LMFDB - Embedded newform 187.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [187,2,Mod(1,187)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(187, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("187.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 187 = 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	187.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$1.49320251780$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	187.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.73205 q^{3} -1.46410 q^{4} -0.267949 q^{5} -1.26795 q^{6} -2.00000 q^{7} -2.53590 q^{8} +O(q^{10})$	\(q+0.732051 q^{2} -1.73205 q^{3} -1.46410 q^{4} -0.267949 q^{5} -1.26795 q^{6} -2.00000 q^{7} -2.53590 q^{8} -0.196152 q^{10} +1.00000 q^{11} +2.53590 q^{12} -6.73205 q^{13} -1.46410 q^{14} +0.464102 q^{15} +1.07180 q^{16} +1.00000 q^{17} +4.19615 q^{19} +0.392305 q^{20} +3.46410 q^{21} +0.732051 q^{22} -0.267949 q^{23} +4.39230 q^{24} -4.92820 q^{25} -4.92820 q^{26} +5.19615 q^{27} +2.92820 q^{28} -4.73205 q^{29} +0.339746 q^{30} +5.73205 q^{31} +5.85641 q^{32} -1.73205 q^{33} +0.732051 q^{34} +0.535898 q^{35} -0.267949 q^{37} +3.07180 q^{38} +11.6603 q^{39} +0.679492 q^{40} +2.53590 q^{41} +2.53590 q^{42} -2.00000 q^{43} -1.46410 q^{44} -0.196152 q^{46} -12.9282 q^{47} -1.85641 q^{48} -3.00000 q^{49} -3.60770 q^{50} -1.73205 q^{51} +9.85641 q^{52} -9.46410 q^{53} +3.80385 q^{54} -0.267949 q^{55} +5.07180 q^{56} -7.26795 q^{57} -3.46410 q^{58} +3.00000 q^{59} -0.679492 q^{60} -10.9282 q^{61} +4.19615 q^{62} +2.14359 q^{64} +1.80385 q^{65} -1.26795 q^{66} +1.00000 q^{67} -1.46410 q^{68} +0.464102 q^{69} +0.392305 q^{70} +3.73205 q^{71} +14.1962 q^{73} -0.196152 q^{74} +8.53590 q^{75} -6.14359 q^{76} -2.00000 q^{77} +8.53590 q^{78} -8.19615 q^{79} -0.287187 q^{80} -9.00000 q^{81} +1.85641 q^{82} -11.8564 q^{83} -5.07180 q^{84} -0.267949 q^{85} -1.46410 q^{86} +8.19615 q^{87} -2.53590 q^{88} +1.92820 q^{89} +13.4641 q^{91} +0.392305 q^{92} -9.92820 q^{93} -9.46410 q^{94} -1.12436 q^{95} -10.1436 q^{96} +12.2679 q^{97} -2.19615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^6 - 4 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8} + 10 q^{10} + 2 q^{11} + 12 q^{12} - 10 q^{13} + 4 q^{14} - 6 q^{15} + 16 q^{16} + 2 q^{17} - 2 q^{19} - 20 q^{20} - 2 q^{22} - 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} + 18 q^{30} + 8 q^{31} - 16 q^{32} - 2 q^{34} + 8 q^{35} - 4 q^{37} + 20 q^{38} + 6 q^{39} + 36 q^{40} + 12 q^{41} + 12 q^{42} - 4 q^{43} + 4 q^{44} + 10 q^{46} - 12 q^{47} + 24 q^{48} - 6 q^{49} - 28 q^{50} - 8 q^{52} - 12 q^{53} + 18 q^{54} - 4 q^{55} + 24 q^{56} - 18 q^{57} + 6 q^{59} - 36 q^{60} - 8 q^{61} - 2 q^{62} + 32 q^{64} + 14 q^{65} - 6 q^{66} + 2 q^{67} + 4 q^{68} - 6 q^{69} - 20 q^{70} + 4 q^{71} + 18 q^{73} + 10 q^{74} + 24 q^{75} - 40 q^{76} - 4 q^{77} + 24 q^{78} - 6 q^{79} - 56 q^{80} - 18 q^{81} - 24 q^{82} + 4 q^{83} - 24 q^{84} - 4 q^{85} + 4 q^{86} + 6 q^{87} - 12 q^{88} - 10 q^{89} + 20 q^{91} - 20 q^{92} - 6 q^{93} - 12 q^{94} + 22 q^{95} - 48 q^{96} + 28 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^6 - 4 * q^7 - 12 * q^8 + 10 * q^10 + 2 * q^11 + 12 * q^12 - 10 * q^13 + 4 * q^14 - 6 * q^15 + 16 * q^16 + 2 * q^17 - 2 * q^19 - 20 * q^20 - 2 * q^22 - 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 8 * q^28 - 6 * q^29 + 18 * q^30 + 8 * q^31 - 16 * q^32 - 2 * q^34 + 8 * q^35 - 4 * q^37 + 20 * q^38 + 6 * q^39 + 36 * q^40 + 12 * q^41 + 12 * q^42 - 4 * q^43 + 4 * q^44 + 10 * q^46 - 12 * q^47 + 24 * q^48 - 6 * q^49 - 28 * q^50 - 8 * q^52 - 12 * q^53 + 18 * q^54 - 4 * q^55 + 24 * q^56 - 18 * q^57 + 6 * q^59 - 36 * q^60 - 8 * q^61 - 2 * q^62 + 32 * q^64 + 14 * q^65 - 6 * q^66 + 2 * q^67 + 4 * q^68 - 6 * q^69 - 20 * q^70 + 4 * q^71 + 18 * q^73 + 10 * q^74 + 24 * q^75 - 40 * q^76 - 4 * q^77 + 24 * q^78 - 6 * q^79 - 56 * q^80 - 18 * q^81 - 24 * q^82 + 4 * q^83 - 24 * q^84 - 4 * q^85 + 4 * q^86 + 6 * q^87 - 12 * q^88 - 10 * q^89 + 20 * q^91 - 20 * q^92 - 6 * q^93 - 12 * q^94 + 22 * q^95 - 48 * q^96 + 28 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	−1.73205	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.46410	−0.732051
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	−1.26795	−0.517638
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−2.53590	−0.896575
$9$	0	0
$10$	−0.196152	−0.0620288
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.53590	0.732051
$13$	−6.73205	−1.86713	−0.933567	−	0.358402i	$-0.883322\pi$
$13$	−6.73205	−1.86713	−0.933567	+	0.358402i	$0.883322\pi$
$14$	−1.46410	−0.391298
$15$	0.464102	0.119831
$16$	1.07180	0.267949
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	0.392305	0.0877220
$21$	3.46410	0.755929
$22$	0.732051	0.156074
$23$	−0.267949	−0.0558713	−0.0279356	−	0.999610i	$-0.508893\pi$
$23$	−0.267949	−0.0558713	−0.0279356	+	0.999610i	$0.508893\pi$
$24$	4.39230	0.896575
$25$	−4.92820	−0.985641
$26$	−4.92820	−0.966500
$27$	5.19615	1.00000
$28$	2.92820	0.553378
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	0.339746	0.0620288
$31$	5.73205	1.02951	0.514753	−	0.857338i	$-0.327883\pi$
$31$	5.73205	1.02951	0.514753	+	0.857338i	$0.327883\pi$
$32$	5.85641	1.03528
$33$	−1.73205	−0.301511
$34$	0.732051	0.125546
$35$	0.535898	0.0905834
$36$	0	0
$37$	−0.267949	−0.0440506	−0.0220253	−	0.999757i	$-0.507011\pi$
$37$	−0.267949	−0.0440506	−0.0220253	+	0.999757i	$0.507011\pi$
$38$	3.07180	0.498311
$39$	11.6603	1.86713
$40$	0.679492	0.107437
$41$	2.53590	0.396041	0.198020	−	0.980198i	$-0.436549\pi$
$41$	2.53590	0.396041	0.198020	+	0.980198i	$0.436549\pi$
$42$	2.53590	0.391298
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−1.46410	−0.220722
$45$	0	0
$46$	−0.196152	−0.0289211
$47$	−12.9282	−1.88577	−0.942886	−	0.333115i	$-0.891900\pi$
$47$	−12.9282	−1.88577	−0.942886	+	0.333115i	$0.891900\pi$
$48$	−1.85641	−0.267949
$49$	−3.00000	−0.428571
$50$	−3.60770	−0.510205
$51$	−1.73205	−0.242536
$52$	9.85641	1.36684
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	3.80385	0.517638
$55$	−0.267949	−0.0361303
$56$	5.07180	0.677747
$57$	−7.26795	−0.962663
$58$	−3.46410	−0.454859
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−0.679492	−0.0877220
$61$	−10.9282	−1.39921	−0.699607	−	0.714528i	$-0.746641\pi$
$61$	−10.9282	−1.39921	−0.699607	+	0.714528i	$0.746641\pi$
$62$	4.19615	0.532912
$63$	0	0
$64$	2.14359	0.267949
$65$	1.80385	0.223740
$66$	−1.26795	−0.156074
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	−1.46410	−0.177548
$69$	0.464102	0.0558713
$70$	0.392305	0.0468894
$71$	3.73205	0.442913	0.221456	−	0.975170i	$-0.428919\pi$
$71$	3.73205	0.442913	0.221456	+	0.975170i	$0.428919\pi$
$72$	0	0
$73$	14.1962	1.66153	0.830767	−	0.556620i	$-0.187902\pi$
$73$	14.1962	1.66153	0.830767	+	0.556620i	$0.187902\pi$
$74$	−0.196152	−0.0228023
$75$	8.53590	0.985641
$76$	−6.14359	−0.704719
$77$	−2.00000	−0.227921
$78$	8.53590	0.966500
$79$	−8.19615	−0.922139	−0.461070	−	0.887364i	$-0.652534\pi$
$79$	−8.19615	−0.922139	−0.461070	+	0.887364i	$0.652534\pi$
$80$	−0.287187	−0.0321085
$81$	−9.00000	−1.00000
$82$	1.85641	0.205006
$83$	−11.8564	−1.30141	−0.650705	−	0.759331i	$-0.725526\pi$
$83$	−11.8564	−1.30141	−0.650705	+	0.759331i	$0.725526\pi$
$84$	−5.07180	−0.553378
$85$	−0.267949	−0.0290632
$86$	−1.46410	−0.157878
$87$	8.19615	0.878720
$88$	−2.53590	−0.270328
$89$	1.92820	0.204389	0.102195	−	0.994764i	$-0.467414\pi$
$89$	1.92820	0.204389	0.102195	+	0.994764i	$0.467414\pi$
$90$	0	0
$91$	13.4641	1.41142
$92$	0.392305	0.0409006
$93$	−9.92820	−1.02951
$94$	−9.46410	−0.976148
$95$	−1.12436	−0.115356
$96$	−10.1436	−1.03528
$97$	12.2679	1.24562	0.622811	−	0.782373i	$-0.285991\pi$
$97$	12.2679	1.24562	0.622811	+	0.782373i	$0.285991\pi$
$98$	−2.19615	−0.221845
$99$	0	0
$100$	7.21539	0.721539
$101$	−15.6603	−1.55825	−0.779127	−	0.626866i	$-0.784337\pi$
$101$	−15.6603	−1.55825	−0.779127	+	0.626866i	$0.784337\pi$
$102$	−1.26795	−0.125546
$103$	15.8564	1.56238	0.781189	−	0.624295i	$-0.214613\pi$
$103$	15.8564	1.56238	0.781189	+	0.624295i	$0.214613\pi$
$104$	17.0718	1.67403
$105$	−0.928203	−0.0905834
$106$	−6.92820	−0.672927
$107$	−14.7321	−1.42420	−0.712101	−	0.702077i	$-0.752256\pi$
$107$	−14.7321	−1.42420	−0.712101	+	0.702077i	$0.752256\pi$
$108$	−7.60770	−0.732051
$109$	8.19615	0.785049	0.392525	−	0.919742i	$-0.371602\pi$
$109$	8.19615	0.785049	0.392525	+	0.919742i	$0.371602\pi$
$110$	−0.196152	−0.0187024
$111$	0.464102	0.0440506
$112$	−2.14359	−0.202551
$113$	−9.73205	−0.915514	−0.457757	−	0.889077i	$-0.651347\pi$
$113$	−9.73205	−0.915514	−0.457757	+	0.889077i	$0.651347\pi$
$114$	−5.32051	−0.498311
$115$	0.0717968	0.00669508
$116$	6.92820	0.643268
$117$	0	0
$118$	2.19615	0.202172
$119$	−2.00000	−0.183340
$120$	−1.17691	−0.107437
$121$	1.00000	0.0909091
$122$	−8.00000	−0.724286
$123$	−4.39230	−0.396041
$124$	−8.39230	−0.753651
$125$	2.66025	0.237940
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−10.1436	−0.896575
$129$	3.46410	0.304997
$130$	1.32051	0.115816
$131$	4.92820	0.430579	0.215290	−	0.976550i	$-0.430930\pi$
$131$	4.92820	0.430579	0.215290	+	0.976550i	$0.430930\pi$
$132$	2.53590	0.220722
$133$	−8.39230	−0.727705
$134$	0.732051	0.0632396
$135$	−1.39230	−0.119831
$136$	−2.53590	−0.217451
$137$	1.39230	0.118953	0.0594763	−	0.998230i	$-0.481057\pi$
$137$	1.39230	0.118953	0.0594763	+	0.998230i	$0.481057\pi$
$138$	0.339746	0.0289211
$139$	−1.66025	−0.140821	−0.0704105	−	0.997518i	$-0.522431\pi$
$139$	−1.66025	−0.140821	−0.0704105	+	0.997518i	$0.522431\pi$
$140$	−0.784610	−0.0663116
$141$	22.3923	1.88577
$142$	2.73205	0.229269
$143$	−6.73205	−0.562962
$144$	0	0
$145$	1.26795	0.105297
$146$	10.3923	0.860073
$147$	5.19615	0.428571
$148$	0.392305	0.0322473
$149$	5.12436	0.419804	0.209902	−	0.977722i	$-0.432686\pi$
$149$	5.12436	0.419804	0.209902	+	0.977722i	$0.432686\pi$
$150$	6.24871	0.510205
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	−10.6410	−0.863100
$153$	0	0
$154$	−1.46410	−0.117981
$155$	−1.53590	−0.123366
$156$	−17.0718	−1.36684
$157$	21.7846	1.73860	0.869301	−	0.494284i	$-0.164570\pi$
$157$	21.7846	1.73860	0.869301	+	0.494284i	$0.164570\pi$
$158$	−6.00000	−0.477334
$159$	16.3923	1.29999
$160$	−1.56922	−0.124058
$161$	0.535898	0.0422347
$162$	−6.58846	−0.517638
$163$	−9.46410	−0.741286	−0.370643	−	0.928775i	$-0.620863\pi$
$163$	−9.46410	−0.741286	−0.370643	+	0.928775i	$0.620863\pi$
$164$	−3.71281	−0.289922
$165$	0.464102	0.0361303
$166$	−8.67949	−0.673659
$167$	15.2679	1.18147	0.590735	−	0.806866i	$-0.298838\pi$
$167$	15.2679	1.18147	0.590735	+	0.806866i	$0.298838\pi$
$168$	−8.78461	−0.677747
$169$	32.3205	2.48619
$170$	−0.196152	−0.0150442
$171$	0	0
$172$	2.92820	0.223273
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	6.00000	0.454859
$175$	9.85641	0.745074
$176$	1.07180	0.0807897
$177$	−5.19615	−0.390567
$178$	1.41154	0.105800
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	−4.66025	−0.346394	−0.173197	−	0.984887i	$-0.555410\pi$
$181$	−4.66025	−0.346394	−0.173197	+	0.984887i	$0.555410\pi$
$182$	9.85641	0.730605
$183$	18.9282	1.39921
$184$	0.679492	0.0500928
$185$	0.0717968	0.00527860
$186$	−7.26795	−0.532912
$187$	1.00000	0.0731272
$188$	18.9282	1.38048
$189$	−10.3923	−0.755929
$190$	−0.823085	−0.0597129
$191$	19.5359	1.41357	0.706784	−	0.707429i	$-0.250145\pi$
$191$	19.5359	1.41357	0.706784	+	0.707429i	$0.250145\pi$
$192$	−3.71281	−0.267949
$193$	−13.8038	−0.993623	−0.496811	−	0.867859i	$-0.665496\pi$
$193$	−13.8038	−0.993623	−0.496811	+	0.867859i	$0.665496\pi$
$194$	8.98076	0.644781
$195$	−3.12436	−0.223740
$196$	4.39230	0.313736
$197$	3.12436	0.222601	0.111301	−	0.993787i	$-0.464498\pi$
$197$	3.12436	0.222601	0.111301	+	0.993787i	$0.464498\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	12.4974	0.883701
$201$	−1.73205	−0.122169
$202$	−11.4641	−0.806611
$203$	9.46410	0.664250
$204$	2.53590	0.177548
$205$	−0.679492	−0.0474578
$206$	11.6077	0.808746
$207$	0	0
$208$	−7.21539	−0.500297
$209$	4.19615	0.290254
$210$	−0.679492	−0.0468894
$211$	5.46410	0.376164	0.188082	−	0.982153i	$-0.439773\pi$
$211$	5.46410	0.376164	0.188082	+	0.982153i	$0.439773\pi$
$212$	13.8564	0.951662
$213$	−6.46410	−0.442913
$214$	−10.7846	−0.737221
$215$	0.535898	0.0365480
$216$	−13.1769	−0.896575
$217$	−11.4641	−0.778234
$218$	6.00000	0.406371
$219$	−24.5885	−1.66153
$220$	0.392305	0.0264492
$221$	−6.73205	−0.452847
$222$	0.339746	0.0228023
$223$	−14.3205	−0.958972	−0.479486	−	0.877549i	$-0.659177\pi$
$223$	−14.3205	−0.958972	−0.479486	+	0.877549i	$0.659177\pi$
$224$	−11.7128	−0.782595
$225$	0	0
$226$	−7.12436	−0.473905
$227$	13.2679	0.880625	0.440312	−	0.897845i	$-0.354868\pi$
$227$	13.2679	0.880625	0.440312	+	0.897845i	$0.354868\pi$
$228$	10.6410	0.704719
$229$	−5.53590	−0.365822	−0.182911	−	0.983129i	$-0.558552\pi$
$229$	−5.53590	−0.365822	−0.182911	+	0.983129i	$0.558552\pi$
$230$	0.0525589	0.00346563
$231$	3.46410	0.227921
$232$	12.0000	0.787839
$233$	−18.9282	−1.24003	−0.620014	−	0.784591i	$-0.712873\pi$
$233$	−18.9282	−1.24003	−0.620014	+	0.784591i	$0.712873\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	−4.39230	−0.285915
$237$	14.1962	0.922139
$238$	−1.46410	−0.0949036
$239$	−6.19615	−0.400796	−0.200398	−	0.979715i	$-0.564223\pi$
$239$	−6.19615	−0.400796	−0.200398	+	0.979715i	$0.564223\pi$
$240$	0.497423	0.0321085
$241$	5.07180	0.326703	0.163352	−	0.986568i	$-0.447770\pi$
$241$	5.07180	0.326703	0.163352	+	0.986568i	$0.447770\pi$
$242$	0.732051	0.0470580
$243$	0	0
$244$	16.0000	1.02430
$245$	0.803848	0.0513559
$246$	−3.21539	−0.205006
$247$	−28.2487	−1.79742
$248$	−14.5359	−0.923030
$249$	20.5359	1.30141
$250$	1.94744	0.123167
$251$	8.32051	0.525186	0.262593	−	0.964907i	$-0.415422\pi$
$251$	8.32051	0.525186	0.262593	+	0.964907i	$0.415422\pi$
$252$	0	0
$253$	−0.267949	−0.0168458
$254$	−2.92820	−0.183732
$255$	0.464102	0.0290632
$256$	−11.7128	−0.732051
$257$	−20.7846	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$257$	−20.7846	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$258$	2.53590	0.157878
$259$	0.535898	0.0332991
$260$	−2.64102	−0.163789
$261$	0	0
$262$	3.60770	0.222884
$263$	−13.1244	−0.809282	−0.404641	−	0.914476i	$-0.632604\pi$
$263$	−13.1244	−0.809282	−0.404641	+	0.914476i	$0.632604\pi$
$264$	4.39230	0.270328
$265$	2.53590	0.155779
$266$	−6.14359	−0.376688
$267$	−3.33975	−0.204389
$268$	−1.46410	−0.0894342
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−1.01924	−0.0620288
$271$	−22.9282	−1.39279	−0.696395	−	0.717659i	$-0.745214\pi$
$271$	−22.9282	−1.39279	−0.696395	+	0.717659i	$0.745214\pi$
$272$	1.07180	0.0649872
$273$	−23.3205	−1.41142
$274$	1.01924	0.0615744
$275$	−4.92820	−0.297182
$276$	−0.679492	−0.0409006
$277$	−2.39230	−0.143740	−0.0718698	−	0.997414i	$-0.522897\pi$
$277$	−2.39230	−0.143740	−0.0718698	+	0.997414i	$0.522897\pi$
$278$	−1.21539	−0.0728943
$279$	0	0
$280$	−1.35898	−0.0812148
$281$	−2.87564	−0.171547	−0.0857733	−	0.996315i	$-0.527336\pi$
$281$	−2.87564	−0.171547	−0.0857733	+	0.996315i	$0.527336\pi$
$282$	16.3923	0.976148
$283$	0.732051	0.0435159	0.0217580	−	0.999763i	$-0.493074\pi$
$283$	0.732051	0.0435159	0.0217580	+	0.999763i	$0.493074\pi$
$284$	−5.46410	−0.324235
$285$	1.94744	0.115356
$286$	−4.92820	−0.291411
$287$	−5.07180	−0.299379
$288$	0	0
$289$	1.00000	0.0588235
$290$	0.928203	0.0545060
$291$	−21.2487	−1.24562
$292$	−20.7846	−1.21633
$293$	13.8564	0.809500	0.404750	−	0.914427i	$-0.367359\pi$
$293$	13.8564	0.809500	0.404750	+	0.914427i	$0.367359\pi$
$294$	3.80385	0.221845
$295$	−0.803848	−0.0468018
$296$	0.679492	0.0394947
$297$	5.19615	0.301511
$298$	3.75129	0.217306
$299$	1.80385	0.104319
$300$	−12.4974	−0.721539
$301$	4.00000	0.230556
$302$	−1.32051	−0.0759867
$303$	27.1244	1.55825
$304$	4.49742	0.257945
$305$	2.92820	0.167668
$306$	0	0
$307$	−20.1962	−1.15266	−0.576328	−	0.817219i	$-0.695515\pi$
$307$	−20.1962	−1.15266	−0.576328	+	0.817219i	$0.695515\pi$
$308$	2.92820	0.166850
$309$	−27.4641	−1.56238
$310$	−1.12436	−0.0638591
$311$	20.3923	1.15634	0.578171	−	0.815916i	$-0.303767\pi$
$311$	20.3923	1.15634	0.578171	+	0.815916i	$0.303767\pi$
$312$	−29.5692	−1.67403
$313$	19.1962	1.08503	0.542515	−	0.840046i	$-0.317472\pi$
$313$	19.1962	1.08503	0.542515	+	0.840046i	$0.317472\pi$
$314$	15.9474	0.899966
$315$	0	0
$316$	12.0000	0.675053
$317$	−33.4449	−1.87845	−0.939225	−	0.343301i	$-0.888455\pi$
$317$	−33.4449	−1.87845	−0.939225	+	0.343301i	$0.888455\pi$
$318$	12.0000	0.672927
$319$	−4.73205	−0.264944
$320$	−0.574374	−0.0321085
$321$	25.5167	1.42420
$322$	0.392305	0.0218623
$323$	4.19615	0.233480
$324$	13.1769	0.732051
$325$	33.1769	1.84032
$326$	−6.92820	−0.383718
$327$	−14.1962	−0.785049
$328$	−6.43078	−0.355080
$329$	25.8564	1.42551
$330$	0.339746	0.0187024
$331$	−3.14359	−0.172788	−0.0863938	−	0.996261i	$-0.527534\pi$
$331$	−3.14359	−0.172788	−0.0863938	+	0.996261i	$0.527534\pi$
$332$	17.3590	0.952698
$333$	0	0
$334$	11.1769	0.611574
$335$	−0.267949	−0.0146396
$336$	3.71281	0.202551
$337$	−30.3923	−1.65557	−0.827787	−	0.561042i	$-0.810401\pi$
$337$	−30.3923	−1.65557	−0.827787	+	0.561042i	$0.810401\pi$
$338$	23.6603	1.28695
$339$	16.8564	0.915514
$340$	0.392305	0.0212757
$341$	5.73205	0.310408
$342$	0	0
$343$	20.0000	1.07990
$344$	5.07180	0.273453
$345$	−0.124356	−0.00669508
$346$	−10.2487	−0.550974
$347$	−25.4641	−1.36698	−0.683492	−	0.729958i	$-0.739540\pi$
$347$	−25.4641	−1.36698	−0.683492	+	0.729958i	$0.739540\pi$
$348$	−12.0000	−0.643268
$349$	−8.53590	−0.456916	−0.228458	−	0.973554i	$-0.573368\pi$
$349$	−8.53590	−0.456916	−0.228458	+	0.973554i	$0.573368\pi$
$350$	7.21539	0.385679
$351$	−34.9808	−1.86713
$352$	5.85641	0.312148
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	−3.80385	−0.202172
$355$	−1.00000	−0.0530745
$356$	−2.82309	−0.149623
$357$	3.46410	0.183340
$358$	−2.19615	−0.116070
$359$	−8.19615	−0.432576	−0.216288	−	0.976330i	$-0.569395\pi$
$359$	−8.19615	−0.432576	−0.216288	+	0.976330i	$0.569395\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	−3.41154	−0.179307
$363$	−1.73205	−0.0909091
$364$	−19.7128	−1.03323
$365$	−3.80385	−0.199102
$366$	13.8564	0.724286
$367$	−20.1244	−1.05048	−0.525241	−	0.850953i	$-0.676025\pi$
$367$	−20.1244	−1.05048	−0.525241	+	0.850953i	$0.676025\pi$
$368$	−0.287187	−0.0149707
$369$	0	0
$370$	0.0525589	0.00273241
$371$	18.9282	0.982703
$372$	14.5359	0.753651
$373$	−2.19615	−0.113712	−0.0568562	−	0.998382i	$-0.518108\pi$
$373$	−2.19615	−0.113712	−0.0568562	+	0.998382i	$0.518108\pi$
$374$	0.732051	0.0378534
$375$	−4.60770	−0.237940
$376$	32.7846	1.69074
$377$	31.8564	1.64069
$378$	−7.60770	−0.391298
$379$	−6.26795	−0.321963	−0.160981	−	0.986957i	$-0.551466\pi$
$379$	−6.26795	−0.321963	−0.160981	+	0.986957i	$0.551466\pi$
$380$	1.64617	0.0844468
$381$	6.92820	0.354943
$382$	14.3013	0.731717
$383$	14.3205	0.731744	0.365872	−	0.930665i	$-0.380771\pi$
$383$	14.3205	0.731744	0.365872	+	0.930665i	$0.380771\pi$
$384$	17.5692	0.896575
$385$	0.535898	0.0273119
$386$	−10.1051	−0.514337
$387$	0	0
$388$	−17.9615	−0.911858
$389$	−3.92820	−0.199168	−0.0995839	−	0.995029i	$-0.531751\pi$
$389$	−3.92820	−0.199168	−0.0995839	+	0.995029i	$0.531751\pi$
$390$	−2.28719	−0.115816
$391$	−0.267949	−0.0135508
$392$	7.60770	0.384247
$393$	−8.53590	−0.430579
$394$	2.28719	0.115227
$395$	2.19615	0.110500
$396$	0	0
$397$	−27.7128	−1.39087	−0.695433	−	0.718591i	$-0.744787\pi$
$397$	−27.7128	−1.39087	−0.695433	+	0.718591i	$0.744787\pi$
$398$	18.1436	0.909456
$399$	14.5359	0.727705
$400$	−5.28203	−0.264102
$401$	12.7846	0.638433	0.319216	−	0.947682i	$-0.396580\pi$
$401$	12.7846	0.638433	0.319216	+	0.947682i	$0.396580\pi$
$402$	−1.26795	−0.0632396
$403$	−38.5885	−1.92223
$404$	22.9282	1.14072
$405$	2.41154	0.119831
$406$	6.92820	0.343841
$407$	−0.267949	−0.0132817
$408$	4.39230	0.217451
$409$	7.66025	0.378775	0.189388	−	0.981902i	$-0.439350\pi$
$409$	7.66025	0.378775	0.189388	+	0.981902i	$0.439350\pi$
$410$	−0.497423	−0.0245660
$411$	−2.41154	−0.118953
$412$	−23.2154	−1.14374
$413$	−6.00000	−0.295241
$414$	0	0
$415$	3.17691	0.155949
$416$	−39.4256	−1.93300
$417$	2.87564	0.140821
$418$	3.07180	0.150246
$419$	−5.85641	−0.286104	−0.143052	−	0.989715i	$-0.545692\pi$
$419$	−5.85641	−0.286104	−0.143052	+	0.989715i	$0.545692\pi$
$420$	1.35898	0.0663116
$421$	−9.85641	−0.480372	−0.240186	−	0.970727i	$-0.577208\pi$
$421$	−9.85641	−0.480372	−0.240186	+	0.970727i	$0.577208\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	24.0000	1.16554
$425$	−4.92820	−0.239053
$426$	−4.73205	−0.229269
$427$	21.8564	1.05771
$428$	21.5692	1.04259
$429$	11.6603	0.562962
$430$	0.392305	0.0189186
$431$	17.6603	0.850665	0.425332	−	0.905037i	$-0.360157\pi$
$431$	17.6603	0.850665	0.425332	+	0.905037i	$0.360157\pi$
$432$	5.56922	0.267949
$433$	39.3923	1.89307	0.946537	−	0.322596i	$-0.104556\pi$
$433$	39.3923	1.89307	0.946537	+	0.322596i	$0.104556\pi$
$434$	−8.39230	−0.402844
$435$	−2.19615	−0.105297
$436$	−12.0000	−0.574696
$437$	−1.12436	−0.0537852
$438$	−18.0000	−0.860073
$439$	−5.80385	−0.277003	−0.138501	−	0.990362i	$-0.544229\pi$
$439$	−5.80385	−0.277003	−0.138501	+	0.990362i	$0.544229\pi$
$440$	0.679492	0.0323935
$441$	0	0
$442$	−4.92820	−0.234411
$443$	14.6077	0.694033	0.347016	−	0.937859i	$-0.387195\pi$
$443$	14.6077	0.694033	0.347016	+	0.937859i	$0.387195\pi$
$444$	−0.679492	−0.0322473
$445$	−0.516660	−0.0244921
$446$	−10.4833	−0.496401
$447$	−8.87564	−0.419804
$448$	−4.28719	−0.202551
$449$	−30.5167	−1.44017	−0.720085	−	0.693886i	$-0.755897\pi$
$449$	−30.5167	−1.44017	−0.720085	+	0.693886i	$0.755897\pi$
$450$	0	0
$451$	2.53590	0.119411
$452$	14.2487	0.670203
$453$	3.12436	0.146795
$454$	9.71281	0.455845
$455$	−3.60770	−0.169131
$456$	18.4308	0.863100
$457$	−1.26795	−0.0593122	−0.0296561	−	0.999560i	$-0.509441\pi$
$457$	−1.26795	−0.0593122	−0.0296561	+	0.999560i	$0.509441\pi$
$458$	−4.05256	−0.189364
$459$	5.19615	0.242536
$460$	−0.105118	−0.00490114
$461$	−26.9282	−1.25417	−0.627086	−	0.778950i	$-0.715752\pi$
$461$	−26.9282	−1.25417	−0.627086	+	0.778950i	$0.715752\pi$
$462$	2.53590	0.117981
$463$	28.0718	1.30461	0.652304	−	0.757958i	$-0.273803\pi$
$463$	28.0718	1.30461	0.652304	+	0.757958i	$0.273803\pi$
$464$	−5.07180	−0.235452
$465$	2.66025	0.123366
$466$	−13.8564	−0.641886
$467$	35.7846	1.65591	0.827957	−	0.560791i	$-0.189503\pi$
$467$	35.7846	1.65591	0.827957	+	0.560791i	$0.189503\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	2.53590	0.116972
$471$	−37.7321	−1.73860
$472$	−7.60770	−0.350173
$473$	−2.00000	−0.0919601
$474$	10.3923	0.477334
$475$	−20.6795	−0.948840
$476$	2.92820	0.134214
$477$	0	0
$478$	−4.53590	−0.207467
$479$	32.7846	1.49797	0.748984	−	0.662589i	$-0.230542\pi$
$479$	32.7846	1.49797	0.748984	+	0.662589i	$0.230542\pi$
$480$	2.71797	0.124058
$481$	1.80385	0.0822484
$482$	3.71281	0.169114
$483$	−0.928203	−0.0422347
$484$	−1.46410	−0.0665501
$485$	−3.28719	−0.149263
$486$	0	0
$487$	39.0526	1.76964	0.884820	−	0.465933i	$-0.154281\pi$
$487$	39.0526	1.76964	0.884820	+	0.465933i	$0.154281\pi$
$488$	27.7128	1.25450
$489$	16.3923	0.741286
$490$	0.588457	0.0265838
$491$	23.1244	1.04359	0.521794	−	0.853072i	$-0.325263\pi$
$491$	23.1244	1.04359	0.521794	+	0.853072i	$0.325263\pi$
$492$	6.43078	0.289922
$493$	−4.73205	−0.213121
$494$	−20.6795	−0.930414
$495$	0	0
$496$	6.14359	0.275855
$497$	−7.46410	−0.334811
$498$	15.0333	0.673659
$499$	−14.3923	−0.644288	−0.322144	−	0.946691i	$-0.604404\pi$
$499$	−14.3923	−0.644288	−0.322144	+	0.946691i	$0.604404\pi$
$500$	−3.89488	−0.174184
$501$	−26.4449	−1.18147
$502$	6.09103	0.271856
$503$	−33.3731	−1.48803	−0.744016	−	0.668162i	$-0.767081\pi$
$503$	−33.3731	−1.48803	−0.744016	+	0.668162i	$0.767081\pi$
$504$	0	0
$505$	4.19615	0.186726
$506$	−0.196152	−0.00872004
$507$	−55.9808	−2.48619
$508$	5.85641	0.259836
$509$	−37.7846	−1.67477	−0.837387	−	0.546611i	$-0.815918\pi$
$509$	−37.7846	−1.67477	−0.837387	+	0.546611i	$0.815918\pi$
$510$	0.339746	0.0150442
$511$	−28.3923	−1.25600
$512$	11.7128	0.517638
$513$	21.8038	0.962663
$514$	−15.2154	−0.671122
$515$	−4.24871	−0.187221
$516$	−5.07180	−0.223273
$517$	−12.9282	−0.568582
$518$	0.392305	0.0172369
$519$	24.2487	1.06440
$520$	−4.57437	−0.200600
$521$	39.0526	1.71092	0.855462	−	0.517866i	$-0.173273\pi$
$521$	39.0526	1.71092	0.855462	+	0.517866i	$0.173273\pi$
$522$	0	0
$523$	−42.2487	−1.84741	−0.923704	−	0.383108i	$-0.874854\pi$
$523$	−42.2487	−1.84741	−0.923704	+	0.383108i	$0.874854\pi$
$524$	−7.21539	−0.315206
$525$	−17.0718	−0.745074
$526$	−9.60770	−0.418915
$527$	5.73205	0.249692
$528$	−1.85641	−0.0807897
$529$	−22.9282	−0.996878
$530$	1.85641	0.0806371
$531$	0	0
$532$	12.2872	0.532717
$533$	−17.0718	−0.739462
$534$	−2.44486	−0.105800
$535$	3.94744	0.170663
$536$	−2.53590	−0.109534
$537$	5.19615	0.224231
$538$	−10.2487	−0.441853
$539$	−3.00000	−0.129219
$540$	2.03848	0.0877220
$541$	23.6603	1.01723	0.508617	−	0.860993i	$-0.330157\pi$
$541$	23.6603	1.01723	0.508617	+	0.860993i	$0.330157\pi$
$542$	−16.7846	−0.720961
$543$	8.07180	0.346394
$544$	5.85641	0.251091
$545$	−2.19615	−0.0940728
$546$	−17.0718	−0.730605
$547$	35.2679	1.50795	0.753974	−	0.656904i	$-0.228134\pi$
$547$	35.2679	1.50795	0.753974	+	0.656904i	$0.228134\pi$
$548$	−2.03848	−0.0870794
$549$	0	0
$550$	−3.60770	−0.153833
$551$	−19.8564	−0.845911
$552$	−1.17691	−0.0500928
$553$	16.3923	0.697072
$554$	−1.75129	−0.0744051
$555$	−0.124356	−0.00527860
$556$	2.43078	0.103088
$557$	0.928203	0.0393292	0.0196646	−	0.999807i	$-0.493740\pi$
$557$	0.928203	0.0393292	0.0196646	+	0.999807i	$0.493740\pi$
$558$	0	0
$559$	13.4641	0.569471
$560$	0.574374	0.0242717
$561$	−1.73205	−0.0731272
$562$	−2.10512	−0.0887990
$563$	−14.7321	−0.620882	−0.310441	−	0.950593i	$-0.600477\pi$
$563$	−14.7321	−0.620882	−0.310441	+	0.950593i	$0.600477\pi$
$564$	−32.7846	−1.38048
$565$	2.60770	0.109707
$566$	0.535898	0.0225255
$567$	18.0000	0.755929
$568$	−9.46410	−0.397105
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	1.42563	0.0597129
$571$	−42.9808	−1.79869	−0.899344	−	0.437241i	$-0.855956\pi$
$571$	−42.9808	−1.79869	−0.899344	+	0.437241i	$0.855956\pi$
$572$	9.85641	0.412117
$573$	−33.8372	−1.41357
$574$	−3.71281	−0.154970
$575$	1.32051	0.0550690
$576$	0	0
$577$	17.3923	0.724051	0.362026	−	0.932168i	$-0.382085\pi$
$577$	17.3923	0.724051	0.362026	+	0.932168i	$0.382085\pi$
$578$	0.732051	0.0304493
$579$	23.9090	0.993623
$580$	−1.85641	−0.0770831
$581$	23.7128	0.983773
$582$	−15.5551	−0.644781
$583$	−9.46410	−0.391963
$584$	−36.0000	−1.48969
$585$	0	0
$586$	10.1436	0.419028
$587$	−31.0718	−1.28247	−0.641235	−	0.767344i	$-0.721578\pi$
$587$	−31.0718	−1.28247	−0.641235	+	0.767344i	$0.721578\pi$
$588$	−7.60770	−0.313736
$589$	24.0526	0.991068
$590$	−0.588457	−0.0242264
$591$	−5.41154	−0.222601
$592$	−0.287187	−0.0118033
$593$	27.1244	1.11386	0.556932	−	0.830558i	$-0.311978\pi$
$593$	27.1244	1.11386	0.556932	+	0.830558i	$0.311978\pi$
$594$	3.80385	0.156074
$595$	0.535898	0.0219697
$596$	−7.50258	−0.307318
$597$	−42.9282	−1.75693
$598$	1.32051	0.0539996
$599$	−36.2487	−1.48108	−0.740541	−	0.672011i	$-0.765431\pi$
$599$	−36.2487	−1.48108	−0.740541	+	0.672011i	$0.765431\pi$
$600$	−21.6462	−0.883701
$601$	−23.4641	−0.957121	−0.478560	−	0.878055i	$-0.658841\pi$
$601$	−23.4641	−0.957121	−0.478560	+	0.878055i	$0.658841\pi$
$602$	2.92820	0.119345
$603$	0	0
$604$	2.64102	0.107461
$605$	−0.267949	−0.0108937
$606$	19.8564	0.806611
$607$	−18.7846	−0.762444	−0.381222	−	0.924484i	$-0.624497\pi$
$607$	−18.7846	−0.762444	−0.381222	+	0.924484i	$0.624497\pi$
$608$	24.5744	0.996622
$609$	−16.3923	−0.664250
$610$	2.14359	0.0867916
$611$	87.0333	3.52099
$612$	0	0
$613$	38.6410	1.56070	0.780348	−	0.625346i	$-0.215042\pi$
$613$	38.6410	1.56070	0.780348	+	0.625346i	$0.215042\pi$
$614$	−14.7846	−0.596658
$615$	1.17691	0.0474578
$616$	5.07180	0.204349
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−20.1051	−0.808746
$619$	25.0526	1.00695	0.503474	−	0.864011i	$-0.332055\pi$
$619$	25.0526	1.00695	0.503474	+	0.864011i	$0.332055\pi$
$620$	2.24871	0.0903104
$621$	−1.39230	−0.0558713
$622$	14.9282	0.598566
$623$	−3.85641	−0.154504
$624$	12.4974	0.500297
$625$	23.9282	0.957128
$626$	14.0526	0.561653
$627$	−7.26795	−0.290254
$628$	−31.8949	−1.27274
$629$	−0.267949	−0.0106838
$630$	0	0
$631$	26.6077	1.05924	0.529618	−	0.848236i	$-0.322335\pi$
$631$	26.6077	1.05924	0.529618	+	0.848236i	$0.322335\pi$
$632$	20.7846	0.826767
$633$	−9.46410	−0.376164
$634$	−24.4833	−0.972358
$635$	1.07180	0.0425330
$636$	−24.0000	−0.951662
$637$	20.1962	0.800201
$638$	−3.46410	−0.137145
$639$	0	0
$640$	2.71797	0.107437
$641$	39.8372	1.57347	0.786737	−	0.617289i	$-0.211769\pi$
$641$	39.8372	1.57347	0.786737	+	0.617289i	$0.211769\pi$
$642$	18.6795	0.737221
$643$	27.9808	1.10345	0.551727	−	0.834025i	$-0.313969\pi$
$643$	27.9808	1.10345	0.551727	+	0.834025i	$0.313969\pi$
$644$	−0.784610	−0.0309180
$645$	−0.928203	−0.0365480
$646$	3.07180	0.120858
$647$	−19.2487	−0.756745	−0.378372	−	0.925653i	$-0.623516\pi$
$647$	−19.2487	−0.756745	−0.378372	+	0.925653i	$0.623516\pi$
$648$	22.8231	0.896575
$649$	3.00000	0.117760
$650$	24.2872	0.952622
$651$	19.8564	0.778234
$652$	13.8564	0.542659
$653$	−30.3731	−1.18859	−0.594295	−	0.804247i	$-0.702569\pi$
$653$	−30.3731	−1.18859	−0.594295	+	0.804247i	$0.702569\pi$
$654$	−10.3923	−0.406371
$655$	−1.32051	−0.0515965
$656$	2.71797	0.106119
$657$	0	0
$658$	18.9282	0.737898
$659$	−10.8756	−0.423655	−0.211827	−	0.977307i	$-0.567941\pi$
$659$	−10.8756	−0.423655	−0.211827	+	0.977307i	$0.567941\pi$
$660$	−0.679492	−0.0264492
$661$	−27.5359	−1.07102	−0.535511	−	0.844528i	$-0.679881\pi$
$661$	−27.5359	−1.07102	−0.535511	+	0.844528i	$0.679881\pi$
$662$	−2.30127	−0.0894414
$663$	11.6603	0.452847
$664$	30.0666	1.16681
$665$	2.24871	0.0872013
$666$	0	0
$667$	1.26795	0.0490952
$668$	−22.3538	−0.864896
$669$	24.8038	0.958972
$670$	−0.196152	−0.00757803
$671$	−10.9282	−0.421879
$672$	20.2872	0.782595
$673$	−0.196152	−0.00756112	−0.00378056	−	0.999993i	$-0.501203\pi$
$673$	−0.196152	−0.00756112	−0.00378056	+	0.999993i	$0.501203\pi$
$674$	−22.2487	−0.856988
$675$	−25.6077	−0.985641
$676$	−47.3205	−1.82002
$677$	10.0526	0.386351	0.193176	−	0.981164i	$-0.438121\pi$
$677$	10.0526	0.386351	0.193176	+	0.981164i	$0.438121\pi$
$678$	12.3397	0.473905
$679$	−24.5359	−0.941601
$680$	0.679492	0.0260573
$681$	−22.9808	−0.880625
$682$	4.19615	0.160679
$683$	5.46410	0.209078	0.104539	−	0.994521i	$-0.466663\pi$
$683$	5.46410	0.209078	0.104539	+	0.994521i	$0.466663\pi$
$684$	0	0
$685$	−0.373067	−0.0142542
$686$	14.6410	0.558997
$687$	9.58846	0.365822
$688$	−2.14359	−0.0817237
$689$	63.7128	2.42726
$690$	−0.0910347	−0.00346563
$691$	20.5167	0.780491	0.390245	−	0.920711i	$-0.372390\pi$
$691$	20.5167	0.780491	0.390245	+	0.920711i	$0.372390\pi$
$692$	20.4974	0.779195
$693$	0	0
$694$	−18.6410	−0.707603
$695$	0.444864	0.0168746
$696$	−20.7846	−0.787839
$697$	2.53590	0.0960540
$698$	−6.24871	−0.236517
$699$	32.7846	1.24003
$700$	−14.4308	−0.545432
$701$	−40.6410	−1.53499	−0.767495	−	0.641055i	$-0.778497\pi$
$701$	−40.6410	−1.53499	−0.767495	+	0.641055i	$0.778497\pi$
$702$	−25.6077	−0.966500
$703$	−1.12436	−0.0424059
$704$	2.14359	0.0807897
$705$	−6.00000	−0.225973
$706$	15.3731	0.578573
$707$	31.3205	1.17793
$708$	7.60770	0.285915
$709$	34.6603	1.30169	0.650847	−	0.759209i	$-0.274414\pi$
$709$	34.6603	1.30169	0.650847	+	0.759209i	$0.274414\pi$
$710$	−0.732051	−0.0274734
$711$	0	0
$712$	−4.88973	−0.183250
$713$	−1.53590	−0.0575198
$714$	2.53590	0.0949036
$715$	1.80385	0.0674601
$716$	4.39230	0.164148
$717$	10.7321	0.400796
$718$	−6.00000	−0.223918
$719$	−44.3731	−1.65484	−0.827418	−	0.561586i	$-0.810191\pi$
$719$	−44.3731	−1.65484	−0.827418	+	0.561586i	$0.810191\pi$
$720$	0	0
$721$	−31.7128	−1.18105
$722$	−1.01924	−0.0379321
$723$	−8.78461	−0.326703
$724$	6.82309	0.253578
$725$	23.3205	0.866102
$726$	−1.26795	−0.0470580
$727$	−10.0718	−0.373542	−0.186771	−	0.982403i	$-0.559802\pi$
$727$	−10.0718	−0.373542	−0.186771	+	0.982403i	$0.559802\pi$
$728$	−34.1436	−1.26545
$729$	27.0000	1.00000
$730$	−2.78461	−0.103063
$731$	−2.00000	−0.0739727
$732$	−27.7128	−1.02430
$733$	−0.392305	−0.0144901	−0.00724506	−	0.999974i	$-0.502306\pi$
$733$	−0.392305	−0.0144901	−0.00724506	+	0.999974i	$0.502306\pi$
$734$	−14.7321	−0.543770
$735$	−1.39230	−0.0513559
$736$	−1.56922	−0.0578422
$737$	1.00000	0.0368355
$738$	0	0
$739$	−12.0526	−0.443361	−0.221680	−	0.975119i	$-0.571154\pi$
$739$	−12.0526	−0.443361	−0.221680	+	0.975119i	$0.571154\pi$
$740$	−0.105118	−0.00386421
$741$	48.9282	1.79742
$742$	13.8564	0.508685
$743$	−16.3923	−0.601375	−0.300688	−	0.953723i	$-0.597216\pi$
$743$	−16.3923	−0.601375	−0.300688	+	0.953723i	$0.597216\pi$
$744$	25.1769	0.923030
$745$	−1.37307	−0.0503053
$746$	−1.60770	−0.0588619
$747$	0	0
$748$	−1.46410	−0.0535329
$749$	29.4641	1.07659
$750$	−3.37307	−0.123167
$751$	−5.05256	−0.184370	−0.0921852	−	0.995742i	$-0.529385\pi$
$751$	−5.05256	−0.184370	−0.0921852	+	0.995742i	$0.529385\pi$
$752$	−13.8564	−0.505291
$753$	−14.4115	−0.525186
$754$	23.3205	0.849283
$755$	0.483340	0.0175905
$756$	15.2154	0.553378
$757$	39.5692	1.43817	0.719084	−	0.694923i	$-0.244562\pi$
$757$	39.5692	1.43817	0.719084	+	0.694923i	$0.244562\pi$
$758$	−4.58846	−0.166660
$759$	0.464102	0.0168458
$760$	2.85125	0.103426
$761$	42.0526	1.52440	0.762202	−	0.647339i	$-0.224118\pi$
$761$	42.0526	1.52440	0.762202	+	0.647339i	$0.224118\pi$
$762$	5.07180	0.183732
$763$	−16.3923	−0.593441
$764$	−28.6025	−1.03480
$765$	0	0
$766$	10.4833	0.378778
$767$	−20.1962	−0.729241
$768$	20.2872	0.732051
$769$	−49.5692	−1.78751	−0.893756	−	0.448554i	$-0.851939\pi$
$769$	−49.5692	−1.78751	−0.893756	+	0.448554i	$0.851939\pi$
$770$	0.392305	0.0141377
$771$	36.0000	1.29651
$772$	20.2102	0.727382
$773$	−8.67949	−0.312180	−0.156090	−	0.987743i	$-0.549889\pi$
$773$	−8.67949	−0.312180	−0.156090	+	0.987743i	$0.549889\pi$
$774$	0	0
$775$	−28.2487	−1.01472
$776$	−31.1103	−1.11679
$777$	−0.928203	−0.0332991
$778$	−2.87564	−0.103097
$779$	10.6410	0.381254
$780$	4.57437	0.163789
$781$	3.73205	0.133543
$782$	−0.196152	−0.00701440
$783$	−24.5885	−0.878720
$784$	−3.21539	−0.114835
$785$	−5.83717	−0.208337
$786$	−6.24871	−0.222884
$787$	−17.1244	−0.610417	−0.305209	−	0.952285i	$-0.598726\pi$
$787$	−17.1244	−0.610417	−0.305209	+	0.952285i	$0.598726\pi$
$788$	−4.57437	−0.162955
$789$	22.7321	0.809282
$790$	1.60770	0.0571992
$791$	19.4641	0.692064
$792$	0	0
$793$	73.5692	2.61252
$794$	−20.2872	−0.719965
$795$	−4.39230	−0.155779
$796$	−36.2872	−1.28617
$797$	−15.3923	−0.545223	−0.272612	−	0.962124i	$-0.587887\pi$
$797$	−15.3923	−0.545223	−0.272612	+	0.962124i	$0.587887\pi$
$798$	10.6410	0.376688
$799$	−12.9282	−0.457367
$800$	−28.8616	−1.02041
$801$	0	0
$802$	9.35898	0.330477
$803$	14.1962	0.500971
$804$	2.53590	0.0894342
$805$	−0.143594	−0.00506101
$806$	−28.2487	−0.995018
$807$	24.2487	0.853595
$808$	39.7128	1.39709
$809$	−33.9090	−1.19218	−0.596088	−	0.802919i	$-0.703279\pi$
$809$	−33.9090	−1.19218	−0.596088	+	0.802919i	$0.703279\pi$
$810$	1.76537	0.0620288
$811$	2.73205	0.0959353	0.0479676	−	0.998849i	$-0.484726\pi$
$811$	2.73205	0.0959353	0.0479676	+	0.998849i	$0.484726\pi$
$812$	−13.8564	−0.486265
$813$	39.7128	1.39279
$814$	−0.196152	−0.00687514
$815$	2.53590	0.0888286
$816$	−1.85641	−0.0649872
$817$	−8.39230	−0.293610
$818$	5.60770	0.196068
$819$	0	0
$820$	0.994845	0.0347415
$821$	34.0526	1.18844	0.594221	−	0.804302i	$-0.297460\pi$
$821$	34.0526	1.18844	0.594221	+	0.804302i	$0.297460\pi$
$822$	−1.76537	−0.0615744
$823$	36.2679	1.26422	0.632111	−	0.774878i	$-0.282189\pi$
$823$	36.2679	1.26422	0.632111	+	0.774878i	$0.282189\pi$
$824$	−40.2102	−1.40079
$825$	8.53590	0.297182
$826$	−4.39230	−0.152828
$827$	−12.3397	−0.429095	−0.214548	−	0.976714i	$-0.568828\pi$
$827$	−12.3397	−0.429095	−0.214548	+	0.976714i	$0.568828\pi$
$828$	0	0
$829$	−0.0717968	−0.00249360	−0.00124680	−	0.999999i	$-0.500397\pi$
$829$	−0.0717968	−0.00249360	−0.00124680	+	0.999999i	$0.500397\pi$
$830$	2.32566	0.0807249
$831$	4.14359	0.143740
$832$	−14.4308	−0.500297
$833$	−3.00000	−0.103944
$834$	2.10512	0.0728943
$835$	−4.09103	−0.141576
$836$	−6.14359	−0.212481
$837$	29.7846	1.02951
$838$	−4.28719	−0.148098
$839$	−50.5167	−1.74403	−0.872014	−	0.489480i	$-0.837187\pi$
$839$	−50.5167	−1.74403	−0.872014	+	0.489480i	$0.837187\pi$
$840$	2.35383	0.0812148
$841$	−6.60770	−0.227852
$842$	−7.21539	−0.248659
$843$	4.98076	0.171547
$844$	−8.00000	−0.275371
$845$	−8.66025	−0.297922
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−10.1436	−0.348332
$849$	−1.26795	−0.0435159
$850$	−3.60770	−0.123743
$851$	0.0717968	0.00246116
$852$	9.46410	0.324235
$853$	11.1244	0.380891	0.190445	−	0.981698i	$-0.439007\pi$
$853$	11.1244	0.380891	0.190445	+	0.981698i	$0.439007\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	37.3590	1.27690
$857$	48.4449	1.65485	0.827423	−	0.561580i	$-0.189806\pi$
$857$	48.4449	1.65485	0.827423	+	0.561580i	$0.189806\pi$
$858$	8.53590	0.291411
$859$	44.8564	1.53048	0.765240	−	0.643745i	$-0.222620\pi$
$859$	44.8564	1.53048	0.765240	+	0.643745i	$0.222620\pi$
$860$	−0.784610	−0.0267550
$861$	8.78461	0.299379
$862$	12.9282	0.440336
$863$	32.6410	1.11111	0.555557	−	0.831479i	$-0.312505\pi$
$863$	32.6410	1.11111	0.555557	+	0.831479i	$0.312505\pi$
$864$	30.4308	1.03528
$865$	3.75129	0.127548
$866$	28.8372	0.979927
$867$	−1.73205	−0.0588235
$868$	16.7846	0.569707
$869$	−8.19615	−0.278035
$870$	−1.60770	−0.0545060
$871$	−6.73205	−0.228107
$872$	−20.7846	−0.703856
$873$	0	0
$874$	−0.823085	−0.0278413
$875$	−5.32051	−0.179866
$876$	36.0000	1.21633
$877$	−18.5885	−0.627688	−0.313844	−	0.949475i	$-0.601617\pi$
$877$	−18.5885	−0.627688	−0.313844	+	0.949475i	$0.601617\pi$
$878$	−4.24871	−0.143387
$879$	−24.0000	−0.809500
$880$	−0.287187	−0.00968107
$881$	−42.9090	−1.44564	−0.722820	−	0.691036i	$-0.757154\pi$
$881$	−42.9090	−1.44564	−0.722820	+	0.691036i	$0.757154\pi$
$882$	0	0
$883$	−6.92820	−0.233153	−0.116576	−	0.993182i	$-0.537192\pi$
$883$	−6.92820	−0.233153	−0.116576	+	0.993182i	$0.537192\pi$
$884$	9.85641	0.331507
$885$	1.39230	0.0468018
$886$	10.6936	0.359258
$887$	−33.7128	−1.13197	−0.565983	−	0.824417i	$-0.691503\pi$
$887$	−33.7128	−1.13197	−0.565983	+	0.824417i	$0.691503\pi$
$888$	−1.17691	−0.0394947
$889$	8.00000	0.268311
$890$	−0.378222	−0.0126780
$891$	−9.00000	−0.301511
$892$	20.9667	0.702016
$893$	−54.2487	−1.81536
$894$	−6.49742	−0.217306
$895$	0.803848	0.0268697
$896$	20.2872	0.677747
$897$	−3.12436	−0.104319
$898$	−22.3397	−0.745487
$899$	−27.1244	−0.904648
$900$	0	0
$901$	−9.46410	−0.315295
$902$	1.85641	0.0618116
$903$	−6.92820	−0.230556
$904$	24.6795	0.820828
$905$	1.24871	0.0415086
$906$	2.28719	0.0759867
$907$	14.9282	0.495683	0.247841	−	0.968801i	$-0.420279\pi$
$907$	14.9282	0.495683	0.247841	+	0.968801i	$0.420279\pi$
$908$	−19.4256	−0.644662
$909$	0	0
$910$	−2.64102	−0.0875488
$911$	−26.3923	−0.874416	−0.437208	−	0.899360i	$-0.644033\pi$
$911$	−26.3923	−0.874416	−0.437208	+	0.899360i	$0.644033\pi$
$912$	−7.78976	−0.257945
$913$	−11.8564	−0.392390
$914$	−0.928203	−0.0307022
$915$	−5.07180	−0.167668
$916$	8.10512	0.267801
$917$	−9.85641	−0.325487
$918$	3.80385	0.125546
$919$	−41.3205	−1.36304	−0.681519	−	0.731801i	$-0.738680\pi$
$919$	−41.3205	−1.36304	−0.681519	+	0.731801i	$0.738680\pi$
$920$	−0.182069	−0.00600265
$921$	34.9808	1.15266
$922$	−19.7128	−0.649207
$923$	−25.1244	−0.826978
$924$	−5.07180	−0.166850
$925$	1.32051	0.0434180
$926$	20.5500	0.675314
$927$	0	0
$928$	−27.7128	−0.909718
$929$	−45.0333	−1.47750	−0.738748	−	0.673982i	$-0.764582\pi$
$929$	−45.0333	−1.47750	−0.738748	+	0.673982i	$0.764582\pi$
$930$	1.94744	0.0638591
$931$	−12.5885	−0.412570
$932$	27.7128	0.907763
$933$	−35.3205	−1.15634
$934$	26.1962	0.857164
$935$	−0.267949	−0.00876288
$936$	0	0
$937$	−36.9808	−1.20811	−0.604054	−	0.796943i	$-0.706449\pi$
$937$	−36.9808	−1.20811	−0.604054	+	0.796943i	$0.706449\pi$
$938$	−1.46410	−0.0478046
$939$	−33.2487	−1.08503
$940$	−5.07180	−0.165424
$941$	−24.2487	−0.790485	−0.395243	−	0.918577i	$-0.629340\pi$
$941$	−24.2487	−0.790485	−0.395243	+	0.918577i	$0.629340\pi$
$942$	−27.6218	−0.899966
$943$	−0.679492	−0.0221273
$944$	3.21539	0.104652
$945$	2.78461	0.0905834
$946$	−1.46410	−0.0476020
$947$	−5.73205	−0.186267	−0.0931333	−	0.995654i	$-0.529688\pi$
$947$	−5.73205	−0.186267	−0.0931333	+	0.995654i	$0.529688\pi$
$948$	−20.7846	−0.675053
$949$	−95.5692	−3.10231
$950$	−15.1384	−0.491156
$951$	57.9282	1.87845
$952$	5.07180	0.164378
$953$	−13.2154	−0.428088	−0.214044	−	0.976824i	$-0.568664\pi$
$953$	−13.2154	−0.428088	−0.214044	+	0.976824i	$0.568664\pi$
$954$	0	0
$955$	−5.23463	−0.169389
$956$	9.07180	0.293403
$957$	8.19615	0.264944
$958$	24.0000	0.775405
$959$	−2.78461	−0.0899197
$960$	0.994845	0.0321085
$961$	1.85641	0.0598841
$962$	1.32051	0.0425749
$963$	0	0
$964$	−7.42563	−0.239163
$965$	3.69873	0.119066
$966$	−0.679492	−0.0218623
$967$	−23.9090	−0.768860	−0.384430	−	0.923154i	$-0.625602\pi$
$967$	−23.9090	−0.768860	−0.384430	+	0.923154i	$0.625602\pi$
$968$	−2.53590	−0.0815069
$969$	−7.26795	−0.233480
$970$	−2.40639	−0.0772645
$971$	−1.14359	−0.0366997	−0.0183498	−	0.999832i	$-0.505841\pi$
$971$	−1.14359	−0.0366997	−0.0183498	+	0.999832i	$0.505841\pi$
$972$	0	0
$973$	3.32051	0.106451
$974$	28.5885	0.916033
$975$	−57.4641	−1.84032
$976$	−11.7128	−0.374918
$977$	−12.2154	−0.390805	−0.195402	−	0.980723i	$-0.562601\pi$
$977$	−12.2154	−0.390805	−0.195402	+	0.980723i	$0.562601\pi$
$978$	12.0000	0.383718
$979$	1.92820	0.0616256
$980$	−1.17691	−0.0375952
$981$	0	0
$982$	16.9282	0.540201
$983$	−14.6603	−0.467589	−0.233795	−	0.972286i	$-0.575114\pi$
$983$	−14.6603	−0.467589	−0.233795	+	0.972286i	$0.575114\pi$
$984$	11.1384	0.355080
$985$	−0.837169	−0.0266744
$986$	−3.46410	−0.110319
$987$	−44.7846	−1.42551
$988$	41.3590	1.31580
$989$	0.535898	0.0170406
$990$	0	0
$991$	46.6410	1.48160	0.740800	−	0.671725i	$-0.234446\pi$
$991$	46.6410	1.48160	0.740800	+	0.671725i	$0.234446\pi$
$992$	33.5692	1.06582
$993$	5.44486	0.172788
$994$	−5.46410	−0.173311
$995$	−6.64102	−0.210534
$996$	−30.0666	−0.952698
$997$	39.2295	1.24241	0.621205	−	0.783648i	$-0.286643\pi$
$997$	39.2295	1.24241	0.621205	+	0.783648i	$0.286643\pi$
$998$	−10.5359	−0.333508
$999$	−1.39230	−0.0440506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.a.c.1.2	✓	2
3.2	odd	2		1683.2.a.r.1.1		2
4.3	odd	2		2992.2.a.j.1.2		2
5.4	even	2		4675.2.a.v.1.1		2
7.6	odd	2		9163.2.a.h.1.2		2
11.10	odd	2		2057.2.a.o.1.1		2
17.16	even	2		3179.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.a.c.1.2	✓	2	1.1	even	1	trivial
1683.2.a.r.1.1		2	3.2	odd	2
2057.2.a.o.1.1		2	11.10	odd	2
2992.2.a.j.1.2		2	4.3	odd	2
3179.2.a.k.1.2		2	17.16	even	2
4675.2.a.v.1.1		2	5.4	even	2
9163.2.a.h.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.73205 q^{3} -1.46410 q^{4} -0.267949 q^{5} -1.26795 q^{6} -2.00000 q^{7} -2.53590 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -1.73205 q^{3} -1.46410 q^{4} -0.267949 q^{5} -1.26795 q^{6} -2.00000 q^{7} -2.53590 q^{8} -0.196152 q^{10} +1.00000 q^{11} +2.53590 q^{12} -6.73205 q^{13} -1.46410 q^{14} +0.464102 q^{15} +1.07180 q^{16} +1.00000 q^{17} +4.19615 q^{19} +0.392305 q^{20} +3.46410 q^{21} +0.732051 q^{22} -0.267949 q^{23} +4.39230 q^{24} -4.92820 q^{25} -4.92820 q^{26} +5.19615 q^{27} +2.92820 q^{28} -4.73205 q^{29} +0.339746 q^{30} +5.73205 q^{31} +5.85641 q^{32} -1.73205 q^{33} +0.732051 q^{34} +0.535898 q^{35} -0.267949 q^{37} +3.07180 q^{38} +11.6603 q^{39} +0.679492 q^{40} +2.53590 q^{41} +2.53590 q^{42} -2.00000 q^{43} -1.46410 q^{44} -0.196152 q^{46} -12.9282 q^{47} -1.85641 q^{48} -3.00000 q^{49} -3.60770 q^{50} -1.73205 q^{51} +9.85641 q^{52} -9.46410 q^{53} +3.80385 q^{54} -0.267949 q^{55} +5.07180 q^{56} -7.26795 q^{57} -3.46410 q^{58} +3.00000 q^{59} -0.679492 q^{60} -10.9282 q^{61} +4.19615 q^{62} +2.14359 q^{64} +1.80385 q^{65} -1.26795 q^{66} +1.00000 q^{67} -1.46410 q^{68} +0.464102 q^{69} +0.392305 q^{70} +3.73205 q^{71} +14.1962 q^{73} -0.196152 q^{74} +8.53590 q^{75} -6.14359 q^{76} -2.00000 q^{77} +8.53590 q^{78} -8.19615 q^{79} -0.287187 q^{80} -9.00000 q^{81} +1.85641 q^{82} -11.8564 q^{83} -5.07180 q^{84} -0.267949 q^{85} -1.46410 q^{86} +8.19615 q^{87} -2.53590 q^{88} +1.92820 q^{89} +13.4641 q^{91} +0.392305 q^{92} -9.92820 q^{93} -9.46410 q^{94} -1.12436 q^{95} -10.1436 q^{96} +12.2679 q^{97} -2.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^6 - 4 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 4 q^{7} - 12 q^{8} + 10 q^{10} + 2 q^{11} + 12 q^{12} - 10 q^{13} + 4 q^{14} - 6 q^{15} + 16 q^{16} + 2 q^{17} - 2 q^{19} - 20 q^{20} - 2 q^{22} - 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} + 18 q^{30} + 8 q^{31} - 16 q^{32} - 2 q^{34} + 8 q^{35} - 4 q^{37} + 20 q^{38} + 6 q^{39} + 36 q^{40} + 12 q^{41} + 12 q^{42} - 4 q^{43} + 4 q^{44} + 10 q^{46} - 12 q^{47} + 24 q^{48} - 6 q^{49} - 28 q^{50} - 8 q^{52} - 12 q^{53} + 18 q^{54} - 4 q^{55} + 24 q^{56} - 18 q^{57} + 6 q^{59} - 36 q^{60} - 8 q^{61} - 2 q^{62} + 32 q^{64} + 14 q^{65} - 6 q^{66} + 2 q^{67} + 4 q^{68} - 6 q^{69} - 20 q^{70} + 4 q^{71} + 18 q^{73} + 10 q^{74} + 24 q^{75} - 40 q^{76} - 4 q^{77} + 24 q^{78} - 6 q^{79} - 56 q^{80} - 18 q^{81} - 24 q^{82} + 4 q^{83} - 24 q^{84} - 4 q^{85} + 4 q^{86} + 6 q^{87} - 12 q^{88} - 10 q^{89} + 20 q^{91} - 20 q^{92} - 6 q^{93} - 12 q^{94} + 22 q^{95} - 48 q^{96} + 28 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 6 * q^6 - 4 * q^7 - 12 * q^8 + 10 * q^10 + 2 * q^11 + 12 * q^12 - 10 * q^13 + 4 * q^14 - 6 * q^15 + 16 * q^16 + 2 * q^17 - 2 * q^19 - 20 * q^20 - 2 * q^22 - 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 8 * q^28 - 6 * q^29 + 18 * q^30 + 8 * q^31 - 16 * q^32 - 2 * q^34 + 8 * q^35 - 4 * q^37 + 20 * q^38 + 6 * q^39 + 36 * q^40 + 12 * q^41 + 12 * q^42 - 4 * q^43 + 4 * q^44 + 10 * q^46 - 12 * q^47 + 24 * q^48 - 6 * q^49 - 28 * q^50 - 8 * q^52 - 12 * q^53 + 18 * q^54 - 4 * q^55 + 24 * q^56 - 18 * q^57 + 6 * q^59 - 36 * q^60 - 8 * q^61 - 2 * q^62 + 32 * q^64 + 14 * q^65 - 6 * q^66 + 2 * q^67 + 4 * q^68 - 6 * q^69 - 20 * q^70 + 4 * q^71 + 18 * q^73 + 10 * q^74 + 24 * q^75 - 40 * q^76 - 4 * q^77 + 24 * q^78 - 6 * q^79 - 56 * q^80 - 18 * q^81 - 24 * q^82 + 4 * q^83 - 24 * q^84 - 4 * q^85 + 4 * q^86 + 6 * q^87 - 12 * q^88 - 10 * q^89 + 20 * q^91 - 20 * q^92 - 6 * q^93 - 12 * q^94 + 22 * q^95 - 48 * q^96 + 28 * q^97 + 6 * q^98

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