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

$f(q)$	$=$	$q+0.850702 q^{2} -7.27631 q^{4} -12.0519 q^{5} -14.5326 q^{7} -12.9956 q^{8} +O(q^{10})$	\(q+0.850702 q^{2} -7.27631 q^{4} -12.0519 q^{5} -14.5326 q^{7} -12.9956 q^{8} -10.2525 q^{10} +62.1036 q^{11} -35.3151 q^{13} -12.3629 q^{14} +47.1551 q^{16} -42.4200 q^{17} -55.1989 q^{19} +87.6930 q^{20} +52.8317 q^{22} -40.9974 q^{23} +20.2474 q^{25} -30.0426 q^{26} +105.744 q^{28} +223.019 q^{29} +107.746 q^{31} +144.080 q^{32} -36.0868 q^{34} +175.145 q^{35} -7.24755 q^{37} -46.9578 q^{38} +156.621 q^{40} -102.577 q^{41} +246.621 q^{43} -451.885 q^{44} -34.8766 q^{46} +384.920 q^{47} -131.803 q^{49} +17.2245 q^{50} +256.963 q^{52} +646.786 q^{53} -748.464 q^{55} +188.860 q^{56} +189.723 q^{58} +180.551 q^{59} -104.757 q^{61} +91.6601 q^{62} -254.672 q^{64} +425.612 q^{65} +801.715 q^{67} +308.661 q^{68} +148.996 q^{70} -944.820 q^{71} -585.288 q^{73} -6.16551 q^{74} +401.644 q^{76} -902.528 q^{77} +611.307 q^{79} -568.306 q^{80} -87.2623 q^{82} -221.844 q^{83} +511.240 q^{85} +209.801 q^{86} -807.073 q^{88} +1360.05 q^{89} +513.220 q^{91} +298.310 q^{92} +327.453 q^{94} +665.249 q^{95} +544.064 q^{97} -112.125 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.850702	0.300769	0.150384	−	0.988628i	$-0.451949\pi$
$2$	0.850702	0.300769	0.150384	+	0.988628i	$0.451949\pi$
$3$	0	0
$3$	0	0
$4$	−7.27631	−0.909538
$5$	−12.0519	−1.07795	−0.538976	−	0.842321i	$-0.681189\pi$
$5$	−12.0519	−1.07795	−0.538976	+	0.842321i	$0.681189\pi$
$6$	0	0
$7$	−14.5326	−0.784688	−0.392344	−	0.919819i	$-0.628336\pi$
$7$	−14.5326	−0.784688	−0.392344	+	0.919819i	$0.628336\pi$
$8$	−12.9956	−0.574329
$9$	0	0
$10$	−10.2525	−0.324214
$11$	62.1036	1.70227	0.851134	−	0.524949i	$-0.175916\pi$
$11$	62.1036	1.70227	0.851134	+	0.524949i	$0.175916\pi$
$12$	0	0
$13$	−35.3151	−0.753433	−0.376717	−	0.926329i	$-0.622947\pi$
$13$	−35.3151	−0.753433	−0.376717	+	0.926329i	$0.622947\pi$
$14$	−12.3629	−0.236009
$15$	0	0
$16$	47.1551	0.736798
$17$	−42.4200	−0.605197	−0.302599	−	0.953118i	$-0.597854\pi$
$17$	−42.4200	−0.605197	−0.302599	+	0.953118i	$0.597854\pi$
$18$	0	0
$19$	−55.1989	−0.666499	−0.333250	−	0.942839i	$-0.608145\pi$
$19$	−55.1989	−0.666499	−0.333250	+	0.942839i	$0.608145\pi$
$20$	87.6930	0.980438
$21$	0	0
$22$	52.8317	0.511989
$23$	−40.9974	−0.371676	−0.185838	−	0.982580i	$-0.559500\pi$
$23$	−40.9974	−0.371676	−0.185838	+	0.982580i	$0.559500\pi$
$24$	0	0
$25$	20.2474	0.161979
$26$	−30.0426	−0.226609
$27$	0	0
$28$	105.744	0.713704
$29$	223.019	1.42805	0.714027	−	0.700118i	$-0.246869\pi$
$29$	223.019	1.42805	0.714027	+	0.700118i	$0.246869\pi$
$30$	0	0
$31$	107.746	0.624252	0.312126	−	0.950041i	$-0.398959\pi$
$31$	107.746	0.624252	0.312126	+	0.950041i	$0.398959\pi$
$32$	144.080	0.795935
$33$	0	0
$34$	−36.0868	−0.182024
$35$	175.145	0.845855
$36$	0	0
$37$	−7.24755	−0.0322024	−0.0161012	−	0.999870i	$-0.505125\pi$
$37$	−7.24755	−0.0322024	−0.0161012	+	0.999870i	$0.505125\pi$
$38$	−46.9578	−0.200462
$39$	0	0
$40$	156.621	0.619099
$41$	−102.577	−0.390727	−0.195364	−	0.980731i	$-0.562589\pi$
$41$	−102.577	−0.390727	−0.195364	+	0.980731i	$0.562589\pi$
$42$	0	0
$43$	246.621	0.874637	0.437318	−	0.899307i	$-0.355928\pi$
$43$	246.621	0.874637	0.437318	+	0.899307i	$0.355928\pi$
$44$	−451.885	−1.54828
$45$	0	0
$46$	−34.8766	−0.111789
$47$	384.920	1.19460	0.597302	−	0.802016i	$-0.296239\pi$
$47$	384.920	1.19460	0.597302	+	0.802016i	$0.296239\pi$
$48$	0	0
$49$	−131.803	−0.384265
$50$	17.2245	0.0487182
$51$	0	0
$52$	256.963	0.685277
$53$	646.786	1.67628	0.838140	−	0.545455i	$-0.183643\pi$
$53$	646.786	1.67628	0.838140	+	0.545455i	$0.183643\pi$
$54$	0	0
$55$	−748.464	−1.83496
$56$	188.860	0.450669
$57$	0	0
$58$	189.723	0.429514
$59$	180.551	0.398403	0.199202	−	0.979959i	$-0.436165\pi$
$59$	180.551	0.398403	0.199202	+	0.979959i	$0.436165\pi$
$60$	0	0
$61$	−104.757	−0.219881	−0.109940	−	0.993938i	$-0.535066\pi$
$61$	−104.757	−0.219881	−0.109940	+	0.993938i	$0.535066\pi$
$62$	91.6601	0.187755
$63$	0	0
$64$	−254.672	−0.497406
$65$	425.612	0.812165
$66$	0	0
$67$	801.715	1.46187	0.730934	−	0.682448i	$-0.239085\pi$
$67$	801.715	1.46187	0.730934	+	0.682448i	$0.239085\pi$
$68$	308.661	0.550450
$69$	0	0
$70$	148.996	0.254407
$71$	−944.820	−1.57929	−0.789644	−	0.613565i	$-0.789735\pi$
$71$	−944.820	−1.57929	−0.789644	+	0.613565i	$0.789735\pi$
$72$	0	0
$73$	−585.288	−0.938394	−0.469197	−	0.883094i	$-0.655457\pi$
$73$	−585.288	−0.938394	−0.469197	+	0.883094i	$0.655457\pi$
$74$	−6.16551	−0.00968548
$75$	0	0
$76$	401.644	0.606207
$77$	−902.528	−1.33575
$78$	0	0
$79$	611.307	0.870599	0.435300	−	0.900286i	$-0.356642\pi$
$79$	611.307	0.870599	0.435300	+	0.900286i	$0.356642\pi$
$80$	−568.306	−0.794232
$81$	0	0
$82$	−87.2623	−0.117518
$83$	−221.844	−0.293380	−0.146690	−	0.989182i	$-0.546862\pi$
$83$	−221.844	−0.293380	−0.146690	+	0.989182i	$0.546862\pi$
$84$	0	0
$85$	511.240	0.652373
$86$	209.801	0.263063
$87$	0	0
$88$	−807.073	−0.977662
$89$	1360.05	1.61984	0.809918	−	0.586543i	$-0.199511\pi$
$89$	1360.05	1.61984	0.809918	+	0.586543i	$0.199511\pi$
$90$	0	0
$91$	513.220	0.591210
$92$	298.310	0.338054
$93$	0	0
$94$	327.453	0.359300
$95$	665.249	0.718454
$96$	0	0
$97$	544.064	0.569498	0.284749	−	0.958602i	$-0.408090\pi$
$97$	544.064	0.569498	0.284749	+	0.958602i	$0.408090\pi$
$98$	−112.125	−0.115575
$99$	0	0
$100$	−147.326	−0.147326
$101$	−951.672	−0.937573	−0.468787	−	0.883311i	$-0.655309\pi$
$101$	−951.672	−0.937573	−0.468787	+	0.883311i	$0.655309\pi$
$102$	0	0
$103$	543.136	0.519580	0.259790	−	0.965665i	$-0.416347\pi$
$103$	543.136	0.519580	0.259790	+	0.965665i	$0.416347\pi$
$104$	458.940	0.432719
$105$	0	0
$106$	550.222	0.504172
$107$	−1353.91	−1.22324	−0.611622	−	0.791150i	$-0.709483\pi$
$107$	−1353.91	−1.22324	−0.611622	+	0.791150i	$0.709483\pi$
$108$	0	0
$109$	455.095	0.399910	0.199955	−	0.979805i	$-0.435920\pi$
$109$	455.095	0.399910	0.199955	+	0.979805i	$0.435920\pi$
$110$	−636.720	−0.551899
$111$	0	0
$112$	−685.287	−0.578156
$113$	686.829	0.571783	0.285891	−	0.958262i	$-0.407710\pi$
$113$	686.829	0.571783	0.285891	+	0.958262i	$0.407710\pi$
$114$	0	0
$115$	494.095	0.400649
$116$	−1622.75	−1.29887
$117$	0	0
$118$	153.595	0.119827
$119$	616.474	0.474891
$120$	0	0
$121$	2525.86	1.89771
$122$	−89.1167	−0.0661332
$123$	0	0
$124$	−783.996	−0.567781
$125$	1262.46	0.903346
$126$	0	0
$127$	−1891.03	−1.32128	−0.660638	−	0.750705i	$-0.729714\pi$
$127$	−1891.03	−1.32128	−0.660638	+	0.750705i	$0.729714\pi$
$128$	−1369.29	−0.945539
$129$	0	0
$130$	362.069	0.244274
$131$	−2194.03	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$131$	−2194.03	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$132$	0	0
$133$	802.184	0.522994
$134$	682.021	0.439684
$135$	0	0
$136$	551.273	0.347583
$137$	−2476.77	−1.54456	−0.772281	−	0.635281i	$-0.780884\pi$
$137$	−2476.77	−1.54456	−0.772281	+	0.635281i	$0.780884\pi$
$138$	0	0
$139$	−315.899	−0.192764	−0.0963821	−	0.995344i	$-0.530727\pi$
$139$	−315.899	−0.192764	−0.0963821	+	0.995344i	$0.530727\pi$
$140$	−1274.41	−0.769338
$141$	0	0
$142$	−803.760	−0.475000
$143$	−2193.19	−1.28255
$144$	0	0
$145$	−2687.79	−1.53937
$146$	−497.905	−0.282239
$147$	0	0
$148$	52.7354	0.0292894
$149$	−2517.12	−1.38397	−0.691983	−	0.721914i	$-0.743263\pi$
$149$	−2517.12	−1.38397	−0.691983	+	0.721914i	$0.743263\pi$
$150$	0	0
$151$	746.571	0.402352	0.201176	−	0.979555i	$-0.435524\pi$
$151$	746.571	0.402352	0.201176	+	0.979555i	$0.435524\pi$
$152$	717.342	0.382790
$153$	0	0
$154$	−767.783	−0.401751
$155$	−1298.54	−0.672913
$156$	0	0
$157$	−2567.69	−1.30525	−0.652625	−	0.757681i	$-0.726332\pi$
$157$	−2567.69	−1.30525	−0.652625	+	0.757681i	$0.726332\pi$
$158$	520.040	0.261849
$159$	0	0
$160$	−1736.43	−0.857979
$161$	595.800	0.291650
$162$	0	0
$163$	−1100.62	−0.528879	−0.264439	−	0.964402i	$-0.585187\pi$
$163$	−1100.62	−0.528879	−0.264439	+	0.964402i	$0.585187\pi$
$164$	746.381	0.355381
$165$	0	0
$166$	−188.723	−0.0882396
$167$	2224.18	1.03061	0.515305	−	0.857007i	$-0.327678\pi$
$167$	2224.18	1.03061	0.515305	+	0.857007i	$0.327678\pi$
$168$	0	0
$169$	−949.847	−0.432338
$170$	434.913	0.196213
$171$	0	0
$172$	−1794.49	−0.795516
$173$	−1216.88	−0.534784	−0.267392	−	0.963588i	$-0.586162\pi$
$173$	−1216.88	−0.534784	−0.267392	+	0.963588i	$0.586162\pi$
$174$	0	0
$175$	−294.247	−0.127103
$176$	2928.50	1.25423
$177$	0	0
$178$	1157.00	0.487196
$179$	−1379.19	−0.575897	−0.287949	−	0.957646i	$-0.592973\pi$
$179$	−1379.19	−0.575897	−0.287949	+	0.957646i	$0.592973\pi$
$180$	0	0
$181$	−2559.96	−1.05127	−0.525636	−	0.850709i	$-0.676173\pi$
$181$	−2559.96	−1.05127	−0.525636	+	0.850709i	$0.676173\pi$
$182$	436.598	0.177817
$183$	0	0
$184$	532.785	0.213464
$185$	87.3465	0.0347127
$186$	0	0
$187$	−2634.43	−1.03021
$188$	−2800.80	−1.08654
$189$	0	0
$190$	565.929	0.216088
$191$	975.622	0.369600	0.184800	−	0.982776i	$-0.440836\pi$
$191$	975.622	0.369600	0.184800	+	0.982776i	$0.440836\pi$
$192$	0	0
$193$	−3373.33	−1.25812	−0.629062	−	0.777355i	$-0.716561\pi$
$193$	−3373.33	−1.25812	−0.629062	+	0.777355i	$0.716561\pi$
$194$	462.836	0.171287
$195$	0	0
$196$	959.038	0.349504
$197$	5145.70	1.86099	0.930497	−	0.366299i	$-0.119375\pi$
$197$	5145.70	1.86099	0.930497	+	0.366299i	$0.119375\pi$
$198$	0	0
$199$	−4117.73	−1.46682	−0.733412	−	0.679784i	$-0.762073\pi$
$199$	−4117.73	−1.46682	−0.733412	+	0.679784i	$0.762073\pi$
$200$	−263.126	−0.0930292
$201$	0	0
$202$	−809.589	−0.281993
$203$	−3241.05	−1.12058
$204$	0	0
$205$	1236.24	0.421185
$206$	462.047	0.156273
$207$	0	0
$208$	−1665.28	−0.555128
$209$	−3428.05	−1.13456
$210$	0	0
$211$	812.312	0.265033	0.132516	−	0.991181i	$-0.457694\pi$
$211$	812.312	0.265033	0.132516	+	0.991181i	$0.457694\pi$
$212$	−4706.21	−1.52464
$213$	0	0
$214$	−1151.77	−0.367913
$215$	−2972.25	−0.942816
$216$	0	0
$217$	−1565.84	−0.489843
$218$	387.150	0.120280
$219$	0	0
$220$	5446.05	1.66897
$221$	1498.06	0.455976
$222$	0	0
$223$	−3388.99	−1.01768	−0.508842	−	0.860860i	$-0.669926\pi$
$223$	−3388.99	−1.01768	−0.508842	+	0.860860i	$0.669926\pi$
$224$	−2093.85	−0.624560
$225$	0	0
$226$	584.287	0.171974
$227$	3108.22	0.908808	0.454404	−	0.890796i	$-0.349852\pi$
$227$	3108.22	0.908808	0.454404	+	0.890796i	$0.349852\pi$
$228$	0	0
$229$	2097.69	0.605323	0.302662	−	0.953098i	$-0.402125\pi$
$229$	2097.69	0.605323	0.302662	+	0.953098i	$0.402125\pi$
$230$	420.328	0.120503
$231$	0	0
$232$	−2898.26	−0.820173
$233$	−23.8827	−0.00671504	−0.00335752	−	0.999994i	$-0.501069\pi$
$233$	−23.8827	−0.00671504	−0.00335752	+	0.999994i	$0.501069\pi$
$234$	0	0
$235$	−4639.01	−1.28773
$236$	−1313.75	−0.362363
$237$	0	0
$238$	524.435	0.142832
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	3824.54	1.02224	0.511121	−	0.859509i	$-0.329230\pi$
$241$	3824.54	1.02224	0.511121	+	0.859509i	$0.329230\pi$
$242$	2148.75	0.570773
$243$	0	0
$244$	762.241	0.199990
$245$	1588.47	0.414219
$246$	0	0
$247$	1949.35	0.502163
$248$	−1400.23	−0.358526
$249$	0	0
$250$	1073.98	0.271698
$251$	−66.2006	−0.0166476	−0.00832380	−	0.999965i	$-0.502650\pi$
$251$	−66.2006	−0.0166476	−0.00832380	+	0.999965i	$0.502650\pi$
$252$	0	0
$253$	−2546.09	−0.632692
$254$	−1608.71	−0.397398
$255$	0	0
$256$	872.519	0.213017
$257$	4815.73	1.16886	0.584430	−	0.811444i	$-0.301318\pi$
$257$	4815.73	1.16886	0.584430	+	0.811444i	$0.301318\pi$
$258$	0	0
$259$	105.326	0.0252689
$260$	−3096.88	−0.738695
$261$	0	0
$262$	−1866.46	−0.440117
$263$	−1507.49	−0.353444	−0.176722	−	0.984261i	$-0.556549\pi$
$263$	−1507.49	−0.353444	−0.176722	+	0.984261i	$0.556549\pi$
$264$	0	0
$265$	−7794.97	−1.80695
$266$	682.420	0.157300
$267$	0	0
$268$	−5833.53	−1.32962
$269$	−7585.66	−1.71935	−0.859677	−	0.510838i	$-0.829335\pi$
$269$	−7585.66	−1.71935	−0.859677	+	0.510838i	$0.829335\pi$
$270$	0	0
$271$	2915.97	0.653626	0.326813	−	0.945089i	$-0.394025\pi$
$271$	2915.97	0.653626	0.326813	+	0.945089i	$0.394025\pi$
$272$	−2000.32	−0.445908
$273$	0	0
$274$	−2107.00	−0.464556
$275$	1257.43	0.275731
$276$	0	0
$277$	2915.05	0.632305	0.316153	−	0.948708i	$-0.397609\pi$
$277$	2915.05	0.632305	0.316153	+	0.948708i	$0.397609\pi$
$278$	−268.736	−0.0579774
$279$	0	0
$280$	−2276.11	−0.485799
$281$	3007.50	0.638479	0.319239	−	0.947674i	$-0.396573\pi$
$281$	3007.50	0.638479	0.319239	+	0.947674i	$0.396573\pi$
$282$	0	0
$283$	1196.39	0.251299	0.125650	−	0.992075i	$-0.459898\pi$
$283$	1196.39	0.251299	0.125650	+	0.992075i	$0.459898\pi$
$284$	6874.80	1.43642
$285$	0	0
$286$	−1865.75	−0.385749
$287$	1490.71	0.306599
$288$	0	0
$289$	−3113.55	−0.633736
$290$	−2286.51	−0.462995
$291$	0	0
$292$	4258.73	0.853505
$293$	−5041.63	−1.00524	−0.502620	−	0.864508i	$-0.667630\pi$
$293$	−5041.63	−1.00524	−0.502620	+	0.864508i	$0.667630\pi$
$294$	0	0
$295$	−2175.98	−0.429459
$296$	94.1862	0.0184948
$297$	0	0
$298$	−2141.32	−0.416253
$299$	1447.83	0.280033
$300$	0	0
$301$	−3584.05	−0.686317
$302$	635.109	0.121015
$303$	0	0
$304$	−2602.91	−0.491075
$305$	1262.51	0.237021
$306$	0	0
$307$	−5388.54	−1.00176	−0.500880	−	0.865517i	$-0.666990\pi$
$307$	−5388.54	−1.00176	−0.500880	+	0.865517i	$0.666990\pi$
$308$	6567.07	1.21491
$309$	0	0
$310$	−1104.67	−0.202391
$311$	3786.86	0.690460	0.345230	−	0.938518i	$-0.387801\pi$
$311$	3786.86	0.690460	0.345230	+	0.938518i	$0.387801\pi$
$312$	0	0
$313$	6126.64	1.10638	0.553192	−	0.833054i	$-0.313410\pi$
$313$	6126.64	1.10638	0.553192	+	0.833054i	$0.313410\pi$
$314$	−2184.34	−0.392578
$315$	0	0
$316$	−4448.05	−0.791844
$317$	10234.9	1.81341	0.906705	−	0.421764i	$-0.138589\pi$
$317$	10234.9	1.81341	0.906705	+	0.421764i	$0.138589\pi$
$318$	0	0
$319$	13850.3	2.43093
$320$	3069.27	0.536179
$321$	0	0
$322$	506.848	0.0877191
$323$	2341.53	0.403364
$324$	0	0
$325$	−715.037	−0.122040
$326$	−936.300	−0.159070
$327$	0	0
$328$	1333.05	0.224406
$329$	−5593.90	−0.937392
$330$	0	0
$331$	−9728.69	−1.61552	−0.807760	−	0.589511i	$-0.799320\pi$
$331$	−9728.69	−1.61552	−0.807760	+	0.589511i	$0.799320\pi$
$332$	1614.21	0.266841
$333$	0	0
$334$	1892.11	0.309975
$335$	−9662.16	−1.57582
$336$	0	0
$337$	95.3964	0.0154201	0.00771005	−	0.999970i	$-0.497546\pi$
$337$	95.3964	0.0154201	0.00771005	+	0.999970i	$0.497546\pi$
$338$	−808.037	−0.130034
$339$	0	0
$340$	−3719.94	−0.593359
$341$	6691.44	1.06264
$342$	0	0
$343$	6900.13	1.08622
$344$	−3204.99	−0.502330
$345$	0	0
$346$	−1035.20	−0.160846
$347$	2041.21	0.315786	0.157893	−	0.987456i	$-0.449530\pi$
$347$	2041.21	0.315786	0.157893	+	0.987456i	$0.449530\pi$
$348$	0	0
$349$	2626.90	0.402908	0.201454	−	0.979498i	$-0.435433\pi$
$349$	2626.90	0.402908	0.201454	+	0.979498i	$0.435433\pi$
$350$	−250.317	−0.0382286
$351$	0	0
$352$	8947.86	1.35489
$353$	2372.68	0.357749	0.178874	−	0.983872i	$-0.442754\pi$
$353$	2372.68	0.357749	0.178874	+	0.983872i	$0.442754\pi$
$354$	0	0
$355$	11386.8	1.70240
$356$	−9896.17	−1.47330
$357$	0	0
$358$	−1173.28	−0.173212
$359$	1611.95	0.236979	0.118490	−	0.992955i	$-0.462195\pi$
$359$	1611.95	0.236979	0.118490	+	0.992955i	$0.462195\pi$
$360$	0	0
$361$	−3812.09	−0.555779
$362$	−2177.76	−0.316190
$363$	0	0
$364$	−3734.35	−0.537728
$365$	7053.81	1.01154
$366$	0	0
$367$	−785.036	−0.111658	−0.0558291	−	0.998440i	$-0.517780\pi$
$367$	−785.036	−0.111658	−0.0558291	+	0.998440i	$0.517780\pi$
$368$	−1933.24	−0.273850
$369$	0	0
$370$	74.3059	0.0104405
$371$	−9399.49	−1.31536
$372$	0	0
$373$	−8083.21	−1.12207	−0.561036	−	0.827792i	$-0.689597\pi$
$373$	−8083.21	−1.12207	−0.561036	+	0.827792i	$0.689597\pi$
$374$	−2241.12	−0.309854
$375$	0	0
$376$	−5002.27	−0.686096
$377$	−7875.93	−1.07594
$378$	0	0
$379$	−7032.61	−0.953143	−0.476571	−	0.879136i	$-0.658121\pi$
$379$	−7032.61	−0.953143	−0.476571	+	0.879136i	$0.658121\pi$
$380$	−4840.56	−0.653461
$381$	0	0
$382$	829.963	0.111164
$383$	−89.1753	−0.0118972	−0.00594862	−	0.999982i	$-0.501894\pi$
$383$	−89.1753	−0.0118972	−0.00594862	+	0.999982i	$0.501894\pi$
$384$	0	0
$385$	10877.1	1.43987
$386$	−2869.70	−0.378404
$387$	0	0
$388$	−3958.77	−0.517980
$389$	1946.32	0.253682	0.126841	−	0.991923i	$-0.459516\pi$
$389$	1946.32	0.253682	0.126841	+	0.991923i	$0.459516\pi$
$390$	0	0
$391$	1739.11	0.224937
$392$	1712.86	0.220695
$393$	0	0
$394$	4377.46	0.559729
$395$	−7367.38	−0.938464
$396$	0	0
$397$	−176.710	−0.0223396	−0.0111698	−	0.999938i	$-0.503556\pi$
$397$	−176.710	−0.0223396	−0.0111698	+	0.999938i	$0.503556\pi$
$398$	−3502.96	−0.441175
$399$	0	0
$400$	954.766	0.119346
$401$	11502.9	1.43249	0.716245	−	0.697849i	$-0.245859\pi$
$401$	11502.9	1.43249	0.716245	+	0.697849i	$0.245859\pi$
$402$	0	0
$403$	−3805.07	−0.470332
$404$	6924.65	0.852759
$405$	0	0
$406$	−2757.17	−0.337034
$407$	−450.099	−0.0548172
$408$	0	0
$409$	−10652.0	−1.28779	−0.643897	−	0.765112i	$-0.722684\pi$
$409$	−10652.0	−1.28779	−0.643897	+	0.765112i	$0.722684\pi$
$410$	1051.67	0.126679
$411$	0	0
$412$	−3952.02	−0.472578
$413$	−2623.89	−0.312622
$414$	0	0
$415$	2673.64	0.316250
$416$	−5088.18	−0.599684
$417$	0	0
$418$	−2916.25	−0.341240
$419$	−6569.34	−0.765950	−0.382975	−	0.923759i	$-0.625100\pi$
$419$	−6569.34	−0.765950	−0.382975	+	0.923759i	$0.625100\pi$
$420$	0	0
$421$	6947.95	0.804329	0.402164	−	0.915567i	$-0.368258\pi$
$421$	6947.95	0.804329	0.402164	+	0.915567i	$0.368258\pi$
$422$	691.036	0.0797135
$423$	0	0
$424$	−8405.36	−0.962737
$425$	−858.893	−0.0980292
$426$	0	0
$427$	1522.39	0.172538
$428$	9851.44	1.11259
$429$	0	0
$430$	−2528.50	−0.283569
$431$	5492.96	0.613890	0.306945	−	0.951727i	$-0.400693\pi$
$431$	5492.96	0.613890	0.306945	+	0.951727i	$0.400693\pi$
$432$	0	0
$433$	9062.69	1.00583	0.502916	−	0.864335i	$-0.332260\pi$
$433$	9062.69	1.00583	0.502916	+	0.864335i	$0.332260\pi$
$434$	−1332.06	−0.147329
$435$	0	0
$436$	−3311.41	−0.363734
$437$	2263.01	0.247722
$438$	0	0
$439$	−15083.4	−1.63985	−0.819923	−	0.572474i	$-0.805984\pi$
$439$	−15083.4	−1.63985	−0.819923	+	0.572474i	$0.805984\pi$
$440$	9726.73	1.05387
$441$	0	0
$442$	1274.41	0.137143
$443$	−11155.9	−1.19646	−0.598229	−	0.801325i	$-0.704129\pi$
$443$	−11155.9	−1.19646	−0.598229	+	0.801325i	$0.704129\pi$
$444$	0	0
$445$	−16391.2	−1.74610
$446$	−2883.02	−0.306087
$447$	0	0
$448$	3701.05	0.390308
$449$	12830.6	1.34858	0.674291	−	0.738466i	$-0.264449\pi$
$449$	12830.6	1.34858	0.674291	+	0.738466i	$0.264449\pi$
$450$	0	0
$451$	−6370.39	−0.665122
$452$	−4997.58	−0.520058
$453$	0	0
$454$	2644.17	0.273341
$455$	−6185.26	−0.637296
$456$	0	0
$457$	15027.0	1.53815	0.769074	−	0.639160i	$-0.220718\pi$
$457$	15027.0	1.53815	0.769074	+	0.639160i	$0.220718\pi$
$458$	1784.51	0.182062
$459$	0	0
$460$	−3595.19	−0.364405
$461$	3814.66	0.385394	0.192697	−	0.981258i	$-0.438277\pi$
$461$	3814.66	0.385394	0.192697	+	0.981258i	$0.438277\pi$
$462$	0	0
$463$	7670.51	0.769933	0.384966	−	0.922931i	$-0.374213\pi$
$463$	7670.51	0.769933	0.384966	+	0.922931i	$0.374213\pi$
$464$	10516.5	1.05219
$465$	0	0
$466$	−20.3170	−0.00201967
$467$	14948.2	1.48120	0.740602	−	0.671944i	$-0.234540\pi$
$467$	14948.2	1.48120	0.740602	+	0.671944i	$0.234540\pi$
$468$	0	0
$469$	−11651.0	−1.14711
$470$	−3946.41	−0.387307
$471$	0	0
$472$	−2346.37	−0.228815
$473$	15316.1	1.48887
$474$	0	0
$475$	−1117.63	−0.107959
$476$	−4485.65	−0.431932
$477$	0	0
$478$	−203.318	−0.0194551
$479$	−18131.4	−1.72953	−0.864766	−	0.502176i	$-0.832533\pi$
$479$	−18131.4	−1.72953	−0.864766	+	0.502176i	$0.832533\pi$
$480$	0	0
$481$	255.948	0.0242624
$482$	3253.54	0.307458
$483$	0	0
$484$	−18378.9	−1.72604
$485$	−6556.98	−0.613891
$486$	0	0
$487$	10084.6	0.938351	0.469175	−	0.883105i	$-0.344551\pi$
$487$	10084.6	0.938351	0.469175	+	0.883105i	$0.344551\pi$
$488$	1361.37	0.126284
$489$	0	0
$490$	1351.31	0.124584
$491$	−14926.9	−1.37198	−0.685990	−	0.727611i	$-0.740630\pi$
$491$	−14926.9	−1.37198	−0.685990	+	0.727611i	$0.740630\pi$
$492$	0	0
$493$	−9460.46	−0.864255
$494$	1658.32	0.151035
$495$	0	0
$496$	5080.79	0.459948
$497$	13730.7	1.23925
$498$	0	0
$499$	−8645.91	−0.775639	−0.387820	−	0.921735i	$-0.626772\pi$
$499$	−8645.91	−0.775639	−0.387820	+	0.921735i	$0.626772\pi$
$500$	−9186.08	−0.821628
$501$	0	0
$502$	−56.3170	−0.00500707
$503$	15493.8	1.37343	0.686716	−	0.726926i	$-0.259052\pi$
$503$	15493.8	1.37343	0.686716	+	0.726926i	$0.259052\pi$
$504$	0	0
$505$	11469.4	1.01066
$506$	−2165.96	−0.190294
$507$	0	0
$508$	13759.7	1.20175
$509$	−10325.4	−0.899148	−0.449574	−	0.893243i	$-0.648424\pi$
$509$	−10325.4	−0.899148	−0.449574	+	0.893243i	$0.648424\pi$
$510$	0	0
$511$	8505.76	0.736346
$512$	11696.5	1.00961
$513$	0	0
$514$	4096.75	0.351556
$515$	−6545.80	−0.560082
$516$	0	0
$517$	23904.9	2.03354
$518$	89.6010	0.00760008
$519$	0	0
$520$	−5531.08	−0.466450
$521$	−6879.06	−0.578459	−0.289229	−	0.957260i	$-0.593399\pi$
$521$	−6879.06	−0.578459	−0.289229	+	0.957260i	$0.593399\pi$
$522$	0	0
$523$	5886.32	0.492143	0.246072	−	0.969252i	$-0.420860\pi$
$523$	5886.32	0.492143	0.246072	+	0.969252i	$0.420860\pi$
$524$	15964.4	1.33093
$525$	0	0
$526$	−1282.42	−0.106305
$527$	−4570.60	−0.377796
$528$	0	0
$529$	−10486.2	−0.861857
$530$	−6631.20	−0.543473
$531$	0	0
$532$	−5836.94	−0.475683
$533$	3622.51	0.294387
$534$	0	0
$535$	16317.1	1.31860
$536$	−10418.8	−0.839593
$537$	0	0
$538$	−6453.14	−0.517128
$539$	−8185.43	−0.654122
$540$	0	0
$541$	2332.25	0.185345	0.0926723	−	0.995697i	$-0.470459\pi$
$541$	2332.25	0.185345	0.0926723	+	0.995697i	$0.470459\pi$
$542$	2480.62	0.196590
$543$	0	0
$544$	−6111.85	−0.481698
$545$	−5484.74	−0.431084
$546$	0	0
$547$	−6913.99	−0.540440	−0.270220	−	0.962799i	$-0.587096\pi$
$547$	−6913.99	−0.540440	−0.270220	+	0.962799i	$0.587096\pi$
$548$	18021.8	1.40484
$549$	0	0
$550$	1069.70	0.0829314
$551$	−12310.4	−0.951797
$552$	0	0
$553$	−8883.89	−0.683149
$554$	2479.84	0.190178
$555$	0	0
$556$	2298.58	0.175326
$557$	11240.3	0.855056	0.427528	−	0.904002i	$-0.359385\pi$
$557$	11240.3	0.855056	0.427528	+	0.904002i	$0.359385\pi$
$558$	0	0
$559$	−8709.44	−0.658981
$560$	8258.98	0.623225
$561$	0	0
$562$	2558.49	0.192034
$563$	−4785.02	−0.358196	−0.179098	−	0.983831i	$-0.557318\pi$
$563$	−4785.02	−0.358196	−0.179098	+	0.983831i	$0.557318\pi$
$564$	0	0
$565$	−8277.57	−0.616354
$566$	1017.77	0.0755830
$567$	0	0
$568$	12278.5	0.907031
$569$	11676.3	0.860274	0.430137	−	0.902764i	$-0.358465\pi$
$569$	11676.3	0.860274	0.430137	+	0.902764i	$0.358465\pi$
$570$	0	0
$571$	21690.3	1.58969	0.794844	−	0.606813i	$-0.207552\pi$
$571$	21690.3	1.58969	0.794844	+	0.606813i	$0.207552\pi$
$572$	15958.3	1.16652
$573$	0	0
$574$	1268.15	0.0922153
$575$	−830.090	−0.0602037
$576$	0	0
$577$	−4857.37	−0.350459	−0.175230	−	0.984528i	$-0.556067\pi$
$577$	−4857.37	−0.350459	−0.175230	+	0.984528i	$0.556067\pi$
$578$	−2648.70	−0.190608
$579$	0	0
$580$	19557.2	1.40012
$581$	3223.98	0.230212
$582$	0	0
$583$	40167.7	2.85348
$584$	7606.16	0.538947
$585$	0	0
$586$	−4288.92	−0.302344
$587$	6723.99	0.472792	0.236396	−	0.971657i	$-0.424034\pi$
$587$	6723.99	0.472792	0.236396	+	0.971657i	$0.424034\pi$
$588$	0	0
$589$	−5947.48	−0.416064
$590$	−1851.11	−0.129168
$591$	0	0
$592$	−341.759	−0.0237267
$593$	12075.8	0.836245	0.418122	−	0.908391i	$-0.362688\pi$
$593$	12075.8	0.836245	0.418122	+	0.908391i	$0.362688\pi$
$594$	0	0
$595$	−7429.65	−0.511909
$596$	18315.4	1.25877
$597$	0	0
$598$	1231.67	0.0842252
$599$	−11022.8	−0.751883	−0.375942	−	0.926643i	$-0.622681\pi$
$599$	−11022.8	−0.751883	−0.375942	+	0.926643i	$0.622681\pi$
$600$	0	0
$601$	−23343.9	−1.58439	−0.792195	−	0.610268i	$-0.791062\pi$
$601$	−23343.9	−1.58439	−0.792195	+	0.610268i	$0.791062\pi$
$602$	−3048.96	−0.206423
$603$	0	0
$604$	−5432.28	−0.365954
$605$	−30441.3	−2.04564
$606$	0	0
$607$	−11561.1	−0.773068	−0.386534	−	0.922275i	$-0.626328\pi$
$607$	−11561.1	−0.773068	−0.386534	+	0.922275i	$0.626328\pi$
$608$	−7953.03	−0.530490
$609$	0	0
$610$	1074.02	0.0712883
$611$	−13593.5	−0.900055
$612$	0	0
$613$	−26252.0	−1.72971	−0.864853	−	0.502025i	$-0.832589\pi$
$613$	−26252.0	−1.72971	−0.864853	+	0.502025i	$0.832589\pi$
$614$	−4584.04	−0.301298
$615$	0	0
$616$	11728.9	0.767159
$617$	3899.08	0.254410	0.127205	−	0.991876i	$-0.459399\pi$
$617$	3899.08	0.254410	0.127205	+	0.991876i	$0.459399\pi$
$618$	0	0
$619$	−6151.33	−0.399423	−0.199712	−	0.979855i	$-0.564001\pi$
$619$	−6151.33	−0.399423	−0.199712	+	0.979855i	$0.564001\pi$
$620$	9448.61	0.612041
$621$	0	0
$622$	3221.49	0.207669
$623$	−19765.2	−1.27107
$624$	0	0
$625$	−17746.0	−1.13574
$626$	5211.95	0.332766
$627$	0	0
$628$	18683.3	1.18717
$629$	307.441	0.0194888
$630$	0	0
$631$	22027.7	1.38971	0.694856	−	0.719149i	$-0.255468\pi$
$631$	22027.7	1.38971	0.694856	+	0.719149i	$0.255468\pi$
$632$	−7944.29	−0.500011
$633$	0	0
$634$	8706.88	0.545417
$635$	22790.5	1.42427
$636$	0	0
$637$	4654.63	0.289518
$638$	11782.5	0.731147
$639$	0	0
$640$	16502.5	1.01924
$641$	13588.5	0.837309	0.418654	−	0.908146i	$-0.362502\pi$
$641$	13588.5	0.837309	0.418654	+	0.908146i	$0.362502\pi$
$642$	0	0
$643$	−27356.4	−1.67781	−0.838906	−	0.544277i	$-0.816804\pi$
$643$	−27356.4	−1.67781	−0.838906	+	0.544277i	$0.816804\pi$
$644$	−4335.22	−0.265267
$645$	0	0
$646$	1991.95	0.121319
$647$	−28177.4	−1.71216	−0.856080	−	0.516844i	$-0.827107\pi$
$647$	−28177.4	−1.71216	−0.856080	+	0.516844i	$0.827107\pi$
$648$	0	0
$649$	11212.9	0.678189
$650$	−608.283	−0.0367059
$651$	0	0
$652$	8008.45	0.481035
$653$	−17727.4	−1.06237	−0.531183	−	0.847257i	$-0.678252\pi$
$653$	−17727.4	−1.06237	−0.531183	+	0.847257i	$0.678252\pi$
$654$	0	0
$655$	26442.1	1.57737
$656$	−4837.02	−0.287887
$657$	0	0
$658$	−4758.74	−0.281938
$659$	−5107.62	−0.301919	−0.150960	−	0.988540i	$-0.548236\pi$
$659$	−5107.62	−0.301919	−0.150960	+	0.988540i	$0.548236\pi$
$660$	0	0
$661$	−16549.5	−0.973830	−0.486915	−	0.873449i	$-0.661878\pi$
$661$	−16549.5	−0.973830	−0.486915	+	0.873449i	$0.661878\pi$
$662$	−8276.22	−0.485898
$663$	0	0
$664$	2883.00	0.168497
$665$	−9667.81	−0.563762
$666$	0	0
$667$	−9143.20	−0.530774
$668$	−16183.8	−0.937380
$669$	0	0
$670$	−8219.62	−0.473958
$671$	−6505.77	−0.374296
$672$	0	0
$673$	−7343.84	−0.420630	−0.210315	−	0.977634i	$-0.567449\pi$
$673$	−7343.84	−0.420630	−0.210315	+	0.977634i	$0.567449\pi$
$674$	81.1539	0.00463788
$675$	0	0
$676$	6911.37	0.393228
$677$	−9797.84	−0.556221	−0.278110	−	0.960549i	$-0.589708\pi$
$677$	−9797.84	−0.556221	−0.278110	+	0.960549i	$0.589708\pi$
$678$	0	0
$679$	−7906.67	−0.446878
$680$	−6643.86	−0.374677
$681$	0	0
$682$	5692.42	0.319610
$683$	−13524.4	−0.757683	−0.378841	−	0.925462i	$-0.623677\pi$
$683$	−13524.4	−0.757683	−0.378841	+	0.925462i	$0.623677\pi$
$684$	0	0
$685$	29849.7	1.66496
$686$	5869.96	0.326700
$687$	0	0
$688$	11629.4	0.644431
$689$	−22841.3	−1.26297
$690$	0	0
$691$	−20718.8	−1.14064	−0.570320	−	0.821423i	$-0.693181\pi$
$691$	−20718.8	−1.14064	−0.570320	+	0.821423i	$0.693181\pi$
$692$	8854.39	0.486407
$693$	0	0
$694$	1736.46	0.0949785
$695$	3807.17	0.207790
$696$	0	0
$697$	4351.31	0.236467
$698$	2234.71	0.121182
$699$	0	0
$700$	2141.03	0.115605
$701$	28520.1	1.53665	0.768323	−	0.640062i	$-0.221091\pi$
$701$	28520.1	1.53665	0.768323	+	0.640062i	$0.221091\pi$
$702$	0	0
$703$	400.057	0.0214629
$704$	−15816.0	−0.846718
$705$	0	0
$706$	2018.45	0.107600
$707$	13830.3	0.735702
$708$	0	0
$709$	−10116.5	−0.535870	−0.267935	−	0.963437i	$-0.586341\pi$
$709$	−10116.5	−0.535870	−0.267935	+	0.963437i	$0.586341\pi$
$710$	9686.81	0.512027
$711$	0	0
$712$	−17674.7	−0.930319
$713$	−4417.32	−0.232020
$714$	0	0
$715$	26432.0	1.38252
$716$	10035.4	0.523800
$717$	0	0
$718$	1371.29	0.0712759
$719$	−24269.0	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$719$	−24269.0	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$720$	0	0
$721$	−7893.19	−0.407708
$722$	−3242.95	−0.167161
$723$	0	0
$724$	18627.1	0.956173
$725$	4515.55	0.231315
$726$	0	0
$727$	3881.52	0.198016	0.0990079	−	0.995087i	$-0.468433\pi$
$727$	3881.52	0.198016	0.0990079	+	0.995087i	$0.468433\pi$
$728$	−6669.60	−0.339549
$729$	0	0
$730$	6000.69	0.304240
$731$	−10461.7	−0.529328
$732$	0	0
$733$	26017.6	1.31103	0.655513	−	0.755184i	$-0.272452\pi$
$733$	26017.6	1.31103	0.655513	+	0.755184i	$0.272452\pi$
$734$	−667.832	−0.0335833
$735$	0	0
$736$	−5906.89	−0.295830
$737$	49789.4	2.48849
$738$	0	0
$739$	−21871.1	−1.08869	−0.544344	−	0.838862i	$-0.683221\pi$
$739$	−21871.1	−1.08869	−0.544344	+	0.838862i	$0.683221\pi$
$740$	−635.560	−0.0315725
$741$	0	0
$742$	−7996.17	−0.395618
$743$	−21219.7	−1.04775	−0.523874	−	0.851796i	$-0.675514\pi$
$743$	−21219.7	−1.04775	−0.523874	+	0.851796i	$0.675514\pi$
$744$	0	0
$745$	30336.0	1.49185
$746$	−6876.40	−0.337484
$747$	0	0
$748$	19168.9	0.937013
$749$	19675.8	0.959865
$750$	0	0
$751$	−26500.9	−1.28766	−0.643829	−	0.765169i	$-0.722655\pi$
$751$	−26500.9	−1.28766	−0.643829	+	0.765169i	$0.722655\pi$
$752$	18150.9	0.880182
$753$	0	0
$754$	−6700.07	−0.323610
$755$	−8997.57	−0.433715
$756$	0	0
$757$	−3928.10	−0.188599	−0.0942993	−	0.995544i	$-0.530061\pi$
$757$	−3928.10	−0.188599	−0.0942993	+	0.995544i	$0.530061\pi$
$758$	−5982.66	−0.286675
$759$	0	0
$760$	−8645.30	−0.412629
$761$	23423.9	1.11579	0.557894	−	0.829912i	$-0.311609\pi$
$761$	23423.9	1.11579	0.557894	+	0.829912i	$0.311609\pi$
$762$	0	0
$763$	−6613.73	−0.313805
$764$	−7098.92	−0.336165
$765$	0	0
$766$	−75.8616	−0.00357832
$767$	−6376.18	−0.300170
$768$	0	0
$769$	−32966.7	−1.54591	−0.772957	−	0.634458i	$-0.781223\pi$
$769$	−32966.7	−1.54591	−0.772957	+	0.634458i	$0.781223\pi$
$770$	9253.21	0.433068
$771$	0	0
$772$	24545.4	1.14431
$773$	−39545.3	−1.84003	−0.920016	−	0.391881i	$-0.871825\pi$
$773$	−39545.3	−1.84003	−0.920016	+	0.391881i	$0.871825\pi$
$774$	0	0
$775$	2181.58	0.101116
$776$	−7070.43	−0.327079
$777$	0	0
$778$	1655.74	0.0762995
$779$	5662.13	0.260419
$780$	0	0
$781$	−58676.7	−2.68837
$782$	1479.46	0.0676541
$783$	0	0
$784$	−6215.17	−0.283126
$785$	30945.5	1.40700
$786$	0	0
$787$	41818.0	1.89409	0.947047	−	0.321095i	$-0.104051\pi$
$787$	41818.0	1.89409	0.947047	+	0.321095i	$0.104051\pi$
$788$	−37441.7	−1.69265
$789$	0	0
$790$	−6267.45	−0.282260
$791$	−9981.43	−0.448671
$792$	0	0
$793$	3699.49	0.165665
$794$	−150.327	−0.00671904
$795$	0	0
$796$	29961.8	1.33413
$797$	17804.0	0.791280	0.395640	−	0.918406i	$-0.370523\pi$
$797$	17804.0	0.791280	0.395640	+	0.918406i	$0.370523\pi$
$798$	0	0
$799$	−16328.3	−0.722972
$800$	2917.23	0.128925
$801$	0	0
$802$	9785.56	0.430848
$803$	−36348.5	−1.59740
$804$	0	0
$805$	−7180.50	−0.314384
$806$	−3236.98	−0.141461
$807$	0	0
$808$	12367.5	0.538476
$809$	13093.8	0.569040	0.284520	−	0.958670i	$-0.408166\pi$
$809$	13093.8	0.569040	0.284520	+	0.958670i	$0.408166\pi$
$810$	0	0
$811$	29611.0	1.28210	0.641050	−	0.767499i	$-0.278499\pi$
$811$	29611.0	1.28210	0.641050	+	0.767499i	$0.278499\pi$
$812$	23582.9	1.01921
$813$	0	0
$814$	−382.900	−0.0164873
$815$	13264.5	0.570105
$816$	0	0
$817$	−13613.2	−0.582945
$818$	−9061.69	−0.387328
$819$	0	0
$820$	−8995.27	−0.383084
$821$	−26817.1	−1.13998	−0.569989	−	0.821652i	$-0.693053\pi$
$821$	−26817.1	−1.13998	−0.569989	+	0.821652i	$0.693053\pi$
$822$	0	0
$823$	−16959.2	−0.718300	−0.359150	−	0.933280i	$-0.616933\pi$
$823$	−16959.2	−0.718300	−0.359150	+	0.933280i	$0.616933\pi$
$824$	−7058.37	−0.298410
$825$	0	0
$826$	−2232.15	−0.0940270
$827$	−44402.3	−1.86701	−0.933507	−	0.358559i	$-0.883268\pi$
$827$	−44402.3	−1.86701	−0.933507	+	0.358559i	$0.883268\pi$
$828$	0	0
$829$	24996.6	1.04725	0.523625	−	0.851949i	$-0.324579\pi$
$829$	24996.6	1.04725	0.523625	+	0.851949i	$0.324579\pi$
$830$	2274.47	0.0951180
$831$	0	0
$832$	8993.75	0.374762
$833$	5591.08	0.232556
$834$	0	0
$835$	−26805.5	−1.11095
$836$	24943.5	1.03193
$837$	0	0
$838$	−5588.55	−0.230374
$839$	−12910.9	−0.531266	−0.265633	−	0.964074i	$-0.585581\pi$
$839$	−12910.9	−0.531266	−0.265633	+	0.964074i	$0.585581\pi$
$840$	0	0
$841$	25348.4	1.03934
$842$	5910.64	0.241917
$843$	0	0
$844$	−5910.63	−0.241057
$845$	11447.4	0.466039
$846$	0	0
$847$	−36707.3	−1.48911
$848$	30499.2	1.23508
$849$	0	0
$850$	−730.662	−0.0294841
$851$	297.131	0.0119689
$852$	0	0
$853$	−24009.0	−0.963719	−0.481859	−	0.876249i	$-0.660038\pi$
$853$	−24009.0	−0.963719	−0.481859	+	0.876249i	$0.660038\pi$
$854$	1295.10	0.0518939
$855$	0	0
$856$	17594.8	0.702545
$857$	−41433.4	−1.65150	−0.825752	−	0.564034i	$-0.809249\pi$
$857$	−41433.4	−1.65150	−0.825752	+	0.564034i	$0.809249\pi$
$858$	0	0
$859$	−24053.5	−0.955408	−0.477704	−	0.878521i	$-0.658531\pi$
$859$	−24053.5	−0.955408	−0.477704	+	0.878521i	$0.658531\pi$
$860$	21627.0	0.857527
$861$	0	0
$862$	4672.88	0.184639
$863$	−7547.91	−0.297722	−0.148861	−	0.988858i	$-0.547561\pi$
$863$	−7547.91	−0.297722	−0.148861	+	0.988858i	$0.547561\pi$
$864$	0	0
$865$	14665.7	0.576471
$866$	7709.65	0.302523
$867$	0	0
$868$	11393.5	0.445531
$869$	37964.3	1.48199
$870$	0	0
$871$	−28312.6	−1.10142
$872$	−5914.23	−0.229680
$873$	0	0
$874$	1925.15	0.0745070
$875$	−18346.9	−0.708845
$876$	0	0
$877$	10477.2	0.403411	0.201706	−	0.979446i	$-0.435352\pi$
$877$	10477.2	0.403411	0.201706	+	0.979446i	$0.435352\pi$
$878$	−12831.5	−0.493214
$879$	0	0
$880$	−35293.9	−1.35200
$881$	18214.7	0.696559	0.348280	−	0.937391i	$-0.386766\pi$
$881$	18214.7	0.696559	0.348280	+	0.937391i	$0.386766\pi$
$882$	0	0
$883$	3970.24	0.151313	0.0756563	−	0.997134i	$-0.475895\pi$
$883$	3970.24	0.151313	0.0756563	+	0.997134i	$0.475895\pi$
$884$	−10900.4	−0.414728
$885$	0	0
$886$	−9490.31	−0.359857
$887$	4868.88	0.184308	0.0921538	−	0.995745i	$-0.470625\pi$
$887$	4868.88	0.184308	0.0921538	+	0.995745i	$0.470625\pi$
$888$	0	0
$889$	27481.7	1.03679
$890$	−13944.0	−0.525173
$891$	0	0
$892$	24659.3	0.925622
$893$	−21247.2	−0.796203
$894$	0	0
$895$	16621.8	0.620789
$896$	19899.3	0.741953
$897$	0	0
$898$	10915.0	0.405611
$899$	24029.5	0.891466
$900$	0	0
$901$	−27436.6	−1.01448
$902$	−5419.31	−0.200048
$903$	0	0
$904$	−8925.74	−0.328391
$905$	30852.3	1.13322
$906$	0	0
$907$	18031.6	0.660121	0.330060	−	0.943960i	$-0.392931\pi$
$907$	18031.6	0.660121	0.330060	+	0.943960i	$0.392931\pi$
$908$	−22616.3	−0.826596
$909$	0	0
$910$	−5261.82	−0.191679
$911$	46616.5	1.69536	0.847680	−	0.530508i	$-0.177999\pi$
$911$	46616.5	1.69536	0.847680	+	0.530508i	$0.177999\pi$
$912$	0	0
$913$	−13777.3	−0.499412
$914$	12783.5	0.462627
$915$	0	0
$916$	−15263.4	−0.550564
$917$	31885.0	1.14824
$918$	0	0
$919$	−35979.2	−1.29145	−0.645726	−	0.763569i	$-0.723445\pi$
$919$	−35979.2	−1.29145	−0.645726	+	0.763569i	$0.723445\pi$
$920$	−6421.06	−0.230104
$921$	0	0
$922$	3245.14	0.115914
$923$	33366.4	1.18989
$924$	0	0
$925$	−146.744	−0.00521612
$926$	6525.32	0.231572
$927$	0	0
$928$	32132.5	1.13664
$929$	7229.35	0.255315	0.127657	−	0.991818i	$-0.459254\pi$
$929$	7229.35	0.255315	0.127657	+	0.991818i	$0.459254\pi$
$930$	0	0
$931$	7275.37	0.256112
$932$	173.778	0.00610759
$933$	0	0
$934$	12716.5	0.445500
$935$	31749.8	1.11051
$936$	0	0
$937$	11277.7	0.393197	0.196599	−	0.980484i	$-0.437010\pi$
$937$	11277.7	0.393197	0.196599	+	0.980484i	$0.437010\pi$
$938$	−9911.55	−0.345015
$939$	0	0
$940$	33754.8	1.17124
$941$	−16046.1	−0.555884	−0.277942	−	0.960598i	$-0.589652\pi$
$941$	−16046.1	−0.555884	−0.277942	+	0.960598i	$0.589652\pi$
$942$	0	0
$943$	4205.39	0.145224
$944$	8513.92	0.293543
$945$	0	0
$946$	13029.4	0.447804
$947$	42897.8	1.47201	0.736004	−	0.676977i	$-0.236710\pi$
$947$	42897.8	1.47201	0.736004	+	0.676977i	$0.236710\pi$
$948$	0	0
$949$	20669.5	0.707017
$950$	−950.772	−0.0324706
$951$	0	0
$952$	−8011.44	−0.272744
$953$	−27086.0	−0.920675	−0.460337	−	0.887744i	$-0.652272\pi$
$953$	−27086.0	−0.920675	−0.460337	+	0.887744i	$0.652272\pi$
$954$	0	0
$955$	−11758.1	−0.398410
$956$	1739.04	0.0588331
$957$	0	0
$958$	−15424.4	−0.520189
$959$	35994.0	1.21200
$960$	0	0
$961$	−18181.7	−0.610309
$962$	217.735	0.00729737
$963$	0	0
$964$	−27828.5	−0.929768
$965$	40655.0	1.35620
$966$	0	0
$967$	18127.4	0.602832	0.301416	−	0.953493i	$-0.402541\pi$
$967$	18127.4	0.602832	0.301416	+	0.953493i	$0.402541\pi$
$968$	−32825.0	−1.08991
$969$	0	0
$970$	−5578.04	−0.184639
$971$	20150.9	0.665986	0.332993	−	0.942929i	$-0.391941\pi$
$971$	20150.9	0.665986	0.332993	+	0.942929i	$0.391941\pi$
$972$	0	0
$973$	4590.84	0.151260
$974$	8578.99	0.282227
$975$	0	0
$976$	−4939.81	−0.162008
$977$	−13134.8	−0.430112	−0.215056	−	0.976602i	$-0.568993\pi$
$977$	−13134.8	−0.430112	−0.215056	+	0.976602i	$0.568993\pi$
$978$	0	0
$979$	84464.2	2.75739
$980$	−11558.2	−0.376748
$981$	0	0
$982$	−12698.4	−0.412648
$983$	7263.72	0.235683	0.117842	−	0.993032i	$-0.462402\pi$
$983$	7263.72	0.235683	0.117842	+	0.993032i	$0.462402\pi$
$984$	0	0
$985$	−62015.2	−2.00606
$986$	−8048.03	−0.259941
$987$	0	0
$988$	−14184.1	−0.456736
$989$	−10110.8	−0.325082
$990$	0	0
$991$	−14456.1	−0.463384	−0.231692	−	0.972789i	$-0.574426\pi$
$991$	−14456.1	−0.463384	−0.231692	+	0.972789i	$0.574426\pi$
$992$	15524.1	0.496864
$993$	0	0
$994$	11680.7	0.372727
$995$	49626.3	1.58116
$996$	0	0
$997$	53773.4	1.70814	0.854072	−	0.520154i	$-0.174126\pi$
$997$	53773.4	1.70814	0.854072	+	0.520154i	$0.174126\pi$
$998$	−7355.09	−0.233288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.12		22
3.2	odd	2		239.4.a.a.1.11	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.11	✓	22	3.2	odd	2
2151.4.a.a.1.12		22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.850702 q^{2} -7.27631 q^{4} -12.0519 q^{5} -14.5326 q^{7} -12.9956 q^{8} +O(q^{10})\)	\(q+0.850702 q^{2} -7.27631 q^{4} -12.0519 q^{5} -14.5326 q^{7} -12.9956 q^{8} -10.2525 q^{10} +62.1036 q^{11} -35.3151 q^{13} -12.3629 q^{14} +47.1551 q^{16} -42.4200 q^{17} -55.1989 q^{19} +87.6930 q^{20} +52.8317 q^{22} -40.9974 q^{23} +20.2474 q^{25} -30.0426 q^{26} +105.744 q^{28} +223.019 q^{29} +107.746 q^{31} +144.080 q^{32} -36.0868 q^{34} +175.145 q^{35} -7.24755 q^{37} -46.9578 q^{38} +156.621 q^{40} -102.577 q^{41} +246.621 q^{43} -451.885 q^{44} -34.8766 q^{46} +384.920 q^{47} -131.803 q^{49} +17.2245 q^{50} +256.963 q^{52} +646.786 q^{53} -748.464 q^{55} +188.860 q^{56} +189.723 q^{58} +180.551 q^{59} -104.757 q^{61} +91.6601 q^{62} -254.672 q^{64} +425.612 q^{65} +801.715 q^{67} +308.661 q^{68} +148.996 q^{70} -944.820 q^{71} -585.288 q^{73} -6.16551 q^{74} +401.644 q^{76} -902.528 q^{77} +611.307 q^{79} -568.306 q^{80} -87.2623 q^{82} -221.844 q^{83} +511.240 q^{85} +209.801 q^{86} -807.073 q^{88} +1360.05 q^{89} +513.220 q^{91} +298.310 q^{92} +327.453 q^{94} +665.249 q^{95} +544.064 q^{97} -112.125 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

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