LMFDB - Embedded newform 269.4.b.a.268.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [269,4,Mod(268,269)] 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("269.268"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(269, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 269 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	269.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$15.8715137915$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			268.8
Character	$\chi$	$=$	269.268
Dual form			269.4.b.a.268.59

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.73556i q^{2} -8.07694i q^{3} -14.4255 q^{4} +20.1267 q^{5} -38.2489 q^{6} -21.3096i q^{7} +30.4286i q^{8} -38.2370 q^{9} -95.3113i q^{10} +26.0646 q^{11} +116.514i q^{12} -33.5638 q^{13} -100.913 q^{14} -162.562i q^{15} +28.6921 q^{16} +27.8099i q^{17} +181.074i q^{18} +95.2488i q^{19} -290.339 q^{20} -172.116 q^{21} -123.431i q^{22} -99.1262 q^{23} +245.770 q^{24} +280.084 q^{25} +158.943i q^{26} +90.7607i q^{27} +307.403i q^{28} -190.323i q^{29} -769.824 q^{30} -11.5174i q^{31} +107.556i q^{32} -210.522i q^{33} +131.696 q^{34} -428.892i q^{35} +551.590 q^{36} +189.094 q^{37} +451.057 q^{38} +271.093i q^{39} +612.427i q^{40} -291.037 q^{41} +815.068i q^{42} +491.737 q^{43} -375.996 q^{44} -769.585 q^{45} +469.418i q^{46} +348.232 q^{47} -231.744i q^{48} -111.099 q^{49} -1326.36i q^{50} +224.619 q^{51} +484.176 q^{52} +370.560 q^{53} +429.803 q^{54} +524.595 q^{55} +648.421 q^{56} +769.319 q^{57} -901.288 q^{58} -440.792i q^{59} +2345.05i q^{60} -859.957 q^{61} -54.5412 q^{62} +814.816i q^{63} +738.873 q^{64} -675.528 q^{65} -996.942 q^{66} +167.415 q^{67} -401.173i q^{68} +800.637i q^{69} -2031.05 q^{70} +242.493i q^{71} -1163.50i q^{72} +656.323 q^{73} -895.465i q^{74} -2262.22i q^{75} -1374.02i q^{76} -555.427i q^{77} +1283.78 q^{78} +1114.95 q^{79} +577.477 q^{80} -299.330 q^{81} +1378.22i q^{82} -422.201i q^{83} +2482.87 q^{84} +559.722i q^{85} -2328.65i q^{86} -1537.23 q^{87} +793.110i q^{88} -1502.11 q^{89} +3644.42i q^{90} +715.231i q^{91} +1429.95 q^{92} -93.0251 q^{93} -1649.08i q^{94} +1917.04i q^{95} +868.720 q^{96} -1348.20 q^{97} +526.117i q^{98} -996.633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) $ 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/269\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	4.73556i		−	1.67427i	−0.546993	−	0.837137i	$-0.684228\pi$
$2$		−	4.73556i		−	1.67427i	0.546993	−	0.837137i	$-0.315772\pi$
$3$		−	8.07694i		−	1.55441i	−0.629248	−	0.777204i	$-0.716637\pi$
$3$		−	8.07694i		−	1.55441i	0.629248	−	0.777204i	$-0.283363\pi$
$4$	−14.4255			−1.80319
$5$	20.1267			1.80019			0.900094	−	0.435697i	$-0.143498\pi$
$5$	20.1267			1.80019			0.900094	+	0.435697i	$0.143498\pi$
$6$	−38.2489			−2.60251
$7$		−	21.3096i		−	1.15061i	−0.817939	−	0.575305i	$-0.804883\pi$
$7$		−	21.3096i		−	1.15061i	0.817939	−	0.575305i	$-0.195117\pi$
$8$			30.4286i			1.34477i
$9$	−38.2370			−1.41619
$10$		−	95.3113i		−	3.01401i
$11$	26.0646			0.714434			0.357217	−	0.934021i	$-0.383726\pi$
$11$	26.0646			0.714434			0.357217	+	0.934021i	$0.383726\pi$
$12$			116.514i			2.80290i
$13$	−33.5638			−0.716070			−0.358035	−	0.933708i	$-0.616553\pi$
$13$	−33.5638			−0.716070			−0.358035	+	0.933708i	$0.616553\pi$
$14$	−100.913			−1.92644
$15$		−	162.562i		−	2.79823i
$16$	28.6921			0.448314
$17$			27.8099i			0.396759i	0.980125	+	0.198379i	$0.0635678\pi$
$17$			27.8099i			0.396759i	−0.980125	+	0.198379i	$0.936432\pi$
$18$			181.074i			2.37108i
$19$			95.2488i			1.15008i	0.818124	+	0.575042i	$0.195014\pi$
$19$			95.2488i			1.15008i	−0.818124	+	0.575042i	$0.804986\pi$
$20$	−290.339			−3.24609
$21$	−172.116			−1.78852
$22$		−	123.431i		−	1.19616i
$23$	−99.1262			−0.898663			−0.449331	−	0.893365i	$-0.648338\pi$
$23$	−99.1262			−0.898663			−0.449331	+	0.893365i	$0.648338\pi$
$24$	245.770			2.09032
$25$	280.084			2.24067
$26$			158.943i			1.19890i
$27$			90.7607i			0.646922i
$28$			307.403i			2.07477i
$29$		−	190.323i		−	1.21869i	−0.792903	−	0.609347i	$-0.791432\pi$
$29$		−	190.323i		−	1.21869i	0.792903	−	0.609347i	$-0.208568\pi$
$30$	−769.824			−4.68500
$31$		−	11.5174i		−	0.0667284i	−0.999443	−	0.0333642i	$-0.989378\pi$
$31$		−	11.5174i		−	0.0667284i	0.999443	−	0.0333642i	$-0.0106221\pi$
$32$			107.556i			0.594166i
$33$		−	210.522i		−	1.11052i
$34$	131.696			0.664283
$35$		−	428.892i		−	2.07131i
$36$	551.590			2.55366
$37$	189.094			0.840184			0.420092	−	0.907482i	$-0.361998\pi$
$37$	189.094			0.840184			0.420092	+	0.907482i	$0.361998\pi$
$38$	451.057			1.92555
$39$			271.093i			1.11307i
$40$			612.427i			2.42083i
$41$	−291.037			−1.10859			−0.554297	−	0.832319i	$-0.687013\pi$
$41$	−291.037			−1.10859			−0.554297	+	0.832319i	$0.687013\pi$
$42$			815.068i			2.99447i
$43$	491.737			1.74393			0.871967	−	0.489564i	$-0.162844\pi$
$43$	491.737			1.74393			0.871967	+	0.489564i	$0.162844\pi$
$44$	−375.996			−1.28826
$45$	−769.585			−2.54940
$46$			469.418i			1.50461i
$47$	348.232			1.08074			0.540372	−	0.841427i	$-0.318284\pi$
$47$	348.232			1.08074			0.540372	+	0.841427i	$0.318284\pi$
$48$		−	231.744i		−	0.696863i
$49$	−111.099			−0.323904
$50$		−	1326.36i		−	3.75150i
$51$	224.619			0.616725
$52$	484.176			1.29121
$53$	370.560			0.960383			0.480191	−	0.877164i	$-0.340567\pi$
$53$	370.560			0.960383			0.480191	+	0.877164i	$0.340567\pi$
$54$	429.803			1.08313
$55$	524.595			1.28612
$56$	648.421			1.54730
$57$	769.319			1.78770
$58$	−901.288			−2.04043
$59$		−	440.792i		−	0.972647i	−0.873779	−	0.486323i	$-0.838338\pi$
$59$		−	440.792i		−	0.972647i	0.873779	−	0.486323i	$-0.161662\pi$
$60$			2345.05i			5.04574i
$61$	−859.957			−1.80502			−0.902510	−	0.430669i	$-0.858278\pi$
$61$	−859.957			−1.80502			−0.902510	+	0.430669i	$0.858278\pi$
$62$	−54.5412			−0.111722
$63$			814.816i			1.62948i
$64$	738.873			1.44311
$65$	−675.528			−1.28906
$66$	−996.942			−1.85932
$67$	167.415			0.305269			0.152635	−	0.988283i	$-0.451224\pi$
$67$	167.415			0.305269			0.152635	+	0.988283i	$0.451224\pi$
$68$		−	401.173i		−	0.715433i
$69$			800.637i			1.39689i
$70$	−2031.05			−3.46795
$71$			242.493i			0.405333i	0.979248	+	0.202666i	$0.0649607\pi$
$71$			242.493i			0.405333i	−0.979248	+	0.202666i	$0.935039\pi$
$72$		−	1163.50i		−	1.90444i
$73$	656.323			1.05228			0.526142	−	0.850397i	$-0.323638\pi$
$73$	656.323			1.05228			0.526142	+	0.850397i	$0.323638\pi$
$74$		−	895.465i		−	1.40670i
$75$		−	2262.22i		−	3.48292i
$76$		−	1374.02i		−	2.07382i
$77$		−	555.427i		−	0.822036i
$78$	1283.78			1.86358
$79$	1114.95			1.58786			0.793932	−	0.608006i	$-0.208030\pi$
$79$	1114.95			1.58786			0.793932	+	0.608006i	$0.208030\pi$
$80$	577.477			0.807049
$81$	−299.330			−0.410604
$82$			1378.22i			1.85609i
$83$		−	422.201i		−	0.558344i	−0.960241	−	0.279172i	$-0.909940\pi$
$83$		−	422.201i		−	0.558344i	0.960241	−	0.279172i	$-0.0900600\pi$
$84$	2482.87			3.22505
$85$			559.722i			0.714240i
$86$		−	2328.65i		−	2.91982i
$87$	−1537.23			−1.89435
$88$			793.110i			0.960747i
$89$	−1502.11			−1.78902			−0.894512	−	0.447044i	$-0.852477\pi$
$89$	−1502.11			−1.78902			−0.894512	+	0.447044i	$0.852477\pi$
$90$			3644.42i			4.26839i
$91$			715.231i			0.823918i
$92$	1429.95			1.62046
$93$	−93.0251			−0.103723
$94$		−	1649.08i		−	1.80946i
$95$			1917.04i			2.07036i
$96$	868.720			0.923577
$97$	−1348.20			−1.41122			−0.705612	−	0.708598i	$-0.749328\pi$
$97$	−1348.20			−1.41122			−0.705612	+	0.708598i	$0.749328\pi$
$98$			526.117i			0.542305i
$99$	−996.633			−1.01177
$100$	−4040.37			−4.04037
$101$			1846.55i			1.81920i	0.415489	+	0.909598i	$0.363610\pi$
$101$			1846.55i			1.81920i	−0.415489	+	0.909598i	$0.636390\pi$
$102$		−	1063.70i		−	1.03257i
$103$	−264.012			−0.252562			−0.126281	−	0.991994i	$-0.540304\pi$
$103$	−264.012			−0.252562			−0.126281	+	0.991994i	$0.540304\pi$
$104$		−	1021.30i		−	0.962947i
$105$	−3464.14			−3.21967
$106$		−	1754.81i		−	1.60794i
$107$			1915.10i			1.73027i	0.501536	+	0.865137i	$0.332769\pi$
$107$			1915.10i			1.73027i	−0.501536	+	0.865137i	$0.667231\pi$
$108$		−	1309.27i		−	1.16653i
$109$		−	1069.72i		−	0.940007i	−0.882664	−	0.470004i	$-0.844253\pi$
$109$		−	1069.72i		−	0.940007i	0.882664	−	0.470004i	$-0.155747\pi$
$110$		−	2484.25i		−	2.15331i
$111$		−	1527.30i		−	1.30599i
$112$		−	611.417i		−	0.515834i
$113$		−	1089.89i		−	0.907330i	−0.891172	−	0.453665i	$-0.850116\pi$
$113$		−	1089.89i		−	0.907330i	0.891172	−	0.453665i	$-0.149884\pi$
$114$		−	3643.16i		−	2.99310i
$115$	−1995.08			−1.61776
$116$			2745.52i			2.19754i
$117$	1283.38			1.01409
$118$	−2087.40			−1.62848
$119$	592.618			0.456515
$120$	4946.54			3.76296
$121$	−651.636			−0.489584
$122$			4072.38i			3.02210i
$123$			2350.69i			1.72321i
$124$			166.144i			0.120324i
$125$	3121.33			2.23344
$126$	3858.61			2.72819
$127$	−889.154			−0.621257			−0.310629	−	0.950531i	$-0.600540\pi$
$127$	−889.154			−0.621257			−0.310629	+	0.950531i	$0.600540\pi$
$128$		−	2638.53i		−	1.82200i
$129$		−	3971.73i		−	2.71079i
$130$			3199.01i			2.15824i
$131$	2524.48			1.68370			0.841849	−	0.539713i	$-0.181467\pi$
$131$	2524.48			1.68370			0.841849	+	0.539713i	$0.181467\pi$
$132$			3036.90i			2.00249i
$133$	2029.71			1.32330
$134$		−	792.806i		−	0.511105i
$135$			1826.71i			1.16458i
$136$	−846.217			−0.533548
$137$		−	1900.21i		−	1.18501i	−0.805567	−	0.592505i	$-0.798139\pi$
$137$		−	1900.21i		−	1.18501i	0.805567	−	0.592505i	$-0.201861\pi$
$138$	3791.47			2.33878
$139$			798.987i			0.487548i	0.969832	+	0.243774i	$0.0783855\pi$
$139$			798.987i			0.487548i	−0.969832	+	0.243774i	$0.921615\pi$
$140$			6187.00i			3.73498i
$141$		−	2812.65i		−	1.67992i
$142$	1148.34			0.678638
$143$	−874.827			−0.511585
$144$	−1097.10			−0.634895
$145$		−	3830.58i		−	2.19388i
$146$		−	3108.06i		−	1.76181i
$147$			897.342i			0.503480i
$148$	−2727.78			−1.51501
$149$	1454.92			0.799942			0.399971	−	0.916528i	$-0.369020\pi$
$149$	1454.92			0.799942			0.399971	+	0.916528i	$0.369020\pi$
$150$	−10712.9			−5.83137
$151$	2386.38			1.28610			0.643048	−	0.765826i	$-0.277670\pi$
$151$	2386.38			1.28610			0.643048	+	0.765826i	$0.277670\pi$
$152$	−2898.29			−1.54659
$153$		−	1063.37i		−	0.561884i
$154$	−2630.26			−1.37631
$155$		−	231.807i		−	0.120124i
$156$		−	3910.66i		−	2.00707i
$157$		−	721.836i		−	0.366935i	−0.983026	−	0.183467i	$-0.941268\pi$
$157$		−	721.836i		−	0.366935i	0.983026	−	0.183467i	$-0.0587322\pi$
$158$		−	5279.90i		−	2.65852i
$159$		−	2992.99i		−	1.49283i
$160$			2164.74i			1.06961i
$161$			2112.34i			1.03401i
$162$			1417.50i			0.687464i
$163$			90.2490i			0.0433671i	0.999765	+	0.0216836i	$0.00690264\pi$
$163$			90.2490i			0.0433671i	−0.999765	+	0.0216836i	$0.993097\pi$
$164$	4198.37			1.99901
$165$		−	4237.12i		−	1.99915i
$166$	−1999.36			−0.934821
$167$			464.752i			0.215351i	0.994186	+	0.107675i	$0.0343407\pi$
$167$			464.752i			0.215351i	−0.994186	+	0.107675i	$0.965659\pi$
$168$		−	5237.26i		−	2.40514i
$169$	−1070.47			−0.487243
$170$	2650.60			1.19583
$171$		−	3642.03i		−	1.62873i
$172$	−7093.57			−3.14465
$173$	860.130			0.378003			0.189001	−	0.981977i	$-0.439475\pi$
$173$	860.130			0.378003			0.189001	+	0.981977i	$0.439475\pi$
$174$			7279.65i			3.17166i
$175$		−	5968.48i		−	2.57814i
$176$	747.848			0.320291
$177$	−3560.25			−1.51189
$178$			7113.32i			2.99532i
$179$		−	90.0971i		−	0.0376211i	−0.999823	−	0.0188105i	$-0.994012\pi$
$179$		−	90.0971i		−	0.0376211i	0.999823	−	0.0188105i	$-0.00598793\pi$
$180$	11101.7			4.59706
$181$		−	1188.19i		−	0.487943i	−0.969782	−	0.243971i	$-0.921550\pi$
$181$		−	1188.19i		−	0.487943i	0.969782	−	0.243971i	$-0.0784503\pi$
$182$	3387.02			1.37946
$183$			6945.82i			2.80574i
$184$		−	3016.27i		−	1.20849i
$185$	3805.83			1.51249
$186$			440.526i			0.173661i
$187$			724.855i			0.283458i
$188$	−5023.44			−1.94879
$189$	1934.07			0.744355
$190$	9078.29			3.46636
$191$	−1982.60			−0.751079			−0.375539	−	0.926806i	$-0.622543\pi$
$191$	−1982.60			−0.751079			−0.375539	+	0.926806i	$0.622543\pi$
$192$		−	5967.83i		−	2.24318i
$193$		−	3140.27i		−	1.17120i	−0.810601	−	0.585599i	$-0.800859\pi$
$193$		−	3140.27i		−	1.17120i	0.810601	−	0.585599i	$-0.199141\pi$
$194$			6384.47i			2.36278i
$195$			5456.20i			2.00373i
$196$	1602.67			0.584062
$197$			2919.11i			1.05572i	0.849330	+	0.527862i	$0.177006\pi$
$197$			2919.11i			1.05572i	−0.849330	+	0.527862i	$0.822994\pi$
$198$			4719.62i			1.69398i
$199$	−152.119			−0.0541882			−0.0270941	−	0.999633i	$-0.508625\pi$
$199$	−152.119			−0.0541882			−0.0270941	+	0.999633i	$0.508625\pi$
$200$			8522.57i			3.01318i
$201$		−	1352.21i		−	0.474513i
$202$	8744.46			3.04583
$203$	−4055.71			−1.40224
$204$	−3240.26			−1.11207
$205$	−5857.61			−1.99568
$206$			1250.25i			0.422858i
$207$	3790.29			1.27267
$208$	−963.014			−0.321024
$209$			2482.62i			0.821659i
$210$			16404.6i			5.39061i
$211$	2308.19			0.753092			0.376546	−	0.926398i	$-0.377112\pi$
$211$	2308.19			0.753092			0.376546	+	0.926398i	$0.377112\pi$
$212$	−5345.53			−1.73176
$213$	1958.60			0.630053
$214$	9069.06			2.89695
$215$	9897.04			3.13941
$216$	−2761.72			−0.869959
$217$	−245.430			−0.0767783
$218$	−5065.74			−1.57383
$219$		−	5301.08i		−	1.63568i
$220$	−7567.57			−2.31912
$221$		−	933.406i		−	0.284107i
$222$	−7232.62			−2.18658
$223$			4027.41i			1.20939i	0.796455	+	0.604697i	$0.206706\pi$
$223$			4027.41i			1.20939i	−0.796455	+	0.604697i	$0.793294\pi$
$224$	2291.97			0.683654
$225$	−10709.6			−3.17321
$226$	−5161.25			−1.51912
$227$		−	2007.31i		−	0.586915i	−0.955972	−	0.293457i	$-0.905194\pi$
$227$		−	2007.31i		−	0.586915i	0.955972	−	0.293457i	$-0.0948059\pi$
$228$	−11097.9			−3.22357
$229$			5831.16i			1.68268i	0.540505	+	0.841341i	$0.318233\pi$
$229$			5831.16i			1.68268i	−0.540505	+	0.841341i	$0.681767\pi$
$230$			9447.85i			2.70858i
$231$	−4486.15			−1.27778
$232$	5791.27			1.63886
$233$	−900.698			−0.253248			−0.126624	−	0.991951i	$-0.540414\pi$
$233$	−900.698			−0.253248			−0.126624	+	0.991951i	$0.540414\pi$
$234$		−	6077.52i		−	1.69786i
$235$	7008.77			1.94554
$236$			6358.66i			1.75387i
$237$		−	9005.36i		−	2.46819i
$238$		−	2806.38i		−	0.764331i
$239$	−581.982			−0.157512			−0.0787559	−	0.996894i	$-0.525095\pi$
$239$	−581.982			−0.157512			−0.0787559	+	0.996894i	$0.525095\pi$
$240$		−	4664.25i		−	1.25448i
$241$		−	159.184i		−	0.0425475i	−0.999774	−	0.0212737i	$-0.993228\pi$
$241$		−	159.184i		−	0.0425475i	0.999774	−	0.0212737i	$-0.00677215\pi$
$242$			3085.86i			0.819697i
$243$			4868.21i			1.28517i
$244$	12405.4			3.25480
$245$	−2236.06			−0.583089
$246$	11131.8			2.88512
$247$		−	3196.91i		−	0.823541i
$248$	350.457			0.0897340
$249$	−3410.09			−0.867895
$250$		−	14781.3i		−	3.73940i
$251$		−	33.6382i		−	0.00845907i	−0.999991	−	0.00422954i	$-0.998654\pi$
$251$		−	33.6382i		−	0.00845907i	0.999991	−	0.00422954i	$-0.00134631\pi$
$252$		−	11754.2i		−	2.93826i
$253$	−2583.69			−0.642036
$254$			4210.65i			1.04016i
$255$	4520.84			1.11022
$256$	−6583.96			−1.60741
$257$		−	372.660i		−	0.0904509i	−0.998977	−	0.0452255i	$-0.985599\pi$
$257$		−	372.660i		−	0.0904509i	0.998977	−	0.0452255i	$-0.0144006\pi$
$258$	−18808.4			−4.53860
$259$		−	4029.51i		−	0.966724i
$260$	9744.86			2.32443
$261$			7277.39i			1.72590i
$262$		−	11954.8i		−	2.81897i
$263$	−6103.61			−1.43104			−0.715522	−	0.698590i	$-0.753811\pi$
$263$	−6103.61			−1.43104			−0.715522	+	0.698590i	$0.753811\pi$
$264$	6405.90			1.49339
$265$	7458.14			1.72887
$266$		−	9611.84i		−	2.21556i
$267$			12132.4i			2.78087i
$268$	−2415.06			−0.550460
$269$	−789.764	−	4340.67i	−0.179007	−	0.983848i
$269$	−789.764	−	4340.67i	−0.179007	−	0.983848i
$270$	8650.51			1.94983
$271$		−	2198.78i		−	0.492866i	−0.969160	−	0.246433i	$-0.920742\pi$
$271$		−	2198.78i		−	0.492866i	0.969160	−	0.246433i	$-0.0792584\pi$
$272$			797.925i			0.177872i
$273$	5776.88			1.28071
$274$	−8998.59			−1.98403
$275$	7300.29			1.60081
$276$		−	11549.6i		−	2.51886i
$277$			7730.15i			1.67675i	0.545093	+	0.838375i	$0.316494\pi$
$277$			7730.15i			1.67675i	−0.545093	+	0.838375i	$0.683506\pi$
$278$	3783.65			0.816289
$279$			440.389i			0.0944997i
$280$	13050.6			2.78543
$281$		−	3967.92i		−	0.842371i	−0.906974	−	0.421186i	$-0.861614\pi$
$281$		−	3967.92i		−	0.842371i	0.906974	−	0.421186i	$-0.138386\pi$
$282$	−13319.5			−2.81264
$283$	−1905.18			−0.400181			−0.200091	−	0.979777i	$-0.564124\pi$
$283$	−1905.18			−0.400181			−0.200091	+	0.979777i	$0.564124\pi$
$284$		−	3498.10i		−	0.730894i
$285$	15483.9			3.21819
$286$			4142.80i			0.856534i
$287$			6201.88i			1.27556i
$288$		−	4112.60i		−	0.841450i
$289$	4139.61			0.842583
$290$	−18139.9			−3.67315
$291$			10889.3i			2.19362i
$292$	−9467.82			−1.89747
$293$	3512.20			0.700290			0.350145	−	0.936696i	$-0.386132\pi$
$293$	3512.20			0.700290			0.350145	+	0.936696i	$0.386132\pi$
$294$	4249.42			0.842963
$295$		−	8871.68i		−	1.75095i
$296$			5753.85i			1.12985i
$297$			2365.64i			0.462183i
$298$		−	6889.84i		−	1.33932i
$299$	3327.05			0.643506
$300$			32633.8i			6.28038i
$301$		−	10478.7i		−	2.00659i
$302$		−	11300.8i		−	2.15328i
$303$	14914.5			2.82777
$304$			2732.89i			0.515598i
$305$	−17308.1			−3.24937
$306$	−5035.65			−0.940748
$307$	−3554.13			−0.660733			−0.330366	−	0.943853i	$-0.607172\pi$
$307$	−3554.13			−0.660733			−0.330366	+	0.943853i	$0.607172\pi$
$308$			8012.34i			1.48229i
$309$			2132.41i			0.392585i
$310$	−1097.73			−0.201120
$311$		−	3198.40i		−	0.583166i	−0.956546	−	0.291583i	$-0.905818\pi$
$311$		−	3198.40i		−	0.583166i	0.956546	−	0.291583i	$-0.0941819\pi$
$312$	−8248.97			−1.49681
$313$	−8821.58			−1.59305			−0.796526	−	0.604605i	$-0.793331\pi$
$313$	−8821.58			−1.59305			−0.796526	+	0.604605i	$0.793331\pi$
$314$	−3418.30			−0.614349
$315$			16399.5i			2.93336i
$316$	−16083.7			−2.86323
$317$			4506.14i			0.798392i	0.916866	+	0.399196i	$0.130711\pi$
$317$			4506.14i			0.798392i	−0.916866	+	0.399196i	$0.869289\pi$
$318$	−14173.5			−2.49940
$319$		−	4960.70i		−	0.870677i
$320$	14871.1			2.59787
$321$	15468.1			2.68955
$322$	10003.1			1.73122
$323$	−2648.86			−0.456305
$324$	4318.01			0.740399
$325$	−9400.68			−1.60448
$326$	427.380			0.0726085
$327$	−8640.09			−1.46116
$328$		−	8855.84i		−	1.49080i
$329$		−	7420.70i		−	1.24351i
$330$	−20065.2			−3.34712
$331$	4847.54			0.804970			0.402485	−	0.915427i	$-0.368147\pi$
$331$	4847.54			0.804970			0.402485	+	0.915427i	$0.368147\pi$
$332$			6090.48i			1.00680i
$333$	−7230.38			−1.18986
$334$	2200.86			0.360556
$335$	3369.52			0.549542
$336$	−4938.38			−0.801817
$337$			11260.0i			1.82010i	0.414501	+	0.910049i	$0.363956\pi$
$337$			11260.0i			1.82010i	−0.414501	+	0.910049i	$0.636044\pi$
$338$			5069.29i			0.815778i
$339$	−8802.99			−1.41036
$340$		−	8074.30i		−	1.28791i
$341$		−	300.196i		−	0.0476730i
$342$	−17247.1			−2.72694
$343$		−	4941.71i		−	0.777923i
$344$			14962.9i			2.34518i
$345$			16114.2i			2.51466i
$346$		−	4073.20i		−	0.632880i
$347$	7308.68			1.13069			0.565347	−	0.824853i	$-0.308742\pi$
$347$	7308.68			1.13069			0.565347	+	0.824853i	$0.308742\pi$
$348$	22175.4			3.41588
$349$	3383.84			0.519005			0.259502	−	0.965742i	$-0.416441\pi$
$349$	3383.84			0.519005			0.259502	+	0.965742i	$0.416441\pi$
$350$	−28264.1			−4.31652
$351$		−	3046.27i		−	0.463242i
$352$			2803.40i			0.424493i
$353$	5010.02			0.755400			0.377700	−	0.925928i	$-0.376715\pi$
$353$	5010.02			0.755400			0.377700	+	0.925928i	$0.376715\pi$
$354$			16859.8i			2.53132i
$355$			4880.59i			0.729675i
$356$	21668.7			3.22596
$357$		−	4786.55i		−	0.709610i
$358$	−426.660			−0.0629880
$359$			537.756i			0.0790576i	0.999218	+	0.0395288i	$0.0125857\pi$
$359$			537.756i			0.0790576i	−0.999218	+	0.0395288i	$0.987414\pi$
$360$		−	23417.4i		−	3.42835i
$361$	−2213.34			−0.322691
$362$	−5626.76			−0.816950
$363$			5263.22i			0.761013i
$364$		−	10317.6i		−	1.48568i
$365$	13209.6			1.89431
$366$	32892.4			4.69757
$367$		−	1596.36i		−	0.227055i	−0.993535	−	0.113528i	$-0.963785\pi$
$367$		−	1596.36i		−	0.227055i	0.993535	−	0.113528i	$-0.0362150\pi$
$368$	−2844.14			−0.402883
$369$	11128.4			1.56997
$370$		−	18022.8i		−	2.53232i
$371$		−	7896.48i		−	1.10503i
$372$	1341.94			0.187033
$373$		−	821.716i		−	0.114067i	−0.998372	−	0.0570333i	$-0.981836\pi$
$373$		−	821.716i		−	0.114067i	0.998372	−	0.0570333i	$-0.0181641\pi$
$374$	3432.60			0.474586
$375$		−	25210.8i		−	3.47168i
$376$			10596.2i			1.45335i
$377$			6387.97i			0.872671i
$378$		−	9158.93i		−	1.24626i
$379$			3784.87i			0.512970i	0.966548	+	0.256485i	$0.0825644\pi$
$379$			3784.87i			0.512970i	−0.966548	+	0.256485i	$0.917436\pi$
$380$		−	27654.4i		−	3.73327i
$381$			7181.65i			0.965688i
$382$			9388.74i			1.25751i
$383$		−	9689.78i		−	1.29275i	−0.763018	−	0.646377i	$-0.776283\pi$
$383$		−	9689.78i		−	1.29275i	0.763018	−	0.646377i	$-0.223717\pi$
$384$	−21311.3			−2.83213
$385$		−	11178.9i		−	1.47982i
$386$	−14870.9			−1.96091
$387$	−18802.5			−2.46973
$388$	19448.5			2.54471
$389$	2542.40			0.331375			0.165687	−	0.986178i	$-0.447016\pi$
$389$	2542.40			0.331375			0.165687	+	0.986178i	$0.447016\pi$
$390$	25838.2			3.35479
$391$		−	2756.69i		−	0.356552i
$392$		−	3380.59i		−	0.435576i
$393$		−	20390.0i		−	2.61715i
$394$	13823.6			1.76757
$395$	22440.2			2.85845
$396$	14377.0			1.82442
$397$			10257.7i			1.29678i	0.761310	+	0.648388i	$0.224556\pi$
$397$			10257.7i			1.29678i	−0.761310	+	0.648388i	$0.775444\pi$
$398$			720.371i			0.0907260i
$399$		−	16393.9i		−	2.05695i
$400$	8036.20			1.00452
$401$			10975.1i			1.36676i	0.730062	+	0.683380i	$0.239491\pi$
$401$			10975.1i			1.36676i	−0.730062	+	0.683380i	$0.760509\pi$
$402$	−6403.45			−0.794466
$403$			386.566i			0.0477822i
$404$		−	26637.5i		−	3.28036i
$405$	−6024.53			−0.739164
$406$			19206.1i			2.34774i
$407$	4928.65			0.600256
$408$			6834.84i			0.829351i
$409$		−	6904.41i		−	0.834721i	−0.908741	−	0.417361i	$-0.862955\pi$
$409$		−	6904.41i		−	0.834721i	0.908741	−	0.417361i	$-0.137045\pi$
$410$			27739.1i			3.34131i
$411$	−15347.9			−1.84199
$412$	3808.52			0.455419
$413$	−9393.09			−1.11914
$414$		−	17949.2i		−	2.13080i
$415$		−	8497.51i		−	1.00512i
$416$		−	3609.97i		−	0.425465i
$417$	6453.37			0.757849
$418$	11756.6			1.37568
$419$	−43.0412			−0.00501838			−0.00250919	−	0.999997i	$-0.500799\pi$
$419$	−43.0412			−0.00501838			−0.00250919	+	0.999997i	$0.500799\pi$
$420$	49972.1			5.80568
$421$	−14164.0			−1.63969			−0.819845	−	0.572585i	$-0.805941\pi$
$421$	−14164.0			−1.63969			−0.819845	+	0.572585i	$0.805941\pi$
$422$		−	10930.6i		−	1.26088i
$423$	−13315.4			−1.53053
$424$			11275.6i			1.29149i
$425$			7789.12i			0.889007i
$426$		−	9275.09i		−	1.05488i
$427$			18325.3i			2.07687i
$428$		−	27626.3i		−	3.12002i
$429$			7065.93i			0.795213i
$430$		−	46868.1i		−	5.25623i
$431$			13563.6i			1.51586i	0.652334	+	0.757931i	$0.273790\pi$
$431$			13563.6i			1.51586i	−0.652334	+	0.757931i	$0.726210\pi$
$432$			2604.11i			0.290024i
$433$	17947.8			1.99196			0.995979	−	0.0895844i	$-0.0285539\pi$
$433$	17947.8			1.99196			0.995979	+	0.0895844i	$0.0285539\pi$
$434$			1162.25i			0.128548i
$435$	−30939.4			−3.41018
$436$			15431.3i			1.69502i
$437$		−	9441.66i		−	1.03354i
$438$	−25103.6			−2.73858
$439$	6275.04			0.682212			0.341106	−	0.940025i	$-0.389198\pi$
$439$	6275.04			0.682212			0.341106	+	0.940025i	$0.389198\pi$
$440$			15962.7i			1.72952i
$441$	4248.10			0.458709
$442$	−4420.20			−0.475673
$443$		−	12827.4i		−	1.37573i	−0.725837	−	0.687866i	$-0.758548\pi$
$443$		−	12827.4i		−	1.37573i	0.725837	−	0.687866i	$-0.241452\pi$
$444$			22032.1i			2.35495i
$445$	−30232.5			−3.22058
$446$	19072.0			2.02486
$447$		−	11751.3i		−	1.24344i
$448$		−	15745.1i		−	1.66046i
$449$	1291.11			0.135704			0.0678519	−	0.997695i	$-0.478385\pi$
$449$	1291.11			0.135704			0.0678519	+	0.997695i	$0.478385\pi$
$450$			50715.9i			5.31282i
$451$	−7585.77			−0.792017
$452$			15722.3i			1.63609i
$453$		−	19274.6i		−	1.99912i
$454$	−9505.73			−0.982656
$455$			14395.2i			1.48321i
$456$			23409.3i			2.40404i
$457$	10581.2			1.08308			0.541541	−	0.840674i	$-0.317841\pi$
$457$	10581.2			1.08308			0.541541	+	0.840674i	$0.317841\pi$
$458$	27613.8			2.81727
$459$	−2524.05			−0.256672
$460$	28780.2			2.91714
$461$			15374.5i			1.55328i	0.629943	+	0.776642i	$0.283078\pi$
$461$			15374.5i			1.55328i	−0.629943	+	0.776642i	$0.716922\pi$
$462$			21244.4i			2.13935i
$463$			5319.23i			0.533922i	0.963707	+	0.266961i	$0.0860194\pi$
$463$			5319.23i			0.533922i	−0.963707	+	0.266961i	$0.913981\pi$
$464$		−	5460.77i		−	0.546358i
$465$	−1872.29			−0.186721
$466$			4265.31i			0.424006i
$467$			659.354i			0.0653346i	0.999466	+	0.0326673i	$0.0104002\pi$
$467$			659.354i			0.0653346i	−0.999466	+	0.0326673i	$0.989600\pi$
$468$	−18513.4			−1.82860
$469$		−	3567.56i		−	0.351246i
$470$		−	33190.5i		−	3.25737i
$471$	−5830.23			−0.570367
$472$	13412.7			1.30798
$473$	12816.9			1.24593
$474$	−42645.5			−4.13243
$475$			26677.7i			2.57696i
$476$	−8548.85			−0.823184
$477$	−14169.1			−1.36008
$478$			2756.01i			0.263718i
$479$			8836.54i			0.842905i	0.906850	+	0.421453i	$0.138480\pi$
$479$			8836.54i			0.842905i	−0.906850	+	0.421453i	$0.861520\pi$
$480$	17484.5			1.66261
$481$	−6346.70			−0.601631
$482$	−753.825			−0.0712361
$483$	17061.3			1.60728
$484$	9400.20			0.882814
$485$	−27134.8			−2.54047
$486$	23053.7			2.15172
$487$	−5443.00			−0.506460			−0.253230	−	0.967406i	$-0.581493\pi$
$487$	−5443.00			−0.506460			−0.253230	+	0.967406i	$0.581493\pi$
$488$		−	26167.3i		−	2.42733i
$489$	728.936			0.0674102
$490$			10589.0i			0.976250i
$491$	19392.6			1.78243			0.891217	−	0.453578i	$-0.149853\pi$
$491$	19392.6			1.78243			0.891217	+	0.453578i	$0.149853\pi$
$492$		−	33910.0i		−	3.10728i
$493$	5292.88			0.483528
$494$	−15139.2			−1.37883
$495$	−20058.9			−1.82138
$496$		−	330.457i		−	0.0299152i
$497$	5167.43			0.466380
$498$			16148.7i			1.45309i
$499$			8271.96i			0.742091i	0.928615	+	0.371046i	$0.121001\pi$
$499$			8271.96i			0.742091i	−0.928615	+	0.371046i	$0.878999\pi$
$500$	−45026.9			−4.02733
$501$	3753.77			0.334743
$502$	−159.296			−0.0141628
$503$		−	15629.3i		−	1.38544i	−0.721206	−	0.692720i	$-0.756412\pi$
$503$		−	15629.3i		−	1.38544i	0.721206	−	0.692720i	$-0.243588\pi$
$504$	−24793.7			−2.19127
$505$			37165.0i			3.27489i
$506$			12235.2i			1.07494i
$507$			8646.15i			0.757375i
$508$	12826.5			1.12025
$509$		−	5128.61i		−	0.446605i	−0.974749	−	0.223302i	$-0.928316\pi$
$509$		−	5128.61i		−	0.446605i	0.974749	−	0.223302i	$-0.0716837\pi$
$510$		−	21408.7i		−	1.85881i
$511$		−	13986.0i		−	1.21077i
$512$			10070.5i			0.869250i
$513$	−8644.85			−0.744014
$514$	−1764.75			−0.151440
$515$	−5313.70			−0.454659
$516$			57294.4i			4.88807i
$517$	9076.55			0.772120
$518$	−19082.0			−1.61856
$519$		−	6947.22i		−	0.587571i
$520$		−	20555.4i		−	1.73349i
$521$			11732.0i			0.986539i	0.869877	+	0.493269i	$0.164198\pi$
$521$			11732.0i			0.986539i	−0.869877	+	0.493269i	$0.835802\pi$
$522$	34462.5			2.88963
$523$			2994.44i			0.250359i	0.992134	+	0.125180i	$0.0399507\pi$
$523$			2994.44i			0.250359i	−0.992134	+	0.125180i	$0.960049\pi$
$524$	−36417.0			−3.03603
$525$	−48207.1			−4.00749
$526$			28904.0i			2.39596i
$527$	320.297			0.0264751
$528$		−	6040.33i		−	0.497863i
$529$	−2340.99			−0.192405
$530$		−	35318.5i		−	2.89460i
$531$			16854.5i			1.37745i
$532$	−29279.8			−2.38616
$533$	9768.30			0.793831
$534$	57453.9			4.65594
$535$			38544.6i			3.11482i
$536$			5094.22i			0.410516i
$537$	−727.709			−0.0584785
$538$	−20555.5	+	3739.98i	−1.64723	+	0.299706i
$539$	−2895.76			−0.231408
$540$		−	26351.3i		−	2.09997i
$541$		−	21174.0i		−	1.68270i	−0.540492	−	0.841349i	$-0.681762\pi$
$541$		−	21174.0i		−	1.68270i	0.540492	−	0.841349i	$-0.318238\pi$
$542$	−10412.5			−0.825192
$543$	−9596.96			−0.758462
$544$	−2991.11			−0.235741
$545$		−	21530.0i		−	1.69219i
$546$		−	27356.8i		−	2.14425i
$547$	−1945.85			−0.152100			−0.0760499	−	0.997104i	$-0.524231\pi$
$547$	−1945.85			−0.152100			−0.0760499	+	0.997104i	$0.524231\pi$
$548$			27411.6i			2.13680i
$549$	32882.2			2.55624
$550$		−	34571.0i		−	2.68020i
$551$	18128.1			1.40160
$552$	−24362.3			−1.87849
$553$		−	23759.1i		−	1.82701i
$554$	36606.6			2.80734
$555$		−	30739.5i		−	2.35102i
$556$		−	11525.8i		−	0.879143i
$557$			20363.0i			1.54903i	0.632558	+	0.774513i	$0.282005\pi$
$557$			20363.0i			1.54903i	−0.632558	+	0.774513i	$0.717995\pi$
$558$	2085.49			0.158218
$559$	−16504.5			−1.24878
$560$		−	12305.8i		−	0.928599i
$561$	5854.61			0.440610
$562$	−18790.3			−1.41036
$563$	−8233.13			−0.616314			−0.308157	−	0.951335i	$-0.599712\pi$
$563$	−8233.13			−0.616314			−0.308157	+	0.951335i	$0.599712\pi$
$564$			40574.1i			3.02921i
$565$		−	21935.9i		−	1.63336i
$566$			9022.11i			0.670013i
$567$			6378.61i			0.472445i
$568$	−7378.72			−0.545078
$569$		−	21399.5i		−	1.57665i	−0.615260	−	0.788324i	$-0.710949\pi$
$569$		−	21399.5i		−	1.57665i	0.615260	−	0.788324i	$-0.289051\pi$
$570$		−	73324.8i		−	5.38814i
$571$		−	2418.21i		−	0.177231i	−0.996066	−	0.0886154i	$-0.971756\pi$
$571$		−	2418.21i		−	0.177231i	0.996066	−	0.0886154i	$-0.0282442\pi$
$572$	12619.9			0.922487
$573$			16013.4i			1.16748i
$574$	29369.4			2.13564
$575$	−27763.7			−2.01361
$576$	−28252.3			−2.04371
$577$			11128.1i			0.802896i	0.915882	+	0.401448i	$0.131493\pi$
$577$			11128.1i			0.802896i	−0.915882	+	0.401448i	$0.868507\pi$
$578$		−	19603.4i		−	1.41071i
$579$	−25363.8			−1.82052
$580$			55258.2i			3.95599i
$581$	−8996.93			−0.642437
$582$	51567.0			3.67272
$583$	9658.50			0.686130
$584$			19971.0i			1.41508i
$585$	25830.2			1.82555
$586$		−	16632.2i		−	1.17248i
$587$	21614.3			1.51979			0.759896	−	0.650044i	$-0.225250\pi$
$587$	21614.3			1.51979			0.759896	+	0.650044i	$0.225250\pi$
$588$		−	12944.7i		−	0.907872i
$589$	1097.02			0.0767432
$590$	−42012.4			−2.93156
$591$	23577.5			1.64103
$592$	5425.49			0.376666
$593$	−12329.1			−0.853785			−0.426892	−	0.904302i	$-0.640392\pi$
$593$	−12329.1			−0.853785			−0.426892	+	0.904302i	$0.640392\pi$
$594$	11202.6			0.773822
$595$	11927.5			0.821812
$596$	−20988.0			−1.44245
$597$			1228.66i			0.0842307i
$598$		−	15755.5i		−	1.07741i
$599$	6683.58			0.455899			0.227950	−	0.973673i	$-0.426798\pi$
$599$	6683.58			0.455899			0.227950	+	0.973673i	$0.426798\pi$
$600$	68836.3			4.68372
$601$			18283.4i			1.24092i	0.784237	+	0.620461i	$0.213055\pi$
$601$			18283.4i			1.24092i	−0.784237	+	0.620461i	$0.786945\pi$
$602$	−49622.6			−3.35958
$603$	−6401.47			−0.432318
$604$	−34424.8			−2.31908
$605$	−13115.3			−0.881342
$606$		−	70628.5i		−	4.73447i
$607$		−	1327.48i		−	0.0887656i	−0.999015	−	0.0443828i	$-0.985868\pi$
$607$		−	1327.48i		−	0.0887656i	0.999015	−	0.0443828i	$-0.0141321\pi$
$608$	−10244.5			−0.683341
$609$			32757.8i			2.17966i
$610$			81963.6i			5.44034i
$611$	−11688.0			−0.773888
$612$			15339.7i			1.01319i
$613$			10405.5i			0.685604i	0.939408	+	0.342802i	$0.111376\pi$
$613$			10405.5i			0.685604i	−0.939408	+	0.342802i	$0.888624\pi$
$614$			16830.8i			1.10625i
$615$			47311.6i			3.10209i
$616$	16900.8			1.10545
$617$	−13704.5			−0.894201			−0.447101	−	0.894484i	$-0.647543\pi$
$617$	−13704.5			−0.894201			−0.447101	+	0.894484i	$0.647543\pi$
$618$	10098.2			0.657295
$619$	−8999.60			−0.584369			−0.292185	−	0.956362i	$-0.594382\pi$
$619$	−8999.60			−0.584369			−0.292185	+	0.956362i	$0.594382\pi$
$620$			3343.94i			0.216606i
$621$		−	8996.76i		−	0.581365i
$622$	−15146.2			−0.976379
$623$			32009.3i			2.05847i
$624$			7778.21i			0.499003i
$625$	27811.6			1.77994
$626$			41775.1i			2.66720i
$627$	20052.0			1.27719
$628$			10412.9i			0.661655i
$629$			5258.68i			0.333350i
$630$	77661.1			4.91126
$631$	−7133.29			−0.450035			−0.225017	−	0.974355i	$-0.572244\pi$
$631$	−7133.29			−0.450035			−0.225017	+	0.974355i	$0.572244\pi$
$632$			33926.3i			2.13531i
$633$		−	18643.1i		−	1.17061i
$634$	21339.1			1.33673
$635$	−17895.7			−1.11838
$636$			43175.5i			2.69186i
$637$	3728.91			0.231938
$638$	−23491.7			−1.45775
$639$		−	9272.21i		−	0.574027i
$640$		−	53105.0i		−	3.27994i
$641$	−5372.22			−0.331029			−0.165515	−	0.986207i	$-0.552929\pi$
$641$	−5372.22			−0.331029			−0.165515	+	0.986207i	$0.552929\pi$
$642$		−	73250.3i		−	4.50305i
$643$	−11188.4			−0.686199			−0.343100	−	0.939299i	$-0.611477\pi$
$643$	−11188.4			−0.686199			−0.343100	+	0.939299i	$0.611477\pi$
$644$		−	30471.7i		−	1.86452i
$645$		−	79937.8i		−	4.87992i
$646$			12543.9i			0.763980i
$647$		−	10964.4i		−	0.666234i	−0.942886	−	0.333117i	$-0.891900\pi$
$647$		−	10964.4i		−	0.666234i	0.942886	−	0.333117i	$-0.108100\pi$
$648$		−	9108.20i		−	0.552167i
$649$		−	11489.1i		−	0.694892i
$650$			44517.5i			2.68634i
$651$			1982.33i			0.119345i
$652$		−	1301.89i		−	0.0781994i
$653$	21489.9			1.28785			0.643923	−	0.765090i	$-0.277306\pi$
$653$	21489.9			1.28785			0.643923	+	0.765090i	$0.277306\pi$
$654$			40915.7i			2.44637i
$655$	50809.4			3.03097
$656$	−8350.45			−0.496998
$657$	−25095.8			−1.49023
$658$	−35141.2			−2.08198
$659$	1174.60			0.0694326			0.0347163	−	0.999397i	$-0.488947\pi$
$659$	1174.60			0.0694326			0.0347163	+	0.999397i	$0.488947\pi$
$660$			61122.8i			3.60485i
$661$		−	8495.64i		−	0.499913i	−0.968257	−	0.249956i	$-0.919584\pi$
$661$		−	8495.64i		−	0.499913i	0.968257	−	0.249956i	$-0.0804163\pi$
$662$		−	22955.8i		−	1.34774i
$663$	−7539.07			−0.441619
$664$	12847.0			0.750843
$665$	40851.5			2.38218
$666$			34239.9i			1.99215i
$667$			18866.0i			1.09520i
$668$		−	6704.30i		−	0.388319i
$669$	32529.1			1.87989
$670$		−	15956.6i		−	0.920084i
$671$	−22414.5			−1.28957
$672$		−	18512.1i		−	1.06268i
$673$		−	26636.0i		−	1.52562i	−0.646623	−	0.762810i	$-0.723819\pi$
$673$		−	26636.0i		−	1.52562i	0.646623	−	0.762810i	$-0.276181\pi$
$674$	53322.6			3.04734
$675$			25420.6i			1.44954i
$676$	15442.2			0.878594
$677$			23389.0i			1.32779i	0.747826	+	0.663895i	$0.231098\pi$
$677$			23389.0i			1.32779i	−0.747826	+	0.663895i	$0.768902\pi$
$678$			41687.1i			2.36133i
$679$			28729.6i			1.62377i
$680$	−17031.6			−0.960486
$681$	−16212.9			−0.912305
$682$	−1421.59			−0.0798177
$683$		−	4715.32i		−	0.264168i	−0.991239	−	0.132084i	$-0.957833\pi$
$683$		−	4715.32i		−	0.264168i	0.991239	−	0.132084i	$-0.0421669\pi$
$684$			52538.3i			2.93692i
$685$		−	38245.1i		−	2.13324i
$686$	−23401.8			−1.30246
$687$	47098.0			2.61557
$688$	14109.0			0.781830
$689$	−12437.4			−0.687702
$690$	76309.7			4.21023
$691$			6719.64i			0.369938i	0.982744	+	0.184969i	$0.0592185\pi$
$691$			6719.64i			0.369938i	−0.982744	+	0.184969i	$0.940781\pi$
$692$	−12407.9			−0.681612
$693$			21237.9i			1.16415i
$694$		−	34610.7i		−	1.89309i
$695$			16081.0i			0.877677i
$696$		−	46775.7i		−	2.54746i
$697$		−	8093.72i		−	0.439844i
$698$		−	16024.4i		−	0.868956i
$699$			7274.89i			0.393650i
$700$			86098.6i			4.64889i
$701$			18659.0i			1.00534i	0.864480	+	0.502668i	$0.167648\pi$
$701$			18659.0i			1.00534i	−0.864480	+	0.502668i	$0.832352\pi$
$702$	−14425.8			−0.775594
$703$			18010.9i			0.966281i
$704$	19258.4			1.03101
$705$		−	56609.4i		−	3.02416i
$706$		−	23725.2i		−	1.26475i
$707$	39349.3			2.09319
$708$	51358.5			2.72623
$709$		−	26439.1i		−	1.40048i	−0.713906	−	0.700241i	$-0.753076\pi$
$709$		−	26439.1i		−	1.40048i	0.713906	−	0.700241i	$-0.246924\pi$
$710$	23112.3			1.22168
$711$	−42632.2			−2.24871
$712$		−	45707.0i		−	2.40582i
$713$			1141.67i			0.0599663i
$714$	−22667.0			−1.18808
$715$	−17607.4			−0.920949
$716$			1299.70i			0.0678381i
$717$			4700.64i			0.244838i
$718$	2546.58			0.132364
$719$		−	3726.01i		−	0.193264i	−0.995320	−	0.0966319i	$-0.969193\pi$
$719$		−	3726.01i		−	0.193264i	0.995320	−	0.0966319i	$-0.0308070\pi$
$720$	−22081.0			−1.14293
$721$			5626.00i			0.290601i
$722$			10481.4i			0.540274i
$723$	−1285.72			−0.0661361
$724$			17140.3i			0.879855i
$725$		−	53306.5i		−	2.73070i
$726$	24924.3			1.27414
$727$	1288.72			0.0657441			0.0328720	−	0.999460i	$-0.489535\pi$
$727$	1288.72			0.0657441			0.0328720	+	0.999460i	$0.489535\pi$
$728$	−21763.5			−1.10798
$729$	31238.4			1.58707
$730$		−	62554.9i		−	3.17159i
$731$			13675.2i			0.691921i
$732$		−	100197.i		−	5.05929i
$733$		−	6011.52i		−	0.302920i	−0.988463	−	0.151460i	$-0.951602\pi$
$733$		−	6011.52i		−	0.302920i	0.988463	−	0.151460i	$-0.0483975\pi$
$734$	−7559.66			−0.380153
$735$			18060.5i			0.906358i
$736$		−	10661.6i		−	0.533955i
$737$	4363.62			0.218095
$738$		−	52699.1i		−	2.62857i
$739$		−	24120.5i		−	1.20066i	−0.799753	−	0.600329i	$-0.795036\pi$
$739$		−	24120.5i		−	1.20066i	0.799753	−	0.600329i	$-0.204964\pi$
$740$	−54901.2			−2.72731
$741$	−25821.3			−1.28012
$742$	−37394.3			−1.85012
$743$	−11342.9			−0.560070			−0.280035	−	0.959990i	$-0.590346\pi$
$743$	−11342.9			−0.560070			−0.280035	+	0.959990i	$0.590346\pi$
$744$		−	2830.62i		−	0.139483i
$745$	29282.6			1.44004
$746$	−3891.29			−0.190979
$747$			16143.7i			0.790719i
$748$		−	10456.4i		−	0.511130i
$749$	40809.9			1.99087
$750$	−119387.			−5.81255
$751$	−35093.2			−1.70515			−0.852575	−	0.522605i	$-0.824960\pi$
$751$	−35093.2			−1.70515			−0.852575	+	0.522605i	$0.824960\pi$
$752$	9991.51			0.484512
$753$	−271.694			−0.0131489
$754$	30250.6			1.46109
$755$	48029.9			2.31521
$756$	−27900.1			−1.34222
$757$			7493.88i			0.359802i	0.983685	+	0.179901i	$0.0575777\pi$
$757$			7493.88i			0.359802i	−0.983685	+	0.179901i	$0.942422\pi$
$758$	17923.5			0.858852
$759$			20868.3i			0.997986i
$760$	−58333.0			−2.78416
$761$			9128.51i			0.434834i	0.976079	+	0.217417i	$0.0697631\pi$
$761$			9128.51i			0.434834i	−0.976079	+	0.217417i	$0.930237\pi$
$762$	34009.1			1.61683
$763$	−22795.4			−1.08158
$764$	28600.1			1.35434
$765$		−	21402.1i		−	1.01150i
$766$	−45886.6			−2.16442
$767$			14794.6i			0.696484i
$768$			53178.2i			2.49857i
$769$	18181.6			0.852593			0.426297	−	0.904583i	$-0.359818\pi$
$769$	18181.6			0.852593			0.426297	+	0.904583i	$0.359818\pi$
$770$	−52938.4			−2.47762
$771$	−3009.95			−0.140598
$772$			45300.1i			2.11190i
$773$	8098.95			0.376842			0.188421	−	0.982088i	$-0.439663\pi$
$773$	8098.95			0.376842			0.188421	+	0.982088i	$0.439663\pi$
$774$			89040.6i			4.13501i
$775$		−	3225.83i		−	0.149516i
$776$		−	41023.8i		−	1.89777i
$777$	−32546.1			−1.50268
$778$		−	12039.7i		−	0.554812i
$779$		−	27720.9i		−	1.27497i
$780$		−	78708.7i		−	3.61311i
$781$			6320.49i			0.289584i
$782$	−13054.5			−0.596966
$783$	17273.9			0.788401
$784$	−3187.67			−0.145211
$785$		−	14528.2i		−	0.660551i
$786$	−96558.3			−4.38183
$787$	−29197.4			−1.32246			−0.661230	−	0.750183i	$-0.729965\pi$
$787$	−29197.4			−1.32246			−0.661230	+	0.750183i	$0.729965\pi$
$788$		−	42109.7i		−	1.90368i
$789$			49298.5i			2.22443i
$790$		−	106267.i		−	4.78583i
$791$	−23225.1			−1.04398
$792$		−	30326.1i		−	1.36060i
$793$	28863.4			1.29252
$794$	48576.0			2.17116
$795$		−	60239.0i		−	2.68737i
$796$	2194.41			0.0977119
$797$			8661.06i			0.384932i	0.981304	+	0.192466i	$0.0616485\pi$
$797$			8661.06i			0.384932i	−0.981304	+	0.192466i	$0.938352\pi$
$798$	−77634.3			−3.44389
$799$			9684.32i			0.428794i
$800$			30124.6i			1.33133i
$801$	57436.1			2.53359
$802$	51973.3			2.28833
$803$	17106.8			0.751788
$804$			19506.3i			0.855640i
$805$			42514.5i			1.86141i
$806$	1830.61			0.0800005
$807$	−35059.3	+	6378.88i	−1.52930	+	0.278249i
$808$	−56188.0			−2.44639
$809$			33501.6i			1.45594i	0.685610	+	0.727969i	$0.259535\pi$
$809$			33501.6i			1.45594i	−0.685610	+	0.727969i	$0.740465\pi$
$810$			28529.6i			1.23756i
$811$	5160.34			0.223433			0.111716	−	0.993740i	$-0.464365\pi$
$811$	5160.34			0.223433			0.111716	+	0.993740i	$0.464365\pi$
$812$	58505.9			2.52852
$813$	−17759.4			−0.766114
$814$		−	23339.9i		−	1.00499i
$815$			1816.41i			0.0780690i
$816$	6444.79			0.276486
$817$			46837.4i			2.00567i
$818$	−32696.3			−1.39755
$819$		−	27348.3i		−	1.16682i
$820$	84499.3			3.59859
$821$	40003.3			1.70052			0.850260	−	0.526363i	$-0.176445\pi$
$821$	40003.3			1.70052			0.850260	+	0.526363i	$0.176445\pi$
$822$			72681.1i			3.08399i
$823$	−8931.00			−0.378269			−0.189134	−	0.981951i	$-0.560568\pi$
$823$	−8931.00			−0.378269			−0.189134	+	0.981951i	$0.560568\pi$
$824$		−	8033.52i		−	0.339637i
$825$		−	58964.0i		−	2.48832i
$826$			44481.6i			1.87374i
$827$	−1638.18			−0.0688817			−0.0344409	−	0.999407i	$-0.510965\pi$
$827$	−1638.18			−0.0688817			−0.0344409	+	0.999407i	$0.510965\pi$
$828$	−54677.0			−2.29488
$829$			315.738i			0.0132280i	0.999978	+	0.00661401i	$0.00210532\pi$
$829$			315.738i			0.0132280i	−0.999978	+	0.00661401i	$0.997895\pi$
$830$	−40240.5			−1.68285
$831$	62436.0			2.60636
$832$	−24799.4			−1.03337
$833$		−	3089.66i		−	0.128512i
$834$		−	30560.3i		−	1.26885i
$835$			9353.92i			0.387671i
$836$		−	35813.2i		−	1.48161i
$837$	1045.32			0.0431681
$838$			203.824i			0.00840214i
$839$		−	11648.8i		−	0.479332i	−0.970855	−	0.239666i	$-0.922962\pi$
$839$		−	11648.8i		−	0.479332i	0.970855	−	0.239666i	$-0.0770380\pi$
$840$		−	105409.i		−	4.32970i
$841$	−11833.9			−0.485217
$842$			67074.4i			2.74529i
$843$	−32048.7			−1.30939
$844$	−33296.9			−1.35797
$845$	−21545.1			−0.877129
$846$			63055.8i			2.56253i
$847$			13886.1i			0.563320i
$848$	10632.1			0.430553
$849$			15388.1i			0.622045i
$850$	36885.9			1.48844
$851$	−18744.1			−0.755042
$852$	−28253.9			−1.13611
$853$			6167.91i			0.247579i	0.992308	+	0.123790i	$0.0395048\pi$
$853$			6167.91i			0.247579i	−0.992308	+	0.123790i	$0.960495\pi$
$854$	86780.8			3.47726
$855$		−	73302.1i		−	2.93202i
$856$	−58273.7			−2.32681
$857$		−	21518.7i		−	0.857720i	−0.903371	−	0.428860i	$-0.858915\pi$
$857$		−	21518.7i		−	0.857720i	0.903371	−	0.428860i	$-0.141085\pi$
$858$	33461.1			1.33140
$859$	5904.86			0.234542			0.117271	−	0.993100i	$-0.462585\pi$
$859$	5904.86			0.234542			0.117271	+	0.993100i	$0.462585\pi$
$860$	−142770.			−5.66096
$861$	50092.2			1.98274
$862$	64231.4			2.53797
$863$	−9362.93			−0.369314			−0.184657	−	0.982803i	$-0.559117\pi$
$863$	−9362.93			−0.369314			−0.184657	+	0.982803i	$0.559117\pi$
$864$	−9761.82			−0.384379
$865$	17311.6			0.680476
$866$		−	84993.1i		−	3.33508i
$867$		−	33435.4i		−	1.30972i
$868$	3540.47			0.138446
$869$	29060.7			1.13443
$870$			146515.i			5.70958i
$871$	−5619.10			−0.218594
$872$	32550.1			1.26409
$873$	51551.0			1.99856
$874$	−44711.6			−1.73042
$875$		−	66514.4i		−	2.56982i
$876$			76471.0i			2.94945i
$877$	23208.3			0.893602			0.446801	−	0.894633i	$-0.352563\pi$
$877$	23208.3			0.893602			0.446801	+	0.894633i	$0.352563\pi$
$878$		−	29715.8i		−	1.14221i
$879$		−	28367.8i		−	1.08854i
$880$	15051.7			0.576583
$881$		−	3624.39i		−	0.138603i	−0.997596	−	0.0693013i	$-0.977923\pi$
$881$		−	3624.39i		−	0.138603i	0.997596	−	0.0693013i	$-0.0220770\pi$
$882$		−	20117.2i		−	0.768004i
$883$			11140.3i			0.424575i	0.977207	+	0.212288i	$0.0680914\pi$
$883$			11140.3i			0.424575i	−0.977207	+	0.212288i	$0.931909\pi$
$884$			13464.9i			0.512300i
$885$	−71656.1			−2.72169
$886$	−60745.1			−2.30335
$887$	36092.6			1.36626			0.683130	−	0.730297i	$-0.260618\pi$
$887$	36092.6			1.36626			0.683130	+	0.730297i	$0.260618\pi$
$888$	46473.5			1.75625
$889$			18947.5i			0.714825i
$890$			143168.i			5.39213i
$891$	−7801.93			−0.293350
$892$		−	58097.5i		−	2.18077i
$893$			33168.7i			1.24294i
$894$	−55648.9			−2.08185
$895$		−	1813.36i		−	0.0677250i
$896$	−56226.1			−2.09641
$897$		−	26872.4i		−	1.00027i
$898$		−	6114.11i		−	0.227205i
$899$	−2192.02			−0.0813215
$900$	154492.			5.72191
$901$			10305.2i			0.381040i
$902$			35922.9i			1.32605i
$903$	−84636.0			−3.11906
$904$	33163.8			1.22015
$905$		−	23914.4i		−	0.878388i
$906$	−91276.1			−3.34707
$907$	11723.4			0.429184			0.214592	−	0.976704i	$-0.431158\pi$
$907$	11723.4			0.429184			0.214592	+	0.976704i	$0.431158\pi$
$908$			28956.5i			1.05832i
$909$		−	70606.6i		−	2.57632i
$910$	68169.5			2.48329
$911$		−	40622.2i		−	1.47736i	−0.674057	−	0.738680i	$-0.735450\pi$
$911$		−	40622.2i		−	1.47736i	0.674057	−	0.738680i	$-0.264550\pi$
$912$	22073.4			0.801450
$913$		−	11004.5i		−	0.398900i
$914$		−	50108.0i		−	1.81338i
$915$			139797.i			5.05085i
$916$		−	84117.7i		−	3.03420i
$917$		−	53795.6i		−	1.93728i
$918$			11952.8i			0.429739i
$919$		−	21883.1i		−	0.785480i	−0.919650	−	0.392740i	$-0.871527\pi$
$919$		−	21883.1i		−	0.785480i	0.919650	−	0.392740i	$-0.128473\pi$
$920$		−	60707.6i		−	2.17551i
$921$			28706.5i			1.02705i
$922$	72807.1			2.60062
$923$		−	8138.98i		−	0.290247i
$924$	64715.2			2.30408
$925$	52962.1			1.88258
$926$	25189.6			0.893931
$927$	10095.0			0.357675
$928$	20470.3			0.724107
$929$			40912.2i			1.44487i	0.691439	+	0.722435i	$0.256977\pi$
$929$			40912.2i			1.44487i	−0.691439	+	0.722435i	$0.743023\pi$
$930$			8866.34i			0.312622i
$931$		−	10582.1i		−	0.372517i
$932$	12993.1			0.456655
$933$	−25833.3			−0.906477
$934$	3122.41			0.109388
$935$			14588.9i			0.510277i
$936$			39051.4i			1.36371i
$937$		−	44502.6i		−	1.55159i	−0.630988	−	0.775793i	$-0.717350\pi$
$937$		−	44502.6i		−	1.55159i	0.630988	−	0.775793i	$-0.282650\pi$
$938$	−16894.4			−0.588082
$939$			71251.4i			2.47625i
$940$	−101105.			−3.50818
$941$		−	21540.3i		−	0.746221i	−0.927787	−	0.373111i	$-0.878291\pi$
$941$		−	21540.3i		−	0.746221i	0.927787	−	0.373111i	$-0.121709\pi$
$942$			27609.4i			0.954950i
$943$	28849.4			0.996252
$944$		−	12647.2i		−	0.436051i
$945$	38926.5			1.33998
$946$		−	60695.4i		−	2.08602i
$947$			20740.1i			0.711681i	0.934547	+	0.355840i	$0.115805\pi$
$947$			20740.1i			0.711681i	−0.934547	+	0.355840i	$0.884195\pi$
$948$			129907.i			4.45063i
$949$	−22028.7			−0.753510
$950$	126334.			4.31454
$951$	36395.9			1.24103
$952$			18032.5i			0.613906i
$953$			34601.4i			1.17613i	0.808815	+	0.588063i	$0.200109\pi$
$953$			34601.4i			1.17613i	−0.808815	+	0.588063i	$0.799891\pi$
$954$			67098.6i			2.27715i
$955$	−39903.2			−1.35208
$956$	8395.41			0.284024
$957$	−40067.3			−1.35339
$958$	41846.0			1.41125
$959$	−40492.8			−1.36348
$960$		−	120113.i		−	4.03815i
$961$	29658.4			0.995547
$962$			30055.2i			1.00730i
$963$		−	73227.5i		−	2.45039i
$964$			2296.32i			0.0767213i
$965$		−	63203.2i		−	2.10838i
$966$		−	80794.6i		−	2.69102i
$967$			56344.0i			1.87373i	0.349686	+	0.936867i	$0.386288\pi$
$967$			56344.0i			1.87373i	−0.349686	+	0.936867i	$0.613712\pi$
$968$		−	19828.4i		−	0.658375i
$969$			21394.7i			0.709285i
$970$			128498.i			4.25344i
$971$	−37400.7			−1.23609			−0.618046	−	0.786142i	$-0.712076\pi$
$971$	−37400.7			−1.23609			−0.618046	+	0.786142i	$0.712076\pi$
$972$		−	70226.6i		−	2.31741i
$973$	17026.1			0.560978
$974$			25775.7i			0.847952i
$975$			75928.8i			2.49402i
$976$	−24674.0			−0.809215
$977$	2699.81			0.0884080			0.0442040	−	0.999023i	$-0.485925\pi$
$977$	2699.81			0.0884080			0.0442040	+	0.999023i	$0.485925\pi$
$978$		−	3451.92i		−	0.112863i
$979$	−39151.9			−1.27814
$980$	32256.4			1.05142
$981$			40903.0i			1.33122i
$982$		−	91834.8i		−	2.98428i
$983$	28578.6			0.927278			0.463639	−	0.886024i	$-0.346543\pi$
$983$	28578.6			0.927278			0.463639	+	0.886024i	$0.346543\pi$
$984$	−71528.1			−2.31731
$985$			58752.0i			1.90050i
$986$		−	25064.7i		−	0.809558i
$987$	−59936.5			−1.93293
$988$			46117.2i			1.48500i
$989$	−48744.0			−1.56721
$990$			94990.3i			3.04949i
$991$			24680.3i			0.791116i	0.918441	+	0.395558i	$0.129449\pi$
$991$			24680.3i			0.791116i	−0.918441	+	0.395558i	$0.870551\pi$
$992$	1238.76			0.0396477
$993$		−	39153.3i		−	1.25125i
$994$		−	24470.7i		−	0.780848i
$995$	−3061.66			−0.0975490
$996$	49192.5			1.56498
$997$	−41312.7			−1.31232			−0.656162	−	0.754620i	$-0.727821\pi$
$997$	−41312.7			−1.31232			−0.656162	+	0.754620i	$0.727821\pi$
$998$	39172.4			1.24246
$999$			17162.3i			0.543534i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	269.4.b.a.268.8	✓	66
269.268	even	2	inner	269.4.b.a.268.59	yes	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.4.b.a.268.8	✓	66	1.1	even	1	trivial
269.4.b.a.268.59	yes	66	269.268	even	2	inner

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:	\( N \)	\(=\)	\( 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	269.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.73556i q^{2} -8.07694i q^{3} -14.4255 q^{4} +20.1267 q^{5} -38.2489 q^{6} -21.3096i q^{7} +30.4286i q^{8} -38.2370 q^{9} -95.3113i q^{10} +26.0646 q^{11} +116.514i q^{12} -33.5638 q^{13} -100.913 q^{14} -162.562i q^{15} +28.6921 q^{16} +27.8099i q^{17} +181.074i q^{18} +95.2488i q^{19} -290.339 q^{20} -172.116 q^{21} -123.431i q^{22} -99.1262 q^{23} +245.770 q^{24} +280.084 q^{25} +158.943i q^{26} +90.7607i q^{27} +307.403i q^{28} -190.323i q^{29} -769.824 q^{30} -11.5174i q^{31} +107.556i q^{32} -210.522i q^{33} +131.696 q^{34} -428.892i q^{35} +551.590 q^{36} +189.094 q^{37} +451.057 q^{38} +271.093i q^{39} +612.427i q^{40} -291.037 q^{41} +815.068i q^{42} +491.737 q^{43} -375.996 q^{44} -769.585 q^{45} +469.418i q^{46} +348.232 q^{47} -231.744i q^{48} -111.099 q^{49} -1326.36i q^{50} +224.619 q^{51} +484.176 q^{52} +370.560 q^{53} +429.803 q^{54} +524.595 q^{55} +648.421 q^{56} +769.319 q^{57} -901.288 q^{58} -440.792i q^{59} +2345.05i q^{60} -859.957 q^{61} -54.5412 q^{62} +814.816i q^{63} +738.873 q^{64} -675.528 q^{65} -996.942 q^{66} +167.415 q^{67} -401.173i q^{68} +800.637i q^{69} -2031.05 q^{70} +242.493i q^{71} -1163.50i q^{72} +656.323 q^{73} -895.465i q^{74} -2262.22i q^{75} -1374.02i q^{76} -555.427i q^{77} +1283.78 q^{78} +1114.95 q^{79} +577.477 q^{80} -299.330 q^{81} +1378.22i q^{82} -422.201i q^{83} +2482.87 q^{84} +559.722i q^{85} -2328.65i q^{86} -1537.23 q^{87} +793.110i q^{88} -1502.11 q^{89} +3644.42i q^{90} +715.231i q^{91} +1429.95 q^{92} -93.0251 q^{93} -1649.08i q^{94} +1917.04i q^{95} +868.720 q^{96} -1348.20 q^{97} +526.117i q^{98} -996.633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) \) 66 * q - 258 * q^4 - 34 * q^5 + 48 * q^6 - 564 * q^9 + 22 * q^11 + 118 * q^13 + 60 * q^14 + 1030 * q^16 + 144 * q^20 - 64 * q^21 - 80 * q^23 - 778 * q^24 + 1676 * q^25 - 1146 * q^30 + 308 * q^34 + 2030 * q^36 + 470 * q^37 + 696 * q^38 + 76 * q^41 + 426 * q^43 + 698 * q^44 + 1254 * q^45 - 1280 * q^47 - 3502 * q^49 - 640 * q^51 - 1924 * q^52 - 618 * q^53 - 3766 * q^54 + 176 * q^55 + 3164 * q^56 + 1668 * q^57 - 2170 * q^58 - 1026 * q^61 + 516 * q^62 - 2936 * q^64 - 900 * q^65 - 5296 * q^66 + 862 * q^67 - 1776 * q^70 + 3560 * q^73 + 8818 * q^78 + 2064 * q^79 - 3966 * q^80 + 7446 * q^81 - 3304 * q^84 - 776 * q^87 - 892 * q^89 - 6402 * q^92 + 7388 * q^93 + 8632 * q^96 + 4552 * q^97 - 3982 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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