pi$
$139$	−1594.19	−0.972787	−0.486394	+	0.873740i	$0.661688\pi$
$140$	−517.704	−0.312528
$141$	0	0
$142$	3038.27	1.79553
$143$	−2504.05	−1.46433
$144$	0	0
$145$	−584.526	−0.334774
$146$	216.015	0.122449
$147$	0	0
$148$	−769.475	−0.427368
$149$	−1184.56	−0.651294	−0.325647	−	0.945491i	$-0.605582\pi$
$149$	−1184.56	−0.651294	−0.325647	+	0.945491i	$0.605582\pi$
$150$	0	0
$151$	−1033.36	−0.556910	−0.278455	−	0.960449i	$-0.589822\pi$
$151$	−1033.36	−0.556910	−0.278455	+	0.960449i	$0.589822\pi$
$152$	1670.91	0.891638
$153$	0	0
$154$	−3741.32	−1.95769
$155$	−1462.64	−0.757949
$156$	0	0
$157$	−506.044	−0.257240	−0.128620	−	0.991694i	$-0.541055\pi$
$157$	−506.044	−0.257240	−0.128620	+	0.991694i	$0.541055\pi$
$158$	2672.28	1.34554
$159$	0	0
$160$	−1066.25	−0.526842
$161$	−1420.68	−0.695434
$162$	0	0
$163$	333.977	0.160485	0.0802427	−	0.996775i	$-0.474430\pi$
$163$	333.977	0.160485	0.0802427	+	0.996775i	$0.474430\pi$
$164$	−2251.52	−1.07204
$165$	0	0
$166$	−2852.20	−1.33357
$167$	1453.23	0.673379	0.336689	−	0.941616i	$-0.390693\pi$
$167$	1453.23	0.673379	0.336689	+	0.941616i	$0.390693\pi$
$168$	0	0
$169$	−18.9298	−0.00861618
$170$	−332.466	−0.149994
$171$	0	0
$172$	1954.38	0.866395
$173$	3764.64	1.65445	0.827226	−	0.561869i	$-0.189918\pi$
$173$	3764.64	1.65445	0.827226	+	0.561869i	$0.189918\pi$
$174$	0	0
$175$	−1814.01	−0.783581
$176$	−4271.67	−1.82948
$177$	0	0
$178$	272.052	0.114557
$179$	−3717.06	−1.55210	−0.776051	−	0.630670i	$-0.782780\pi$
$179$	−3717.06	−1.55210	−0.776051	+	0.630670i	$0.782780\pi$
$180$	0	0
$181$	−3654.09	−1.50059	−0.750294	−	0.661105i	$-0.770088\pi$
$181$	−3654.09	−1.50059	−0.750294	+	0.661105i	$0.770088\pi$
$182$	3254.28	1.32540
$183$	0	0
$184$	−868.869	−0.348119
$185$	950.441	0.377718
$186$	0	0
$187$	−879.701	−0.344011
$188$	−2031.38	−0.788053
$189$	0	0
$190$	2822.55	1.07773
$191$	−2494.70	−0.945078	−0.472539	−	0.881310i	$-0.656662\pi$
$191$	−2494.70	−0.945078	−0.472539	+	0.881310i	$0.656662\pi$
$192$	0	0
$193$	2175.37	0.811329	0.405664	−	0.914022i	$-0.367040\pi$
$193$	2175.37	0.811329	0.405664	+	0.914022i	$0.367040\pi$
$194$	−2156.63	−0.798127
$195$	0	0
$196$	195.238	0.0711508
$197$	1135.62	0.410709	0.205355	−	0.978688i	$-0.434165\pi$
$197$	1135.62	0.410709	0.205355	+	0.978688i	$0.434165\pi$
$198$	0	0
$199$	5509.25	1.96251	0.981256	−	0.192706i	$-0.0617265\pi$
$199$	5509.25	1.96251	0.981256	+	0.192706i	$0.0617265\pi$
$200$	−1109.43	−0.392243
$201$	0	0
$202$	−4176.46	−1.45473
$203$	2010.04	0.694961
$204$	0	0
$205$	2781.04	0.947493
$206$	−6825.69	−2.30859
$207$	0	0
$208$	3715.58	1.23860
$209$	7468.44	2.47178
$210$	0	0
$211$	−3503.75	−1.14317	−0.571584	−	0.820544i	$-0.693671\pi$
$211$	−3503.75	−1.14317	−0.571584	+	0.820544i	$0.693671\pi$
$212$	−3534.82	−1.14515
$213$	0	0
$214$	3241.45	1.03542
$215$	−2414.01	−0.765740
$216$	0	0
$217$	5029.66	1.57344
$218$	3942.61	1.22490
$219$	0	0
$220$	−1415.20	−0.433693
$221$	765.183	0.232904
$222$	0	0
$223$	−2005.35	−0.602190	−0.301095	−	0.953594i	$-0.597352\pi$
$223$	−2005.35	−0.602190	−0.301095	+	0.953594i	$0.597352\pi$
$224$	3666.58	1.09368
$225$	0	0
$226$	2831.83	0.833497
$227$	−3006.14	−0.878963	−0.439481	−	0.898252i	$-0.644838\pi$
$227$	−3006.14	−0.878963	−0.439481	+	0.898252i	$0.644838\pi$
$228$	0	0
$229$	949.396	0.273965	0.136982	−	0.990574i	$-0.456260\pi$
$229$	949.396	0.273965	0.136982	+	0.990574i	$0.456260\pi$
$230$	−1467.72	−0.420775
$231$	0	0
$232$	1229.32	0.347882
$233$	−1217.31	−0.342270	−0.171135	−	0.985248i	$-0.554743\pi$
$233$	−1217.31	−0.342270	−0.171135	+	0.985248i	$0.554743\pi$
$234$	0	0
$235$	2509.13	0.696499
$236$	−2302.92	−0.635199
$237$	0	0
$238$	1143.27	0.311374
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−4212.94	−1.12606	−0.563028	−	0.826438i	$-0.690364\pi$
$241$	−4212.94	−1.12606	−0.563028	+	0.826438i	$0.690364\pi$
$242$	−5498.75	−1.46063
$243$	0	0
$244$	−508.261	−0.133353
$245$	−241.154	−0.0628847
$246$	0	0
$247$	−6496.20	−1.67346
$248$	3076.08	0.787627
$249$	0	0
$250$	−4408.79	−1.11534
$251$	−3574.73	−0.898944	−0.449472	−	0.893294i	$-0.648388\pi$
$251$	−3574.73	−0.898944	−0.449472	+	0.893294i	$0.648388\pi$
$252$	0	0
$253$	−3883.56	−0.965049
$254$	−6760.19	−1.66997
$255$	0	0
$256$	5185.65	1.26603
$257$	−2586.73	−0.627842	−0.313921	−	0.949449i	$-0.601643\pi$
$257$	−2586.73	−0.627842	−0.313921	+	0.949449i	$0.601643\pi$
$258$	0	0
$259$	−3268.33	−0.784109
$260$	1230.97	0.293620
$261$	0	0
$262$	−658.634	−0.155308
$263$	−6001.44	−1.40709	−0.703545	−	0.710650i	$-0.748401\pi$
$263$	−6001.44	−1.40709	−0.703545	+	0.710650i	$0.748401\pi$
$264$	0	0
$265$	4366.15	1.01211
$266$	−9706.05	−2.23728
$267$	0	0
$268$	−811.273	−0.184912
$269$	2402.19	0.544476	0.272238	−	0.962230i	$-0.412236\pi$
$269$	2402.19	0.544476	0.272238	+	0.962230i	$0.412236\pi$
$270$	0	0
$271$	8464.58	1.89737	0.948684	−	0.316227i	$-0.102416\pi$
$271$	8464.58	1.89737	0.948684	+	0.316227i	$0.102416\pi$
$272$	1305.33	0.290983
$273$	0	0
$274$	−9633.44	−2.12401
$275$	−4958.79	−1.08737
$276$	0	0
$277$	1336.30	0.289856	0.144928	−	0.989442i	$-0.453705\pi$
$277$	1336.30	0.289856	0.144928	+	0.989442i	$0.453705\pi$
$278$	5663.54	1.22186
$279$	0	0
$280$	−1344.84	−0.287035
$281$	−642.356	−0.136369	−0.0681846	−	0.997673i	$-0.521721\pi$
$281$	−642.356	−0.136369	−0.0681846	+	0.997673i	$0.521721\pi$
$282$	0	0
$283$	−486.073	−0.102099	−0.0510496	−	0.998696i	$-0.516257\pi$
$283$	−486.073	−0.102099	−0.0510496	+	0.998696i	$0.516257\pi$
$284$	−3952.01	−0.825735
$285$	0	0
$286$	8895.90	1.83925
$287$	−9563.30	−1.96691
$288$	0	0
$289$	−4644.18	−0.945284
$290$	2076.59	0.420489
$291$	0	0
$292$	−280.980	−0.0563121
$293$	−3402.70	−0.678456	−0.339228	−	0.940704i	$-0.610166\pi$
$293$	−3402.70	−0.678456	−0.339228	+	0.940704i	$0.610166\pi$
$294$	0	0
$295$	2844.52	0.561404
$296$	−1998.87	−0.392507
$297$	0	0
$298$	4208.27	0.818050
$299$	3378.00	0.653361
$300$	0	0
$301$	8301.18	1.58961
$302$	3671.11	0.699500
$303$	0	0
$304$	−11081.9	−2.09076
$305$	627.794	0.117860
$306$	0	0
$307$	−1456.86	−0.270838	−0.135419	−	0.990788i	$-0.543238\pi$
$307$	−1456.86	−0.270838	−0.135419	+	0.990788i	$0.543238\pi$
$308$	4866.51	0.900308
$309$	0	0
$310$	5196.19	0.952013
$311$	3961.26	0.722258	0.361129	−	0.932516i	$-0.382391\pi$
$311$	3961.26	0.722258	0.361129	+	0.932516i	$0.382391\pi$
$312$	0	0
$313$	816.016	0.147361	0.0736804	−	0.997282i	$-0.476526\pi$
$313$	816.016	0.147361	0.0736804	+	0.997282i	$0.476526\pi$
$314$	1797.78	0.323103
$315$	0	0
$316$	−3475.95	−0.618790
$317$	1038.47	0.183995	0.0919976	−	0.995759i	$-0.470675\pi$
$317$	1038.47	0.183995	0.0919976	+	0.995759i	$0.470675\pi$
$318$	0	0
$319$	5494.65	0.964392
$320$	152.588	0.0266560
$321$	0	0
$322$	5047.11	0.873492
$323$	−2282.20	−0.393142
$324$	0	0
$325$	4313.26	0.736174
$326$	−1186.49	−0.201576
$327$	0	0
$328$	−5848.80	−0.984592
$329$	−8628.26	−1.44587
$330$	0	0
$331$	3941.12	0.654452	0.327226	−	0.944946i	$-0.393886\pi$
$331$	3941.12	0.654452	0.327226	+	0.944946i	$0.393886\pi$
$332$	3709.98	0.613288
$333$	0	0
$334$	−5162.76	−0.845789
$335$	1002.07	0.163429
$336$	0	0
$337$	6841.99	1.10596	0.552978	−	0.833196i	$-0.313491\pi$
$337$	6841.99	1.10596	0.552978	+	0.833196i	$0.313491\pi$
$338$	67.2500	0.0108222
$339$	0	0
$340$	432.453	0.0689796
$341$	13749.1	2.18344
$342$	0	0
$343$	−5903.06	−0.929257
$344$	5076.91	0.795722
$345$	0	0
$346$	−13374.3	−2.07805
$347$	−3929.06	−0.607847	−0.303924	−	0.952696i	$-0.598297\pi$
$347$	−3929.06	−0.607847	−0.303924	+	0.952696i	$0.598297\pi$
$348$	0	0
$349$	−2666.15	−0.408927	−0.204464	−	0.978874i	$-0.565545\pi$
$349$	−2666.15	−0.408927	−0.204464	+	0.978874i	$0.565545\pi$
$350$	6444.49	0.984207
$351$	0	0
$352$	10023.0	1.51769
$353$	−2623.25	−0.395529	−0.197764	−	0.980250i	$-0.563368\pi$
$353$	−2623.25	−0.395529	−0.197764	+	0.980250i	$0.563368\pi$
$354$	0	0
$355$	4881.45	0.729804
$356$	−353.871	−0.0526829
$357$	0	0
$358$	13205.3	1.94950
$359$	8508.51	1.25087	0.625435	−	0.780277i	$-0.284922\pi$
$359$	8508.51	1.25087	0.625435	+	0.780277i	$0.284922\pi$
$360$	0	0
$361$	12516.2	1.82479
$362$	12981.6	1.88479
$363$	0	0
$364$	−4232.99	−0.609530
$365$	347.062	0.0497700
$366$	0	0
$367$	3435.39	0.488627	0.244313	−	0.969696i	$-0.421437\pi$
$367$	3435.39	0.488627	0.244313	+	0.969696i	$0.421437\pi$
$368$	5762.55	0.816288
$369$	0	0
$370$	−3376.55	−0.474428
$371$	−15014.1	−2.10106
$372$	0	0
$373$	−11997.8	−1.66547	−0.832737	−	0.553669i	$-0.813227\pi$
$373$	−11997.8	−1.66547	−0.832737	+	0.553669i	$0.813227\pi$
$374$	3125.24	0.432091
$375$	0	0
$376$	−5276.95	−0.723771
$377$	−4779.36	−0.652916
$378$	0	0
$379$	−221.426	−0.0300103	−0.0150051	−	0.999887i	$-0.504776\pi$
$379$	−221.426	−0.0300103	−0.0150051	+	0.999887i	$0.504776\pi$
$380$	−3671.42	−0.495631
$381$	0	0
$382$	8862.68	1.18705
$383$	4120.45	0.549727	0.274863	−	0.961483i	$-0.411367\pi$
$383$	4120.45	0.549727	0.274863	+	0.961483i	$0.411367\pi$
$384$	0	0
$385$	−6011.01	−0.795713
$386$	−7728.23	−1.01906
$387$	0	0
$388$	2805.22	0.367045
$389$	−8703.96	−1.13447	−0.567235	−	0.823556i	$-0.691987\pi$
$389$	−8703.96	−1.13447	−0.567235	+	0.823556i	$0.691987\pi$
$390$	0	0
$391$	1186.73	0.153493
$392$	507.171	0.0653469
$393$	0	0
$394$	−4034.42	−0.515866
$395$	4293.43	0.546901
$396$	0	0
$397$	−12830.5	−1.62202	−0.811012	−	0.585030i	$-0.801083\pi$
$397$	−12830.5	−1.62202	−0.811012	+	0.585030i	$0.801083\pi$
$398$	−19572.2	−2.46499
$399$	0	0
$400$	7358.02	0.919752
$401$	3221.15	0.401138	0.200569	−	0.979680i	$-0.435721\pi$
$401$	3221.15	0.401138	0.200569	+	0.979680i	$0.435721\pi$
$402$	0	0
$403$	−11959.2	−1.47824
$404$	5432.51	0.669004
$405$	0	0
$406$	−7140.89	−0.872898
$407$	−8934.31	−1.08810
$408$	0	0
$409$	−3351.34	−0.405166	−0.202583	−	0.979265i	$-0.564934\pi$
$409$	−3351.34	−0.405166	−0.202583	+	0.979265i	$0.564934\pi$
$410$	−9879.95	−1.19009
$411$	0	0
$412$	8878.49	1.06168
$413$	−9781.59	−1.16543
$414$	0	0
$415$	−4582.50	−0.542039
$416$	−8718.18	−1.02751
$417$	0	0
$418$	−26532.5	−3.10465
$419$	−9813.39	−1.14419	−0.572095	−	0.820187i	$-0.693869\pi$
$419$	−9813.39	−1.14419	−0.572095	+	0.820187i	$0.693869\pi$
$420$	0	0
$421$	8489.75	0.982815	0.491407	−	0.870930i	$-0.336483\pi$
$421$	8489.75	0.982815	0.491407	+	0.870930i	$0.336483\pi$
$422$	12447.5	1.43586
$423$	0	0
$424$	−9182.45	−1.05174
$425$	1515.30	0.172948
$426$	0	0
$427$	−2158.83	−0.244668
$428$	−4216.30	−0.476174
$429$	0	0
$430$	8576.04	0.961798
$431$	−12803.0	−1.43086	−0.715428	−	0.698687i	$-0.753768\pi$
$431$	−12803.0	−1.43086	−0.715428	+	0.698687i	$0.753768\pi$
$432$	0	0
$433$	−9516.03	−1.05615	−0.528073	−	0.849199i	$-0.677085\pi$
$433$	−9516.03	−1.05615	−0.528073	+	0.849199i	$0.677085\pi$
$434$	−17868.4	−1.97629
$435$	0	0
$436$	−5128.33	−0.563308
$437$	−10075.1	−1.10287
$438$	0	0
$439$	−17217.7	−1.87188	−0.935942	−	0.352154i	$-0.885449\pi$
$439$	−17217.7	−1.87188	−0.935942	+	0.352154i	$0.885449\pi$
$440$	−3676.27	−0.398316
$441$	0	0
$442$	−2718.40	−0.292536
$443$	−1799.91	−0.193040	−0.0965198	−	0.995331i	$-0.530771\pi$
$443$	−1799.91	−0.193040	−0.0965198	+	0.995331i	$0.530771\pi$
$444$	0	0
$445$	437.095	0.0465624
$446$	7124.24	0.756373
$447$	0	0
$448$	−524.712	−0.0553355
$449$	−6622.64	−0.696084	−0.348042	−	0.937479i	$-0.613153\pi$
$449$	−6622.64	−0.696084	−0.348042	+	0.937479i	$0.613153\pi$
$450$	0	0
$451$	−26142.2	−2.72947
$452$	−3683.48	−0.383311
$453$	0	0
$454$	10679.6	1.10401
$455$	5228.50	0.538717
$456$	0	0
$457$	9029.78	0.924278	0.462139	−	0.886807i	$-0.347082\pi$
$457$	9029.78	0.924278	0.462139	+	0.886807i	$0.347082\pi$
$458$	−3372.84	−0.344110
$459$	0	0
$460$	1909.12	0.193507
$461$	−12247.5	−1.23736	−0.618681	−	0.785642i	$-0.712333\pi$
$461$	−12247.5	−1.23736	−0.618681	+	0.785642i	$0.712333\pi$
$462$	0	0
$463$	10118.1	1.01561	0.507806	−	0.861471i	$-0.330457\pi$
$463$	10118.1	1.01561	0.507806	+	0.861471i	$0.330457\pi$
$464$	−8153.14	−0.815732
$465$	0	0
$466$	4324.64	0.429904
$467$	4155.48	0.411762	0.205881	−	0.978577i	$-0.433994\pi$
$467$	4155.48	0.411762	0.205881	+	0.978577i	$0.433994\pi$
$468$	0	0
$469$	−3445.86	−0.339265
$470$	−8913.95	−0.874829
$471$	0	0
$472$	−5982.31	−0.583386
$473$	22692.1	2.20589
$474$	0	0
$475$	−12864.5	−1.24266
$476$	−1487.10	−0.143196
$477$	0	0
$478$	−849.074	−0.0812463
$479$	6501.62	0.620180	0.310090	−	0.950707i	$-0.399641\pi$
$479$	6501.62	0.620180	0.310090	+	0.950707i	$0.399641\pi$
$480$	0	0
$481$	7771.24	0.736670
$482$	14966.9	1.41437
$483$	0	0
$484$	7152.48	0.671720
$485$	−3464.96	−0.324403
$486$	0	0
$487$	−179.841	−0.0167339	−0.00836693	−	0.999965i	$-0.502663\pi$
$487$	−179.841	−0.0167339	−0.00836693	+	0.999965i	$0.502663\pi$
$488$	−1320.32	−0.122475
$489$	0	0
$490$	856.726	0.0789856
$491$	−10136.9	−0.931712	−0.465856	−	0.884860i	$-0.654254\pi$
$491$	−10136.9	−0.931712	−0.465856	+	0.884860i	$0.654254\pi$
$492$	0	0
$493$	−1679.05	−0.153388
$494$	23078.5	2.10192
$495$	0	0
$496$	−20401.3	−1.84687
$497$	−16786.1	−1.51501
$498$	0	0
$499$	3379.46	0.303177	0.151589	−	0.988444i	$-0.451561\pi$
$499$	3379.46	0.303177	0.151589	+	0.988444i	$0.451561\pi$
$500$	5734.71	0.512928
$501$	0	0
$502$	12699.6	1.12911
$503$	20053.4	1.77761	0.888805	−	0.458285i	$-0.151536\pi$
$503$	20053.4	1.77761	0.888805	+	0.458285i	$0.151536\pi$
$504$	0	0
$505$	−6710.14	−0.591282
$506$	13796.8	1.21214
$507$	0	0
$508$	8793.29	0.767990
$509$	−13499.8	−1.17558	−0.587788	−	0.809015i	$-0.700001\pi$
$509$	−13499.8	−1.17558	−0.587788	+	0.809015i	$0.700001\pi$
$510$	0	0
$511$	−1193.46	−0.103318
$512$	−7226.80	−0.623794
$513$	0	0
$514$	9189.63	0.788594
$515$	−10966.5	−0.938337
$516$	0	0
$517$	−23586.2	−2.00642
$518$	11611.1	0.984870
$519$	0	0
$520$	3197.69	0.269670
$521$	−11011.0	−0.925911	−0.462955	−	0.886382i	$-0.653211\pi$
$521$	−11011.0	−0.925911	−0.462955	+	0.886382i	$0.653211\pi$
$522$	0	0
$523$	−6433.53	−0.537894	−0.268947	−	0.963155i	$-0.586676\pi$
$523$	−6433.53	−0.537894	−0.268947	+	0.963155i	$0.586676\pi$
$524$	856.716	0.0714233
$525$	0	0
$526$	21320.8	1.76736
$527$	−4201.42	−0.347281
$528$	0	0
$529$	−6928.01	−0.569410
$530$	−15511.2	−1.27125
$531$	0	0
$532$	12625.1	1.02889
$533$	22739.1	1.84791
$534$	0	0
$535$	5207.89	0.420854
$536$	−2107.45	−0.169828
$537$	0	0
$538$	−8534.05	−0.683883
$539$	2266.89	0.181154
$540$	0	0
$541$	−10806.9	−0.858822	−0.429411	−	0.903109i	$-0.641279\pi$
$541$	−10806.9	−0.858822	−0.429411	+	0.903109i	$0.641279\pi$
$542$	−30071.4	−2.38316
$543$	0	0
$544$	−3062.80	−0.241391
$545$	6334.41	0.497865
$546$	0	0
$547$	−18131.7	−1.41729	−0.708643	−	0.705567i	$-0.750692\pi$
$547$	−18131.7	−1.41729	−0.708643	+	0.705567i	$0.750692\pi$
$548$	12530.7	0.976794
$549$	0	0
$550$	17616.7	1.36578
$551$	14254.7	1.10212
$552$	0	0
$553$	−14764.0	−1.13532
$554$	−4747.34	−0.364070
$555$	0	0
$556$	−7366.82	−0.561911
$557$	14112.3	1.07353	0.536767	−	0.843731i	$-0.319645\pi$
$557$	14112.3	1.07353	0.536767	+	0.843731i	$0.319645\pi$
$558$	0	0
$559$	−19738.1	−1.49344
$560$	8919.34	0.673055
$561$	0	0
$562$	2282.04	0.171285
$563$	10409.1	0.779202	0.389601	−	0.920984i	$-0.372613\pi$
$563$	10409.1	0.779202	0.389601	+	0.920984i	$0.372613\pi$
$564$	0	0
$565$	4549.77	0.338779
$566$	1726.83	0.128240
$567$	0	0
$568$	−10266.2	−0.758379
$569$	22025.3	1.62276	0.811379	−	0.584521i	$-0.198717\pi$
$569$	22025.3	1.62276	0.811379	+	0.584521i	$0.198717\pi$
$570$	0	0
$571$	−16262.2	−1.19186	−0.595929	−	0.803037i	$-0.703216\pi$
$571$	−16262.2	−1.19186	−0.595929	+	0.803037i	$0.703216\pi$
$572$	−11571.3	−0.845839
$573$	0	0
$574$	33974.7	2.47052
$575$	6689.50	0.485167
$576$	0	0
$577$	19118.8	1.37942	0.689711	−	0.724085i	$-0.257738\pi$
$577$	19118.8	1.37942	0.689711	+	0.724085i	$0.257738\pi$
$578$	16499.0	1.18731
$579$	0	0
$580$	−2701.12	−0.193376
$581$	15758.1	1.12522
$582$	0	0
$583$	−41042.5	−2.91562
$584$	−729.906	−0.0517187
$585$	0	0
$586$	12088.5	0.852166
$587$	−12216.2	−0.858975	−0.429487	−	0.903073i	$-0.641306\pi$
$587$	−12216.2	−0.858975	−0.429487	+	0.903073i	$0.641306\pi$
$588$	0	0
$589$	35669.0	2.49527
$590$	−10105.5	−0.705145
$591$	0	0
$592$	13257.0	0.920372
$593$	−11338.2	−0.785164	−0.392582	−	0.919717i	$-0.628418\pi$
$593$	−11338.2	−0.785164	−0.392582	+	0.919717i	$0.628418\pi$
$594$	0	0
$595$	1836.84	0.126560
$596$	−5473.89	−0.376207
$597$	0	0
$598$	−12000.7	−0.820645
$599$	−11705.2	−0.798435	−0.399218	−	0.916856i	$-0.630718\pi$
$599$	−11705.2	−0.798435	−0.399218	+	0.916856i	$0.630718\pi$
$600$	0	0
$601$	−18935.0	−1.28515	−0.642576	−	0.766222i	$-0.722134\pi$
$601$	−18935.0	−1.28515	−0.642576	+	0.766222i	$0.722134\pi$
$602$	−29490.9	−1.99661
$603$	0	0
$604$	−4775.18	−0.321688
$605$	−8834.60	−0.593682
$606$	0	0
$607$	11841.6	0.791825	0.395912	−	0.918288i	$-0.370428\pi$
$607$	11841.6	0.791825	0.395912	+	0.918288i	$0.370428\pi$
$608$	26002.4	1.73444
$609$	0	0
$610$	−2230.31	−0.148037
$611$	20515.8	1.35840
$612$	0	0
$613$	13416.0	0.883962	0.441981	−	0.897024i	$-0.354276\pi$
$613$	13416.0	0.883962	0.441981	+	0.897024i	$0.354276\pi$
$614$	5175.64	0.340182
$615$	0	0
$616$	12641.8	0.826869
$617$	−15006.2	−0.979139	−0.489569	−	0.871964i	$-0.662846\pi$
$617$	−15006.2	−0.979139	−0.489569	+	0.871964i	$0.662846\pi$
$618$	0	0
$619$	19524.5	1.26778	0.633890	−	0.773423i	$-0.281457\pi$
$619$	19524.5	1.26778	0.633890	+	0.773423i	$0.281457\pi$
$620$	−6758.92	−0.437814
$621$	0	0
$622$	−14072.8	−0.907183
$623$	−1503.06	−0.0966594
$624$	0	0
$625$	4469.19	0.286028
$626$	−2898.99	−0.185091
$627$	0	0
$628$	−2338.45	−0.148590
$629$	2730.13	0.173064
$630$	0	0
$631$	−23824.6	−1.50308	−0.751539	−	0.659689i	$-0.770688\pi$
$631$	−23824.6	−1.50308	−0.751539	+	0.659689i	$0.770688\pi$
$632$	−9029.52	−0.568315
$633$	0	0
$634$	−3689.29	−0.231105
$635$	−10861.3	−0.678768
$636$	0	0
$637$	−1971.79	−0.122645
$638$	−19520.3	−1.21131
$639$	0	0
$640$	7987.94	0.493361
$641$	1259.89	0.0776328	0.0388164	−	0.999246i	$-0.487641\pi$
$641$	1259.89	0.0776328	0.0388164	+	0.999246i	$0.487641\pi$
$642$	0	0
$643$	−24351.6	−1.49352	−0.746760	−	0.665094i	$-0.768392\pi$
$643$	−24351.6	−1.49352	−0.746760	+	0.665094i	$0.768392\pi$
$644$	−6565.00	−0.401704
$645$	0	0
$646$	8107.75	0.493801
$647$	−852.292	−0.0517883	−0.0258942	−	0.999665i	$-0.508243\pi$
$647$	−852.292	−0.0517883	−0.0258942	+	0.999665i	$0.508243\pi$
$648$	0	0
$649$	−26739.0	−1.61725
$650$	−15323.3	−0.924662
$651$	0	0
$652$	1543.32	0.0927012
$653$	22118.7	1.32553	0.662766	−	0.748826i	$-0.269382\pi$
$653$	22118.7	1.32553	0.662766	+	0.748826i	$0.269382\pi$
$654$	0	0
$655$	−1058.20	−0.0631256
$656$	38790.7	2.30872
$657$	0	0
$658$	30652.9	1.81607
$659$	−766.897	−0.0453324	−0.0226662	−	0.999743i	$-0.507216\pi$
$659$	−766.897	−0.0453324	−0.0226662	+	0.999743i	$0.507216\pi$
$660$	0	0
$661$	28950.6	1.70355	0.851776	−	0.523907i	$-0.175526\pi$
$661$	28950.6	1.70355	0.851776	+	0.523907i	$0.175526\pi$
$662$	−14001.3	−0.822016
$663$	0	0
$664$	9637.46	0.563262
$665$	−15594.3	−0.909353
$666$	0	0
$667$	−7412.37	−0.430297
$668$	6715.43	0.388964
$669$	0	0
$670$	−3559.96	−0.205273
$671$	−5901.38	−0.339523
$672$	0	0
$673$	−12371.2	−0.708581	−0.354290	−	0.935135i	$-0.615278\pi$
$673$	−12371.2	−0.708581	−0.354290	+	0.935135i	$0.615278\pi$
$674$	−24306.9	−1.38912
$675$	0	0
$676$	−87.4752	−0.00497697
$677$	4310.04	0.244680	0.122340	−	0.992488i	$-0.460960\pi$
$677$	4310.04	0.244680	0.122340	+	0.992488i	$0.460960\pi$
$678$	0	0
$679$	11915.1	0.673432
$680$	1123.39	0.0633529
$681$	0	0
$682$	−48845.1	−2.74249
$683$	2271.67	0.127267	0.0636334	−	0.997973i	$-0.479731\pi$
$683$	2271.67	0.127267	0.0636334	+	0.997973i	$0.479731\pi$
$684$	0	0
$685$	−15477.6	−0.863313
$686$	20971.3	1.16718
$687$	0	0
$688$	−33671.3	−1.86585
$689$	35699.6	1.97395
$690$	0	0
$691$	23141.4	1.27401	0.637005	−	0.770859i	$-0.280173\pi$
$691$	23141.4	1.27401	0.637005	+	0.770859i	$0.280173\pi$
$692$	17396.6	0.955662
$693$	0	0
$694$	13958.4	0.763479
$695$	9099.35	0.496630
$696$	0	0
$697$	7988.51	0.434127
$698$	9471.78	0.513628
$699$	0	0
$700$	−8382.64	−0.452620
$701$	10239.1	0.551675	0.275837	−	0.961204i	$-0.411045\pi$
$701$	10239.1	0.551675	0.275837	+	0.961204i	$0.411045\pi$
$702$	0	0
$703$	−23178.1	−1.24350
$704$	−1434.35	−0.0767887
$705$	0	0
$706$	9319.40	0.496799
$707$	23074.5	1.22745
$708$	0	0
$709$	20890.1	1.10655	0.553274	−	0.832999i	$-0.313378\pi$
$709$	20890.1	1.10655	0.553274	+	0.832999i	$0.313378\pi$
$710$	−17341.9	−0.916661
$711$	0	0
$712$	−919.254	−0.0483855
$713$	−18547.7	−0.974220
$714$	0	0
$715$	14292.6	0.747573
$716$	−17176.7	−0.896541
$717$	0	0
$718$	−30227.4	−1.57114
$719$	−2533.73	−0.131422	−0.0657108	−	0.997839i	$-0.520931\pi$
$719$	−2533.73	−0.131422	−0.0657108	+	0.997839i	$0.520931\pi$
$720$	0	0
$721$	37711.2	1.94790
$722$	−44465.4	−2.29201
$723$	0	0
$724$	−16885.7	−0.866784
$725$	−9464.62	−0.484837
$726$	0	0
$727$	23985.6	1.22363	0.611814	−	0.791001i	$-0.290440\pi$
$727$	23985.6	1.22363	0.611814	+	0.791001i	$0.290440\pi$
$728$	−10996.1	−0.559810
$729$	0	0
$730$	−1232.98	−0.0625130
$731$	−6934.22	−0.350850
$732$	0	0
$733$	−14379.3	−0.724571	−0.362285	−	0.932067i	$-0.618003\pi$
$733$	−14379.3	−0.724571	−0.362285	+	0.932067i	$0.618003\pi$
$734$	−12204.6	−0.613734
$735$	0	0
$736$	−13521.2	−0.677169
$737$	−9419.62	−0.470795
$738$	0	0
$739$	−32219.7	−1.60382	−0.801908	−	0.597447i	$-0.796182\pi$
$739$	−32219.7	−1.60382	−0.801908	+	0.597447i	$0.796182\pi$
$740$	4392.03	0.218181
$741$	0	0
$742$	53339.2	2.63901
$743$	−14744.4	−0.728021	−0.364010	−	0.931395i	$-0.618593\pi$
$743$	−14744.4	−0.728021	−0.364010	+	0.931395i	$0.618593\pi$
$744$	0	0
$745$	6761.25	0.332501
$746$	42623.5	2.09190
$747$	0	0
$748$	−4065.14	−0.198711
$749$	−17908.6	−0.873655
$750$	0	0
$751$	−21034.6	−1.02206	−0.511028	−	0.859564i	$-0.670735\pi$
$751$	−21034.6	−1.02206	−0.511028	+	0.859564i	$0.670735\pi$
$752$	34998.0	1.69714
$753$	0	0
$754$	16979.2	0.820087
$755$	5898.22	0.284315
$756$	0	0
$757$	8988.99	0.431586	0.215793	−	0.976439i	$-0.430766\pi$
$757$	8988.99	0.431586	0.215793	+	0.976439i	$0.430766\pi$
$758$	786.641	0.0376940
$759$	0	0
$760$	−9537.28	−0.455202
$761$	33696.6	1.60513	0.802564	−	0.596566i	$-0.203469\pi$
$761$	33696.6	1.60513	0.802564	+	0.596566i	$0.203469\pi$
$762$	0	0
$763$	−21782.5	−1.03352
$764$	−11528.1	−0.545905
$765$	0	0
$766$	−14638.4	−0.690477
$767$	23258.1	1.09492
$768$	0	0
$769$	−23816.8	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$769$	−23816.8	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$770$	21354.8	0.999446
$771$	0	0
$772$	10052.5	0.468648
$773$	29990.6	1.39545	0.697727	−	0.716363i	$-0.254195\pi$
$773$	29990.6	1.39545	0.697727	+	0.716363i	$0.254195\pi$
$774$	0	0
$775$	−23683.0	−1.09770
$776$	7287.15	0.337105
$777$	0	0
$778$	30921.8	1.42494
$779$	−67820.4	−3.11928
$780$	0	0
$781$	−45886.4	−2.10236
$782$	−4216.00	−0.192793
$783$	0	0
$784$	−3363.68	−0.153229
$785$	2888.41	0.131327
$786$	0	0
$787$	−15665.0	−0.709525	−0.354763	−	0.934956i	$-0.615438\pi$
$787$	−15665.0	−0.709525	−0.354763	+	0.934956i	$0.615438\pi$
$788$	5247.75	0.237238
$789$	0	0
$790$	−15252.9	−0.686928
$791$	−15645.5	−0.703275
$792$	0	0
$793$	5133.14	0.229865
$794$	45581.7	2.03732
$795$	0	0
$796$	25458.4	1.13361
$797$	9677.03	0.430085	0.215043	−	0.976605i	$-0.431011\pi$
$797$	9677.03	0.430085	0.215043	+	0.976605i	$0.431011\pi$
$798$	0	0
$799$	7207.44	0.319125
$800$	−17264.7	−0.763000
$801$	0	0
$802$	−11443.5	−0.503844
$803$	−3262.44	−0.143374
$804$	0	0
$805$	8108.96	0.355035
$806$	42486.5	1.85673
$807$	0	0
$808$	14112.1	0.614433
$809$	9157.44	0.397971	0.198985	−	0.980002i	$-0.436235\pi$
$809$	9157.44	0.397971	0.198985	+	0.980002i	$0.436235\pi$
$810$	0	0
$811$	−8583.47	−0.371648	−0.185824	−	0.982583i	$-0.559495\pi$
$811$	−8583.47	−0.371648	−0.185824	+	0.982583i	$0.559495\pi$
$812$	9288.48	0.401431
$813$	0	0
$814$	31740.1	1.36670
$815$	−1906.28	−0.0819315
$816$	0	0
$817$	58869.8	2.52092
$818$	11906.0	0.508904
$819$	0	0
$820$	12851.3	0.547301
$821$	28825.7	1.22536	0.612681	−	0.790330i	$-0.290091\pi$
$821$	28825.7	1.22536	0.612681	+	0.790330i	$0.290091\pi$
$822$	0	0
$823$	−9379.82	−0.397278	−0.198639	−	0.980073i	$-0.563652\pi$
$823$	−9379.82	−0.397278	−0.198639	+	0.980073i	$0.563652\pi$
$824$	23063.7	0.975077
$825$	0	0
$826$	34750.2	1.46382
$827$	−26612.6	−1.11900	−0.559499	−	0.828831i	$-0.689006\pi$
$827$	−26612.6	−1.11900	−0.559499	+	0.828831i	$0.689006\pi$
$828$	0	0
$829$	−37188.8	−1.55805	−0.779024	−	0.626994i	$-0.784285\pi$
$829$	−37188.8	−1.55805	−0.779024	+	0.626994i	$0.784285\pi$
$830$	16279.8	0.680821
$831$	0	0
$832$	1247.63	0.0519877
$833$	−692.712	−0.0288128
$834$	0	0
$835$	−8294.77	−0.343775
$836$	34512.0	1.42778
$837$	0	0
$838$	34863.2	1.43715
$839$	3063.10	0.126043	0.0630213	−	0.998012i	$-0.479926\pi$
$839$	3063.10	0.126043	0.0630213	+	0.998012i	$0.479926\pi$
$840$	0	0
$841$	−13901.6	−0.569995
$842$	−30160.8	−1.23445
$843$	0	0
$844$	−16191.0	−0.660328
$845$	108.048	0.00439876
$846$	0	0
$847$	30380.0	1.23243
$848$	60900.3	2.46618
$849$	0	0
$850$	−5383.27	−0.217229
$851$	12052.5	0.485494
$852$	0	0
$853$	21972.5	0.881976	0.440988	−	0.897513i	$-0.354628\pi$
$853$	21972.5	0.881976	0.440988	+	0.897513i	$0.354628\pi$
$854$	7669.48	0.307312
$855$	0	0
$856$	−10952.7	−0.437332
$857$	−3576.85	−0.142571	−0.0712853	−	0.997456i	$-0.522710\pi$
$857$	−3576.85	−0.142571	−0.0712853	+	0.997456i	$0.522710\pi$
$858$	0	0
$859$	31586.8	1.25463	0.627315	−	0.778765i	$-0.284154\pi$
$859$	31586.8	1.25463	0.627315	+	0.778765i	$0.284154\pi$
$860$	−11155.2	−0.442315
$861$	0	0
$862$	45484.1	1.79721
$863$	8188.52	0.322990	0.161495	−	0.986874i	$-0.448368\pi$
$863$	8188.52	0.322990	0.161495	+	0.986874i	$0.448368\pi$
$864$	0	0
$865$	−21487.9	−0.844636
$866$	33806.7	1.32656
$867$	0	0
$868$	23242.3	0.908864
$869$	−40359.0	−1.57547
$870$	0	0
$871$	8193.38	0.318739
$872$	−13321.9	−0.517359
$873$	0	0
$874$	35792.7	1.38525
$875$	24358.1	0.941089
$876$	0	0
$877$	−5897.45	−0.227073	−0.113536	−	0.993534i	$-0.536218\pi$
$877$	−5897.45	−0.227073	−0.113536	+	0.993534i	$0.536218\pi$
$878$	61167.9	2.35116
$879$	0	0
$880$	24381.9	0.933993
$881$	−13588.4	−0.519642	−0.259821	−	0.965657i	$-0.583664\pi$
$881$	−13588.4	−0.519642	−0.259821	+	0.965657i	$0.583664\pi$
$882$	0	0
$883$	−12771.7	−0.486751	−0.243376	−	0.969932i	$-0.578255\pi$
$883$	−12771.7	−0.486751	−0.243376	+	0.969932i	$0.578255\pi$
$884$	3535.94	0.134532
$885$	0	0
$886$	6394.39	0.242465
$887$	5928.78	0.224429	0.112215	−	0.993684i	$-0.464206\pi$
$887$	5928.78	0.224429	0.112215	+	0.993684i	$0.464206\pi$
$888$	0	0
$889$	37349.3	1.40906
$890$	−1552.83	−0.0584841
$891$	0	0
$892$	−9266.82	−0.347843
$893$	−61189.3	−2.29297
$894$	0	0
$895$	21216.3	0.792384
$896$	−27468.5	−1.02417
$897$	0	0
$898$	23527.7	0.874308
$899$	26242.2	0.973557
$900$	0	0
$901$	12541.7	0.463735
$902$	92873.2	3.42831
$903$	0	0
$904$	−9568.62	−0.352044
$905$	20856.9	0.766084
$906$	0	0
$907$	35917.4	1.31491	0.657453	−	0.753496i	$-0.271634\pi$
$907$	35917.4	1.31491	0.657453	+	0.753496i	$0.271634\pi$
$908$	−13891.5	−0.507715
$909$	0	0
$910$	−18574.8	−0.676648
$911$	21121.1	0.768138	0.384069	−	0.923304i	$-0.374522\pi$
$911$	21121.1	0.768138	0.384069	+	0.923304i	$0.374522\pi$
$912$	0	0
$913$	43076.3	1.56146
$914$	−32079.3	−1.16093
$915$	0	0
$916$	4387.20	0.158250
$917$	3638.88	0.131043
$918$	0	0
$919$	−37482.1	−1.34540	−0.672699	−	0.739917i	$-0.734865\pi$
$919$	−37482.1	−1.34540	−0.672699	+	0.739917i	$0.734865\pi$
$920$	4959.35	0.177723
$921$	0	0
$922$	43510.7	1.55417
$923$	39913.0	1.42335
$924$	0	0
$925$	15389.5	0.547031
$926$	−35945.7	−1.27565
$927$	0	0
$928$	19130.4	0.676708
$929$	−8547.79	−0.301877	−0.150939	−	0.988543i	$-0.548230\pi$
$929$	−8547.79	−0.301877	−0.150939	+	0.988543i	$0.548230\pi$
$930$	0	0
$931$	5880.95	0.207025
$932$	−5625.26	−0.197705
$933$	0	0
$934$	−14762.8	−0.517189
$935$	5021.18	0.175626
$936$	0	0
$937$	7197.62	0.250945	0.125473	−	0.992097i	$-0.459955\pi$
$937$	7197.62	0.250945	0.125473	+	0.992097i	$0.459955\pi$
$938$	12241.8	0.426129
$939$	0	0
$940$	11594.8	0.402319
$941$	−22139.1	−0.766966	−0.383483	−	0.923548i	$-0.625275\pi$
$941$	−22139.1	−0.766966	−0.383483	+	0.923548i	$0.625275\pi$
$942$	0	0
$943$	35266.3	1.21785
$944$	39676.1	1.36795
$945$	0	0
$946$	−80616.2	−2.77068
$947$	10550.2	0.362023	0.181012	−	0.983481i	$-0.442063\pi$
$947$	10550.2	0.362023	0.181012	+	0.983481i	$0.442063\pi$
$948$	0	0
$949$	2837.74	0.0970673
$950$	45702.6	1.56083
$951$	0	0
$952$	−3863.05	−0.131515
$953$	−49443.4	−1.68062	−0.840310	−	0.542107i	$-0.817627\pi$
$953$	−49443.4	−1.68062	−0.840310	+	0.542107i	$0.817627\pi$
$954$	0	0
$955$	14239.3	0.482484
$956$	1104.43	0.0373638
$957$	0	0
$958$	−23097.7	−0.778970
$959$	53223.7	1.79216
$960$	0	0
$961$	35874.1	1.20419
$962$	−27608.2	−0.925285
$963$	0	0
$964$	−19468.2	−0.650443
$965$	−12416.6	−0.414202
$966$	0	0
$967$	53236.7	1.77040	0.885199	−	0.465212i	$-0.154022\pi$
$967$	53236.7	1.77040	0.885199	+	0.465212i	$0.154022\pi$
$968$	18580.1	0.616927
$969$	0	0
$970$	12309.6	0.407462
$971$	−2874.21	−0.0949927	−0.0474963	−	0.998871i	$-0.515124\pi$
$971$	−2874.21	−0.0949927	−0.0474963	+	0.998871i	$0.515124\pi$
$972$	0	0
$973$	−31290.4	−1.03096
$974$	638.906	0.0210184
$975$	0	0
$976$	8756.65	0.287186
$977$	493.436	0.0161581	0.00807903	−	0.999967i	$-0.497428\pi$
$977$	493.436	0.0161581	0.00807903	+	0.999967i	$0.497428\pi$
$978$	0	0
$979$	−4108.76	−0.134133
$980$	−1114.38	−0.0363241
$981$	0	0
$982$	36012.4	1.17027
$983$	−28461.1	−0.923467	−0.461733	−	0.887019i	$-0.652772\pi$
$983$	−28461.1	−0.923467	−0.461733	+	0.887019i	$0.652772\pi$
$984$	0	0
$985$	−6481.93	−0.209676
$986$	5965.00	0.192661
$987$	0	0
$988$	−30019.2	−0.966638
$989$	−30612.1	−0.984233
$990$	0	0
$991$	−37443.0	−1.20022	−0.600109	−	0.799919i	$-0.704876\pi$
$991$	−37443.0	−1.20022	−0.600109	+	0.799919i	$0.704876\pi$
$992$	47869.3	1.53211
$993$	0	0
$994$	59634.4	1.90291
$995$	−31445.8	−1.00191
$996$	0	0
$997$	−57551.7	−1.82817	−0.914083	−	0.405527i	$-0.867088\pi$
$997$	−57551.7	−1.82817	−0.914083	+	0.405527i	$0.867088\pi$
$998$	−12005.9	−0.380802
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.13	✓	59
3.2	odd	2		2151.4.a.h.1.47	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.13	✓	59	1.1	even	1	trivial
2151.4.a.h.1.47	yes	59	3.2	odd	2

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-3.55261 q^{2} +4.62104 q^{4} -5.70782 q^{5} +19.6278 q^{7} +12.0041 q^{8} +O(q^{10})\)	\(q-3.55261 q^{2} +4.62104 q^{4} -5.70782 q^{5} +19.6278 q^{7} +12.0041 q^{8} +20.2777 q^{10} +53.6545 q^{11} -46.6698 q^{13} -69.7299 q^{14} -79.6143 q^{16} -16.3957 q^{17} +139.195 q^{19} -26.3761 q^{20} -190.614 q^{22} -72.3809 q^{23} -92.4208 q^{25} +165.800 q^{26} +90.7008 q^{28} +102.408 q^{29} +256.252 q^{31} +186.806 q^{32} +58.2474 q^{34} -112.032 q^{35} -166.515 q^{37} -494.506 q^{38} -68.5174 q^{40} -487.233 q^{41} +422.930 q^{43} +247.940 q^{44} +257.141 q^{46} -439.594 q^{47} +42.2497 q^{49} +328.335 q^{50} -215.663 q^{52} -764.941 q^{53} -306.250 q^{55} +235.614 q^{56} -363.815 q^{58} -498.354 q^{59} -109.988 q^{61} -910.364 q^{62} -26.7331 q^{64} +266.383 q^{65} -175.561 q^{67} -75.7651 q^{68} +398.006 q^{70} -855.221 q^{71} -60.8046 q^{73} +591.565 q^{74} +643.226 q^{76} +1053.12 q^{77} -752.201 q^{79} +454.424 q^{80} +1730.95 q^{82} +802.845 q^{83} +93.5836 q^{85} -1502.51 q^{86} +644.075 q^{88} -76.5782 q^{89} -916.025 q^{91} -334.475 q^{92} +1561.71 q^{94} -794.500 q^{95} +607.054 q^{97} -150.097 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more