LMFDB - Embedded newform 2254.4.a.x.1.8

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			1.8
Root			$-2.74955$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.74955 q^{3} +4.00000 q^{4} -14.1785 q^{5} +7.49911 q^{6} +8.00000 q^{8} -12.9409 q^{9} -28.3569 q^{10} +48.6127 q^{11} +14.9982 q^{12} +60.3778 q^{13} -53.1629 q^{15} +16.0000 q^{16} +29.0238 q^{17} -25.8817 q^{18} +0.266041 q^{19} -56.7138 q^{20} +97.2254 q^{22} -23.0000 q^{23} +29.9964 q^{24} +76.0287 q^{25} +120.756 q^{26} -149.760 q^{27} -67.2909 q^{29} -106.326 q^{30} +89.5691 q^{31} +32.0000 q^{32} +182.276 q^{33} +58.0476 q^{34} -51.7634 q^{36} -241.316 q^{37} +0.532083 q^{38} +226.390 q^{39} -113.428 q^{40} +106.382 q^{41} -215.270 q^{43} +194.451 q^{44} +183.481 q^{45} -46.0000 q^{46} +216.082 q^{47} +59.9928 q^{48} +152.057 q^{50} +108.826 q^{51} +241.511 q^{52} +44.3423 q^{53} -299.521 q^{54} -689.253 q^{55} +0.997537 q^{57} -134.582 q^{58} +228.015 q^{59} -212.652 q^{60} +375.759 q^{61} +179.138 q^{62} +64.0000 q^{64} -856.064 q^{65} +364.552 q^{66} +1019.88 q^{67} +116.095 q^{68} -86.2397 q^{69} -243.002 q^{71} -103.527 q^{72} +46.2389 q^{73} -482.632 q^{74} +285.073 q^{75} +1.06417 q^{76} +452.779 q^{78} -214.895 q^{79} -226.855 q^{80} -212.131 q^{81} +212.763 q^{82} +829.464 q^{83} -411.513 q^{85} -430.540 q^{86} -252.311 q^{87} +388.902 q^{88} -995.496 q^{89} +366.963 q^{90} -92.0000 q^{92} +335.844 q^{93} +432.165 q^{94} -3.77206 q^{95} +119.986 q^{96} +1095.74 q^{97} -629.090 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.74955	0.721602	0.360801	−	0.932643i	$-0.382503\pi$
$3$	3.74955	0.721602	0.360801	+	0.932643i	$0.382503\pi$
$4$	4.00000	0.500000
$5$	−14.1785	−1.26816	−0.634080	−	0.773268i	$-0.718621\pi$
$5$	−14.1785	−1.26816	−0.634080	+	0.773268i	$0.718621\pi$
$6$	7.49911	0.510249
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−12.9409	−0.479291
$10$	−28.3569	−0.896724
$11$	48.6127	1.33248	0.666240	−	0.745737i	$-0.267903\pi$
$11$	48.6127	1.33248	0.666240	+	0.745737i	$0.267903\pi$
$12$	14.9982	0.360801
$13$	60.3778	1.28814	0.644069	−	0.764968i	$-0.277245\pi$
$13$	60.3778	1.28814	0.644069	+	0.764968i	$0.277245\pi$
$14$	0	0
$15$	−53.1629	−0.915106
$16$	16.0000	0.250000
$17$	29.0238	0.414077	0.207038	−	0.978333i	$-0.433618\pi$
$17$	29.0238	0.414077	0.207038	+	0.978333i	$0.433618\pi$
$18$	−25.8817	−0.338910
$19$	0.266041	0.00321232	0.00160616	−	0.999999i	$-0.499489\pi$
$19$	0.266041	0.00321232	0.00160616	+	0.999999i	$0.499489\pi$
$20$	−56.7138	−0.634080
$21$	0	0
$22$	97.2254	0.942205
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	29.9964	0.255125
$25$	76.0287	0.608229
$26$	120.756	0.910851
$27$	−149.760	−1.06746
$28$	0	0
$29$	−67.2909	−0.430883	−0.215441	−	0.976517i	$-0.569119\pi$
$29$	−67.2909	−0.430883	−0.215441	+	0.976517i	$0.569119\pi$
$30$	−106.326	−0.647078
$31$	89.5691	0.518938	0.259469	−	0.965751i	$-0.416452\pi$
$31$	89.5691	0.518938	0.259469	+	0.965751i	$0.416452\pi$
$32$	32.0000	0.176777
$33$	182.276	0.961520
$34$	58.0476	0.292796
$35$	0	0
$36$	−51.7634	−0.239645
$37$	−241.316	−1.07222	−0.536109	−	0.844148i	$-0.680107\pi$
$37$	−241.316	−1.07222	−0.536109	+	0.844148i	$0.680107\pi$
$38$	0.532083	0.00227145
$39$	226.390	0.929522
$40$	−113.428	−0.448362
$41$	106.382	0.405220	0.202610	−	0.979260i	$-0.435058\pi$
$41$	106.382	0.405220	0.202610	+	0.979260i	$0.435058\pi$
$42$	0	0
$43$	−215.270	−0.763450	−0.381725	−	0.924276i	$-0.624670\pi$
$43$	−215.270	−0.763450	−0.381725	+	0.924276i	$0.624670\pi$
$44$	194.451	0.666240
$45$	183.481	0.607817
$46$	−46.0000	−0.147442
$47$	216.082	0.670614	0.335307	−	0.942109i	$-0.391160\pi$
$47$	216.082	0.670614	0.335307	+	0.942109i	$0.391160\pi$
$48$	59.9928	0.180400
$49$	0	0
$50$	152.057	0.430083
$51$	108.826	0.298798
$52$	241.511	0.644069
$53$	44.3423	0.114922	0.0574611	−	0.998348i	$-0.481699\pi$
$53$	44.3423	0.114922	0.0574611	+	0.998348i	$0.481699\pi$
$54$	−299.521	−0.754807
$55$	−689.253	−1.68980
$56$	0	0
$57$	0.997537	0.00231802
$58$	−134.582	−0.304680
$59$	228.015	0.503135	0.251568	−	0.967840i	$-0.419054\pi$
$59$	228.015	0.503135	0.251568	+	0.967840i	$0.419054\pi$
$60$	−212.652	−0.457553
$61$	375.759	0.788706	0.394353	−	0.918959i	$-0.370969\pi$
$61$	375.759	0.788706	0.394353	+	0.918959i	$0.370969\pi$
$62$	179.138	0.366945
$63$	0	0
$64$	64.0000	0.125000
$65$	−856.064	−1.63356
$66$	364.552	0.679897
$67$	1019.88	1.85967	0.929833	−	0.367983i	$-0.119952\pi$
$67$	1019.88	1.85967	0.929833	+	0.367983i	$0.119952\pi$
$68$	116.095	0.207038
$69$	−86.2397	−0.150464
$70$	0	0
$71$	−243.002	−0.406183	−0.203092	−	0.979160i	$-0.565099\pi$
$71$	−243.002	−0.406183	−0.203092	+	0.979160i	$0.565099\pi$
$72$	−103.527	−0.169455
$73$	46.2389	0.0741350	0.0370675	−	0.999313i	$-0.488198\pi$
$73$	46.2389	0.0741350	0.0370675	+	0.999313i	$0.488198\pi$
$74$	−482.632	−0.758173
$75$	285.073	0.438899
$76$	1.06417	0.00160616
$77$	0	0
$78$	452.779	0.657272
$79$	−214.895	−0.306045	−0.153023	−	0.988223i	$-0.548901\pi$
$79$	−214.895	−0.306045	−0.153023	+	0.988223i	$0.548901\pi$
$80$	−226.855	−0.317040
$81$	−212.131	−0.290989
$82$	212.763	0.286534
$83$	829.464	1.09693	0.548467	−	0.836172i	$-0.315212\pi$
$83$	829.464	1.09693	0.548467	+	0.836172i	$0.315212\pi$
$84$	0	0
$85$	−411.513	−0.525115
$86$	−430.540	−0.539840
$87$	−252.311	−0.310926
$88$	388.902	0.471103
$89$	−995.496	−1.18564	−0.592822	−	0.805333i	$-0.701986\pi$
$89$	−995.496	−1.18564	−0.592822	+	0.805333i	$0.701986\pi$
$90$	366.963	0.429792
$91$	0	0
$92$	−92.0000	−0.104257
$93$	335.844	0.374467
$94$	432.165	0.474196
$95$	−3.77206	−0.00407374
$96$	119.986	0.127562
$97$	1095.74	1.14697	0.573483	−	0.819217i	$-0.305592\pi$
$97$	1095.74	1.14697	0.573483	+	0.819217i	$0.305592\pi$
$98$	0	0
$99$	−629.090	−0.638645
$100$	304.115	0.304115
$101$	1447.69	1.42624	0.713120	−	0.701042i	$-0.247281\pi$
$101$	1447.69	1.42624	0.713120	+	0.701042i	$0.247281\pi$
$102$	217.652	0.211282
$103$	1601.06	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$103$	1601.06	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$104$	483.022	0.455425
$105$	0	0
$106$	88.6845	0.0812623
$107$	1276.99	1.15375	0.576875	−	0.816833i	$-0.304272\pi$
$107$	1276.99	1.15375	0.576875	+	0.816833i	$0.304272\pi$
$108$	−599.041	−0.533729
$109$	−557.328	−0.489747	−0.244873	−	0.969555i	$-0.578746\pi$
$109$	−557.328	−0.489747	−0.244873	+	0.969555i	$0.578746\pi$
$110$	−1378.51	−1.19487
$111$	−904.827	−0.773715
$112$	0	0
$113$	−559.393	−0.465693	−0.232846	−	0.972514i	$-0.574804\pi$
$113$	−559.393	−0.465693	−0.232846	+	0.972514i	$0.574804\pi$
$114$	1.99507	0.00163909
$115$	326.105	0.264430
$116$	−269.164	−0.215441
$117$	−781.340	−0.617393
$118$	456.029	0.355770
$119$	0	0
$120$	−425.303	−0.323539
$121$	1032.19	0.775502
$122$	751.519	0.557699
$123$	398.884	0.292407
$124$	358.276	0.259469
$125$	694.338	0.496828
$126$	0	0
$127$	1160.62	0.810932	0.405466	−	0.914110i	$-0.367109\pi$
$127$	1160.62	0.810932	0.405466	+	0.914110i	$0.367109\pi$
$128$	128.000	0.0883883
$129$	−807.165	−0.550907
$130$	−1712.13	−1.15510
$131$	−2661.57	−1.77513	−0.887567	−	0.460679i	$-0.847606\pi$
$131$	−2661.57	−1.77513	−0.887567	+	0.460679i	$0.847606\pi$
$132$	729.103	0.480760
$133$	0	0
$134$	2039.75	1.31498
$135$	2123.37	1.35371
$136$	232.190	0.146398
$137$	2994.06	1.86715	0.933575	−	0.358382i	$-0.116672\pi$
$137$	2994.06	1.86715	0.933575	+	0.358382i	$0.116672\pi$
$138$	−172.479	−0.106394
$139$	1480.91	0.903665	0.451833	−	0.892103i	$-0.350770\pi$
$139$	1480.91	0.903665	0.451833	+	0.892103i	$0.350770\pi$
$140$	0	0
$141$	810.212	0.483916
$142$	−486.003	−0.287215
$143$	2935.13	1.71642
$144$	−207.054	−0.119823
$145$	954.081	0.546428
$146$	92.4778	0.0524213
$147$	0	0
$148$	−965.264	−0.536109
$149$	1176.82	0.647041	0.323521	−	0.946221i	$-0.395134\pi$
$149$	1176.82	0.647041	0.323521	+	0.946221i	$0.395134\pi$
$150$	570.147	0.310349
$151$	221.550	0.119401	0.0597004	−	0.998216i	$-0.480985\pi$
$151$	221.550	0.119401	0.0597004	+	0.998216i	$0.480985\pi$
$152$	2.12833	0.00113573
$153$	−375.593	−0.198463
$154$	0	0
$155$	−1269.95	−0.658096
$156$	905.559	0.464761
$157$	−2091.05	−1.06296	−0.531478	−	0.847072i	$-0.678363\pi$
$157$	−2091.05	−1.06296	−0.531478	+	0.847072i	$0.678363\pi$
$158$	−429.790	−0.216407
$159$	166.264	0.0829281
$160$	−453.711	−0.224181
$161$	0	0
$162$	−424.262	−0.205761
$163$	702.545	0.337593	0.168796	−	0.985651i	$-0.446012\pi$
$163$	702.545	0.337593	0.168796	+	0.985651i	$0.446012\pi$
$164$	425.527	0.202610
$165$	−2584.39	−1.21936
$166$	1658.93	0.775649
$167$	2797.35	1.29620	0.648101	−	0.761555i	$-0.275564\pi$
$167$	2797.35	1.29620	0.648101	+	0.761555i	$0.275564\pi$
$168$	0	0
$169$	1448.48	0.659298
$170$	−823.025	−0.371313
$171$	−3.44280	−0.00153964
$172$	−861.079	−0.381725
$173$	461.133	0.202655	0.101328	−	0.994853i	$-0.467691\pi$
$173$	461.133	0.202655	0.101328	+	0.994853i	$0.467691\pi$
$174$	−504.622	−0.219858
$175$	0	0
$176$	777.803	0.333120
$177$	854.953	0.363063
$178$	−1990.99	−0.838377
$179$	36.1898	0.0151115	0.00755573	−	0.999971i	$-0.497595\pi$
$179$	36.1898	0.0151115	0.00755573	+	0.999971i	$0.497595\pi$
$180$	733.925	0.303909
$181$	3882.22	1.59427	0.797136	−	0.603799i	$-0.206347\pi$
$181$	3882.22	1.59427	0.797136	+	0.603799i	$0.206347\pi$
$182$	0	0
$183$	1408.93	0.569132
$184$	−184.000	−0.0737210
$185$	3421.49	1.35974
$186$	671.688	0.264788
$187$	1410.92	0.551749
$188$	864.329	0.335307
$189$	0	0
$190$	−7.54412	−0.00288057
$191$	3908.40	1.48064	0.740318	−	0.672257i	$-0.234675\pi$
$191$	3908.40	1.48064	0.740318	+	0.672257i	$0.234675\pi$
$192$	239.971	0.0902002
$193$	−1399.25	−0.521866	−0.260933	−	0.965357i	$-0.584030\pi$
$193$	−1399.25	−0.521866	−0.260933	+	0.965357i	$0.584030\pi$
$194$	2191.48	0.811027
$195$	−3209.86	−1.17878
$196$	0	0
$197$	2441.53	0.883004	0.441502	−	0.897260i	$-0.354446\pi$
$197$	2441.53	0.883004	0.441502	+	0.897260i	$0.354446\pi$
$198$	−1258.18	−0.451591
$199$	2188.86	0.779719	0.389860	−	0.920874i	$-0.372524\pi$
$199$	2188.86	0.779719	0.389860	+	0.920874i	$0.372524\pi$
$200$	608.229	0.215042
$201$	3824.08	1.34194
$202$	2895.38	1.00850
$203$	0	0
$204$	435.305	0.149399
$205$	−1508.33	−0.513884
$206$	3202.11	1.08302
$207$	297.640	0.0999391
$208$	966.045	0.322034
$209$	12.9330	0.00428035
$210$	0	0
$211$	−5185.63	−1.69191	−0.845956	−	0.533252i	$-0.820970\pi$
$211$	−5185.63	−1.69191	−0.845956	+	0.533252i	$0.820970\pi$
$212$	177.369	0.0574611
$213$	−911.148	−0.293102
$214$	2553.98	0.815824
$215$	3052.19	0.968176
$216$	−1198.08	−0.377404
$217$	0	0
$218$	−1114.66	−0.346303
$219$	173.375	0.0534959
$220$	−2757.01	−0.844899
$221$	1752.39	0.533388
$222$	−1809.65	−0.547099
$223$	−1016.81	−0.305339	−0.152669	−	0.988277i	$-0.548787\pi$
$223$	−1016.81	−0.305339	−0.152669	+	0.988277i	$0.548787\pi$
$224$	0	0
$225$	−983.876	−0.291519
$226$	−1118.79	−0.329295
$227$	−2411.55	−0.705112	−0.352556	−	0.935791i	$-0.614687\pi$
$227$	−2411.55	−0.705112	−0.352556	+	0.935791i	$0.614687\pi$
$228$	3.99015	0.00115901
$229$	4621.34	1.33357	0.666783	−	0.745252i	$-0.267671\pi$
$229$	4621.34	1.33357	0.666783	+	0.745252i	$0.267671\pi$
$230$	652.209	0.186980
$231$	0	0
$232$	−538.327	−0.152340
$233$	−3487.53	−0.980584	−0.490292	−	0.871558i	$-0.663110\pi$
$233$	−3487.53	−0.980584	−0.490292	+	0.871558i	$0.663110\pi$
$234$	−1562.68	−0.436563
$235$	−3063.71	−0.850445
$236$	912.058	0.251568
$237$	−805.760	−0.220843
$238$	0	0
$239$	−6164.99	−1.66854	−0.834268	−	0.551359i	$-0.814109\pi$
$239$	−6164.99	−1.66854	−0.834268	+	0.551359i	$0.814109\pi$
$240$	−850.606	−0.228777
$241$	−1225.09	−0.327449	−0.163725	−	0.986506i	$-0.552351\pi$
$241$	−1225.09	−0.327449	−0.163725	+	0.986506i	$0.552351\pi$
$242$	2064.39	0.548363
$243$	3248.13	0.857481
$244$	1503.04	0.394353
$245$	0	0
$246$	797.767	0.206763
$247$	16.0630	0.00413791
$248$	716.553	0.183472
$249$	3110.12	0.791549
$250$	1388.68	0.351310
$251$	1499.55	0.377094	0.188547	−	0.982064i	$-0.439622\pi$
$251$	1499.55	0.377094	0.188547	+	0.982064i	$0.439622\pi$
$252$	0	0
$253$	−1118.09	−0.277841
$254$	2321.24	0.573416
$255$	−1542.99	−0.378924
$256$	256.000	0.0625000
$257$	3362.45	0.816124	0.408062	−	0.912954i	$-0.366205\pi$
$257$	3362.45	0.816124	0.408062	+	0.912954i	$0.366205\pi$
$258$	−1614.33	−0.389550
$259$	0	0
$260$	−3424.26	−0.816782
$261$	870.802	0.206518
$262$	−5323.14	−1.25521
$263$	−6287.94	−1.47426	−0.737131	−	0.675750i	$-0.763820\pi$
$263$	−6287.94	−1.47426	−0.737131	+	0.675750i	$0.763820\pi$
$264$	1458.21	0.339949
$265$	−628.705	−0.145740
$266$	0	0
$267$	−3732.66	−0.855563
$268$	4079.50	0.929833
$269$	2213.82	0.501781	0.250891	−	0.968015i	$-0.419277\pi$
$269$	2213.82	0.501781	0.250891	+	0.968015i	$0.419277\pi$
$270$	4246.74	0.957216
$271$	−2651.01	−0.594233	−0.297117	−	0.954841i	$-0.596025\pi$
$271$	−2651.01	−0.594233	−0.297117	+	0.954841i	$0.596025\pi$
$272$	464.381	0.103519
$273$	0	0
$274$	5988.11	1.32027
$275$	3695.96	0.810453
$276$	−344.959	−0.0752322
$277$	−2337.19	−0.506960	−0.253480	−	0.967341i	$-0.581575\pi$
$277$	−2337.19	−0.506960	−0.253480	+	0.967341i	$0.581575\pi$
$278$	2961.83	0.638988
$279$	−1159.10	−0.248722
$280$	0	0
$281$	1693.43	0.359506	0.179753	−	0.983712i	$-0.442470\pi$
$281$	1693.43	0.359506	0.179753	+	0.983712i	$0.442470\pi$
$282$	1620.42	0.342180
$283$	−110.893	−0.0232930	−0.0116465	−	0.999932i	$-0.503707\pi$
$283$	−110.893	−0.0232930	−0.0116465	+	0.999932i	$0.503707\pi$
$284$	−972.007	−0.203092
$285$	−14.1435	−0.00293962
$286$	5870.25	1.21369
$287$	0	0
$288$	−414.107	−0.0847275
$289$	−4070.62	−0.828541
$290$	1908.16	0.386383
$291$	4108.54	0.827653
$292$	184.956	0.0370675
$293$	−7665.00	−1.52831	−0.764153	−	0.645035i	$-0.776843\pi$
$293$	−7665.00	−1.52831	−0.764153	+	0.645035i	$0.776843\pi$
$294$	0	0
$295$	−3232.90	−0.638056
$296$	−1930.53	−0.379087
$297$	−7280.25	−1.42237
$298$	2353.65	0.457527
$299$	−1388.69	−0.268595
$300$	1140.29	0.219450
$301$	0	0
$302$	443.101	0.0844291
$303$	5428.18	1.02918
$304$	4.25666	0.000803080 0
$305$	−5327.69	−1.00021
$306$	−751.185	−0.140335
$307$	−3951.57	−0.734620	−0.367310	−	0.930099i	$-0.619721\pi$
$307$	−3951.57	−0.734620	−0.367310	+	0.930099i	$0.619721\pi$
$308$	0	0
$309$	6003.25	1.10522
$310$	−2539.90	−0.465344
$311$	6938.06	1.26502	0.632510	−	0.774552i	$-0.282025\pi$
$311$	6938.06	1.26502	0.632510	+	0.774552i	$0.282025\pi$
$312$	1811.12	0.328636
$313$	4221.98	0.762430	0.381215	−	0.924486i	$-0.375506\pi$
$313$	4221.98	0.762430	0.381215	+	0.924486i	$0.375506\pi$
$314$	−4182.10	−0.751623
$315$	0	0
$316$	−859.580	−0.153023
$317$	−1689.93	−0.299420	−0.149710	−	0.988730i	$-0.547834\pi$
$317$	−1689.93	−0.299420	−0.149710	+	0.988730i	$0.547834\pi$
$318$	332.527	0.0586390
$319$	−3271.19	−0.574143
$320$	−907.421	−0.158520
$321$	4788.14	0.832548
$322$	0	0
$323$	7.72153	0.00133015
$324$	−848.525	−0.145495
$325$	4590.44	0.783483
$326$	1405.09	0.238714
$327$	−2089.73	−0.353402
$328$	851.053	0.143267
$329$	0	0
$330$	−5168.78	−0.862218
$331$	3728.75	0.619186	0.309593	−	0.950869i	$-0.399807\pi$
$331$	3728.75	0.619186	0.309593	+	0.950869i	$0.399807\pi$
$332$	3317.85	0.548467
$333$	3122.83	0.513905
$334$	5594.71	0.916553
$335$	−14460.3	−2.35835
$336$	0	0
$337$	837.300	0.135343	0.0676716	−	0.997708i	$-0.478443\pi$
$337$	837.300	0.135343	0.0676716	+	0.997708i	$0.478443\pi$
$338$	2896.96	0.466194
$339$	−2097.47	−0.336045
$340$	−1646.05	−0.262558
$341$	4354.19	0.691474
$342$	−6.88561	−0.00108869
$343$	0	0
$344$	−1722.16	−0.269920
$345$	1222.75	0.190813
$346$	922.267	0.143299
$347$	1107.14	0.171281	0.0856405	−	0.996326i	$-0.472706\pi$
$347$	1107.14	0.171281	0.0856405	+	0.996326i	$0.472706\pi$
$348$	−1009.24	−0.155463
$349$	6461.90	0.991111	0.495555	−	0.868576i	$-0.334965\pi$
$349$	6461.90	0.991111	0.495555	+	0.868576i	$0.334965\pi$
$350$	0	0
$351$	−9042.20	−1.37503
$352$	1555.61	0.235551
$353$	−4969.58	−0.749304	−0.374652	−	0.927166i	$-0.622238\pi$
$353$	−4969.58	−0.749304	−0.374652	+	0.927166i	$0.622238\pi$
$354$	1709.91	0.256724
$355$	3445.39	0.515105
$356$	−3981.98	−0.592822
$357$	0	0
$358$	72.3796	0.0106854
$359$	−1290.16	−0.189671	−0.0948356	−	0.995493i	$-0.530233\pi$
$359$	−1290.16	−0.189671	−0.0948356	+	0.995493i	$0.530233\pi$
$360$	1467.85	0.214896
$361$	−6858.93	−0.999990
$362$	7764.45	1.12732
$363$	3870.26	0.559604
$364$	0	0
$365$	−655.596	−0.0940150
$366$	2817.86	0.402437
$367$	10709.3	1.52322	0.761610	−	0.648035i	$-0.224409\pi$
$367$	10709.3	1.52322	0.761610	+	0.648035i	$0.224409\pi$
$368$	−368.000	−0.0521286
$369$	−1376.67	−0.194218
$370$	6842.98	0.961485
$371$	0	0
$372$	1343.38	0.187233
$373$	−10316.3	−1.43206	−0.716031	−	0.698068i	$-0.754043\pi$
$373$	−10316.3	−1.43206	−0.716031	+	0.698068i	$0.754043\pi$
$374$	2821.85	0.390145
$375$	2603.46	0.358512
$376$	1728.66	0.237098
$377$	−4062.88	−0.555037
$378$	0	0
$379$	−12606.5	−1.70859	−0.854293	−	0.519792i	$-0.826009\pi$
$379$	−12606.5	−1.70859	−0.854293	+	0.519792i	$0.826009\pi$
$380$	−15.0882	−0.00203687
$381$	4351.81	0.585170
$382$	7816.79	1.04697
$383$	−14702.1	−1.96147	−0.980734	−	0.195349i	$-0.937416\pi$
$383$	−14702.1	−1.96147	−0.980734	+	0.195349i	$0.937416\pi$
$384$	479.943	0.0637812
$385$	0	0
$386$	−2798.50	−0.369015
$387$	2785.77	0.365914
$388$	4382.97	0.573483
$389$	14410.1	1.87820	0.939100	−	0.343645i	$-0.111662\pi$
$389$	14410.1	1.87820	0.939100	+	0.343645i	$0.111662\pi$
$390$	−6419.71	−0.833525
$391$	−667.547	−0.0863409
$392$	0	0
$393$	−9979.70	−1.28094
$394$	4883.06	0.624378
$395$	3046.88	0.388114
$396$	−2516.36	−0.319323
$397$	9966.02	1.25990	0.629950	−	0.776636i	$-0.283075\pi$
$397$	9966.02	1.25990	0.629950	+	0.776636i	$0.283075\pi$
$398$	4377.72	0.551345
$399$	0	0
$400$	1216.46	0.152057
$401$	8234.71	1.02549	0.512746	−	0.858541i	$-0.328628\pi$
$401$	8234.71	1.02549	0.512746	+	0.858541i	$0.328628\pi$
$402$	7648.15	0.948893
$403$	5407.98	0.668464
$404$	5790.75	0.713120
$405$	3007.69	0.369021
$406$	0	0
$407$	−11731.0	−1.42871
$408$	870.610	0.105641
$409$	−5061.02	−0.611861	−0.305930	−	0.952054i	$-0.598968\pi$
$409$	−5061.02	−0.611861	−0.305930	+	0.952054i	$0.598968\pi$
$410$	−3016.66	−0.363371
$411$	11226.4	1.34734
$412$	6404.23	0.765810
$413$	0	0
$414$	595.279	0.0706676
$415$	−11760.5	−1.39109
$416$	1932.09	0.227713
$417$	5552.76	0.652086
$418$	25.8660	0.00302667
$419$	−10902.9	−1.27122	−0.635611	−	0.772010i	$-0.719252\pi$
$419$	−10902.9	−1.27122	−0.635611	+	0.772010i	$0.719252\pi$
$420$	0	0
$421$	−16125.7	−1.86680	−0.933398	−	0.358843i	$-0.883171\pi$
$421$	−16125.7	−1.86680	−0.933398	+	0.358843i	$0.883171\pi$
$422$	−10371.3	−1.19636
$423$	−2796.29	−0.321419
$424$	354.738	0.0406311
$425$	2206.64	0.251854
$426$	−1822.30	−0.207255
$427$	0	0
$428$	5107.96	0.576875
$429$	11005.4	1.23857
$430$	6104.39	0.684604
$431$	−14451.5	−1.61509	−0.807545	−	0.589807i	$-0.799204\pi$
$431$	−14451.5	−1.61509	−0.807545	+	0.589807i	$0.799204\pi$
$432$	−2396.17	−0.266865
$433$	132.229	0.0146755	0.00733776	−	0.999973i	$-0.497664\pi$
$433$	132.229	0.0146755	0.00733776	+	0.999973i	$0.497664\pi$
$434$	0	0
$435$	3577.38	0.394304
$436$	−2229.31	−0.244873
$437$	−6.11895	−0.000669815 0
$438$	346.750	0.0378273
$439$	6294.88	0.684370	0.342185	−	0.939633i	$-0.388833\pi$
$439$	6294.88	0.684370	0.342185	+	0.939633i	$0.388833\pi$
$440$	−5514.02	−0.597434
$441$	0	0
$442$	3504.78	0.377162
$443$	4345.97	0.466102	0.233051	−	0.972465i	$-0.425129\pi$
$443$	4345.97	0.466102	0.233051	+	0.972465i	$0.425129\pi$
$444$	−3619.31	−0.386858
$445$	14114.6	1.50359
$446$	−2033.62	−0.215907
$447$	4412.56	0.466906
$448$	0	0
$449$	14238.4	1.49656	0.748278	−	0.663385i	$-0.230881\pi$
$449$	14238.4	1.49656	0.748278	+	0.663385i	$0.230881\pi$
$450$	−1967.75	−0.206135
$451$	5171.50	0.539947
$452$	−2237.57	−0.232846
$453$	830.715	0.0861598
$454$	−4823.11	−0.498590
$455$	0	0
$456$	7.98029	0.000819543 0
$457$	11994.8	1.22778	0.613888	−	0.789393i	$-0.289605\pi$
$457$	11994.8	1.22778	0.613888	+	0.789393i	$0.289605\pi$
$458$	9242.68	0.942974
$459$	−4346.61	−0.442010
$460$	1304.42	0.132215
$461$	−2960.34	−0.299082	−0.149541	−	0.988756i	$-0.547780\pi$
$461$	−2960.34	−0.299082	−0.149541	+	0.988756i	$0.547780\pi$
$462$	0	0
$463$	−2534.97	−0.254449	−0.127225	−	0.991874i	$-0.540607\pi$
$463$	−2534.97	−0.254449	−0.127225	+	0.991874i	$0.540607\pi$
$464$	−1076.65	−0.107721
$465$	−4761.75	−0.474884
$466$	−6975.07	−0.693377
$467$	−14876.0	−1.47404	−0.737021	−	0.675870i	$-0.763768\pi$
$467$	−14876.0	−1.47404	−0.737021	+	0.675870i	$0.763768\pi$
$468$	−3125.36	−0.308696
$469$	0	0
$470$	−6127.43	−0.601356
$471$	−7840.50	−0.767030
$472$	1824.12	0.177885
$473$	−10464.8	−1.01728
$474$	−1611.52	−0.156159
$475$	20.2268	0.00195383
$476$	0	0
$477$	−573.827	−0.0550812
$478$	−12330.0	−1.17983
$479$	5444.59	0.519352	0.259676	−	0.965696i	$-0.416384\pi$
$479$	5444.59	0.519352	0.259676	+	0.965696i	$0.416384\pi$
$480$	−1701.21	−0.161769
$481$	−14570.1	−1.38117
$482$	−2450.19	−0.231542
$483$	0	0
$484$	4128.77	0.387751
$485$	−15535.9	−1.45454
$486$	6496.26	0.606330
$487$	−20465.3	−1.90426	−0.952129	−	0.305698i	$-0.901110\pi$
$487$	−20465.3	−1.90426	−0.952129	+	0.305698i	$0.901110\pi$
$488$	3006.08	0.278850
$489$	2634.23	0.243607
$490$	0	0
$491$	−6247.24	−0.574203	−0.287102	−	0.957900i	$-0.592692\pi$
$491$	−6247.24	−0.574203	−0.287102	+	0.957900i	$0.592692\pi$
$492$	1595.53	0.146204
$493$	−1953.04	−0.178419
$494$	32.1260	0.00292595
$495$	8919.52	0.809904
$496$	1433.11	0.129735
$497$	0	0
$498$	6220.24	0.559710
$499$	−19087.4	−1.71236	−0.856182	−	0.516674i	$-0.827170\pi$
$499$	−19087.4	−1.71236	−0.856182	+	0.516674i	$0.827170\pi$
$500$	2777.35	0.248414
$501$	10488.8	0.935341
$502$	2999.09	0.266646
$503$	4291.01	0.380371	0.190186	−	0.981748i	$-0.439091\pi$
$503$	4291.01	0.380371	0.190186	+	0.981748i	$0.439091\pi$
$504$	0	0
$505$	−20526.0	−1.80870
$506$	−2236.18	−0.196463
$507$	5431.15	0.475751
$508$	4642.48	0.405466
$509$	−7164.92	−0.623928	−0.311964	−	0.950094i	$-0.600987\pi$
$509$	−7164.92	−0.623928	−0.311964	+	0.950094i	$0.600987\pi$
$510$	−3085.98	−0.267940
$511$	0	0
$512$	512.000	0.0441942
$513$	−39.8425	−0.00342902
$514$	6724.90	0.577087
$515$	−22700.5	−1.94234
$516$	−3228.66	−0.275453
$517$	10504.3	0.893579
$518$	0	0
$519$	1729.04	0.146236
$520$	−6848.51	−0.577552
$521$	10113.7	0.850459	0.425229	−	0.905086i	$-0.360193\pi$
$521$	10113.7	0.850459	0.425229	+	0.905086i	$0.360193\pi$
$522$	1741.60	0.146030
$523$	−13719.7	−1.14707	−0.573537	−	0.819180i	$-0.694429\pi$
$523$	−13719.7	−1.14707	−0.573537	+	0.819180i	$0.694429\pi$
$524$	−10646.3	−0.887567
$525$	0	0
$526$	−12575.9	−1.04246
$527$	2599.63	0.214880
$528$	2916.41	0.240380
$529$	529.000	0.0434783
$530$	−1257.41	−0.103054
$531$	−2950.70	−0.241148
$532$	0	0
$533$	6423.09	0.521979
$534$	−7465.33	−0.604974
$535$	−18105.7	−1.46314
$536$	8159.00	0.657491
$537$	135.696	0.0109045
$538$	4427.65	0.354813
$539$	0	0
$540$	8493.48	0.676854
$541$	3358.70	0.266916	0.133458	−	0.991054i	$-0.457392\pi$
$541$	3358.70	0.266916	0.133458	+	0.991054i	$0.457392\pi$
$542$	−5302.01	−0.420186
$543$	14556.6	1.15043
$544$	928.761	0.0731991
$545$	7902.06	0.621077
$546$	0	0
$547$	13242.2	1.03509	0.517545	−	0.855656i	$-0.326846\pi$
$547$	13242.2	1.03509	0.517545	+	0.855656i	$0.326846\pi$
$548$	11976.2	0.933575
$549$	−4862.65	−0.378020
$550$	7391.92	0.573077
$551$	−17.9022	−0.00138413
$552$	−689.918	−0.0531972
$553$	0	0
$554$	−4674.37	−0.358475
$555$	12829.1	0.981194
$556$	5923.66	0.451833
$557$	21134.9	1.60774	0.803872	−	0.594803i	$-0.202770\pi$
$557$	21134.9	1.60774	0.803872	+	0.594803i	$0.202770\pi$
$558$	−2318.20	−0.175873
$559$	−12997.5	−0.983428
$560$	0	0
$561$	5290.34	0.398143
$562$	3386.85	0.254209
$563$	−19331.5	−1.44711	−0.723557	−	0.690265i	$-0.757494\pi$
$563$	−19331.5	−1.44711	−0.723557	+	0.690265i	$0.757494\pi$
$564$	3240.85	0.241958
$565$	7931.33	0.590573
$566$	−221.786	−0.0164706
$567$	0	0
$568$	−1944.01	−0.143607
$569$	25141.5	1.85235	0.926173	−	0.377100i	$-0.123079\pi$
$569$	25141.5	1.85235	0.926173	+	0.377100i	$0.123079\pi$
$570$	−28.2871	−0.00207862
$571$	26174.1	1.91831	0.959153	−	0.282889i	$-0.0912928\pi$
$571$	26174.1	1.91831	0.959153	+	0.282889i	$0.0912928\pi$
$572$	11740.5	0.858209
$573$	14654.7	1.06843
$574$	0	0
$575$	−1748.66	−0.126825
$576$	−828.215	−0.0599114
$577$	−13690.1	−0.987738	−0.493869	−	0.869536i	$-0.664418\pi$
$577$	−13690.1	−0.987738	−0.493869	+	0.869536i	$0.664418\pi$
$578$	−8141.24	−0.585867
$579$	−5246.56	−0.376579
$580$	3816.32	0.273214
$581$	0	0
$582$	8217.08	0.585239
$583$	2155.60	0.153132
$584$	369.911	0.0262107
$585$	11078.2	0.782953
$586$	−15330.0	−1.08068
$587$	17982.6	1.26443	0.632217	−	0.774791i	$-0.282145\pi$
$587$	17982.6	1.26443	0.632217	+	0.774791i	$0.282145\pi$
$588$	0	0
$589$	23.8291	0.00166700
$590$	−6465.79	−0.451174
$591$	9154.64	0.637177
$592$	−3861.06	−0.268055
$593$	4545.73	0.314790	0.157395	−	0.987536i	$-0.449690\pi$
$593$	4545.73	0.314790	0.157395	+	0.987536i	$0.449690\pi$
$594$	−14560.5	−1.00577
$595$	0	0
$596$	4707.29	0.323521
$597$	8207.24	0.562647
$598$	−2777.38	−0.189926
$599$	−16482.1	−1.12427	−0.562136	−	0.827045i	$-0.690020\pi$
$599$	−16482.1	−1.12427	−0.562136	+	0.827045i	$0.690020\pi$
$600$	2280.59	0.155174
$601$	−15905.7	−1.07955	−0.539774	−	0.841810i	$-0.681490\pi$
$601$	−15905.7	−1.07955	−0.539774	+	0.841810i	$0.681490\pi$
$602$	0	0
$603$	−13198.1	−0.891321
$604$	886.201	0.0597004
$605$	−14634.9	−0.983461
$606$	10856.4	0.727739
$607$	13004.0	0.869549	0.434774	−	0.900539i	$-0.356828\pi$
$607$	13004.0	0.869549	0.434774	+	0.900539i	$0.356828\pi$
$608$	8.51333	0.000567863 0
$609$	0	0
$610$	−10655.4	−0.707252
$611$	13046.6	0.863843
$612$	−1502.37	−0.0992316
$613$	−4549.63	−0.299768	−0.149884	−	0.988704i	$-0.547890\pi$
$613$	−4549.63	−0.299768	−0.149884	+	0.988704i	$0.547890\pi$
$614$	−7903.15	−0.519454
$615$	−5655.55	−0.370819
$616$	0	0
$617$	13522.4	0.882323	0.441162	−	0.897428i	$-0.354567\pi$
$617$	13522.4	0.882323	0.441162	+	0.897428i	$0.354567\pi$
$618$	12006.5	0.781508
$619$	−13513.1	−0.877445	−0.438722	−	0.898623i	$-0.644569\pi$
$619$	−13513.1	−0.877445	−0.438722	+	0.898623i	$0.644569\pi$
$620$	−5079.81	−0.329048
$621$	3444.49	0.222581
$622$	13876.1	0.894504
$623$	0	0
$624$	3622.24	0.232381
$625$	−19348.2	−1.23829
$626$	8443.96	0.539119
$627$	48.4929	0.00308871
$628$	−8364.20	−0.531478
$629$	−7003.90	−0.443981
$630$	0	0
$631$	−8568.92	−0.540608	−0.270304	−	0.962775i	$-0.587124\pi$
$631$	−8568.92	−0.540608	−0.270304	+	0.962775i	$0.587124\pi$
$632$	−1719.16	−0.108203
$633$	−19443.8	−1.22089
$634$	−3379.87	−0.211722
$635$	−16455.8	−1.02839
$636$	665.055	0.0414640
$637$	0	0
$638$	−6542.38	−0.405980
$639$	3144.65	0.194680
$640$	−1814.84	−0.112091
$641$	17958.0	1.10655	0.553275	−	0.832999i	$-0.313378\pi$
$641$	17958.0	1.10655	0.553275	+	0.832999i	$0.313378\pi$
$642$	9576.28	0.588700
$643$	−232.314	−0.0142481	−0.00712407	−	0.999975i	$-0.502268\pi$
$643$	−232.314	−0.0142481	−0.00712407	+	0.999975i	$0.502268\pi$
$644$	0	0
$645$	11444.4	0.698638
$646$	15.4431	0.000940556 0
$647$	20467.9	1.24370	0.621851	−	0.783136i	$-0.286381\pi$
$647$	20467.9	1.24370	0.621851	+	0.783136i	$0.286381\pi$
$648$	−1697.05	−0.102880
$649$	11084.4	0.670417
$650$	9180.89	0.554006
$651$	0	0
$652$	2810.18	0.168796
$653$	−1224.64	−0.0733904	−0.0366952	−	0.999327i	$-0.511683\pi$
$653$	−1224.64	−0.0733904	−0.0366952	+	0.999327i	$0.511683\pi$
$654$	−4179.47	−0.249893
$655$	37737.0	2.25115
$656$	1702.11	0.101305
$657$	−598.371	−0.0355322
$658$	0	0
$659$	16443.6	0.972008	0.486004	−	0.873957i	$-0.338454\pi$
$659$	16443.6	0.972008	0.486004	+	0.873957i	$0.338454\pi$
$660$	−10337.6	−0.609680
$661$	−8782.98	−0.516821	−0.258410	−	0.966035i	$-0.583199\pi$
$661$	−8782.98	−0.516821	−0.258410	+	0.966035i	$0.583199\pi$
$662$	7457.50	0.437831
$663$	6570.69	0.384893
$664$	6635.71	0.387825
$665$	0	0
$666$	6245.67	0.363386
$667$	1547.69	0.0898453
$668$	11189.4	0.648101
$669$	−3812.58	−0.220333
$670$	−28920.5	−1.66761
$671$	18266.7	1.05093
$672$	0	0
$673$	26555.7	1.52102	0.760510	−	0.649326i	$-0.224949\pi$
$673$	26555.7	1.52102	0.760510	+	0.649326i	$0.224949\pi$
$674$	1674.60	0.0957021
$675$	−11386.1	−0.649260
$676$	5793.91	0.329649
$677$	20532.9	1.16565	0.582824	−	0.812598i	$-0.301948\pi$
$677$	20532.9	1.16565	0.582824	+	0.812598i	$0.301948\pi$
$678$	−4194.95	−0.237620
$679$	0	0
$680$	−3292.10	−0.185656
$681$	−9042.25	−0.508810
$682$	8708.39	0.488946
$683$	12060.7	0.675678	0.337839	−	0.941204i	$-0.390304\pi$
$683$	12060.7	0.675678	0.337839	+	0.941204i	$0.390304\pi$
$684$	−13.7712	−0.000769818 0
$685$	−42451.1	−2.36784
$686$	0	0
$687$	17328.0	0.962304
$688$	−3444.32	−0.190862
$689$	2677.29	0.148036
$690$	2445.49	0.134925
$691$	4575.07	0.251872	0.125936	−	0.992038i	$-0.459807\pi$
$691$	4575.07	0.251872	0.125936	+	0.992038i	$0.459807\pi$
$692$	1844.53	0.101328
$693$	0	0
$694$	2214.28	0.121114
$695$	−20997.1	−1.14599
$696$	−2018.49	−0.109929
$697$	3087.60	0.167792
$698$	12923.8	0.700821
$699$	−13076.7	−0.707591
$700$	0	0
$701$	9743.67	0.524984	0.262492	−	0.964934i	$-0.415456\pi$
$701$	9743.67	0.524984	0.262492	+	0.964934i	$0.415456\pi$
$702$	−18084.4	−0.972296
$703$	−64.2001	−0.00344431
$704$	3111.21	0.166560
$705$	−11487.6	−0.613683
$706$	−9939.16	−0.529838
$707$	0	0
$708$	3419.81	0.181532
$709$	28402.2	1.50446	0.752232	−	0.658898i	$-0.228977\pi$
$709$	28402.2	1.50446	0.752232	+	0.658898i	$0.228977\pi$
$710$	6890.78	0.364234
$711$	2780.92	0.146685
$712$	−7963.97	−0.419189
$713$	−2060.09	−0.108206
$714$	0	0
$715$	−41615.6	−2.17669
$716$	144.759	0.00755573
$717$	−23116.0	−1.20402
$718$	−2580.32	−0.134118
$719$	36134.8	1.87427	0.937136	−	0.348965i	$-0.113467\pi$
$719$	36134.8	1.87427	0.937136	+	0.348965i	$0.113467\pi$
$720$	2935.70	0.151954
$721$	0	0
$722$	−13717.9	−0.707099
$723$	−4593.56	−0.236288
$724$	15528.9	0.797136
$725$	−5116.04	−0.262076
$726$	7740.53	0.395700
$727$	16156.1	0.824202	0.412101	−	0.911138i	$-0.364795\pi$
$727$	16156.1	0.824202	0.412101	+	0.911138i	$0.364795\pi$
$728$	0	0
$729$	17906.6	0.909749
$730$	−1311.19	−0.0664786
$731$	−6247.94	−0.316127
$732$	5635.72	0.284566
$733$	−38195.1	−1.92465	−0.962325	−	0.271903i	$-0.912347\pi$
$733$	−38195.1	−1.92465	−0.962325	+	0.271903i	$0.912347\pi$
$734$	21418.6	1.07708
$735$	0	0
$736$	−736.000	−0.0368605
$737$	49578.9	2.47797
$738$	−2753.34	−0.137333
$739$	−31790.2	−1.58244	−0.791219	−	0.611533i	$-0.790553\pi$
$739$	−31790.2	−1.58244	−0.791219	+	0.611533i	$0.790553\pi$
$740$	13686.0	0.679872
$741$	60.2291	0.00298592
$742$	0	0
$743$	−1657.10	−0.0818210	−0.0409105	−	0.999163i	$-0.513026\pi$
$743$	−1657.10	−0.0818210	−0.0409105	+	0.999163i	$0.513026\pi$
$744$	2686.75	0.132394
$745$	−16685.5	−0.820552
$746$	−20632.7	−1.01262
$747$	−10734.0	−0.525750
$748$	5643.70	0.275874
$749$	0	0
$750$	5206.91	0.253506
$751$	12336.6	0.599425	0.299712	−	0.954030i	$-0.403109\pi$
$751$	12336.6	0.599425	0.299712	+	0.954030i	$0.403109\pi$
$752$	3457.32	0.167653
$753$	5622.63	0.272112
$754$	−8125.75	−0.392470
$755$	−3141.24	−0.151419
$756$	0	0
$757$	19801.6	0.950727	0.475363	−	0.879790i	$-0.342317\pi$
$757$	19801.6	0.950727	0.475363	+	0.879790i	$0.342317\pi$
$758$	−25213.1	−1.20815
$759$	−4192.34	−0.200491
$760$	−30.1765	−0.00144028
$761$	2043.78	0.0973546	0.0486773	−	0.998815i	$-0.484499\pi$
$761$	2043.78	0.0973546	0.0486773	+	0.998815i	$0.484499\pi$
$762$	8703.61	0.413778
$763$	0	0
$764$	15633.6	0.740318
$765$	5325.32	0.251683
$766$	−29404.2	−1.38697
$767$	13767.0	0.648107
$768$	959.885	0.0451001
$769$	−20117.6	−0.943378	−0.471689	−	0.881765i	$-0.656356\pi$
$769$	−20117.6	−0.943378	−0.471689	+	0.881765i	$0.656356\pi$
$770$	0	0
$771$	12607.7	0.588917
$772$	−5597.00	−0.260933
$773$	−7575.56	−0.352489	−0.176244	−	0.984346i	$-0.556395\pi$
$773$	−7575.56	−0.352489	−0.176244	+	0.984346i	$0.556395\pi$
$774$	5571.55	0.258741
$775$	6809.82	0.315633
$776$	8765.93	0.405514
$777$	0	0
$778$	28820.1	1.32809
$779$	28.3019	0.00130170
$780$	−12839.4	−0.589391
$781$	−11813.0	−0.541231
$782$	−1335.09	−0.0610523
$783$	10077.5	0.459950
$784$	0	0
$785$	29647.9	1.34800
$786$	−19959.4	−0.905761
$787$	2961.31	0.134129	0.0670644	−	0.997749i	$-0.478637\pi$
$787$	2961.31	0.134129	0.0670644	+	0.997749i	$0.478637\pi$
$788$	9766.12	0.441502
$789$	−23577.0	−1.06383
$790$	6093.76	0.274438
$791$	0	0
$792$	−5032.72	−0.225795
$793$	22687.5	1.01596
$794$	19932.0	0.890883
$795$	−2357.36	−0.105166
$796$	8755.44	0.389860
$797$	−31231.3	−1.38804	−0.694022	−	0.719954i	$-0.744163\pi$
$797$	−31231.3	−1.38804	−0.694022	+	0.719954i	$0.744163\pi$
$798$	0	0
$799$	6271.53	0.277685
$800$	2432.92	0.107521
$801$	12882.6	0.568268
$802$	16469.4	0.725132
$803$	2247.80	0.0987834
$804$	15296.3	0.670969
$805$	0	0
$806$	10816.0	0.472675
$807$	8300.85	0.362086
$808$	11581.5	0.504252
$809$	−43376.0	−1.88507	−0.942534	−	0.334111i	$-0.891564\pi$
$809$	−43376.0	−1.88507	−0.942534	+	0.334111i	$0.891564\pi$
$810$	6015.39	0.260937
$811$	33430.7	1.44748	0.723742	−	0.690071i	$-0.242421\pi$
$811$	33430.7	1.44748	0.723742	+	0.690071i	$0.242421\pi$
$812$	0	0
$813$	−9940.09	−0.428800
$814$	−23462.0	−1.01025
$815$	−9961.01	−0.428121
$816$	1741.22	0.0746996
$817$	−57.2707	−0.00245245
$818$	−10122.0	−0.432651
$819$	0	0
$820$	−6033.31	−0.256942
$821$	−7382.93	−0.313844	−0.156922	−	0.987611i	$-0.550157\pi$
$821$	−7382.93	−0.313844	−0.156922	+	0.987611i	$0.550157\pi$
$822$	22452.7	0.952712
$823$	−13896.8	−0.588594	−0.294297	−	0.955714i	$-0.595085\pi$
$823$	−13896.8	−0.588594	−0.294297	+	0.955714i	$0.595085\pi$
$824$	12808.5	0.541509
$825$	13858.2	0.584825
$826$	0	0
$827$	6254.03	0.262967	0.131484	−	0.991318i	$-0.458026\pi$
$827$	6254.03	0.262967	0.131484	+	0.991318i	$0.458026\pi$
$828$	1190.56	0.0499695
$829$	9397.49	0.393713	0.196857	−	0.980432i	$-0.436927\pi$
$829$	9397.49	0.393713	0.196857	+	0.980432i	$0.436927\pi$
$830$	−23521.0	−0.983647
$831$	−8763.40	−0.365823
$832$	3864.18	0.161017
$833$	0	0
$834$	11105.5	0.461095
$835$	−39662.1	−1.64379
$836$	51.7320	0.00214018
$837$	−13413.9	−0.553945
$838$	−21805.8	−0.898889
$839$	−17384.8	−0.715362	−0.357681	−	0.933844i	$-0.616432\pi$
$839$	−17384.8	−0.715362	−0.357681	+	0.933844i	$0.616432\pi$
$840$	0	0
$841$	−19860.9	−0.814340
$842$	−32251.5	−1.32002
$843$	6349.59	0.259420
$844$	−20742.5	−0.845956
$845$	−20537.2	−0.836096
$846$	−5592.58	−0.227278
$847$	0	0
$848$	709.476	0.0287306
$849$	−415.799	−0.0168082
$850$	4413.28	0.178087
$851$	5550.27	0.223573
$852$	−3644.59	−0.146551
$853$	−32263.7	−1.29506	−0.647530	−	0.762040i	$-0.724198\pi$
$853$	−32263.7	−1.29506	−0.647530	+	0.762040i	$0.724198\pi$
$854$	0	0
$855$	48.8137	0.00195250
$856$	10215.9	0.407912
$857$	−42951.1	−1.71200	−0.855999	−	0.516977i	$-0.827057\pi$
$857$	−42951.1	−1.71200	−0.855999	+	0.516977i	$0.827057\pi$
$858$	22010.8	0.875801
$859$	−15702.3	−0.623698	−0.311849	−	0.950132i	$-0.600948\pi$
$859$	−15702.3	−0.623698	−0.311849	+	0.950132i	$0.600948\pi$
$860$	12208.8	0.484088
$861$	0	0
$862$	−28903.0	−1.14204
$863$	23146.9	0.913011	0.456505	−	0.889721i	$-0.349101\pi$
$863$	23146.9	0.913011	0.456505	+	0.889721i	$0.349101\pi$
$864$	−4792.33	−0.188702
$865$	−6538.16	−0.256999
$866$	264.457	0.0103772
$867$	−15263.0	−0.597876
$868$	0	0
$869$	−10446.6	−0.407799
$870$	7154.76	0.278815
$871$	61577.8	2.39550
$872$	−4458.63	−0.173152
$873$	−14179.8	−0.549730
$874$	−12.2379	−0.000473631 0
$875$	0	0
$876$	693.501	0.0267480
$877$	42380.2	1.63179	0.815893	−	0.578203i	$-0.196246\pi$
$877$	42380.2	1.63179	0.815893	+	0.578203i	$0.196246\pi$
$878$	12589.8	0.483922
$879$	−28740.3	−1.10283
$880$	−11028.0	−0.422449
$881$	10114.4	0.386791	0.193396	−	0.981121i	$-0.438050\pi$
$881$	10114.4	0.386791	0.193396	+	0.981121i	$0.438050\pi$
$882$	0	0
$883$	−5552.09	−0.211600	−0.105800	−	0.994387i	$-0.533740\pi$
$883$	−5552.09	−0.211600	−0.105800	+	0.994387i	$0.533740\pi$
$884$	7009.57	0.266694
$885$	−12121.9	−0.460422
$886$	8691.93	0.329584
$887$	14504.4	0.549054	0.274527	−	0.961579i	$-0.411479\pi$
$887$	14504.4	0.549054	0.274527	+	0.961579i	$0.411479\pi$
$888$	−7238.62	−0.273550
$889$	0	0
$890$	28229.2	1.06320
$891$	−10312.3	−0.387737
$892$	−4067.23	−0.152669
$893$	57.4869	0.00215423
$894$	8825.12	0.330152
$895$	−513.115	−0.0191638
$896$	0	0
$897$	−5206.96	−0.193819
$898$	28476.9	1.05822
$899$	−6027.18	−0.223602
$900$	−3935.50	−0.145759
$901$	1286.98	0.0475866
$902$	10343.0	0.381801
$903$	0	0
$904$	−4475.15	−0.164647
$905$	−55043.9	−2.02179
$906$	1661.43	0.0609242
$907$	−39573.3	−1.44874	−0.724372	−	0.689409i	$-0.757870\pi$
$907$	−39573.3	−1.44874	−0.724372	+	0.689409i	$0.757870\pi$
$908$	−9646.22	−0.352556
$909$	−18734.3	−0.683584
$910$	0	0
$911$	−40575.1	−1.47565	−0.737823	−	0.674994i	$-0.764146\pi$
$911$	−40575.1	−1.47565	−0.737823	+	0.674994i	$0.764146\pi$
$912$	15.9606	0.000579504 0
$913$	40322.5	1.46164
$914$	23989.6	0.868168
$915$	−19976.4	−0.721750
$916$	18485.4	0.666783
$917$	0	0
$918$	−8693.22	−0.312548
$919$	−14409.5	−0.517220	−0.258610	−	0.965982i	$-0.583264\pi$
$919$	−14409.5	−0.517220	−0.258610	+	0.965982i	$0.583264\pi$
$920$	2608.84	0.0934900
$921$	−14816.6	−0.530103
$922$	−5920.68	−0.211483
$923$	−14671.9	−0.523220
$924$	0	0
$925$	−18346.9	−0.652155
$926$	−5069.94	−0.179923
$927$	−20719.0	−0.734091
$928$	−2153.31	−0.0761701
$929$	−18626.7	−0.657827	−0.328913	−	0.944360i	$-0.606682\pi$
$929$	−18626.7	−0.657827	−0.328913	+	0.944360i	$0.606682\pi$
$930$	−9523.50	−0.335793
$931$	0	0
$932$	−13950.1	−0.490292
$933$	26014.6	0.912840
$934$	−29751.9	−1.04231
$935$	−20004.7	−0.699706
$936$	−6250.72	−0.218281
$937$	−13307.6	−0.463969	−0.231985	−	0.972719i	$-0.574522\pi$
$937$	−13307.6	−0.463969	−0.231985	+	0.972719i	$0.574522\pi$
$938$	0	0
$939$	15830.5	0.550171
$940$	−12254.9	−0.425223
$941$	−20348.0	−0.704917	−0.352458	−	0.935827i	$-0.614654\pi$
$941$	−20348.0	−0.704917	−0.352458	+	0.935827i	$0.614654\pi$
$942$	−15681.0	−0.542372
$943$	−2446.78	−0.0844942
$944$	3648.23	0.125784
$945$	0	0
$946$	−20929.7	−0.719326
$947$	8864.55	0.304181	0.152090	−	0.988367i	$-0.451400\pi$
$947$	8864.55	0.304181	0.152090	+	0.988367i	$0.451400\pi$
$948$	−3223.04	−0.110421
$949$	2791.80	0.0954961
$950$	40.4536	0.00138156
$951$	−6336.50	−0.216062
$952$	0	0
$953$	48651.3	1.65369	0.826847	−	0.562427i	$-0.190132\pi$
$953$	48651.3	1.65369	0.826847	+	0.562427i	$0.190132\pi$
$954$	−1147.65	−0.0389483
$955$	−55415.0	−1.87768
$956$	−24660.0	−0.834268
$957$	−12265.5	−0.414302
$958$	10889.2	0.367237
$959$	0	0
$960$	−3402.42	−0.114388
$961$	−21768.4	−0.730703
$962$	−29140.3	−0.976631
$963$	−16525.3	−0.552982
$964$	−4900.38	−0.163725
$965$	19839.2	0.661810
$966$	0	0
$967$	−38135.4	−1.26820	−0.634101	−	0.773251i	$-0.718629\pi$
$967$	−38135.4	−1.26820	−0.634101	+	0.773251i	$0.718629\pi$
$968$	8257.55	0.274181
$969$	28.9523	0.000959836 0
$970$	−31071.9	−1.02851
$971$	−51160.8	−1.69086	−0.845431	−	0.534084i	$-0.820656\pi$
$971$	−51160.8	−1.69086	−0.845431	+	0.534084i	$0.820656\pi$
$972$	12992.5	0.428740
$973$	0	0
$974$	−40930.7	−1.34651
$975$	17212.1	0.565363
$976$	6012.15	0.197176
$977$	46005.8	1.50650	0.753252	−	0.657732i	$-0.228484\pi$
$977$	46005.8	1.50650	0.753252	+	0.657732i	$0.228484\pi$
$978$	5268.46	0.172256
$979$	−48393.7	−1.57985
$980$	0	0
$981$	7212.31	0.234731
$982$	−12494.5	−0.406023
$983$	4371.78	0.141849	0.0709247	−	0.997482i	$-0.477405\pi$
$983$	4371.78	0.141849	0.0709247	+	0.997482i	$0.477405\pi$
$984$	3191.07	0.103382
$985$	−34617.1	−1.11979
$986$	−3906.07	−0.126161
$987$	0	0
$988$	64.2520	0.00206896
$989$	4951.20	0.159190
$990$	17839.0	0.572689
$991$	20836.2	0.667895	0.333947	−	0.942592i	$-0.391619\pi$
$991$	20836.2	0.667895	0.333947	+	0.942592i	$0.391619\pi$
$992$	2866.21	0.0917362
$993$	13981.1	0.446806
$994$	0	0
$995$	−31034.7	−0.988809
$996$	12440.5	0.395775
$997$	16190.9	0.514315	0.257158	−	0.966369i	$-0.417214\pi$
$997$	16190.9	0.514315	0.257158	+	0.966369i	$0.417214\pi$
$998$	−38174.8	−1.21082
$999$	36139.6	1.14455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.8		11
7.3	odd	6		322.4.e.b.93.8	✓	22
7.5	odd	6		322.4.e.b.277.8	yes	22
7.6	odd	2		2254.4.a.w.1.4		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.8	✓	22	7.3	odd	6
322.4.e.b.277.8	yes	22	7.5	odd	6
2254.4.a.w.1.4		11	7.6	odd	2
2254.4.a.x.1.8		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.74955 q^{3} +4.00000 q^{4} -14.1785 q^{5} +7.49911 q^{6} +8.00000 q^{8} -12.9409 q^{9} -28.3569 q^{10} +48.6127 q^{11} +14.9982 q^{12} +60.3778 q^{13} -53.1629 q^{15} +16.0000 q^{16} +29.0238 q^{17} -25.8817 q^{18} +0.266041 q^{19} -56.7138 q^{20} +97.2254 q^{22} -23.0000 q^{23} +29.9964 q^{24} +76.0287 q^{25} +120.756 q^{26} -149.760 q^{27} -67.2909 q^{29} -106.326 q^{30} +89.5691 q^{31} +32.0000 q^{32} +182.276 q^{33} +58.0476 q^{34} -51.7634 q^{36} -241.316 q^{37} +0.532083 q^{38} +226.390 q^{39} -113.428 q^{40} +106.382 q^{41} -215.270 q^{43} +194.451 q^{44} +183.481 q^{45} -46.0000 q^{46} +216.082 q^{47} +59.9928 q^{48} +152.057 q^{50} +108.826 q^{51} +241.511 q^{52} +44.3423 q^{53} -299.521 q^{54} -689.253 q^{55} +0.997537 q^{57} -134.582 q^{58} +228.015 q^{59} -212.652 q^{60} +375.759 q^{61} +179.138 q^{62} +64.0000 q^{64} -856.064 q^{65} +364.552 q^{66} +1019.88 q^{67} +116.095 q^{68} -86.2397 q^{69} -243.002 q^{71} -103.527 q^{72} +46.2389 q^{73} -482.632 q^{74} +285.073 q^{75} +1.06417 q^{76} +452.779 q^{78} -214.895 q^{79} -226.855 q^{80} -212.131 q^{81} +212.763 q^{82} +829.464 q^{83} -411.513 q^{85} -430.540 q^{86} -252.311 q^{87} +388.902 q^{88} -995.496 q^{89} +366.963 q^{90} -92.0000 q^{92} +335.844 q^{93} +432.165 q^{94} -3.77206 q^{95} +119.986 q^{96} +1095.74 q^{97} -629.090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 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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