LMFDB - Embedded newform 2151.4.a.g.1.54

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			1.54
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)

$f(q)$	$=$	$q+4.50896 q^{2} +12.3307 q^{4} +9.52144 q^{5} -13.3640 q^{7} +19.5269 q^{8} +O(q^{10})$	\(q+4.50896 q^{2} +12.3307 q^{4} +9.52144 q^{5} -13.3640 q^{7} +19.5269 q^{8} +42.9318 q^{10} -2.94125 q^{11} +33.8562 q^{13} -60.2578 q^{14} -10.5995 q^{16} -28.2145 q^{17} -135.304 q^{19} +117.406 q^{20} -13.2620 q^{22} -84.3598 q^{23} -34.3422 q^{25} +152.656 q^{26} -164.788 q^{28} -107.051 q^{29} -131.876 q^{31} -204.008 q^{32} -127.218 q^{34} -127.245 q^{35} -139.270 q^{37} -610.080 q^{38} +185.924 q^{40} -173.525 q^{41} +476.983 q^{43} -36.2677 q^{44} -380.375 q^{46} -323.326 q^{47} -164.403 q^{49} -154.847 q^{50} +417.471 q^{52} +601.871 q^{53} -28.0049 q^{55} -260.958 q^{56} -482.690 q^{58} +430.427 q^{59} -651.074 q^{61} -594.622 q^{62} -835.068 q^{64} +322.360 q^{65} +1013.18 q^{67} -347.905 q^{68} -573.741 q^{70} +935.109 q^{71} -271.689 q^{73} -627.964 q^{74} -1668.39 q^{76} +39.3069 q^{77} -301.342 q^{79} -100.923 q^{80} -782.415 q^{82} -689.562 q^{83} -268.643 q^{85} +2150.70 q^{86} -57.4335 q^{88} -650.347 q^{89} -452.455 q^{91} -1040.22 q^{92} -1457.87 q^{94} -1288.29 q^{95} +211.167 q^{97} -741.287 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.50896	1.59416	0.797079	−	0.603876i	$-0.206378\pi$
$2$	4.50896	1.59416	0.797079	+	0.603876i	$0.206378\pi$
$3$	0	0
$3$	0	0
$4$	12.3307	1.54134
$5$	9.52144	0.851624	0.425812	−	0.904812i	$-0.359989\pi$
$5$	9.52144	0.851624	0.425812	+	0.904812i	$0.359989\pi$
$6$	0	0
$7$	−13.3640	−0.721589	−0.360794	−	0.932645i	$-0.617494\pi$
$7$	−13.3640	−0.721589	−0.360794	+	0.932645i	$0.617494\pi$
$8$	19.5269	0.862976
$9$	0	0
$10$	42.9318	1.35762
$11$	−2.94125	−0.0806200	−0.0403100	−	0.999187i	$-0.512835\pi$
$11$	−2.94125	−0.0806200	−0.0403100	+	0.999187i	$0.512835\pi$
$12$	0	0
$13$	33.8562	0.722310	0.361155	−	0.932506i	$-0.382383\pi$
$13$	33.8562	0.722310	0.361155	+	0.932506i	$0.382383\pi$
$14$	−60.2578	−1.15033
$15$	0	0
$16$	−10.5995	−0.165618
$17$	−28.2145	−0.402531	−0.201265	−	0.979537i	$-0.564505\pi$
$17$	−28.2145	−0.402531	−0.201265	+	0.979537i	$0.564505\pi$
$18$	0	0
$19$	−135.304	−1.63373	−0.816865	−	0.576829i	$-0.804290\pi$
$19$	−135.304	−1.63373	−0.816865	+	0.576829i	$0.804290\pi$
$20$	117.406	1.31264
$21$	0	0
$22$	−13.2620	−0.128521
$23$	−84.3598	−0.764793	−0.382397	−	0.923998i	$-0.624901\pi$
$23$	−84.3598	−0.764793	−0.382397	+	0.923998i	$0.624901\pi$
$24$	0	0
$25$	−34.3422	−0.274737
$26$	152.656	1.15147
$27$	0	0
$28$	−164.788	−1.11221
$29$	−107.051	−0.685481	−0.342741	−	0.939430i	$-0.611355\pi$
$29$	−107.051	−0.685481	−0.342741	+	0.939430i	$0.611355\pi$
$30$	0	0
$31$	−131.876	−0.764051	−0.382025	−	0.924152i	$-0.624773\pi$
$31$	−131.876	−0.764051	−0.382025	+	0.924152i	$0.624773\pi$
$32$	−204.008	−1.12700
$33$	0	0
$34$	−127.218	−0.641697
$35$	−127.245	−0.614522
$36$	0	0
$37$	−139.270	−0.618808	−0.309404	−	0.950931i	$-0.600129\pi$
$37$	−139.270	−0.618808	−0.309404	+	0.950931i	$0.600129\pi$
$38$	−610.080	−2.60442
$39$	0	0
$40$	185.924	0.734931
$41$	−173.525	−0.660975	−0.330488	−	0.943810i	$-0.607213\pi$
$41$	−173.525	−0.660975	−0.330488	+	0.943810i	$0.607213\pi$
$42$	0	0
$43$	476.983	1.69161	0.845805	−	0.533492i	$-0.179121\pi$
$43$	476.983	1.69161	0.845805	+	0.533492i	$0.179121\pi$
$44$	−36.2677	−0.124263
$45$	0	0
$46$	−380.375	−1.21920
$47$	−323.326	−1.00345	−0.501723	−	0.865028i	$-0.667301\pi$
$47$	−323.326	−1.00345	−0.501723	+	0.865028i	$0.667301\pi$
$48$	0	0
$49$	−164.403	−0.479310
$50$	−154.847	−0.437974
$51$	0	0
$52$	417.471	1.11332
$53$	601.871	1.55987	0.779937	−	0.625858i	$-0.215251\pi$
$53$	601.871	1.55987	0.779937	+	0.625858i	$0.215251\pi$
$54$	0	0
$55$	−28.0049	−0.0686579
$56$	−260.958	−0.622714
$57$	0	0
$58$	−482.690	−1.09276
$59$	430.427	0.949776	0.474888	−	0.880046i	$-0.342489\pi$
$59$	430.427	0.949776	0.474888	+	0.880046i	$0.342489\pi$
$60$	0	0
$61$	−651.074	−1.36658	−0.683291	−	0.730147i	$-0.739452\pi$
$61$	−651.074	−1.36658	−0.683291	+	0.730147i	$0.739452\pi$
$62$	−594.622	−1.21802
$63$	0	0
$64$	−835.068	−1.63099
$65$	322.360	0.615136
$66$	0	0
$67$	1013.18	1.84745	0.923727	−	0.383052i	$-0.125127\pi$
$67$	1013.18	1.84745	0.923727	+	0.383052i	$0.125127\pi$
$68$	−347.905	−0.620436
$69$	0	0
$70$	−573.741	−0.979645
$71$	935.109	1.56306	0.781528	−	0.623870i	$-0.214440\pi$
$71$	935.109	1.56306	0.781528	+	0.623870i	$0.214440\pi$
$72$	0	0
$73$	−271.689	−0.435600	−0.217800	−	0.975993i	$-0.569888\pi$
$73$	−271.689	−0.435600	−0.217800	+	0.975993i	$0.569888\pi$
$74$	−627.964	−0.986477
$75$	0	0
$76$	−1668.39	−2.51813
$77$	39.3069	0.0581745
$78$	0	0
$79$	−301.342	−0.429159	−0.214580	−	0.976706i	$-0.568838\pi$
$79$	−301.342	−0.429159	−0.214580	+	0.976706i	$0.568838\pi$
$80$	−100.923	−0.141044
$81$	0	0
$82$	−782.415	−1.05370
$83$	−689.562	−0.911919	−0.455960	−	0.890001i	$-0.650704\pi$
$83$	−689.562	−0.911919	−0.455960	+	0.890001i	$0.650704\pi$
$84$	0	0
$85$	−268.643	−0.342805
$86$	2150.70	2.69669
$87$	0	0
$88$	−57.4335	−0.0695731
$89$	−650.347	−0.774568	−0.387284	−	0.921960i	$-0.626587\pi$
$89$	−650.347	−0.774568	−0.387284	+	0.921960i	$0.626587\pi$
$90$	0	0
$91$	−452.455	−0.521210
$92$	−1040.22	−1.17880
$93$	0	0
$94$	−1457.87	−1.59965
$95$	−1288.29	−1.39132
$96$	0	0
$97$	211.167	0.221039	0.110520	−	0.993874i	$-0.464749\pi$
$97$	211.167	0.221039	0.110520	+	0.993874i	$0.464749\pi$
$98$	−741.287	−0.764095
$99$	0	0
$100$	−423.463	−0.423463
$101$	526.291	0.518494	0.259247	−	0.965811i	$-0.416526\pi$
$101$	526.291	0.518494	0.259247	+	0.965811i	$0.416526\pi$
$102$	0	0
$103$	−1107.17	−1.05916	−0.529578	−	0.848261i	$-0.677650\pi$
$103$	−1107.17	−1.05916	−0.529578	+	0.848261i	$0.677650\pi$
$104$	661.107	0.623336
$105$	0	0
$106$	2713.81	2.48668
$107$	−284.273	−0.256838	−0.128419	−	0.991720i	$-0.540990\pi$
$107$	−284.273	−0.256838	−0.128419	+	0.991720i	$0.540990\pi$
$108$	0	0
$109$	1109.29	0.974780	0.487390	−	0.873184i	$-0.337949\pi$
$109$	1109.29	0.974780	0.487390	+	0.873184i	$0.337949\pi$
$110$	−126.273	−0.109451
$111$	0	0
$112$	141.652	0.119508
$113$	891.630	0.742279	0.371140	−	0.928577i	$-0.378967\pi$
$113$	891.630	0.742279	0.371140	+	0.928577i	$0.378967\pi$
$114$	0	0
$115$	−803.227	−0.651316
$116$	−1320.02	−1.05656
$117$	0	0
$118$	1940.78	1.51409
$119$	377.059	0.290462
$120$	0	0
$121$	−1322.35	−0.993500
$122$	−2935.66	−2.17855
$123$	0	0
$124$	−1626.12	−1.17766
$125$	−1517.17	−1.08560
$126$	0	0
$127$	−1211.31	−0.846347	−0.423174	−	0.906049i	$-0.639084\pi$
$127$	−1211.31	−0.846347	−0.423174	+	0.906049i	$0.639084\pi$
$128$	−2133.22	−1.47306
$129$	0	0
$130$	1453.51	0.980623
$131$	−1226.76	−0.818188	−0.409094	−	0.912492i	$-0.634155\pi$
$131$	−1226.76	−0.818188	−0.409094	+	0.912492i	$0.634155\pi$
$132$	0	0
$133$	1808.21	1.17888
$134$	4568.38	2.94513
$135$	0	0
$136$	−550.943	−0.347374
$137$	1736.26	1.08276	0.541382	−	0.840776i	$-0.317901\pi$
$137$	1736.26	1.08276	0.541382	+	0.840776i	$0.317901\pi$
$138$	0	0
$139$	318.169	0.194149	0.0970745	−	0.995277i	$-0.469051\pi$
$139$	318.169	0.194149	0.0970745	+	0.995277i	$0.469051\pi$
$140$	−1569.01	−0.947185
$141$	0	0
$142$	4216.37	2.49176
$143$	−99.5796	−0.0582326
$144$	0	0
$145$	−1019.28	−0.583772
$146$	−1225.03	−0.694415
$147$	0	0
$148$	−1717.30	−0.953791
$149$	1076.64	0.591956	0.295978	−	0.955195i	$-0.404355\pi$
$149$	1076.64	0.591956	0.295978	+	0.955195i	$0.404355\pi$
$150$	0	0
$151$	−1156.90	−0.623491	−0.311745	−	0.950166i	$-0.600914\pi$
$151$	−1156.90	−0.623491	−0.311745	+	0.950166i	$0.600914\pi$
$152$	−2642.07	−1.40987
$153$	0	0
$154$	177.233	0.0927393
$155$	−1255.65	−0.650684
$156$	0	0
$157$	−2167.66	−1.10190	−0.550950	−	0.834538i	$-0.685735\pi$
$157$	−2167.66	−1.10190	−0.550950	+	0.834538i	$0.685735\pi$
$158$	−1358.74	−0.684147
$159$	0	0
$160$	−1942.45	−0.959777
$161$	1127.39	0.551866
$162$	0	0
$163$	−1150.19	−0.552696	−0.276348	−	0.961058i	$-0.589124\pi$
$163$	−1150.19	−0.552696	−0.276348	+	0.961058i	$0.589124\pi$
$164$	−2139.68	−1.01879
$165$	0	0
$166$	−3109.21	−1.45374
$167$	2654.37	1.22995	0.614974	−	0.788547i	$-0.289166\pi$
$167$	2654.37	1.22995	0.614974	+	0.788547i	$0.289166\pi$
$168$	0	0
$169$	−1050.76	−0.478269
$170$	−1211.30	−0.546485
$171$	0	0
$172$	5881.53	2.60734
$173$	−849.869	−0.373493	−0.186747	−	0.982408i	$-0.559794\pi$
$173$	−849.869	−0.373493	−0.186747	+	0.982408i	$0.559794\pi$
$174$	0	0
$175$	458.949	0.198247
$176$	31.1758	0.0133521
$177$	0	0
$178$	−2932.38	−1.23478
$179$	−2819.54	−1.17733	−0.588666	−	0.808376i	$-0.700347\pi$
$179$	−2819.54	−1.17733	−0.588666	+	0.808376i	$0.700347\pi$
$180$	0	0
$181$	1234.91	0.507128	0.253564	−	0.967319i	$-0.418397\pi$
$181$	1234.91	0.507128	0.253564	+	0.967319i	$0.418397\pi$
$182$	−2040.10	−0.830891
$183$	0	0
$184$	−1647.29	−0.659998
$185$	−1326.05	−0.526991
$186$	0	0
$187$	82.9859	0.0324520
$188$	−3986.84	−1.54665
$189$	0	0
$190$	−5808.84	−2.21799
$191$	2491.75	0.943963	0.471981	−	0.881609i	$-0.343539\pi$
$191$	2491.75	0.943963	0.471981	+	0.881609i	$0.343539\pi$
$192$	0	0
$193$	−4237.89	−1.58057	−0.790284	−	0.612740i	$-0.790067\pi$
$193$	−4237.89	−1.58057	−0.790284	+	0.612740i	$0.790067\pi$
$194$	952.144	0.352371
$195$	0	0
$196$	−2027.21	−0.738778
$197$	−5180.33	−1.87352	−0.936760	−	0.349973i	$-0.886191\pi$
$197$	−5180.33	−1.87352	−0.936760	+	0.349973i	$0.886191\pi$
$198$	0	0
$199$	4099.35	1.46028	0.730139	−	0.683299i	$-0.239455\pi$
$199$	4099.35	1.46028	0.730139	+	0.683299i	$0.239455\pi$
$200$	−670.597	−0.237092
$201$	0	0
$202$	2373.02	0.826561
$203$	1430.64	0.494636
$204$	0	0
$205$	−1652.20	−0.562902
$206$	−4992.20	−1.68846
$207$	0	0
$208$	−358.860	−0.119627
$209$	397.963	0.131711
$210$	0	0
$211$	3788.99	1.23623	0.618116	−	0.786087i	$-0.287896\pi$
$211$	3788.99	1.23623	0.618116	+	0.786087i	$0.287896\pi$
$212$	7421.49	2.40429
$213$	0	0
$214$	−1281.77	−0.409440
$215$	4541.57	1.44061
$216$	0	0
$217$	1762.39	0.551331
$218$	5001.76	1.55395
$219$	0	0
$220$	−345.320	−0.105825
$221$	−955.237	−0.290752
$222$	0	0
$223$	−4451.92	−1.33687	−0.668437	−	0.743769i	$-0.733036\pi$
$223$	−4451.92	−1.33687	−0.668437	+	0.743769i	$0.733036\pi$
$224$	2726.37	0.813228
$225$	0	0
$226$	4020.32	1.18331
$227$	1524.03	0.445611	0.222805	−	0.974863i	$-0.428479\pi$
$227$	1524.03	0.445611	0.222805	+	0.974863i	$0.428479\pi$
$228$	0	0
$229$	−3437.46	−0.991939	−0.495969	−	0.868340i	$-0.665187\pi$
$229$	−3437.46	−0.991939	−0.495969	+	0.868340i	$0.665187\pi$
$230$	−3621.72	−1.03830
$231$	0	0
$232$	−2090.39	−0.591554
$233$	−3451.71	−0.970510	−0.485255	−	0.874373i	$-0.661273\pi$
$233$	−3451.71	−0.970510	−0.485255	+	0.874373i	$0.661273\pi$
$234$	0	0
$235$	−3078.53	−0.854559
$236$	5307.46	1.46392
$237$	0	0
$238$	1700.14	0.463042
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	2828.41	0.755990	0.377995	−	0.925808i	$-0.376614\pi$
$241$	2828.41	0.755990	0.377995	+	0.925808i	$0.376614\pi$
$242$	−5962.42	−1.58380
$243$	0	0
$244$	−8028.19	−2.10636
$245$	−1565.36	−0.408191
$246$	0	0
$247$	−4580.88	−1.18006
$248$	−2575.13	−0.659358
$249$	0	0
$250$	−6840.84	−1.73061
$251$	7281.21	1.83102	0.915509	−	0.402297i	$-0.131788\pi$
$251$	7281.21	1.83102	0.915509	+	0.402297i	$0.131788\pi$
$252$	0	0
$253$	248.123	0.0616576
$254$	−5461.73	−1.34921
$255$	0	0
$256$	−2938.05	−0.717298
$257$	5028.04	1.22039	0.610196	−	0.792250i	$-0.291091\pi$
$257$	5028.04	1.22039	0.610196	+	0.792250i	$0.291091\pi$
$258$	0	0
$259$	1861.21	0.446525
$260$	3974.92	0.948131
$261$	0	0
$262$	−5531.41	−1.30432
$263$	7977.18	1.87032	0.935160	−	0.354225i	$-0.115255\pi$
$263$	7977.18	1.87032	0.935160	+	0.354225i	$0.115255\pi$
$264$	0	0
$265$	5730.68	1.32843
$266$	8153.12	1.87932
$267$	0	0
$268$	12493.2	2.84755
$269$	−1734.11	−0.393051	−0.196525	−	0.980499i	$-0.562966\pi$
$269$	−1734.11	−0.393051	−0.196525	+	0.980499i	$0.562966\pi$
$270$	0	0
$271$	6903.99	1.54756	0.773778	−	0.633457i	$-0.218365\pi$
$271$	6903.99	1.54756	0.773778	+	0.633457i	$0.218365\pi$
$272$	299.060	0.0666662
$273$	0	0
$274$	7828.73	1.72610
$275$	101.009	0.0221493
$276$	0	0
$277$	−3699.16	−0.802387	−0.401193	−	0.915993i	$-0.631404\pi$
$277$	−3699.16	−0.802387	−0.401193	+	0.915993i	$0.631404\pi$
$278$	1434.61	0.309504
$279$	0	0
$280$	−2484.70	−0.530318
$281$	−7660.30	−1.62625	−0.813123	−	0.582091i	$-0.802235\pi$
$281$	−7660.30	−1.62625	−0.813123	+	0.582091i	$0.802235\pi$
$282$	0	0
$283$	−2988.83	−0.627801	−0.313900	−	0.949456i	$-0.601636\pi$
$283$	−2988.83	−0.627801	−0.313900	+	0.949456i	$0.601636\pi$
$284$	11530.5	2.40920
$285$	0	0
$286$	−449.000	−0.0928319
$287$	2318.98	0.476952
$288$	0	0
$289$	−4116.94	−0.837969
$290$	−4595.91	−0.930624
$291$	0	0
$292$	−3350.12	−0.671406
$293$	−2649.48	−0.528275	−0.264137	−	0.964485i	$-0.585087\pi$
$293$	−2649.48	−0.528275	−0.264137	+	0.964485i	$0.585087\pi$
$294$	0	0
$295$	4098.28	0.808852
$296$	−2719.52	−0.534016
$297$	0	0
$298$	4854.50	0.943670
$299$	−2856.10	−0.552417
$300$	0	0
$301$	−6374.41	−1.22065
$302$	−5216.41	−0.993943
$303$	0	0
$304$	1434.16	0.270574
$305$	−6199.16	−1.16381
$306$	0	0
$307$	−3164.53	−0.588304	−0.294152	−	0.955759i	$-0.595037\pi$
$307$	−3164.53	−0.588304	−0.294152	+	0.955759i	$0.595037\pi$
$308$	484.681	0.0896665
$309$	0	0
$310$	−5661.66	−1.03729
$311$	8788.76	1.60246	0.801230	−	0.598357i	$-0.204180\pi$
$311$	8788.76	1.60246	0.801230	+	0.598357i	$0.204180\pi$
$312$	0	0
$313$	6066.85	1.09559	0.547794	−	0.836613i	$-0.315468\pi$
$313$	6066.85	1.09559	0.547794	+	0.836613i	$0.315468\pi$
$314$	−9773.90	−1.75660
$315$	0	0
$316$	−3715.75	−0.661479
$317$	−341.345	−0.0604789	−0.0302395	−	0.999543i	$-0.509627\pi$
$317$	−341.345	−0.0604789	−0.0302395	+	0.999543i	$0.509627\pi$
$318$	0	0
$319$	314.865	0.0552635
$320$	−7951.05	−1.38899
$321$	0	0
$322$	5083.33	0.879761
$323$	3817.54	0.657627
$324$	0	0
$325$	−1162.70	−0.198445
$326$	−5186.14	−0.881085
$327$	0	0
$328$	−3388.40	−0.570406
$329$	4320.94	0.724076
$330$	0	0
$331$	−1591.92	−0.264351	−0.132175	−	0.991226i	$-0.542196\pi$
$331$	−1591.92	−0.264351	−0.132175	+	0.991226i	$0.542196\pi$
$332$	−8502.78	−1.40557
$333$	0	0
$334$	11968.4	1.96073
$335$	9646.91	1.57333
$336$	0	0
$337$	7329.03	1.18468	0.592341	−	0.805688i	$-0.298204\pi$
$337$	7329.03	1.18468	0.592341	+	0.805688i	$0.298204\pi$
$338$	−4737.82	−0.762436
$339$	0	0
$340$	−3312.55	−0.528378
$341$	387.879	0.0615978
$342$	0	0
$343$	6780.94	1.06745
$344$	9314.01	1.45982
$345$	0	0
$346$	−3832.02	−0.595407
$347$	4172.50	0.645509	0.322755	−	0.946483i	$-0.395391\pi$
$347$	4172.50	0.645509	0.322755	+	0.946483i	$0.395391\pi$
$348$	0	0
$349$	9998.71	1.53358	0.766789	−	0.641899i	$-0.221853\pi$
$349$	9998.71	1.53358	0.766789	+	0.641899i	$0.221853\pi$
$350$	2069.38	0.316037
$351$	0	0
$352$	600.039	0.0908585
$353$	−12562.4	−1.89413	−0.947063	−	0.321048i	$-0.895965\pi$
$353$	−12562.4	−1.89413	−0.947063	+	0.321048i	$0.895965\pi$
$354$	0	0
$355$	8903.59	1.33114
$356$	−8019.22	−1.19387
$357$	0	0
$358$	−12713.2	−1.87685
$359$	−7210.95	−1.06011	−0.530055	−	0.847963i	$-0.677829\pi$
$359$	−7210.95	−1.06011	−0.530055	+	0.847963i	$0.677829\pi$
$360$	0	0
$361$	11448.2	1.66908
$362$	5568.16	0.808441
$363$	0	0
$364$	−5579.08	−0.803361
$365$	−2586.87	−0.370967
$366$	0	0
$367$	10963.6	1.55938	0.779692	−	0.626163i	$-0.215376\pi$
$367$	10963.6	1.55938	0.779692	+	0.626163i	$0.215376\pi$
$368$	894.174	0.126663
$369$	0	0
$370$	−5979.12	−0.840107
$371$	−8043.41	−1.12559
$372$	0	0
$373$	5992.61	0.831865	0.415932	−	0.909396i	$-0.363455\pi$
$373$	5992.61	0.831865	0.415932	+	0.909396i	$0.363455\pi$
$374$	374.180	0.0517337
$375$	0	0
$376$	−6313.57	−0.865951
$377$	−3624.36	−0.495130
$378$	0	0
$379$	−5911.11	−0.801144	−0.400572	−	0.916265i	$-0.631188\pi$
$379$	−5911.11	−0.801144	−0.400572	+	0.916265i	$0.631188\pi$
$380$	−15885.5	−2.14450
$381$	0	0
$382$	11235.2	1.50482
$383$	−11200.1	−1.49426	−0.747129	−	0.664679i	$-0.768568\pi$
$383$	−11200.1	−1.49426	−0.747129	+	0.664679i	$0.768568\pi$
$384$	0	0
$385$	374.258	0.0495428
$386$	−19108.4	−2.51967
$387$	0	0
$388$	2603.84	0.340696
$389$	554.700	0.0722992	0.0361496	−	0.999346i	$-0.488491\pi$
$389$	554.700	0.0722992	0.0361496	+	0.999346i	$0.488491\pi$
$390$	0	0
$391$	2380.17	0.307853
$392$	−3210.29	−0.413633
$393$	0	0
$394$	−23357.9	−2.98668
$395$	−2869.21	−0.365482
$396$	0	0
$397$	−6188.23	−0.782314	−0.391157	−	0.920324i	$-0.627925\pi$
$397$	−6188.23	−0.782314	−0.391157	+	0.920324i	$0.627925\pi$
$398$	18483.8	2.32791
$399$	0	0
$400$	364.011	0.0455013
$401$	−401.406	−0.0499881	−0.0249941	−	0.999688i	$-0.507957\pi$
$401$	−401.406	−0.0499881	−0.0249941	+	0.999688i	$0.507957\pi$
$402$	0	0
$403$	−4464.81	−0.551881
$404$	6489.53	0.799174
$405$	0	0
$406$	6450.68	0.788527
$407$	409.629	0.0498883
$408$	0	0
$409$	13706.7	1.65709	0.828546	−	0.559921i	$-0.189168\pi$
$409$	13706.7	1.65709	0.828546	+	0.559921i	$0.189168\pi$
$410$	−7449.72	−0.897354
$411$	0	0
$412$	−13652.2	−1.63252
$413$	−5752.23	−0.685348
$414$	0	0
$415$	−6565.63	−0.776612
$416$	−6906.94	−0.814040
$417$	0	0
$418$	1794.40	0.209969
$419$	5143.37	0.599690	0.299845	−	0.953988i	$-0.403065\pi$
$419$	5143.37	0.599690	0.299845	+	0.953988i	$0.403065\pi$
$420$	0	0
$421$	12319.4	1.42615	0.713077	−	0.701086i	$-0.247301\pi$
$421$	12319.4	1.42615	0.713077	+	0.701086i	$0.247301\pi$
$422$	17084.4	1.97075
$423$	0	0
$424$	11752.7	1.34613
$425$	968.948	0.110590
$426$	0	0
$427$	8700.96	0.986110
$428$	−3505.28	−0.395874
$429$	0	0
$430$	20477.7	2.29657
$431$	−4201.09	−0.469512	−0.234756	−	0.972054i	$-0.575429\pi$
$431$	−4201.09	−0.469512	−0.234756	+	0.972054i	$0.575429\pi$
$432$	0	0
$433$	−15710.6	−1.74365	−0.871826	−	0.489816i	$-0.837064\pi$
$433$	−15710.6	−1.74365	−0.871826	+	0.489816i	$0.837064\pi$
$434$	7946.54	0.878908
$435$	0	0
$436$	13678.4	1.50247
$437$	11414.2	1.24947
$438$	0	0
$439$	−3398.57	−0.369487	−0.184743	−	0.982787i	$-0.559145\pi$
$439$	−3398.57	−0.369487	−0.184743	+	0.982787i	$0.559145\pi$
$440$	−546.850	−0.0592501
$441$	0	0
$442$	−4307.12	−0.463504
$443$	−17894.5	−1.91917	−0.959585	−	0.281421i	$-0.909194\pi$
$443$	−17894.5	−1.91917	−0.959585	+	0.281421i	$0.909194\pi$
$444$	0	0
$445$	−6192.24	−0.659641
$446$	−20073.5	−2.13119
$447$	0	0
$448$	11159.9	1.17691
$449$	−18018.6	−1.89387	−0.946936	−	0.321423i	$-0.895839\pi$
$449$	−18018.6	−1.89387	−0.946936	+	0.321423i	$0.895839\pi$
$450$	0	0
$451$	510.379	0.0532878
$452$	10994.4	1.14410
$453$	0	0
$454$	6871.80	0.710374
$455$	−4308.02	−0.443875
$456$	0	0
$457$	676.858	0.0692824	0.0346412	−	0.999400i	$-0.488971\pi$
$457$	676.858	0.0692824	0.0346412	+	0.999400i	$0.488971\pi$
$458$	−15499.4	−1.58131
$459$	0	0
$460$	−9904.35	−1.00390
$461$	4958.46	0.500951	0.250476	−	0.968123i	$-0.419413\pi$
$461$	4958.46	0.500951	0.250476	+	0.968123i	$0.419413\pi$
$462$	0	0
$463$	7946.43	0.797629	0.398814	−	0.917032i	$-0.369422\pi$
$463$	7946.43	0.797629	0.398814	+	0.917032i	$0.369422\pi$
$464$	1134.69	0.113528
$465$	0	0
$466$	−15563.6	−1.54715
$467$	−1079.41	−0.106958	−0.0534788	−	0.998569i	$-0.517031\pi$
$467$	−1079.41	−0.106958	−0.0534788	+	0.998569i	$0.517031\pi$
$468$	0	0
$469$	−13540.1	−1.33310
$470$	−13881.0	−1.36230
$471$	0	0
$472$	8404.91	0.819634
$473$	−1402.93	−0.136378
$474$	0	0
$475$	4646.63	0.448847
$476$	4649.40	0.447699
$477$	0	0
$478$	1077.64	0.103117
$479$	929.575	0.0886709	0.0443355	−	0.999017i	$-0.485883\pi$
$479$	929.575	0.0886709	0.0443355	+	0.999017i	$0.485883\pi$
$480$	0	0
$481$	−4715.16	−0.446971
$482$	12753.2	1.20517
$483$	0	0
$484$	−16305.5	−1.53132
$485$	2010.62	0.188242
$486$	0	0
$487$	−386.533	−0.0359661	−0.0179830	−	0.999838i	$-0.505724\pi$
$487$	−386.533	−0.0359661	−0.0179830	+	0.999838i	$0.505724\pi$
$488$	−12713.5	−1.17933
$489$	0	0
$490$	−7058.12	−0.650721
$491$	−9376.47	−0.861821	−0.430911	−	0.902395i	$-0.641808\pi$
$491$	−9376.47	−0.861821	−0.430911	+	0.902395i	$0.641808\pi$
$492$	0	0
$493$	3020.41	0.275927
$494$	−20655.0	−1.88120
$495$	0	0
$496$	1397.82	0.126540
$497$	−12496.8	−1.12788
$498$	0	0
$499$	4239.68	0.380349	0.190174	−	0.981750i	$-0.439095\pi$
$499$	4239.68	0.380349	0.190174	+	0.981750i	$0.439095\pi$
$500$	−18707.7	−1.67327
$501$	0	0
$502$	32830.6	2.91893
$503$	−10396.2	−0.921557	−0.460779	−	0.887515i	$-0.652430\pi$
$503$	−10396.2	−0.921557	−0.460779	+	0.887515i	$0.652430\pi$
$504$	0	0
$505$	5011.05	0.441562
$506$	1118.78	0.0982919
$507$	0	0
$508$	−14936.3	−1.30451
$509$	1506.46	0.131184	0.0655922	−	0.997847i	$-0.479106\pi$
$509$	1506.46	0.131184	0.0655922	+	0.997847i	$0.479106\pi$
$510$	0	0
$511$	3630.86	0.314324
$512$	3818.20	0.329574
$513$	0	0
$514$	22671.2	1.94550
$515$	−10541.9	−0.902003
$516$	0	0
$517$	950.984	0.0808979
$518$	8392.11	0.711831
$519$	0	0
$520$	6294.70	0.530847
$521$	−20112.3	−1.69124	−0.845619	−	0.533787i	$-0.820768\pi$
$521$	−20112.3	−1.69124	−0.845619	+	0.533787i	$0.820768\pi$
$522$	0	0
$523$	−22514.3	−1.88238	−0.941188	−	0.337884i	$-0.890289\pi$
$523$	−22514.3	−1.88238	−0.941188	+	0.337884i	$0.890289\pi$
$524$	−15126.8	−1.26110
$525$	0	0
$526$	35968.8	2.98158
$527$	3720.81	0.307554
$528$	0	0
$529$	−5050.42	−0.415092
$530$	25839.4	2.11772
$531$	0	0
$532$	22296.4	1.81705
$533$	−5874.89	−0.477429
$534$	0	0
$535$	−2706.69	−0.218729
$536$	19784.2	1.59431
$537$	0	0
$538$	−7819.04	−0.626585
$539$	483.551	0.0386419
$540$	0	0
$541$	−7007.02	−0.556849	−0.278425	−	0.960458i	$-0.589812\pi$
$541$	−7007.02	−0.556849	−0.278425	+	0.960458i	$0.589812\pi$
$542$	31129.8	2.46705
$543$	0	0
$544$	5755.99	0.453651
$545$	10562.1	0.830146
$546$	0	0
$547$	18716.5	1.46300	0.731501	−	0.681841i	$-0.238820\pi$
$547$	18716.5	1.46300	0.731501	+	0.681841i	$0.238820\pi$
$548$	21409.3	1.66891
$549$	0	0
$550$	455.445	0.0353095
$551$	14484.5	1.11989
$552$	0	0
$553$	4027.13	0.309677
$554$	−16679.4	−1.27913
$555$	0	0
$556$	3923.24	0.299249
$557$	−15671.3	−1.19213	−0.596063	−	0.802937i	$-0.703269\pi$
$557$	−15671.3	−1.19213	−0.596063	+	0.802937i	$0.703269\pi$
$558$	0	0
$559$	16148.8	1.22187
$560$	1348.73	0.101776
$561$	0	0
$562$	−34540.0	−2.59249
$563$	19848.8	1.48584	0.742919	−	0.669381i	$-0.233441\pi$
$563$	19848.8	1.48584	0.742919	+	0.669381i	$0.233441\pi$
$564$	0	0
$565$	8489.61	0.632142
$566$	−13476.5	−1.00081
$567$	0	0
$568$	18259.8	1.34888
$569$	−6192.68	−0.456258	−0.228129	−	0.973631i	$-0.573261\pi$
$569$	−6192.68	−0.456258	−0.228129	+	0.973631i	$0.573261\pi$
$570$	0	0
$571$	19889.3	1.45769	0.728847	−	0.684677i	$-0.240057\pi$
$571$	19889.3	1.45769	0.728847	+	0.684677i	$0.240057\pi$
$572$	−1227.89	−0.0897561
$573$	0	0
$574$	10456.2	0.760337
$575$	2897.10	0.210117
$576$	0	0
$577$	−812.698	−0.0586362	−0.0293181	−	0.999570i	$-0.509334\pi$
$577$	−812.698	−0.0586362	−0.0293181	+	0.999570i	$0.509334\pi$
$578$	−18563.1	−1.33585
$579$	0	0
$580$	−12568.5	−0.899789
$581$	9215.32	0.658031
$582$	0	0
$583$	−1770.25	−0.125757
$584$	−5305.25	−0.375912
$585$	0	0
$586$	−11946.4	−0.842153
$587$	−12737.5	−0.895625	−0.447813	−	0.894127i	$-0.647797\pi$
$587$	−12737.5	−0.895625	−0.447813	+	0.894127i	$0.647797\pi$
$588$	0	0
$589$	17843.3	1.24825
$590$	18479.0	1.28944
$591$	0	0
$592$	1476.20	0.102485
$593$	2724.99	0.188705	0.0943525	−	0.995539i	$-0.469922\pi$
$593$	2724.99	0.188705	0.0943525	+	0.995539i	$0.469922\pi$
$594$	0	0
$595$	3590.15	0.247364
$596$	13275.7	0.912403
$597$	0	0
$598$	−12878.1	−0.880640
$599$	−2202.04	−0.150205	−0.0751027	−	0.997176i	$-0.523928\pi$
$599$	−2202.04	−0.150205	−0.0751027	+	0.997176i	$0.523928\pi$
$600$	0	0
$601$	27419.9	1.86103	0.930516	−	0.366251i	$-0.119359\pi$
$601$	27419.9	1.86103	0.930516	+	0.366251i	$0.119359\pi$
$602$	−28741.9	−1.94590
$603$	0	0
$604$	−14265.4	−0.961010
$605$	−12590.7	−0.846088
$606$	0	0
$607$	19222.9	1.28539	0.642697	−	0.766120i	$-0.277815\pi$
$607$	19222.9	1.28539	0.642697	+	0.766120i	$0.277815\pi$
$608$	27603.1	1.84121
$609$	0	0
$610$	−27951.7	−1.85530
$611$	−10946.6	−0.724799
$612$	0	0
$613$	−9914.28	−0.653236	−0.326618	−	0.945156i	$-0.605909\pi$
$613$	−9914.28	−0.653236	−0.326618	+	0.945156i	$0.605909\pi$
$614$	−14268.7	−0.937850
$615$	0	0
$616$	767.543	0.0502032
$617$	11380.3	0.742552	0.371276	−	0.928523i	$-0.378920\pi$
$617$	11380.3	0.742552	0.371276	+	0.928523i	$0.378920\pi$
$618$	0	0
$619$	−9959.91	−0.646725	−0.323362	−	0.946275i	$-0.604813\pi$
$619$	−9959.91	−0.646725	−0.323362	+	0.946275i	$0.604813\pi$
$620$	−15483.0	−1.00292
$621$	0	0
$622$	39628.1	2.55457
$623$	8691.24	0.558920
$624$	0	0
$625$	−10152.8	−0.649782
$626$	27355.2	1.74654
$627$	0	0
$628$	−26728.8	−1.69840
$629$	3929.44	0.249089
$630$	0	0
$631$	20723.6	1.30744	0.653720	−	0.756737i	$-0.273208\pi$
$631$	20723.6	1.30744	0.653720	+	0.756737i	$0.273208\pi$
$632$	−5884.27	−0.370354
$633$	0	0
$634$	−1539.11	−0.0964129
$635$	−11533.4	−0.720769
$636$	0	0
$637$	−5566.07	−0.346210
$638$	1419.71	0.0880987
$639$	0	0
$640$	−20311.3	−1.25449
$641$	8283.61	0.510426	0.255213	−	0.966885i	$-0.417854\pi$
$641$	8283.61	0.510426	0.255213	+	0.966885i	$0.417854\pi$
$642$	0	0
$643$	−10978.6	−0.673337	−0.336668	−	0.941623i	$-0.609300\pi$
$643$	−10978.6	−0.673337	−0.336668	+	0.941623i	$0.609300\pi$
$644$	13901.4	0.850612
$645$	0	0
$646$	17213.1	1.04836
$647$	−19305.5	−1.17307	−0.586536	−	0.809923i	$-0.699509\pi$
$647$	−19305.5	−1.17307	−0.586536	+	0.809923i	$0.699509\pi$
$648$	0	0
$649$	−1265.99	−0.0765710
$650$	−5242.54	−0.316353
$651$	0	0
$652$	−14182.6	−0.851891
$653$	−12072.8	−0.723497	−0.361748	−	0.932276i	$-0.617820\pi$
$653$	−12072.8	−0.723497	−0.361748	+	0.932276i	$0.617820\pi$
$654$	0	0
$655$	−11680.5	−0.696788
$656$	1839.28	0.109469
$657$	0	0
$658$	19482.9	1.15429
$659$	24979.0	1.47655	0.738274	−	0.674501i	$-0.235641\pi$
$659$	24979.0	1.47655	0.738274	+	0.674501i	$0.235641\pi$
$660$	0	0
$661$	−5431.43	−0.319604	−0.159802	−	0.987149i	$-0.551086\pi$
$661$	−5431.43	−0.319604	−0.159802	+	0.987149i	$0.551086\pi$
$662$	−7177.92	−0.421416
$663$	0	0
$664$	−13465.0	−0.786964
$665$	17216.7	1.00396
$666$	0	0
$667$	9030.84	0.524251
$668$	32730.2	1.89577
$669$	0	0
$670$	43497.5	2.50814
$671$	1914.97	0.110174
$672$	0	0
$673$	439.624	0.0251802	0.0125901	−	0.999921i	$-0.495992\pi$
$673$	439.624	0.0251802	0.0125901	+	0.999921i	$0.495992\pi$
$674$	33046.3	1.88857
$675$	0	0
$676$	−12956.6	−0.737174
$677$	−24517.3	−1.39184	−0.695922	−	0.718117i	$-0.745004\pi$
$677$	−24517.3	−1.39184	−0.695922	+	0.718117i	$0.745004\pi$
$678$	0	0
$679$	−2822.04	−0.159499
$680$	−5245.77	−0.295832
$681$	0	0
$682$	1748.93	0.0981966
$683$	15501.9	0.868466	0.434233	−	0.900801i	$-0.357019\pi$
$683$	15501.9	0.868466	0.434233	+	0.900801i	$0.357019\pi$
$684$	0	0
$685$	16531.7	0.922108
$686$	30575.0	1.70169
$687$	0	0
$688$	−5055.79	−0.280160
$689$	20377.1	1.12671
$690$	0	0
$691$	23007.6	1.26664	0.633322	−	0.773888i	$-0.281691\pi$
$691$	23007.6	1.26664	0.633322	+	0.773888i	$0.281691\pi$
$692$	−10479.5	−0.575679
$693$	0	0
$694$	18813.6	1.02904
$695$	3029.42	0.165342
$696$	0	0
$697$	4895.91	0.266063
$698$	45083.8	2.44476
$699$	0	0
$700$	5659.16	0.305566
$701$	33032.5	1.77977	0.889886	−	0.456184i	$-0.150784\pi$
$701$	33032.5	1.77977	0.889886	+	0.456184i	$0.150784\pi$
$702$	0	0
$703$	18843.8	1.01097
$704$	2456.14	0.131491
$705$	0	0
$706$	−56643.1	−3.01953
$707$	−7033.36	−0.374140
$708$	0	0
$709$	−7523.49	−0.398520	−0.199260	−	0.979947i	$-0.563854\pi$
$709$	−7523.49	−0.398520	−0.199260	+	0.979947i	$0.563854\pi$
$710$	40145.9	2.12204
$711$	0	0
$712$	−12699.3	−0.668434
$713$	11125.0	0.584341
$714$	0	0
$715$	−948.141	−0.0495923
$716$	−34766.9	−1.81467
$717$	0	0
$718$	−32513.9	−1.68998
$719$	−16846.3	−0.873800	−0.436900	−	0.899510i	$-0.643924\pi$
$719$	−16846.3	−0.873800	−0.436900	+	0.899510i	$0.643924\pi$
$720$	0	0
$721$	14796.3	0.764276
$722$	51619.4	2.66077
$723$	0	0
$724$	15227.3	0.781655
$725$	3676.38	0.188327
$726$	0	0
$727$	2903.64	0.148129	0.0740647	−	0.997253i	$-0.476403\pi$
$727$	2903.64	0.148129	0.0740647	+	0.997253i	$0.476403\pi$
$728$	−8835.05	−0.449792
$729$	0	0
$730$	−11664.1	−0.591380
$731$	−13457.8	−0.680925
$732$	0	0
$733$	−25017.0	−1.26060	−0.630302	−	0.776350i	$-0.717069\pi$
$733$	−25017.0	−1.26060	−0.630302	+	0.776350i	$0.717069\pi$
$734$	49434.3	2.48590
$735$	0	0
$736$	17210.1	0.861919
$737$	−2980.01	−0.148942
$738$	0	0
$739$	28998.8	1.44349	0.721745	−	0.692159i	$-0.243340\pi$
$739$	28998.8	1.44349	0.721745	+	0.692159i	$0.243340\pi$
$740$	−16351.2	−0.812271
$741$	0	0
$742$	−36267.4	−1.79436
$743$	17862.0	0.881958	0.440979	−	0.897518i	$-0.354631\pi$
$743$	17862.0	0.881958	0.440979	+	0.897518i	$0.354631\pi$
$744$	0	0
$745$	10251.1	0.504123
$746$	27020.4	1.32612
$747$	0	0
$748$	1023.27	0.0500195
$749$	3799.03	0.185332
$750$	0	0
$751$	−15428.4	−0.749656	−0.374828	−	0.927094i	$-0.622298\pi$
$751$	−15428.4	−0.749656	−0.374828	+	0.927094i	$0.622298\pi$
$752$	3427.11	0.166188
$753$	0	0
$754$	−16342.1	−0.789314
$755$	−11015.3	−0.530980
$756$	0	0
$757$	27126.6	1.30242	0.651211	−	0.758896i	$-0.274261\pi$
$757$	27126.6	1.30242	0.651211	+	0.758896i	$0.274261\pi$
$758$	−26653.0	−1.27715
$759$	0	0
$760$	−25156.3	−1.20068
$761$	−17958.1	−0.855426	−0.427713	−	0.903914i	$-0.640681\pi$
$761$	−17958.1	−0.855426	−0.427713	+	0.903914i	$0.640681\pi$
$762$	0	0
$763$	−14824.6	−0.703391
$764$	30725.0	1.45496
$765$	0	0
$766$	−50501.0	−2.38208
$767$	14572.6	0.686032
$768$	0	0
$769$	15157.7	0.710796	0.355398	−	0.934715i	$-0.384345\pi$
$769$	15157.7	0.710796	0.355398	+	0.934715i	$0.384345\pi$
$770$	1687.51	0.0789790
$771$	0	0
$772$	−52256.1	−2.43619
$773$	−33668.1	−1.56657	−0.783285	−	0.621663i	$-0.786457\pi$
$773$	−33668.1	−1.56657	−0.783285	+	0.621663i	$0.786457\pi$
$774$	0	0
$775$	4528.90	0.209913
$776$	4123.45	0.190751
$777$	0	0
$778$	2501.12	0.115256
$779$	23478.6	1.07986
$780$	0	0
$781$	−2750.39	−0.126014
$782$	10732.1	0.490766
$783$	0	0
$784$	1742.60	0.0793821
$785$	−20639.3	−0.938405
$786$	0	0
$787$	22259.6	1.00822	0.504110	−	0.863640i	$-0.331821\pi$
$787$	22259.6	1.00822	0.504110	+	0.863640i	$0.331821\pi$
$788$	−63877.1	−2.88772
$789$	0	0
$790$	−12937.1	−0.582636
$791$	−11915.8	−0.535620
$792$	0	0
$793$	−22042.9	−0.987095
$794$	−27902.5	−1.24713
$795$	0	0
$796$	50547.8	2.25078
$797$	−24064.3	−1.06951	−0.534757	−	0.845006i	$-0.679597\pi$
$797$	−24064.3	−1.06951	−0.534757	+	0.845006i	$0.679597\pi$
$798$	0	0
$799$	9122.50	0.403918
$800$	7006.08	0.309628
$801$	0	0
$802$	−1809.92	−0.0796889
$803$	799.105	0.0351181
$804$	0	0
$805$	10734.3	0.469982
$806$	−20131.6	−0.879785
$807$	0	0
$808$	10276.8	0.447448
$809$	29725.2	1.29182	0.645910	−	0.763414i	$-0.276478\pi$
$809$	29725.2	1.29182	0.645910	+	0.763414i	$0.276478\pi$
$810$	0	0
$811$	24720.6	1.07036	0.535178	−	0.844739i	$-0.320245\pi$
$811$	24720.6	1.07036	0.535178	+	0.844739i	$0.320245\pi$
$812$	17640.7	0.762400
$813$	0	0
$814$	1847.00	0.0795298
$815$	−10951.4	−0.470689
$816$	0	0
$817$	−64537.7	−2.76363
$818$	61802.7	2.64167
$819$	0	0
$820$	−20372.8	−0.867622
$821$	95.7985	0.00407234	0.00203617	−	0.999998i	$-0.499352\pi$
$821$	95.7985	0.00407234	0.00203617	+	0.999998i	$0.499352\pi$
$822$	0	0
$823$	16210.6	0.686595	0.343298	−	0.939227i	$-0.388456\pi$
$823$	16210.6	0.686595	0.343298	+	0.939227i	$0.388456\pi$
$824$	−21619.7	−0.914027
$825$	0	0
$826$	−25936.6	−1.09255
$827$	324.820	0.0136579	0.00682895	−	0.999977i	$-0.497826\pi$
$827$	324.820	0.0136579	0.00682895	+	0.999977i	$0.497826\pi$
$828$	0	0
$829$	21670.0	0.907877	0.453938	−	0.891033i	$-0.350019\pi$
$829$	21670.0	0.907877	0.453938	+	0.891033i	$0.350019\pi$
$830$	−29604.1	−1.23804
$831$	0	0
$832$	−28272.2	−1.17808
$833$	4638.56	0.192937
$834$	0	0
$835$	25273.4	1.04745
$836$	4907.16	0.203012
$837$	0	0
$838$	23191.2	0.956000
$839$	−19402.4	−0.798386	−0.399193	−	0.916867i	$-0.630710\pi$
$839$	−19402.4	−0.798386	−0.399193	+	0.916867i	$0.630710\pi$
$840$	0	0
$841$	−12929.0	−0.530115
$842$	55547.6	2.27351
$843$	0	0
$844$	46720.9	1.90545
$845$	−10004.7	−0.407305
$846$	0	0
$847$	17671.9	0.716899
$848$	−6379.54	−0.258343
$849$	0	0
$850$	4368.94	0.176298
$851$	11748.8	0.473260
$852$	0	0
$853$	40326.6	1.61871	0.809353	−	0.587323i	$-0.199818\pi$
$853$	40326.6	1.61871	0.809353	+	0.587323i	$0.199818\pi$
$854$	39232.2	1.57201
$855$	0	0
$856$	−5550.97	−0.221645
$857$	−33139.5	−1.32092	−0.660458	−	0.750863i	$-0.729638\pi$
$857$	−33139.5	−1.32092	−0.660458	+	0.750863i	$0.729638\pi$
$858$	0	0
$859$	−38149.5	−1.51530	−0.757651	−	0.652659i	$-0.773653\pi$
$859$	−38149.5	−1.51530	−0.757651	+	0.652659i	$0.773653\pi$
$860$	56000.7	2.22047
$861$	0	0
$862$	−18942.6	−0.748476
$863$	3945.03	0.155609	0.0778044	−	0.996969i	$-0.475209\pi$
$863$	3945.03	0.155609	0.0778044	+	0.996969i	$0.475209\pi$
$864$	0	0
$865$	−8091.98	−0.318076
$866$	−70838.2	−2.77965
$867$	0	0
$868$	21731.5	0.849786
$869$	886.321	0.0345988
$870$	0	0
$871$	34302.4	1.33443
$872$	21661.1	0.841212
$873$	0	0
$874$	51466.3	1.99184
$875$	20275.4	0.783354
$876$	0	0
$877$	−13862.9	−0.533771	−0.266886	−	0.963728i	$-0.585995\pi$
$877$	−13862.9	−0.533771	−0.266886	+	0.963728i	$0.585995\pi$
$878$	−15324.0	−0.589020
$879$	0	0
$880$	296.839	0.0113710
$881$	5631.94	0.215374	0.107687	−	0.994185i	$-0.465655\pi$
$881$	5631.94	0.215374	0.107687	+	0.994185i	$0.465655\pi$
$882$	0	0
$883$	−17981.2	−0.685294	−0.342647	−	0.939464i	$-0.611323\pi$
$883$	−17981.2	−0.685294	−0.342647	+	0.939464i	$0.611323\pi$
$884$	−11778.7	−0.448147
$885$	0	0
$886$	−80685.4	−3.05946
$887$	−14979.6	−0.567041	−0.283521	−	0.958966i	$-0.591502\pi$
$887$	−14979.6	−0.567041	−0.283521	+	0.958966i	$0.591502\pi$
$888$	0	0
$889$	16187.9	0.610715
$890$	−27920.5	−1.05157
$891$	0	0
$892$	−54895.3	−2.06057
$893$	43747.4	1.63936
$894$	0	0
$895$	−26846.1	−1.00264
$896$	28508.4	1.06294
$897$	0	0
$898$	−81244.9	−3.01913
$899$	14117.5	0.523743
$900$	0	0
$901$	−16981.5	−0.627898
$902$	2301.28	0.0849492
$903$	0	0
$904$	17410.8	0.640569
$905$	11758.1	0.431882
$906$	0	0
$907$	−53497.6	−1.95850	−0.979249	−	0.202661i	$-0.935041\pi$
$907$	−53497.6	−1.95850	−0.979249	+	0.202661i	$0.935041\pi$
$908$	18792.4	0.686836
$909$	0	0
$910$	−19424.7	−0.707607
$911$	23406.8	0.851265	0.425633	−	0.904896i	$-0.360052\pi$
$911$	23406.8	0.851265	0.425633	+	0.904896i	$0.360052\pi$
$912$	0	0
$913$	2028.17	0.0735189
$914$	3051.92	0.110447
$915$	0	0
$916$	−42386.3	−1.52891
$917$	16394.4	0.590395
$918$	0	0
$919$	9513.83	0.341493	0.170747	−	0.985315i	$-0.445382\pi$
$919$	9513.83	0.341493	0.170747	+	0.985315i	$0.445382\pi$
$920$	−15684.5	−0.562070
$921$	0	0
$922$	22357.5	0.798595
$923$	31659.3	1.12901
$924$	0	0
$925$	4782.84	0.170010
$926$	35830.1	1.27155
$927$	0	0
$928$	21839.4	0.772535
$929$	33262.4	1.17471	0.587353	−	0.809331i	$-0.300170\pi$
$929$	33262.4	1.17471	0.587353	+	0.809331i	$0.300170\pi$
$930$	0	0
$931$	22244.4	0.783063
$932$	−42562.0	−1.49588
$933$	0	0
$934$	−4867.02	−0.170507
$935$	790.146	0.0276369
$936$	0	0
$937$	25966.8	0.905333	0.452666	−	0.891680i	$-0.350473\pi$
$937$	25966.8	0.905333	0.452666	+	0.891680i	$0.350473\pi$
$938$	−61051.8	−2.12517
$939$	0	0
$940$	−37960.5	−1.31716
$941$	4390.79	0.152110	0.0760552	−	0.997104i	$-0.475767\pi$
$941$	4390.79	0.152110	0.0760552	+	0.997104i	$0.475767\pi$
$942$	0	0
$943$	14638.5	0.505509
$944$	−4562.32	−0.157300
$945$	0	0
$946$	−6325.73	−0.217407
$947$	54191.6	1.85955	0.929774	−	0.368131i	$-0.120002\pi$
$947$	54191.6	1.85955	0.929774	+	0.368131i	$0.120002\pi$
$948$	0	0
$949$	−9198.36	−0.314638
$950$	20951.5	0.715532
$951$	0	0
$952$	7362.80	0.250662
$953$	42172.3	1.43347	0.716734	−	0.697346i	$-0.245636\pi$
$953$	42172.3	1.43347	0.716734	+	0.697346i	$0.245636\pi$
$954$	0	0
$955$	23725.1	0.803901
$956$	2947.04	0.0997008
$957$	0	0
$958$	4191.42	0.141355
$959$	−23203.4	−0.781311
$960$	0	0
$961$	−12399.8	−0.416226
$962$	−21260.5	−0.712541
$963$	0	0
$964$	34876.2	1.16524
$965$	−40350.8	−1.34605
$966$	0	0
$967$	−53176.0	−1.76838	−0.884190	−	0.467127i	$-0.845289\pi$
$967$	−53176.0	−1.76838	−0.884190	+	0.467127i	$0.845289\pi$
$968$	−25821.4	−0.857367
$969$	0	0
$970$	9065.78	0.300087
$971$	−10197.8	−0.337036	−0.168518	−	0.985699i	$-0.553898\pi$
$971$	−10197.8	−0.337036	−0.168518	+	0.985699i	$0.553898\pi$
$972$	0	0
$973$	−4252.01	−0.140096
$974$	−1742.86	−0.0573356
$975$	0	0
$976$	6901.07	0.226330
$977$	−4263.95	−0.139627	−0.0698136	−	0.997560i	$-0.522240\pi$
$977$	−4263.95	−0.139627	−0.0698136	+	0.997560i	$0.522240\pi$
$978$	0	0
$979$	1912.83	0.0624457
$980$	−19301.9	−0.629160
$981$	0	0
$982$	−42278.1	−1.37388
$983$	−9411.52	−0.305372	−0.152686	−	0.988275i	$-0.548792\pi$
$983$	−9411.52	−0.305372	−0.152686	+	0.988275i	$0.548792\pi$
$984$	0	0
$985$	−49324.2	−1.59553
$986$	13618.9	0.439872
$987$	0	0
$988$	−56485.5	−1.81887
$989$	−40238.2	−1.29373
$990$	0	0
$991$	−47766.3	−1.53113	−0.765563	−	0.643361i	$-0.777539\pi$
$991$	−47766.3	−1.53113	−0.765563	+	0.643361i	$0.777539\pi$
$992$	26903.7	0.861083
$993$	0	0
$994$	−56347.6	−1.79802
$995$	39031.7	1.24361
$996$	0	0
$997$	−50485.2	−1.60369	−0.801847	−	0.597530i	$-0.796149\pi$
$997$	−50485.2	−1.60369	−0.801847	+	0.597530i	$0.796149\pi$
$998$	19116.5	0.606336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.54	✓	59
3.2	odd	2		2151.4.a.h.1.6	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.54	✓	59	1.1	even	1	trivial
2151.4.a.h.1.6	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)

\(f(q)\)	\(=\)	\(q+4.50896 q^{2} +12.3307 q^{4} +9.52144 q^{5} -13.3640 q^{7} +19.5269 q^{8} +O(q^{10})\)	\(q+4.50896 q^{2} +12.3307 q^{4} +9.52144 q^{5} -13.3640 q^{7} +19.5269 q^{8} +42.9318 q^{10} -2.94125 q^{11} +33.8562 q^{13} -60.2578 q^{14} -10.5995 q^{16} -28.2145 q^{17} -135.304 q^{19} +117.406 q^{20} -13.2620 q^{22} -84.3598 q^{23} -34.3422 q^{25} +152.656 q^{26} -164.788 q^{28} -107.051 q^{29} -131.876 q^{31} -204.008 q^{32} -127.218 q^{34} -127.245 q^{35} -139.270 q^{37} -610.080 q^{38} +185.924 q^{40} -173.525 q^{41} +476.983 q^{43} -36.2677 q^{44} -380.375 q^{46} -323.326 q^{47} -164.403 q^{49} -154.847 q^{50} +417.471 q^{52} +601.871 q^{53} -28.0049 q^{55} -260.958 q^{56} -482.690 q^{58} +430.427 q^{59} -651.074 q^{61} -594.622 q^{62} -835.068 q^{64} +322.360 q^{65} +1013.18 q^{67} -347.905 q^{68} -573.741 q^{70} +935.109 q^{71} -271.689 q^{73} -627.964 q^{74} -1668.39 q^{76} +39.3069 q^{77} -301.342 q^{79} -100.923 q^{80} -782.415 q^{82} -689.562 q^{83} -268.643 q^{85} +2150.70 q^{86} -57.4335 q^{88} -650.347 q^{89} -452.455 q^{91} -1040.22 q^{92} -1457.87 q^{94} -1288.29 q^{95} +211.167 q^{97} -741.287 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

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