LMFDB - Embedded newform 2151.4.a.f.1.18

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-0.412743 q^{2} -7.82964 q^{4} +15.2997 q^{5} +4.25811 q^{7} +6.53358 q^{8} +O(q^{10})$	\(q-0.412743 q^{2} -7.82964 q^{4} +15.2997 q^{5} +4.25811 q^{7} +6.53358 q^{8} -6.31484 q^{10} +16.2790 q^{11} -18.4697 q^{13} -1.75751 q^{14} +59.9405 q^{16} +134.548 q^{17} +161.774 q^{19} -119.791 q^{20} -6.71903 q^{22} +67.9891 q^{23} +109.080 q^{25} +7.62323 q^{26} -33.3395 q^{28} -176.152 q^{29} +257.784 q^{31} -77.0087 q^{32} -55.5340 q^{34} +65.1478 q^{35} -286.234 q^{37} -66.7709 q^{38} +99.9617 q^{40} +33.2981 q^{41} +58.3439 q^{43} -127.458 q^{44} -28.0620 q^{46} +411.778 q^{47} -324.868 q^{49} -45.0222 q^{50} +144.611 q^{52} +71.8529 q^{53} +249.063 q^{55} +27.8207 q^{56} +72.7055 q^{58} +193.423 q^{59} -492.108 q^{61} -106.399 q^{62} -447.739 q^{64} -282.580 q^{65} +483.316 q^{67} -1053.47 q^{68} -26.8893 q^{70} -1128.10 q^{71} +390.348 q^{73} +118.141 q^{74} -1266.63 q^{76} +69.3177 q^{77} +411.724 q^{79} +917.070 q^{80} -13.7436 q^{82} +1014.97 q^{83} +2058.55 q^{85} -24.0811 q^{86} +106.360 q^{88} +652.775 q^{89} -78.6459 q^{91} -532.330 q^{92} -169.959 q^{94} +2475.08 q^{95} +1607.19 q^{97} +134.087 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * 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 $

$2$	−0.412743	−0.145927	−0.0729634	−	0.997335i	$-0.523246\pi$
$2$	−0.412743	−0.145927	−0.0729634	+	0.997335i	$0.523246\pi$
$3$	0	0
$3$	0	0
$4$	−7.82964	−0.978705
$5$	15.2997	1.36845	0.684223	−	0.729273i	$-0.260142\pi$
$5$	15.2997	1.36845	0.684223	+	0.729273i	$0.260142\pi$
$6$	0	0
$7$	4.25811	0.229917	0.114958	−	0.993370i	$-0.463327\pi$
$7$	4.25811	0.229917	0.114958	+	0.993370i	$0.463327\pi$
$8$	6.53358	0.288746
$9$	0	0
$10$	−6.31484	−0.199693
$11$	16.2790	0.446208	0.223104	−	0.974795i	$-0.428381\pi$
$11$	16.2790	0.446208	0.223104	+	0.974795i	$0.428381\pi$
$12$	0	0
$13$	−18.4697	−0.394043	−0.197022	−	0.980399i	$-0.563127\pi$
$13$	−18.4697	−0.394043	−0.197022	+	0.980399i	$0.563127\pi$
$14$	−1.75751	−0.0335510
$15$	0	0
$16$	59.9405	0.936570
$17$	134.548	1.91958	0.959788	−	0.280726i	$-0.0905752\pi$
$17$	134.548	1.91958	0.959788	+	0.280726i	$0.0905752\pi$
$18$	0	0
$19$	161.774	1.95334	0.976668	−	0.214754i	$-0.0688951\pi$
$19$	161.774	1.95334	0.976668	+	0.214754i	$0.0688951\pi$
$20$	−119.791	−1.33930
$21$	0	0
$22$	−6.71903	−0.0651138
$23$	67.9891	0.616379	0.308189	−	0.951325i	$-0.400277\pi$
$23$	67.9891	0.616379	0.308189	+	0.951325i	$0.400277\pi$
$24$	0	0
$25$	109.080	0.872643
$26$	7.62323	0.0575015
$27$	0	0
$28$	−33.3395	−0.225021
$29$	−176.152	−1.12795	−0.563976	−	0.825791i	$-0.690729\pi$
$29$	−176.152	−1.12795	−0.563976	+	0.825791i	$0.690729\pi$
$30$	0	0
$31$	257.784	1.49353	0.746763	−	0.665090i	$-0.231607\pi$
$31$	257.784	1.49353	0.746763	+	0.665090i	$0.231607\pi$
$32$	−77.0087	−0.425417
$33$	0	0
$34$	−55.5340	−0.280118
$35$	65.1478	0.314628
$36$	0	0
$37$	−286.234	−1.27180	−0.635900	−	0.771771i	$-0.719371\pi$
$37$	−286.234	−1.27180	−0.635900	+	0.771771i	$0.719371\pi$
$38$	−66.7709	−0.285044
$39$	0	0
$40$	99.9617	0.395133
$41$	33.2981	0.126836	0.0634182	−	0.997987i	$-0.479800\pi$
$41$	33.2981	0.126836	0.0634182	+	0.997987i	$0.479800\pi$
$42$	0	0
$43$	58.3439	0.206915	0.103458	−	0.994634i	$-0.467009\pi$
$43$	58.3439	0.206915	0.103458	+	0.994634i	$0.467009\pi$
$44$	−127.458	−0.436707
$45$	0	0
$46$	−28.0620	−0.0899461
$47$	411.778	1.27796	0.638978	−	0.769225i	$-0.279357\pi$
$47$	411.778	1.27796	0.638978	+	0.769225i	$0.279357\pi$
$48$	0	0
$49$	−324.868	−0.947138
$50$	−45.0222	−0.127342
$51$	0	0
$52$	144.611	0.385652
$53$	71.8529	0.186222	0.0931109	−	0.995656i	$-0.470319\pi$
$53$	71.8529	0.186222	0.0931109	+	0.995656i	$0.470319\pi$
$54$	0	0
$55$	249.063	0.610612
$56$	27.8207	0.0663875
$57$	0	0
$58$	72.7055	0.164598
$59$	193.423	0.426805	0.213403	−	0.976964i	$-0.431545\pi$
$59$	193.423	0.426805	0.213403	+	0.976964i	$0.431545\pi$
$60$	0	0
$61$	−492.108	−1.03292	−0.516459	−	0.856312i	$-0.672750\pi$
$61$	−492.108	−1.03292	−0.516459	+	0.856312i	$0.672750\pi$
$62$	−106.399	−0.217946
$63$	0	0
$64$	−447.739	−0.874490
$65$	−282.580	−0.539227
$66$	0	0
$67$	483.316	0.881291	0.440646	−	0.897681i	$-0.354750\pi$
$67$	483.316	0.881291	0.440646	+	0.897681i	$0.354750\pi$
$68$	−1053.47	−1.87870
$69$	0	0
$70$	−26.8893	−0.0459127
$71$	−1128.10	−1.88564	−0.942821	−	0.333298i	$-0.891838\pi$
$71$	−1128.10	−1.88564	−0.942821	+	0.333298i	$0.891838\pi$
$72$	0	0
$73$	390.348	0.625846	0.312923	−	0.949778i	$-0.398692\pi$
$73$	390.348	0.625846	0.312923	+	0.949778i	$0.398692\pi$
$74$	118.141	0.185590
$75$	0	0
$76$	−1266.63	−1.91174
$77$	69.3177	0.102591
$78$	0	0
$79$	411.724	0.586362	0.293181	−	0.956057i	$-0.405286\pi$
$79$	411.724	0.586362	0.293181	+	0.956057i	$0.405286\pi$
$80$	917.070	1.28164
$81$	0	0
$82$	−13.7436	−0.0185088
$83$	1014.97	1.34225	0.671127	−	0.741342i	$-0.265811\pi$
$83$	1014.97	1.34225	0.671127	+	0.741342i	$0.265811\pi$
$84$	0	0
$85$	2058.55	2.62684
$86$	−24.0811	−0.0301945
$87$	0	0
$88$	106.360	0.128841
$89$	652.775	0.777460	0.388730	−	0.921352i	$-0.372914\pi$
$89$	652.775	0.777460	0.388730	+	0.921352i	$0.372914\pi$
$90$	0	0
$91$	−78.6459	−0.0905971
$92$	−532.330	−0.603253
$93$	0	0
$94$	−169.959	−0.186488
$95$	2475.08	2.67303
$96$	0	0
$97$	1607.19	1.68233	0.841164	−	0.540780i	$-0.181871\pi$
$97$	1607.19	1.68233	0.841164	+	0.540780i	$0.181871\pi$
$98$	134.087	0.138213
$99$	0	0
$100$	−854.060	−0.854060
$101$	−695.565	−0.685260	−0.342630	−	0.939470i	$-0.611318\pi$
$101$	−695.565	−0.685260	−0.342630	+	0.939470i	$0.611318\pi$
$102$	0	0
$103$	−173.802	−0.166264	−0.0831321	−	0.996539i	$-0.526492\pi$
$103$	−173.802	−0.166264	−0.0831321	+	0.996539i	$0.526492\pi$
$104$	−120.673	−0.113778
$105$	0	0
$106$	−29.6568	−0.0271747
$107$	−280.670	−0.253583	−0.126792	−	0.991929i	$-0.540468\pi$
$107$	−280.670	−0.253583	−0.126792	+	0.991929i	$0.540468\pi$
$108$	0	0
$109$	−1807.95	−1.58872	−0.794359	−	0.607448i	$-0.792193\pi$
$109$	−1807.95	−1.58872	−0.794359	+	0.607448i	$0.792193\pi$
$110$	−102.799	−0.0891046
$111$	0	0
$112$	255.233	0.215333
$113$	−1093.65	−0.910457	−0.455229	−	0.890375i	$-0.650442\pi$
$113$	−1093.65	−0.910457	−0.455229	+	0.890375i	$0.650442\pi$
$114$	0	0
$115$	1040.21	0.843480
$116$	1379.21	1.10393
$117$	0	0
$118$	−79.8340	−0.0622823
$119$	572.923	0.441342
$120$	0	0
$121$	−1066.00	−0.800898
$122$	203.114	0.150730
$123$	0	0
$124$	−2018.35	−1.46172
$125$	−243.565	−0.174281
$126$	0	0
$127$	−446.897	−0.312250	−0.156125	−	0.987737i	$-0.549900\pi$
$127$	−446.897	−0.312250	−0.156125	+	0.987737i	$0.549900\pi$
$128$	800.870	0.553028
$129$	0	0
$130$	116.633	0.0786876
$131$	−1866.31	−1.24474	−0.622368	−	0.782725i	$-0.713829\pi$
$131$	−1866.31	−1.24474	−0.622368	+	0.782725i	$0.713829\pi$
$132$	0	0
$133$	688.850	0.449104
$134$	−199.486	−0.128604
$135$	0	0
$136$	879.083	0.554270
$137$	−1622.93	−1.01209	−0.506046	−	0.862506i	$-0.668893\pi$
$137$	−1622.93	−1.01209	−0.506046	+	0.862506i	$0.668893\pi$
$138$	0	0
$139$	1312.27	0.800759	0.400380	−	0.916349i	$-0.368878\pi$
$139$	1312.27	0.800759	0.400380	+	0.916349i	$0.368878\pi$
$140$	−510.084	−0.307928
$141$	0	0
$142$	465.615	0.275166
$143$	−300.667	−0.175825
$144$	0	0
$145$	−2695.07	−1.54354
$146$	−161.114	−0.0913277
$147$	0	0
$148$	2241.11	1.24472
$149$	296.172	0.162842	0.0814208	−	0.996680i	$-0.474054\pi$
$149$	296.172	0.162842	0.0814208	+	0.996680i	$0.474054\pi$
$150$	0	0
$151$	2494.32	1.34427	0.672135	−	0.740429i	$-0.265378\pi$
$151$	2494.32	1.34427	0.672135	+	0.740429i	$0.265378\pi$
$152$	1056.96	0.564018
$153$	0	0
$154$	−28.6104	−0.0149707
$155$	3944.01	2.04381
$156$	0	0
$157$	968.817	0.492484	0.246242	−	0.969208i	$-0.420804\pi$
$157$	968.817	0.492484	0.246242	+	0.969208i	$0.420804\pi$
$158$	−169.936	−0.0855659
$159$	0	0
$160$	−1178.21	−0.582160
$161$	289.505	0.141716
$162$	0	0
$163$	−3130.58	−1.50433	−0.752165	−	0.658975i	$-0.770990\pi$
$163$	−3130.58	−1.50433	−0.752165	+	0.658975i	$0.770990\pi$
$164$	−260.713	−0.124136
$165$	0	0
$166$	−418.921	−0.195871
$167$	−1894.64	−0.877914	−0.438957	−	0.898508i	$-0.644652\pi$
$167$	−1894.64	−0.877914	−0.438957	+	0.898508i	$0.644652\pi$
$168$	0	0
$169$	−1855.87	−0.844730
$170$	−849.652	−0.383326
$171$	0	0
$172$	−456.812	−0.202509
$173$	425.982	0.187207	0.0936035	−	0.995610i	$-0.470161\pi$
$173$	425.982	0.187207	0.0936035	+	0.995610i	$0.470161\pi$
$174$	0	0
$175$	464.477	0.200635
$176$	975.769	0.417905
$177$	0	0
$178$	−269.428	−0.113452
$179$	1279.17	0.534134	0.267067	−	0.963678i	$-0.413946\pi$
$179$	1279.17	0.534134	0.267067	+	0.963678i	$0.413946\pi$
$180$	0	0
$181$	30.5114	0.0125298	0.00626491	−	0.999980i	$-0.498006\pi$
$181$	30.5114	0.0125298	0.00626491	+	0.999980i	$0.498006\pi$
$182$	32.4606	0.0132205
$183$	0	0
$184$	444.212	0.177977
$185$	−4379.29	−1.74039
$186$	0	0
$187$	2190.31	0.856531
$188$	−3224.07	−1.25074
$189$	0	0
$190$	−1021.57	−0.390067
$191$	−4355.19	−1.64990	−0.824948	−	0.565208i	$-0.808796\pi$
$191$	−4355.19	−1.64990	−0.824948	+	0.565208i	$0.808796\pi$
$192$	0	0
$193$	1237.43	0.461513	0.230756	−	0.973012i	$-0.425880\pi$
$193$	1237.43	0.461513	0.230756	+	0.973012i	$0.425880\pi$
$194$	−663.359	−0.245497
$195$	0	0
$196$	2543.60	0.926969
$197$	−942.924	−0.341018	−0.170509	−	0.985356i	$-0.554541\pi$
$197$	−942.924	−0.341018	−0.170509	+	0.985356i	$0.554541\pi$
$198$	0	0
$199$	−380.277	−0.135463	−0.0677314	−	0.997704i	$-0.521576\pi$
$199$	−380.277	−0.135463	−0.0677314	+	0.997704i	$0.521576\pi$
$200$	712.685	0.251972
$201$	0	0
$202$	287.090	0.0999979
$203$	−750.075	−0.259335
$204$	0	0
$205$	509.451	0.173569
$206$	71.7356	0.0242624
$207$	0	0
$208$	−1107.08	−0.369049
$209$	2633.51	0.871595
$210$	0	0
$211$	3211.80	1.04791	0.523956	−	0.851745i	$-0.324456\pi$
$211$	3211.80	1.04791	0.523956	+	0.851745i	$0.324456\pi$
$212$	−562.582	−0.182256
$213$	0	0
$214$	115.845	0.0370046
$215$	892.644	0.283153
$216$	0	0
$217$	1097.67	0.343387
$218$	746.220	0.231837
$219$	0	0
$220$	−1950.07	−0.597609
$221$	−2485.06	−0.756396
$222$	0	0
$223$	6313.31	1.89583	0.947916	−	0.318521i	$-0.103186\pi$
$223$	6313.31	1.89583	0.947916	+	0.318521i	$0.103186\pi$
$224$	−327.912	−0.0978104
$225$	0	0
$226$	451.396	0.132860
$227$	4616.29	1.34975	0.674877	−	0.737930i	$-0.264197\pi$
$227$	4616.29	1.34975	0.674877	+	0.737930i	$0.264197\pi$
$228$	0	0
$229$	6634.48	1.91449	0.957247	−	0.289273i	$-0.0934134\pi$
$229$	6634.48	1.91449	0.957247	+	0.289273i	$0.0934134\pi$
$230$	−429.340	−0.123086
$231$	0	0
$232$	−1150.90	−0.325692
$233$	3538.83	0.995007	0.497503	−	0.867462i	$-0.334250\pi$
$233$	3538.83	0.995007	0.497503	+	0.867462i	$0.334250\pi$
$234$	0	0
$235$	6300.07	1.74881
$236$	−1514.43	−0.417717
$237$	0	0
$238$	−236.470	−0.0644037
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	2596.05	0.693885	0.346942	−	0.937886i	$-0.387220\pi$
$241$	2596.05	0.693885	0.346942	+	0.937886i	$0.387220\pi$
$242$	439.982	0.116872
$243$	0	0
$244$	3853.03	1.01092
$245$	−4970.39	−1.29611
$246$	0	0
$247$	−2987.90	−0.769699
$248$	1684.25	0.431250
$249$	0	0
$250$	100.530	0.0254323
$251$	−5746.38	−1.44505	−0.722527	−	0.691343i	$-0.757019\pi$
$251$	−5746.38	−1.44505	−0.722527	+	0.691343i	$0.757019\pi$
$252$	0	0
$253$	1106.79	0.275033
$254$	184.454	0.0455656
$255$	0	0
$256$	3251.36	0.793788
$257$	−6729.28	−1.63331	−0.816656	−	0.577125i	$-0.804175\pi$
$257$	−6729.28	−1.63331	−0.816656	+	0.577125i	$0.804175\pi$
$258$	0	0
$259$	−1218.82	−0.292408
$260$	2212.50	0.527744
$261$	0	0
$262$	770.307	0.181640
$263$	6375.42	1.49477	0.747386	−	0.664390i	$-0.231308\pi$
$263$	6375.42	1.49477	0.747386	+	0.664390i	$0.231308\pi$
$264$	0	0
$265$	1099.33	0.254834
$266$	−284.318	−0.0655364
$267$	0	0
$268$	−3784.20	−0.862524
$269$	1823.95	0.413414	0.206707	−	0.978403i	$-0.433725\pi$
$269$	1823.95	0.413414	0.206707	+	0.978403i	$0.433725\pi$
$270$	0	0
$271$	3135.94	0.702934	0.351467	−	0.936200i	$-0.385683\pi$
$271$	3135.94	0.702934	0.351467	+	0.936200i	$0.385683\pi$
$272$	8064.89	1.79782
$273$	0	0
$274$	669.855	0.147691
$275$	1775.72	0.389381
$276$	0	0
$277$	−4824.27	−1.04644	−0.523218	−	0.852199i	$-0.675268\pi$
$277$	−4824.27	−1.04644	−0.523218	+	0.852199i	$0.675268\pi$
$278$	−541.632	−0.116852
$279$	0	0
$280$	425.648	0.0908477
$281$	7112.26	1.50990	0.754950	−	0.655782i	$-0.227661\pi$
$281$	7112.26	1.50990	0.754950	+	0.655782i	$0.227661\pi$
$282$	0	0
$283$	−3307.28	−0.694691	−0.347346	−	0.937737i	$-0.612917\pi$
$283$	−3307.28	−0.694691	−0.347346	+	0.937737i	$0.612917\pi$
$284$	8832.61	1.84549
$285$	0	0
$286$	124.098	0.0256576
$287$	141.787	0.0291618
$288$	0	0
$289$	13190.3	2.68477
$290$	1112.37	0.225244
$291$	0	0
$292$	−3056.29	−0.612519
$293$	3506.86	0.699226	0.349613	−	0.936894i	$-0.386313\pi$
$293$	3506.86	0.699226	0.349613	+	0.936894i	$0.386313\pi$
$294$	0	0
$295$	2959.31	0.584060
$296$	−1870.13	−0.367228
$297$	0	0
$298$	−122.243	−0.0237629
$299$	−1255.74	−0.242880
$300$	0	0
$301$	248.435	0.0475733
$302$	−1029.51	−0.196165
$303$	0	0
$304$	9696.78	1.82944
$305$	−7529.10	−1.41349
$306$	0	0
$307$	−8419.56	−1.56524	−0.782622	−	0.622498i	$-0.786118\pi$
$307$	−8419.56	−1.56524	−0.782622	+	0.622498i	$0.786118\pi$
$308$	−542.733	−0.100406
$309$	0	0
$310$	−1627.86	−0.298247
$311$	2368.37	0.431827	0.215913	−	0.976413i	$-0.430727\pi$
$311$	2368.37	0.431827	0.215913	+	0.976413i	$0.430727\pi$
$312$	0	0
$313$	5463.42	0.986616	0.493308	−	0.869855i	$-0.335788\pi$
$313$	5463.42	0.986616	0.493308	+	0.869855i	$0.335788\pi$
$314$	−399.873	−0.0718666
$315$	0	0
$316$	−3223.65	−0.573875
$317$	5229.07	0.926478	0.463239	−	0.886233i	$-0.346687\pi$
$317$	5229.07	0.926478	0.463239	+	0.886233i	$0.346687\pi$
$318$	0	0
$319$	−2867.57	−0.503302
$320$	−6850.26	−1.19669
$321$	0	0
$322$	−119.491	−0.0206801
$323$	21766.4	3.74958
$324$	0	0
$325$	−2014.68	−0.343859
$326$	1292.13	0.219522
$327$	0	0
$328$	217.556	0.0366235
$329$	1753.40	0.293824
$330$	0	0
$331$	−1089.49	−0.180919	−0.0904593	−	0.995900i	$-0.528834\pi$
$331$	−1089.49	−0.180919	−0.0904593	+	0.995900i	$0.528834\pi$
$332$	−7946.83	−1.31367
$333$	0	0
$334$	782.000	0.128111
$335$	7394.59	1.20600
$336$	0	0
$337$	9809.59	1.58564	0.792822	−	0.609453i	$-0.208611\pi$
$337$	9809.59	1.58564	0.792822	+	0.609453i	$0.208611\pi$
$338$	765.999	0.123269
$339$	0	0
$340$	−16117.7	−2.57090
$341$	4196.45	0.666424
$342$	0	0
$343$	−2843.86	−0.447679
$344$	381.195	0.0597460
$345$	0	0
$346$	−175.821	−0.0273185
$347$	−2545.48	−0.393800	−0.196900	−	0.980424i	$-0.563087\pi$
$347$	−2545.48	−0.393800	−0.196900	+	0.980424i	$0.563087\pi$
$348$	0	0
$349$	−2847.10	−0.436682	−0.218341	−	0.975873i	$-0.570064\pi$
$349$	−2847.10	−0.436682	−0.218341	+	0.975873i	$0.570064\pi$
$350$	−191.710	−0.0292780
$351$	0	0
$352$	−1253.62	−0.189825
$353$	6141.65	0.926026	0.463013	−	0.886351i	$-0.346768\pi$
$353$	6141.65	0.926026	0.463013	+	0.886351i	$0.346768\pi$
$354$	0	0
$355$	−17259.6	−2.58040
$356$	−5110.99	−0.760905
$357$	0	0
$358$	−527.970	−0.0779444
$359$	8114.47	1.19294	0.596470	−	0.802635i	$-0.296569\pi$
$359$	8114.47	1.19294	0.596470	+	0.802635i	$0.296569\pi$
$360$	0	0
$361$	19311.7	2.81552
$362$	−12.5934	−0.00182844
$363$	0	0
$364$	615.770	0.0886679
$365$	5972.20	0.856436
$366$	0	0
$367$	−5249.54	−0.746659	−0.373329	−	0.927699i	$-0.621784\pi$
$367$	−5249.54	−0.746659	−0.373329	+	0.927699i	$0.621784\pi$
$368$	4075.30	0.577281
$369$	0	0
$370$	1807.52	0.253970
$371$	305.958	0.0428155
$372$	0	0
$373$	2034.61	0.282434	0.141217	−	0.989979i	$-0.454898\pi$
$373$	2034.61	0.282434	0.141217	+	0.989979i	$0.454898\pi$
$374$	−904.036	−0.124991
$375$	0	0
$376$	2690.38	0.369005
$377$	3253.47	0.444462
$378$	0	0
$379$	−416.190	−0.0564070	−0.0282035	−	0.999602i	$-0.508979\pi$
$379$	−416.190	−0.0564070	−0.0282035	+	0.999602i	$0.508979\pi$
$380$	−19379.0	−2.61611
$381$	0	0
$382$	1797.57	0.240764
$383$	−5407.09	−0.721382	−0.360691	−	0.932685i	$-0.617459\pi$
$383$	−5407.09	−0.721382	−0.360691	+	0.932685i	$0.617459\pi$
$384$	0	0
$385$	1060.54	0.140390
$386$	−510.740	−0.0673471
$387$	0	0
$388$	−12583.8	−1.64650
$389$	−4010.55	−0.522732	−0.261366	−	0.965240i	$-0.584173\pi$
$389$	−4010.55	−0.522732	−0.261366	+	0.965240i	$0.584173\pi$
$390$	0	0
$391$	9147.83	1.18319
$392$	−2122.55	−0.273483
$393$	0	0
$394$	389.185	0.0497636
$395$	6299.25	0.802404
$396$	0	0
$397$	−7958.88	−1.00616	−0.503079	−	0.864241i	$-0.667799\pi$
$397$	−7958.88	−1.00616	−0.503079	+	0.864241i	$0.667799\pi$
$398$	156.957	0.0197677
$399$	0	0
$400$	6538.33	0.817291
$401$	−1436.63	−0.178908	−0.0894540	−	0.995991i	$-0.528512\pi$
$401$	−1436.63	−0.178908	−0.0894540	+	0.995991i	$0.528512\pi$
$402$	0	0
$403$	−4761.18	−0.588514
$404$	5446.03	0.670668
$405$	0	0
$406$	309.589	0.0378439
$407$	−4659.60	−0.567488
$408$	0	0
$409$	238.380	0.0288194	0.0144097	−	0.999896i	$-0.495413\pi$
$409$	238.380	0.0288194	0.0144097	+	0.999896i	$0.495413\pi$
$410$	−210.272	−0.0253283
$411$	0	0
$412$	1360.81	0.162724
$413$	823.617	0.0981296
$414$	0	0
$415$	15528.7	1.83680
$416$	1422.32	0.167633
$417$	0	0
$418$	−1086.96	−0.127189
$419$	5476.58	0.638540	0.319270	−	0.947664i	$-0.396562\pi$
$419$	5476.58	0.638540	0.319270	+	0.947664i	$0.396562\pi$
$420$	0	0
$421$	−12944.0	−1.49846	−0.749230	−	0.662310i	$-0.769576\pi$
$421$	−12944.0	−1.49846	−0.749230	+	0.662310i	$0.769576\pi$
$422$	−1325.65	−0.152918
$423$	0	0
$424$	469.456	0.0537708
$425$	14676.6	1.67510
$426$	0	0
$427$	−2095.45	−0.237485
$428$	2197.55	0.248183
$429$	0	0
$430$	−368.433	−0.0413195
$431$	−12822.2	−1.43301	−0.716503	−	0.697584i	$-0.754258\pi$
$431$	−12822.2	−1.43301	−0.716503	+	0.697584i	$0.754258\pi$
$432$	0	0
$433$	15672.2	1.73939	0.869695	−	0.493590i	$-0.164316\pi$
$433$	15672.2	1.73939	0.869695	+	0.493590i	$0.164316\pi$
$434$	−453.057	−0.0501093
$435$	0	0
$436$	14155.6	1.55489
$437$	10998.8	1.20399
$438$	0	0
$439$	−5965.32	−0.648541	−0.324270	−	0.945964i	$-0.605119\pi$
$439$	−5965.32	−0.648541	−0.324270	+	0.945964i	$0.605119\pi$
$440$	1627.27	0.176312
$441$	0	0
$442$	1025.69	0.110378
$443$	−9360.62	−1.00392	−0.501960	−	0.864891i	$-0.667387\pi$
$443$	−9360.62	−1.00392	−0.501960	+	0.864891i	$0.667387\pi$
$444$	0	0
$445$	9987.25	1.06391
$446$	−2605.78	−0.276653
$447$	0	0
$448$	−1906.52	−0.201060
$449$	12653.4	1.32996	0.664979	−	0.746862i	$-0.268440\pi$
$449$	12653.4	1.32996	0.664979	+	0.746862i	$0.268440\pi$
$450$	0	0
$451$	542.059	0.0565955
$452$	8562.87	0.891069
$453$	0	0
$454$	−1905.34	−0.196965
$455$	−1203.26	−0.123977
$456$	0	0
$457$	7693.65	0.787514	0.393757	−	0.919215i	$-0.371175\pi$
$457$	7693.65	0.787514	0.393757	+	0.919215i	$0.371175\pi$
$458$	−2738.34	−0.279376
$459$	0	0
$460$	−8144.49	−0.825519
$461$	−15274.5	−1.54317	−0.771587	−	0.636123i	$-0.780537\pi$
$461$	−15274.5	−1.54317	−0.771587	+	0.636123i	$0.780537\pi$
$462$	0	0
$463$	6114.26	0.613723	0.306861	−	0.951754i	$-0.400721\pi$
$463$	6114.26	0.613723	0.306861	+	0.951754i	$0.400721\pi$
$464$	−10558.6	−1.05641
$465$	0	0
$466$	−1460.63	−0.145198
$467$	−10072.5	−0.998071	−0.499035	−	0.866582i	$-0.666312\pi$
$467$	−10072.5	−0.998071	−0.499035	+	0.866582i	$0.666312\pi$
$468$	0	0
$469$	2058.02	0.202623
$470$	−2600.31	−0.255199
$471$	0	0
$472$	1263.74	0.123238
$473$	949.779	0.0923274
$474$	0	0
$475$	17646.3	1.70456
$476$	−4485.78	−0.431944
$477$	0	0
$478$	−98.6457	−0.00943922
$479$	−6208.85	−0.592254	−0.296127	−	0.955149i	$-0.595695\pi$
$479$	−6208.85	−0.592254	−0.296127	+	0.955149i	$0.595695\pi$
$480$	0	0
$481$	5286.65	0.501145
$482$	−1071.50	−0.101256
$483$	0	0
$484$	8346.36	0.783843
$485$	24589.6	2.30217
$486$	0	0
$487$	−7162.96	−0.666499	−0.333249	−	0.942839i	$-0.608145\pi$
$487$	−7162.96	−0.666499	−0.333249	+	0.942839i	$0.608145\pi$
$488$	−3215.23	−0.298251
$489$	0	0
$490$	2051.49	0.189137
$491$	11880.1	1.09194	0.545968	−	0.837806i	$-0.316162\pi$
$491$	11880.1	1.09194	0.545968	+	0.837806i	$0.316162\pi$
$492$	0	0
$493$	−23701.0	−2.16519
$494$	1233.24	0.112320
$495$	0	0
$496$	15451.7	1.39879
$497$	−4803.57	−0.433541
$498$	0	0
$499$	−12912.9	−1.15844	−0.579220	−	0.815171i	$-0.696643\pi$
$499$	−12912.9	−1.15844	−0.579220	+	0.815171i	$0.696643\pi$
$500$	1907.03	0.170570
$501$	0	0
$502$	2371.78	0.210872
$503$	−7692.08	−0.681855	−0.340927	−	0.940090i	$-0.610741\pi$
$503$	−7692.08	−0.681855	−0.340927	+	0.940090i	$0.610741\pi$
$504$	0	0
$505$	−10641.9	−0.937741
$506$	−456.821	−0.0401347
$507$	0	0
$508$	3499.05	0.305600
$509$	−2011.47	−0.175161	−0.0875804	−	0.996157i	$-0.527913\pi$
$509$	−2011.47	−0.175161	−0.0875804	+	0.996157i	$0.527913\pi$
$510$	0	0
$511$	1662.15	0.143892
$512$	−7748.94	−0.668863
$513$	0	0
$514$	2777.47	0.238344
$515$	−2659.11	−0.227523
$516$	0	0
$517$	6703.32	0.570235
$518$	503.059	0.0426702
$519$	0	0
$520$	−1846.26	−0.155700
$521$	−825.418	−0.0694093	−0.0347046	−	0.999398i	$-0.511049\pi$
$521$	−825.418	−0.0694093	−0.0347046	+	0.999398i	$0.511049\pi$
$522$	0	0
$523$	14927.4	1.24805	0.624025	−	0.781405i	$-0.285496\pi$
$523$	14927.4	1.24805	0.624025	+	0.781405i	$0.285496\pi$
$524$	14612.5	1.21823
$525$	0	0
$526$	−2631.41	−0.218127
$527$	34684.4	2.86694
$528$	0	0
$529$	−7544.48	−0.620078
$530$	−453.740	−0.0371872
$531$	0	0
$532$	−5393.45	−0.439541
$533$	−615.005	−0.0499791
$534$	0	0
$535$	−4294.17	−0.347015
$536$	3157.79	0.254469
$537$	0	0
$538$	−752.824	−0.0603282
$539$	−5288.52	−0.422621
$540$	0	0
$541$	−1087.79	−0.0864468	−0.0432234	−	0.999065i	$-0.513763\pi$
$541$	−1087.79	−0.0864468	−0.0432234	+	0.999065i	$0.513763\pi$
$542$	−1294.34	−0.102577
$543$	0	0
$544$	−10361.4	−0.816620
$545$	−27661.1	−2.17408
$546$	0	0
$547$	12629.1	0.987169	0.493584	−	0.869698i	$-0.335686\pi$
$547$	12629.1	0.987169	0.493584	+	0.869698i	$0.335686\pi$
$548$	12707.0	0.990540
$549$	0	0
$550$	−732.915	−0.0568211
$551$	−28496.7	−2.20327
$552$	0	0
$553$	1753.17	0.134814
$554$	1991.19	0.152703
$555$	0	0
$556$	−10274.6	−0.783707
$557$	−3111.70	−0.236709	−0.118354	−	0.992971i	$-0.537762\pi$
$557$	−3111.70	−0.236709	−0.118354	+	0.992971i	$0.537762\pi$
$558$	0	0
$559$	−1077.59	−0.0815336
$560$	3904.99	0.294671
$561$	0	0
$562$	−2935.54	−0.220335
$563$	−14980.3	−1.12139	−0.560695	−	0.828022i	$-0.689466\pi$
$563$	−14980.3	−1.12139	−0.560695	+	0.828022i	$0.689466\pi$
$564$	0	0
$565$	−16732.5	−1.24591
$566$	1365.06	0.101374
$567$	0	0
$568$	−7370.52	−0.544472
$569$	6716.76	0.494870	0.247435	−	0.968904i	$-0.420412\pi$
$569$	6716.76	0.494870	0.247435	+	0.968904i	$0.420412\pi$
$570$	0	0
$571$	−8345.27	−0.611626	−0.305813	−	0.952092i	$-0.598928\pi$
$571$	−8345.27	−0.611626	−0.305813	+	0.952092i	$0.598928\pi$
$572$	2354.12	0.172081
$573$	0	0
$574$	−58.5218	−0.00425549
$575$	7416.27	0.537878
$576$	0	0
$577$	17850.6	1.28792	0.643961	−	0.765058i	$-0.277290\pi$
$577$	17850.6	1.28792	0.643961	+	0.765058i	$0.277290\pi$
$578$	−5444.20	−0.391780
$579$	0	0
$580$	21101.4	1.51067
$581$	4321.85	0.308606
$582$	0	0
$583$	1169.69	0.0830937
$584$	2550.37	0.180711
$585$	0	0
$586$	−1447.43	−0.102036
$587$	15914.5	1.11902	0.559509	−	0.828824i	$-0.310990\pi$
$587$	15914.5	1.11902	0.559509	+	0.828824i	$0.310990\pi$
$588$	0	0
$589$	41702.6	2.91736
$590$	−1221.43	−0.0852300
$591$	0	0
$592$	−17157.0	−1.19113
$593$	−17229.8	−1.19316	−0.596579	−	0.802555i	$-0.703474\pi$
$593$	−17229.8	−1.19316	−0.596579	+	0.802555i	$0.703474\pi$
$594$	0	0
$595$	8765.54	0.603953
$596$	−2318.92	−0.159374
$597$	0	0
$598$	518.296	0.0354427
$599$	−12512.4	−0.853494	−0.426747	−	0.904371i	$-0.640341\pi$
$599$	−12512.4	−0.853494	−0.426747	+	0.904371i	$0.640341\pi$
$600$	0	0
$601$	−5387.84	−0.365682	−0.182841	−	0.983143i	$-0.558529\pi$
$601$	−5387.84	−0.365682	−0.182841	+	0.983143i	$0.558529\pi$
$602$	−102.540	−0.00694222
$603$	0	0
$604$	−19529.6	−1.31564
$605$	−16309.4	−1.09599
$606$	0	0
$607$	−8669.05	−0.579680	−0.289840	−	0.957075i	$-0.593602\pi$
$607$	−8669.05	−0.579680	−0.289840	+	0.957075i	$0.593602\pi$
$608$	−12458.0	−0.830982
$609$	0	0
$610$	3107.59	0.206266
$611$	−7605.40	−0.503570
$612$	0	0
$613$	−17438.5	−1.14900	−0.574498	−	0.818506i	$-0.694803\pi$
$613$	−17438.5	−1.14900	−0.574498	+	0.818506i	$0.694803\pi$
$614$	3475.12	0.228411
$615$	0	0
$616$	452.893	0.0296227
$617$	−8531.50	−0.556670	−0.278335	−	0.960484i	$-0.589782\pi$
$617$	−8531.50	−0.556670	−0.278335	+	0.960484i	$0.589782\pi$
$618$	0	0
$619$	25417.1	1.65040	0.825202	−	0.564838i	$-0.191061\pi$
$619$	25417.1	1.65040	0.825202	+	0.564838i	$0.191061\pi$
$620$	−30880.2	−2.00029
$621$	0	0
$622$	−977.530	−0.0630151
$623$	2779.59	0.178751
$624$	0	0
$625$	−17361.5	−1.11114
$626$	−2254.99	−0.143974
$627$	0	0
$628$	−7585.49	−0.481997
$629$	−38512.4	−2.44132
$630$	0	0
$631$	−6793.70	−0.428610	−0.214305	−	0.976767i	$-0.568749\pi$
$631$	−6793.70	−0.428610	−0.214305	+	0.976767i	$0.568749\pi$
$632$	2690.03	0.169310
$633$	0	0
$634$	−2158.26	−0.135198
$635$	−6837.39	−0.427297
$636$	0	0
$637$	6000.21	0.373214
$638$	1183.57	0.0734452
$639$	0	0
$640$	12253.1	0.756789
$641$	21616.4	1.33197	0.665987	−	0.745963i	$-0.268011\pi$
$641$	21616.4	1.33197	0.665987	+	0.745963i	$0.268011\pi$
$642$	0	0
$643$	−10785.1	−0.661468	−0.330734	−	0.943724i	$-0.607296\pi$
$643$	−10785.1	−0.661468	−0.330734	+	0.943724i	$0.607296\pi$
$644$	−2266.72	−0.138698
$645$	0	0
$646$	−8983.93	−0.547164
$647$	17023.1	1.03439	0.517193	−	0.855869i	$-0.326977\pi$
$647$	17023.1	1.03439	0.517193	+	0.855869i	$0.326977\pi$
$648$	0	0
$649$	3148.72	0.190444
$650$	831.545	0.0501783
$651$	0	0
$652$	24511.3	1.47230
$653$	18741.5	1.12314	0.561572	−	0.827428i	$-0.310197\pi$
$653$	18741.5	1.12314	0.561572	+	0.827428i	$0.310197\pi$
$654$	0	0
$655$	−28554.0	−1.70335
$656$	1995.91	0.118791
$657$	0	0
$658$	−723.703	−0.0428767
$659$	−18984.9	−1.12222	−0.561112	−	0.827740i	$-0.689626\pi$
$659$	−18984.9	−1.12222	−0.561112	+	0.827740i	$0.689626\pi$
$660$	0	0
$661$	2812.93	0.165523	0.0827613	−	0.996569i	$-0.473626\pi$
$661$	2812.93	0.165523	0.0827613	+	0.996569i	$0.473626\pi$
$662$	449.682	0.0264009
$663$	0	0
$664$	6631.37	0.387571
$665$	10539.2	0.614575
$666$	0	0
$667$	−11976.4	−0.695245
$668$	14834.4	0.859219
$669$	0	0
$670$	−3052.07	−0.175988
$671$	−8011.02	−0.460897
$672$	0	0
$673$	−1800.14	−0.103106	−0.0515530	−	0.998670i	$-0.516417\pi$
$673$	−1800.14	−0.103106	−0.0515530	+	0.998670i	$0.516417\pi$
$674$	−4048.84	−0.231388
$675$	0	0
$676$	14530.8	0.826742
$677$	383.877	0.0217926	0.0108963	−	0.999941i	$-0.496532\pi$
$677$	383.877	0.0217926	0.0108963	+	0.999941i	$0.496532\pi$
$678$	0	0
$679$	6843.62	0.386795
$680$	13449.7	0.758488
$681$	0	0
$682$	−1732.06	−0.0972492
$683$	1041.26	0.0583350	0.0291675	−	0.999575i	$-0.490714\pi$
$683$	1041.26	0.0583350	0.0291675	+	0.999575i	$0.490714\pi$
$684$	0	0
$685$	−24830.4	−1.38499
$686$	1173.78	0.0653284
$687$	0	0
$688$	3497.16	0.193791
$689$	−1327.10	−0.0733794
$690$	0	0
$691$	1215.27	0.0669044	0.0334522	−	0.999440i	$-0.489350\pi$
$691$	1215.27	0.0669044	0.0334522	+	0.999440i	$0.489350\pi$
$692$	−3335.29	−0.183220
$693$	0	0
$694$	1050.63	0.0574659
$695$	20077.4	1.09580
$696$	0	0
$697$	4480.21	0.243472
$698$	1175.12	0.0637236
$699$	0	0
$700$	−3636.69	−0.196363
$701$	−31620.1	−1.70367	−0.851837	−	0.523808i	$-0.824511\pi$
$701$	−31620.1	−1.70367	−0.851837	+	0.523808i	$0.824511\pi$
$702$	0	0
$703$	−46305.1	−2.48425
$704$	−7288.73	−0.390205
$705$	0	0
$706$	−2534.93	−0.135132
$707$	−2961.80	−0.157553
$708$	0	0
$709$	−15461.0	−0.818969	−0.409484	−	0.912317i	$-0.634291\pi$
$709$	−15461.0	−0.818969	−0.409484	+	0.912317i	$0.634291\pi$
$710$	7123.76	0.376549
$711$	0	0
$712$	4264.96	0.224489
$713$	17526.5	0.920578
$714$	0	0
$715$	−4600.11	−0.240608
$716$	−10015.5	−0.522759
$717$	0	0
$718$	−3349.20	−0.174082
$719$	4823.69	0.250199	0.125100	−	0.992144i	$-0.460075\pi$
$719$	4823.69	0.250199	0.125100	+	0.992144i	$0.460075\pi$
$720$	0	0
$721$	−740.068	−0.0382269
$722$	−7970.76	−0.410860
$723$	0	0
$724$	−238.894	−0.0122630
$725$	−19214.7	−0.984299
$726$	0	0
$727$	−4961.87	−0.253130	−0.126565	−	0.991958i	$-0.540395\pi$
$727$	−4961.87	−0.253130	−0.126565	+	0.991958i	$0.540395\pi$
$728$	−513.839	−0.0261596
$729$	0	0
$730$	−2464.99	−0.124977
$731$	7850.08	0.397190
$732$	0	0
$733$	5307.77	0.267458	0.133729	−	0.991018i	$-0.457305\pi$
$733$	5307.77	0.267458	0.133729	+	0.991018i	$0.457305\pi$
$734$	2166.71	0.108958
$735$	0	0
$736$	−5235.75	−0.262218
$737$	7867.89	0.393240
$738$	0	0
$739$	−15832.7	−0.788111	−0.394055	−	0.919087i	$-0.628928\pi$
$739$	−15832.7	−0.788111	−0.394055	+	0.919087i	$0.628928\pi$
$740$	34288.3	1.70333
$741$	0	0
$742$	−126.282	−0.00624792
$743$	−15595.3	−0.770033	−0.385017	−	0.922910i	$-0.625804\pi$
$743$	−15595.3	−0.770033	−0.385017	+	0.922910i	$0.625804\pi$
$744$	0	0
$745$	4531.34	0.222840
$746$	−839.771	−0.0412148
$747$	0	0
$748$	−17149.3	−0.838292
$749$	−1195.13	−0.0583030
$750$	0	0
$751$	29970.2	1.45623	0.728113	−	0.685457i	$-0.240397\pi$
$751$	29970.2	1.45623	0.728113	+	0.685457i	$0.240397\pi$
$752$	24682.2	1.19690
$753$	0	0
$754$	−1342.85	−0.0648589
$755$	38162.3	1.83956
$756$	0	0
$757$	23367.7	1.12195	0.560973	−	0.827834i	$-0.310427\pi$
$757$	23367.7	1.12195	0.560973	+	0.827834i	$0.310427\pi$
$758$	171.780	0.00823130
$759$	0	0
$760$	16171.2	0.771828
$761$	−156.995	−0.00747838	−0.00373919	−	0.999993i	$-0.501190\pi$
$761$	−156.995	−0.00747838	−0.00373919	+	0.999993i	$0.501190\pi$
$762$	0	0
$763$	−7698.47	−0.365273
$764$	34099.6	1.61476
$765$	0	0
$766$	2231.74	0.105269
$767$	−3572.45	−0.168180
$768$	0	0
$769$	39952.0	1.87348	0.936741	−	0.350024i	$-0.113826\pi$
$769$	39952.0	1.87348	0.936741	+	0.350024i	$0.113826\pi$
$770$	−437.730	−0.0204866
$771$	0	0
$772$	−9688.61	−0.451685
$773$	34478.7	1.60429	0.802143	−	0.597133i	$-0.203693\pi$
$773$	34478.7	1.60429	0.802143	+	0.597133i	$0.203693\pi$
$774$	0	0
$775$	28119.1	1.30332
$776$	10500.7	0.485766
$777$	0	0
$778$	1655.33	0.0762807
$779$	5386.76	0.247754
$780$	0	0
$781$	−18364.3	−0.841390
$782$	−3775.70	−0.172658
$783$	0	0
$784$	−19472.8	−0.887061
$785$	14822.6	0.673938
$786$	0	0
$787$	40427.7	1.83112	0.915559	−	0.402183i	$-0.131749\pi$
$787$	40427.7	1.83112	0.915559	+	0.402183i	$0.131749\pi$
$788$	7382.75	0.333756
$789$	0	0
$790$	−2599.97	−0.117092
$791$	−4656.88	−0.209329
$792$	0	0
$793$	9089.08	0.407015
$794$	3284.97	0.146825
$795$	0	0
$796$	2977.43	0.132578
$797$	−7136.45	−0.317172	−0.158586	−	0.987345i	$-0.550693\pi$
$797$	−7136.45	−0.317172	−0.158586	+	0.987345i	$0.550693\pi$
$798$	0	0
$799$	55404.1	2.45314
$800$	−8400.13	−0.371237
$801$	0	0
$802$	592.961	0.0261075
$803$	6354.46	0.279258
$804$	0	0
$805$	4429.34	0.193930
$806$	1965.14	0.0858800
$807$	0	0
$808$	−4544.53	−0.197866
$809$	6891.43	0.299493	0.149746	−	0.988724i	$-0.452154\pi$
$809$	6891.43	0.299493	0.149746	+	0.988724i	$0.452154\pi$
$810$	0	0
$811$	−3450.04	−0.149380	−0.0746901	−	0.997207i	$-0.523797\pi$
$811$	−3450.04	−0.149380	−0.0746901	+	0.997207i	$0.523797\pi$
$812$	5872.82	0.253812
$813$	0	0
$814$	1923.22	0.0828117
$815$	−47896.9	−2.05859
$816$	0	0
$817$	9438.50	0.404175
$818$	−98.3897	−0.00420552
$819$	0	0
$820$	−3988.82	−0.169873
$821$	27575.2	1.17221	0.586104	−	0.810236i	$-0.300661\pi$
$821$	27575.2	1.17221	0.586104	+	0.810236i	$0.300661\pi$
$822$	0	0
$823$	−7692.71	−0.325821	−0.162911	−	0.986641i	$-0.552088\pi$
$823$	−7692.71	−0.325821	−0.162911	+	0.986641i	$0.552088\pi$
$824$	−1135.55	−0.0480081
$825$	0	0
$826$	−339.942	−0.0143197
$827$	−8379.61	−0.352343	−0.176171	−	0.984360i	$-0.556371\pi$
$827$	−8379.61	−0.352343	−0.176171	+	0.984360i	$0.556371\pi$
$828$	0	0
$829$	−20770.1	−0.870176	−0.435088	−	0.900388i	$-0.643283\pi$
$829$	−20770.1	−0.870176	−0.435088	+	0.900388i	$0.643283\pi$
$830$	−6409.36	−0.268039
$831$	0	0
$832$	8269.58	0.344587
$833$	−43710.5	−1.81810
$834$	0	0
$835$	−28987.4	−1.20138
$836$	−20619.4	−0.853035
$837$	0	0
$838$	−2260.42	−0.0931801
$839$	−16916.1	−0.696076	−0.348038	−	0.937480i	$-0.613152\pi$
$839$	−16916.1	−0.696076	−0.348038	+	0.937480i	$0.613152\pi$
$840$	0	0
$841$	6640.52	0.272275
$842$	5342.55	0.218665
$843$	0	0
$844$	−25147.2	−1.02560
$845$	−28394.3	−1.15597
$846$	0	0
$847$	−4539.13	−0.184140
$848$	4306.89	0.174410
$849$	0	0
$850$	−6057.67	−0.244443
$851$	−19460.8	−0.783911
$852$	0	0
$853$	34997.3	1.40479	0.702394	−	0.711789i	$-0.252115\pi$
$853$	34997.3	1.40479	0.702394	+	0.711789i	$0.252115\pi$
$854$	864.885	0.0346554
$855$	0	0
$856$	−1833.78	−0.0732212
$857$	15759.6	0.628164	0.314082	−	0.949396i	$-0.398303\pi$
$857$	15759.6	0.628164	0.314082	+	0.949396i	$0.398303\pi$
$858$	0	0
$859$	−14662.0	−0.582376	−0.291188	−	0.956666i	$-0.594051\pi$
$859$	−14662.0	−0.582376	−0.291188	+	0.956666i	$0.594051\pi$
$860$	−6989.08	−0.277123
$861$	0	0
$862$	5292.29	0.209114
$863$	−19914.7	−0.785521	−0.392761	−	0.919641i	$-0.628480\pi$
$863$	−19914.7	−0.785521	−0.392761	+	0.919641i	$0.628480\pi$
$864$	0	0
$865$	6517.39	0.256182
$866$	−6468.58	−0.253823
$867$	0	0
$868$	−8594.39	−0.336074
$869$	6702.44	0.261640
$870$	0	0
$871$	−8926.69	−0.347267
$872$	−11812.4	−0.458736
$873$	0	0
$874$	−4539.70	−0.175695
$875$	−1037.13	−0.0400702
$876$	0	0
$877$	−24939.0	−0.960240	−0.480120	−	0.877203i	$-0.659407\pi$
$877$	−24939.0	−0.960240	−0.480120	+	0.877203i	$0.659407\pi$
$878$	2462.15	0.0946395
$879$	0	0
$880$	14929.0	0.571881
$881$	−36544.1	−1.39750	−0.698752	−	0.715364i	$-0.746261\pi$
$881$	−36544.1	−1.39750	−0.698752	+	0.715364i	$0.746261\pi$
$882$	0	0
$883$	23736.9	0.904656	0.452328	−	0.891852i	$-0.350594\pi$
$883$	23736.9	0.904656	0.452328	+	0.891852i	$0.350594\pi$
$884$	19457.2	0.740289
$885$	0	0
$886$	3863.53	0.146499
$887$	−24749.3	−0.936868	−0.468434	−	0.883499i	$-0.655182\pi$
$887$	−24749.3	−0.936868	−0.468434	+	0.883499i	$0.655182\pi$
$888$	0	0
$889$	−1902.94	−0.0717914
$890$	−4122.17	−0.155253
$891$	0	0
$892$	−49431.0	−1.85546
$893$	66614.8	2.49628
$894$	0	0
$895$	19571.0	0.730933
$896$	3410.20	0.127150
$897$	0	0
$898$	−5222.61	−0.194077
$899$	−45409.1	−1.68463
$900$	0	0
$901$	9667.69	0.357467
$902$	−223.731	−0.00825880
$903$	0	0
$904$	−7145.43	−0.262891
$905$	466.815	0.0171464
$906$	0	0
$907$	14386.4	0.526671	0.263336	−	0.964704i	$-0.415177\pi$
$907$	14386.4	0.526671	0.263336	+	0.964704i	$0.415177\pi$
$908$	−36143.9	−1.32101
$909$	0	0
$910$	496.637	0.0180916
$911$	45217.0	1.64446	0.822232	−	0.569152i	$-0.192728\pi$
$911$	45217.0	1.64446	0.822232	+	0.569152i	$0.192728\pi$
$912$	0	0
$913$	16522.6	0.598925
$914$	−3175.50	−0.114919
$915$	0	0
$916$	−51945.6	−1.87373
$917$	−7946.97	−0.286185
$918$	0	0
$919$	3652.87	0.131117	0.0655587	−	0.997849i	$-0.479117\pi$
$919$	3652.87	0.131117	0.0655587	+	0.997849i	$0.479117\pi$
$920$	6796.31	0.243552
$921$	0	0
$922$	6304.44	0.225191
$923$	20835.6	0.743025
$924$	0	0
$925$	−31222.5	−1.10983
$926$	−2523.62	−0.0895586
$927$	0	0
$928$	13565.2	0.479850
$929$	−10029.4	−0.354202	−0.177101	−	0.984193i	$-0.556672\pi$
$929$	−10029.4	−0.354202	−0.177101	+	0.984193i	$0.556672\pi$
$930$	0	0
$931$	−52555.1	−1.85008
$932$	−27707.8	−0.973818
$933$	0	0
$934$	4157.35	0.145645
$935$	33511.0	1.17212
$936$	0	0
$937$	18935.1	0.660173	0.330086	−	0.943951i	$-0.392922\pi$
$937$	18935.1	0.660173	0.330086	+	0.943951i	$0.392922\pi$
$938$	−849.433	−0.0295682
$939$	0	0
$940$	−49327.3	−1.71157
$941$	−36838.0	−1.27618	−0.638089	−	0.769963i	$-0.720275\pi$
$941$	−36838.0	−1.27618	−0.638089	+	0.769963i	$0.720275\pi$
$942$	0	0
$943$	2263.91	0.0781793
$944$	11593.9	0.399733
$945$	0	0
$946$	−392.015	−0.0134730
$947$	−23130.4	−0.793703	−0.396852	−	0.917883i	$-0.629897\pi$
$947$	−23130.4	−0.793703	−0.396852	+	0.917883i	$0.629897\pi$
$948$	0	0
$949$	−7209.60	−0.246611
$950$	−7283.40	−0.248742
$951$	0	0
$952$	3743.24	0.127436
$953$	21422.8	0.728178	0.364089	−	0.931364i	$-0.381380\pi$
$953$	21422.8	0.728178	0.364089	+	0.931364i	$0.381380\pi$
$954$	0	0
$955$	−66633.0	−2.25779
$956$	−1871.28	−0.0633072
$957$	0	0
$958$	2562.66	0.0864257
$959$	−6910.64	−0.232697
$960$	0	0
$961$	36661.5	1.23062
$962$	−2182.03	−0.0731304
$963$	0	0
$964$	−20326.1	−0.679109
$965$	18932.2	0.631555
$966$	0	0
$967$	−2205.71	−0.0733515	−0.0366758	−	0.999327i	$-0.511677\pi$
$967$	−2205.71	−0.0733515	−0.0366758	+	0.999327i	$0.511677\pi$
$968$	−6964.76	−0.231256
$969$	0	0
$970$	−10149.2	−0.335949
$971$	−43461.6	−1.43640	−0.718202	−	0.695835i	$-0.755035\pi$
$971$	−43461.6	−1.43640	−0.718202	+	0.695835i	$0.755035\pi$
$972$	0	0
$973$	5587.81	0.184108
$974$	2956.46	0.0972600
$975$	0	0
$976$	−29497.2	−0.967400
$977$	−32410.6	−1.06132	−0.530659	−	0.847586i	$-0.678055\pi$
$977$	−32410.6	−1.06132	−0.530659	+	0.847586i	$0.678055\pi$
$978$	0	0
$979$	10626.5	0.346909
$980$	38916.3	1.26851
$981$	0	0
$982$	−4903.42	−0.159343
$983$	18851.1	0.611654	0.305827	−	0.952087i	$-0.401067\pi$
$983$	18851.1	0.611654	0.305827	+	0.952087i	$0.401067\pi$
$984$	0	0
$985$	−14426.4	−0.466664
$986$	9782.42	0.315959
$987$	0	0
$988$	23394.2	0.753309
$989$	3966.75	0.127538
$990$	0	0
$991$	−19523.1	−0.625802	−0.312901	−	0.949786i	$-0.601301\pi$
$991$	−19523.1	−0.625802	−0.312901	+	0.949786i	$0.601301\pi$
$992$	−19851.6	−0.635371
$993$	0	0
$994$	1982.64	0.0632652
$995$	−5818.11	−0.185373
$996$	0	0
$997$	28901.4	0.918070	0.459035	−	0.888418i	$-0.348195\pi$
$997$	28901.4	0.918070	0.459035	+	0.888418i	$0.348195\pi$
$998$	5329.72	0.169047
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.18		37
3.2	odd	2		239.4.a.b.1.20	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.20	✓	37	3.2	odd	2
2151.4.a.f.1.18		37	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-0.412743 q^{2} -7.82964 q^{4} +15.2997 q^{5} +4.25811 q^{7} +6.53358 q^{8} +O(q^{10})\)	\(q-0.412743 q^{2} -7.82964 q^{4} +15.2997 q^{5} +4.25811 q^{7} +6.53358 q^{8} -6.31484 q^{10} +16.2790 q^{11} -18.4697 q^{13} -1.75751 q^{14} +59.9405 q^{16} +134.548 q^{17} +161.774 q^{19} -119.791 q^{20} -6.71903 q^{22} +67.9891 q^{23} +109.080 q^{25} +7.62323 q^{26} -33.3395 q^{28} -176.152 q^{29} +257.784 q^{31} -77.0087 q^{32} -55.5340 q^{34} +65.1478 q^{35} -286.234 q^{37} -66.7709 q^{38} +99.9617 q^{40} +33.2981 q^{41} +58.3439 q^{43} -127.458 q^{44} -28.0620 q^{46} +411.778 q^{47} -324.868 q^{49} -45.0222 q^{50} +144.611 q^{52} +71.8529 q^{53} +249.063 q^{55} +27.8207 q^{56} +72.7055 q^{58} +193.423 q^{59} -492.108 q^{61} -106.399 q^{62} -447.739 q^{64} -282.580 q^{65} +483.316 q^{67} -1053.47 q^{68} -26.8893 q^{70} -1128.10 q^{71} +390.348 q^{73} +118.141 q^{74} -1266.63 q^{76} +69.3177 q^{77} +411.724 q^{79} +917.070 q^{80} -13.7436 q^{82} +1014.97 q^{83} +2058.55 q^{85} -24.0811 q^{86} +106.360 q^{88} +652.775 q^{89} -78.6459 q^{91} -532.330 q^{92} -169.959 q^{94} +2475.08 q^{95} +1607.19 q^{97} +134.087 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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