LMFDB - Embedded newform 2151.4.a.g.1.5

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			1.5
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)

$f(q)$	$=$	$q-4.88066 q^{2} +15.8208 q^{4} -22.1582 q^{5} +19.1765 q^{7} -38.1708 q^{8} +O(q^{10})$	\(q-4.88066 q^{2} +15.8208 q^{4} -22.1582 q^{5} +19.1765 q^{7} -38.1708 q^{8} +108.146 q^{10} -38.0469 q^{11} -46.3765 q^{13} -93.5939 q^{14} +59.7320 q^{16} -14.9451 q^{17} -94.8378 q^{19} -350.561 q^{20} +185.694 q^{22} -67.5214 q^{23} +365.984 q^{25} +226.348 q^{26} +303.388 q^{28} +287.137 q^{29} +18.7137 q^{31} +13.8349 q^{32} +72.9421 q^{34} -424.916 q^{35} +256.123 q^{37} +462.871 q^{38} +845.795 q^{40} -235.846 q^{41} -55.1789 q^{43} -601.934 q^{44} +329.549 q^{46} -251.798 q^{47} +24.7380 q^{49} -1786.25 q^{50} -733.715 q^{52} -57.5527 q^{53} +843.050 q^{55} -731.982 q^{56} -1401.42 q^{58} +486.414 q^{59} +28.1776 q^{61} -91.3351 q^{62} -545.379 q^{64} +1027.62 q^{65} +155.784 q^{67} -236.444 q^{68} +2073.87 q^{70} +13.7025 q^{71} -859.846 q^{73} -1250.05 q^{74} -1500.41 q^{76} -729.606 q^{77} +236.911 q^{79} -1323.55 q^{80} +1151.08 q^{82} +595.613 q^{83} +331.157 q^{85} +269.309 q^{86} +1452.28 q^{88} -36.2036 q^{89} -889.339 q^{91} -1068.24 q^{92} +1228.94 q^{94} +2101.43 q^{95} +695.709 q^{97} -120.738 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.88066	−1.72557	−0.862787	−	0.505568i	$-0.831283\pi$
$2$	−4.88066	−1.72557	−0.862787	+	0.505568i	$0.831283\pi$
$3$	0	0
$3$	0	0
$4$	15.8208	1.97760
$5$	−22.1582	−1.98189	−0.990943	−	0.134280i	$-0.957128\pi$
$5$	−22.1582	−1.98189	−0.990943	+	0.134280i	$0.957128\pi$
$6$	0	0
$7$	19.1765	1.03543	0.517717	−	0.855552i	$-0.326782\pi$
$7$	19.1765	1.03543	0.517717	+	0.855552i	$0.326782\pi$
$8$	−38.1708	−1.68693
$9$	0	0
$10$	108.146	3.41989
$11$	−38.0469	−1.04287	−0.521435	−	0.853291i	$-0.674603\pi$
$11$	−38.0469	−1.04287	−0.521435	+	0.853291i	$0.674603\pi$
$12$	0	0
$13$	−46.3765	−0.989426	−0.494713	−	0.869056i	$-0.664727\pi$
$13$	−46.3765	−0.989426	−0.494713	+	0.869056i	$0.664727\pi$
$14$	−93.5939	−1.78672
$15$	0	0
$16$	59.7320	0.933312
$17$	−14.9451	−0.213219	−0.106610	−	0.994301i	$-0.533999\pi$
$17$	−14.9451	−0.213219	−0.106610	+	0.994301i	$0.533999\pi$
$18$	0	0
$19$	−94.8378	−1.14512	−0.572560	−	0.819863i	$-0.694050\pi$
$19$	−94.8378	−1.14512	−0.572560	+	0.819863i	$0.694050\pi$
$20$	−350.561	−3.91939
$21$	0	0
$22$	185.694	1.79955
$23$	−67.5214	−0.612138	−0.306069	−	0.952009i	$-0.599014\pi$
$23$	−67.5214	−0.612138	−0.306069	+	0.952009i	$0.599014\pi$
$24$	0	0
$25$	365.984	2.92788
$26$	226.348	1.70733
$27$	0	0
$28$	303.388	2.04768
$29$	287.137	1.83862	0.919312	−	0.393530i	$-0.128746\pi$
$29$	287.137	1.83862	0.919312	+	0.393530i	$0.128746\pi$
$30$	0	0
$31$	18.7137	0.108422	0.0542109	−	0.998530i	$-0.482736\pi$
$31$	18.7137	0.108422	0.0542109	+	0.998530i	$0.482736\pi$
$32$	13.8349	0.0764279
$33$	0	0
$34$	72.9421	0.367925
$35$	−424.916	−2.05211
$36$	0	0
$37$	256.123	1.13801	0.569005	−	0.822334i	$-0.307329\pi$
$37$	256.123	1.13801	0.569005	+	0.822334i	$0.307329\pi$
$38$	462.871	1.97599
$39$	0	0
$40$	845.795	3.34330
$41$	−235.846	−0.898363	−0.449182	−	0.893440i	$-0.648284\pi$
$41$	−235.846	−0.898363	−0.449182	+	0.893440i	$0.648284\pi$
$42$	0	0
$43$	−55.1789	−0.195691	−0.0978454	−	0.995202i	$-0.531195\pi$
$43$	−55.1789	−0.195691	−0.0978454	+	0.995202i	$0.531195\pi$
$44$	−601.934	−2.06238
$45$	0	0
$46$	329.549	1.05629
$47$	−251.798	−0.781456	−0.390728	−	0.920506i	$-0.627777\pi$
$47$	−251.798	−0.781456	−0.390728	+	0.920506i	$0.627777\pi$
$48$	0	0
$49$	24.7380	0.0721225
$50$	−1786.25	−5.05226
$51$	0	0
$52$	−733.715	−1.95669
$53$	−57.5527	−0.149160	−0.0745799	−	0.997215i	$-0.523762\pi$
$53$	−57.5527	−0.149160	−0.0745799	+	0.997215i	$0.523762\pi$
$54$	0	0
$55$	843.050	2.06685
$56$	−731.982	−1.74670
$57$	0	0
$58$	−1401.42	−3.17268
$59$	486.414	1.07332	0.536658	−	0.843800i	$-0.319686\pi$
$59$	486.414	1.07332	0.536658	+	0.843800i	$0.319686\pi$
$60$	0	0
$61$	28.1776	0.0591437	0.0295719	−	0.999563i	$-0.490586\pi$
$61$	28.1776	0.0591437	0.0295719	+	0.999563i	$0.490586\pi$
$62$	−91.3351	−0.187090
$63$	0	0
$64$	−545.379	−1.06519
$65$	1027.62	1.96093
$66$	0	0
$67$	155.784	0.284060	0.142030	−	0.989862i	$-0.454637\pi$
$67$	155.784	0.284060	0.142030	+	0.989862i	$0.454637\pi$
$68$	−236.444	−0.421663
$69$	0	0
$70$	2073.87	3.54107
$71$	13.7025	0.0229040	0.0114520	−	0.999934i	$-0.496355\pi$
$71$	13.7025	0.0229040	0.0114520	+	0.999934i	$0.496355\pi$
$72$	0	0
$73$	−859.846	−1.37859	−0.689297	−	0.724479i	$-0.742081\pi$
$73$	−859.846	−1.37859	−0.689297	+	0.724479i	$0.742081\pi$
$74$	−1250.05	−1.96372
$75$	0	0
$76$	−1500.41	−2.26459
$77$	−729.606	−1.07982
$78$	0	0
$79$	236.911	0.337400	0.168700	−	0.985667i	$-0.446043\pi$
$79$	236.911	0.337400	0.168700	+	0.985667i	$0.446043\pi$
$80$	−1323.55	−1.84972
$81$	0	0
$82$	1151.08	1.55019
$83$	595.613	0.787675	0.393837	−	0.919180i	$-0.371147\pi$
$83$	595.613	0.787675	0.393837	+	0.919180i	$0.371147\pi$
$84$	0	0
$85$	331.157	0.422576
$86$	269.309	0.337679
$87$	0	0
$88$	1452.28	1.75925
$89$	−36.2036	−0.0431189	−0.0215594	−	0.999768i	$-0.506863\pi$
$89$	−36.2036	−0.0431189	−0.0215594	+	0.999768i	$0.506863\pi$
$90$	0	0
$91$	−889.339	−1.02448
$92$	−1068.24	−1.21057
$93$	0	0
$94$	1228.94	1.34846
$95$	2101.43	2.26950
$96$	0	0
$97$	695.709	0.728233	0.364116	−	0.931353i	$-0.381371\pi$
$97$	695.709	0.728233	0.364116	+	0.931353i	$0.381371\pi$
$98$	−120.738	−0.124453
$99$	0	0
$100$	5790.18	5.79018
$101$	1365.38	1.34516	0.672578	−	0.740027i	$-0.265187\pi$
$101$	1365.38	1.34516	0.672578	+	0.740027i	$0.265187\pi$
$102$	0	0
$103$	1155.74	1.10562	0.552809	−	0.833308i	$-0.313556\pi$
$103$	1155.74	1.10562	0.552809	+	0.833308i	$0.313556\pi$
$104$	1770.23	1.66909
$105$	0	0
$106$	280.895	0.257386
$107$	−94.1753	−0.0850866	−0.0425433	−	0.999095i	$-0.513546\pi$
$107$	−94.1753	−0.0850866	−0.0425433	+	0.999095i	$0.513546\pi$
$108$	0	0
$109$	−1262.68	−1.10957	−0.554783	−	0.831995i	$-0.687199\pi$
$109$	−1262.68	−1.10957	−0.554783	+	0.831995i	$0.687199\pi$
$110$	−4114.64	−3.56650
$111$	0	0
$112$	1145.45	0.966383
$113$	−103.549	−0.0862039	−0.0431020	−	0.999071i	$-0.513724\pi$
$113$	−103.549	−0.0862039	−0.0431020	+	0.999071i	$0.513724\pi$
$114$	0	0
$115$	1496.15	1.21319
$116$	4542.75	3.63607
$117$	0	0
$118$	−2374.02	−1.85209
$119$	−286.595	−0.220774
$120$	0	0
$121$	116.567	0.0875783
$122$	−137.525	−0.102057
$123$	0	0
$124$	296.066	0.214415
$125$	−5339.78	−3.82083
$126$	0	0
$127$	38.6285	0.0269899	0.0134950	−	0.999909i	$-0.495704\pi$
$127$	38.6285	0.0269899	0.0134950	+	0.999909i	$0.495704\pi$
$128$	2551.13	1.76164
$129$	0	0
$130$	−5015.46	−3.38373
$131$	2601.84	1.73530	0.867648	−	0.497178i	$-0.165631\pi$
$131$	2601.84	1.73530	0.867648	+	0.497178i	$0.165631\pi$
$132$	0	0
$133$	−1818.66	−1.18570
$134$	−760.328	−0.490167
$135$	0	0
$136$	570.467	0.359685
$137$	2217.76	1.38304	0.691519	−	0.722358i	$-0.256942\pi$
$137$	2217.76	1.38304	0.691519	+	0.722358i	$0.256942\pi$
$138$	0	0
$139$	267.808	0.163418	0.0817092	−	0.996656i	$-0.473962\pi$
$139$	267.808	0.163418	0.0817092	+	0.996656i	$0.473962\pi$
$140$	−6722.52	−4.05826
$141$	0	0
$142$	−66.8771	−0.0395225
$143$	1764.48	1.03184
$144$	0	0
$145$	−6362.44	−3.64394
$146$	4196.62	2.37887
$147$	0	0
$148$	4052.08	2.25053
$149$	2100.82	1.15507	0.577536	−	0.816365i	$-0.304014\pi$
$149$	2100.82	1.15507	0.577536	+	0.816365i	$0.304014\pi$
$150$	0	0
$151$	659.216	0.355273	0.177637	−	0.984096i	$-0.443155\pi$
$151$	659.216	0.355273	0.177637	+	0.984096i	$0.443155\pi$
$152$	3620.03	1.93173
$153$	0	0
$154$	3560.96	1.86331
$155$	−414.661	−0.214880
$156$	0	0
$157$	2282.60	1.16033	0.580163	−	0.814501i	$-0.302989\pi$
$157$	2282.60	1.16033	0.580163	+	0.814501i	$0.302989\pi$
$158$	−1156.28	−0.582209
$159$	0	0
$160$	−306.557	−0.151471
$161$	−1294.82	−0.633828
$162$	0	0
$163$	−176.692	−0.0849053	−0.0424527	−	0.999098i	$-0.513517\pi$
$163$	−176.692	−0.0849053	−0.0424527	+	0.999098i	$0.513517\pi$
$164$	−3731.27	−1.77661
$165$	0	0
$166$	−2906.98	−1.35919
$167$	−3048.62	−1.41263	−0.706314	−	0.707898i	$-0.749643\pi$
$167$	−3048.62	−1.41263	−0.706314	+	0.707898i	$0.749643\pi$
$168$	0	0
$169$	−46.2169	−0.0210364
$170$	−1616.26	−0.729186
$171$	0	0
$172$	−872.976	−0.386999
$173$	−3220.09	−1.41514	−0.707568	−	0.706645i	$-0.750208\pi$
$173$	−3220.09	−1.41514	−0.707568	+	0.706645i	$0.750208\pi$
$174$	0	0
$175$	7018.30	3.03162
$176$	−2272.62	−0.973324
$177$	0	0
$178$	176.698	0.0744048
$179$	3094.02	1.29194	0.645972	−	0.763361i	$-0.276452\pi$
$179$	3094.02	1.29194	0.645972	+	0.763361i	$0.276452\pi$
$180$	0	0
$181$	478.914	0.196671	0.0983353	−	0.995153i	$-0.468648\pi$
$181$	478.914	0.196671	0.0983353	+	0.995153i	$0.468648\pi$
$182$	4340.56	1.76782
$183$	0	0
$184$	2577.34	1.03263
$185$	−5675.21	−2.25541
$186$	0	0
$187$	568.616	0.222360
$188$	−3983.65	−1.54541
$189$	0	0
$190$	−10256.4	−3.91619
$191$	−5245.42	−1.98715	−0.993574	−	0.113180i	$-0.963896\pi$
$191$	−5245.42	−1.98715	−0.993574	+	0.113180i	$0.963896\pi$
$192$	0	0
$193$	1335.85	0.498221	0.249110	−	0.968475i	$-0.419862\pi$
$193$	1335.85	0.498221	0.249110	+	0.968475i	$0.419862\pi$
$194$	−3395.52	−1.25662
$195$	0	0
$196$	391.376	0.142630
$197$	3848.28	1.39177	0.695884	−	0.718154i	$-0.255013\pi$
$197$	3848.28	1.39177	0.695884	+	0.718154i	$0.255013\pi$
$198$	0	0
$199$	261.544	0.0931676	0.0465838	−	0.998914i	$-0.485167\pi$
$199$	261.544	0.0931676	0.0465838	+	0.998914i	$0.485167\pi$
$200$	−13969.9	−4.93911
$201$	0	0
$202$	−6663.97	−2.32116
$203$	5506.29	1.90377
$204$	0	0
$205$	5225.91	1.78045
$206$	−5640.78	−1.90782
$207$	0	0
$208$	−2770.16	−0.923443
$209$	3608.28	1.19421
$210$	0	0
$211$	3224.48	1.05205	0.526024	−	0.850470i	$-0.323682\pi$
$211$	3224.48	1.05205	0.526024	+	0.850470i	$0.323682\pi$
$212$	−910.531	−0.294979
$213$	0	0
$214$	459.637	0.146823
$215$	1222.66	0.387837
$216$	0	0
$217$	358.863	0.112264
$218$	6162.70	1.91464
$219$	0	0
$220$	13337.7	4.08741
$221$	693.103	0.210965
$222$	0	0
$223$	−3565.09	−1.07057	−0.535283	−	0.844673i	$-0.679795\pi$
$223$	−3565.09	−1.07057	−0.535283	+	0.844673i	$0.679795\pi$
$224$	265.305	0.0791360
$225$	0	0
$226$	505.386	0.148751
$227$	5926.84	1.73294	0.866471	−	0.499227i	$-0.166382\pi$
$227$	5926.84	1.73294	0.866471	+	0.499227i	$0.166382\pi$
$228$	0	0
$229$	1103.42	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$229$	1103.42	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$230$	−7302.20	−2.09345
$231$	0	0
$232$	−10960.3	−3.10162
$233$	1535.32	0.431684	0.215842	−	0.976428i	$-0.430750\pi$
$233$	1535.32	0.431684	0.215842	+	0.976428i	$0.430750\pi$
$234$	0	0
$235$	5579.37	1.54876
$236$	7695.47	2.12260
$237$	0	0
$238$	1398.77	0.380962
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	1856.18	0.496129	0.248064	−	0.968744i	$-0.420206\pi$
$241$	1856.18	0.496129	0.248064	+	0.968744i	$0.420206\pi$
$242$	−568.923	−0.151123
$243$	0	0
$244$	445.793	0.116963
$245$	−548.149	−0.142939
$246$	0	0
$247$	4398.25	1.13301
$248$	−714.316	−0.182900
$249$	0	0
$250$	26061.6	6.59313
$251$	2221.39	0.558616	0.279308	−	0.960202i	$-0.409895\pi$
$251$	2221.39	0.558616	0.279308	+	0.960202i	$0.409895\pi$
$252$	0	0
$253$	2568.98	0.638381
$254$	−188.532	−0.0465731
$255$	0	0
$256$	−8088.17	−1.97465
$257$	−3346.82	−0.812330	−0.406165	−	0.913800i	$-0.633134\pi$
$257$	−3346.82	−0.812330	−0.406165	+	0.913800i	$0.633134\pi$
$258$	0	0
$259$	4911.54	1.17833
$260$	16257.8	3.87794
$261$	0	0
$262$	−12698.7	−2.99438
$263$	−1590.35	−0.372872	−0.186436	−	0.982467i	$-0.559694\pi$
$263$	−1590.35	−0.372872	−0.186436	+	0.982467i	$0.559694\pi$
$264$	0	0
$265$	1275.26	0.295618
$266$	8876.24	2.04600
$267$	0	0
$268$	2464.63	0.561758
$269$	−2064.68	−0.467978	−0.233989	−	0.972239i	$-0.575178\pi$
$269$	−2064.68	−0.467978	−0.233989	+	0.972239i	$0.575178\pi$
$270$	0	0
$271$	4124.24	0.924465	0.462232	−	0.886759i	$-0.347049\pi$
$271$	4124.24	0.924465	0.462232	+	0.886759i	$0.347049\pi$
$272$	−892.702	−0.199000
$273$	0	0
$274$	−10824.1	−2.38653
$275$	−13924.6	−3.05339
$276$	0	0
$277$	5478.18	1.18828	0.594138	−	0.804363i	$-0.297493\pi$
$277$	5478.18	1.18828	0.594138	+	0.804363i	$0.297493\pi$
$278$	−1307.08	−0.281990
$279$	0	0
$280$	16219.4	3.46176
$281$	−2244.60	−0.476518	−0.238259	−	0.971202i	$-0.576577\pi$
$281$	−2244.60	−0.476518	−0.238259	+	0.971202i	$0.576577\pi$
$282$	0	0
$283$	−8956.77	−1.88136	−0.940680	−	0.339295i	$-0.889811\pi$
$283$	−8956.77	−1.88136	−0.940680	+	0.339295i	$0.889811\pi$
$284$	216.784	0.0452950
$285$	0	0
$286$	−8611.84	−1.78052
$287$	−4522.69	−0.930195
$288$	0	0
$289$	−4689.64	−0.954538
$290$	31052.9	6.28789
$291$	0	0
$292$	−13603.5	−2.72631
$293$	−8894.07	−1.77337	−0.886685	−	0.462375i	$-0.846997\pi$
$293$	−8894.07	−1.77337	−0.886685	+	0.462375i	$0.846997\pi$
$294$	0	0
$295$	−10778.0	−2.12719
$296$	−9776.41	−1.91974
$297$	0	0
$298$	−10253.4	−1.99316
$299$	3131.41	0.605665
$300$	0	0
$301$	−1058.14	−0.202625
$302$	−3217.41	−0.613050
$303$	0	0
$304$	−5664.85	−1.06875
$305$	−624.363	−0.117216
$306$	0	0
$307$	9150.39	1.70111	0.850555	−	0.525887i	$-0.176266\pi$
$307$	9150.39	1.70111	0.850555	+	0.525887i	$0.176266\pi$
$308$	−11543.0	−2.13546
$309$	0	0
$310$	2023.82	0.370791
$311$	9363.27	1.70721	0.853605	−	0.520921i	$-0.174411\pi$
$311$	9363.27	1.70721	0.853605	+	0.520921i	$0.174411\pi$
$312$	0	0
$313$	5700.12	1.02936	0.514680	−	0.857382i	$-0.327911\pi$
$313$	5700.12	1.02936	0.514680	+	0.857382i	$0.327911\pi$
$314$	−11140.6	−2.00223
$315$	0	0
$316$	3748.13	0.667244
$317$	−4301.90	−0.762204	−0.381102	−	0.924533i	$-0.624455\pi$
$317$	−4301.90	−0.762204	−0.381102	+	0.924533i	$0.624455\pi$
$318$	0	0
$319$	−10924.7	−1.91745
$320$	12084.6	2.11109
$321$	0	0
$322$	6319.59	1.09372
$323$	1417.36	0.244162
$324$	0	0
$325$	−16973.1	−2.89692
$326$	862.372	0.146510
$327$	0	0
$328$	9002.41	1.51547
$329$	−4828.60	−0.809146
$330$	0	0
$331$	−4533.88	−0.752884	−0.376442	−	0.926440i	$-0.622853\pi$
$331$	−4533.88	−0.752884	−0.376442	+	0.926440i	$0.622853\pi$
$332$	9423.09	1.55771
$333$	0	0
$334$	14879.3	2.43759
$335$	−3451.89	−0.562975
$336$	0	0
$337$	−7552.02	−1.22073	−0.610363	−	0.792122i	$-0.708976\pi$
$337$	−7552.02	−1.22073	−0.610363	+	0.792122i	$0.708976\pi$
$338$	225.569	0.0362998
$339$	0	0
$340$	5239.17	0.835688
$341$	−711.997	−0.113070
$342$	0	0
$343$	−6103.15	−0.960755
$344$	2106.22	0.330116
$345$	0	0
$346$	15716.1	2.44192
$347$	10681.0	1.65241	0.826204	−	0.563371i	$-0.190496\pi$
$347$	10681.0	1.65241	0.826204	+	0.563371i	$0.190496\pi$
$348$	0	0
$349$	−7989.89	−1.22547	−0.612735	−	0.790289i	$-0.709931\pi$
$349$	−7989.89	−1.22547	−0.612735	+	0.790289i	$0.709931\pi$
$350$	−34253.9	−5.23128
$351$	0	0
$352$	−526.376	−0.0797044
$353$	−10714.4	−1.61549	−0.807745	−	0.589532i	$-0.799312\pi$
$353$	−10714.4	−1.61549	−0.807745	+	0.589532i	$0.799312\pi$
$354$	0	0
$355$	−303.622	−0.0453931
$356$	−572.772	−0.0852720
$357$	0	0
$358$	−15100.9	−2.22935
$359$	−11652.8	−1.71312	−0.856562	−	0.516045i	$-0.827404\pi$
$359$	−11652.8	−1.71312	−0.856562	+	0.516045i	$0.827404\pi$
$360$	0	0
$361$	2135.21	0.311300
$362$	−2337.41	−0.339369
$363$	0	0
$364$	−14070.1	−2.02602
$365$	19052.6	2.73222
$366$	0	0
$367$	−1903.43	−0.270731	−0.135365	−	0.990796i	$-0.543221\pi$
$367$	−1903.43	−0.270731	−0.135365	+	0.990796i	$0.543221\pi$
$368$	−4033.19	−0.571316
$369$	0	0
$370$	27698.8	3.89187
$371$	−1103.66	−0.154445
$372$	0	0
$373$	1241.00	0.172269	0.0861346	−	0.996284i	$-0.472548\pi$
$373$	1241.00	0.172269	0.0861346	+	0.996284i	$0.472548\pi$
$374$	−2775.22	−0.383698
$375$	0	0
$376$	9611.31	1.31826
$377$	−13316.4	−1.81918
$378$	0	0
$379$	730.505	0.0990067	0.0495034	−	0.998774i	$-0.484236\pi$
$379$	730.505	0.0990067	0.0495034	+	0.998774i	$0.484236\pi$
$380$	33246.4	4.48817
$381$	0	0
$382$	25601.1	3.42897
$383$	−590.313	−0.0787562	−0.0393781	−	0.999224i	$-0.512538\pi$
$383$	−590.313	−0.0787562	−0.0393781	+	0.999224i	$0.512538\pi$
$384$	0	0
$385$	16166.7	2.14009
$386$	−6519.83	−0.859716
$387$	0	0
$388$	11006.7	1.44016
$389$	7050.37	0.918941	0.459471	−	0.888193i	$-0.348039\pi$
$389$	7050.37	0.918941	0.459471	+	0.888193i	$0.348039\pi$
$390$	0	0
$391$	1009.12	0.130520
$392$	−944.269	−0.121665
$393$	0	0
$394$	−18782.1	−2.40160
$395$	−5249.52	−0.668689
$396$	0	0
$397$	5039.69	0.637115	0.318558	−	0.947903i	$-0.396802\pi$
$397$	5039.69	0.637115	0.318558	+	0.947903i	$0.396802\pi$
$398$	−1276.51	−0.160768
$399$	0	0
$400$	21861.0	2.73262
$401$	12329.7	1.53546	0.767728	−	0.640776i	$-0.221387\pi$
$401$	12329.7	1.53546	0.767728	+	0.640776i	$0.221387\pi$
$402$	0	0
$403$	−867.875	−0.107275
$404$	21601.5	2.66018
$405$	0	0
$406$	−26874.3	−3.28510
$407$	−9744.68	−1.18680
$408$	0	0
$409$	−9205.82	−1.11295	−0.556477	−	0.830863i	$-0.687847\pi$
$409$	−9205.82	−1.11295	−0.556477	+	0.830863i	$0.687847\pi$
$410$	−25505.9	−3.07230
$411$	0	0
$412$	18284.8	2.18647
$413$	9327.71	1.11135
$414$	0	0
$415$	−13197.7	−1.56108
$416$	−641.616	−0.0756198
$417$	0	0
$418$	−17610.8	−2.06070
$419$	−13015.6	−1.51755	−0.758774	−	0.651354i	$-0.774201\pi$
$419$	−13015.6	−1.51755	−0.758774	+	0.651354i	$0.774201\pi$
$420$	0	0
$421$	−4749.27	−0.549798	−0.274899	−	0.961473i	$-0.588644\pi$
$421$	−4749.27	−0.549798	−0.274899	+	0.961473i	$0.588644\pi$
$422$	−15737.6	−1.81539
$423$	0	0
$424$	2196.83	0.251622
$425$	−5469.68	−0.624279
$426$	0	0
$427$	540.347	0.0612394
$428$	−1489.93	−0.168268
$429$	0	0
$430$	−5967.40	−0.669241
$431$	−1146.30	−0.128109	−0.0640547	−	0.997946i	$-0.520403\pi$
$431$	−1146.30	−0.128109	−0.0640547	+	0.997946i	$0.520403\pi$
$432$	0	0
$433$	13887.9	1.54136	0.770679	−	0.637224i	$-0.219917\pi$
$433$	13887.9	1.54136	0.770679	+	0.637224i	$0.219917\pi$
$434$	−1751.49	−0.193719
$435$	0	0
$436$	−19976.6	−2.19428
$437$	6403.58	0.700972
$438$	0	0
$439$	693.250	0.0753690	0.0376845	−	0.999290i	$-0.488002\pi$
$439$	693.250	0.0753690	0.0376845	+	0.999290i	$0.488002\pi$
$440$	−32179.9	−3.48663
$441$	0	0
$442$	−3382.80	−0.364035
$443$	−7626.87	−0.817977	−0.408988	−	0.912540i	$-0.634118\pi$
$443$	−7626.87	−0.817977	−0.408988	+	0.912540i	$0.634118\pi$
$444$	0	0
$445$	802.207	0.0854567
$446$	17400.0	1.84734
$447$	0	0
$448$	−10458.5	−1.10294
$449$	−2351.73	−0.247183	−0.123591	−	0.992333i	$-0.539441\pi$
$449$	−2351.73	−0.247183	−0.123591	+	0.992333i	$0.539441\pi$
$450$	0	0
$451$	8973.19	0.936876
$452$	−1638.23	−0.170477
$453$	0	0
$454$	−28926.9	−2.99032
$455$	19706.1	2.03041
$456$	0	0
$457$	9770.57	1.00010	0.500052	−	0.865995i	$-0.333314\pi$
$457$	9770.57	1.00010	0.500052	+	0.865995i	$0.333314\pi$
$458$	−5385.41	−0.549440
$459$	0	0
$460$	23670.3	2.39921
$461$	−994.886	−0.100513	−0.0502565	−	0.998736i	$-0.516004\pi$
$461$	−994.886	−0.100513	−0.0502565	+	0.998736i	$0.516004\pi$
$462$	0	0
$463$	−6404.12	−0.642818	−0.321409	−	0.946940i	$-0.604156\pi$
$463$	−6404.12	−0.642818	−0.321409	+	0.946940i	$0.604156\pi$
$464$	17151.3	1.71601
$465$	0	0
$466$	−7493.39	−0.744902
$467$	−12608.6	−1.24937	−0.624684	−	0.780877i	$-0.714772\pi$
$467$	−12608.6	−1.24937	−0.624684	+	0.780877i	$0.714772\pi$
$468$	0	0
$469$	2987.39	0.294125
$470$	−27231.0	−2.67250
$471$	0	0
$472$	−18566.8	−1.81061
$473$	2099.39	0.204080
$474$	0	0
$475$	−34709.2	−3.35277
$476$	−4534.17	−0.436604
$477$	0	0
$478$	−1166.48	−0.111618
$479$	−18325.7	−1.74806	−0.874031	−	0.485869i	$-0.838503\pi$
$479$	−18325.7	−1.74806	−0.874031	+	0.485869i	$0.838503\pi$
$480$	0	0
$481$	−11878.1	−1.12598
$482$	−9059.37	−0.856106
$483$	0	0
$484$	1844.18	0.173195
$485$	−15415.6	−1.44327
$486$	0	0
$487$	15037.3	1.39919	0.699596	−	0.714539i	$-0.253364\pi$
$487$	15037.3	1.39919	0.699596	+	0.714539i	$0.253364\pi$
$488$	−1075.56	−0.0997712
$489$	0	0
$490$	2675.33	0.246651
$491$	13985.7	1.28547	0.642734	−	0.766090i	$-0.277800\pi$
$491$	13985.7	1.28547	0.642734	+	0.766090i	$0.277800\pi$
$492$	0	0
$493$	−4291.31	−0.392030
$494$	−21466.3	−1.95509
$495$	0	0
$496$	1117.80	0.101191
$497$	262.765	0.0237156
$498$	0	0
$499$	−18121.1	−1.62567	−0.812835	−	0.582494i	$-0.802077\pi$
$499$	−18121.1	−1.62567	−0.812835	+	0.582494i	$0.802077\pi$
$500$	−84479.7	−7.55609
$501$	0	0
$502$	−10841.8	−0.963933
$503$	−19016.4	−1.68569	−0.842844	−	0.538158i	$-0.819120\pi$
$503$	−19016.4	−1.68569	−0.842844	+	0.538158i	$0.819120\pi$
$504$	0	0
$505$	−30254.4	−2.66595
$506$	−12538.3	−1.10157
$507$	0	0
$508$	611.134	0.0533754
$509$	14889.0	1.29655	0.648275	−	0.761406i	$-0.275491\pi$
$509$	14889.0	1.29655	0.648275	+	0.761406i	$0.275491\pi$
$510$	0	0
$511$	−16488.8	−1.42744
$512$	19066.5	1.64576
$513$	0	0
$514$	16334.7	1.40174
$515$	−25609.1	−2.19121
$516$	0	0
$517$	9580.12	0.814958
$518$	−23971.5	−2.03330
$519$	0	0
$520$	−39225.0	−3.30795
$521$	13551.3	1.13952	0.569761	−	0.821810i	$-0.307036\pi$
$521$	13551.3	1.13952	0.569761	+	0.821810i	$0.307036\pi$
$522$	0	0
$523$	−2753.51	−0.230216	−0.115108	−	0.993353i	$-0.536721\pi$
$523$	−2753.51	−0.230216	−0.115108	+	0.993353i	$0.536721\pi$
$524$	41163.3	3.43173
$525$	0	0
$526$	7761.97	0.643418
$527$	−279.678	−0.0231176
$528$	0	0
$529$	−7607.87	−0.625287
$530$	−6224.12	−0.510110
$531$	0	0
$532$	−28772.7	−2.34484
$533$	10937.7	0.888864
$534$	0	0
$535$	2086.75	0.168632
$536$	−5946.40	−0.479189
$537$	0	0
$538$	10077.0	0.807530
$539$	−941.204	−0.0752144
$540$	0	0
$541$	−7317.61	−0.581532	−0.290766	−	0.956794i	$-0.593910\pi$
$541$	−7317.61	−0.581532	−0.290766	+	0.956794i	$0.593910\pi$
$542$	−20129.0	−1.59523
$543$	0	0
$544$	−206.765	−0.0162959
$545$	27978.6	2.19903
$546$	0	0
$547$	−7121.66	−0.556673	−0.278337	−	0.960484i	$-0.589783\pi$
$547$	−7121.66	−0.556673	−0.278337	+	0.960484i	$0.589783\pi$
$548$	35086.8	2.73510
$549$	0	0
$550$	67961.1	5.26886
$551$	−27231.5	−2.10544
$552$	0	0
$553$	4543.13	0.349355
$554$	−26737.2	−2.05046
$555$	0	0
$556$	4236.94	0.323177
$557$	−5445.83	−0.414268	−0.207134	−	0.978313i	$-0.566414\pi$
$557$	−5445.83	−0.414268	−0.207134	+	0.978313i	$0.566414\pi$
$558$	0	0
$559$	2559.01	0.193622
$560$	−25381.1	−1.91526
$561$	0	0
$562$	10955.1	0.822267
$563$	4660.13	0.348847	0.174424	−	0.984671i	$-0.444194\pi$
$563$	4660.13	0.348847	0.174424	+	0.984671i	$0.444194\pi$
$564$	0	0
$565$	2294.45	0.170846
$566$	43714.9	3.24642
$567$	0	0
$568$	−523.034	−0.0386374
$569$	560.305	0.0412816	0.0206408	−	0.999787i	$-0.493429\pi$
$569$	560.305	0.0412816	0.0206408	+	0.999787i	$0.493429\pi$
$570$	0	0
$571$	−2317.40	−0.169843	−0.0849214	−	0.996388i	$-0.527064\pi$
$571$	−2317.40	−0.169843	−0.0849214	+	0.996388i	$0.527064\pi$
$572$	27915.6	2.04058
$573$	0	0
$574$	22073.7	1.60512
$575$	−24711.8	−1.79226
$576$	0	0
$577$	−9845.12	−0.710325	−0.355163	−	0.934804i	$-0.615575\pi$
$577$	−9845.12	−0.710325	−0.355163	+	0.934804i	$0.615575\pi$
$578$	22888.5	1.64712
$579$	0	0
$580$	−100659.	−7.20628
$581$	11421.8	0.815585
$582$	0	0
$583$	2189.70	0.155554
$584$	32821.0	2.32559
$585$	0	0
$586$	43408.9	3.06008
$587$	−9842.27	−0.692051	−0.346025	−	0.938225i	$-0.612469\pi$
$587$	−9842.27	−0.692051	−0.346025	+	0.938225i	$0.612469\pi$
$588$	0	0
$589$	−1774.76	−0.124156
$590$	52603.9	3.67063
$591$	0	0
$592$	15298.7	1.06212
$593$	16680.2	1.15510	0.577550	−	0.816356i	$-0.304009\pi$
$593$	16680.2	1.15510	0.577550	+	0.816356i	$0.304009\pi$
$594$	0	0
$595$	6350.42	0.437550
$596$	33236.7	2.28428
$597$	0	0
$598$	−15283.3	−1.04512
$599$	−18730.2	−1.27762	−0.638811	−	0.769364i	$-0.720573\pi$
$599$	−18730.2	−1.27762	−0.638811	+	0.769364i	$0.720573\pi$
$600$	0	0
$601$	22204.5	1.50705	0.753527	−	0.657417i	$-0.228351\pi$
$601$	22204.5	1.50705	0.753527	+	0.657417i	$0.228351\pi$
$602$	5164.41	0.349644
$603$	0	0
$604$	10429.3	0.702589
$605$	−2582.91	−0.173570
$606$	0	0
$607$	−13820.9	−0.924173	−0.462087	−	0.886835i	$-0.652899\pi$
$607$	−13820.9	−0.924173	−0.462087	+	0.886835i	$0.652899\pi$
$608$	−1312.07	−0.0875191
$609$	0	0
$610$	3047.30	0.202265
$611$	11677.5	0.773193
$612$	0	0
$613$	−19299.2	−1.27159	−0.635796	−	0.771857i	$-0.719328\pi$
$613$	−19299.2	−1.27159	−0.635796	+	0.771857i	$0.719328\pi$
$614$	−44659.9	−2.93539
$615$	0	0
$616$	27849.6	1.82158
$617$	−12670.3	−0.826719	−0.413359	−	0.910568i	$-0.635645\pi$
$617$	−12670.3	−0.826719	−0.413359	+	0.910568i	$0.635645\pi$
$618$	0	0
$619$	−19760.9	−1.28313	−0.641565	−	0.767069i	$-0.721714\pi$
$619$	−19760.9	−1.28313	−0.641565	+	0.767069i	$0.721714\pi$
$620$	−6560.28	−0.424947
$621$	0	0
$622$	−45698.9	−2.94592
$623$	−694.259	−0.0446467
$624$	0	0
$625$	72571.6	4.64458
$626$	−27820.3	−1.77624
$627$	0	0
$628$	36112.6	2.29466
$629$	−3827.79	−0.242645
$630$	0	0
$631$	−18921.7	−1.19376	−0.596879	−	0.802331i	$-0.703593\pi$
$631$	−18921.7	−1.19376	−0.596879	+	0.802331i	$0.703593\pi$
$632$	−9043.09	−0.569169
$633$	0	0
$634$	20996.1	1.31524
$635$	−855.936	−0.0534910
$636$	0	0
$637$	−1147.26	−0.0713598
$638$	53319.7	3.30869
$639$	0	0
$640$	−56528.4	−3.49138
$641$	−15666.3	−0.965337	−0.482668	−	0.875803i	$-0.660332\pi$
$641$	−15666.3	−0.965337	−0.482668	+	0.875803i	$0.660332\pi$
$642$	0	0
$643$	9696.30	0.594688	0.297344	−	0.954770i	$-0.403899\pi$
$643$	9696.30	0.594688	0.297344	+	0.954770i	$0.403899\pi$
$644$	−20485.2	−1.25346
$645$	0	0
$646$	−6917.66	−0.421319
$647$	13944.4	0.847310	0.423655	−	0.905824i	$-0.360747\pi$
$647$	13944.4	0.847310	0.423655	+	0.905824i	$0.360747\pi$
$648$	0	0
$649$	−18506.5	−1.11933
$650$	82839.9	4.99884
$651$	0	0
$652$	−2795.41	−0.167909
$653$	−20298.4	−1.21644	−0.608221	−	0.793767i	$-0.708117\pi$
$653$	−20298.4	−1.21644	−0.608221	+	0.793767i	$0.708117\pi$
$654$	0	0
$655$	−57652.0	−3.43916
$656$	−14087.5	−0.838453
$657$	0	0
$658$	23566.7	1.39624
$659$	17439.6	1.03088	0.515441	−	0.856925i	$-0.327628\pi$
$659$	17439.6	1.03088	0.515441	+	0.856925i	$0.327628\pi$
$660$	0	0
$661$	−4471.87	−0.263140	−0.131570	−	0.991307i	$-0.542002\pi$
$661$	−4471.87	−0.263140	−0.131570	+	0.991307i	$0.542002\pi$
$662$	22128.3	1.29916
$663$	0	0
$664$	−22735.0	−1.32875
$665$	40298.1	2.34991
$666$	0	0
$667$	−19387.9	−1.12549
$668$	−48231.6	−2.79362
$669$	0	0
$670$	16847.5	0.971455
$671$	−1072.07	−0.0616793
$672$	0	0
$673$	25280.3	1.44797	0.723984	−	0.689816i	$-0.242309\pi$
$673$	25280.3	1.44797	0.723984	+	0.689816i	$0.242309\pi$
$674$	36858.8	2.10645
$675$	0	0
$676$	−731.189	−0.0416016
$677$	16973.4	0.963573	0.481787	−	0.876289i	$-0.339988\pi$
$677$	16973.4	0.963573	0.481787	+	0.876289i	$0.339988\pi$
$678$	0	0
$679$	13341.3	0.754036
$680$	−12640.5	−0.712855
$681$	0	0
$682$	3475.02	0.195110
$683$	11194.2	0.627135	0.313568	−	0.949566i	$-0.398476\pi$
$683$	11194.2	0.627135	0.313568	+	0.949566i	$0.398476\pi$
$684$	0	0
$685$	−49141.5	−2.74102
$686$	29787.4	1.65785
$687$	0	0
$688$	−3295.95	−0.182641
$689$	2669.09	0.147583
$690$	0	0
$691$	−10596.9	−0.583392	−0.291696	−	0.956511i	$-0.594220\pi$
$691$	−10596.9	−0.583392	−0.291696	+	0.956511i	$0.594220\pi$
$692$	−50944.4	−2.79858
$693$	0	0
$694$	−52130.3	−2.85135
$695$	−5934.13	−0.323877
$696$	0	0
$697$	3524.74	0.191548
$698$	38995.9	2.11464
$699$	0	0
$700$	111035.	5.99534
$701$	359.802	0.0193859	0.00969296	−	0.999953i	$-0.496915\pi$
$701$	359.802	0.0193859	0.00969296	+	0.999953i	$0.496915\pi$
$702$	0	0
$703$	−24290.1	−1.30316
$704$	20750.0	1.11086
$705$	0	0
$706$	52293.1	2.78765
$707$	26183.3	1.39282
$708$	0	0
$709$	−21610.1	−1.14469	−0.572343	−	0.820014i	$-0.693966\pi$
$709$	−21610.1	−1.14469	−0.572343	+	0.820014i	$0.693966\pi$
$710$	1481.87	0.0783292
$711$	0	0
$712$	1381.92	0.0727384
$713$	−1263.57	−0.0663691
$714$	0	0
$715$	−39097.7	−2.04500
$716$	48950.0	2.55495
$717$	0	0
$718$	56873.3	2.95612
$719$	12467.6	0.646678	0.323339	−	0.946283i	$-0.395195\pi$
$719$	12467.6	0.646678	0.323339	+	0.946283i	$0.395195\pi$
$720$	0	0
$721$	22163.1	1.14479
$722$	−10421.2	−0.537171
$723$	0	0
$724$	7576.81	0.388936
$725$	105088.	5.38326
$726$	0	0
$727$	26998.5	1.37733	0.688666	−	0.725079i	$-0.258197\pi$
$727$	26998.5	1.37733	0.688666	+	0.725079i	$0.258197\pi$
$728$	33946.8	1.72823
$729$	0	0
$730$	−92989.3	−4.71464
$731$	824.656	0.0417250
$732$	0	0
$733$	−13933.1	−0.702086	−0.351043	−	0.936359i	$-0.614173\pi$
$733$	−13933.1	−0.702086	−0.351043	+	0.936359i	$0.614173\pi$
$734$	9289.99	0.467166
$735$	0	0
$736$	−934.153	−0.0467844
$737$	−5927.10	−0.296238
$738$	0	0
$739$	16084.6	0.800651	0.400325	−	0.916373i	$-0.368897\pi$
$739$	16084.6	0.800651	0.400325	+	0.916373i	$0.368897\pi$
$740$	−89786.6	−4.46030
$741$	0	0
$742$	5386.58	0.266506
$743$	24337.3	1.20168	0.600840	−	0.799369i	$-0.294833\pi$
$743$	24337.3	1.20168	0.600840	+	0.799369i	$0.294833\pi$
$744$	0	0
$745$	−46550.3	−2.28922
$746$	−6056.89	−0.297263
$747$	0	0
$748$	8995.97	0.439740
$749$	−1805.95	−0.0881015
$750$	0	0
$751$	14637.0	0.711198	0.355599	−	0.934639i	$-0.384277\pi$
$751$	14637.0	0.711198	0.355599	+	0.934639i	$0.384277\pi$
$752$	−15040.4	−0.729343
$753$	0	0
$754$	64993.0	3.13913
$755$	−14607.0	−0.704111
$756$	0	0
$757$	4285.60	0.205763	0.102882	−	0.994694i	$-0.467194\pi$
$757$	4285.60	0.205763	0.102882	+	0.994694i	$0.467194\pi$
$758$	−3565.35	−0.170843
$759$	0	0
$760$	−80213.3	−3.82848
$761$	5966.50	0.284212	0.142106	−	0.989851i	$-0.454613\pi$
$761$	5966.50	0.284212	0.142106	+	0.989851i	$0.454613\pi$
$762$	0	0
$763$	−24213.8	−1.14888
$764$	−82986.9	−3.92979
$765$	0	0
$766$	2881.12	0.135900
$767$	−22558.2	−1.06197
$768$	0	0
$769$	−33819.4	−1.58590	−0.792951	−	0.609286i	$-0.791456\pi$
$769$	−33819.4	−1.58590	−0.792951	+	0.609286i	$0.791456\pi$
$770$	−78904.3	−3.69288
$771$	0	0
$772$	21134.3	0.985283
$773$	−37869.3	−1.76205	−0.881024	−	0.473072i	$-0.843145\pi$
$773$	−37869.3	−1.76205	−0.881024	+	0.473072i	$0.843145\pi$
$774$	0	0
$775$	6848.91	0.317445
$776$	−26555.8	−1.22848
$777$	0	0
$778$	−34410.5	−1.58570
$779$	22367.1	1.02873
$780$	0	0
$781$	−521.337	−0.0238859
$782$	−4925.15	−0.225221
$783$	0	0
$784$	1477.65	0.0673128
$785$	−50578.2	−2.29963
$786$	0	0
$787$	30856.0	1.39758	0.698791	−	0.715326i	$-0.253722\pi$
$787$	30856.0	1.39758	0.698791	+	0.715326i	$0.253722\pi$
$788$	60883.0	2.75237
$789$	0	0
$790$	25621.1	1.15387
$791$	−1985.70	−0.0892584
$792$	0	0
$793$	−1306.78	−0.0585184
$794$	−24597.0	−1.09939
$795$	0	0
$796$	4137.84	0.184249
$797$	35794.6	1.59085	0.795427	−	0.606050i	$-0.207247\pi$
$797$	35794.6	1.59085	0.795427	+	0.606050i	$0.207247\pi$
$798$	0	0
$799$	3763.15	0.166621
$800$	5063.37	0.223771
$801$	0	0
$802$	−60177.2	−2.64954
$803$	32714.5	1.43770
$804$	0	0
$805$	28690.9	1.25618
$806$	4235.80	0.185111
$807$	0	0
$808$	−52117.7	−2.26918
$809$	−40689.7	−1.76832	−0.884161	−	0.467182i	$-0.845269\pi$
$809$	−40689.7	−1.76832	−0.884161	+	0.467182i	$0.845269\pi$
$810$	0	0
$811$	−36231.2	−1.56874	−0.784370	−	0.620293i	$-0.787014\pi$
$811$	−36231.2	−1.56874	−0.784370	+	0.620293i	$0.787014\pi$
$812$	87114.1	3.76491
$813$	0	0
$814$	47560.5	2.04790
$815$	3915.17	0.168273
$816$	0	0
$817$	5233.05	0.224089
$818$	44930.5	1.92049
$819$	0	0
$820$	82678.2	3.52103
$821$	11905.4	0.506091	0.253046	−	0.967454i	$-0.418568\pi$
$821$	11905.4	0.506091	0.253046	+	0.967454i	$0.418568\pi$
$822$	0	0
$823$	11971.3	0.507038	0.253519	−	0.967330i	$-0.418412\pi$
$823$	11971.3	0.507038	0.253519	+	0.967330i	$0.418412\pi$
$824$	−44115.6	−1.86510
$825$	0	0
$826$	−45525.4	−1.91771
$827$	−26508.0	−1.11460	−0.557300	−	0.830312i	$-0.688162\pi$
$827$	−26508.0	−1.11460	−0.557300	+	0.830312i	$0.688162\pi$
$828$	0	0
$829$	−33315.3	−1.39577	−0.697883	−	0.716212i	$-0.745874\pi$
$829$	−33315.3	−1.39577	−0.697883	+	0.716212i	$0.745874\pi$
$830$	64413.4	2.69376
$831$	0	0
$832$	25292.8	1.05393
$833$	−369.713	−0.0153779
$834$	0	0
$835$	67551.7	2.79967
$836$	57086.0	2.36168
$837$	0	0
$838$	63524.6	2.61864
$839$	11942.9	0.491437	0.245719	−	0.969341i	$-0.420976\pi$
$839$	11942.9	0.491437	0.245719	+	0.969341i	$0.420976\pi$
$840$	0	0
$841$	58058.9	2.38054
$842$	23179.6	0.948718
$843$	0	0
$844$	51013.9	2.08053
$845$	1024.08	0.0416917
$846$	0	0
$847$	2235.34	0.0906815
$848$	−3437.74	−0.139213
$849$	0	0
$850$	26695.7	1.07724
$851$	−17293.8	−0.696619
$852$	0	0
$853$	−35537.7	−1.42648	−0.713240	−	0.700920i	$-0.752773\pi$
$853$	−35537.7	−1.42648	−0.713240	+	0.700920i	$0.752773\pi$
$854$	−2637.25	−0.105673
$855$	0	0
$856$	3594.74	0.143535
$857$	−12798.3	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$857$	−12798.3	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$858$	0	0
$859$	−45668.3	−1.81395	−0.906975	−	0.421185i	$-0.861614\pi$
$859$	−45668.3	−1.81395	−0.906975	+	0.421185i	$0.861614\pi$
$860$	19343.6	0.766988
$861$	0	0
$862$	5594.68	0.221062
$863$	−9571.66	−0.377547	−0.188773	−	0.982021i	$-0.560451\pi$
$863$	−9571.66	−0.377547	−0.188773	+	0.982021i	$0.560451\pi$
$864$	0	0
$865$	71351.2	2.80464
$866$	−67781.9	−2.65972
$867$	0	0
$868$	5677.50	0.222013
$869$	−9013.74	−0.351865
$870$	0	0
$871$	−7224.72	−0.281057
$872$	48197.4	1.87176
$873$	0	0
$874$	−31253.7	−1.20958
$875$	−102398.	−3.95622
$876$	0	0
$877$	41029.5	1.57978	0.789891	−	0.613247i	$-0.210137\pi$
$877$	41029.5	1.57978	0.789891	+	0.613247i	$0.210137\pi$
$878$	−3383.52	−0.130055
$879$	0	0
$880$	50357.0	1.92902
$881$	−19868.6	−0.759808	−0.379904	−	0.925026i	$-0.624043\pi$
$881$	−19868.6	−0.759808	−0.379904	+	0.925026i	$0.624043\pi$
$882$	0	0
$883$	−34966.8	−1.33265	−0.666323	−	0.745663i	$-0.732133\pi$
$883$	−34966.8	−1.33265	−0.666323	+	0.745663i	$0.732133\pi$
$884$	10965.5	0.417204
$885$	0	0
$886$	37224.2	1.41148
$887$	41395.0	1.56698	0.783488	−	0.621407i	$-0.213439\pi$
$887$	41395.0	1.56698	0.783488	+	0.621407i	$0.213439\pi$
$888$	0	0
$889$	740.758	0.0279463
$890$	−3915.30	−0.147462
$891$	0	0
$892$	−56402.7	−2.11715
$893$	23879.9	0.894861
$894$	0	0
$895$	−68557.9	−2.56049
$896$	48921.8	1.82406
$897$	0	0
$898$	11478.0	0.426532
$899$	5373.40	0.199347
$900$	0	0
$901$	860.132	0.0318037
$902$	−43795.1	−1.61665
$903$	0	0
$904$	3952.54	0.145420
$905$	−10611.8	−0.389779
$906$	0	0
$907$	41497.8	1.51920	0.759598	−	0.650393i	$-0.225396\pi$
$907$	41497.8	1.51920	0.759598	+	0.650393i	$0.225396\pi$
$908$	93767.5	3.42707
$909$	0	0
$910$	−96178.9	−3.50363
$911$	−46700.7	−1.69842	−0.849212	−	0.528052i	$-0.822923\pi$
$911$	−46700.7	−1.69842	−0.849212	+	0.528052i	$0.822923\pi$
$912$	0	0
$913$	−22661.2	−0.821443
$914$	−47686.8	−1.72575
$915$	0	0
$916$	17457.0	0.629689
$917$	49894.2	1.79678
$918$	0	0
$919$	5838.59	0.209573	0.104786	−	0.994495i	$-0.466584\pi$
$919$	5838.59	0.209573	0.104786	+	0.994495i	$0.466584\pi$
$920$	−57109.2	−2.04656
$921$	0	0
$922$	4855.70	0.173442
$923$	−635.473	−0.0226618
$924$	0	0
$925$	93737.0	3.33195
$926$	31256.3	1.10923
$927$	0	0
$928$	3972.53	0.140522
$929$	−49754.6	−1.75715	−0.878577	−	0.477600i	$-0.841507\pi$
$929$	−49754.6	−1.75715	−0.878577	+	0.477600i	$0.841507\pi$
$930$	0	0
$931$	−2346.10	−0.0825889
$932$	24290.1	0.853700
$933$	0	0
$934$	61538.1	2.15588
$935$	−12599.5	−0.440692
$936$	0	0
$937$	2453.10	0.0855275	0.0427638	−	0.999085i	$-0.486384\pi$
$937$	2453.10	0.0855275	0.0427638	+	0.999085i	$0.486384\pi$
$938$	−14580.4	−0.507535
$939$	0	0
$940$	88270.3	3.06283
$941$	−44631.5	−1.54617	−0.773085	−	0.634303i	$-0.781287\pi$
$941$	−44631.5	−1.54617	−0.773085	+	0.634303i	$0.781287\pi$
$942$	0	0
$943$	15924.6	0.549922
$944$	29054.5	1.00174
$945$	0	0
$946$	−10246.4	−0.352155
$947$	56708.7	1.94592	0.972959	−	0.230977i	$-0.0741922\pi$
$947$	56708.7	1.94592	0.972959	+	0.230977i	$0.0741922\pi$
$948$	0	0
$949$	39876.7	1.36402
$950$	169404.	5.78545
$951$	0	0
$952$	10939.6	0.372430
$953$	21931.6	0.745472	0.372736	−	0.927937i	$-0.378420\pi$
$953$	21931.6	0.745472	0.372736	+	0.927937i	$0.378420\pi$
$954$	0	0
$955$	116229.	3.93830
$956$	3781.18	0.127921
$957$	0	0
$958$	89441.4	3.01641
$959$	42528.9	1.43204
$960$	0	0
$961$	−29440.8	−0.988245
$962$	57972.9	1.94295
$963$	0	0
$964$	29366.3	0.981145
$965$	−29600.0	−0.987417
$966$	0	0
$967$	−21948.8	−0.729914	−0.364957	−	0.931024i	$-0.618916\pi$
$967$	−21948.8	−0.729914	−0.364957	+	0.931024i	$0.618916\pi$
$968$	−4449.45	−0.147738
$969$	0	0
$970$	75238.5	2.49048
$971$	17886.1	0.591136	0.295568	−	0.955322i	$-0.404491\pi$
$971$	17886.1	0.591136	0.295568	+	0.955322i	$0.404491\pi$
$972$	0	0
$973$	5135.61	0.169209
$974$	−73392.0	−2.41441
$975$	0	0
$976$	1683.10	0.0551996
$977$	8552.17	0.280049	0.140025	−	0.990148i	$-0.455282\pi$
$977$	8552.17	0.280049	0.140025	+	0.990148i	$0.455282\pi$
$978$	0	0
$979$	1377.44	0.0449674
$980$	−8672.17	−0.282676
$981$	0	0
$982$	−68259.3	−2.21817
$983$	−24269.3	−0.787459	−0.393729	−	0.919226i	$-0.628815\pi$
$983$	−24269.3	−0.787459	−0.393729	+	0.919226i	$0.628815\pi$
$984$	0	0
$985$	−85270.8	−2.75833
$986$	20944.4	0.676476
$987$	0	0
$988$	69583.9	2.24065
$989$	3725.76	0.119790
$990$	0	0
$991$	34874.6	1.11789	0.558945	−	0.829205i	$-0.311206\pi$
$991$	34874.6	1.11789	0.558945	+	0.829205i	$0.311206\pi$
$992$	258.902	0.00828645
$993$	0	0
$994$	−1282.47	−0.0409230
$995$	−5795.33	−0.184648
$996$	0	0
$997$	−60140.2	−1.91039	−0.955195	−	0.295978i	$-0.904354\pi$
$997$	−60140.2	−1.91039	−0.955195	+	0.295978i	$0.904354\pi$
$998$	88442.7	2.80521
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.88066 q^{2} +15.8208 q^{4} -22.1582 q^{5} +19.1765 q^{7} -38.1708 q^{8} +O(q^{10})\)	\(q-4.88066 q^{2} +15.8208 q^{4} -22.1582 q^{5} +19.1765 q^{7} -38.1708 q^{8} +108.146 q^{10} -38.0469 q^{11} -46.3765 q^{13} -93.5939 q^{14} +59.7320 q^{16} -14.9451 q^{17} -94.8378 q^{19} -350.561 q^{20} +185.694 q^{22} -67.5214 q^{23} +365.984 q^{25} +226.348 q^{26} +303.388 q^{28} +287.137 q^{29} +18.7137 q^{31} +13.8349 q^{32} +72.9421 q^{34} -424.916 q^{35} +256.123 q^{37} +462.871 q^{38} +845.795 q^{40} -235.846 q^{41} -55.1789 q^{43} -601.934 q^{44} +329.549 q^{46} -251.798 q^{47} +24.7380 q^{49} -1786.25 q^{50} -733.715 q^{52} -57.5527 q^{53} +843.050 q^{55} -731.982 q^{56} -1401.42 q^{58} +486.414 q^{59} +28.1776 q^{61} -91.3351 q^{62} -545.379 q^{64} +1027.62 q^{65} +155.784 q^{67} -236.444 q^{68} +2073.87 q^{70} +13.7025 q^{71} -859.846 q^{73} -1250.05 q^{74} -1500.41 q^{76} -729.606 q^{77} +236.911 q^{79} -1323.55 q^{80} +1151.08 q^{82} +595.613 q^{83} +331.157 q^{85} +269.309 q^{86} +1452.28 q^{88} -36.2036 q^{89} -889.339 q^{91} -1068.24 q^{92} +1228.94 q^{94} +2101.43 q^{95} +695.709 q^{97} -120.738 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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

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