LMFDB - Embedded newform 1058.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1058,4,Mod(1,1058)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1058.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1058 = 2 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1058.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-12,12,24,0,-24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.4240207861$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 75x^{4} + 102x^{3} + 1209x^{2} - 844x - 2078 $ x^6 - 2x^5 - 75x^4 + 102x^3 + 1209x^2 - 844*x - 2078
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.85963$ of defining polynomial
Character	$\chi$	$=$	1058.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -5.21166 q^{3} +4.00000 q^{4} +4.33058 q^{5} +10.4233 q^{6} -9.04844 q^{7} -8.00000 q^{8} +0.161433 q^{9} -8.66116 q^{10} +46.5516 q^{11} -20.8467 q^{12} -32.9791 q^{13} +18.0969 q^{14} -22.5695 q^{15} +16.0000 q^{16} +65.4116 q^{17} -0.322867 q^{18} -16.9296 q^{19} +17.3223 q^{20} +47.1574 q^{21} -93.1031 q^{22} +41.6933 q^{24} -106.246 q^{25} +65.9582 q^{26} +139.874 q^{27} -36.1938 q^{28} +22.4230 q^{29} +45.1391 q^{30} -251.263 q^{31} -32.0000 q^{32} -242.611 q^{33} -130.823 q^{34} -39.1850 q^{35} +0.645734 q^{36} -51.9548 q^{37} +33.8592 q^{38} +171.876 q^{39} -34.6446 q^{40} +373.388 q^{41} -94.3148 q^{42} +433.075 q^{43} +186.206 q^{44} +0.699101 q^{45} -218.465 q^{47} -83.3866 q^{48} -261.126 q^{49} +212.492 q^{50} -340.903 q^{51} -131.916 q^{52} +423.615 q^{53} -279.747 q^{54} +201.595 q^{55} +72.3875 q^{56} +88.2313 q^{57} -44.8459 q^{58} -382.178 q^{59} -90.2781 q^{60} -191.489 q^{61} +502.525 q^{62} -1.46072 q^{63} +64.0000 q^{64} -142.819 q^{65} +485.222 q^{66} -579.602 q^{67} +261.646 q^{68} +78.3700 q^{70} -1053.27 q^{71} -1.29147 q^{72} -600.364 q^{73} +103.910 q^{74} +553.719 q^{75} -67.7183 q^{76} -421.219 q^{77} -343.752 q^{78} -240.591 q^{79} +69.2893 q^{80} -733.333 q^{81} -746.775 q^{82} +750.461 q^{83} +188.630 q^{84} +283.270 q^{85} -866.150 q^{86} -116.861 q^{87} -372.412 q^{88} +1533.04 q^{89} -1.39820 q^{90} +298.409 q^{91} +1309.50 q^{93} +436.930 q^{94} -73.3149 q^{95} +166.773 q^{96} -1666.70 q^{97} +522.252 q^{98} +7.51498 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 12 q^{3} + 24 q^{4} - 24 q^{6} - 48 q^{8} + 90 q^{9} + 48 q^{12} + 44 q^{13} + 96 q^{16} - 180 q^{18} - 96 q^{24} - 98 q^{25} - 88 q^{26} + 984 q^{27} + 580 q^{29} - 268 q^{31} - 192 q^{32}+ \cdots - 1092 q^{98}+O(q^{100}) $ 6 * q - 12 * q^2 + 12 * q^3 + 24 * q^4 - 24 * q^6 - 48 * q^8 + 90 * q^9 + 48 * q^12 + 44 * q^13 + 96 * q^16 - 180 * q^18 - 96 * q^24 - 98 * q^25 - 88 * q^26 + 984 * q^27 + 580 * q^29 - 268 * q^31 - 192 * q^32 - 1136 * q^35 + 360 * q^36 + 484 * q^39 - 48 * q^41 + 1324 * q^47 + 192 * q^48 + 546 * q^49 + 196 * q^50 + 176 * q^52 - 1968 * q^54 + 436 * q^55 - 1160 * q^58 + 644 * q^59 + 536 * q^62 + 384 * q^64 + 2272 * q^70 - 528 * q^71 - 720 * q^72 + 1268 * q^73 + 4184 * q^75 - 3440 * q^77 - 968 * q^78 + 4254 * q^81 + 96 * q^82 + 1440 * q^85 + 3068 * q^87 + 5776 * q^93 - 2648 * q^94 + 1092 * q^95 - 384 * q^96 - 1092 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−5.21166	−1.00299	−0.501493	−	0.865162i	$-0.667216\pi$
$3$	−5.21166	−1.00299	−0.501493	+	0.865162i	$0.667216\pi$
$4$	4.00000	0.500000
$5$	4.33058	0.387339	0.193669	−	0.981067i	$-0.437961\pi$
$5$	4.33058	0.387339	0.193669	+	0.981067i	$0.437961\pi$
$6$	10.4233	0.709218
$7$	−9.04844	−0.488570	−0.244285	−	0.969704i	$-0.578553\pi$
$7$	−9.04844	−0.488570	−0.244285	+	0.969704i	$0.578553\pi$
$8$	−8.00000	−0.353553
$9$	0.161433	0.00597902
$10$	−8.66116	−0.273890
$11$	46.5516	1.27598	0.637992	−	0.770043i	$-0.279765\pi$
$11$	46.5516	1.27598	0.637992	+	0.770043i	$0.279765\pi$
$12$	−20.8467	−0.501493
$13$	−32.9791	−0.703597	−0.351798	−	0.936076i	$-0.614430\pi$
$13$	−32.9791	−0.703597	−0.351798	+	0.936076i	$0.614430\pi$
$14$	18.0969	0.345471
$15$	−22.5695	−0.388495
$16$	16.0000	0.250000
$17$	65.4116	0.933214	0.466607	−	0.884465i	$-0.345476\pi$
$17$	65.4116	0.933214	0.466607	+	0.884465i	$0.345476\pi$
$18$	−0.322867	−0.00422780
$19$	−16.9296	−0.204416	−0.102208	−	0.994763i	$-0.532591\pi$
$19$	−16.9296	−0.204416	−0.102208	+	0.994763i	$0.532591\pi$
$20$	17.3223	0.193669
$21$	47.1574	0.490028
$22$	−93.1031	−0.902257
$23$	0	0
$23$	0	0
$24$	41.6933	0.354609
$25$	−106.246	−0.849969
$26$	65.9582	0.497518
$27$	139.874	0.996988
$28$	−36.1938	−0.244285
$29$	22.4230	0.143581	0.0717904	−	0.997420i	$-0.477129\pi$
$29$	22.4230	0.143581	0.0717904	+	0.997420i	$0.477129\pi$
$30$	45.1391	0.274708
$31$	−251.263	−1.45574	−0.727872	−	0.685712i	$-0.759491\pi$
$31$	−251.263	−1.45574	−0.727872	+	0.685712i	$0.759491\pi$
$32$	−32.0000	−0.176777
$33$	−242.611	−1.27979
$34$	−130.823	−0.659882
$35$	−39.1850	−0.189242
$36$	0.645734	0.00298951
$37$	−51.9548	−0.230846	−0.115423	−	0.993316i	$-0.536822\pi$
$37$	−51.9548	−0.230846	−0.115423	+	0.993316i	$0.536822\pi$
$38$	33.8592	0.144544
$39$	171.876	0.705697
$40$	−34.6446	−0.136945
$41$	373.388	1.42228	0.711139	−	0.703052i	$-0.248180\pi$
$41$	373.388	1.42228	0.711139	+	0.703052i	$0.248180\pi$
$42$	−94.3148	−0.346502
$43$	433.075	1.53589	0.767946	−	0.640515i	$-0.221279\pi$
$43$	433.075	1.53589	0.767946	+	0.640515i	$0.221279\pi$
$44$	186.206	0.637992
$45$	0.699101	0.00231591
$46$	0	0
$47$	−218.465	−0.678009	−0.339004	−	0.940785i	$-0.610090\pi$
$47$	−218.465	−0.678009	−0.339004	+	0.940785i	$0.610090\pi$
$48$	−83.3866	−0.250746
$49$	−261.126	−0.761300
$50$	212.492	0.601019
$51$	−340.903	−0.935999
$52$	−131.916	−0.351798
$53$	423.615	1.09789	0.548944	−	0.835859i	$-0.315030\pi$
$53$	423.615	1.09789	0.548944	+	0.835859i	$0.315030\pi$
$54$	−279.747	−0.704977
$55$	201.595	0.494238
$56$	72.3875	0.172735
$57$	88.2313	0.205027
$58$	−44.8459	−0.101527
$59$	−382.178	−0.843312	−0.421656	−	0.906756i	$-0.638551\pi$
$59$	−382.178	−0.843312	−0.421656	+	0.906756i	$0.638551\pi$
$60$	−90.2781	−0.194248
$61$	−191.489	−0.401929	−0.200964	−	0.979599i	$-0.564408\pi$
$61$	−191.489	−0.401929	−0.200964	+	0.979599i	$0.564408\pi$
$62$	502.525	1.02937
$63$	−1.46072	−0.00292117
$64$	64.0000	0.125000
$65$	−142.819	−0.272530
$66$	485.222	0.904950
$67$	−579.602	−1.05686	−0.528431	−	0.848976i	$-0.677219\pi$
$67$	−579.602	−1.05686	−0.528431	+	0.848976i	$0.677219\pi$
$68$	261.646	0.466607
$69$	0	0
$70$	78.3700	0.133814
$71$	−1053.27	−1.76057	−0.880283	−	0.474450i	$-0.842647\pi$
$71$	−1053.27	−1.76057	−0.880283	+	0.474450i	$0.842647\pi$
$72$	−1.29147	−0.00211390
$73$	−600.364	−0.962566	−0.481283	−	0.876565i	$-0.659829\pi$
$73$	−600.364	−0.962566	−0.481283	+	0.876565i	$0.659829\pi$
$74$	103.910	0.163233
$75$	553.719	0.852506
$76$	−67.7183	−0.102208
$77$	−421.219	−0.623407
$78$	−343.752	−0.499003
$79$	−240.591	−0.342640	−0.171320	−	0.985215i	$-0.554803\pi$
$79$	−240.591	−0.342640	−0.171320	+	0.985215i	$0.554803\pi$
$80$	69.2893	0.0968347
$81$	−733.333	−1.00594
$82$	−746.775	−1.00570
$83$	750.461	0.992456	0.496228	−	0.868192i	$-0.334718\pi$
$83$	750.461	0.992456	0.496228	+	0.868192i	$0.334718\pi$
$84$	188.630	0.245014
$85$	283.270	0.361470
$86$	−866.150	−1.08604
$87$	−116.861	−0.144009
$88$	−372.412	−0.451128
$89$	1533.04	1.82586	0.912932	−	0.408113i	$-0.133813\pi$
$89$	1533.04	1.82586	0.912932	+	0.408113i	$0.133813\pi$
$90$	−1.39820	−0.00163759
$91$	298.409	0.343756
$92$	0	0
$93$	1309.50	1.46009
$94$	436.930	0.479425
$95$	−73.3149	−0.0791784
$96$	166.773	0.177304
$97$	−1666.70	−1.74462	−0.872308	−	0.488957i	$-0.837377\pi$
$97$	−1666.70	−1.74462	−0.872308	+	0.488957i	$0.837377\pi$
$98$	522.252	0.538320
$99$	7.51498	0.00762913
$100$	−424.984	−0.424984
$101$	715.317	0.704720	0.352360	−	0.935865i	$-0.385379\pi$
$101$	715.317	0.704720	0.352360	+	0.935865i	$0.385379\pi$
$102$	681.806	0.661851
$103$	637.684	0.610028	0.305014	−	0.952348i	$-0.401339\pi$
$103$	637.684	0.610028	0.305014	+	0.952348i	$0.401339\pi$
$104$	263.833	0.248759
$105$	204.219	0.189807
$106$	−847.231	−0.776324
$107$	990.139	0.894582	0.447291	−	0.894388i	$-0.352389\pi$
$107$	990.139	0.894582	0.447291	+	0.894388i	$0.352389\pi$
$108$	559.494	0.498494
$109$	−1280.38	−1.12512	−0.562561	−	0.826756i	$-0.690184\pi$
$109$	−1280.38	−1.12512	−0.562561	+	0.826756i	$0.690184\pi$
$110$	−403.191	−0.349479
$111$	270.771	0.231536
$112$	−144.775	−0.122142
$113$	−1236.77	−1.02961	−0.514803	−	0.857309i	$-0.672135\pi$
$113$	−1236.77	−1.02961	−0.514803	+	0.857309i	$0.672135\pi$
$114$	−176.463	−0.144976
$115$	0	0
$116$	89.6919	0.0717904
$117$	−5.32393	−0.00420682
$118$	764.357	0.596311
$119$	−591.872	−0.455940
$120$	180.556	0.137354
$121$	836.048	0.628135
$122$	382.978	0.284206
$123$	−1945.97	−1.42652
$124$	−1005.05	−0.727872
$125$	−1001.43	−0.716565
$126$	2.92144	0.00206558
$127$	2650.20	1.85171	0.925855	−	0.377878i	$-0.123346\pi$
$127$	2650.20	1.85171	0.925855	+	0.377878i	$0.123346\pi$
$128$	−128.000	−0.0883883
$129$	−2257.04	−1.54048
$130$	285.637	0.192708
$131$	−417.264	−0.278294	−0.139147	−	0.990272i	$-0.544436\pi$
$131$	−417.264	−0.278294	−0.139147	+	0.990272i	$0.544436\pi$
$132$	−970.444	−0.639896
$133$	153.186	0.0998717
$134$	1159.20	0.747314
$135$	605.734	0.386172
$136$	−523.292	−0.329941
$137$	1960.98	1.22290	0.611451	−	0.791282i	$-0.290586\pi$
$137$	1960.98	1.22290	0.611451	+	0.791282i	$0.290586\pi$
$138$	0	0
$139$	1759.38	1.07359	0.536793	−	0.843714i	$-0.319635\pi$
$139$	1759.38	1.07359	0.536793	+	0.843714i	$0.319635\pi$
$140$	−156.740	−0.0946210
$141$	1138.57	0.680033
$142$	2106.54	1.24491
$143$	−1535.23	−0.897778
$144$	2.58294	0.00149475
$145$	97.1045	0.0556144
$146$	1200.73	0.680637
$147$	1360.90	0.763572
$148$	−207.819	−0.115423
$149$	2882.34	1.58477	0.792383	−	0.610024i	$-0.208840\pi$
$149$	2882.34	1.58477	0.792383	+	0.610024i	$0.208840\pi$
$150$	−1107.44	−0.602813
$151$	3032.12	1.63411	0.817056	−	0.576559i	$-0.195605\pi$
$151$	3032.12	1.63411	0.817056	+	0.576559i	$0.195605\pi$
$152$	135.437	0.0722721
$153$	10.5596	0.00557970
$154$	842.438	0.440815
$155$	−1088.11	−0.563867
$156$	687.504	0.352849
$157$	2791.44	1.41899	0.709493	−	0.704712i	$-0.248924\pi$
$157$	2791.44	1.41899	0.709493	+	0.704712i	$0.248924\pi$
$158$	481.181	0.242283
$159$	−2207.74	−1.10116
$160$	−138.579	−0.0684725
$161$	0	0
$162$	1466.67	0.711309
$163$	1678.20	0.806423	0.403212	−	0.915107i	$-0.367894\pi$
$163$	1678.20	0.806423	0.403212	+	0.915107i	$0.367894\pi$
$164$	1493.55	0.711139
$165$	−1050.65	−0.495714
$166$	−1500.92	−0.701772
$167$	2918.31	1.35225	0.676125	−	0.736787i	$-0.263658\pi$
$167$	2918.31	1.35225	0.676125	+	0.736787i	$0.263658\pi$
$168$	−377.259	−0.173251
$169$	−1109.38	−0.504952
$170$	−566.540	−0.255598
$171$	−2.73300	−0.00122221
$172$	1732.30	0.767946
$173$	153.954	0.0676583	0.0338292	−	0.999428i	$-0.489230\pi$
$173$	153.954	0.0676583	0.0338292	+	0.999428i	$0.489230\pi$
$174$	233.722	0.101830
$175$	961.361	0.415269
$176$	744.825	0.318996
$177$	1991.79	0.845829
$178$	−3066.08	−1.29108
$179$	1031.48	0.430707	0.215354	−	0.976536i	$-0.430910\pi$
$179$	1031.48	0.430707	0.215354	+	0.976536i	$0.430910\pi$
$180$	2.79640	0.00115795
$181$	−72.7337	−0.0298688	−0.0149344	−	0.999888i	$-0.504754\pi$
$181$	−72.7337	−0.0298688	−0.0149344	+	0.999888i	$0.504754\pi$
$182$	−596.819	−0.243072
$183$	997.976	0.403128
$184$	0	0
$185$	−224.995	−0.0894158
$186$	−2618.99	−1.03244
$187$	3045.01	1.19077
$188$	−873.861	−0.339004
$189$	−1265.64	−0.487098
$190$	146.630	0.0559876
$191$	−2802.09	−1.06153	−0.530764	−	0.847520i	$-0.678095\pi$
$191$	−2802.09	−1.06153	−0.530764	+	0.847520i	$0.678095\pi$
$192$	−333.546	−0.125373
$193$	1327.92	0.495265	0.247632	−	0.968854i	$-0.420347\pi$
$193$	1327.92	0.495265	0.247632	+	0.968854i	$0.420347\pi$
$194$	3333.40	1.23363
$195$	744.323	0.273344
$196$	−1044.50	−0.380650
$197$	1832.22	0.662640	0.331320	−	0.943518i	$-0.392506\pi$
$197$	1832.22	0.662640	0.331320	+	0.943518i	$0.392506\pi$
$198$	−15.0300	−0.00539461
$199$	1378.55	0.491068	0.245534	−	0.969388i	$-0.421037\pi$
$199$	1378.55	0.491068	0.245534	+	0.969388i	$0.421037\pi$
$200$	849.969	0.300509
$201$	3020.69	1.06002
$202$	−1430.63	−0.498312
$203$	−202.893	−0.0701492
$204$	−1363.61	−0.468000
$205$	1616.99	0.550903
$206$	−1275.37	−0.431355
$207$	0	0
$208$	−527.666	−0.175899
$209$	−788.098	−0.260832
$210$	−408.438	−0.134214
$211$	−476.743	−0.155547	−0.0777733	−	0.996971i	$-0.524781\pi$
$211$	−476.743	−0.155547	−0.0777733	+	0.996971i	$0.524781\pi$
$212$	1694.46	0.548944
$213$	5489.29	1.76582
$214$	−1980.28	−0.632565
$215$	1875.47	0.594910
$216$	−1118.99	−0.352489
$217$	2273.53	0.711233
$218$	2560.76	0.795582
$219$	3128.90	0.965440
$220$	806.381	0.247119
$221$	−2157.21	−0.656606
$222$	−541.542	−0.163720
$223$	4015.20	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$223$	4015.20	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$224$	289.550	0.0863677
$225$	−17.1517	−0.00508198
$226$	2473.54	0.728041
$227$	4393.17	1.28452	0.642258	−	0.766489i	$-0.277998\pi$
$227$	4393.17	1.28452	0.642258	+	0.766489i	$0.277998\pi$
$228$	352.925	0.102513
$229$	4311.28	1.24409	0.622047	−	0.782980i	$-0.286301\pi$
$229$	4311.28	1.24409	0.622047	+	0.782980i	$0.286301\pi$
$230$	0	0
$231$	2195.25	0.625268
$232$	−179.384	−0.0507635
$233$	3508.07	0.986358	0.493179	−	0.869928i	$-0.335835\pi$
$233$	3508.07	0.986358	0.493179	+	0.869928i	$0.335835\pi$
$234$	10.6479	0.00297467
$235$	−946.081	−0.262619
$236$	−1528.71	−0.421656
$237$	1253.88	0.343663
$238$	1183.74	0.322398
$239$	3364.14	0.910494	0.455247	−	0.890365i	$-0.349551\pi$
$239$	3364.14	0.910494	0.455247	+	0.890365i	$0.349551\pi$
$240$	−361.112	−0.0971238
$241$	−6462.14	−1.72723	−0.863616	−	0.504151i	$-0.831806\pi$
$241$	−6462.14	−1.72723	−0.863616	+	0.504151i	$0.831806\pi$
$242$	−1672.10	−0.444158
$243$	45.2964	0.0119579
$244$	−765.956	−0.200964
$245$	−1130.83	−0.294881
$246$	3891.94	1.00870
$247$	558.322	0.143827
$248$	2010.10	0.514684
$249$	−3911.15	−0.995419
$250$	2002.86	0.506688
$251$	−388.032	−0.0975793	−0.0487896	−	0.998809i	$-0.515536\pi$
$251$	−388.032	−0.0975793	−0.0487896	+	0.998809i	$0.515536\pi$
$252$	−5.84288	−0.00146058
$253$	0	0
$254$	−5300.40	−1.30936
$255$	−1476.31	−0.362549
$256$	256.000	0.0625000
$257$	−930.487	−0.225845	−0.112923	−	0.993604i	$-0.536021\pi$
$257$	−930.487	−0.225845	−0.112923	+	0.993604i	$0.536021\pi$
$258$	4514.08	1.08928
$259$	470.110	0.112785
$260$	−571.275	−0.136265
$261$	3.61982	0.000858472 0
$262$	834.527	0.196783
$263$	−1149.87	−0.269596	−0.134798	−	0.990873i	$-0.543039\pi$
$263$	−1149.87	−0.269596	−0.134798	+	0.990873i	$0.543039\pi$
$264$	1940.89	0.452475
$265$	1834.50	0.425255
$266$	−306.372	−0.0706199
$267$	−7989.68	−1.83131
$268$	−2318.41	−0.528431
$269$	−3814.34	−0.864553	−0.432276	−	0.901741i	$-0.642289\pi$
$269$	−3814.34	−0.864553	−0.432276	+	0.901741i	$0.642289\pi$
$270$	−1211.47	−0.273065
$271$	4169.39	0.934586	0.467293	−	0.884103i	$-0.345229\pi$
$271$	4169.39	0.934586	0.467293	+	0.884103i	$0.345229\pi$
$272$	1046.58	0.233303
$273$	−1555.21	−0.344782
$274$	−3921.95	−0.864722
$275$	−4945.92	−1.08455
$276$	0	0
$277$	−4272.56	−0.926763	−0.463381	−	0.886159i	$-0.653364\pi$
$277$	−4272.56	−0.926763	−0.463381	+	0.886159i	$0.653364\pi$
$278$	−3518.76	−0.759140
$279$	−40.5622	−0.00870393
$280$	313.480	0.0669072
$281$	−7639.10	−1.62174	−0.810872	−	0.585223i	$-0.801007\pi$
$281$	−7639.10	−1.62174	−0.810872	+	0.585223i	$0.801007\pi$
$282$	−2277.13	−0.480856
$283$	1217.49	0.255732	0.127866	−	0.991791i	$-0.459187\pi$
$283$	1217.49	0.255732	0.127866	+	0.991791i	$0.459187\pi$
$284$	−4213.08	−0.880283
$285$	382.093	0.0794148
$286$	3070.46	0.634825
$287$	−3378.58	−0.694881
$288$	−5.16587	−0.00105695
$289$	−634.329	−0.129112
$290$	−194.209	−0.0393253
$291$	8686.28	1.74982
$292$	−2401.46	−0.481283
$293$	5320.54	1.06085	0.530425	−	0.847732i	$-0.322032\pi$
$293$	5320.54	1.06085	0.530425	+	0.847732i	$0.322032\pi$
$294$	−2721.80	−0.539927
$295$	−1655.05	−0.326647
$296$	415.639	0.0816166
$297$	6511.33	1.27214
$298$	−5764.67	−1.12060
$299$	0	0
$300$	2214.88	0.426253
$301$	−3918.65	−0.750390
$302$	−6064.25	−1.15549
$303$	−3727.99	−0.706823
$304$	−270.873	−0.0511041
$305$	−829.258	−0.155683
$306$	−21.1192	−0.00394544
$307$	6898.95	1.28255	0.641276	−	0.767310i	$-0.278405\pi$
$307$	6898.95	1.28255	0.641276	+	0.767310i	$0.278405\pi$
$308$	−1684.88	−0.311704
$309$	−3323.39	−0.611849
$310$	2176.23	0.398714
$311$	629.934	0.114856	0.0574281	−	0.998350i	$-0.481710\pi$
$311$	629.934	0.114856	0.0574281	+	0.998350i	$0.481710\pi$
$312$	−1375.01	−0.249502
$313$	−1369.81	−0.247367	−0.123684	−	0.992322i	$-0.539471\pi$
$313$	−1369.81	−0.247367	−0.123684	+	0.992322i	$0.539471\pi$
$314$	−5582.87	−1.00337
$315$	−6.32577	−0.00113148
$316$	−962.363	−0.171320
$317$	2583.87	0.457807	0.228903	−	0.973449i	$-0.426486\pi$
$317$	2583.87	0.457807	0.228903	+	0.973449i	$0.426486\pi$
$318$	4415.48	0.778641
$319$	1043.82	0.183207
$320$	277.157	0.0484174
$321$	−5160.27	−0.897253
$322$	0	0
$323$	−1107.39	−0.190764
$324$	−2933.33	−0.502972
$325$	3503.90	0.598035
$326$	−3356.40	−0.570227
$327$	6672.92	1.12848
$328$	−2987.10	−0.502851
$329$	1976.77	0.331255
$330$	2101.29	0.350522
$331$	1028.64	0.170813	0.0854065	−	0.996346i	$-0.472781\pi$
$331$	1028.64	0.170813	0.0854065	+	0.996346i	$0.472781\pi$
$332$	3001.85	0.496228
$333$	−8.38725	−0.00138024
$334$	−5836.62	−0.956185
$335$	−2510.02	−0.409364
$336$	754.519	0.122507
$337$	−8650.70	−1.39832	−0.699159	−	0.714966i	$-0.746442\pi$
$337$	−8650.70	−1.39832	−0.699159	+	0.714966i	$0.746442\pi$
$338$	2218.76	0.357055
$339$	6445.63	1.03268
$340$	1133.08	0.180735
$341$	−11696.7	−1.85751
$342$	5.46600	0.000864232 0
$343$	5466.39	0.860518
$344$	−3464.60	−0.543019
$345$	0	0
$346$	−307.908	−0.0478417
$347$	12314.1	1.90506	0.952531	−	0.304440i	$-0.0984693\pi$
$347$	12314.1	1.90506	0.952531	+	0.304440i	$0.0984693\pi$
$348$	−467.444	−0.0720047
$349$	2938.78	0.450743	0.225372	−	0.974273i	$-0.427640\pi$
$349$	2938.78	0.450743	0.225372	+	0.974273i	$0.427640\pi$
$350$	−1922.72	−0.293639
$351$	−4612.91	−0.701478
$352$	−1489.65	−0.225564
$353$	−9491.13	−1.43105	−0.715527	−	0.698585i	$-0.753813\pi$
$353$	−9491.13	−1.43105	−0.715527	+	0.698585i	$0.753813\pi$
$354$	−3983.57	−0.598091
$355$	−4561.27	−0.681935
$356$	6132.16	0.912932
$357$	3084.64	0.457301
$358$	−2062.96	−0.304556
$359$	11320.5	1.66426	0.832132	−	0.554578i	$-0.187120\pi$
$359$	11320.5	1.66426	0.832132	+	0.554578i	$0.187120\pi$
$360$	−5.59281	−0.000818796 0
$361$	−6572.39	−0.958214
$362$	145.467	0.0211204
$363$	−4357.20	−0.630010
$364$	1193.64	0.171878
$365$	−2599.93	−0.372839
$366$	−1995.95	−0.285055
$367$	6529.40	0.928698	0.464349	−	0.885652i	$-0.346288\pi$
$367$	6529.40	0.928698	0.464349	+	0.885652i	$0.346288\pi$
$368$	0	0
$369$	60.2773	0.00850382
$370$	449.989	0.0632265
$371$	−3833.06	−0.536395
$372$	5237.98	0.730045
$373$	2942.44	0.408456	0.204228	−	0.978923i	$-0.434532\pi$
$373$	2942.44	0.408456	0.204228	+	0.978923i	$0.434532\pi$
$374$	−6090.02	−0.841998
$375$	5219.11	0.718704
$376$	1747.72	0.239712
$377$	−739.490	−0.101023
$378$	2531.27	0.344430
$379$	−1474.30	−0.199815	−0.0999075	−	0.994997i	$-0.531855\pi$
$379$	−1474.30	−0.199815	−0.0999075	+	0.994997i	$0.531855\pi$
$380$	−293.260	−0.0395892
$381$	−13812.0	−1.85724
$382$	5604.17	0.750613
$383$	950.773	0.126847	0.0634233	−	0.997987i	$-0.479798\pi$
$383$	950.773	0.126847	0.0634233	+	0.997987i	$0.479798\pi$
$384$	667.093	0.0886522
$385$	−1824.12	−0.241470
$386$	−2655.85	−0.350205
$387$	69.9128	0.00918312
$388$	−6666.80	−0.872308
$389$	5589.10	0.728480	0.364240	−	0.931305i	$-0.381329\pi$
$389$	5589.10	0.728480	0.364240	+	0.931305i	$0.381329\pi$
$390$	−1488.65	−0.193283
$391$	0	0
$392$	2089.01	0.269160
$393$	2174.64	0.279125
$394$	−3664.43	−0.468557
$395$	−1041.90	−0.132718
$396$	30.0599	0.00381457
$397$	7496.86	0.947750	0.473875	−	0.880592i	$-0.342855\pi$
$397$	7496.86	0.947750	0.473875	+	0.880592i	$0.342855\pi$
$398$	−2757.09	−0.347238
$399$	−798.355	−0.100170
$400$	−1699.94	−0.212492
$401$	1773.23	0.220826	0.110413	−	0.993886i	$-0.464783\pi$
$401$	1773.23	0.220826	0.110413	+	0.993886i	$0.464783\pi$
$402$	−6041.39	−0.749545
$403$	8286.42	1.02426
$404$	2861.27	0.352360
$405$	−3175.76	−0.389641
$406$	405.786	0.0496030
$407$	−2418.58	−0.294556
$408$	2727.22	0.330926
$409$	−2200.53	−0.266038	−0.133019	−	0.991114i	$-0.542467\pi$
$409$	−2200.53	−0.266038	−0.133019	+	0.991114i	$0.542467\pi$
$410$	−3233.97	−0.389547
$411$	−10220.0	−1.22655
$412$	2550.74	0.305014
$413$	3458.12	0.412017
$414$	0	0
$415$	3249.93	0.384417
$416$	1055.33	0.124380
$417$	−9169.29	−1.07679
$418$	1576.20	0.184436
$419$	836.942	0.0975830	0.0487915	−	0.998809i	$-0.484463\pi$
$419$	836.942	0.0975830	0.0487915	+	0.998809i	$0.484463\pi$
$420$	816.876	0.0949035
$421$	12156.2	1.40726	0.703628	−	0.710569i	$-0.251562\pi$
$421$	12156.2	1.40726	0.703628	+	0.710569i	$0.251562\pi$
$422$	953.486	0.109988
$423$	−35.2676	−0.00405383
$424$	−3388.92	−0.388162
$425$	−6949.72	−0.793202
$426$	−10978.6	−1.24862
$427$	1732.68	0.196370
$428$	3960.55	0.447291
$429$	8001.09	0.900458
$430$	−3750.93	−0.420665
$431$	5695.64	0.636541	0.318270	−	0.948000i	$-0.396898\pi$
$431$	5695.64	0.636541	0.318270	+	0.948000i	$0.396898\pi$
$432$	2237.98	0.249247
$433$	−513.619	−0.0570045	−0.0285023	−	0.999594i	$-0.509074\pi$
$433$	−513.619	−0.0570045	−0.0285023	+	0.999594i	$0.509074\pi$
$434$	−4547.07	−0.502918
$435$	−506.076	−0.0557804
$436$	−5121.53	−0.562561
$437$	0	0
$438$	−6257.79	−0.682669
$439$	16785.3	1.82487	0.912435	−	0.409221i	$-0.134200\pi$
$439$	16785.3	1.82487	0.912435	+	0.409221i	$0.134200\pi$
$440$	−1612.76	−0.174740
$441$	−42.1544	−0.00455182
$442$	4314.43	0.464291
$443$	−13552.0	−1.45344	−0.726720	−	0.686934i	$-0.758956\pi$
$443$	−13552.0	−1.45344	−0.726720	+	0.686934i	$0.758956\pi$
$444$	1083.08	0.115768
$445$	6638.95	0.707228
$446$	−8030.39	−0.852579
$447$	−15021.8	−1.58950
$448$	−579.100	−0.0610712
$449$	−5732.64	−0.602539	−0.301270	−	0.953539i	$-0.597410\pi$
$449$	−5732.64	−0.602539	−0.301270	+	0.953539i	$0.597410\pi$
$450$	34.3033	0.00359350
$451$	17381.8	1.81480
$452$	−4947.08	−0.514803
$453$	−15802.4	−1.63899
$454$	−8786.34	−0.908290
$455$	1292.29	0.133150
$456$	−705.850	−0.0724878
$457$	−3360.90	−0.344018	−0.172009	−	0.985095i	$-0.555026\pi$
$457$	−3360.90	−0.344018	−0.172009	+	0.985095i	$0.555026\pi$
$458$	−8622.57	−0.879708
$459$	9149.35	0.930403
$460$	0	0
$461$	−8453.93	−0.854097	−0.427048	−	0.904229i	$-0.640447\pi$
$461$	−8453.93	−0.854097	−0.427048	+	0.904229i	$0.640447\pi$
$462$	−4390.50	−0.442131
$463$	−3741.11	−0.375516	−0.187758	−	0.982215i	$-0.560122\pi$
$463$	−3741.11	−0.375516	−0.187758	+	0.982215i	$0.560122\pi$
$464$	358.768	0.0358952
$465$	5670.88	0.565550
$466$	−7016.14	−0.697460
$467$	−8272.99	−0.819760	−0.409880	−	0.912139i	$-0.634429\pi$
$467$	−8272.99	−0.819760	−0.409880	+	0.912139i	$0.634429\pi$
$468$	−21.2957	−0.00210341
$469$	5244.50	0.516350
$470$	1892.16	0.185700
$471$	−14548.0	−1.42322
$472$	3057.43	0.298156
$473$	20160.3	1.95977
$474$	−2507.76	−0.243006
$475$	1798.70	0.173748
$476$	−2367.49	−0.227970
$477$	68.3857	0.00656429
$478$	−6728.28	−0.643817
$479$	11451.7	1.09236	0.546182	−	0.837667i	$-0.316081\pi$
$479$	11451.7	1.09236	0.546182	+	0.837667i	$0.316081\pi$
$480$	722.225	0.0686769
$481$	1713.42	0.162423
$482$	12924.3	1.22134
$483$	0	0
$484$	3344.19	0.314067
$485$	−7217.78	−0.675758
$486$	−90.5928	−0.00845550
$487$	16249.7	1.51200	0.755999	−	0.654573i	$-0.227151\pi$
$487$	16249.7	1.51200	0.755999	+	0.654573i	$0.227151\pi$
$488$	1531.91	0.142103
$489$	−8746.23	−0.808830
$490$	2261.65	0.208512
$491$	7470.12	0.686602	0.343301	−	0.939225i	$-0.388455\pi$
$491$	7470.12	0.686602	0.343301	+	0.939225i	$0.388455\pi$
$492$	−7783.88	−0.713261
$493$	1466.72	0.133992
$494$	−1116.64	−0.101701
$495$	32.5442	0.00295506
$496$	−4020.20	−0.363936
$497$	9530.45	0.860159
$498$	7822.30	0.703867
$499$	−11438.5	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$499$	−11438.5	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$500$	−4005.72	−0.358282
$501$	−15209.3	−1.35629
$502$	776.065	0.0689990
$503$	−5288.25	−0.468770	−0.234385	−	0.972144i	$-0.575308\pi$
$503$	−5288.25	−0.468770	−0.234385	+	0.972144i	$0.575308\pi$
$504$	11.6858	0.00103279
$505$	3097.74	0.272965
$506$	0	0
$507$	5781.71	0.506459
$508$	10600.8	0.925855
$509$	−20444.9	−1.78036	−0.890180	−	0.455608i	$-0.849422\pi$
$509$	−20444.9	−1.78036	−0.890180	+	0.455608i	$0.849422\pi$
$510$	2952.62	0.256361
$511$	5432.36	0.470281
$512$	−512.000	−0.0441942
$513$	−2368.00	−0.203801
$514$	1860.97	0.159697
$515$	2761.54	0.236288
$516$	−9028.16	−0.770238
$517$	−10169.9	−0.865128
$518$	−940.220	−0.0797508
$519$	−802.356	−0.0678603
$520$	1142.55	0.0963540
$521$	−15469.5	−1.30083	−0.650415	−	0.759579i	$-0.725405\pi$
$521$	−15469.5	−1.30083	−0.650415	+	0.759579i	$0.725405\pi$
$522$	−7.23964	−0.000607031 0
$523$	−3611.57	−0.301956	−0.150978	−	0.988537i	$-0.548242\pi$
$523$	−3611.57	−0.301956	−0.150978	+	0.988537i	$0.548242\pi$
$524$	−1669.05	−0.139147
$525$	−5010.29	−0.416509
$526$	2299.73	0.190633
$527$	−16435.5	−1.35852
$528$	−3881.78	−0.319948
$529$	0	0
$530$	−3669.00	−0.300700
$531$	−61.6964	−0.00504218
$532$	612.745	0.0499358
$533$	−12314.0	−1.00071
$534$	15979.4	1.29493
$535$	4287.88	0.346507
$536$	4636.82	0.373657
$537$	−5375.74	−0.431993
$538$	7628.69	0.611331
$539$	−12155.8	−0.971406
$540$	2422.93	0.193086
$541$	−20809.9	−1.65377	−0.826883	−	0.562374i	$-0.809888\pi$
$541$	−20809.9	−1.65377	−0.826883	+	0.562374i	$0.809888\pi$
$542$	−8338.79	−0.660852
$543$	379.064	0.0299580
$544$	−2093.17	−0.164970
$545$	−5544.80	−0.435804
$546$	3110.42	0.243798
$547$	−15166.8	−1.18553	−0.592765	−	0.805375i	$-0.701964\pi$
$547$	−15166.8	−1.18553	−0.592765	+	0.805375i	$0.701964\pi$
$548$	7843.91	0.611451
$549$	−30.9127	−0.00240314
$550$	9891.84	0.766890
$551$	−379.611	−0.0293503
$552$	0	0
$553$	2176.97	0.167404
$554$	8545.12	0.655320
$555$	1172.60	0.0896827
$556$	7037.51	0.536793
$557$	12474.4	0.948935	0.474468	−	0.880273i	$-0.342641\pi$
$557$	12474.4	0.948935	0.474468	+	0.880273i	$0.342641\pi$
$558$	81.1244	0.00615460
$559$	−14282.4	−1.08065
$560$	−626.960	−0.0473105
$561$	−15869.6	−1.19432
$562$	15278.2	1.14675
$563$	10739.4	0.803929	0.401964	−	0.915655i	$-0.368328\pi$
$563$	10739.4	0.803929	0.401964	+	0.915655i	$0.368328\pi$
$564$	4554.27	0.340016
$565$	−5355.93	−0.398806
$566$	−2434.98	−0.180830
$567$	6635.51	0.491473
$568$	8426.16	0.622454
$569$	13583.2	1.00077	0.500386	−	0.865802i	$-0.333191\pi$
$569$	13583.2	1.00077	0.500386	+	0.865802i	$0.333191\pi$
$570$	−764.185	−0.0561547
$571$	−5262.91	−0.385720	−0.192860	−	0.981226i	$-0.561776\pi$
$571$	−5262.91	−0.385720	−0.192860	+	0.981226i	$0.561776\pi$
$572$	−6140.91	−0.448889
$573$	14603.5	1.06470
$574$	6757.15	0.491355
$575$	0	0
$576$	10.3317	0.000747377 0
$577$	22690.5	1.63712	0.818559	−	0.574423i	$-0.194774\pi$
$577$	22690.5	1.63712	0.818559	+	0.574423i	$0.194774\pi$
$578$	1268.66	0.0912962
$579$	−6920.69	−0.496743
$580$	388.418	0.0278072
$581$	−6790.50	−0.484884
$582$	−17372.6	−1.23731
$583$	19720.0	1.40089
$584$	4802.92	0.340319
$585$	−23.0557	−0.00162946
$586$	−10641.1	−0.750134
$587$	−450.330	−0.0316646	−0.0158323	−	0.999875i	$-0.505040\pi$
$587$	−450.330	−0.0316646	−0.0158323	+	0.999875i	$0.505040\pi$
$588$	5443.60	0.381786
$589$	4253.77	0.297578
$590$	3310.11	0.230975
$591$	−9548.90	−0.664618
$592$	−831.277	−0.0577116
$593$	21707.5	1.50324	0.751619	−	0.659598i	$-0.229273\pi$
$593$	21707.5	1.50324	0.751619	+	0.659598i	$0.229273\pi$
$594$	−13022.7	−0.899539
$595$	−2563.15	−0.176603
$596$	11529.3	0.792383
$597$	−7184.52	−0.492534
$598$	0	0
$599$	5528.42	0.377103	0.188552	−	0.982063i	$-0.439621\pi$
$599$	5528.42	0.377103	0.188552	+	0.982063i	$0.439621\pi$
$600$	−4429.75	−0.301406
$601$	391.648	0.0265818	0.0132909	−	0.999912i	$-0.495769\pi$
$601$	391.648	0.0265818	0.0132909	+	0.999912i	$0.495769\pi$
$602$	7837.30	0.530606
$603$	−93.5672	−0.00631899
$604$	12128.5	0.817056
$605$	3620.57	0.243301
$606$	7455.98	0.499800
$607$	689.357	0.0460958	0.0230479	−	0.999734i	$-0.492663\pi$
$607$	689.357	0.0460958	0.0230479	+	0.999734i	$0.492663\pi$
$608$	541.746	0.0361361
$609$	1057.41	0.0703586
$610$	1658.52	0.110084
$611$	7204.79	0.477045
$612$	42.2385	0.00278985
$613$	−13835.3	−0.911590	−0.455795	−	0.890085i	$-0.650645\pi$
$613$	−13835.3	−0.911590	−0.455795	+	0.890085i	$0.650645\pi$
$614$	−13797.9	−0.906902
$615$	−8427.18	−0.552548
$616$	3369.75	0.220408
$617$	−14646.6	−0.955670	−0.477835	−	0.878450i	$-0.658578\pi$
$617$	−14646.6	−0.955670	−0.477835	+	0.878450i	$0.658578\pi$
$618$	6646.79	0.432642
$619$	−17606.2	−1.14322	−0.571610	−	0.820526i	$-0.693681\pi$
$619$	−17606.2	−1.14322	−0.571610	+	0.820526i	$0.693681\pi$
$620$	−4352.45	−0.281933
$621$	0	0
$622$	−1259.87	−0.0812156
$623$	−13871.6	−0.892061
$624$	2750.02	0.176424
$625$	8943.99	0.572415
$626$	2739.61	0.174915
$627$	4107.30	0.261611
$628$	11165.7	0.709493
$629$	−3398.45	−0.215429
$630$	12.6515	0.000800078 0
$631$	9503.45	0.599567	0.299783	−	0.954007i	$-0.403086\pi$
$631$	9503.45	0.599567	0.299783	+	0.954007i	$0.403086\pi$
$632$	1924.73	0.121142
$633$	2484.62	0.156011
$634$	−5167.75	−0.323718
$635$	11476.9	0.717240
$636$	−8830.96	−0.550582
$637$	8611.69	0.535648
$638$	−2087.65	−0.129547
$639$	−170.033	−0.0105265
$640$	−554.314	−0.0342362
$641$	−170.214	−0.0104884	−0.00524419	−	0.999986i	$-0.501669\pi$
$641$	−170.214	−0.0104884	−0.00524419	+	0.999986i	$0.501669\pi$
$642$	10320.5	0.634454
$643$	−9664.25	−0.592723	−0.296362	−	0.955076i	$-0.595773\pi$
$643$	−9664.25	−0.592723	−0.296362	+	0.955076i	$0.595773\pi$
$644$	0	0
$645$	−9774.30	−0.596686
$646$	2214.78	0.134891
$647$	12000.3	0.729179	0.364589	−	0.931168i	$-0.381209\pi$
$647$	12000.3	0.729179	0.364589	+	0.931168i	$0.381209\pi$
$648$	5866.66	0.355655
$649$	−17791.0	−1.07605
$650$	−7007.80	−0.422875
$651$	−11848.9	−0.713356
$652$	6712.81	0.403212
$653$	12388.9	0.742444	0.371222	−	0.928544i	$-0.378939\pi$
$653$	12388.9	0.742444	0.371222	+	0.928544i	$0.378939\pi$
$654$	−13345.8	−0.797957
$655$	−1806.99	−0.107794
$656$	5974.20	0.355569
$657$	−96.9189	−0.00575520
$658$	−3953.54	−0.234232
$659$	−13558.0	−0.801431	−0.400716	−	0.916202i	$-0.631238\pi$
$659$	−13558.0	−0.801431	−0.400716	+	0.916202i	$0.631238\pi$
$660$	−4202.59	−0.247857
$661$	25480.9	1.49938	0.749692	−	0.661787i	$-0.230202\pi$
$661$	25480.9	1.49938	0.749692	+	0.661787i	$0.230202\pi$
$662$	−2057.28	−0.120783
$663$	11242.7	0.658566
$664$	−6003.69	−0.350886
$665$	663.385	0.0386842
$666$	16.7745	0.000975974 0
$667$	0	0
$668$	11673.2	0.676125
$669$	−20925.8	−1.20933
$670$	5020.03	0.289464
$671$	−8914.11	−0.512854
$672$	−1509.04	−0.0866256
$673$	−6133.05	−0.351281	−0.175640	−	0.984454i	$-0.556200\pi$
$673$	−6133.05	−0.351281	−0.175640	+	0.984454i	$0.556200\pi$
$674$	17301.4	0.988761
$675$	−14861.0	−0.847409
$676$	−4437.51	−0.252476
$677$	−9489.27	−0.538704	−0.269352	−	0.963042i	$-0.586809\pi$
$677$	−9489.27	−0.538704	−0.269352	+	0.963042i	$0.586809\pi$
$678$	−12891.3	−0.730215
$679$	15081.0	0.852366
$680$	−2266.16	−0.127799
$681$	−22895.7	−1.28835
$682$	23393.3	1.31346
$683$	966.384	0.0541400	0.0270700	−	0.999634i	$-0.491382\pi$
$683$	966.384	0.0541400	0.0270700	+	0.999634i	$0.491382\pi$
$684$	−10.9320	−0.000611105 0
$685$	8492.17	0.473678
$686$	−10932.8	−0.608478
$687$	−22469.0	−1.24781
$688$	6929.20	0.383973
$689$	−13970.5	−0.772470
$690$	0	0
$691$	−8288.05	−0.456284	−0.228142	−	0.973628i	$-0.573265\pi$
$691$	−8288.05	−0.456284	−0.228142	+	0.973628i	$0.573265\pi$
$692$	615.815	0.0338292
$693$	−67.9988	−0.00372736
$694$	−24628.2	−1.34708
$695$	7619.13	0.415842
$696$	934.888	0.0509150
$697$	24423.9	1.32729
$698$	−5877.56	−0.318724
$699$	−18282.9	−0.989302
$700$	3845.44	0.207634
$701$	33338.2	1.79625	0.898123	−	0.439745i	$-0.144931\pi$
$701$	33338.2	1.79625	0.898123	+	0.439745i	$0.144931\pi$
$702$	9225.81	0.496020
$703$	879.573	0.0471888
$704$	2979.30	0.159498
$705$	4930.66	0.263403
$706$	18982.3	1.01191
$707$	−6472.50	−0.344305
$708$	7967.14	0.422915
$709$	22451.8	1.18927	0.594636	−	0.803995i	$-0.297296\pi$
$709$	22451.8	1.18927	0.594636	+	0.803995i	$0.297296\pi$
$710$	9122.54	0.482201
$711$	−38.8394	−0.00204865
$712$	−12264.3	−0.645540
$713$	0	0
$714$	−6169.28	−0.323361
$715$	−6648.43	−0.347744
$716$	4125.93	0.215354
$717$	−17532.8	−0.913212
$718$	−22640.9	−1.17681
$719$	−14426.9	−0.748306	−0.374153	−	0.927367i	$-0.622067\pi$
$719$	−14426.9	−0.748306	−0.374153	+	0.927367i	$0.622067\pi$
$720$	11.1856	0.000578977 0
$721$	−5770.04	−0.298041
$722$	13144.8	0.677560
$723$	33678.5	1.73239
$724$	−290.935	−0.0149344
$725$	−2382.35	−0.122039
$726$	8714.40	0.445484
$727$	−18589.4	−0.948339	−0.474169	−	0.880434i	$-0.657252\pi$
$727$	−18589.4	−0.948339	−0.474169	+	0.880434i	$0.657252\pi$
$728$	−2387.27	−0.121536
$729$	19563.9	0.993950
$730$	5199.85	0.263637
$731$	28328.1	1.43331
$732$	3991.90	0.201564
$733$	26534.1	1.33705	0.668525	−	0.743689i	$-0.266926\pi$
$733$	26534.1	1.33705	0.668525	+	0.743689i	$0.266926\pi$
$734$	−13058.8	−0.656689
$735$	5893.49	0.295761
$736$	0	0
$737$	−26981.4	−1.34854
$738$	−120.555	−0.00601311
$739$	−27669.3	−1.37731	−0.688655	−	0.725090i	$-0.741798\pi$
$739$	−27669.3	−1.37731	−0.688655	+	0.725090i	$0.741798\pi$
$740$	−899.978	−0.0447079
$741$	−2909.79	−0.144256
$742$	7666.11	0.379288
$743$	−17614.1	−0.869717	−0.434859	−	0.900499i	$-0.643202\pi$
$743$	−17614.1	−0.869717	−0.434859	+	0.900499i	$0.643202\pi$
$744$	−10476.0	−0.516220
$745$	12482.2	0.613842
$746$	−5884.89	−0.288822
$747$	121.150	0.00593391
$748$	12180.0	0.595383
$749$	−8959.21	−0.437066
$750$	−10438.2	−0.508200
$751$	−21905.3	−1.06436	−0.532182	−	0.846630i	$-0.678628\pi$
$751$	−21905.3	−1.06436	−0.532182	+	0.846630i	$0.678628\pi$
$752$	−3495.44	−0.169502
$753$	2022.29	0.0978705
$754$	1478.98	0.0714340
$755$	13130.9	0.632955
$756$	−5062.55	−0.243549
$757$	32332.5	1.55237	0.776184	−	0.630506i	$-0.217153\pi$
$757$	32332.5	1.55237	0.776184	+	0.630506i	$0.217153\pi$
$758$	2948.61	0.141291
$759$	0	0
$760$	586.519	0.0279938
$761$	−23474.6	−1.11821	−0.559103	−	0.829099i	$-0.688854\pi$
$761$	−23474.6	−1.11821	−0.559103	+	0.829099i	$0.688854\pi$
$762$	27623.9	1.31327
$763$	11585.5	0.549701
$764$	−11208.3	−0.530764
$765$	45.7293	0.00216124
$766$	−1901.55	−0.0896941
$767$	12603.9	0.593351
$768$	−1334.19	−0.0626866
$769$	27320.3	1.28114	0.640569	−	0.767900i	$-0.278698\pi$
$769$	27320.3	1.28114	0.640569	+	0.767900i	$0.278698\pi$
$770$	3648.24	0.170745
$771$	4849.38	0.226519
$772$	5311.70	0.247632
$773$	−17902.8	−0.833013	−0.416506	−	0.909133i	$-0.636746\pi$
$773$	−17902.8	−0.833013	−0.416506	+	0.909133i	$0.636746\pi$
$774$	−139.826	−0.00649345
$775$	26695.7	1.23734
$776$	13333.6	0.616815
$777$	−2450.05	−0.113121
$778$	−11178.2	−0.515113
$779$	−6321.30	−0.290737
$780$	2977.29	0.136672
$781$	−49031.3	−2.24645
$782$	0	0
$783$	3136.38	0.143148
$784$	−4178.01	−0.190325
$785$	12088.5	0.549629
$786$	−4349.27	−0.197371
$787$	−450.226	−0.0203924	−0.0101962	−	0.999948i	$-0.503246\pi$
$787$	−450.226	−0.0203924	−0.0101962	+	0.999948i	$0.503246\pi$
$788$	7328.87	0.331320
$789$	5992.72	0.270401
$790$	2083.79	0.0938457
$791$	11190.8	0.503034
$792$	−60.1198	−0.00269731
$793$	6315.13	0.282796
$794$	−14993.7	−0.670161
$795$	−9560.80	−0.426524
$796$	5514.18	0.245534
$797$	28787.8	1.27944	0.639722	−	0.768606i	$-0.279049\pi$
$797$	28787.8	1.27944	0.639722	+	0.768606i	$0.279049\pi$
$798$	1596.71	0.0708307
$799$	−14290.1	−0.632727
$800$	3399.87	0.150255
$801$	247.484	0.0109169
$802$	−3546.47	−0.156147
$803$	−27947.9	−1.22822
$804$	12082.8	0.530008
$805$	0	0
$806$	−16572.8	−0.724259
$807$	19879.1	0.867133
$808$	−5722.54	−0.249156
$809$	−11782.9	−0.512071	−0.256036	−	0.966667i	$-0.582416\pi$
$809$	−11782.9	−0.512071	−0.256036	+	0.966667i	$0.582416\pi$
$810$	6351.51	0.275518
$811$	31774.6	1.37578	0.687890	−	0.725815i	$-0.258537\pi$
$811$	31774.6	1.37578	0.687890	+	0.725815i	$0.258537\pi$
$812$	−811.572	−0.0350746
$813$	−21729.5	−0.937375
$814$	4837.16	0.208283
$815$	7267.59	0.312359
$816$	−5454.45	−0.234000
$817$	−7331.78	−0.313961
$818$	4401.07	0.188117
$819$	48.1733	0.00205532
$820$	6467.94	0.275452
$821$	34804.0	1.47950	0.739749	−	0.672883i	$-0.234944\pi$
$821$	34804.0	1.47950	0.739749	+	0.672883i	$0.234944\pi$
$822$	20439.9	0.867304
$823$	−40133.7	−1.69984	−0.849922	−	0.526908i	$-0.823351\pi$
$823$	−40133.7	−1.69984	−0.849922	+	0.526908i	$0.823351\pi$
$824$	−5101.47	−0.215677
$825$	25776.5	1.08778
$826$	−6916.24	−0.291340
$827$	30628.1	1.28784	0.643919	−	0.765093i	$-0.277307\pi$
$827$	30628.1	1.28784	0.643919	+	0.765093i	$0.277307\pi$
$828$	0	0
$829$	23165.8	0.970543	0.485271	−	0.874364i	$-0.338721\pi$
$829$	23165.8	0.970543	0.485271	+	0.874364i	$0.338721\pi$
$830$	−6499.87	−0.271824
$831$	22267.1	0.929529
$832$	−2110.66	−0.0879496
$833$	−17080.6	−0.710455
$834$	18338.6	0.761406
$835$	12638.0	0.523779
$836$	−3152.39	−0.130416
$837$	−35145.0	−1.45136
$838$	−1673.88	−0.0690016
$839$	31553.9	1.29840	0.649202	−	0.760616i	$-0.275103\pi$
$839$	31553.9	1.29840	0.649202	+	0.760616i	$0.275103\pi$
$840$	−1633.75	−0.0671069
$841$	−23886.2	−0.979385
$842$	−24312.3	−0.995080
$843$	39812.4	1.62659
$844$	−1906.97	−0.0777733
$845$	−4804.25	−0.195587
$846$	70.5352	0.00286649
$847$	−7564.92	−0.306888
$848$	6777.84	0.274472
$849$	−6345.15	−0.256496
$850$	13899.4	0.560879
$851$	0	0
$852$	21957.1	0.882910
$853$	−42509.2	−1.70632	−0.853158	−	0.521653i	$-0.825315\pi$
$853$	−42509.2	−1.70632	−0.853158	+	0.521653i	$0.825315\pi$
$854$	−3465.35	−0.138855
$855$	−11.8355	−0.000473409 0
$856$	−7921.11	−0.316283
$857$	−7026.51	−0.280071	−0.140036	−	0.990146i	$-0.544722\pi$
$857$	−7026.51	−0.280071	−0.140036	+	0.990146i	$0.544722\pi$
$858$	−16002.2	−0.636720
$859$	15146.7	0.601627	0.300813	−	0.953683i	$-0.402742\pi$
$859$	15146.7	0.601627	0.300813	+	0.953683i	$0.402742\pi$
$860$	7501.86	0.297455
$861$	17608.0	0.696956
$862$	−11391.3	−0.450102
$863$	−24338.2	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$863$	−24338.2	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$864$	−4475.95	−0.176244
$865$	666.709	0.0262067
$866$	1027.24	0.0403083
$867$	3305.91	0.129498
$868$	9094.14	0.355616
$869$	−11199.9	−0.437203
$870$	1012.15	0.0394427
$871$	19114.8	0.743604
$872$	10243.1	0.397791
$873$	−269.061	−0.0104311
$874$	0	0
$875$	9061.37	0.350092
$876$	12515.6	0.482720
$877$	42546.2	1.63818	0.819089	−	0.573666i	$-0.194479\pi$
$877$	42546.2	1.63818	0.819089	+	0.573666i	$0.194479\pi$
$878$	−33570.6	−1.29038
$879$	−27728.8	−1.06402
$880$	3225.52	0.123560
$881$	−23917.9	−0.914660	−0.457330	−	0.889297i	$-0.651194\pi$
$881$	−23917.9	−0.914660	−0.457330	+	0.889297i	$0.651194\pi$
$882$	84.3089	0.00321863
$883$	8266.56	0.315053	0.157527	−	0.987515i	$-0.449648\pi$
$883$	8266.56	0.315053	0.157527	+	0.987515i	$0.449648\pi$
$884$	−8628.86	−0.328303
$885$	8625.59	0.327622
$886$	27104.0	1.02774
$887$	19604.7	0.742122	0.371061	−	0.928608i	$-0.378994\pi$
$887$	19604.7	0.742122	0.371061	+	0.928608i	$0.378994\pi$
$888$	−2166.17	−0.0818602
$889$	−23980.2	−0.904690
$890$	−13277.9	−0.500086
$891$	−34137.8	−1.28357
$892$	16060.8	0.602864
$893$	3698.52	0.138596
$894$	30043.5	1.12394
$895$	4466.92	0.166830
$896$	1158.20	0.0431839
$897$	0	0
$898$	11465.3	0.426060
$899$	−5634.05	−0.209017
$900$	−68.6067	−0.00254099
$901$	27709.3	1.02456
$902$	−34763.6	−1.28326
$903$	20422.7	0.752630
$904$	9894.15	0.364021
$905$	−314.979	−0.0115694
$906$	31604.8	1.15894
$907$	−1501.34	−0.0549628	−0.0274814	−	0.999622i	$-0.508749\pi$
$907$	−1501.34	−0.0549628	−0.0274814	+	0.999622i	$0.508749\pi$
$908$	17572.7	0.642258
$909$	115.476	0.00421353
$910$	−2584.57	−0.0941513
$911$	−16524.2	−0.600957	−0.300478	−	0.953789i	$-0.597146\pi$
$911$	−16524.2	−0.600957	−0.300478	+	0.953789i	$0.597146\pi$
$912$	1411.70	0.0512566
$913$	34935.1	1.26636
$914$	6721.79	0.243257
$915$	4321.82	0.156147
$916$	17245.1	0.622047
$917$	3775.58	0.135966
$918$	−18298.7	−0.657894
$919$	−13312.6	−0.477848	−0.238924	−	0.971038i	$-0.576795\pi$
$919$	−13312.6	−0.477848	−0.238924	+	0.971038i	$0.576795\pi$
$920$	0	0
$921$	−35955.0	−1.28638
$922$	16907.9	0.603938
$923$	34735.9	1.23873
$924$	8781.00	0.312634
$925$	5520.00	0.196212
$926$	7482.21	0.265530
$927$	102.944	0.00364737
$928$	−717.535	−0.0253817
$929$	31441.3	1.11039	0.555197	−	0.831719i	$-0.312643\pi$
$929$	31441.3	1.11039	0.555197	+	0.831719i	$0.312643\pi$
$930$	−11341.8	−0.399904
$931$	4420.75	0.155622
$932$	14032.3	0.493179
$933$	−3283.00	−0.115199
$934$	16546.0	0.579658
$935$	13186.7	0.461230
$936$	42.5915	0.00148733
$937$	−50500.2	−1.76069	−0.880346	−	0.474332i	$-0.842690\pi$
$937$	−50500.2	−1.76069	−0.880346	+	0.474332i	$0.842690\pi$
$938$	−10489.0	−0.365115
$939$	7138.97	0.248106
$940$	−3784.32	−0.131310
$941$	−33474.1	−1.15964	−0.579822	−	0.814743i	$-0.696878\pi$
$941$	−33474.1	−1.15964	−0.579822	+	0.814743i	$0.696878\pi$
$942$	29096.1	1.00637
$943$	0	0
$944$	−6114.85	−0.210828
$945$	−5480.94	−0.188672
$946$	−40320.6	−1.38577
$947$	12214.7	0.419138	0.209569	−	0.977794i	$-0.432794\pi$
$947$	12214.7	0.419138	0.209569	+	0.977794i	$0.432794\pi$
$948$	5015.51	0.171831
$949$	19799.5	0.677259
$950$	−3597.40	−0.122858
$951$	−13466.3	−0.459173
$952$	4734.98	0.161199
$953$	−3098.85	−0.105332	−0.0526661	−	0.998612i	$-0.516772\pi$
$953$	−3098.85	−0.105332	−0.0526661	+	0.998612i	$0.516772\pi$
$954$	−136.771	−0.00464165
$955$	−12134.7	−0.411171
$956$	13456.6	0.455247
$957$	−5440.06	−0.183754
$958$	−22903.4	−0.772418
$959$	−17743.8	−0.597473
$960$	−1444.45	−0.0485619
$961$	33341.9	1.11919
$962$	−3426.85	−0.114850
$963$	159.842	0.00534872
$964$	−25848.5	−0.863616
$965$	5750.68	0.191835
$966$	0	0
$967$	35481.7	1.17995	0.589977	−	0.807420i	$-0.299137\pi$
$967$	35481.7	1.17995	0.589977	+	0.807420i	$0.299137\pi$
$968$	−6688.38	−0.222079
$969$	5771.34	0.191334
$970$	14435.6	0.477833
$971$	46079.9	1.52294	0.761469	−	0.648201i	$-0.224478\pi$
$971$	46079.9	1.52294	0.761469	+	0.648201i	$0.224478\pi$
$972$	181.186	0.00597894
$973$	−15919.6	−0.524522
$974$	−32499.4	−1.06914
$975$	−18261.1	−0.599820
$976$	−3063.82	−0.100482
$977$	−24804.7	−0.812256	−0.406128	−	0.913816i	$-0.633121\pi$
$977$	−24804.7	−0.812256	−0.406128	+	0.913816i	$0.633121\pi$
$978$	17492.5	0.571929
$979$	71365.4	2.32977
$980$	−4523.30	−0.147440
$981$	−206.697	−0.00672713
$982$	−14940.2	−0.485501
$983$	−50657.7	−1.64367	−0.821836	−	0.569724i	$-0.807050\pi$
$983$	−50657.7	−1.64367	−0.821836	+	0.569724i	$0.807050\pi$
$984$	15567.8	0.504352
$985$	7934.56	0.256666
$986$	−2933.44	−0.0947463
$987$	−10302.3	−0.332243
$988$	2233.29	0.0719134
$989$	0	0
$990$	−65.0885	−0.00208954
$991$	16887.0	0.541304	0.270652	−	0.962677i	$-0.412761\pi$
$991$	16887.0	0.541304	0.270652	+	0.962677i	$0.412761\pi$
$992$	8040.40	0.257342
$993$	−5360.92	−0.171323
$994$	−19060.9	−0.608224
$995$	5969.90	0.190210
$996$	−15644.6	−0.497709
$997$	−2976.93	−0.0945639	−0.0472820	−	0.998882i	$-0.515056\pi$
$997$	−2976.93	−0.0945639	−0.0472820	+	0.998882i	$0.515056\pi$
$998$	22877.0	0.725609
$999$	−7267.11	−0.230151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1058.4.a.p.1.2	yes	6
23.22	odd	2	inner	1058.4.a.p.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.4.a.p.1.1	✓	6	23.22	odd	2	inner
1058.4.a.p.1.2	yes	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -5.21166 q^{3} +4.00000 q^{4} +4.33058 q^{5} +10.4233 q^{6} -9.04844 q^{7} -8.00000 q^{8} +0.161433 q^{9} -8.66116 q^{10} +46.5516 q^{11} -20.8467 q^{12} -32.9791 q^{13} +18.0969 q^{14} -22.5695 q^{15} +16.0000 q^{16} +65.4116 q^{17} -0.322867 q^{18} -16.9296 q^{19} +17.3223 q^{20} +47.1574 q^{21} -93.1031 q^{22} +41.6933 q^{24} -106.246 q^{25} +65.9582 q^{26} +139.874 q^{27} -36.1938 q^{28} +22.4230 q^{29} +45.1391 q^{30} -251.263 q^{31} -32.0000 q^{32} -242.611 q^{33} -130.823 q^{34} -39.1850 q^{35} +0.645734 q^{36} -51.9548 q^{37} +33.8592 q^{38} +171.876 q^{39} -34.6446 q^{40} +373.388 q^{41} -94.3148 q^{42} +433.075 q^{43} +186.206 q^{44} +0.699101 q^{45} -218.465 q^{47} -83.3866 q^{48} -261.126 q^{49} +212.492 q^{50} -340.903 q^{51} -131.916 q^{52} +423.615 q^{53} -279.747 q^{54} +201.595 q^{55} +72.3875 q^{56} +88.2313 q^{57} -44.8459 q^{58} -382.178 q^{59} -90.2781 q^{60} -191.489 q^{61} +502.525 q^{62} -1.46072 q^{63} +64.0000 q^{64} -142.819 q^{65} +485.222 q^{66} -579.602 q^{67} +261.646 q^{68} +78.3700 q^{70} -1053.27 q^{71} -1.29147 q^{72} -600.364 q^{73} +103.910 q^{74} +553.719 q^{75} -67.7183 q^{76} -421.219 q^{77} -343.752 q^{78} -240.591 q^{79} +69.2893 q^{80} -733.333 q^{81} -746.775 q^{82} +750.461 q^{83} +188.630 q^{84} +283.270 q^{85} -866.150 q^{86} -116.861 q^{87} -372.412 q^{88} +1533.04 q^{89} -1.39820 q^{90} +298.409 q^{91} +1309.50 q^{93} +436.930 q^{94} -73.3149 q^{95} +166.773 q^{96} -1666.70 q^{97} +522.252 q^{98} +7.51498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 12 q^{3} + 24 q^{4} - 24 q^{6} - 48 q^{8} + 90 q^{9} + 48 q^{12} + 44 q^{13} + 96 q^{16} - 180 q^{18} - 96 q^{24} - 98 q^{25} - 88 q^{26} + 984 q^{27} + 580 q^{29} - 268 q^{31} - 192 q^{32}+ \cdots - 1092 q^{98}+O(q^{100}) \) 6 * q - 12 * q^2 + 12 * q^3 + 24 * q^4 - 24 * q^6 - 48 * q^8 + 90 * q^9 + 48 * q^12 + 44 * q^13 + 96 * q^16 - 180 * q^18 - 96 * q^24 - 98 * q^25 - 88 * q^26 + 984 * q^27 + 580 * q^29 - 268 * q^31 - 192 * q^32 - 1136 * q^35 + 360 * q^36 + 484 * q^39 - 48 * q^41 + 1324 * q^47 + 192 * q^48 + 546 * q^49 + 196 * q^50 + 176 * q^52 - 1968 * q^54 + 436 * q^55 - 1160 * q^58 + 644 * q^59 + 536 * q^62 + 384 * q^64 + 2272 * q^70 - 528 * q^71 - 720 * q^72 + 1268 * q^73 + 4184 * q^75 - 3440 * q^77 - 968 * q^78 + 4254 * q^81 + 96 * q^82 + 1440 * q^85 + 3068 * q^87 + 5776 * q^93 - 2648 * q^94 + 1092 * q^95 - 384 * q^96 - 1092 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more