LMFDB - Embedded newform 2254.4.a.x.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,4,Mod(1,2254)] 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("2254.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,22,6,44,27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.10
Root			$-6.47210$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +7.47210 q^{3} +4.00000 q^{4} +5.30939 q^{5} +14.9442 q^{6} +8.00000 q^{8} +28.8323 q^{9} +10.6188 q^{10} +15.0723 q^{11} +29.8884 q^{12} -19.5834 q^{13} +39.6723 q^{15} +16.0000 q^{16} +114.919 q^{17} +57.6646 q^{18} -51.8349 q^{19} +21.2376 q^{20} +30.1447 q^{22} -23.0000 q^{23} +59.7768 q^{24} -96.8104 q^{25} -39.1667 q^{26} +13.6912 q^{27} +227.585 q^{29} +79.3446 q^{30} +178.024 q^{31} +32.0000 q^{32} +112.622 q^{33} +229.838 q^{34} +115.329 q^{36} +20.7398 q^{37} -103.670 q^{38} -146.329 q^{39} +42.4751 q^{40} +107.492 q^{41} +138.795 q^{43} +60.2894 q^{44} +153.082 q^{45} -46.0000 q^{46} -21.4507 q^{47} +119.554 q^{48} -193.621 q^{50} +858.687 q^{51} -78.3334 q^{52} +429.203 q^{53} +27.3825 q^{54} +80.0249 q^{55} -387.316 q^{57} +455.170 q^{58} -128.564 q^{59} +158.689 q^{60} -225.931 q^{61} +356.049 q^{62} +64.0000 q^{64} -103.976 q^{65} +225.244 q^{66} +478.716 q^{67} +459.676 q^{68} -171.858 q^{69} +652.010 q^{71} +230.659 q^{72} +516.418 q^{73} +41.4796 q^{74} -723.377 q^{75} -207.340 q^{76} -292.658 q^{78} -349.397 q^{79} +84.9502 q^{80} -676.170 q^{81} +214.984 q^{82} -373.014 q^{83} +610.150 q^{85} +277.589 q^{86} +1700.54 q^{87} +120.579 q^{88} -896.694 q^{89} +306.164 q^{90} -92.0000 q^{92} +1330.22 q^{93} -42.9014 q^{94} -275.211 q^{95} +239.107 q^{96} +1294.92 q^{97} +434.570 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.47210	1.43801	0.719003	−	0.695007i	$-0.244599\pi$
$3$	7.47210	1.43801	0.719003	+	0.695007i	$0.244599\pi$
$4$	4.00000	0.500000
$5$	5.30939	0.474886	0.237443	−	0.971401i	$-0.423691\pi$
$5$	5.30939	0.474886	0.237443	+	0.971401i	$0.423691\pi$
$6$	14.9442	1.01682
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	28.8323	1.06786
$10$	10.6188	0.335795
$11$	15.0723	0.413135	0.206567	−	0.978432i	$-0.433771\pi$
$11$	15.0723	0.413135	0.206567	+	0.978432i	$0.433771\pi$
$12$	29.8884	0.719003
$13$	−19.5834	−0.417804	−0.208902	−	0.977937i	$-0.566989\pi$
$13$	−19.5834	−0.417804	−0.208902	+	0.977937i	$0.566989\pi$
$14$	0	0
$15$	39.6723	0.682889
$16$	16.0000	0.250000
$17$	114.919	1.63953	0.819763	−	0.572702i	$-0.194105\pi$
$17$	114.919	1.63953	0.819763	+	0.572702i	$0.194105\pi$
$18$	57.6646	0.755093
$19$	−51.8349	−0.625881	−0.312940	−	0.949773i	$-0.601314\pi$
$19$	−51.8349	−0.625881	−0.312940	+	0.949773i	$0.601314\pi$
$20$	21.2376	0.237443
$21$	0	0
$22$	30.1447	0.292130
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	59.7768	0.508412
$25$	−96.8104	−0.774483
$26$	−39.1667	−0.295432
$27$	13.6912	0.0975881
$28$	0	0
$29$	227.585	1.45729	0.728646	−	0.684890i	$-0.240150\pi$
$29$	227.585	1.45729	0.728646	+	0.684890i	$0.240150\pi$
$30$	79.3446	0.482876
$31$	178.024	1.03142	0.515712	−	0.856762i	$-0.327528\pi$
$31$	178.024	1.03142	0.515712	+	0.856762i	$0.327528\pi$
$32$	32.0000	0.176777
$33$	112.622	0.594090
$34$	229.838	1.15932
$35$	0	0
$36$	115.329	0.533932
$37$	20.7398	0.0921513	0.0460756	−	0.998938i	$-0.485328\pi$
$37$	20.7398	0.0921513	0.0460756	+	0.998938i	$0.485328\pi$
$38$	−103.670	−0.442565
$39$	−146.329	−0.600804
$40$	42.4751	0.167898
$41$	107.492	0.409449	0.204724	−	0.978820i	$-0.434370\pi$
$41$	107.492	0.409449	0.204724	+	0.978820i	$0.434370\pi$
$42$	0	0
$43$	138.795	0.492232	0.246116	−	0.969240i	$-0.420846\pi$
$43$	138.795	0.492232	0.246116	+	0.969240i	$0.420846\pi$
$44$	60.2894	0.206567
$45$	153.082	0.507113
$46$	−46.0000	−0.147442
$47$	−21.4507	−0.0665725	−0.0332863	−	0.999446i	$-0.510597\pi$
$47$	−21.4507	−0.0665725	−0.0332863	+	0.999446i	$0.510597\pi$
$48$	119.554	0.359502
$49$	0	0
$50$	−193.621	−0.547642
$51$	858.687	2.35765
$52$	−78.3334	−0.208902
$53$	429.203	1.11237	0.556185	−	0.831059i	$-0.312265\pi$
$53$	429.203	1.11237	0.556185	+	0.831059i	$0.312265\pi$
$54$	27.3825	0.0690052
$55$	80.0249	0.196192
$56$	0	0
$57$	−387.316	−0.900021
$58$	455.170	1.03046
$59$	−128.564	−0.283689	−0.141844	−	0.989889i	$-0.545303\pi$
$59$	−128.564	−0.283689	−0.141844	+	0.989889i	$0.545303\pi$
$60$	158.689	0.341445
$61$	−225.931	−0.474222	−0.237111	−	0.971483i	$-0.576200\pi$
$61$	−225.931	−0.474222	−0.237111	+	0.971483i	$0.576200\pi$
$62$	356.049	0.729326
$63$	0	0
$64$	64.0000	0.125000
$65$	−103.976	−0.198409
$66$	225.244	0.420085
$67$	478.716	0.872902	0.436451	−	0.899728i	$-0.356235\pi$
$67$	478.716	0.872902	0.436451	+	0.899728i	$0.356235\pi$
$68$	459.676	0.819763
$69$	−171.858	−0.299845
$70$	0	0
$71$	652.010	1.08985	0.544925	−	0.838485i	$-0.316558\pi$
$71$	652.010	1.08985	0.544925	+	0.838485i	$0.316558\pi$
$72$	230.659	0.377547
$73$	516.418	0.827975	0.413987	−	0.910283i	$-0.364136\pi$
$73$	516.418	0.827975	0.413987	+	0.910283i	$0.364136\pi$
$74$	41.4796	0.0651608
$75$	−723.377	−1.11371
$76$	−207.340	−0.312940
$77$	0	0
$78$	−292.658	−0.424833
$79$	−349.397	−0.497597	−0.248799	−	0.968555i	$-0.580036\pi$
$79$	−349.397	−0.497597	−0.248799	+	0.968555i	$0.580036\pi$
$80$	84.9502	0.118722
$81$	−676.170	−0.927531
$82$	214.984	0.289524
$83$	−373.014	−0.493297	−0.246649	−	0.969105i	$-0.579329\pi$
$83$	−373.014	−0.493297	−0.246649	+	0.969105i	$0.579329\pi$
$84$	0	0
$85$	610.150	0.778588
$86$	277.589	0.348061
$87$	1700.54	2.09560
$88$	120.579	0.146065
$89$	−896.694	−1.06797	−0.533985	−	0.845494i	$-0.679306\pi$
$89$	−896.694	−1.06797	−0.533985	+	0.845494i	$0.679306\pi$
$90$	306.164	0.358583
$91$	0	0
$92$	−92.0000	−0.104257
$93$	1330.22	1.48319
$94$	−42.9014	−0.0470739
$95$	−275.211	−0.297222
$96$	239.107	0.254206
$97$	1294.92	1.35546	0.677729	−	0.735312i	$-0.262964\pi$
$97$	1294.92	1.35546	0.677729	+	0.735312i	$0.262964\pi$
$98$	0	0
$99$	434.570	0.441171
$100$	−387.242	−0.387242
$101$	−309.230	−0.304649	−0.152325	−	0.988331i	$-0.548676\pi$
$101$	−309.230	−0.304649	−0.152325	+	0.988331i	$0.548676\pi$
$102$	1717.37	1.66711
$103$	−617.628	−0.590842	−0.295421	−	0.955367i	$-0.595460\pi$
$103$	−617.628	−0.590842	−0.295421	+	0.955367i	$0.595460\pi$
$104$	−156.667	−0.147716
$105$	0	0
$106$	858.407	0.786564
$107$	−1225.20	−1.10696	−0.553481	−	0.832862i	$-0.686701\pi$
$107$	−1225.20	−1.10696	−0.553481	+	0.832862i	$0.686701\pi$
$108$	54.7649	0.0487941
$109$	157.245	0.138177	0.0690886	−	0.997611i	$-0.477991\pi$
$109$	157.245	0.138177	0.0690886	+	0.997611i	$0.477991\pi$
$110$	160.050	0.138729
$111$	154.970	0.132514
$112$	0	0
$113$	−1226.97	−1.02145	−0.510723	−	0.859745i	$-0.670622\pi$
$113$	−1226.97	−1.02145	−0.510723	+	0.859745i	$0.670622\pi$
$114$	−774.631	−0.636411
$115$	−122.116	−0.0990206
$116$	910.340	0.728646
$117$	−564.634	−0.446157
$118$	−257.128	−0.200598
$119$	0	0
$120$	317.378	0.241438
$121$	−1103.82	−0.829320
$122$	−451.862	−0.335325
$123$	803.190	0.588790
$124$	712.097	0.515712
$125$	−1177.68	−0.842677
$126$	0	0
$127$	1229.19	0.858840	0.429420	−	0.903105i	$-0.358718\pi$
$127$	1229.19	0.858840	0.429420	+	0.903105i	$0.358718\pi$
$128$	128.000	0.0883883
$129$	1037.09	0.707834
$130$	−207.951	−0.140296
$131$	2420.93	1.61464	0.807321	−	0.590113i	$-0.200917\pi$
$131$	2420.93	1.61464	0.807321	+	0.590113i	$0.200917\pi$
$132$	450.488	0.297045
$133$	0	0
$134$	957.432	0.617235
$135$	72.6921	0.0463432
$136$	919.352	0.579660
$137$	3202.71	1.99727	0.998635	−	0.0522373i	$-0.0166352\pi$
$137$	3202.71	1.99727	0.998635	+	0.0522373i	$0.0166352\pi$
$138$	−343.717	−0.212023
$139$	1710.35	1.04367	0.521833	−	0.853048i	$-0.325248\pi$
$139$	1710.35	1.04367	0.521833	+	0.853048i	$0.325248\pi$
$140$	0	0
$141$	−160.282	−0.0957317
$142$	1304.02	0.770641
$143$	−295.167	−0.172609
$144$	461.317	0.266966
$145$	1208.34	0.692048
$146$	1032.84	0.585466
$147$	0	0
$148$	82.9591	0.0460756
$149$	−2647.44	−1.45561	−0.727807	−	0.685782i	$-0.759461\pi$
$149$	−2647.44	−1.45561	−0.727807	+	0.685782i	$0.759461\pi$
$150$	−1446.75	−0.787513
$151$	−1010.81	−0.544761	−0.272381	−	0.962190i	$-0.587811\pi$
$151$	−1010.81	−0.544761	−0.272381	+	0.962190i	$0.587811\pi$
$152$	−414.679	−0.221282
$153$	3313.38	1.75079
$154$	0	0
$155$	945.200	0.489809
$156$	−585.315	−0.300402
$157$	−76.7891	−0.0390347	−0.0195173	−	0.999810i	$-0.506213\pi$
$157$	−76.7891	−0.0390347	−0.0195173	+	0.999810i	$0.506213\pi$
$158$	−698.793	−0.351854
$159$	3207.05	1.59960
$160$	169.900	0.0839488
$161$	0	0
$162$	−1352.34	−0.655864
$163$	2149.22	1.03276	0.516379	−	0.856360i	$-0.327280\pi$
$163$	2149.22	1.03276	0.516379	+	0.856360i	$0.327280\pi$
$164$	429.967	0.204724
$165$	597.954	0.282125
$166$	−746.029	−0.348814
$167$	2071.34	0.959791	0.479896	−	0.877326i	$-0.340675\pi$
$167$	2071.34	0.959791	0.479896	+	0.877326i	$0.340675\pi$
$168$	0	0
$169$	−1813.49	−0.825440
$170$	1220.30	0.550545
$171$	−1494.52	−0.668355
$172$	555.179	0.246116
$173$	−3121.85	−1.37197	−0.685983	−	0.727618i	$-0.740627\pi$
$173$	−3121.85	−1.37197	−0.685983	+	0.727618i	$0.740627\pi$
$174$	3401.08	1.48181
$175$	0	0
$176$	241.157	0.103284
$177$	−960.645	−0.407946
$178$	−1793.39	−0.755169
$179$	−3069.29	−1.28162	−0.640808	−	0.767701i	$-0.721401\pi$
$179$	−3069.29	−1.28162	−0.640808	+	0.767701i	$0.721401\pi$
$180$	612.328	0.253557
$181$	−1108.70	−0.455300	−0.227650	−	0.973743i	$-0.573104\pi$
$181$	−1108.70	−0.455300	−0.227650	+	0.973743i	$0.573104\pi$
$182$	0	0
$183$	−1688.18	−0.681934
$184$	−184.000	−0.0737210
$185$	110.116	0.0437614
$186$	2660.43	1.04878
$187$	1732.10	0.677345
$188$	−85.8028	−0.0332863
$189$	0	0
$190$	−550.423	−0.210168
$191$	−3180.62	−1.20493	−0.602465	−	0.798146i	$-0.705815\pi$
$191$	−3180.62	−1.20493	−0.602465	+	0.798146i	$0.705815\pi$
$192$	478.215	0.179751
$193$	−812.325	−0.302966	−0.151483	−	0.988460i	$-0.548405\pi$
$193$	−812.325	−0.302966	−0.151483	+	0.988460i	$0.548405\pi$
$194$	2589.84	0.958454
$195$	−776.917	−0.285314
$196$	0	0
$197$	−3772.40	−1.36433	−0.682164	−	0.731199i	$-0.738961\pi$
$197$	−3772.40	−1.36433	−0.682164	+	0.731199i	$0.738961\pi$
$198$	869.141	0.311955
$199$	1524.28	0.542980	0.271490	−	0.962441i	$-0.412484\pi$
$199$	1524.28	0.542980	0.271490	+	0.962441i	$0.412484\pi$
$200$	−774.483	−0.273821
$201$	3577.01	1.25524
$202$	−618.460	−0.215419
$203$	0	0
$204$	3434.75	1.17883
$205$	570.716	0.194441
$206$	−1235.26	−0.417788
$207$	−663.143	−0.222665
$208$	−313.334	−0.104451
$209$	−781.273	−0.258573
$210$	0	0
$211$	910.563	0.297089	0.148544	−	0.988906i	$-0.452541\pi$
$211$	910.563	0.297089	0.148544	+	0.988906i	$0.452541\pi$
$212$	1716.81	0.556185
$213$	4871.89	1.56721
$214$	−2450.41	−0.782740
$215$	736.915	0.233754
$216$	109.530	0.0345026
$217$	0	0
$218$	314.489	0.0977060
$219$	3858.73	1.19063
$220$	320.100	0.0980959
$221$	−2250.50	−0.685000
$222$	309.939	0.0937017
$223$	−1289.30	−0.387166	−0.193583	−	0.981084i	$-0.562011\pi$
$223$	−1289.30	−0.387166	−0.193583	+	0.981084i	$0.562011\pi$
$224$	0	0
$225$	−2791.27	−0.827042
$226$	−2453.94	−0.722272
$227$	−1413.94	−0.413419	−0.206710	−	0.978402i	$-0.566276\pi$
$227$	−1413.94	−0.413419	−0.206710	+	0.978402i	$0.566276\pi$
$228$	−1549.26	−0.450010
$229$	799.165	0.230613	0.115306	−	0.993330i	$-0.463215\pi$
$229$	799.165	0.230613	0.115306	+	0.993330i	$0.463215\pi$
$230$	−244.232	−0.0700181
$231$	0	0
$232$	1820.68	0.515231
$233$	1554.41	0.437050	0.218525	−	0.975831i	$-0.429875\pi$
$233$	1554.41	0.437050	0.218525	+	0.975831i	$0.429875\pi$
$234$	−1129.27	−0.315481
$235$	−113.890	−0.0316144
$236$	−514.257	−0.141844
$237$	−2610.73	−0.715548
$238$	0	0
$239$	−6967.81	−1.88582	−0.942908	−	0.333052i	$-0.891922\pi$
$239$	−6967.81	−1.88582	−0.942908	+	0.333052i	$0.891922\pi$
$240$	634.757	0.170722
$241$	5058.24	1.35199	0.675996	−	0.736905i	$-0.263714\pi$
$241$	5058.24	1.35199	0.675996	+	0.736905i	$0.263714\pi$
$242$	−2207.65	−0.586418
$243$	−5422.08	−1.43138
$244$	−903.725	−0.237111
$245$	0	0
$246$	1606.38	0.416337
$247$	1015.10	0.261495
$248$	1424.19	0.364663
$249$	−2787.20	−0.709365
$250$	−2355.35	−0.595863
$251$	3363.38	0.845796	0.422898	−	0.906177i	$-0.361013\pi$
$251$	3363.38	0.845796	0.422898	+	0.906177i	$0.361013\pi$
$252$	0	0
$253$	−346.664	−0.0861445
$254$	2458.37	0.607292
$255$	4559.10	1.11962
$256$	256.000	0.0625000
$257$	7020.29	1.70394	0.851972	−	0.523588i	$-0.175407\pi$
$257$	7020.29	1.70394	0.851972	+	0.523588i	$0.175407\pi$
$258$	2074.18	0.500514
$259$	0	0
$260$	−415.903	−0.0992046
$261$	6561.80	1.55619
$262$	4841.87	1.14172
$263$	−1118.57	−0.262258	−0.131129	−	0.991365i	$-0.541860\pi$
$263$	−1118.57	−0.262258	−0.131129	+	0.991365i	$0.541860\pi$
$264$	900.976	0.210043
$265$	2278.81	0.528249
$266$	0	0
$267$	−6700.19	−1.53575
$268$	1914.86	0.436451
$269$	2785.87	0.631439	0.315720	−	0.948853i	$-0.397754\pi$
$269$	2785.87	0.631439	0.315720	+	0.948853i	$0.397754\pi$
$270$	145.384	0.0327696
$271$	2689.54	0.602870	0.301435	−	0.953487i	$-0.402534\pi$
$271$	2689.54	0.602870	0.301435	+	0.953487i	$0.402534\pi$
$272$	1838.70	0.409882
$273$	0	0
$274$	6405.42	1.41228
$275$	−1459.16	−0.319966
$276$	−687.433	−0.149923
$277$	−830.836	−0.180217	−0.0901085	−	0.995932i	$-0.528721\pi$
$277$	−830.836	−0.180217	−0.0901085	+	0.995932i	$0.528721\pi$
$278$	3420.69	0.737984
$279$	5132.85	1.10142
$280$	0	0
$281$	−8850.37	−1.87889	−0.939447	−	0.342695i	$-0.888660\pi$
$281$	−8850.37	−1.87889	−0.939447	+	0.342695i	$0.888660\pi$
$282$	−320.564	−0.0676925
$283$	557.575	0.117118	0.0585590	−	0.998284i	$-0.481349\pi$
$283$	557.575	0.117118	0.0585590	+	0.998284i	$0.481349\pi$
$284$	2608.04	0.544925
$285$	−2056.41	−0.427407
$286$	−590.334	−0.122053
$287$	0	0
$288$	922.634	0.188773
$289$	8293.38	1.68805
$290$	2416.67	0.489352
$291$	9675.79	1.94916
$292$	2065.67	0.413987
$293$	263.506	0.0525400	0.0262700	−	0.999655i	$-0.491637\pi$
$293$	263.506	0.0525400	0.0262700	+	0.999655i	$0.491637\pi$
$294$	0	0
$295$	−682.597	−0.134720
$296$	165.918	0.0325804
$297$	206.359	0.0403170
$298$	−5294.88	−1.02928
$299$	450.417	0.0871181
$300$	−2893.51	−0.556856
$301$	0	0
$302$	−2021.63	−0.385204
$303$	−2310.60	−0.438087
$304$	−829.358	−0.156470
$305$	−1199.56	−0.225201
$306$	6626.76	1.23800
$307$	−6200.77	−1.15276	−0.576379	−	0.817182i	$-0.695535\pi$
$307$	−6200.77	−1.15276	−0.576379	+	0.817182i	$0.695535\pi$
$308$	0	0
$309$	−4614.98	−0.849635
$310$	1890.40	0.346347
$311$	−8424.24	−1.53600	−0.767999	−	0.640452i	$-0.778747\pi$
$311$	−8424.24	−1.53600	−0.767999	+	0.640452i	$0.778747\pi$
$312$	−1170.63	−0.212416
$313$	2421.11	0.437218	0.218609	−	0.975813i	$-0.429848\pi$
$313$	2421.11	0.437218	0.218609	+	0.975813i	$0.429848\pi$
$314$	−153.578	−0.0276017
$315$	0	0
$316$	−1397.59	−0.248799
$317$	−1032.26	−0.182895	−0.0914474	−	0.995810i	$-0.529149\pi$
$317$	−1032.26	−0.182895	−0.0914474	+	0.995810i	$0.529149\pi$
$318$	6414.10	1.13108
$319$	3430.24	0.602058
$320$	339.801	0.0593608
$321$	−9154.85	−1.59182
$322$	0	0
$323$	−5956.81	−1.02615
$324$	−2704.68	−0.463766
$325$	1895.87	0.323582
$326$	4298.43	0.730270
$327$	1174.95	0.198700
$328$	859.934	0.144762
$329$	0	0
$330$	1195.91	0.199493
$331$	1815.35	0.301453	0.150726	−	0.988576i	$-0.451839\pi$
$331$	1815.35	0.301453	0.150726	+	0.988576i	$0.451839\pi$
$332$	−1492.06	−0.246649
$333$	597.976	0.0984050
$334$	4142.68	0.678675
$335$	2541.69	0.414529
$336$	0	0
$337$	−8160.36	−1.31906	−0.659530	−	0.751678i	$-0.729245\pi$
$337$	−8160.36	−1.31906	−0.659530	+	0.751678i	$0.729245\pi$
$338$	−3626.98	−0.583674
$339$	−9168.03	−1.46885
$340$	2440.60	0.389294
$341$	2683.24	0.426117
$342$	−2989.04	−0.472599
$343$	0	0
$344$	1110.36	0.174030
$345$	−912.463	−0.142392
$346$	−6243.70	−0.970126
$347$	−5693.79	−0.880861	−0.440431	−	0.897787i	$-0.645174\pi$
$347$	−5693.79	−0.880861	−0.440431	+	0.897787i	$0.645174\pi$
$348$	6802.15	1.04780
$349$	−4560.60	−0.699494	−0.349747	−	0.936844i	$-0.613733\pi$
$349$	−4560.60	−0.699494	−0.349747	+	0.936844i	$0.613733\pi$
$350$	0	0
$351$	−268.120	−0.0407727
$352$	482.315	0.0730326
$353$	−8580.49	−1.29375	−0.646874	−	0.762597i	$-0.723924\pi$
$353$	−8580.49	−1.29375	−0.646874	+	0.762597i	$0.723924\pi$
$354$	−1921.29	−0.288462
$355$	3461.78	0.517555
$356$	−3586.78	−0.533985
$357$	0	0
$358$	−6138.58	−0.906240
$359$	1989.15	0.292433	0.146217	−	0.989253i	$-0.453290\pi$
$359$	1989.15	0.292433	0.146217	+	0.989253i	$0.453290\pi$
$360$	1224.66	0.179292
$361$	−4172.15	−0.608273
$362$	−2217.41	−0.321945
$363$	−8247.89	−1.19257
$364$	0	0
$365$	2741.86	0.393194
$366$	−3376.36	−0.482200
$367$	114.888	0.0163409	0.00817046	−	0.999967i	$-0.497399\pi$
$367$	114.888	0.0163409	0.00817046	+	0.999967i	$0.497399\pi$
$368$	−368.000	−0.0521286
$369$	3099.24	0.437235
$370$	220.231	0.0309440
$371$	0	0
$372$	5320.87	0.741597
$373$	12568.7	1.74473	0.872363	−	0.488858i	$-0.162586\pi$
$373$	12568.7	1.74473	0.872363	+	0.488858i	$0.162586\pi$
$374$	3464.20	0.478956
$375$	−8799.73	−1.21178
$376$	−171.606	−0.0235369
$377$	−4456.88	−0.608862
$378$	0	0
$379$	−12075.5	−1.63662	−0.818308	−	0.574781i	$-0.805087\pi$
$379$	−12075.5	−1.63662	−0.818308	+	0.574781i	$0.805087\pi$
$380$	−1100.85	−0.148611
$381$	9184.61	1.23502
$382$	−6361.24	−0.852014
$383$	12228.0	1.63139	0.815695	−	0.578482i	$-0.196355\pi$
$383$	12228.0	1.63139	0.815695	+	0.578482i	$0.196355\pi$
$384$	956.429	0.127103
$385$	0	0
$386$	−1624.65	−0.214229
$387$	4001.77	0.525637
$388$	5179.69	0.677729
$389$	3137.56	0.408948	0.204474	−	0.978872i	$-0.434452\pi$
$389$	3137.56	0.408948	0.204474	+	0.978872i	$0.434452\pi$
$390$	−1553.83	−0.201747
$391$	−2643.14	−0.341865
$392$	0	0
$393$	18089.5	2.32187
$394$	−7544.81	−0.964725
$395$	−1855.08	−0.236302
$396$	1738.28	0.220586
$397$	−15489.8	−1.95821	−0.979106	−	0.203352i	$-0.934816\pi$
$397$	−15489.8	−1.95821	−0.979106	+	0.203352i	$0.934816\pi$
$398$	3048.55	0.383945
$399$	0	0
$400$	−1548.97	−0.193621
$401$	−6052.52	−0.753737	−0.376869	−	0.926267i	$-0.622999\pi$
$401$	−6052.52	−0.753737	−0.376869	+	0.926267i	$0.622999\pi$
$402$	7154.03	0.887588
$403$	−3486.32	−0.430932
$404$	−1236.92	−0.152325
$405$	−3590.05	−0.440472
$406$	0	0
$407$	312.597	0.0380709
$408$	6869.49	0.833555
$409$	−11921.8	−1.44130	−0.720652	−	0.693297i	$-0.756157\pi$
$409$	−11921.8	−1.44130	−0.720652	+	0.693297i	$0.756157\pi$
$410$	1141.43	0.137491
$411$	23931.0	2.87209
$412$	−2470.51	−0.295421
$413$	0	0
$414$	−1326.29	−0.157448
$415$	−1980.48	−0.234260
$416$	−626.668	−0.0738579
$417$	12779.9	1.50080
$418$	−1562.55	−0.182839
$419$	7468.53	0.870792	0.435396	−	0.900239i	$-0.356608\pi$
$419$	7468.53	0.870792	0.435396	+	0.900239i	$0.356608\pi$
$420$	0	0
$421$	6039.59	0.699172	0.349586	−	0.936904i	$-0.386322\pi$
$421$	6039.59	0.699172	0.349586	+	0.936904i	$0.386322\pi$
$422$	1821.13	0.210073
$423$	−618.474	−0.0710903
$424$	3433.63	0.393282
$425$	−11125.4	−1.26979
$426$	9743.78	1.10819
$427$	0	0
$428$	−4900.81	−0.553481
$429$	−2205.52	−0.248213
$430$	1473.83	0.165289
$431$	13552.1	1.51458	0.757289	−	0.653079i	$-0.226523\pi$
$431$	13552.1	1.51458	0.757289	+	0.653079i	$0.226523\pi$
$432$	219.060	0.0243970
$433$	−2807.93	−0.311641	−0.155821	−	0.987785i	$-0.549802\pi$
$433$	−2807.93	−0.311641	−0.155821	+	0.987785i	$0.549802\pi$
$434$	0	0
$435$	9028.82	0.995169
$436$	628.979	0.0690886
$437$	1192.20	0.130505
$438$	7717.45	0.841905
$439$	15898.7	1.72849	0.864243	−	0.503075i	$-0.167798\pi$
$439$	15898.7	1.72849	0.864243	+	0.503075i	$0.167798\pi$
$440$	640.199	0.0693643
$441$	0	0
$442$	−4501.00	−0.484368
$443$	−9194.64	−0.986118	−0.493059	−	0.869996i	$-0.664121\pi$
$443$	−9194.64	−0.986118	−0.493059	+	0.869996i	$0.664121\pi$
$444$	619.879	0.0662571
$445$	−4760.90	−0.507164
$446$	−2578.61	−0.273768
$447$	−19781.9	−2.09318
$448$	0	0
$449$	−11231.7	−1.18053	−0.590265	−	0.807209i	$-0.700977\pi$
$449$	−11231.7	−1.18053	−0.590265	+	0.807209i	$0.700977\pi$
$450$	−5582.54	−0.584807
$451$	1620.15	0.169157
$452$	−4907.87	−0.510723
$453$	−7552.91	−0.783370
$454$	−2827.87	−0.292332
$455$	0	0
$456$	−3098.52	−0.318205
$457$	−9751.96	−0.998200	−0.499100	−	0.866544i	$-0.666336\pi$
$457$	−9751.96	−0.998200	−0.499100	+	0.866544i	$0.666336\pi$
$458$	1598.33	0.163068
$459$	1573.38	0.159998
$460$	−488.464	−0.0495103
$461$	6728.85	0.679813	0.339906	−	0.940459i	$-0.389605\pi$
$461$	6728.85	0.679813	0.339906	+	0.940459i	$0.389605\pi$
$462$	0	0
$463$	−7665.18	−0.769398	−0.384699	−	0.923042i	$-0.625695\pi$
$463$	−7665.18	−0.769398	−0.384699	+	0.923042i	$0.625695\pi$
$464$	3641.36	0.364323
$465$	7062.63	0.704348
$466$	3108.82	0.309041
$467$	16755.9	1.66033	0.830163	−	0.557521i	$-0.188248\pi$
$467$	16755.9	1.66033	0.830163	+	0.557521i	$0.188248\pi$
$468$	−2258.53	−0.223079
$469$	0	0
$470$	−227.780	−0.0223547
$471$	−573.776	−0.0561321
$472$	−1028.51	−0.100299
$473$	2091.96	0.203358
$474$	−5221.45	−0.505969
$475$	5018.16	0.484734
$476$	0	0
$477$	12374.9	1.18786
$478$	−13935.6	−1.33347
$479$	−5518.57	−0.526409	−0.263205	−	0.964740i	$-0.584779\pi$
$479$	−5518.57	−0.526409	−0.263205	+	0.964740i	$0.584779\pi$
$480$	1269.51	0.120719
$481$	−406.154	−0.0385011
$482$	10116.5	0.956003
$483$	0	0
$484$	−4415.30	−0.414660
$485$	6875.24	0.643688
$486$	−10844.2	−1.01214
$487$	3953.92	0.367904	0.183952	−	0.982935i	$-0.441111\pi$
$487$	3953.92	0.367904	0.183952	+	0.982935i	$0.441111\pi$
$488$	−1807.45	−0.167663
$489$	16059.2	1.48511
$490$	0	0
$491$	12730.6	1.17011	0.585053	−	0.810995i	$-0.301074\pi$
$491$	12730.6	1.17011	0.585053	+	0.810995i	$0.301074\pi$
$492$	3212.76	0.294395
$493$	26153.8	2.38927
$494$	2030.20	0.184905
$495$	2307.30	0.209506
$496$	2848.39	0.257856
$497$	0	0
$498$	−5574.40	−0.501596
$499$	−18019.9	−1.61659	−0.808297	−	0.588775i	$-0.799611\pi$
$499$	−18019.9	−1.61659	−0.808297	+	0.588775i	$0.799611\pi$
$500$	−4710.71	−0.421339
$501$	15477.3	1.38019
$502$	6726.77	0.598068
$503$	10200.1	0.904175	0.452087	−	0.891974i	$-0.350680\pi$
$503$	10200.1	0.904175	0.452087	+	0.891974i	$0.350680\pi$
$504$	0	0
$505$	−1641.82	−0.144674
$506$	−693.328	−0.0609134
$507$	−13550.6	−1.18699
$508$	4916.75	0.429420
$509$	−12372.6	−1.07741	−0.538707	−	0.842493i	$-0.681087\pi$
$509$	−12372.6	−1.07741	−0.538707	+	0.842493i	$0.681087\pi$
$510$	9118.20	0.791688
$511$	0	0
$512$	512.000	0.0441942
$513$	−709.683	−0.0610785
$514$	14040.6	1.20487
$515$	−3279.23	−0.280583
$516$	4148.35	0.353917
$517$	−323.312	−0.0275034
$518$	0	0
$519$	−23326.8	−1.97290
$520$	−831.805	−0.0701482
$521$	−11660.6	−0.980538	−0.490269	−	0.871571i	$-0.663101\pi$
$521$	−11660.6	−0.980538	−0.490269	+	0.871571i	$0.663101\pi$
$522$	13123.6	1.10039
$523$	3985.38	0.333210	0.166605	−	0.986024i	$-0.446720\pi$
$523$	3985.38	0.333210	0.166605	+	0.986024i	$0.446720\pi$
$524$	9683.74	0.807321
$525$	0	0
$526$	−2237.13	−0.185444
$527$	20458.4	1.69105
$528$	1801.95	0.148523
$529$	529.000	0.0434783
$530$	4557.61	0.373528
$531$	−3706.80	−0.302941
$532$	0	0
$533$	−2105.05	−0.171069
$534$	−13400.4	−1.08594
$535$	−6505.08	−0.525681
$536$	3829.73	0.308618
$537$	−22934.0	−1.84297
$538$	5571.73	0.446495
$539$	0	0
$540$	290.768	0.0231716
$541$	8567.19	0.680836	0.340418	−	0.940274i	$-0.389431\pi$
$541$	8567.19	0.680836	0.340418	+	0.940274i	$0.389431\pi$
$542$	5379.07	0.426293
$543$	−8284.34	−0.654724
$544$	3677.41	0.289830
$545$	834.873	0.0656184
$546$	0	0
$547$	−14381.3	−1.12413	−0.562067	−	0.827091i	$-0.689994\pi$
$547$	−14381.3	−1.12413	−0.562067	+	0.827091i	$0.689994\pi$
$548$	12810.8	0.998635
$549$	−6514.12	−0.506404
$550$	−2918.32	−0.226250
$551$	−11796.8	−0.912091
$552$	−1374.87	−0.106011
$553$	0	0
$554$	−1661.67	−0.127433
$555$	822.794	0.0629291
$556$	6841.38	0.521833
$557$	−1759.65	−0.133857	−0.0669287	−	0.997758i	$-0.521320\pi$
$557$	−1759.65	−0.133857	−0.0669287	+	0.997758i	$0.521320\pi$
$558$	10265.7	0.778821
$559$	−2718.07	−0.205656
$560$	0	0
$561$	12942.4	0.974027
$562$	−17700.7	−1.32858
$563$	12597.1	0.942989	0.471495	−	0.881869i	$-0.343715\pi$
$563$	12597.1	0.942989	0.471495	+	0.881869i	$0.343715\pi$
$564$	−641.128	−0.0478659
$565$	−6514.45	−0.485071
$566$	1115.15	0.0828149
$567$	0	0
$568$	5216.08	0.385320
$569$	−6429.70	−0.473721	−0.236860	−	0.971544i	$-0.576118\pi$
$569$	−6429.70	−0.473721	−0.236860	+	0.971544i	$0.576118\pi$
$570$	−4112.82	−0.302223
$571$	−5090.14	−0.373057	−0.186529	−	0.982450i	$-0.559724\pi$
$571$	−5090.14	−0.373057	−0.186529	+	0.982450i	$0.559724\pi$
$572$	−1180.67	−0.0863046
$573$	−23765.9	−1.73270
$574$	0	0
$575$	2226.64	0.161491
$576$	1845.27	0.133483
$577$	12748.9	0.919834	0.459917	−	0.887962i	$-0.347879\pi$
$577$	12748.9	0.919834	0.459917	+	0.887962i	$0.347879\pi$
$578$	16586.8	1.19363
$579$	−6069.78	−0.435667
$580$	4833.35	0.346024
$581$	0	0
$582$	19351.6	1.37826
$583$	6469.10	0.459559
$584$	4131.34	0.292733
$585$	−2997.86	−0.211874
$586$	527.013	0.0371514
$587$	−7455.98	−0.524261	−0.262131	−	0.965032i	$-0.584425\pi$
$587$	−7455.98	−0.524261	−0.262131	+	0.965032i	$0.584425\pi$
$588$	0	0
$589$	−9227.87	−0.645548
$590$	−1365.19	−0.0952613
$591$	−28187.8	−1.96191
$592$	331.836	0.0230378
$593$	−9005.60	−0.623635	−0.311817	−	0.950142i	$-0.600938\pi$
$593$	−9005.60	−0.623635	−0.311817	+	0.950142i	$0.600938\pi$
$594$	412.718	0.0285084
$595$	0	0
$596$	−10589.8	−0.727807
$597$	11389.5	0.780809
$598$	900.835	0.0616018
$599$	−14196.3	−0.968359	−0.484179	−	0.874969i	$-0.660882\pi$
$599$	−14196.3	−0.968359	−0.484179	+	0.874969i	$0.660882\pi$
$600$	−5787.02	−0.393757
$601$	−4208.80	−0.285658	−0.142829	−	0.989747i	$-0.545620\pi$
$601$	−4208.80	−0.285658	−0.142829	+	0.989747i	$0.545620\pi$
$602$	0	0
$603$	13802.5	0.932140
$604$	−4043.26	−0.272381
$605$	−5860.63	−0.393832
$606$	−4621.20	−0.309775
$607$	13431.8	0.898155	0.449078	−	0.893493i	$-0.351753\pi$
$607$	13431.8	0.898155	0.449078	+	0.893493i	$0.351753\pi$
$608$	−1658.72	−0.110641
$609$	0	0
$610$	−2399.11	−0.159241
$611$	420.077	0.0278142
$612$	13253.5	0.875395
$613$	27774.2	1.83000	0.914999	−	0.403456i	$-0.132191\pi$
$613$	27774.2	1.83000	0.914999	+	0.403456i	$0.132191\pi$
$614$	−12401.5	−0.815123
$615$	4264.44	0.279608
$616$	0	0
$617$	10560.5	0.689061	0.344530	−	0.938775i	$-0.388038\pi$
$617$	10560.5	0.689061	0.344530	+	0.938775i	$0.388038\pi$
$618$	−9229.96	−0.600782
$619$	20438.9	1.32715	0.663577	−	0.748108i	$-0.269038\pi$
$619$	20438.9	1.32715	0.663577	+	0.748108i	$0.269038\pi$
$620$	3780.80	0.244904
$621$	−314.898	−0.0203485
$622$	−16848.5	−1.08611
$623$	0	0
$624$	−2341.26	−0.150201
$625$	5848.55	0.374308
$626$	4842.22	0.309160
$627$	−5837.75	−0.371830
$628$	−307.157	−0.0195173
$629$	2383.39	0.151085
$630$	0	0
$631$	−15693.6	−0.990099	−0.495049	−	0.868865i	$-0.664850\pi$
$631$	−15693.6	−0.990099	−0.495049	+	0.868865i	$0.664850\pi$
$632$	−2795.17	−0.175927
$633$	6803.82	0.427216
$634$	−2064.53	−0.129326
$635$	6526.23	0.407851
$636$	12828.2	0.799798
$637$	0	0
$638$	6860.48	0.425719
$639$	18799.0	1.16381
$640$	679.602	0.0419744
$641$	−13898.9	−0.856434	−0.428217	−	0.903676i	$-0.640858\pi$
$641$	−13898.9	−0.856434	−0.428217	+	0.903676i	$0.640858\pi$
$642$	−18309.7	−1.12559
$643$	−18032.4	−1.10595	−0.552977	−	0.833196i	$-0.686508\pi$
$643$	−18032.4	−1.10595	−0.552977	+	0.833196i	$0.686508\pi$
$644$	0	0
$645$	5506.30	0.336140
$646$	−11913.6	−0.725597
$647$	2023.00	0.122925	0.0614624	−	0.998109i	$-0.480424\pi$
$647$	2023.00	0.122925	0.0614624	+	0.998109i	$0.480424\pi$
$648$	−5409.36	−0.327932
$649$	−1937.76	−0.117202
$650$	3791.75	0.228807
$651$	0	0
$652$	8596.86	0.516379
$653$	−153.043	−0.00917154	−0.00458577	−	0.999989i	$-0.501460\pi$
$653$	−153.043	−0.00917154	−0.00458577	+	0.999989i	$0.501460\pi$
$654$	2349.90	0.140502
$655$	12853.7	0.766771
$656$	1719.87	0.102362
$657$	14889.5	0.884164
$658$	0	0
$659$	−5171.29	−0.305683	−0.152841	−	0.988251i	$-0.548842\pi$
$659$	−5171.29	−0.305683	−0.152841	+	0.988251i	$0.548842\pi$
$660$	2391.82	0.141063
$661$	4426.43	0.260466	0.130233	−	0.991483i	$-0.458427\pi$
$661$	4426.43	0.260466	0.130233	+	0.991483i	$0.458427\pi$
$662$	3630.71	0.213159
$663$	−16816.0	−0.985035
$664$	−2984.12	−0.174407
$665$	0	0
$666$	1195.95	0.0695828
$667$	−5234.46	−0.303866
$668$	8285.36	0.479896
$669$	−9633.80	−0.556748
$670$	5083.38	0.293116
$671$	−3405.31	−0.195917
$672$	0	0
$673$	19846.7	1.13675	0.568377	−	0.822768i	$-0.307571\pi$
$673$	19846.7	1.13675	0.568377	+	0.822768i	$0.307571\pi$
$674$	−16320.7	−0.932717
$675$	−1325.45	−0.0755804
$676$	−7253.97	−0.412720
$677$	23737.7	1.34758	0.673792	−	0.738921i	$-0.264664\pi$
$677$	23737.7	1.34758	0.673792	+	0.738921i	$0.264664\pi$
$678$	−18336.1	−1.03863
$679$	0	0
$680$	4881.20	0.275273
$681$	−10565.1	−0.594500
$682$	5366.49	0.301310
$683$	−4733.62	−0.265193	−0.132597	−	0.991170i	$-0.542332\pi$
$683$	−4733.62	−0.265193	−0.132597	+	0.991170i	$0.542332\pi$
$684$	−5978.08	−0.334178
$685$	17004.4	0.948475
$686$	0	0
$687$	5971.45	0.331623
$688$	2220.72	0.123058
$689$	−8405.24	−0.464752
$690$	−1824.93	−0.100687
$691$	275.544	0.0151696	0.00758480	−	0.999971i	$-0.497586\pi$
$691$	275.544	0.0151696	0.00758480	+	0.999971i	$0.497586\pi$
$692$	−12487.4	−0.685983
$693$	0	0
$694$	−11387.6	−0.622863
$695$	9080.89	0.495623
$696$	13604.3	0.740905
$697$	12352.9	0.671302
$698$	−9121.21	−0.494617
$699$	11614.7	0.628481
$700$	0	0
$701$	15236.5	0.820933	0.410467	−	0.911876i	$-0.365366\pi$
$701$	15236.5	0.820933	0.410467	+	0.911876i	$0.365366\pi$
$702$	−536.241	−0.0288306
$703$	−1075.04	−0.0576757
$704$	964.630	0.0516418
$705$	−850.999	−0.0454617
$706$	−17161.0	−0.914819
$707$	0	0
$708$	−3842.58	−0.203973
$709$	20262.7	1.07332	0.536659	−	0.843799i	$-0.319686\pi$
$709$	20262.7	1.07332	0.536659	+	0.843799i	$0.319686\pi$
$710$	6923.55	0.365967
$711$	−10073.9	−0.531366
$712$	−7173.55	−0.377585
$713$	−4094.56	−0.215067
$714$	0	0
$715$	−1567.16	−0.0819697
$716$	−12277.2	−0.640808
$717$	−52064.2	−2.71182
$718$	3978.30	0.206781
$719$	−4018.57	−0.208439	−0.104219	−	0.994554i	$-0.533234\pi$
$719$	−4018.57	−0.208439	−0.104219	+	0.994554i	$0.533234\pi$
$720$	2449.31	0.126778
$721$	0	0
$722$	−8344.29	−0.430114
$723$	37795.7	1.94417
$724$	−4434.81	−0.227650
$725$	−22032.6	−1.12865
$726$	−16495.8	−0.843273
$727$	−31658.7	−1.61507	−0.807536	−	0.589819i	$-0.799199\pi$
$727$	−31658.7	−1.61507	−0.807536	+	0.589819i	$0.799199\pi$
$728$	0	0
$729$	−22257.7	−1.13081
$730$	5483.73	0.278030
$731$	15950.2	0.807028
$732$	−6752.72	−0.340967
$733$	−35997.8	−1.81393	−0.906963	−	0.421210i	$-0.861605\pi$
$733$	−35997.8	−1.81393	−0.906963	+	0.421210i	$0.861605\pi$
$734$	229.776	0.0115548
$735$	0	0
$736$	−736.000	−0.0368605
$737$	7215.37	0.360626
$738$	6198.47	0.309172
$739$	8160.48	0.406209	0.203104	−	0.979157i	$-0.434897\pi$
$739$	8160.48	0.406209	0.203104	+	0.979157i	$0.434897\pi$
$740$	440.462	0.0218807
$741$	7584.94	0.376032
$742$	0	0
$743$	−2264.67	−0.111821	−0.0559104	−	0.998436i	$-0.517806\pi$
$743$	−2264.67	−0.111821	−0.0559104	+	0.998436i	$0.517806\pi$
$744$	10641.7	0.524388
$745$	−14056.3	−0.691251
$746$	25137.4	1.23371
$747$	−10754.9	−0.526774
$748$	6928.39	0.338673
$749$	0	0
$750$	−17599.5	−0.856855
$751$	−20177.1	−0.980391	−0.490195	−	0.871613i	$-0.663075\pi$
$751$	−20177.1	−0.980391	−0.490195	+	0.871613i	$0.663075\pi$
$752$	−343.211	−0.0166431
$753$	25131.5	1.21626
$754$	−8913.76	−0.430530
$755$	−5366.81	−0.258700
$756$	0	0
$757$	2404.49	0.115446	0.0577231	−	0.998333i	$-0.481616\pi$
$757$	2404.49	0.115446	0.0577231	+	0.998333i	$0.481616\pi$
$758$	−24151.0	−1.15726
$759$	−2590.31	−0.123876
$760$	−2201.69	−0.105084
$761$	−36396.7	−1.73374	−0.866871	−	0.498532i	$-0.833873\pi$
$761$	−36396.7	−1.73374	−0.866871	+	0.498532i	$0.833873\pi$
$762$	18369.2	0.873290
$763$	0	0
$764$	−12722.5	−0.602465
$765$	17592.0	0.831426
$766$	24456.0	1.15357
$767$	2517.72	0.118526
$768$	1912.86	0.0898754
$769$	29050.2	1.36226	0.681129	−	0.732163i	$-0.261489\pi$
$769$	29050.2	1.36226	0.681129	+	0.732163i	$0.261489\pi$
$770$	0	0
$771$	52456.3	2.45028
$772$	−3249.30	−0.151483
$773$	30769.4	1.43169	0.715847	−	0.698257i	$-0.246041\pi$
$773$	30769.4	1.43169	0.715847	+	0.698257i	$0.246041\pi$
$774$	8003.54	0.371681
$775$	−17234.6	−0.798820
$776$	10359.4	0.479227
$777$	0	0
$778$	6275.12	0.289170
$779$	−5571.82	−0.256266
$780$	−3107.67	−0.142657
$781$	9827.32	0.450255
$782$	−5286.27	−0.241735
$783$	3115.92	0.142214
$784$	0	0
$785$	−407.703	−0.0185370
$786$	36178.9	1.64181
$787$	−11172.5	−0.506043	−0.253022	−	0.967461i	$-0.581424\pi$
$787$	−11172.5	−0.506043	−0.253022	+	0.967461i	$0.581424\pi$
$788$	−15089.6	−0.682164
$789$	−8358.04	−0.377128
$790$	−3710.16	−0.167091
$791$	0	0
$792$	3476.56	0.155978
$793$	4424.49	0.198132
$794$	−30979.6	−1.38466
$795$	17027.5	0.759626
$796$	6097.10	0.271490
$797$	7065.60	0.314023	0.157012	−	0.987597i	$-0.449814\pi$
$797$	7065.60	0.314023	0.157012	+	0.987597i	$0.449814\pi$
$798$	0	0
$799$	−2465.09	−0.109147
$800$	−3097.93	−0.136911
$801$	−25853.8	−1.14045
$802$	−12105.0	−0.532973
$803$	7783.63	0.342065
$804$	14308.1	0.627620
$805$	0	0
$806$	−6972.63	−0.304715
$807$	20816.3	0.908014
$808$	−2473.84	−0.107710
$809$	37869.3	1.64575	0.822875	−	0.568222i	$-0.192368\pi$
$809$	37869.3	1.64575	0.822875	+	0.568222i	$0.192368\pi$
$810$	−7180.10	−0.311460
$811$	−33563.4	−1.45323	−0.726616	−	0.687044i	$-0.758908\pi$
$811$	−33563.4	−1.45323	−0.726616	+	0.687044i	$0.758908\pi$
$812$	0	0
$813$	20096.5	0.866931
$814$	625.194	0.0269202
$815$	11411.0	0.490442
$816$	13739.0	0.589413
$817$	−7194.41	−0.308079
$818$	−23843.5	−1.01916
$819$	0	0
$820$	2282.86	0.0972207
$821$	−17903.5	−0.761069	−0.380534	−	0.924767i	$-0.624260\pi$
$821$	−17903.5	−0.761069	−0.380534	+	0.924767i	$0.624260\pi$
$822$	47861.9	2.03087
$823$	6635.99	0.281064	0.140532	−	0.990076i	$-0.455119\pi$
$823$	6635.99	0.281064	0.140532	+	0.990076i	$0.455119\pi$
$824$	−4941.02	−0.208894
$825$	−10903.0	−0.460113
$826$	0	0
$827$	30086.7	1.26508	0.632538	−	0.774529i	$-0.282013\pi$
$827$	30086.7	1.26508	0.632538	+	0.774529i	$0.282013\pi$
$828$	−2652.57	−0.111332
$829$	−18965.3	−0.794562	−0.397281	−	0.917697i	$-0.630046\pi$
$829$	−18965.3	−0.794562	−0.397281	+	0.917697i	$0.630046\pi$
$830$	−3960.96	−0.165647
$831$	−6208.09	−0.259153
$832$	−1253.34	−0.0522254
$833$	0	0
$834$	25559.8	1.06123
$835$	10997.5	0.455791
$836$	−3125.09	−0.129287
$837$	2437.37	0.100655
$838$	14937.1	0.615743
$839$	5909.42	0.243166	0.121583	−	0.992581i	$-0.461203\pi$
$839$	5909.42	0.243166	0.121583	+	0.992581i	$0.461203\pi$
$840$	0	0
$841$	27405.9	1.12370
$842$	12079.2	0.494389
$843$	−66130.9	−2.70186
$844$	3642.25	0.148544
$845$	−9628.53	−0.391990
$846$	−1236.95	−0.0502685
$847$	0	0
$848$	6867.25	0.278092
$849$	4166.26	0.168416
$850$	−22250.7	−0.897874
$851$	−477.015	−0.0192149
$852$	19487.6	0.783606
$853$	4235.96	0.170031	0.0850156	−	0.996380i	$-0.472906\pi$
$853$	4235.96	0.170031	0.0850156	+	0.996380i	$0.472906\pi$
$854$	0	0
$855$	−7934.98	−0.317393
$856$	−9801.63	−0.391370
$857$	−22565.0	−0.899424	−0.449712	−	0.893174i	$-0.648473\pi$
$857$	−22565.0	−0.899424	−0.449712	+	0.893174i	$0.648473\pi$
$858$	−4411.04	−0.175513
$859$	3340.91	0.132701	0.0663505	−	0.997796i	$-0.478864\pi$
$859$	3340.91	0.132701	0.0663505	+	0.997796i	$0.478864\pi$
$860$	2947.66	0.116877
$861$	0	0
$862$	27104.3	1.07097
$863$	−49888.4	−1.96781	−0.983905	−	0.178691i	$-0.942814\pi$
$863$	−49888.4	−1.96781	−0.983905	+	0.178691i	$0.942814\pi$
$864$	438.119	0.0172513
$865$	−16575.1	−0.651527
$866$	−5615.86	−0.220364
$867$	61969.0	2.42742
$868$	0	0
$869$	−5266.22	−0.205575
$870$	18057.6	0.703691
$871$	−9374.86	−0.364702
$872$	1257.96	0.0488530
$873$	37335.6	1.44744
$874$	2384.40	0.0922811
$875$	0	0
$876$	15434.9	0.595316
$877$	41188.0	1.58588	0.792942	−	0.609297i	$-0.208548\pi$
$877$	41188.0	1.58588	0.792942	+	0.609297i	$0.208548\pi$
$878$	31797.5	1.22222
$879$	1968.95	0.0755528
$880$	1280.40	0.0490480
$881$	−19351.1	−0.740018	−0.370009	−	0.929028i	$-0.620645\pi$
$881$	−19351.1	−0.740018	−0.370009	+	0.929028i	$0.620645\pi$
$882$	0	0
$883$	−31330.1	−1.19404	−0.597022	−	0.802225i	$-0.703649\pi$
$883$	−31330.1	−1.19404	−0.597022	+	0.802225i	$0.703649\pi$
$884$	−9002.00	−0.342500
$885$	−5100.44	−0.193728
$886$	−18389.3	−0.697291
$887$	18838.7	0.713124	0.356562	−	0.934272i	$-0.383949\pi$
$887$	18838.7	0.713124	0.356562	+	0.934272i	$0.383949\pi$
$888$	1239.76	0.0468508
$889$	0	0
$890$	−9521.79	−0.358619
$891$	−10191.5	−0.383195
$892$	−5157.21	−0.193583
$893$	1111.90	0.0416665
$894$	−39563.9	−1.48010
$895$	−16296.0	−0.608622
$896$	0	0
$897$	3365.56	0.125276
$898$	−22463.5	−0.834761
$899$	40515.7	1.50309
$900$	−11165.1	−0.413521
$901$	49323.6	1.82376
$902$	3240.31	0.119612
$903$	0	0
$904$	−9815.75	−0.361136
$905$	−5886.53	−0.216215
$906$	−15105.8	−0.553926
$907$	−28754.5	−1.05268	−0.526339	−	0.850275i	$-0.676436\pi$
$907$	−28754.5	−1.05268	−0.526339	+	0.850275i	$0.676436\pi$
$908$	−5655.74	−0.206710
$909$	−8915.82	−0.325324
$910$	0	0
$911$	7679.37	0.279285	0.139643	−	0.990202i	$-0.455405\pi$
$911$	7679.37	0.279285	0.139643	+	0.990202i	$0.455405\pi$
$912$	−6197.05	−0.225005
$913$	−5622.20	−0.203798
$914$	−19503.9	−0.705834
$915$	−8963.21	−0.323841
$916$	3196.66	0.115306
$917$	0	0
$918$	3146.77	0.113136
$919$	45937.0	1.64888	0.824440	−	0.565950i	$-0.191490\pi$
$919$	45937.0	1.64888	0.824440	+	0.565950i	$0.191490\pi$
$920$	−976.927	−0.0350091
$921$	−46332.8	−1.65767
$922$	13457.7	0.480700
$923$	−12768.6	−0.455344
$924$	0	0
$925$	−2007.83	−0.0713696
$926$	−15330.4	−0.544046
$927$	−17807.6	−0.630938
$928$	7282.72	0.257615
$929$	35684.8	1.26026	0.630129	−	0.776490i	$-0.283002\pi$
$929$	35684.8	1.26026	0.630129	+	0.776490i	$0.283002\pi$
$930$	14125.3	0.498049
$931$	0	0
$932$	6217.64	0.218525
$933$	−62946.8	−2.20877
$934$	33511.9	1.17403
$935$	9196.38	0.321662
$936$	−4517.07	−0.157740
$937$	−4161.99	−0.145108	−0.0725540	−	0.997364i	$-0.523115\pi$
$937$	−4161.99	−0.145108	−0.0725540	+	0.997364i	$0.523115\pi$
$938$	0	0
$939$	18090.8	0.628722
$940$	−455.561	−0.0158072
$941$	30661.7	1.06221	0.531106	−	0.847305i	$-0.321776\pi$
$941$	30661.7	1.06221	0.531106	+	0.847305i	$0.321776\pi$
$942$	−1147.55	−0.0396914
$943$	−2472.31	−0.0853760
$944$	−2057.03	−0.0709222
$945$	0	0
$946$	4183.92	0.143796
$947$	9165.36	0.314503	0.157251	−	0.987559i	$-0.449737\pi$
$947$	9165.36	0.314503	0.157251	+	0.987559i	$0.449737\pi$
$948$	−10442.9	−0.357774
$949$	−10113.2	−0.345931
$950$	10036.3	0.342759
$951$	−7713.18	−0.263004
$952$	0	0
$953$	−27150.4	−0.922862	−0.461431	−	0.887176i	$-0.652664\pi$
$953$	−27150.4	−0.922862	−0.461431	+	0.887176i	$0.652664\pi$
$954$	24749.8	0.839943
$955$	−16887.1	−0.572204
$956$	−27871.2	−0.942908
$957$	25631.1	0.865763
$958$	−11037.1	−0.372227
$959$	0	0
$960$	2539.03	0.0853612
$961$	1901.68	0.0638340
$962$	−812.309	−0.0272244
$963$	−35325.5	−1.18208
$964$	20233.0	0.675996
$965$	−4312.95	−0.143874
$966$	0	0
$967$	−34024.7	−1.13150	−0.565750	−	0.824577i	$-0.691413\pi$
$967$	−34024.7	−1.13150	−0.565750	+	0.824577i	$0.691413\pi$
$968$	−8830.60	−0.293209
$969$	−44509.9	−1.47561
$970$	13750.5	0.455156
$971$	−18794.4	−0.621154	−0.310577	−	0.950548i	$-0.600522\pi$
$971$	−18794.4	−0.621154	−0.310577	+	0.950548i	$0.600522\pi$
$972$	−21688.3	−0.715692
$973$	0	0
$974$	7907.83	0.260147
$975$	14166.2	0.465313
$976$	−3614.90	−0.118555
$977$	−60427.0	−1.97874	−0.989371	−	0.145411i	$-0.953549\pi$
$977$	−60427.0	−1.97874	−0.989371	+	0.145411i	$0.953549\pi$
$978$	32118.3	1.05013
$979$	−13515.3	−0.441216
$980$	0	0
$981$	4533.73	0.147554
$982$	25461.1	0.827390
$983$	−38938.6	−1.26343	−0.631713	−	0.775203i	$-0.717648\pi$
$983$	−38938.6	−1.26343	−0.631713	+	0.775203i	$0.717648\pi$
$984$	6425.52	0.208169
$985$	−20029.1	−0.647900
$986$	52307.7	1.68947
$987$	0	0
$988$	4060.40	0.130748
$989$	−3192.28	−0.102638
$990$	4614.61	0.148143
$991$	−31599.6	−1.01291	−0.506455	−	0.862267i	$-0.669044\pi$
$991$	−31599.6	−1.01291	−0.506455	+	0.862267i	$0.669044\pi$
$992$	5696.78	0.182332
$993$	13564.5	0.433491
$994$	0	0
$995$	8092.97	0.257854
$996$	−11148.8	−0.354682
$997$	−38706.6	−1.22954	−0.614770	−	0.788707i	$-0.710751\pi$
$997$	−38706.6	−1.22954	−0.614770	+	0.788707i	$0.710751\pi$
$998$	−36039.8	−1.14310
$999$	283.953	0.00899287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.10		11
7.3	odd	6		322.4.e.b.93.10	✓	22
7.5	odd	6		322.4.e.b.277.10	yes	22
7.6	odd	2		2254.4.a.w.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.10	✓	22	7.3	odd	6
322.4.e.b.277.10	yes	22	7.5	odd	6
2254.4.a.w.1.2		11	7.6	odd	2
2254.4.a.x.1.10		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +7.47210 q^{3} +4.00000 q^{4} +5.30939 q^{5} +14.9442 q^{6} +8.00000 q^{8} +28.8323 q^{9} +10.6188 q^{10} +15.0723 q^{11} +29.8884 q^{12} -19.5834 q^{13} +39.6723 q^{15} +16.0000 q^{16} +114.919 q^{17} +57.6646 q^{18} -51.8349 q^{19} +21.2376 q^{20} +30.1447 q^{22} -23.0000 q^{23} +59.7768 q^{24} -96.8104 q^{25} -39.1667 q^{26} +13.6912 q^{27} +227.585 q^{29} +79.3446 q^{30} +178.024 q^{31} +32.0000 q^{32} +112.622 q^{33} +229.838 q^{34} +115.329 q^{36} +20.7398 q^{37} -103.670 q^{38} -146.329 q^{39} +42.4751 q^{40} +107.492 q^{41} +138.795 q^{43} +60.2894 q^{44} +153.082 q^{45} -46.0000 q^{46} -21.4507 q^{47} +119.554 q^{48} -193.621 q^{50} +858.687 q^{51} -78.3334 q^{52} +429.203 q^{53} +27.3825 q^{54} +80.0249 q^{55} -387.316 q^{57} +455.170 q^{58} -128.564 q^{59} +158.689 q^{60} -225.931 q^{61} +356.049 q^{62} +64.0000 q^{64} -103.976 q^{65} +225.244 q^{66} +478.716 q^{67} +459.676 q^{68} -171.858 q^{69} +652.010 q^{71} +230.659 q^{72} +516.418 q^{73} +41.4796 q^{74} -723.377 q^{75} -207.340 q^{76} -292.658 q^{78} -349.397 q^{79} +84.9502 q^{80} -676.170 q^{81} +214.984 q^{82} -373.014 q^{83} +610.150 q^{85} +277.589 q^{86} +1700.54 q^{87} +120.579 q^{88} -896.694 q^{89} +306.164 q^{90} -92.0000 q^{92} +1330.22 q^{93} -42.9014 q^{94} -275.211 q^{95} +239.107 q^{96} +1294.92 q^{97} +434.570 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 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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