LMFDB - Embedded newform 2151.4.a.g.1.6

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			1.6
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)

$f(q)$	$=$	$q-4.81368 q^{2} +15.1715 q^{4} -10.3877 q^{5} +20.9834 q^{7} -34.5214 q^{8} +O(q^{10})$	\(q-4.81368 q^{2} +15.1715 q^{4} -10.3877 q^{5} +20.9834 q^{7} -34.5214 q^{8} +50.0029 q^{10} -6.43436 q^{11} +30.5626 q^{13} -101.007 q^{14} +44.8030 q^{16} +52.9744 q^{17} +47.8804 q^{19} -157.597 q^{20} +30.9730 q^{22} -28.3430 q^{23} -17.0964 q^{25} -147.119 q^{26} +318.350 q^{28} -211.958 q^{29} +268.710 q^{31} +60.5041 q^{32} -255.002 q^{34} -217.969 q^{35} -239.346 q^{37} -230.481 q^{38} +358.597 q^{40} +104.560 q^{41} +2.82750 q^{43} -97.6191 q^{44} +136.434 q^{46} -457.167 q^{47} +97.3033 q^{49} +82.2968 q^{50} +463.682 q^{52} +536.716 q^{53} +66.8380 q^{55} -724.378 q^{56} +1020.30 q^{58} -141.045 q^{59} -737.733 q^{61} -1293.48 q^{62} -649.672 q^{64} -317.474 q^{65} +268.192 q^{67} +803.702 q^{68} +1049.23 q^{70} -745.293 q^{71} -435.710 q^{73} +1152.14 q^{74} +726.419 q^{76} -135.015 q^{77} +1384.51 q^{79} -465.399 q^{80} -503.319 q^{82} -1127.30 q^{83} -550.280 q^{85} -13.6107 q^{86} +222.124 q^{88} +208.529 q^{89} +641.308 q^{91} -430.006 q^{92} +2200.66 q^{94} -497.366 q^{95} -253.519 q^{97} -468.387 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.81368	−1.70189	−0.850947	−	0.525252i	$-0.823971\pi$
$2$	−4.81368	−1.70189	−0.850947	+	0.525252i	$0.823971\pi$
$3$	0	0
$3$	0	0
$4$	15.1715	1.89644
$5$	−10.3877	−0.929101	−0.464550	−	0.885547i	$-0.653784\pi$
$5$	−10.3877	−0.929101	−0.464550	+	0.885547i	$0.653784\pi$
$6$	0	0
$7$	20.9834	1.13300	0.566499	−	0.824063i	$-0.308298\pi$
$7$	20.9834	1.13300	0.566499	+	0.824063i	$0.308298\pi$
$8$	−34.5214	−1.52565
$9$	0	0
$10$	50.0029	1.58123
$11$	−6.43436	−0.176367	−0.0881834	−	0.996104i	$-0.528106\pi$
$11$	−6.43436	−0.176367	−0.0881834	+	0.996104i	$0.528106\pi$
$12$	0	0
$13$	30.5626	0.652042	0.326021	−	0.945363i	$-0.394292\pi$
$13$	30.5626	0.652042	0.326021	+	0.945363i	$0.394292\pi$
$14$	−101.007	−1.92824
$15$	0	0
$16$	44.8030	0.700047
$17$	52.9744	0.755775	0.377887	−	0.925852i	$-0.376651\pi$
$17$	52.9744	0.755775	0.377887	+	0.925852i	$0.376651\pi$
$18$	0	0
$19$	47.8804	0.578133	0.289066	−	0.957309i	$-0.406655\pi$
$19$	47.8804	0.578133	0.289066	+	0.957309i	$0.406655\pi$
$20$	−157.597	−1.76199
$21$	0	0
$22$	30.9730	0.300157
$23$	−28.3430	−0.256953	−0.128476	−	0.991713i	$-0.541009\pi$
$23$	−28.3430	−0.256953	−0.128476	+	0.991713i	$0.541009\pi$
$24$	0	0
$25$	−17.0964	−0.136771
$26$	−147.119	−1.10971
$27$	0	0
$28$	318.350	2.14866
$29$	−211.958	−1.35723	−0.678614	−	0.734495i	$-0.737419\pi$
$29$	−211.958	−1.35723	−0.678614	+	0.734495i	$0.737419\pi$
$30$	0	0
$31$	268.710	1.55683	0.778414	−	0.627751i	$-0.216024\pi$
$31$	268.710	1.55683	0.778414	+	0.627751i	$0.216024\pi$
$32$	60.5041	0.334241
$33$	0	0
$34$	−255.002	−1.28625
$35$	−217.969	−1.05267
$36$	0	0
$37$	−239.346	−1.06347	−0.531733	−	0.846912i	$-0.678459\pi$
$37$	−239.346	−1.06347	−0.531733	+	0.846912i	$0.678459\pi$
$38$	−230.481	−0.983920
$39$	0	0
$40$	358.597	1.41748
$41$	104.560	0.398281	0.199141	−	0.979971i	$-0.436185\pi$
$41$	104.560	0.398281	0.199141	+	0.979971i	$0.436185\pi$
$42$	0	0
$43$	2.82750	0.0100277	0.00501384	−	0.999987i	$-0.498404\pi$
$43$	2.82750	0.0100277	0.00501384	+	0.999987i	$0.498404\pi$
$44$	−97.6191	−0.334469
$45$	0	0
$46$	136.434	0.437306
$47$	−457.167	−1.41882	−0.709412	−	0.704794i	$-0.751039\pi$
$47$	−457.167	−1.41882	−0.709412	+	0.704794i	$0.751039\pi$
$48$	0	0
$49$	97.3033	0.283683
$50$	82.2968	0.232770
$51$	0	0
$52$	463.682	1.23656
$53$	536.716	1.39101	0.695505	−	0.718521i	$-0.255180\pi$
$53$	536.716	1.39101	0.695505	+	0.718521i	$0.255180\pi$
$54$	0	0
$55$	66.8380	0.163862
$56$	−724.378	−1.72855
$57$	0	0
$58$	1020.30	2.30986
$59$	−141.045	−0.311230	−0.155615	−	0.987818i	$-0.549736\pi$
$59$	−141.045	−0.311230	−0.155615	+	0.987818i	$0.549736\pi$
$60$	0	0
$61$	−737.733	−1.54848	−0.774238	−	0.632895i	$-0.781867\pi$
$61$	−737.733	−1.54848	−0.774238	+	0.632895i	$0.781867\pi$
$62$	−1293.48	−2.64956
$63$	0	0
$64$	−649.672	−1.26889
$65$	−317.474	−0.605813
$66$	0	0
$67$	268.192	0.489028	0.244514	−	0.969646i	$-0.421372\pi$
$67$	268.192	0.489028	0.244514	+	0.969646i	$0.421372\pi$
$68$	803.702	1.43328
$69$	0	0
$70$	1049.23	1.79153
$71$	−745.293	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$71$	−745.293	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$72$	0	0
$73$	−435.710	−0.698575	−0.349287	−	0.937016i	$-0.613576\pi$
$73$	−435.710	−0.698575	−0.349287	+	0.937016i	$0.613576\pi$
$74$	1152.14	1.80991
$75$	0	0
$76$	726.419	1.09639
$77$	−135.015	−0.199823
$78$	0	0
$79$	1384.51	1.97177	0.985885	−	0.167422i	$-0.0535443\pi$
$79$	1384.51	1.97177	0.985885	+	0.167422i	$0.0535443\pi$
$80$	−465.399	−0.650415
$81$	0	0
$82$	−503.319	−0.677832
$83$	−1127.30	−1.49081	−0.745405	−	0.666612i	$-0.767744\pi$
$83$	−1127.30	−1.49081	−0.745405	+	0.666612i	$0.767744\pi$
$84$	0	0
$85$	−550.280	−0.702191
$86$	−13.6107	−0.0170660
$87$	0	0
$88$	222.124	0.269073
$89$	208.529	0.248359	0.124180	−	0.992260i	$-0.460370\pi$
$89$	208.529	0.248359	0.124180	+	0.992260i	$0.460370\pi$
$90$	0	0
$91$	641.308	0.738762
$92$	−430.006	−0.487296
$93$	0	0
$94$	2200.66	2.41469
$95$	−497.366	−0.537144
$96$	0	0
$97$	−253.519	−0.265370	−0.132685	−	0.991158i	$-0.542360\pi$
$97$	−253.519	−0.265370	−0.132685	+	0.991158i	$0.542360\pi$
$98$	−468.387	−0.482798
$99$	0	0
$100$	−259.379	−0.259379
$101$	1054.97	1.03934	0.519672	−	0.854366i	$-0.326054\pi$
$101$	1054.97	1.03934	0.519672	+	0.854366i	$0.326054\pi$
$102$	0	0
$103$	−1825.38	−1.74621	−0.873105	−	0.487532i	$-0.837897\pi$
$103$	−1825.38	−1.74621	−0.873105	+	0.487532i	$0.837897\pi$
$104$	−1055.07	−0.994786
$105$	0	0
$106$	−2583.58	−2.36735
$107$	4.50462	0.00406989	0.00203494	−	0.999998i	$-0.499352\pi$
$107$	4.50462	0.00406989	0.00203494	+	0.999998i	$0.499352\pi$
$108$	0	0
$109$	836.054	0.734674	0.367337	−	0.930088i	$-0.380270\pi$
$109$	836.054	0.734674	0.367337	+	0.930088i	$0.380270\pi$
$110$	−321.737	−0.278876
$111$	0	0
$112$	940.120	0.793152
$113$	928.893	0.773300	0.386650	−	0.922227i	$-0.373632\pi$
$113$	928.893	0.773300	0.386650	+	0.922227i	$0.373632\pi$
$114$	0	0
$115$	294.417	0.238735
$116$	−3215.73	−2.57390
$117$	0	0
$118$	678.948	0.529680
$119$	1111.58	0.856291
$120$	0	0
$121$	−1289.60	−0.968895
$122$	3551.21	2.63534
$123$	0	0
$124$	4076.74	2.95243
$125$	1476.05	1.05618
$126$	0	0
$127$	−801.818	−0.560235	−0.280117	−	0.959966i	$-0.590373\pi$
$127$	−801.818	−0.560235	−0.280117	+	0.959966i	$0.590373\pi$
$128$	2643.28	1.82527
$129$	0	0
$130$	1528.22	1.03103
$131$	−867.345	−0.578476	−0.289238	−	0.957257i	$-0.593402\pi$
$131$	−867.345	−0.578476	−0.289238	+	0.957257i	$0.593402\pi$
$132$	0	0
$133$	1004.69	0.655023
$134$	−1290.99	−0.832273
$135$	0	0
$136$	−1828.75	−1.15305
$137$	−1070.78	−0.667757	−0.333879	−	0.942616i	$-0.608358\pi$
$137$	−1070.78	−0.667757	−0.333879	+	0.942616i	$0.608358\pi$
$138$	0	0
$139$	324.630	0.198092	0.0990458	−	0.995083i	$-0.468421\pi$
$139$	324.630	0.198092	0.0990458	+	0.995083i	$0.468421\pi$
$140$	−3306.92	−1.99632
$141$	0	0
$142$	3587.60	2.12017
$143$	−196.651	−0.114999
$144$	0	0
$145$	2201.75	1.26100
$146$	2097.37	1.18890
$147$	0	0
$148$	−3631.25	−2.01680
$149$	−225.532	−0.124002	−0.0620011	−	0.998076i	$-0.519748\pi$
$149$	−225.532	−0.124002	−0.0620011	+	0.998076i	$0.519748\pi$
$150$	0	0
$151$	−86.7460	−0.0467503	−0.0233751	−	0.999727i	$-0.507441\pi$
$151$	−86.7460	−0.0467503	−0.0233751	+	0.999727i	$0.507441\pi$
$152$	−1652.90	−0.882026
$153$	0	0
$154$	649.919	0.340077
$155$	−2791.27	−1.44645
$156$	0	0
$157$	−3392.64	−1.72460	−0.862300	−	0.506398i	$-0.830977\pi$
$157$	−3392.64	−1.72460	−0.862300	+	0.506398i	$0.830977\pi$
$158$	−6664.60	−3.35574
$159$	0	0
$160$	−628.496	−0.310544
$161$	−594.732	−0.291127
$162$	0	0
$163$	1944.97	0.934612	0.467306	−	0.884096i	$-0.345225\pi$
$163$	1944.97	0.934612	0.467306	+	0.884096i	$0.345225\pi$
$164$	1586.34	0.755317
$165$	0	0
$166$	5426.46	2.53720
$167$	813.916	0.377142	0.188571	−	0.982060i	$-0.439614\pi$
$167$	813.916	0.377142	0.188571	+	0.982060i	$0.439614\pi$
$168$	0	0
$169$	−1262.93	−0.574841
$170$	2648.87	1.19505
$171$	0	0
$172$	42.8975	0.0190169
$173$	1328.39	0.583788	0.291894	−	0.956451i	$-0.405715\pi$
$173$	1328.39	0.583788	0.291894	+	0.956451i	$0.405715\pi$
$174$	0	0
$175$	−358.741	−0.154962
$176$	−288.279	−0.123465
$177$	0	0
$178$	−1003.79	−0.422681
$179$	2382.45	0.994819	0.497410	−	0.867516i	$-0.334285\pi$
$179$	2382.45	0.994819	0.497410	+	0.867516i	$0.334285\pi$
$180$	0	0
$181$	2869.39	1.17834	0.589172	−	0.808007i	$-0.299454\pi$
$181$	2869.39	1.17834	0.589172	+	0.808007i	$0.299454\pi$
$182$	−3087.05	−1.25729
$183$	0	0
$184$	978.440	0.392019
$185$	2486.25	0.988068
$186$	0	0
$187$	−340.856	−0.133293
$188$	−6935.93	−2.69072
$189$	0	0
$190$	2394.16	0.914161
$191$	1132.26	0.428940	0.214470	−	0.976731i	$-0.431198\pi$
$191$	1132.26	0.428940	0.214470	+	0.976731i	$0.431198\pi$
$192$	0	0
$193$	1445.30	0.539043	0.269521	−	0.962994i	$-0.413134\pi$
$193$	1445.30	0.539043	0.269521	+	0.962994i	$0.413134\pi$
$194$	1220.36	0.451632
$195$	0	0
$196$	1476.24	0.537988
$197$	−3222.98	−1.16562	−0.582811	−	0.812607i	$-0.698047\pi$
$197$	−3222.98	−1.16562	−0.582811	+	0.812607i	$0.698047\pi$
$198$	0	0
$199$	−3396.25	−1.20982	−0.604909	−	0.796295i	$-0.706791\pi$
$199$	−3396.25	−1.20982	−0.604909	+	0.796295i	$0.706791\pi$
$200$	590.194	0.208665
$201$	0	0
$202$	−5078.30	−1.76885
$203$	−4447.60	−1.53774
$204$	0	0
$205$	−1086.13	−0.370044
$206$	8786.78	2.97186
$207$	0	0
$208$	1369.30	0.456460
$209$	−308.080	−0.101963
$210$	0	0
$211$	3875.97	1.26461	0.632305	−	0.774720i	$-0.282109\pi$
$211$	3875.97	1.26461	0.632305	+	0.774720i	$0.282109\pi$
$212$	8142.80	2.63797
$213$	0	0
$214$	−21.6838	−0.00692652
$215$	−29.3711	−0.00931672
$216$	0	0
$217$	5638.44	1.76388
$218$	−4024.50	−1.25034
$219$	0	0
$220$	1014.03	0.310756
$221$	1619.04	0.492797
$222$	0	0
$223$	4467.57	1.34157	0.670786	−	0.741651i	$-0.265957\pi$
$223$	4467.57	1.34157	0.670786	+	0.741651i	$0.265957\pi$
$224$	1269.58	0.378694
$225$	0	0
$226$	−4471.39	−1.31607
$227$	3452.39	1.00944	0.504721	−	0.863283i	$-0.331596\pi$
$227$	3452.39	1.00944	0.504721	+	0.863283i	$0.331596\pi$
$228$	0	0
$229$	2661.40	0.767993	0.383996	−	0.923335i	$-0.374548\pi$
$229$	2661.40	0.767993	0.383996	+	0.923335i	$0.374548\pi$
$230$	−1417.23	−0.406302
$231$	0	0
$232$	7317.10	2.07065
$233$	−4533.88	−1.27478	−0.637392	−	0.770540i	$-0.719987\pi$
$233$	−4533.88	−1.27478	−0.637392	+	0.770540i	$0.719987\pi$
$234$	0	0
$235$	4748.90	1.31823
$236$	−2139.87	−0.590229
$237$	0	0
$238$	−5350.80	−1.45732
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	6097.88	1.62987	0.814935	−	0.579552i	$-0.196773\pi$
$241$	6097.88	1.62987	0.814935	+	0.579552i	$0.196773\pi$
$242$	6207.72	1.64896
$243$	0	0
$244$	−11192.5	−2.93659
$245$	−1010.75	−0.263570
$246$	0	0
$247$	1463.35	0.376967
$248$	−9276.25	−2.37517
$249$	0	0
$250$	−7105.23	−1.79750
$251$	259.843	0.0653433	0.0326716	−	0.999466i	$-0.489598\pi$
$251$	259.843	0.0653433	0.0326716	+	0.999466i	$0.489598\pi$
$252$	0	0
$253$	182.369	0.0453179
$254$	3859.70	0.953460
$255$	0	0
$256$	−7526.53	−1.83753
$257$	5271.41	1.27946	0.639731	−	0.768599i	$-0.279046\pi$
$257$	5271.41	1.27946	0.639731	+	0.768599i	$0.279046\pi$
$258$	0	0
$259$	−5022.30	−1.20491
$260$	−4816.57	−1.14889
$261$	0	0
$262$	4175.12	0.984504
$263$	2814.06	0.659781	0.329890	−	0.944019i	$-0.392988\pi$
$263$	2814.06	0.659781	0.329890	+	0.944019i	$0.392988\pi$
$264$	0	0
$265$	−5575.22	−1.29239
$266$	−4836.28	−1.11478
$267$	0	0
$268$	4068.88	0.927412
$269$	−2397.37	−0.543385	−0.271692	−	0.962384i	$-0.587583\pi$
$269$	−2397.37	−0.543385	−0.271692	+	0.962384i	$0.587583\pi$
$270$	0	0
$271$	4328.22	0.970187	0.485093	−	0.874462i	$-0.338786\pi$
$271$	4328.22	0.970187	0.485093	+	0.874462i	$0.338786\pi$
$272$	2373.41	0.529078
$273$	0	0
$274$	5154.38	1.13645
$275$	110.005	0.0241219
$276$	0	0
$277$	3043.01	0.660060	0.330030	−	0.943970i	$-0.392941\pi$
$277$	3043.01	0.660060	0.330030	+	0.943970i	$0.392941\pi$
$278$	−1562.66	−0.337131
$279$	0	0
$280$	7524.59	1.60600
$281$	−6849.53	−1.45412	−0.727061	−	0.686572i	$-0.759115\pi$
$281$	−6849.53	−1.45412	−0.727061	+	0.686572i	$0.759115\pi$
$282$	0	0
$283$	−4843.24	−1.01732	−0.508659	−	0.860968i	$-0.669859\pi$
$283$	−4843.24	−1.01732	−0.508659	+	0.860968i	$0.669859\pi$
$284$	−11307.2	−2.36254
$285$	0	0
$286$	946.616	0.195715
$287$	2194.03	0.451252
$288$	0	0
$289$	−2106.72	−0.428805
$290$	−10598.5	−2.14609
$291$	0	0
$292$	−6610.38	−1.32481
$293$	2730.83	0.544494	0.272247	−	0.962227i	$-0.412233\pi$
$293$	2730.83	0.544494	0.272247	+	0.962227i	$0.412233\pi$
$294$	0	0
$295$	1465.13	0.289164
$296$	8262.58	1.62247
$297$	0	0
$298$	1085.64	0.211039
$299$	−866.235	−0.167544
$300$	0	0
$301$	59.3306	0.0113613
$302$	417.568	0.0795640
$303$	0	0
$304$	2145.19	0.404720
$305$	7663.32	1.43869
$306$	0	0
$307$	−7199.12	−1.33836	−0.669178	−	0.743102i	$-0.733354\pi$
$307$	−7199.12	−1.33836	−0.669178	+	0.743102i	$0.733354\pi$
$308$	−2048.38	−0.378953
$309$	0	0
$310$	13436.3	2.46170
$311$	−9532.54	−1.73807	−0.869037	−	0.494748i	$-0.835260\pi$
$311$	−9532.54	−1.73807	−0.869037	+	0.494748i	$0.835260\pi$
$312$	0	0
$313$	−2655.50	−0.479545	−0.239772	−	0.970829i	$-0.577073\pi$
$313$	−2655.50	−0.479545	−0.239772	+	0.970829i	$0.577073\pi$
$314$	16331.1	2.93508
$315$	0	0
$316$	21005.2	3.73935
$317$	5571.42	0.987136	0.493568	−	0.869707i	$-0.335692\pi$
$317$	5571.42	0.987136	0.493568	+	0.869707i	$0.335692\pi$
$318$	0	0
$319$	1363.82	0.239370
$320$	6748.57	1.17893
$321$	0	0
$322$	2862.85	0.495467
$323$	2536.43	0.436938
$324$	0	0
$325$	−522.512	−0.0891807
$326$	−9362.46	−1.59061
$327$	0	0
$328$	−3609.56	−0.607637
$329$	−9592.93	−1.60752
$330$	0	0
$331$	−6288.20	−1.04420	−0.522101	−	0.852884i	$-0.674852\pi$
$331$	−6288.20	−1.04420	−0.522101	+	0.852884i	$0.674852\pi$
$332$	−17102.9	−2.82723
$333$	0	0
$334$	−3917.93	−0.641856
$335$	−2785.89	−0.454356
$336$	0	0
$337$	4540.87	0.733998	0.366999	−	0.930221i	$-0.380385\pi$
$337$	4540.87	0.733998	0.366999	+	0.930221i	$0.380385\pi$
$338$	6079.32	0.978318
$339$	0	0
$340$	−8348.58	−1.33166
$341$	−1728.98	−0.274573
$342$	0	0
$343$	−5155.55	−0.811585
$344$	−97.6094	−0.0152987
$345$	0	0
$346$	−6394.43	−0.993545
$347$	8598.02	1.33016	0.665080	−	0.746772i	$-0.268397\pi$
$347$	8598.02	1.33016	0.665080	+	0.746772i	$0.268397\pi$
$348$	0	0
$349$	−1311.65	−0.201177	−0.100589	−	0.994928i	$-0.532073\pi$
$349$	−1311.65	−0.201177	−0.100589	+	0.994928i	$0.532073\pi$
$350$	1726.87	0.263728
$351$	0	0
$352$	−389.305	−0.0589490
$353$	2896.55	0.436736	0.218368	−	0.975866i	$-0.429927\pi$
$353$	2896.55	0.436736	0.218368	+	0.975866i	$0.429927\pi$
$354$	0	0
$355$	7741.85	1.15745
$356$	3163.70	0.470999
$357$	0	0
$358$	−11468.4	−1.69308
$359$	−8951.39	−1.31598	−0.657989	−	0.753027i	$-0.728593\pi$
$359$	−8951.39	−1.31598	−0.657989	+	0.753027i	$0.728593\pi$
$360$	0	0
$361$	−4566.47	−0.665763
$362$	−13812.3	−2.00542
$363$	0	0
$364$	9729.62	1.40102
$365$	4526.01	0.649047
$366$	0	0
$367$	−4286.97	−0.609750	−0.304875	−	0.952392i	$-0.598615\pi$
$367$	−4286.97	−0.609750	−0.304875	+	0.952392i	$0.598615\pi$
$368$	−1269.85	−0.179879
$369$	0	0
$370$	−11968.0	−1.68159
$371$	11262.1	1.57601
$372$	0	0
$373$	−10467.1	−1.45299	−0.726496	−	0.687170i	$-0.758853\pi$
$373$	−10467.1	−1.45299	−0.726496	+	0.687170i	$0.758853\pi$
$374$	1640.77	0.226851
$375$	0	0
$376$	15782.1	2.16462
$377$	−6478.00	−0.884970
$378$	0	0
$379$	−12073.9	−1.63640	−0.818199	−	0.574935i	$-0.805027\pi$
$379$	−12073.9	−1.63640	−0.818199	+	0.574935i	$0.805027\pi$
$380$	−7545.80	−1.01866
$381$	0	0
$382$	−5450.34	−0.730009
$383$	−1253.27	−0.167204	−0.0836020	−	0.996499i	$-0.526642\pi$
$383$	−1253.27	−0.167204	−0.0836020	+	0.996499i	$0.526642\pi$
$384$	0	0
$385$	1402.49	0.185656
$386$	−6957.24	−0.917394
$387$	0	0
$388$	−3846.27	−0.503259
$389$	1018.96	0.132811	0.0664054	−	0.997793i	$-0.478847\pi$
$389$	1018.96	0.132811	0.0664054	+	0.997793i	$0.478847\pi$
$390$	0	0
$391$	−1501.45	−0.194198
$392$	−3359.05	−0.432800
$393$	0	0
$394$	15514.4	1.98377
$395$	−14381.9	−1.83197
$396$	0	0
$397$	1428.01	0.180528	0.0902641	−	0.995918i	$-0.471229\pi$
$397$	1428.01	0.180528	0.0902641	+	0.995918i	$0.471229\pi$
$398$	16348.5	2.05898
$399$	0	0
$400$	−765.972	−0.0957465
$401$	8068.98	1.00485	0.502426	−	0.864620i	$-0.332441\pi$
$401$	8068.98	1.00485	0.502426	+	0.864620i	$0.332441\pi$
$402$	0	0
$403$	8212.47	1.01512
$404$	16005.5	1.97105
$405$	0	0
$406$	21409.3	2.61706
$407$	1540.04	0.187560
$408$	0	0
$409$	−1259.78	−0.152303	−0.0761515	−	0.997096i	$-0.524263\pi$
$409$	−1259.78	−0.152303	−0.0761515	+	0.997096i	$0.524263\pi$
$410$	5228.31	0.629775
$411$	0	0
$412$	−27693.7	−3.31158
$413$	−2959.61	−0.352622
$414$	0	0
$415$	11710.0	1.38511
$416$	1849.16	0.217939
$417$	0	0
$418$	1483.00	0.173531
$419$	8647.46	1.00825	0.504124	−	0.863631i	$-0.331815\pi$
$419$	8647.46	1.00825	0.504124	+	0.863631i	$0.331815\pi$
$420$	0	0
$421$	1701.27	0.196947	0.0984737	−	0.995140i	$-0.468604\pi$
$421$	1701.27	0.196947	0.0984737	+	0.995140i	$0.468604\pi$
$422$	−18657.7	−2.15223
$423$	0	0
$424$	−18528.2	−2.12219
$425$	−905.672	−0.103368
$426$	0	0
$427$	−15480.2	−1.75442
$428$	68.3420	0.00771830
$429$	0	0
$430$	141.383	0.0158561
$431$	−16419.6	−1.83504	−0.917521	−	0.397687i	$-0.869813\pi$
$431$	−16419.6	−1.83504	−0.917521	+	0.397687i	$0.869813\pi$
$432$	0	0
$433$	−9528.69	−1.05755	−0.528775	−	0.848762i	$-0.677349\pi$
$433$	−9528.69	−1.05755	−0.528775	+	0.848762i	$0.677349\pi$
$434$	−27141.7	−3.00194
$435$	0	0
$436$	12684.2	1.39327
$437$	−1357.07	−0.148553
$438$	0	0
$439$	−8431.98	−0.916712	−0.458356	−	0.888769i	$-0.651562\pi$
$439$	−8431.98	−0.916712	−0.458356	+	0.888769i	$0.651562\pi$
$440$	−2307.34	−0.249996
$441$	0	0
$442$	−7793.52	−0.838688
$443$	−13933.1	−1.49432	−0.747159	−	0.664645i	$-0.768583\pi$
$443$	−13933.1	−1.49432	−0.747159	+	0.664645i	$0.768583\pi$
$444$	0	0
$445$	−2166.12	−0.230751
$446$	−21505.4	−2.28321
$447$	0	0
$448$	−13632.3	−1.43765
$449$	−4531.33	−0.476274	−0.238137	−	0.971232i	$-0.576537\pi$
$449$	−4531.33	−0.476274	−0.238137	+	0.971232i	$0.576537\pi$
$450$	0	0
$451$	−672.777	−0.0702436
$452$	14092.7	1.46652
$453$	0	0
$454$	−16618.7	−1.71796
$455$	−6661.69	−0.686384
$456$	0	0
$457$	−9924.91	−1.01590	−0.507951	−	0.861386i	$-0.669597\pi$
$457$	−9924.91	−1.01590	−0.507951	+	0.861386i	$0.669597\pi$
$458$	−12811.1	−1.30704
$459$	0	0
$460$	4466.76	0.452747
$461$	−17191.1	−1.73681	−0.868405	−	0.495856i	$-0.834854\pi$
$461$	−17191.1	−1.73681	−0.868405	+	0.495856i	$0.834854\pi$
$462$	0	0
$463$	−8456.43	−0.848820	−0.424410	−	0.905470i	$-0.639518\pi$
$463$	−8456.43	−0.848820	−0.424410	+	0.905470i	$0.639518\pi$
$464$	−9496.37	−0.950124
$465$	0	0
$466$	21824.7	2.16955
$467$	−3656.45	−0.362313	−0.181157	−	0.983454i	$-0.557984\pi$
$467$	−3656.45	−0.362313	−0.181157	+	0.983454i	$0.557984\pi$
$468$	0	0
$469$	5627.58	0.554067
$470$	−22859.7	−2.24349
$471$	0	0
$472$	4869.09	0.474827
$473$	−18.1932	−0.00176855
$474$	0	0
$475$	−818.584	−0.0790720
$476$	16864.4	1.62390
$477$	0	0
$478$	−1150.47	−0.110086
$479$	−6977.41	−0.665566	−0.332783	−	0.943003i	$-0.607988\pi$
$479$	−6977.41	−0.665566	−0.332783	+	0.943003i	$0.607988\pi$
$480$	0	0
$481$	−7315.05	−0.693425
$482$	−29353.2	−2.77386
$483$	0	0
$484$	−19565.2	−1.83745
$485$	2633.47	0.246556
$486$	0	0
$487$	−2021.37	−0.188084	−0.0940420	−	0.995568i	$-0.529979\pi$
$487$	−2021.37	−0.188084	−0.0940420	+	0.995568i	$0.529979\pi$
$488$	25467.6	2.36243
$489$	0	0
$490$	4865.45	0.448568
$491$	−2608.90	−0.239792	−0.119896	−	0.992786i	$-0.538256\pi$
$491$	−2608.90	−0.239792	−0.119896	+	0.992786i	$0.538256\pi$
$492$	0	0
$493$	−11228.3	−1.02576
$494$	−7044.11	−0.641557
$495$	0	0
$496$	12039.0	1.08985
$497$	−15638.8	−1.41146
$498$	0	0
$499$	−12569.5	−1.12763	−0.563816	−	0.825901i	$-0.690667\pi$
$499$	−12569.5	−1.12763	−0.563816	+	0.825901i	$0.690667\pi$
$500$	22393.9	2.00297
$501$	0	0
$502$	−1250.80	−0.111207
$503$	−15614.7	−1.38415	−0.692074	−	0.721827i	$-0.743303\pi$
$503$	−15614.7	−1.38415	−0.692074	+	0.721827i	$0.743303\pi$
$504$	0	0
$505$	−10958.7	−0.965655
$506$	−877.866	−0.0771263
$507$	0	0
$508$	−12164.8	−1.06245
$509$	14092.7	1.22721	0.613603	−	0.789615i	$-0.289720\pi$
$509$	14092.7	1.22721	0.613603	+	0.789615i	$0.289720\pi$
$510$	0	0
$511$	−9142.67	−0.791484
$512$	15084.1	1.30201
$513$	0	0
$514$	−25374.9	−2.17751
$515$	18961.4	1.62241
$516$	0	0
$517$	2941.58	0.250233
$518$	24175.7	2.05062
$519$	0	0
$520$	10959.7	0.924257
$521$	4352.21	0.365977	0.182988	−	0.983115i	$-0.441423\pi$
$521$	4352.21	0.365977	0.182988	+	0.983115i	$0.441423\pi$
$522$	0	0
$523$	12111.5	1.01261	0.506307	−	0.862353i	$-0.331010\pi$
$523$	12111.5	1.01261	0.506307	+	0.862353i	$0.331010\pi$
$524$	−13159.0	−1.09704
$525$	0	0
$526$	−13546.0	−1.12288
$527$	14234.7	1.17661
$528$	0	0
$529$	−11363.7	−0.933975
$530$	26837.3	2.19951
$531$	0	0
$532$	15242.7	1.24221
$533$	3195.63	0.259696
$534$	0	0
$535$	−46.7925	−0.00378134
$536$	−9258.38	−0.746084
$537$	0	0
$538$	11540.2	0.924783
$539$	−626.085	−0.0500323
$540$	0	0
$541$	−3200.22	−0.254322	−0.127161	−	0.991882i	$-0.540586\pi$
$541$	−3200.22	−0.254322	−0.127161	+	0.991882i	$0.540586\pi$
$542$	−20834.7	−1.65115
$543$	0	0
$544$	3205.17	0.252611
$545$	−8684.65	−0.682586
$546$	0	0
$547$	−9472.29	−0.740413	−0.370206	−	0.928950i	$-0.620713\pi$
$547$	−9472.29	−0.740413	−0.370206	+	0.928950i	$0.620713\pi$
$548$	−16245.3	−1.26636
$549$	0	0
$550$	−529.527	−0.0410530
$551$	−10148.6	−0.784658
$552$	0	0
$553$	29051.8	2.23401
$554$	−14648.1	−1.12335
$555$	0	0
$556$	4925.13	0.375669
$557$	20744.3	1.57804	0.789018	−	0.614370i	$-0.210590\pi$
$557$	20744.3	1.57804	0.789018	+	0.614370i	$0.210590\pi$
$558$	0	0
$559$	86.4159	0.00653846
$560$	−9765.65	−0.736918
$561$	0	0
$562$	32971.4	2.47476
$563$	16016.4	1.19895	0.599475	−	0.800393i	$-0.295376\pi$
$563$	16016.4	1.19895	0.599475	+	0.800393i	$0.295376\pi$
$564$	0	0
$565$	−9649.03	−0.718474
$566$	23313.8	1.73137
$567$	0	0
$568$	25728.6	1.90061
$569$	−14727.4	−1.08507	−0.542533	−	0.840034i	$-0.682535\pi$
$569$	−14727.4	−1.08507	−0.542533	+	0.840034i	$0.682535\pi$
$570$	0	0
$571$	−17445.4	−1.27858	−0.639290	−	0.768966i	$-0.720771\pi$
$571$	−17445.4	−1.27858	−0.639290	+	0.768966i	$0.720771\pi$
$572$	−2983.50	−0.218088
$573$	0	0
$574$	−10561.3	−0.767982
$575$	484.563	0.0351438
$576$	0	0
$577$	−12053.5	−0.869662	−0.434831	−	0.900512i	$-0.643192\pi$
$577$	−12053.5	−0.869662	−0.434831	+	0.900512i	$0.643192\pi$
$578$	10141.1	0.729780
$579$	0	0
$580$	33403.9	2.39142
$581$	−23654.6	−1.68908
$582$	0	0
$583$	−3453.42	−0.245328
$584$	15041.3	1.06578
$585$	0	0
$586$	−13145.4	−0.926671
$587$	3261.56	0.229334	0.114667	−	0.993404i	$-0.463420\pi$
$587$	3261.56	0.229334	0.114667	+	0.993404i	$0.463420\pi$
$588$	0	0
$589$	12865.9	0.900053
$590$	−7052.68	−0.492126
$591$	0	0
$592$	−10723.4	−0.744477
$593$	−12088.0	−0.837088	−0.418544	−	0.908196i	$-0.637459\pi$
$593$	−12088.0	−0.837088	−0.418544	+	0.908196i	$0.637459\pi$
$594$	0	0
$595$	−11546.7	−0.795580
$596$	−3421.67	−0.235163
$597$	0	0
$598$	4169.78	0.285142
$599$	10152.5	0.692521	0.346260	−	0.938138i	$-0.387451\pi$
$599$	10152.5	0.692521	0.346260	+	0.938138i	$0.387451\pi$
$600$	0	0
$601$	−16361.4	−1.11047	−0.555237	−	0.831692i	$-0.687372\pi$
$601$	−16361.4	−1.11047	−0.555237	+	0.831692i	$0.687372\pi$
$602$	−285.599	−0.0193358
$603$	0	0
$604$	−1316.07	−0.0886591
$605$	13395.9	0.900201
$606$	0	0
$607$	−21515.2	−1.43867	−0.719336	−	0.694662i	$-0.755554\pi$
$607$	−21515.2	−1.43867	−0.719336	+	0.694662i	$0.755554\pi$
$608$	2896.96	0.193236
$609$	0	0
$610$	−36888.8	−2.44850
$611$	−13972.2	−0.925133
$612$	0	0
$613$	−6010.19	−0.396002	−0.198001	−	0.980202i	$-0.563445\pi$
$613$	−6010.19	−0.396002	−0.198001	+	0.980202i	$0.563445\pi$
$614$	34654.2	2.27774
$615$	0	0
$616$	4660.91	0.304859
$617$	9930.86	0.647977	0.323988	−	0.946061i	$-0.394976\pi$
$617$	9930.86	0.647977	0.323988	+	0.946061i	$0.394976\pi$
$618$	0	0
$619$	20418.0	1.32580	0.662899	−	0.748709i	$-0.269326\pi$
$619$	20418.0	1.32580	0.662899	+	0.748709i	$0.269326\pi$
$620$	−42347.8	−2.74311
$621$	0	0
$622$	45886.6	2.95801
$623$	4375.64	0.281391
$624$	0	0
$625$	−13195.7	−0.844522
$626$	12782.7	0.816134
$627$	0	0
$628$	−51471.5	−3.27060
$629$	−12679.2	−0.803741
$630$	0	0
$631$	16967.0	1.07044	0.535220	−	0.844713i	$-0.320229\pi$
$631$	16967.0	1.07044	0.535220	+	0.844713i	$0.320229\pi$
$632$	−47795.4	−3.00823
$633$	0	0
$634$	−26819.1	−1.68000
$635$	8329.01	0.520515
$636$	0	0
$637$	2973.84	0.184973
$638$	−6564.97	−0.407382
$639$	0	0
$640$	−27457.5	−1.69586
$641$	−9876.01	−0.608548	−0.304274	−	0.952585i	$-0.598414\pi$
$641$	−9876.01	−0.608548	−0.304274	+	0.952585i	$0.598414\pi$
$642$	0	0
$643$	13864.9	0.850354	0.425177	−	0.905110i	$-0.360212\pi$
$643$	13864.9	0.850354	0.425177	+	0.905110i	$0.360212\pi$
$644$	−9022.99	−0.552105
$645$	0	0
$646$	−12209.6	−0.743622
$647$	27291.6	1.65834	0.829168	−	0.558999i	$-0.188814\pi$
$647$	27291.6	1.65834	0.829168	+	0.558999i	$0.188814\pi$
$648$	0	0
$649$	907.538	0.0548906
$650$	2515.21	0.151776
$651$	0	0
$652$	29508.2	1.77244
$653$	7450.48	0.446493	0.223246	−	0.974762i	$-0.428335\pi$
$653$	7450.48	0.446493	0.223246	+	0.974762i	$0.428335\pi$
$654$	0	0
$655$	9009.69	0.537462
$656$	4684.61	0.278816
$657$	0	0
$658$	46177.3	2.73583
$659$	−17932.7	−1.06003	−0.530015	−	0.847988i	$-0.677814\pi$
$659$	−17932.7	−1.06003	−0.530015	+	0.847988i	$0.677814\pi$
$660$	0	0
$661$	24199.3	1.42397	0.711985	−	0.702194i	$-0.247796\pi$
$661$	24199.3	1.42397	0.711985	+	0.702194i	$0.247796\pi$
$662$	30269.4	1.77712
$663$	0	0
$664$	38916.0	2.27445
$665$	−10436.4	−0.608582
$666$	0	0
$667$	6007.52	0.348744
$668$	12348.4	0.715228
$669$	0	0
$670$	13410.4	0.773266
$671$	4746.84	0.273100
$672$	0	0
$673$	−24900.0	−1.42619	−0.713094	−	0.701068i	$-0.752707\pi$
$673$	−24900.0	−1.42619	−0.713094	+	0.701068i	$0.752707\pi$
$674$	−21858.3	−1.24919
$675$	0	0
$676$	−19160.5	−1.09015
$677$	−7225.62	−0.410196	−0.205098	−	0.978741i	$-0.565751\pi$
$677$	−7225.62	−0.410196	−0.205098	+	0.978741i	$0.565751\pi$
$678$	0	0
$679$	−5319.69	−0.300664
$680$	18996.5	1.07130
$681$	0	0
$682$	8322.74	0.467293
$683$	8050.84	0.451035	0.225517	−	0.974239i	$-0.427593\pi$
$683$	8050.84	0.451035	0.225517	+	0.974239i	$0.427593\pi$
$684$	0	0
$685$	11122.9	0.620414
$686$	24817.2	1.38123
$687$	0	0
$688$	126.681	0.00701984
$689$	16403.4	0.906998
$690$	0	0
$691$	1943.64	0.107003	0.0535017	−	0.998568i	$-0.482962\pi$
$691$	1943.64	0.107003	0.0535017	+	0.998568i	$0.482962\pi$
$692$	20153.7	1.10712
$693$	0	0
$694$	−41388.1	−2.26379
$695$	−3372.14	−0.184047
$696$	0	0
$697$	5539.00	0.301011
$698$	6313.86	0.342383
$699$	0	0
$700$	−5442.65	−0.293876
$701$	−12464.4	−0.671573	−0.335786	−	0.941938i	$-0.609002\pi$
$701$	−12464.4	−0.671573	−0.335786	+	0.941938i	$0.609002\pi$
$702$	0	0
$703$	−11460.0	−0.614825
$704$	4180.22	0.223790
$705$	0	0
$706$	−13943.1	−0.743278
$707$	22136.9	1.17757
$708$	0	0
$709$	8838.71	0.468187	0.234094	−	0.972214i	$-0.424788\pi$
$709$	8838.71	0.468187	0.234094	+	0.972214i	$0.424788\pi$
$710$	−37266.8	−1.96986
$711$	0	0
$712$	−7198.71	−0.378909
$713$	−7616.03	−0.400032
$714$	0	0
$715$	2042.75	0.106845
$716$	36145.4	1.88662
$717$	0	0
$718$	43089.1	2.23966
$719$	−20574.9	−1.06720	−0.533598	−	0.845738i	$-0.679161\pi$
$719$	−20574.9	−1.06720	−0.533598	+	0.845738i	$0.679161\pi$
$720$	0	0
$721$	−38302.6	−1.97845
$722$	21981.5	1.13306
$723$	0	0
$724$	43533.1	2.23466
$725$	3623.73	0.185630
$726$	0	0
$727$	−10363.9	−0.528713	−0.264357	−	0.964425i	$-0.585160\pi$
$727$	−10363.9	−0.528713	−0.264357	+	0.964425i	$0.585160\pi$
$728$	−22138.9	−1.12709
$729$	0	0
$730$	−21786.8	−1.10461
$731$	149.785	0.00757866
$732$	0	0
$733$	−12139.1	−0.611689	−0.305845	−	0.952081i	$-0.598939\pi$
$733$	−12139.1	−0.611689	−0.305845	+	0.952081i	$0.598939\pi$
$734$	20636.1	1.03773
$735$	0	0
$736$	−1714.87	−0.0858842
$737$	−1725.65	−0.0862482
$738$	0	0
$739$	−31589.1	−1.57243	−0.786213	−	0.617956i	$-0.787961\pi$
$739$	−31589.1	−1.57243	−0.786213	+	0.617956i	$0.787961\pi$
$740$	37720.2	1.87381
$741$	0	0
$742$	−54212.3	−2.68220
$743$	21829.4	1.07785	0.538925	−	0.842353i	$-0.318830\pi$
$743$	21829.4	1.07785	0.538925	+	0.842353i	$0.318830\pi$
$744$	0	0
$745$	2342.76	0.115211
$746$	50385.3	2.47284
$747$	0	0
$748$	−5171.31	−0.252783
$749$	94.5223	0.00461117
$750$	0	0
$751$	1454.81	0.0706883	0.0353441	−	0.999375i	$-0.488747\pi$
$751$	1454.81	0.0706883	0.0353441	+	0.999375i	$0.488747\pi$
$752$	−20482.5	−0.993244
$753$	0	0
$754$	31183.0	1.50613
$755$	901.089	0.0434357
$756$	0	0
$757$	−22583.2	−1.08428	−0.542140	−	0.840288i	$-0.682386\pi$
$757$	−22583.2	−1.08428	−0.542140	+	0.840288i	$0.682386\pi$
$758$	58119.9	2.78498
$759$	0	0
$760$	17169.8	0.819491
$761$	6221.15	0.296342	0.148171	−	0.988962i	$-0.452661\pi$
$761$	6221.15	0.296342	0.148171	+	0.988962i	$0.452661\pi$
$762$	0	0
$763$	17543.3	0.832384
$764$	17178.1	0.813459
$765$	0	0
$766$	6032.85	0.284563
$767$	−4310.72	−0.202935
$768$	0	0
$769$	12916.0	0.605675	0.302838	−	0.953042i	$-0.402066\pi$
$769$	12916.0	0.605675	0.302838	+	0.953042i	$0.402066\pi$
$770$	−6751.14	−0.315966
$771$	0	0
$772$	21927.5	1.02226
$773$	22276.4	1.03652	0.518258	−	0.855224i	$-0.326581\pi$
$773$	22276.4	1.03652	0.518258	+	0.855224i	$0.326581\pi$
$774$	0	0
$775$	−4593.98	−0.212930
$776$	8751.84	0.404862
$777$	0	0
$778$	−4904.95	−0.226030
$779$	5006.38	0.230259
$780$	0	0
$781$	4795.48	0.219713
$782$	7227.50	0.330505
$783$	0	0
$784$	4359.48	0.198592
$785$	35241.6	1.60233
$786$	0	0
$787$	−8703.10	−0.394196	−0.197098	−	0.980384i	$-0.563152\pi$
$787$	−8703.10	−0.394196	−0.197098	+	0.980384i	$0.563152\pi$
$788$	−48897.5	−2.21053
$789$	0	0
$790$	69229.7	3.11782
$791$	19491.3	0.876147
$792$	0	0
$793$	−22547.1	−1.00967
$794$	−6873.97	−0.307240
$795$	0	0
$796$	−51526.3	−2.29435
$797$	33765.7	1.50068	0.750340	−	0.661052i	$-0.229890\pi$
$797$	33765.7	1.50068	0.750340	+	0.661052i	$0.229890\pi$
$798$	0	0
$799$	−24218.1	−1.07231
$800$	−1034.40	−0.0457146
$801$	0	0
$802$	−38841.5	−1.71015
$803$	2803.52	0.123205
$804$	0	0
$805$	6177.87	0.270486
$806$	−39532.2	−1.72762
$807$	0	0
$808$	−36419.2	−1.58567
$809$	24593.7	1.06881	0.534406	−	0.845228i	$-0.320536\pi$
$809$	24593.7	1.06881	0.534406	+	0.845228i	$0.320536\pi$
$810$	0	0
$811$	−34472.3	−1.49258	−0.746292	−	0.665619i	$-0.768168\pi$
$811$	−34472.3	−1.49258	−0.746292	+	0.665619i	$0.768168\pi$
$812$	−67476.9	−2.91623
$813$	0	0
$814$	−7413.26	−0.319207
$815$	−20203.7	−0.868349
$816$	0	0
$817$	135.382	0.00579732
$818$	6064.16	0.259203
$819$	0	0
$820$	−16478.3	−0.701766
$821$	7721.34	0.328230	0.164115	−	0.986441i	$-0.447523\pi$
$821$	7721.34	0.328230	0.164115	+	0.986441i	$0.447523\pi$
$822$	0	0
$823$	4013.26	0.169980	0.0849900	−	0.996382i	$-0.472914\pi$
$823$	4013.26	0.169980	0.0849900	+	0.996382i	$0.472914\pi$
$824$	63014.6	2.66410
$825$	0	0
$826$	14246.6	0.600126
$827$	−45918.0	−1.93074	−0.965372	−	0.260875i	$-0.915989\pi$
$827$	−45918.0	−1.93074	−0.965372	+	0.260875i	$0.915989\pi$
$828$	0	0
$829$	11020.5	0.461709	0.230854	−	0.972988i	$-0.425848\pi$
$829$	11020.5	0.461709	0.230854	+	0.972988i	$0.425848\pi$
$830$	−56368.3	−2.35731
$831$	0	0
$832$	−19855.7	−0.827370
$833$	5154.58	0.214400
$834$	0	0
$835$	−8454.69	−0.350403
$836$	−4674.04	−0.193367
$837$	0	0
$838$	−41626.1	−1.71593
$839$	−22387.3	−0.921209	−0.460604	−	0.887606i	$-0.652367\pi$
$839$	−22387.3	−0.921209	−0.460604	+	0.887606i	$0.652367\pi$
$840$	0	0
$841$	20537.2	0.842070
$842$	−8189.37	−0.335183
$843$	0	0
$844$	58804.4	2.39826
$845$	13118.8	0.534085
$846$	0	0
$847$	−27060.2	−1.09776
$848$	24046.5	0.973773
$849$	0	0
$850$	4359.62	0.175922
$851$	6783.78	0.273261
$852$	0	0
$853$	5881.92	0.236100	0.118050	−	0.993008i	$-0.462336\pi$
$853$	5881.92	0.236100	0.118050	+	0.993008i	$0.462336\pi$
$854$	74516.5	2.98583
$855$	0	0
$856$	−155.506	−0.00620921
$857$	−528.358	−0.0210599	−0.0105300	−	0.999945i	$-0.503352\pi$
$857$	−528.358	−0.0210599	−0.0105300	+	0.999945i	$0.503352\pi$
$858$	0	0
$859$	10702.6	0.425110	0.212555	−	0.977149i	$-0.431822\pi$
$859$	10702.6	0.425110	0.212555	+	0.977149i	$0.431822\pi$
$860$	−445.605	−0.0176686
$861$	0	0
$862$	79038.6	3.12305
$863$	45440.5	1.79237	0.896184	−	0.443683i	$-0.146329\pi$
$863$	45440.5	1.79237	0.896184	+	0.443683i	$0.146329\pi$
$864$	0	0
$865$	−13798.8	−0.542398
$866$	45868.1	1.79984
$867$	0	0
$868$	85543.8	3.34510
$869$	−8908.46	−0.347755
$870$	0	0
$871$	8196.65	0.318867
$872$	−28861.8	−1.12085
$873$	0	0
$874$	6532.51	0.252821
$875$	30972.6	1.19664
$876$	0	0
$877$	18473.9	0.711312	0.355656	−	0.934617i	$-0.384258\pi$
$877$	18473.9	0.711312	0.355656	+	0.934617i	$0.384258\pi$
$878$	40588.9	1.56015
$879$	0	0
$880$	2994.55	0.114711
$881$	26356.6	1.00792	0.503959	−	0.863728i	$-0.331876\pi$
$881$	26356.6	1.00792	0.503959	+	0.863728i	$0.331876\pi$
$882$	0	0
$883$	−17274.3	−0.658352	−0.329176	−	0.944269i	$-0.606771\pi$
$883$	−17274.3	−0.658352	−0.329176	+	0.944269i	$0.606771\pi$
$884$	24563.2	0.934560
$885$	0	0
$886$	67069.6	2.54317
$887$	−46241.1	−1.75042	−0.875211	−	0.483741i	$-0.839278\pi$
$887$	−46241.1	−1.75042	−0.875211	+	0.483741i	$0.839278\pi$
$888$	0	0
$889$	−16824.9	−0.634745
$890$	10427.0	0.392713
$891$	0	0
$892$	67779.8	2.54421
$893$	−21889.4	−0.820268
$894$	0	0
$895$	−24748.1	−0.924288
$896$	55465.0	2.06803
$897$	0	0
$898$	21812.4	0.810567
$899$	−56955.2	−2.11297
$900$	0	0
$901$	28432.2	1.05129
$902$	3238.54	0.119547
$903$	0	0
$904$	−32066.7	−1.17978
$905$	−29806.3	−1.09480
$906$	0	0
$907$	27640.6	1.01190	0.505949	−	0.862563i	$-0.331142\pi$
$907$	27640.6	1.01190	0.505949	+	0.862563i	$0.331142\pi$
$908$	52378.0	1.91435
$909$	0	0
$910$	32067.3	1.16815
$911$	−32728.6	−1.19028	−0.595141	−	0.803621i	$-0.702904\pi$
$911$	−32728.6	−1.19028	−0.595141	+	0.803621i	$0.702904\pi$
$912$	0	0
$913$	7253.46	0.262929
$914$	47775.3	1.72896
$915$	0	0
$916$	40377.5	1.45645
$917$	−18199.9	−0.655411
$918$	0	0
$919$	35550.5	1.27606	0.638032	−	0.770010i	$-0.279749\pi$
$919$	35550.5	1.27606	0.638032	+	0.770010i	$0.279749\pi$
$920$	−10163.7	−0.364225
$921$	0	0
$922$	82752.5	2.95586
$923$	−22778.1	−0.812297
$924$	0	0
$925$	4091.97	0.145452
$926$	40706.5	1.44460
$927$	0	0
$928$	−12824.3	−0.453642
$929$	17878.9	0.631417	0.315708	−	0.948856i	$-0.397758\pi$
$929$	17878.9	0.631417	0.315708	+	0.948856i	$0.397758\pi$
$930$	0	0
$931$	4658.92	0.164006
$932$	−68785.9	−2.41755
$933$	0	0
$934$	17601.0	0.616619
$935$	3540.70	0.123843
$936$	0	0
$937$	14019.2	0.488782	0.244391	−	0.969677i	$-0.421412\pi$
$937$	14019.2	0.488782	0.244391	+	0.969677i	$0.421412\pi$
$938$	−27089.4	−0.942963
$939$	0	0
$940$	72048.1	2.49995
$941$	24290.5	0.841496	0.420748	−	0.907178i	$-0.361768\pi$
$941$	24290.5	0.841496	0.420748	+	0.907178i	$0.361768\pi$
$942$	0	0
$943$	−2963.54	−0.102340
$944$	−6319.26	−0.217876
$945$	0	0
$946$	87.5761	0.00300988
$947$	−37775.8	−1.29625	−0.648125	−	0.761534i	$-0.724447\pi$
$947$	−37775.8	−1.29625	−0.648125	+	0.761534i	$0.724447\pi$
$948$	0	0
$949$	−13316.4	−0.455500
$950$	3940.40	0.134572
$951$	0	0
$952$	−38373.4	−1.30640
$953$	9204.65	0.312873	0.156436	−	0.987688i	$-0.449999\pi$
$953$	9204.65	0.312873	0.156436	+	0.987688i	$0.449999\pi$
$954$	0	0
$955$	−11761.5	−0.398528
$956$	3626.00	0.122671
$957$	0	0
$958$	33587.0	1.13272
$959$	−22468.6	−0.756567
$960$	0	0
$961$	42413.9	1.42371
$962$	35212.3	1.18014
$963$	0	0
$964$	92514.1	3.09095
$965$	−15013.3	−0.500825
$966$	0	0
$967$	−25672.1	−0.853734	−0.426867	−	0.904314i	$-0.640383\pi$
$967$	−25672.1	−0.853734	−0.426867	+	0.904314i	$0.640383\pi$
$968$	44518.8	1.47819
$969$	0	0
$970$	−12676.7	−0.419612
$971$	25953.9	0.857775	0.428888	−	0.903358i	$-0.358906\pi$
$971$	25953.9	0.857775	0.428888	+	0.903358i	$0.358906\pi$
$972$	0	0
$973$	6811.83	0.224437
$974$	9730.22	0.320099
$975$	0	0
$976$	−33052.7	−1.08401
$977$	−8420.89	−0.275750	−0.137875	−	0.990450i	$-0.544027\pi$
$977$	−8420.89	−0.275750	−0.137875	+	0.990450i	$0.544027\pi$
$978$	0	0
$979$	−1341.75	−0.0438023
$980$	−15334.7	−0.499845
$981$	0	0
$982$	12558.4	0.408100
$983$	−52216.7	−1.69426	−0.847128	−	0.531389i	$-0.821670\pi$
$983$	−52216.7	−1.69426	−0.847128	+	0.531389i	$0.821670\pi$
$984$	0	0
$985$	33479.2	1.08298
$986$	54049.7	1.74573
$987$	0	0
$988$	22201.3	0.714895
$989$	−80.1398	−0.00257664
$990$	0	0
$991$	−34385.5	−1.10221	−0.551105	−	0.834436i	$-0.685794\pi$
$991$	−34385.5	−1.10221	−0.551105	+	0.834436i	$0.685794\pi$
$992$	16258.0	0.520356
$993$	0	0
$994$	75280.1	2.40215
$995$	35279.1	1.12404
$996$	0	0
$997$	37442.9	1.18940	0.594698	−	0.803949i	$-0.297272\pi$
$997$	37442.9	1.18940	0.594698	+	0.803949i	$0.297272\pi$
$998$	60505.5	1.91911
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.81368 q^{2} +15.1715 q^{4} -10.3877 q^{5} +20.9834 q^{7} -34.5214 q^{8} +O(q^{10})\)	\(q-4.81368 q^{2} +15.1715 q^{4} -10.3877 q^{5} +20.9834 q^{7} -34.5214 q^{8} +50.0029 q^{10} -6.43436 q^{11} +30.5626 q^{13} -101.007 q^{14} +44.8030 q^{16} +52.9744 q^{17} +47.8804 q^{19} -157.597 q^{20} +30.9730 q^{22} -28.3430 q^{23} -17.0964 q^{25} -147.119 q^{26} +318.350 q^{28} -211.958 q^{29} +268.710 q^{31} +60.5041 q^{32} -255.002 q^{34} -217.969 q^{35} -239.346 q^{37} -230.481 q^{38} +358.597 q^{40} +104.560 q^{41} +2.82750 q^{43} -97.6191 q^{44} +136.434 q^{46} -457.167 q^{47} +97.3033 q^{49} +82.2968 q^{50} +463.682 q^{52} +536.716 q^{53} +66.8380 q^{55} -724.378 q^{56} +1020.30 q^{58} -141.045 q^{59} -737.733 q^{61} -1293.48 q^{62} -649.672 q^{64} -317.474 q^{65} +268.192 q^{67} +803.702 q^{68} +1049.23 q^{70} -745.293 q^{71} -435.710 q^{73} +1152.14 q^{74} +726.419 q^{76} -135.015 q^{77} +1384.51 q^{79} -465.399 q^{80} -503.319 q^{82} -1127.30 q^{83} -550.280 q^{85} -13.6107 q^{86} +222.124 q^{88} +208.529 q^{89} +641.308 q^{91} -430.006 q^{92} +2200.66 q^{94} -497.366 q^{95} -253.519 q^{97} -468.387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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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