LMFDB - Embedded newform 2057.4.a.h.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2057,4,Mod(1,2057)] 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("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-4,11,40,47] 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:	$121.366928882$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 52 x^{8} + 180 x^{7} + 933 x^{6} - 2524 x^{5} - 6654 x^{4} + 11196 x^{3} + 16044 x^{2} + \cdots - 88 $ x^10 - 4x^9 - 52x^8 + 180x^7 + 933x^6 - 2524x^5 - 6654x^4 + 11196x^3 + 16044x^2 - 7880*x - 88
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.62662$ of defining polynomial
Character	$\chi$	$=$	2057.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.62662 q^{2} +0.447567 q^{3} -1.10085 q^{4} +9.81562 q^{5} +1.17559 q^{6} -21.1571 q^{7} -23.9045 q^{8} -26.7997 q^{9} +25.7819 q^{10} -0.492704 q^{12} +31.4725 q^{13} -55.5718 q^{14} +4.39315 q^{15} -53.9813 q^{16} +17.0000 q^{17} -70.3927 q^{18} -164.426 q^{19} -10.8055 q^{20} -9.46923 q^{21} +118.597 q^{23} -10.6989 q^{24} -28.6536 q^{25} +82.6665 q^{26} -24.0790 q^{27} +23.2908 q^{28} -40.9982 q^{29} +11.5391 q^{30} +229.807 q^{31} +49.4474 q^{32} +44.6526 q^{34} -207.670 q^{35} +29.5024 q^{36} +196.941 q^{37} -431.885 q^{38} +14.0861 q^{39} -234.637 q^{40} -328.840 q^{41} -24.8721 q^{42} +34.5048 q^{43} -263.055 q^{45} +311.509 q^{46} +281.576 q^{47} -24.1603 q^{48} +104.623 q^{49} -75.2623 q^{50} +7.60864 q^{51} -34.6465 q^{52} +89.1184 q^{53} -63.2464 q^{54} +505.750 q^{56} -73.5916 q^{57} -107.687 q^{58} +218.326 q^{59} -4.83619 q^{60} -113.552 q^{61} +603.618 q^{62} +567.004 q^{63} +561.730 q^{64} +308.922 q^{65} +38.6827 q^{67} -18.7144 q^{68} +53.0800 q^{69} -545.471 q^{70} -242.093 q^{71} +640.633 q^{72} +345.226 q^{73} +517.290 q^{74} -12.8244 q^{75} +181.008 q^{76} +36.9988 q^{78} +959.412 q^{79} -529.860 q^{80} +712.814 q^{81} -863.739 q^{82} -109.109 q^{83} +10.4242 q^{84} +166.866 q^{85} +90.6312 q^{86} -18.3495 q^{87} +1343.72 q^{89} -690.948 q^{90} -665.868 q^{91} -130.557 q^{92} +102.854 q^{93} +739.594 q^{94} -1613.94 q^{95} +22.1310 q^{96} -1248.68 q^{97} +274.806 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 11 q^{3} + 40 q^{4} + 47 q^{5} - 105 q^{6} + 17 q^{7} - 84 q^{8} + 107 q^{9} + 7 q^{10} + 101 q^{12} - 15 q^{13} + 45 q^{14} + 224 q^{15} + 140 q^{16} + 170 q^{17} - 81 q^{18} - q^{19}+ \cdots - 3281 q^{98}+O(q^{100}) $ 10 * q - 4 * q^2 + 11 * q^3 + 40 * q^4 + 47 * q^5 - 105 * q^6 + 17 * q^7 - 84 * q^8 + 107 * q^9 + 7 * q^10 + 101 * q^12 - 15 * q^13 + 45 * q^14 + 224 * q^15 + 140 * q^16 + 170 * q^17 - 81 * q^18 - q^19 + 289 * q^20 - 16 * q^21 + 340 * q^23 - 663 * q^24 + 591 * q^25 - 277 * q^26 + 776 * q^27 + 127 * q^28 - 452 * q^29 - 594 * q^30 + 298 * q^31 - 536 * q^32 - 68 * q^34 - 440 * q^35 - 15 * q^36 + 619 * q^37 + 349 * q^38 + 771 * q^39 - 379 * q^40 - 367 * q^41 + 292 * q^42 + 385 * q^43 + 1542 * q^45 - 776 * q^46 + 365 * q^47 - 363 * q^48 + 985 * q^49 + 2167 * q^50 + 187 * q^51 - 133 * q^52 + 1624 * q^53 + 576 * q^54 - 2217 * q^56 - 1498 * q^57 - 802 * q^58 - 414 * q^59 - 3420 * q^60 + 1162 * q^61 + 90 * q^62 + 1309 * q^63 - 1044 * q^64 - 2818 * q^65 - 1216 * q^67 + 680 * q^68 - 1048 * q^69 - 1556 * q^70 + 850 * q^71 + 117 * q^72 + 983 * q^73 + 555 * q^74 + 415 * q^75 + 1313 * q^76 - 1939 * q^78 + 1165 * q^79 + 209 * q^80 + 1442 * q^81 + 481 * q^82 + 1391 * q^83 + 3728 * q^84 + 799 * q^85 + 2757 * q^86 + 1434 * q^87 + 2225 * q^89 + 7526 * q^90 - 508 * q^91 + 358 * q^92 + 1266 * q^93 + 5233 * q^94 + 2768 * q^95 + 3209 * q^96 + 700 * q^97 - 3281 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.62662	0.928652	0.464326	−	0.885664i	$-0.346297\pi$
$2$	2.62662	0.928652	0.464326	+	0.885664i	$0.346297\pi$
$3$	0.447567	0.0861344	0.0430672	−	0.999072i	$-0.486287\pi$
$3$	0.447567	0.0861344	0.0430672	+	0.999072i	$0.486287\pi$
$4$	−1.10085	−0.137606
$5$	9.81562	0.877936	0.438968	−	0.898503i	$-0.355344\pi$
$5$	9.81562	0.877936	0.438968	+	0.898503i	$0.355344\pi$
$6$	1.17559	0.0799888
$7$	−21.1571	−1.14238	−0.571188	−	0.820819i	$-0.693517\pi$
$7$	−21.1571	−1.14238	−0.571188	+	0.820819i	$0.693517\pi$
$8$	−23.9045	−1.05644
$9$	−26.7997	−0.992581
$10$	25.7819	0.815296
$11$	0	0
$11$	0	0
$12$	−0.492704	−0.0118526
$13$	31.4725	0.671455	0.335727	−	0.941959i	$-0.391018\pi$
$13$	31.4725	0.671455	0.335727	+	0.941959i	$0.391018\pi$
$14$	−55.5718	−1.06087
$15$	4.39315	0.0756204
$16$	−53.9813	−0.843458
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−70.3927	−0.921762
$19$	−164.426	−1.98536	−0.992681	−	0.120769i	$-0.961464\pi$
$19$	−164.426	−1.98536	−0.992681	+	0.120769i	$0.961464\pi$
$20$	−10.8055	−0.120809
$21$	−9.46923	−0.0983979
$22$	0	0
$23$	118.597	1.07518	0.537589	−	0.843207i	$-0.319335\pi$
$23$	118.597	1.07518	0.537589	+	0.843207i	$0.319335\pi$
$24$	−10.6989	−0.0909958
$25$	−28.6536	−0.229229
$26$	82.6665	0.623547
$27$	−24.0790	−0.171630
$28$	23.2908	0.157198
$29$	−40.9982	−0.262523	−0.131262	−	0.991348i	$-0.541903\pi$
$29$	−40.9982	−0.262523	−0.131262	+	0.991348i	$0.541903\pi$
$30$	11.5391	0.0702250
$31$	229.807	1.33144	0.665720	−	0.746202i	$-0.268125\pi$
$31$	229.807	1.33144	0.665720	+	0.746202i	$0.268125\pi$
$32$	49.4474	0.273161
$33$	0	0
$34$	44.6526	0.225231
$35$	−207.670	−1.00293
$36$	29.5024	0.136585
$37$	196.941	0.875051	0.437526	−	0.899206i	$-0.355855\pi$
$37$	196.941	0.875051	0.437526	+	0.899206i	$0.355855\pi$
$38$	−431.885	−1.84371
$39$	14.0861	0.0578353
$40$	−234.637	−0.927486
$41$	−328.840	−1.25259	−0.626295	−	0.779586i	$-0.715429\pi$
$41$	−328.840	−1.25259	−0.626295	+	0.779586i	$0.715429\pi$
$42$	−24.8721	−0.0913773
$43$	34.5048	0.122371	0.0611853	−	0.998126i	$-0.480512\pi$
$43$	34.5048	0.122371	0.0611853	+	0.998126i	$0.480512\pi$
$44$	0	0
$45$	−263.055	−0.871422
$46$	311.509	0.998466
$47$	281.576	0.873874	0.436937	−	0.899492i	$-0.356063\pi$
$47$	281.576	0.873874	0.436937	+	0.899492i	$0.356063\pi$
$48$	−24.1603	−0.0726507
$49$	104.623	0.305024
$50$	−75.2623	−0.212874
$51$	7.60864	0.0208906
$52$	−34.6465	−0.0923963
$53$	89.1184	0.230969	0.115484	−	0.993309i	$-0.463158\pi$
$53$	89.1184	0.230969	0.115484	+	0.993309i	$0.463158\pi$
$54$	−63.2464	−0.159384
$55$	0	0
$56$	505.750	1.20685
$57$	−73.5916	−0.171008
$58$	−107.687	−0.243793
$59$	218.326	0.481757	0.240879	−	0.970555i	$-0.422564\pi$
$59$	218.326	0.481757	0.240879	+	0.970555i	$0.422564\pi$
$60$	−4.83619	−0.0104058
$61$	−113.552	−0.238342	−0.119171	−	0.992874i	$-0.538024\pi$
$61$	−113.552	−0.238342	−0.119171	+	0.992874i	$0.538024\pi$
$62$	603.618	1.23644
$63$	567.004	1.13390
$64$	561.730	1.09713
$65$	308.922	0.589494
$66$	0	0
$67$	38.6827	0.0705351	0.0352675	−	0.999378i	$-0.488772\pi$
$67$	38.6827	0.0705351	0.0352675	+	0.999378i	$0.488772\pi$
$68$	−18.7144	−0.0333744
$69$	53.0800	0.0926098
$70$	−545.471	−0.931375
$71$	−242.093	−0.404664	−0.202332	−	0.979317i	$-0.564852\pi$
$71$	−242.093	−0.404664	−0.202332	+	0.979317i	$0.564852\pi$
$72$	640.633	1.04860
$73$	345.226	0.553503	0.276751	−	0.960942i	$-0.410742\pi$
$73$	345.226	0.553503	0.276751	+	0.960942i	$0.410742\pi$
$74$	517.290	0.812618
$75$	−12.8244	−0.0197445
$76$	181.008	0.273198
$77$	0	0
$78$	36.9988	0.0537089
$79$	959.412	1.36636	0.683179	−	0.730251i	$-0.260597\pi$
$79$	959.412	1.36636	0.683179	+	0.730251i	$0.260597\pi$
$80$	−529.860	−0.740502
$81$	712.814	0.977798
$82$	−863.739	−1.16322
$83$	−109.109	−0.144292	−0.0721460	−	0.997394i	$-0.522985\pi$
$83$	−109.109	−0.144292	−0.0721460	+	0.997394i	$0.522985\pi$
$84$	10.4242	0.0135401
$85$	166.866	0.212931
$86$	90.6312	0.113640
$87$	−18.3495	−0.0226123
$88$	0	0
$89$	1343.72	1.60039	0.800193	−	0.599742i	$-0.204730\pi$
$89$	1343.72	1.60039	0.800193	+	0.599742i	$0.204730\pi$
$90$	−690.948	−0.809248
$91$	−665.868	−0.767054
$92$	−130.557	−0.147951
$93$	102.854	0.114683
$94$	739.594	0.811525
$95$	−1613.94	−1.74302
$96$	22.1310	0.0235285
$97$	−1248.68	−1.30705	−0.653527	−	0.756903i	$-0.726711\pi$
$97$	−1248.68	−1.30705	−0.653527	+	0.756903i	$0.726711\pi$
$98$	274.806	0.283261
$99$	0	0
$100$	31.5433	0.0315433
$101$	−195.533	−0.192637	−0.0963184	−	0.995351i	$-0.530707\pi$
$101$	−195.533	−0.192637	−0.0963184	+	0.995351i	$0.530707\pi$
$102$	19.9850	0.0194001
$103$	−158.938	−0.152045	−0.0760224	−	0.997106i	$-0.524222\pi$
$103$	−158.938	−0.152045	−0.0760224	+	0.997106i	$0.524222\pi$
$104$	−752.335	−0.709351
$105$	−92.9463	−0.0863870
$106$	234.080	0.214490
$107$	1120.03	1.01194	0.505971	−	0.862550i	$-0.331134\pi$
$107$	1120.03	1.01194	0.505971	+	0.862550i	$0.331134\pi$
$108$	26.5073	0.0236173
$109$	896.710	0.787975	0.393987	−	0.919116i	$-0.371095\pi$
$109$	896.710	0.787975	0.393987	+	0.919116i	$0.371095\pi$
$110$	0	0
$111$	88.1443	0.0753720
$112$	1142.09	0.963547
$113$	1157.69	0.963777	0.481888	−	0.876233i	$-0.339951\pi$
$113$	1157.69	0.963777	0.481888	+	0.876233i	$0.339951\pi$
$114$	−193.297	−0.158807
$115$	1164.10	0.943938
$116$	45.1328	0.0361248
$117$	−843.454	−0.666473
$118$	573.461	0.447385
$119$	−359.671	−0.277067
$120$	−105.016	−0.0798884
$121$	0	0
$122$	−298.258	−0.221336
$123$	−147.178	−0.107891
$124$	−252.983	−0.183214
$125$	−1508.21	−1.07918
$126$	1489.31	1.05300
$127$	486.552	0.339957	0.169978	−	0.985448i	$-0.445630\pi$
$127$	486.552	0.339957	0.169978	+	0.985448i	$0.445630\pi$
$128$	1079.88	0.745690
$129$	15.4432	0.0105403
$130$	811.423	0.547434
$131$	−1906.32	−1.27142	−0.635711	−	0.771927i	$-0.719293\pi$
$131$	−1906.32	−1.27142	−0.635711	+	0.771927i	$0.719293\pi$
$132$	0	0
$133$	3478.77	2.26803
$134$	101.605	0.0655025
$135$	−236.350	−0.150680
$136$	−406.377	−0.256224
$137$	2128.27	1.32723	0.663613	−	0.748076i	$-0.269022\pi$
$137$	2128.27	1.32723	0.663613	+	0.748076i	$0.269022\pi$
$138$	139.421	0.0860023
$139$	286.684	0.174937	0.0874683	−	0.996167i	$-0.472122\pi$
$139$	286.684	0.174937	0.0874683	+	0.996167i	$0.472122\pi$
$140$	228.613	0.138010
$141$	126.024	0.0752706
$142$	−635.886	−0.375792
$143$	0	0
$144$	1446.68	0.837201
$145$	−402.423	−0.230479
$146$	906.780	0.514011
$147$	46.8259	0.0262730
$148$	−216.802	−0.120412
$149$	2026.95	1.11446	0.557229	−	0.830359i	$-0.311865\pi$
$149$	2026.95	1.11446	0.557229	+	0.830359i	$0.311865\pi$
$150$	−33.6849	−0.0183358
$151$	584.797	0.315166	0.157583	−	0.987506i	$-0.449630\pi$
$151$	584.797	0.315166	0.157583	+	0.987506i	$0.449630\pi$
$152$	3930.52	2.09741
$153$	−455.595	−0.240736
$154$	0	0
$155$	2255.70	1.16892
$156$	−15.5066	−0.00795849
$157$	2654.42	1.34934	0.674668	−	0.738121i	$-0.264287\pi$
$157$	2654.42	1.34934	0.674668	+	0.738121i	$0.264287\pi$
$158$	2520.01	1.26887
$159$	39.8865	0.0198943
$160$	485.357	0.239818
$161$	−2509.16	−1.22826
$162$	1872.30	0.908033
$163$	−1359.91	−0.653474	−0.326737	−	0.945115i	$-0.605949\pi$
$163$	−1359.91	−0.653474	−0.326737	+	0.945115i	$0.605949\pi$
$164$	362.003	0.172364
$165$	0	0
$166$	−286.588	−0.133997
$167$	−323.366	−0.149837	−0.0749186	−	0.997190i	$-0.523870\pi$
$167$	−323.366	−0.149837	−0.0749186	+	0.997190i	$0.523870\pi$
$168$	226.357	0.103951
$169$	−1206.48	−0.549149
$170$	438.293	0.197738
$171$	4406.56	1.97063
$172$	−37.9846	−0.0168389
$173$	1975.51	0.868180	0.434090	−	0.900870i	$-0.357070\pi$
$173$	1975.51	0.868180	0.434090	+	0.900870i	$0.357070\pi$
$174$	−48.1971	−0.0209989
$175$	606.228	0.261866
$176$	0	0
$177$	97.7157	0.0414958
$178$	3529.46	1.48620
$179$	−3282.28	−1.37055	−0.685276	−	0.728283i	$-0.740318\pi$
$179$	−3282.28	−1.37055	−0.685276	+	0.728283i	$0.740318\pi$
$180$	289.584	0.119913
$181$	−2541.60	−1.04373	−0.521867	−	0.853027i	$-0.674764\pi$
$181$	−2541.60	−1.04373	−0.521867	+	0.853027i	$0.674764\pi$
$182$	−1748.98	−0.712326
$183$	−50.8222	−0.0205294
$184$	−2834.99	−1.13586
$185$	1933.10	0.768239
$186$	270.159	0.106500
$187$	0	0
$188$	−309.973	−0.120250
$189$	509.441	0.196066
$190$	−4239.22	−1.61866
$191$	3316.38	1.25636	0.628180	−	0.778068i	$-0.283800\pi$
$191$	3316.38	1.25636	0.628180	+	0.778068i	$0.283800\pi$
$192$	251.412	0.0945006
$193$	2669.74	0.995710	0.497855	−	0.867260i	$-0.334121\pi$
$193$	2669.74	0.995710	0.497855	+	0.867260i	$0.334121\pi$
$194$	−3279.81	−1.21380
$195$	138.264	0.0507757
$196$	−115.174	−0.0419732
$197$	−2301.09	−0.832213	−0.416107	−	0.909316i	$-0.636606\pi$
$197$	−2301.09	−0.832213	−0.416107	+	0.909316i	$0.636606\pi$
$198$	0	0
$199$	−462.619	−0.164795	−0.0823975	−	0.996600i	$-0.526258\pi$
$199$	−462.619	−0.164795	−0.0823975	+	0.996600i	$0.526258\pi$
$200$	684.951	0.242167
$201$	17.3131	0.00607549
$202$	−513.593	−0.178892
$203$	867.403	0.299900
$204$	−8.37597	−0.00287468
$205$	−3227.77	−1.09969
$206$	−417.470	−0.141197
$207$	−3178.35	−1.06720
$208$	−1698.93	−0.566344
$209$	0	0
$210$	−244.135	−0.0802234
$211$	5307.17	1.73157	0.865784	−	0.500418i	$-0.166820\pi$
$211$	5307.17	1.73157	0.865784	+	0.500418i	$0.166820\pi$
$212$	−98.1059	−0.0317827
$213$	−108.353	−0.0348554
$214$	2941.91	0.939742
$215$	338.686	0.107434
$216$	575.596	0.181316
$217$	−4862.06	−1.52101
$218$	2355.32	0.731754
$219$	154.512	0.0476756
$220$	0	0
$221$	535.033	0.162852
$222$	231.522	0.0699943
$223$	−4061.80	−1.21972	−0.609861	−	0.792508i	$-0.708775\pi$
$223$	−4061.80	−1.21972	−0.609861	+	0.792508i	$0.708775\pi$
$224$	−1046.16	−0.312052
$225$	767.908	0.227528
$226$	3040.83	0.895013
$227$	−5650.97	−1.65228	−0.826141	−	0.563463i	$-0.809469\pi$
$227$	−5650.97	−1.65228	−0.826141	+	0.563463i	$0.809469\pi$
$228$	81.0133	0.0235317
$229$	−3414.75	−0.985385	−0.492692	−	0.870204i	$-0.663987\pi$
$229$	−3414.75	−0.985385	−0.492692	+	0.870204i	$0.663987\pi$
$230$	3057.65	0.876589
$231$	0	0
$232$	980.042	0.277340
$233$	1890.81	0.531635	0.265818	−	0.964023i	$-0.414358\pi$
$233$	1890.81	0.531635	0.265818	+	0.964023i	$0.414358\pi$
$234$	−2215.44	−0.618921
$235$	2763.84	0.767205
$236$	−240.344	−0.0662927
$237$	429.401	0.117690
$238$	−944.720	−0.257299
$239$	3871.28	1.04775	0.523875	−	0.851795i	$-0.324486\pi$
$239$	3871.28	1.04775	0.523875	+	0.851795i	$0.324486\pi$
$240$	−237.148	−0.0637827
$241$	−2530.30	−0.676312	−0.338156	−	0.941090i	$-0.609803\pi$
$241$	−2530.30	−0.676312	−0.338156	+	0.941090i	$0.609803\pi$
$242$	0	0
$243$	969.165	0.255852
$244$	125.004	0.0327973
$245$	1026.94	0.267791
$246$	−386.581	−0.100193
$247$	−5174.90	−1.33308
$248$	−5493.43	−1.40659
$249$	−48.8335	−0.0124285
$250$	−3961.49	−1.00219
$251$	5750.99	1.44621	0.723106	−	0.690737i	$-0.242714\pi$
$251$	5750.99	1.44621	0.723106	+	0.690737i	$0.242714\pi$
$252$	−624.186	−0.156032
$253$	0	0
$254$	1277.99	0.315702
$255$	74.6835	0.0183406
$256$	−1657.42	−0.404643
$257$	−611.131	−0.148332	−0.0741659	−	0.997246i	$-0.523629\pi$
$257$	−611.131	−0.148332	−0.0741659	+	0.997246i	$0.523629\pi$
$258$	40.5635	0.00978828
$259$	−4166.70	−0.999638
$260$	−340.077	−0.0811180
$261$	1098.74	0.260576
$262$	−5007.20	−1.18071
$263$	3314.85	0.777195	0.388597	−	0.921408i	$-0.372960\pi$
$263$	3314.85	0.777195	0.388597	+	0.921408i	$0.372960\pi$
$264$	0	0
$265$	874.752	0.202776
$266$	9137.43	2.10621
$267$	601.407	0.137848
$268$	−42.5839	−0.00970606
$269$	−5055.32	−1.14583	−0.572915	−	0.819615i	$-0.694188\pi$
$269$	−5055.32	−1.14583	−0.572915	+	0.819615i	$0.694188\pi$
$270$	−620.803	−0.139929
$271$	4359.84	0.977275	0.488637	−	0.872487i	$-0.337494\pi$
$271$	4359.84	0.977275	0.488637	+	0.872487i	$0.337494\pi$
$272$	−917.683	−0.204569
$273$	−298.021	−0.0660697
$274$	5590.15	1.23253
$275$	0	0
$276$	−58.4330	−0.0127437
$277$	−5272.18	−1.14359	−0.571796	−	0.820396i	$-0.693753\pi$
$277$	−5272.18	−1.14359	−0.571796	+	0.820396i	$0.693753\pi$
$278$	753.010	0.162455
$279$	−6158.77	−1.32156
$280$	4964.25	1.05954
$281$	7315.48	1.55304	0.776522	−	0.630090i	$-0.216982\pi$
$281$	7315.48	1.55304	0.776522	+	0.630090i	$0.216982\pi$
$282$	331.018	0.0699002
$283$	3844.65	0.807565	0.403782	−	0.914855i	$-0.367695\pi$
$283$	3844.65	0.807565	0.403782	+	0.914855i	$0.367695\pi$
$284$	266.508	0.0556842
$285$	−722.347	−0.150134
$286$	0	0
$287$	6957.30	1.43093
$288$	−1325.17	−0.271134
$289$	289.000	0.0588235
$290$	−1057.01	−0.214034
$291$	−558.868	−0.112582
$292$	−380.042	−0.0761654
$293$	−7082.74	−1.41221	−0.706106	−	0.708106i	$-0.749550\pi$
$293$	−7082.74	−1.41221	−0.706106	+	0.708106i	$0.749550\pi$
$294$	122.994	0.0243985
$295$	2143.01	0.422952
$296$	−4707.78	−0.924439
$297$	0	0
$298$	5324.03	1.03494
$299$	3732.54	0.721934
$300$	14.1178	0.00271696
$301$	−730.022	−0.139793
$302$	1536.04	0.292680
$303$	−87.5144	−0.0165926
$304$	8875.92	1.67457
$305$	−1114.58	−0.209249
$306$	−1196.68	−0.223560
$307$	−9752.05	−1.81296	−0.906481	−	0.422247i	$-0.861241\pi$
$307$	−9752.05	−1.81296	−0.906481	+	0.422247i	$0.861241\pi$
$308$	0	0
$309$	−71.1354	−0.0130963
$310$	5924.88	1.08552
$311$	5131.62	0.935652	0.467826	−	0.883821i	$-0.345037\pi$
$311$	5131.62	0.935652	0.467826	+	0.883821i	$0.345037\pi$
$312$	−336.721	−0.0610995
$313$	5460.01	0.986000	0.493000	−	0.870029i	$-0.335900\pi$
$313$	5460.01	0.986000	0.493000	+	0.870029i	$0.335900\pi$
$314$	6972.16	1.25306
$315$	5565.49	0.995492
$316$	−1056.17	−0.188019
$317$	−7660.98	−1.35736	−0.678680	−	0.734434i	$-0.737448\pi$
$317$	−7660.98	−1.35736	−0.678680	+	0.734434i	$0.737448\pi$
$318$	104.767	0.0184749
$319$	0	0
$320$	5513.73	0.963209
$321$	501.291	0.0871630
$322$	−6590.62	−1.14062
$323$	−2795.24	−0.481521
$324$	−784.701	−0.134551
$325$	−901.802	−0.153917
$326$	−3571.97	−0.606850
$327$	401.338	0.0678717
$328$	7860.76	1.32329
$329$	−5957.34	−0.998293
$330$	0	0
$331$	−1943.61	−0.322750	−0.161375	−	0.986893i	$-0.551593\pi$
$331$	−1943.61	−0.322750	−0.161375	+	0.986893i	$0.551593\pi$
$332$	120.112	0.0198555
$333$	−5277.96	−0.868559
$334$	−849.361	−0.139147
$335$	379.695	0.0619252
$336$	511.162	0.0829945
$337$	6547.56	1.05836	0.529181	−	0.848509i	$-0.322499\pi$
$337$	6547.56	1.05836	0.529181	+	0.848509i	$0.322499\pi$
$338$	−3168.97	−0.509968
$339$	518.146	0.0830143
$340$	−183.694	−0.0293006
$341$	0	0
$342$	11574.4	1.83003
$343$	5043.36	0.793924
$344$	−824.821	−0.129277
$345$	521.013	0.0813055
$346$	5188.92	0.806237
$347$	−4321.37	−0.668540	−0.334270	−	0.942477i	$-0.608490\pi$
$347$	−4321.37	−0.668540	−0.334270	+	0.942477i	$0.608490\pi$
$348$	20.2000	0.00311159
$349$	4937.21	0.757258	0.378629	−	0.925549i	$-0.376396\pi$
$349$	4937.21	0.757258	0.378629	+	0.925549i	$0.376396\pi$
$350$	1592.33	0.243182
$351$	−757.826	−0.115242
$352$	0	0
$353$	−5817.82	−0.877199	−0.438599	−	0.898683i	$-0.644525\pi$
$353$	−5817.82	−0.877199	−0.438599	+	0.898683i	$0.644525\pi$
$354$	256.662	0.0385352
$355$	−2376.29	−0.355269
$356$	−1479.24	−0.220223
$357$	−160.977	−0.0238650
$358$	−8621.30	−1.27277
$359$	12193.3	1.79258	0.896291	−	0.443466i	$-0.146251\pi$
$359$	12193.3	1.79258	0.896291	+	0.443466i	$0.146251\pi$
$360$	6288.21	0.920605
$361$	20176.8	2.94166
$362$	−6675.84	−0.969266
$363$	0	0
$364$	733.020	0.105551
$365$	3388.61	0.485940
$366$	−133.491	−0.0190647
$367$	7122.35	1.01304	0.506518	−	0.862230i	$-0.330933\pi$
$367$	7122.35	1.01304	0.506518	+	0.862230i	$0.330933\pi$
$368$	−6402.01	−0.906868
$369$	8812.81	1.24330
$370$	5077.52	0.713426
$371$	−1885.49	−0.263853
$372$	−113.227	−0.0157810
$373$	3434.51	0.476762	0.238381	−	0.971172i	$-0.423383\pi$
$373$	3434.51	0.476762	0.238381	+	0.971172i	$0.423383\pi$
$374$	0	0
$375$	−675.023	−0.0929548
$376$	−6730.94	−0.923196
$377$	−1290.32	−0.176272
$378$	1338.11	0.182077
$379$	12549.1	1.70080	0.850399	−	0.526138i	$-0.176361\pi$
$379$	12549.1	1.70080	0.850399	+	0.526138i	$0.176361\pi$
$380$	1776.71	0.239850
$381$	217.765	0.0292820
$382$	8710.88	1.16672
$383$	11780.3	1.57166	0.785831	−	0.618441i	$-0.212235\pi$
$383$	11780.3	1.57166	0.785831	+	0.618441i	$0.212235\pi$
$384$	483.317	0.0642296
$385$	0	0
$386$	7012.40	0.924667
$387$	−924.718	−0.121463
$388$	1374.61	0.179859
$389$	−8032.97	−1.04701	−0.523506	−	0.852022i	$-0.675376\pi$
$389$	−8032.97	−1.04701	−0.523506	+	0.852022i	$0.675376\pi$
$390$	363.166	0.0471529
$391$	2016.14	0.260769
$392$	−2500.96	−0.322239
$393$	−853.208	−0.109513
$394$	−6044.10	−0.772836
$395$	9417.22	1.19957
$396$	0	0
$397$	−6434.07	−0.813392	−0.406696	−	0.913563i	$-0.633319\pi$
$397$	−6434.07	−0.813392	−0.406696	+	0.913563i	$0.633319\pi$
$398$	−1215.13	−0.153037
$399$	1556.99	0.195355
$400$	1546.76	0.193345
$401$	8429.79	1.04978	0.524892	−	0.851169i	$-0.324106\pi$
$401$	8429.79	1.04978	0.524892	+	0.851169i	$0.324106\pi$
$402$	45.4751	0.00564201
$403$	7232.62	0.894001
$404$	215.253	0.0265080
$405$	6996.72	0.858443
$406$	2278.34	0.278503
$407$	0	0
$408$	−181.881	−0.0220697
$409$	−13871.5	−1.67703	−0.838513	−	0.544881i	$-0.816575\pi$
$409$	−13871.5	−1.67703	−0.838513	+	0.544881i	$0.816575\pi$
$410$	−8478.13	−1.02123
$411$	952.542	0.114320
$412$	174.967	0.0209223
$413$	−4619.15	−0.550348
$414$	−8348.33	−0.991059
$415$	−1070.97	−0.126679
$416$	1556.23	0.183415
$417$	128.310	0.0150681
$418$	0	0
$419$	−5981.50	−0.697412	−0.348706	−	0.937232i	$-0.613379\pi$
$419$	−5981.50	−0.697412	−0.348706	+	0.937232i	$0.613379\pi$
$420$	102.320	0.0118874
$421$	2176.05	0.251910	0.125955	−	0.992036i	$-0.459800\pi$
$421$	2176.05	0.251910	0.125955	+	0.992036i	$0.459800\pi$
$422$	13939.9	1.60802
$423$	−7546.15	−0.867391
$424$	−2130.33	−0.244005
$425$	−487.112	−0.0555962
$426$	−284.602	−0.0323686
$427$	2402.43	0.272276
$428$	−1232.99	−0.139249
$429$	0	0
$430$	889.601	0.0997683
$431$	16124.0	1.80201	0.901004	−	0.433811i	$-0.142831\pi$
$431$	16124.0	1.80201	0.901004	+	0.433811i	$0.142831\pi$
$432$	1299.82	0.144762
$433$	11222.2	1.24551	0.622756	−	0.782416i	$-0.286013\pi$
$433$	11222.2	1.24551	0.622756	+	0.782416i	$0.286013\pi$
$434$	−12770.8	−1.41248
$435$	−180.111	−0.0198521
$436$	−987.143	−0.108430
$437$	−19500.3	−2.13462
$438$	405.845	0.0442740
$439$	−11850.4	−1.28836	−0.644178	−	0.764876i	$-0.722800\pi$
$439$	−11850.4	−1.28836	−0.644178	+	0.764876i	$0.722800\pi$
$440$	0	0
$441$	−2803.87	−0.302761
$442$	1405.33	0.151232
$443$	−6913.30	−0.741447	−0.370723	−	0.928743i	$-0.620890\pi$
$443$	−6913.30	−0.741447	−0.370723	+	0.928743i	$0.620890\pi$
$444$	−97.0336	−0.0103716
$445$	13189.5	1.40504
$446$	−10668.8	−1.13270
$447$	907.196	0.0959931
$448$	−11884.6	−1.25333
$449$	2076.79	0.218285	0.109142	−	0.994026i	$-0.465190\pi$
$449$	2076.79	0.218285	0.109142	+	0.994026i	$0.465190\pi$
$450$	2017.01	0.211295
$451$	0	0
$452$	−1274.45	−0.132622
$453$	261.736	0.0271466
$454$	−14843.0	−1.53440
$455$	−6535.90	−0.673424
$456$	1759.17	0.180659
$457$	−7379.60	−0.755368	−0.377684	−	0.925935i	$-0.623279\pi$
$457$	−7379.60	−0.755368	−0.377684	+	0.925935i	$0.623279\pi$
$458$	−8969.26	−0.915079
$459$	−409.343	−0.0416263
$460$	−1281.50	−0.129892
$461$	−8852.42	−0.894357	−0.447178	−	0.894445i	$-0.647571\pi$
$461$	−8852.42	−0.894357	−0.447178	+	0.894445i	$0.647571\pi$
$462$	0	0
$463$	−7239.37	−0.726657	−0.363328	−	0.931661i	$-0.618360\pi$
$463$	−7239.37	−0.726657	−0.363328	+	0.931661i	$0.618360\pi$
$464$	2213.14	0.221427
$465$	1009.58	0.100684
$466$	4966.44	0.493704
$467$	16024.3	1.58782	0.793912	−	0.608032i	$-0.208041\pi$
$467$	16024.3	1.58782	0.793912	+	0.608032i	$0.208041\pi$
$468$	928.516	0.0917108
$469$	−818.415	−0.0805776
$470$	7259.58	0.712467
$471$	1188.03	0.116224
$472$	−5218.98	−0.508947
$473$	0	0
$474$	1127.88	0.109293
$475$	4711.40	0.455102
$476$	395.943	0.0381261
$477$	−2388.34	−0.229255
$478$	10168.4	0.972994
$479$	9515.90	0.907709	0.453855	−	0.891076i	$-0.350049\pi$
$479$	9515.90	0.907709	0.453855	+	0.891076i	$0.350049\pi$
$480$	217.230	0.0206565
$481$	6198.23	0.587557
$482$	−6646.15	−0.628058
$483$	−1123.02	−0.105795
$484$	0	0
$485$	−12256.6	−1.14751
$486$	2545.63	0.237597
$487$	−7281.24	−0.677504	−0.338752	−	0.940876i	$-0.610005\pi$
$487$	−7281.24	−0.677504	−0.338752	+	0.940876i	$0.610005\pi$
$488$	2714.40	0.251794
$489$	−608.651	−0.0562866
$490$	2697.39	0.248685
$491$	−4668.89	−0.429132	−0.214566	−	0.976709i	$-0.568834\pi$
$491$	−4668.89	−0.429132	−0.214566	+	0.976709i	$0.568834\pi$
$492$	162.021	0.0148465
$493$	−696.970	−0.0636713
$494$	−13592.5	−1.23797
$495$	0	0
$496$	−12405.3	−1.12301
$497$	5121.98	0.462278
$498$	−128.267	−0.0115418
$499$	−20500.5	−1.83913	−0.919567	−	0.392934i	$-0.871460\pi$
$499$	−20500.5	−1.83913	−0.919567	+	0.392934i	$0.871460\pi$
$500$	1660.31	0.148502
$501$	−144.728	−0.0129061
$502$	15105.7	1.34303
$503$	−5141.02	−0.455719	−0.227860	−	0.973694i	$-0.573173\pi$
$503$	−5141.02	−0.455719	−0.227860	+	0.973694i	$0.573173\pi$
$504$	−13553.9	−1.19790
$505$	−1919.28	−0.169123
$506$	0	0
$507$	−539.981	−0.0473006
$508$	−535.621	−0.0467802
$509$	10957.5	0.954193	0.477097	−	0.878851i	$-0.341689\pi$
$509$	10957.5	0.954193	0.477097	+	0.878851i	$0.341689\pi$
$510$	196.166	0.0170321
$511$	−7303.99	−0.632308
$512$	−12992.4	−1.12146
$513$	3959.20	0.340747
$514$	−1605.21	−0.137749
$515$	−1560.07	−0.133486
$516$	−17.0007	−0.00145041
$517$	0	0
$518$	−10944.4	−0.928316
$519$	884.173	0.0747801
$520$	−7384.64	−0.622765
$521$	9623.01	0.809197	0.404599	−	0.914494i	$-0.367411\pi$
$521$	9623.01	0.809197	0.404599	+	0.914494i	$0.367411\pi$
$522$	2885.97	0.241984
$523$	21271.6	1.77847	0.889237	−	0.457446i	$-0.151236\pi$
$523$	21271.6	1.77847	0.889237	+	0.457446i	$0.151236\pi$
$524$	2098.58	0.174956
$525$	271.328	0.0225556
$526$	8706.85	0.721743
$527$	3906.73	0.322922
$528$	0	0
$529$	1898.16	0.156009
$530$	2297.64	0.188308
$531$	−5851.08	−0.478183
$532$	−3829.61	−0.312095
$533$	−10349.4	−0.841057
$534$	1579.67	0.128013
$535$	10993.8	0.888420
$536$	−924.692	−0.0745160
$537$	−1469.04	−0.118052
$538$	−13278.4	−1.06408
$539$	0	0
$540$	260.186	0.0207345
$541$	−657.106	−0.0522204	−0.0261102	−	0.999659i	$-0.508312\pi$
$541$	−657.106	−0.0522204	−0.0261102	+	0.999659i	$0.508312\pi$
$542$	11451.7	0.907548
$543$	−1137.54	−0.0899014
$544$	840.606	0.0662512
$545$	8801.77	0.691791
$546$	−782.788	−0.0613557
$547$	24418.1	1.90867	0.954336	−	0.298735i	$-0.0965646\pi$
$547$	24418.1	1.90867	0.954336	+	0.298735i	$0.0965646\pi$
$548$	−2342.90	−0.182634
$549$	3043.16	0.236573
$550$	0	0
$551$	6741.16	0.521204
$552$	−1268.85	−0.0978367
$553$	−20298.4	−1.56090
$554$	−13848.0	−1.06200
$555$	865.191	0.0661718
$556$	−315.596	−0.0240724
$557$	−9529.58	−0.724922	−0.362461	−	0.931999i	$-0.618063\pi$
$557$	−9529.58	−0.724922	−0.362461	+	0.931999i	$0.618063\pi$
$558$	−16176.8	−1.22727
$559$	1085.95	0.0821663
$560$	11210.3	0.845932
$561$	0	0
$562$	19215.0	1.44224
$563$	−1501.45	−0.112395	−0.0561975	−	0.998420i	$-0.517898\pi$
$563$	−1501.45	−0.112395	−0.0561975	+	0.998420i	$0.517898\pi$
$564$	−138.734	−0.0103577
$565$	11363.5	0.846134
$566$	10098.5	0.749947
$567$	−15081.1	−1.11701
$568$	5787.11	0.427503
$569$	8465.64	0.623723	0.311861	−	0.950128i	$-0.399048\pi$
$569$	8465.64	0.623723	0.311861	+	0.950128i	$0.399048\pi$
$570$	−1897.33	−0.139422
$571$	−5622.12	−0.412046	−0.206023	−	0.978547i	$-0.566052\pi$
$571$	−5622.12	−0.412046	−0.206023	+	0.978547i	$0.566052\pi$
$572$	0	0
$573$	1484.30	0.108216
$574$	18274.2	1.32883
$575$	−3398.22	−0.246462
$576$	−15054.2	−1.08899
$577$	−10537.8	−0.760305	−0.380153	−	0.924924i	$-0.624129\pi$
$577$	−10537.8	−0.760305	−0.380153	+	0.924924i	$0.624129\pi$
$578$	759.094	0.0546266
$579$	1194.89	0.0857648
$580$	443.007	0.0317153
$581$	2308.43	0.164836
$582$	−1467.94	−0.104550
$583$	0	0
$584$	−8252.47	−0.584742
$585$	−8279.02	−0.585120
$586$	−18603.7	−1.31145
$587$	−14545.3	−1.02274	−0.511371	−	0.859360i	$-0.670862\pi$
$587$	−14545.3	−1.02274	−0.511371	+	0.859360i	$0.670862\pi$
$588$	−51.5483	−0.00361533
$589$	−37786.3	−2.64339
$590$	5628.88	0.392775
$591$	−1029.89	−0.0716822
$592$	−10631.1	−0.738069
$593$	23802.4	1.64831	0.824156	−	0.566363i	$-0.191650\pi$
$593$	23802.4	1.64831	0.824156	+	0.566363i	$0.191650\pi$
$594$	0	0
$595$	−3530.39	−0.243247
$596$	−2231.37	−0.153356
$597$	−207.053	−0.0141945
$598$	9803.97	0.670425
$599$	1494.23	0.101924	0.0509621	−	0.998701i	$-0.483771\pi$
$599$	1494.23	0.101924	0.0509621	+	0.998701i	$0.483771\pi$
$600$	306.562	0.0208589
$601$	−20323.3	−1.37938	−0.689689	−	0.724106i	$-0.742253\pi$
$601$	−20323.3	−1.37938	−0.689689	+	0.724106i	$0.742253\pi$
$602$	−1917.49	−0.129819
$603$	−1036.69	−0.0700117
$604$	−643.773	−0.0433688
$605$	0	0
$606$	−229.867	−0.0154088
$607$	−18367.5	−1.22819	−0.614096	−	0.789232i	$-0.710479\pi$
$607$	−18367.5	−1.22819	−0.614096	+	0.789232i	$0.710479\pi$
$608$	−8130.43	−0.542323
$609$	388.221	0.0258317
$610$	−2927.59	−0.194319
$611$	8861.91	0.586767
$612$	501.541	0.0331268
$613$	−21692.8	−1.42930	−0.714652	−	0.699480i	$-0.753415\pi$
$613$	−21692.8	−1.42930	−0.714652	+	0.699480i	$0.753415\pi$
$614$	−25615.0	−1.68361
$615$	−1444.64	−0.0947214
$616$	0	0
$617$	−3342.60	−0.218100	−0.109050	−	0.994036i	$-0.534781\pi$
$617$	−3342.60	−0.218100	−0.109050	+	0.994036i	$0.534781\pi$
$618$	−186.846	−0.0121619
$619$	21127.4	1.37186	0.685929	−	0.727668i	$-0.259396\pi$
$619$	21127.4	1.37186	0.685929	+	0.727668i	$0.259396\pi$
$620$	−2483.19	−0.160850
$621$	−2855.69	−0.184533
$622$	13478.8	0.868894
$623$	−28429.3	−1.82824
$624$	−760.385	−0.0487817
$625$	−11222.3	−0.718225
$626$	14341.4	0.915651
$627$	0	0
$628$	−2922.12	−0.185677
$629$	3348.00	0.212231
$630$	14618.5	0.924465
$631$	−21087.0	−1.33036	−0.665182	−	0.746681i	$-0.731646\pi$
$631$	−21087.0	−1.33036	−0.665182	+	0.746681i	$0.731646\pi$
$632$	−22934.3	−1.44348
$633$	2375.32	0.149147
$634$	−20122.5	−1.26051
$635$	4775.81	0.298460
$636$	−43.9090	−0.00273758
$637$	3292.76	0.204810
$638$	0	0
$639$	6488.01	0.401661
$640$	10599.6	0.654668
$641$	18629.7	1.14794	0.573971	−	0.818876i	$-0.305402\pi$
$641$	18629.7	1.14794	0.573971	+	0.818876i	$0.305402\pi$
$642$	1316.70	0.0809440
$643$	−13426.5	−0.823469	−0.411735	−	0.911304i	$-0.635077\pi$
$643$	−13426.5	−0.823469	−0.411735	+	0.911304i	$0.635077\pi$
$644$	2762.21	0.169016
$645$	151.585	0.00925372
$646$	−7342.04	−0.447165
$647$	11631.2	0.706751	0.353376	−	0.935482i	$-0.385034\pi$
$647$	11631.2	0.706751	0.353376	+	0.935482i	$0.385034\pi$
$648$	−17039.5	−1.03298
$649$	0	0
$650$	−2368.69	−0.142935
$651$	−2176.10	−0.131011
$652$	1497.05	0.0899221
$653$	24813.6	1.48703	0.743515	−	0.668719i	$-0.233157\pi$
$653$	24813.6	1.48703	0.743515	+	0.668719i	$0.233157\pi$
$654$	1054.16	0.0630292
$655$	−18711.8	−1.11623
$656$	17751.2	1.05651
$657$	−9251.96	−0.549396
$658$	−15647.7	−0.927067
$659$	28209.0	1.66747	0.833737	−	0.552161i	$-0.186197\pi$
$659$	28209.0	1.66747	0.833737	+	0.552161i	$0.186197\pi$
$660$	0	0
$661$	10453.0	0.615087	0.307544	−	0.951534i	$-0.400493\pi$
$661$	10453.0	0.615087	0.307544	+	0.951534i	$0.400493\pi$
$662$	−5105.12	−0.299723
$663$	239.463	0.0140271
$664$	2608.19	0.152436
$665$	34146.3	1.99118
$666$	−13863.2	−0.806589
$667$	−4862.25	−0.282259
$668$	355.977	0.0206185
$669$	−1817.93	−0.105060
$670$	997.316	0.0575070
$671$	0	0
$672$	−468.229	−0.0268784
$673$	−11513.5	−0.659455	−0.329728	−	0.944076i	$-0.606957\pi$
$673$	−11513.5	−0.659455	−0.329728	+	0.944076i	$0.606957\pi$
$674$	17198.0	0.982850
$675$	689.950	0.0393425
$676$	1328.15	0.0755662
$677$	−3285.96	−0.186543	−0.0932715	−	0.995641i	$-0.529732\pi$
$677$	−3285.96	−0.186543	−0.0932715	+	0.995641i	$0.529732\pi$
$678$	1360.97	0.0770913
$679$	26418.5	1.49315
$680$	−3988.84	−0.224948
$681$	−2529.19	−0.142318
$682$	0	0
$683$	−14822.1	−0.830386	−0.415193	−	0.909733i	$-0.636286\pi$
$683$	−14822.1	−0.830386	−0.415193	+	0.909733i	$0.636286\pi$
$684$	−4850.96	−0.271171
$685$	20890.2	1.16522
$686$	13247.0	0.737279
$687$	−1528.33	−0.0848755
$688$	−1862.62	−0.103215
$689$	2804.78	0.155085
$690$	1368.50	0.0755044
$691$	7389.63	0.406823	0.203412	−	0.979093i	$-0.434797\pi$
$691$	7389.63	0.406823	0.203412	+	0.979093i	$0.434797\pi$
$692$	−2174.74	−0.119467
$693$	0	0
$694$	−11350.6	−0.620841
$695$	2813.98	0.153583
$696$	438.635	0.0238885
$697$	−5590.28	−0.303798
$698$	12968.2	0.703229
$699$	846.264	0.0457920
$700$	−667.365	−0.0360343
$701$	−6812.00	−0.367027	−0.183513	−	0.983017i	$-0.558747\pi$
$701$	−6812.00	−0.367027	−0.183513	+	0.983017i	$0.558747\pi$
$702$	−1990.52	−0.107019
$703$	−32382.2	−1.73729
$704$	0	0
$705$	1237.01	0.0660827
$706$	−15281.2	−0.814612
$707$	4136.92	0.220064
$708$	−107.570	−0.00571008
$709$	−34700.2	−1.83807	−0.919036	−	0.394173i	$-0.871031\pi$
$709$	−34700.2	−1.83807	−0.919036	+	0.394173i	$0.871031\pi$
$710$	−6241.62	−0.329921
$711$	−25711.9	−1.35622
$712$	−32121.0	−1.69071
$713$	27254.4	1.43154
$714$	−422.826	−0.0221623
$715$	0	0
$716$	3613.29	0.188596
$717$	1732.66	0.0902472
$718$	32027.2	1.66468
$719$	18345.8	0.951576	0.475788	−	0.879560i	$-0.342163\pi$
$719$	18345.8	0.951576	0.475788	+	0.879560i	$0.342163\pi$
$720$	14200.1	0.735008
$721$	3362.67	0.173692
$722$	52997.0	2.73178
$723$	−1132.48	−0.0582537
$724$	2797.92	0.143624
$725$	1174.75	0.0601780
$726$	0	0
$727$	−26658.5	−1.35999	−0.679993	−	0.733219i	$-0.738017\pi$
$727$	−26658.5	−1.35999	−0.679993	+	0.733219i	$0.738017\pi$
$728$	15917.2	0.810346
$729$	−18812.2	−0.955760
$730$	8900.60	0.451269
$731$	586.582	0.0296792
$732$	55.9475	0.00282497
$733$	−36988.4	−1.86384	−0.931922	−	0.362658i	$-0.881869\pi$
$733$	−36988.4	−1.86384	−0.931922	+	0.362658i	$0.881869\pi$
$734$	18707.7	0.940757
$735$	459.625	0.0230660
$736$	5864.29	0.293697
$737$	0	0
$738$	23147.9	1.15459
$739$	−13103.7	−0.652271	−0.326135	−	0.945323i	$-0.605747\pi$
$739$	−13103.7	−0.652271	−0.326135	+	0.945323i	$0.605747\pi$
$740$	−2128.05	−0.105714
$741$	−2316.11	−0.114824
$742$	−4952.46	−0.245028
$743$	−2793.90	−0.137952	−0.0689759	−	0.997618i	$-0.521973\pi$
$743$	−2793.90	−0.137952	−0.0689759	+	0.997618i	$0.521973\pi$
$744$	−2458.68	−0.121155
$745$	19895.8	0.978422
$746$	9021.17	0.442746
$747$	2924.08	0.143222
$748$	0	0
$749$	−23696.7	−1.15602
$750$	−1773.03	−0.0863226
$751$	21755.5	1.05708	0.528542	−	0.848907i	$-0.322739\pi$
$751$	21755.5	1.05708	0.528542	+	0.848907i	$0.322739\pi$
$752$	−15199.9	−0.737077
$753$	2573.96	0.124569
$754$	−3389.18	−0.163696
$755$	5740.14	0.276696
$756$	−560.818	−0.0269798
$757$	−33379.5	−1.60264	−0.801319	−	0.598237i	$-0.795868\pi$
$757$	−33379.5	−1.60264	−0.801319	+	0.598237i	$0.795868\pi$
$758$	32961.7	1.57945
$759$	0	0
$760$	38580.5	1.84140
$761$	15410.8	0.734090	0.367045	−	0.930203i	$-0.380370\pi$
$761$	15410.8	0.734090	0.367045	+	0.930203i	$0.380370\pi$
$762$	571.986	0.0271927
$763$	−18971.8	−0.900164
$764$	−3650.83	−0.172883
$765$	−4471.94	−0.211351
$766$	30942.5	1.45953
$767$	6871.28	0.323478
$768$	−741.806	−0.0348537
$769$	7909.87	0.370920	0.185460	−	0.982652i	$-0.440623\pi$
$769$	7909.87	0.370920	0.185460	+	0.982652i	$0.440623\pi$
$770$	0	0
$771$	−273.522	−0.0127765
$772$	−2938.98	−0.137016
$773$	23265.3	1.08253	0.541264	−	0.840853i	$-0.317946\pi$
$773$	23265.3	1.08253	0.541264	+	0.840853i	$0.317946\pi$
$774$	−2428.89	−0.112797
$775$	−6584.82	−0.305205
$776$	29849.1	1.38082
$777$	−1864.88	−0.0861032
$778$	−21099.6	−0.972309
$779$	54069.8	2.48684
$780$	−152.207	−0.00698705
$781$	0	0
$782$	5295.65	0.242164
$783$	987.195	0.0450568
$784$	−5647.70	−0.257275
$785$	26054.8	1.18463
$786$	−2241.06	−0.101700
$787$	−21095.9	−0.955511	−0.477756	−	0.878493i	$-0.658550\pi$
$787$	−21095.9	−0.955511	−0.477756	+	0.878493i	$0.658550\pi$
$788$	2533.16	0.114518
$789$	1483.62	0.0669432
$790$	24735.5	1.11399
$791$	−24493.5	−1.10100
$792$	0	0
$793$	−3573.77	−0.160036
$794$	−16899.9	−0.755358
$795$	391.510	0.0174660
$796$	509.274	0.0226768
$797$	−79.4517	−0.00353115	−0.00176557	−	0.999998i	$-0.500562\pi$
$797$	−79.4517	−0.00353115	−0.00176557	+	0.999998i	$0.500562\pi$
$798$	4089.61	0.181417
$799$	4786.79	0.211946
$800$	−1416.85	−0.0626164
$801$	−36011.4	−1.58851
$802$	22141.9	0.974884
$803$	0	0
$804$	−19.0591	−0.000836025 0
$805$	−24629.0	−1.07833
$806$	18997.4	0.830216
$807$	−2262.60	−0.0986953
$808$	4674.13	0.203509
$809$	22943.5	0.997095	0.498548	−	0.866862i	$-0.333867\pi$
$809$	22943.5	0.997095	0.498548	+	0.866862i	$0.333867\pi$
$810$	18377.7	0.797195
$811$	36919.6	1.59855	0.799274	−	0.600967i	$-0.205218\pi$
$811$	36919.6	1.59855	0.799274	+	0.600967i	$0.205218\pi$
$812$	−954.880	−0.0412681
$813$	1951.32	0.0841769
$814$	0	0
$815$	−13348.3	−0.573708
$816$	−410.725	−0.0176204
$817$	−5673.48	−0.242950
$818$	−36435.3	−1.55737
$819$	17845.0	0.761363
$820$	3553.29	0.151325
$821$	−7061.69	−0.300188	−0.150094	−	0.988672i	$-0.547958\pi$
$821$	−7061.69	−0.300188	−0.150094	+	0.988672i	$0.547958\pi$
$822$	2501.97	0.106163
$823$	17365.1	0.735492	0.367746	−	0.929926i	$-0.380130\pi$
$823$	17365.1	0.735492	0.367746	+	0.929926i	$0.380130\pi$
$824$	3799.33	0.160626
$825$	0	0
$826$	−12132.8	−0.511082
$827$	1350.18	0.0567718	0.0283859	−	0.999597i	$-0.490963\pi$
$827$	1350.18	0.0567718	0.0283859	+	0.999597i	$0.490963\pi$
$828$	3498.89	0.146854
$829$	−27944.8	−1.17076	−0.585382	−	0.810758i	$-0.699055\pi$
$829$	−27944.8	−1.17076	−0.585382	+	0.810758i	$0.699055\pi$
$830$	−2813.04	−0.117641
$831$	−2359.66	−0.0985025
$832$	17679.1	0.736673
$833$	1778.59	0.0739791
$834$	337.023	0.0139930
$835$	−3174.04	−0.131547
$836$	0	0
$837$	−5533.53	−0.228515
$838$	−15711.2	−0.647653
$839$	−2211.80	−0.0910128	−0.0455064	−	0.998964i	$-0.514490\pi$
$839$	−2211.80	−0.0910128	−0.0455064	+	0.998964i	$0.514490\pi$
$840$	2221.84	0.0912626
$841$	−22708.1	−0.931082
$842$	5715.67	0.233937
$843$	3274.17	0.133770
$844$	−5842.40	−0.238274
$845$	−11842.3	−0.482117
$846$	−19820.9	−0.805504
$847$	0	0
$848$	−4810.73	−0.194813
$849$	1720.74	0.0695591
$850$	−1279.46	−0.0516295
$851$	23356.5	0.940837
$852$	119.280	0.00479632
$853$	38390.8	1.54100	0.770501	−	0.637439i	$-0.220006\pi$
$853$	38390.8	1.54100	0.770501	+	0.637439i	$0.220006\pi$
$854$	6310.29	0.252850
$855$	43253.1	1.73009
$856$	−26773.9	−1.06906
$857$	21371.6	0.851855	0.425928	−	0.904757i	$-0.359948\pi$
$857$	21371.6	0.851855	0.425928	+	0.904757i	$0.359948\pi$
$858$	0	0
$859$	24207.0	0.961504	0.480752	−	0.876857i	$-0.340364\pi$
$859$	24207.0	0.961504	0.480752	+	0.876857i	$0.340364\pi$
$860$	−372.842	−0.0147835
$861$	3113.86	0.123252
$862$	42351.7	1.67344
$863$	11773.3	0.464391	0.232196	−	0.972669i	$-0.425409\pi$
$863$	11773.3	0.464391	0.232196	+	0.972669i	$0.425409\pi$
$864$	−1190.64	−0.0468825
$865$	19390.8	0.762206
$866$	29476.6	1.15665
$867$	129.347	0.00506673
$868$	5352.40	0.209300
$869$	0	0
$870$	−473.084	−0.0184357
$871$	1217.44	0.0473611
$872$	−21435.4	−0.832448
$873$	33464.2	1.29736
$874$	−51220.1	−1.98232
$875$	31909.3	1.23283
$876$	−170.094	−0.00656045
$877$	−36559.1	−1.40765	−0.703826	−	0.710372i	$-0.748527\pi$
$877$	−36559.1	−1.40765	−0.703826	+	0.710372i	$0.748527\pi$
$878$	−31126.5	−1.19643
$879$	−3170.00	−0.121640
$880$	0	0
$881$	−39666.6	−1.51692	−0.758458	−	0.651722i	$-0.774047\pi$
$881$	−39666.6	−1.51692	−0.758458	+	0.651722i	$0.774047\pi$
$882$	−7364.70	−0.281159
$883$	15193.5	0.579052	0.289526	−	0.957170i	$-0.406502\pi$
$883$	15193.5	0.579052	0.289526	+	0.957170i	$0.406502\pi$
$884$	−588.991	−0.0224094
$885$	959.140	0.0364307
$886$	−18158.6	−0.688546
$887$	−39304.3	−1.48783	−0.743917	−	0.668272i	$-0.767034\pi$
$887$	−39304.3	−1.48783	−0.743917	+	0.668272i	$0.767034\pi$
$888$	−2107.05	−0.0796260
$889$	−10294.0	−0.388359
$890$	34643.8	1.30479
$891$	0	0
$892$	4471.43	0.167841
$893$	−46298.4	−1.73496
$894$	2382.86	0.0891441
$895$	−32217.6	−1.20326
$896$	−22847.0	−0.851859
$897$	1670.56	0.0621833
$898$	5454.95	0.202711
$899$	−9421.69	−0.349534
$900$	−845.351	−0.0313093
$901$	1515.01	0.0560182
$902$	0	0
$903$	−326.734	−0.0120410
$904$	−27674.1	−1.01817
$905$	−24947.4	−0.916332
$906$	687.482	0.0252098
$907$	−37140.3	−1.35967	−0.679836	−	0.733364i	$-0.737949\pi$
$907$	−37140.3	−1.35967	−0.679836	+	0.733364i	$0.737949\pi$
$908$	6220.87	0.227364
$909$	5240.24	0.191208
$910$	−17167.4	−0.625376
$911$	−3463.44	−0.125959	−0.0629797	−	0.998015i	$-0.520060\pi$
$911$	−3463.44	−0.125959	−0.0629797	+	0.998015i	$0.520060\pi$
$912$	3972.57	0.144238
$913$	0	0
$914$	−19383.4	−0.701473
$915$	−498.851	−0.0180235
$916$	3759.13	0.135595
$917$	40332.3	1.45244
$918$	−1075.19	−0.0386563
$919$	41545.0	1.49123	0.745616	−	0.666376i	$-0.232155\pi$
$919$	41545.0	1.49123	0.745616	+	0.666376i	$0.232155\pi$
$920$	−27827.2	−0.997213
$921$	−4364.70	−0.156158
$922$	−23252.0	−0.830546
$923$	−7619.27	−0.271713
$924$	0	0
$925$	−5643.07	−0.200587
$926$	−19015.1	−0.674811
$927$	4259.49	0.150917
$928$	−2027.25	−0.0717111
$929$	−25841.8	−0.912641	−0.456320	−	0.889816i	$-0.650833\pi$
$929$	−25841.8	−0.912641	−0.456320	+	0.889816i	$0.650833\pi$
$930$	2651.78	0.0935004
$931$	−17202.7	−0.605582
$932$	−2081.49	−0.0731562
$933$	2296.75	0.0805917
$934$	42089.7	1.47454
$935$	0	0
$936$	20162.3	0.704089
$937$	−21136.9	−0.736939	−0.368470	−	0.929640i	$-0.620118\pi$
$937$	−21136.9	−0.736939	−0.368470	+	0.929640i	$0.620118\pi$
$938$	−2149.67	−0.0748285
$939$	2443.72	0.0849285
$940$	−3042.57	−0.105572
$941$	5398.95	0.187036	0.0935180	−	0.995618i	$-0.470189\pi$
$941$	5398.95	0.187036	0.0935180	+	0.995618i	$0.470189\pi$
$942$	3120.51	0.107932
$943$	−38999.3	−1.34676
$944$	−11785.5	−0.406342
$945$	5000.48	0.172133
$946$	0	0
$947$	28724.5	0.985661	0.492831	−	0.870125i	$-0.335962\pi$
$947$	28724.5	0.985661	0.492831	+	0.870125i	$0.335962\pi$
$948$	−472.706	−0.0161949
$949$	10865.1	0.371652
$950$	12375.1	0.422632
$951$	−3428.80	−0.116915
$952$	8597.75	0.292705
$953$	48567.4	1.65084	0.825421	−	0.564518i	$-0.190938\pi$
$953$	48567.4	1.65084	0.825421	+	0.564518i	$0.190938\pi$
$954$	−6273.28	−0.212898
$955$	32552.3	1.10300
$956$	−4261.69	−0.144177
$957$	0	0
$958$	24994.7	0.842946
$959$	−45027.9	−1.51619
$960$	2467.77	0.0829654
$961$	23020.5	0.772732
$962$	16280.4	0.545636
$963$	−30016.6	−1.00443
$964$	2785.48	0.0930647
$965$	26205.1	0.874169
$966$	−2949.75	−0.0982469
$967$	36587.2	1.21672	0.608359	−	0.793662i	$-0.291828\pi$
$967$	36587.2	1.21672	0.608359	+	0.793662i	$0.291828\pi$
$968$	0	0
$969$	−1251.06	−0.0414755
$970$	−32193.4	−1.06564
$971$	−13679.7	−0.452112	−0.226056	−	0.974114i	$-0.572583\pi$
$971$	−13679.7	−0.452112	−0.226056	+	0.974114i	$0.572583\pi$
$972$	−1066.90	−0.0352068
$973$	−6065.40	−0.199844
$974$	−19125.1	−0.629165
$975$	−403.617	−0.0132575
$976$	6129.69	0.201031
$977$	−3140.90	−0.102852	−0.0514260	−	0.998677i	$-0.516377\pi$
$977$	−3140.90	−0.102852	−0.0514260	+	0.998677i	$0.516377\pi$
$978$	−1598.70	−0.0522706
$979$	0	0
$980$	−1130.51	−0.0368497
$981$	−24031.6	−0.782129
$982$	−12263.4	−0.398515
$983$	−22968.9	−0.745265	−0.372632	−	0.927979i	$-0.621545\pi$
$983$	−22968.9	−0.745265	−0.372632	+	0.927979i	$0.621545\pi$
$984$	3518.22	0.113980
$985$	−22586.6	−0.730630
$986$	−1830.68	−0.0591284
$987$	−2666.31	−0.0859874
$988$	5696.78	0.183440
$989$	4092.16	0.131570
$990$	0	0
$991$	7553.19	0.242114	0.121057	−	0.992646i	$-0.461372\pi$
$991$	7553.19	0.242114	0.121057	+	0.992646i	$0.461372\pi$
$992$	11363.4	0.363697
$993$	−869.895	−0.0277999
$994$	13453.5	0.429295
$995$	−4540.89	−0.144679
$996$	53.7583	0.00171024
$997$	−39778.8	−1.26360	−0.631798	−	0.775133i	$-0.717683\pi$
$997$	−39778.8	−1.26360	−0.631798	+	0.775133i	$0.717683\pi$
$998$	−53847.0	−1.70791
$999$	−4742.14	−0.150185

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.h.1.8		10
11.10	odd	2		187.4.a.e.1.3	✓	10
33.32	even	2		1683.4.a.j.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.a.e.1.3	✓	10	11.10	odd	2
1683.4.a.j.1.8		10	33.32	even	2
2057.4.a.h.1.8		10	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:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62662 q^{2} +0.447567 q^{3} -1.10085 q^{4} +9.81562 q^{5} +1.17559 q^{6} -21.1571 q^{7} -23.9045 q^{8} -26.7997 q^{9} +25.7819 q^{10} -0.492704 q^{12} +31.4725 q^{13} -55.5718 q^{14} +4.39315 q^{15} -53.9813 q^{16} +17.0000 q^{17} -70.3927 q^{18} -164.426 q^{19} -10.8055 q^{20} -9.46923 q^{21} +118.597 q^{23} -10.6989 q^{24} -28.6536 q^{25} +82.6665 q^{26} -24.0790 q^{27} +23.2908 q^{28} -40.9982 q^{29} +11.5391 q^{30} +229.807 q^{31} +49.4474 q^{32} +44.6526 q^{34} -207.670 q^{35} +29.5024 q^{36} +196.941 q^{37} -431.885 q^{38} +14.0861 q^{39} -234.637 q^{40} -328.840 q^{41} -24.8721 q^{42} +34.5048 q^{43} -263.055 q^{45} +311.509 q^{46} +281.576 q^{47} -24.1603 q^{48} +104.623 q^{49} -75.2623 q^{50} +7.60864 q^{51} -34.6465 q^{52} +89.1184 q^{53} -63.2464 q^{54} +505.750 q^{56} -73.5916 q^{57} -107.687 q^{58} +218.326 q^{59} -4.83619 q^{60} -113.552 q^{61} +603.618 q^{62} +567.004 q^{63} +561.730 q^{64} +308.922 q^{65} +38.6827 q^{67} -18.7144 q^{68} +53.0800 q^{69} -545.471 q^{70} -242.093 q^{71} +640.633 q^{72} +345.226 q^{73} +517.290 q^{74} -12.8244 q^{75} +181.008 q^{76} +36.9988 q^{78} +959.412 q^{79} -529.860 q^{80} +712.814 q^{81} -863.739 q^{82} -109.109 q^{83} +10.4242 q^{84} +166.866 q^{85} +90.6312 q^{86} -18.3495 q^{87} +1343.72 q^{89} -690.948 q^{90} -665.868 q^{91} -130.557 q^{92} +102.854 q^{93} +739.594 q^{94} -1613.94 q^{95} +22.1310 q^{96} -1248.68 q^{97} +274.806 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 11 q^{3} + 40 q^{4} + 47 q^{5} - 105 q^{6} + 17 q^{7} - 84 q^{8} + 107 q^{9} + 7 q^{10} + 101 q^{12} - 15 q^{13} + 45 q^{14} + 224 q^{15} + 140 q^{16} + 170 q^{17} - 81 q^{18} - q^{19}+ \cdots - 3281 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 + 11 * q^3 + 40 * q^4 + 47 * q^5 - 105 * q^6 + 17 * q^7 - 84 * q^8 + 107 * q^9 + 7 * q^10 + 101 * q^12 - 15 * q^13 + 45 * q^14 + 224 * q^15 + 140 * q^16 + 170 * q^17 - 81 * q^18 - q^19 + 289 * q^20 - 16 * q^21 + 340 * q^23 - 663 * q^24 + 591 * q^25 - 277 * q^26 + 776 * q^27 + 127 * q^28 - 452 * q^29 - 594 * q^30 + 298 * q^31 - 536 * q^32 - 68 * q^34 - 440 * q^35 - 15 * q^36 + 619 * q^37 + 349 * q^38 + 771 * q^39 - 379 * q^40 - 367 * q^41 + 292 * q^42 + 385 * q^43 + 1542 * q^45 - 776 * q^46 + 365 * q^47 - 363 * q^48 + 985 * q^49 + 2167 * q^50 + 187 * q^51 - 133 * q^52 + 1624 * q^53 + 576 * q^54 - 2217 * q^56 - 1498 * q^57 - 802 * q^58 - 414 * q^59 - 3420 * q^60 + 1162 * q^61 + 90 * q^62 + 1309 * q^63 - 1044 * q^64 - 2818 * q^65 - 1216 * q^67 + 680 * q^68 - 1048 * q^69 - 1556 * q^70 + 850 * q^71 + 117 * q^72 + 983 * q^73 + 555 * q^74 + 415 * q^75 + 1313 * q^76 - 1939 * q^78 + 1165 * q^79 + 209 * q^80 + 1442 * q^81 + 481 * q^82 + 1391 * q^83 + 3728 * q^84 + 799 * q^85 + 2757 * q^86 + 1434 * q^87 + 2225 * q^89 + 7526 * q^90 - 508 * q^91 + 358 * q^92 + 1266 * q^93 + 5233 * q^94 + 2768 * q^95 + 3209 * q^96 + 700 * q^97 - 3281 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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