LMFDB - Embedded newform 2151.4.a.d.1.3

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-5.23366 q^{2} +19.3912 q^{4} +13.4597 q^{5} +28.9382 q^{7} -59.6177 q^{8} +O(q^{10})$	\(q-5.23366 q^{2} +19.3912 q^{4} +13.4597 q^{5} +28.9382 q^{7} -59.6177 q^{8} -70.4436 q^{10} -36.5886 q^{11} -14.0985 q^{13} -151.453 q^{14} +156.889 q^{16} +34.8170 q^{17} -115.166 q^{19} +261.000 q^{20} +191.492 q^{22} -175.075 q^{23} +56.1642 q^{25} +73.7870 q^{26} +561.147 q^{28} -163.045 q^{29} +141.688 q^{31} -344.163 q^{32} -182.221 q^{34} +389.500 q^{35} -87.4744 q^{37} +602.738 q^{38} -802.438 q^{40} +517.668 q^{41} +221.419 q^{43} -709.498 q^{44} +916.282 q^{46} -196.711 q^{47} +494.420 q^{49} -293.944 q^{50} -273.388 q^{52} +120.995 q^{53} -492.473 q^{55} -1725.23 q^{56} +853.324 q^{58} +586.110 q^{59} -31.9565 q^{61} -741.548 q^{62} +546.121 q^{64} -189.763 q^{65} +411.888 q^{67} +675.144 q^{68} -2038.51 q^{70} -611.058 q^{71} -949.549 q^{73} +457.811 q^{74} -2233.20 q^{76} -1058.81 q^{77} -383.400 q^{79} +2111.69 q^{80} -2709.30 q^{82} -685.600 q^{83} +468.628 q^{85} -1158.83 q^{86} +2181.33 q^{88} +232.830 q^{89} -407.987 q^{91} -3394.91 q^{92} +1029.52 q^{94} -1550.10 q^{95} -758.134 q^{97} -2587.63 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * 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 $

$2$	−5.23366	−1.85038	−0.925189	−	0.379506i	$-0.876094\pi$
$2$	−5.23366	−1.85038	−0.925189	+	0.379506i	$0.876094\pi$
$3$	0	0
$3$	0	0
$4$	19.3912	2.42390
$5$	13.4597	1.20387	0.601937	−	0.798543i	$-0.294396\pi$
$5$	13.4597	1.20387	0.601937	+	0.798543i	$0.294396\pi$
$6$	0	0
$7$	28.9382	1.56252	0.781258	−	0.624208i	$-0.214578\pi$
$7$	28.9382	1.56252	0.781258	+	0.624208i	$0.214578\pi$
$8$	−59.6177	−2.63476
$9$	0	0
$10$	−70.4436	−2.22762
$11$	−36.5886	−1.00290	−0.501449	−	0.865187i	$-0.667200\pi$
$11$	−36.5886	−1.00290	−0.501449	+	0.865187i	$0.667200\pi$
$12$	0	0
$13$	−14.0985	−0.300787	−0.150394	−	0.988626i	$-0.548054\pi$
$13$	−14.0985	−0.300787	−0.150394	+	0.988626i	$0.548054\pi$
$14$	−151.453	−2.89125
$15$	0	0
$16$	156.889	2.45139
$17$	34.8170	0.496728	0.248364	−	0.968667i	$-0.420107\pi$
$17$	34.8170	0.496728	0.248364	+	0.968667i	$0.420107\pi$
$18$	0	0
$19$	−115.166	−1.39057	−0.695285	−	0.718734i	$-0.744722\pi$
$19$	−115.166	−1.39057	−0.695285	+	0.718734i	$0.744722\pi$
$20$	261.000	2.91807
$21$	0	0
$22$	191.492	1.85574
$23$	−175.075	−1.58720	−0.793601	−	0.608439i	$-0.791796\pi$
$23$	−175.075	−1.58720	−0.793601	+	0.608439i	$0.791796\pi$
$24$	0	0
$25$	56.1642	0.449313
$26$	73.7870	0.556570
$27$	0	0
$28$	561.147	3.78739
$29$	−163.045	−1.04403	−0.522013	−	0.852937i	$-0.674819\pi$
$29$	−163.045	−1.04403	−0.522013	+	0.852937i	$0.674819\pi$
$30$	0	0
$31$	141.688	0.820902	0.410451	−	0.911883i	$-0.365371\pi$
$31$	141.688	0.820902	0.410451	+	0.911883i	$0.365371\pi$
$32$	−344.163	−1.90125
$33$	0	0
$34$	−182.221	−0.919134
$35$	389.500	1.88107
$36$	0	0
$37$	−87.4744	−0.388668	−0.194334	−	0.980935i	$-0.562255\pi$
$37$	−87.4744	−0.388668	−0.194334	+	0.980935i	$0.562255\pi$
$38$	602.738	2.57308
$39$	0	0
$40$	−802.438	−3.17191
$41$	517.668	1.97186	0.985928	−	0.167170i	$-0.0534630\pi$
$41$	517.668	1.97186	0.985928	+	0.167170i	$0.0534630\pi$
$42$	0	0
$43$	221.419	0.785256	0.392628	−	0.919697i	$-0.371566\pi$
$43$	221.419	0.785256	0.392628	+	0.919697i	$0.371566\pi$
$44$	−709.498	−2.43093
$45$	0	0
$46$	916.282	2.93692
$47$	−196.711	−0.610495	−0.305248	−	0.952273i	$-0.598739\pi$
$47$	−196.711	−0.610495	−0.305248	+	0.952273i	$0.598739\pi$
$48$	0	0
$49$	494.420	1.44146
$50$	−293.944	−0.831400
$51$	0	0
$52$	−273.388	−0.729078
$53$	120.995	0.313583	0.156792	−	0.987632i	$-0.449885\pi$
$53$	120.995	0.313583	0.156792	+	0.987632i	$0.449885\pi$
$54$	0	0
$55$	−492.473	−1.20736
$56$	−1725.23	−4.11685
$57$	0	0
$58$	853.324	1.93184
$59$	586.110	1.29331	0.646653	−	0.762784i	$-0.276168\pi$
$59$	586.110	1.29331	0.646653	+	0.762784i	$0.276168\pi$
$60$	0	0
$61$	−31.9565	−0.0670755	−0.0335378	−	0.999437i	$-0.510677\pi$
$61$	−31.9565	−0.0670755	−0.0335378	+	0.999437i	$0.510677\pi$
$62$	−741.548	−1.51898
$63$	0	0
$64$	546.121	1.06664
$65$	−189.763	−0.362110
$66$	0	0
$67$	411.888	0.751046	0.375523	−	0.926813i	$-0.377463\pi$
$67$	411.888	0.751046	0.375523	+	0.926813i	$0.377463\pi$
$68$	675.144	1.20402
$69$	0	0
$70$	−2038.51	−3.48070
$71$	−611.058	−1.02140	−0.510699	−	0.859760i	$-0.670613\pi$
$71$	−611.058	−1.02140	−0.510699	+	0.859760i	$0.670613\pi$
$72$	0	0
$73$	−949.549	−1.52241	−0.761207	−	0.648509i	$-0.775393\pi$
$73$	−949.549	−1.52241	−0.761207	+	0.648509i	$0.775393\pi$
$74$	457.811	0.719182
$75$	0	0
$76$	−2233.20	−3.37060
$77$	−1058.81	−1.56705
$78$	0	0
$79$	−383.400	−0.546024	−0.273012	−	0.962011i	$-0.588020\pi$
$79$	−383.400	−0.546024	−0.273012	+	0.962011i	$0.588020\pi$
$80$	2111.69	2.95117
$81$	0	0
$82$	−2709.30	−3.64868
$83$	−685.600	−0.906679	−0.453339	−	0.891338i	$-0.649767\pi$
$83$	−685.600	−0.906679	−0.453339	+	0.891338i	$0.649767\pi$
$84$	0	0
$85$	468.628	0.597998
$86$	−1158.83	−1.45302
$87$	0	0
$88$	2181.33	2.64239
$89$	232.830	0.277302	0.138651	−	0.990341i	$-0.455723\pi$
$89$	232.830	0.277302	0.138651	+	0.990341i	$0.455723\pi$
$90$	0	0
$91$	−407.987	−0.469985
$92$	−3394.91	−3.84722
$93$	0	0
$94$	1029.52	1.12965
$95$	−1550.10	−1.67407
$96$	0	0
$97$	−758.134	−0.793576	−0.396788	−	0.917910i	$-0.629875\pi$
$97$	−758.134	−0.793576	−0.396788	+	0.917910i	$0.629875\pi$
$98$	−2587.63	−2.66724
$99$	0	0
$100$	1089.09	1.08909
$101$	−810.194	−0.798191	−0.399096	−	0.916909i	$-0.630676\pi$
$101$	−810.194	−0.798191	−0.399096	+	0.916909i	$0.630676\pi$
$102$	0	0
$103$	−139.139	−0.133104	−0.0665522	−	0.997783i	$-0.521200\pi$
$103$	−139.139	−0.133104	−0.0665522	+	0.997783i	$0.521200\pi$
$104$	840.523	0.792501
$105$	0	0
$106$	−633.246	−0.580248
$107$	2092.89	1.89091	0.945453	−	0.325757i	$-0.105619\pi$
$107$	2092.89	1.89091	0.945453	+	0.325757i	$0.105619\pi$
$108$	0	0
$109$	−565.009	−0.496496	−0.248248	−	0.968697i	$-0.579855\pi$
$109$	−565.009	−0.496496	−0.248248	+	0.968697i	$0.579855\pi$
$110$	2577.44	2.23408
$111$	0	0
$112$	4540.09	3.83034
$113$	−858.839	−0.714980	−0.357490	−	0.933917i	$-0.616367\pi$
$113$	−858.839	−0.714980	−0.357490	+	0.933917i	$0.616367\pi$
$114$	0	0
$115$	−2356.46	−1.91079
$116$	−3161.65	−2.53062
$117$	0	0
$118$	−3067.50	−2.39311
$119$	1007.54	0.776145
$120$	0	0
$121$	7.72742	0.00580573
$122$	167.249	0.124115
$123$	0	0
$124$	2747.51	1.98979
$125$	−926.511	−0.662958
$126$	0	0
$127$	−1548.04	−1.08162	−0.540811	−	0.841144i	$-0.681883\pi$
$127$	−1548.04	−1.08162	−0.540811	+	0.841144i	$0.681883\pi$
$128$	−104.903	−0.0724390
$129$	0	0
$130$	993.153	0.670041
$131$	1019.13	0.679709	0.339855	−	0.940478i	$-0.389622\pi$
$131$	1019.13	0.679709	0.339855	+	0.940478i	$0.389622\pi$
$132$	0	0
$133$	−3332.69	−2.17279
$134$	−2155.68	−1.38972
$135$	0	0
$136$	−2075.71	−1.30876
$137$	1967.25	1.22682	0.613408	−	0.789767i	$-0.289798\pi$
$137$	1967.25	1.22682	0.613408	+	0.789767i	$0.289798\pi$
$138$	0	0
$139$	621.401	0.379183	0.189592	−	0.981863i	$-0.439284\pi$
$139$	621.401	0.379183	0.189592	+	0.981863i	$0.439284\pi$
$140$	7552.88	4.55954
$141$	0	0
$142$	3198.07	1.88997
$143$	515.847	0.301659
$144$	0	0
$145$	−2194.55	−1.25688
$146$	4969.62	2.81704
$147$	0	0
$148$	−1696.23	−0.942092
$149$	−520.530	−0.286198	−0.143099	−	0.989708i	$-0.545707\pi$
$149$	−520.530	−0.286198	−0.143099	+	0.989708i	$0.545707\pi$
$150$	0	0
$151$	−19.7111	−0.0106230	−0.00531149	−	0.999986i	$-0.501691\pi$
$151$	−19.7111	−0.0106230	−0.00531149	+	0.999986i	$0.501691\pi$
$152$	6865.92	3.66381
$153$	0	0
$154$	5541.45	2.89963
$155$	1907.08	0.988263
$156$	0	0
$157$	1838.21	0.934426	0.467213	−	0.884145i	$-0.345258\pi$
$157$	1838.21	0.934426	0.467213	+	0.884145i	$0.345258\pi$
$158$	2006.59	1.01035
$159$	0	0
$160$	−4632.34	−2.28887
$161$	−5066.35	−2.48003
$162$	0	0
$163$	−3061.14	−1.47096	−0.735482	−	0.677545i	$-0.763044\pi$
$163$	−3061.14	−1.47096	−0.735482	+	0.677545i	$0.763044\pi$
$164$	10038.2	4.77958
$165$	0	0
$166$	3588.20	1.67770
$167$	3071.32	1.42315	0.711575	−	0.702611i	$-0.247982\pi$
$167$	3071.32	1.42315	0.711575	+	0.702611i	$0.247982\pi$
$168$	0	0
$169$	−1998.23	−0.909527
$170$	−2452.64	−1.10652
$171$	0	0
$172$	4293.57	1.90338
$173$	−2489.79	−1.09419	−0.547096	−	0.837070i	$-0.684267\pi$
$173$	−2489.79	−1.09419	−0.547096	+	0.837070i	$0.684267\pi$
$174$	0	0
$175$	1625.29	0.702060
$176$	−5740.36	−2.45850
$177$	0	0
$178$	−1218.55	−0.513114
$179$	−2324.08	−0.970446	−0.485223	−	0.874390i	$-0.661262\pi$
$179$	−2324.08	−0.970446	−0.485223	+	0.874390i	$0.661262\pi$
$180$	0	0
$181$	−2410.98	−0.990091	−0.495045	−	0.868867i	$-0.664849\pi$
$181$	−2410.98	−0.990091	−0.495045	+	0.868867i	$0.664849\pi$
$182$	2135.26	0.869650
$183$	0	0
$184$	10437.6	4.18189
$185$	−1177.38	−0.467907
$186$	0	0
$187$	−1273.91	−0.498168
$188$	−3814.47	−1.47978
$189$	0	0
$190$	8112.69	3.09767
$191$	2927.14	1.10890	0.554452	−	0.832216i	$-0.312928\pi$
$191$	2927.14	1.10890	0.554452	+	0.832216i	$0.312928\pi$
$192$	0	0
$193$	295.808	0.110325	0.0551625	−	0.998477i	$-0.482432\pi$
$193$	295.808	0.110325	0.0551625	+	0.998477i	$0.482432\pi$
$194$	3967.82	1.46842
$195$	0	0
$196$	9587.40	3.49395
$197$	1643.19	0.594276	0.297138	−	0.954835i	$-0.403968\pi$
$197$	1643.19	0.594276	0.297138	+	0.954835i	$0.403968\pi$
$198$	0	0
$199$	−1060.35	−0.377720	−0.188860	−	0.982004i	$-0.560479\pi$
$199$	−1060.35	−0.377720	−0.188860	+	0.982004i	$0.560479\pi$
$200$	−3348.38	−1.18383
$201$	0	0
$202$	4240.28	1.47696
$203$	−4718.24	−1.63131
$204$	0	0
$205$	6967.66	2.37387
$206$	728.205	0.246293
$207$	0	0
$208$	−2211.91	−0.737348
$209$	4213.75	1.39460
$210$	0	0
$211$	−1904.28	−0.621310	−0.310655	−	0.950523i	$-0.600548\pi$
$211$	−1904.28	−0.621310	−0.310655	+	0.950523i	$0.600548\pi$
$212$	2346.24	0.760095
$213$	0	0
$214$	−10953.5	−3.49889
$215$	2980.23	0.945349
$216$	0	0
$217$	4100.20	1.28267
$218$	2957.06	0.918705
$219$	0	0
$220$	−9549.64	−2.92653
$221$	−490.870	−0.149409
$222$	0	0
$223$	−3676.26	−1.10395	−0.551975	−	0.833861i	$-0.686126\pi$
$223$	−3676.26	−1.10395	−0.551975	+	0.833861i	$0.686126\pi$
$224$	−9959.47	−2.97074
$225$	0	0
$226$	4494.87	1.32298
$227$	−6418.71	−1.87676	−0.938381	−	0.345603i	$-0.887674\pi$
$227$	−6418.71	−1.87676	−0.938381	+	0.345603i	$0.887674\pi$
$228$	0	0
$229$	6252.71	1.80433	0.902163	−	0.431396i	$-0.141979\pi$
$229$	6252.71	1.80433	0.902163	+	0.431396i	$0.141979\pi$
$230$	12332.9	3.53569
$231$	0	0
$232$	9720.39	2.75075
$233$	−4771.19	−1.34151	−0.670753	−	0.741681i	$-0.734029\pi$
$233$	−4771.19	−1.34151	−0.670753	+	0.741681i	$0.734029\pi$
$234$	0	0
$235$	−2647.68	−0.734959
$236$	11365.4	3.13485
$237$	0	0
$238$	−5273.14	−1.43616
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6434.14	−1.71975	−0.859874	−	0.510507i	$-0.829458\pi$
$241$	−6434.14	−1.71975	−0.859874	+	0.510507i	$0.829458\pi$
$242$	−40.4427	−0.0107428
$243$	0	0
$244$	−619.674	−0.162584
$245$	6654.76	1.73533
$246$	0	0
$247$	1623.67	0.418266
$248$	−8447.13	−2.16288
$249$	0	0
$250$	4849.05	1.22672
$251$	−1836.49	−0.461827	−0.230913	−	0.972974i	$-0.574171\pi$
$251$	−1836.49	−0.461827	−0.230913	+	0.972974i	$0.574171\pi$
$252$	0	0
$253$	6405.75	1.59180
$254$	8101.90	2.00141
$255$	0	0
$256$	−3819.94	−0.932602
$257$	−7022.33	−1.70444	−0.852219	−	0.523184i	$-0.824744\pi$
$257$	−7022.33	−1.70444	−0.852219	+	0.523184i	$0.824744\pi$
$258$	0	0
$259$	−2531.35	−0.607300
$260$	−3679.73	−0.877719
$261$	0	0
$262$	−5333.79	−1.25772
$263$	7329.06	1.71836	0.859181	−	0.511672i	$-0.170974\pi$
$263$	7329.06	1.71836	0.859181	+	0.511672i	$0.170974\pi$
$264$	0	0
$265$	1628.56	0.377515
$266$	17442.2	4.02048
$267$	0	0
$268$	7987.00	1.82046
$269$	−1121.03	−0.254090	−0.127045	−	0.991897i	$-0.540549\pi$
$269$	−1121.03	−0.254090	−0.127045	+	0.991897i	$0.540549\pi$
$270$	0	0
$271$	2113.88	0.473835	0.236918	−	0.971530i	$-0.423863\pi$
$271$	2113.88	0.473835	0.236918	+	0.971530i	$0.423863\pi$
$272$	5462.42	1.21768
$273$	0	0
$274$	−10295.9	−2.27007
$275$	−2054.97	−0.450616
$276$	0	0
$277$	−4287.01	−0.929896	−0.464948	−	0.885338i	$-0.653927\pi$
$277$	−4287.01	−0.929896	−0.464948	+	0.885338i	$0.653927\pi$
$278$	−3252.20	−0.701633
$279$	0	0
$280$	−23221.1	−4.95617
$281$	−5755.75	−1.22192	−0.610960	−	0.791662i	$-0.709216\pi$
$281$	−5755.75	−1.22192	−0.610960	+	0.791662i	$0.709216\pi$
$282$	0	0
$283$	−5065.28	−1.06396	−0.531978	−	0.846758i	$-0.678551\pi$
$283$	−5065.28	−1.06396	−0.531978	+	0.846758i	$0.678551\pi$
$284$	−11849.1	−2.47577
$285$	0	0
$286$	−2699.77	−0.558184
$287$	14980.4	3.08106
$288$	0	0
$289$	−3700.77	−0.753262
$290$	11485.5	2.32570
$291$	0	0
$292$	−18412.9	−3.69018
$293$	−7690.35	−1.53336	−0.766681	−	0.642028i	$-0.778093\pi$
$293$	−7690.35	−1.53336	−0.766681	+	0.642028i	$0.778093\pi$
$294$	0	0
$295$	7888.88	1.55698
$296$	5215.02	1.02404
$297$	0	0
$298$	2724.28	0.529574
$299$	2468.30	0.477410
$300$	0	0
$301$	6407.46	1.22698
$302$	103.161	0.0196565
$303$	0	0
$304$	−18068.3	−3.40883
$305$	−430.125	−0.0807505
$306$	0	0
$307$	−776.424	−0.144342	−0.0721708	−	0.997392i	$-0.522993\pi$
$307$	−776.424	−0.144342	−0.0721708	+	0.997392i	$0.522993\pi$
$308$	−20531.6	−3.79836
$309$	0	0
$310$	−9981.04	−1.82866
$311$	582.584	0.106223	0.0531114	−	0.998589i	$-0.483086\pi$
$311$	582.584	0.106223	0.0531114	+	0.998589i	$0.483086\pi$
$312$	0	0
$313$	6239.23	1.12672	0.563358	−	0.826213i	$-0.309509\pi$
$313$	6239.23	1.12672	0.563358	+	0.826213i	$0.309509\pi$
$314$	−9620.55	−1.72904
$315$	0	0
$316$	−7434.59	−1.32351
$317$	−5911.99	−1.04748	−0.523739	−	0.851879i	$-0.675463\pi$
$317$	−5911.99	−1.04748	−0.523739	+	0.851879i	$0.675463\pi$
$318$	0	0
$319$	5965.61	1.04705
$320$	7350.63	1.28410
$321$	0	0
$322$	26515.6	4.58899
$323$	−4009.73	−0.690734
$324$	0	0
$325$	−791.833	−0.135148
$326$	16021.0	2.72184
$327$	0	0
$328$	−30862.2	−5.19536
$329$	−5692.47	−0.953909
$330$	0	0
$331$	10183.2	1.69099	0.845495	−	0.533983i	$-0.179305\pi$
$331$	10183.2	1.69099	0.845495	+	0.533983i	$0.179305\pi$
$332$	−13294.6	−2.19770
$333$	0	0
$334$	−16074.2	−2.63336
$335$	5543.89	0.904165
$336$	0	0
$337$	−8651.56	−1.39846	−0.699230	−	0.714897i	$-0.746473\pi$
$337$	−8651.56	−1.39846	−0.699230	+	0.714897i	$0.746473\pi$
$338$	10458.1	1.68297
$339$	0	0
$340$	9087.26	1.44949
$341$	−5184.18	−0.823282
$342$	0	0
$343$	4381.82	0.689785
$344$	−13200.5	−2.06896
$345$	0	0
$346$	13030.7	2.02467
$347$	−746.378	−0.115469	−0.0577344	−	0.998332i	$-0.518388\pi$
$347$	−746.378	−0.115469	−0.0577344	+	0.998332i	$0.518388\pi$
$348$	0	0
$349$	8285.96	1.27088	0.635440	−	0.772150i	$-0.280819\pi$
$349$	8285.96	1.27088	0.635440	+	0.772150i	$0.280819\pi$
$350$	−8506.22	−1.29908
$351$	0	0
$352$	12592.5	1.90676
$353$	4587.73	0.691728	0.345864	−	0.938285i	$-0.387586\pi$
$353$	4587.73	0.691728	0.345864	+	0.938285i	$0.387586\pi$
$354$	0	0
$355$	−8224.67	−1.22963
$356$	4514.85	0.672153
$357$	0	0
$358$	12163.4	1.79569
$359$	5504.45	0.809231	0.404615	−	0.914487i	$-0.367405\pi$
$359$	5504.45	0.809231	0.404615	+	0.914487i	$0.367405\pi$
$360$	0	0
$361$	6404.14	0.933684
$362$	12618.2	1.83204
$363$	0	0
$364$	−7911.36	−1.13920
$365$	−12780.7	−1.83280
$366$	0	0
$367$	5284.66	0.751654	0.375827	−	0.926690i	$-0.377359\pi$
$367$	5284.66	0.751654	0.375827	+	0.926690i	$0.377359\pi$
$368$	−27467.4	−3.89086
$369$	0	0
$370$	6162.01	0.865805
$371$	3501.38	0.489979
$372$	0	0
$373$	3124.84	0.433775	0.216888	−	0.976197i	$-0.430409\pi$
$373$	3124.84	0.433775	0.216888	+	0.976197i	$0.430409\pi$
$374$	6667.20	0.921798
$375$	0	0
$376$	11727.5	1.60851
$377$	2298.70	0.314030
$378$	0	0
$379$	12536.6	1.69910	0.849552	−	0.527505i	$-0.176873\pi$
$379$	12536.6	1.69910	0.849552	+	0.527505i	$0.176873\pi$
$380$	−30058.3	−4.05778
$381$	0	0
$382$	−15319.7	−2.05189
$383$	−14599.5	−1.94777	−0.973887	−	0.227033i	$-0.927097\pi$
$383$	−14599.5	−1.94777	−0.973887	+	0.227033i	$0.927097\pi$
$384$	0	0
$385$	−14251.3	−1.88653
$386$	−1548.16	−0.204143
$387$	0	0
$388$	−14701.1	−1.92355
$389$	−7503.20	−0.977962	−0.488981	−	0.872294i	$-0.662631\pi$
$389$	−7503.20	−0.977962	−0.488981	+	0.872294i	$0.662631\pi$
$390$	0	0
$391$	−6095.59	−0.788407
$392$	−29476.2	−3.79789
$393$	0	0
$394$	−8599.90	−1.09964
$395$	−5160.46	−0.657344
$396$	0	0
$397$	3958.87	0.500479	0.250239	−	0.968184i	$-0.419491\pi$
$397$	3958.87	0.500479	0.250239	+	0.968184i	$0.419491\pi$
$398$	5549.52	0.698925
$399$	0	0
$400$	8811.55	1.10144
$401$	−5279.60	−0.657483	−0.328741	−	0.944420i	$-0.606624\pi$
$401$	−5279.60	−0.657483	−0.328741	+	0.944420i	$0.606624\pi$
$402$	0	0
$403$	−1997.60	−0.246917
$404$	−15710.6	−1.93474
$405$	0	0
$406$	24693.7	3.01854
$407$	3200.57	0.389794
$408$	0	0
$409$	7984.85	0.965343	0.482672	−	0.875801i	$-0.339666\pi$
$409$	7984.85	0.965343	0.482672	+	0.875801i	$0.339666\pi$
$410$	−36466.4	−4.39255
$411$	0	0
$412$	−2698.07	−0.322632
$413$	16961.0	2.02081
$414$	0	0
$415$	−9227.98	−1.09153
$416$	4852.20	0.571872
$417$	0	0
$418$	−22053.4	−2.58054
$419$	−7478.27	−0.871927	−0.435964	−	0.899964i	$-0.643592\pi$
$419$	−7478.27	−0.871927	−0.435964	+	0.899964i	$0.643592\pi$
$420$	0	0
$421$	−14805.7	−1.71398	−0.856990	−	0.515334i	$-0.827668\pi$
$421$	−14805.7	−1.71398	−0.856990	+	0.515334i	$0.827668\pi$
$422$	9966.38	1.14966
$423$	0	0
$424$	−7213.44	−0.826216
$425$	1955.47	0.223186
$426$	0	0
$427$	−924.763	−0.104807
$428$	40583.6	4.58337
$429$	0	0
$430$	−15597.5	−1.74925
$431$	3025.44	0.338122	0.169061	−	0.985606i	$-0.445927\pi$
$431$	3025.44	0.338122	0.169061	+	0.985606i	$0.445927\pi$
$432$	0	0
$433$	−13290.6	−1.47508	−0.737538	−	0.675306i	$-0.764012\pi$
$433$	−13290.6	−1.47508	−0.737538	+	0.675306i	$0.764012\pi$
$434$	−21459.1	−2.37343
$435$	0	0
$436$	−10956.2	−1.20346
$437$	20162.6	2.20711
$438$	0	0
$439$	−16939.3	−1.84162	−0.920808	−	0.390016i	$-0.872469\pi$
$439$	−16939.3	−1.84162	−0.920808	+	0.390016i	$0.872469\pi$
$440$	29360.1	3.18111
$441$	0	0
$442$	2569.05	0.276464
$443$	4213.17	0.451860	0.225930	−	0.974144i	$-0.427458\pi$
$443$	4213.17	0.451860	0.225930	+	0.974144i	$0.427458\pi$
$444$	0	0
$445$	3133.82	0.333837
$446$	19240.3	2.04272
$447$	0	0
$448$	15803.8	1.66665
$449$	−7809.93	−0.820876	−0.410438	−	0.911889i	$-0.634624\pi$
$449$	−7809.93	−0.820876	−0.410438	+	0.911889i	$0.634624\pi$
$450$	0	0
$451$	−18940.7	−1.97757
$452$	−16653.9	−1.73304
$453$	0	0
$454$	33593.4	3.47272
$455$	−5491.39	−0.565803
$456$	0	0
$457$	996.534	0.102004	0.0510021	−	0.998699i	$-0.483758\pi$
$457$	996.534	0.102004	0.0510021	+	0.998699i	$0.483758\pi$
$458$	−32724.5	−3.33868
$459$	0	0
$460$	−45694.6	−4.63157
$461$	6013.37	0.607528	0.303764	−	0.952747i	$-0.401757\pi$
$461$	6013.37	0.607528	0.303764	+	0.952747i	$0.401757\pi$
$462$	0	0
$463$	6080.85	0.610370	0.305185	−	0.952293i	$-0.401282\pi$
$463$	6080.85	0.610370	0.305185	+	0.952293i	$0.401282\pi$
$464$	−25580.1	−2.55932
$465$	0	0
$466$	24970.8	2.48229
$467$	452.443	0.0448320	0.0224160	−	0.999749i	$-0.492864\pi$
$467$	452.443	0.0448320	0.0224160	+	0.999749i	$0.492864\pi$
$468$	0	0
$469$	11919.3	1.17352
$470$	13857.0	1.35995
$471$	0	0
$472$	−34942.6	−3.40755
$473$	−8101.40	−0.787532
$474$	0	0
$475$	−6468.19	−0.624801
$476$	19537.5	1.88130
$477$	0	0
$478$	1250.84	0.119691
$479$	−18532.6	−1.76780	−0.883898	−	0.467679i	$-0.845090\pi$
$479$	−18532.6	−1.76780	−0.883898	+	0.467679i	$0.845090\pi$
$480$	0	0
$481$	1233.26	0.116906
$482$	33674.1	3.18218
$483$	0	0
$484$	149.844	0.0140725
$485$	−10204.3	−0.955366
$486$	0	0
$487$	14947.8	1.39086	0.695431	−	0.718593i	$-0.255213\pi$
$487$	14947.8	1.39086	0.695431	+	0.718593i	$0.255213\pi$
$488$	1905.17	0.176728
$489$	0	0
$490$	−34828.7	−3.21102
$491$	17920.5	1.64713	0.823564	−	0.567223i	$-0.191982\pi$
$491$	17920.5	1.64713	0.823564	+	0.567223i	$0.191982\pi$
$492$	0	0
$493$	−5676.76	−0.518597
$494$	−8497.73	−0.773950
$495$	0	0
$496$	22229.4	2.01235
$497$	−17682.9	−1.59595
$498$	0	0
$499$	−7885.88	−0.707456	−0.353728	−	0.935348i	$-0.615086\pi$
$499$	−7885.88	−0.707456	−0.353728	+	0.935348i	$0.615086\pi$
$500$	−17966.2	−1.60694
$501$	0	0
$502$	9611.59	0.854554
$503$	−18912.7	−1.67649	−0.838246	−	0.545292i	$-0.816419\pi$
$503$	−18912.7	−1.67649	−0.838246	+	0.545292i	$0.816419\pi$
$504$	0	0
$505$	−10905.0	−0.960922
$506$	−33525.5	−2.94544
$507$	0	0
$508$	−30018.3	−2.62175
$509$	−17490.3	−1.52307	−0.761534	−	0.648125i	$-0.775554\pi$
$509$	−17490.3	−1.52307	−0.761534	+	0.648125i	$0.775554\pi$
$510$	0	0
$511$	−27478.2	−2.37880
$512$	20831.5	1.79811
$513$	0	0
$514$	36752.5	3.15386
$515$	−1872.77	−0.160241
$516$	0	0
$517$	7197.39	0.612265
$518$	13248.2	1.12373
$519$	0	0
$520$	11313.2	0.954071
$521$	−6354.27	−0.534329	−0.267165	−	0.963651i	$-0.586087\pi$
$521$	−6354.27	−0.534329	−0.267165	+	0.963651i	$0.586087\pi$
$522$	0	0
$523$	−2100.61	−0.175628	−0.0878140	−	0.996137i	$-0.527988\pi$
$523$	−2100.61	−0.175628	−0.0878140	+	0.996137i	$0.527988\pi$
$524$	19762.2	1.64755
$525$	0	0
$526$	−38357.8	−3.17962
$527$	4933.16	0.407765
$528$	0	0
$529$	18484.2	1.51921
$530$	−8523.32	−0.698546
$531$	0	0
$532$	−64624.9	−5.26662
$533$	−7298.36	−0.593109
$534$	0	0
$535$	28169.7	2.27641
$536$	−24555.8	−1.97882
$537$	0	0
$538$	5867.07	0.470162
$539$	−18090.1	−1.44564
$540$	0	0
$541$	−6757.46	−0.537017	−0.268508	−	0.963277i	$-0.586531\pi$
$541$	−6757.46	−0.537017	−0.268508	+	0.963277i	$0.586531\pi$
$542$	−11063.4	−0.876774
$543$	0	0
$544$	−11982.7	−0.944405
$545$	−7604.86	−0.597718
$546$	0	0
$547$	8951.41	0.699697	0.349849	−	0.936806i	$-0.386233\pi$
$547$	8951.41	0.699697	0.349849	+	0.936806i	$0.386233\pi$
$548$	38147.4	2.97368
$549$	0	0
$550$	10755.0	0.833810
$551$	18777.2	1.45179
$552$	0	0
$553$	−11094.9	−0.853171
$554$	22436.7	1.72066
$555$	0	0
$556$	12049.7	0.919103
$557$	−2082.46	−0.158414	−0.0792072	−	0.996858i	$-0.525239\pi$
$557$	−2082.46	−0.158414	−0.0792072	+	0.996858i	$0.525239\pi$
$558$	0	0
$559$	−3121.68	−0.236195
$560$	61108.4	4.61125
$561$	0	0
$562$	30123.7	2.26101
$563$	18374.6	1.37549	0.687743	−	0.725954i	$-0.258602\pi$
$563$	18374.6	1.37549	0.687743	+	0.725954i	$0.258602\pi$
$564$	0	0
$565$	−11559.7	−0.860746
$566$	26509.9	1.96872
$567$	0	0
$568$	36429.9	2.69113
$569$	−199.788	−0.0147198	−0.00735988	−	0.999973i	$-0.502343\pi$
$569$	−199.788	−0.0147198	−0.00735988	+	0.999973i	$0.502343\pi$
$570$	0	0
$571$	−8979.24	−0.658090	−0.329045	−	0.944314i	$-0.606727\pi$
$571$	−8979.24	−0.658090	−0.329045	+	0.944314i	$0.606727\pi$
$572$	10002.9	0.731192
$573$	0	0
$574$	−78402.2	−5.70112
$575$	−9832.93	−0.713151
$576$	0	0
$577$	2645.73	0.190890	0.0954448	−	0.995435i	$-0.469573\pi$
$577$	2645.73	0.190890	0.0954448	+	0.995435i	$0.469573\pi$
$578$	19368.6	1.39382
$579$	0	0
$580$	−42554.9	−3.04654
$581$	−19840.0	−1.41670
$582$	0	0
$583$	−4427.04	−0.314492
$584$	56609.9	4.01119
$585$	0	0
$586$	40248.7	2.83730
$587$	−19785.2	−1.39118	−0.695588	−	0.718441i	$-0.744856\pi$
$587$	−19785.2	−1.39118	−0.695588	+	0.718441i	$0.744856\pi$
$588$	0	0
$589$	−16317.6	−1.14152
$590$	−41287.7	−2.88100
$591$	0	0
$592$	−13723.8	−0.952778
$593$	21757.9	1.50673	0.753365	−	0.657603i	$-0.228430\pi$
$593$	21757.9	1.50673	0.753365	+	0.657603i	$0.228430\pi$
$594$	0	0
$595$	13561.2	0.934381
$596$	−10093.7	−0.693715
$597$	0	0
$598$	−12918.3	−0.883389
$599$	−14085.5	−0.960796	−0.480398	−	0.877051i	$-0.659508\pi$
$599$	−14085.5	−0.960796	−0.480398	+	0.877051i	$0.659508\pi$
$600$	0	0
$601$	26870.4	1.82374	0.911868	−	0.410483i	$-0.134640\pi$
$601$	26870.4	1.82374	0.911868	+	0.410483i	$0.134640\pi$
$602$	−33534.4	−2.27037
$603$	0	0
$604$	−382.223	−0.0257490
$605$	104.009	0.00698936
$606$	0	0
$607$	4139.96	0.276830	0.138415	−	0.990374i	$-0.455799\pi$
$607$	4139.96	0.276830	0.138415	+	0.990374i	$0.455799\pi$
$608$	39635.8	2.64382
$609$	0	0
$610$	2251.13	0.149419
$611$	2773.34	0.183629
$612$	0	0
$613$	−8513.26	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$613$	−8513.26	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$614$	4063.54	0.267087
$615$	0	0
$616$	63123.8	4.12878
$617$	13889.9	0.906296	0.453148	−	0.891435i	$-0.350301\pi$
$617$	13889.9	0.906296	0.453148	+	0.891435i	$0.350301\pi$
$618$	0	0
$619$	24762.1	1.60788	0.803938	−	0.594714i	$-0.202735\pi$
$619$	24762.1	1.60788	0.803938	+	0.594714i	$0.202735\pi$
$620$	36980.7	2.39545
$621$	0	0
$622$	−3049.05	−0.196553
$623$	6737.67	0.433289
$624$	0	0
$625$	−19491.1	−1.24743
$626$	−32654.0	−2.08485
$627$	0	0
$628$	35645.0	2.26495
$629$	−3045.60	−0.193062
$630$	0	0
$631$	−25948.4	−1.63707	−0.818533	−	0.574459i	$-0.805212\pi$
$631$	−25948.4	−1.63707	−0.818533	+	0.574459i	$0.805212\pi$
$632$	22857.4	1.43864
$633$	0	0
$634$	30941.4	1.93823
$635$	−20836.1	−1.30214
$636$	0	0
$637$	−6970.61	−0.433572
$638$	−31222.0	−1.93744
$639$	0	0
$640$	−1411.96	−0.0872074
$641$	−26658.9	−1.64269	−0.821345	−	0.570432i	$-0.806776\pi$
$641$	−26658.9	−1.64269	−0.821345	+	0.570432i	$0.806776\pi$
$642$	0	0
$643$	26489.7	1.62465	0.812327	−	0.583203i	$-0.198201\pi$
$643$	26489.7	1.62465	0.812327	+	0.583203i	$0.198201\pi$
$644$	−98242.7	−6.01134
$645$	0	0
$646$	20985.6	1.27812
$647$	12251.5	0.744445	0.372222	−	0.928144i	$-0.378596\pi$
$647$	12251.5	0.744445	0.372222	+	0.928144i	$0.378596\pi$
$648$	0	0
$649$	−21445.0	−1.29706
$650$	4144.19	0.250074
$651$	0	0
$652$	−59359.2	−3.56547
$653$	13637.5	0.817271	0.408636	−	0.912698i	$-0.366005\pi$
$653$	13637.5	0.817271	0.408636	+	0.912698i	$0.366005\pi$
$654$	0	0
$655$	13717.2	0.818285
$656$	81216.5	4.83380
$657$	0	0
$658$	29792.4	1.76509
$659$	−24202.9	−1.43067	−0.715335	−	0.698782i	$-0.753726\pi$
$659$	−24202.9	−1.43067	−0.715335	+	0.698782i	$0.753726\pi$
$660$	0	0
$661$	7094.61	0.417471	0.208735	−	0.977972i	$-0.433065\pi$
$661$	7094.61	0.417471	0.208735	+	0.977972i	$0.433065\pi$
$662$	−53295.3	−3.12897
$663$	0	0
$664$	40873.9	2.38888
$665$	−44857.1	−2.61576
$666$	0	0
$667$	28545.1	1.65708
$668$	59556.6	3.44957
$669$	0	0
$670$	−29014.9	−1.67305
$671$	1169.24	0.0672699
$672$	0	0
$673$	1402.82	0.0803486	0.0401743	−	0.999193i	$-0.487209\pi$
$673$	1402.82	0.0803486	0.0401743	+	0.999193i	$0.487209\pi$
$674$	45279.3	2.58768
$675$	0	0
$676$	−38748.1	−2.20460
$677$	23935.9	1.35884	0.679419	−	0.733751i	$-0.262232\pi$
$677$	23935.9	1.35884	0.679419	+	0.733751i	$0.262232\pi$
$678$	0	0
$679$	−21939.0	−1.23998
$680$	−27938.5	−1.57558
$681$	0	0
$682$	27132.2	1.52338
$683$	−1764.84	−0.0988725	−0.0494362	−	0.998777i	$-0.515742\pi$
$683$	−1764.84	−0.0988725	−0.0494362	+	0.998777i	$0.515742\pi$
$684$	0	0
$685$	26478.7	1.47693
$686$	−22933.0	−1.27636
$687$	0	0
$688$	34738.2	1.92497
$689$	−1705.85	−0.0943219
$690$	0	0
$691$	−17028.6	−0.937480	−0.468740	−	0.883336i	$-0.655292\pi$
$691$	−17028.6	−0.937480	−0.468740	+	0.883336i	$0.655292\pi$
$692$	−48280.0	−2.65221
$693$	0	0
$694$	3906.29	0.213661
$695$	8363.88	0.456489
$696$	0	0
$697$	18023.6	0.979475
$698$	−43365.9	−2.35161
$699$	0	0
$700$	31516.3	1.70172
$701$	−19820.1	−1.06789	−0.533947	−	0.845518i	$-0.679292\pi$
$701$	−19820.1	−1.06789	−0.533947	+	0.845518i	$0.679292\pi$
$702$	0	0
$703$	10074.1	0.540469
$704$	−19981.8	−1.06973
$705$	0	0
$706$	−24010.6	−1.27996
$707$	−23445.6	−1.24719
$708$	0	0
$709$	5237.12	0.277411	0.138705	−	0.990334i	$-0.455706\pi$
$709$	5237.12	0.277411	0.138705	+	0.990334i	$0.455706\pi$
$710$	43045.1	2.27529
$711$	0	0
$712$	−13880.8	−0.730624
$713$	−24806.1	−1.30294
$714$	0	0
$715$	6943.15	0.363160
$716$	−45066.7	−2.35226
$717$	0	0
$718$	−28808.4	−1.49738
$719$	14436.1	0.748785	0.374392	−	0.927270i	$-0.377851\pi$
$719$	14436.1	0.748785	0.374392	+	0.927270i	$0.377851\pi$
$720$	0	0
$721$	−4026.43	−0.207978
$722$	−33517.1	−1.72767
$723$	0	0
$724$	−46751.7	−2.39988
$725$	−9157.31	−0.469095
$726$	0	0
$727$	3650.14	0.186212	0.0931060	−	0.995656i	$-0.470320\pi$
$727$	3650.14	0.186212	0.0931060	+	0.995656i	$0.470320\pi$
$728$	24323.2	1.23830
$729$	0	0
$730$	66889.6	3.39137
$731$	7709.14	0.390058
$732$	0	0
$733$	−3192.45	−0.160868	−0.0804338	−	0.996760i	$-0.525631\pi$
$733$	−3192.45	−0.160868	−0.0804338	+	0.996760i	$0.525631\pi$
$734$	−27658.1	−1.39084
$735$	0	0
$736$	60254.4	3.01767
$737$	−15070.4	−0.753223
$738$	0	0
$739$	−8261.47	−0.411236	−0.205618	−	0.978632i	$-0.565920\pi$
$739$	−8261.47	−0.411236	−0.205618	+	0.978632i	$0.565920\pi$
$740$	−22830.8	−1.13416
$741$	0	0
$742$	−18325.0	−0.906647
$743$	10124.1	0.499886	0.249943	−	0.968261i	$-0.419588\pi$
$743$	10124.1	0.499886	0.249943	+	0.968261i	$0.419588\pi$
$744$	0	0
$745$	−7006.19	−0.344546
$746$	−16354.4	−0.802648
$747$	0	0
$748$	−24702.6	−1.20751
$749$	60564.4	2.95457
$750$	0	0
$751$	−17216.8	−0.836553	−0.418276	−	0.908320i	$-0.637366\pi$
$751$	−17216.8	−0.836553	−0.418276	+	0.908320i	$0.637366\pi$
$752$	−30861.9	−1.49656
$753$	0	0
$754$	−12030.6	−0.581074
$755$	−265.306	−0.0127887
$756$	0	0
$757$	25579.2	1.22813	0.614063	−	0.789257i	$-0.289534\pi$
$757$	25579.2	1.22813	0.614063	+	0.789257i	$0.289534\pi$
$758$	−65612.1	−3.14398
$759$	0	0
$760$	92413.3	4.41077
$761$	30971.5	1.47532	0.737659	−	0.675173i	$-0.235931\pi$
$761$	30971.5	1.47532	0.737659	+	0.675173i	$0.235931\pi$
$762$	0	0
$763$	−16350.3	−0.775783
$764$	56760.9	2.68787
$765$	0	0
$766$	76408.6	3.60412
$767$	−8263.31	−0.389010
$768$	0	0
$769$	38265.8	1.79441	0.897204	−	0.441616i	$-0.145595\pi$
$769$	38265.8	1.79441	0.897204	+	0.441616i	$0.145595\pi$
$770$	74586.4	3.49079
$771$	0	0
$772$	5736.07	0.267417
$773$	18092.7	0.841847	0.420923	−	0.907096i	$-0.361706\pi$
$773$	18092.7	0.841847	0.420923	+	0.907096i	$0.361706\pi$
$774$	0	0
$775$	7957.80	0.368842
$776$	45198.2	2.09088
$777$	0	0
$778$	39269.2	1.80960
$779$	−59617.5	−2.74200
$780$	0	0
$781$	22357.8	1.02436
$782$	31902.2	1.45885
$783$	0	0
$784$	77569.2	3.53358
$785$	24741.7	1.12493
$786$	0	0
$787$	25147.7	1.13903	0.569516	−	0.821980i	$-0.307131\pi$
$787$	25147.7	1.13903	0.569516	+	0.821980i	$0.307131\pi$
$788$	31863.4	1.44047
$789$	0	0
$790$	27008.1	1.21634
$791$	−24853.3	−1.11717
$792$	0	0
$793$	450.540	0.0201755
$794$	−20719.4	−0.926075
$795$	0	0
$796$	−20561.5	−0.915556
$797$	−19911.4	−0.884941	−0.442470	−	0.896783i	$-0.645898\pi$
$797$	−19911.4	−0.884941	−0.442470	+	0.896783i	$0.645898\pi$
$798$	0	0
$799$	−6848.90	−0.303250
$800$	−19329.7	−0.854258
$801$	0	0
$802$	27631.6	1.21659
$803$	34742.7	1.52683
$804$	0	0
$805$	−68191.7	−2.98564
$806$	10454.8	0.456890
$807$	0	0
$808$	48301.9	2.10304
$809$	−2293.83	−0.0996869	−0.0498434	−	0.998757i	$-0.515872\pi$
$809$	−2293.83	−0.0996869	−0.0498434	+	0.998757i	$0.515872\pi$
$810$	0	0
$811$	11434.4	0.495088	0.247544	−	0.968877i	$-0.420377\pi$
$811$	11434.4	0.495088	0.247544	+	0.968877i	$0.420377\pi$
$812$	−91492.4	−3.95413
$813$	0	0
$814$	−16750.7	−0.721267
$815$	−41202.1	−1.77086
$816$	0	0
$817$	−25499.8	−1.09195
$818$	−41790.0	−1.78625
$819$	0	0
$820$	135111.	5.75402
$821$	43225.8	1.83751	0.918753	−	0.394833i	$-0.129198\pi$
$821$	43225.8	1.83751	0.918753	+	0.394833i	$0.129198\pi$
$822$	0	0
$823$	−34427.4	−1.45816	−0.729078	−	0.684430i	$-0.760051\pi$
$823$	−34427.4	−1.45816	−0.729078	+	0.684430i	$0.760051\pi$
$824$	8295.13	0.350697
$825$	0	0
$826$	−88768.1	−3.73927
$827$	−25073.1	−1.05426	−0.527132	−	0.849783i	$-0.676733\pi$
$827$	−25073.1	−1.05426	−0.527132	+	0.849783i	$0.676733\pi$
$828$	0	0
$829$	12627.8	0.529049	0.264525	−	0.964379i	$-0.414785\pi$
$829$	12627.8	0.529049	0.264525	+	0.964379i	$0.414785\pi$
$830$	48296.1	2.01974
$831$	0	0
$832$	−7699.51	−0.320832
$833$	17214.2	0.716012
$834$	0	0
$835$	41339.1	1.71329
$836$	81709.8	3.38037
$837$	0	0
$838$	39138.7	1.61340
$839$	−13760.0	−0.566206	−0.283103	−	0.959089i	$-0.591364\pi$
$839$	−13760.0	−0.566206	−0.283103	+	0.959089i	$0.591364\pi$
$840$	0	0
$841$	2194.80	0.0899913
$842$	77488.0	3.17151
$843$	0	0
$844$	−36926.4	−1.50599
$845$	−26895.6	−1.09496
$846$	0	0
$847$	223.618	0.00907154
$848$	18982.8	0.768717
$849$	0	0
$850$	−10234.3	−0.412979
$851$	15314.6	0.616894
$852$	0	0
$853$	21420.0	0.859798	0.429899	−	0.902877i	$-0.358549\pi$
$853$	21420.0	0.859798	0.429899	+	0.902877i	$0.358549\pi$
$854$	4839.90	0.193932
$855$	0	0
$856$	−124773.	−4.98208
$857$	43409.5	1.73027	0.865135	−	0.501539i	$-0.167233\pi$
$857$	43409.5	1.73027	0.865135	+	0.501539i	$0.167233\pi$
$858$	0	0
$859$	−24374.3	−0.968148	−0.484074	−	0.875027i	$-0.660843\pi$
$859$	−24374.3	−0.968148	−0.484074	+	0.875027i	$0.660843\pi$
$860$	57790.3	2.29143
$861$	0	0
$862$	−15834.1	−0.625653
$863$	−10364.0	−0.408799	−0.204399	−	0.978888i	$-0.565524\pi$
$863$	−10364.0	−0.408799	−0.204399	+	0.978888i	$0.565524\pi$
$864$	0	0
$865$	−33511.9	−1.31727
$866$	69558.7	2.72945
$867$	0	0
$868$	79507.9	3.10907
$869$	14028.1	0.547607
$870$	0	0
$871$	−5807.02	−0.225905
$872$	33684.5	1.30814
$873$	0	0
$874$	−105524.	−4.08400
$875$	−26811.6	−1.03588
$876$	0	0
$877$	23117.9	0.890121	0.445060	−	0.895501i	$-0.353182\pi$
$877$	23117.9	0.890121	0.445060	+	0.895501i	$0.353182\pi$
$878$	88654.6	3.40769
$879$	0	0
$880$	−77263.7	−2.95973
$881$	−35719.0	−1.36595	−0.682976	−	0.730441i	$-0.739315\pi$
$881$	−35719.0	−1.36595	−0.682976	+	0.730441i	$0.739315\pi$
$882$	0	0
$883$	29602.4	1.12820	0.564100	−	0.825707i	$-0.309223\pi$
$883$	29602.4	1.12820	0.564100	+	0.825707i	$0.309223\pi$
$884$	−9518.55	−0.362153
$885$	0	0
$886$	−22050.3	−0.836111
$887$	23130.6	0.875590	0.437795	−	0.899075i	$-0.355759\pi$
$887$	23130.6	0.875590	0.437795	+	0.899075i	$0.355759\pi$
$888$	0	0
$889$	−44797.4	−1.69005
$890$	−16401.4	−0.617725
$891$	0	0
$892$	−71287.2	−2.67586
$893$	22654.4	0.848936
$894$	0	0
$895$	−31281.5	−1.16829
$896$	−3035.70	−0.113187
$897$	0	0
$898$	40874.5	1.51893
$899$	−23101.6	−0.857043
$900$	0	0
$901$	4212.68	0.155766
$902$	99129.4	3.65926
$903$	0	0
$904$	51202.0	1.88380
$905$	−32451.1	−1.19194
$906$	0	0
$907$	30465.1	1.11530	0.557651	−	0.830076i	$-0.311703\pi$
$907$	30465.1	1.11530	0.557651	+	0.830076i	$0.311703\pi$
$908$	−124467.	−4.54908
$909$	0	0
$910$	28740.1	1.04695
$911$	53720.8	1.95373	0.976867	−	0.213850i	$-0.0686002\pi$
$911$	53720.8	1.95373	0.976867	+	0.213850i	$0.0686002\pi$
$912$	0	0
$913$	25085.2	0.909307
$914$	−5215.52	−0.188746
$915$	0	0
$916$	121248.	4.37351
$917$	29491.8	1.06206
$918$	0	0
$919$	41973.9	1.50663	0.753315	−	0.657660i	$-0.228454\pi$
$919$	41973.9	1.50663	0.753315	+	0.657660i	$0.228454\pi$
$920$	140487.	5.03447
$921$	0	0
$922$	−31471.9	−1.12416
$923$	8615.03	0.307223
$924$	0	0
$925$	−4912.93	−0.174634
$926$	−31825.1	−1.12941
$927$	0	0
$928$	56114.3	1.98496
$929$	−54192.3	−1.91388	−0.956938	−	0.290293i	$-0.906247\pi$
$929$	−54192.3	−1.91388	−0.956938	+	0.290293i	$0.906247\pi$
$930$	0	0
$931$	−56940.2	−2.00445
$932$	−92519.1	−3.25168
$933$	0	0
$934$	−2367.93	−0.0829562
$935$	−17146.4	−0.599731
$936$	0	0
$937$	−16668.4	−0.581147	−0.290573	−	0.956853i	$-0.593846\pi$
$937$	−16668.4	−0.581147	−0.290573	+	0.956853i	$0.593846\pi$
$938$	−62381.5	−2.17146
$939$	0	0
$940$	−51341.7	−1.78147
$941$	24233.0	0.839504	0.419752	−	0.907639i	$-0.362117\pi$
$941$	24233.0	0.839504	0.419752	+	0.907639i	$0.362117\pi$
$942$	0	0
$943$	−90630.6	−3.12973
$944$	91954.4	3.17040
$945$	0	0
$946$	42400.0	1.45723
$947$	−7197.74	−0.246985	−0.123493	−	0.992345i	$-0.539410\pi$
$947$	−7197.74	−0.246985	−0.123493	+	0.992345i	$0.539410\pi$
$948$	0	0
$949$	13387.3	0.457923
$950$	33852.3	1.15612
$951$	0	0
$952$	−60067.4	−2.04495
$953$	8774.41	0.298249	0.149124	−	0.988818i	$-0.452355\pi$
$953$	8774.41	0.298249	0.149124	+	0.988818i	$0.452355\pi$
$954$	0	0
$955$	39398.6	1.33498
$956$	−4634.50	−0.156789
$957$	0	0
$958$	96993.2	3.27109
$959$	56928.7	1.91692
$960$	0	0
$961$	−9715.44	−0.326120
$962$	−6454.48	−0.216321
$963$	0	0
$964$	−124766.	−4.16850
$965$	3981.49	0.132817
$966$	0	0
$967$	−9434.11	−0.313734	−0.156867	−	0.987620i	$-0.550139\pi$
$967$	−9434.11	−0.313734	−0.156867	+	0.987620i	$0.550139\pi$
$968$	−460.691	−0.0152967
$969$	0	0
$970$	53405.7	1.76779
$971$	43007.4	1.42139	0.710696	−	0.703499i	$-0.248380\pi$
$971$	43007.4	1.42139	0.710696	+	0.703499i	$0.248380\pi$
$972$	0	0
$973$	17982.2	0.592480
$974$	−78231.8	−2.57362
$975$	0	0
$976$	−5013.63	−0.164429
$977$	−24436.4	−0.800194	−0.400097	−	0.916473i	$-0.631024\pi$
$977$	−24436.4	−0.800194	−0.400097	+	0.916473i	$0.631024\pi$
$978$	0	0
$979$	−8518.92	−0.278106
$980$	129044.	4.20628
$981$	0	0
$982$	−93789.7	−3.04781
$983$	7343.20	0.238262	0.119131	−	0.992879i	$-0.461989\pi$
$983$	7343.20	0.238262	0.119131	+	0.992879i	$0.461989\pi$
$984$	0	0
$985$	22116.9	0.715434
$986$	29710.2	0.959600
$987$	0	0
$988$	31484.9	1.01383
$989$	−38764.8	−1.24636
$990$	0	0
$991$	−57363.2	−1.83875	−0.919376	−	0.393380i	$-0.871306\pi$
$991$	−57363.2	−1.83875	−0.919376	+	0.393380i	$0.871306\pi$
$992$	−48763.9	−1.56074
$993$	0	0
$994$	92546.4	2.95311
$995$	−14272.0	−0.454727
$996$	0	0
$997$	987.760	0.0313768	0.0156884	−	0.999877i	$-0.495006\pi$
$997$	987.760	0.0313768	0.0156884	+	0.999877i	$0.495006\pi$
$998$	41272.0	1.30906
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.3		32
3.2	odd	2		717.4.a.d.1.30	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.30	✓	32	3.2	odd	2
2151.4.a.d.1.3		32	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-5.23366 q^{2} +19.3912 q^{4} +13.4597 q^{5} +28.9382 q^{7} -59.6177 q^{8} +O(q^{10})\)	\(q-5.23366 q^{2} +19.3912 q^{4} +13.4597 q^{5} +28.9382 q^{7} -59.6177 q^{8} -70.4436 q^{10} -36.5886 q^{11} -14.0985 q^{13} -151.453 q^{14} +156.889 q^{16} +34.8170 q^{17} -115.166 q^{19} +261.000 q^{20} +191.492 q^{22} -175.075 q^{23} +56.1642 q^{25} +73.7870 q^{26} +561.147 q^{28} -163.045 q^{29} +141.688 q^{31} -344.163 q^{32} -182.221 q^{34} +389.500 q^{35} -87.4744 q^{37} +602.738 q^{38} -802.438 q^{40} +517.668 q^{41} +221.419 q^{43} -709.498 q^{44} +916.282 q^{46} -196.711 q^{47} +494.420 q^{49} -293.944 q^{50} -273.388 q^{52} +120.995 q^{53} -492.473 q^{55} -1725.23 q^{56} +853.324 q^{58} +586.110 q^{59} -31.9565 q^{61} -741.548 q^{62} +546.121 q^{64} -189.763 q^{65} +411.888 q^{67} +675.144 q^{68} -2038.51 q^{70} -611.058 q^{71} -949.549 q^{73} +457.811 q^{74} -2233.20 q^{76} -1058.81 q^{77} -383.400 q^{79} +2111.69 q^{80} -2709.30 q^{82} -685.600 q^{83} +468.628 q^{85} -1158.83 q^{86} +2181.33 q^{88} +232.830 q^{89} -407.987 q^{91} -3394.91 q^{92} +1029.52 q^{94} -1550.10 q^{95} -758.134 q^{97} -2587.63 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

L-functions

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