LMFDB - Embedded newform 2254.4.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, 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("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 165 x^{9} + 798 x^{8} + 8769 x^{7} - 38472 x^{6} - 184213 x^{5} + 644009 x^{4} + \cdots + 2848203 $ x^11 - 5x^10 - 165x^9 + 798x^8 + 8769x^7 - 38472x^6 - 184213x^5 + 644009x^4 + 1428336x^3 - 2764269x^2 - 1864863x + 2848203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.84446$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -4.84446 q^{3} +4.00000 q^{4} -6.31640 q^{5} -9.68892 q^{6} +8.00000 q^{8} -3.53122 q^{9} -12.6328 q^{10} -56.5325 q^{11} -19.3778 q^{12} -23.2303 q^{13} +30.5995 q^{15} +16.0000 q^{16} -44.2506 q^{17} -7.06245 q^{18} +82.8762 q^{19} -25.2656 q^{20} -113.065 q^{22} -23.0000 q^{23} -38.7557 q^{24} -85.1031 q^{25} -46.4606 q^{26} +147.907 q^{27} -283.571 q^{29} +61.1991 q^{30} -138.459 q^{31} +32.0000 q^{32} +273.869 q^{33} -88.5012 q^{34} -14.1249 q^{36} -221.502 q^{37} +165.752 q^{38} +112.538 q^{39} -50.5312 q^{40} +295.330 q^{41} -260.857 q^{43} -226.130 q^{44} +22.3046 q^{45} -46.0000 q^{46} -227.467 q^{47} -77.5113 q^{48} -170.206 q^{50} +214.370 q^{51} -92.9213 q^{52} +350.955 q^{53} +295.814 q^{54} +357.082 q^{55} -401.490 q^{57} -567.141 q^{58} -555.330 q^{59} +122.398 q^{60} +443.347 q^{61} -276.919 q^{62} +64.0000 q^{64} +146.732 q^{65} +547.739 q^{66} -65.4770 q^{67} -177.002 q^{68} +111.423 q^{69} +728.329 q^{71} -28.2498 q^{72} -170.417 q^{73} -443.005 q^{74} +412.278 q^{75} +331.505 q^{76} +225.077 q^{78} -1079.60 q^{79} -101.062 q^{80} -621.187 q^{81} +590.660 q^{82} -949.211 q^{83} +279.505 q^{85} -521.714 q^{86} +1373.75 q^{87} -452.260 q^{88} +790.169 q^{89} +44.6093 q^{90} -92.0000 q^{92} +670.761 q^{93} -454.933 q^{94} -523.479 q^{95} -155.023 q^{96} +1086.53 q^{97} +199.629 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * q^99

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
$3$	−4.84446	−0.932316	−0.466158	−	0.884701i	$-0.654362\pi$
$3$	−4.84446	−0.932316	−0.466158	+	0.884701i	$0.654362\pi$
$4$	4.00000	0.500000
$5$	−6.31640	−0.564956	−0.282478	−	0.959274i	$-0.591156\pi$
$5$	−6.31640	−0.564956	−0.282478	+	0.959274i	$0.591156\pi$
$6$	−9.68892	−0.659247
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−3.53122	−0.130786
$10$	−12.6328	−0.399484
$11$	−56.5325	−1.54956	−0.774781	−	0.632230i	$-0.782140\pi$
$11$	−56.5325	−1.54956	−0.774781	+	0.632230i	$0.782140\pi$
$12$	−19.3778	−0.466158
$13$	−23.2303	−0.495610	−0.247805	−	0.968810i	$-0.579709\pi$
$13$	−23.2303	−0.495610	−0.247805	+	0.968810i	$0.579709\pi$
$14$	0	0
$15$	30.5995	0.526718
$16$	16.0000	0.250000
$17$	−44.2506	−0.631314	−0.315657	−	0.948873i	$-0.602225\pi$
$17$	−44.2506	−0.631314	−0.315657	+	0.948873i	$0.602225\pi$
$18$	−7.06245	−0.0924797
$19$	82.8762	1.00069	0.500345	−	0.865826i	$-0.333207\pi$
$19$	82.8762	1.00069	0.500345	+	0.865826i	$0.333207\pi$
$20$	−25.2656	−0.282478
$21$	0	0
$22$	−113.065	−1.09571
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−38.7557	−0.329624
$25$	−85.1031	−0.680825
$26$	−46.4606	−0.350449
$27$	147.907	1.05425
$28$	0	0
$29$	−283.571	−1.81578	−0.907892	−	0.419204i	$-0.862309\pi$
$29$	−283.571	−1.81578	−0.907892	+	0.419204i	$0.862309\pi$
$30$	61.1991	0.372446
$31$	−138.459	−0.802195	−0.401097	−	0.916035i	$-0.631371\pi$
$31$	−138.459	−0.802195	−0.401097	+	0.916035i	$0.631371\pi$
$32$	32.0000	0.176777
$33$	273.869	1.44468
$34$	−88.5012	−0.446407
$35$	0	0
$36$	−14.1249	−0.0653930
$37$	−221.502	−0.984183	−0.492091	−	0.870544i	$-0.663767\pi$
$37$	−221.502	−0.984183	−0.492091	+	0.870544i	$0.663767\pi$
$38$	165.752	0.707594
$39$	112.538	0.462065
$40$	−50.5312	−0.199742
$41$	295.330	1.12495	0.562473	−	0.826816i	$-0.309850\pi$
$41$	295.330	1.12495	0.562473	+	0.826816i	$0.309850\pi$
$42$	0	0
$43$	−260.857	−0.925124	−0.462562	−	0.886587i	$-0.653070\pi$
$43$	−260.857	−0.925124	−0.462562	+	0.886587i	$0.653070\pi$
$44$	−226.130	−0.774781
$45$	22.3046	0.0738884
$46$	−46.0000	−0.147442
$47$	−227.467	−0.705945	−0.352973	−	0.935634i	$-0.614829\pi$
$47$	−227.467	−0.705945	−0.352973	+	0.935634i	$0.614829\pi$
$48$	−77.5113	−0.233079
$49$	0	0
$50$	−170.206	−0.481416
$51$	214.370	0.588585
$52$	−92.9213	−0.247805
$53$	350.955	0.909573	0.454787	−	0.890600i	$-0.349716\pi$
$53$	350.955	0.909573	0.454787	+	0.890600i	$0.349716\pi$
$54$	295.814	0.745468
$55$	357.082	0.875435
$56$	0	0
$57$	−401.490	−0.932959
$58$	−567.141	−1.28395
$59$	−555.330	−1.22539	−0.612693	−	0.790321i	$-0.709914\pi$
$59$	−555.330	−1.22539	−0.612693	+	0.790321i	$0.709914\pi$
$60$	122.398	0.263359
$61$	443.347	0.930569	0.465284	−	0.885161i	$-0.345952\pi$
$61$	443.347	0.930569	0.465284	+	0.885161i	$0.345952\pi$
$62$	−276.919	−0.567237
$63$	0	0
$64$	64.0000	0.125000
$65$	146.732	0.279998
$66$	547.739	1.02154
$67$	−65.4770	−0.119392	−0.0596961	−	0.998217i	$-0.519013\pi$
$67$	−65.4770	−0.119392	−0.0596961	+	0.998217i	$0.519013\pi$
$68$	−177.002	−0.315657
$69$	111.423	0.194401
$70$	0	0
$71$	728.329	1.21742	0.608710	−	0.793393i	$-0.291687\pi$
$71$	728.329	1.21742	0.608710	+	0.793393i	$0.291687\pi$
$72$	−28.2498	−0.0462399
$73$	−170.417	−0.273230	−0.136615	−	0.990624i	$-0.543622\pi$
$73$	−170.417	−0.273230	−0.136615	+	0.990624i	$0.543622\pi$
$74$	−443.005	−0.695922
$75$	412.278	0.634744
$76$	331.505	0.500345
$77$	0	0
$78$	225.077	0.326730
$79$	−1079.60	−1.53753	−0.768763	−	0.639533i	$-0.779128\pi$
$79$	−1079.60	−1.53753	−0.768763	+	0.639533i	$0.779128\pi$
$80$	−101.062	−0.141239
$81$	−621.187	−0.852109
$82$	590.660	0.795457
$83$	−949.211	−1.25529	−0.627647	−	0.778498i	$-0.715982\pi$
$83$	−949.211	−1.25529	−0.627647	+	0.778498i	$0.715982\pi$
$84$	0	0
$85$	279.505	0.356665
$86$	−521.714	−0.654162
$87$	1373.75	1.69289
$88$	−452.260	−0.547853
$89$	790.169	0.941098	0.470549	−	0.882374i	$-0.344056\pi$
$89$	790.169	0.941098	0.470549	+	0.882374i	$0.344056\pi$
$90$	44.6093	0.0522470
$91$	0	0
$92$	−92.0000	−0.104257
$93$	670.761	0.747900
$94$	−454.933	−0.499179
$95$	−523.479	−0.565346
$96$	−155.023	−0.164812
$97$	1086.53	1.13733	0.568663	−	0.822571i	$-0.307461\pi$
$97$	1086.53	1.13733	0.568663	+	0.822571i	$0.307461\pi$
$98$	0	0
$99$	199.629	0.202661
$100$	−340.412	−0.340412
$101$	351.378	0.346172	0.173086	−	0.984907i	$-0.444626\pi$
$101$	351.378	0.346172	0.173086	+	0.984907i	$0.444626\pi$
$102$	428.740	0.416192
$103$	−382.567	−0.365975	−0.182988	−	0.983115i	$-0.558577\pi$
$103$	−382.567	−0.365975	−0.182988	+	0.983115i	$0.558577\pi$
$104$	−185.843	−0.175225
$105$	0	0
$106$	701.910	0.643165
$107$	705.845	0.637725	0.318863	−	0.947801i	$-0.396699\pi$
$107$	705.845	0.637725	0.318863	+	0.947801i	$0.396699\pi$
$108$	591.629	0.527125
$109$	1073.97	0.943737	0.471869	−	0.881669i	$-0.343580\pi$
$109$	1073.97	0.943737	0.471869	+	0.881669i	$0.343580\pi$
$110$	714.164	0.619026
$111$	1073.06	0.917570
$112$	0	0
$113$	−2317.97	−1.92970	−0.964850	−	0.262801i	$-0.915354\pi$
$113$	−2317.97	−1.92970	−0.964850	+	0.262801i	$0.915354\pi$
$114$	−802.980	−0.659702
$115$	145.277	0.117802
$116$	−1134.28	−0.907892
$117$	82.0314	0.0648189
$118$	−1110.66	−0.866479
$119$	0	0
$120$	244.796	0.186223
$121$	1864.92	1.40114
$122$	886.693	0.658012
$123$	−1430.71	−1.04881
$124$	−553.838	−0.401097
$125$	1327.10	0.949592
$126$	0	0
$127$	−693.827	−0.484781	−0.242390	−	0.970179i	$-0.577931\pi$
$127$	−693.827	−0.484781	−0.242390	+	0.970179i	$0.577931\pi$
$128$	128.000	0.0883883
$129$	1263.71	0.862508
$130$	293.464	0.197988
$131$	2353.85	1.56990	0.784948	−	0.619561i	$-0.212689\pi$
$131$	2353.85	1.56990	0.784948	+	0.619561i	$0.212689\pi$
$132$	1095.48	0.722341
$133$	0	0
$134$	−130.954	−0.0844231
$135$	−934.242	−0.595605
$136$	−354.005	−0.223203
$137$	−147.598	−0.0920448	−0.0460224	−	0.998940i	$-0.514655\pi$
$137$	−147.598	−0.0920448	−0.0460224	+	0.998940i	$0.514655\pi$
$138$	222.845	0.137463
$139$	699.330	0.426736	0.213368	−	0.976972i	$-0.431557\pi$
$139$	699.330	0.426736	0.213368	+	0.976972i	$0.431557\pi$
$140$	0	0
$141$	1101.95	0.658164
$142$	1456.66	0.860846
$143$	1313.27	0.767979
$144$	−56.4996	−0.0326965
$145$	1791.15	1.02584
$146$	−340.834	−0.193203
$147$	0	0
$148$	−886.009	−0.492091
$149$	2041.51	1.12247	0.561233	−	0.827658i	$-0.310327\pi$
$149$	2041.51	1.12247	0.561233	+	0.827658i	$0.310327\pi$
$150$	824.557	0.448832
$151$	−1075.38	−0.579556	−0.289778	−	0.957094i	$-0.593581\pi$
$151$	−1075.38	−0.579556	−0.289778	+	0.957094i	$0.593581\pi$
$152$	663.009	0.353797
$153$	156.259	0.0825671
$154$	0	0
$155$	874.565	0.453205
$156$	450.153	0.231033
$157$	−3773.23	−1.91807	−0.959034	−	0.283292i	$-0.908573\pi$
$157$	−3773.23	−1.91807	−0.959034	+	0.283292i	$0.908573\pi$
$158$	−2159.20	−1.08720
$159$	−1700.19	−0.848010
$160$	−202.125	−0.0998711
$161$	0	0
$162$	−1242.37	−0.602532
$163$	1490.24	0.716103	0.358051	−	0.933702i	$-0.383441\pi$
$163$	1490.24	0.716103	0.358051	+	0.933702i	$0.383441\pi$
$164$	1181.32	0.562473
$165$	−1729.87	−0.816182
$166$	−1898.42	−0.887627
$167$	−1691.40	−0.783741	−0.391871	−	0.920020i	$-0.628172\pi$
$167$	−1691.40	−0.783741	−0.391871	+	0.920020i	$0.628172\pi$
$168$	0	0
$169$	−1657.35	−0.754371
$170$	559.009	0.252200
$171$	−292.654	−0.130876
$172$	−1043.43	−0.462562
$173$	601.968	0.264548	0.132274	−	0.991213i	$-0.457772\pi$
$173$	601.968	0.264548	0.132274	+	0.991213i	$0.457772\pi$
$174$	2747.49	1.19705
$175$	0	0
$176$	−904.520	−0.387391
$177$	2690.27	1.14245
$178$	1580.34	0.665457
$179$	−2474.99	−1.03346	−0.516731	−	0.856148i	$-0.672851\pi$
$179$	−2474.99	−1.03346	−0.516731	+	0.856148i	$0.672851\pi$
$180$	89.2185	0.0369442
$181$	−2966.31	−1.21815	−0.609073	−	0.793114i	$-0.708458\pi$
$181$	−2966.31	−1.21815	−0.609073	+	0.793114i	$0.708458\pi$
$182$	0	0
$183$	−2147.77	−0.867585
$184$	−184.000	−0.0737210
$185$	1399.10	0.556020
$186$	1341.52	0.528845
$187$	2501.60	0.978261
$188$	−909.867	−0.352973
$189$	0	0
$190$	−1046.96	−0.399760
$191$	1128.82	0.427636	0.213818	−	0.976874i	$-0.431410\pi$
$191$	1128.82	0.427636	0.213818	+	0.976874i	$0.431410\pi$
$192$	−310.045	−0.116540
$193$	4338.48	1.61809	0.809043	−	0.587750i	$-0.199986\pi$
$193$	4338.48	1.61809	0.809043	+	0.587750i	$0.199986\pi$
$194$	2173.06	0.804211
$195$	−710.837	−0.261047
$196$	0	0
$197$	2826.56	1.02225	0.511127	−	0.859505i	$-0.329228\pi$
$197$	2826.56	1.02225	0.511127	+	0.859505i	$0.329228\pi$
$198$	399.258	0.143303
$199$	1480.92	0.527534	0.263767	−	0.964586i	$-0.415035\pi$
$199$	1480.92	0.527534	0.263767	+	0.964586i	$0.415035\pi$
$200$	−680.825	−0.240708
$201$	317.200	0.111311
$202$	702.756	0.244781
$203$	0	0
$204$	857.481	0.294292
$205$	−1865.42	−0.635545
$206$	−765.134	−0.258784
$207$	81.2182	0.0272708
$208$	−371.685	−0.123902
$209$	−4685.20	−1.55063
$210$	0	0
$211$	−2283.99	−0.745197	−0.372599	−	0.927993i	$-0.621533\pi$
$211$	−2283.99	−0.745197	−0.372599	+	0.927993i	$0.621533\pi$
$212$	1403.82	0.454787
$213$	−3528.36	−1.13502
$214$	1411.69	0.450940
$215$	1647.68	0.522655
$216$	1183.26	0.372734
$217$	0	0
$218$	2147.93	0.667323
$219$	825.577	0.254737
$220$	1428.33	0.437717
$221$	1027.96	0.312886
$222$	2146.12	0.648820
$223$	−2292.56	−0.688436	−0.344218	−	0.938890i	$-0.611856\pi$
$223$	−2292.56	−0.688436	−0.344218	+	0.938890i	$0.611856\pi$
$224$	0	0
$225$	300.518	0.0890424
$226$	−4635.94	−1.36450
$227$	1308.74	0.382661	0.191330	−	0.981526i	$-0.438720\pi$
$227$	1308.74	0.382661	0.191330	+	0.981526i	$0.438720\pi$
$228$	−1605.96	−0.466480
$229$	3626.20	1.04640	0.523200	−	0.852210i	$-0.324738\pi$
$229$	3626.20	1.04640	0.523200	+	0.852210i	$0.324738\pi$
$230$	290.554	0.0832982
$231$	0	0
$232$	−2268.56	−0.641977
$233$	5312.41	1.49368	0.746840	−	0.665003i	$-0.231570\pi$
$233$	5312.41	1.49368	0.746840	+	0.665003i	$0.231570\pi$
$234$	164.063	0.0458339
$235$	1436.77	0.398828
$236$	−2221.32	−0.612693
$237$	5230.08	1.43346
$238$	0	0
$239$	6150.60	1.66464	0.832321	−	0.554294i	$-0.187011\pi$
$239$	6150.60	1.66464	0.832321	+	0.554294i	$0.187011\pi$
$240$	489.593	0.131679
$241$	−7422.12	−1.98382	−0.991911	−	0.126938i	$-0.959485\pi$
$241$	−7422.12	−1.98382	−0.991911	+	0.126938i	$0.959485\pi$
$242$	3729.84	0.990758
$243$	−984.179	−0.259815
$244$	1773.39	0.465284
$245$	0	0
$246$	−2861.43	−0.741617
$247$	−1925.24	−0.495952
$248$	−1107.68	−0.283619
$249$	4598.41	1.17033
$250$	2654.19	0.671463
$251$	6503.81	1.63553	0.817763	−	0.575556i	$-0.195214\pi$
$251$	6503.81	1.63553	0.817763	+	0.575556i	$0.195214\pi$
$252$	0	0
$253$	1300.25	0.323106
$254$	−1387.65	−0.342792
$255$	−1354.05	−0.332525
$256$	256.000	0.0625000
$257$	6174.34	1.49862	0.749309	−	0.662220i	$-0.230386\pi$
$257$	6174.34	1.49862	0.749309	+	0.662220i	$0.230386\pi$
$258$	2527.42	0.609886
$259$	0	0
$260$	586.928	0.139999
$261$	1001.35	0.237479
$262$	4707.69	1.11008
$263$	994.102	0.233076	0.116538	−	0.993186i	$-0.462820\pi$
$263$	994.102	0.233076	0.116538	+	0.993186i	$0.462820\pi$
$264$	2190.95	0.510772
$265$	−2216.77	−0.513869
$266$	0	0
$267$	−3827.94	−0.877401
$268$	−261.908	−0.0596961
$269$	1416.70	0.321106	0.160553	−	0.987027i	$-0.448672\pi$
$269$	1416.70	0.321106	0.160553	+	0.987027i	$0.448672\pi$
$270$	−1868.48	−0.421157
$271$	3881.51	0.870054	0.435027	−	0.900417i	$-0.356739\pi$
$271$	3881.51	0.870054	0.435027	+	0.900417i	$0.356739\pi$
$272$	−708.009	−0.157829
$273$	0	0
$274$	−295.196	−0.0650855
$275$	4811.09	1.05498
$276$	445.690	0.0972007
$277$	−718.681	−0.155889	−0.0779446	−	0.996958i	$-0.524836\pi$
$277$	−718.681	−0.155889	−0.0779446	+	0.996958i	$0.524836\pi$
$278$	1398.66	0.301748
$279$	488.931	0.104916
$280$	0	0
$281$	−2879.77	−0.611361	−0.305681	−	0.952134i	$-0.598884\pi$
$281$	−2879.77	−0.611361	−0.305681	+	0.952134i	$0.598884\pi$
$282$	2203.91	0.465393
$283$	7291.08	1.53148	0.765742	−	0.643148i	$-0.222372\pi$
$283$	7291.08	1.53148	0.765742	+	0.643148i	$0.222372\pi$
$284$	2913.32	0.608710
$285$	2535.97	0.527081
$286$	2626.53	0.543043
$287$	0	0
$288$	−112.999	−0.0231199
$289$	−2954.89	−0.601442
$290$	3582.29	0.725377
$291$	−5263.66	−1.06035
$292$	−681.667	−0.136615
$293$	5269.85	1.05074	0.525372	−	0.850873i	$-0.323926\pi$
$293$	5269.85	1.05074	0.525372	+	0.850873i	$0.323926\pi$
$294$	0	0
$295$	3507.69	0.692290
$296$	−1772.02	−0.347961
$297$	−8361.56	−1.63363
$298$	4083.03	0.793703
$299$	534.297	0.103342
$300$	1649.11	0.317372
$301$	0	0
$302$	−2150.75	−0.409808
$303$	−1702.24	−0.322742
$304$	1326.02	0.250172
$305$	−2800.35	−0.525731
$306$	312.518	0.0583838
$307$	10384.3	1.93050	0.965250	−	0.261329i	$-0.0841609\pi$
$307$	10384.3	1.93050	0.965250	+	0.261329i	$0.0841609\pi$
$308$	0	0
$309$	1853.33	0.341205
$310$	1749.13	0.320464
$311$	4311.74	0.786162	0.393081	−	0.919504i	$-0.371409\pi$
$311$	4311.74	0.786162	0.393081	+	0.919504i	$0.371409\pi$
$312$	900.306	0.163365
$313$	−3490.76	−0.630382	−0.315191	−	0.949028i	$-0.602069\pi$
$313$	−3490.76	−0.630382	−0.315191	+	0.949028i	$0.602069\pi$
$314$	−7546.46	−1.35628
$315$	0	0
$316$	−4318.40	−0.768763
$317$	2113.37	0.374444	0.187222	−	0.982318i	$-0.440052\pi$
$317$	2113.37	0.374444	0.187222	+	0.982318i	$0.440052\pi$
$318$	−3400.37	−0.599634
$319$	16031.0	2.81367
$320$	−404.250	−0.0706195
$321$	−3419.44	−0.594562
$322$	0	0
$323$	−3667.32	−0.631750
$324$	−2484.75	−0.426054
$325$	1976.97	0.337423
$326$	2980.48	0.506361
$327$	−5202.79	−0.879862
$328$	2362.64	0.397728
$329$	0	0
$330$	−3459.74	−0.577128
$331$	−3761.19	−0.624573	−0.312287	−	0.949988i	$-0.601095\pi$
$331$	−3761.19	−0.624573	−0.312287	+	0.949988i	$0.601095\pi$
$332$	−3796.84	−0.627647
$333$	782.174	0.128717
$334$	−3382.81	−0.554189
$335$	413.579	0.0674514
$336$	0	0
$337$	−4773.47	−0.771595	−0.385797	−	0.922584i	$-0.626074\pi$
$337$	−4773.47	−0.771595	−0.385797	+	0.922584i	$0.626074\pi$
$338$	−3314.71	−0.533421
$339$	11229.3	1.79909
$340$	1118.02	0.178332
$341$	7827.45	1.24305
$342$	−585.309	−0.0925435
$343$	0	0
$344$	−2086.86	−0.327081
$345$	−703.790	−0.109828
$346$	1203.94	0.187063
$347$	6185.18	0.956882	0.478441	−	0.878120i	$-0.341202\pi$
$347$	6185.18	0.956882	0.478441	+	0.878120i	$0.341202\pi$
$348$	5494.98	0.846443
$349$	−2878.98	−0.441571	−0.220786	−	0.975322i	$-0.570862\pi$
$349$	−2878.98	−0.441571	−0.220786	+	0.975322i	$0.570862\pi$
$350$	0	0
$351$	−3435.93	−0.522497
$352$	−1809.04	−0.273927
$353$	−1241.11	−0.187133	−0.0935664	−	0.995613i	$-0.529827\pi$
$353$	−1241.11	−0.187133	−0.0935664	+	0.995613i	$0.529827\pi$
$354$	5380.55	0.807833
$355$	−4600.42	−0.687789
$356$	3160.68	0.470549
$357$	0	0
$358$	−4949.99	−0.730768
$359$	6971.50	1.02491	0.512453	−	0.858715i	$-0.328737\pi$
$359$	6971.50	1.02491	0.512453	+	0.858715i	$0.328737\pi$
$360$	178.437	0.0261235
$361$	9.46054	0.00137929
$362$	−5932.63	−0.861359
$363$	−9034.54	−1.30631
$364$	0	0
$365$	1076.42	0.154363
$366$	−4295.55	−0.613475
$367$	650.459	0.0925169	0.0462585	−	0.998930i	$-0.485270\pi$
$367$	650.459	0.0925169	0.0462585	+	0.998930i	$0.485270\pi$
$368$	−368.000	−0.0521286
$369$	−1042.88	−0.147127
$370$	2798.20	0.393166
$371$	0	0
$372$	2683.04	0.373950
$373$	12553.3	1.74259	0.871295	−	0.490760i	$-0.163281\pi$
$373$	12553.3	1.74259	0.871295	+	0.490760i	$0.163281\pi$
$374$	5003.19	0.691735
$375$	−6429.06	−0.885320
$376$	−1819.73	−0.249589
$377$	6587.43	0.899921
$378$	0	0
$379$	−6923.71	−0.938383	−0.469191	−	0.883097i	$-0.655455\pi$
$379$	−6923.71	−0.938383	−0.469191	+	0.883097i	$0.655455\pi$
$380$	−2093.92	−0.282673
$381$	3361.21	0.451969
$382$	2257.64	0.302384
$383$	3031.64	0.404464	0.202232	−	0.979338i	$-0.435180\pi$
$383$	3031.64	0.404464	0.202232	+	0.979338i	$0.435180\pi$
$384$	−620.091	−0.0824059
$385$	0	0
$386$	8676.96	1.14416
$387$	921.145	0.120993
$388$	4346.13	0.568663
$389$	−9306.37	−1.21299	−0.606493	−	0.795089i	$-0.707424\pi$
$389$	−9306.37	−1.21299	−0.606493	+	0.795089i	$0.707424\pi$
$390$	−1421.67	−0.184588
$391$	1017.76	0.131638
$392$	0	0
$393$	−11403.1	−1.46364
$394$	5653.12	0.722843
$395$	6819.19	0.868635
$396$	798.515	0.101331
$397$	−11227.5	−1.41937	−0.709687	−	0.704517i	$-0.751164\pi$
$397$	−11227.5	−1.41937	−0.709687	+	0.704517i	$0.751164\pi$
$398$	2961.83	0.373023
$399$	0	0
$400$	−1361.65	−0.170206
$401$	−15951.2	−1.98645	−0.993225	−	0.116209i	$-0.962926\pi$
$401$	−15951.2	−1.98645	−0.993225	+	0.116209i	$0.962926\pi$
$402$	634.401	0.0787090
$403$	3216.46	0.397576
$404$	1405.51	0.173086
$405$	3923.67	0.481404
$406$	0	0
$407$	12522.1	1.52505
$408$	1714.96	0.208096
$409$	−5785.98	−0.699507	−0.349754	−	0.936842i	$-0.613735\pi$
$409$	−5785.98	−0.699507	−0.349754	+	0.936842i	$0.613735\pi$
$410$	−3730.84	−0.449398
$411$	715.032	0.0858149
$412$	−1530.27	−0.182988
$413$	0	0
$414$	162.436	0.0192834
$415$	5995.60	0.709186
$416$	−743.370	−0.0876123
$417$	−3387.87	−0.397853
$418$	−9370.39	−1.09646
$419$	−261.906	−0.0305368	−0.0152684	−	0.999883i	$-0.504860\pi$
$419$	−261.906	−0.0305368	−0.0152684	+	0.999883i	$0.504860\pi$
$420$	0	0
$421$	7111.00	0.823204	0.411602	−	0.911364i	$-0.364969\pi$
$421$	7111.00	0.823204	0.411602	+	0.911364i	$0.364969\pi$
$422$	−4567.99	−0.526934
$423$	803.236	0.0923278
$424$	2807.64	0.321583
$425$	3765.86	0.429814
$426$	−7056.72	−0.802581
$427$	0	0
$428$	2823.38	0.318863
$429$	−6362.07	−0.715999
$430$	3295.36	0.369573
$431$	−2615.18	−0.292271	−0.146136	−	0.989265i	$-0.546684\pi$
$431$	−2615.18	−0.292271	−0.146136	+	0.989265i	$0.546684\pi$
$432$	2366.52	0.263563
$433$	−4709.78	−0.522719	−0.261360	−	0.965242i	$-0.584171\pi$
$433$	−4709.78	−0.522719	−0.261360	+	0.965242i	$0.584171\pi$
$434$	0	0
$435$	−8677.13	−0.956406
$436$	4295.87	0.471869
$437$	−1906.15	−0.208658
$438$	1651.15	0.180126
$439$	−1969.59	−0.214131	−0.107065	−	0.994252i	$-0.534145\pi$
$439$	−1969.59	−0.214131	−0.107065	+	0.994252i	$0.534145\pi$
$440$	2856.66	0.309513
$441$	0	0
$442$	2055.91	0.221244
$443$	5419.90	0.581281	0.290640	−	0.956832i	$-0.406132\pi$
$443$	5419.90	0.581281	0.290640	+	0.956832i	$0.406132\pi$
$444$	4292.24	0.458785
$445$	−4991.03	−0.531679
$446$	−4585.12	−0.486798
$447$	−9890.03	−1.04649
$448$	0	0
$449$	−7456.34	−0.783712	−0.391856	−	0.920027i	$-0.628167\pi$
$449$	−7456.34	−0.783712	−0.391856	+	0.920027i	$0.628167\pi$
$450$	601.036	0.0629625
$451$	−16695.7	−1.74317
$452$	−9271.87	−0.964850
$453$	5209.62	0.540330
$454$	2617.47	0.270582
$455$	0	0
$456$	−3211.92	−0.329851
$457$	−12489.8	−1.27844	−0.639222	−	0.769022i	$-0.720743\pi$
$457$	−12489.8	−1.27844	−0.639222	+	0.769022i	$0.720743\pi$
$458$	7252.39	0.739917
$459$	−6544.98	−0.665563
$460$	581.109	0.0589008
$461$	−12091.4	−1.22159	−0.610796	−	0.791788i	$-0.709151\pi$
$461$	−12091.4	−1.22159	−0.610796	+	0.791788i	$0.709151\pi$
$462$	0	0
$463$	4380.17	0.439662	0.219831	−	0.975538i	$-0.429449\pi$
$463$	4380.17	0.439662	0.219831	+	0.975538i	$0.429449\pi$
$464$	−4537.13	−0.453946
$465$	−4236.79	−0.422530
$466$	10624.8	1.05619
$467$	6496.45	0.643726	0.321863	−	0.946786i	$-0.395691\pi$
$467$	6496.45	0.643726	0.321863	+	0.946786i	$0.395691\pi$
$468$	328.126	0.0324094
$469$	0	0
$470$	2873.54	0.282014
$471$	18279.3	1.78825
$472$	−4442.64	−0.433240
$473$	14746.9	1.43354
$474$	10460.2	1.01361
$475$	−7053.02	−0.681294
$476$	0	0
$477$	−1239.30	−0.118959
$478$	12301.2	1.17708
$479$	−1408.88	−0.134391	−0.0671956	−	0.997740i	$-0.521405\pi$
$479$	−1408.88	−0.134391	−0.0671956	+	0.997740i	$0.521405\pi$
$480$	979.185	0.0931115
$481$	5145.57	0.487771
$482$	−14844.2	−1.40277
$483$	0	0
$484$	7459.69	0.700572
$485$	−6862.97	−0.642539
$486$	−1968.36	−0.183717
$487$	6880.05	0.640174	0.320087	−	0.947388i	$-0.396288\pi$
$487$	6880.05	0.640174	0.320087	+	0.947388i	$0.396288\pi$
$488$	3546.77	0.329006
$489$	−7219.41	−0.667634
$490$	0	0
$491$	−9109.25	−0.837260	−0.418630	−	0.908157i	$-0.637490\pi$
$491$	−9109.25	−0.837260	−0.418630	+	0.908157i	$0.637490\pi$
$492$	−5722.85	−0.524403
$493$	12548.2	1.14633
$494$	−3850.48	−0.350691
$495$	−1260.94	−0.114495
$496$	−2215.35	−0.200549
$497$	0	0
$498$	9196.83	0.827549
$499$	−5933.57	−0.532311	−0.266155	−	0.963930i	$-0.585753\pi$
$499$	−5933.57	−0.532311	−0.266155	+	0.963930i	$0.585753\pi$
$500$	5308.38	0.474796
$501$	8193.94	0.730695
$502$	13007.6	1.15649
$503$	−4624.92	−0.409970	−0.204985	−	0.978765i	$-0.565715\pi$
$503$	−4624.92	−0.409970	−0.204985	+	0.978765i	$0.565715\pi$
$504$	0	0
$505$	−2219.44	−0.195572
$506$	2600.49	0.228470
$507$	8028.98	0.703312
$508$	−2775.31	−0.242390
$509$	−17185.7	−1.49654	−0.748272	−	0.663392i	$-0.769116\pi$
$509$	−17185.7	−1.49654	−0.748272	+	0.663392i	$0.769116\pi$
$510$	−2708.10	−0.235130
$511$	0	0
$512$	512.000	0.0441942
$513$	12258.0	1.05498
$514$	12348.7	1.05968
$515$	2416.45	0.206760
$516$	5054.85	0.431254
$517$	12859.3	1.09391
$518$	0	0
$519$	−2916.21	−0.246642
$520$	1173.86	0.0989942
$521$	10642.4	0.894921	0.447461	−	0.894304i	$-0.352328\pi$
$521$	10642.4	0.894921	0.447461	+	0.894304i	$0.352328\pi$
$522$	2002.70	0.167923
$523$	8829.91	0.738251	0.369125	−	0.929380i	$-0.379657\pi$
$523$	8829.91	0.738251	0.369125	+	0.929380i	$0.379657\pi$
$524$	9415.38	0.784948
$525$	0	0
$526$	1988.20	0.164810
$527$	6126.91	0.506437
$528$	4381.91	0.361171
$529$	529.000	0.0434783
$530$	−4433.55	−0.363360
$531$	1960.99	0.160264
$532$	0	0
$533$	−6860.60	−0.557534
$534$	−7655.88	−0.620417
$535$	−4458.40	−0.360287
$536$	−523.816	−0.0422115
$537$	11990.0	0.963513
$538$	2833.39	0.227056
$539$	0	0
$540$	−3736.97	−0.297803
$541$	21851.0	1.73650	0.868250	−	0.496128i	$-0.165245\pi$
$541$	21851.0	1.73650	0.868250	+	0.496128i	$0.165245\pi$
$542$	7763.01	0.615221
$543$	14370.2	1.13570
$544$	−1416.02	−0.111602
$545$	−6783.60	−0.533170
$546$	0	0
$547$	−20378.5	−1.59291	−0.796455	−	0.604698i	$-0.793294\pi$
$547$	−20378.5	−1.59291	−0.796455	+	0.604698i	$0.793294\pi$
$548$	−590.392	−0.0460224
$549$	−1565.56	−0.121705
$550$	9622.18	0.745984
$551$	−23501.2	−1.81704
$552$	891.380	0.0687313
$553$	0	0
$554$	−1437.36	−0.110230
$555$	−6777.87	−0.518387
$556$	2797.32	0.213368
$557$	−19264.1	−1.46543	−0.732717	−	0.680534i	$-0.761748\pi$
$557$	−19264.1	−1.46543	−0.732717	+	0.680534i	$0.761748\pi$
$558$	977.862	0.0741868
$559$	6059.79	0.458501
$560$	0	0
$561$	−12118.9	−0.912049
$562$	−5759.53	−0.432298
$563$	14546.1	1.08889	0.544444	−	0.838797i	$-0.316741\pi$
$563$	14546.1	1.08889	0.544444	+	0.838797i	$0.316741\pi$
$564$	4407.81	0.329082
$565$	14641.2	1.09020
$566$	14582.2	1.08292
$567$	0	0
$568$	5826.64	0.430423
$569$	3350.10	0.246825	0.123413	−	0.992355i	$-0.460616\pi$
$569$	3350.10	0.246825	0.123413	+	0.992355i	$0.460616\pi$
$570$	5071.95	0.372703
$571$	1863.65	0.136588	0.0682938	−	0.997665i	$-0.478244\pi$
$571$	1863.65	0.136588	0.0682938	+	0.997665i	$0.478244\pi$
$572$	5253.07	0.383989
$573$	−5468.52	−0.398692
$574$	0	0
$575$	1957.37	0.141962
$576$	−225.998	−0.0163483
$577$	−22307.6	−1.60949	−0.804746	−	0.593619i	$-0.797699\pi$
$577$	−22307.6	−1.60949	−0.804746	+	0.593619i	$0.797699\pi$
$578$	−5909.77	−0.425284
$579$	−21017.6	−1.50857
$580$	7164.58	0.512919
$581$	0	0
$582$	−10527.3	−0.749779
$583$	−19840.4	−1.40944
$584$	−1363.33	−0.0966014
$585$	−518.144	−0.0366198
$586$	10539.7	0.742988
$587$	−1239.13	−0.0871287	−0.0435643	−	0.999051i	$-0.513871\pi$
$587$	−1239.13	−0.0871287	−0.0435643	+	0.999051i	$0.513871\pi$
$588$	0	0
$589$	−11475.0	−0.802748
$590$	7015.38	0.489523
$591$	−13693.1	−0.953064
$592$	−3544.04	−0.246046
$593$	13012.3	0.901101	0.450550	−	0.892751i	$-0.351228\pi$
$593$	13012.3	0.901101	0.450550	+	0.892751i	$0.351228\pi$
$594$	−16723.1	−1.15515
$595$	0	0
$596$	8166.05	0.561233
$597$	−7174.23	−0.491829
$598$	1068.59	0.0730737
$599$	−5387.70	−0.367505	−0.183752	−	0.982973i	$-0.558824\pi$
$599$	−5387.70	−0.367505	−0.183752	+	0.982973i	$0.558824\pi$
$600$	3298.23	0.224416
$601$	−19623.8	−1.33190	−0.665949	−	0.745998i	$-0.731973\pi$
$601$	−19623.8	−1.33190	−0.665949	+	0.745998i	$0.731973\pi$
$602$	0	0
$603$	231.214	0.0156148
$604$	−4301.51	−0.289778
$605$	−11779.6	−0.791585
$606$	−3404.47	−0.228213
$607$	−7417.23	−0.495974	−0.247987	−	0.968763i	$-0.579769\pi$
$607$	−7417.23	−0.495974	−0.247987	+	0.968763i	$0.579769\pi$
$608$	2652.04	0.176899
$609$	0	0
$610$	−5600.71	−0.371748
$611$	5284.12	0.349874
$612$	625.035	0.0412836
$613$	−292.381	−0.0192645	−0.00963225	−	0.999954i	$-0.503066\pi$
$613$	−292.381	−0.0192645	−0.00963225	+	0.999954i	$0.503066\pi$
$614$	20768.6	1.36507
$615$	9036.96	0.592529
$616$	0	0
$617$	−4411.01	−0.287813	−0.143906	−	0.989591i	$-0.545966\pi$
$617$	−4411.01	−0.287813	−0.143906	+	0.989591i	$0.545966\pi$
$618$	3706.66	0.241268
$619$	−15225.2	−0.988613	−0.494307	−	0.869288i	$-0.664578\pi$
$619$	−15225.2	−0.988613	−0.494307	+	0.869288i	$0.664578\pi$
$620$	3498.26	0.226602
$621$	−3401.87	−0.219826
$622$	8623.48	0.555900
$623$	0	0
$624$	1800.61	0.115516
$625$	2255.42	0.144347
$626$	−6981.53	−0.445747
$627$	22697.2	1.44568
$628$	−15092.9	−0.959034
$629$	9801.61	0.621329
$630$	0	0
$631$	17188.8	1.08443	0.542216	−	0.840239i	$-0.317586\pi$
$631$	17188.8	1.08443	0.542216	+	0.840239i	$0.317586\pi$
$632$	−8636.81	−0.543598
$633$	11064.7	0.694759
$634$	4226.74	0.264772
$635$	4382.49	0.273880
$636$	−6800.75	−0.424005
$637$	0	0
$638$	32061.9	1.98957
$639$	−2571.89	−0.159222
$640$	−808.499	−0.0499355
$641$	24494.6	1.50933	0.754663	−	0.656113i	$-0.227800\pi$
$641$	24494.6	1.50933	0.754663	+	0.656113i	$0.227800\pi$
$642$	−6838.87	−0.420419
$643$	16904.2	1.03676	0.518381	−	0.855150i	$-0.326535\pi$
$643$	16904.2	1.03676	0.518381	+	0.855150i	$0.326535\pi$
$644$	0	0
$645$	−7982.11	−0.487279
$646$	−7334.64	−0.446714
$647$	11895.5	0.722813	0.361406	−	0.932408i	$-0.382297\pi$
$647$	11895.5	0.722813	0.361406	+	0.932408i	$0.382297\pi$
$648$	−4969.50	−0.301266
$649$	31394.2	1.89881
$650$	3953.94	0.238594
$651$	0	0
$652$	5960.97	0.358051
$653$	−19053.6	−1.14184	−0.570922	−	0.821004i	$-0.693414\pi$
$653$	−19053.6	−1.14184	−0.570922	+	0.821004i	$0.693414\pi$
$654$	−10405.6	−0.622156
$655$	−14867.8	−0.886923
$656$	4725.28	0.281236
$657$	601.780	0.0357347
$658$	0	0
$659$	23676.8	1.39957	0.699786	−	0.714352i	$-0.253279\pi$
$659$	23676.8	1.39957	0.699786	+	0.714352i	$0.253279\pi$
$660$	−6919.47	−0.408091
$661$	7506.93	0.441733	0.220867	−	0.975304i	$-0.429111\pi$
$661$	7506.93	0.441733	0.220867	+	0.975304i	$0.429111\pi$
$662$	−7522.38	−0.441640
$663$	−4979.89	−0.291708
$664$	−7593.69	−0.443814
$665$	0	0
$666$	1564.35	0.0910169
$667$	6522.12	0.378617
$668$	−6765.62	−0.391871
$669$	11106.2	0.641840
$670$	827.158	0.0476953
$671$	−25063.5	−1.44197
$672$	0	0
$673$	−17638.7	−1.01028	−0.505142	−	0.863036i	$-0.668560\pi$
$673$	−17638.7	−1.01028	−0.505142	+	0.863036i	$0.668560\pi$
$674$	−9546.94	−0.545600
$675$	−12587.4	−0.717760
$676$	−6629.41	−0.377185
$677$	−5989.95	−0.340048	−0.170024	−	0.985440i	$-0.554384\pi$
$677$	−5989.95	−0.340048	−0.170024	+	0.985440i	$0.554384\pi$
$678$	22458.6	1.27215
$679$	0	0
$680$	2236.04	0.126100
$681$	−6340.12	−0.356761
$682$	15654.9	0.878970
$683$	−25823.5	−1.44672	−0.723360	−	0.690471i	$-0.757403\pi$
$683$	−25823.5	−1.44672	−0.723360	+	0.690471i	$0.757403\pi$
$684$	−1170.62	−0.0654381
$685$	932.288	0.0520013
$686$	0	0
$687$	−17567.0	−0.975577
$688$	−4173.71	−0.231281
$689$	−8152.79	−0.450793
$690$	−1407.58	−0.0776603
$691$	−23226.7	−1.27871	−0.639353	−	0.768913i	$-0.720798\pi$
$691$	−23226.7	−1.27871	−0.639353	+	0.768913i	$0.720798\pi$
$692$	2407.87	0.132274
$693$	0	0
$694$	12370.4	0.676617
$695$	−4417.25	−0.241087
$696$	10990.0	0.598525
$697$	−13068.5	−0.710194
$698$	−5757.97	−0.312238
$699$	−25735.8	−1.39258
$700$	0	0
$701$	5370.34	0.289351	0.144675	−	0.989479i	$-0.453786\pi$
$701$	5370.34	0.289351	0.144675	+	0.989479i	$0.453786\pi$
$702$	−6871.86	−0.369461
$703$	−18357.3	−0.984861
$704$	−3618.08	−0.193695
$705$	−6960.38	−0.371834
$706$	−2482.23	−0.132323
$707$	0	0
$708$	10761.1	0.571224
$709$	18320.9	0.970460	0.485230	−	0.874387i	$-0.338736\pi$
$709$	18320.9	0.970460	0.485230	+	0.874387i	$0.338736\pi$
$710$	−9200.84	−0.486340
$711$	3812.31	0.201087
$712$	6321.35	0.332729
$713$	3184.57	0.167269
$714$	0	0
$715$	−8295.12	−0.433874
$716$	−9899.97	−0.516731
$717$	−29796.3	−1.55197
$718$	13943.0	0.724719
$719$	20962.5	1.08730	0.543649	−	0.839313i	$-0.317042\pi$
$719$	20962.5	1.08730	0.543649	+	0.839313i	$0.317042\pi$
$720$	356.874	0.0184721
$721$	0	0
$722$	18.9211	0.000975304 0
$723$	35956.2	1.84955
$724$	−11865.3	−0.609073
$725$	24132.7	1.23623
$726$	−18069.1	−0.923700
$727$	−7257.65	−0.370250	−0.185125	−	0.982715i	$-0.559269\pi$
$727$	−7257.65	−0.370250	−0.185125	+	0.982715i	$0.559269\pi$
$728$	0	0
$729$	21539.9	1.09434
$730$	2152.84	0.109151
$731$	11543.1	0.584044
$732$	−8591.10	−0.433792
$733$	−1856.09	−0.0935285	−0.0467642	−	0.998906i	$-0.514891\pi$
$733$	−1856.09	−0.0935285	−0.0467642	+	0.998906i	$0.514891\pi$
$734$	1300.92	0.0654193
$735$	0	0
$736$	−736.000	−0.0368605
$737$	3701.57	0.185006
$738$	−2085.75	−0.104035
$739$	−29285.2	−1.45774	−0.728871	−	0.684651i	$-0.759955\pi$
$739$	−29285.2	−1.45774	−0.728871	+	0.684651i	$0.759955\pi$
$740$	5596.39	0.278010
$741$	9326.74	0.462384
$742$	0	0
$743$	−27200.5	−1.34305	−0.671527	−	0.740980i	$-0.734361\pi$
$743$	−27200.5	−1.34305	−0.671527	+	0.740980i	$0.734361\pi$
$744$	5366.09	0.264422
$745$	−12895.0	−0.634144
$746$	25106.6	1.23220
$747$	3351.88	0.164175
$748$	10006.4	0.489131
$749$	0	0
$750$	−12858.1	−0.626016
$751$	−22318.3	−1.08443	−0.542215	−	0.840239i	$-0.682414\pi$
$751$	−22318.3	−1.08443	−0.542215	+	0.840239i	$0.682414\pi$
$752$	−3639.47	−0.176486
$753$	−31507.4	−1.52483
$754$	13174.9	0.636340
$755$	6792.52	0.327424
$756$	0	0
$757$	−13446.1	−0.645585	−0.322792	−	0.946470i	$-0.604622\pi$
$757$	−13446.1	−0.645585	−0.322792	+	0.946470i	$0.604622\pi$
$758$	−13847.4	−0.663537
$759$	−6298.99	−0.301237
$760$	−4187.83	−0.199880
$761$	−35160.1	−1.67484	−0.837419	−	0.546561i	$-0.815937\pi$
$761$	−35160.1	−1.67484	−0.837419	+	0.546561i	$0.815937\pi$
$762$	6722.43	0.319590
$763$	0	0
$764$	4515.28	0.213818
$765$	−986.993	−0.0466468
$766$	6063.29	0.285999
$767$	12900.5	0.607314
$768$	−1240.18	−0.0582698
$769$	−3592.07	−0.168444	−0.0842219	−	0.996447i	$-0.526840\pi$
$769$	−3592.07	−0.168444	−0.0842219	+	0.996447i	$0.526840\pi$
$770$	0	0
$771$	−29911.4	−1.39719
$772$	17353.9	0.809043
$773$	−37891.1	−1.76306	−0.881532	−	0.472125i	$-0.843487\pi$
$773$	−37891.1	−1.76306	−0.881532	+	0.472125i	$0.843487\pi$
$774$	1842.29	0.0855552
$775$	11783.3	0.546154
$776$	8692.25	0.402105
$777$	0	0
$778$	−18612.7	−0.857711
$779$	24475.8	1.12572
$780$	−2843.35	−0.130523
$781$	−41174.3	−1.88647
$782$	2035.53	0.0930822
$783$	−41942.1	−1.91429
$784$	0	0
$785$	23833.2	1.08362
$786$	−22806.2	−1.03495
$787$	6333.43	0.286865	0.143432	−	0.989660i	$-0.454186\pi$
$787$	6333.43	0.286865	0.143432	+	0.989660i	$0.454186\pi$
$788$	11306.2	0.511127
$789$	−4815.89	−0.217301
$790$	13638.4	0.614218
$791$	0	0
$792$	1597.03	0.0716515
$793$	−10299.1	−0.461199
$794$	−22455.0	−1.00365
$795$	10739.1	0.479088
$796$	5923.66	0.263767
$797$	−27020.4	−1.20089	−0.600447	−	0.799665i	$-0.705010\pi$
$797$	−27020.4	−1.20089	−0.600447	+	0.799665i	$0.705010\pi$
$798$	0	0
$799$	10065.5	0.445673
$800$	−2723.30	−0.120354
$801$	−2790.26	−0.123083
$802$	−31902.5	−1.40463
$803$	9634.09	0.423387
$804$	1268.80	0.0556557
$805$	0	0
$806$	6432.91	0.281129
$807$	−6863.12	−0.299372
$808$	2811.02	0.122390
$809$	14330.7	0.622795	0.311397	−	0.950280i	$-0.399203\pi$
$809$	14330.7	0.622795	0.311397	+	0.950280i	$0.399203\pi$
$810$	7847.34	0.340404
$811$	−18949.5	−0.820478	−0.410239	−	0.911978i	$-0.634555\pi$
$811$	−18949.5	−0.820478	−0.410239	+	0.911978i	$0.634555\pi$
$812$	0	0
$813$	−18803.8	−0.811166
$814$	25044.2	1.07837
$815$	−9412.97	−0.404567
$816$	3429.92	0.147146
$817$	−21618.8	−0.925762
$818$	−11572.0	−0.494626
$819$	0	0
$820$	−7461.69	−0.317772
$821$	1447.98	0.0615529	0.0307765	−	0.999526i	$-0.490202\pi$
$821$	1447.98	0.0615529	0.0307765	+	0.999526i	$0.490202\pi$
$822$	1430.06	0.0606803
$823$	−21854.2	−0.925623	−0.462812	−	0.886457i	$-0.653159\pi$
$823$	−21854.2	−0.925623	−0.462812	+	0.886457i	$0.653159\pi$
$824$	−3060.54	−0.129392
$825$	−23307.1	−0.983575
$826$	0	0
$827$	−19235.3	−0.808801	−0.404400	−	0.914582i	$-0.632520\pi$
$827$	−19235.3	−0.808801	−0.404400	+	0.914582i	$0.632520\pi$
$828$	324.873	0.0136354
$829$	−27546.8	−1.15409	−0.577044	−	0.816713i	$-0.695794\pi$
$829$	−27546.8	−1.15409	−0.577044	+	0.816713i	$0.695794\pi$
$830$	11991.2	0.501470
$831$	3481.62	0.145338
$832$	−1486.74	−0.0619512
$833$	0	0
$834$	−6775.75	−0.281325
$835$	10683.6	0.442780
$836$	−18740.8	−0.775315
$837$	−20479.1	−0.845714
$838$	−523.812	−0.0215928
$839$	32163.6	1.32349	0.661747	−	0.749727i	$-0.269815\pi$
$839$	32163.6	1.32349	0.661747	+	0.749727i	$0.269815\pi$
$840$	0	0
$841$	56023.3	2.29707
$842$	14222.0	0.582093
$843$	13950.9	0.569982
$844$	−9135.97	−0.372599
$845$	10468.5	0.426186
$846$	1606.47	0.0652856
$847$	0	0
$848$	5615.28	0.227393
$849$	−35321.4	−1.42783
$850$	7531.72	0.303925
$851$	5094.55	0.205216
$852$	−14113.4	−0.567510
$853$	28761.0	1.15446	0.577231	−	0.816581i	$-0.304133\pi$
$853$	28761.0	1.15446	0.577231	+	0.816581i	$0.304133\pi$
$854$	0	0
$855$	1848.52	0.0739393
$856$	5646.76	0.225470
$857$	−7937.48	−0.316382	−0.158191	−	0.987409i	$-0.550566\pi$
$857$	−7937.48	−0.316382	−0.158191	+	0.987409i	$0.550566\pi$
$858$	−12724.1	−0.506288
$859$	26715.0	1.06112	0.530560	−	0.847647i	$-0.321982\pi$
$859$	26715.0	1.06112	0.530560	+	0.847647i	$0.321982\pi$
$860$	6590.71	0.261327
$861$	0	0
$862$	−5230.37	−0.206667
$863$	13718.7	0.541124	0.270562	−	0.962703i	$-0.412791\pi$
$863$	13718.7	0.541124	0.270562	+	0.962703i	$0.412791\pi$
$864$	4733.03	0.186367
$865$	−3802.27	−0.149458
$866$	−9419.55	−0.369618
$867$	14314.8	0.560734
$868$	0	0
$869$	61032.5	2.38249
$870$	−17354.3	−0.676281
$871$	1521.05	0.0591720
$872$	8591.73	0.333661
$873$	−3836.79	−0.148746
$874$	−3812.30	−0.147544
$875$	0	0
$876$	3302.31	0.127368
$877$	102.571	0.00394935	0.00197468	−	0.999998i	$-0.499371\pi$
$877$	102.571	0.00394935	0.00197468	+	0.999998i	$0.499371\pi$
$878$	−3939.18	−0.151413
$879$	−25529.6	−0.979625
$880$	5713.31	0.218859
$881$	18369.0	0.702460	0.351230	−	0.936289i	$-0.385764\pi$
$881$	18369.0	0.702460	0.351230	+	0.936289i	$0.385764\pi$
$882$	0	0
$883$	−7764.26	−0.295909	−0.147955	−	0.988994i	$-0.547269\pi$
$883$	−7764.26	−0.295909	−0.147955	+	0.988994i	$0.547269\pi$
$884$	4111.82	0.156443
$885$	−16992.8	−0.645433
$886$	10839.8	0.411028
$887$	−48834.1	−1.84858	−0.924290	−	0.381692i	$-0.875341\pi$
$887$	−48834.1	−1.84858	−0.924290	+	0.381692i	$0.875341\pi$
$888$	8584.47	0.324410
$889$	0	0
$890$	−9982.05	−0.375954
$891$	35117.3	1.32040
$892$	−9170.24	−0.344218
$893$	−18851.6	−0.706432
$894$	−19780.1	−0.739982
$895$	15633.1	0.583861
$896$	0	0
$897$	−2588.38	−0.0963473
$898$	−14912.7	−0.554168
$899$	39263.0	1.45661
$900$	1202.07	0.0445212
$901$	−15530.0	−0.574227
$902$	−33391.5	−1.23261
$903$	0	0
$904$	−18543.7	−0.682252
$905$	18736.4	0.688199
$906$	10419.2	0.382071
$907$	−19713.5	−0.721693	−0.360846	−	0.932625i	$-0.617512\pi$
$907$	−19713.5	−0.721693	−0.360846	+	0.932625i	$0.617512\pi$
$908$	5234.95	0.191330
$909$	−1240.79	−0.0452745
$910$	0	0
$911$	10896.3	0.396279	0.198139	−	0.980174i	$-0.436510\pi$
$911$	10896.3	0.396279	0.198139	+	0.980174i	$0.436510\pi$
$912$	−6423.84	−0.233240
$913$	53661.3	1.94516
$914$	−24979.6	−0.903996
$915$	13566.2	0.490147
$916$	14504.8	0.523200
$917$	0	0
$918$	−13090.0	−0.470624
$919$	−32343.4	−1.16095	−0.580474	−	0.814279i	$-0.697133\pi$
$919$	−32343.4	−1.16095	−0.580474	+	0.814279i	$0.697133\pi$
$920$	1162.22	0.0416491
$921$	−50306.3	−1.79984
$922$	−24182.9	−0.863797
$923$	−16919.3	−0.603365
$924$	0	0
$925$	18850.5	0.670056
$926$	8760.33	0.310888
$927$	1350.93	0.0478645
$928$	−9074.26	−0.320988
$929$	12322.3	0.435178	0.217589	−	0.976041i	$-0.430181\pi$
$929$	12322.3	0.435178	0.217589	+	0.976041i	$0.430181\pi$
$930$	−8473.59	−0.298774
$931$	0	0
$932$	21249.6	0.746840
$933$	−20888.0	−0.732952
$934$	12992.9	0.455183
$935$	−15801.1	−0.552675
$936$	656.252	0.0229169
$937$	15953.1	0.556206	0.278103	−	0.960551i	$-0.410294\pi$
$937$	15953.1	0.556206	0.278103	+	0.960551i	$0.410294\pi$
$938$	0	0
$939$	16910.9	0.587716
$940$	5747.09	0.199414
$941$	19049.1	0.659919	0.329960	−	0.943995i	$-0.392965\pi$
$941$	19049.1	0.659919	0.329960	+	0.943995i	$0.392965\pi$
$942$	36558.5	1.26448
$943$	−6792.59	−0.234567
$944$	−8885.28	−0.306347
$945$	0	0
$946$	29493.8	1.01366
$947$	−12341.6	−0.423495	−0.211747	−	0.977324i	$-0.567915\pi$
$947$	−12341.6	−0.423495	−0.211747	+	0.977324i	$0.567915\pi$
$948$	20920.3	0.716731
$949$	3958.84	0.135415
$950$	−14106.0	−0.481748
$951$	−10238.1	−0.349100
$952$	0	0
$953$	15410.7	0.523821	0.261911	−	0.965092i	$-0.415647\pi$
$953$	15410.7	0.523821	0.261911	+	0.965092i	$0.415647\pi$
$954$	−2478.60	−0.0841171
$955$	−7130.07	−0.241596
$956$	24602.4	0.832321
$957$	−77661.3	−2.62323
$958$	−2817.76	−0.0950289
$959$	0	0
$960$	1958.37	0.0658397
$961$	−10620.0	−0.356483
$962$	10291.1	0.344906
$963$	−2492.50	−0.0834056
$964$	−29688.5	−0.991911
$965$	−27403.6	−0.914147
$966$	0	0
$967$	47289.3	1.57262	0.786309	−	0.617834i	$-0.211989\pi$
$967$	47289.3	1.57262	0.786309	+	0.617834i	$0.211989\pi$
$968$	14919.4	0.495379
$969$	17766.2	0.588991
$970$	−13725.9	−0.454344
$971$	−11989.4	−0.396249	−0.198124	−	0.980177i	$-0.563485\pi$
$971$	−11989.4	−0.396249	−0.198124	+	0.980177i	$0.563485\pi$
$972$	−3936.72	−0.129908
$973$	0	0
$974$	13760.1	0.452671
$975$	−9577.35	−0.314585
$976$	7093.54	0.232642
$977$	−45588.0	−1.49282	−0.746411	−	0.665485i	$-0.768225\pi$
$977$	−45588.0	−1.49282	−0.746411	+	0.665485i	$0.768225\pi$
$978$	−14438.8	−0.472089
$979$	−44670.2	−1.45829
$980$	0	0
$981$	−3792.42	−0.123428
$982$	−18218.5	−0.592032
$983$	−31079.1	−1.00841	−0.504206	−	0.863584i	$-0.668215\pi$
$983$	−31079.1	−1.00841	−0.504206	+	0.863584i	$0.668215\pi$
$984$	−11445.7	−0.370809
$985$	−17853.7	−0.577529
$986$	25096.3	0.810578
$987$	0	0
$988$	−7700.96	−0.247976
$989$	5999.71	0.192902
$990$	−2521.87	−0.0809600
$991$	56741.2	1.81881	0.909407	−	0.415908i	$-0.136536\pi$
$991$	56741.2	1.81881	0.909407	+	0.415908i	$0.136536\pi$
$992$	−4430.70	−0.141809
$993$	18220.9	0.582300
$994$	0	0
$995$	−9354.06	−0.298034
$996$	18393.7	0.585166
$997$	48676.0	1.54622	0.773112	−	0.634269i	$-0.218699\pi$
$997$	48676.0	1.54622	0.773112	+	0.634269i	$0.218699\pi$
$998$	−11867.1	−0.376401
$999$	−32761.8	−1.03758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.3		11
7.3	odd	6		322.4.e.b.93.3	✓	22
7.5	odd	6		322.4.e.b.277.3	yes	22
7.6	odd	2		2254.4.a.w.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.3	✓	22	7.3	odd	6
322.4.e.b.277.3	yes	22	7.5	odd	6
2254.4.a.w.1.9		11	7.6	odd	2
2254.4.a.x.1.3		11	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -4.84446 q^{3} +4.00000 q^{4} -6.31640 q^{5} -9.68892 q^{6} +8.00000 q^{8} -3.53122 q^{9} -12.6328 q^{10} -56.5325 q^{11} -19.3778 q^{12} -23.2303 q^{13} +30.5995 q^{15} +16.0000 q^{16} -44.2506 q^{17} -7.06245 q^{18} +82.8762 q^{19} -25.2656 q^{20} -113.065 q^{22} -23.0000 q^{23} -38.7557 q^{24} -85.1031 q^{25} -46.4606 q^{26} +147.907 q^{27} -283.571 q^{29} +61.1991 q^{30} -138.459 q^{31} +32.0000 q^{32} +273.869 q^{33} -88.5012 q^{34} -14.1249 q^{36} -221.502 q^{37} +165.752 q^{38} +112.538 q^{39} -50.5312 q^{40} +295.330 q^{41} -260.857 q^{43} -226.130 q^{44} +22.3046 q^{45} -46.0000 q^{46} -227.467 q^{47} -77.5113 q^{48} -170.206 q^{50} +214.370 q^{51} -92.9213 q^{52} +350.955 q^{53} +295.814 q^{54} +357.082 q^{55} -401.490 q^{57} -567.141 q^{58} -555.330 q^{59} +122.398 q^{60} +443.347 q^{61} -276.919 q^{62} +64.0000 q^{64} +146.732 q^{65} +547.739 q^{66} -65.4770 q^{67} -177.002 q^{68} +111.423 q^{69} +728.329 q^{71} -28.2498 q^{72} -170.417 q^{73} -443.005 q^{74} +412.278 q^{75} +331.505 q^{76} +225.077 q^{78} -1079.60 q^{79} -101.062 q^{80} -621.187 q^{81} +590.660 q^{82} -949.211 q^{83} +279.505 q^{85} -521.714 q^{86} +1373.75 q^{87} -452.260 q^{88} +790.169 q^{89} +44.6093 q^{90} -92.0000 q^{92} +670.761 q^{93} -454.933 q^{94} -523.479 q^{95} -155.023 q^{96} +1086.53 q^{97} +199.629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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