LMFDB - Embedded newform 2151.4.a.g.1.41

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			1.41
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)

$f(q)$	$=$	$q+2.12161 q^{2} -3.49876 q^{4} -4.52878 q^{5} +14.7886 q^{7} -24.3959 q^{8} +O(q^{10})$	\(q+2.12161 q^{2} -3.49876 q^{4} -4.52878 q^{5} +14.7886 q^{7} -24.3959 q^{8} -9.60831 q^{10} +61.2887 q^{11} -8.40363 q^{13} +31.3757 q^{14} -23.7686 q^{16} -127.628 q^{17} -43.6873 q^{19} +15.8451 q^{20} +130.031 q^{22} +165.626 q^{23} -104.490 q^{25} -17.8292 q^{26} -51.7418 q^{28} -212.984 q^{29} +213.376 q^{31} +144.740 q^{32} -270.778 q^{34} -66.9744 q^{35} +56.0146 q^{37} -92.6875 q^{38} +110.484 q^{40} +328.931 q^{41} +33.0560 q^{43} -214.435 q^{44} +351.394 q^{46} +259.011 q^{47} -124.297 q^{49} -221.688 q^{50} +29.4023 q^{52} -150.446 q^{53} -277.563 q^{55} -360.782 q^{56} -451.870 q^{58} -807.354 q^{59} -437.380 q^{61} +452.702 q^{62} +497.230 q^{64} +38.0582 q^{65} +641.805 q^{67} +446.541 q^{68} -142.094 q^{70} -417.719 q^{71} +236.933 q^{73} +118.841 q^{74} +152.851 q^{76} +906.375 q^{77} -67.8200 q^{79} +107.643 q^{80} +697.864 q^{82} -13.7102 q^{83} +578.000 q^{85} +70.1320 q^{86} -1495.19 q^{88} -1497.63 q^{89} -124.278 q^{91} -579.486 q^{92} +549.521 q^{94} +197.850 q^{95} +471.860 q^{97} -263.710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.12161	0.750103	0.375052	−	0.927004i	$-0.377625\pi$
$2$	2.12161	0.750103	0.375052	+	0.927004i	$0.377625\pi$
$3$	0	0
$3$	0	0
$4$	−3.49876	−0.437345
$5$	−4.52878	−0.405066	−0.202533	−	0.979275i	$-0.564917\pi$
$5$	−4.52878	−0.405066	−0.202533	+	0.979275i	$0.564917\pi$
$6$	0	0
$7$	14.7886	0.798510	0.399255	−	0.916840i	$-0.369269\pi$
$7$	14.7886	0.798510	0.399255	+	0.916840i	$0.369269\pi$
$8$	−24.3959	−1.07816
$9$	0	0
$10$	−9.60831	−0.303842
$11$	61.2887	1.67993	0.839965	−	0.542640i	$-0.182575\pi$
$11$	61.2887	1.67993	0.839965	+	0.542640i	$0.182575\pi$
$12$	0	0
$13$	−8.40363	−0.179288	−0.0896441	−	0.995974i	$-0.528573\pi$
$13$	−8.40363	−0.179288	−0.0896441	+	0.995974i	$0.528573\pi$
$14$	31.3757	0.598965
$15$	0	0
$16$	−23.7686	−0.371384
$17$	−127.628	−1.82085	−0.910423	−	0.413678i	$-0.864244\pi$
$17$	−127.628	−1.82085	−0.910423	+	0.413678i	$0.864244\pi$
$18$	0	0
$19$	−43.6873	−0.527503	−0.263751	−	0.964591i	$-0.584960\pi$
$19$	−43.6873	−0.527503	−0.263751	+	0.964591i	$0.584960\pi$
$20$	15.8451	0.177154
$21$	0	0
$22$	130.031	1.26012
$23$	165.626	1.50154	0.750770	−	0.660564i	$-0.229683\pi$
$23$	165.626	1.50154	0.750770	+	0.660564i	$0.229683\pi$
$24$	0	0
$25$	−104.490	−0.835921
$26$	−17.8292	−0.134485
$27$	0	0
$28$	−51.7418	−0.349225
$29$	−212.984	−1.36380	−0.681899	−	0.731446i	$-0.738846\pi$
$29$	−212.984	−1.36380	−0.681899	+	0.731446i	$0.738846\pi$
$30$	0	0
$31$	213.376	1.23624	0.618122	−	0.786082i	$-0.287894\pi$
$31$	213.376	1.23624	0.618122	+	0.786082i	$0.287894\pi$
$32$	144.740	0.799581
$33$	0	0
$34$	−270.778	−1.36582
$35$	−66.9744	−0.323450
$36$	0	0
$37$	56.0146	0.248885	0.124443	−	0.992227i	$-0.460286\pi$
$37$	56.0146	0.248885	0.124443	+	0.992227i	$0.460286\pi$
$38$	−92.6875	−0.395682
$39$	0	0
$40$	110.484	0.436725
$41$	328.931	1.25294	0.626468	−	0.779447i	$-0.284500\pi$
$41$	328.931	1.25294	0.626468	+	0.779447i	$0.284500\pi$
$42$	0	0
$43$	33.0560	0.117232	0.0586161	−	0.998281i	$-0.481331\pi$
$43$	33.0560	0.117232	0.0586161	+	0.998281i	$0.481331\pi$
$44$	−214.435	−0.734710
$45$	0	0
$46$	351.394	1.12631
$47$	259.011	0.803843	0.401922	−	0.915674i	$-0.368342\pi$
$47$	259.011	0.803843	0.401922	+	0.915674i	$0.368342\pi$
$48$	0	0
$49$	−124.297	−0.362381
$50$	−221.688	−0.627027
$51$	0	0
$52$	29.4023	0.0784108
$53$	−150.446	−0.389913	−0.194957	−	0.980812i	$-0.562457\pi$
$53$	−150.446	−0.389913	−0.194957	+	0.980812i	$0.562457\pi$
$54$	0	0
$55$	−277.563	−0.680483
$56$	−360.782	−0.860920
$57$	0	0
$58$	−451.870	−1.02299
$59$	−807.354	−1.78150	−0.890750	−	0.454494i	$-0.849820\pi$
$59$	−807.354	−1.78150	−0.890750	+	0.454494i	$0.849820\pi$
$60$	0	0
$61$	−437.380	−0.918044	−0.459022	−	0.888425i	$-0.651800\pi$
$61$	−437.380	−0.918044	−0.459022	+	0.888425i	$0.651800\pi$
$62$	452.702	0.927310
$63$	0	0
$64$	497.230	0.971152
$65$	38.0582	0.0726236
$66$	0	0
$67$	641.805	1.17028	0.585141	−	0.810931i	$-0.301039\pi$
$67$	641.805	1.17028	0.585141	+	0.810931i	$0.301039\pi$
$68$	446.541	0.796338
$69$	0	0
$70$	−142.094	−0.242621
$71$	−417.719	−0.698226	−0.349113	−	0.937081i	$-0.613517\pi$
$71$	−417.719	−0.698226	−0.349113	+	0.937081i	$0.613517\pi$
$72$	0	0
$73$	236.933	0.379875	0.189937	−	0.981796i	$-0.439171\pi$
$73$	236.933	0.379875	0.189937	+	0.981796i	$0.439171\pi$
$74$	118.841	0.186690
$75$	0	0
$76$	152.851	0.230701
$77$	906.375	1.34144
$78$	0	0
$79$	−67.8200	−0.0965867	−0.0482934	−	0.998833i	$-0.515378\pi$
$79$	−67.8200	−0.0965867	−0.0482934	+	0.998833i	$0.515378\pi$
$80$	107.643	0.150435
$81$	0	0
$82$	697.864	0.939832
$83$	−13.7102	−0.0181312	−0.00906559	−	0.999959i	$-0.502886\pi$
$83$	−13.7102	−0.0181312	−0.00906559	+	0.999959i	$0.502886\pi$
$84$	0	0
$85$	578.000	0.737563
$86$	70.1320	0.0879363
$87$	0	0
$88$	−1495.19	−1.81123
$89$	−1497.63	−1.78369	−0.891847	−	0.452337i	$-0.850590\pi$
$89$	−1497.63	−1.78369	−0.891847	+	0.452337i	$0.850590\pi$
$90$	0	0
$91$	−124.278	−0.143163
$92$	−579.486	−0.656691
$93$	0	0
$94$	549.521	0.602966
$95$	197.850	0.213674
$96$	0	0
$97$	471.860	0.493919	0.246959	−	0.969026i	$-0.420569\pi$
$97$	471.860	0.493919	0.246959	+	0.969026i	$0.420569\pi$
$98$	−263.710	−0.271823
$99$	0	0
$100$	365.586	0.365586
$101$	1253.92	1.23534	0.617670	−	0.786437i	$-0.288077\pi$
$101$	1253.92	1.23534	0.617670	+	0.786437i	$0.288077\pi$
$102$	0	0
$103$	−664.452	−0.635635	−0.317817	−	0.948152i	$-0.602950\pi$
$103$	−664.452	−0.635635	−0.317817	+	0.948152i	$0.602950\pi$
$104$	205.014	0.193301
$105$	0	0
$106$	−319.189	−0.292475
$107$	−753.874	−0.681119	−0.340560	−	0.940223i	$-0.610616\pi$
$107$	−753.874	−0.681119	−0.340560	+	0.940223i	$0.610616\pi$
$108$	0	0
$109$	−873.770	−0.767816	−0.383908	−	0.923371i	$-0.625422\pi$
$109$	−873.770	−0.767816	−0.383908	+	0.923371i	$0.625422\pi$
$110$	−588.881	−0.510433
$111$	0	0
$112$	−351.505	−0.296554
$113$	473.333	0.394048	0.197024	−	0.980399i	$-0.436872\pi$
$113$	473.333	0.394048	0.197024	+	0.980399i	$0.436872\pi$
$114$	0	0
$115$	−750.084	−0.608223
$116$	745.180	0.596450
$117$	0	0
$118$	−1712.89	−1.33631
$119$	−1887.44	−1.45396
$120$	0	0
$121$	2425.31	1.82217
$122$	−927.950	−0.688628
$123$	0	0
$124$	−746.553	−0.540665
$125$	1039.31	0.743670
$126$	0	0
$127$	−2576.49	−1.80021	−0.900104	−	0.435675i	$-0.856510\pi$
$127$	−2576.49	−1.80021	−0.900104	+	0.435675i	$0.856510\pi$
$128$	−102.987	−0.0711163
$129$	0	0
$130$	80.7447	0.0544752
$131$	2100.65	1.40103	0.700514	−	0.713639i	$-0.252954\pi$
$131$	2100.65	1.40103	0.700514	+	0.713639i	$0.252954\pi$
$132$	0	0
$133$	−646.075	−0.421216
$134$	1361.66	0.877833
$135$	0	0
$136$	3113.61	1.96316
$137$	−1943.15	−1.21179	−0.605893	−	0.795546i	$-0.707184\pi$
$137$	−1943.15	−1.21179	−0.605893	+	0.795546i	$0.707184\pi$
$138$	0	0
$139$	−412.166	−0.251507	−0.125754	−	0.992062i	$-0.540135\pi$
$139$	−412.166	−0.251507	−0.125754	+	0.992062i	$0.540135\pi$
$140$	234.327	0.141459
$141$	0	0
$142$	−886.237	−0.523742
$143$	−515.047	−0.301192
$144$	0	0
$145$	964.558	0.552429
$146$	502.679	0.284945
$147$	0	0
$148$	−195.982	−0.108849
$149$	1864.74	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$149$	1864.74	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$150$	0	0
$151$	−2491.43	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$151$	−2491.43	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$152$	1065.79	0.568731
$153$	0	0
$154$	1922.98	1.00622
$155$	−966.335	−0.500761
$156$	0	0
$157$	−1467.06	−0.745761	−0.372880	−	0.927879i	$-0.621630\pi$
$157$	−1467.06	−0.745761	−0.372880	+	0.927879i	$0.621630\pi$
$158$	−143.888	−0.0724500
$159$	0	0
$160$	−655.494	−0.323883
$161$	2449.38	1.19899
$162$	0	0
$163$	−1756.39	−0.843997	−0.421998	−	0.906597i	$-0.638671\pi$
$163$	−1756.39	−0.843997	−0.421998	+	0.906597i	$0.638671\pi$
$164$	−1150.85	−0.547966
$165$	0	0
$166$	−29.0877	−0.0136003
$167$	−2433.18	−1.12746	−0.563728	−	0.825961i	$-0.690633\pi$
$167$	−2433.18	−1.12746	−0.563728	+	0.825961i	$0.690633\pi$
$168$	0	0
$169$	−2126.38	−0.967856
$170$	1226.29	0.553249
$171$	0	0
$172$	−115.655	−0.0512710
$173$	−2724.13	−1.19718	−0.598589	−	0.801056i	$-0.704272\pi$
$173$	−2724.13	−1.19718	−0.598589	+	0.801056i	$0.704272\pi$
$174$	0	0
$175$	−1545.26	−0.667492
$176$	−1456.75	−0.623900
$177$	0	0
$178$	−3177.40	−1.33795
$179$	−3760.40	−1.57020	−0.785099	−	0.619370i	$-0.787388\pi$
$179$	−3760.40	−1.57020	−0.785099	+	0.619370i	$0.787388\pi$
$180$	0	0
$181$	967.404	0.397274	0.198637	−	0.980073i	$-0.436349\pi$
$181$	967.404	0.397274	0.198637	+	0.980073i	$0.436349\pi$
$182$	−263.670	−0.107387
$183$	0	0
$184$	−4040.60	−1.61890
$185$	−253.678	−0.100815
$186$	0	0
$187$	−7822.17	−3.05890
$188$	−906.218	−0.351557
$189$	0	0
$190$	419.761	0.160277
$191$	−835.684	−0.316586	−0.158293	−	0.987392i	$-0.550599\pi$
$191$	−835.684	−0.316586	−0.158293	+	0.987392i	$0.550599\pi$
$192$	0	0
$193$	−1302.81	−0.485897	−0.242948	−	0.970039i	$-0.578115\pi$
$193$	−1302.81	−0.485897	−0.242948	+	0.970039i	$0.578115\pi$
$194$	1001.10	0.370490
$195$	0	0
$196$	434.885	0.158486
$197$	−3977.01	−1.43833	−0.719164	−	0.694841i	$-0.755475\pi$
$197$	−3977.01	−1.43833	−0.719164	+	0.694841i	$0.755475\pi$
$198$	0	0
$199$	2651.64	0.944571	0.472285	−	0.881446i	$-0.343429\pi$
$199$	2651.64	0.944571	0.472285	+	0.881446i	$0.343429\pi$
$200$	2549.13	0.901255
$201$	0	0
$202$	2660.33	0.926633
$203$	−3149.74	−1.08901
$204$	0	0
$205$	−1489.66	−0.507522
$206$	−1409.71	−0.476792
$207$	0	0
$208$	199.742	0.0665848
$209$	−2677.54	−0.886168
$210$	0	0
$211$	−1128.60	−0.368228	−0.184114	−	0.982905i	$-0.558942\pi$
$211$	−1128.60	−0.368228	−0.184114	+	0.982905i	$0.558942\pi$
$212$	526.376	0.170527
$213$	0	0
$214$	−1599.43	−0.510910
$215$	−149.703	−0.0474869
$216$	0	0
$217$	3155.54	0.987153
$218$	−1853.80	−0.575942
$219$	0	0
$220$	971.127	0.297606
$221$	1072.54	0.326456
$222$	0	0
$223$	−2515.64	−0.755426	−0.377713	−	0.925923i	$-0.623289\pi$
$223$	−2515.64	−0.755426	−0.377713	+	0.925923i	$0.623289\pi$
$224$	2140.50	0.638473
$225$	0	0
$226$	1004.23	0.295577
$227$	−346.735	−0.101382	−0.0506908	−	0.998714i	$-0.516142\pi$
$227$	−346.735	−0.101382	−0.0506908	+	0.998714i	$0.516142\pi$
$228$	0	0
$229$	1416.59	0.408782	0.204391	−	0.978889i	$-0.434479\pi$
$229$	1416.59	0.408782	0.204391	+	0.978889i	$0.434479\pi$
$230$	−1591.39	−0.456230
$231$	0	0
$232$	5195.94	1.47039
$233$	−2713.73	−0.763014	−0.381507	−	0.924366i	$-0.624595\pi$
$233$	−2713.73	−0.763014	−0.381507	+	0.924366i	$0.624595\pi$
$234$	0	0
$235$	−1173.00	−0.325610
$236$	2824.74	0.779130
$237$	0	0
$238$	−4004.43	−1.09062
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	5659.55	1.51271	0.756355	−	0.654161i	$-0.226978\pi$
$241$	5659.55	1.51271	0.756355	+	0.654161i	$0.226978\pi$
$242$	5145.56	1.36681
$243$	0	0
$244$	1530.29	0.401502
$245$	562.913	0.146789
$246$	0	0
$247$	367.132	0.0945750
$248$	−5205.51	−1.33286
$249$	0	0
$250$	2205.01	0.557829
$251$	−3002.94	−0.755154	−0.377577	−	0.925978i	$-0.623243\pi$
$251$	−3002.94	−0.755154	−0.377577	+	0.925978i	$0.623243\pi$
$252$	0	0
$253$	10151.0	2.52248
$254$	−5466.31	−1.35034
$255$	0	0
$256$	−4196.34	−1.02450
$257$	−6875.65	−1.66884	−0.834418	−	0.551132i	$-0.814196\pi$
$257$	−6875.65	−1.66884	−0.834418	+	0.551132i	$0.814196\pi$
$258$	0	0
$259$	828.379	0.198737
$260$	−133.156	−0.0317616
$261$	0	0
$262$	4456.76	1.05092
$263$	1797.45	0.421429	0.210714	−	0.977548i	$-0.432421\pi$
$263$	1797.45	0.421429	0.210714	+	0.977548i	$0.432421\pi$
$264$	0	0
$265$	681.339	0.157941
$266$	−1370.72	−0.315956
$267$	0	0
$268$	−2245.52	−0.511817
$269$	−8271.17	−1.87473	−0.937365	−	0.348349i	$-0.886742\pi$
$269$	−8271.17	−1.87473	−0.937365	+	0.348349i	$0.886742\pi$
$270$	0	0
$271$	−4182.05	−0.937422	−0.468711	−	0.883352i	$-0.655281\pi$
$271$	−4182.05	−0.937422	−0.468711	+	0.883352i	$0.655281\pi$
$272$	3033.54	0.676233
$273$	0	0
$274$	−4122.62	−0.908965
$275$	−6404.07	−1.40429
$276$	0	0
$277$	4445.28	0.964228	0.482114	−	0.876108i	$-0.339869\pi$
$277$	4445.28	0.964228	0.482114	+	0.876108i	$0.339869\pi$
$278$	−874.458	−0.188656
$279$	0	0
$280$	1633.90	0.348730
$281$	542.974	0.115271	0.0576355	−	0.998338i	$-0.481644\pi$
$281$	542.974	0.115271	0.0576355	+	0.998338i	$0.481644\pi$
$282$	0	0
$283$	7379.58	1.55007	0.775036	−	0.631917i	$-0.217731\pi$
$283$	7379.58	1.55007	0.775036	+	0.631917i	$0.217731\pi$
$284$	1461.50	0.305366
$285$	0	0
$286$	−1092.73	−0.225925
$287$	4864.43	1.00048
$288$	0	0
$289$	11376.0	2.31548
$290$	2046.42	0.414379
$291$	0	0
$292$	−828.971	−0.166136
$293$	−5272.35	−1.05124	−0.525621	−	0.850719i	$-0.676167\pi$
$293$	−5272.35	−1.05124	−0.525621	+	0.850719i	$0.676167\pi$
$294$	0	0
$295$	3656.33	0.721626
$296$	−1366.53	−0.268337
$297$	0	0
$298$	3956.25	0.769059
$299$	−1391.86	−0.269208
$300$	0	0
$301$	488.852	0.0936112
$302$	−5285.85	−1.00717
$303$	0	0
$304$	1038.39	0.195906
$305$	1980.80	0.371869
$306$	0	0
$307$	−5750.06	−1.06897	−0.534484	−	0.845179i	$-0.679494\pi$
$307$	−5750.06	−1.06897	−0.534484	+	0.845179i	$0.679494\pi$
$308$	−3171.19	−0.586673
$309$	0	0
$310$	−2050.19	−0.375622
$311$	−7963.67	−1.45202	−0.726010	−	0.687684i	$-0.758628\pi$
$311$	−7963.67	−1.45202	−0.726010	+	0.687684i	$0.758628\pi$
$312$	0	0
$313$	6104.69	1.10242	0.551210	−	0.834367i	$-0.314166\pi$
$313$	6104.69	1.10242	0.551210	+	0.834367i	$0.314166\pi$
$314$	−3112.54	−0.559398
$315$	0	0
$316$	237.286	0.0422417
$317$	4949.91	0.877018	0.438509	−	0.898727i	$-0.355507\pi$
$317$	4949.91	0.877018	0.438509	+	0.898727i	$0.355507\pi$
$318$	0	0
$319$	−13053.5	−2.29109
$320$	−2251.84	−0.393381
$321$	0	0
$322$	5196.63	0.899370
$323$	5575.73	0.960502
$324$	0	0
$325$	878.096	0.149871
$326$	−3726.39	−0.633085
$327$	0	0
$328$	−8024.57	−1.35086
$329$	3830.42	0.641877
$330$	0	0
$331$	−2722.03	−0.452013	−0.226007	−	0.974126i	$-0.572567\pi$
$331$	−2722.03	−0.452013	−0.226007	+	0.974126i	$0.572567\pi$
$332$	47.9687	0.00792958
$333$	0	0
$334$	−5162.26	−0.845708
$335$	−2906.59	−0.474042
$336$	0	0
$337$	6453.19	1.04311	0.521554	−	0.853218i	$-0.325352\pi$
$337$	6453.19	1.04311	0.521554	+	0.853218i	$0.325352\pi$
$338$	−4511.35	−0.725992
$339$	0	0
$340$	−2022.28	−0.322570
$341$	13077.6	2.07680
$342$	0	0
$343$	−6910.67	−1.08788
$344$	−806.431	−0.126395
$345$	0	0
$346$	−5779.55	−0.898007
$347$	−2579.75	−0.399101	−0.199551	−	0.979888i	$-0.563948\pi$
$347$	−2579.75	−0.399101	−0.199551	+	0.979888i	$0.563948\pi$
$348$	0	0
$349$	−3787.07	−0.580852	−0.290426	−	0.956897i	$-0.593797\pi$
$349$	−3787.07	−0.580852	−0.290426	+	0.956897i	$0.593797\pi$
$350$	−3278.45	−0.500688
$351$	0	0
$352$	8870.90	1.34324
$353$	−10562.0	−1.59252	−0.796262	−	0.604952i	$-0.793192\pi$
$353$	−10562.0	−1.59252	−0.796262	+	0.604952i	$0.793192\pi$
$354$	0	0
$355$	1891.76	0.282828
$356$	5239.86	0.780090
$357$	0	0
$358$	−7978.11	−1.17781
$359$	7215.46	1.06077	0.530386	−	0.847756i	$-0.322047\pi$
$359$	7215.46	1.06077	0.530386	+	0.847756i	$0.322047\pi$
$360$	0	0
$361$	−4950.42	−0.721741
$362$	2052.46	0.297996
$363$	0	0
$364$	434.819	0.0626118
$365$	−1073.02	−0.153875
$366$	0	0
$367$	6325.12	0.899642	0.449821	−	0.893119i	$-0.351488\pi$
$367$	6325.12	0.899642	0.449821	+	0.893119i	$0.351488\pi$
$368$	−3936.70	−0.557648
$369$	0	0
$370$	−538.206	−0.0756217
$371$	−2224.90	−0.311350
$372$	0	0
$373$	−1950.11	−0.270704	−0.135352	−	0.990798i	$-0.543217\pi$
$373$	−1950.11	−0.270704	−0.135352	+	0.990798i	$0.543217\pi$
$374$	−16595.6	−2.29449
$375$	0	0
$376$	−6318.81	−0.866670
$377$	1789.84	0.244513
$378$	0	0
$379$	11140.9	1.50994	0.754972	−	0.655757i	$-0.227651\pi$
$379$	11140.9	1.50994	0.754972	+	0.655757i	$0.227651\pi$
$380$	−692.230	−0.0934491
$381$	0	0
$382$	−1773.00	−0.237472
$383$	−3581.16	−0.477777	−0.238888	−	0.971047i	$-0.576783\pi$
$383$	−3581.16	−0.477777	−0.238888	+	0.971047i	$0.576783\pi$
$384$	0	0
$385$	−4104.77	−0.543373
$386$	−2764.05	−0.364473
$387$	0	0
$388$	−1650.93	−0.216013
$389$	2412.86	0.314491	0.157245	−	0.987560i	$-0.449739\pi$
$389$	2412.86	0.314491	0.157245	+	0.987560i	$0.449739\pi$
$390$	0	0
$391$	−21138.6	−2.73407
$392$	3032.33	0.390704
$393$	0	0
$394$	−8437.68	−1.07889
$395$	307.142	0.0391240
$396$	0	0
$397$	13611.6	1.72078	0.860388	−	0.509640i	$-0.170221\pi$
$397$	13611.6	1.72078	0.860388	+	0.509640i	$0.170221\pi$
$398$	5625.75	0.708525
$399$	0	0
$400$	2483.58	0.310448
$401$	947.640	0.118012	0.0590061	−	0.998258i	$-0.481207\pi$
$401$	947.640	0.118012	0.0590061	+	0.998258i	$0.481207\pi$
$402$	0	0
$403$	−1793.14	−0.221644
$404$	−4387.15	−0.540270
$405$	0	0
$406$	−6682.53	−0.816868
$407$	3433.07	0.418110
$408$	0	0
$409$	−2212.57	−0.267493	−0.133746	−	0.991016i	$-0.542701\pi$
$409$	−2212.57	−0.267493	−0.133746	+	0.991016i	$0.542701\pi$
$410$	−3160.47	−0.380694
$411$	0	0
$412$	2324.76	0.277992
$413$	−11939.6	−1.42255
$414$	0	0
$415$	62.0904	0.00734433
$416$	−1216.34	−0.143355
$417$	0	0
$418$	−5680.70	−0.664718
$419$	−7516.47	−0.876380	−0.438190	−	0.898882i	$-0.644380\pi$
$419$	−7516.47	−0.876380	−0.438190	+	0.898882i	$0.644380\pi$
$420$	0	0
$421$	−5276.93	−0.610883	−0.305441	−	0.952211i	$-0.598804\pi$
$421$	−5276.93	−0.610883	−0.305441	+	0.952211i	$0.598804\pi$
$422$	−2394.45	−0.276209
$423$	0	0
$424$	3670.28	0.420388
$425$	13335.9	1.52208
$426$	0	0
$427$	−6468.24	−0.733068
$428$	2637.62	0.297884
$429$	0	0
$430$	−317.612	−0.0356200
$431$	−2177.53	−0.243359	−0.121680	−	0.992569i	$-0.538828\pi$
$431$	−2177.53	−0.243359	−0.121680	+	0.992569i	$0.538828\pi$
$432$	0	0
$433$	1636.35	0.181611	0.0908057	−	0.995869i	$-0.471056\pi$
$433$	1636.35	0.181611	0.0908057	+	0.995869i	$0.471056\pi$
$434$	6694.84	0.740467
$435$	0	0
$436$	3057.11	0.335801
$437$	−7235.76	−0.792067
$438$	0	0
$439$	7565.28	0.822486	0.411243	−	0.911526i	$-0.365095\pi$
$439$	7565.28	0.822486	0.411243	+	0.911526i	$0.365095\pi$
$440$	6771.40	0.733668
$441$	0	0
$442$	2275.51	0.244876
$443$	−1414.15	−0.151667	−0.0758334	−	0.997121i	$-0.524162\pi$
$443$	−1414.15	−0.151667	−0.0758334	+	0.997121i	$0.524162\pi$
$444$	0	0
$445$	6782.45	0.722514
$446$	−5337.22	−0.566647
$447$	0	0
$448$	7353.34	0.775475
$449$	6478.28	0.680911	0.340455	−	0.940261i	$-0.389419\pi$
$449$	6478.28	0.680911	0.340455	+	0.940261i	$0.389419\pi$
$450$	0	0
$451$	20159.8	2.10485
$452$	−1656.08	−0.172335
$453$	0	0
$454$	−735.637	−0.0760466
$455$	562.828	0.0579907
$456$	0	0
$457$	7137.95	0.730633	0.365317	−	0.930883i	$-0.380961\pi$
$457$	7137.95	0.730633	0.365317	+	0.930883i	$0.380961\pi$
$458$	3005.46	0.306628
$459$	0	0
$460$	2624.36	0.266003
$461$	−1376.00	−0.139016	−0.0695082	−	0.997581i	$-0.522143\pi$
$461$	−1376.00	−0.139016	−0.0695082	+	0.997581i	$0.522143\pi$
$462$	0	0
$463$	9939.58	0.997692	0.498846	−	0.866691i	$-0.333757\pi$
$463$	9939.58	0.997692	0.498846	+	0.866691i	$0.333757\pi$
$464$	5062.33	0.506493
$465$	0	0
$466$	−5757.48	−0.572340
$467$	−4325.57	−0.428616	−0.214308	−	0.976766i	$-0.568750\pi$
$467$	−4325.57	−0.428616	−0.214308	+	0.976766i	$0.568750\pi$
$468$	0	0
$469$	9491.41	0.934483
$470$	−2488.66	−0.244241
$471$	0	0
$472$	19696.1	1.92074
$473$	2025.96	0.196942
$474$	0	0
$475$	4564.89	0.440951
$476$	6603.72	0.635884
$477$	0	0
$478$	507.065	0.0485201
$479$	−2362.70	−0.225375	−0.112687	−	0.993631i	$-0.535946\pi$
$479$	−2362.70	−0.225375	−0.112687	+	0.993631i	$0.535946\pi$
$480$	0	0
$481$	−470.726	−0.0446222
$482$	12007.4	1.13469
$483$	0	0
$484$	−8485.56	−0.796916
$485$	−2136.95	−0.200070
$486$	0	0
$487$	−3588.96	−0.333945	−0.166972	−	0.985962i	$-0.553399\pi$
$487$	−3588.96	−0.333945	−0.166972	+	0.985962i	$0.553399\pi$
$488$	10670.3	0.989796
$489$	0	0
$490$	1194.28	0.110107
$491$	20576.6	1.89126	0.945632	−	0.325238i	$-0.105445\pi$
$491$	20576.6	1.89126	0.945632	+	0.325238i	$0.105445\pi$
$492$	0	0
$493$	27182.8	2.48327
$494$	778.911	0.0709411
$495$	0	0
$496$	−5071.66	−0.459121
$497$	−6177.48	−0.557541
$498$	0	0
$499$	14629.4	1.31243	0.656213	−	0.754576i	$-0.272157\pi$
$499$	14629.4	1.31243	0.656213	+	0.754576i	$0.272157\pi$
$500$	−3636.30	−0.325240
$501$	0	0
$502$	−6371.06	−0.566443
$503$	−5101.70	−0.452233	−0.226117	−	0.974100i	$-0.572603\pi$
$503$	−5101.70	−0.452233	−0.226117	+	0.974100i	$0.572603\pi$
$504$	0	0
$505$	−5678.71	−0.500395
$506$	21536.5	1.89212
$507$	0	0
$508$	9014.52	0.787312
$509$	756.470	0.0658741	0.0329371	−	0.999457i	$-0.489514\pi$
$509$	756.470	0.0658741	0.0329371	+	0.999457i	$0.489514\pi$
$510$	0	0
$511$	3503.91	0.303334
$512$	−8079.11	−0.697362
$513$	0	0
$514$	−14587.5	−1.25180
$515$	3009.16	0.257474
$516$	0	0
$517$	15874.5	1.35040
$518$	1757.50	0.149074
$519$	0	0
$520$	−928.464	−0.0782997
$521$	4918.37	0.413585	0.206792	−	0.978385i	$-0.433698\pi$
$521$	4918.37	0.413585	0.206792	+	0.978385i	$0.433698\pi$
$522$	0	0
$523$	−9376.44	−0.783945	−0.391972	−	0.919977i	$-0.628207\pi$
$523$	−9376.44	−0.783945	−0.391972	+	0.919977i	$0.628207\pi$
$524$	−7349.67	−0.612733
$525$	0	0
$526$	3813.50	0.316115
$527$	−27232.9	−2.25101
$528$	0	0
$529$	15265.0	1.25462
$530$	1445.54	0.118472
$531$	0	0
$532$	2260.46	0.184217
$533$	−2764.21	−0.224637
$534$	0	0
$535$	3414.13	0.275898
$536$	−15657.4	−1.26175
$537$	0	0
$538$	−17548.2	−1.40624
$539$	−7617.99	−0.608776
$540$	0	0
$541$	−581.756	−0.0462322	−0.0231161	−	0.999733i	$-0.507359\pi$
$541$	−581.756	−0.0462322	−0.0231161	+	0.999733i	$0.507359\pi$
$542$	−8872.69	−0.703163
$543$	0	0
$544$	−18472.9	−1.45591
$545$	3957.11	0.311017
$546$	0	0
$547$	11639.1	0.909784	0.454892	−	0.890547i	$-0.349678\pi$
$547$	11639.1	0.909784	0.454892	+	0.890547i	$0.349678\pi$
$548$	6798.63	0.529969
$549$	0	0
$550$	−13586.9	−1.05336
$551$	9304.70	0.719408
$552$	0	0
$553$	−1002.96	−0.0771255
$554$	9431.17	0.723271
$555$	0	0
$556$	1442.07	0.109995
$557$	−12074.7	−0.918533	−0.459267	−	0.888298i	$-0.651888\pi$
$557$	−12074.7	−0.918533	−0.459267	+	0.888298i	$0.651888\pi$
$558$	0	0
$559$	−277.790	−0.0210184
$560$	1591.89	0.120124
$561$	0	0
$562$	1151.98	0.0864651
$563$	22157.6	1.65867	0.829335	−	0.558751i	$-0.188719\pi$
$563$	22157.6	1.65867	0.829335	+	0.558751i	$0.188719\pi$
$564$	0	0
$565$	−2143.62	−0.159616
$566$	15656.6	1.16271
$567$	0	0
$568$	10190.6	0.752798
$569$	−20839.2	−1.53537	−0.767683	−	0.640830i	$-0.778590\pi$
$569$	−20839.2	−1.53537	−0.767683	+	0.640830i	$0.778590\pi$
$570$	0	0
$571$	15783.4	1.15677	0.578384	−	0.815765i	$-0.303683\pi$
$571$	15783.4	1.15677	0.578384	+	0.815765i	$0.303683\pi$
$572$	1802.03	0.131725
$573$	0	0
$574$	10320.4	0.750465
$575$	−17306.3	−1.25517
$576$	0	0
$577$	6568.74	0.473934	0.236967	−	0.971518i	$-0.423847\pi$
$577$	6568.74	0.473934	0.236967	+	0.971518i	$0.423847\pi$
$578$	24135.4	1.73685
$579$	0	0
$580$	−3374.76	−0.241602
$581$	−202.755	−0.0144779
$582$	0	0
$583$	−9220.67	−0.655028
$584$	−5780.19	−0.409565
$585$	0	0
$586$	−11185.9	−0.788540
$587$	−21007.0	−1.47709	−0.738546	−	0.674204i	$-0.764487\pi$
$587$	−21007.0	−1.47709	−0.738546	+	0.674204i	$0.764487\pi$
$588$	0	0
$589$	−9321.84	−0.652122
$590$	7757.31	0.541294
$591$	0	0
$592$	−1331.39	−0.0924320
$593$	4418.62	0.305988	0.152994	−	0.988227i	$-0.451108\pi$
$593$	4418.62	0.305988	0.152994	+	0.988227i	$0.451108\pi$
$594$	0	0
$595$	8547.82	0.588952
$596$	−6524.27	−0.448397
$597$	0	0
$598$	−2952.99	−0.201934
$599$	6648.83	0.453529	0.226764	−	0.973950i	$-0.427185\pi$
$599$	6648.83	0.453529	0.226764	+	0.973950i	$0.427185\pi$
$600$	0	0
$601$	−14071.6	−0.955064	−0.477532	−	0.878614i	$-0.658468\pi$
$601$	−14071.6	−0.955064	−0.477532	+	0.878614i	$0.658468\pi$
$602$	1037.15	0.0702181
$603$	0	0
$604$	8716.92	0.587229
$605$	−10983.7	−0.738099
$606$	0	0
$607$	7017.00	0.469212	0.234606	−	0.972091i	$-0.424620\pi$
$607$	7017.00	0.469212	0.234606	+	0.972091i	$0.424620\pi$
$608$	−6323.28	−0.421781
$609$	0	0
$610$	4202.48	0.278940
$611$	−2176.63	−0.144120
$612$	0	0
$613$	21273.7	1.40169	0.700845	−	0.713314i	$-0.252807\pi$
$613$	21273.7	1.40169	0.700845	+	0.713314i	$0.252807\pi$
$614$	−12199.4	−0.801836
$615$	0	0
$616$	−22111.9	−1.44629
$617$	−1330.68	−0.0868254	−0.0434127	−	0.999057i	$-0.513823\pi$
$617$	−1330.68	−0.0868254	−0.0434127	+	0.999057i	$0.513823\pi$
$618$	0	0
$619$	19837.9	1.28813	0.644066	−	0.764970i	$-0.277246\pi$
$619$	19837.9	1.28813	0.644066	+	0.764970i	$0.277246\pi$
$620$	3380.98	0.219005
$621$	0	0
$622$	−16895.8	−1.08917
$623$	−22147.9	−1.42430
$624$	0	0
$625$	8354.46	0.534686
$626$	12951.8	0.826929
$627$	0	0
$628$	5132.91	0.326155
$629$	−7149.05	−0.453182
$630$	0	0
$631$	−20162.9	−1.27206	−0.636032	−	0.771663i	$-0.719425\pi$
$631$	−20162.9	−1.27206	−0.636032	+	0.771663i	$0.719425\pi$
$632$	1654.53	0.104136
$633$	0	0
$634$	10501.8	0.657854
$635$	11668.4	0.729204
$636$	0	0
$637$	1044.54	0.0649707
$638$	−27694.5	−1.71855
$639$	0	0
$640$	466.407	0.0288068
$641$	4425.57	0.272698	0.136349	−	0.990661i	$-0.456463\pi$
$641$	4425.57	0.272698	0.136349	+	0.990661i	$0.456463\pi$
$642$	0	0
$643$	1555.17	0.0953809	0.0476905	−	0.998862i	$-0.484814\pi$
$643$	1555.17	0.0953809	0.0476905	+	0.998862i	$0.484814\pi$
$644$	−8569.79	−0.524375
$645$	0	0
$646$	11829.5	0.720475
$647$	−14141.3	−0.859277	−0.429639	−	0.903001i	$-0.641359\pi$
$647$	−14141.3	−0.859277	−0.429639	+	0.903001i	$0.641359\pi$
$648$	0	0
$649$	−49481.7	−2.99280
$650$	1862.98	0.112419
$651$	0	0
$652$	6145.21	0.369118
$653$	−13309.6	−0.797618	−0.398809	−	0.917034i	$-0.630576\pi$
$653$	−13309.6	−0.797618	−0.398809	+	0.917034i	$0.630576\pi$
$654$	0	0
$655$	−9513.38	−0.567509
$656$	−7818.22	−0.465321
$657$	0	0
$658$	8126.66	0.481474
$659$	23689.3	1.40031	0.700155	−	0.713991i	$-0.253114\pi$
$659$	23689.3	1.40031	0.700155	+	0.713991i	$0.253114\pi$
$660$	0	0
$661$	216.801	0.0127573	0.00637867	−	0.999980i	$-0.497970\pi$
$661$	216.801	0.0127573	0.00637867	+	0.999980i	$0.497970\pi$
$662$	−5775.10	−0.339057
$663$	0	0
$664$	334.472	0.0195483
$665$	2925.93	0.170621
$666$	0	0
$667$	−35275.7	−2.04780
$668$	8513.11	0.493087
$669$	0	0
$670$	−6166.66	−0.355581
$671$	−26806.4	−1.54225
$672$	0	0
$673$	3912.27	0.224082	0.112041	−	0.993704i	$-0.464261\pi$
$673$	3912.27	0.224082	0.112041	+	0.993704i	$0.464261\pi$
$674$	13691.2	0.782439
$675$	0	0
$676$	7439.69	0.423287
$677$	−25774.0	−1.46318	−0.731590	−	0.681744i	$-0.761222\pi$
$677$	−25774.0	−1.46318	−0.731590	+	0.681744i	$0.761222\pi$
$678$	0	0
$679$	6978.16	0.394399
$680$	−14100.8	−0.795209
$681$	0	0
$682$	27745.5	1.55782
$683$	18936.2	1.06087	0.530435	−	0.847725i	$-0.322029\pi$
$683$	18936.2	1.06087	0.530435	+	0.847725i	$0.322029\pi$
$684$	0	0
$685$	8800.11	0.490854
$686$	−14661.8	−0.816019
$687$	0	0
$688$	−785.694	−0.0435382
$689$	1264.30	0.0699069
$690$	0	0
$691$	17599.7	0.968922	0.484461	−	0.874813i	$-0.339016\pi$
$691$	17599.7	0.968922	0.484461	+	0.874813i	$0.339016\pi$
$692$	9531.08	0.523580
$693$	0	0
$694$	−5473.23	−0.299367
$695$	1866.61	0.101877
$696$	0	0
$697$	−41980.9	−2.28140
$698$	−8034.70	−0.435699
$699$	0	0
$700$	5406.51	0.291924
$701$	14300.1	0.770482	0.385241	−	0.922816i	$-0.374118\pi$
$701$	14300.1	0.770482	0.385241	+	0.922816i	$0.374118\pi$
$702$	0	0
$703$	−2447.13	−0.131288
$704$	30474.6	1.63147
$705$	0	0
$706$	−22408.6	−1.19456
$707$	18543.7	0.986432
$708$	0	0
$709$	955.389	0.0506070	0.0253035	−	0.999680i	$-0.491945\pi$
$709$	955.389	0.0506070	0.0253035	+	0.999680i	$0.491945\pi$
$710$	4013.57	0.212150
$711$	0	0
$712$	36536.1	1.92310
$713$	35340.7	1.85627
$714$	0	0
$715$	2332.54	0.122003
$716$	13156.7	0.686719
$717$	0	0
$718$	15308.4	0.795689
$719$	34178.1	1.77278	0.886390	−	0.462940i	$-0.153205\pi$
$719$	34178.1	1.77278	0.886390	+	0.462940i	$0.153205\pi$
$720$	0	0
$721$	−9826.32	−0.507561
$722$	−10502.9	−0.541380
$723$	0	0
$724$	−3384.72	−0.173746
$725$	22254.7	1.14003
$726$	0	0
$727$	−12282.6	−0.626599	−0.313299	−	0.949654i	$-0.601434\pi$
$727$	−12282.6	−0.626599	−0.313299	+	0.949654i	$0.601434\pi$
$728$	3031.88	0.154353
$729$	0	0
$730$	−2276.52	−0.115422
$731$	−4218.87	−0.213462
$732$	0	0
$733$	−4269.39	−0.215134	−0.107567	−	0.994198i	$-0.534306\pi$
$733$	−4269.39	−0.215134	−0.107567	+	0.994198i	$0.534306\pi$
$734$	13419.5	0.674824
$735$	0	0
$736$	23972.6	1.20060
$737$	39335.4	1.96599
$738$	0	0
$739$	−5321.85	−0.264909	−0.132454	−	0.991189i	$-0.542286\pi$
$739$	−5321.85	−0.264909	−0.132454	+	0.991189i	$0.542286\pi$
$740$	887.559	0.0440909
$741$	0	0
$742$	−4720.37	−0.233545
$743$	−27587.5	−1.36216	−0.681080	−	0.732209i	$-0.738490\pi$
$743$	−27587.5	−1.36216	−0.681080	+	0.732209i	$0.738490\pi$
$744$	0	0
$745$	−8444.99	−0.415303
$746$	−4137.37	−0.203056
$747$	0	0
$748$	27367.9	1.33779
$749$	−11148.8	−0.543881
$750$	0	0
$751$	32157.3	1.56250	0.781249	−	0.624220i	$-0.214583\pi$
$751$	32157.3	1.56250	0.781249	+	0.624220i	$0.214583\pi$
$752$	−6156.33	−0.298535
$753$	0	0
$754$	3797.34	0.183410
$755$	11283.1	0.543888
$756$	0	0
$757$	6817.36	0.327320	0.163660	−	0.986517i	$-0.447670\pi$
$757$	6817.36	0.327320	0.163660	+	0.986517i	$0.447670\pi$
$758$	23636.6	1.13261
$759$	0	0
$760$	−4826.74	−0.230374
$761$	−20761.9	−0.988986	−0.494493	−	0.869182i	$-0.664646\pi$
$761$	−20761.9	−0.988986	−0.494493	+	0.869182i	$0.664646\pi$
$762$	0	0
$763$	−12921.8	−0.613109
$764$	2923.86	0.138457
$765$	0	0
$766$	−7597.82	−0.358382
$767$	6784.70	0.319402
$768$	0	0
$769$	−18990.3	−0.890515	−0.445258	−	0.895403i	$-0.646888\pi$
$769$	−18990.3	−0.890515	−0.445258	+	0.895403i	$0.646888\pi$
$770$	−8708.74	−0.407586
$771$	0	0
$772$	4558.21	0.212505
$773$	9384.47	0.436657	0.218329	−	0.975875i	$-0.429940\pi$
$773$	9384.47	0.436657	0.218329	+	0.975875i	$0.429940\pi$
$774$	0	0
$775$	−22295.7	−1.03340
$776$	−11511.5	−0.532522
$777$	0	0
$778$	5119.16	0.235901
$779$	−14370.1	−0.660928
$780$	0	0
$781$	−25601.4	−1.17297
$782$	−44847.8	−2.05084
$783$	0	0
$784$	2954.36	0.134583
$785$	6644.01	0.302083
$786$	0	0
$787$	−41767.3	−1.89180	−0.945898	−	0.324464i	$-0.894816\pi$
$787$	−41767.3	−1.89180	−0.945898	+	0.324464i	$0.894816\pi$
$788$	13914.6	0.629045
$789$	0	0
$790$	651.636	0.0293471
$791$	6999.94	0.314651
$792$	0	0
$793$	3675.57	0.164595
$794$	28878.6	1.29076
$795$	0	0
$796$	−9277.44	−0.413103
$797$	9656.69	0.429181	0.214591	−	0.976704i	$-0.431158\pi$
$797$	9656.69	0.429181	0.214591	+	0.976704i	$0.431158\pi$
$798$	0	0
$799$	−33057.1	−1.46368
$800$	−15123.9	−0.668387
$801$	0	0
$802$	2010.53	0.0885214
$803$	14521.3	0.638164
$804$	0	0
$805$	−11092.7	−0.485673
$806$	−3804.34	−0.166256
$807$	0	0
$808$	−30590.4	−1.33189
$809$	−28931.7	−1.25734	−0.628668	−	0.777674i	$-0.716400\pi$
$809$	−28931.7	−1.25734	−0.628668	+	0.777674i	$0.716400\pi$
$810$	0	0
$811$	−33280.0	−1.44096	−0.720480	−	0.693476i	$-0.756079\pi$
$811$	−33280.0	−1.44096	−0.720480	+	0.693476i	$0.756079\pi$
$812$	11020.2	0.476272
$813$	0	0
$814$	7283.63	0.313626
$815$	7954.33	0.341875
$816$	0	0
$817$	−1444.13	−0.0618404
$818$	−4694.22	−0.200647
$819$	0	0
$820$	5211.95	0.221962
$821$	7615.15	0.323716	0.161858	−	0.986814i	$-0.448251\pi$
$821$	7615.15	0.323716	0.161858	+	0.986814i	$0.448251\pi$
$822$	0	0
$823$	4847.61	0.205318	0.102659	−	0.994717i	$-0.467265\pi$
$823$	4847.61	0.205318	0.102659	+	0.994717i	$0.467265\pi$
$824$	16209.9	0.685314
$825$	0	0
$826$	−25331.3	−1.06706
$827$	−36284.3	−1.52567	−0.762834	−	0.646595i	$-0.776193\pi$
$827$	−36284.3	−1.52567	−0.762834	+	0.646595i	$0.776193\pi$
$828$	0	0
$829$	−26834.3	−1.12424	−0.562118	−	0.827057i	$-0.690013\pi$
$829$	−26834.3	−1.12424	−0.562118	+	0.827057i	$0.690013\pi$
$830$	131.732	0.00550901
$831$	0	0
$832$	−4178.53	−0.174116
$833$	15863.8	0.659841
$834$	0	0
$835$	11019.3	0.456694
$836$	9368.07	0.387561
$837$	0	0
$838$	−15947.0	−0.657376
$839$	22500.9	0.925883	0.462942	−	0.886389i	$-0.346794\pi$
$839$	22500.9	0.925883	0.462942	+	0.886389i	$0.346794\pi$
$840$	0	0
$841$	20973.2	0.859945
$842$	−11195.6	−0.458225
$843$	0	0
$844$	3948.71	0.161043
$845$	9629.90	0.392046
$846$	0	0
$847$	35866.9	1.45502
$848$	3575.90	0.144808
$849$	0	0
$850$	28293.6	1.14172
$851$	9277.48	0.373711
$852$	0	0
$853$	−11019.1	−0.442308	−0.221154	−	0.975239i	$-0.570982\pi$
$853$	−11019.1	−0.442308	−0.221154	+	0.975239i	$0.570982\pi$
$854$	−13723.1	−0.549877
$855$	0	0
$856$	18391.4	0.734353
$857$	−22920.3	−0.913586	−0.456793	−	0.889573i	$-0.651002\pi$
$857$	−22920.3	−0.913586	−0.456793	+	0.889573i	$0.651002\pi$
$858$	0	0
$859$	−29518.9	−1.17249	−0.586246	−	0.810133i	$-0.699395\pi$
$859$	−29518.9	−1.17249	−0.586246	+	0.810133i	$0.699395\pi$
$860$	523.776	0.0207681
$861$	0	0
$862$	−4619.87	−0.182544
$863$	−22986.6	−0.906688	−0.453344	−	0.891336i	$-0.649769\pi$
$863$	−22986.6	−0.906688	−0.453344	+	0.891336i	$0.649769\pi$
$864$	0	0
$865$	12337.0	0.484936
$866$	3471.69	0.136227
$867$	0	0
$868$	−11040.5	−0.431727
$869$	−4156.60	−0.162259
$870$	0	0
$871$	−5393.49	−0.209818
$872$	21316.4	0.827827
$873$	0	0
$874$	−15351.5	−0.594132
$875$	15370.0	0.593828
$876$	0	0
$877$	1336.66	0.0514660	0.0257330	−	0.999669i	$-0.491808\pi$
$877$	1336.66	0.0514660	0.0257330	+	0.999669i	$0.491808\pi$
$878$	16050.6	0.616949
$879$	0	0
$880$	6597.28	0.252721
$881$	18598.2	0.711227	0.355613	−	0.934633i	$-0.384272\pi$
$881$	18598.2	0.711227	0.355613	+	0.934633i	$0.384272\pi$
$882$	0	0
$883$	43727.7	1.66654	0.833269	−	0.552867i	$-0.186466\pi$
$883$	43727.7	1.66654	0.833269	+	0.552867i	$0.186466\pi$
$884$	−3752.56	−0.142774
$885$	0	0
$886$	−3000.28	−0.113766
$887$	−18665.8	−0.706579	−0.353289	−	0.935514i	$-0.614937\pi$
$887$	−18665.8	−0.706579	−0.353289	+	0.935514i	$0.614937\pi$
$888$	0	0
$889$	−38102.7	−1.43748
$890$	14389.7	0.541960
$891$	0	0
$892$	8801.64	0.330382
$893$	−11315.5	−0.424030
$894$	0	0
$895$	17030.0	0.636035
$896$	−1523.04	−0.0567871
$897$	0	0
$898$	13744.4	0.510753
$899$	−45445.8	−1.68599
$900$	0	0
$901$	19201.2	0.709972
$902$	42771.2	1.57885
$903$	0	0
$904$	−11547.4	−0.424846
$905$	−4381.16	−0.160922
$906$	0	0
$907$	−21624.4	−0.791649	−0.395825	−	0.918326i	$-0.629541\pi$
$907$	−21624.4	−0.791649	−0.395825	+	0.918326i	$0.629541\pi$
$908$	1213.14	0.0443387
$909$	0	0
$910$	1194.10	0.0434990
$911$	−15510.6	−0.564093	−0.282046	−	0.959401i	$-0.591013\pi$
$911$	−15510.6	−0.564093	−0.282046	+	0.959401i	$0.591013\pi$
$912$	0	0
$913$	−840.279	−0.0304591
$914$	15144.0	0.548050
$915$	0	0
$916$	−4956.31	−0.178779
$917$	31065.7	1.11874
$918$	0	0
$919$	348.853	0.0125219	0.00626093	−	0.999980i	$-0.498007\pi$
$919$	348.853	0.0125219	0.00626093	+	0.999980i	$0.498007\pi$
$920$	18299.0	0.655760
$921$	0	0
$922$	−2919.33	−0.104277
$923$	3510.35	0.125184
$924$	0	0
$925$	−5852.98	−0.208048
$926$	21087.9	0.748372
$927$	0	0
$928$	−30827.2	−1.09047
$929$	−25420.2	−0.897749	−0.448875	−	0.893595i	$-0.648175\pi$
$929$	−25420.2	−0.897749	−0.448875	+	0.893595i	$0.648175\pi$
$930$	0	0
$931$	5430.19	0.191157
$932$	9494.69	0.333701
$933$	0	0
$934$	−9177.19	−0.321506
$935$	35424.9	1.23906
$936$	0	0
$937$	37546.0	1.30904	0.654521	−	0.756044i	$-0.272870\pi$
$937$	37546.0	1.30904	0.654521	+	0.756044i	$0.272870\pi$
$938$	20137.1	0.700959
$939$	0	0
$940$	4104.06	0.142404
$941$	35076.2	1.21514	0.607572	−	0.794265i	$-0.292144\pi$
$941$	35076.2	1.21514	0.607572	+	0.794265i	$0.292144\pi$
$942$	0	0
$943$	54479.5	1.88133
$944$	19189.7	0.661621
$945$	0	0
$946$	4298.30	0.147727
$947$	42212.3	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$947$	42212.3	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$948$	0	0
$949$	−1991.09	−0.0681071
$950$	9684.93	0.330759
$951$	0	0
$952$	46045.9	1.56760
$953$	−25142.4	−0.854610	−0.427305	−	0.904108i	$-0.640537\pi$
$953$	−25142.4	−0.854610	−0.427305	+	0.904108i	$0.640537\pi$
$954$	0	0
$955$	3784.63	0.128238
$956$	−836.204	−0.0282895
$957$	0	0
$958$	−5012.73	−0.169054
$959$	−28736.5	−0.967624
$960$	0	0
$961$	15738.5	0.528298
$962$	−998.698	−0.0334712
$963$	0	0
$964$	−19801.4	−0.661577
$965$	5900.12	0.196820
$966$	0	0
$967$	36960.6	1.22913	0.614566	−	0.788865i	$-0.289331\pi$
$967$	36960.6	1.22913	0.614566	+	0.788865i	$0.289331\pi$
$968$	−59167.5	−1.96458
$969$	0	0
$970$	−4533.78	−0.150073
$971$	−14076.6	−0.465231	−0.232615	−	0.972569i	$-0.574728\pi$
$971$	−14076.6	−0.465231	−0.232615	+	0.972569i	$0.574728\pi$
$972$	0	0
$973$	−6095.37	−0.200831
$974$	−7614.37	−0.250493
$975$	0	0
$976$	10395.9	0.340947
$977$	40542.1	1.32759	0.663795	−	0.747915i	$-0.268945\pi$
$977$	40542.1	1.32759	0.663795	+	0.747915i	$0.268945\pi$
$978$	0	0
$979$	−91788.0	−2.99648
$980$	−1969.50	−0.0641972
$981$	0	0
$982$	43655.7	1.41864
$983$	20946.5	0.679644	0.339822	−	0.940490i	$-0.389633\pi$
$983$	20946.5	0.679644	0.339822	+	0.940490i	$0.389633\pi$
$984$	0	0
$985$	18011.0	0.582618
$986$	57671.3	1.86271
$987$	0	0
$988$	−1284.51	−0.0413619
$989$	5474.93	0.176029
$990$	0	0
$991$	−14016.6	−0.449295	−0.224648	−	0.974440i	$-0.572123\pi$
$991$	−14016.6	−0.449295	−0.224648	+	0.974440i	$0.572123\pi$
$992$	30884.0	0.988477
$993$	0	0
$994$	−13106.2	−0.418213
$995$	−12008.7	−0.382614
$996$	0	0
$997$	−31149.2	−0.989474	−0.494737	−	0.869043i	$-0.664736\pi$
$997$	−31149.2	−0.989474	−0.494737	+	0.869043i	$0.664736\pi$
$998$	31037.8	0.984454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.41	✓	59
3.2	odd	2		2151.4.a.h.1.19	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.41	✓	59	1.1	even	1	trivial
2151.4.a.h.1.19	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)

\(f(q)\)	\(=\)	\(q+2.12161 q^{2} -3.49876 q^{4} -4.52878 q^{5} +14.7886 q^{7} -24.3959 q^{8} +O(q^{10})\)	\(q+2.12161 q^{2} -3.49876 q^{4} -4.52878 q^{5} +14.7886 q^{7} -24.3959 q^{8} -9.60831 q^{10} +61.2887 q^{11} -8.40363 q^{13} +31.3757 q^{14} -23.7686 q^{16} -127.628 q^{17} -43.6873 q^{19} +15.8451 q^{20} +130.031 q^{22} +165.626 q^{23} -104.490 q^{25} -17.8292 q^{26} -51.7418 q^{28} -212.984 q^{29} +213.376 q^{31} +144.740 q^{32} -270.778 q^{34} -66.9744 q^{35} +56.0146 q^{37} -92.6875 q^{38} +110.484 q^{40} +328.931 q^{41} +33.0560 q^{43} -214.435 q^{44} +351.394 q^{46} +259.011 q^{47} -124.297 q^{49} -221.688 q^{50} +29.4023 q^{52} -150.446 q^{53} -277.563 q^{55} -360.782 q^{56} -451.870 q^{58} -807.354 q^{59} -437.380 q^{61} +452.702 q^{62} +497.230 q^{64} +38.0582 q^{65} +641.805 q^{67} +446.541 q^{68} -142.094 q^{70} -417.719 q^{71} +236.933 q^{73} +118.841 q^{74} +152.851 q^{76} +906.375 q^{77} -67.8200 q^{79} +107.643 q^{80} +697.864 q^{82} -13.7102 q^{83} +578.000 q^{85} +70.1320 q^{86} -1495.19 q^{88} -1497.63 q^{89} -124.278 q^{91} -579.486 q^{92} +549.521 q^{94} +197.850 q^{95} +471.860 q^{97} -263.710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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

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