LMFDB - Embedded newform 2151.4.a.a.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:	$22$
Twist minimal:	no (minimal twist has level 239)
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)

$f(q)$	$=$	$q-3.32790 q^{2} +3.07490 q^{4} -7.52905 q^{5} -10.0131 q^{7} +16.3902 q^{8} +O(q^{10})$	\(q-3.32790 q^{2} +3.07490 q^{4} -7.52905 q^{5} -10.0131 q^{7} +16.3902 q^{8} +25.0559 q^{10} +7.51891 q^{11} -44.8420 q^{13} +33.3225 q^{14} -79.1442 q^{16} -89.2454 q^{17} -24.5927 q^{19} -23.1511 q^{20} -25.0222 q^{22} +177.431 q^{23} -68.3133 q^{25} +149.229 q^{26} -30.7893 q^{28} +172.228 q^{29} +235.607 q^{31} +132.262 q^{32} +296.999 q^{34} +75.3891 q^{35} -136.687 q^{37} +81.8420 q^{38} -123.403 q^{40} +231.712 q^{41} -11.0506 q^{43} +23.1199 q^{44} -590.473 q^{46} -53.8741 q^{47} -242.738 q^{49} +227.340 q^{50} -137.885 q^{52} -263.280 q^{53} -56.6103 q^{55} -164.117 q^{56} -573.156 q^{58} -773.202 q^{59} +68.7188 q^{61} -784.075 q^{62} +193.000 q^{64} +337.618 q^{65} +151.767 q^{67} -274.420 q^{68} -250.887 q^{70} -142.667 q^{71} +713.268 q^{73} +454.881 q^{74} -75.6201 q^{76} -75.2875 q^{77} +699.219 q^{79} +595.881 q^{80} -771.113 q^{82} +807.735 q^{83} +671.933 q^{85} +36.7753 q^{86} +123.237 q^{88} +888.191 q^{89} +449.007 q^{91} +545.583 q^{92} +179.287 q^{94} +185.160 q^{95} +1395.05 q^{97} +807.807 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	−3.32790	−1.17659	−0.588295	−	0.808647i	$-0.700200\pi$
$2$	−3.32790	−1.17659	−0.588295	+	0.808647i	$0.700200\pi$
$3$	0	0
$3$	0	0
$4$	3.07490	0.384362
$5$	−7.52905	−0.673419	−0.336710	−	0.941609i	$-0.609314\pi$
$5$	−7.52905	−0.673419	−0.336710	+	0.941609i	$0.609314\pi$
$6$	0	0
$7$	−10.0131	−0.540656	−0.270328	−	0.962768i	$-0.587132\pi$
$7$	−10.0131	−0.540656	−0.270328	+	0.962768i	$0.587132\pi$
$8$	16.3902	0.724353
$9$	0	0
$10$	25.0559	0.792338
$11$	7.51891	0.206094	0.103047	−	0.994676i	$-0.467141\pi$
$11$	7.51891	0.206094	0.103047	+	0.994676i	$0.467141\pi$
$12$	0	0
$13$	−44.8420	−0.956687	−0.478343	−	0.878173i	$-0.658763\pi$
$13$	−44.8420	−0.956687	−0.478343	+	0.878173i	$0.658763\pi$
$14$	33.3225	0.636130
$15$	0	0
$16$	−79.1442	−1.23663
$17$	−89.2454	−1.27325	−0.636623	−	0.771175i	$-0.719669\pi$
$17$	−89.2454	−1.27325	−0.636623	+	0.771175i	$0.719669\pi$
$18$	0	0
$19$	−24.5927	−0.296945	−0.148473	−	0.988917i	$-0.547436\pi$
$19$	−24.5927	−0.296945	−0.148473	+	0.988917i	$0.547436\pi$
$20$	−23.1511	−0.258837
$21$	0	0
$22$	−25.0222	−0.242488
$23$	177.431	1.60856	0.804282	−	0.594248i	$-0.202550\pi$
$23$	177.431	1.60856	0.804282	+	0.594248i	$0.202550\pi$
$24$	0	0
$25$	−68.3133	−0.546507
$26$	149.229	1.12563
$27$	0	0
$28$	−30.7893	−0.207808
$29$	172.228	1.10282	0.551412	−	0.834233i	$-0.314089\pi$
$29$	172.228	1.10282	0.551412	+	0.834233i	$0.314089\pi$
$30$	0	0
$31$	235.607	1.36504	0.682519	−	0.730867i	$-0.260884\pi$
$31$	235.607	1.36504	0.682519	+	0.730867i	$0.260884\pi$
$32$	132.262	0.730651
$33$	0	0
$34$	296.999	1.49809
$35$	75.3891	0.364088
$36$	0	0
$37$	−136.687	−0.607330	−0.303665	−	0.952779i	$-0.598210\pi$
$37$	−136.687	−0.607330	−0.303665	+	0.952779i	$0.598210\pi$
$38$	81.8420	0.349382
$39$	0	0
$40$	−123.403	−0.487793
$41$	231.712	0.882617	0.441308	−	0.897356i	$-0.354515\pi$
$41$	231.712	0.882617	0.441308	+	0.897356i	$0.354515\pi$
$42$	0	0
$43$	−11.0506	−0.0391907	−0.0195954	−	0.999808i	$-0.506238\pi$
$43$	−11.0506	−0.0391907	−0.0195954	+	0.999808i	$0.506238\pi$
$44$	23.1199	0.0792149
$45$	0	0
$46$	−590.473	−1.89262
$47$	−53.8741	−0.167199	−0.0835994	−	0.996499i	$-0.526642\pi$
$47$	−53.8741	−0.167199	−0.0835994	+	0.996499i	$0.526642\pi$
$48$	0	0
$49$	−242.738	−0.707691
$50$	227.340	0.643014
$51$	0	0
$52$	−137.885	−0.367714
$53$	−263.280	−0.682344	−0.341172	−	0.940001i	$-0.610824\pi$
$53$	−263.280	−0.682344	−0.341172	+	0.940001i	$0.610824\pi$
$54$	0	0
$55$	−56.6103	−0.138788
$56$	−164.117	−0.391626
$57$	0	0
$58$	−573.156	−1.29757
$59$	−773.202	−1.70614	−0.853070	−	0.521796i	$-0.825262\pi$
$59$	−773.202	−1.70614	−0.853070	+	0.521796i	$0.825262\pi$
$60$	0	0
$61$	68.7188	0.144238	0.0721192	−	0.997396i	$-0.477024\pi$
$61$	68.7188	0.144238	0.0721192	+	0.997396i	$0.477024\pi$
$62$	−784.075	−1.60609
$63$	0	0
$64$	193.000	0.376952
$65$	337.618	0.644251
$66$	0	0
$67$	151.767	0.276737	0.138368	−	0.990381i	$-0.455814\pi$
$67$	151.767	0.276737	0.138368	+	0.990381i	$0.455814\pi$
$68$	−274.420	−0.489388
$69$	0	0
$70$	−250.887	−0.428382
$71$	−142.667	−0.238471	−0.119236	−	0.992866i	$-0.538044\pi$
$71$	−142.667	−0.238471	−0.119236	+	0.992866i	$0.538044\pi$
$72$	0	0
$73$	713.268	1.14358	0.571792	−	0.820398i	$-0.306248\pi$
$73$	713.268	1.14358	0.571792	+	0.820398i	$0.306248\pi$
$74$	454.881	0.714578
$75$	0	0
$76$	−75.6201	−0.114135
$77$	−75.2875	−0.111426
$78$	0	0
$79$	699.219	0.995801	0.497900	−	0.867234i	$-0.334105\pi$
$79$	699.219	0.995801	0.497900	+	0.867234i	$0.334105\pi$
$80$	595.881	0.832769
$81$	0	0
$82$	−771.113	−1.03848
$83$	807.735	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$83$	807.735	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$84$	0	0
$85$	671.933	0.857428
$86$	36.7753	0.0461114
$87$	0	0
$88$	123.237	0.149285
$89$	888.191	1.05784	0.528921	−	0.848671i	$-0.322597\pi$
$89$	888.191	1.05784	0.528921	+	0.848671i	$0.322597\pi$
$90$	0	0
$91$	449.007	0.517239
$92$	545.583	0.618271
$93$	0	0
$94$	179.287	0.196724
$95$	185.160	0.199968
$96$	0	0
$97$	1395.05	1.46026	0.730132	−	0.683307i	$-0.239459\pi$
$97$	1395.05	1.46026	0.730132	+	0.683307i	$0.239459\pi$
$98$	807.807	0.832661
$99$	0	0
$100$	−210.057	−0.210057
$101$	−1001.48	−0.986647	−0.493323	−	0.869846i	$-0.664218\pi$
$101$	−1001.48	−0.986647	−0.493323	+	0.869846i	$0.664218\pi$
$102$	0	0
$103$	−1685.04	−1.61196	−0.805980	−	0.591943i	$-0.798361\pi$
$103$	−1685.04	−1.61196	−0.805980	+	0.591943i	$0.798361\pi$
$104$	−734.970	−0.692978
$105$	0	0
$106$	876.168	0.802839
$107$	462.375	0.417753	0.208876	−	0.977942i	$-0.433019\pi$
$107$	462.375	0.417753	0.208876	+	0.977942i	$0.433019\pi$
$108$	0	0
$109$	−1012.45	−0.889683	−0.444842	−	0.895609i	$-0.646740\pi$
$109$	−1012.45	−0.889683	−0.444842	+	0.895609i	$0.646740\pi$
$110$	188.393	0.163296
$111$	0	0
$112$	792.478	0.668591
$113$	2037.04	1.69583	0.847913	−	0.530135i	$-0.177859\pi$
$113$	2037.04	1.69583	0.847913	+	0.530135i	$0.177859\pi$
$114$	0	0
$115$	−1335.89	−1.08324
$116$	529.583	0.423884
$117$	0	0
$118$	2573.14	2.00743
$119$	893.622	0.688388
$120$	0	0
$121$	−1274.47	−0.957525
$122$	−228.689	−0.169709
$123$	0	0
$124$	724.467	0.524670
$125$	1455.47	1.04145
$126$	0	0
$127$	1354.32	0.946273	0.473137	−	0.880989i	$-0.343122\pi$
$127$	1354.32	0.946273	0.473137	+	0.880989i	$0.343122\pi$
$128$	−1700.38	−1.17417
$129$	0	0
$130$	−1123.56	−0.758019
$131$	1176.43	0.784619	0.392309	−	0.919833i	$-0.371676\pi$
$131$	1176.43	0.784619	0.392309	+	0.919833i	$0.371676\pi$
$132$	0	0
$133$	246.249	0.160545
$134$	−505.067	−0.325605
$135$	0	0
$136$	−1462.75	−0.922279
$137$	2083.78	1.29949	0.649743	−	0.760154i	$-0.274877\pi$
$137$	2083.78	1.29949	0.649743	+	0.760154i	$0.274877\pi$
$138$	0	0
$139$	−1151.73	−0.702794	−0.351397	−	0.936227i	$-0.614293\pi$
$139$	−1151.73	−0.702794	−0.351397	+	0.936227i	$0.614293\pi$
$140$	231.814	0.139942
$141$	0	0
$142$	474.781	0.280583
$143$	−337.163	−0.197168
$144$	0	0
$145$	−1296.71	−0.742662
$146$	−2373.68	−1.34553
$147$	0	0
$148$	−420.299	−0.233435
$149$	1159.57	0.637556	0.318778	−	0.947829i	$-0.396727\pi$
$149$	1159.57	0.637556	0.318778	+	0.947829i	$0.396727\pi$
$150$	0	0
$151$	1470.45	0.792472	0.396236	−	0.918149i	$-0.370316\pi$
$151$	1470.45	0.792472	0.396236	+	0.918149i	$0.370316\pi$
$152$	−403.080	−0.215093
$153$	0	0
$154$	250.549	0.131103
$155$	−1773.90	−0.919243
$156$	0	0
$157$	146.355	0.0743973	0.0371986	−	0.999308i	$-0.488157\pi$
$157$	146.355	0.0743973	0.0371986	+	0.999308i	$0.488157\pi$
$158$	−2326.93	−1.17165
$159$	0	0
$160$	−995.807	−0.492034
$161$	−1776.64	−0.869680
$162$	0	0
$163$	2985.36	1.43455	0.717273	−	0.696792i	$-0.245390\pi$
$163$	2985.36	1.43455	0.717273	+	0.696792i	$0.245390\pi$
$164$	712.490	0.339245
$165$	0	0
$166$	−2688.06	−1.25683
$167$	−2782.57	−1.28935	−0.644676	−	0.764456i	$-0.723008\pi$
$167$	−2782.57	−1.28935	−0.644676	+	0.764456i	$0.723008\pi$
$168$	0	0
$169$	−186.198	−0.0847509
$170$	−2236.12	−1.00884
$171$	0	0
$172$	−33.9795	−0.0150634
$173$	1977.52	0.869064	0.434532	−	0.900656i	$-0.356914\pi$
$173$	1977.52	0.869064	0.434532	+	0.900656i	$0.356914\pi$
$174$	0	0
$175$	684.028	0.295472
$176$	−595.078	−0.254862
$177$	0	0
$178$	−2955.81	−1.24465
$179$	2980.27	1.24445	0.622223	−	0.782840i	$-0.286230\pi$
$179$	2980.27	1.24445	0.622223	+	0.782840i	$0.286230\pi$
$180$	0	0
$181$	1375.32	0.564790	0.282395	−	0.959298i	$-0.408871\pi$
$181$	1375.32	0.564790	0.282395	+	0.959298i	$0.408871\pi$
$182$	−1494.25	−0.608577
$183$	0	0
$184$	2908.14	1.16517
$185$	1029.12	0.408988
$186$	0	0
$187$	−671.028	−0.262409
$188$	−165.657	−0.0642649
$189$	0	0
$190$	−616.193	−0.235281
$191$	−2728.83	−1.03378	−0.516888	−	0.856053i	$-0.672910\pi$
$191$	−2728.83	−1.03378	−0.516888	+	0.856053i	$0.672910\pi$
$192$	0	0
$193$	−1456.68	−0.543286	−0.271643	−	0.962398i	$-0.587567\pi$
$193$	−1456.68	−0.543286	−0.271643	+	0.962398i	$0.587567\pi$
$194$	−4642.57	−1.71813
$195$	0	0
$196$	−746.395	−0.272010
$197$	5456.06	1.97324	0.986619	−	0.163042i	$-0.0521306\pi$
$197$	5456.06	1.97324	0.986619	+	0.163042i	$0.0521306\pi$
$198$	0	0
$199$	−5020.62	−1.78846	−0.894228	−	0.447612i	$-0.852274\pi$
$199$	−5020.62	−1.78846	−0.894228	+	0.447612i	$0.852274\pi$
$200$	−1119.67	−0.395864
$201$	0	0
$202$	3332.83	1.16088
$203$	−1724.53	−0.596248
$204$	0	0
$205$	−1744.57	−0.594371
$206$	5607.64	1.89661
$207$	0	0
$208$	3548.98	1.18307
$209$	−184.910	−0.0611987
$210$	0	0
$211$	−5312.39	−1.73327	−0.866634	−	0.498944i	$-0.833721\pi$
$211$	−5312.39	−1.73327	−0.866634	+	0.498944i	$0.833721\pi$
$212$	−809.559	−0.262267
$213$	0	0
$214$	−1538.74	−0.491523
$215$	83.2006	0.0263918
$216$	0	0
$217$	−2359.15	−0.738017
$218$	3369.34	1.04679
$219$	0	0
$220$	−174.071	−0.0533448
$221$	4001.94	1.21810
$222$	0	0
$223$	−2750.09	−0.825829	−0.412915	−	0.910770i	$-0.635489\pi$
$223$	−2750.09	−0.825829	−0.412915	+	0.910770i	$0.635489\pi$
$224$	−1324.35	−0.395031
$225$	0	0
$226$	−6779.05	−1.99529
$227$	−4620.82	−1.35108	−0.675539	−	0.737324i	$-0.736089\pi$
$227$	−4620.82	−1.35108	−0.675539	+	0.737324i	$0.736089\pi$
$228$	0	0
$229$	4992.19	1.44058	0.720291	−	0.693672i	$-0.244008\pi$
$229$	4992.19	1.44058	0.720291	+	0.693672i	$0.244008\pi$
$230$	4445.70	1.27453
$231$	0	0
$232$	2822.85	0.798833
$233$	−471.023	−0.132437	−0.0662183	−	0.997805i	$-0.521093\pi$
$233$	−471.023	−0.132437	−0.0662183	+	0.997805i	$0.521093\pi$
$234$	0	0
$235$	405.621	0.112595
$236$	−2377.52	−0.655776
$237$	0	0
$238$	−2973.88	−0.809950
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−799.298	−0.213640	−0.106820	−	0.994278i	$-0.534067\pi$
$241$	−799.298	−0.213640	−0.106820	+	0.994278i	$0.534067\pi$
$242$	4241.29	1.12661
$243$	0	0
$244$	211.303	0.0554398
$245$	1827.59	0.476572
$246$	0	0
$247$	1102.79	0.284083
$248$	3861.65	0.988770
$249$	0	0
$250$	−4843.64	−1.22536
$251$	1993.85	0.501398	0.250699	−	0.968065i	$-0.419340\pi$
$251$	1993.85	0.501398	0.250699	+	0.968065i	$0.419340\pi$
$252$	0	0
$253$	1334.09	0.331516
$254$	−4507.05	−1.11337
$255$	0	0
$256$	4114.69	1.00456
$257$	−1771.03	−0.429860	−0.214930	−	0.976629i	$-0.568952\pi$
$257$	−1771.03	−0.429860	−0.214930	+	0.976629i	$0.568952\pi$
$258$	0	0
$259$	1368.66	0.328357
$260$	1038.14	0.247626
$261$	0	0
$262$	−3915.03	−0.923174
$263$	−1181.08	−0.276914	−0.138457	−	0.990368i	$-0.544214\pi$
$263$	−1181.08	−0.276914	−0.138457	+	0.990368i	$0.544214\pi$
$264$	0	0
$265$	1982.25	0.459504
$266$	−819.492	−0.188896
$267$	0	0
$268$	466.670	0.106367
$269$	3719.61	0.843080	0.421540	−	0.906810i	$-0.361490\pi$
$269$	3719.61	0.843080	0.421540	+	0.906810i	$0.361490\pi$
$270$	0	0
$271$	−6754.67	−1.51409	−0.757043	−	0.653365i	$-0.773357\pi$
$271$	−6754.67	−1.51409	−0.757043	+	0.653365i	$0.773357\pi$
$272$	7063.25	1.57453
$273$	0	0
$274$	−6934.61	−1.52896
$275$	−513.642	−0.112632
$276$	0	0
$277$	−747.367	−0.162112	−0.0810558	−	0.996710i	$-0.525829\pi$
$277$	−747.367	−0.162112	−0.0810558	+	0.996710i	$0.525829\pi$
$278$	3832.84	0.826900
$279$	0	0
$280$	1235.65	0.263728
$281$	−7565.28	−1.60607	−0.803037	−	0.595929i	$-0.796784\pi$
$281$	−7565.28	−1.60607	−0.803037	+	0.595929i	$0.796784\pi$
$282$	0	0
$283$	−1437.42	−0.301928	−0.150964	−	0.988539i	$-0.548238\pi$
$283$	−1437.42	−0.301928	−0.150964	+	0.988539i	$0.548238\pi$
$284$	−438.687	−0.0916594
$285$	0	0
$286$	1122.04	0.231985
$287$	−2320.15	−0.477192
$288$	0	0
$289$	3051.73	0.621155
$290$	4315.32	0.873808
$291$	0	0
$292$	2193.23	0.439551
$293$	3423.44	0.682591	0.341296	−	0.939956i	$-0.389134\pi$
$293$	3423.44	0.682591	0.341296	+	0.939956i	$0.389134\pi$
$294$	0	0
$295$	5821.48	1.14895
$296$	−2240.33	−0.439921
$297$	0	0
$298$	−3858.94	−0.750142
$299$	−7956.37	−1.53889
$300$	0	0
$301$	110.651	0.0211887
$302$	−4893.50	−0.932414
$303$	0	0
$304$	1946.37	0.367211
$305$	−517.388	−0.0971329
$306$	0	0
$307$	3289.42	0.611522	0.305761	−	0.952108i	$-0.401089\pi$
$307$	3289.42	0.611522	0.305761	+	0.952108i	$0.401089\pi$
$308$	−231.502	−0.0428280
$309$	0	0
$310$	5903.34	1.08157
$311$	1317.39	0.240201	0.120101	−	0.992762i	$-0.461678\pi$
$311$	1317.39	0.240201	0.120101	+	0.992762i	$0.461678\pi$
$312$	0	0
$313$	675.257	0.121942	0.0609709	−	0.998140i	$-0.480580\pi$
$313$	675.257	0.121942	0.0609709	+	0.998140i	$0.480580\pi$
$314$	−487.053	−0.0875350
$315$	0	0
$316$	2150.03	0.382748
$317$	−1976.63	−0.350217	−0.175108	−	0.984549i	$-0.556028\pi$
$317$	−1976.63	−0.350217	−0.175108	+	0.984549i	$0.556028\pi$
$318$	0	0
$319$	1294.96	0.227285
$320$	−1453.10	−0.253847
$321$	0	0
$322$	5912.46	1.02326
$323$	2194.79	0.378084
$324$	0	0
$325$	3063.30	0.522836
$326$	−9934.96	−1.68787
$327$	0	0
$328$	3797.81	0.639326
$329$	539.446	0.0903971
$330$	0	0
$331$	3318.49	0.551060	0.275530	−	0.961292i	$-0.411147\pi$
$331$	3318.49	0.551060	0.275530	+	0.961292i	$0.411147\pi$
$332$	2483.70	0.410575
$333$	0	0
$334$	9260.11	1.51704
$335$	−1142.67	−0.186360
$336$	0	0
$337$	−10177.6	−1.64513	−0.822566	−	0.568669i	$-0.807459\pi$
$337$	−10177.6	−1.64513	−0.822566	+	0.568669i	$0.807459\pi$
$338$	619.647	0.0997170
$339$	0	0
$340$	2066.13	0.329563
$341$	1771.50	0.281327
$342$	0	0
$343$	5865.05	0.923274
$344$	−181.122	−0.0283879
$345$	0	0
$346$	−6580.98	−1.02253
$347$	−2211.35	−0.342108	−0.171054	−	0.985262i	$-0.554717\pi$
$347$	−2211.35	−0.342108	−0.171054	+	0.985262i	$0.554717\pi$
$348$	0	0
$349$	−2751.22	−0.421975	−0.210988	−	0.977489i	$-0.567668\pi$
$349$	−2751.22	−0.421975	−0.210988	+	0.977489i	$0.567668\pi$
$350$	−2276.37	−0.347650
$351$	0	0
$352$	994.465	0.150583
$353$	−9835.91	−1.48304	−0.741520	−	0.670931i	$-0.765895\pi$
$353$	−9835.91	−1.48304	−0.741520	+	0.670931i	$0.765895\pi$
$354$	0	0
$355$	1074.15	0.160591
$356$	2731.10	0.406595
$357$	0	0
$358$	−9918.03	−1.46420
$359$	−6288.54	−0.924502	−0.462251	−	0.886749i	$-0.652958\pi$
$359$	−6288.54	−0.924502	−0.462251	+	0.886749i	$0.652958\pi$
$360$	0	0
$361$	−6254.20	−0.911824
$362$	−4576.94	−0.664526
$363$	0	0
$364$	1380.65	0.198807
$365$	−5370.23	−0.770112
$366$	0	0
$367$	−7788.11	−1.10773	−0.553864	−	0.832607i	$-0.686847\pi$
$367$	−7788.11	−1.10773	−0.553864	+	0.832607i	$0.686847\pi$
$368$	−14042.7	−1.98920
$369$	0	0
$370$	−3424.82	−0.481211
$371$	2636.24	0.368914
$372$	0	0
$373$	1548.56	0.214964	0.107482	−	0.994207i	$-0.465721\pi$
$373$	1548.56	0.214964	0.107482	+	0.994207i	$0.465721\pi$
$374$	2233.11	0.308747
$375$	0	0
$376$	−883.009	−0.121111
$377$	−7723.03	−1.05506
$378$	0	0
$379$	9423.83	1.27723	0.638614	−	0.769527i	$-0.279508\pi$
$379$	9423.83	1.27723	0.638614	+	0.769527i	$0.279508\pi$
$380$	569.348	0.0768604
$381$	0	0
$382$	9081.27	1.21633
$383$	−8800.37	−1.17409	−0.587047	−	0.809553i	$-0.699710\pi$
$383$	−8800.37	−1.17409	−0.587047	+	0.809553i	$0.699710\pi$
$384$	0	0
$385$	566.844	0.0750365
$386$	4847.68	0.639224
$387$	0	0
$388$	4289.63	0.561270
$389$	1060.27	0.138195	0.0690975	−	0.997610i	$-0.477988\pi$
$389$	1060.27	0.138195	0.0690975	+	0.997610i	$0.477988\pi$
$390$	0	0
$391$	−15834.9	−2.04810
$392$	−3978.53	−0.512618
$393$	0	0
$394$	−18157.2	−2.32169
$395$	−5264.46	−0.670591
$396$	0	0
$397$	−2558.90	−0.323495	−0.161747	−	0.986832i	$-0.551713\pi$
$397$	−2558.90	−0.323495	−0.161747	+	0.986832i	$0.551713\pi$
$398$	16708.1	2.10428
$399$	0	0
$400$	5406.60	0.675825
$401$	10818.3	1.34723	0.673615	−	0.739082i	$-0.264740\pi$
$401$	10818.3	1.34723	0.673615	+	0.739082i	$0.264740\pi$
$402$	0	0
$403$	−10565.1	−1.30591
$404$	−3079.46	−0.379230
$405$	0	0
$406$	5739.06	0.701539
$407$	−1027.74	−0.125167
$408$	0	0
$409$	497.075	0.0600948	0.0300474	−	0.999548i	$-0.490434\pi$
$409$	497.075	0.0600948	0.0300474	+	0.999548i	$0.490434\pi$
$410$	5805.75	0.699331
$411$	0	0
$412$	−5181.33	−0.619577
$413$	7742.14	0.922436
$414$	0	0
$415$	−6081.48	−0.719345
$416$	−5930.88	−0.699004
$417$	0	0
$418$	615.363	0.0720057
$419$	−12936.7	−1.50835	−0.754177	−	0.656671i	$-0.771964\pi$
$419$	−12936.7	−1.50835	−0.754177	+	0.656671i	$0.771964\pi$
$420$	0	0
$421$	−13968.7	−1.61709	−0.808544	−	0.588435i	$-0.799744\pi$
$421$	−13968.7	−1.61709	−0.808544	+	0.588435i	$0.799744\pi$
$422$	17679.1	2.03935
$423$	0	0
$424$	−4315.21	−0.494258
$425$	6096.65	0.695837
$426$	0	0
$427$	−688.088	−0.0779834
$428$	1421.76	0.160568
$429$	0	0
$430$	−276.883	−0.0310523
$431$	−15440.8	−1.72565	−0.862827	−	0.505500i	$-0.831308\pi$
$431$	−15440.8	−1.72565	−0.862827	+	0.505500i	$0.831308\pi$
$432$	0	0
$433$	1916.27	0.212679	0.106340	−	0.994330i	$-0.466087\pi$
$433$	1916.27	0.212679	0.106340	+	0.994330i	$0.466087\pi$
$434$	7851.01	0.868343
$435$	0	0
$436$	−3113.19	−0.341961
$437$	−4363.52	−0.477655
$438$	0	0
$439$	4389.03	0.477169	0.238584	−	0.971122i	$-0.423317\pi$
$439$	4389.03	0.477169	0.238584	+	0.971122i	$0.423317\pi$
$440$	−927.855	−0.100531
$441$	0	0
$442$	−13318.0	−1.43320
$443$	−8485.05	−0.910016	−0.455008	−	0.890487i	$-0.650364\pi$
$443$	−8485.05	−0.910016	−0.455008	+	0.890487i	$0.650364\pi$
$444$	0	0
$445$	−6687.24	−0.712372
$446$	9152.03	0.971662
$447$	0	0
$448$	−1932.52	−0.203802
$449$	6903.35	0.725588	0.362794	−	0.931869i	$-0.381823\pi$
$449$	6903.35	0.725588	0.362794	+	0.931869i	$0.381823\pi$
$450$	0	0
$451$	1742.22	0.181902
$452$	6263.69	0.651812
$453$	0	0
$454$	15377.6	1.58966
$455$	−3380.60	−0.348318
$456$	0	0
$457$	−3608.44	−0.369357	−0.184678	−	0.982799i	$-0.559124\pi$
$457$	−3608.44	−0.369357	−0.184678	+	0.982799i	$0.559124\pi$
$458$	−16613.5	−1.69497
$459$	0	0
$460$	−4107.73	−0.416356
$461$	−7410.34	−0.748664	−0.374332	−	0.927295i	$-0.622128\pi$
$461$	−7410.34	−0.748664	−0.374332	+	0.927295i	$0.622128\pi$
$462$	0	0
$463$	−903.330	−0.0906724	−0.0453362	−	0.998972i	$-0.514436\pi$
$463$	−903.330	−0.0906724	−0.0453362	+	0.998972i	$0.514436\pi$
$464$	−13630.8	−1.36378
$465$	0	0
$466$	1567.52	0.155823
$467$	7902.89	0.783088	0.391544	−	0.920159i	$-0.371941\pi$
$467$	7902.89	0.783088	0.391544	+	0.920159i	$0.371941\pi$
$468$	0	0
$469$	−1519.66	−0.149619
$470$	−1349.87	−0.132478
$471$	0	0
$472$	−12673.0	−1.23585
$473$	−83.0885	−0.00807698
$474$	0	0
$475$	1680.01	0.162282
$476$	2747.80	0.264591
$477$	0	0
$478$	795.367	0.0761072
$479$	14850.0	1.41652	0.708259	−	0.705953i	$-0.249481\pi$
$479$	14850.0	1.41652	0.708259	+	0.705953i	$0.249481\pi$
$480$	0	0
$481$	6129.32	0.581025
$482$	2659.98	0.251367
$483$	0	0
$484$	−3918.85	−0.368037
$485$	−10503.4	−0.983369
$486$	0	0
$487$	5561.87	0.517520	0.258760	−	0.965942i	$-0.416686\pi$
$487$	5561.87	0.517520	0.258760	+	0.965942i	$0.416686\pi$
$488$	1126.32	0.104479
$489$	0	0
$490$	−6082.02	−0.560730
$491$	12076.8	1.11002	0.555010	−	0.831844i	$-0.312714\pi$
$491$	12076.8	1.11002	0.555010	+	0.831844i	$0.312714\pi$
$492$	0	0
$493$	−15370.5	−1.40417
$494$	−3669.96	−0.334249
$495$	0	0
$496$	−18646.9	−1.68805
$497$	1428.54	0.128931
$498$	0	0
$499$	−5916.65	−0.530793	−0.265397	−	0.964139i	$-0.585503\pi$
$499$	−5916.65	−0.530793	−0.265397	+	0.964139i	$0.585503\pi$
$500$	4475.41	0.400293
$501$	0	0
$502$	−6635.34	−0.589940
$503$	−4576.47	−0.405676	−0.202838	−	0.979212i	$-0.565016\pi$
$503$	−4576.47	−0.405676	−0.202838	+	0.979212i	$0.565016\pi$
$504$	0	0
$505$	7540.22	0.664427
$506$	−4439.71	−0.390058
$507$	0	0
$508$	4164.41	0.363712
$509$	−11838.0	−1.03087	−0.515433	−	0.856930i	$-0.672369\pi$
$509$	−11838.0	−1.03087	−0.515433	+	0.856930i	$0.672369\pi$
$510$	0	0
$511$	−7142.02	−0.618286
$512$	−90.2293	−0.00778830
$513$	0	0
$514$	5893.82	0.505769
$515$	12686.8	1.08552
$516$	0	0
$517$	−405.074	−0.0344587
$518$	−4554.76	−0.386341
$519$	0	0
$520$	5533.63	0.466665
$521$	−11405.7	−0.959102	−0.479551	−	0.877514i	$-0.659200\pi$
$521$	−11405.7	−0.959102	−0.479551	+	0.877514i	$0.659200\pi$
$522$	0	0
$523$	18991.4	1.58783	0.793917	−	0.608027i	$-0.208039\pi$
$523$	18991.4	1.58783	0.793917	+	0.608027i	$0.208039\pi$
$524$	3617.40	0.301578
$525$	0	0
$526$	3930.51	0.325814
$527$	−21026.8	−1.73803
$528$	0	0
$529$	19314.8	1.58748
$530$	−6596.71	−0.540647
$531$	0	0
$532$	757.192	0.0617075
$533$	−10390.4	−0.844388
$534$	0	0
$535$	−3481.25	−0.281323
$536$	2487.50	0.200455
$537$	0	0
$538$	−12378.5	−0.991959
$539$	−1825.12	−0.145851
$540$	0	0
$541$	−545.101	−0.0433192	−0.0216596	−	0.999765i	$-0.506895\pi$
$541$	−545.101	−0.0433192	−0.0216596	+	0.999765i	$0.506895\pi$
$542$	22478.9	1.78146
$543$	0	0
$544$	−11803.8	−0.930298
$545$	7622.82	0.599130
$546$	0	0
$547$	−23247.9	−1.81720	−0.908598	−	0.417671i	$-0.862847\pi$
$547$	−23247.9	−1.81720	−0.908598	+	0.417671i	$0.862847\pi$
$548$	6407.42	0.499473
$549$	0	0
$550$	1709.35	0.132521
$551$	−4235.55	−0.327478
$552$	0	0
$553$	−7001.34	−0.538386
$554$	2487.16	0.190739
$555$	0	0
$556$	−3541.45	−0.270128
$557$	2171.98	0.165224	0.0826119	−	0.996582i	$-0.473674\pi$
$557$	2171.98	0.165224	0.0826119	+	0.996582i	$0.473674\pi$
$558$	0	0
$559$	495.531	0.0374932
$560$	−5966.61	−0.450242
$561$	0	0
$562$	25176.5	1.88969
$563$	−2190.24	−0.163957	−0.0819784	−	0.996634i	$-0.526124\pi$
$563$	−2190.24	−0.163957	−0.0819784	+	0.996634i	$0.526124\pi$
$564$	0	0
$565$	−15337.0	−1.14200
$566$	4783.58	0.355245
$567$	0	0
$568$	−2338.34	−0.172737
$569$	−21378.4	−1.57509	−0.787547	−	0.616255i	$-0.788649\pi$
$569$	−21378.4	−1.57509	−0.787547	+	0.616255i	$0.788649\pi$
$570$	0	0
$571$	−6167.71	−0.452033	−0.226016	−	0.974124i	$-0.572570\pi$
$571$	−6167.71	−0.452033	−0.226016	+	0.974124i	$0.572570\pi$
$572$	−1036.74	−0.0757838
$573$	0	0
$574$	7721.22	0.561459
$575$	−12120.9	−0.879091
$576$	0	0
$577$	−13829.5	−0.997801	−0.498900	−	0.866659i	$-0.666263\pi$
$577$	−13829.5	−0.997801	−0.498900	+	0.866659i	$0.666263\pi$
$578$	−10155.9	−0.730844
$579$	0	0
$580$	−3987.26	−0.285451
$581$	−8087.93	−0.577528
$582$	0	0
$583$	−1979.58	−0.140627
$584$	11690.6	0.828359
$585$	0	0
$586$	−11392.8	−0.803130
$587$	−5975.81	−0.420184	−0.210092	−	0.977682i	$-0.567376\pi$
$587$	−5975.81	−0.420184	−0.210092	+	0.977682i	$0.567376\pi$
$588$	0	0
$589$	−5794.21	−0.405342
$590$	−19373.3	−1.35184
$591$	0	0
$592$	10818.0	0.751042
$593$	17002.9	1.17745	0.588723	−	0.808335i	$-0.299631\pi$
$593$	17002.9	1.17745	0.588723	+	0.808335i	$0.299631\pi$
$594$	0	0
$595$	−6728.13	−0.463574
$596$	3565.57	0.245053
$597$	0	0
$598$	26478.0	1.81064
$599$	2510.13	0.171221	0.0856104	−	0.996329i	$-0.472716\pi$
$599$	2510.13	0.171221	0.0856104	+	0.996329i	$0.472716\pi$
$600$	0	0
$601$	−25423.5	−1.72553	−0.862766	−	0.505603i	$-0.831270\pi$
$601$	−25423.5	−1.72553	−0.862766	+	0.505603i	$0.831270\pi$
$602$	−368.234	−0.0249304
$603$	0	0
$604$	4521.48	0.304597
$605$	9595.52	0.644816
$606$	0	0
$607$	16459.4	1.10060	0.550302	−	0.834966i	$-0.314513\pi$
$607$	16459.4	1.10060	0.550302	+	0.834966i	$0.314513\pi$
$608$	−3252.68	−0.216963
$609$	0	0
$610$	1721.81	0.114286
$611$	2415.82	0.159957
$612$	0	0
$613$	18458.2	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$613$	18458.2	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$614$	−10946.9	−0.719510
$615$	0	0
$616$	−1233.98	−0.0807118
$617$	−14734.9	−0.961433	−0.480716	−	0.876876i	$-0.659623\pi$
$617$	−14734.9	−0.961433	−0.480716	+	0.876876i	$0.659623\pi$
$618$	0	0
$619$	−2177.68	−0.141403	−0.0707014	−	0.997498i	$-0.522524\pi$
$619$	−2177.68	−0.141403	−0.0707014	+	0.997498i	$0.522524\pi$
$620$	−5454.55	−0.353323
$621$	0	0
$622$	−4384.15	−0.282618
$623$	−8893.54	−0.571929
$624$	0	0
$625$	−2419.12	−0.154824
$626$	−2247.19	−0.143475
$627$	0	0
$628$	450.026	0.0285955
$629$	12198.7	0.773281
$630$	0	0
$631$	9478.86	0.598015	0.299007	−	0.954251i	$-0.403344\pi$
$631$	9478.86	0.598015	0.299007	+	0.954251i	$0.403344\pi$
$632$	11460.4	0.721311
$633$	0	0
$634$	6578.03	0.412061
$635$	−10196.8	−0.637238
$636$	0	0
$637$	10884.8	0.677038
$638$	−4309.51	−0.267422
$639$	0	0
$640$	12802.2	0.790708
$641$	2869.40	0.176809	0.0884043	−	0.996085i	$-0.471823\pi$
$641$	2869.40	0.176809	0.0884043	+	0.996085i	$0.471823\pi$
$642$	0	0
$643$	14833.4	0.909757	0.454879	−	0.890553i	$-0.349683\pi$
$643$	14833.4	0.909757	0.454879	+	0.890553i	$0.349683\pi$
$644$	−5462.98	−0.334272
$645$	0	0
$646$	−7304.02	−0.444850
$647$	−954.068	−0.0579726	−0.0289863	−	0.999580i	$-0.509228\pi$
$647$	−954.068	−0.0579726	−0.0289863	+	0.999580i	$0.509228\pi$
$648$	0	0
$649$	−5813.63	−0.351626
$650$	−10194.4	−0.615163
$651$	0	0
$652$	9179.67	0.551386
$653$	−11185.7	−0.670334	−0.335167	−	0.942159i	$-0.608793\pi$
$653$	−11185.7	−0.670334	−0.335167	+	0.942159i	$0.608793\pi$
$654$	0	0
$655$	−8857.40	−0.528377
$656$	−18338.6	−1.09147
$657$	0	0
$658$	−1795.22	−0.106360
$659$	−14434.0	−0.853213	−0.426607	−	0.904437i	$-0.640291\pi$
$659$	−14434.0	−0.853213	−0.426607	+	0.904437i	$0.640291\pi$
$660$	0	0
$661$	−20753.6	−1.22121	−0.610606	−	0.791935i	$-0.709074\pi$
$661$	−20753.6	−1.22121	−0.610606	+	0.791935i	$0.709074\pi$
$662$	−11043.6	−0.648371
$663$	0	0
$664$	13239.0	0.773752
$665$	−1854.02	−0.108114
$666$	0	0
$667$	30558.6	1.77396
$668$	−8556.13	−0.495579
$669$	0	0
$670$	3802.67	0.219269
$671$	516.691	0.0297267
$672$	0	0
$673$	16071.7	0.920533	0.460267	−	0.887781i	$-0.347754\pi$
$673$	16071.7	0.920533	0.460267	+	0.887781i	$0.347754\pi$
$674$	33870.0	1.93565
$675$	0	0
$676$	−572.539	−0.0325751
$677$	−1871.17	−0.106226	−0.0531129	−	0.998589i	$-0.516914\pi$
$677$	−1871.17	−0.106226	−0.0531129	+	0.998589i	$0.516914\pi$
$678$	0	0
$679$	−13968.7	−0.789501
$680$	11013.1	0.621080
$681$	0	0
$682$	−5895.39	−0.331006
$683$	16958.2	0.950053	0.475027	−	0.879971i	$-0.342438\pi$
$683$	16958.2	0.950053	0.475027	+	0.879971i	$0.342438\pi$
$684$	0	0
$685$	−15688.9	−0.875098
$686$	−19518.3	−1.08631
$687$	0	0
$688$	874.591	0.0484643
$689$	11806.0	0.652790
$690$	0	0
$691$	−5896.46	−0.324619	−0.162310	−	0.986740i	$-0.551894\pi$
$691$	−5896.46	−0.324619	−0.162310	+	0.986740i	$0.551894\pi$
$692$	6080.67	0.334035
$693$	0	0
$694$	7359.14	0.402520
$695$	8671.43	0.473275
$696$	0	0
$697$	−20679.2	−1.12379
$698$	9155.77	0.496492
$699$	0	0
$700$	2103.32	0.113568
$701$	6.45827	0.000347968 0	0.000173984	−	1.00000i	$-0.499945\pi$
$701$	6.45827	0.000347968 0	0.000173984		1.00000i	$0.499945\pi$
$702$	0	0
$703$	3361.51	0.180344
$704$	1451.15	0.0776877
$705$	0	0
$706$	32732.9	1.74493
$707$	10027.9	0.533437
$708$	0	0
$709$	−5988.65	−0.317219	−0.158610	−	0.987341i	$-0.550701\pi$
$709$	−5988.65	−0.317219	−0.158610	+	0.987341i	$0.550701\pi$
$710$	−3574.65	−0.188950
$711$	0	0
$712$	14557.6	0.766251
$713$	41804.0	2.19575
$714$	0	0
$715$	2538.52	0.132776
$716$	9164.03	0.478318
$717$	0	0
$718$	20927.6	1.08776
$719$	24718.3	1.28211	0.641054	−	0.767496i	$-0.278497\pi$
$719$	24718.3	1.28211	0.641054	+	0.767496i	$0.278497\pi$
$720$	0	0
$721$	16872.5	0.871516
$722$	20813.3	1.07284
$723$	0	0
$724$	4228.98	0.217084
$725$	−11765.4	−0.602700
$726$	0	0
$727$	−13505.7	−0.688993	−0.344496	−	0.938788i	$-0.611950\pi$
$727$	−13505.7	−0.688993	−0.344496	+	0.938788i	$0.611950\pi$
$728$	7359.33	0.374663
$729$	0	0
$730$	17871.6	0.906105
$731$	986.215	0.0498994
$732$	0	0
$733$	−9329.49	−0.470113	−0.235056	−	0.971982i	$-0.575527\pi$
$733$	−9329.49	−0.470113	−0.235056	+	0.971982i	$0.575527\pi$
$734$	25918.0	1.30334
$735$	0	0
$736$	23467.4	1.17530
$737$	1141.13	0.0570338
$738$	0	0
$739$	17831.8	0.887621	0.443810	−	0.896121i	$-0.353626\pi$
$739$	17831.8	0.887621	0.443810	+	0.896121i	$0.353626\pi$
$740$	3164.45	0.157200
$741$	0	0
$742$	−8773.15	−0.434060
$743$	3453.44	0.170517	0.0852587	−	0.996359i	$-0.472828\pi$
$743$	3453.44	0.170517	0.0852587	+	0.996359i	$0.472828\pi$
$744$	0	0
$745$	−8730.49	−0.429343
$746$	−5153.45	−0.252924
$747$	0	0
$748$	−2063.34	−0.100860
$749$	−4629.81	−0.225861
$750$	0	0
$751$	−11151.4	−0.541837	−0.270918	−	0.962602i	$-0.587327\pi$
$751$	−11151.4	−0.541837	−0.270918	+	0.962602i	$0.587327\pi$
$752$	4263.82	0.206763
$753$	0	0
$754$	25701.4	1.24137
$755$	−11071.1	−0.533666
$756$	0	0
$757$	−39210.5	−1.88260	−0.941302	−	0.337567i	$-0.890396\pi$
$757$	−39210.5	−1.88260	−0.941302	+	0.337567i	$0.890396\pi$
$758$	−31361.5	−1.50277
$759$	0	0
$760$	3034.81	0.144848
$761$	8738.79	0.416269	0.208135	−	0.978100i	$-0.433261\pi$
$761$	8738.79	0.416269	0.208135	+	0.978100i	$0.433261\pi$
$762$	0	0
$763$	10137.8	0.481013
$764$	−8390.88	−0.397345
$765$	0	0
$766$	29286.7	1.38143
$767$	34671.9	1.63224
$768$	0	0
$769$	−3048.73	−0.142965	−0.0714824	−	0.997442i	$-0.522773\pi$
$769$	−3048.73	−0.142965	−0.0714824	+	0.997442i	$0.522773\pi$
$770$	−1886.40	−0.0882871
$771$	0	0
$772$	−4479.14	−0.208819
$773$	−213.850	−0.00995041	−0.00497520	−	0.999988i	$-0.501584\pi$
$773$	−213.850	−0.00995041	−0.00497520	+	0.999988i	$0.501584\pi$
$774$	0	0
$775$	−16095.1	−0.746003
$776$	22865.1	1.05775
$777$	0	0
$778$	−3528.47	−0.162599
$779$	−5698.42	−0.262089
$780$	0	0
$781$	−1072.70	−0.0491475
$782$	52697.0	2.40977
$783$	0	0
$784$	19211.3	0.875150
$785$	−1101.91	−0.0501005
$786$	0	0
$787$	1657.52	0.0750752	0.0375376	−	0.999295i	$-0.488049\pi$
$787$	1657.52	0.0750752	0.0375376	+	0.999295i	$0.488049\pi$
$788$	16776.8	0.758439
$789$	0	0
$790$	17519.6	0.789011
$791$	−20397.1	−0.916859
$792$	0	0
$793$	−3081.49	−0.137991
$794$	8515.74	0.380620
$795$	0	0
$796$	−15437.9	−0.687415
$797$	−41248.4	−1.83324	−0.916621	−	0.399757i	$-0.869094\pi$
$797$	−41248.4	−1.83324	−0.916621	+	0.399757i	$0.869094\pi$
$798$	0	0
$799$	4808.01	0.212885
$800$	−9035.25	−0.399305
$801$	0	0
$802$	−36002.2	−1.58514
$803$	5363.00	0.235686
$804$	0	0
$805$	13376.4	0.585659
$806$	35159.5	1.53652
$807$	0	0
$808$	−16414.5	−0.714680
$809$	−18909.7	−0.821791	−0.410895	−	0.911683i	$-0.634784\pi$
$809$	−18909.7	−0.821791	−0.410895	+	0.911683i	$0.634784\pi$
$810$	0	0
$811$	−6745.72	−0.292077	−0.146038	−	0.989279i	$-0.546652\pi$
$811$	−6745.72	−0.292077	−0.146038	+	0.989279i	$0.546652\pi$
$812$	−5302.76	−0.229175
$813$	0	0
$814$	3420.21	0.147270
$815$	−22476.9	−0.966051
$816$	0	0
$817$	271.764	0.0116375
$818$	−1654.21	−0.0707069
$819$	0	0
$820$	−5364.38	−0.228454
$821$	−1070.60	−0.0455104	−0.0227552	−	0.999741i	$-0.507244\pi$
$821$	−1070.60	−0.0455104	−0.0227552	+	0.999741i	$0.507244\pi$
$822$	0	0
$823$	−22848.7	−0.967747	−0.483874	−	0.875138i	$-0.660771\pi$
$823$	−22848.7	−0.967747	−0.483874	+	0.875138i	$0.660771\pi$
$824$	−27618.2	−1.16763
$825$	0	0
$826$	−25765.1	−1.08533
$827$	−32751.8	−1.37714	−0.688568	−	0.725171i	$-0.741761\pi$
$827$	−32751.8	−1.37714	−0.688568	+	0.725171i	$0.741761\pi$
$828$	0	0
$829$	43044.8	1.80339	0.901694	−	0.432374i	$-0.142324\pi$
$829$	43044.8	1.80339	0.901694	+	0.432374i	$0.142324\pi$
$830$	20238.5	0.846374
$831$	0	0
$832$	−8654.48	−0.360625
$833$	21663.2	0.901064
$834$	0	0
$835$	20950.1	0.868275
$836$	−568.581	−0.0235225
$837$	0	0
$838$	43052.1	1.77471
$839$	−6795.31	−0.279619	−0.139809	−	0.990178i	$-0.544649\pi$
$839$	−6795.31	−0.279619	−0.139809	+	0.990178i	$0.544649\pi$
$840$	0	0
$841$	5273.37	0.216219
$842$	46486.5	1.90265
$843$	0	0
$844$	−16335.1	−0.666203
$845$	1401.89	0.0570729
$846$	0	0
$847$	12761.3	0.517692
$848$	20837.1	0.843806
$849$	0	0
$850$	−20289.0	−0.818715
$851$	−24252.6	−0.976930
$852$	0	0
$853$	−19239.8	−0.772286	−0.386143	−	0.922439i	$-0.626193\pi$
$853$	−19239.8	−0.772286	−0.386143	+	0.922439i	$0.626193\pi$
$854$	2289.89	0.0917545
$855$	0	0
$856$	7578.44	0.302600
$857$	47707.1	1.90157	0.950783	−	0.309858i	$-0.100281\pi$
$857$	47707.1	1.90157	0.950783	+	0.309858i	$0.100281\pi$
$858$	0	0
$859$	−39202.3	−1.55712	−0.778559	−	0.627572i	$-0.784049\pi$
$859$	−39202.3	−1.55712	−0.778559	+	0.627572i	$0.784049\pi$
$860$	255.833	0.0101440
$861$	0	0
$862$	51385.4	2.03039
$863$	−18919.4	−0.746263	−0.373132	−	0.927778i	$-0.621716\pi$
$863$	−18919.4	−0.746263	−0.373132	+	0.927778i	$0.621716\pi$
$864$	0	0
$865$	−14888.9	−0.585244
$866$	−6377.15	−0.250236
$867$	0	0
$868$	−7254.15	−0.283666
$869$	5257.36	0.205229
$870$	0	0
$871$	−6805.55	−0.264750
$872$	−16594.3	−0.644444
$873$	0	0
$874$	14521.3	0.562004
$875$	−14573.7	−0.563065
$876$	0	0
$877$	14270.0	0.549446	0.274723	−	0.961523i	$-0.411414\pi$
$877$	14270.0	0.549446	0.274723	+	0.961523i	$0.411414\pi$
$878$	−14606.2	−0.561432
$879$	0	0
$880$	4480.37	0.171629
$881$	37145.1	1.42049	0.710244	−	0.703956i	$-0.248585\pi$
$881$	37145.1	1.42049	0.710244	+	0.703956i	$0.248585\pi$
$882$	0	0
$883$	38337.1	1.46110	0.730548	−	0.682862i	$-0.239265\pi$
$883$	38337.1	1.46110	0.730548	+	0.682862i	$0.239265\pi$
$884$	12305.6	0.468191
$885$	0	0
$886$	28237.4	1.07071
$887$	23266.3	0.880729	0.440365	−	0.897819i	$-0.354849\pi$
$887$	23266.3	0.880729	0.440365	+	0.897819i	$0.354849\pi$
$888$	0	0
$889$	−13561.0	−0.511609
$890$	22254.4	0.838169
$891$	0	0
$892$	−8456.26	−0.317418
$893$	1324.91	0.0496489
$894$	0	0
$895$	−22438.6	−0.838034
$896$	17026.0	0.634822
$897$	0	0
$898$	−22973.6	−0.853719
$899$	40578.0	1.50540
$900$	0	0
$901$	23496.5	0.868792
$902$	−5797.93	−0.214024
$903$	0	0
$904$	33387.5	1.22838
$905$	−10354.9	−0.380341
$906$	0	0
$907$	9638.93	0.352873	0.176436	−	0.984312i	$-0.443543\pi$
$907$	9638.93	0.352873	0.176436	+	0.984312i	$0.443543\pi$
$908$	−14208.6	−0.519304
$909$	0	0
$910$	11250.3	0.409828
$911$	31015.7	1.12799	0.563993	−	0.825780i	$-0.309265\pi$
$911$	31015.7	1.12799	0.563993	+	0.825780i	$0.309265\pi$
$912$	0	0
$913$	6073.28	0.220149
$914$	12008.5	0.434581
$915$	0	0
$916$	15350.5	0.553706
$917$	−11779.7	−0.424209
$918$	0	0
$919$	−10459.3	−0.375432	−0.187716	−	0.982223i	$-0.560108\pi$
$919$	−10459.3	−0.375432	−0.187716	+	0.982223i	$0.560108\pi$
$920$	−21895.5	−0.784646
$921$	0	0
$922$	24660.9	0.880870
$923$	6397.47	0.228142
$924$	0	0
$925$	9337.55	0.331910
$926$	3006.19	0.106684
$927$	0	0
$928$	22779.2	0.805778
$929$	5508.56	0.194542	0.0972712	−	0.995258i	$-0.468989\pi$
$929$	5508.56	0.194542	0.0972712	+	0.995258i	$0.468989\pi$
$930$	0	0
$931$	5969.59	0.210145
$932$	−1448.35	−0.0509037
$933$	0	0
$934$	−26300.0	−0.921373
$935$	5052.20	0.176711
$936$	0	0
$937$	−242.254	−0.00844620	−0.00422310	−	0.999991i	$-0.501344\pi$
$937$	−242.254	−0.00844620	−0.00422310	+	0.999991i	$0.501344\pi$
$938$	5057.28	0.176041
$939$	0	0
$940$	1247.24	0.0432772
$941$	−31959.5	−1.10717	−0.553587	−	0.832791i	$-0.686742\pi$
$941$	−31959.5	−1.10717	−0.553587	+	0.832791i	$0.686742\pi$
$942$	0	0
$943$	41112.9	1.41975
$944$	61194.4	2.10986
$945$	0	0
$946$	276.510	0.00950329
$947$	−24526.0	−0.841593	−0.420797	−	0.907155i	$-0.638249\pi$
$947$	−24526.0	−0.841593	−0.420797	+	0.907155i	$0.638249\pi$
$948$	0	0
$949$	−31984.3	−1.09405
$950$	−5590.90	−0.190940
$951$	0	0
$952$	14646.7	0.498636
$953$	47989.0	1.63118	0.815590	−	0.578630i	$-0.196412\pi$
$953$	47989.0	1.63118	0.815590	+	0.578630i	$0.196412\pi$
$954$	0	0
$955$	20545.5	0.696165
$956$	−734.901	−0.0248623
$957$	0	0
$958$	−49419.1	−1.66666
$959$	−20865.1	−0.702575
$960$	0	0
$961$	25719.5	0.863331
$962$	−20397.7	−0.683627
$963$	0	0
$964$	−2457.76	−0.0821153
$965$	10967.4	0.365859
$966$	0	0
$967$	7694.21	0.255873	0.127937	−	0.991782i	$-0.459165\pi$
$967$	7694.21	0.255873	0.127937	+	0.991782i	$0.459165\pi$
$968$	−20888.8	−0.693586
$969$	0	0
$970$	34954.2	1.15702
$971$	−292.834	−0.00967815	−0.00483907	−	0.999988i	$-0.501540\pi$
$971$	−292.834	−0.00967815	−0.00483907	+	0.999988i	$0.501540\pi$
$972$	0	0
$973$	11532.4	0.379970
$974$	−18509.3	−0.608909
$975$	0	0
$976$	−5438.70	−0.178369
$977$	−11394.8	−0.373136	−0.186568	−	0.982442i	$-0.559736\pi$
$977$	−11394.8	−0.373136	−0.186568	+	0.982442i	$0.559736\pi$
$978$	0	0
$979$	6678.22	0.218015
$980$	5619.65	0.183177
$981$	0	0
$982$	−40190.4	−1.30604
$983$	−18239.5	−0.591812	−0.295906	−	0.955217i	$-0.595621\pi$
$983$	−18239.5	−0.591812	−0.295906	+	0.955217i	$0.595621\pi$
$984$	0	0
$985$	−41078.9	−1.32882
$986$	51151.5	1.65213
$987$	0	0
$988$	3390.96	0.109191
$989$	−1960.72	−0.0630408
$990$	0	0
$991$	−37787.7	−1.21127	−0.605634	−	0.795743i	$-0.707081\pi$
$991$	−37787.7	−1.21127	−0.605634	+	0.795743i	$0.707081\pi$
$992$	31161.8	0.997367
$993$	0	0
$994$	−4754.03	−0.151699
$995$	37800.6	1.20438
$996$	0	0
$997$	13542.5	0.430187	0.215094	−	0.976593i	$-0.430994\pi$
$997$	13542.5	0.430187	0.215094	+	0.976593i	$0.430994\pi$
$998$	19690.0	0.624525
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.5		22
3.2	odd	2		239.4.a.a.1.18	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.18	✓	22	3.2	odd	2
2151.4.a.a.1.5		22	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)

\(f(q)\)	\(=\)	\(q-3.32790 q^{2} +3.07490 q^{4} -7.52905 q^{5} -10.0131 q^{7} +16.3902 q^{8} +O(q^{10})\)	\(q-3.32790 q^{2} +3.07490 q^{4} -7.52905 q^{5} -10.0131 q^{7} +16.3902 q^{8} +25.0559 q^{10} +7.51891 q^{11} -44.8420 q^{13} +33.3225 q^{14} -79.1442 q^{16} -89.2454 q^{17} -24.5927 q^{19} -23.1511 q^{20} -25.0222 q^{22} +177.431 q^{23} -68.3133 q^{25} +149.229 q^{26} -30.7893 q^{28} +172.228 q^{29} +235.607 q^{31} +132.262 q^{32} +296.999 q^{34} +75.3891 q^{35} -136.687 q^{37} +81.8420 q^{38} -123.403 q^{40} +231.712 q^{41} -11.0506 q^{43} +23.1199 q^{44} -590.473 q^{46} -53.8741 q^{47} -242.738 q^{49} +227.340 q^{50} -137.885 q^{52} -263.280 q^{53} -56.6103 q^{55} -164.117 q^{56} -573.156 q^{58} -773.202 q^{59} +68.7188 q^{61} -784.075 q^{62} +193.000 q^{64} +337.618 q^{65} +151.767 q^{67} -274.420 q^{68} -250.887 q^{70} -142.667 q^{71} +713.268 q^{73} +454.881 q^{74} -75.6201 q^{76} -75.2875 q^{77} +699.219 q^{79} +595.881 q^{80} -771.113 q^{82} +807.735 q^{83} +671.933 q^{85} +36.7753 q^{86} +123.237 q^{88} +888.191 q^{89} +449.007 q^{91} +545.583 q^{92} +179.287 q^{94} +185.160 q^{95} +1395.05 q^{97} +807.807 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

Modular forms

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