LMFDB - Embedded newform 2151.4.a.f.1.8

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.91247 q^{2} +7.30741 q^{4} +2.39092 q^{5} +22.0359 q^{7} +2.70973 q^{8} +O(q^{10})$	\(q-3.91247 q^{2} +7.30741 q^{4} +2.39092 q^{5} +22.0359 q^{7} +2.70973 q^{8} -9.35442 q^{10} -39.1022 q^{11} -5.99505 q^{13} -86.2149 q^{14} -69.0610 q^{16} -69.7319 q^{17} +101.142 q^{19} +17.4715 q^{20} +152.986 q^{22} -194.152 q^{23} -119.283 q^{25} +23.4555 q^{26} +161.026 q^{28} -252.233 q^{29} +146.433 q^{31} +248.521 q^{32} +272.824 q^{34} +52.6862 q^{35} -288.929 q^{37} -395.716 q^{38} +6.47875 q^{40} +131.470 q^{41} -385.479 q^{43} -285.736 q^{44} +759.612 q^{46} -55.0867 q^{47} +142.582 q^{49} +466.693 q^{50} -43.8083 q^{52} -352.449 q^{53} -93.4904 q^{55} +59.7113 q^{56} +986.852 q^{58} +772.084 q^{59} +656.185 q^{61} -572.915 q^{62} -419.844 q^{64} -14.3337 q^{65} -135.462 q^{67} -509.560 q^{68} -206.133 q^{70} +1175.44 q^{71} +453.756 q^{73} +1130.43 q^{74} +739.088 q^{76} -861.653 q^{77} +588.978 q^{79} -165.120 q^{80} -514.371 q^{82} +1155.74 q^{83} -166.724 q^{85} +1508.17 q^{86} -105.956 q^{88} -1590.19 q^{89} -132.107 q^{91} -1418.75 q^{92} +215.525 q^{94} +241.824 q^{95} +444.125 q^{97} -557.847 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.91247	−1.38327	−0.691633	−	0.722249i	$-0.743109\pi$
$2$	−3.91247	−1.38327	−0.691633	+	0.722249i	$0.743109\pi$
$3$	0	0
$3$	0	0
$4$	7.30741	0.913427
$5$	2.39092	0.213851	0.106925	−	0.994267i	$-0.465899\pi$
$5$	2.39092	0.213851	0.106925	+	0.994267i	$0.465899\pi$
$6$	0	0
$7$	22.0359	1.18983	0.594914	−	0.803789i	$-0.297186\pi$
$7$	22.0359	1.18983	0.594914	+	0.803789i	$0.297186\pi$
$8$	2.70973	0.119754
$9$	0	0
$10$	−9.35442	−0.295813
$11$	−39.1022	−1.07180	−0.535898	−	0.844283i	$-0.680027\pi$
$11$	−39.1022	−1.07180	−0.535898	+	0.844283i	$0.680027\pi$
$12$	0	0
$13$	−5.99505	−0.127902	−0.0639511	−	0.997953i	$-0.520370\pi$
$13$	−5.99505	−0.127902	−0.0639511	+	0.997953i	$0.520370\pi$
$14$	−86.2149	−1.64585
$15$	0	0
$16$	−69.0610	−1.07908
$17$	−69.7319	−0.994852	−0.497426	−	0.867506i	$-0.665721\pi$
$17$	−69.7319	−0.994852	−0.497426	+	0.867506i	$0.665721\pi$
$18$	0	0
$19$	101.142	1.22124	0.610622	−	0.791922i	$-0.290920\pi$
$19$	101.142	1.22124	0.610622	+	0.791922i	$0.290920\pi$
$20$	17.4715	0.195337
$21$	0	0
$22$	152.986	1.48258
$23$	−194.152	−1.76015	−0.880074	−	0.474836i	$-0.842507\pi$
$23$	−194.152	−1.76015	−0.880074	+	0.474836i	$0.842507\pi$
$24$	0	0
$25$	−119.283	−0.954268
$26$	23.4555	0.176923
$27$	0	0
$28$	161.026	1.08682
$29$	−252.233	−1.61512	−0.807559	−	0.589787i	$-0.799212\pi$
$29$	−252.233	−1.61512	−0.807559	+	0.589787i	$0.799212\pi$
$30$	0	0
$31$	146.433	0.848393	0.424196	−	0.905570i	$-0.360557\pi$
$31$	146.433	0.848393	0.424196	+	0.905570i	$0.360557\pi$
$32$	248.521	1.37290
$33$	0	0
$34$	272.824	1.37615
$35$	52.6862	0.254446
$36$	0	0
$37$	−288.929	−1.28378	−0.641888	−	0.766798i	$-0.721849\pi$
$37$	−288.929	−1.28378	−0.641888	+	0.766798i	$0.721849\pi$
$38$	−395.716	−1.68931
$39$	0	0
$40$	6.47875	0.0256095
$41$	131.470	0.500784	0.250392	−	0.968145i	$-0.419441\pi$
$41$	131.470	0.500784	0.250392	+	0.968145i	$0.419441\pi$
$42$	0	0
$43$	−385.479	−1.36709	−0.683546	−	0.729908i	$-0.739563\pi$
$43$	−385.479	−1.36709	−0.683546	+	0.729908i	$0.739563\pi$
$44$	−285.736	−0.979007
$45$	0	0
$46$	759.612	2.43476
$47$	−55.0867	−0.170962	−0.0854810	−	0.996340i	$-0.527243\pi$
$47$	−55.0867	−0.170962	−0.0854810	+	0.996340i	$0.527243\pi$
$48$	0	0
$49$	142.582	0.415691
$50$	466.693	1.32001
$51$	0	0
$52$	−43.8083	−0.116829
$53$	−352.449	−0.913444	−0.456722	−	0.889609i	$-0.650977\pi$
$53$	−352.449	−0.913444	−0.456722	+	0.889609i	$0.650977\pi$
$54$	0	0
$55$	−93.4904	−0.229204
$56$	59.7113	0.142487
$57$	0	0
$58$	986.852	2.23414
$59$	772.084	1.70367	0.851837	−	0.523807i	$-0.175489\pi$
$59$	772.084	1.70367	0.851837	+	0.523807i	$0.175489\pi$
$60$	0	0
$61$	656.185	1.37731	0.688655	−	0.725089i	$-0.258202\pi$
$61$	656.185	1.37731	0.688655	+	0.725089i	$0.258202\pi$
$62$	−572.915	−1.17355
$63$	0	0
$64$	−419.844	−0.820007
$65$	−14.3337	−0.0273520
$66$	0	0
$67$	−135.462	−0.247004	−0.123502	−	0.992344i	$-0.539413\pi$
$67$	−135.462	−0.247004	−0.123502	+	0.992344i	$0.539413\pi$
$68$	−509.560	−0.908724
$69$	0	0
$70$	−206.133	−0.351966
$71$	1175.44	1.96478	0.982390	−	0.186840i	$-0.0598245\pi$
$71$	1175.44	1.96478	0.982390	+	0.186840i	$0.0598245\pi$
$72$	0	0
$73$	453.756	0.727508	0.363754	−	0.931495i	$-0.381495\pi$
$73$	453.756	0.727508	0.363754	+	0.931495i	$0.381495\pi$
$74$	1130.43	1.77580
$75$	0	0
$76$	739.088	1.11552
$77$	−861.653	−1.27525
$78$	0	0
$79$	588.978	0.838799	0.419400	−	0.907802i	$-0.362241\pi$
$79$	588.978	0.838799	0.419400	+	0.907802i	$0.362241\pi$
$80$	−165.120	−0.230762
$81$	0	0
$82$	−514.371	−0.692717
$83$	1155.74	1.52842	0.764208	−	0.644970i	$-0.223130\pi$
$83$	1155.74	1.52842	0.764208	+	0.644970i	$0.223130\pi$
$84$	0	0
$85$	−166.724	−0.212750
$86$	1508.17	1.89105
$87$	0	0
$88$	−105.956	−0.128352
$89$	−1590.19	−1.89393	−0.946963	−	0.321343i	$-0.895866\pi$
$89$	−1590.19	−1.89393	−0.946963	+	0.321343i	$0.895866\pi$
$90$	0	0
$91$	−132.107	−0.152182
$92$	−1418.75	−1.60777
$93$	0	0
$94$	215.525	0.236486
$95$	241.824	0.261164
$96$	0	0
$97$	444.125	0.464888	0.232444	−	0.972610i	$-0.425328\pi$
$97$	444.125	0.464888	0.232444	+	0.972610i	$0.425328\pi$
$98$	−557.847	−0.575011
$99$	0	0
$100$	−871.654	−0.871654
$101$	28.2464	0.0278280	0.0139140	−	0.999903i	$-0.495571\pi$
$101$	28.2464	0.0278280	0.0139140	+	0.999903i	$0.495571\pi$
$102$	0	0
$103$	1345.39	1.28704	0.643521	−	0.765428i	$-0.277473\pi$
$103$	1345.39	1.28704	0.643521	+	0.765428i	$0.277473\pi$
$104$	−16.2449	−0.0153168
$105$	0	0
$106$	1378.94	1.26354
$107$	−561.592	−0.507394	−0.253697	−	0.967284i	$-0.581647\pi$
$107$	−561.592	−0.507394	−0.253697	+	0.967284i	$0.581647\pi$
$108$	0	0
$109$	1004.63	0.882811	0.441406	−	0.897308i	$-0.354480\pi$
$109$	1004.63	0.882811	0.441406	+	0.897308i	$0.354480\pi$
$110$	365.778	0.317051
$111$	0	0
$112$	−1521.82	−1.28392
$113$	−378.412	−0.315027	−0.157513	−	0.987517i	$-0.550348\pi$
$113$	−378.412	−0.315027	−0.157513	+	0.987517i	$0.550348\pi$
$114$	0	0
$115$	−464.202	−0.376409
$116$	−1843.17	−1.47529
$117$	0	0
$118$	−3020.75	−2.35664
$119$	−1536.61	−1.18370
$120$	0	0
$121$	197.981	0.148746
$122$	−2567.30	−1.90519
$123$	0	0
$124$	1070.05	0.774944
$125$	−584.063	−0.417922
$126$	0	0
$127$	−241.258	−0.168569	−0.0842843	−	0.996442i	$-0.526860\pi$
$127$	−241.258	−0.168569	−0.0842843	+	0.996442i	$0.526860\pi$
$128$	−345.545	−0.238611
$129$	0	0
$130$	56.0802	0.0378351
$131$	336.134	0.224184	0.112092	−	0.993698i	$-0.464245\pi$
$131$	336.134	0.224184	0.112092	+	0.993698i	$0.464245\pi$
$132$	0	0
$133$	2228.76	1.45307
$134$	529.989	0.341672
$135$	0	0
$136$	−188.954	−0.119138
$137$	771.147	0.480902	0.240451	−	0.970661i	$-0.422705\pi$
$137$	771.147	0.480902	0.240451	+	0.970661i	$0.422705\pi$
$138$	0	0
$139$	2253.93	1.37537	0.687684	−	0.726011i	$-0.258628\pi$
$139$	2253.93	1.37537	0.687684	+	0.726011i	$0.258628\pi$
$140$	385.000	0.232417
$141$	0	0
$142$	−4598.88	−2.71782
$143$	234.420	0.137085
$144$	0	0
$145$	−603.069	−0.345394
$146$	−1775.31	−1.00634
$147$	0	0
$148$	−2111.33	−1.17264
$149$	2962.22	1.62869	0.814345	−	0.580382i	$-0.197097\pi$
$149$	2962.22	1.62869	0.814345	+	0.580382i	$0.197097\pi$
$150$	0	0
$151$	−1453.14	−0.783147	−0.391574	−	0.920147i	$-0.628069\pi$
$151$	−1453.14	−0.783147	−0.391574	+	0.920147i	$0.628069\pi$
$152$	274.068	0.146249
$153$	0	0
$154$	3371.19	1.76401
$155$	350.111	0.181429
$156$	0	0
$157$	−2224.49	−1.13079	−0.565393	−	0.824822i	$-0.691275\pi$
$157$	−2224.49	−1.13079	−0.565393	+	0.824822i	$0.691275\pi$
$158$	−2304.36	−1.16028
$159$	0	0
$160$	594.196	0.293596
$161$	−4278.31	−2.09427
$162$	0	0
$163$	125.611	0.0603596	0.0301798	−	0.999544i	$-0.490392\pi$
$163$	125.611	0.0603596	0.0301798	+	0.999544i	$0.490392\pi$
$164$	960.704	0.457429
$165$	0	0
$166$	−4521.78	−2.11421
$167$	1442.07	0.668207	0.334104	−	0.942536i	$-0.391566\pi$
$167$	1442.07	0.668207	0.334104	+	0.942536i	$0.391566\pi$
$168$	0	0
$169$	−2161.06	−0.983641
$170$	652.302	0.294290
$171$	0	0
$172$	−2816.85	−1.24874
$173$	651.983	0.286528	0.143264	−	0.989685i	$-0.454240\pi$
$173$	651.983	0.286528	0.143264	+	0.989685i	$0.454240\pi$
$174$	0	0
$175$	−2628.52	−1.13541
$176$	2700.44	1.15655
$177$	0	0
$178$	6221.55	2.61980
$179$	4564.86	1.90611	0.953055	−	0.302799i	$-0.0979210\pi$
$179$	4564.86	1.90611	0.953055	+	0.302799i	$0.0979210\pi$
$180$	0	0
$181$	−1046.67	−0.429827	−0.214913	−	0.976633i	$-0.568947\pi$
$181$	−1046.67	−0.429827	−0.214913	+	0.976633i	$0.568947\pi$
$182$	516.863	0.210508
$183$	0	0
$184$	−526.098	−0.210785
$185$	−690.809	−0.274537
$186$	0	0
$187$	2726.67	1.06628
$188$	−402.541	−0.156161
$189$	0	0
$190$	−946.127	−0.361259
$191$	−4314.96	−1.63466	−0.817329	−	0.576171i	$-0.804546\pi$
$191$	−4314.96	−1.63466	−0.817329	+	0.576171i	$0.804546\pi$
$192$	0	0
$193$	430.135	0.160424	0.0802119	−	0.996778i	$-0.474440\pi$
$193$	430.135	0.160424	0.0802119	+	0.996778i	$0.474440\pi$
$194$	−1737.63	−0.643063
$195$	0	0
$196$	1041.91	0.379703
$197$	−91.6105	−0.0331319	−0.0165659	−	0.999863i	$-0.505273\pi$
$197$	−91.6105	−0.0331319	−0.0165659	+	0.999863i	$0.505273\pi$
$198$	0	0
$199$	1383.19	0.492721	0.246360	−	0.969178i	$-0.420765\pi$
$199$	1383.19	0.492721	0.246360	+	0.969178i	$0.420765\pi$
$200$	−323.225	−0.114277
$201$	0	0
$202$	−110.513	−0.0384935
$203$	−5558.18	−1.92171
$204$	0	0
$205$	314.334	0.107093
$206$	−5263.80	−1.78032
$207$	0	0
$208$	414.025	0.138017
$209$	−3954.88	−1.30892
$210$	0	0
$211$	−455.541	−0.148629	−0.0743145	−	0.997235i	$-0.523677\pi$
$211$	−455.541	−0.148629	−0.0743145	+	0.997235i	$0.523677\pi$
$212$	−2575.49	−0.834364
$213$	0	0
$214$	2197.21	0.701861
$215$	−921.650	−0.292354
$216$	0	0
$217$	3226.79	1.00944
$218$	−3930.60	−1.22116
$219$	0	0
$220$	−683.173	−0.209361
$221$	418.047	0.127244
$222$	0	0
$223$	1678.36	0.503997	0.251998	−	0.967728i	$-0.418912\pi$
$223$	1678.36	0.503997	0.251998	+	0.967728i	$0.418912\pi$
$224$	5476.40	1.63351
$225$	0	0
$226$	1480.53	0.435766
$227$	2369.91	0.692937	0.346469	−	0.938062i	$-0.387381\pi$
$227$	2369.91	0.692937	0.346469	+	0.938062i	$0.387381\pi$
$228$	0	0
$229$	−3856.62	−1.11289	−0.556446	−	0.830884i	$-0.687835\pi$
$229$	−3856.62	−1.11289	−0.556446	+	0.830884i	$0.687835\pi$
$230$	1816.18	0.520674
$231$	0	0
$232$	−683.481	−0.193417
$233$	4104.94	1.15418	0.577089	−	0.816681i	$-0.304189\pi$
$233$	4104.94	1.15418	0.577089	+	0.816681i	$0.304189\pi$
$234$	0	0
$235$	−131.708	−0.0365604
$236$	5641.94	1.55618
$237$	0	0
$238$	6011.93	1.63738
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	5639.55	1.50737	0.753684	−	0.657237i	$-0.228275\pi$
$241$	5639.55	1.50737	0.753684	+	0.657237i	$0.228275\pi$
$242$	−774.594	−0.205755
$243$	0	0
$244$	4795.01	1.25807
$245$	340.903	0.0888958
$246$	0	0
$247$	−606.353	−0.156200
$248$	396.794	0.101598
$249$	0	0
$250$	2285.13	0.578097
$251$	4979.92	1.25231	0.626155	−	0.779699i	$-0.284628\pi$
$251$	4979.92	1.25231	0.626155	+	0.779699i	$0.284628\pi$
$252$	0	0
$253$	7591.76	1.88652
$254$	943.915	0.233175
$255$	0	0
$256$	4710.68	1.15007
$257$	6418.63	1.55791	0.778956	−	0.627079i	$-0.215750\pi$
$257$	6418.63	1.55791	0.778956	+	0.627079i	$0.215750\pi$
$258$	0	0
$259$	−6366.83	−1.52747
$260$	−104.742	−0.0249840
$261$	0	0
$262$	−1315.11	−0.310107
$263$	3230.77	0.757482	0.378741	−	0.925503i	$-0.376357\pi$
$263$	3230.77	0.757482	0.378741	+	0.925503i	$0.376357\pi$
$264$	0	0
$265$	−842.678	−0.195341
$266$	−8719.97	−2.00998
$267$	0	0
$268$	−989.873	−0.225620
$269$	−4988.24	−1.13063	−0.565313	−	0.824876i	$-0.691245\pi$
$269$	−4988.24	−1.13063	−0.565313	+	0.824876i	$0.691245\pi$
$270$	0	0
$271$	2933.56	0.657569	0.328784	−	0.944405i	$-0.393361\pi$
$271$	2933.56	0.657569	0.328784	+	0.944405i	$0.393361\pi$
$272$	4815.76	1.07352
$273$	0	0
$274$	−3017.09	−0.665215
$275$	4664.24	1.02278
$276$	0	0
$277$	549.935	0.119287	0.0596433	−	0.998220i	$-0.481004\pi$
$277$	549.935	0.119287	0.0596433	+	0.998220i	$0.481004\pi$
$278$	−8818.44	−1.90250
$279$	0	0
$280$	142.765	0.0304709
$281$	−6605.10	−1.40223	−0.701116	−	0.713048i	$-0.747314\pi$
$281$	−6605.10	−1.40223	−0.701116	+	0.713048i	$0.747314\pi$
$282$	0	0
$283$	−1764.55	−0.370642	−0.185321	−	0.982678i	$-0.559332\pi$
$283$	−1764.55	−0.370642	−0.185321	+	0.982678i	$0.559332\pi$
$284$	8589.45	1.79468
$285$	0	0
$286$	−917.160	−0.189625
$287$	2897.06	0.595846
$288$	0	0
$289$	−50.4567	−0.0102700
$290$	2359.49	0.477772
$291$	0	0
$292$	3315.78	0.664526
$293$	−5417.54	−1.08019	−0.540095	−	0.841604i	$-0.681612\pi$
$293$	−5417.54	−1.08019	−0.540095	+	0.841604i	$0.681612\pi$
$294$	0	0
$295$	1845.99	0.364332
$296$	−782.919	−0.153737
$297$	0	0
$298$	−11589.6	−2.25291
$299$	1163.95	0.225127
$300$	0	0
$301$	−8494.38	−1.62660
$302$	5685.38	1.08330
$303$	0	0
$304$	−6984.99	−1.31782
$305$	1568.89	0.294539
$306$	0	0
$307$	−1086.44	−0.201975	−0.100987	−	0.994888i	$-0.532200\pi$
$307$	−1086.44	−0.201975	−0.100987	+	0.994888i	$0.532200\pi$
$308$	−6296.45	−1.16485
$309$	0	0
$310$	−1369.80	−0.250965
$311$	−1077.87	−0.196528	−0.0982640	−	0.995160i	$-0.531329\pi$
$311$	−1077.87	−0.196528	−0.0982640	+	0.995160i	$0.531329\pi$
$312$	0	0
$313$	−2430.75	−0.438958	−0.219479	−	0.975617i	$-0.570436\pi$
$313$	−2430.75	−0.438958	−0.219479	+	0.975617i	$0.570436\pi$
$314$	8703.23	1.56418
$315$	0	0
$316$	4303.90	0.766182
$317$	2433.10	0.431093	0.215546	−	0.976494i	$-0.430847\pi$
$317$	2433.10	0.431093	0.215546	+	0.976494i	$0.430847\pi$
$318$	0	0
$319$	9862.84	1.73108
$320$	−1003.81	−0.175359
$321$	0	0
$322$	16738.8	2.89694
$323$	−7052.85	−1.21496
$324$	0	0
$325$	715.111	0.122053
$326$	−491.449	−0.0834934
$327$	0	0
$328$	356.247	0.0599709
$329$	−1213.89	−0.203415
$330$	0	0
$331$	−10590.8	−1.75869	−0.879343	−	0.476189i	$-0.842018\pi$
$331$	−10590.8	−1.75869	−0.879343	+	0.476189i	$0.842018\pi$
$332$	8445.44	1.39610
$333$	0	0
$334$	−5642.05	−0.924309
$335$	−323.878	−0.0528220
$336$	0	0
$337$	895.468	0.144746	0.0723728	−	0.997378i	$-0.476943\pi$
$337$	895.468	0.144746	0.0723728	+	0.997378i	$0.476943\pi$
$338$	8455.08	1.36064
$339$	0	0
$340$	−1218.32	−0.194331
$341$	−5725.86	−0.909304
$342$	0	0
$343$	−4416.40	−0.695227
$344$	−1044.54	−0.163715
$345$	0	0
$346$	−2550.86	−0.396345
$347$	2911.05	0.450356	0.225178	−	0.974318i	$-0.427704\pi$
$347$	2911.05	0.450356	0.225178	+	0.974318i	$0.427704\pi$
$348$	0	0
$349$	4260.06	0.653398	0.326699	−	0.945128i	$-0.394064\pi$
$349$	4260.06	0.653398	0.326699	+	0.945128i	$0.394064\pi$
$350$	10284.0	1.57058
$351$	0	0
$352$	−9717.73	−1.47147
$353$	2031.89	0.306365	0.153182	−	0.988198i	$-0.451048\pi$
$353$	2031.89	0.306365	0.153182	+	0.988198i	$0.451048\pi$
$354$	0	0
$355$	2810.40	0.420170
$356$	−11620.1	−1.72996
$357$	0	0
$358$	−17859.9	−2.63666
$359$	6538.90	0.961309	0.480655	−	0.876910i	$-0.340399\pi$
$359$	6538.90	0.961309	0.480655	+	0.876910i	$0.340399\pi$
$360$	0	0
$361$	3370.76	0.491436
$362$	4095.08	0.594565
$363$	0	0
$364$	−965.357	−0.139007
$365$	1084.90	0.155578
$366$	0	0
$367$	9747.25	1.38638	0.693191	−	0.720754i	$-0.256204\pi$
$367$	9747.25	1.38638	0.693191	+	0.720754i	$0.256204\pi$
$368$	13408.3	1.89934
$369$	0	0
$370$	2702.77	0.379757
$371$	−7766.53	−1.08684
$372$	0	0
$373$	3510.30	0.487283	0.243642	−	0.969865i	$-0.421658\pi$
$373$	3510.30	0.487283	0.243642	+	0.969865i	$0.421658\pi$
$374$	−10668.0	−1.47495
$375$	0	0
$376$	−149.270	−0.0204734
$377$	1512.15	0.206577
$378$	0	0
$379$	2936.69	0.398014	0.199007	−	0.979998i	$-0.436228\pi$
$379$	2936.69	0.398014	0.199007	+	0.979998i	$0.436228\pi$
$380$	1767.10	0.238554
$381$	0	0
$382$	16882.2	2.26117
$383$	−4773.42	−0.636842	−0.318421	−	0.947949i	$-0.603153\pi$
$383$	−4773.42	−0.636842	−0.318421	+	0.947949i	$0.603153\pi$
$384$	0	0
$385$	−2060.15	−0.272714
$386$	−1682.89	−0.221909
$387$	0	0
$388$	3245.41	0.424641
$389$	−3916.40	−0.510461	−0.255230	−	0.966880i	$-0.582151\pi$
$389$	−3916.40	−0.510461	−0.255230	+	0.966880i	$0.582151\pi$
$390$	0	0
$391$	13538.6	1.75109
$392$	386.358	0.0497807
$393$	0	0
$394$	358.423	0.0458302
$395$	1408.20	0.179378
$396$	0	0
$397$	−14892.7	−1.88273	−0.941363	−	0.337396i	$-0.890454\pi$
$397$	−14892.7	−1.88273	−0.941363	+	0.337396i	$0.890454\pi$
$398$	−5411.67	−0.681565
$399$	0	0
$400$	8237.84	1.02973
$401$	8757.97	1.09065	0.545327	−	0.838224i	$-0.316406\pi$
$401$	8757.97	1.09065	0.545327	+	0.838224i	$0.316406\pi$
$402$	0	0
$403$	−877.875	−0.108511
$404$	206.408	0.0254188
$405$	0	0
$406$	21746.2	2.65824
$407$	11297.8	1.37595
$408$	0	0
$409$	11849.8	1.43260	0.716301	−	0.697792i	$-0.245834\pi$
$409$	11849.8	1.43260	0.716301	+	0.697792i	$0.245834\pi$
$410$	−1229.82	−0.148138
$411$	0	0
$412$	9831.33	1.17562
$413$	17013.6	2.02708
$414$	0	0
$415$	2763.28	0.326853
$416$	−1489.90	−0.175597
$417$	0	0
$418$	15473.4	1.81059
$419$	15018.0	1.75102	0.875508	−	0.483204i	$-0.160527\pi$
$419$	15018.0	1.75102	0.875508	+	0.483204i	$0.160527\pi$
$420$	0	0
$421$	4366.03	0.505433	0.252717	−	0.967540i	$-0.418676\pi$
$421$	4366.03	0.505433	0.252717	+	0.967540i	$0.418676\pi$
$422$	1782.29	0.205594
$423$	0	0
$424$	−955.039	−0.109389
$425$	8317.87	0.949355
$426$	0	0
$427$	14459.6	1.63876
$428$	−4103.79	−0.463467
$429$	0	0
$430$	3605.93	0.404403
$431$	9707.32	1.08488	0.542442	−	0.840093i	$-0.317500\pi$
$431$	9707.32	1.08488	0.542442	+	0.840093i	$0.317500\pi$
$432$	0	0
$433$	−5084.84	−0.564347	−0.282173	−	0.959363i	$-0.591055\pi$
$433$	−5084.84	−0.564347	−0.282173	+	0.959363i	$0.591055\pi$
$434$	−12624.7	−1.39633
$435$	0	0
$436$	7341.27	0.806383
$437$	−19636.9	−2.14957
$438$	0	0
$439$	−914.886	−0.0994650	−0.0497325	−	0.998763i	$-0.515837\pi$
$439$	−914.886	−0.0994650	−0.0497325	+	0.998763i	$0.515837\pi$
$440$	−253.333	−0.0274482
$441$	0	0
$442$	−1635.59	−0.176012
$443$	15982.7	1.71413	0.857066	−	0.515207i	$-0.172285\pi$
$443$	15982.7	1.71413	0.857066	+	0.515207i	$0.172285\pi$
$444$	0	0
$445$	−3802.02	−0.405018
$446$	−6566.53	−0.697162
$447$	0	0
$448$	−9251.64	−0.975668
$449$	−11330.3	−1.19089	−0.595444	−	0.803397i	$-0.703024\pi$
$449$	−11330.3	−1.19089	−0.595444	+	0.803397i	$0.703024\pi$
$450$	0	0
$451$	−5140.75	−0.536738
$452$	−2765.22	−0.287754
$453$	0	0
$454$	−9272.21	−0.958517
$455$	−315.857	−0.0325442
$456$	0	0
$457$	893.345	0.0914418	0.0457209	−	0.998954i	$-0.485442\pi$
$457$	893.345	0.0914418	0.0457209	+	0.998954i	$0.485442\pi$
$458$	15088.9	1.53943
$459$	0	0
$460$	−3392.12	−0.343822
$461$	5456.89	0.551308	0.275654	−	0.961257i	$-0.411106\pi$
$461$	5456.89	0.551308	0.275654	+	0.961257i	$0.411106\pi$
$462$	0	0
$463$	−5481.23	−0.550182	−0.275091	−	0.961418i	$-0.588708\pi$
$463$	−5481.23	−0.550182	−0.275091	+	0.961418i	$0.588708\pi$
$464$	17419.4	1.74284
$465$	0	0
$466$	−16060.4	−1.59654
$467$	8349.12	0.827304	0.413652	−	0.910435i	$-0.364253\pi$
$467$	8349.12	0.827304	0.413652	+	0.910435i	$0.364253\pi$
$468$	0	0
$469$	−2985.02	−0.293892
$470$	515.304	0.0505727
$471$	0	0
$472$	2092.14	0.204022
$473$	15073.1	1.46524
$474$	0	0
$475$	−12064.6	−1.16539
$476$	−11228.6	−1.08123
$477$	0	0
$478$	−935.080	−0.0894761
$479$	4382.11	0.418004	0.209002	−	0.977915i	$-0.432979\pi$
$479$	4382.11	0.418004	0.209002	+	0.977915i	$0.432979\pi$
$480$	0	0
$481$	1732.15	0.164198
$482$	−22064.6	−2.08509
$483$	0	0
$484$	1446.73	0.135869
$485$	1061.87	0.0994166
$486$	0	0
$487$	1854.59	0.172566	0.0862828	−	0.996271i	$-0.472501\pi$
$487$	1854.59	0.172566	0.0862828	+	0.996271i	$0.472501\pi$
$488$	1778.08	0.164938
$489$	0	0
$490$	−1333.77	−0.122967
$491$	11785.9	1.08328	0.541642	−	0.840609i	$-0.317803\pi$
$491$	11785.9	1.08328	0.541642	+	0.840609i	$0.317803\pi$
$492$	0	0
$493$	17588.7	1.60680
$494$	2372.34	0.216066
$495$	0	0
$496$	−10112.8	−0.915482
$497$	25902.0	2.33775
$498$	0	0
$499$	17765.7	1.59380	0.796898	−	0.604114i	$-0.206473\pi$
$499$	17765.7	1.59380	0.796898	+	0.604114i	$0.206473\pi$
$500$	−4267.99	−0.381741
$501$	0	0
$502$	−19483.8	−1.73228
$503$	454.779	0.0403133	0.0201566	−	0.999797i	$-0.493584\pi$
$503$	454.779	0.0403133	0.0201566	+	0.999797i	$0.493584\pi$
$504$	0	0
$505$	67.5351	0.00595104
$506$	−29702.5	−2.60956
$507$	0	0
$508$	−1762.97	−0.153975
$509$	−11369.7	−0.990088	−0.495044	−	0.868868i	$-0.664848\pi$
$509$	−11369.7	−0.990088	−0.495044	+	0.868868i	$0.664848\pi$
$510$	0	0
$511$	9998.93	0.865610
$512$	−15666.0	−1.35224
$513$	0	0
$514$	−25112.7	−2.15501
$515$	3216.73	0.275235
$516$	0	0
$517$	2154.01	0.183236
$518$	24910.0	2.11290
$519$	0	0
$520$	−38.8404	−0.00327551
$521$	5203.86	0.437591	0.218796	−	0.975771i	$-0.429787\pi$
$521$	5203.86	0.437591	0.218796	+	0.975771i	$0.429787\pi$
$522$	0	0
$523$	−18951.9	−1.58453	−0.792265	−	0.610178i	$-0.791098\pi$
$523$	−18951.9	−1.58453	−0.792265	+	0.610178i	$0.791098\pi$
$524$	2456.27	0.204776
$525$	0	0
$526$	−12640.3	−1.04780
$527$	−10211.1	−0.844025
$528$	0	0
$529$	25527.9	2.09812
$530$	3296.95	0.270208
$531$	0	0
$532$	16286.5	1.32727
$533$	−788.168	−0.0640513
$534$	0	0
$535$	−1342.72	−0.108507
$536$	−367.063	−0.0295797
$537$	0	0
$538$	19516.3	1.56396
$539$	−5575.27	−0.445536
$540$	0	0
$541$	−4050.94	−0.321929	−0.160964	−	0.986960i	$-0.551460\pi$
$541$	−4050.94	−0.321929	−0.160964	+	0.986960i	$0.551460\pi$
$542$	−11477.5	−0.909593
$543$	0	0
$544$	−17329.9	−1.36583
$545$	2402.00	0.188790
$546$	0	0
$547$	−1312.96	−0.102629	−0.0513146	−	0.998683i	$-0.516341\pi$
$547$	−1312.96	−0.102629	−0.0513146	+	0.998683i	$0.516341\pi$
$548$	5635.09	0.439268
$549$	0	0
$550$	−18248.7	−1.41478
$551$	−25511.4	−1.97245
$552$	0	0
$553$	12978.7	0.998027
$554$	−2151.60	−0.165005
$555$	0	0
$556$	16470.4	1.25630
$557$	17090.6	1.30009	0.650046	−	0.759894i	$-0.274749\pi$
$557$	17090.6	1.30009	0.650046	+	0.759894i	$0.274749\pi$
$558$	0	0
$559$	2310.96	0.174854
$560$	−3638.56	−0.274567
$561$	0	0
$562$	25842.2	1.93966
$563$	1901.70	0.142357	0.0711784	−	0.997464i	$-0.477324\pi$
$563$	1901.70	0.142357	0.0711784	+	0.997464i	$0.477324\pi$
$564$	0	0
$565$	−904.755	−0.0673688
$566$	6903.74	0.512696
$567$	0	0
$568$	3185.13	0.235290
$569$	13568.8	0.999711	0.499855	−	0.866109i	$-0.333386\pi$
$569$	13568.8	0.999711	0.499855	+	0.866109i	$0.333386\pi$
$570$	0	0
$571$	3789.91	0.277763	0.138882	−	0.990309i	$-0.455649\pi$
$571$	3789.91	0.277763	0.138882	+	0.990309i	$0.455649\pi$
$572$	1713.00	0.125217
$573$	0	0
$574$	−11334.6	−0.824214
$575$	23159.1	1.67965
$576$	0	0
$577$	6406.05	0.462197	0.231098	−	0.972930i	$-0.425768\pi$
$577$	6406.05	0.462197	0.231098	+	0.972930i	$0.425768\pi$
$578$	197.410	0.0142062
$579$	0	0
$580$	−4406.87	−0.315492
$581$	25467.7	1.81855
$582$	0	0
$583$	13781.5	0.979026
$584$	1229.55	0.0871221
$585$	0	0
$586$	21195.9	1.49419
$587$	−20627.1	−1.45038	−0.725189	−	0.688550i	$-0.758248\pi$
$587$	−20627.1	−1.45038	−0.725189	+	0.688550i	$0.758248\pi$
$588$	0	0
$589$	14810.6	1.03609
$590$	−7222.40	−0.503968
$591$	0	0
$592$	19953.8	1.38529
$593$	5080.50	0.351823	0.175912	−	0.984406i	$-0.443713\pi$
$593$	5080.50	0.351823	0.175912	+	0.984406i	$0.443713\pi$
$594$	0	0
$595$	−3673.91	−0.253136
$596$	21646.2	1.48769
$597$	0	0
$598$	−4553.92	−0.311411
$599$	−19962.9	−1.36171	−0.680853	−	0.732420i	$-0.738391\pi$
$599$	−19962.9	−1.36171	−0.680853	+	0.732420i	$0.738391\pi$
$600$	0	0
$601$	14806.2	1.00492	0.502459	−	0.864601i	$-0.332429\pi$
$601$	14806.2	1.00492	0.502459	+	0.864601i	$0.332429\pi$
$602$	33234.0	2.25003
$603$	0	0
$604$	−10618.7	−0.715348
$605$	473.358	0.0318095
$606$	0	0
$607$	−1505.78	−0.100688	−0.0503441	−	0.998732i	$-0.516032\pi$
$607$	−1505.78	−0.100688	−0.0503441	+	0.998732i	$0.516032\pi$
$608$	25136.0	1.67664
$609$	0	0
$610$	−6138.23	−0.407426
$611$	330.248	0.0218664
$612$	0	0
$613$	−155.161	−0.0102233	−0.00511166	−	0.999987i	$-0.501627\pi$
$613$	−155.161	−0.0102233	−0.00511166	+	0.999987i	$0.501627\pi$
$614$	4250.66	0.279385
$615$	0	0
$616$	−2334.84	−0.152717
$617$	3855.11	0.251541	0.125770	−	0.992059i	$-0.459860\pi$
$617$	3855.11	0.251541	0.125770	+	0.992059i	$0.459860\pi$
$618$	0	0
$619$	−6526.08	−0.423756	−0.211878	−	0.977296i	$-0.567958\pi$
$619$	−6526.08	−0.423756	−0.211878	+	0.977296i	$0.567958\pi$
$620$	2558.40	0.165722
$621$	0	0
$622$	4217.12	0.271851
$623$	−35041.2	−2.25345
$624$	0	0
$625$	13514.0	0.864895
$626$	9510.22	0.607196
$627$	0	0
$628$	−16255.2	−1.03289
$629$	20147.6	1.27717
$630$	0	0
$631$	−1156.46	−0.0729602	−0.0364801	−	0.999334i	$-0.511615\pi$
$631$	−1156.46	−0.0729602	−0.0364801	+	0.999334i	$0.511615\pi$
$632$	1595.97	0.100450
$633$	0	0
$634$	−9519.42	−0.596316
$635$	−576.830	−0.0360485
$636$	0	0
$637$	−854.786	−0.0531678
$638$	−38588.1	−2.39454
$639$	0	0
$640$	−826.172	−0.0510271
$641$	−23404.6	−1.44216	−0.721081	−	0.692851i	$-0.756354\pi$
$641$	−23404.6	−1.44216	−0.721081	+	0.692851i	$0.756354\pi$
$642$	0	0
$643$	−7822.95	−0.479793	−0.239897	−	0.970798i	$-0.577114\pi$
$643$	−7822.95	−0.479793	−0.239897	+	0.970798i	$0.577114\pi$
$644$	−31263.4	−1.91297
$645$	0	0
$646$	27594.0	1.68061
$647$	−24643.7	−1.49744	−0.748719	−	0.662888i	$-0.769331\pi$
$647$	−24643.7	−1.49744	−0.748719	+	0.662888i	$0.769331\pi$
$648$	0	0
$649$	−30190.2	−1.82599
$650$	−2797.85	−0.168832
$651$	0	0
$652$	917.891	0.0551340
$653$	−14912.1	−0.893656	−0.446828	−	0.894620i	$-0.647446\pi$
$653$	−14912.1	−0.893656	−0.446828	+	0.894620i	$0.647446\pi$
$654$	0	0
$655$	803.670	0.0479420
$656$	−9079.43	−0.540385
$657$	0	0
$658$	4749.29	0.281378
$659$	−23817.4	−1.40788	−0.703942	−	0.710258i	$-0.748578\pi$
$659$	−23817.4	−1.40788	−0.703942	+	0.710258i	$0.748578\pi$
$660$	0	0
$661$	−28302.3	−1.66541	−0.832703	−	0.553720i	$-0.813208\pi$
$661$	−28302.3	−1.66541	−0.832703	+	0.553720i	$0.813208\pi$
$662$	41436.3	2.43273
$663$	0	0
$664$	3131.73	0.183034
$665$	5328.80	0.310740
$666$	0	0
$667$	48971.4	2.84285
$668$	10537.8	0.610358
$669$	0	0
$670$	1267.16	0.0730669
$671$	−25658.3	−1.47619
$672$	0	0
$673$	−9269.54	−0.530928	−0.265464	−	0.964121i	$-0.585525\pi$
$673$	−9269.54	−0.530928	−0.265464	+	0.964121i	$0.585525\pi$
$674$	−3503.49	−0.200222
$675$	0	0
$676$	−15791.8	−0.898484
$677$	18415.2	1.04543	0.522713	−	0.852509i	$-0.324920\pi$
$677$	18415.2	1.04543	0.522713	+	0.852509i	$0.324920\pi$
$678$	0	0
$679$	9786.71	0.553136
$680$	−451.776	−0.0254777
$681$	0	0
$682$	22402.2	1.25781
$683$	19230.9	1.07738	0.538689	−	0.842504i	$-0.318920\pi$
$683$	19230.9	1.07738	0.538689	+	0.842504i	$0.318920\pi$
$684$	0	0
$685$	1843.75	0.102841
$686$	17279.0	0.961685
$687$	0	0
$688$	26621.5	1.47520
$689$	2112.95	0.116832
$690$	0	0
$691$	7472.35	0.411377	0.205689	−	0.978618i	$-0.434057\pi$
$691$	7472.35	0.411377	0.205689	+	0.978618i	$0.434057\pi$
$692$	4764.31	0.261722
$693$	0	0
$694$	−11389.4	−0.622963
$695$	5388.98	0.294123
$696$	0	0
$697$	−9167.64	−0.498205
$698$	−16667.4	−0.903823
$699$	0	0
$700$	−19207.7	−1.03712
$701$	−11848.3	−0.638380	−0.319190	−	0.947691i	$-0.603411\pi$
$701$	−11848.3	−0.638380	−0.319190	+	0.947691i	$0.603411\pi$
$702$	0	0
$703$	−29223.0	−1.56780
$704$	16416.8	0.878880
$705$	0	0
$706$	−7949.72	−0.423784
$707$	622.436	0.0331105
$708$	0	0
$709$	−1089.48	−0.0577096	−0.0288548	−	0.999584i	$-0.509186\pi$
$709$	−1089.48	−0.0577096	−0.0288548	+	0.999584i	$0.509186\pi$
$710$	−10995.6	−0.581207
$711$	0	0
$712$	−4308.97	−0.226805
$713$	−28430.2	−1.49330
$714$	0	0
$715$	560.480	0.0293157
$716$	33357.3	1.74109
$717$	0	0
$718$	−25583.3	−1.32975
$719$	6751.30	0.350182	0.175091	−	0.984552i	$-0.443978\pi$
$719$	6751.30	0.350182	0.175091	+	0.984552i	$0.443978\pi$
$720$	0	0
$721$	29647.0	1.53136
$722$	−13188.0	−0.679787
$723$	0	0
$724$	−7648.48	−0.392615
$725$	30087.2	1.54125
$726$	0	0
$727$	−18562.2	−0.946952	−0.473476	−	0.880807i	$-0.657001\pi$
$727$	−18562.2	−0.946952	−0.473476	+	0.880807i	$0.657001\pi$
$728$	−357.972	−0.0182244
$729$	0	0
$730$	−4244.62	−0.215206
$731$	26880.2	1.36005
$732$	0	0
$733$	26628.0	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$733$	26628.0	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$734$	−38135.8	−1.91774
$735$	0	0
$736$	−48250.8	−2.41651
$737$	5296.84	0.264738
$738$	0	0
$739$	30749.5	1.53063	0.765317	−	0.643653i	$-0.222582\pi$
$739$	30749.5	1.53063	0.765317	+	0.643653i	$0.222582\pi$
$740$	−5048.02	−0.250769
$741$	0	0
$742$	30386.3	1.50339
$743$	−9757.41	−0.481783	−0.240891	−	0.970552i	$-0.577440\pi$
$743$	−9757.41	−0.481783	−0.240891	+	0.970552i	$0.577440\pi$
$744$	0	0
$745$	7082.45	0.348296
$746$	−13733.9	−0.674042
$747$	0	0
$748$	19924.9	0.973967
$749$	−12375.2	−0.603712
$750$	0	0
$751$	24497.8	1.19033	0.595164	−	0.803604i	$-0.297087\pi$
$751$	24497.8	1.19033	0.595164	+	0.803604i	$0.297087\pi$
$752$	3804.34	0.184481
$753$	0	0
$754$	−5916.23	−0.285751
$755$	−3474.36	−0.167477
$756$	0	0
$757$	−40303.4	−1.93508	−0.967539	−	0.252723i	$-0.918674\pi$
$757$	−40303.4	−1.93508	−0.967539	+	0.252723i	$0.918674\pi$
$758$	−11489.7	−0.550560
$759$	0	0
$760$	655.275	0.0312754
$761$	4432.83	0.211156	0.105578	−	0.994411i	$-0.466331\pi$
$761$	4432.83	0.211156	0.105578	+	0.994411i	$0.466331\pi$
$762$	0	0
$763$	22138.0	1.05039
$764$	−31531.2	−1.49314
$765$	0	0
$766$	18675.9	0.880922
$767$	−4628.69	−0.217904
$768$	0	0
$769$	−14418.4	−0.676124	−0.338062	−	0.941124i	$-0.609771\pi$
$769$	−14418.4	−0.676124	−0.338062	+	0.941124i	$0.609771\pi$
$770$	8060.26	0.377236
$771$	0	0
$772$	3143.17	0.146535
$773$	−32829.3	−1.52754	−0.763769	−	0.645490i	$-0.776653\pi$
$773$	−32829.3	−1.52754	−0.763769	+	0.645490i	$0.776653\pi$
$774$	0	0
$775$	−17467.1	−0.809594
$776$	1203.46	0.0556722
$777$	0	0
$778$	15322.8	0.706103
$779$	13297.1	0.611579
$780$	0	0
$781$	−45962.4	−2.10584
$782$	−52969.2	−2.42222
$783$	0	0
$784$	−9846.85	−0.448563
$785$	−5318.58	−0.241819
$786$	0	0
$787$	−32650.6	−1.47887	−0.739434	−	0.673229i	$-0.764907\pi$
$787$	−32650.6	−1.47887	−0.739434	+	0.673229i	$0.764907\pi$
$788$	−669.436	−0.0302635
$789$	0	0
$790$	−5509.54	−0.248128
$791$	−8338.67	−0.374828
$792$	0	0
$793$	−3933.86	−0.176161
$794$	58267.1	2.60431
$795$	0	0
$796$	10107.5	0.450064
$797$	13771.3	0.612050	0.306025	−	0.952023i	$-0.401001\pi$
$797$	13771.3	0.612050	0.306025	+	0.952023i	$0.401001\pi$
$798$	0	0
$799$	3841.30	0.170082
$800$	−29644.5	−1.31011
$801$	0	0
$802$	−34265.3	−1.50866
$803$	−17742.8	−0.779740
$804$	0	0
$805$	−10229.1	−0.447862
$806$	3434.66	0.150100
$807$	0	0
$808$	76.5401	0.00333251
$809$	−34001.0	−1.47764	−0.738820	−	0.673903i	$-0.764617\pi$
$809$	−34001.0	−1.47764	−0.738820	+	0.673903i	$0.764617\pi$
$810$	0	0
$811$	3174.40	0.137445	0.0687227	−	0.997636i	$-0.478108\pi$
$811$	3174.40	0.137445	0.0687227	+	0.997636i	$0.478108\pi$
$812$	−40615.9	−1.75534
$813$	0	0
$814$	−44202.2	−1.90330
$815$	300.326	0.0129079
$816$	0	0
$817$	−38988.2	−1.66955
$818$	−46361.9	−1.98167
$819$	0	0
$820$	2296.97	0.0978216
$821$	31067.6	1.32067	0.660334	−	0.750972i	$-0.270415\pi$
$821$	31067.6	1.32067	0.660334	+	0.750972i	$0.270415\pi$
$822$	0	0
$823$	−4460.33	−0.188915	−0.0944576	−	0.995529i	$-0.530112\pi$
$823$	−4460.33	−0.188915	−0.0944576	+	0.995529i	$0.530112\pi$
$824$	3645.64	0.154129
$825$	0	0
$826$	−66565.1	−2.80399
$827$	−7393.38	−0.310874	−0.155437	−	0.987846i	$-0.549679\pi$
$827$	−7393.38	−0.310874	−0.155437	+	0.987846i	$0.549679\pi$
$828$	0	0
$829$	18851.4	0.789789	0.394894	−	0.918727i	$-0.370781\pi$
$829$	18851.4	0.789789	0.394894	+	0.918727i	$0.370781\pi$
$830$	−10811.2	−0.452125
$831$	0	0
$832$	2516.99	0.104881
$833$	−9942.52	−0.413551
$834$	0	0
$835$	3447.88	0.142897
$836$	−28900.0	−1.19561
$837$	0	0
$838$	−58757.3	−2.42212
$839$	−16978.9	−0.698663	−0.349331	−	0.936999i	$-0.613591\pi$
$839$	−16978.9	−0.698663	−0.349331	+	0.936999i	$0.613591\pi$
$840$	0	0
$841$	39232.3	1.60861
$842$	−17082.0	−0.699149
$843$	0	0
$844$	−3328.82	−0.135762
$845$	−5166.93	−0.210352
$846$	0	0
$847$	4362.69	0.176982
$848$	24340.5	0.985678
$849$	0	0
$850$	−32543.4	−1.31321
$851$	56096.1	2.25964
$852$	0	0
$853$	27627.5	1.10896	0.554482	−	0.832196i	$-0.312916\pi$
$853$	27627.5	1.10896	0.554482	+	0.832196i	$0.312916\pi$
$854$	−56572.9	−2.26684
$855$	0	0
$856$	−1521.76	−0.0607625
$857$	−31877.7	−1.27062	−0.635309	−	0.772258i	$-0.719127\pi$
$857$	−31877.7	−1.27062	−0.635309	+	0.772258i	$0.719127\pi$
$858$	0	0
$859$	−22898.5	−0.909530	−0.454765	−	0.890612i	$-0.650277\pi$
$859$	−22898.5	−0.909530	−0.454765	+	0.890612i	$0.650277\pi$
$860$	−6734.88	−0.267044
$861$	0	0
$862$	−37979.6	−1.50068
$863$	27052.4	1.06706	0.533532	−	0.845780i	$-0.320865\pi$
$863$	27052.4	1.06706	0.533532	+	0.845780i	$0.320865\pi$
$864$	0	0
$865$	1558.84	0.0612743
$866$	19894.3	0.780642
$867$	0	0
$868$	23579.5	0.922051
$869$	−23030.3	−0.899022
$870$	0	0
$871$	812.099	0.0315923
$872$	2722.28	0.105720
$873$	0	0
$874$	76828.9	2.97343
$875$	−12870.4	−0.497255
$876$	0	0
$877$	17159.8	0.660712	0.330356	−	0.943856i	$-0.392831\pi$
$877$	17159.8	0.660712	0.330356	+	0.943856i	$0.392831\pi$
$878$	3579.46	0.137587
$879$	0	0
$880$	6456.54	0.247329
$881$	3825.51	0.146294	0.0731468	−	0.997321i	$-0.476696\pi$
$881$	3825.51	0.146294	0.0731468	+	0.997321i	$0.476696\pi$
$882$	0	0
$883$	−41723.4	−1.59015	−0.795075	−	0.606511i	$-0.792569\pi$
$883$	−41723.4	−1.59015	−0.795075	+	0.606511i	$0.792569\pi$
$884$	3054.84	0.116228
$885$	0	0
$886$	−62531.8	−2.37110
$887$	27451.9	1.03917	0.519586	−	0.854418i	$-0.326086\pi$
$887$	27451.9	1.03917	0.519586	+	0.854418i	$0.326086\pi$
$888$	0	0
$889$	−5316.35	−0.200568
$890$	14875.3	0.560247
$891$	0	0
$892$	12264.5	0.460364
$893$	−5571.59	−0.208786
$894$	0	0
$895$	10914.2	0.407623
$896$	−7614.40	−0.283906
$897$	0	0
$898$	44329.4	1.64732
$899$	−36935.2	−1.37025
$900$	0	0
$901$	24576.9	0.908742
$902$	20113.0	0.742451
$903$	0	0
$904$	−1025.39	−0.0377258
$905$	−2502.52	−0.0919188
$906$	0	0
$907$	21127.2	0.773447	0.386723	−	0.922196i	$-0.373607\pi$
$907$	21127.2	0.773447	0.386723	+	0.922196i	$0.373607\pi$
$908$	17317.9	0.632947
$909$	0	0
$910$	1235.78	0.0450173
$911$	33233.2	1.20863	0.604316	−	0.796745i	$-0.293446\pi$
$911$	33233.2	1.20863	0.604316	+	0.796745i	$0.293446\pi$
$912$	0	0
$913$	−45191.8	−1.63815
$914$	−3495.18	−0.126488
$915$	0	0
$916$	−28181.9	−1.01655
$917$	7407.01	0.266741
$918$	0	0
$919$	1874.05	0.0672678	0.0336339	−	0.999434i	$-0.489292\pi$
$919$	1874.05	0.0672678	0.0336339	+	0.999434i	$0.489292\pi$
$920$	−1257.86	−0.0450765
$921$	0	0
$922$	−21349.9	−0.762605
$923$	−7046.85	−0.251300
$924$	0	0
$925$	34464.5	1.22507
$926$	21445.1	0.761048
$927$	0	0
$928$	−62685.2	−2.21739
$929$	−24055.6	−0.849556	−0.424778	−	0.905298i	$-0.639648\pi$
$929$	−24055.6	−0.849556	−0.424778	+	0.905298i	$0.639648\pi$
$930$	0	0
$931$	14421.1	0.507660
$932$	29996.5	1.05426
$933$	0	0
$934$	−32665.7	−1.14438
$935$	6519.27	0.228024
$936$	0	0
$937$	−269.220	−0.00938639	−0.00469319	−	0.999989i	$-0.501494\pi$
$937$	−269.220	−0.00938639	−0.00469319	+	0.999989i	$0.501494\pi$
$938$	11678.8	0.406531
$939$	0	0
$940$	−962.445	−0.0333952
$941$	−42886.2	−1.48571	−0.742853	−	0.669455i	$-0.766528\pi$
$941$	−42886.2	−1.48571	−0.742853	+	0.669455i	$0.766528\pi$
$942$	0	0
$943$	−25525.1	−0.881454
$944$	−53320.9	−1.83840
$945$	0	0
$946$	−58972.9	−2.02682
$947$	−24485.9	−0.840216	−0.420108	−	0.907474i	$-0.638008\pi$
$947$	−24485.9	−0.840216	−0.420108	+	0.907474i	$0.638008\pi$
$948$	0	0
$949$	−2720.29	−0.0930499
$950$	47202.4	1.61205
$951$	0	0
$952$	−4163.78	−0.141753
$953$	−14568.8	−0.495203	−0.247602	−	0.968862i	$-0.579642\pi$
$953$	−14568.8	−0.495203	−0.247602	+	0.968862i	$0.579642\pi$
$954$	0	0
$955$	−10316.8	−0.349573
$956$	1746.47	0.0590847
$957$	0	0
$958$	−17144.9	−0.578211
$959$	16992.9	0.572190
$960$	0	0
$961$	−8348.33	−0.280230
$962$	−6776.97	−0.227129
$963$	0	0
$964$	41210.5	1.37687
$965$	1028.42	0.0343068
$966$	0	0
$967$	53116.2	1.76639	0.883197	−	0.469003i	$-0.155387\pi$
$967$	53116.2	1.76639	0.883197	+	0.469003i	$0.155387\pi$
$968$	536.474	0.0178129
$969$	0	0
$970$	−4154.53	−0.137520
$971$	5452.96	0.180220	0.0901101	−	0.995932i	$-0.471278\pi$
$971$	5452.96	0.180220	0.0901101	+	0.995932i	$0.471278\pi$
$972$	0	0
$973$	49667.5	1.63645
$974$	−7256.02	−0.238704
$975$	0	0
$976$	−45316.8	−1.48622
$977$	6249.86	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$977$	6249.86	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$978$	0	0
$979$	62179.8	2.02990
$980$	2491.12	0.0811998
$981$	0	0
$982$	−46112.2	−1.49847
$983$	−644.222	−0.0209028	−0.0104514	−	0.999945i	$-0.503327\pi$
$983$	−644.222	−0.0209028	−0.0104514	+	0.999945i	$0.503327\pi$
$984$	0	0
$985$	−219.034	−0.00708528
$986$	−68815.1	−2.22264
$987$	0	0
$988$	−4430.87	−0.142677
$989$	74841.3	2.40628
$990$	0	0
$991$	2433.76	0.0780131	0.0390065	−	0.999239i	$-0.487581\pi$
$991$	2433.76	0.0780131	0.0390065	+	0.999239i	$0.487581\pi$
$992$	36391.8	1.16476
$993$	0	0
$994$	−101341.	−3.23373
$995$	3307.09	0.105369
$996$	0	0
$997$	6103.54	0.193883	0.0969413	−	0.995290i	$-0.469094\pi$
$997$	6103.54	0.193883	0.0969413	+	0.995290i	$0.469094\pi$
$998$	−69507.9	−2.20464
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.8		37
3.2	odd	2		239.4.a.b.1.30	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.30	✓	37	3.2	odd	2
2151.4.a.f.1.8		37	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-3.91247 q^{2} +7.30741 q^{4} +2.39092 q^{5} +22.0359 q^{7} +2.70973 q^{8} +O(q^{10})\)	\(q-3.91247 q^{2} +7.30741 q^{4} +2.39092 q^{5} +22.0359 q^{7} +2.70973 q^{8} -9.35442 q^{10} -39.1022 q^{11} -5.99505 q^{13} -86.2149 q^{14} -69.0610 q^{16} -69.7319 q^{17} +101.142 q^{19} +17.4715 q^{20} +152.986 q^{22} -194.152 q^{23} -119.283 q^{25} +23.4555 q^{26} +161.026 q^{28} -252.233 q^{29} +146.433 q^{31} +248.521 q^{32} +272.824 q^{34} +52.6862 q^{35} -288.929 q^{37} -395.716 q^{38} +6.47875 q^{40} +131.470 q^{41} -385.479 q^{43} -285.736 q^{44} +759.612 q^{46} -55.0867 q^{47} +142.582 q^{49} +466.693 q^{50} -43.8083 q^{52} -352.449 q^{53} -93.4904 q^{55} +59.7113 q^{56} +986.852 q^{58} +772.084 q^{59} +656.185 q^{61} -572.915 q^{62} -419.844 q^{64} -14.3337 q^{65} -135.462 q^{67} -509.560 q^{68} -206.133 q^{70} +1175.44 q^{71} +453.756 q^{73} +1130.43 q^{74} +739.088 q^{76} -861.653 q^{77} +588.978 q^{79} -165.120 q^{80} -514.371 q^{82} +1155.74 q^{83} -166.724 q^{85} +1508.17 q^{86} -105.956 q^{88} -1590.19 q^{89} -132.107 q^{91} -1418.75 q^{92} +215.525 q^{94} +241.824 q^{95} +444.125 q^{97} -557.847 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

L-functions

Modular forms

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Database

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