LMFDB - Embedded newform 269.4.b.a.268.6

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			268.6
Character	$\chi$	$=$	269.268
Dual form			269.4.b.a.268.61

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.95781i q^{2} +7.62155i q^{3} -16.5798 q^{4} +15.1989 q^{5} +37.7862 q^{6} +12.6208i q^{7} +42.5372i q^{8} -31.0881 q^{9} -75.3533i q^{10} +1.57260 q^{11} -126.364i q^{12} -37.6604 q^{13} +62.5715 q^{14} +115.839i q^{15} +78.2526 q^{16} +88.9897i q^{17} +154.129i q^{18} +21.2465i q^{19} -251.996 q^{20} -96.1902 q^{21} -7.79665i q^{22} -103.287 q^{23} -324.200 q^{24} +106.007 q^{25} +186.713i q^{26} -31.1576i q^{27} -209.251i q^{28} +3.06706i q^{29} +574.309 q^{30} +198.950i q^{31} -47.6633i q^{32} +11.9857i q^{33} +441.194 q^{34} +191.823i q^{35} +515.436 q^{36} +372.008 q^{37} +105.336 q^{38} -287.031i q^{39} +646.520i q^{40} +173.906 q^{41} +476.892i q^{42} -264.672 q^{43} -26.0735 q^{44} -472.506 q^{45} +512.078i q^{46} -25.0784 q^{47} +596.406i q^{48} +183.715 q^{49} -525.564i q^{50} -678.240 q^{51} +624.403 q^{52} -571.345 q^{53} -154.473 q^{54} +23.9018 q^{55} -536.854 q^{56} -161.932 q^{57} +15.2059 q^{58} +14.3309i q^{59} -1920.60i q^{60} +768.923 q^{61} +986.354 q^{62} -392.357i q^{63} +389.715 q^{64} -572.397 q^{65} +59.4226 q^{66} -674.938 q^{67} -1475.44i q^{68} -787.209i q^{69} +951.020 q^{70} +759.850i q^{71} -1322.40i q^{72} +356.197 q^{73} -1844.34i q^{74} +807.940i q^{75} -352.264i q^{76} +19.8475i q^{77} -1423.04 q^{78} -606.660 q^{79} +1189.35 q^{80} -601.909 q^{81} -862.191i q^{82} +260.514i q^{83} +1594.82 q^{84} +1352.55i q^{85} +1312.19i q^{86} -23.3758 q^{87} +66.8941i q^{88} +596.578 q^{89} +2342.59i q^{90} -475.304i q^{91} +1712.49 q^{92} -1516.31 q^{93} +124.334i q^{94} +322.924i q^{95} +363.268 q^{96} +869.474 q^{97} -910.824i q^{98} -48.8892 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	4.95781i		−	1.75285i	−0.481539	−	0.876425i	$-0.659922\pi$
$2$		−	4.95781i		−	1.75285i	0.481539	−	0.876425i	$-0.340078\pi$
$3$			7.62155i			1.46677i	0.679814	+	0.733384i	$0.262060\pi$
$3$			7.62155i			1.46677i	−0.679814	+	0.733384i	$0.737940\pi$
$4$	−16.5798			−2.07248
$5$	15.1989			1.35943			0.679717	−	0.733475i	$-0.262103\pi$
$5$	15.1989			1.35943			0.679717	+	0.733475i	$0.262103\pi$
$6$	37.7862			2.57102
$7$			12.6208i			0.681460i	0.940161	+	0.340730i	$0.110674\pi$
$7$			12.6208i			0.681460i	−0.940161	+	0.340730i	$0.889326\pi$
$8$			42.5372i			1.87990i
$9$	−31.0881			−1.15141
$10$		−	75.3533i		−	2.38288i
$11$	1.57260			0.0431052			0.0215526	−	0.999768i	$-0.493139\pi$
$11$	1.57260			0.0431052			0.0215526	+	0.999768i	$0.493139\pi$
$12$		−	126.364i		−	3.03985i
$13$	−37.6604			−0.803470			−0.401735	−	0.915756i	$-0.631593\pi$
$13$	−37.6604			−0.803470			−0.401735	+	0.915756i	$0.631593\pi$
$14$	62.5715			1.19450
$15$			115.839i			1.99397i
$16$	78.2526			1.22270
$17$			88.9897i			1.26960i	0.772677	+	0.634799i	$0.218917\pi$
$17$			88.9897i			1.26960i	−0.772677	+	0.634799i	$0.781083\pi$
$18$			154.129i			2.01825i
$19$			21.2465i			0.256542i	0.991739	+	0.128271i	$0.0409427\pi$
$19$			21.2465i			0.256542i	−0.991739	+	0.128271i	$0.959057\pi$
$20$	−251.996			−2.81740
$21$	−96.1902			−0.999544
$22$		−	7.79665i		−	0.0755569i
$23$	−103.287			−0.936386			−0.468193	−	0.883626i	$-0.655095\pi$
$23$	−103.287			−0.936386			−0.468193	+	0.883626i	$0.655095\pi$
$24$	−324.200			−2.75738
$25$	106.007			0.848058
$26$			186.713i			1.40836i
$27$		−	31.1576i		−	0.222085i
$28$		−	209.251i		−	1.41231i
$29$			3.06706i			0.0196393i	0.999952	+	0.00981965i	$0.00312574\pi$
$29$			3.06706i			0.0196393i	−0.999952	+	0.00981965i	$0.996874\pi$
$30$	574.309			3.49514
$31$			198.950i			1.15266i	0.817218	+	0.576329i	$0.195515\pi$
$31$			198.950i			1.15266i	−0.817218	+	0.576329i	$0.804485\pi$
$32$		−	47.6633i		−	0.263305i
$33$			11.9857i			0.0632253i
$34$	441.194			2.22541
$35$			191.823i			0.926399i
$36$	515.436			2.38628
$37$	372.008			1.65291			0.826456	−	0.563001i	$-0.190353\pi$
$37$	372.008			1.65291			0.826456	+	0.563001i	$0.190353\pi$
$38$	105.336			0.449679
$39$		−	287.031i		−	1.17850i
$40$			646.520i			2.55559i
$41$	173.906			0.662427			0.331214	−	0.943556i	$-0.392542\pi$
$41$	173.906			0.662427			0.331214	+	0.943556i	$0.392542\pi$
$42$			476.892i			1.75205i
$43$	−264.672			−0.938655			−0.469327	−	0.883024i	$-0.655504\pi$
$43$	−264.672			−0.938655			−0.469327	+	0.883024i	$0.655504\pi$
$44$	−26.0735			−0.0893347
$45$	−472.506			−1.56527
$46$			512.078i			1.64134i
$47$	−25.0784			−0.0778311			−0.0389156	−	0.999243i	$-0.512390\pi$
$47$	−25.0784			−0.0778311			−0.0389156	+	0.999243i	$0.512390\pi$
$48$			596.406i			1.79341i
$49$	183.715			0.535613
$50$		−	525.564i		−	1.48652i
$51$	−678.240			−1.86221
$52$	624.403			1.66518
$53$	−571.345			−1.48076			−0.740380	−	0.672188i	$-0.765355\pi$
$53$	−571.345			−1.48076			−0.740380	+	0.672188i	$0.765355\pi$
$54$	−154.473			−0.389281
$55$	23.9018			0.0585986
$56$	−536.854			−1.28107
$57$	−161.932			−0.376287
$58$	15.2059			0.0344247
$59$			14.3309i			0.0316225i	0.999875	+	0.0158113i	$0.00503309\pi$
$59$			14.3309i			0.0316225i	−0.999875	+	0.0158113i	$0.994967\pi$
$60$		−	1920.60i		−	4.13247i
$61$	768.923			1.61394			0.806971	−	0.590590i	$-0.201105\pi$
$61$	768.923			1.61394			0.806971	+	0.590590i	$0.201105\pi$
$62$	986.354			2.02044
$63$		−	392.357i		−	0.784640i
$64$	389.715			0.761162
$65$	−572.397			−1.09226
$66$	59.4226			0.110824
$67$	−674.938			−1.23070			−0.615349	−	0.788255i	$-0.710985\pi$
$67$	−674.938			−1.23070			−0.615349	+	0.788255i	$0.710985\pi$
$68$		−	1475.44i		−	2.63122i
$69$		−	787.209i		−	1.37346i
$70$	951.020			1.62384
$71$			759.850i			1.27011i	0.772468	+	0.635053i	$0.219022\pi$
$71$			759.850i			1.27011i	−0.772468	+	0.635053i	$0.780978\pi$
$72$		−	1322.40i		−	2.16453i
$73$	356.197			0.571092			0.285546	−	0.958365i	$-0.407825\pi$
$73$	356.197			0.571092			0.285546	+	0.958365i	$0.407825\pi$
$74$		−	1844.34i		−	2.89731i
$75$			807.940i			1.24391i
$76$		−	352.264i		−	0.531677i
$77$			19.8475i			0.0293744i
$78$	−1423.04			−2.06574
$79$	−606.660			−0.863982			−0.431991	−	0.901878i	$-0.642189\pi$
$79$	−606.660			−0.863982			−0.431991	+	0.901878i	$0.642189\pi$
$80$	1189.35			1.66217
$81$	−601.909			−0.825664
$82$		−	862.191i		−	1.16114i
$83$			260.514i			0.344519i	0.985052	+	0.172260i	$0.0551068\pi$
$83$			260.514i			0.344519i	−0.985052	+	0.172260i	$0.944893\pi$
$84$	1594.82			2.07154
$85$			1352.55i			1.72593i
$86$			1312.19i			1.64532i
$87$	−23.3758			−0.0288063
$88$			66.8941i			0.0810333i
$89$	596.578			0.710529			0.355265	−	0.934766i	$-0.384391\pi$
$89$	596.578			0.710529			0.355265	+	0.934766i	$0.384391\pi$
$90$			2342.59i			2.74368i
$91$		−	475.304i		−	0.547532i
$92$	1712.49			1.94064
$93$	−1516.31			−1.69068
$94$			124.334i			0.136426i
$95$			322.924i			0.348751i
$96$	363.268			0.386207
$97$	869.474			0.910121			0.455061	−	0.890460i	$-0.349618\pi$
$97$	869.474			0.910121			0.455061	+	0.890460i	$0.349618\pi$
$98$		−	910.824i		−	0.938848i
$99$	−48.8892			−0.0496318
$100$	−1757.58			−1.75758
$101$			56.5618i			0.0557238i	0.999612	+	0.0278619i	$0.00886987\pi$
$101$			56.5618i			0.0557238i	−0.999612	+	0.0278619i	$0.991130\pi$
$102$			3362.58i			3.26417i
$103$	1708.12			1.63404			0.817019	−	0.576610i	$-0.195625\pi$
$103$	1708.12			1.63404			0.817019	+	0.576610i	$0.195625\pi$
$104$		−	1601.97i		−	1.51044i
$105$	−1461.99			−1.35881
$106$			2832.62i			2.59555i
$107$		−	584.870i		−	0.528425i	−0.964464	−	0.264213i	$-0.914888\pi$
$107$		−	584.870i		−	0.528425i	0.964464	−	0.264213i	$-0.0851121\pi$
$108$			516.588i			0.460266i
$109$		−	601.021i		−	0.528141i	−0.964503	−	0.264070i	$-0.914935\pi$
$109$		−	601.021i		−	0.528141i	0.964503	−	0.264070i	$-0.0850651\pi$
$110$		−	118.501i		−	0.102715i
$111$			2835.28i			2.42444i
$112$			987.611i			0.833218i
$113$		−	1248.01i		−	1.03896i	−0.854482	−	0.519482i	$-0.826125\pi$
$113$		−	1248.01i		−	1.03896i	0.854482	−	0.519482i	$-0.173875\pi$
$114$			802.826i			0.659575i
$115$	−1569.85			−1.27295
$116$		−	50.8515i		−	0.0407021i
$117$	1170.79			0.925124
$118$	71.0501			0.0554296
$119$	−1123.12			−0.865180
$120$	−4927.49			−3.74847
$121$	−1328.53			−0.998142
$122$		−	3812.17i		−	2.82900i
$123$			1325.43i			0.971628i
$124$		−	3298.55i		−	2.38886i
$125$	−288.669			−0.206555
$126$	−1945.23			−1.37536
$127$	−1013.30			−0.708002			−0.354001	−	0.935245i	$-0.615179\pi$
$127$	−1013.30			−0.708002			−0.354001	+	0.935245i	$0.615179\pi$
$128$		−	2313.44i		−	1.59751i
$129$		−	2017.21i		−	1.37679i
$130$			2837.83i			1.91457i
$131$	2432.15			1.62212			0.811062	−	0.584960i	$-0.198890\pi$
$131$	2432.15			1.62212			0.811062	+	0.584960i	$0.198890\pi$
$132$		−	198.720i		−	0.131033i
$133$	−268.149			−0.174823
$134$			3346.21i			2.15723i
$135$		−	473.562i		−	0.301909i
$136$	−3785.38			−2.38671
$137$		−	1929.53i		−	1.20329i	−0.798763	−	0.601646i	$-0.794512\pi$
$137$		−	1929.53i		−	1.20329i	0.798763	−	0.601646i	$-0.205488\pi$
$138$	−3902.83			−2.40747
$139$			2384.10i			1.45480i	0.686216	+	0.727398i	$0.259270\pi$
$139$			2384.10i			1.45480i	−0.686216	+	0.727398i	$0.740730\pi$
$140$		−	3180.39i		−	1.91994i
$141$		−	191.136i		−	0.114160i
$142$	3767.19			2.22631
$143$	−59.2247			−0.0346337
$144$	−2432.72			−1.40783
$145$			46.6161i			0.0266983i
$146$		−	1765.96i		−	1.00104i
$147$			1400.19i			0.785620i
$148$	−6167.84			−3.42563
$149$	102.351			0.0562746			0.0281373	−	0.999604i	$-0.491042\pi$
$149$	102.351			0.0562746			0.0281373	+	0.999604i	$0.491042\pi$
$150$	4005.61			2.18038
$151$	2028.44			1.09319			0.546597	−	0.837396i	$-0.315923\pi$
$151$	2028.44			1.09319			0.546597	+	0.837396i	$0.315923\pi$
$152$	−903.769			−0.482272
$153$		−	2766.52i		−	1.46183i
$154$	98.4001			0.0514890
$155$			3023.82i			1.56696i
$156$			4758.92i			2.44243i
$157$			1113.52i			0.566043i	0.959113	+	0.283022i	$0.0913369\pi$
$157$			1113.52i			0.566043i	−0.959113	+	0.283022i	$0.908663\pi$
$158$			3007.70i			1.51443i
$159$		−	4354.54i		−	2.17193i
$160$		−	724.431i		−	0.357945i
$161$		−	1303.57i		−	0.638109i
$162$			2984.15i			1.44726i
$163$		−	3851.70i		−	1.85085i	−0.378930	−	0.925425i	$-0.623708\pi$
$163$		−	3851.70i		−	1.85085i	0.378930	−	0.925425i	$-0.376292\pi$
$164$	−2883.33			−1.37287
$165$			182.169i			0.0859506i
$166$	1291.58			0.603890
$167$		−	3411.59i		−	1.58082i	−0.612580	−	0.790409i	$-0.709868\pi$
$167$		−	3411.59i		−	1.58082i	0.612580	−	0.790409i	$-0.290132\pi$
$168$		−	4091.66i		−	1.87904i
$169$	−778.697			−0.354436
$170$	6705.67			3.02530
$171$		−	660.514i		−	0.295385i
$172$	4388.23			1.94534
$173$	−3490.18			−1.53384			−0.766918	−	0.641745i	$-0.778211\pi$
$173$	−3490.18			−1.53384			−0.766918	+	0.641745i	$0.778211\pi$
$174$			115.893i			0.0504931i
$175$			1337.90i			0.577918i
$176$	123.060			0.0527045
$177$	−109.224			−0.0463830
$178$		−	2957.72i		−	1.24545i
$179$			526.934i			0.220027i	0.993930	+	0.110014i	$0.0350895\pi$
$179$			526.934i			0.220027i	−0.993930	+	0.110014i	$0.964911\pi$
$180$	7834.07			3.24398
$181$			1697.10i			0.696929i	0.937322	+	0.348465i	$0.113297\pi$
$181$			1697.10i			0.696929i	−0.937322	+	0.348465i	$0.886703\pi$
$182$	−2356.47			−0.959742
$183$			5860.39i			2.36728i
$184$		−	4393.55i		−	1.76031i
$185$	5654.12			2.24702
$186$			7517.55i			2.96351i
$187$			139.945i			0.0547263i
$188$	415.796			0.161303
$189$	393.234			0.151342
$190$	1601.00			0.611308
$191$	4632.42			1.75492			0.877462	−	0.479647i	$-0.159235\pi$
$191$	4632.42			1.75492			0.877462	+	0.479647i	$0.159235\pi$
$192$			2970.24i			1.11645i
$193$		−	2750.88i		−	1.02597i	−0.858397	−	0.512986i	$-0.828539\pi$
$193$		−	2750.88i		−	1.02597i	0.858397	−	0.512986i	$-0.171461\pi$
$194$		−	4310.69i		−	1.59531i
$195$		−	4362.56i		−	1.60210i
$196$	−3045.97			−1.11005
$197$		−	1211.75i		−	0.438243i	−0.975698	−	0.219121i	$-0.929681\pi$
$197$		−	1211.75i		−	0.438243i	0.975698	−	0.219121i	$-0.0703191\pi$
$198$			242.383i			0.0869970i
$199$	−777.501			−0.276963			−0.138481	−	0.990365i	$-0.544222\pi$
$199$	−777.501			−0.276963			−0.138481	+	0.990365i	$0.544222\pi$
$200$			4509.26i			1.59426i
$201$		−	5144.07i		−	1.80515i
$202$	280.422			0.0976755
$203$	−38.7088			−0.0133834
$204$	11245.1			3.85939
$205$	2643.18			0.900525
$206$		−	8468.53i		−	2.86422i
$207$	3211.00			1.07817
$208$	−2947.02			−0.982399
$209$			33.4123i			0.0110583i
$210$			7248.25i			2.38179i
$211$	2930.70			0.956199			0.478099	−	0.878306i	$-0.341326\pi$
$211$	2930.70			0.956199			0.478099	+	0.878306i	$0.341326\pi$
$212$	9472.82			3.06885
$213$	−5791.24			−1.86295
$214$	−2899.67			−0.926250
$215$	−4022.73			−1.27604
$216$	1325.36			0.417496
$217$	−2510.91			−0.785490
$218$	−2979.74			−0.925751
$219$			2714.78i			0.837660i
$220$	−396.289			−0.121444
$221$		−	3351.38i		−	1.02008i
$222$	14056.8			4.24968
$223$		−	5039.29i		−	1.51325i	−0.653847	−	0.756627i	$-0.726846\pi$
$223$		−	5039.29i		−	1.51325i	0.653847	−	0.756627i	$-0.273154\pi$
$224$	601.549			0.179432
$225$	−3295.56			−0.976463
$226$	−6187.39			−1.82115
$227$			5168.68i			1.51127i	0.654995	+	0.755633i	$0.272671\pi$
$227$			5168.68i			1.51127i	−0.654995	+	0.755633i	$0.727329\pi$
$228$	2684.80			0.779848
$229$		−	4397.95i		−	1.26910i	−0.772881	−	0.634551i	$-0.781185\pi$
$229$		−	4397.95i		−	1.26910i	0.772881	−	0.634551i	$-0.218815\pi$
$230$			7783.04i			2.23130i
$231$	−151.269			−0.0430855
$232$	−130.464			−0.0369199
$233$	3417.64			0.960932			0.480466	−	0.877013i	$-0.340467\pi$
$233$	3417.64			0.960932			0.480466	+	0.877013i	$0.340467\pi$
$234$		−	5804.55i		−	1.62160i
$235$	−381.165			−0.105806
$236$		−	237.605i		−	0.0655371i
$237$		−	4623.69i		−	1.26726i
$238$			5568.22i			1.51653i
$239$	−5256.30			−1.42260			−0.711301	−	0.702888i	$-0.751893\pi$
$239$	−5256.30			−1.42260			−0.711301	+	0.702888i	$0.751893\pi$
$240$			9064.73i			2.43802i
$241$		−	4285.44i		−	1.14543i	−0.819753	−	0.572717i	$-0.805889\pi$
$241$		−	4285.44i		−	1.14543i	0.819753	−	0.572717i	$-0.194111\pi$
$242$			6586.58i			1.74959i
$243$		−	5428.74i		−	1.43314i
$244$	−12748.6			−3.34487
$245$	2792.27			0.728129
$246$	6571.24			1.70312
$247$		−	800.152i		−	0.206123i
$248$	−8462.76			−2.16688
$249$	−1985.52			−0.505330
$250$			1431.17i			0.362059i
$251$			3564.90i			0.896471i	0.893915	+	0.448236i	$0.147947\pi$
$251$			3564.90i			0.896471i	−0.893915	+	0.448236i	$0.852053\pi$
$252$			6505.22i			1.62615i
$253$	−162.430			−0.0403631
$254$			5023.77i			1.24102i
$255$	−10308.5			−2.53155
$256$	−8351.86			−2.03903
$257$			7296.09i			1.77088i	0.464750	+	0.885442i	$0.346144\pi$
$257$			7296.09i			1.77088i	−0.464750	+	0.885442i	$0.653856\pi$
$258$	−10001.0			−2.41330
$259$			4695.05i			1.12639i
$260$	9490.26			2.26369
$261$		−	95.3492i		−	0.0226129i
$262$		−	12058.1i		−	2.84334i
$263$	2868.66			0.672582			0.336291	−	0.941758i	$-0.390827\pi$
$263$	2868.66			0.672582			0.336291	+	0.941758i	$0.390827\pi$
$264$	−509.837			−0.118857
$265$	−8683.84			−2.01300
$266$			1329.43i			0.306438i
$267$			4546.85i			1.04218i
$268$	11190.4			2.55060
$269$	3534.27	−	2640.84i	0.801072	−	0.598569i
$269$	3534.27	−	2640.84i	0.801072	−	0.598569i
$270$	−2347.83			−0.529201
$271$		−	528.768i		−	0.118525i	−0.998242	−	0.0592627i	$-0.981125\pi$
$271$		−	528.768i		−	0.118525i	0.998242	−	0.0592627i	$-0.0188750\pi$
$272$			6963.67i			1.55233i
$273$	3622.56			0.803103
$274$	−9566.25			−2.10919
$275$	166.707			0.0365557
$276$			13051.8i			2.84647i
$277$		−	1236.95i		−	0.268308i	−0.990961	−	0.134154i	$-0.957168\pi$
$277$		−	1236.95i		−	0.268308i	0.990961	−	0.134154i	$-0.0428316\pi$
$278$	11819.9			2.55004
$279$		−	6184.96i		−	1.32718i
$280$	−8159.61			−1.74154
$281$			2220.29i			0.471356i	0.971831	+	0.235678i	$0.0757311\pi$
$281$			2220.29i			0.471356i	−0.971831	+	0.235678i	$0.924269\pi$
$282$	−947.618			−0.200106
$283$	−1832.16			−0.384843			−0.192422	−	0.981312i	$-0.561634\pi$
$283$	−1832.16			−0.384843			−0.192422	+	0.981312i	$0.561634\pi$
$284$		−	12598.2i		−	2.63227i
$285$	−2461.19			−0.511537
$286$			293.625i			0.0607077i
$287$			2194.83i			0.451417i
$288$			1481.76i			0.303172i
$289$	−3006.17			−0.611880
$290$	231.114			0.0467981
$291$			6626.75i			1.33494i
$292$	−5905.70			−1.18358
$293$	4447.80			0.886837			0.443418	−	0.896315i	$-0.353766\pi$
$293$	4447.80			0.886837			0.443418	+	0.896315i	$0.353766\pi$
$294$	6941.89			1.37707
$295$			217.815i			0.0429887i
$296$			15824.2i			3.10731i
$297$		−	48.9985i		−	0.00957300i
$298$		−	507.437i		−	0.0986410i
$299$	3889.84			0.752358
$300$		−	13395.5i		−	2.57797i
$301$		−	3340.38i		−	0.639655i
$302$		−	10056.6i		−	1.91620i
$303$	−431.089			−0.0817340
$304$			1662.60i			0.313672i
$305$	11686.8			2.19405
$306$	−13715.9			−2.56237
$307$	8062.05			1.49878			0.749390	−	0.662128i	$-0.230347\pi$
$307$	8062.05			1.49878			0.749390	+	0.662128i	$0.230347\pi$
$308$		−	329.068i		−	0.0608780i
$309$			13018.5i			2.39676i
$310$	14991.5			2.74665
$311$			2892.88i			0.527460i	0.964597	+	0.263730i	$0.0849528\pi$
$311$			2892.88i			0.527460i	−0.964597	+	0.263730i	$0.915047\pi$
$312$	12209.5			2.21547
$313$	7447.54			1.34492			0.672459	−	0.740134i	$-0.265238\pi$
$313$	7447.54			1.34492			0.672459	+	0.740134i	$0.265238\pi$
$314$	5520.63			0.992189
$315$		−	5963.40i		−	1.06667i
$316$	10058.3			1.79059
$317$		−	869.733i		−	0.154098i	−0.997027	−	0.0770490i	$-0.975450\pi$
$317$		−	869.733i		−	0.154098i	0.997027	−	0.0770490i	$-0.0245498\pi$
$318$	−21589.0			−3.80707
$319$			4.82327i			0.000846555i
$320$	5923.25			1.03475
$321$	4457.62			0.775078
$322$	−6462.84			−1.11851
$323$	−1890.72			−0.325705
$324$	9979.56			1.71117
$325$	−3992.27			−0.681389
$326$	−19096.0			−3.24426
$327$	4580.71			0.774660
$328$			7397.47i			1.24530i
$329$		−	316.510i		−	0.0530388i
$330$	903.159			0.150658
$331$	−213.953			−0.0355285			−0.0177642	−	0.999842i	$-0.505655\pi$
$331$	−213.953			−0.0355285			−0.0177642	+	0.999842i	$0.505655\pi$
$332$		−	4319.28i		−	0.714010i
$333$	−11565.0			−1.90318
$334$	−16914.0			−2.77094
$335$	−10258.3			−1.67305
$336$	−7527.13			−1.22214
$337$			8612.67i			1.39217i	0.717958	+	0.696086i	$0.245077\pi$
$337$			8612.67i			1.39217i	−0.717958	+	0.696086i	$0.754923\pi$
$338$			3860.63i			0.621274i
$339$	9511.77			1.52392
$340$		−	22425.0i		−	3.57697i
$341$			312.868i			0.0496855i
$342$	−3274.70			−0.517765
$343$			6647.57i			1.04646i
$344$		−	11258.4i		−	1.76457i
$345$		−	11964.7i		−	1.86713i
$346$			17303.7i			2.68858i
$347$	−11076.3			−1.71356			−0.856781	−	0.515681i	$-0.827539\pi$
$347$	−11076.3			−1.71356			−0.856781	+	0.515681i	$0.827539\pi$
$348$	387.567			0.0597005
$349$	1975.79			0.303041			0.151521	−	0.988454i	$-0.451583\pi$
$349$	1975.79			0.303041			0.151521	+	0.988454i	$0.451583\pi$
$350$	6633.04			1.01300
$351$			1173.41i			0.178438i
$352$		−	74.9553i		−	0.0113498i
$353$	2710.41			0.408670			0.204335	−	0.978901i	$-0.434497\pi$
$353$	2710.41			0.408670			0.204335	+	0.978901i	$0.434497\pi$
$354$			541.512i			0.0813024i
$355$			11548.9i			1.72663i
$356$	−9891.17			−1.47256
$357$		−	8559.94i		−	1.26902i
$358$	2612.44			0.385675
$359$			2803.49i			0.412151i	0.978536	+	0.206076i	$0.0660693\pi$
$359$			2803.49i			0.412151i	−0.978536	+	0.206076i	$0.933931\pi$
$360$		−	20099.1i		−	2.94254i
$361$	6407.58			0.934186
$362$	8413.88			1.22161
$363$		−	10125.4i		−	1.46404i
$364$			7880.47i			1.13475i
$365$	5413.82			0.776362
$366$	29054.7			4.14949
$367$			6737.38i			0.958279i	0.877739	+	0.479140i	$0.159051\pi$
$367$			6737.38i			0.958279i	−0.877739	+	0.479140i	$0.840949\pi$
$368$	−8082.49			−1.14492
$369$	−5406.40			−0.762726
$370$		−	28032.1i		−	3.93869i
$371$		−	7210.84i		−	1.00908i
$372$	25140.1			3.50391
$373$			3588.48i			0.498135i	0.968486	+	0.249067i	$0.0801240\pi$
$373$			3588.48i			0.498135i	−0.968486	+	0.249067i	$0.919876\pi$
$374$	693.822			0.0959269
$375$		−	2200.11i		−	0.302968i
$376$		−	1066.77i		−	0.146314i
$377$		−	115.507i		−	0.0157796i
$378$		−	1949.58i		−	0.265279i
$379$			8436.11i			1.14336i	0.820476	+	0.571681i	$0.193708\pi$
$379$			8436.11i			1.14336i	−0.820476	+	0.571681i	$0.806292\pi$
$380$		−	5354.04i		−	0.722780i
$381$		−	7722.95i		−	1.03847i
$382$		−	22966.7i		−	3.07612i
$383$		−	6519.78i		−	0.869831i	−0.900471	−	0.434916i	$-0.856778\pi$
$383$		−	6519.78i		−	0.869831i	0.900471	−	0.434916i	$-0.143222\pi$
$384$	17632.0			2.34318
$385$			301.661i			0.0399326i
$386$	−13638.3			−1.79837
$387$	8228.16			1.08078
$388$	−14415.8			−1.88621
$389$	7945.96			1.03567			0.517836	−	0.855480i	$-0.326738\pi$
$389$	7945.96			1.03567			0.517836	+	0.855480i	$0.326738\pi$
$390$	−21628.7			−2.80824
$391$		−	9191.50i		−	1.18883i
$392$			7814.73i			1.00690i
$393$			18536.8i			2.37928i
$394$	−6007.64			−0.768174
$395$	−9220.58			−1.17453
$396$	810.575			0.102861
$397$			6697.98i			0.846755i	0.905953	+	0.423378i	$0.139156\pi$
$397$			6697.98i			0.846755i	−0.905953	+	0.423378i	$0.860844\pi$
$398$			3854.70i			0.485474i
$399$		−	2043.71i		−	0.256425i
$400$	8295.34			1.03692
$401$			2642.93i			0.329132i	0.986366	+	0.164566i	$0.0526223\pi$
$401$			2642.93i			0.329132i	−0.986366	+	0.164566i	$0.947378\pi$
$402$	−25503.3			−3.16415
$403$		−	7492.52i		−	0.926126i
$404$		−	937.786i		−	0.115487i
$405$	−9148.37			−1.12243
$406$			191.911i			0.0234591i
$407$	585.020			0.0712491
$408$		−	28850.4i		−	3.50076i
$409$			11419.6i			1.38060i	0.723524	+	0.690299i	$0.242521\pi$
$409$			11419.6i			1.38060i	−0.723524	+	0.690299i	$0.757479\pi$
$410$		−	13104.4i		−	1.57849i
$411$	14706.0			1.76495
$412$	−28320.4			−3.38651
$413$	−180.868			−0.0215495
$414$		−	15919.5i		−	1.88986i
$415$			3959.53i			0.468351i
$416$			1795.02i			0.211558i
$417$	−18170.5			−2.13385
$418$	165.652			0.0193835
$419$	15450.7			1.80147			0.900734	−	0.434372i	$-0.143030\pi$
$419$	15450.7			1.80147			0.900734	+	0.434372i	$0.143030\pi$
$420$	24239.5			2.81611
$421$	−2426.07			−0.280854			−0.140427	−	0.990091i	$-0.544848\pi$
$421$	−2426.07			−0.280854			−0.140427	+	0.990091i	$0.544848\pi$
$422$		−	14529.9i		−	1.67607i
$423$	779.640			0.0896156
$424$		−	24303.4i		−	2.78368i
$425$			9433.56i			1.07669i
$426$			28711.8i			3.26548i
$427$			9704.43i			1.09984i
$428$			9697.05i			1.09515i
$429$		−	451.384i		−	0.0507996i
$430$			19943.9i			2.23670i
$431$			21.2328i			0.00237297i	0.999999	+	0.00118648i	$0.000377670\pi$
$431$			21.2328i			0.00237297i	−0.999999	+	0.00118648i	$0.999622\pi$
$432$		−	2438.16i		−	0.271542i
$433$	−4284.70			−0.475542			−0.237771	−	0.971321i	$-0.576417\pi$
$433$	−4284.70			−0.475542			−0.237771	+	0.971321i	$0.576417\pi$
$434$			12448.6i			1.37685i
$435$	−355.287			−0.0391602
$436$			9964.83i			1.09456i
$437$		−	2194.50i		−	0.240222i
$438$	13459.3			1.46829
$439$	−12210.0			−1.32745			−0.663724	−	0.747977i	$-0.731025\pi$
$439$	−12210.0			−1.32745			−0.663724	+	0.747977i	$0.731025\pi$
$440$			1016.72i			0.110159i
$441$	−5711.35			−0.616710
$442$	−16615.5			−1.78805
$443$			11792.6i			1.26475i	0.774662	+	0.632375i	$0.217920\pi$
$443$			11792.6i			1.26475i	−0.774662	+	0.632375i	$0.782080\pi$
$444$		−	47008.5i		−	5.02461i
$445$	9067.34			0.965917
$446$	−24983.8			−2.65251
$447$			780.074i			0.0825419i
$448$			4918.52i			0.518702i
$449$	1464.96			0.153977			0.0769886	−	0.997032i	$-0.475470\pi$
$449$	1464.96			0.153977			0.0769886	+	0.997032i	$0.475470\pi$
$450$			16338.8i			1.71159i
$451$	273.484			0.0285540
$452$			20691.8i			2.15323i
$453$			15459.9i			1.60346i
$454$	25625.3			2.64902
$455$		−	7224.12i		−	0.744334i
$456$		−	6888.12i		−	0.707381i
$457$	−7117.41			−0.728530			−0.364265	−	0.931295i	$-0.618680\pi$
$457$	−7117.41			−0.728530			−0.364265	+	0.931295i	$0.618680\pi$
$458$	−21804.2			−2.22455
$459$	2772.71			0.281958
$460$	26028.0			2.63817
$461$		−	9909.12i		−	1.00111i	−0.865704	−	0.500557i	$-0.833129\pi$
$461$		−	9909.12i		−	1.00111i	0.865704	−	0.500557i	$-0.166871\pi$
$462$			749.961i			0.0755224i
$463$			11770.3i			1.18145i	0.806871	+	0.590727i	$0.201159\pi$
$463$			11770.3i			1.18145i	−0.806871	+	0.590727i	$0.798841\pi$
$464$			240.006i			0.0240129i
$465$	−23046.2			−2.29837
$466$		−	16944.0i		−	1.68437i
$467$		−	8192.86i		−	0.811820i	−0.913913	−	0.405910i	$-0.866955\pi$
$467$		−	8192.86i		−	0.811820i	0.913913	−	0.405910i	$-0.133045\pi$
$468$	−19411.5			−1.91730
$469$		−	8518.26i		−	0.838671i
$470$			1889.74i			0.185462i
$471$	−8486.78			−0.830255
$472$	−609.599			−0.0594471
$473$	−416.224			−0.0404609
$474$	−22923.4			−2.22132
$475$			2252.29i			0.217562i
$476$	18621.2			1.79307
$477$	17762.0			1.70496
$478$			26059.7i			2.49361i
$479$			633.723i			0.0604500i	0.999543	+	0.0302250i	$0.00962238\pi$
$479$			633.723i			0.0604500i	−0.999543	+	0.0302250i	$0.990378\pi$
$480$	5521.29			0.525023
$481$	−14010.0			−1.32807
$482$	−21246.4			−2.00777
$483$	9935.22			0.935959
$484$	22026.8			2.06863
$485$	13215.1			1.23725
$486$	−26914.6			−2.51208
$487$	−1241.82			−0.115549			−0.0577745	−	0.998330i	$-0.518400\pi$
$487$	−1241.82			−0.115549			−0.0577745	+	0.998330i	$0.518400\pi$
$488$			32707.9i			3.03405i
$489$	29356.0			2.71477
$490$		−	13843.5i		−	1.27630i
$491$	−4282.65			−0.393632			−0.196816	−	0.980440i	$-0.563060\pi$
$491$	−4282.65			−0.393632			−0.196816	+	0.980440i	$0.563060\pi$
$492$		−	21975.5i		−	2.01368i
$493$	−272.937			−0.0249340
$494$	−3967.00			−0.361303
$495$	−743.063			−0.0674711
$496$			15568.3i			1.40935i
$497$	−9589.92			−0.865527
$498$			9843.82i			0.885768i
$499$		−	8735.12i		−	0.783642i	−0.920041	−	0.391821i	$-0.871845\pi$
$499$		−	8735.12i		−	0.783642i	0.920041	−	0.391821i	$-0.128155\pi$
$500$	4786.09			0.428081
$501$	26001.6			2.31869
$502$	17674.1			1.57138
$503$		−	280.586i		−	0.0248722i	−0.999923	−	0.0124361i	$-0.996041\pi$
$503$		−	280.586i		−	0.0248722i	0.999923	−	0.0124361i	$-0.00395863\pi$
$504$	16689.8			1.47504
$505$			859.678i			0.0757528i
$506$			805.295i			0.0707504i
$507$		−	5934.88i		−	0.519876i
$508$	16800.4			1.46732
$509$		−	18953.0i		−	1.65044i	−0.564809	−	0.825222i	$-0.691050\pi$
$509$		−	18953.0i		−	1.65044i	0.564809	−	0.825222i	$-0.308950\pi$
$510$			51107.6i			4.43742i
$511$			4495.50i			0.389176i
$512$			22899.4i			1.97660i
$513$	661.991			0.0569739
$514$	36172.6			3.10409
$515$	25961.6			2.22137
$516$			33445.1i			2.85337i
$517$	−39.4383			−0.00335492
$518$	23277.1			1.97440
$519$		−	26600.6i		−	2.24978i
$520$		−	24348.2i		−	2.05334i
$521$			8311.93i			0.698949i	0.936946	+	0.349474i	$0.113640\pi$
$521$			8311.93i			0.698949i	−0.936946	+	0.349474i	$0.886360\pi$
$522$	−472.723			−0.0396370
$523$			19480.5i			1.62873i	0.580354	+	0.814364i	$0.302914\pi$
$523$			19480.5i			1.62873i	−0.580354	+	0.814364i	$0.697086\pi$
$524$	−40324.7			−3.36182
$525$	−10196.9			−0.847671
$526$		−	14222.3i		−	1.17893i
$527$	−17704.5			−1.46341
$528$			937.909i			0.0773054i
$529$	−1498.74			−0.123181
$530$			43052.8i			3.52848i
$531$		−	445.522i		−	0.0364105i
$532$	4445.86			0.362317
$533$	−6549.35			−0.532240
$534$	22542.4			1.82679
$535$		−	8889.39i		−	0.718359i
$536$		−	28710.0i		−	2.31359i
$537$	−4016.06			−0.322729
$538$	−13092.8	−	17522.2i	−1.04920	−	1.40416i
$539$	288.911			0.0230877
$540$			7851.59i			0.625701i
$541$			21391.5i			1.69999i	0.526793	+	0.849994i	$0.323394\pi$
$541$			21391.5i			1.69999i	−0.526793	+	0.849994i	$0.676606\pi$
$542$	−2621.53			−0.207757
$543$	−12934.5			−1.02223
$544$	4241.54			0.334291
$545$		−	9134.87i		−	0.717972i
$546$		−	17959.9i		−	1.40772i
$547$	1297.46			0.101418			0.0507089	−	0.998713i	$-0.483852\pi$
$547$	1297.46			0.101418			0.0507089	+	0.998713i	$0.483852\pi$
$548$			31991.3i			2.49380i
$549$	−23904.4			−1.85831
$550$		−	826.502i		−	0.0640766i
$551$	−65.1645			−0.00503830
$552$	33485.7			2.58197
$553$		−	7656.54i		−	0.588769i
$554$	−6132.57			−0.470303
$555$			43093.2i			3.29587i
$556$		−	39528.0i		−	3.01504i
$557$		−	20021.4i		−	1.52304i	−0.648142	−	0.761520i	$-0.724454\pi$
$557$		−	20021.4i		−	1.52304i	0.648142	−	0.761520i	$-0.275546\pi$
$558$	−30663.9			−2.32635
$559$	9967.66			0.754181
$560$			15010.6i			1.13270i
$561$	−1066.60			−0.0802708
$562$	11007.7			0.826217
$563$	7131.14			0.533822			0.266911	−	0.963721i	$-0.413997\pi$
$563$	7131.14			0.533822			0.266911	+	0.963721i	$0.413997\pi$
$564$			3169.01i			0.236595i
$565$		−	18968.4i		−	1.41240i
$566$			9083.50i			0.674572i
$567$		−	7596.58i		−	0.562657i
$568$	−32321.9			−2.38767
$569$		−	24027.7i		−	1.77029i	−0.465314	−	0.885145i	$-0.654059\pi$
$569$		−	24027.7i		−	1.77029i	0.465314	−	0.885145i	$-0.345941\pi$
$570$			12202.1i			0.896648i
$571$		−	4919.44i		−	0.360547i	−0.983617	−	0.180273i	$-0.942302\pi$
$571$		−	4919.44i		−	0.360547i	0.983617	−	0.180273i	$-0.0576982\pi$
$572$	981.937			0.0717777
$573$			35306.3i			2.57407i
$574$	10881.6			0.791267
$575$	−10949.2			−0.794110
$576$	−12115.5			−0.876411
$577$			4128.25i			0.297853i	0.988848	+	0.148927i	$0.0475819\pi$
$577$			4128.25i			0.297853i	−0.988848	+	0.148927i	$0.952418\pi$
$578$			14904.0i			1.07253i
$579$	20966.0			1.50486
$580$		−	772.888i		−	0.0553317i
$581$	−3287.90			−0.234776
$582$	32854.1			2.33994
$583$	−898.498			−0.0638285
$584$			15151.6i			1.07360i
$585$	17794.7			1.25764
$586$		−	22051.3i		−	1.55449i
$587$	−3913.26			−0.275157			−0.137579	−	0.990491i	$-0.543932\pi$
$587$	−3913.26			−0.275157			−0.137579	+	0.990491i	$0.543932\pi$
$588$		−	23215.0i		−	1.62818i
$589$	−4226.99			−0.295705
$590$	1079.88			0.0753528
$591$	9235.44			0.642801
$592$	29110.6			2.02101
$593$	−14080.1			−0.975043			−0.487522	−	0.873111i	$-0.662099\pi$
$593$	−14080.1			−0.975043			−0.487522	+	0.873111i	$0.662099\pi$
$594$	−242.925			−0.0167800
$595$	−17070.2			−1.17615
$596$	−1696.96			−0.116628
$597$		−	5925.77i		−	0.406240i
$598$		−	19285.1i		−	1.31877i
$599$	9638.90			0.657487			0.328744	−	0.944419i	$-0.393375\pi$
$599$	9638.90			0.657487			0.328744	+	0.944419i	$0.393375\pi$
$600$	−34367.5			−2.33841
$601$			15042.9i			1.02099i	0.859882	+	0.510494i	$0.170537\pi$
$601$			15042.9i			1.02099i	−0.859882	+	0.510494i	$0.829463\pi$
$602$	−16561.0			−1.12122
$603$	20982.5			1.41704
$604$	−33631.3			−2.26562
$605$	−20192.2			−1.35691
$606$			2137.25i			0.143267i
$607$			6015.15i			0.402220i	0.979569	+	0.201110i	$0.0644549\pi$
$607$			6015.15i			0.402220i	−0.979569	+	0.201110i	$0.935545\pi$
$608$	1012.68			0.0675486
$609$		−	295.022i		−	0.0196303i
$610$		−	57940.9i		−	3.84583i
$611$	944.462			0.0625349
$612$			45868.5i			3.02961i
$613$		−	21381.6i		−	1.40880i	−0.709803	−	0.704400i	$-0.751216\pi$
$613$		−	21381.6i		−	1.40880i	0.709803	−	0.704400i	$-0.248784\pi$
$614$		−	39970.1i		−	2.62714i
$615$			20145.1i			1.32086i
$616$	−844.257			−0.0552209
$617$	−20117.5			−1.31264			−0.656320	−	0.754483i	$-0.727888\pi$
$617$	−20117.5			−1.31264			−0.656320	+	0.754483i	$0.727888\pi$
$618$	64543.3			4.20115
$619$	6554.83			0.425623			0.212812	−	0.977093i	$-0.431738\pi$
$619$	6554.83			0.425623			0.212812	+	0.977093i	$0.431738\pi$
$620$		−	50134.5i		−	3.24750i
$621$			3218.18i			0.207957i
$622$	14342.3			0.924558
$623$			7529.30i			0.484197i
$624$		−	22460.9i		−	1.44095i
$625$	−17638.4			−1.12886
$626$		−	36923.4i		−	2.35744i
$627$	−254.654			−0.0162199
$628$		−	18462.0i		−	1.17311i
$629$			33104.9i			2.09854i
$630$	−29565.4			−1.86970
$631$	15300.6			0.965308			0.482654	−	0.875811i	$-0.339673\pi$
$631$	15300.6			0.965308			0.482654	+	0.875811i	$0.339673\pi$
$632$		−	25805.6i		−	1.62420i
$633$			22336.5i			1.40252i
$634$	−4311.97			−0.270111
$635$	−15401.1			−0.962481
$636$			72197.6i			4.50129i
$637$	−6918.78			−0.430348
$638$	23.9128			0.00148388
$639$		−	23622.3i		−	1.46241i
$640$		−	35161.8i		−	2.17171i
$641$	18026.6			1.11078			0.555389	−	0.831591i	$-0.312570\pi$
$641$	18026.6			1.11078			0.555389	+	0.831591i	$0.312570\pi$
$642$		−	22100.0i		−	1.35859i
$643$	881.112			0.0540399			0.0270200	−	0.999635i	$-0.491398\pi$
$643$	881.112			0.0540399			0.0270200	+	0.999635i	$0.491398\pi$
$644$			21613.0i			1.32247i
$645$		−	30659.5i		−	1.87165i
$646$			9373.84i			0.570911i
$647$			8262.95i			0.502087i	0.967976	+	0.251043i	$0.0807737\pi$
$647$			8262.95i			0.502087i	−0.967976	+	0.251043i	$0.919226\pi$
$648$		−	25603.5i		−	1.55216i
$649$			22.5369i			0.00136310i
$650$			19792.9i			1.19437i
$651$		−	19137.0i		−	1.15213i
$652$			63860.7i			3.83585i
$653$	−12517.5			−0.750147			−0.375074	−	0.926995i	$-0.622383\pi$
$653$	−12517.5			−0.750147			−0.375074	+	0.926995i	$0.622383\pi$
$654$		−	22710.3i		−	1.35786i
$655$	36966.1			2.20517
$656$	13608.6			0.809947
$657$	−11073.5			−0.657562
$658$	−1569.19			−0.0929690
$659$	6482.32			0.383180			0.191590	−	0.981475i	$-0.438636\pi$
$659$	6482.32			0.383180			0.191590	+	0.981475i	$0.438636\pi$
$660$		−	3020.34i		−	0.178131i
$661$			10969.5i			0.645485i	0.946487	+	0.322742i	$0.104605\pi$
$661$			10969.5i			0.645485i	−0.946487	+	0.322742i	$0.895395\pi$
$662$			1060.74i			0.0622760i
$663$	25542.8			1.49623
$664$	−11081.5			−0.647661
$665$	−4075.57			−0.237660
$666$			57337.2i			3.33599i
$667$		−	316.789i		−	0.0183900i
$668$			56563.6i			3.27621i
$669$	38407.2			2.21959
$670$			50858.8i			2.93261i
$671$	1209.21			0.0695693
$672$			4584.74i			0.263185i
$673$			9756.00i			0.558791i	0.960176	+	0.279395i	$0.0901340\pi$
$673$			9756.00i			0.558791i	−0.960176	+	0.279395i	$0.909866\pi$
$674$	42700.0			2.44027
$675$		−	3302.93i		−	0.188341i
$676$	12910.7			0.734563
$677$		−	15924.3i		−	0.904017i	−0.892014	−	0.452009i	$-0.850708\pi$
$677$		−	15924.3i		−	0.904017i	0.892014	−	0.452009i	$-0.149292\pi$
$678$		−	47157.5i		−	2.67120i
$679$			10973.5i			0.620211i
$680$	−57533.6			−3.24458
$681$	−39393.4			−2.21668
$682$	1551.14			0.0870913
$683$		−	12104.3i		−	0.678120i	−0.940765	−	0.339060i	$-0.889891\pi$
$683$		−	12104.3i		−	0.678120i	0.940765	−	0.339060i	$-0.110109\pi$
$684$			10951.2i			0.612179i
$685$		−	29326.8i		−	1.63580i
$686$	32957.4			1.83428
$687$	33519.2			1.86148
$688$	−20711.3			−1.14769
$689$	21517.1			1.18975
$690$	−59318.8			−3.27280
$691$			22889.9i			1.26016i	0.776530	+	0.630081i	$0.216978\pi$
$691$			22889.9i			1.26016i	−0.776530	+	0.630081i	$0.783022\pi$
$692$	57866.7			3.17885
$693$		−	617.021i		−	0.0338221i
$694$			54914.1i			3.00362i
$695$			36235.7i			1.97770i
$696$		−	994.342i		−	0.0541529i
$697$			15475.8i			0.841016i
$698$		−	9795.57i		−	0.531186i
$699$			26047.8i			1.40947i
$700$		−	22182.1i		−	1.19772i
$701$			12290.9i			0.662225i	0.943591	+	0.331112i	$0.107424\pi$
$701$			12290.9i			0.662225i	−0.943591	+	0.331112i	$0.892576\pi$
$702$	5817.53			0.312775
$703$			7903.89i			0.424041i
$704$	612.866			0.0328100
$705$		−	2905.07i		−	0.155193i
$706$		−	13437.7i		−	0.716337i
$707$	−713.855			−0.0379735
$708$	1810.92			0.0961278
$709$			14451.0i			0.765472i	0.923858	+	0.382736i	$0.125018\pi$
$709$			14451.0i			0.765472i	−0.923858	+	0.382736i	$0.874982\pi$
$710$	57257.2			3.02651
$711$	18859.9			0.994798
$712$			25376.8i			1.33572i
$713$		−	20549.0i		−	1.07933i
$714$	−42438.5			−2.22440
$715$	−900.152			−0.0470822
$716$		−	8736.48i		−	0.456002i
$717$		−	40061.2i		−	2.08663i
$718$	13899.1			0.722439
$719$			11389.9i			0.590780i	0.955377	+	0.295390i	$0.0954496\pi$
$719$			11389.9i			0.590780i	−0.955377	+	0.295390i	$0.904550\pi$
$720$	−36974.8			−1.91384
$721$			21557.9i			1.11353i
$722$		−	31767.6i		−	1.63749i
$723$	32661.7			1.68009
$724$		−	28137.6i		−	1.44437i
$725$			325.131i			0.0166553i
$726$	−50200.0			−2.56625
$727$	31400.5			1.60190			0.800948	−	0.598734i	$-0.204329\pi$
$727$	31400.5			1.60190			0.800948	+	0.598734i	$0.204329\pi$
$728$	20218.1			1.02930
$729$	25123.9			1.27643
$730$		−	26840.7i		−	1.36085i
$731$		−	23553.1i		−	1.19171i
$732$		−	97164.4i		−	4.90615i
$733$		−	17991.6i		−	0.906594i	−0.891359	−	0.453297i	$-0.850248\pi$
$733$		−	17991.6i		−	0.906594i	0.891359	−	0.453297i	$-0.149752\pi$
$734$	33402.6			1.67972
$735$			21281.5i			1.06800i
$736$			4923.01i			0.246555i
$737$	−1061.41			−0.0530494
$738$			26803.9i			1.33694i
$739$		−	21335.2i		−	1.06201i	−0.847367	−	0.531007i	$-0.821814\pi$
$739$		−	21335.2i		−	1.06201i	0.847367	−	0.531007i	$-0.178186\pi$
$740$	−93744.5			−4.65691
$741$	6098.40			0.302335
$742$	−35750.0			−1.76876
$743$	−17346.2			−0.856486			−0.428243	−	0.903664i	$-0.640867\pi$
$743$	−17346.2			−0.856486			−0.428243	+	0.903664i	$0.640867\pi$
$744$		−	64499.4i		−	3.17831i
$745$	1555.63			0.0765016
$746$	17791.0			0.873155
$747$		−	8098.88i		−	0.396683i
$748$		−	2320.27i		−	0.113419i
$749$	7381.53			0.360101
$750$	−10907.7			−0.531057
$751$	15225.3			0.739785			0.369893	−	0.929074i	$-0.379394\pi$
$751$	15225.3			0.739785			0.369893	+	0.929074i	$0.379394\pi$
$752$	−1962.45			−0.0951638
$753$	−27170.1			−1.31492
$754$	−572.660			−0.0276592
$755$	30830.1			1.48612
$756$	−6519.77			−0.313653
$757$		−	4050.23i		−	0.194463i	−0.995262	−	0.0972313i	$-0.969001\pi$
$757$		−	4050.23i		−	0.194463i	0.995262	−	0.0972313i	$-0.0309987\pi$
$758$	41824.6			2.00414
$759$		−	1237.97i		−	0.0592033i
$760$	−13736.3			−0.655616
$761$		−	2021.92i		−	0.0963133i	−0.998840	−	0.0481567i	$-0.984665\pi$
$761$		−	2021.92i		−	0.0963133i	0.998840	−	0.0481567i	$-0.0153347\pi$
$762$	−38288.9			−1.82029
$763$	7585.37			0.359907
$764$	−76804.9			−3.63705
$765$		−	42048.1i		−	1.98726i
$766$	−32323.8			−1.52468
$767$		−	539.709i		−	0.0254078i
$768$		−	63654.2i		−	2.99078i
$769$	−17308.8			−0.811666			−0.405833	−	0.913947i	$-0.633019\pi$
$769$	−17308.8			−0.811666			−0.405833	+	0.913947i	$0.633019\pi$
$770$	1495.57			0.0699958
$771$	−55607.5			−2.59748
$772$			45609.2i			2.12631i
$773$	−27725.7			−1.29007			−0.645036	−	0.764153i	$-0.723157\pi$
$773$	−27725.7			−1.29007			−0.645036	+	0.764153i	$0.723157\pi$
$774$		−	40793.6i		−	1.89444i
$775$			21090.1i			0.977521i
$776$			36985.0i			1.71093i
$777$	−35783.5			−1.65216
$778$		−	39394.5i		−	1.81538i
$779$			3694.89i			0.169940i
$780$			72330.5i			3.32032i
$781$			1194.94i			0.0547482i
$782$	−45569.7			−2.08385
$783$	95.5624			0.00436159
$784$	14376.2			0.654891
$785$			16924.4i			0.769498i
$786$	91901.8			4.17052
$787$	−37504.4			−1.69871			−0.849356	−	0.527821i	$-0.823009\pi$
$787$	−37504.4			−1.69871			−0.849356	+	0.527821i	$0.823009\pi$
$788$			20090.7i			0.908250i
$789$			21863.6i			0.986522i
$790$			45713.8i			2.05877i
$791$	15750.9			0.708012
$792$		−	2079.61i		−	0.0933026i
$793$	−28957.9			−1.29675
$794$	33207.3			1.48423
$795$		−	66184.3i		−	2.95260i
$796$	12890.8			0.574000
$797$			7766.70i			0.345183i	0.984993	+	0.172591i	$0.0552140\pi$
$797$			7766.70i			0.345183i	−0.984993	+	0.172591i	$0.944786\pi$
$798$	−10132.3			−0.449474
$799$		−	2231.72i		−	0.0988142i
$800$		−	5052.65i		−	0.223298i
$801$	−18546.5			−0.818111
$802$	13103.2			0.576918
$803$	560.156			0.0246170
$804$			85288.0i			3.74114i
$805$		−	19812.8i		−	0.867467i
$806$	−37146.4			−1.62336
$807$	20127.3	+	26936.6i	0.877962	+	1.17499i
$808$	−2405.98			−0.104755
$809$		−	24390.6i		−	1.05998i	−0.848003	−	0.529992i	$-0.822195\pi$
$809$		−	24390.6i		−	1.05998i	0.848003	−	0.529992i	$-0.177805\pi$
$810$			45355.8i			1.96746i
$811$	−3547.50			−0.153600			−0.0767999	−	0.997047i	$-0.524470\pi$
$811$	−3547.50			−0.153600			−0.0767999	+	0.997047i	$0.524470\pi$
$812$	641.787			0.0277368
$813$	4030.04			0.173849
$814$		−	2900.42i		−	0.124889i
$815$		−	58541.7i		−	2.51611i
$816$	−53074.0			−2.27691
$817$		−	5623.37i		−	0.240804i
$818$	56616.3			2.41998
$819$			14776.3i			0.630435i
$820$	−43823.5			−1.86632
$821$	3546.45			0.150758			0.0753788	−	0.997155i	$-0.475983\pi$
$821$	3546.45			0.150758			0.0753788	+	0.997155i	$0.475983\pi$
$822$		−	72909.7i		−	3.09369i
$823$	−46466.1			−1.96805			−0.984027	−	0.178020i	$-0.943031\pi$
$823$	−46466.1			−1.96805			−0.984027	+	0.178020i	$0.943031\pi$
$824$			72658.7i			3.07183i
$825$			1270.57i			0.0536188i
$826$			896.709i			0.0377730i
$827$	−25300.0			−1.06381			−0.531903	−	0.846806i	$-0.678523\pi$
$827$	−25300.0			−1.06381			−0.531903	+	0.846806i	$0.678523\pi$
$828$	−53237.9			−2.23448
$829$			32621.9i			1.36671i	0.730086	+	0.683356i	$0.239480\pi$
$829$			32621.9i			1.36671i	−0.730086	+	0.683356i	$0.760520\pi$
$830$	19630.6			0.820949
$831$	9427.50			0.393546
$832$	−14676.8			−0.611571
$833$			16348.8i			0.680013i
$834$			90086.0i			3.74031i
$835$		−	51852.5i		−	2.14902i
$836$		−	553.971i		−	0.0229181i
$837$	6198.79			0.255988
$838$		−	76601.5i		−	3.15770i
$839$		−	24155.3i		−	0.993962i	−0.867762	−	0.496981i	$-0.834442\pi$
$839$		−	24155.3i		−	0.993962i	0.867762	−	0.496981i	$-0.165558\pi$
$840$		−	62188.9i		−	2.55443i
$841$	24379.6			0.999614
$842$			12028.0i			0.492295i
$843$	−16922.0			−0.691371
$844$	−48590.6			−1.98170
$845$	−11835.4			−0.481833
$846$		−	3865.30i		−	0.157083i
$847$		−	16767.1i		−	0.680194i
$848$	−44709.2			−1.81052
$849$		−	13963.9i		−	0.564476i
$850$	46769.7			1.88728
$851$	−38423.7			−1.54776
$852$	96017.8			3.86093
$853$			2537.26i			0.101846i	0.998703	+	0.0509228i	$0.0162162\pi$
$853$			2537.26i			0.101846i	−0.998703	+	0.0509228i	$0.983784\pi$
$854$	48112.7			1.92785
$855$		−	10039.1i		−	0.401556i
$856$	24878.7			0.993385
$857$		−	46754.8i		−	1.86361i	−0.362958	−	0.931806i	$-0.618233\pi$
$857$		−	46754.8i		−	1.86361i	0.362958	−	0.931806i	$-0.381767\pi$
$858$	−2237.88			−0.0890441
$859$	16319.3			0.648206			0.324103	−	0.946022i	$-0.394938\pi$
$859$	16319.3			0.648206			0.324103	+	0.946022i	$0.394938\pi$
$860$	66696.3			2.64456
$861$	−16728.0			−0.662125
$862$	105.268			0.00415945
$863$	47353.7			1.86783			0.933916	−	0.357491i	$-0.116368\pi$
$863$	47353.7			1.86783			0.933916	+	0.357491i	$0.116368\pi$
$864$	−1485.07			−0.0584760
$865$	−53047.0			−2.08515
$866$			21242.7i			0.833554i
$867$		−	22911.7i		−	0.897487i
$868$	41630.4			1.62791
$869$	−954.034			−0.0372421
$870$			1761.44i			0.0686420i
$871$	25418.4			0.988828
$872$	25565.8			0.992851
$873$	−27030.3			−1.04792
$874$	−10879.9			−0.421073
$875$		−	3643.24i		−	0.140759i
$876$		−	45010.6i		−	1.73604i
$877$	−10575.0			−0.407176			−0.203588	−	0.979057i	$-0.565260\pi$
$877$	−10575.0			−0.407176			−0.203588	+	0.979057i	$0.565260\pi$
$878$			60534.7i			2.32682i
$879$			33899.1i			1.30078i
$880$	1870.38			0.0716483
$881$			17895.8i			0.684363i	0.939634	+	0.342181i	$0.111166\pi$
$881$			17895.8i			0.684363i	−0.939634	+	0.342181i	$0.888834\pi$
$882$			28315.8i			1.08100i
$883$			27973.4i			1.06612i	0.846078	+	0.533059i	$0.178957\pi$
$883$			27973.4i			1.06612i	−0.846078	+	0.533059i	$0.821043\pi$
$884$			55565.5i			2.11410i
$885$	−1660.09			−0.0630545
$886$	58465.6			2.21692
$887$	−45094.1			−1.70700			−0.853501	−	0.521091i	$-0.825525\pi$
$887$	−45094.1			−1.70700			−0.853501	+	0.521091i	$0.825525\pi$
$888$	−120605.			−4.55770
$889$		−	12788.7i		−	0.482475i
$890$		−	44954.1i		−	1.69311i
$891$	−946.563			−0.0355904
$892$			83550.6i			3.13619i
$893$		−	532.829i		−	0.0199669i
$894$	3867.46			0.144684
$895$			8008.83i			0.299112i
$896$	29197.5			1.08864
$897$			29646.6i			1.10354i
$898$		−	7262.99i		−	0.269899i
$899$	−610.191			−0.0226374
$900$	54639.9			2.02370
$901$		−	50843.9i		−	1.87997i
$902$		−	1355.88i		−	0.0500509i
$903$	25458.9			0.938227
$904$	53086.9			1.95314
$905$			25794.0i			0.947429i
$906$	76647.1			2.81063
$907$	11487.3			0.420540			0.210270	−	0.977643i	$-0.432566\pi$
$907$	11487.3			0.420540			0.210270	+	0.977643i	$0.432566\pi$
$908$		−	85695.9i		−	3.13207i
$909$		−	1758.40i		−	0.0641610i
$910$	−35815.8			−1.30470
$911$		−	47182.7i		−	1.71595i	−0.513688	−	0.857977i	$-0.671721\pi$
$911$		−	47182.7i		−	1.71595i	0.513688	−	0.857977i	$-0.328279\pi$
$912$	−12671.6			−0.460085
$913$			409.684i			0.0148506i
$914$			35286.7i			1.27700i
$915$			89071.6i			3.21816i
$916$			72917.3i			2.63019i
$917$			30695.7i			1.10541i
$918$		−	13746.5i		−	0.494230i
$919$		−	13554.8i		−	0.486542i	−0.969958	−	0.243271i	$-0.921780\pi$
$919$		−	13554.8i		−	0.486542i	0.969958	−	0.243271i	$-0.0782204\pi$
$920$		−	66777.3i		−	2.39302i
$921$			61445.4i			2.19836i
$922$	−49127.5			−1.75480
$923$		−	28616.2i		−	1.02049i
$924$	2508.01			0.0892939
$925$	39435.6			1.40177
$926$	58355.0			2.07091
$927$	−53102.2			−1.88145
$928$	146.186			0.00517112
$929$		−	20410.2i		−	0.720813i	−0.932795	−	0.360407i	$-0.882638\pi$
$929$		−	20410.2i		−	0.720813i	0.932795	−	0.360407i	$-0.117362\pi$
$930$			114259.i			4.02870i
$931$			3903.31i			0.137407i
$932$	−56664.0			−1.99151
$933$	−22048.2			−0.773662
$934$	−40618.6			−1.42300
$935$			2127.02i			0.0743967i
$936$			49802.1i			1.73914i
$937$			18184.7i			0.634011i	0.948424	+	0.317005i	$0.102677\pi$
$937$			18184.7i			0.634011i	−0.948424	+	0.317005i	$0.897323\pi$
$938$	−42231.9			−1.47006
$939$			56761.8i			1.97268i
$940$	6319.65			0.219281
$941$		−	43923.7i		−	1.52165i	−0.648957	−	0.760825i	$-0.724795\pi$
$941$		−	43923.7i		−	1.52165i	0.648957	−	0.760825i	$-0.275205\pi$
$942$			42075.8i			1.45531i
$943$	−17962.2			−0.620288
$944$			1121.43i			0.0386648i
$945$	5976.74			0.205739
$946$			2063.56i			0.0709218i
$947$		−	42431.9i		−	1.45602i	−0.685567	−	0.728010i	$-0.740446\pi$
$947$		−	42431.9i		−	1.45602i	0.685567	−	0.728010i	$-0.259554\pi$
$948$			76660.1i			2.62638i
$949$	−13414.5			−0.458855
$950$	11166.4			0.381354
$951$	6628.72			0.226026
$952$		−	47774.5i		−	1.62645i
$953$			27182.9i			0.923968i	0.886888	+	0.461984i	$0.152862\pi$
$953$			27182.9i			0.923968i	−0.886888	+	0.461984i	$0.847138\pi$
$954$		−	88060.8i		−	2.98855i
$955$	70407.8			2.38570
$956$	87148.6			2.94831
$957$	−36.7608			−0.00124170
$958$	3141.88			0.105960
$959$	24352.3			0.819995
$960$			45144.4i			1.51774i
$961$	−9789.95			−0.328621
$962$			69458.7i			2.32790i
$963$			18182.5i			0.608435i
$964$			71052.0i			2.37389i
$965$		−	41810.4i		−	1.39474i
$966$		−	49256.9i		−	1.64060i
$967$		−	18215.0i		−	0.605743i	−0.953031	−	0.302871i	$-0.902055\pi$
$967$		−	18215.0i		−	0.605743i	0.953031	−	0.302871i	$-0.0979453\pi$
$968$		−	56511.8i		−	1.87640i
$969$		−	14410.2i		−	0.477734i
$970$		−	65517.8i		−	2.16871i
$971$	10719.6			0.354281			0.177141	−	0.984186i	$-0.443315\pi$
$971$	10719.6			0.354281			0.177141	+	0.984186i	$0.443315\pi$
$972$			90007.6i			2.97016i
$973$	−30089.3			−0.991384
$974$			6156.72i			0.202540i
$975$		−	30427.3i		−	0.999440i
$976$	60170.2			1.97336
$977$	−10526.5			−0.344701			−0.172350	−	0.985036i	$-0.555136\pi$
$977$	−10526.5			−0.344701			−0.172350	+	0.985036i	$0.555136\pi$
$978$		−	145541.i		−	4.75858i
$979$	938.179			0.0306275
$980$	−46295.4			−1.50903
$981$			18684.6i			0.608107i
$982$			21232.5i			0.689977i
$983$	58312.0			1.89203			0.946015	−	0.324124i	$-0.105070\pi$
$983$	58312.0			1.89203			0.946015	+	0.324124i	$0.105070\pi$
$984$	−56380.2			−1.82656
$985$		−	18417.3i		−	0.595762i
$986$			1353.17i			0.0437056i
$987$	2412.30			0.0777956
$988$			13266.4i			0.427187i
$989$	27337.3			0.878943
$990$			3683.96i			0.118267i
$991$			3595.96i			0.115267i	0.998338	+	0.0576335i	$0.0183555\pi$
$991$			3595.96i			0.115267i	−0.998338	+	0.0576335i	$0.981645\pi$
$992$	9482.59			0.303501
$993$		−	1630.65i		−	0.0521120i
$994$			47545.0i			1.51714i
$995$	−11817.2			−0.376512
$996$	32919.6			1.04729
$997$	−6659.93			−0.211557			−0.105778	−	0.994390i	$-0.533733\pi$
$997$	−6659.93			−0.211557			−0.105778	+	0.994390i	$0.533733\pi$
$998$	−43307.0			−1.37361
$999$		−	11590.9i		−	0.367087i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	269.4.b.a.268.6	✓	66
269.268	even	2	inner	269.4.b.a.268.61	yes	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.4.b.a.268.6	✓	66	1.1	even	1	trivial
269.4.b.a.268.61	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)

\(f(q)\)	\(=\)	\(q-4.95781i q^{2} +7.62155i q^{3} -16.5798 q^{4} +15.1989 q^{5} +37.7862 q^{6} +12.6208i q^{7} +42.5372i q^{8} -31.0881 q^{9} -75.3533i q^{10} +1.57260 q^{11} -126.364i q^{12} -37.6604 q^{13} +62.5715 q^{14} +115.839i q^{15} +78.2526 q^{16} +88.9897i q^{17} +154.129i q^{18} +21.2465i q^{19} -251.996 q^{20} -96.1902 q^{21} -7.79665i q^{22} -103.287 q^{23} -324.200 q^{24} +106.007 q^{25} +186.713i q^{26} -31.1576i q^{27} -209.251i q^{28} +3.06706i q^{29} +574.309 q^{30} +198.950i q^{31} -47.6633i q^{32} +11.9857i q^{33} +441.194 q^{34} +191.823i q^{35} +515.436 q^{36} +372.008 q^{37} +105.336 q^{38} -287.031i q^{39} +646.520i q^{40} +173.906 q^{41} +476.892i q^{42} -264.672 q^{43} -26.0735 q^{44} -472.506 q^{45} +512.078i q^{46} -25.0784 q^{47} +596.406i q^{48} +183.715 q^{49} -525.564i q^{50} -678.240 q^{51} +624.403 q^{52} -571.345 q^{53} -154.473 q^{54} +23.9018 q^{55} -536.854 q^{56} -161.932 q^{57} +15.2059 q^{58} +14.3309i q^{59} -1920.60i q^{60} +768.923 q^{61} +986.354 q^{62} -392.357i q^{63} +389.715 q^{64} -572.397 q^{65} +59.4226 q^{66} -674.938 q^{67} -1475.44i q^{68} -787.209i q^{69} +951.020 q^{70} +759.850i q^{71} -1322.40i q^{72} +356.197 q^{73} -1844.34i q^{74} +807.940i q^{75} -352.264i q^{76} +19.8475i q^{77} -1423.04 q^{78} -606.660 q^{79} +1189.35 q^{80} -601.909 q^{81} -862.191i q^{82} +260.514i q^{83} +1594.82 q^{84} +1352.55i q^{85} +1312.19i q^{86} -23.3758 q^{87} +66.8941i q^{88} +596.578 q^{89} +2342.59i q^{90} -475.304i q^{91} +1712.49 q^{92} -1516.31 q^{93} +124.334i q^{94} +322.924i q^{95} +363.268 q^{96} +869.474 q^{97} -910.824i q^{98} -48.8892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) \) 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

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