LMFDB - Embedded newform 2151.4.a.d.1.12

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.25922 q^{2} -2.89594 q^{4} -14.4181 q^{5} +23.8295 q^{7} +24.6163 q^{8} +O(q^{10})$	\(q-2.25922 q^{2} -2.89594 q^{4} -14.4181 q^{5} +23.8295 q^{7} +24.6163 q^{8} +32.5737 q^{10} +48.0836 q^{11} +36.7224 q^{13} -53.8359 q^{14} -32.4460 q^{16} -119.463 q^{17} +5.19174 q^{19} +41.7541 q^{20} -108.631 q^{22} -27.4037 q^{23} +82.8829 q^{25} -82.9639 q^{26} -69.0086 q^{28} -59.8779 q^{29} -48.1413 q^{31} -123.628 q^{32} +269.894 q^{34} -343.576 q^{35} +84.8700 q^{37} -11.7293 q^{38} -354.921 q^{40} +136.581 q^{41} +118.792 q^{43} -139.247 q^{44} +61.9109 q^{46} -169.492 q^{47} +224.843 q^{49} -187.250 q^{50} -106.346 q^{52} -239.539 q^{53} -693.276 q^{55} +586.593 q^{56} +135.277 q^{58} -38.1913 q^{59} -800.350 q^{61} +108.762 q^{62} +538.870 q^{64} -529.469 q^{65} +614.366 q^{67} +345.959 q^{68} +776.214 q^{70} +478.314 q^{71} -869.565 q^{73} -191.740 q^{74} -15.0350 q^{76} +1145.81 q^{77} +618.757 q^{79} +467.811 q^{80} -308.566 q^{82} -693.854 q^{83} +1722.44 q^{85} -268.377 q^{86} +1183.64 q^{88} -411.081 q^{89} +875.075 q^{91} +79.3595 q^{92} +382.919 q^{94} -74.8553 q^{95} +467.113 q^{97} -507.969 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * 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 $

$2$	−2.25922	−0.798754	−0.399377	−	0.916787i	$-0.630774\pi$
$2$	−2.25922	−0.798754	−0.399377	+	0.916787i	$0.630774\pi$
$3$	0	0
$3$	0	0
$4$	−2.89594	−0.361992
$5$	−14.4181	−1.28960	−0.644799	−	0.764352i	$-0.723059\pi$
$5$	−14.4181	−1.28960	−0.644799	+	0.764352i	$0.723059\pi$
$6$	0	0
$7$	23.8295	1.28667	0.643335	−	0.765585i	$-0.277550\pi$
$7$	23.8295	1.28667	0.643335	+	0.765585i	$0.277550\pi$
$8$	24.6163	1.08790
$9$	0	0
$10$	32.5737	1.03007
$11$	48.0836	1.31798	0.658989	−	0.752153i	$-0.270985\pi$
$11$	48.0836	1.31798	0.658989	+	0.752153i	$0.270985\pi$
$12$	0	0
$13$	36.7224	0.783459	0.391729	−	0.920080i	$-0.371877\pi$
$13$	36.7224	0.783459	0.391729	+	0.920080i	$0.371877\pi$
$14$	−53.8359	−1.02773
$15$	0	0
$16$	−32.4460	−0.506969
$17$	−119.463	−1.70436	−0.852180	−	0.523249i	$-0.824720\pi$
$17$	−119.463	−1.70436	−0.852180	+	0.523249i	$0.824720\pi$
$18$	0	0
$19$	5.19174	0.0626877	0.0313439	−	0.999509i	$-0.490021\pi$
$19$	5.19174	0.0626877	0.0313439	+	0.999509i	$0.490021\pi$
$20$	41.7541	0.466825
$21$	0	0
$22$	−108.631	−1.05274
$23$	−27.4037	−0.248438	−0.124219	−	0.992255i	$-0.539643\pi$
$23$	−27.4037	−0.248438	−0.124219	+	0.992255i	$0.539643\pi$
$24$	0	0
$25$	82.8829	0.663063
$26$	−82.9639	−0.625791
$27$	0	0
$28$	−69.0086	−0.465765
$29$	−59.8779	−0.383415	−0.191708	−	0.981452i	$-0.561403\pi$
$29$	−59.8779	−0.383415	−0.191708	+	0.981452i	$0.561403\pi$
$30$	0	0
$31$	−48.1413	−0.278917	−0.139458	−	0.990228i	$-0.544536\pi$
$31$	−48.1413	−0.278917	−0.139458	+	0.990228i	$0.544536\pi$
$32$	−123.628	−0.682953
$33$	0	0
$34$	269.894	1.36136
$35$	−343.576	−1.65929
$36$	0	0
$37$	84.8700	0.377096	0.188548	−	0.982064i	$-0.439622\pi$
$37$	84.8700	0.377096	0.188548	+	0.982064i	$0.439622\pi$
$38$	−11.7293	−0.0500721
$39$	0	0
$40$	−354.921	−1.40295
$41$	136.581	0.520253	0.260127	−	0.965574i	$-0.416236\pi$
$41$	136.581	0.520253	0.260127	+	0.965574i	$0.416236\pi$
$42$	0	0
$43$	118.792	0.421293	0.210647	−	0.977562i	$-0.432443\pi$
$43$	118.792	0.421293	0.210647	+	0.977562i	$0.432443\pi$
$44$	−139.247	−0.477098
$45$	0	0
$46$	61.9109	0.198441
$47$	−169.492	−0.526020	−0.263010	−	0.964793i	$-0.584715\pi$
$47$	−169.492	−0.526020	−0.263010	+	0.964793i	$0.584715\pi$
$48$	0	0
$49$	224.843	0.655518
$50$	−187.250	−0.529624
$51$	0	0
$52$	−106.346	−0.283606
$53$	−239.539	−0.620815	−0.310408	−	0.950604i	$-0.600466\pi$
$53$	−239.539	−0.620815	−0.310408	+	0.950604i	$0.600466\pi$
$54$	0	0
$55$	−693.276	−1.69966
$56$	586.593	1.39976
$57$	0	0
$58$	135.277	0.306254
$59$	−38.1913	−0.0842726	−0.0421363	−	0.999112i	$-0.513416\pi$
$59$	−38.1913	−0.0842726	−0.0421363	+	0.999112i	$0.513416\pi$
$60$	0	0
$61$	−800.350	−1.67991	−0.839953	−	0.542659i	$-0.817418\pi$
$61$	−800.350	−1.67991	−0.839953	+	0.542659i	$0.817418\pi$
$62$	108.762	0.222786
$63$	0	0
$64$	538.870	1.05248
$65$	−529.469	−1.01035
$66$	0	0
$67$	614.366	1.12025	0.560125	−	0.828408i	$-0.310753\pi$
$67$	614.366	1.12025	0.560125	+	0.828408i	$0.310753\pi$
$68$	345.959	0.616965
$69$	0	0
$70$	776.214	1.32536
$71$	478.314	0.799514	0.399757	−	0.916621i	$-0.369095\pi$
$71$	478.314	0.799514	0.399757	+	0.916621i	$0.369095\pi$
$72$	0	0
$73$	−869.565	−1.39418	−0.697088	−	0.716985i	$-0.745521\pi$
$73$	−869.565	−1.39418	−0.697088	+	0.716985i	$0.745521\pi$
$74$	−191.740	−0.301207
$75$	0	0
$76$	−15.0350	−0.0226925
$77$	1145.81	1.69580
$78$	0	0
$79$	618.757	0.881210	0.440605	−	0.897701i	$-0.354764\pi$
$79$	618.757	0.881210	0.440605	+	0.897701i	$0.354764\pi$
$80$	467.811	0.653786
$81$	0	0
$82$	−308.566	−0.415554
$83$	−693.854	−0.917595	−0.458797	−	0.888541i	$-0.651720\pi$
$83$	−693.854	−0.917595	−0.458797	+	0.888541i	$0.651720\pi$
$84$	0	0
$85$	1722.44	2.19794
$86$	−268.377	−0.336510
$87$	0	0
$88$	1183.64	1.43382
$89$	−411.081	−0.489601	−0.244801	−	0.969573i	$-0.578722\pi$
$89$	−411.081	−0.489601	−0.244801	+	0.969573i	$0.578722\pi$
$90$	0	0
$91$	875.075	1.00805
$92$	79.3595	0.0899326
$93$	0	0
$94$	382.919	0.420160
$95$	−74.8553	−0.0808420
$96$	0	0
$97$	467.113	0.488950	0.244475	−	0.969656i	$-0.421384\pi$
$97$	467.113	0.488950	0.244475	+	0.969656i	$0.421384\pi$
$98$	−507.969	−0.523598
$99$	0	0
$100$	−240.024	−0.240024
$101$	8.71699	0.00858785	0.00429392	−	0.999991i	$-0.498633\pi$
$101$	8.71699	0.00858785	0.00429392	+	0.999991i	$0.498633\pi$
$102$	0	0
$103$	1602.71	1.53320	0.766600	−	0.642124i	$-0.221947\pi$
$103$	1602.71	1.53320	0.766600	+	0.642124i	$0.221947\pi$
$104$	903.970	0.852322
$105$	0	0
$106$	541.170	0.495878
$107$	399.920	0.361324	0.180662	−	0.983545i	$-0.442176\pi$
$107$	399.920	0.361324	0.180662	+	0.983545i	$0.442176\pi$
$108$	0	0
$109$	−824.317	−0.724360	−0.362180	−	0.932108i	$-0.617967\pi$
$109$	−824.317	−0.724360	−0.362180	+	0.932108i	$0.617967\pi$
$110$	1566.26	1.35761
$111$	0	0
$112$	−773.171	−0.652302
$113$	997.550	0.830457	0.415228	−	0.909717i	$-0.363702\pi$
$113$	997.550	0.830457	0.415228	+	0.909717i	$0.363702\pi$
$114$	0	0
$115$	395.111	0.320385
$116$	173.403	0.138793
$117$	0	0
$118$	86.2825	0.0673131
$119$	−2846.75	−2.19295
$120$	0	0
$121$	981.032	0.737064
$122$	1808.16	1.34183
$123$	0	0
$124$	139.414	0.100966
$125$	607.251	0.434513
$126$	0	0
$127$	1664.10	1.16272	0.581359	−	0.813647i	$-0.302521\pi$
$127$	1664.10	1.16272	0.581359	+	0.813647i	$0.302521\pi$
$128$	−228.403	−0.157720
$129$	0	0
$130$	1196.19	0.807018
$131$	−546.192	−0.364283	−0.182141	−	0.983272i	$-0.558303\pi$
$131$	−546.192	−0.364283	−0.182141	+	0.983272i	$0.558303\pi$
$132$	0	0
$133$	123.716	0.0806584
$134$	−1387.99	−0.894804
$135$	0	0
$136$	−2940.74	−1.85417
$137$	−94.0942	−0.0586789	−0.0293395	−	0.999570i	$-0.509340\pi$
$137$	−94.0942	−0.0586789	−0.0293395	+	0.999570i	$0.509340\pi$
$138$	0	0
$139$	540.666	0.329919	0.164959	−	0.986300i	$-0.447251\pi$
$139$	540.666	0.329919	0.164959	+	0.986300i	$0.447251\pi$
$140$	994.976	0.600649
$141$	0	0
$142$	−1080.62	−0.638615
$143$	1765.75	1.03258
$144$	0	0
$145$	863.328	0.494452
$146$	1964.54	1.11360
$147$	0	0
$148$	−245.778	−0.136506
$149$	3063.17	1.68419	0.842096	−	0.539328i	$-0.181322\pi$
$149$	3063.17	1.68419	0.842096	+	0.539328i	$0.181322\pi$
$150$	0	0
$151$	−905.549	−0.488030	−0.244015	−	0.969771i	$-0.578465\pi$
$151$	−905.549	−0.488030	−0.244015	+	0.969771i	$0.578465\pi$
$152$	127.801	0.0681978
$153$	0	0
$154$	−2588.62	−1.35453
$155$	694.108	0.359691
$156$	0	0
$157$	−3312.46	−1.68384	−0.841921	−	0.539601i	$-0.818575\pi$
$157$	−3312.46	−1.68384	−0.841921	+	0.539601i	$0.818575\pi$
$158$	−1397.91	−0.703870
$159$	0	0
$160$	1782.48	0.880735
$161$	−653.016	−0.319657
$162$	0	0
$163$	−381.186	−0.183170	−0.0915852	−	0.995797i	$-0.529193\pi$
$163$	−381.186	−0.183170	−0.0915852	+	0.995797i	$0.529193\pi$
$164$	−395.531	−0.188328
$165$	0	0
$166$	1567.57	0.732932
$167$	−1780.12	−0.824850	−0.412425	−	0.910991i	$-0.635318\pi$
$167$	−1780.12	−0.824850	−0.412425	+	0.910991i	$0.635318\pi$
$168$	0	0
$169$	−848.464	−0.386192
$170$	−3891.37	−1.75561
$171$	0	0
$172$	−344.014	−0.152505
$173$	−3148.36	−1.38362	−0.691808	−	0.722081i	$-0.743186\pi$
$173$	−3148.36	−1.38362	−0.691808	+	0.722081i	$0.743186\pi$
$174$	0	0
$175$	1975.05	0.853143
$176$	−1560.12	−0.668174
$177$	0	0
$178$	928.721	0.391071
$179$	920.992	0.384571	0.192285	−	0.981339i	$-0.438410\pi$
$179$	920.992	0.384571	0.192285	+	0.981339i	$0.438410\pi$
$180$	0	0
$181$	−2570.28	−1.05551	−0.527755	−	0.849397i	$-0.676966\pi$
$181$	−2570.28	−1.05551	−0.527755	+	0.849397i	$0.676966\pi$
$182$	−1976.98	−0.805186
$183$	0	0
$184$	−674.578	−0.270275
$185$	−1223.67	−0.486302
$186$	0	0
$187$	−5744.23	−2.24631
$188$	490.838	0.190415
$189$	0	0
$190$	169.114	0.0645728
$191$	−520.764	−0.197283	−0.0986417	−	0.995123i	$-0.531450\pi$
$191$	−520.764	−0.197283	−0.0986417	+	0.995123i	$0.531450\pi$
$192$	0	0
$193$	−2831.16	−1.05591	−0.527957	−	0.849271i	$-0.677042\pi$
$193$	−2831.16	−1.05591	−0.527957	+	0.849271i	$0.677042\pi$
$194$	−1055.31	−0.390550
$195$	0	0
$196$	−651.131	−0.237293
$197$	3835.56	1.38717	0.693584	−	0.720376i	$-0.256031\pi$
$197$	3835.56	1.38717	0.693584	+	0.720376i	$0.256031\pi$
$198$	0	0
$199$	−1122.39	−0.399820	−0.199910	−	0.979814i	$-0.564065\pi$
$199$	−1122.39	−0.399820	−0.199910	+	0.979814i	$0.564065\pi$
$200$	2040.27	0.721344
$201$	0	0
$202$	−19.6936	−0.00685958
$203$	−1426.86	−0.493329
$204$	0	0
$205$	−1969.25	−0.670918
$206$	−3620.87	−1.22465
$207$	0	0
$208$	−1191.50	−0.397189
$209$	249.638	0.0826210
$210$	0	0
$211$	−1895.16	−0.618333	−0.309166	−	0.951008i	$-0.600050\pi$
$211$	−1895.16	−0.618333	−0.309166	+	0.951008i	$0.600050\pi$
$212$	693.690	0.224730
$213$	0	0
$214$	−903.506	−0.288609
$215$	−1712.76	−0.543299
$216$	0	0
$217$	−1147.18	−0.358874
$218$	1862.31	0.578585
$219$	0	0
$220$	2007.69	0.615264
$221$	−4386.98	−1.33530
$222$	0	0
$223$	−4002.58	−1.20194	−0.600969	−	0.799272i	$-0.705219\pi$
$223$	−4002.58	−1.20194	−0.600969	+	0.799272i	$0.705219\pi$
$224$	−2945.98	−0.878735
$225$	0	0
$226$	−2253.68	−0.663330
$227$	−1756.01	−0.513437	−0.256718	−	0.966486i	$-0.582641\pi$
$227$	−1756.01	−0.513437	−0.256718	+	0.966486i	$0.582641\pi$
$228$	0	0
$229$	2499.83	0.721369	0.360684	−	0.932688i	$-0.382543\pi$
$229$	2499.83	0.721369	0.360684	+	0.932688i	$0.382543\pi$
$230$	−892.641	−0.255909
$231$	0	0
$232$	−1473.97	−0.417116
$233$	5752.08	1.61730	0.808650	−	0.588290i	$-0.200198\pi$
$233$	5752.08	1.61730	0.808650	+	0.588290i	$0.200198\pi$
$234$	0	0
$235$	2443.76	0.678354
$236$	110.600	0.0305061
$237$	0	0
$238$	6431.42	1.75163
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6650.47	−1.77757	−0.888785	−	0.458324i	$-0.848450\pi$
$241$	−6650.47	−1.77757	−0.888785	+	0.458324i	$0.848450\pi$
$242$	−2216.36	−0.588732
$243$	0	0
$244$	2317.76	0.608113
$245$	−3241.82	−0.845355
$246$	0	0
$247$	190.653	0.0491133
$248$	−1185.06	−0.303433
$249$	0	0
$250$	−1371.91	−0.347069
$251$	−1841.94	−0.463195	−0.231597	−	0.972812i	$-0.574395\pi$
$251$	−1841.94	−0.463195	−0.231597	+	0.972812i	$0.574395\pi$
$252$	0	0
$253$	−1317.67	−0.327435
$254$	−3759.57	−0.928726
$255$	0	0
$256$	−3794.95	−0.926501
$257$	4506.26	1.09375	0.546873	−	0.837215i	$-0.315818\pi$
$257$	4506.26	1.09375	0.546873	+	0.837215i	$0.315818\pi$
$258$	0	0
$259$	2022.40	0.485197
$260$	1533.31	0.365738
$261$	0	0
$262$	1233.97	0.290972
$263$	5159.62	1.20972	0.604859	−	0.796333i	$-0.293230\pi$
$263$	5159.62	1.20972	0.604859	+	0.796333i	$0.293230\pi$
$264$	0	0
$265$	3453.71	0.800602
$266$	−279.502	−0.0644262
$267$	0	0
$268$	−1779.17	−0.405522
$269$	−3296.89	−0.747268	−0.373634	−	0.927576i	$-0.621888\pi$
$269$	−3296.89	−0.747268	−0.373634	+	0.927576i	$0.621888\pi$
$270$	0	0
$271$	−1496.57	−0.335461	−0.167730	−	0.985833i	$-0.553644\pi$
$271$	−1496.57	−0.335461	−0.167730	+	0.985833i	$0.553644\pi$
$272$	3876.11	0.864058
$273$	0	0
$274$	212.579	0.0468700
$275$	3985.31	0.873902
$276$	0	0
$277$	33.8175	0.00733537	0.00366769	−	0.999993i	$-0.498833\pi$
$277$	33.8175	0.00733537	0.00366769	+	0.999993i	$0.498833\pi$
$278$	−1221.48	−0.263524
$279$	0	0
$280$	−8457.58	−1.80513
$281$	831.221	0.176464	0.0882321	−	0.996100i	$-0.471878\pi$
$281$	831.221	0.176464	0.0882321	+	0.996100i	$0.471878\pi$
$282$	0	0
$283$	−4983.37	−1.04675	−0.523376	−	0.852102i	$-0.675328\pi$
$283$	−4983.37	−1.04675	−0.523376	+	0.852102i	$0.675328\pi$
$284$	−1385.17	−0.289418
$285$	0	0
$286$	−3989.20	−0.824778
$287$	3254.65	0.669394
$288$	0	0
$289$	9358.49	1.90484
$290$	−1950.44	−0.394945
$291$	0	0
$292$	2518.21	0.504681
$293$	−9099.52	−1.81433	−0.907167	−	0.420771i	$-0.861759\pi$
$293$	−9099.52	−1.81433	−0.907167	+	0.420771i	$0.861759\pi$
$294$	0	0
$295$	550.648	0.108678
$296$	2089.18	0.410241
$297$	0	0
$298$	−6920.36	−1.34525
$299$	−1006.33	−0.194641
$300$	0	0
$301$	2830.75	0.542065
$302$	2045.83	0.389816
$303$	0	0
$304$	−168.451	−0.0317808
$305$	11539.6	2.16640
$306$	0	0
$307$	2255.32	0.419276	0.209638	−	0.977779i	$-0.432771\pi$
$307$	2255.32	0.419276	0.209638	+	0.977779i	$0.432771\pi$
$308$	−3318.18	−0.613867
$309$	0	0
$310$	−1568.14	−0.287304
$311$	−965.052	−0.175958	−0.0879792	−	0.996122i	$-0.528041\pi$
$311$	−965.052	−0.175958	−0.0879792	+	0.996122i	$0.528041\pi$
$312$	0	0
$313$	2032.58	0.367055	0.183527	−	0.983015i	$-0.441248\pi$
$313$	2032.58	0.367055	0.183527	+	0.983015i	$0.441248\pi$
$314$	7483.57	1.34497
$315$	0	0
$316$	−1791.88	−0.318991
$317$	4854.11	0.860044	0.430022	−	0.902818i	$-0.358506\pi$
$317$	4854.11	0.860044	0.430022	+	0.902818i	$0.358506\pi$
$318$	0	0
$319$	−2879.14	−0.505333
$320$	−7769.51	−1.35728
$321$	0	0
$322$	1475.30	0.255328
$323$	−620.223	−0.106842
$324$	0	0
$325$	3043.66	0.519483
$326$	861.182	0.146308
$327$	0	0
$328$	3362.12	0.565982
$329$	−4038.90	−0.676813
$330$	0	0
$331$	−4368.24	−0.725378	−0.362689	−	0.931910i	$-0.618141\pi$
$331$	−4368.24	−0.725378	−0.362689	+	0.931910i	$0.618141\pi$
$332$	2009.36	0.332162
$333$	0	0
$334$	4021.68	0.658852
$335$	−8858.01	−1.44467
$336$	0	0
$337$	2239.06	0.361927	0.180963	−	0.983490i	$-0.442078\pi$
$337$	2239.06	0.361927	0.180963	+	0.983490i	$0.442078\pi$
$338$	1916.86	0.308472
$339$	0	0
$340$	−4988.08	−0.795637
$341$	−2314.80	−0.367606
$342$	0	0
$343$	−2815.62	−0.443234
$344$	2924.22	0.458324
$345$	0	0
$346$	7112.84	1.10517
$347$	4761.14	0.736575	0.368288	−	0.929712i	$-0.379944\pi$
$347$	4761.14	0.736575	0.368288	+	0.929712i	$0.379944\pi$
$348$	0	0
$349$	457.127	0.0701131	0.0350565	−	0.999385i	$-0.488839\pi$
$349$	457.127	0.0701131	0.0350565	+	0.999385i	$0.488839\pi$
$350$	−4462.07	−0.681451
$351$	0	0
$352$	−5944.46	−0.900116
$353$	5873.64	0.885616	0.442808	−	0.896616i	$-0.353982\pi$
$353$	5873.64	0.885616	0.442808	+	0.896616i	$0.353982\pi$
$354$	0	0
$355$	−6896.41	−1.03105
$356$	1190.47	0.177232
$357$	0	0
$358$	−2080.72	−0.307177
$359$	4593.02	0.675238	0.337619	−	0.941283i	$-0.390378\pi$
$359$	4593.02	0.675238	0.337619	+	0.941283i	$0.390378\pi$
$360$	0	0
$361$	−6832.05	−0.996070
$362$	5806.82	0.843093
$363$	0	0
$364$	−2534.16	−0.364907
$365$	12537.5	1.79793
$366$	0	0
$367$	−11055.2	−1.57242	−0.786209	−	0.617961i	$-0.787959\pi$
$367$	−11055.2	−1.57242	−0.786209	+	0.617961i	$0.787959\pi$
$368$	889.142	0.125950
$369$	0	0
$370$	2764.53	0.388435
$371$	−5708.08	−0.798784
$372$	0	0
$373$	3243.07	0.450188	0.225094	−	0.974337i	$-0.427731\pi$
$373$	3243.07	0.450188	0.225094	+	0.974337i	$0.427731\pi$
$374$	12977.5	1.79425
$375$	0	0
$376$	−4172.26	−0.572255
$377$	−2198.86	−0.300390
$378$	0	0
$379$	−8411.84	−1.14007	−0.570036	−	0.821620i	$-0.693071\pi$
$379$	−8411.84	−1.14007	−0.570036	+	0.821620i	$0.693071\pi$
$380$	216.776	0.0292642
$381$	0	0
$382$	1176.52	0.157581
$383$	−14035.5	−1.87253	−0.936264	−	0.351296i	$-0.885741\pi$
$383$	−14035.5	−1.87253	−0.936264	+	0.351296i	$0.885741\pi$
$384$	0	0
$385$	−16520.4	−2.18690
$386$	6396.20	0.843415
$387$	0	0
$388$	−1352.73	−0.176996
$389$	−11497.0	−1.49851	−0.749253	−	0.662283i	$-0.769588\pi$
$389$	−11497.0	−1.49851	−0.749253	+	0.662283i	$0.769588\pi$
$390$	0	0
$391$	3273.74	0.423428
$392$	5534.79	0.713136
$393$	0	0
$394$	−8665.35	−1.10801
$395$	−8921.33	−1.13641
$396$	0	0
$397$	2360.15	0.298368	0.149184	−	0.988809i	$-0.452335\pi$
$397$	2360.15	0.298368	0.149184	+	0.988809i	$0.452335\pi$
$398$	2535.72	0.319358
$399$	0	0
$400$	−2689.22	−0.336152
$401$	8293.53	1.03282	0.516408	−	0.856343i	$-0.327269\pi$
$401$	8293.53	1.03282	0.516408	+	0.856343i	$0.327269\pi$
$402$	0	0
$403$	−1767.86	−0.218520
$404$	−25.2439	−0.00310874
$405$	0	0
$406$	3223.58	0.394048
$407$	4080.85	0.497003
$408$	0	0
$409$	10753.1	1.30002	0.650010	−	0.759926i	$-0.274765\pi$
$409$	10753.1	1.30002	0.650010	+	0.759926i	$0.274765\pi$
$410$	4448.96	0.535898
$411$	0	0
$412$	−4641.35	−0.555007
$413$	−910.078	−0.108431
$414$	0	0
$415$	10004.1	1.18333
$416$	−4539.91	−0.535066
$417$	0	0
$418$	−563.985	−0.0659938
$419$	3099.12	0.361342	0.180671	−	0.983544i	$-0.442173\pi$
$419$	3099.12	0.361342	0.180671	+	0.983544i	$0.442173\pi$
$420$	0	0
$421$	8758.71	1.01395	0.506976	−	0.861960i	$-0.330763\pi$
$421$	8758.71	1.01395	0.506976	+	0.861960i	$0.330763\pi$
$422$	4281.58	0.493895
$423$	0	0
$424$	−5896.56	−0.675383
$425$	−9901.47	−1.13010
$426$	0	0
$427$	−19071.9	−2.16148
$428$	−1158.14	−0.130797
$429$	0	0
$430$	3869.50	0.433962
$431$	−15108.8	−1.68855	−0.844277	−	0.535907i	$-0.819970\pi$
$431$	−15108.8	−1.68855	−0.844277	+	0.535907i	$0.819970\pi$
$432$	0	0
$433$	−17005.2	−1.88734	−0.943670	−	0.330888i	$-0.892652\pi$
$433$	−17005.2	−1.88734	−0.943670	+	0.330888i	$0.892652\pi$
$434$	2591.73	0.286652
$435$	0	0
$436$	2387.17	0.262213
$437$	−142.273	−0.0155740
$438$	0	0
$439$	8568.79	0.931585	0.465793	−	0.884894i	$-0.345769\pi$
$439$	8568.79	0.931585	0.465793	+	0.884894i	$0.345769\pi$
$440$	−17065.9	−1.84905
$441$	0	0
$442$	9911.15	1.06657
$443$	−6074.16	−0.651449	−0.325725	−	0.945465i	$-0.605608\pi$
$443$	−6074.16	−0.651449	−0.325725	+	0.945465i	$0.605608\pi$
$444$	0	0
$445$	5927.02	0.631389
$446$	9042.69	0.960053
$447$	0	0
$448$	12841.0	1.35419
$449$	8809.62	0.925950	0.462975	−	0.886371i	$-0.346782\pi$
$449$	8809.62	0.925950	0.462975	+	0.886371i	$0.346782\pi$
$450$	0	0
$451$	6567.31	0.685682
$452$	−2888.84	−0.300619
$453$	0	0
$454$	3967.20	0.410110
$455$	−12617.0	−1.29998
$456$	0	0
$457$	−3730.07	−0.381806	−0.190903	−	0.981609i	$-0.561142\pi$
$457$	−3730.07	−0.381806	−0.190903	+	0.981609i	$0.561142\pi$
$458$	−5647.66	−0.576196
$459$	0	0
$460$	−1144.22	−0.115977
$461$	17516.0	1.76964	0.884818	−	0.465938i	$-0.154283\pi$
$461$	17516.0	1.76964	0.884818	+	0.465938i	$0.154283\pi$
$462$	0	0
$463$	−4066.82	−0.408210	−0.204105	−	0.978949i	$-0.565428\pi$
$463$	−4066.82	−0.408210	−0.204105	+	0.978949i	$0.565428\pi$
$464$	1942.80	0.194380
$465$	0	0
$466$	−12995.2	−1.29182
$467$	−14819.9	−1.46848	−0.734241	−	0.678889i	$-0.762462\pi$
$467$	−14819.9	−1.46848	−0.734241	+	0.678889i	$0.762462\pi$
$468$	0	0
$469$	14640.0	1.44139
$470$	−5520.98	−0.541838
$471$	0	0
$472$	−940.128	−0.0916799
$473$	5711.95	0.555255
$474$	0	0
$475$	430.306	0.0415659
$476$	8244.00	0.793830
$477$	0	0
$478$	539.953	0.0516671
$479$	1455.99	0.138885	0.0694426	−	0.997586i	$-0.477878\pi$
$479$	1455.99	0.138885	0.0694426	+	0.997586i	$0.477878\pi$
$480$	0	0
$481$	3116.63	0.295439
$482$	15024.9	1.41984
$483$	0	0
$484$	−2841.01	−0.266811
$485$	−6734.90	−0.630549
$486$	0	0
$487$	16076.8	1.49591	0.747956	−	0.663749i	$-0.231035\pi$
$487$	16076.8	1.49591	0.747956	+	0.663749i	$0.231035\pi$
$488$	−19701.6	−1.82756
$489$	0	0
$490$	7323.96	0.675231
$491$	−14019.9	−1.28861	−0.644306	−	0.764768i	$-0.722854\pi$
$491$	−14019.9	−1.28861	−0.644306	+	0.764768i	$0.722854\pi$
$492$	0	0
$493$	7153.21	0.653478
$494$	−430.727	−0.0392294
$495$	0	0
$496$	1561.99	0.141402
$497$	11398.0	1.02871
$498$	0	0
$499$	743.987	0.0667443	0.0333722	−	0.999443i	$-0.489375\pi$
$499$	743.987	0.0667443	0.0333722	+	0.999443i	$0.489375\pi$
$500$	−1758.56	−0.157291
$501$	0	0
$502$	4161.33	0.369979
$503$	−16156.5	−1.43217	−0.716085	−	0.698013i	$-0.754068\pi$
$503$	−16156.5	−1.43217	−0.716085	+	0.698013i	$0.754068\pi$
$504$	0	0
$505$	−125.683	−0.0110749
$506$	2976.90	0.261540
$507$	0	0
$508$	−4819.14	−0.420895
$509$	17640.5	1.53615	0.768076	−	0.640358i	$-0.221214\pi$
$509$	17640.5	1.53615	0.768076	+	0.640358i	$0.221214\pi$
$510$	0	0
$511$	−20721.3	−1.79384
$512$	10400.8	0.897766
$513$	0	0
$514$	−10180.6	−0.873635
$515$	−23108.1	−1.97721
$516$	0	0
$517$	−8149.77	−0.693282
$518$	−4569.05	−0.387553
$519$	0	0
$520$	−13033.6	−1.09915
$521$	−9166.53	−0.770812	−0.385406	−	0.922747i	$-0.625939\pi$
$521$	−9166.53	−0.770812	−0.385406	+	0.922747i	$0.625939\pi$
$522$	0	0
$523$	−10004.1	−0.836425	−0.418213	−	0.908349i	$-0.637343\pi$
$523$	−10004.1	−0.836425	−0.418213	+	0.908349i	$0.637343\pi$
$524$	1581.74	0.131868
$525$	0	0
$526$	−11656.7	−0.966266
$527$	5751.12	0.475375
$528$	0	0
$529$	−11416.0	−0.938279
$530$	−7802.67	−0.639484
$531$	0	0
$532$	−358.275	−0.0291977
$533$	5015.59	0.407597
$534$	0	0
$535$	−5766.10	−0.465963
$536$	15123.4	1.21872
$537$	0	0
$538$	7448.40	0.596883
$539$	10811.2	0.863958
$540$	0	0
$541$	20442.1	1.62454	0.812268	−	0.583284i	$-0.198232\pi$
$541$	20442.1	1.62454	0.812268	+	0.583284i	$0.198232\pi$
$542$	3381.07	0.267951
$543$	0	0
$544$	14769.0	1.16400
$545$	11885.1	0.934133
$546$	0	0
$547$	6459.91	0.504947	0.252473	−	0.967604i	$-0.418756\pi$
$547$	6459.91	0.504947	0.252473	+	0.967604i	$0.418756\pi$
$548$	272.491	0.0212413
$549$	0	0
$550$	−9003.67	−0.698032
$551$	−310.870	−0.0240354
$552$	0	0
$553$	14744.6	1.13383
$554$	−76.4011	−0.00585916
$555$	0	0
$556$	−1565.74	−0.119428
$557$	−16149.7	−1.22852	−0.614258	−	0.789105i	$-0.710545\pi$
$557$	−16149.7	−1.22852	−0.614258	+	0.789105i	$0.710545\pi$
$558$	0	0
$559$	4362.33	0.330066
$560$	11147.7	0.841207
$561$	0	0
$562$	−1877.91	−0.140952
$563$	13990.9	1.04733	0.523666	−	0.851924i	$-0.324564\pi$
$563$	13990.9	1.04733	0.523666	+	0.851924i	$0.324564\pi$
$564$	0	0
$565$	−14382.8	−1.07096
$566$	11258.5	0.836097
$567$	0	0
$568$	11774.3	0.869788
$569$	4744.46	0.349557	0.174779	−	0.984608i	$-0.444079\pi$
$569$	4744.46	0.349557	0.174779	+	0.984608i	$0.444079\pi$
$570$	0	0
$571$	−24411.1	−1.78909	−0.894546	−	0.446975i	$-0.852501\pi$
$571$	−24411.1	−1.78909	−0.894546	+	0.446975i	$0.852501\pi$
$572$	−5113.49	−0.373786
$573$	0	0
$574$	−7352.97	−0.534681
$575$	−2271.30	−0.164730
$576$	0	0
$577$	3630.64	0.261951	0.130975	−	0.991386i	$-0.458189\pi$
$577$	3630.64	0.261951	0.130975	+	0.991386i	$0.458189\pi$
$578$	−21142.9	−1.52150
$579$	0	0
$580$	−2500.14	−0.178988
$581$	−16534.2	−1.18064
$582$	0	0
$583$	−11517.9	−0.818220
$584$	−21405.5	−1.51672
$585$	0	0
$586$	20557.8	1.44921
$587$	−39.1901	−0.00275562	−0.00137781	−	0.999999i	$-0.500439\pi$
$587$	−39.1901	−0.00275562	−0.00137781	+	0.999999i	$0.500439\pi$
$588$	0	0
$589$	−249.937	−0.0174847
$590$	−1244.03	−0.0868068
$591$	0	0
$592$	−2753.69	−0.191176
$593$	−18815.5	−1.30297	−0.651484	−	0.758662i	$-0.725853\pi$
$593$	−18815.5	−1.30297	−0.651484	+	0.758662i	$0.725853\pi$
$594$	0	0
$595$	41044.8	2.82802
$596$	−8870.75	−0.609664
$597$	0	0
$598$	2273.52	0.155470
$599$	1183.18	0.0807066	0.0403533	−	0.999185i	$-0.487152\pi$
$599$	1183.18	0.0807066	0.0403533	+	0.999185i	$0.487152\pi$
$600$	0	0
$601$	10002.4	0.678880	0.339440	−	0.940628i	$-0.389762\pi$
$601$	10002.4	0.678880	0.339440	+	0.940628i	$0.389762\pi$
$602$	−6395.28	−0.432977
$603$	0	0
$604$	2622.41	0.176663
$605$	−14144.7	−0.950516
$606$	0	0
$607$	−15024.5	−1.00466	−0.502328	−	0.864677i	$-0.667523\pi$
$607$	−15024.5	−1.00466	−0.502328	+	0.864677i	$0.667523\pi$
$608$	−641.843	−0.0428128
$609$	0	0
$610$	−26070.4	−1.73042
$611$	−6224.15	−0.412115
$612$	0	0
$613$	−13849.4	−0.912514	−0.456257	−	0.889848i	$-0.650810\pi$
$613$	−13849.4	−0.912514	−0.456257	+	0.889848i	$0.650810\pi$
$614$	−5095.25	−0.334898
$615$	0	0
$616$	28205.5	1.84486
$617$	−15420.8	−1.00619	−0.503096	−	0.864231i	$-0.667806\pi$
$617$	−15420.8	−1.00619	−0.503096	+	0.864231i	$0.667806\pi$
$618$	0	0
$619$	8573.83	0.556723	0.278361	−	0.960476i	$-0.410209\pi$
$619$	8573.83	0.556723	0.278361	+	0.960476i	$0.410209\pi$
$620$	−2010.09	−0.130205
$621$	0	0
$622$	2180.26	0.140547
$623$	−9795.83	−0.629955
$624$	0	0
$625$	−19115.8	−1.22341
$626$	−4592.03	−0.293186
$627$	0	0
$628$	9592.68	0.609538
$629$	−10138.9	−0.642707
$630$	0	0
$631$	−6587.20	−0.415582	−0.207791	−	0.978173i	$-0.566627\pi$
$631$	−6587.20	−0.415582	−0.207791	+	0.978173i	$0.566627\pi$
$632$	15231.5	0.958666
$633$	0	0
$634$	−10966.5	−0.686964
$635$	−23993.3	−1.49944
$636$	0	0
$637$	8256.77	0.513572
$638$	6504.61	0.403636
$639$	0	0
$640$	3293.14	0.203395
$641$	−15999.8	−0.985888	−0.492944	−	0.870061i	$-0.664079\pi$
$641$	−15999.8	−0.985888	−0.492944	+	0.870061i	$0.664079\pi$
$642$	0	0
$643$	−7881.05	−0.483357	−0.241678	−	0.970356i	$-0.577698\pi$
$643$	−7881.05	−0.483357	−0.241678	+	0.970356i	$0.577698\pi$
$644$	1891.09	0.115714
$645$	0	0
$646$	1401.22	0.0853408
$647$	23810.5	1.44681	0.723407	−	0.690422i	$-0.242575\pi$
$647$	23810.5	1.44681	0.723407	+	0.690422i	$0.242575\pi$
$648$	0	0
$649$	−1836.38	−0.111069
$650$	−6876.29	−0.414939
$651$	0	0
$652$	1103.89	0.0663063
$653$	9315.57	0.558264	0.279132	−	0.960253i	$-0.409953\pi$
$653$	9315.57	0.558264	0.279132	+	0.960253i	$0.409953\pi$
$654$	0	0
$655$	7875.08	0.469778
$656$	−4431.52	−0.263752
$657$	0	0
$658$	9124.74	0.540607
$659$	−12286.1	−0.726247	−0.363124	−	0.931741i	$-0.618290\pi$
$659$	−12286.1	−0.726247	−0.363124	+	0.931741i	$0.618290\pi$
$660$	0	0
$661$	23919.8	1.40752	0.703761	−	0.710437i	$-0.251503\pi$
$661$	23919.8	1.40752	0.703761	+	0.710437i	$0.251503\pi$
$662$	9868.80	0.579399
$663$	0	0
$664$	−17080.1	−0.998248
$665$	−1783.76	−0.104017
$666$	0	0
$667$	1640.88	0.0952549
$668$	5155.13	0.298590
$669$	0	0
$670$	20012.2	1.15394
$671$	−38483.7	−2.21408
$672$	0	0
$673$	15004.4	0.859399	0.429700	−	0.902972i	$-0.358619\pi$
$673$	15004.4	0.859399	0.429700	+	0.902972i	$0.358619\pi$
$674$	−5058.52	−0.289090
$675$	0	0
$676$	2457.10	0.139799
$677$	−26596.8	−1.50989	−0.754947	−	0.655786i	$-0.772338\pi$
$677$	−26596.8	−1.50989	−0.754947	+	0.655786i	$0.772338\pi$
$678$	0	0
$679$	11131.0	0.629117
$680$	42400.1	2.39113
$681$	0	0
$682$	5229.65	0.293627
$683$	−24684.8	−1.38292	−0.691461	−	0.722414i	$-0.743033\pi$
$683$	−24684.8	−1.38292	−0.691461	+	0.722414i	$0.743033\pi$
$684$	0	0
$685$	1356.66	0.0756722
$686$	6361.10	0.354035
$687$	0	0
$688$	−3854.33	−0.213583
$689$	−8796.45	−0.486383
$690$	0	0
$691$	28730.0	1.58168	0.790839	−	0.612025i	$-0.209645\pi$
$691$	28730.0	1.58168	0.790839	+	0.612025i	$0.209645\pi$
$692$	9117.47	0.500859
$693$	0	0
$694$	−10756.5	−0.588342
$695$	−7795.40	−0.425462
$696$	0	0
$697$	−16316.4	−0.886699
$698$	−1032.75	−0.0560031
$699$	0	0
$700$	−5719.63	−0.308831
$701$	−34273.6	−1.84664	−0.923321	−	0.384029i	$-0.874536\pi$
$701$	−34273.6	−1.84664	−0.923321	+	0.384029i	$0.874536\pi$
$702$	0	0
$703$	440.623	0.0236393
$704$	25910.8	1.38715
$705$	0	0
$706$	−13269.8	−0.707389
$707$	207.721	0.0110497
$708$	0	0
$709$	−29945.6	−1.58622	−0.793111	−	0.609077i	$-0.791540\pi$
$709$	−29945.6	−1.58622	−0.793111	+	0.609077i	$0.791540\pi$
$710$	15580.5	0.823556
$711$	0	0
$712$	−10119.3	−0.532635
$713$	1319.25	0.0692935
$714$	0	0
$715$	−25458.8	−1.33161
$716$	−2667.14	−0.139212
$717$	0	0
$718$	−10376.6	−0.539349
$719$	−15604.0	−0.809363	−0.404682	−	0.914458i	$-0.632618\pi$
$719$	−15604.0	−0.809363	−0.404682	+	0.914458i	$0.632618\pi$
$720$	0	0
$721$	38191.7	1.97272
$722$	15435.1	0.795615
$723$	0	0
$724$	7443.37	0.382087
$725$	−4962.85	−0.254228
$726$	0	0
$727$	−19780.9	−1.00912	−0.504562	−	0.863376i	$-0.668346\pi$
$727$	−19780.9	−1.00912	−0.504562	+	0.863376i	$0.668346\pi$
$728$	21541.1	1.09666
$729$	0	0
$730$	−28325.0	−1.43610
$731$	−14191.3	−0.718036
$732$	0	0
$733$	7308.63	0.368282	0.184141	−	0.982900i	$-0.441050\pi$
$733$	7308.63	0.368282	0.184141	+	0.982900i	$0.441050\pi$
$734$	24976.1	1.25597
$735$	0	0
$736$	3387.86	0.169671
$737$	29540.9	1.47646
$738$	0	0
$739$	−22800.3	−1.13495	−0.567473	−	0.823392i	$-0.692079\pi$
$739$	−22800.3	−1.13495	−0.567473	+	0.823392i	$0.692079\pi$
$740$	3543.67	0.176037
$741$	0	0
$742$	12895.8	0.638032
$743$	−22439.9	−1.10800	−0.553998	−	0.832518i	$-0.686899\pi$
$743$	−22439.9	−1.10800	−0.553998	+	0.832518i	$0.686899\pi$
$744$	0	0
$745$	−44165.2	−2.17193
$746$	−7326.81	−0.359589
$747$	0	0
$748$	16634.9	0.813146
$749$	9529.87	0.464905
$750$	0	0
$751$	434.671	0.0211203	0.0105602	−	0.999944i	$-0.496639\pi$
$751$	434.671	0.0211203	0.0105602	+	0.999944i	$0.496639\pi$
$752$	5499.34	0.266676
$753$	0	0
$754$	4967.70	0.239938
$755$	13056.3	0.629363
$756$	0	0
$757$	−32231.2	−1.54751	−0.773754	−	0.633487i	$-0.781623\pi$
$757$	−32231.2	−1.54751	−0.773754	+	0.633487i	$0.781623\pi$
$758$	19004.2	0.910637
$759$	0	0
$760$	−1842.66	−0.0879477
$761$	−3846.78	−0.183240	−0.0916200	−	0.995794i	$-0.529204\pi$
$761$	−3846.78	−0.183240	−0.0916200	+	0.995794i	$0.529204\pi$
$762$	0	0
$763$	−19643.0	−0.932012
$764$	1508.10	0.0714151
$765$	0	0
$766$	31709.1	1.49569
$767$	−1402.48	−0.0660241
$768$	0	0
$769$	−6428.44	−0.301451	−0.150725	−	0.988576i	$-0.548161\pi$
$769$	−6428.44	−0.301451	−0.150725	+	0.988576i	$0.548161\pi$
$770$	37323.1	1.74680
$771$	0	0
$772$	8198.86	0.382233
$773$	32217.1	1.49905	0.749527	−	0.661973i	$-0.230281\pi$
$773$	32217.1	1.49905	0.749527	+	0.661973i	$0.230281\pi$
$774$	0	0
$775$	−3990.09	−0.184939
$776$	11498.6	0.531927
$777$	0	0
$778$	25974.1	1.19694
$779$	709.094	0.0326135
$780$	0	0
$781$	22999.1	1.05374
$782$	−7396.09	−0.338214
$783$	0	0
$784$	−7295.25	−0.332328
$785$	47759.5	2.17148
$786$	0	0
$787$	37160.4	1.68313	0.841567	−	0.540153i	$-0.181634\pi$
$787$	37160.4	1.68313	0.841567	+	0.540153i	$0.181634\pi$
$788$	−11107.5	−0.502144
$789$	0	0
$790$	20155.2	0.907709
$791$	23771.1	1.06852
$792$	0	0
$793$	−29390.8	−1.31614
$794$	−5332.08	−0.238323
$795$	0	0
$796$	3250.37	0.144732
$797$	−9981.04	−0.443597	−0.221798	−	0.975093i	$-0.571193\pi$
$797$	−9981.04	−0.443597	−0.221798	+	0.975093i	$0.571193\pi$
$798$	0	0
$799$	20248.1	0.896527
$800$	−10246.6	−0.452841
$801$	0	0
$802$	−18736.9	−0.824965
$803$	−41811.8	−1.83749
$804$	0	0
$805$	9415.27	0.412230
$806$	3993.99	0.174544
$807$	0	0
$808$	214.580	0.00934269
$809$	−5855.18	−0.254459	−0.127229	−	0.991873i	$-0.540608\pi$
$809$	−5855.18	−0.254459	−0.127229	+	0.991873i	$0.540608\pi$
$810$	0	0
$811$	25857.5	1.11958	0.559790	−	0.828634i	$-0.310882\pi$
$811$	25857.5	1.11958	0.559790	+	0.828634i	$0.310882\pi$
$812$	4132.09	0.178581
$813$	0	0
$814$	−9219.53	−0.396983
$815$	5495.99	0.236216
$816$	0	0
$817$	616.737	0.0264099
$818$	−24293.6	−1.03840
$819$	0	0
$820$	5702.82	0.242867
$821$	−18705.2	−0.795146	−0.397573	−	0.917570i	$-0.630148\pi$
$821$	−18705.2	−0.795146	−0.397573	+	0.917570i	$0.630148\pi$
$822$	0	0
$823$	11464.7	0.485583	0.242792	−	0.970078i	$-0.421937\pi$
$823$	11464.7	0.485583	0.242792	+	0.970078i	$0.421937\pi$
$824$	39452.8	1.66796
$825$	0	0
$826$	2056.06	0.0866097
$827$	44661.5	1.87791	0.938955	−	0.344039i	$-0.111795\pi$
$827$	44661.5	1.87791	0.938955	+	0.344039i	$0.111795\pi$
$828$	0	0
$829$	28377.6	1.18890	0.594448	−	0.804134i	$-0.297371\pi$
$829$	28377.6	1.18890	0.594448	+	0.804134i	$0.297371\pi$
$830$	−22601.4	−0.945188
$831$	0	0
$832$	19788.6	0.824575
$833$	−26860.5	−1.11724
$834$	0	0
$835$	25666.1	1.06373
$836$	−722.935	−0.0299082
$837$	0	0
$838$	−7001.60	−0.288623
$839$	−27402.8	−1.12759	−0.563796	−	0.825914i	$-0.690659\pi$
$839$	−27402.8	−1.12759	−0.563796	+	0.825914i	$0.690659\pi$
$840$	0	0
$841$	−20803.6	−0.852993
$842$	−19787.8	−0.809898
$843$	0	0
$844$	5488.27	0.223832
$845$	12233.3	0.498033
$846$	0	0
$847$	23377.4	0.948357
$848$	7772.09	0.314734
$849$	0	0
$850$	22369.6	0.902670
$851$	−2325.75	−0.0936848
$852$	0	0
$853$	35700.3	1.43301	0.716503	−	0.697584i	$-0.245741\pi$
$853$	35700.3	1.43301	0.716503	+	0.697584i	$0.245741\pi$
$854$	43087.6	1.72649
$855$	0	0
$856$	9844.55	0.393084
$857$	16514.6	0.658257	0.329129	−	0.944285i	$-0.393245\pi$
$857$	16514.6	0.658257	0.329129	+	0.944285i	$0.393245\pi$
$858$	0	0
$859$	1849.98	0.0734814	0.0367407	−	0.999325i	$-0.488302\pi$
$859$	1849.98	0.0734814	0.0367407	+	0.999325i	$0.488302\pi$
$860$	4960.05	0.196670
$861$	0	0
$862$	34134.1	1.34874
$863$	−26721.9	−1.05402	−0.527012	−	0.849858i	$-0.676688\pi$
$863$	−26721.9	−1.05402	−0.527012	+	0.849858i	$0.676688\pi$
$864$	0	0
$865$	45393.6	1.78431
$866$	38418.5	1.50752
$867$	0	0
$868$	3322.16	0.129910
$869$	29752.1	1.16141
$870$	0	0
$871$	22561.0	0.877670
$872$	−20291.6	−0.788029
$873$	0	0
$874$	321.426	0.0124398
$875$	14470.5	0.559075
$876$	0	0
$877$	24465.9	0.942022	0.471011	−	0.882127i	$-0.343889\pi$
$877$	24465.9	0.942022	0.471011	+	0.882127i	$0.343889\pi$
$878$	−19358.8	−0.744107
$879$	0	0
$880$	22494.1	0.861676
$881$	−33429.1	−1.27838	−0.639191	−	0.769048i	$-0.720731\pi$
$881$	−33429.1	−1.27838	−0.639191	+	0.769048i	$0.720731\pi$
$882$	0	0
$883$	−41193.2	−1.56994	−0.784972	−	0.619531i	$-0.787323\pi$
$883$	−41193.2	−1.56994	−0.784972	+	0.619531i	$0.787323\pi$
$884$	12704.4	0.483367
$885$	0	0
$886$	13722.8	0.520348
$887$	33244.0	1.25843	0.629213	−	0.777233i	$-0.283377\pi$
$887$	33244.0	1.25843	0.629213	+	0.777233i	$0.283377\pi$
$888$	0	0
$889$	39654.7	1.49603
$890$	−13390.4	−0.504324
$891$	0	0
$892$	11591.2	0.435093
$893$	−879.958	−0.0329750
$894$	0	0
$895$	−13279.0	−0.495941
$896$	−5442.71	−0.202933
$897$	0	0
$898$	−19902.8	−0.739606
$899$	2882.60	0.106941
$900$	0	0
$901$	28616.1	1.05809
$902$	−14837.0	−0.547691
$903$	0	0
$904$	24556.0	0.903451
$905$	37058.7	1.36118
$906$	0	0
$907$	−6511.21	−0.238370	−0.119185	−	0.992872i	$-0.538028\pi$
$907$	−6511.21	−0.238370	−0.119185	+	0.992872i	$0.538028\pi$
$908$	5085.29	0.185860
$909$	0	0
$910$	28504.4	1.03837
$911$	25846.9	0.940005	0.470003	−	0.882665i	$-0.344253\pi$
$911$	25846.9	0.940005	0.470003	+	0.882665i	$0.344253\pi$
$912$	0	0
$913$	−33363.0	−1.20937
$914$	8427.04	0.304969
$915$	0	0
$916$	−7239.35	−0.261130
$917$	−13015.5	−0.468711
$918$	0	0
$919$	44039.8	1.58078	0.790391	−	0.612603i	$-0.209877\pi$
$919$	44039.8	1.58078	0.790391	+	0.612603i	$0.209877\pi$
$920$	9726.16	0.348546
$921$	0	0
$922$	−39572.5	−1.41350
$923$	17564.9	0.626386
$924$	0	0
$925$	7034.27	0.250038
$926$	9187.84	0.326060
$927$	0	0
$928$	7402.56	0.261855
$929$	6621.33	0.233841	0.116921	−	0.993141i	$-0.462698\pi$
$929$	6621.33	0.233841	0.116921	+	0.993141i	$0.462698\pi$
$930$	0	0
$931$	1167.33	0.0410930
$932$	−16657.7	−0.585450
$933$	0	0
$934$	33481.3	1.17296
$935$	82821.1	2.89683
$936$	0	0
$937$	−22510.6	−0.784835	−0.392418	−	0.919787i	$-0.628361\pi$
$937$	−22510.6	−0.784835	−0.392418	+	0.919787i	$0.628361\pi$
$938$	−33074.9	−1.15132
$939$	0	0
$940$	−7076.97	−0.245559
$941$	−21390.2	−0.741021	−0.370510	−	0.928828i	$-0.620817\pi$
$941$	−21390.2	−0.741021	−0.370510	+	0.928828i	$0.620817\pi$
$942$	0	0
$943$	−3742.83	−0.129251
$944$	1239.16	0.0427236
$945$	0	0
$946$	−12904.5	−0.443512
$947$	−20055.1	−0.688176	−0.344088	−	0.938937i	$-0.611812\pi$
$947$	−20055.1	−0.688176	−0.344088	+	0.938937i	$0.611812\pi$
$948$	0	0
$949$	−31932.5	−1.09228
$950$	−972.156	−0.0332009
$951$	0	0
$952$	−70076.3	−2.38570
$953$	44437.7	1.51047	0.755235	−	0.655454i	$-0.227523\pi$
$953$	44437.7	1.51047	0.755235	+	0.655454i	$0.227523\pi$
$954$	0	0
$955$	7508.45	0.254416
$956$	692.129	0.0234153
$957$	0	0
$958$	−3289.40	−0.110935
$959$	−2242.21	−0.0755004
$960$	0	0
$961$	−27473.4	−0.922205
$962$	−7041.14	−0.235983
$963$	0	0
$964$	19259.4	0.643467
$965$	40820.1	1.36170
$966$	0	0
$967$	−28437.2	−0.945687	−0.472844	−	0.881146i	$-0.656772\pi$
$967$	−28437.2	−0.945687	−0.472844	+	0.881146i	$0.656772\pi$
$968$	24149.4	0.801849
$969$	0	0
$970$	15215.6	0.503653
$971$	27709.5	0.915798	0.457899	−	0.889004i	$-0.348602\pi$
$971$	27709.5	0.915798	0.457899	+	0.889004i	$0.348602\pi$
$972$	0	0
$973$	12883.8	0.424496
$974$	−36321.0	−1.19487
$975$	0	0
$976$	25968.2	0.851661
$977$	18475.5	0.604997	0.302499	−	0.953150i	$-0.402179\pi$
$977$	18475.5	0.604997	0.302499	+	0.953150i	$0.402179\pi$
$978$	0	0
$979$	−19766.2	−0.645283
$980$	9388.10	0.306012
$981$	0	0
$982$	31673.9	1.02928
$983$	−45930.8	−1.49030	−0.745151	−	0.666896i	$-0.767622\pi$
$983$	−45930.8	−1.49030	−0.745151	+	0.666896i	$0.767622\pi$
$984$	0	0
$985$	−55301.6	−1.78889
$986$	−16160.7	−0.521968
$987$	0	0
$988$	−552.120	−0.0177786
$989$	−3255.34	−0.104665
$990$	0	0
$991$	36354.5	1.16533	0.582664	−	0.812713i	$-0.302011\pi$
$991$	36354.5	1.16533	0.582664	+	0.812713i	$0.302011\pi$
$992$	5951.59	0.190487
$993$	0	0
$994$	−25750.5	−0.821686
$995$	16182.8	0.515607
$996$	0	0
$997$	−16242.2	−0.515943	−0.257971	−	0.966153i	$-0.583054\pi$
$997$	−16242.2	−0.515943	−0.257971	+	0.966153i	$0.583054\pi$
$998$	−1680.83	−0.0533123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.12		32
3.2	odd	2		717.4.a.d.1.21	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.21	✓	32	3.2	odd	2
2151.4.a.d.1.12		32	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)

\(f(q)\)	\(=\)	\(q-2.25922 q^{2} -2.89594 q^{4} -14.4181 q^{5} +23.8295 q^{7} +24.6163 q^{8} +O(q^{10})\)	\(q-2.25922 q^{2} -2.89594 q^{4} -14.4181 q^{5} +23.8295 q^{7} +24.6163 q^{8} +32.5737 q^{10} +48.0836 q^{11} +36.7224 q^{13} -53.8359 q^{14} -32.4460 q^{16} -119.463 q^{17} +5.19174 q^{19} +41.7541 q^{20} -108.631 q^{22} -27.4037 q^{23} +82.8829 q^{25} -82.9639 q^{26} -69.0086 q^{28} -59.8779 q^{29} -48.1413 q^{31} -123.628 q^{32} +269.894 q^{34} -343.576 q^{35} +84.8700 q^{37} -11.7293 q^{38} -354.921 q^{40} +136.581 q^{41} +118.792 q^{43} -139.247 q^{44} +61.9109 q^{46} -169.492 q^{47} +224.843 q^{49} -187.250 q^{50} -106.346 q^{52} -239.539 q^{53} -693.276 q^{55} +586.593 q^{56} +135.277 q^{58} -38.1913 q^{59} -800.350 q^{61} +108.762 q^{62} +538.870 q^{64} -529.469 q^{65} +614.366 q^{67} +345.959 q^{68} +776.214 q^{70} +478.314 q^{71} -869.565 q^{73} -191.740 q^{74} -15.0350 q^{76} +1145.81 q^{77} +618.757 q^{79} +467.811 q^{80} -308.566 q^{82} -693.854 q^{83} +1722.44 q^{85} -268.377 q^{86} +1183.64 q^{88} -411.081 q^{89} +875.075 q^{91} +79.3595 q^{92} +382.919 q^{94} -74.8553 q^{95} +467.113 q^{97} -507.969 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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