LMFDB - Embedded newform 2151.4.a.e.1.8

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

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11857 q^{2} +1.72551 q^{4} +0.660207 q^{5} -24.6492 q^{7} +19.5675 q^{8} +O(q^{10})$	\(q-3.11857 q^{2} +1.72551 q^{4} +0.660207 q^{5} -24.6492 q^{7} +19.5675 q^{8} -2.05890 q^{10} -27.1706 q^{11} +16.3666 q^{13} +76.8704 q^{14} -74.8267 q^{16} -14.7407 q^{17} -127.999 q^{19} +1.13919 q^{20} +84.7334 q^{22} -159.659 q^{23} -124.564 q^{25} -51.0405 q^{26} -42.5324 q^{28} -229.035 q^{29} +244.139 q^{31} +76.8127 q^{32} +45.9700 q^{34} -16.2736 q^{35} +49.5205 q^{37} +399.176 q^{38} +12.9186 q^{40} +219.421 q^{41} +10.3175 q^{43} -46.8829 q^{44} +497.908 q^{46} -337.310 q^{47} +264.584 q^{49} +388.462 q^{50} +28.2407 q^{52} -636.552 q^{53} -17.9382 q^{55} -482.323 q^{56} +714.263 q^{58} -154.976 q^{59} -213.808 q^{61} -761.367 q^{62} +359.067 q^{64} +10.8053 q^{65} -116.121 q^{67} -25.4352 q^{68} +50.7504 q^{70} -922.188 q^{71} +529.548 q^{73} -154.433 q^{74} -220.864 q^{76} +669.733 q^{77} -266.253 q^{79} -49.4011 q^{80} -684.282 q^{82} -1511.55 q^{83} -9.73192 q^{85} -32.1760 q^{86} -531.659 q^{88} +96.6959 q^{89} -403.424 q^{91} -275.492 q^{92} +1051.93 q^{94} -84.5060 q^{95} -1226.82 q^{97} -825.125 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.11857	−1.10258	−0.551291	−	0.834313i	$-0.685865\pi$
$2$	−3.11857	−1.10258	−0.551291	+	0.834313i	$0.685865\pi$
$3$	0	0
$3$	0	0
$4$	1.72551	0.215688
$5$	0.660207	0.0590507	0.0295253	−	0.999564i	$-0.490600\pi$
$5$	0.660207	0.0590507	0.0295253	+	0.999564i	$0.490600\pi$
$6$	0	0
$7$	−24.6492	−1.33093	−0.665466	−	0.746428i	$-0.731767\pi$
$7$	−24.6492	−1.33093	−0.665466	+	0.746428i	$0.731767\pi$
$8$	19.5675	0.864769
$9$	0	0
$10$	−2.05890	−0.0651082
$11$	−27.1706	−0.744748	−0.372374	−	0.928083i	$-0.621456\pi$
$11$	−27.1706	−0.744748	−0.372374	+	0.928083i	$0.621456\pi$
$12$	0	0
$13$	16.3666	0.349175	0.174588	−	0.984642i	$-0.444141\pi$
$13$	16.3666	0.349175	0.174588	+	0.984642i	$0.444141\pi$
$14$	76.8704	1.46746
$15$	0	0
$16$	−74.8267	−1.16917
$17$	−14.7407	−0.210303	−0.105151	−	0.994456i	$-0.533533\pi$
$17$	−14.7407	−0.210303	−0.105151	+	0.994456i	$0.533533\pi$
$18$	0	0
$19$	−127.999	−1.54553	−0.772765	−	0.634692i	$-0.781127\pi$
$19$	−127.999	−1.54553	−0.772765	+	0.634692i	$0.781127\pi$
$20$	1.13919	0.0127365
$21$	0	0
$22$	84.7334	0.821146
$23$	−159.659	−1.44744	−0.723721	−	0.690092i	$-0.757570\pi$
$23$	−159.659	−1.44744	−0.723721	+	0.690092i	$0.757570\pi$
$24$	0	0
$25$	−124.564	−0.996513
$26$	−51.0405	−0.384995
$27$	0	0
$28$	−42.5324	−0.287066
$29$	−229.035	−1.46658	−0.733288	−	0.679918i	$-0.762016\pi$
$29$	−229.035	−1.46658	−0.733288	+	0.679918i	$0.762016\pi$
$30$	0	0
$31$	244.139	1.41447	0.707237	−	0.706976i	$-0.249941\pi$
$31$	244.139	1.41447	0.707237	+	0.706976i	$0.249941\pi$
$32$	76.8127	0.424334
$33$	0	0
$34$	45.9700	0.231876
$35$	−16.2736	−0.0785925
$36$	0	0
$37$	49.5205	0.220030	0.110015	−	0.993930i	$-0.464910\pi$
$37$	49.5205	0.220030	0.110015	+	0.993930i	$0.464910\pi$
$38$	399.176	1.70407
$39$	0	0
$40$	12.9186	0.0510652
$41$	219.421	0.835802	0.417901	−	0.908493i	$-0.362766\pi$
$41$	219.421	0.835802	0.417901	+	0.908493i	$0.362766\pi$
$42$	0	0
$43$	10.3175	0.0365909	0.0182955	−	0.999833i	$-0.494176\pi$
$43$	10.3175	0.0365909	0.0182955	+	0.999833i	$0.494176\pi$
$44$	−46.8829	−0.160633
$45$	0	0
$46$	497.908	1.59593
$47$	−337.310	−1.04684	−0.523422	−	0.852074i	$-0.675345\pi$
$47$	−337.310	−1.04684	−0.523422	+	0.852074i	$0.675345\pi$
$48$	0	0
$49$	264.584	0.771382
$50$	388.462	1.09874
$51$	0	0
$52$	28.2407	0.0753130
$53$	−636.552	−1.64976	−0.824878	−	0.565310i	$-0.808756\pi$
$53$	−636.552	−1.64976	−0.824878	+	0.565310i	$0.808756\pi$
$54$	0	0
$55$	−17.9382	−0.0439779
$56$	−482.323	−1.15095
$57$	0	0
$58$	714.263	1.61702
$59$	−154.976	−0.341969	−0.170985	−	0.985274i	$-0.554695\pi$
$59$	−154.976	−0.341969	−0.170985	+	0.985274i	$0.554695\pi$
$60$	0	0
$61$	−213.808	−0.448775	−0.224387	−	0.974500i	$-0.572038\pi$
$61$	−213.808	−0.448775	−0.224387	+	0.974500i	$0.572038\pi$
$62$	−761.367	−1.55958
$63$	0	0
$64$	359.067	0.701303
$65$	10.8053	0.0206190
$66$	0	0
$67$	−116.121	−0.211737	−0.105869	−	0.994380i	$-0.533762\pi$
$67$	−116.121	−0.211737	−0.105869	+	0.994380i	$0.533762\pi$
$68$	−25.4352	−0.0453598
$69$	0	0
$70$	50.7504	0.0866547
$71$	−922.188	−1.54146	−0.770730	−	0.637162i	$-0.780108\pi$
$71$	−922.188	−1.54146	−0.770730	+	0.637162i	$0.780108\pi$
$72$	0	0
$73$	529.548	0.849026	0.424513	−	0.905422i	$-0.360445\pi$
$73$	529.548	0.849026	0.424513	+	0.905422i	$0.360445\pi$
$74$	−154.433	−0.242602
$75$	0	0
$76$	−220.864	−0.333353
$77$	669.733	0.991210
$78$	0	0
$79$	−266.253	−0.379187	−0.189594	−	0.981863i	$-0.560717\pi$
$79$	−266.253	−0.379187	−0.189594	+	0.981863i	$0.560717\pi$
$80$	−49.4011	−0.0690401
$81$	0	0
$82$	−684.282	−0.921540
$83$	−1511.55	−1.99897	−0.999486	−	0.0320643i	$-0.989792\pi$
$83$	−1511.55	−1.99897	−0.999486	+	0.0320643i	$0.989792\pi$
$84$	0	0
$85$	−9.73192	−0.0124185
$86$	−32.1760	−0.0403445
$87$	0	0
$88$	−531.659	−0.644035
$89$	96.6959	0.115166	0.0575829	−	0.998341i	$-0.481661\pi$
$89$	96.6959	0.115166	0.0575829	+	0.998341i	$0.481661\pi$
$90$	0	0
$91$	−403.424	−0.464729
$92$	−275.492	−0.312196
$93$	0	0
$94$	1051.93	1.15423
$95$	−84.5060	−0.0912646
$96$	0	0
$97$	−1226.82	−1.28418	−0.642088	−	0.766631i	$-0.721931\pi$
$97$	−1226.82	−1.28418	−0.642088	+	0.766631i	$0.721931\pi$
$98$	−825.125	−0.850512
$99$	0	0
$100$	−214.936	−0.214936
$101$	13.0458	0.0128525	0.00642627	−	0.999979i	$-0.497954\pi$
$101$	13.0458	0.0128525	0.00642627	+	0.999979i	$0.497954\pi$
$102$	0	0
$103$	−1053.54	−1.00785	−0.503926	−	0.863747i	$-0.668111\pi$
$103$	−1053.54	−1.00785	−0.503926	+	0.863747i	$0.668111\pi$
$104$	320.253	0.301956
$105$	0	0
$106$	1985.13	1.81899
$107$	1855.23	1.67618	0.838091	−	0.545530i	$-0.183672\pi$
$107$	1855.23	1.67618	0.838091	+	0.545530i	$0.183672\pi$
$108$	0	0
$109$	−70.2818	−0.0617594	−0.0308797	−	0.999523i	$-0.509831\pi$
$109$	−70.2818	−0.0617594	−0.0308797	+	0.999523i	$0.509831\pi$
$110$	55.9416	0.0484893
$111$	0	0
$112$	1844.42	1.55608
$113$	952.243	0.792739	0.396370	−	0.918091i	$-0.370270\pi$
$113$	952.243	0.792739	0.396370	+	0.918091i	$0.370270\pi$
$114$	0	0
$115$	−105.408	−0.0854725
$116$	−395.201	−0.316323
$117$	0	0
$118$	483.305	0.377049
$119$	363.347	0.279899
$120$	0	0
$121$	−592.761	−0.445350
$122$	666.775	0.494811
$123$	0	0
$124$	421.264	0.305085
$125$	−164.764	−0.117895
$126$	0	0
$127$	1597.98	1.11652	0.558259	−	0.829667i	$-0.311469\pi$
$127$	1597.98	1.11652	0.558259	+	0.829667i	$0.311469\pi$
$128$	−1734.28	−1.19758
$129$	0	0
$130$	−33.6973	−0.0227342
$131$	−1471.88	−0.981668	−0.490834	−	0.871253i	$-0.663308\pi$
$131$	−1471.88	−0.981668	−0.490834	+	0.871253i	$0.663308\pi$
$132$	0	0
$133$	3155.08	2.05700
$134$	362.131	0.233458
$135$	0	0
$136$	−288.439	−0.181863
$137$	−1855.77	−1.15729	−0.578647	−	0.815578i	$-0.696419\pi$
$137$	−1855.77	−1.15729	−0.578647	+	0.815578i	$0.696419\pi$
$138$	0	0
$139$	−244.630	−0.149275	−0.0746374	−	0.997211i	$-0.523780\pi$
$139$	−244.630	−0.149275	−0.0746374	+	0.997211i	$0.523780\pi$
$140$	−28.0801	−0.0169515
$141$	0	0
$142$	2875.91	1.69959
$143$	−444.690	−0.260048
$144$	0	0
$145$	−151.210	−0.0866024
$146$	−1651.43	−0.936121
$147$	0	0
$148$	85.4479	0.0474579
$149$	−1067.25	−0.586797	−0.293399	−	0.955990i	$-0.594786\pi$
$149$	−1067.25	−0.586797	−0.293399	+	0.955990i	$0.594786\pi$
$150$	0	0
$151$	1976.07	1.06497	0.532485	−	0.846440i	$-0.321258\pi$
$151$	1976.07	1.06497	0.532485	+	0.846440i	$0.321258\pi$
$152$	−2504.62	−1.33653
$153$	0	0
$154$	−2088.61	−1.09289
$155$	161.182	0.0835257
$156$	0	0
$157$	−2385.67	−1.21272	−0.606361	−	0.795190i	$-0.707371\pi$
$157$	−2385.67	−1.21272	−0.606361	+	0.795190i	$0.707371\pi$
$158$	830.329	0.418085
$159$	0	0
$160$	50.7123	0.0250572
$161$	3935.47	1.92645
$162$	0	0
$163$	−2292.03	−1.10138	−0.550692	−	0.834709i	$-0.685636\pi$
$163$	−2292.03	−1.10138	−0.550692	+	0.834709i	$0.685636\pi$
$164$	378.613	0.180273
$165$	0	0
$166$	4713.89	2.20403
$167$	547.942	0.253898	0.126949	−	0.991909i	$-0.459481\pi$
$167$	547.942	0.253898	0.126949	+	0.991909i	$0.459481\pi$
$168$	0	0
$169$	−1929.13	−0.878077
$170$	30.3497	0.0136924
$171$	0	0
$172$	17.8030	0.00789223
$173$	1412.72	0.620850	0.310425	−	0.950598i	$-0.399529\pi$
$173$	1412.72	0.620850	0.310425	+	0.950598i	$0.399529\pi$
$174$	0	0
$175$	3070.41	1.32629
$176$	2033.08	0.870735
$177$	0	0
$178$	−301.553	−0.126980
$179$	−4383.79	−1.83050	−0.915251	−	0.402885i	$-0.868008\pi$
$179$	−4383.79	−1.83050	−0.915251	+	0.402885i	$0.868008\pi$
$180$	0	0
$181$	−1928.58	−0.791992	−0.395996	−	0.918252i	$-0.629601\pi$
$181$	−1928.58	−0.791992	−0.395996	+	0.918252i	$0.629601\pi$
$182$	1258.11	0.512402
$183$	0	0
$184$	−3124.12	−1.25170
$185$	32.6938	0.0129929
$186$	0	0
$187$	400.513	0.156623
$188$	−582.030	−0.225792
$189$	0	0
$190$	263.538	0.100627
$191$	−4791.52	−1.81519	−0.907597	−	0.419843i	$-0.862085\pi$
$191$	−4791.52	−1.81519	−0.907597	+	0.419843i	$0.862085\pi$
$192$	0	0
$193$	−1215.90	−0.453483	−0.226741	−	0.973955i	$-0.572807\pi$
$193$	−1215.90	−0.453483	−0.226741	+	0.973955i	$0.572807\pi$
$194$	3825.94	1.41591
$195$	0	0
$196$	456.541	0.166378
$197$	−4391.96	−1.58840	−0.794198	−	0.607659i	$-0.792109\pi$
$197$	−4391.96	−1.58840	−0.794198	+	0.607659i	$0.792109\pi$
$198$	0	0
$199$	4804.76	1.71156	0.855780	−	0.517339i	$-0.173077\pi$
$199$	4804.76	1.71156	0.855780	+	0.517339i	$0.173077\pi$
$200$	−2437.41	−0.861753
$201$	0	0
$202$	−40.6843	−0.0141710
$203$	5645.53	1.95192
$204$	0	0
$205$	144.863	0.0493547
$206$	3285.55	1.11124
$207$	0	0
$208$	−1224.66	−0.408244
$209$	3477.81	1.15103
$210$	0	0
$211$	498.351	0.162597	0.0812983	−	0.996690i	$-0.474093\pi$
$211$	498.351	0.162597	0.0812983	+	0.996690i	$0.474093\pi$
$212$	−1098.37	−0.355833
$213$	0	0
$214$	−5785.66	−1.84813
$215$	6.81171	0.00216072
$216$	0	0
$217$	−6017.84	−1.88257
$218$	219.179	0.0680949
$219$	0	0
$220$	−30.9524	−0.00948551
$221$	−241.255	−0.0734326
$222$	0	0
$223$	−251.420	−0.0754993	−0.0377496	−	0.999287i	$-0.512019\pi$
$223$	−251.420	−0.0754993	−0.0377496	+	0.999287i	$0.512019\pi$
$224$	−1893.37	−0.564760
$225$	0	0
$226$	−2969.64	−0.874060
$227$	3398.06	0.993555	0.496777	−	0.867878i	$-0.334517\pi$
$227$	3398.06	0.993555	0.496777	+	0.867878i	$0.334517\pi$
$228$	0	0
$229$	5376.83	1.55158	0.775789	−	0.630993i	$-0.217352\pi$
$229$	5376.83	1.55158	0.775789	+	0.630993i	$0.217352\pi$
$230$	328.722	0.0942405
$231$	0	0
$232$	−4481.64	−1.26825
$233$	531.206	0.149358	0.0746791	−	0.997208i	$-0.476207\pi$
$233$	531.206	0.149358	0.0746791	+	0.997208i	$0.476207\pi$
$234$	0	0
$235$	−222.694	−0.0618168
$236$	−267.412	−0.0737587
$237$	0	0
$238$	−1133.12	−0.308612
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	3450.31	0.922214	0.461107	−	0.887344i	$-0.347452\pi$
$241$	3450.31	0.922214	0.461107	+	0.887344i	$0.347452\pi$
$242$	1848.57	0.491035
$243$	0	0
$244$	−368.926	−0.0967954
$245$	174.680	0.0455506
$246$	0	0
$247$	−2094.92	−0.539661
$248$	4777.19	1.22319
$249$	0	0
$250$	513.828	0.129989
$251$	−1570.99	−0.395060	−0.197530	−	0.980297i	$-0.563292\pi$
$251$	−1570.99	−0.395060	−0.197530	+	0.980297i	$0.563292\pi$
$252$	0	0
$253$	4338.02	1.07798
$254$	−4983.42	−1.23105
$255$	0	0
$256$	2535.94	0.619126
$257$	3017.18	0.732321	0.366161	−	0.930552i	$-0.380672\pi$
$257$	3017.18	0.732321	0.366161	+	0.930552i	$0.380672\pi$
$258$	0	0
$259$	−1220.64	−0.292846
$260$	18.6447	0.00444728
$261$	0	0
$262$	4590.16	1.08237
$263$	1378.43	0.323184	0.161592	−	0.986858i	$-0.448337\pi$
$263$	1378.43	0.323184	0.161592	+	0.986858i	$0.448337\pi$
$264$	0	0
$265$	−420.256	−0.0974193
$266$	−9839.36	−2.26801
$267$	0	0
$268$	−200.367	−0.0456692
$269$	767.489	0.173958	0.0869789	−	0.996210i	$-0.472279\pi$
$269$	767.489	0.173958	0.0869789	+	0.996210i	$0.472279\pi$
$270$	0	0
$271$	−6459.87	−1.44800	−0.724002	−	0.689798i	$-0.757699\pi$
$271$	−6459.87	−1.44800	−0.724002	+	0.689798i	$0.757699\pi$
$272$	1103.00	0.245879
$273$	0	0
$274$	5787.36	1.27601
$275$	3384.48	0.742151
$276$	0	0
$277$	313.358	0.0679705	0.0339853	−	0.999422i	$-0.489180\pi$
$277$	313.358	0.0679705	0.0339853	+	0.999422i	$0.489180\pi$
$278$	762.895	0.164588
$279$	0	0
$280$	−318.433	−0.0679643
$281$	4741.64	1.00663	0.503314	−	0.864104i	$-0.332114\pi$
$281$	4741.64	1.00663	0.503314	+	0.864104i	$0.332114\pi$
$282$	0	0
$283$	−2872.20	−0.603302	−0.301651	−	0.953418i	$-0.597538\pi$
$283$	−2872.20	−0.603302	−0.301651	+	0.953418i	$0.597538\pi$
$284$	−1591.24	−0.332475
$285$	0	0
$286$	1386.80	0.286724
$287$	−5408.57	−1.11240
$288$	0	0
$289$	−4695.71	−0.955773
$290$	471.561	0.0954863
$291$	0	0
$292$	913.738	0.183125
$293$	−3385.57	−0.675042	−0.337521	−	0.941318i	$-0.609588\pi$
$293$	−3385.57	−0.675042	−0.337521	+	0.941318i	$0.609588\pi$
$294$	0	0
$295$	−102.316	−0.0201935
$296$	968.992	0.190275
$297$	0	0
$298$	3328.31	0.646993
$299$	−2613.08	−0.505411
$300$	0	0
$301$	−254.319	−0.0487001
$302$	−6162.53	−1.17422
$303$	0	0
$304$	9577.77	1.80698
$305$	−141.157	−0.0265005
$306$	0	0
$307$	840.925	0.156333	0.0781663	−	0.996940i	$-0.475093\pi$
$307$	840.925	0.156333	0.0781663	+	0.996940i	$0.475093\pi$
$308$	1155.63	0.213792
$309$	0	0
$310$	−502.659	−0.0920940
$311$	1849.80	0.337275	0.168638	−	0.985678i	$-0.446063\pi$
$311$	1849.80	0.337275	0.168638	+	0.985678i	$0.446063\pi$
$312$	0	0
$313$	9474.94	1.71104	0.855519	−	0.517771i	$-0.173238\pi$
$313$	9474.94	1.71104	0.855519	+	0.517771i	$0.173238\pi$
$314$	7439.89	1.33713
$315$	0	0
$316$	−459.421	−0.0817862
$317$	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
$319$	6223.01	1.09223
$320$	237.059	0.0414124
$321$	0	0
$322$	−12273.0	−2.12407
$323$	1886.80	0.325029
$324$	0	0
$325$	−2038.69	−0.347958
$326$	7147.86	1.21437
$327$	0	0
$328$	4293.52	0.722775
$329$	8314.42	1.39328
$330$	0	0
$331$	−10571.8	−1.75553	−0.877765	−	0.479091i	$-0.840966\pi$
$331$	−10571.8	−1.75553	−0.877765	+	0.479091i	$0.840966\pi$
$332$	−2608.20	−0.431155
$333$	0	0
$334$	−1708.80	−0.279944
$335$	−76.6636	−0.0125032
$336$	0	0
$337$	7123.72	1.15149	0.575747	−	0.817628i	$-0.304711\pi$
$337$	7123.72	1.15149	0.575747	+	0.817628i	$0.304711\pi$
$338$	6016.15	0.968152
$339$	0	0
$340$	−16.7925	−0.00267853
$341$	−6633.40	−1.05343
$342$	0	0
$343$	1932.89	0.304276
$344$	201.888	0.0316427
$345$	0	0
$346$	−4405.67	−0.684538
$347$	2722.04	0.421114	0.210557	−	0.977582i	$-0.432472\pi$
$347$	2722.04	0.421114	0.210557	+	0.977582i	$0.432472\pi$
$348$	0	0
$349$	−7882.83	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$349$	−7882.83	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$350$	−9575.30	−1.46235
$351$	0	0
$352$	−2087.04	−0.316022
$353$	−2761.85	−0.416427	−0.208213	−	0.978083i	$-0.566765\pi$
$353$	−2761.85	−0.416427	−0.208213	+	0.978083i	$0.566765\pi$
$354$	0	0
$355$	−608.835	−0.0910242
$356$	166.849	0.0248399
$357$	0	0
$358$	13671.2	2.01828
$359$	−9106.56	−1.33879	−0.669395	−	0.742906i	$-0.733447\pi$
$359$	−9106.56	−1.33879	−0.669395	+	0.742906i	$0.733447\pi$
$360$	0	0
$361$	9524.84	1.38866
$362$	6014.43	0.873237
$363$	0	0
$364$	−696.110	−0.100237
$365$	349.611	0.0501355
$366$	0	0
$367$	8896.73	1.26541	0.632705	−	0.774393i	$-0.281945\pi$
$367$	8896.73	1.26541	0.632705	+	0.774393i	$0.281945\pi$
$368$	11946.7	1.69230
$369$	0	0
$370$	−101.958	−0.0143258
$371$	15690.5	2.19572
$372$	0	0
$373$	8133.79	1.12909	0.564547	−	0.825401i	$-0.309051\pi$
$373$	8133.79	1.12909	0.564547	+	0.825401i	$0.309051\pi$
$374$	−1249.03	−0.172689
$375$	0	0
$376$	−6600.30	−0.905278
$377$	−3748.53	−0.512093
$378$	0	0
$379$	6555.05	0.888417	0.444209	−	0.895923i	$-0.353485\pi$
$379$	6555.05	0.888417	0.444209	+	0.895923i	$0.353485\pi$
$380$	−145.816	−0.0196847
$381$	0	0
$382$	14942.7	2.00140
$383$	−13229.1	−1.76495	−0.882473	−	0.470363i	$-0.844123\pi$
$383$	−13229.1	−1.76495	−0.882473	+	0.470363i	$0.844123\pi$
$384$	0	0
$385$	442.162	0.0585316
$386$	3791.86	0.500002
$387$	0	0
$388$	−2116.89	−0.276981
$389$	8009.68	1.04398	0.521988	−	0.852953i	$-0.325190\pi$
$389$	8009.68	1.04398	0.521988	+	0.852953i	$0.325190\pi$
$390$	0	0
$391$	2353.49	0.304401
$392$	5177.24	0.667067
$393$	0	0
$394$	13696.7	1.75134
$395$	−175.782	−0.0223913
$396$	0	0
$397$	6464.22	0.817203	0.408602	−	0.912713i	$-0.366017\pi$
$397$	6464.22	0.817203	0.408602	+	0.912713i	$0.366017\pi$
$398$	−14984.0	−1.88714
$399$	0	0
$400$	9320.72	1.16509
$401$	13412.8	1.67033	0.835165	−	0.549999i	$-0.185372\pi$
$401$	13412.8	1.67033	0.835165	+	0.549999i	$0.185372\pi$
$402$	0	0
$403$	3995.73	0.493900
$404$	22.5106	0.00277214
$405$	0	0
$406$	−17606.0	−2.15215
$407$	−1345.50	−0.163867
$408$	0	0
$409$	−5966.35	−0.721313	−0.360656	−	0.932699i	$-0.617447\pi$
$409$	−5966.35	−0.721313	−0.360656	+	0.932699i	$0.617447\pi$
$410$	−451.768	−0.0544176
$411$	0	0
$412$	−1817.89	−0.217382
$413$	3820.04	0.455138
$414$	0	0
$415$	−997.938	−0.118041
$416$	1257.16	0.148167
$417$	0	0
$418$	−10845.8	−1.26911
$419$	−13679.5	−1.59496	−0.797482	−	0.603343i	$-0.793835\pi$
$419$	−13679.5	−1.59496	−0.797482	+	0.603343i	$0.793835\pi$
$420$	0	0
$421$	−3188.51	−0.369117	−0.184558	−	0.982822i	$-0.559086\pi$
$421$	−3188.51	−0.369117	−0.184558	+	0.982822i	$0.559086\pi$
$422$	−1554.14	−0.179276
$423$	0	0
$424$	−12455.7	−1.42666
$425$	1836.16	0.209569
$426$	0	0
$427$	5270.19	0.597289
$428$	3201.20	0.361533
$429$	0	0
$430$	−21.2428	−0.00238237
$431$	1367.80	0.152864	0.0764322	−	0.997075i	$-0.475647\pi$
$431$	1367.80	0.152864	0.0764322	+	0.997075i	$0.475647\pi$
$432$	0	0
$433$	10893.3	1.20901	0.604503	−	0.796603i	$-0.293372\pi$
$433$	10893.3	1.20901	0.604503	+	0.796603i	$0.293372\pi$
$434$	18767.1	2.07569
$435$	0	0
$436$	−121.272	−0.0133208
$437$	20436.2	2.23707
$438$	0	0
$439$	−1436.35	−0.156157	−0.0780786	−	0.996947i	$-0.524879\pi$
$439$	−1436.35	−0.156157	−0.0780786	+	0.996947i	$0.524879\pi$
$440$	−351.005	−0.0380307
$441$	0	0
$442$	752.373	0.0809655
$443$	−7343.14	−0.787546	−0.393773	−	0.919208i	$-0.628830\pi$
$443$	−7343.14	−0.787546	−0.393773	+	0.919208i	$0.628830\pi$
$444$	0	0
$445$	63.8393	0.00680061
$446$	784.072	0.0832442
$447$	0	0
$448$	−8850.73	−0.933387
$449$	12523.9	1.31634	0.658172	−	0.752868i	$-0.271330\pi$
$449$	12523.9	1.31634	0.658172	+	0.752868i	$0.271330\pi$
$450$	0	0
$451$	−5961.80	−0.622462
$452$	1643.10	0.170984
$453$	0	0
$454$	−10597.1	−1.09548
$455$	−266.343	−0.0274426
$456$	0	0
$457$	15316.1	1.56774	0.783870	−	0.620925i	$-0.213243\pi$
$457$	15316.1	1.56774	0.783870	+	0.620925i	$0.213243\pi$
$458$	−16768.1	−1.71074
$459$	0	0
$460$	−181.882	−0.0184354
$461$	−1284.64	−0.129787	−0.0648936	−	0.997892i	$-0.520671\pi$
$461$	−1284.64	−0.129787	−0.0648936	+	0.997892i	$0.520671\pi$
$462$	0	0
$463$	−4151.21	−0.416680	−0.208340	−	0.978056i	$-0.566806\pi$
$463$	−4151.21	−0.416680	−0.208340	+	0.978056i	$0.566806\pi$
$464$	17137.9	1.71467
$465$	0	0
$466$	−1656.61	−0.164680
$467$	−3625.99	−0.359295	−0.179647	−	0.983731i	$-0.557496\pi$
$467$	−3625.99	−0.359295	−0.179647	+	0.983731i	$0.557496\pi$
$468$	0	0
$469$	2862.28	0.281808
$470$	694.488	0.0681582
$471$	0	0
$472$	−3032.49	−0.295724
$473$	−280.333	−0.0272510
$474$	0	0
$475$	15944.1	1.54014
$476$	626.957	0.0603709
$477$	0	0
$478$	−745.339	−0.0713201
$479$	−2571.22	−0.245265	−0.122632	−	0.992452i	$-0.539134\pi$
$479$	−2571.22	−0.245265	−0.122632	+	0.992452i	$0.539134\pi$
$480$	0	0
$481$	810.483	0.0768292
$482$	−10760.0	−1.01682
$483$	0	0
$484$	−1022.81	−0.0960567
$485$	−809.957	−0.0758314
$486$	0	0
$487$	6128.08	0.570205	0.285102	−	0.958497i	$-0.407972\pi$
$487$	6128.08	0.570205	0.285102	+	0.958497i	$0.407972\pi$
$488$	−4183.68	−0.388086
$489$	0	0
$490$	−544.753	−0.0502233
$491$	726.299	0.0667564	0.0333782	−	0.999443i	$-0.489373\pi$
$491$	726.299	0.0667564	0.0333782	+	0.999443i	$0.489373\pi$
$492$	0	0
$493$	3376.14	0.308425
$494$	6533.15	0.595021
$495$	0	0
$496$	−18268.1	−1.65376
$497$	22731.2	2.05158
$498$	0	0
$499$	9545.91	0.856380	0.428190	−	0.903689i	$-0.359151\pi$
$499$	9545.91	0.856380	0.428190	+	0.903689i	$0.359151\pi$
$500$	−284.301	−0.0254287
$501$	0	0
$502$	4899.25	0.435586
$503$	−4592.58	−0.407104	−0.203552	−	0.979064i	$-0.565249\pi$
$503$	−4592.58	−0.407104	−0.203552	+	0.979064i	$0.565249\pi$
$504$	0	0
$505$	8.61293	0.000758951 0
$506$	−13528.4	−1.18856
$507$	0	0
$508$	2757.32	0.240820
$509$	−8037.18	−0.699885	−0.349943	−	0.936771i	$-0.613799\pi$
$509$	−8037.18	−0.699885	−0.349943	+	0.936771i	$0.613799\pi$
$510$	0	0
$511$	−13052.9	−1.13000
$512$	5965.71	0.514941
$513$	0	0
$514$	−9409.30	−0.807445
$515$	−695.556	−0.0595143
$516$	0	0
$517$	9164.89	0.779635
$518$	3806.66	0.322886
$519$	0	0
$520$	211.433	0.0178307
$521$	20137.6	1.69337	0.846684	−	0.532096i	$-0.178595\pi$
$521$	20137.6	1.69337	0.846684	+	0.532096i	$0.178595\pi$
$522$	0	0
$523$	−3946.75	−0.329980	−0.164990	−	0.986295i	$-0.552759\pi$
$523$	−3946.75	−0.329980	−0.164990	+	0.986295i	$0.552759\pi$
$524$	−2539.73	−0.211734
$525$	0	0
$526$	−4298.72	−0.356337
$527$	−3598.79	−0.297468
$528$	0	0
$529$	13324.0	1.09509
$530$	1310.60	0.107413
$531$	0	0
$532$	5444.11	0.443670
$533$	3591.18	0.291841
$534$	0	0
$535$	1224.83	0.0989797
$536$	−2272.19	−0.183104
$537$	0	0
$538$	−2393.47	−0.191803
$539$	−7188.89	−0.574485
$540$	0	0
$541$	20832.3	1.65555	0.827775	−	0.561060i	$-0.189606\pi$
$541$	20832.3	1.65555	0.827775	+	0.561060i	$0.189606\pi$
$542$	20145.6	1.59654
$543$	0	0
$544$	−1132.27	−0.0892387
$545$	−46.4005	−0.00364694
$546$	0	0
$547$	1376.99	0.107634	0.0538171	−	0.998551i	$-0.482861\pi$
$547$	1376.99	0.107634	0.0538171	+	0.998551i	$0.482861\pi$
$548$	−3202.14	−0.249615
$549$	0	0
$550$	−10554.7	−0.818283
$551$	29316.3	2.26664
$552$	0	0
$553$	6562.92	0.504672
$554$	−977.229	−0.0749431
$555$	0	0
$556$	−422.110	−0.0321968
$557$	−15003.2	−1.14130	−0.570651	−	0.821193i	$-0.693309\pi$
$557$	−15003.2	−1.14130	−0.570651	+	0.821193i	$0.693309\pi$
$558$	0	0
$559$	168.863	0.0127767
$560$	1217.70	0.0918877
$561$	0	0
$562$	−14787.2	−1.10989
$563$	3492.65	0.261452	0.130726	−	0.991419i	$-0.458269\pi$
$563$	3492.65	0.261452	0.130726	+	0.991419i	$0.458269\pi$
$564$	0	0
$565$	628.677	0.0468118
$566$	8957.16	0.665190
$567$	0	0
$568$	−18044.9	−1.33301
$569$	−23088.4	−1.70109	−0.850543	−	0.525906i	$-0.823727\pi$
$569$	−23088.4	−1.70109	−0.850543	+	0.525906i	$0.823727\pi$
$570$	0	0
$571$	18697.5	1.37035	0.685173	−	0.728381i	$-0.259727\pi$
$571$	18697.5	1.37035	0.685173	+	0.728381i	$0.259727\pi$
$572$	−767.315	−0.0560892
$573$	0	0
$574$	16867.0	1.22651
$575$	19887.8	1.44240
$576$	0	0
$577$	2168.48	0.156456	0.0782280	−	0.996935i	$-0.475074\pi$
$577$	2168.48	0.156456	0.0782280	+	0.996935i	$0.475074\pi$
$578$	14643.9	1.05382
$579$	0	0
$580$	−260.914	−0.0186791
$581$	37258.6	2.66050
$582$	0	0
$583$	17295.5	1.22865
$584$	10361.9	0.734211
$585$	0	0
$586$	10558.2	0.744289
$587$	−633.214	−0.0445239	−0.0222619	−	0.999752i	$-0.507087\pi$
$587$	−633.214	−0.0445239	−0.0222619	+	0.999752i	$0.507087\pi$
$588$	0	0
$589$	−31249.7	−2.18611
$590$	319.081	0.0222650
$591$	0	0
$592$	−3705.46	−0.257252
$593$	−7239.39	−0.501326	−0.250663	−	0.968074i	$-0.580649\pi$
$593$	−7239.39	−0.501326	−0.250663	+	0.968074i	$0.580649\pi$
$594$	0	0
$595$	239.884	0.0165282
$596$	−1841.55	−0.126565
$597$	0	0
$598$	8149.07	0.557258
$599$	8371.89	0.571062	0.285531	−	0.958369i	$-0.407830\pi$
$599$	8371.89	0.571062	0.285531	+	0.958369i	$0.407830\pi$
$600$	0	0
$601$	20639.4	1.40083	0.700414	−	0.713737i	$-0.252999\pi$
$601$	20639.4	1.40083	0.700414	+	0.713737i	$0.252999\pi$
$602$	793.114	0.0536959
$603$	0	0
$604$	3409.72	0.229701
$605$	−391.345	−0.0262982
$606$	0	0
$607$	−4104.54	−0.274462	−0.137231	−	0.990539i	$-0.543820\pi$
$607$	−4104.54	−0.274462	−0.137231	+	0.990539i	$0.543820\pi$
$608$	−9831.98	−0.655821
$609$	0	0
$610$	440.209	0.0292189
$611$	−5520.61	−0.365532
$612$	0	0
$613$	27181.4	1.79094	0.895470	−	0.445122i	$-0.146840\pi$
$613$	27181.4	1.79094	0.895470	+	0.445122i	$0.146840\pi$
$614$	−2622.49	−0.172370
$615$	0	0
$616$	13105.0	0.857167
$617$	−23040.0	−1.50333	−0.751667	−	0.659543i	$-0.770750\pi$
$617$	−23040.0	−1.50333	−0.751667	+	0.659543i	$0.770750\pi$
$618$	0	0
$619$	−3094.33	−0.200924	−0.100462	−	0.994941i	$-0.532032\pi$
$619$	−3094.33	−0.200924	−0.100462	+	0.994941i	$0.532032\pi$
$620$	278.121	0.0180155
$621$	0	0
$622$	−5768.74	−0.371874
$623$	−2383.48	−0.153278
$624$	0	0
$625$	15461.7	0.989551
$626$	−29548.3	−1.88656
$627$	0	0
$628$	−4116.49	−0.261570
$629$	−729.968	−0.0462730
$630$	0	0
$631$	−4145.71	−0.261550	−0.130775	−	0.991412i	$-0.541747\pi$
$631$	−4145.71	−0.261550	−0.130775	+	0.991412i	$0.541747\pi$
$632$	−5209.90	−0.327909
$633$	0	0
$634$	−16307.2	−1.02152
$635$	1055.00	0.0659312
$636$	0	0
$637$	4330.34	0.269348
$638$	−19406.9	−1.20427
$639$	0	0
$640$	−1144.98	−0.0707178
$641$	22809.7	1.40550	0.702752	−	0.711435i	$-0.251954\pi$
$641$	22809.7	1.40550	0.702752	+	0.711435i	$0.251954\pi$
$642$	0	0
$643$	6567.15	0.402773	0.201386	−	0.979512i	$-0.435455\pi$
$643$	6567.15	0.402773	0.201386	+	0.979512i	$0.435455\pi$
$644$	6790.67	0.415512
$645$	0	0
$646$	−5884.13	−0.358372
$647$	−4595.13	−0.279217	−0.139608	−	0.990207i	$-0.544584\pi$
$647$	−4595.13	−0.279217	−0.139608	+	0.990207i	$0.544584\pi$
$648$	0	0
$649$	4210.79	0.254681
$650$	6357.81	0.383652
$651$	0	0
$652$	−3954.91	−0.237555
$653$	−12184.0	−0.730164	−0.365082	−	0.930975i	$-0.618959\pi$
$653$	−12184.0	−0.730164	−0.365082	+	0.930975i	$0.618959\pi$
$654$	0	0
$655$	−971.743	−0.0579681
$656$	−16418.6	−0.977192
$657$	0	0
$658$	−25929.1	−1.53620
$659$	−13578.8	−0.802662	−0.401331	−	0.915933i	$-0.631452\pi$
$659$	−13578.8	−0.802662	−0.401331	+	0.915933i	$0.631452\pi$
$660$	0	0
$661$	−15030.5	−0.884448	−0.442224	−	0.896905i	$-0.645810\pi$
$661$	−15030.5	−0.884448	−0.442224	+	0.896905i	$0.645810\pi$
$662$	32969.1	1.93562
$663$	0	0
$664$	−29577.3	−1.72865
$665$	2083.01	0.121467
$666$	0	0
$667$	36567.5	2.12279
$668$	945.477	0.0547629
$669$	0	0
$670$	239.081	0.0137858
$671$	5809.27	0.334224
$672$	0	0
$673$	−9263.40	−0.530576	−0.265288	−	0.964169i	$-0.585467\pi$
$673$	−9263.40	−0.530576	−0.265288	+	0.964169i	$0.585467\pi$
$674$	−22215.8	−1.26962
$675$	0	0
$676$	−3328.73	−0.189391
$677$	−14736.2	−0.836570	−0.418285	−	0.908316i	$-0.637369\pi$
$677$	−14736.2	−0.836570	−0.418285	+	0.908316i	$0.637369\pi$
$678$	0	0
$679$	30240.2	1.70915
$680$	−190.429	−0.0107391
$681$	0	0
$682$	20686.8	1.16149
$683$	−24664.8	−1.38180	−0.690902	−	0.722948i	$-0.742786\pi$
$683$	−24664.8	−1.38180	−0.690902	+	0.722948i	$0.742786\pi$
$684$	0	0
$685$	−1225.19	−0.0683390
$686$	−6027.88	−0.335489
$687$	0	0
$688$	−772.028	−0.0427809
$689$	−10418.2	−0.576055
$690$	0	0
$691$	−9868.43	−0.543289	−0.271644	−	0.962398i	$-0.587567\pi$
$691$	−9868.43	−0.543289	−0.271644	+	0.962398i	$0.587567\pi$
$692$	2437.65	0.133910
$693$	0	0
$694$	−8488.88	−0.464313
$695$	−161.506	−0.00881478
$696$	0	0
$697$	−3234.43	−0.175771
$698$	24583.2	1.33308
$699$	0	0
$700$	5298.01	0.286065
$701$	5389.23	0.290368	0.145184	−	0.989405i	$-0.453623\pi$
$701$	5389.23	0.290368	0.145184	+	0.989405i	$0.453623\pi$
$702$	0	0
$703$	−6338.60	−0.340064
$704$	−9756.06	−0.522294
$705$	0	0
$706$	8613.05	0.459145
$707$	−321.569	−0.0171059
$708$	0	0
$709$	457.173	0.0242165	0.0121082	−	0.999927i	$-0.496146\pi$
$709$	457.173	0.0242165	0.0121082	+	0.999927i	$0.496146\pi$
$710$	1898.70	0.100362
$711$	0	0
$712$	1892.10	0.0995917
$713$	−38979.0	−2.04737
$714$	0	0
$715$	−293.587	−0.0153560
$716$	−7564.25	−0.394818
$717$	0	0
$718$	28399.5	1.47613
$719$	−19617.0	−1.01751	−0.508756	−	0.860911i	$-0.669894\pi$
$719$	−19617.0	−1.01751	−0.508756	+	0.860911i	$0.669894\pi$
$720$	0	0
$721$	25969.0	1.34138
$722$	−29703.9	−1.53112
$723$	0	0
$724$	−3327.78	−0.170823
$725$	28529.5	1.46146
$726$	0	0
$727$	1026.58	0.0523711	0.0261855	−	0.999657i	$-0.491664\pi$
$727$	1026.58	0.0523711	0.0261855	+	0.999657i	$0.491664\pi$
$728$	−7893.99	−0.401883
$729$	0	0
$730$	−1090.29	−0.0552786
$731$	−152.088	−0.00769518
$732$	0	0
$733$	−32818.2	−1.65371	−0.826853	−	0.562418i	$-0.809871\pi$
$733$	−32818.2	−1.65371	−0.826853	+	0.562418i	$0.809871\pi$
$734$	−27745.1	−1.39522
$735$	0	0
$736$	−12263.8	−0.614200
$737$	3155.06	0.157691
$738$	0	0
$739$	−30401.7	−1.51332	−0.756660	−	0.653809i	$-0.773170\pi$
$739$	−30401.7	−1.51332	−0.756660	+	0.653809i	$0.773170\pi$
$740$	56.4133	0.00280242
$741$	0	0
$742$	−48932.0	−2.42096
$743$	−28686.3	−1.41642	−0.708208	−	0.706004i	$-0.750496\pi$
$743$	−28686.3	−1.41642	−0.708208	+	0.706004i	$0.750496\pi$
$744$	0	0
$745$	−704.608	−0.0346508
$746$	−25365.8	−1.24492
$747$	0	0
$748$	691.088	0.0337817
$749$	−45729.9	−2.23089
$750$	0	0
$751$	14134.7	0.686795	0.343398	−	0.939190i	$-0.388422\pi$
$751$	14134.7	0.686795	0.343398	+	0.939190i	$0.388422\pi$
$752$	25239.8	1.22394
$753$	0	0
$754$	11690.1	0.564624
$755$	1304.62	0.0628872
$756$	0	0
$757$	−20252.9	−0.972398	−0.486199	−	0.873848i	$-0.661617\pi$
$757$	−20252.9	−0.972398	−0.486199	+	0.873848i	$0.661617\pi$
$758$	−20442.4	−0.979553
$759$	0	0
$760$	−1653.57	−0.0789227
$761$	29869.1	1.42280	0.711402	−	0.702786i	$-0.248061\pi$
$761$	29869.1	1.42280	0.711402	+	0.702786i	$0.248061\pi$
$762$	0	0
$763$	1732.39	0.0821977
$764$	−8267.79	−0.391516
$765$	0	0
$766$	41255.9	1.94600
$767$	−2536.44	−0.119407
$768$	0	0
$769$	−14902.4	−0.698821	−0.349410	−	0.936970i	$-0.613618\pi$
$769$	−14902.4	−0.698821	−0.349410	+	0.936970i	$0.613618\pi$
$770$	−1378.92	−0.0645359
$771$	0	0
$772$	−2098.04	−0.0978109
$773$	24845.3	1.15604	0.578022	−	0.816021i	$-0.303825\pi$
$773$	24845.3	1.15604	0.578022	+	0.816021i	$0.303825\pi$
$774$	0	0
$775$	−30411.0	−1.40954
$776$	−24005.8	−1.11051
$777$	0	0
$778$	−24978.8	−1.15107
$779$	−28085.8	−1.29176
$780$	0	0
$781$	25056.4	1.14800
$782$	−7339.52	−0.335628
$783$	0	0
$784$	−19797.9	−0.901874
$785$	−1575.04	−0.0716120
$786$	0	0
$787$	−1929.10	−0.0873760	−0.0436880	−	0.999045i	$-0.513911\pi$
$787$	−1929.10	−0.0873760	−0.0436880	+	0.999045i	$0.513911\pi$
$788$	−7578.35	−0.342598
$789$	0	0
$790$	548.189	0.0246882
$791$	−23472.1	−1.05508
$792$	0	0
$793$	−3499.31	−0.156701
$794$	−20159.1	−0.901034
$795$	0	0
$796$	8290.64	0.369163
$797$	−30699.3	−1.36440	−0.682199	−	0.731166i	$-0.738976\pi$
$797$	−30699.3	−1.36440	−0.682199	+	0.731166i	$0.738976\pi$
$798$	0	0
$799$	4972.18	0.220154
$800$	−9568.11	−0.422855
$801$	0	0
$802$	−41828.8	−1.84168
$803$	−14388.1	−0.632311
$804$	0	0
$805$	2598.22	0.113758
$806$	−12461.0	−0.544565
$807$	0	0
$808$	255.274	0.0111145
$809$	−1299.04	−0.0564547	−0.0282274	−	0.999602i	$-0.508986\pi$
$809$	−1299.04	−0.0564547	−0.0282274	+	0.999602i	$0.508986\pi$
$810$	0	0
$811$	−44420.8	−1.92333	−0.961667	−	0.274220i	$-0.911580\pi$
$811$	−44420.8	−1.92333	−0.961667	+	0.274220i	$0.911580\pi$
$812$	9741.40	0.421005
$813$	0	0
$814$	4196.04	0.180677
$815$	−1513.21	−0.0650374
$816$	0	0
$817$	−1320.64	−0.0565524
$818$	18606.5	0.795307
$819$	0	0
$820$	249.963	0.0106452
$821$	13347.3	0.567388	0.283694	−	0.958915i	$-0.408440\pi$
$821$	13347.3	0.567388	0.283694	+	0.958915i	$0.408440\pi$
$822$	0	0
$823$	22743.1	0.963272	0.481636	−	0.876371i	$-0.340043\pi$
$823$	22743.1	0.963272	0.481636	+	0.876371i	$0.340043\pi$
$824$	−20615.2	−0.871558
$825$	0	0
$826$	−11913.1	−0.501827
$827$	42160.0	1.77273	0.886364	−	0.462988i	$-0.153223\pi$
$827$	42160.0	1.77273	0.886364	+	0.462988i	$0.153223\pi$
$828$	0	0
$829$	23741.5	0.994662	0.497331	−	0.867561i	$-0.334313\pi$
$829$	23741.5	0.994662	0.497331	+	0.867561i	$0.334313\pi$
$830$	3112.14	0.130150
$831$	0	0
$832$	5876.71	0.244878
$833$	−3900.16	−0.162224
$834$	0	0
$835$	361.755	0.0149929
$836$	6000.99	0.248264
$837$	0	0
$838$	42660.7	1.75858
$839$	42049.6	1.73029	0.865145	−	0.501522i	$-0.167227\pi$
$839$	42049.6	1.73029	0.865145	+	0.501522i	$0.167227\pi$
$840$	0	0
$841$	28068.0	1.15085
$842$	9943.59	0.406982
$843$	0	0
$844$	859.907	0.0350702
$845$	−1273.63	−0.0518510
$846$	0	0
$847$	14611.1	0.592731
$848$	47631.1	1.92884
$849$	0	0
$850$	−5726.21	−0.231068
$851$	−7906.39	−0.318481
$852$	0	0
$853$	−22446.8	−0.901012	−0.450506	−	0.892773i	$-0.648756\pi$
$853$	−22446.8	−0.901012	−0.450506	+	0.892773i	$0.648756\pi$
$854$	−16435.5	−0.658561
$855$	0	0
$856$	36302.1	1.44951
$857$	−2186.19	−0.0871400	−0.0435700	−	0.999050i	$-0.513873\pi$
$857$	−2186.19	−0.0871400	−0.0435700	+	0.999050i	$0.513873\pi$
$858$	0	0
$859$	11510.4	0.457195	0.228597	−	0.973521i	$-0.426586\pi$
$859$	11510.4	0.457195	0.228597	+	0.973521i	$0.426586\pi$
$860$	11.7536	0.000466042 0
$861$	0	0
$862$	−4265.58	−0.168546
$863$	5804.81	0.228966	0.114483	−	0.993425i	$-0.463479\pi$
$863$	5804.81	0.228966	0.114483	+	0.993425i	$0.463479\pi$
$864$	0	0
$865$	932.687	0.0366616
$866$	−33971.6	−1.33303
$867$	0	0
$868$	−10383.8	−0.406048
$869$	7234.24	0.282399
$870$	0	0
$871$	−1900.50	−0.0739334
$872$	−1375.24	−0.0534076
$873$	0	0
$874$	−63731.9	−2.46655
$875$	4061.30	0.156911
$876$	0	0
$877$	−19351.9	−0.745115	−0.372558	−	0.928009i	$-0.621519\pi$
$877$	−19351.9	−0.745115	−0.372558	+	0.928009i	$0.621519\pi$
$878$	4479.35	0.172176
$879$	0	0
$880$	1342.25	0.0514175
$881$	−9779.74	−0.373993	−0.186997	−	0.982361i	$-0.559875\pi$
$881$	−9779.74	−0.373993	−0.186997	+	0.982361i	$0.559875\pi$
$882$	0	0
$883$	−23777.0	−0.906181	−0.453091	−	0.891464i	$-0.649679\pi$
$883$	−23777.0	−0.906181	−0.453091	+	0.891464i	$0.649679\pi$
$884$	−416.288	−0.0158385
$885$	0	0
$886$	22900.1	0.868335
$887$	2870.26	0.108652	0.0543258	−	0.998523i	$-0.482699\pi$
$887$	2870.26	0.108652	0.0543258	+	0.998523i	$0.482699\pi$
$888$	0	0
$889$	−39389.0	−1.48601
$890$	−199.088	−0.00749824
$891$	0	0
$892$	−433.827	−0.0162843
$893$	43175.4	1.61793
$894$	0	0
$895$	−2894.21	−0.108092
$896$	42748.6	1.59390
$897$	0	0
$898$	−39056.7	−1.45138
$899$	−55916.5	−2.07444
$900$	0	0
$901$	9383.23	0.346948
$902$	18592.3	0.686316
$903$	0	0
$904$	18633.0	0.685536
$905$	−1273.26	−0.0467677
$906$	0	0
$907$	−13724.3	−0.502433	−0.251216	−	0.967931i	$-0.580831\pi$
$907$	−13724.3	−0.502433	−0.251216	+	0.967931i	$0.580831\pi$
$908$	5863.37	0.214298
$909$	0	0
$910$	830.611	0.0302577
$911$	29887.9	1.08697	0.543485	−	0.839419i	$-0.317105\pi$
$911$	29887.9	1.08697	0.543485	+	0.839419i	$0.317105\pi$
$912$	0	0
$913$	41069.8	1.48873
$914$	−47764.4	−1.72856
$915$	0	0
$916$	9277.76	0.334657
$917$	36280.6	1.30653
$918$	0	0
$919$	−11804.2	−0.423704	−0.211852	−	0.977302i	$-0.567949\pi$
$919$	−11804.2	−0.423704	−0.211852	+	0.977302i	$0.567949\pi$
$920$	−2062.57	−0.0739139
$921$	0	0
$922$	4006.26	0.143101
$923$	−15093.1	−0.538240
$924$	0	0
$925$	−6168.48	−0.219263
$926$	12945.8	0.459424
$927$	0	0
$928$	−17592.8	−0.622319
$929$	16263.3	0.574362	0.287181	−	0.957876i	$-0.407282\pi$
$929$	16263.3	0.574362	0.287181	+	0.957876i	$0.407282\pi$
$930$	0	0
$931$	−33866.6	−1.19219
$932$	916.599	0.0322148
$933$	0	0
$934$	11307.9	0.396152
$935$	264.422	0.00924867
$936$	0	0
$937$	−55368.4	−1.93042	−0.965211	−	0.261470i	$-0.915793\pi$
$937$	−55368.4	−1.93042	−0.965211	+	0.261470i	$0.915793\pi$
$938$	−8926.24	−0.310717
$939$	0	0
$940$	−384.260	−0.0133332
$941$	50310.4	1.74290	0.871452	−	0.490480i	$-0.163179\pi$
$941$	50310.4	1.74290	0.871452	+	0.490480i	$0.163179\pi$
$942$	0	0
$943$	−35032.6	−1.20978
$944$	11596.4	0.399819
$945$	0	0
$946$	874.241	0.0300465
$947$	42285.0	1.45098	0.725490	−	0.688232i	$-0.241613\pi$
$947$	42285.0	1.45098	0.725490	+	0.688232i	$0.241613\pi$
$948$	0	0
$949$	8666.90	0.296459
$950$	−49723.0	−1.69813
$951$	0	0
$952$	7109.78	0.242048
$953$	−37108.8	−1.26136	−0.630678	−	0.776044i	$-0.717223\pi$
$953$	−37108.8	−1.26136	−0.630678	+	0.776044i	$0.717223\pi$
$954$	0	0
$955$	−3163.39	−0.107188
$956$	412.396	0.0139517
$957$	0	0
$958$	8018.53	0.270425
$959$	45743.3	1.54028
$960$	0	0
$961$	29813.0	1.00074
$962$	−2527.55	−0.0847105
$963$	0	0
$964$	5953.52	0.198911
$965$	−802.743	−0.0267785
$966$	0	0
$967$	−8564.91	−0.284828	−0.142414	−	0.989807i	$-0.545486\pi$
$967$	−8564.91	−0.284828	−0.142414	+	0.989807i	$0.545486\pi$
$968$	−11598.8	−0.385125
$969$	0	0
$970$	2525.91	0.0836104
$971$	26338.3	0.870481	0.435240	−	0.900314i	$-0.356663\pi$
$971$	26338.3	0.870481	0.435240	+	0.900314i	$0.356663\pi$
$972$	0	0
$973$	6029.93	0.198675
$974$	−19110.9	−0.628698
$975$	0	0
$976$	15998.5	0.524693
$977$	−2099.04	−0.0687351	−0.0343676	−	0.999409i	$-0.510942\pi$
$977$	−2099.04	−0.0687351	−0.0343676	+	0.999409i	$0.510942\pi$
$978$	0	0
$979$	−2627.28	−0.0857695
$980$	301.411	0.00982473
$981$	0	0
$982$	−2265.02	−0.0736045
$983$	−27194.6	−0.882375	−0.441187	−	0.897415i	$-0.645443\pi$
$983$	−27194.6	−0.882375	−0.441187	+	0.897415i	$0.645443\pi$
$984$	0	0
$985$	−2899.60	−0.0937959
$986$	−10528.7	−0.340064
$987$	0	0
$988$	−3614.79	−0.116399
$989$	−1647.29	−0.0529633
$990$	0	0
$991$	28481.7	0.912967	0.456483	−	0.889732i	$-0.349109\pi$
$991$	28481.7	0.912967	0.456483	+	0.889732i	$0.349109\pi$
$992$	18753.0	0.600210
$993$	0	0
$994$	−70889.0	−2.26203
$995$	3172.14	0.101069
$996$	0	0
$997$	−50629.4	−1.60827	−0.804137	−	0.594443i	$-0.797372\pi$
$997$	−50629.4	−1.60827	−0.804137	+	0.594443i	$0.797372\pi$
$998$	−29769.6	−0.944229
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.8		32
3.2	odd	2		717.4.a.c.1.25	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.25	✓	32	3.2	odd	2
2151.4.a.e.1.8		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)

\(f(q)\)	\(=\)	\(q-3.11857 q^{2} +1.72551 q^{4} +0.660207 q^{5} -24.6492 q^{7} +19.5675 q^{8} +O(q^{10})\)	\(q-3.11857 q^{2} +1.72551 q^{4} +0.660207 q^{5} -24.6492 q^{7} +19.5675 q^{8} -2.05890 q^{10} -27.1706 q^{11} +16.3666 q^{13} +76.8704 q^{14} -74.8267 q^{16} -14.7407 q^{17} -127.999 q^{19} +1.13919 q^{20} +84.7334 q^{22} -159.659 q^{23} -124.564 q^{25} -51.0405 q^{26} -42.5324 q^{28} -229.035 q^{29} +244.139 q^{31} +76.8127 q^{32} +45.9700 q^{34} -16.2736 q^{35} +49.5205 q^{37} +399.176 q^{38} +12.9186 q^{40} +219.421 q^{41} +10.3175 q^{43} -46.8829 q^{44} +497.908 q^{46} -337.310 q^{47} +264.584 q^{49} +388.462 q^{50} +28.2407 q^{52} -636.552 q^{53} -17.9382 q^{55} -482.323 q^{56} +714.263 q^{58} -154.976 q^{59} -213.808 q^{61} -761.367 q^{62} +359.067 q^{64} +10.8053 q^{65} -116.121 q^{67} -25.4352 q^{68} +50.7504 q^{70} -922.188 q^{71} +529.548 q^{73} -154.433 q^{74} -220.864 q^{76} +669.733 q^{77} -266.253 q^{79} -49.4011 q^{80} -684.282 q^{82} -1511.55 q^{83} -9.73192 q^{85} -32.1760 q^{86} -531.659 q^{88} +96.6959 q^{89} -403.424 q^{91} -275.492 q^{92} +1051.93 q^{94} -84.5060 q^{95} -1226.82 q^{97} -825.125 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

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Varieties

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Database

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