LMFDB - Embedded newform 2151.4.a.g.1.55

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			1.55
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.09369 q^{2} +17.9457 q^{4} +15.2658 q^{5} -21.0984 q^{7} +50.6602 q^{8} +O(q^{10})$	\(q+5.09369 q^{2} +17.9457 q^{4} +15.2658 q^{5} -21.0984 q^{7} +50.6602 q^{8} +77.7594 q^{10} -65.8349 q^{11} -72.1125 q^{13} -107.469 q^{14} +114.482 q^{16} -51.9062 q^{17} +10.1429 q^{19} +273.955 q^{20} -335.343 q^{22} +9.17600 q^{23} +108.045 q^{25} -367.319 q^{26} -378.625 q^{28} -32.6852 q^{29} +212.392 q^{31} +177.853 q^{32} -264.394 q^{34} -322.084 q^{35} -53.7021 q^{37} +51.6646 q^{38} +773.369 q^{40} -49.4038 q^{41} -259.609 q^{43} -1181.45 q^{44} +46.7397 q^{46} -276.825 q^{47} +102.141 q^{49} +550.349 q^{50} -1294.11 q^{52} -226.029 q^{53} -1005.02 q^{55} -1068.85 q^{56} -166.488 q^{58} +894.934 q^{59} -118.405 q^{61} +1081.86 q^{62} -9.92451 q^{64} -1100.86 q^{65} -270.762 q^{67} -931.492 q^{68} -1640.60 q^{70} -314.378 q^{71} +338.874 q^{73} -273.542 q^{74} +182.021 q^{76} +1389.01 q^{77} -716.571 q^{79} +1747.66 q^{80} -251.648 q^{82} +632.265 q^{83} -792.391 q^{85} -1322.37 q^{86} -3335.21 q^{88} -187.187 q^{89} +1521.46 q^{91} +164.670 q^{92} -1410.06 q^{94} +154.839 q^{95} -483.286 q^{97} +520.277 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.09369	1.80089	0.900446	−	0.434969i	$-0.143241\pi$
$2$	5.09369	1.80089	0.900446	+	0.434969i	$0.143241\pi$
$3$	0	0
$3$	0	0
$4$	17.9457	2.24321
$5$	15.2658	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$5$	15.2658	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$6$	0	0
$7$	−21.0984	−1.13921	−0.569603	−	0.821920i	$-0.692903\pi$
$7$	−21.0984	−1.13921	−0.569603	+	0.821920i	$0.692903\pi$
$8$	50.6602	2.23888
$9$	0	0
$10$	77.7594	2.45897
$11$	−65.8349	−1.80454	−0.902272	−	0.431167i	$-0.858102\pi$
$11$	−65.8349	−1.80454	−0.902272	+	0.431167i	$0.858102\pi$
$12$	0	0
$13$	−72.1125	−1.53849	−0.769247	−	0.638952i	$-0.779368\pi$
$13$	−72.1125	−1.53849	−0.769247	+	0.638952i	$0.779368\pi$
$14$	−107.469	−2.05158
$15$	0	0
$16$	114.482	1.78878
$17$	−51.9062	−0.740536	−0.370268	−	0.928925i	$-0.620734\pi$
$17$	−51.9062	−0.740536	−0.370268	+	0.928925i	$0.620734\pi$
$18$	0	0
$19$	10.1429	0.122470	0.0612351	−	0.998123i	$-0.480496\pi$
$19$	10.1429	0.122470	0.0612351	+	0.998123i	$0.480496\pi$
$20$	273.955	3.06292
$21$	0	0
$22$	−335.343	−3.24979
$23$	9.17600	0.0831882	0.0415941	−	0.999135i	$-0.486756\pi$
$23$	9.17600	0.0831882	0.0415941	+	0.999135i	$0.486756\pi$
$24$	0	0
$25$	108.045	0.864363
$26$	−367.319	−2.77066
$27$	0	0
$28$	−378.625	−2.55548
$29$	−32.6852	−0.209292	−0.104646	−	0.994510i	$-0.533371\pi$
$29$	−32.6852	−0.209292	−0.104646	+	0.994510i	$0.533371\pi$
$30$	0	0
$31$	212.392	1.23054	0.615270	−	0.788317i	$-0.289047\pi$
$31$	212.392	1.23054	0.615270	+	0.788317i	$0.289047\pi$
$32$	177.853	0.982510
$33$	0	0
$34$	−264.394	−1.33362
$35$	−322.084	−1.55549
$36$	0	0
$37$	−53.7021	−0.238610	−0.119305	−	0.992858i	$-0.538067\pi$
$37$	−53.7021	−0.238610	−0.119305	+	0.992858i	$0.538067\pi$
$38$	51.6646	0.220556
$39$	0	0
$40$	773.369	3.05701
$41$	−49.4038	−0.188185	−0.0940925	−	0.995563i	$-0.529995\pi$
$41$	−49.4038	−0.188185	−0.0940925	+	0.995563i	$0.529995\pi$
$42$	0	0
$43$	−259.609	−0.920697	−0.460348	−	0.887738i	$-0.652276\pi$
$43$	−259.609	−0.920697	−0.460348	+	0.887738i	$0.652276\pi$
$44$	−1181.45	−4.04797
$45$	0	0
$46$	46.7397	0.149813
$47$	−276.825	−0.859131	−0.429565	−	0.903036i	$-0.641333\pi$
$47$	−276.825	−0.859131	−0.429565	+	0.903036i	$0.641333\pi$
$48$	0	0
$49$	102.141	0.297788
$50$	550.349	1.55662
$51$	0	0
$52$	−1294.11	−3.45116
$53$	−226.029	−0.585800	−0.292900	−	0.956143i	$-0.594620\pi$
$53$	−226.029	−0.585800	−0.292900	+	0.956143i	$0.594620\pi$
$54$	0	0
$55$	−1005.02	−2.46395
$56$	−1068.85	−2.55055
$57$	0	0
$58$	−166.488	−0.376913
$59$	894.934	1.97475	0.987377	−	0.158389i	$-0.0506298\pi$
$59$	894.934	1.97475	0.987377	+	0.158389i	$0.0506298\pi$
$60$	0	0
$61$	−118.405	−0.248528	−0.124264	−	0.992249i	$-0.539657\pi$
$61$	−118.405	−0.248528	−0.124264	+	0.992249i	$0.539657\pi$
$62$	1081.86	2.21607
$63$	0	0
$64$	−9.92451	−0.0193838
$65$	−1100.86	−2.10068
$66$	0	0
$67$	−270.762	−0.493715	−0.246857	−	0.969052i	$-0.579398\pi$
$67$	−270.762	−0.493715	−0.246857	+	0.969052i	$0.579398\pi$
$68$	−931.492	−1.66118
$69$	0	0
$70$	−1640.60	−2.80127
$71$	−314.378	−0.525490	−0.262745	−	0.964865i	$-0.584628\pi$
$71$	−314.378	−0.525490	−0.262745	+	0.964865i	$0.584628\pi$
$72$	0	0
$73$	338.874	0.543319	0.271659	−	0.962393i	$-0.412428\pi$
$73$	338.874	0.543319	0.271659	+	0.962393i	$0.412428\pi$
$74$	−273.542	−0.429710
$75$	0	0
$76$	182.021	0.274726
$77$	1389.01	2.05575
$78$	0	0
$79$	−716.571	−1.02051	−0.510257	−	0.860022i	$-0.670450\pi$
$79$	−716.571	−1.02051	−0.510257	+	0.860022i	$0.670450\pi$
$80$	1747.66	2.44243
$81$	0	0
$82$	−251.648	−0.338901
$83$	632.265	0.836146	0.418073	−	0.908413i	$-0.362706\pi$
$83$	632.265	0.836146	0.418073	+	0.908413i	$0.362706\pi$
$84$	0	0
$85$	−792.391	−1.01114
$86$	−1322.37	−1.65807
$87$	0	0
$88$	−3335.21	−4.04017
$89$	−187.187	−0.222941	−0.111471	−	0.993768i	$-0.535556\pi$
$89$	−187.187	−0.222941	−0.111471	+	0.993768i	$0.535556\pi$
$90$	0	0
$91$	1521.46	1.75266
$92$	164.670	0.186609
$93$	0	0
$94$	−1410.06	−1.54720
$95$	154.839	0.167223
$96$	0	0
$97$	−483.286	−0.505879	−0.252939	−	0.967482i	$-0.581397\pi$
$97$	−483.286	−0.505879	−0.252939	+	0.967482i	$0.581397\pi$
$98$	520.277	0.536284
$99$	0	0
$100$	1938.95	1.93895
$101$	883.661	0.870570	0.435285	−	0.900293i	$-0.356648\pi$
$101$	883.661	0.870570	0.435285	+	0.900293i	$0.356648\pi$
$102$	0	0
$103$	−303.031	−0.289889	−0.144945	−	0.989440i	$-0.546300\pi$
$103$	−303.031	−0.289889	−0.144945	+	0.989440i	$0.546300\pi$
$104$	−3653.23	−3.44451
$105$	0	0
$106$	−1151.32	−1.05496
$107$	−287.390	−0.259655	−0.129827	−	0.991537i	$-0.541442\pi$
$107$	−287.390	−0.259655	−0.129827	+	0.991537i	$0.541442\pi$
$108$	0	0
$109$	−1904.73	−1.67376	−0.836880	−	0.547386i	$-0.815623\pi$
$109$	−1904.73	−1.67376	−0.836880	+	0.547386i	$0.815623\pi$
$110$	−5119.28	−4.43731
$111$	0	0
$112$	−2415.38	−2.03779
$113$	−874.738	−0.728217	−0.364108	−	0.931357i	$-0.618626\pi$
$113$	−874.738	−0.728217	−0.364108	+	0.931357i	$0.618626\pi$
$114$	0	0
$115$	140.079	0.113587
$116$	−586.557	−0.469487
$117$	0	0
$118$	4558.52	3.55632
$119$	1095.14	0.843622
$120$	0	0
$121$	3003.24	2.25638
$122$	−603.119	−0.447572
$123$	0	0
$124$	3811.52	2.76036
$125$	−258.827	−0.185202
$126$	0	0
$127$	1384.96	0.967677	0.483839	−	0.875157i	$-0.339242\pi$
$127$	1384.96	0.967677	0.483839	+	0.875157i	$0.339242\pi$
$128$	−1473.38	−1.01742
$129$	0	0
$130$	−5607.42	−3.78311
$131$	−717.700	−0.478670	−0.239335	−	0.970937i	$-0.576929\pi$
$131$	−717.700	−0.478670	−0.239335	+	0.970937i	$0.576929\pi$
$132$	0	0
$133$	−213.998	−0.139519
$134$	−1379.18	−0.889127
$135$	0	0
$136$	−2629.58	−1.65797
$137$	1873.04	1.16806	0.584032	−	0.811731i	$-0.301474\pi$
$137$	1873.04	1.16806	0.584032	+	0.811731i	$0.301474\pi$
$138$	0	0
$139$	1957.96	1.19476	0.597381	−	0.801958i	$-0.296208\pi$
$139$	1957.96	1.19476	0.597381	+	0.801958i	$0.296208\pi$
$140$	−5780.01	−3.48929
$141$	0	0
$142$	−1601.34	−0.946351
$143$	4747.52	2.77628
$144$	0	0
$145$	−498.966	−0.285771
$146$	1726.12	0.978458
$147$	0	0
$148$	−963.720	−0.535252
$149$	−238.440	−0.131099	−0.0655497	−	0.997849i	$-0.520880\pi$
$149$	−238.440	−0.131099	−0.0655497	+	0.997849i	$0.520880\pi$
$150$	0	0
$151$	2937.31	1.58301	0.791507	−	0.611161i	$-0.209297\pi$
$151$	2937.31	1.58301	0.791507	+	0.611161i	$0.209297\pi$
$152$	513.840	0.274197
$153$	0	0
$154$	7075.19	3.70217
$155$	3242.34	1.68020
$156$	0	0
$157$	−3524.14	−1.79145	−0.895723	−	0.444613i	$-0.853341\pi$
$157$	−3524.14	−1.79145	−0.895723	+	0.444613i	$0.853341\pi$
$158$	−3649.99	−1.83783
$159$	0	0
$160$	2715.08	1.34154
$161$	−193.599	−0.0947684
$162$	0	0
$163$	−1629.25	−0.782901	−0.391451	−	0.920199i	$-0.628027\pi$
$163$	−1629.25	−0.782901	−0.391451	+	0.920199i	$0.628027\pi$
$164$	−886.585	−0.422138
$165$	0	0
$166$	3220.56	1.50581
$167$	3327.84	1.54201	0.771005	−	0.636829i	$-0.219754\pi$
$167$	3327.84	1.54201	0.771005	+	0.636829i	$0.219754\pi$
$168$	0	0
$169$	3003.22	1.36696
$170$	−4036.19	−1.82095
$171$	0	0
$172$	−4658.85	−2.06532
$173$	4303.92	1.89145	0.945725	−	0.324969i	$-0.105354\pi$
$173$	4303.92	1.89145	0.945725	+	0.324969i	$0.105354\pi$
$174$	0	0
$175$	−2279.58	−0.984686
$176$	−7536.90	−3.22793
$177$	0	0
$178$	−953.473	−0.401493
$179$	−812.139	−0.339118	−0.169559	−	0.985520i	$-0.554234\pi$
$179$	−812.139	−0.339118	−0.169559	+	0.985520i	$0.554234\pi$
$180$	0	0
$181$	2852.30	1.17132	0.585662	−	0.810555i	$-0.300834\pi$
$181$	2852.30	1.17132	0.585662	+	0.810555i	$0.300834\pi$
$182$	7749.83	3.15635
$183$	0	0
$184$	464.858	0.186249
$185$	−819.806	−0.325802
$186$	0	0
$187$	3417.24	1.33633
$188$	−4967.82	−1.92721
$189$	0	0
$190$	788.703	0.301150
$191$	470.301	0.178166	0.0890831	−	0.996024i	$-0.471606\pi$
$191$	470.301	0.178166	0.0890831	+	0.996024i	$0.471606\pi$
$192$	0	0
$193$	597.809	0.222960	0.111480	−	0.993767i	$-0.464441\pi$
$193$	597.809	0.222960	0.111480	+	0.993767i	$0.464441\pi$
$194$	−2461.71	−0.911033
$195$	0	0
$196$	1833.00	0.668001
$197$	3383.20	1.22357	0.611784	−	0.791025i	$-0.290452\pi$
$197$	3383.20	1.22357	0.611784	+	0.791025i	$0.290452\pi$
$198$	0	0
$199$	−2613.38	−0.930944	−0.465472	−	0.885063i	$-0.654115\pi$
$199$	−2613.38	−0.930944	−0.465472	+	0.885063i	$0.654115\pi$
$200$	5473.59	1.93521
$201$	0	0
$202$	4501.09	1.56780
$203$	689.604	0.238427
$204$	0	0
$205$	−754.190	−0.256951
$206$	−1543.55	−0.522059
$207$	0	0
$208$	−8255.57	−2.75202
$209$	−667.755	−0.221003
$210$	0	0
$211$	−5044.71	−1.64593	−0.822967	−	0.568089i	$-0.807683\pi$
$211$	−5044.71	−1.64593	−0.822967	+	0.568089i	$0.807683\pi$
$212$	−4056.24	−1.31407
$213$	0	0
$214$	−1463.88	−0.467610
$215$	−3963.14	−1.25713
$216$	0	0
$217$	−4481.12	−1.40184
$218$	−9702.09	−3.01426
$219$	0	0
$220$	−18035.8	−5.52716
$221$	3743.09	1.13931
$222$	0	0
$223$	1710.18	0.513553	0.256777	−	0.966471i	$-0.417340\pi$
$223$	1710.18	0.513553	0.256777	+	0.966471i	$0.417340\pi$
$224$	−3752.42	−1.11928
$225$	0	0
$226$	−4455.65	−1.31144
$227$	−4856.18	−1.41989	−0.709947	−	0.704255i	$-0.751281\pi$
$227$	−4856.18	−1.41989	−0.709947	+	0.704255i	$0.751281\pi$
$228$	0	0
$229$	−6598.10	−1.90399	−0.951997	−	0.306108i	$-0.900973\pi$
$229$	−6598.10	−1.90399	−0.951997	+	0.306108i	$0.900973\pi$
$230$	713.520	0.204557
$231$	0	0
$232$	−1655.84	−0.468582
$233$	3530.52	0.992670	0.496335	−	0.868131i	$-0.334679\pi$
$233$	3530.52	0.992670	0.496335	+	0.868131i	$0.334679\pi$
$234$	0	0
$235$	−4225.97	−1.17307
$236$	16060.2	4.42979
$237$	0	0
$238$	5578.29	1.51927
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−894.168	−0.238998	−0.119499	−	0.992834i	$-0.538129\pi$
$241$	−894.168	−0.238998	−0.119499	+	0.992834i	$0.538129\pi$
$242$	15297.6	4.06349
$243$	0	0
$244$	−2124.86	−0.557501
$245$	1559.27	0.406605
$246$	0	0
$247$	−731.428	−0.188420
$248$	10759.8	2.75503
$249$	0	0
$250$	−1318.38	−0.333528
$251$	−6830.85	−1.71777	−0.858884	−	0.512171i	$-0.828842\pi$
$251$	−6830.85	−1.71777	−0.858884	+	0.512171i	$0.828842\pi$
$252$	0	0
$253$	−604.102	−0.150117
$254$	7054.54	1.74268
$255$	0	0
$256$	−7425.54	−1.81288
$257$	−5522.18	−1.34033	−0.670164	−	0.742213i	$-0.733776\pi$
$257$	−5522.18	−1.34033	−0.670164	+	0.742213i	$0.733776\pi$
$258$	0	0
$259$	1133.03	0.271826
$260$	−19755.6	−4.71228
$261$	0	0
$262$	−3655.74	−0.862032
$263$	1758.65	0.412330	0.206165	−	0.978517i	$-0.433902\pi$
$263$	1758.65	0.412330	0.206165	+	0.978517i	$0.433902\pi$
$264$	0	0
$265$	−3450.51	−0.799862
$266$	−1090.04	−0.251258
$267$	0	0
$268$	−4859.01	−1.10751
$269$	4098.64	0.928990	0.464495	−	0.885576i	$-0.346236\pi$
$269$	4098.64	0.928990	0.464495	+	0.885576i	$0.346236\pi$
$270$	0	0
$271$	708.798	0.158880	0.0794398	−	0.996840i	$-0.474687\pi$
$271$	708.798	0.158880	0.0794398	+	0.996840i	$0.474687\pi$
$272$	−5942.32	−1.32465
$273$	0	0
$274$	9540.69	2.10356
$275$	−7113.16	−1.55978
$276$	0	0
$277$	−1802.14	−0.390904	−0.195452	−	0.980713i	$-0.562617\pi$
$277$	−1802.14	−0.390904	−0.195452	+	0.980713i	$0.562617\pi$
$278$	9973.23	2.15164
$279$	0	0
$280$	−16316.8	−3.48256
$281$	−5937.56	−1.26052	−0.630258	−	0.776386i	$-0.717051\pi$
$281$	−5937.56	−1.26052	−0.630258	+	0.776386i	$0.717051\pi$
$282$	0	0
$283$	2616.37	0.549566	0.274783	−	0.961506i	$-0.411394\pi$
$283$	2616.37	0.549566	0.274783	+	0.961506i	$0.411394\pi$
$284$	−5641.73	−1.17878
$285$	0	0
$286$	24182.4	4.99978
$287$	1042.34	0.214381
$288$	0	0
$289$	−2218.74	−0.451607
$290$	−2541.58	−0.514643
$291$	0	0
$292$	6081.33	1.21878
$293$	3627.53	0.723285	0.361642	−	0.932317i	$-0.382216\pi$
$293$	3627.53	0.723285	0.361642	+	0.932317i	$0.382216\pi$
$294$	0	0
$295$	13661.9	2.69636
$296$	−2720.56	−0.534220
$297$	0	0
$298$	−1214.54	−0.236096
$299$	−661.705	−0.127985
$300$	0	0
$301$	5477.32	1.04886
$302$	14961.8	2.85083
$303$	0	0
$304$	1161.17	0.219072
$305$	−1807.55	−0.339345
$306$	0	0
$307$	−351.968	−0.0654328	−0.0327164	−	0.999465i	$-0.510416\pi$
$307$	−351.968	−0.0654328	−0.0327164	+	0.999465i	$0.510416\pi$
$308$	24926.7	4.61147
$309$	0	0
$310$	16515.5	3.02585
$311$	−9713.56	−1.77108	−0.885540	−	0.464564i	$-0.846211\pi$
$311$	−9713.56	−1.77108	−0.885540	+	0.464564i	$0.846211\pi$
$312$	0	0
$313$	1189.54	0.214814	0.107407	−	0.994215i	$-0.465745\pi$
$313$	1189.54	0.214814	0.107407	+	0.994215i	$0.465745\pi$
$314$	−17950.9	−3.22620
$315$	0	0
$316$	−12859.3	−2.28922
$317$	7847.06	1.39033	0.695165	−	0.718850i	$-0.255331\pi$
$317$	7847.06	1.39033	0.695165	+	0.718850i	$0.255331\pi$
$318$	0	0
$319$	2151.83	0.377677
$320$	−151.506	−0.0264670
$321$	0	0
$322$	−986.132	−0.170668
$323$	−526.478	−0.0906936
$324$	0	0
$325$	−7791.42	−1.32982
$326$	−8298.90	−1.40992
$327$	0	0
$328$	−2502.81	−0.421324
$329$	5840.57	0.978726
$330$	0	0
$331$	7868.75	1.30666	0.653332	−	0.757071i	$-0.273371\pi$
$331$	7868.75	1.30666	0.653332	+	0.757071i	$0.273371\pi$
$332$	11346.4	1.87565
$333$	0	0
$334$	16951.0	2.77699
$335$	−4133.41	−0.674126
$336$	0	0
$337$	−3434.71	−0.555194	−0.277597	−	0.960698i	$-0.589538\pi$
$337$	−3434.71	−0.555194	−0.277597	+	0.960698i	$0.589538\pi$
$338$	15297.5	2.46175
$339$	0	0
$340$	−14220.0	−2.26820
$341$	−13982.8	−2.22056
$342$	0	0
$343$	5081.73	0.799963
$344$	−13151.8	−2.06133
$345$	0	0
$346$	21922.8	3.40629
$347$	6802.75	1.05242	0.526211	−	0.850354i	$-0.323612\pi$
$347$	6802.75	1.05242	0.526211	+	0.850354i	$0.323612\pi$
$348$	0	0
$349$	5753.22	0.882415	0.441208	−	0.897405i	$-0.354550\pi$
$349$	5753.22	0.882415	0.441208	+	0.897405i	$0.354550\pi$
$350$	−11611.5	−1.77331
$351$	0	0
$352$	−11709.0	−1.77298
$353$	−10610.4	−1.59981	−0.799907	−	0.600123i	$-0.795118\pi$
$353$	−10610.4	−1.59981	−0.799907	+	0.600123i	$0.795118\pi$
$354$	0	0
$355$	−4799.24	−0.717513
$356$	−3359.20	−0.500104
$357$	0	0
$358$	−4136.79	−0.610715
$359$	931.791	0.136986	0.0684931	−	0.997652i	$-0.478181\pi$
$359$	931.791	0.136986	0.0684931	+	0.997652i	$0.478181\pi$
$360$	0	0
$361$	−6756.12	−0.985001
$362$	14528.7	2.10943
$363$	0	0
$364$	27303.6	3.93158
$365$	5173.20	0.741856
$366$	0	0
$367$	−82.0412	−0.0116690	−0.00583449	−	0.999983i	$-0.501857\pi$
$367$	−82.0412	−0.0116690	−0.00583449	+	0.999983i	$0.501857\pi$
$368$	1050.49	0.148805
$369$	0	0
$370$	−4175.84	−0.586734
$371$	4768.84	0.667347
$372$	0	0
$373$	−14168.5	−1.96680	−0.983400	−	0.181448i	$-0.941922\pi$
$373$	−14168.5	−1.96680	−0.983400	+	0.181448i	$0.941922\pi$
$374$	17406.4	2.40658
$375$	0	0
$376$	−14024.0	−1.92349
$377$	2357.01	0.321995
$378$	0	0
$379$	6954.02	0.942490	0.471245	−	0.882002i	$-0.343805\pi$
$379$	6954.02	0.942490	0.471245	+	0.882002i	$0.343805\pi$
$380$	2778.69	0.375116
$381$	0	0
$382$	2395.57	0.320858
$383$	1197.77	0.159799	0.0798994	−	0.996803i	$-0.474540\pi$
$383$	1197.77	0.159799	0.0798994	+	0.996803i	$0.474540\pi$
$384$	0	0
$385$	21204.4	2.80695
$386$	3045.05	0.401526
$387$	0	0
$388$	−8672.89	−1.13479
$389$	−6557.56	−0.854708	−0.427354	−	0.904084i	$-0.640554\pi$
$389$	−6557.56	−0.854708	−0.427354	+	0.904084i	$0.640554\pi$
$390$	0	0
$391$	−476.292	−0.0616038
$392$	5174.50	0.666714
$393$	0	0
$394$	17233.0	2.20351
$395$	−10939.0	−1.39343
$396$	0	0
$397$	10980.4	1.38814	0.694072	−	0.719906i	$-0.255815\pi$
$397$	10980.4	1.38814	0.694072	+	0.719906i	$0.255815\pi$
$398$	−13311.8	−1.67653
$399$	0	0
$400$	12369.2	1.54615
$401$	7274.21	0.905877	0.452939	−	0.891542i	$-0.350376\pi$
$401$	7274.21	0.905877	0.452939	+	0.891542i	$0.350376\pi$
$402$	0	0
$403$	−15316.1	−1.89318
$404$	15857.9	1.95287
$405$	0	0
$406$	3512.63	0.429381
$407$	3535.47	0.430582
$408$	0	0
$409$	3143.20	0.380003	0.190001	−	0.981784i	$-0.439151\pi$
$409$	3143.20	0.380003	0.190001	+	0.981784i	$0.439151\pi$
$410$	−3841.61	−0.462741
$411$	0	0
$412$	−5438.10	−0.650282
$413$	−18881.7	−2.24965
$414$	0	0
$415$	9652.05	1.14169
$416$	−12825.5	−1.51159
$417$	0	0
$418$	−3401.34	−0.398002
$419$	−8638.09	−1.00716	−0.503578	−	0.863950i	$-0.667983\pi$
$419$	−8638.09	−1.00716	−0.503578	+	0.863950i	$0.667983\pi$
$420$	0	0
$421$	−15140.2	−1.75270	−0.876352	−	0.481671i	$-0.840030\pi$
$421$	−15140.2	−1.75270	−0.876352	+	0.481671i	$0.840030\pi$
$422$	−25696.2	−2.96415
$423$	0	0
$424$	−11450.6	−1.31154
$425$	−5608.22	−0.640091
$426$	0	0
$427$	2498.15	0.283125
$428$	−5157.42	−0.582460
$429$	0	0
$430$	−20187.0	−2.26396
$431$	−17159.1	−1.91770	−0.958848	−	0.283921i	$-0.908365\pi$
$431$	−17159.1	−1.91770	−0.958848	+	0.283921i	$0.908365\pi$
$432$	0	0
$433$	−2686.61	−0.298176	−0.149088	−	0.988824i	$-0.547634\pi$
$433$	−2686.61	−0.298176	−0.149088	+	0.988824i	$0.547634\pi$
$434$	−22825.5	−2.52456
$435$	0	0
$436$	−34181.6	−3.75459
$437$	93.0710	0.0101881
$438$	0	0
$439$	6276.20	0.682339	0.341169	−	0.940002i	$-0.389177\pi$
$439$	6276.20	0.682339	0.341169	+	0.940002i	$0.389177\pi$
$440$	−50914.7	−5.51651
$441$	0	0
$442$	19066.1	2.05177
$443$	−6764.78	−0.725518	−0.362759	−	0.931883i	$-0.618165\pi$
$443$	−6764.78	−0.725518	−0.362759	+	0.931883i	$0.618165\pi$
$444$	0	0
$445$	−2857.56	−0.304408
$446$	8711.15	0.924854
$447$	0	0
$448$	209.391	0.0220821
$449$	−11701.9	−1.22995	−0.614975	−	0.788546i	$-0.710834\pi$
$449$	−11701.9	−1.22995	−0.614975	+	0.788546i	$0.710834\pi$
$450$	0	0
$451$	3252.50	0.339588
$452$	−15697.8	−1.63354
$453$	0	0
$454$	−24735.9	−2.55707
$455$	23226.3	2.39311
$456$	0	0
$457$	18533.1	1.89703	0.948515	−	0.316731i	$-0.102585\pi$
$457$	18533.1	1.89703	0.948515	+	0.316731i	$0.102585\pi$
$458$	−33608.7	−3.42889
$459$	0	0
$460$	2513.82	0.254798
$461$	−4084.28	−0.412634	−0.206317	−	0.978485i	$-0.566148\pi$
$461$	−4084.28	−0.412634	−0.206317	+	0.978485i	$0.566148\pi$
$462$	0	0
$463$	1285.93	0.129076	0.0645378	−	0.997915i	$-0.479443\pi$
$463$	1285.93	0.129076	0.0645378	+	0.997915i	$0.479443\pi$
$464$	−3741.86	−0.374378
$465$	0	0
$466$	17983.4	1.78769
$467$	2968.54	0.294149	0.147075	−	0.989125i	$-0.453014\pi$
$467$	2968.54	0.294149	0.147075	+	0.989125i	$0.453014\pi$
$468$	0	0
$469$	5712.65	0.562442
$470$	−21525.8	−2.11257
$471$	0	0
$472$	45337.5	4.42125
$473$	17091.3	1.66144
$474$	0	0
$475$	1095.89	0.105859
$476$	19653.0	1.89242
$477$	0	0
$478$	1217.39	0.116490
$479$	−7000.52	−0.667771	−0.333885	−	0.942614i	$-0.608360\pi$
$479$	−7000.52	−0.667771	−0.333885	+	0.942614i	$0.608360\pi$
$480$	0	0
$481$	3872.59	0.367100
$482$	−4554.61	−0.430409
$483$	0	0
$484$	53895.2	5.06153
$485$	−7377.76	−0.690736
$486$	0	0
$487$	−5513.98	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$487$	−5513.98	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$488$	−5998.42	−0.556426
$489$	0	0
$490$	7942.45	0.732252
$491$	18652.2	1.71439	0.857193	−	0.514995i	$-0.172206\pi$
$491$	18652.2	1.71439	0.857193	+	0.514995i	$0.172206\pi$
$492$	0	0
$493$	1696.56	0.154989
$494$	−3725.67	−0.339323
$495$	0	0
$496$	24315.0	2.20116
$497$	6632.87	0.598641
$498$	0	0
$499$	6533.66	0.586146	0.293073	−	0.956090i	$-0.405322\pi$
$499$	6533.66	0.586146	0.293073	+	0.956090i	$0.405322\pi$
$500$	−4644.83	−0.415446
$501$	0	0
$502$	−34794.2	−3.09351
$503$	−8908.49	−0.789681	−0.394841	−	0.918750i	$-0.629200\pi$
$503$	−8908.49	−0.789681	−0.394841	+	0.918750i	$0.629200\pi$
$504$	0	0
$505$	13489.8	1.18869
$506$	−3077.11	−0.270344
$507$	0	0
$508$	24854.0	2.17070
$509$	−13951.4	−1.21490	−0.607451	−	0.794357i	$-0.707808\pi$
$509$	−13951.4	−1.21490	−0.607451	+	0.794357i	$0.707808\pi$
$510$	0	0
$511$	−7149.70	−0.618951
$512$	−26036.4	−2.24737
$513$	0	0
$514$	−28128.3	−2.41378
$515$	−4626.02	−0.395819
$516$	0	0
$517$	18224.8	1.55034
$518$	5771.28	0.489528
$519$	0	0
$520$	−55769.6	−4.70319
$521$	−11335.0	−0.953155	−0.476577	−	0.879132i	$-0.658123\pi$
$521$	−11335.0	−0.953155	−0.476577	+	0.879132i	$0.658123\pi$
$522$	0	0
$523$	17354.8	1.45100	0.725499	−	0.688223i	$-0.241609\pi$
$523$	17354.8	1.45100	0.725499	+	0.688223i	$0.241609\pi$
$524$	−12879.6	−1.07376
$525$	0	0
$526$	8957.99	0.742561
$527$	−11024.5	−0.911258
$528$	0	0
$529$	−12082.8	−0.993080
$530$	−17575.8	−1.44046
$531$	0	0
$532$	−3840.34	−0.312970
$533$	3562.64	0.289521
$534$	0	0
$535$	−4387.25	−0.354537
$536$	−13716.9	−1.10537
$537$	0	0
$538$	20877.2	1.67301
$539$	−6724.47	−0.537372
$540$	0	0
$541$	−5050.61	−0.401373	−0.200686	−	0.979656i	$-0.564317\pi$
$541$	−5050.61	−0.401373	−0.200686	+	0.979656i	$0.564317\pi$
$542$	3610.39	0.286125
$543$	0	0
$544$	−9231.70	−0.727584
$545$	−29077.2	−2.28538
$546$	0	0
$547$	13897.3	1.08630	0.543149	−	0.839636i	$-0.317232\pi$
$547$	13897.3	1.08630	0.543149	+	0.839636i	$0.317232\pi$
$548$	33613.0	2.62021
$549$	0	0
$550$	−36232.2	−2.80899
$551$	−331.521	−0.0256321
$552$	0	0
$553$	15118.5	1.16257
$554$	−9179.56	−0.703975
$555$	0	0
$556$	35136.9	2.68010
$557$	105.819	0.00804975	0.00402487	−	0.999992i	$-0.498719\pi$
$557$	105.819	0.00804975	0.00402487	+	0.999992i	$0.498719\pi$
$558$	0	0
$559$	18721.0	1.41649
$560$	−36872.8	−2.78243
$561$	0	0
$562$	−30244.1	−2.27005
$563$	19919.9	1.49116	0.745580	−	0.666416i	$-0.232173\pi$
$563$	19919.9	1.49116	0.745580	+	0.666416i	$0.232173\pi$
$564$	0	0
$565$	−13353.6	−0.994319
$566$	13327.0	0.989708
$567$	0	0
$568$	−15926.4	−1.17651
$569$	−2973.81	−0.219101	−0.109551	−	0.993981i	$-0.534941\pi$
$569$	−2973.81	−0.219101	−0.109551	+	0.993981i	$0.534941\pi$
$570$	0	0
$571$	−22422.7	−1.64336	−0.821680	−	0.569948i	$-0.806963\pi$
$571$	−22422.7	−1.64336	−0.821680	+	0.569948i	$0.806963\pi$
$572$	85197.5	6.22778
$573$	0	0
$574$	5309.36	0.386077
$575$	991.424	0.0719048
$576$	0	0
$577$	4267.84	0.307925	0.153962	−	0.988077i	$-0.450797\pi$
$577$	4267.84	0.307925	0.153962	+	0.988077i	$0.450797\pi$
$578$	−11301.6	−0.813295
$579$	0	0
$580$	−8954.28	−0.641045
$581$	−13339.8	−0.952542
$582$	0	0
$583$	14880.6	1.05710
$584$	17167.4	1.21643
$585$	0	0
$586$	18477.5	1.30256
$587$	−8210.74	−0.577331	−0.288666	−	0.957430i	$-0.593212\pi$
$587$	−8210.74	−0.577331	−0.288666	+	0.957430i	$0.593212\pi$
$588$	0	0
$589$	2154.26	0.150704
$590$	69589.5	4.85585
$591$	0	0
$592$	−6147.91	−0.426820
$593$	−1843.22	−0.127642	−0.0638211	−	0.997961i	$-0.520329\pi$
$593$	−1843.22	−0.127642	−0.0638211	+	0.997961i	$0.520329\pi$
$594$	0	0
$595$	16718.2	1.15190
$596$	−4278.97	−0.294083
$597$	0	0
$598$	−3370.52	−0.230486
$599$	5234.62	0.357063	0.178532	−	0.983934i	$-0.442865\pi$
$599$	5234.62	0.357063	0.178532	+	0.983934i	$0.442865\pi$
$600$	0	0
$601$	17417.4	1.18215	0.591073	−	0.806618i	$-0.298704\pi$
$601$	17417.4	1.18215	0.591073	+	0.806618i	$0.298704\pi$
$602$	27899.8	1.88889
$603$	0	0
$604$	52712.0	3.55103
$605$	45846.9	3.08090
$606$	0	0
$607$	10280.6	0.687444	0.343722	−	0.939071i	$-0.388312\pi$
$607$	10280.6	0.687444	0.343722	+	0.939071i	$0.388312\pi$
$608$	1803.94	0.120328
$609$	0	0
$610$	−9207.10	−0.611123
$611$	19962.6	1.32177
$612$	0	0
$613$	−1205.97	−0.0794596	−0.0397298	−	0.999210i	$-0.512650\pi$
$613$	−1205.97	−0.0794596	−0.0397298	+	0.999210i	$0.512650\pi$
$614$	−1792.82	−0.117837
$615$	0	0
$616$	70367.5	4.60258
$617$	−8577.63	−0.559680	−0.279840	−	0.960047i	$-0.590281\pi$
$617$	−8577.63	−0.559680	−0.279840	+	0.960047i	$0.590281\pi$
$618$	0	0
$619$	17638.8	1.14533	0.572667	−	0.819788i	$-0.305909\pi$
$619$	17638.8	1.14533	0.572667	+	0.819788i	$0.305909\pi$
$620$	58185.9	3.76904
$621$	0	0
$622$	−49477.9	−3.18952
$623$	3949.34	0.253976
$624$	0	0
$625$	−17456.9	−1.11724
$626$	6059.16	0.386858
$627$	0	0
$628$	−63243.1	−4.01859
$629$	2787.47	0.176699
$630$	0	0
$631$	−28112.5	−1.77360	−0.886799	−	0.462156i	$-0.847076\pi$
$631$	−28112.5	−1.77360	−0.886799	+	0.462156i	$0.847076\pi$
$632$	−36301.6	−2.28481
$633$	0	0
$634$	39970.5	2.50383
$635$	21142.5	1.32128
$636$	0	0
$637$	−7365.67	−0.458145
$638$	10960.7	0.680156
$639$	0	0
$640$	−22492.3	−1.38920
$641$	2280.32	0.140511	0.0702554	−	0.997529i	$-0.477619\pi$
$641$	2280.32	0.140511	0.0702554	+	0.997529i	$0.477619\pi$
$642$	0	0
$643$	15435.3	0.946672	0.473336	−	0.880882i	$-0.343050\pi$
$643$	15435.3	0.946672	0.473336	+	0.880882i	$0.343050\pi$
$644$	−3474.26	−0.212585
$645$	0	0
$646$	−2681.72	−0.163329
$647$	26247.4	1.59489	0.797443	−	0.603394i	$-0.206185\pi$
$647$	26247.4	1.59489	0.797443	+	0.603394i	$0.206185\pi$
$648$	0	0
$649$	−58917.9	−3.56353
$650$	−39687.1	−2.39485
$651$	0	0
$652$	−29238.0	−1.75621
$653$	5590.95	0.335055	0.167527	−	0.985867i	$-0.446422\pi$
$653$	5590.95	0.335055	0.167527	+	0.985867i	$0.446422\pi$
$654$	0	0
$655$	−10956.3	−0.653584
$656$	−5655.84	−0.336621
$657$	0	0
$658$	29750.0	1.76258
$659$	9664.49	0.571282	0.285641	−	0.958337i	$-0.407793\pi$
$659$	9664.49	0.571282	0.285641	+	0.958337i	$0.407793\pi$
$660$	0	0
$661$	−20072.6	−1.18114	−0.590570	−	0.806986i	$-0.701097\pi$
$661$	−20072.6	−1.18114	−0.590570	+	0.806986i	$0.701097\pi$
$662$	40081.0	2.35316
$663$	0	0
$664$	32030.7	1.87203
$665$	−3266.86	−0.190501
$666$	0	0
$667$	−299.919	−0.0174107
$668$	59720.3	3.45905
$669$	0	0
$670$	−21054.3	−1.21403
$671$	7795.19	0.448480
$672$	0	0
$673$	−17473.0	−1.00079	−0.500396	−	0.865796i	$-0.666812\pi$
$673$	−17473.0	−1.00079	−0.500396	+	0.865796i	$0.666812\pi$
$674$	−17495.3	−0.999844
$675$	0	0
$676$	53894.8	3.06638
$677$	9879.28	0.560844	0.280422	−	0.959877i	$-0.409526\pi$
$677$	9879.28	0.560844	0.280422	+	0.959877i	$0.409526\pi$
$678$	0	0
$679$	10196.5	0.576300
$680$	−40142.7	−2.26383
$681$	0	0
$682$	−71224.1	−3.99899
$683$	−20749.8	−1.16248	−0.581238	−	0.813734i	$-0.697431\pi$
$683$	−20749.8	−1.16248	−0.581238	+	0.813734i	$0.697431\pi$
$684$	0	0
$685$	28593.5	1.59489
$686$	25884.7	1.44065
$687$	0	0
$688$	−29720.5	−1.64692
$689$	16299.5	0.901250
$690$	0	0
$691$	−328.098	−0.0180628	−0.00903142	−	0.999959i	$-0.502875\pi$
$691$	−328.098	−0.0180628	−0.00903142	+	0.999959i	$0.502875\pi$
$692$	77236.7	4.24292
$693$	0	0
$694$	34651.1	1.89530
$695$	29889.9	1.63135
$696$	0	0
$697$	2564.37	0.139358
$698$	29305.1	1.58913
$699$	0	0
$700$	−40908.6	−2.20886
$701$	−25860.5	−1.39335	−0.696675	−	0.717387i	$-0.745338\pi$
$701$	−25860.5	−1.39335	−0.696675	+	0.717387i	$0.745338\pi$
$702$	0	0
$703$	−544.693	−0.0292226
$704$	653.379	0.0349789
$705$	0	0
$706$	−54046.1	−2.88109
$707$	−18643.8	−0.991758
$708$	0	0
$709$	4489.98	0.237834	0.118917	−	0.992904i	$-0.462058\pi$
$709$	4489.98	0.237834	0.118917	+	0.992904i	$0.462058\pi$
$710$	−24445.8	−1.29216
$711$	0	0
$712$	−9482.93	−0.499140
$713$	1948.91	0.102366
$714$	0	0
$715$	72474.9	3.79078
$716$	−14574.4	−0.760713
$717$	0	0
$718$	4746.26	0.246697
$719$	−12971.5	−0.672817	−0.336409	−	0.941716i	$-0.609212\pi$
$719$	−12971.5	−0.672817	−0.336409	+	0.941716i	$0.609212\pi$
$720$	0	0
$721$	6393.47	0.330243
$722$	−34413.6	−1.77388
$723$	0	0
$724$	51186.4	2.62753
$725$	−3531.48	−0.180905
$726$	0	0
$727$	36346.8	1.85424	0.927118	−	0.374770i	$-0.122278\pi$
$727$	36346.8	1.85424	0.927118	+	0.374770i	$0.122278\pi$
$728$	77077.3	3.92400
$729$	0	0
$730$	26350.7	1.33600
$731$	13475.3	0.681809
$732$	0	0
$733$	5914.92	0.298053	0.149026	−	0.988833i	$-0.452386\pi$
$733$	5914.92	0.298053	0.149026	+	0.988833i	$0.452386\pi$
$734$	−417.892	−0.0210146
$735$	0	0
$736$	1631.98	0.0817333
$737$	17825.6	0.890930
$738$	0	0
$739$	−17782.1	−0.885148	−0.442574	−	0.896732i	$-0.645935\pi$
$739$	−17782.1	−0.885148	−0.442574	+	0.896732i	$0.645935\pi$
$740$	−14712.0	−0.730842
$741$	0	0
$742$	24291.0	1.20182
$743$	−30917.1	−1.52657	−0.763283	−	0.646064i	$-0.776414\pi$
$743$	−30917.1	−1.52657	−0.763283	+	0.646064i	$0.776414\pi$
$744$	0	0
$745$	−3639.99	−0.179005
$746$	−72169.9	−3.54199
$747$	0	0
$748$	61324.7	2.99767
$749$	6063.47	0.295800
$750$	0	0
$751$	20251.5	0.984006	0.492003	−	0.870594i	$-0.336265\pi$
$751$	20251.5	0.984006	0.492003	+	0.870594i	$0.336265\pi$
$752$	−31691.5	−1.53679
$753$	0	0
$754$	12005.9	0.579878
$755$	44840.5	2.16147
$756$	0	0
$757$	21481.8	1.03140	0.515701	−	0.856769i	$-0.327532\pi$
$757$	21481.8	1.03140	0.515701	+	0.856769i	$0.327532\pi$
$758$	35421.6	1.69732
$759$	0	0
$760$	7844.18	0.374393
$761$	−16871.2	−0.803654	−0.401827	−	0.915716i	$-0.631625\pi$
$761$	−16871.2	−0.803654	−0.401827	+	0.915716i	$0.631625\pi$
$762$	0	0
$763$	40186.7	1.90676
$764$	8439.86	0.399664
$765$	0	0
$766$	6101.05	0.287780
$767$	−64536.0	−3.03815
$768$	0	0
$769$	−29154.0	−1.36713	−0.683563	−	0.729891i	$-0.739571\pi$
$769$	−29154.0	−1.36713	−0.683563	+	0.729891i	$0.739571\pi$
$770$	108009.	5.05501
$771$	0	0
$772$	10728.1	0.500146
$773$	−39247.5	−1.82618	−0.913089	−	0.407760i	$-0.866310\pi$
$773$	−39247.5	−1.82618	−0.913089	+	0.407760i	$0.866310\pi$
$774$	0	0
$775$	22948.0	1.06363
$776$	−24483.4	−1.13260
$777$	0	0
$778$	−33402.2	−1.53924
$779$	−501.097	−0.0230471
$780$	0	0
$781$	20697.1	0.948270
$782$	−2426.08	−0.110942
$783$	0	0
$784$	11693.3	0.532677
$785$	−53798.9	−2.44607
$786$	0	0
$787$	2925.52	0.132508	0.0662539	−	0.997803i	$-0.478895\pi$
$787$	2925.52	0.132508	0.0662539	+	0.997803i	$0.478895\pi$
$788$	60713.8	2.74472
$789$	0	0
$790$	−55720.1	−2.50941
$791$	18455.6	0.829588
$792$	0	0
$793$	8538.49	0.382359
$794$	55931.0	2.49990
$795$	0	0
$796$	−46898.9	−2.08830
$797$	36424.5	1.61885	0.809423	−	0.587226i	$-0.199780\pi$
$797$	36424.5	1.61885	0.809423	+	0.587226i	$0.199780\pi$
$798$	0	0
$799$	14369.0	0.636217
$800$	19216.2	0.849245
$801$	0	0
$802$	37052.6	1.63139
$803$	−22309.8	−0.980442
$804$	0	0
$805$	−2955.44	−0.129398
$806$	−78015.6	−3.40941
$807$	0	0
$808$	44766.4	1.94911
$809$	−9574.01	−0.416074	−0.208037	−	0.978121i	$-0.566707\pi$
$809$	−9574.01	−0.416074	−0.208037	+	0.978121i	$0.566707\pi$
$810$	0	0
$811$	28563.4	1.23674	0.618371	−	0.785886i	$-0.287793\pi$
$811$	28563.4	1.23674	0.618371	+	0.785886i	$0.287793\pi$
$812$	12375.4	0.534842
$813$	0	0
$814$	18008.6	0.775431
$815$	−24871.9	−1.06899
$816$	0	0
$817$	−2633.18	−0.112758
$818$	16010.5	0.684344
$819$	0	0
$820$	−13534.5	−0.576395
$821$	−17217.3	−0.731899	−0.365949	−	0.930635i	$-0.619256\pi$
$821$	−17217.3	−0.731899	−0.365949	+	0.930635i	$0.619256\pi$
$822$	0	0
$823$	−32703.7	−1.38515	−0.692575	−	0.721346i	$-0.743524\pi$
$823$	−32703.7	−1.38515	−0.692575	+	0.721346i	$0.743524\pi$
$824$	−15351.6	−0.649028
$825$	0	0
$826$	−96177.3	−4.05137
$827$	26106.9	1.09773	0.548867	−	0.835909i	$-0.315059\pi$
$827$	26106.9	1.09773	0.548867	+	0.835909i	$0.315059\pi$
$828$	0	0
$829$	−6260.31	−0.262279	−0.131140	−	0.991364i	$-0.541864\pi$
$829$	−6260.31	−0.262279	−0.131140	+	0.991364i	$0.541864\pi$
$830$	49164.5	2.05606
$831$	0	0
$832$	715.681	0.0298219
$833$	−5301.77	−0.220523
$834$	0	0
$835$	50802.2	2.10549
$836$	−11983.3	−0.495756
$837$	0	0
$838$	−43999.8	−1.81378
$839$	−28092.5	−1.15597	−0.577987	−	0.816046i	$-0.696162\pi$
$839$	−28092.5	−1.15597	−0.577987	+	0.816046i	$0.696162\pi$
$840$	0	0
$841$	−23320.7	−0.956197
$842$	−77119.5	−3.15643
$843$	0	0
$844$	−90530.7	−3.69217
$845$	45846.6	1.86647
$846$	0	0
$847$	−63363.5	−2.57048
$848$	−25876.2	−1.04787
$849$	0	0
$850$	−28566.6	−1.15273
$851$	−492.770	−0.0198495
$852$	0	0
$853$	15997.1	0.642123	0.321061	−	0.947058i	$-0.395960\pi$
$853$	15997.1	0.642123	0.321061	+	0.947058i	$0.395960\pi$
$854$	12724.8	0.509877
$855$	0	0
$856$	−14559.3	−0.581338
$857$	−29799.1	−1.18777	−0.593885	−	0.804550i	$-0.702407\pi$
$857$	−29799.1	−1.18777	−0.593885	+	0.804550i	$0.702407\pi$
$858$	0	0
$859$	12684.4	0.503824	0.251912	−	0.967750i	$-0.418941\pi$
$859$	12684.4	0.503824	0.251912	+	0.967750i	$0.418941\pi$
$860$	−71121.2	−2.82002
$861$	0	0
$862$	−87403.3	−3.45356
$863$	17526.2	0.691308	0.345654	−	0.938362i	$-0.387657\pi$
$863$	17526.2	0.691308	0.345654	+	0.938362i	$0.387657\pi$
$864$	0	0
$865$	65702.8	2.58262
$866$	−13684.7	−0.536982
$867$	0	0
$868$	−80416.8	−3.14461
$869$	47175.4	1.84156
$870$	0	0
$871$	19525.4	0.759577
$872$	−96493.8	−3.74736
$873$	0	0
$874$	474.075	0.0183476
$875$	5460.83	0.210983
$876$	0	0
$877$	−20669.9	−0.795863	−0.397932	−	0.917415i	$-0.630272\pi$
$877$	−20669.9	−0.795863	−0.397932	+	0.917415i	$0.630272\pi$
$878$	31969.0	1.22882
$879$	0	0
$880$	−115057.	−4.40747
$881$	−12261.9	−0.468914	−0.234457	−	0.972126i	$-0.575331\pi$
$881$	−12261.9	−0.468914	−0.234457	+	0.972126i	$0.575331\pi$
$882$	0	0
$883$	36489.9	1.39069	0.695347	−	0.718674i	$-0.255251\pi$
$883$	36489.9	1.39069	0.695347	+	0.718674i	$0.255251\pi$
$884$	67172.3	2.55571
$885$	0	0
$886$	−34457.7	−1.30658
$887$	4328.08	0.163836	0.0819182	−	0.996639i	$-0.473895\pi$
$887$	4328.08	0.163836	0.0819182	+	0.996639i	$0.473895\pi$
$888$	0	0
$889$	−29220.3	−1.10238
$890$	−14555.5	−0.548206
$891$	0	0
$892$	30690.4	1.15201
$893$	−2807.80	−0.105218
$894$	0	0
$895$	−12398.0	−0.463038
$896$	31085.9	1.15905
$897$	0	0
$898$	−59605.9	−2.21501
$899$	−6942.06	−0.257543
$900$	0	0
$901$	11732.3	0.433806
$902$	16567.2	0.611561
$903$	0	0
$904$	−44314.4	−1.63039
$905$	43542.7	1.59935
$906$	0	0
$907$	47757.6	1.74836	0.874182	−	0.485599i	$-0.161398\pi$
$907$	47757.6	1.74836	0.874182	+	0.485599i	$0.161398\pi$
$908$	−87147.4	−3.18512
$909$	0	0
$910$	118308.	4.30973
$911$	−33658.7	−1.22411	−0.612054	−	0.790816i	$-0.709656\pi$
$911$	−33658.7	−1.22411	−0.612054	+	0.790816i	$0.709656\pi$
$912$	0	0
$913$	−41625.1	−1.50886
$914$	94402.0	3.41635
$915$	0	0
$916$	−118407.	−4.27106
$917$	15142.3	0.545303
$918$	0	0
$919$	21314.5	0.765072	0.382536	−	0.923941i	$-0.375051\pi$
$919$	21314.5	0.765072	0.382536	+	0.923941i	$0.375051\pi$
$920$	7096.44	0.254307
$921$	0	0
$922$	−20804.1	−0.743108
$923$	22670.6	0.808464
$924$	0	0
$925$	−5802.26	−0.206245
$926$	6550.10	0.232451
$927$	0	0
$928$	−5813.17	−0.205632
$929$	37123.4	1.31106	0.655532	−	0.755167i	$-0.272444\pi$
$929$	37123.4	1.31106	0.655532	+	0.755167i	$0.272444\pi$
$930$	0	0
$931$	1036.01	0.0364702
$932$	63357.6	2.22677
$933$	0	0
$934$	15120.8	0.529731
$935$	52167.0	1.82465
$936$	0	0
$937$	−9025.33	−0.314669	−0.157334	−	0.987545i	$-0.550290\pi$
$937$	−9025.33	−0.314669	−0.157334	+	0.987545i	$0.550290\pi$
$938$	29098.4	1.01290
$939$	0	0
$940$	−75837.8	−2.63144
$941$	−2003.99	−0.0694241	−0.0347121	−	0.999397i	$-0.511051\pi$
$941$	−2003.99	−0.0694241	−0.0347121	+	0.999397i	$0.511051\pi$
$942$	0	0
$943$	−453.330	−0.0156548
$944$	102454.	3.53240
$945$	0	0
$946$	87057.9	2.99207
$947$	−27399.7	−0.940200	−0.470100	−	0.882613i	$-0.655782\pi$
$947$	−27399.7	−0.940200	−0.470100	+	0.882613i	$0.655782\pi$
$948$	0	0
$949$	−24437.1	−0.835892
$950$	5582.12	0.190640
$951$	0	0
$952$	55479.8	1.88877
$953$	−33468.6	−1.13762	−0.568811	−	0.822468i	$-0.692596\pi$
$953$	−33468.6	−1.13762	−0.568811	+	0.822468i	$0.692596\pi$
$954$	0	0
$955$	7179.53	0.243271
$956$	4289.02	0.145101
$957$	0	0
$958$	−35658.5	−1.20258
$959$	−39518.1	−1.33066
$960$	0	0
$961$	15319.3	0.514227
$962$	19725.8	0.661107
$963$	0	0
$964$	−16046.4	−0.536122
$965$	9126.05	0.304433
$966$	0	0
$967$	−610.226	−0.0202932	−0.0101466	−	0.999949i	$-0.503230\pi$
$967$	−610.226	−0.0202932	−0.0101466	+	0.999949i	$0.503230\pi$
$968$	152145.	5.05177
$969$	0	0
$970$	−37580.0	−1.24394
$971$	−43937.4	−1.45213	−0.726066	−	0.687625i	$-0.758653\pi$
$971$	−43937.4	−1.45213	−0.726066	+	0.687625i	$0.758653\pi$
$972$	0	0
$973$	−41309.8	−1.36108
$974$	−28086.5	−0.923972
$975$	0	0
$976$	−13555.2	−0.444562
$977$	7713.22	0.252577	0.126289	−	0.991994i	$-0.459693\pi$
$977$	7713.22	0.252577	0.126289	+	0.991994i	$0.459693\pi$
$978$	0	0
$979$	12323.5	0.402308
$980$	27982.2	0.912100
$981$	0	0
$982$	95008.7	3.08742
$983$	44381.4	1.44003	0.720013	−	0.693961i	$-0.244136\pi$
$983$	44381.4	1.44003	0.720013	+	0.693961i	$0.244136\pi$
$984$	0	0
$985$	51647.3	1.67068
$986$	8641.77	0.279118
$987$	0	0
$988$	−13126.0	−0.422665
$989$	−2382.17	−0.0765911
$990$	0	0
$991$	30319.3	0.971871	0.485936	−	0.873995i	$-0.338479\pi$
$991$	30319.3	0.971871	0.485936	+	0.873995i	$0.338479\pi$
$992$	37774.6	1.20902
$993$	0	0
$994$	33785.8	1.07809
$995$	−39895.4	−1.27113
$996$	0	0
$997$	−37153.6	−1.18021	−0.590103	−	0.807328i	$-0.700913\pi$
$997$	−37153.6	−1.18021	−0.590103	+	0.807328i	$0.700913\pi$
$998$	33280.4	1.05559
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.55	✓	59
3.2	odd	2		2151.4.a.h.1.5	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.55	✓	59	1.1	even	1	trivial
2151.4.a.h.1.5	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)

\(f(q)\)	\(=\)	\(q+5.09369 q^{2} +17.9457 q^{4} +15.2658 q^{5} -21.0984 q^{7} +50.6602 q^{8} +O(q^{10})\)	\(q+5.09369 q^{2} +17.9457 q^{4} +15.2658 q^{5} -21.0984 q^{7} +50.6602 q^{8} +77.7594 q^{10} -65.8349 q^{11} -72.1125 q^{13} -107.469 q^{14} +114.482 q^{16} -51.9062 q^{17} +10.1429 q^{19} +273.955 q^{20} -335.343 q^{22} +9.17600 q^{23} +108.045 q^{25} -367.319 q^{26} -378.625 q^{28} -32.6852 q^{29} +212.392 q^{31} +177.853 q^{32} -264.394 q^{34} -322.084 q^{35} -53.7021 q^{37} +51.6646 q^{38} +773.369 q^{40} -49.4038 q^{41} -259.609 q^{43} -1181.45 q^{44} +46.7397 q^{46} -276.825 q^{47} +102.141 q^{49} +550.349 q^{50} -1294.11 q^{52} -226.029 q^{53} -1005.02 q^{55} -1068.85 q^{56} -166.488 q^{58} +894.934 q^{59} -118.405 q^{61} +1081.86 q^{62} -9.92451 q^{64} -1100.86 q^{65} -270.762 q^{67} -931.492 q^{68} -1640.60 q^{70} -314.378 q^{71} +338.874 q^{73} -273.542 q^{74} +182.021 q^{76} +1389.01 q^{77} -716.571 q^{79} +1747.66 q^{80} -251.648 q^{82} +632.265 q^{83} -792.391 q^{85} -1322.37 q^{86} -3335.21 q^{88} -187.187 q^{89} +1521.46 q^{91} +164.670 q^{92} -1410.06 q^{94} +154.839 q^{95} -483.286 q^{97} +520.277 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more