LMFDB - Embedded newform 2151.4.a.c.1.9

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:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-2.01135 q^{2} -3.95446 q^{4} +6.74089 q^{5} +7.15060 q^{7} +24.0446 q^{8} +O(q^{10})$	\(q-2.01135 q^{2} -3.95446 q^{4} +6.74089 q^{5} +7.15060 q^{7} +24.0446 q^{8} -13.5583 q^{10} -55.3745 q^{11} -49.0766 q^{13} -14.3824 q^{14} -16.7266 q^{16} -107.911 q^{17} -146.466 q^{19} -26.6566 q^{20} +111.378 q^{22} +154.811 q^{23} -79.5604 q^{25} +98.7105 q^{26} -28.2768 q^{28} -211.264 q^{29} -128.031 q^{31} -158.714 q^{32} +217.048 q^{34} +48.2014 q^{35} -106.118 q^{37} +294.594 q^{38} +162.082 q^{40} +245.145 q^{41} -330.715 q^{43} +218.976 q^{44} -311.379 q^{46} +146.336 q^{47} -291.869 q^{49} +160.024 q^{50} +194.072 q^{52} -245.642 q^{53} -373.273 q^{55} +171.934 q^{56} +424.927 q^{58} +650.133 q^{59} +121.237 q^{61} +257.515 q^{62} +453.043 q^{64} -330.820 q^{65} +126.883 q^{67} +426.731 q^{68} -96.9501 q^{70} +1045.89 q^{71} +773.622 q^{73} +213.442 q^{74} +579.192 q^{76} -395.961 q^{77} -941.822 q^{79} -112.752 q^{80} -493.074 q^{82} +607.780 q^{83} -727.419 q^{85} +665.185 q^{86} -1331.46 q^{88} +1253.44 q^{89} -350.928 q^{91} -612.193 q^{92} -294.333 q^{94} -987.309 q^{95} -119.925 q^{97} +587.051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	−2.01135	−0.711121	−0.355560	−	0.934653i	$-0.615710\pi$
$2$	−2.01135	−0.711121	−0.355560	+	0.934653i	$0.615710\pi$
$3$	0	0
$3$	0	0
$4$	−3.95446	−0.494307
$5$	6.74089	0.602923	0.301462	−	0.953478i	$-0.402525\pi$
$5$	6.74089	0.602923	0.301462	+	0.953478i	$0.402525\pi$
$6$	0	0
$7$	7.15060	0.386096	0.193048	−	0.981189i	$-0.438163\pi$
$7$	7.15060	0.386096	0.193048	+	0.981189i	$0.438163\pi$
$8$	24.0446	1.06263
$9$	0	0
$10$	−13.5583	−0.428751
$11$	−55.3745	−1.51782	−0.758911	−	0.651194i	$-0.774268\pi$
$11$	−55.3745	−1.51782	−0.758911	+	0.651194i	$0.774268\pi$
$12$	0	0
$13$	−49.0766	−1.04703	−0.523516	−	0.852016i	$-0.675380\pi$
$13$	−49.0766	−1.04703	−0.523516	+	0.852016i	$0.675380\pi$
$14$	−14.3824	−0.274561
$15$	0	0
$16$	−16.7266	−0.261353
$17$	−107.911	−1.53955	−0.769776	−	0.638314i	$-0.779632\pi$
$17$	−107.911	−1.53955	−0.769776	+	0.638314i	$0.779632\pi$
$18$	0	0
$19$	−146.466	−1.76850	−0.884251	−	0.467012i	$-0.845330\pi$
$19$	−146.466	−1.76850	−0.884251	+	0.467012i	$0.845330\pi$
$20$	−26.6566	−0.298029
$21$	0	0
$22$	111.378	1.07935
$23$	154.811	1.40349	0.701745	−	0.712428i	$-0.252404\pi$
$23$	154.811	1.40349	0.701745	+	0.712428i	$0.252404\pi$
$24$	0	0
$25$	−79.5604	−0.636483
$26$	98.7105	0.744566
$27$	0	0
$28$	−28.2768	−0.190850
$29$	−211.264	−1.35279	−0.676393	−	0.736541i	$-0.736458\pi$
$29$	−211.264	−1.35279	−0.676393	+	0.736541i	$0.736458\pi$
$30$	0	0
$31$	−128.031	−0.741774	−0.370887	−	0.928678i	$-0.620946\pi$
$31$	−128.031	−0.741774	−0.370887	+	0.928678i	$0.620946\pi$
$32$	−158.714	−0.876779
$33$	0	0
$34$	217.048	1.09481
$35$	48.2014	0.232786
$36$	0	0
$37$	−106.118	−0.471507	−0.235753	−	0.971813i	$-0.575756\pi$
$37$	−106.118	−0.471507	−0.235753	+	0.971813i	$0.575756\pi$
$38$	294.594	1.25762
$39$	0	0
$40$	162.082	0.640686
$41$	245.145	0.933786	0.466893	−	0.884314i	$-0.345373\pi$
$41$	245.145	0.933786	0.466893	+	0.884314i	$0.345373\pi$
$42$	0	0
$43$	−330.715	−1.17287	−0.586437	−	0.809995i	$-0.699470\pi$
$43$	−330.715	−1.17287	−0.586437	+	0.809995i	$0.699470\pi$
$44$	218.976	0.750270
$45$	0	0
$46$	−311.379	−0.998051
$47$	146.336	0.454154	0.227077	−	0.973877i	$-0.427083\pi$
$47$	146.336	0.454154	0.227077	+	0.973877i	$0.427083\pi$
$48$	0	0
$49$	−291.869	−0.850930
$50$	160.024	0.452617
$51$	0	0
$52$	194.072	0.517555
$53$	−245.642	−0.636633	−0.318317	−	0.947984i	$-0.603118\pi$
$53$	−245.642	−0.636633	−0.318317	+	0.947984i	$0.603118\pi$
$54$	0	0
$55$	−373.273	−0.915130
$56$	171.934	0.410279
$57$	0	0
$58$	424.927	0.961994
$59$	650.133	1.43458	0.717289	−	0.696776i	$-0.245383\pi$
$59$	650.133	1.43458	0.717289	+	0.696776i	$0.245383\pi$
$60$	0	0
$61$	121.237	0.254471	0.127236	−	0.991873i	$-0.459390\pi$
$61$	121.237	0.254471	0.127236	+	0.991873i	$0.459390\pi$
$62$	257.515	0.527491
$63$	0	0
$64$	453.043	0.884849
$65$	−330.820	−0.631280
$66$	0	0
$67$	126.883	0.231361	0.115680	−	0.993286i	$-0.463095\pi$
$67$	126.883	0.231361	0.115680	+	0.993286i	$0.463095\pi$
$68$	426.731	0.761011
$69$	0	0
$70$	−96.9501	−0.165539
$71$	1045.89	1.74822	0.874112	−	0.485724i	$-0.161444\pi$
$71$	1045.89	1.74822	0.874112	+	0.485724i	$0.161444\pi$
$72$	0	0
$73$	773.622	1.24035	0.620175	−	0.784463i	$-0.287062\pi$
$73$	773.622	1.24035	0.620175	+	0.784463i	$0.287062\pi$
$74$	213.442	0.335298
$75$	0	0
$76$	579.192	0.874183
$77$	−395.961	−0.586025
$78$	0	0
$79$	−941.822	−1.34131	−0.670654	−	0.741771i	$-0.733986\pi$
$79$	−941.822	−1.34131	−0.670654	+	0.741771i	$0.733986\pi$
$80$	−112.752	−0.157576
$81$	0	0
$82$	−493.074	−0.664035
$83$	607.780	0.803766	0.401883	−	0.915691i	$-0.368356\pi$
$83$	607.780	0.803766	0.401883	+	0.915691i	$0.368356\pi$
$84$	0	0
$85$	−727.419	−0.928232
$86$	665.185	0.834055
$87$	0	0
$88$	−1331.46	−1.61289
$89$	1253.44	1.49285	0.746426	−	0.665468i	$-0.231768\pi$
$89$	1253.44	1.49285	0.746426	+	0.665468i	$0.231768\pi$
$90$	0	0
$91$	−350.928	−0.404255
$92$	−612.193	−0.693756
$93$	0	0
$94$	−294.333	−0.322958
$95$	−987.309	−1.06627
$96$	0	0
$97$	−119.925	−0.125532	−0.0627658	−	0.998028i	$-0.519992\pi$
$97$	−119.925	−0.125532	−0.0627658	+	0.998028i	$0.519992\pi$
$98$	587.051	0.605114
$99$	0	0
$100$	314.618	0.314618
$101$	−1370.82	−1.35051	−0.675256	−	0.737583i	$-0.735967\pi$
$101$	−1370.82	−1.35051	−0.675256	+	0.737583i	$0.735967\pi$
$102$	0	0
$103$	910.491	0.871004	0.435502	−	0.900188i	$-0.356571\pi$
$103$	910.491	0.871004	0.435502	+	0.900188i	$0.356571\pi$
$104$	−1180.03	−1.11261
$105$	0	0
$106$	494.074	0.452723
$107$	−615.223	−0.555849	−0.277924	−	0.960603i	$-0.589646\pi$
$107$	−615.223	−0.555849	−0.277924	+	0.960603i	$0.589646\pi$
$108$	0	0
$109$	−1165.69	−1.02434	−0.512168	−	0.858885i	$-0.671157\pi$
$109$	−1165.69	−1.02434	−0.512168	+	0.858885i	$0.671157\pi$
$110$	750.784	0.650768
$111$	0	0
$112$	−119.605	−0.100907
$113$	1288.16	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$113$	1288.16	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$114$	0	0
$115$	1043.56	0.846197
$116$	835.435	0.668691
$117$	0	0
$118$	−1307.65	−1.02016
$119$	−771.632	−0.594415
$120$	0	0
$121$	1735.34	1.30378
$122$	−243.850	−0.180960
$123$	0	0
$124$	506.292	0.366664
$125$	−1378.92	−0.986674
$126$	0	0
$127$	809.460	0.565575	0.282787	−	0.959183i	$-0.408741\pi$
$127$	809.460	0.565575	0.282787	+	0.959183i	$0.408741\pi$
$128$	358.483	0.247545
$129$	0	0
$130$	665.396	0.448916
$131$	−1656.97	−1.10512	−0.552559	−	0.833474i	$-0.686349\pi$
$131$	−1656.97	−1.10512	−0.552559	+	0.833474i	$0.686349\pi$
$132$	0	0
$133$	−1047.32	−0.682812
$134$	−255.206	−0.164525
$135$	0	0
$136$	−2594.69	−1.63598
$137$	−2406.38	−1.50066	−0.750330	−	0.661063i	$-0.770106\pi$
$137$	−2406.38	−1.50066	−0.750330	+	0.661063i	$0.770106\pi$
$138$	0	0
$139$	−2748.06	−1.67689	−0.838443	−	0.544990i	$-0.816534\pi$
$139$	−2748.06	−1.67689	−0.838443	+	0.544990i	$0.816534\pi$
$140$	−190.610	−0.115068
$141$	0	0
$142$	−2103.65	−1.24320
$143$	2717.59	1.58921
$144$	0	0
$145$	−1424.11	−0.815626
$146$	−1556.03	−0.882039
$147$	0	0
$148$	419.641	0.233069
$149$	−1394.75	−0.766860	−0.383430	−	0.923570i	$-0.625257\pi$
$149$	−1394.75	−0.766860	−0.383430	+	0.923570i	$0.625257\pi$
$150$	0	0
$151$	2258.35	1.21710	0.608549	−	0.793516i	$-0.291752\pi$
$151$	2258.35	1.21710	0.608549	+	0.793516i	$0.291752\pi$
$152$	−3521.71	−1.87927
$153$	0	0
$154$	796.417	0.416735
$155$	−863.041	−0.447233
$156$	0	0
$157$	2906.90	1.47768	0.738840	−	0.673881i	$-0.235374\pi$
$157$	2906.90	1.47768	0.738840	+	0.673881i	$0.235374\pi$
$158$	1894.34	0.953832
$159$	0	0
$160$	−1069.87	−0.528631
$161$	1106.99	0.541882
$162$	0	0
$163$	−480.002	−0.230654	−0.115327	−	0.993328i	$-0.536792\pi$
$163$	−480.002	−0.230654	−0.115327	+	0.993328i	$0.536792\pi$
$164$	−969.416	−0.461577
$165$	0	0
$166$	−1222.46	−0.571575
$167$	−1161.32	−0.538119	−0.269059	−	0.963124i	$-0.586713\pi$
$167$	−1161.32	−0.538119	−0.269059	+	0.963124i	$0.586713\pi$
$168$	0	0
$169$	211.517	0.0962754
$170$	1463.10	0.660085
$171$	0	0
$172$	1307.80	0.579760
$173$	−2218.05	−0.974770	−0.487385	−	0.873187i	$-0.662049\pi$
$173$	−2218.05	−0.974770	−0.487385	+	0.873187i	$0.662049\pi$
$174$	0	0
$175$	−568.905	−0.245744
$176$	926.227	0.396688
$177$	0	0
$178$	−2521.10	−1.06160
$179$	−1808.75	−0.755263	−0.377632	−	0.925956i	$-0.623261\pi$
$179$	−1808.75	−0.755263	−0.377632	+	0.925956i	$0.623261\pi$
$180$	0	0
$181$	−898.033	−0.368786	−0.184393	−	0.982853i	$-0.559032\pi$
$181$	−898.033	−0.368786	−0.184393	+	0.982853i	$0.559032\pi$
$182$	705.839	0.287474
$183$	0	0
$184$	3722.37	1.49140
$185$	−715.332	−0.284282
$186$	0	0
$187$	5975.54	2.33676
$188$	−578.678	−0.224492
$189$	0	0
$190$	1985.83	0.758247
$191$	3244.92	1.22929	0.614644	−	0.788805i	$-0.289300\pi$
$191$	3244.92	1.22929	0.614644	+	0.788805i	$0.289300\pi$
$192$	0	0
$193$	−4184.50	−1.56066	−0.780328	−	0.625371i	$-0.784948\pi$
$193$	−4184.50	−1.56066	−0.780328	+	0.625371i	$0.784948\pi$
$194$	241.212	0.0892681
$195$	0	0
$196$	1154.18	0.420621
$197$	3491.52	1.26274	0.631372	−	0.775480i	$-0.282492\pi$
$197$	3491.52	1.26274	0.631372	+	0.775480i	$0.282492\pi$
$198$	0	0
$199$	−359.328	−0.128001	−0.0640003	−	0.997950i	$-0.520386\pi$
$199$	−359.328	−0.128001	−0.0640003	+	0.997950i	$0.520386\pi$
$200$	−1913.00	−0.676348
$201$	0	0
$202$	2757.21	0.960378
$203$	−1510.67	−0.522305
$204$	0	0
$205$	1652.50	0.563002
$206$	−1831.32	−0.619389
$207$	0	0
$208$	820.886	0.273645
$209$	8110.46	2.68427
$210$	0	0
$211$	2893.44	0.944040	0.472020	−	0.881588i	$-0.343525\pi$
$211$	2893.44	0.944040	0.472020	+	0.881588i	$0.343525\pi$
$212$	971.382	0.314692
$213$	0	0
$214$	1237.43	0.395276
$215$	−2229.31	−0.707153
$216$	0	0
$217$	−915.497	−0.286396
$218$	2344.61	0.728427
$219$	0	0
$220$	1476.09	0.452355
$221$	5295.93	1.61196
$222$	0	0
$223$	2810.03	0.843826	0.421913	−	0.906636i	$-0.361359\pi$
$223$	2810.03	0.843826	0.421913	+	0.906636i	$0.361359\pi$
$224$	−1134.90	−0.338521
$225$	0	0
$226$	−2590.94	−0.762595
$227$	3407.89	0.996429	0.498214	−	0.867054i	$-0.333989\pi$
$227$	3407.89	0.996429	0.498214	+	0.867054i	$0.333989\pi$
$228$	0	0
$229$	2728.66	0.787401	0.393700	−	0.919239i	$-0.371195\pi$
$229$	2728.66	0.787401	0.393700	+	0.919239i	$0.371195\pi$
$230$	−2098.97	−0.601748
$231$	0	0
$232$	−5079.77	−1.43751
$233$	6079.49	1.70936	0.854679	−	0.519157i	$-0.173754\pi$
$233$	6079.49	1.70936	0.854679	+	0.519157i	$0.173754\pi$
$234$	0	0
$235$	986.432	0.273820
$236$	−2570.92	−0.709122
$237$	0	0
$238$	1552.02	0.422701
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−975.805	−0.260818	−0.130409	−	0.991460i	$-0.541629\pi$
$241$	−975.805	−0.260818	−0.130409	+	0.991460i	$0.541629\pi$
$242$	−3490.37	−0.927147
$243$	0	0
$244$	−479.425	−0.125787
$245$	−1967.46	−0.513045
$246$	0	0
$247$	7188.04	1.85168
$248$	−3078.45	−0.788234
$249$	0	0
$250$	2773.49	0.701644
$251$	−1287.90	−0.323871	−0.161935	−	0.986801i	$-0.551774\pi$
$251$	−1287.90	−0.323871	−0.161935	+	0.986801i	$0.551774\pi$
$252$	0	0
$253$	−8572.57	−2.13025
$254$	−1628.11	−0.402192
$255$	0	0
$256$	−4345.38	−1.06088
$257$	−879.568	−0.213486	−0.106743	−	0.994287i	$-0.534042\pi$
$257$	−879.568	−0.213486	−0.106743	+	0.994287i	$0.534042\pi$
$258$	0	0
$259$	−758.810	−0.182047
$260$	1308.21	0.312046
$261$	0	0
$262$	3332.76	0.785873
$263$	−293.800	−0.0688839	−0.0344420	−	0.999407i	$-0.510965\pi$
$263$	−293.800	−0.0688839	−0.0344420	+	0.999407i	$0.510965\pi$
$264$	0	0
$265$	−1655.85	−0.383841
$266$	2106.53	0.485562
$267$	0	0
$268$	−501.752	−0.114363
$269$	−4273.57	−0.968640	−0.484320	−	0.874891i	$-0.660933\pi$
$269$	−4273.57	−0.968640	−0.484320	+	0.874891i	$0.660933\pi$
$270$	0	0
$271$	−3251.91	−0.728928	−0.364464	−	0.931218i	$-0.618748\pi$
$271$	−3251.91	−0.728928	−0.364464	+	0.931218i	$0.618748\pi$
$272$	1804.99	0.402367
$273$	0	0
$274$	4840.07	1.06715
$275$	4405.62	0.966068
$276$	0	0
$277$	521.183	0.113050	0.0565250	−	0.998401i	$-0.481998\pi$
$277$	521.183	0.113050	0.0565250	+	0.998401i	$0.481998\pi$
$278$	5527.31	1.19247
$279$	0	0
$280$	1158.99	0.247367
$281$	6682.97	1.41876	0.709382	−	0.704824i	$-0.248974\pi$
$281$	6682.97	1.41876	0.709382	+	0.704824i	$0.248974\pi$
$282$	0	0
$283$	3020.96	0.634549	0.317275	−	0.948334i	$-0.397232\pi$
$283$	3020.96	0.634549	0.317275	+	0.948334i	$0.397232\pi$
$284$	−4135.92	−0.864160
$285$	0	0
$286$	−5466.04	−1.13012
$287$	1752.94	0.360531
$288$	0	0
$289$	6731.89	1.37022
$290$	2864.38	0.580008
$291$	0	0
$292$	−3059.25	−0.613114
$293$	35.7616	0.00713043	0.00356522	−	0.999994i	$-0.498865\pi$
$293$	35.7616	0.00713043	0.00356522	+	0.999994i	$0.498865\pi$
$294$	0	0
$295$	4382.47	0.864941
$296$	−2551.58	−0.501039
$297$	0	0
$298$	2805.33	0.545330
$299$	−7597.59	−1.46950
$300$	0	0
$301$	−2364.81	−0.452842
$302$	−4542.34	−0.865504
$303$	0	0
$304$	2449.87	0.462203
$305$	817.242	0.153427
$306$	0	0
$307$	−4743.33	−0.881811	−0.440906	−	0.897554i	$-0.645343\pi$
$307$	−4743.33	−0.881811	−0.440906	+	0.897554i	$0.645343\pi$
$308$	1565.81	0.289677
$309$	0	0
$310$	1735.88	0.318037
$311$	−8608.50	−1.56959	−0.784797	−	0.619754i	$-0.787233\pi$
$311$	−8608.50	−1.56959	−0.784797	+	0.619754i	$0.787233\pi$
$312$	0	0
$313$	8626.71	1.55786	0.778930	−	0.627111i	$-0.215763\pi$
$313$	8626.71	1.55786	0.778930	+	0.627111i	$0.215763\pi$
$314$	−5846.80	−1.05081
$315$	0	0
$316$	3724.40	0.663018
$317$	−1975.41	−0.350000	−0.175000	−	0.984568i	$-0.555993\pi$
$317$	−1975.41	−0.350000	−0.175000	+	0.984568i	$0.555993\pi$
$318$	0	0
$319$	11698.6	2.05329
$320$	3053.91	0.533496
$321$	0	0
$322$	−2226.55	−0.385344
$323$	15805.3	2.72270
$324$	0	0
$325$	3904.56	0.666418
$326$	965.453	0.164023
$327$	0	0
$328$	5894.43	0.992272
$329$	1046.39	0.175347
$330$	0	0
$331$	3712.29	0.616453	0.308226	−	0.951313i	$-0.400265\pi$
$331$	3712.29	0.616453	0.308226	+	0.951313i	$0.400265\pi$
$332$	−2403.44	−0.397307
$333$	0	0
$334$	2335.83	0.382667
$335$	855.301	0.139493
$336$	0	0
$337$	5570.33	0.900401	0.450200	−	0.892928i	$-0.351353\pi$
$337$	5570.33	0.900401	0.450200	+	0.892928i	$0.351353\pi$
$338$	−425.435	−0.0684634
$339$	0	0
$340$	2876.55	0.458832
$341$	7089.64	1.12588
$342$	0	0
$343$	−4539.69	−0.714637
$344$	−7951.93	−1.24633
$345$	0	0
$346$	4461.28	0.693179
$347$	7497.77	1.15995	0.579973	−	0.814635i	$-0.303063\pi$
$347$	7497.77	1.15995	0.579973	+	0.814635i	$0.303063\pi$
$348$	0	0
$349$	−3536.20	−0.542374	−0.271187	−	0.962527i	$-0.587416\pi$
$349$	−3536.20	−0.542374	−0.271187	+	0.962527i	$0.587416\pi$
$350$	1144.27	0.174754
$351$	0	0
$352$	8788.71	1.33079
$353$	−1841.34	−0.277634	−0.138817	−	0.990318i	$-0.544330\pi$
$353$	−1841.34	−0.277634	−0.138817	+	0.990318i	$0.544330\pi$
$354$	0	0
$355$	7050.21	1.05405
$356$	−4956.66	−0.737928
$357$	0	0
$358$	3638.03	0.537083
$359$	−2393.11	−0.351820	−0.175910	−	0.984406i	$-0.556287\pi$
$359$	−2393.11	−0.351820	−0.175910	+	0.984406i	$0.556287\pi$
$360$	0	0
$361$	14593.2	2.12760
$362$	1806.26	0.262251
$363$	0	0
$364$	1387.73	0.199826
$365$	5214.90	0.747836
$366$	0	0
$367$	4474.64	0.636443	0.318221	−	0.948016i	$-0.396914\pi$
$367$	4474.64	0.636443	0.318221	+	0.948016i	$0.396914\pi$
$368$	−2589.46	−0.366807
$369$	0	0
$370$	1438.79	0.202159
$371$	−1756.49	−0.245802
$372$	0	0
$373$	−1385.41	−0.192316	−0.0961579	−	0.995366i	$-0.530655\pi$
$373$	−1385.41	−0.192316	−0.0961579	+	0.995366i	$0.530655\pi$
$374$	−12018.9	−1.66172
$375$	0	0
$376$	3518.59	0.482599
$377$	10368.1	1.41641
$378$	0	0
$379$	−13427.6	−1.81987	−0.909936	−	0.414750i	$-0.863869\pi$
$379$	−13427.6	−1.81987	−0.909936	+	0.414750i	$0.863869\pi$
$380$	3904.27	0.527065
$381$	0	0
$382$	−6526.68	−0.874172
$383$	9516.01	1.26957	0.634785	−	0.772689i	$-0.281089\pi$
$383$	9516.01	1.26957	0.634785	+	0.772689i	$0.281089\pi$
$384$	0	0
$385$	−2669.13	−0.353328
$386$	8416.50	1.10981
$387$	0	0
$388$	474.239	0.0620512
$389$	337.138	0.0439423	0.0219711	−	0.999759i	$-0.493006\pi$
$389$	337.138	0.0439423	0.0219711	+	0.999759i	$0.493006\pi$
$390$	0	0
$391$	−16705.9	−2.16075
$392$	−7017.88	−0.904226
$393$	0	0
$394$	−7022.68	−0.897963
$395$	−6348.72	−0.808706
$396$	0	0
$397$	−3268.80	−0.413240	−0.206620	−	0.978421i	$-0.566246\pi$
$397$	−3268.80	−0.413240	−0.206620	+	0.978421i	$0.566246\pi$
$398$	722.736	0.0910239
$399$	0	0
$400$	1330.78	0.166347
$401$	−13229.3	−1.64748	−0.823739	−	0.566970i	$-0.808116\pi$
$401$	−13229.3	−1.64748	−0.823739	+	0.566970i	$0.808116\pi$
$402$	0	0
$403$	6283.32	0.776661
$404$	5420.85	0.667568
$405$	0	0
$406$	3038.48	0.371422
$407$	5876.25	0.715663
$408$	0	0
$409$	−8426.96	−1.01879	−0.509396	−	0.860532i	$-0.670131\pi$
$409$	−8426.96	−1.01879	−0.509396	+	0.860532i	$0.670131\pi$
$410$	−3323.75	−0.400362
$411$	0	0
$412$	−3600.50	−0.430543
$413$	4648.84	0.553885
$414$	0	0
$415$	4096.98	0.484609
$416$	7789.15	0.918016
$417$	0	0
$418$	−16313.0	−1.90884
$419$	−1343.42	−0.156636	−0.0783181	−	0.996928i	$-0.524955\pi$
$419$	−1343.42	−0.156636	−0.0783181	+	0.996928i	$0.524955\pi$
$420$	0	0
$421$	6500.60	0.752541	0.376270	−	0.926510i	$-0.377206\pi$
$421$	6500.60	0.752541	0.376270	+	0.926510i	$0.377206\pi$
$422$	−5819.72	−0.671326
$423$	0	0
$424$	−5906.38	−0.676508
$425$	8585.48	0.979899
$426$	0	0
$427$	866.914	0.0982504
$428$	2432.87	0.274760
$429$	0	0
$430$	4483.94	0.502871
$431$	−9752.07	−1.08989	−0.544943	−	0.838473i	$-0.683449\pi$
$431$	−9752.07	−1.08989	−0.544943	+	0.838473i	$0.683449\pi$
$432$	0	0
$433$	−3070.02	−0.340729	−0.170364	−	0.985381i	$-0.554494\pi$
$433$	−3070.02	−0.340729	−0.170364	+	0.985381i	$0.554494\pi$
$434$	1841.39	0.203662
$435$	0	0
$436$	4609.66	0.506337
$437$	−22674.5	−2.48208
$438$	0	0
$439$	13661.9	1.48530	0.742652	−	0.669678i	$-0.233568\pi$
$439$	13661.9	1.48530	0.742652	+	0.669678i	$0.233568\pi$
$440$	−8975.22	−0.972447
$441$	0	0
$442$	−10652.0	−1.14630
$443$	8598.25	0.922157	0.461078	−	0.887359i	$-0.347463\pi$
$443$	8598.25	0.922157	0.461078	+	0.887359i	$0.347463\pi$
$444$	0	0
$445$	8449.27	0.900076
$446$	−5651.96	−0.600063
$447$	0	0
$448$	3239.53	0.341637
$449$	−8665.84	−0.910838	−0.455419	−	0.890277i	$-0.650510\pi$
$449$	−8665.84	−0.910838	−0.455419	+	0.890277i	$0.650510\pi$
$450$	0	0
$451$	−13574.8	−1.41732
$452$	−5093.96	−0.530087
$453$	0	0
$454$	−6854.46	−0.708581
$455$	−2365.56	−0.243735
$456$	0	0
$457$	−5108.88	−0.522940	−0.261470	−	0.965212i	$-0.584207\pi$
$457$	−5108.88	−0.522940	−0.261470	+	0.965212i	$0.584207\pi$
$458$	−5488.29	−0.559937
$459$	0	0
$460$	−4126.72	−0.418281
$461$	−1434.32	−0.144908	−0.0724541	−	0.997372i	$-0.523083\pi$
$461$	−1434.32	−0.144908	−0.0724541	+	0.997372i	$0.523083\pi$
$462$	0	0
$463$	9917.89	0.995515	0.497758	−	0.867316i	$-0.334157\pi$
$463$	9917.89	0.995515	0.497758	+	0.867316i	$0.334157\pi$
$464$	3533.73	0.353555
$465$	0	0
$466$	−12228.0	−1.21556
$467$	15406.4	1.52660	0.763302	−	0.646042i	$-0.223577\pi$
$467$	15406.4	1.52660	0.763302	+	0.646042i	$0.223577\pi$
$468$	0	0
$469$	907.286	0.0893275
$470$	−1984.06	−0.194719
$471$	0	0
$472$	15632.2	1.52443
$473$	18313.2	1.78021
$474$	0	0
$475$	11652.9	1.12562
$476$	3051.39	0.293824
$477$	0	0
$478$	−480.713	−0.0459986
$479$	−14716.1	−1.40375	−0.701874	−	0.712302i	$-0.747653\pi$
$479$	−14716.1	−1.40375	−0.701874	+	0.712302i	$0.747653\pi$
$480$	0	0
$481$	5207.93	0.493683
$482$	1962.69	0.185473
$483$	0	0
$484$	−6862.31	−0.644469
$485$	−808.403	−0.0756859
$486$	0	0
$487$	−5923.12	−0.551134	−0.275567	−	0.961282i	$-0.588866\pi$
$487$	−5923.12	−0.551134	−0.275567	+	0.961282i	$0.588866\pi$
$488$	2915.09	0.270410
$489$	0	0
$490$	3957.25	0.364837
$491$	7010.72	0.644378	0.322189	−	0.946675i	$-0.395581\pi$
$491$	7010.72	0.644378	0.322189	+	0.946675i	$0.395581\pi$
$492$	0	0
$493$	22797.8	2.08268
$494$	−14457.7	−1.31677
$495$	0	0
$496$	2141.52	0.193865
$497$	7478.72	0.674983
$498$	0	0
$499$	−10724.4	−0.962109	−0.481054	−	0.876691i	$-0.659746\pi$
$499$	−10724.4	−0.962109	−0.481054	+	0.876691i	$0.659746\pi$
$500$	5452.88	0.487720
$501$	0	0
$502$	2590.42	0.230311
$503$	−6763.29	−0.599523	−0.299761	−	0.954014i	$-0.596907\pi$
$503$	−6763.29	−0.599523	−0.299761	+	0.954014i	$0.596907\pi$
$504$	0	0
$505$	−9240.55	−0.814256
$506$	17242.5	1.51486
$507$	0	0
$508$	−3200.98	−0.279568
$509$	9201.80	0.801302	0.400651	−	0.916231i	$-0.368784\pi$
$509$	9201.80	0.801302	0.400651	+	0.916231i	$0.368784\pi$
$510$	0	0
$511$	5531.86	0.478895
$512$	5872.23	0.506872
$513$	0	0
$514$	1769.12	0.151814
$515$	6137.52	0.525148
$516$	0	0
$517$	−8103.26	−0.689325
$518$	1526.24	0.129457
$519$	0	0
$520$	−7954.45	−0.670819
$521$	−14260.1	−1.19913	−0.599567	−	0.800325i	$-0.704660\pi$
$521$	−14260.1	−1.19913	−0.599567	+	0.800325i	$0.704660\pi$
$522$	0	0
$523$	−12998.5	−1.08678	−0.543389	−	0.839481i	$-0.682859\pi$
$523$	−12998.5	−1.08678	−0.543389	+	0.839481i	$0.682859\pi$
$524$	6552.43	0.546268
$525$	0	0
$526$	590.935	0.0489848
$527$	13816.0	1.14200
$528$	0	0
$529$	11799.4	0.969786
$530$	3330.49	0.272957
$531$	0	0
$532$	4141.57	0.337519
$533$	−12030.9	−0.977704
$534$	0	0
$535$	−4147.15	−0.335134
$536$	3050.84	0.245852
$537$	0	0
$538$	8595.66	0.688820
$539$	16162.1	1.29156
$540$	0	0
$541$	−7751.60	−0.616021	−0.308011	−	0.951383i	$-0.599663\pi$
$541$	−7751.60	−0.616021	−0.308011	+	0.951383i	$0.599663\pi$
$542$	6540.74	0.518356
$543$	0	0
$544$	17127.1	1.34985
$545$	−7857.77	−0.617596
$546$	0	0
$547$	−2182.84	−0.170625	−0.0853123	−	0.996354i	$-0.527189\pi$
$547$	−2182.84	−0.170625	−0.0853123	+	0.996354i	$0.527189\pi$
$548$	9515.91	0.741787
$549$	0	0
$550$	−8861.26	−0.686991
$551$	30942.9	2.39240
$552$	0	0
$553$	−6734.60	−0.517874
$554$	−1048.28	−0.0803923
$555$	0	0
$556$	10867.1	0.828897
$557$	16957.8	1.28999	0.644997	−	0.764185i	$-0.276859\pi$
$557$	16957.8	1.28999	0.644997	+	0.764185i	$0.276859\pi$
$558$	0	0
$559$	16230.4	1.22804
$560$	−806.246	−0.0608395
$561$	0	0
$562$	−13441.8	−1.00891
$563$	−16814.1	−1.25867	−0.629333	−	0.777136i	$-0.716672\pi$
$563$	−16814.1	−1.25867	−0.629333	+	0.777136i	$0.716672\pi$
$564$	0	0
$565$	8683.31	0.646566
$566$	−6076.22	−0.451241
$567$	0	0
$568$	25148.0	1.85772
$569$	20111.1	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$569$	20111.1	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$570$	0	0
$571$	−4633.78	−0.339611	−0.169805	−	0.985478i	$-0.554314\pi$
$571$	−4633.78	−0.339611	−0.169805	+	0.985478i	$0.554314\pi$
$572$	−10746.6	−0.785557
$573$	0	0
$574$	−3525.77	−0.256381
$575$	−12316.8	−0.893299
$576$	0	0
$577$	11864.8	0.856045	0.428022	−	0.903768i	$-0.359210\pi$
$577$	11864.8	0.856045	0.428022	+	0.903768i	$0.359210\pi$
$578$	−13540.2	−0.974391
$579$	0	0
$580$	5631.57	0.403170
$581$	4346.00	0.310331
$582$	0	0
$583$	13602.3	0.966296
$584$	18601.5	1.31804
$585$	0	0
$586$	−71.9293	−0.00507060
$587$	−2920.48	−0.205351	−0.102676	−	0.994715i	$-0.532740\pi$
$587$	−2920.48	−0.205351	−0.102676	+	0.994715i	$0.532740\pi$
$588$	0	0
$589$	18752.1	1.31183
$590$	−8814.70	−0.615077
$591$	0	0
$592$	1775.00	0.123230
$593$	−4942.18	−0.342244	−0.171122	−	0.985250i	$-0.554739\pi$
$593$	−4942.18	−0.342244	−0.171122	+	0.985250i	$0.554739\pi$
$594$	0	0
$595$	−5201.48	−0.358387
$596$	5515.47	0.379064
$597$	0	0
$598$	15281.4	1.04499
$599$	−313.245	−0.0213670	−0.0106835	−	0.999943i	$-0.503401\pi$
$599$	−313.245	−0.0213670	−0.0106835	+	0.999943i	$0.503401\pi$
$600$	0	0
$601$	−8262.68	−0.560802	−0.280401	−	0.959883i	$-0.590467\pi$
$601$	−8262.68	−0.560802	−0.280401	+	0.959883i	$0.590467\pi$
$602$	4756.47	0.322026
$603$	0	0
$604$	−8930.55	−0.601620
$605$	11697.7	0.786081
$606$	0	0
$607$	−5574.49	−0.372754	−0.186377	−	0.982478i	$-0.559675\pi$
$607$	−5574.49	−0.372754	−0.186377	+	0.982478i	$0.559675\pi$
$608$	23246.2	1.55059
$609$	0	0
$610$	−1643.76	−0.109105
$611$	−7181.66	−0.475514
$612$	0	0
$613$	−14290.0	−0.941546	−0.470773	−	0.882254i	$-0.656025\pi$
$613$	−14290.0	−0.941546	−0.470773	+	0.882254i	$0.656025\pi$
$614$	9540.51	0.627074
$615$	0	0
$616$	−9520.74	−0.622730
$617$	6240.24	0.407168	0.203584	−	0.979057i	$-0.434741\pi$
$617$	6240.24	0.407168	0.203584	+	0.979057i	$0.434741\pi$
$618$	0	0
$619$	−17419.6	−1.13110	−0.565551	−	0.824713i	$-0.691337\pi$
$619$	−17419.6	−1.13110	−0.565551	+	0.824713i	$0.691337\pi$
$620$	3412.86	0.221070
$621$	0	0
$622$	17314.7	1.11617
$623$	8962.82	0.576385
$624$	0	0
$625$	649.916	0.0415946
$626$	−17351.4	−1.10783
$627$	0	0
$628$	−11495.2	−0.730428
$629$	11451.4	0.725909
$630$	0	0
$631$	−18932.8	−1.19446	−0.597228	−	0.802072i	$-0.703731\pi$
$631$	−18932.8	−1.19446	−0.597228	+	0.802072i	$0.703731\pi$
$632$	−22645.8	−1.42532
$633$	0	0
$634$	3973.25	0.248892
$635$	5456.48	0.340998
$636$	0	0
$637$	14323.9	0.890950
$638$	−23530.1	−1.46013
$639$	0	0
$640$	2416.49	0.149250
$641$	15564.5	0.959067	0.479534	−	0.877523i	$-0.340806\pi$
$641$	15564.5	0.959067	0.479534	+	0.877523i	$0.340806\pi$
$642$	0	0
$643$	4681.54	0.287126	0.143563	−	0.989641i	$-0.454144\pi$
$643$	4681.54	0.287126	0.143563	+	0.989641i	$0.454144\pi$
$644$	−4377.55	−0.267856
$645$	0	0
$646$	−31790.1	−1.93617
$647$	20196.3	1.22720	0.613601	−	0.789617i	$-0.289721\pi$
$647$	20196.3	1.22720	0.613601	+	0.789617i	$0.289721\pi$
$648$	0	0
$649$	−36000.8	−2.17743
$650$	−7853.45	−0.473904
$651$	0	0
$652$	1898.15	0.114014
$653$	25667.3	1.53819	0.769097	−	0.639132i	$-0.220706\pi$
$653$	25667.3	1.53819	0.769097	+	0.639132i	$0.220706\pi$
$654$	0	0
$655$	−11169.5	−0.666302
$656$	−4100.45	−0.244048
$657$	0	0
$658$	−2104.65	−0.124693
$659$	−12899.1	−0.762487	−0.381244	−	0.924475i	$-0.624504\pi$
$659$	−12899.1	−0.762487	−0.381244	+	0.924475i	$0.624504\pi$
$660$	0	0
$661$	28681.4	1.68771	0.843854	−	0.536572i	$-0.180281\pi$
$661$	28681.4	1.68771	0.843854	+	0.536572i	$0.180281\pi$
$662$	−7466.73	−0.438373
$663$	0	0
$664$	14613.9	0.854108
$665$	−7059.85	−0.411683
$666$	0	0
$667$	−32706.0	−1.89862
$668$	4592.40	0.265996
$669$	0	0
$670$	−1720.31	−0.0991962
$671$	−6713.41	−0.386242
$672$	0	0
$673$	−9745.84	−0.558209	−0.279105	−	0.960261i	$-0.590038\pi$
$673$	−9745.84	−0.558209	−0.279105	+	0.960261i	$0.590038\pi$
$674$	−11203.9	−0.640294
$675$	0	0
$676$	−836.435	−0.0475896
$677$	−5177.70	−0.293937	−0.146968	−	0.989141i	$-0.546952\pi$
$677$	−5177.70	−0.293937	−0.146968	+	0.989141i	$0.546952\pi$
$678$	0	0
$679$	−857.538	−0.0484673
$680$	−17490.5	−0.986369
$681$	0	0
$682$	−14259.8	−0.800637
$683$	−4890.88	−0.274003	−0.137002	−	0.990571i	$-0.543747\pi$
$683$	−4890.88	−0.274003	−0.137002	+	0.990571i	$0.543747\pi$
$684$	0	0
$685$	−16221.1	−0.904783
$686$	9130.93	0.508193
$687$	0	0
$688$	5531.74	0.306534
$689$	12055.3	0.666575
$690$	0	0
$691$	−22953.6	−1.26367	−0.631835	−	0.775103i	$-0.717698\pi$
$691$	−22953.6	−1.26367	−0.631835	+	0.775103i	$0.717698\pi$
$692$	8771.19	0.481836
$693$	0	0
$694$	−15080.7	−0.824862
$695$	−18524.3	−1.01103
$696$	0	0
$697$	−26454.0	−1.43761
$698$	7112.55	0.385693
$699$	0	0
$700$	2249.71	0.121473
$701$	30782.7	1.65856	0.829278	−	0.558837i	$-0.188752\pi$
$701$	30782.7	1.65856	0.829278	+	0.558837i	$0.188752\pi$
$702$	0	0
$703$	15542.7	0.833860
$704$	−25087.0	−1.34304
$705$	0	0
$706$	3703.59	0.197431
$707$	−9802.20	−0.521428
$708$	0	0
$709$	9306.46	0.492964	0.246482	−	0.969147i	$-0.420725\pi$
$709$	9306.46	0.492964	0.246482	+	0.969147i	$0.420725\pi$
$710$	−14180.5	−0.749554
$711$	0	0
$712$	30138.4	1.58635
$713$	−19820.5	−1.04107
$714$	0	0
$715$	18319.0	0.958170
$716$	7152.62	0.373332
$717$	0	0
$718$	4813.38	0.250187
$719$	−5329.17	−0.276418	−0.138209	−	0.990403i	$-0.544135\pi$
$719$	−5329.17	−0.276418	−0.138209	+	0.990403i	$0.544135\pi$
$720$	0	0
$721$	6510.56	0.336291
$722$	−29352.1	−1.51298
$723$	0	0
$724$	3551.23	0.182294
$725$	16808.3	0.861025
$726$	0	0
$727$	15424.1	0.786863	0.393432	−	0.919354i	$-0.371288\pi$
$727$	15424.1	0.786863	0.393432	+	0.919354i	$0.371288\pi$
$728$	−8437.93	−0.429575
$729$	0	0
$730$	−10489.0	−0.531802
$731$	35688.0	1.80570
$732$	0	0
$733$	2001.68	0.100865	0.0504324	−	0.998727i	$-0.483940\pi$
$733$	2001.68	0.100865	0.0504324	+	0.998727i	$0.483940\pi$
$734$	−9000.09	−0.452588
$735$	0	0
$736$	−24570.6	−1.23055
$737$	−7026.06	−0.351164
$738$	0	0
$739$	−14786.3	−0.736024	−0.368012	−	0.929821i	$-0.619961\pi$
$739$	−14786.3	−0.736024	−0.368012	+	0.929821i	$0.619961\pi$
$740$	2828.75	0.140523
$741$	0	0
$742$	3532.92	0.174795
$743$	−8632.31	−0.426230	−0.213115	−	0.977027i	$-0.568361\pi$
$743$	−8632.31	−0.426230	−0.213115	+	0.977027i	$0.568361\pi$
$744$	0	0
$745$	−9401.83	−0.462358
$746$	2786.55	0.136760
$747$	0	0
$748$	−23630.0	−1.15508
$749$	−4399.21	−0.214611
$750$	0	0
$751$	14904.8	0.724213	0.362106	−	0.932137i	$-0.382058\pi$
$751$	14904.8	0.724213	0.362106	+	0.932137i	$0.382058\pi$
$752$	−2447.70	−0.118695
$753$	0	0
$754$	−20854.0	−1.00724
$755$	15223.3	0.733817
$756$	0	0
$757$	19521.9	0.937301	0.468650	−	0.883384i	$-0.344740\pi$
$757$	19521.9	0.937301	0.468650	+	0.883384i	$0.344740\pi$
$758$	27007.7	1.29415
$759$	0	0
$760$	−23739.5	−1.13305
$761$	26566.0	1.26546	0.632730	−	0.774372i	$-0.281934\pi$
$761$	26566.0	1.26546	0.632730	+	0.774372i	$0.281934\pi$
$762$	0	0
$763$	−8335.37	−0.395492
$764$	−12831.9	−0.607646
$765$	0	0
$766$	−19140.1	−0.902818
$767$	−31906.3	−1.50205
$768$	0	0
$769$	−13882.6	−0.650999	−0.325499	−	0.945542i	$-0.605532\pi$
$769$	−13882.6	−0.650999	−0.325499	+	0.945542i	$0.605532\pi$
$770$	5368.56	0.251259
$771$	0	0
$772$	16547.4	0.771443
$773$	−33287.2	−1.54884	−0.774422	−	0.632669i	$-0.781959\pi$
$773$	−33287.2	−1.54884	−0.774422	+	0.632669i	$0.781959\pi$
$774$	0	0
$775$	10186.2	0.472127
$776$	−2883.56	−0.133394
$777$	0	0
$778$	−678.103	−0.0312483
$779$	−35905.4	−1.65140
$780$	0	0
$781$	−57915.5	−2.65349
$782$	33601.4	1.53655
$783$	0	0
$784$	4881.97	0.222393
$785$	19595.1	0.890928
$786$	0	0
$787$	7640.06	0.346047	0.173023	−	0.984918i	$-0.444646\pi$
$787$	7640.06	0.346047	0.173023	+	0.984918i	$0.444646\pi$
$788$	−13807.1	−0.624183
$789$	0	0
$790$	12769.5	0.575087
$791$	9211.09	0.414044
$792$	0	0
$793$	−5949.88	−0.266439
$794$	6574.71	0.293864
$795$	0	0
$796$	1420.95	0.0632716
$797$	83.4496	0.00370883	0.00185441	−	0.999998i	$-0.499410\pi$
$797$	83.4496	0.00370883	0.00185441	+	0.999998i	$0.499410\pi$
$798$	0	0
$799$	−15791.3	−0.699193
$800$	12627.4	0.558055
$801$	0	0
$802$	26608.7	1.17156
$803$	−42838.9	−1.88263
$804$	0	0
$805$	7462.10	0.326714
$806$	−12638.0	−0.552300
$807$	0	0
$808$	−32960.9	−1.43510
$809$	−42908.7	−1.86476	−0.932380	−	0.361480i	$-0.882271\pi$
$809$	−42908.7	−1.86476	−0.932380	+	0.361480i	$0.882271\pi$
$810$	0	0
$811$	−21610.1	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$811$	−21610.1	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$812$	5973.86	0.258179
$813$	0	0
$814$	−11819.2	−0.508923
$815$	−3235.64	−0.139067
$816$	0	0
$817$	48438.4	2.07423
$818$	16949.6	0.724485
$819$	0	0
$820$	−6534.73	−0.278296
$821$	13743.6	0.584231	0.292115	−	0.956383i	$-0.405641\pi$
$821$	13743.6	0.584231	0.292115	+	0.956383i	$0.405641\pi$
$822$	0	0
$823$	40983.9	1.73585	0.867927	−	0.496692i	$-0.165452\pi$
$823$	40983.9	1.73585	0.867927	+	0.496692i	$0.165452\pi$
$824$	21892.4	0.925557
$825$	0	0
$826$	−9350.46	−0.393879
$827$	−43287.0	−1.82012	−0.910059	−	0.414478i	$-0.863964\pi$
$827$	−43287.0	−1.82012	−0.910059	+	0.414478i	$0.863964\pi$
$828$	0	0
$829$	30943.6	1.29640	0.648199	−	0.761471i	$-0.275522\pi$
$829$	30943.6	1.29640	0.648199	+	0.761471i	$0.275522\pi$
$830$	−8240.47	−0.344616
$831$	0	0
$832$	−22233.8	−0.926465
$833$	31496.0	1.31005
$834$	0	0
$835$	−7828.35	−0.324444
$836$	−32072.5	−1.32685
$837$	0	0
$838$	2702.10	0.111387
$839$	24422.1	1.00494	0.502470	−	0.864594i	$-0.332425\pi$
$839$	24422.1	1.00494	0.502470	+	0.864594i	$0.332425\pi$
$840$	0	0
$841$	20243.5	0.830028
$842$	−13075.0	−0.535147
$843$	0	0
$844$	−11442.0	−0.466646
$845$	1425.81	0.0580467
$846$	0	0
$847$	12408.7	0.503386
$848$	4108.76	0.166386
$849$	0	0
$850$	−17268.4	−0.696827
$851$	−16428.3	−0.661755
$852$	0	0
$853$	−35700.8	−1.43303	−0.716514	−	0.697573i	$-0.754263\pi$
$853$	−35700.8	−1.43303	−0.716514	+	0.697573i	$0.754263\pi$
$854$	−1743.67	−0.0698679
$855$	0	0
$856$	−14792.8	−0.590663
$857$	−14137.6	−0.563513	−0.281756	−	0.959486i	$-0.590917\pi$
$857$	−14137.6	−0.563513	−0.281756	+	0.959486i	$0.590917\pi$
$858$	0	0
$859$	−41798.7	−1.66025	−0.830123	−	0.557580i	$-0.811730\pi$
$859$	−41798.7	−1.66025	−0.830123	+	0.557580i	$0.811730\pi$
$860$	8815.73	0.349551
$861$	0	0
$862$	19614.9	0.775041
$863$	37592.4	1.48280	0.741402	−	0.671061i	$-0.234161\pi$
$863$	37592.4	1.48280	0.741402	+	0.671061i	$0.234161\pi$
$864$	0	0
$865$	−14951.6	−0.587712
$866$	6174.89	0.242299
$867$	0	0
$868$	3620.29	0.141568
$869$	52152.9	2.03587
$870$	0	0
$871$	−6226.97	−0.242242
$872$	−28028.5	−1.08849
$873$	0	0
$874$	45606.4	1.76506
$875$	−9860.10	−0.380951
$876$	0	0
$877$	31890.0	1.22788	0.613939	−	0.789354i	$-0.289584\pi$
$877$	31890.0	1.22788	0.613939	+	0.789354i	$0.289584\pi$
$878$	−27479.0	−1.05623
$879$	0	0
$880$	6243.59	0.239172
$881$	32763.5	1.25293	0.626465	−	0.779450i	$-0.284501\pi$
$881$	32763.5	1.25293	0.626465	+	0.779450i	$0.284501\pi$
$882$	0	0
$883$	21782.7	0.830178	0.415089	−	0.909781i	$-0.363750\pi$
$883$	21782.7	0.830178	0.415089	+	0.909781i	$0.363750\pi$
$884$	−20942.5	−0.796803
$885$	0	0
$886$	−17294.1	−0.655765
$887$	26894.7	1.01808	0.509039	−	0.860743i	$-0.330001\pi$
$887$	26894.7	1.01808	0.509039	+	0.860743i	$0.330001\pi$
$888$	0	0
$889$	5788.13	0.218366
$890$	−16994.5	−0.640063
$891$	0	0
$892$	−11112.1	−0.417110
$893$	−21433.1	−0.803172
$894$	0	0
$895$	−12192.6	−0.455366
$896$	2563.37	0.0955760
$897$	0	0
$898$	17430.1	0.647716
$899$	27048.3	1.00346
$900$	0	0
$901$	26507.6	0.980130
$902$	27303.7	1.00789
$903$	0	0
$904$	30973.2	1.13955
$905$	−6053.54	−0.222350
$906$	0	0
$907$	−38107.9	−1.39510	−0.697548	−	0.716538i	$-0.745726\pi$
$907$	−38107.9	−1.39510	−0.697548	+	0.716538i	$0.745726\pi$
$908$	−13476.3	−0.492542
$909$	0	0
$910$	4757.98	0.173325
$911$	−22821.2	−0.829967	−0.414983	−	0.909829i	$-0.636213\pi$
$911$	−22821.2	−0.829967	−0.414983	+	0.909829i	$0.636213\pi$
$912$	0	0
$913$	−33655.5	−1.21997
$914$	10275.8	0.371873
$915$	0	0
$916$	−10790.4	−0.389218
$917$	−11848.4	−0.426682
$918$	0	0
$919$	35906.9	1.28886	0.644429	−	0.764664i	$-0.277095\pi$
$919$	35906.9	1.28886	0.644429	+	0.764664i	$0.277095\pi$
$920$	25092.1	0.899197
$921$	0	0
$922$	2884.91	0.103047
$923$	−51328.6	−1.83045
$924$	0	0
$925$	8442.82	0.300106
$926$	−19948.4	−0.707932
$927$	0	0
$928$	33530.6	1.18609
$929$	−21625.1	−0.763719	−0.381860	−	0.924220i	$-0.624716\pi$
$929$	−21625.1	−0.763719	−0.381860	+	0.924220i	$0.624716\pi$
$930$	0	0
$931$	42748.8	1.50487
$932$	−24041.1	−0.844948
$933$	0	0
$934$	−30987.7	−1.08560
$935$	40280.5	1.40889
$936$	0	0
$937$	−8975.79	−0.312942	−0.156471	−	0.987683i	$-0.550012\pi$
$937$	−8975.79	−0.312942	−0.156471	+	0.987683i	$0.550012\pi$
$938$	−1824.87	−0.0635226
$939$	0	0
$940$	−3900.80	−0.135351
$941$	41596.0	1.44101	0.720505	−	0.693449i	$-0.243910\pi$
$941$	41596.0	1.44101	0.720505	+	0.693449i	$0.243910\pi$
$942$	0	0
$943$	37951.1	1.31056
$944$	−10874.5	−0.374932
$945$	0	0
$946$	−36834.3	−1.26595
$947$	22951.7	0.787572	0.393786	−	0.919202i	$-0.371165\pi$
$947$	22951.7	0.787572	0.393786	+	0.919202i	$0.371165\pi$
$948$	0	0
$949$	−37966.8	−1.29869
$950$	−23438.0	−0.800453
$951$	0	0
$952$	−18553.6	−0.631645
$953$	−44443.8	−1.51068	−0.755339	−	0.655335i	$-0.772528\pi$
$953$	−44443.8	−1.51068	−0.755339	+	0.655335i	$0.772528\pi$
$954$	0	0
$955$	21873.6	0.741167
$956$	−945.115	−0.0319741
$957$	0	0
$958$	29599.2	0.998234
$959$	−17207.0	−0.579399
$960$	0	0
$961$	−13399.1	−0.449771
$962$	−10475.0	−0.351068
$963$	0	0
$964$	3858.78	0.128924
$965$	−28207.2	−0.940956
$966$	0	0
$967$	−34758.2	−1.15589	−0.577946	−	0.816075i	$-0.696146\pi$
$967$	−34758.2	−1.15589	−0.577946	+	0.816075i	$0.696146\pi$
$968$	41725.5	1.38544
$969$	0	0
$970$	1625.98	0.0538218
$971$	−45117.8	−1.49114	−0.745570	−	0.666427i	$-0.767823\pi$
$971$	−45117.8	−1.49114	−0.745570	+	0.666427i	$0.767823\pi$
$972$	0	0
$973$	−19650.3	−0.647439
$974$	11913.5	0.391923
$975$	0	0
$976$	−2027.88	−0.0665069
$977$	43014.1	1.40854	0.704269	−	0.709933i	$-0.251275\pi$
$977$	43014.1	1.40854	0.704269	+	0.709933i	$0.251275\pi$
$978$	0	0
$979$	−69408.4	−2.26588
$980$	7780.22	0.253602
$981$	0	0
$982$	−14101.0	−0.458230
$983$	−28614.4	−0.928440	−0.464220	−	0.885720i	$-0.653665\pi$
$983$	−28614.4	−0.928440	−0.464220	+	0.885720i	$0.653665\pi$
$984$	0	0
$985$	23535.9	0.761337
$986$	−45854.5	−1.48104
$987$	0	0
$988$	−28424.8	−0.915297
$989$	−51198.3	−1.64612
$990$	0	0
$991$	−26457.0	−0.848067	−0.424033	−	0.905647i	$-0.639386\pi$
$991$	−26457.0	−0.848067	−0.424033	+	0.905647i	$0.639386\pi$
$992$	20320.3	0.650372
$993$	0	0
$994$	−15042.4	−0.479994
$995$	−2422.19	−0.0771745
$996$	0	0
$997$	−28570.9	−0.907572	−0.453786	−	0.891111i	$-0.649927\pi$
$997$	−28570.9	−0.907572	−0.453786	+	0.891111i	$0.649927\pi$
$998$	21570.7	0.684176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.9		28
3.2	odd	2		717.4.a.a.1.20	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.20	✓	28	3.2	odd	2
2151.4.a.c.1.9		28	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)

\(f(q)\)	\(=\)	\(q-2.01135 q^{2} -3.95446 q^{4} +6.74089 q^{5} +7.15060 q^{7} +24.0446 q^{8} +O(q^{10})\)	\(q-2.01135 q^{2} -3.95446 q^{4} +6.74089 q^{5} +7.15060 q^{7} +24.0446 q^{8} -13.5583 q^{10} -55.3745 q^{11} -49.0766 q^{13} -14.3824 q^{14} -16.7266 q^{16} -107.911 q^{17} -146.466 q^{19} -26.6566 q^{20} +111.378 q^{22} +154.811 q^{23} -79.5604 q^{25} +98.7105 q^{26} -28.2768 q^{28} -211.264 q^{29} -128.031 q^{31} -158.714 q^{32} +217.048 q^{34} +48.2014 q^{35} -106.118 q^{37} +294.594 q^{38} +162.082 q^{40} +245.145 q^{41} -330.715 q^{43} +218.976 q^{44} -311.379 q^{46} +146.336 q^{47} -291.869 q^{49} +160.024 q^{50} +194.072 q^{52} -245.642 q^{53} -373.273 q^{55} +171.934 q^{56} +424.927 q^{58} +650.133 q^{59} +121.237 q^{61} +257.515 q^{62} +453.043 q^{64} -330.820 q^{65} +126.883 q^{67} +426.731 q^{68} -96.9501 q^{70} +1045.89 q^{71} +773.622 q^{73} +213.442 q^{74} +579.192 q^{76} -395.961 q^{77} -941.822 q^{79} -112.752 q^{80} -493.074 q^{82} +607.780 q^{83} -727.419 q^{85} +665.185 q^{86} -1331.46 q^{88} +1253.44 q^{89} -350.928 q^{91} -612.193 q^{92} -294.333 q^{94} -987.309 q^{95} -119.925 q^{97} +587.051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

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