LMFDB - Embedded newform 1369.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1369,2,Mod(1,1369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1369, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1369.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1369 = 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1369.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:	$10.9315200367$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 37)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.73205 q^{3} -1.00000 q^{4} -0.267949 q^{5} +2.73205 q^{6} +3.46410 q^{7} -3.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.73205 q^{3} -1.00000 q^{4} -0.267949 q^{5} +2.73205 q^{6} +3.46410 q^{7} -3.00000 q^{8} +4.46410 q^{9} -0.267949 q^{10} +4.73205 q^{11} -2.73205 q^{12} -3.46410 q^{13} +3.46410 q^{14} -0.732051 q^{15} -1.00000 q^{16} +3.73205 q^{17} +4.46410 q^{18} -1.26795 q^{19} +0.267949 q^{20} +9.46410 q^{21} +4.73205 q^{22} -0.732051 q^{23} -8.19615 q^{24} -4.92820 q^{25} -3.46410 q^{26} +4.00000 q^{27} -3.46410 q^{28} -0.267949 q^{29} -0.732051 q^{30} +8.19615 q^{31} +5.00000 q^{32} +12.9282 q^{33} +3.73205 q^{34} -0.928203 q^{35} -4.46410 q^{36} -1.26795 q^{38} -9.46410 q^{39} +0.803848 q^{40} -3.00000 q^{41} +9.46410 q^{42} +3.46410 q^{43} -4.73205 q^{44} -1.19615 q^{45} -0.732051 q^{46} -2.19615 q^{47} -2.73205 q^{48} +5.00000 q^{49} -4.92820 q^{50} +10.1962 q^{51} +3.46410 q^{52} -2.53590 q^{53} +4.00000 q^{54} -1.26795 q^{55} -10.3923 q^{56} -3.46410 q^{57} -0.267949 q^{58} -13.4641 q^{59} +0.732051 q^{60} +4.26795 q^{61} +8.19615 q^{62} +15.4641 q^{63} +7.00000 q^{64} +0.928203 q^{65} +12.9282 q^{66} +7.26795 q^{67} -3.73205 q^{68} -2.00000 q^{69} -0.928203 q^{70} +6.00000 q^{71} -13.3923 q^{72} -13.4641 q^{75} +1.26795 q^{76} +16.3923 q^{77} -9.46410 q^{78} -3.80385 q^{79} +0.267949 q^{80} -2.46410 q^{81} -3.00000 q^{82} -8.19615 q^{83} -9.46410 q^{84} -1.00000 q^{85} +3.46410 q^{86} -0.732051 q^{87} -14.1962 q^{88} -9.19615 q^{89} -1.19615 q^{90} -12.0000 q^{91} +0.732051 q^{92} +22.3923 q^{93} -2.19615 q^{94} +0.339746 q^{95} +13.6603 q^{96} -4.26795 q^{97} +5.00000 q^{98} +21.1244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} - 4 q^{10} + 6 q^{11} - 2 q^{12} + 2 q^{15} - 2 q^{16} + 4 q^{17} + 2 q^{18} - 6 q^{19} + 4 q^{20} + 12 q^{21} + 6 q^{22} + 2 q^{23} - 6 q^{24} + 4 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 6 q^{31} + 10 q^{32} + 12 q^{33} + 4 q^{34} + 12 q^{35} - 2 q^{36} - 6 q^{38} - 12 q^{39} + 12 q^{40} - 6 q^{41} + 12 q^{42} - 6 q^{44} + 8 q^{45} + 2 q^{46} + 6 q^{47} - 2 q^{48} + 10 q^{49} + 4 q^{50} + 10 q^{51} - 12 q^{53} + 8 q^{54} - 6 q^{55} - 4 q^{58} - 20 q^{59} - 2 q^{60} + 12 q^{61} + 6 q^{62} + 24 q^{63} + 14 q^{64} - 12 q^{65} + 12 q^{66} + 18 q^{67} - 4 q^{68} - 4 q^{69} + 12 q^{70} + 12 q^{71} - 6 q^{72} - 20 q^{75} + 6 q^{76} + 12 q^{77} - 12 q^{78} - 18 q^{79} + 4 q^{80} + 2 q^{81} - 6 q^{82} - 6 q^{83} - 12 q^{84} - 2 q^{85} + 2 q^{87} - 18 q^{88} - 8 q^{89} + 8 q^{90} - 24 q^{91} - 2 q^{92} + 24 q^{93} + 6 q^{94} + 18 q^{95} + 10 q^{96} - 12 q^{97} + 10 q^{98} + 18 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 - 4 * q^10 + 6 * q^11 - 2 * q^12 + 2 * q^15 - 2 * q^16 + 4 * q^17 + 2 * q^18 - 6 * q^19 + 4 * q^20 + 12 * q^21 + 6 * q^22 + 2 * q^23 - 6 * q^24 + 4 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 6 * q^31 + 10 * q^32 + 12 * q^33 + 4 * q^34 + 12 * q^35 - 2 * q^36 - 6 * q^38 - 12 * q^39 + 12 * q^40 - 6 * q^41 + 12 * q^42 - 6 * q^44 + 8 * q^45 + 2 * q^46 + 6 * q^47 - 2 * q^48 + 10 * q^49 + 4 * q^50 + 10 * q^51 - 12 * q^53 + 8 * q^54 - 6 * q^55 - 4 * q^58 - 20 * q^59 - 2 * q^60 + 12 * q^61 + 6 * q^62 + 24 * q^63 + 14 * q^64 - 12 * q^65 + 12 * q^66 + 18 * q^67 - 4 * q^68 - 4 * q^69 + 12 * q^70 + 12 * q^71 - 6 * q^72 - 20 * q^75 + 6 * q^76 + 12 * q^77 - 12 * q^78 - 18 * q^79 + 4 * q^80 + 2 * q^81 - 6 * q^82 - 6 * q^83 - 12 * q^84 - 2 * q^85 + 2 * q^87 - 18 * q^88 - 8 * q^89 + 8 * q^90 - 24 * q^91 - 2 * q^92 + 24 * q^93 + 6 * q^94 + 18 * q^95 + 10 * q^96 - 12 * q^97 + 10 * q^98 + 18 * q^99

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	−1.00000	−0.500000
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	2.73205	1.11536
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	−3.00000	−1.06066
$9$	4.46410	1.48803
$10$	−0.267949	−0.0847330
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	−2.73205	−0.788675
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	3.46410	0.925820
$15$	−0.732051	−0.189015
$16$	−1.00000	−0.250000
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	4.46410	1.05220
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	0.267949	0.0599153
$21$	9.46410	2.06524
$22$	4.73205	1.00888
$23$	−0.732051	−0.152643	−0.0763216	−	0.997083i	$-0.524318\pi$
$23$	−0.732051	−0.152643	−0.0763216	+	0.997083i	$0.524318\pi$
$24$	−8.19615	−1.67303
$25$	−4.92820	−0.985641
$26$	−3.46410	−0.679366
$27$	4.00000	0.769800
$28$	−3.46410	−0.654654
$29$	−0.267949	−0.0497569	−0.0248785	−	0.999690i	$-0.507920\pi$
$29$	−0.267949	−0.0497569	−0.0248785	+	0.999690i	$0.507920\pi$
$30$	−0.732051	−0.133654
$31$	8.19615	1.47207	0.736036	−	0.676942i	$-0.236695\pi$
$31$	8.19615	1.47207	0.736036	+	0.676942i	$0.236695\pi$
$32$	5.00000	0.883883
$33$	12.9282	2.25051
$34$	3.73205	0.640041
$35$	−0.928203	−0.156895
$36$	−4.46410	−0.744017
$37$	0	0
$37$	0	0
$38$	−1.26795	−0.205689
$39$	−9.46410	−1.51547
$40$	0.803848	0.127099
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	9.46410	1.46034
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	−4.73205	−0.713384
$45$	−1.19615	−0.178312
$46$	−0.732051	−0.107935
$47$	−2.19615	−0.320342	−0.160171	−	0.987089i	$-0.551205\pi$
$47$	−2.19615	−0.320342	−0.160171	+	0.987089i	$0.551205\pi$
$48$	−2.73205	−0.394338
$49$	5.00000	0.714286
$50$	−4.92820	−0.696953
$51$	10.1962	1.42775
$52$	3.46410	0.480384
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	4.00000	0.544331
$55$	−1.26795	−0.170970
$56$	−10.3923	−1.38873
$57$	−3.46410	−0.458831
$58$	−0.267949	−0.0351835
$59$	−13.4641	−1.75288	−0.876438	−	0.481514i	$-0.840087\pi$
$59$	−13.4641	−1.75288	−0.876438	+	0.481514i	$0.840087\pi$
$60$	0.732051	0.0945074
$61$	4.26795	0.546455	0.273227	−	0.961949i	$-0.411909\pi$
$61$	4.26795	0.546455	0.273227	+	0.961949i	$0.411909\pi$
$62$	8.19615	1.04091
$63$	15.4641	1.94829
$64$	7.00000	0.875000
$65$	0.928203	0.115129
$66$	12.9282	1.59135
$67$	7.26795	0.887921	0.443961	−	0.896046i	$-0.353573\pi$
$67$	7.26795	0.887921	0.443961	+	0.896046i	$0.353573\pi$
$68$	−3.73205	−0.452578
$69$	−2.00000	−0.240772
$70$	−0.928203	−0.110942
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−13.3923	−1.57830
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−13.4641	−1.55470
$76$	1.26795	0.145444
$77$	16.3923	1.86808
$78$	−9.46410	−1.07160
$79$	−3.80385	−0.427966	−0.213983	−	0.976837i	$-0.568644\pi$
$79$	−3.80385	−0.427966	−0.213983	+	0.976837i	$0.568644\pi$
$80$	0.267949	0.0299576
$81$	−2.46410	−0.273789
$82$	−3.00000	−0.331295
$83$	−8.19615	−0.899645	−0.449822	−	0.893118i	$-0.648513\pi$
$83$	−8.19615	−0.899645	−0.449822	+	0.893118i	$0.648513\pi$
$84$	−9.46410	−1.03262
$85$	−1.00000	−0.108465
$86$	3.46410	0.373544
$87$	−0.732051	−0.0784841
$88$	−14.1962	−1.51331
$89$	−9.19615	−0.974790	−0.487395	−	0.873182i	$-0.662053\pi$
$89$	−9.19615	−0.974790	−0.487395	+	0.873182i	$0.662053\pi$
$90$	−1.19615	−0.126086
$91$	−12.0000	−1.25794
$92$	0.732051	0.0763216
$93$	22.3923	2.32197
$94$	−2.19615	−0.226516
$95$	0.339746	0.0348572
$96$	13.6603	1.39419
$97$	−4.26795	−0.433345	−0.216672	−	0.976244i	$-0.569520\pi$
$97$	−4.26795	−0.433345	−0.216672	+	0.976244i	$0.569520\pi$
$98$	5.00000	0.505076
$99$	21.1244	2.12308
$100$	4.92820	0.492820
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	10.1962	1.00957
$103$	12.9282	1.27385	0.636927	−	0.770924i	$-0.280205\pi$
$103$	12.9282	1.27385	0.636927	+	0.770924i	$0.280205\pi$
$104$	10.3923	1.01905
$105$	−2.53590	−0.247478
$106$	−2.53590	−0.246308
$107$	−12.9282	−1.24982	−0.624908	−	0.780698i	$-0.714864\pi$
$107$	−12.9282	−1.24982	−0.624908	+	0.780698i	$0.714864\pi$
$108$	−4.00000	−0.384900
$109$	−6.80385	−0.651690	−0.325845	−	0.945423i	$-0.605649\pi$
$109$	−6.80385	−0.651690	−0.325845	+	0.945423i	$0.605649\pi$
$110$	−1.26795	−0.120894
$111$	0	0
$112$	−3.46410	−0.327327
$113$	4.53590	0.426701	0.213351	−	0.976976i	$-0.431562\pi$
$113$	4.53590	0.426701	0.213351	+	0.976976i	$0.431562\pi$
$114$	−3.46410	−0.324443
$115$	0.196152	0.0182913
$116$	0.267949	0.0248785
$117$	−15.4641	−1.42966
$118$	−13.4641	−1.23947
$119$	12.9282	1.18513
$120$	2.19615	0.200480
$121$	11.3923	1.03566
$122$	4.26795	0.386402
$123$	−8.19615	−0.739022
$124$	−8.19615	−0.736036
$125$	2.66025	0.237940
$126$	15.4641	1.37765
$127$	−12.1962	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$127$	−12.1962	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$128$	−3.00000	−0.265165
$129$	9.46410	0.833268
$130$	0.928203	0.0814088
$131$	10.5885	0.925118	0.462559	−	0.886589i	$-0.346931\pi$
$131$	10.5885	0.925118	0.462559	+	0.886589i	$0.346931\pi$
$132$	−12.9282	−1.12526
$133$	−4.39230	−0.380861
$134$	7.26795	0.627855
$135$	−1.07180	−0.0922456
$136$	−11.1962	−0.960062
$137$	−6.46410	−0.552265	−0.276133	−	0.961120i	$-0.589053\pi$
$137$	−6.46410	−0.552265	−0.276133	+	0.961120i	$0.589053\pi$
$138$	−2.00000	−0.170251
$139$	−12.5885	−1.06774	−0.533870	−	0.845567i	$-0.679263\pi$
$139$	−12.5885	−1.06774	−0.533870	+	0.845567i	$0.679263\pi$
$140$	0.928203	0.0784475
$141$	−6.00000	−0.505291
$142$	6.00000	0.503509
$143$	−16.3923	−1.37079
$144$	−4.46410	−0.372008
$145$	0.0717968	0.00596240
$146$	0	0
$147$	13.6603	1.12668
$148$	0	0
$149$	5.53590	0.453518	0.226759	−	0.973951i	$-0.427187\pi$
$149$	5.53590	0.453518	0.226759	+	0.973951i	$0.427187\pi$
$150$	−13.4641	−1.09934
$151$	−16.5885	−1.34995	−0.674975	−	0.737841i	$-0.735845\pi$
$151$	−16.5885	−1.34995	−0.674975	+	0.737841i	$0.735845\pi$
$152$	3.80385	0.308533
$153$	16.6603	1.34690
$154$	16.3923	1.32093
$155$	−2.19615	−0.176399
$156$	9.46410	0.757735
$157$	−14.3205	−1.14290	−0.571450	−	0.820637i	$-0.693619\pi$
$157$	−14.3205	−1.14290	−0.571450	+	0.820637i	$0.693619\pi$
$158$	−3.80385	−0.302618
$159$	−6.92820	−0.549442
$160$	−1.33975	−0.105916
$161$	−2.53590	−0.199857
$162$	−2.46410	−0.193598
$163$	−22.3923	−1.75390	−0.876950	−	0.480581i	$-0.840426\pi$
$163$	−22.3923	−1.75390	−0.876950	+	0.480581i	$0.840426\pi$
$164$	3.00000	0.234261
$165$	−3.46410	−0.269680
$166$	−8.19615	−0.636145
$167$	−10.9282	−0.845650	−0.422825	−	0.906211i	$-0.638961\pi$
$167$	−10.9282	−0.845650	−0.422825	+	0.906211i	$0.638961\pi$
$168$	−28.3923	−2.19051
$169$	−1.00000	−0.0769231
$170$	−1.00000	−0.0766965
$171$	−5.66025	−0.432850
$172$	−3.46410	−0.264135
$173$	−9.92820	−0.754827	−0.377414	−	0.926045i	$-0.623187\pi$
$173$	−9.92820	−0.754827	−0.377414	+	0.926045i	$0.623187\pi$
$174$	−0.732051	−0.0554966
$175$	−17.0718	−1.29051
$176$	−4.73205	−0.356692
$177$	−36.7846	−2.76490
$178$	−9.19615	−0.689281
$179$	−1.46410	−0.109432	−0.0547160	−	0.998502i	$-0.517425\pi$
$179$	−1.46410	−0.109432	−0.0547160	+	0.998502i	$0.517425\pi$
$180$	1.19615	0.0891559
$181$	9.92820	0.737958	0.368979	−	0.929438i	$-0.379708\pi$
$181$	9.92820	0.737958	0.368979	+	0.929438i	$0.379708\pi$
$182$	−12.0000	−0.889499
$183$	11.6603	0.861951
$184$	2.19615	0.161903
$185$	0	0
$186$	22.3923	1.64188
$187$	17.6603	1.29145
$188$	2.19615	0.160171
$189$	13.8564	1.00791
$190$	0.339746	0.0246478
$191$	−6.19615	−0.448338	−0.224169	−	0.974550i	$-0.571967\pi$
$191$	−6.19615	−0.448338	−0.224169	+	0.974550i	$0.571967\pi$
$192$	19.1244	1.38018
$193$	−14.6603	−1.05527	−0.527634	−	0.849472i	$-0.676921\pi$
$193$	−14.6603	−1.05527	−0.527634	+	0.849472i	$0.676921\pi$
$194$	−4.26795	−0.306421
$195$	2.53590	0.181599
$196$	−5.00000	−0.357143
$197$	27.9282	1.98980	0.994901	−	0.100856i	$-0.0321581\pi$
$197$	27.9282	1.98980	0.994901	+	0.100856i	$0.0321581\pi$
$198$	21.1244	1.50124
$199$	11.0718	0.784859	0.392429	−	0.919782i	$-0.371635\pi$
$199$	11.0718	0.784859	0.392429	+	0.919782i	$0.371635\pi$
$200$	14.7846	1.04543
$201$	19.8564	1.40056
$202$	9.00000	0.633238
$203$	−0.928203	−0.0651471
$204$	−10.1962	−0.713873
$205$	0.803848	0.0561432
$206$	12.9282	0.900751
$207$	−3.26795	−0.227138
$208$	3.46410	0.240192
$209$	−6.00000	−0.415029
$210$	−2.53590	−0.174994
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	2.53590	0.174166
$213$	16.3923	1.12318
$214$	−12.9282	−0.883754
$215$	−0.928203	−0.0633029
$216$	−12.0000	−0.816497
$217$	28.3923	1.92740
$218$	−6.80385	−0.460815
$219$	0	0
$220$	1.26795	0.0854851
$221$	−12.9282	−0.869645
$222$	0	0
$223$	4.58846	0.307266	0.153633	−	0.988128i	$-0.450903\pi$
$223$	4.58846	0.307266	0.153633	+	0.988128i	$0.450903\pi$
$224$	17.3205	1.15728
$225$	−22.0000	−1.46667
$226$	4.53590	0.301723
$227$	−23.1244	−1.53482	−0.767409	−	0.641158i	$-0.778454\pi$
$227$	−23.1244	−1.53482	−0.767409	+	0.641158i	$0.778454\pi$
$228$	3.46410	0.229416
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0.196152	0.0129339
$231$	44.7846	2.94661
$232$	0.803848	0.0527752
$233$	10.8564	0.711227	0.355613	−	0.934633i	$-0.384272\pi$
$233$	10.8564	0.711227	0.355613	+	0.934633i	$0.384272\pi$
$234$	−15.4641	−1.01092
$235$	0.588457	0.0383867
$236$	13.4641	0.876438
$237$	−10.3923	−0.675053
$238$	12.9282	0.838011
$239$	16.9282	1.09499	0.547497	−	0.836808i	$-0.315581\pi$
$239$	16.9282	1.09499	0.547497	+	0.836808i	$0.315581\pi$
$240$	0.732051	0.0472537
$241$	15.4641	0.996130	0.498065	−	0.867140i	$-0.334044\pi$
$241$	15.4641	0.996130	0.498065	+	0.867140i	$0.334044\pi$
$242$	11.3923	0.732325
$243$	−18.7321	−1.20166
$244$	−4.26795	−0.273227
$245$	−1.33975	−0.0855932
$246$	−8.19615	−0.522568
$247$	4.39230	0.279476
$248$	−24.5885	−1.56137
$249$	−22.3923	−1.41905
$250$	2.66025	0.168249
$251$	−25.7128	−1.62298	−0.811489	−	0.584367i	$-0.801343\pi$
$251$	−25.7128	−1.62298	−0.811489	+	0.584367i	$0.801343\pi$
$252$	−15.4641	−0.974147
$253$	−3.46410	−0.217786
$254$	−12.1962	−0.765255
$255$	−2.73205	−0.171088
$256$	−17.0000	−1.06250
$257$	19.5885	1.22189	0.610947	−	0.791671i	$-0.290789\pi$
$257$	19.5885	1.22189	0.610947	+	0.791671i	$0.290789\pi$
$258$	9.46410	0.589209
$259$	0	0
$260$	−0.928203	−0.0575647
$261$	−1.19615	−0.0740400
$262$	10.5885	0.654157
$263$	7.60770	0.469111	0.234555	−	0.972103i	$-0.424637\pi$
$263$	7.60770	0.469111	0.234555	+	0.972103i	$0.424637\pi$
$264$	−38.7846	−2.38703
$265$	0.679492	0.0417409
$266$	−4.39230	−0.269309
$267$	−25.1244	−1.53759
$268$	−7.26795	−0.443961
$269$	14.5359	0.886269	0.443135	−	0.896455i	$-0.353866\pi$
$269$	14.5359	0.886269	0.443135	+	0.896455i	$0.353866\pi$
$270$	−1.07180	−0.0652275
$271$	−11.6603	−0.708310	−0.354155	−	0.935187i	$-0.615231\pi$
$271$	−11.6603	−0.708310	−0.354155	+	0.935187i	$0.615231\pi$
$272$	−3.73205	−0.226289
$273$	−32.7846	−1.98421
$274$	−6.46410	−0.390511
$275$	−23.3205	−1.40628
$276$	2.00000	0.120386
$277$	−10.2679	−0.616941	−0.308471	−	0.951234i	$-0.599817\pi$
$277$	−10.2679	−0.616941	−0.308471	+	0.951234i	$0.599817\pi$
$278$	−12.5885	−0.755005
$279$	36.5885	2.19049
$280$	2.78461	0.166412
$281$	33.1962	1.98032	0.990158	−	0.139953i	$-0.0446950\pi$
$281$	33.1962	1.98032	0.990158	+	0.139953i	$0.0446950\pi$
$282$	−6.00000	−0.357295
$283$	10.3923	0.617758	0.308879	−	0.951101i	$-0.400046\pi$
$283$	10.3923	0.617758	0.308879	+	0.951101i	$0.400046\pi$
$284$	−6.00000	−0.356034
$285$	0.928203	0.0549820
$286$	−16.3923	−0.969297
$287$	−10.3923	−0.613438
$288$	22.3205	1.31525
$289$	−3.07180	−0.180694
$290$	0.0717968	0.00421605
$291$	−11.6603	−0.683536
$292$	0	0
$293$	28.8564	1.68581	0.842905	−	0.538063i	$-0.180844\pi$
$293$	28.8564	1.68581	0.842905	+	0.538063i	$0.180844\pi$
$294$	13.6603	0.796682
$295$	3.60770	0.210048
$296$	0	0
$297$	18.9282	1.09833
$298$	5.53590	0.320686
$299$	2.53590	0.146655
$300$	13.4641	0.777350
$301$	12.0000	0.691669
$302$	−16.5885	−0.954558
$303$	24.5885	1.41257
$304$	1.26795	0.0727219
$305$	−1.14359	−0.0654820
$306$	16.6603	0.952403
$307$	14.3923	0.821412	0.410706	−	0.911768i	$-0.365282\pi$
$307$	14.3923	0.821412	0.410706	+	0.911768i	$0.365282\pi$
$308$	−16.3923	−0.934038
$309$	35.3205	2.00931
$310$	−2.19615	−0.124733
$311$	30.3923	1.72339	0.861695	−	0.507427i	$-0.169403\pi$
$311$	30.3923	1.72339	0.861695	+	0.507427i	$0.169403\pi$
$312$	28.3923	1.60740
$313$	−15.5885	−0.881112	−0.440556	−	0.897725i	$-0.645219\pi$
$313$	−15.5885	−0.881112	−0.440556	+	0.897725i	$0.645219\pi$
$314$	−14.3205	−0.808153
$315$	−4.14359	−0.233465
$316$	3.80385	0.213983
$317$	15.9282	0.894617	0.447309	−	0.894380i	$-0.352383\pi$
$317$	15.9282	0.894617	0.447309	+	0.894380i	$0.352383\pi$
$318$	−6.92820	−0.388514
$319$	−1.26795	−0.0709915
$320$	−1.87564	−0.104852
$321$	−35.3205	−1.97140
$322$	−2.53590	−0.141320
$323$	−4.73205	−0.263298
$324$	2.46410	0.136895
$325$	17.0718	0.946973
$326$	−22.3923	−1.24020
$327$	−18.5885	−1.02794
$328$	9.00000	0.496942
$329$	−7.60770	−0.419426
$330$	−3.46410	−0.190693
$331$	32.7846	1.80201	0.901003	−	0.433814i	$-0.142832\pi$
$331$	32.7846	1.80201	0.901003	+	0.433814i	$0.142832\pi$
$332$	8.19615	0.449822
$333$	0	0
$334$	−10.9282	−0.597965
$335$	−1.94744	−0.106400
$336$	−9.46410	−0.516309
$337$	−29.3923	−1.60110	−0.800550	−	0.599265i	$-0.795459\pi$
$337$	−29.3923	−1.60110	−0.800550	+	0.599265i	$0.795459\pi$
$338$	−1.00000	−0.0543928
$339$	12.3923	0.673058
$340$	1.00000	0.0542326
$341$	38.7846	2.10030
$342$	−5.66025	−0.306071
$343$	−6.92820	−0.374088
$344$	−10.3923	−0.560316
$345$	0.535898	0.0288518
$346$	−9.92820	−0.533744
$347$	−12.7321	−0.683492	−0.341746	−	0.939792i	$-0.611018\pi$
$347$	−12.7321	−0.683492	−0.341746	+	0.939792i	$0.611018\pi$
$348$	0.732051	0.0392420
$349$	8.07180	0.432073	0.216037	−	0.976385i	$-0.430687\pi$
$349$	8.07180	0.432073	0.216037	+	0.976385i	$0.430687\pi$
$350$	−17.0718	−0.912526
$351$	−13.8564	−0.739600
$352$	23.6603	1.26110
$353$	−12.2679	−0.652957	−0.326479	−	0.945205i	$-0.605862\pi$
$353$	−12.2679	−0.652957	−0.326479	+	0.945205i	$0.605862\pi$
$354$	−36.7846	−1.95508
$355$	−1.60770	−0.0853276
$356$	9.19615	0.487395
$357$	35.3205	1.86936
$358$	−1.46410	−0.0773802
$359$	16.3923	0.865153	0.432576	−	0.901597i	$-0.357605\pi$
$359$	16.3923	0.865153	0.432576	+	0.901597i	$0.357605\pi$
$360$	3.58846	0.189128
$361$	−17.3923	−0.915384
$362$	9.92820	0.521815
$363$	31.1244	1.63361
$364$	12.0000	0.628971
$365$	0	0
$366$	11.6603	0.609491
$367$	−20.3923	−1.06447	−0.532235	−	0.846597i	$-0.678648\pi$
$367$	−20.3923	−1.06447	−0.532235	+	0.846597i	$0.678648\pi$
$368$	0.732051	0.0381608
$369$	−13.3923	−0.697176
$370$	0	0
$371$	−8.78461	−0.456074
$372$	−22.3923	−1.16099
$373$	−17.5359	−0.907974	−0.453987	−	0.891008i	$-0.649999\pi$
$373$	−17.5359	−0.907974	−0.453987	+	0.891008i	$0.649999\pi$
$374$	17.6603	0.913190
$375$	7.26795	0.375315
$376$	6.58846	0.339774
$377$	0.928203	0.0478049
$378$	13.8564	0.712697
$379$	8.53590	0.438460	0.219230	−	0.975673i	$-0.429646\pi$
$379$	8.53590	0.438460	0.219230	+	0.975673i	$0.429646\pi$
$380$	−0.339746	−0.0174286
$381$	−33.3205	−1.70706
$382$	−6.19615	−0.317023
$383$	15.8564	0.810225	0.405112	−	0.914267i	$-0.367232\pi$
$383$	15.8564	0.810225	0.405112	+	0.914267i	$0.367232\pi$
$384$	−8.19615	−0.418258
$385$	−4.39230	−0.223853
$386$	−14.6603	−0.746187
$387$	15.4641	0.786084
$388$	4.26795	0.216672
$389$	−31.1962	−1.58171	−0.790854	−	0.612005i	$-0.790363\pi$
$389$	−31.1962	−1.58171	−0.790854	+	0.612005i	$0.790363\pi$
$390$	2.53590	0.128410
$391$	−2.73205	−0.138166
$392$	−15.0000	−0.757614
$393$	28.9282	1.45923
$394$	27.9282	1.40700
$395$	1.01924	0.0512834
$396$	−21.1244	−1.06154
$397$	21.2487	1.06644	0.533221	−	0.845976i	$-0.320981\pi$
$397$	21.2487	1.06644	0.533221	+	0.845976i	$0.320981\pi$
$398$	11.0718	0.554979
$399$	−12.0000	−0.600751
$400$	4.92820	0.246410
$401$	5.60770	0.280035	0.140017	−	0.990149i	$-0.455284\pi$
$401$	5.60770	0.280035	0.140017	+	0.990149i	$0.455284\pi$
$402$	19.8564	0.990348
$403$	−28.3923	−1.41432
$404$	−9.00000	−0.447767
$405$	0.660254	0.0328083
$406$	−0.928203	−0.0460660
$407$	0	0
$408$	−30.5885	−1.51435
$409$	−8.66025	−0.428222	−0.214111	−	0.976809i	$-0.568685\pi$
$409$	−8.66025	−0.428222	−0.214111	+	0.976809i	$0.568685\pi$
$410$	0.803848	0.0396992
$411$	−17.6603	−0.871116
$412$	−12.9282	−0.636927
$413$	−46.6410	−2.29505
$414$	−3.26795	−0.160611
$415$	2.19615	0.107805
$416$	−17.3205	−0.849208
$417$	−34.3923	−1.68420
$418$	−6.00000	−0.293470
$419$	17.3205	0.846162	0.423081	−	0.906092i	$-0.360949\pi$
$419$	17.3205	0.846162	0.423081	+	0.906092i	$0.360949\pi$
$420$	2.53590	0.123739
$421$	−16.2679	−0.792851	−0.396426	−	0.918067i	$-0.629750\pi$
$421$	−16.2679	−0.792851	−0.396426	+	0.918067i	$0.629750\pi$
$422$	−10.0000	−0.486792
$423$	−9.80385	−0.476679
$424$	7.60770	0.369462
$425$	−18.3923	−0.892158
$426$	16.3923	0.794210
$427$	14.7846	0.715477
$428$	12.9282	0.624908
$429$	−44.7846	−2.16222
$430$	−0.928203	−0.0447619
$431$	−11.5167	−0.554738	−0.277369	−	0.960763i	$-0.589462\pi$
$431$	−11.5167	−0.554738	−0.277369	+	0.960763i	$0.589462\pi$
$432$	−4.00000	−0.192450
$433$	27.0000	1.29754	0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000	1.29754	0.648769	+	0.760986i	$0.275284\pi$
$434$	28.3923	1.36287
$435$	0.196152	0.00940479
$436$	6.80385	0.325845
$437$	0.928203	0.0444020
$438$	0	0
$439$	14.5359	0.693761	0.346880	−	0.937909i	$-0.387241\pi$
$439$	14.5359	0.693761	0.346880	+	0.937909i	$0.387241\pi$
$440$	3.80385	0.181341
$441$	22.3205	1.06288
$442$	−12.9282	−0.614932
$443$	−33.4641	−1.58993	−0.794964	−	0.606657i	$-0.792510\pi$
$443$	−33.4641	−1.58993	−0.794964	+	0.606657i	$0.792510\pi$
$444$	0	0
$445$	2.46410	0.116810
$446$	4.58846	0.217270
$447$	15.1244	0.715357
$448$	24.2487	1.14564
$449$	32.2487	1.52191	0.760955	−	0.648804i	$-0.224731\pi$
$449$	32.2487	1.52191	0.760955	+	0.648804i	$0.224731\pi$
$450$	−22.0000	−1.03709
$451$	−14.1962	−0.668471
$452$	−4.53590	−0.213351
$453$	−45.3205	−2.12934
$454$	−23.1244	−1.08528
$455$	3.21539	0.150740
$456$	10.3923	0.486664
$457$	36.1244	1.68983	0.844913	−	0.534904i	$-0.179652\pi$
$457$	36.1244	1.68983	0.844913	+	0.534904i	$0.179652\pi$
$458$	7.00000	0.327089
$459$	14.9282	0.696789
$460$	−0.196152	−0.00914565
$461$	−16.5359	−0.770154	−0.385077	−	0.922885i	$-0.625825\pi$
$461$	−16.5359	−0.770154	−0.385077	+	0.922885i	$0.625825\pi$
$462$	44.7846	2.08357
$463$	15.4641	0.718678	0.359339	−	0.933207i	$-0.383002\pi$
$463$	15.4641	0.718678	0.359339	+	0.933207i	$0.383002\pi$
$464$	0.267949	0.0124392
$465$	−6.00000	−0.278243
$466$	10.8564	0.502913
$467$	−3.32051	−0.153655	−0.0768274	−	0.997044i	$-0.524479\pi$
$467$	−3.32051	−0.153655	−0.0768274	+	0.997044i	$0.524479\pi$
$468$	15.4641	0.714828
$469$	25.1769	1.16256
$470$	0.588457	0.0271435
$471$	−39.1244	−1.80276
$472$	40.3923	1.85921
$473$	16.3923	0.753719
$474$	−10.3923	−0.477334
$475$	6.24871	0.286711
$476$	−12.9282	−0.592563
$477$	−11.3205	−0.518330
$478$	16.9282	0.774278
$479$	−39.3205	−1.79660	−0.898300	−	0.439383i	$-0.855197\pi$
$479$	−39.3205	−1.79660	−0.898300	+	0.439383i	$0.855197\pi$
$480$	−3.66025	−0.167067
$481$	0	0
$482$	15.4641	0.704371
$483$	−6.92820	−0.315244
$484$	−11.3923	−0.517832
$485$	1.14359	0.0519279
$486$	−18.7321	−0.849703
$487$	32.4449	1.47022	0.735109	−	0.677949i	$-0.237131\pi$
$487$	32.4449	1.47022	0.735109	+	0.677949i	$0.237131\pi$
$488$	−12.8038	−0.579603
$489$	−61.1769	−2.76652
$490$	−1.33975	−0.0605236
$491$	31.2679	1.41110	0.705551	−	0.708659i	$-0.250699\pi$
$491$	31.2679	1.41110	0.705551	+	0.708659i	$0.250699\pi$
$492$	8.19615	0.369511
$493$	−1.00000	−0.0450377
$494$	4.39230	0.197619
$495$	−5.66025	−0.254409
$496$	−8.19615	−0.368018
$497$	20.7846	0.932317
$498$	−22.3923	−1.00342
$499$	12.5885	0.563537	0.281768	−	0.959482i	$-0.409079\pi$
$499$	12.5885	0.563537	0.281768	+	0.959482i	$0.409079\pi$
$500$	−2.66025	−0.118970
$501$	−29.8564	−1.33389
$502$	−25.7128	−1.14762
$503$	−8.98076	−0.400432	−0.200216	−	0.979752i	$-0.564164\pi$
$503$	−8.98076	−0.400432	−0.200216	+	0.979752i	$0.564164\pi$
$504$	−46.3923	−2.06648
$505$	−2.41154	−0.107312
$506$	−3.46410	−0.153998
$507$	−2.73205	−0.121335
$508$	12.1962	0.541117
$509$	5.78461	0.256398	0.128199	−	0.991748i	$-0.459080\pi$
$509$	5.78461	0.256398	0.128199	+	0.991748i	$0.459080\pi$
$510$	−2.73205	−0.120977
$511$	0	0
$512$	−11.0000	−0.486136
$513$	−5.07180	−0.223925
$514$	19.5885	0.864010
$515$	−3.46410	−0.152647
$516$	−9.46410	−0.416634
$517$	−10.3923	−0.457053
$518$	0	0
$519$	−27.1244	−1.19063
$520$	−2.78461	−0.122113
$521$	−6.92820	−0.303530	−0.151765	−	0.988417i	$-0.548496\pi$
$521$	−6.92820	−0.303530	−0.151765	+	0.988417i	$0.548496\pi$
$522$	−1.19615	−0.0523542
$523$	−16.0526	−0.701929	−0.350965	−	0.936389i	$-0.614146\pi$
$523$	−16.0526	−0.701929	−0.350965	+	0.936389i	$0.614146\pi$
$524$	−10.5885	−0.462559
$525$	−46.6410	−2.03558
$526$	7.60770	0.331711
$527$	30.5885	1.33245
$528$	−12.9282	−0.562628
$529$	−22.4641	−0.976700
$530$	0.679492	0.0295152
$531$	−60.1051	−2.60834
$532$	4.39230	0.190431
$533$	10.3923	0.450141
$534$	−25.1244	−1.08724
$535$	3.46410	0.149766
$536$	−21.8038	−0.941783
$537$	−4.00000	−0.172613
$538$	14.5359	0.626687
$539$	23.6603	1.01912
$540$	1.07180	0.0461228
$541$	−4.26795	−0.183493	−0.0917467	−	0.995782i	$-0.529245\pi$
$541$	−4.26795	−0.183493	−0.0917467	+	0.995782i	$0.529245\pi$
$542$	−11.6603	−0.500851
$543$	27.1244	1.16402
$544$	18.6603	0.800052
$545$	1.82309	0.0780924
$546$	−32.7846	−1.40305
$547$	−3.80385	−0.162641	−0.0813204	−	0.996688i	$-0.525914\pi$
$547$	−3.80385	−0.162641	−0.0813204	+	0.996688i	$0.525914\pi$
$548$	6.46410	0.276133
$549$	19.0526	0.813143
$550$	−23.3205	−0.994390
$551$	0.339746	0.0144737
$552$	6.00000	0.255377
$553$	−13.1769	−0.560339
$554$	−10.2679	−0.436243
$555$	0	0
$556$	12.5885	0.533870
$557$	17.7321	0.751331	0.375666	−	0.926755i	$-0.377414\pi$
$557$	17.7321	0.751331	0.375666	+	0.926755i	$0.377414\pi$
$558$	36.5885	1.54891
$559$	−12.0000	−0.507546
$560$	0.928203	0.0392237
$561$	48.2487	2.03706
$562$	33.1962	1.40030
$563$	34.5885	1.45773	0.728865	−	0.684658i	$-0.240048\pi$
$563$	34.5885	1.45773	0.728865	+	0.684658i	$0.240048\pi$
$564$	6.00000	0.252646
$565$	−1.21539	−0.0511319
$566$	10.3923	0.436821
$567$	−8.53590	−0.358474
$568$	−18.0000	−0.755263
$569$	36.5167	1.53086	0.765429	−	0.643520i	$-0.222527\pi$
$569$	36.5167	1.53086	0.765429	+	0.643520i	$0.222527\pi$
$570$	0.928203	0.0388782
$571$	−31.5167	−1.31893	−0.659466	−	0.751735i	$-0.729217\pi$
$571$	−31.5167	−1.31893	−0.659466	+	0.751735i	$0.729217\pi$
$572$	16.3923	0.685397
$573$	−16.9282	−0.707186
$574$	−10.3923	−0.433766
$575$	3.60770	0.150451
$576$	31.2487	1.30203
$577$	12.9282	0.538208	0.269104	−	0.963111i	$-0.413272\pi$
$577$	12.9282	0.538208	0.269104	+	0.963111i	$0.413272\pi$
$578$	−3.07180	−0.127770
$579$	−40.0526	−1.66453
$580$	−0.0717968	−0.00298120
$581$	−28.3923	−1.17791
$582$	−11.6603	−0.483333
$583$	−12.0000	−0.496989
$584$	0	0
$585$	4.14359	0.171317
$586$	28.8564	1.19205
$587$	−16.3397	−0.674413	−0.337207	−	0.941431i	$-0.609482\pi$
$587$	−16.3397	−0.674413	−0.337207	+	0.941431i	$0.609482\pi$
$588$	−13.6603	−0.563339
$589$	−10.3923	−0.428207
$590$	3.60770	0.148526
$591$	76.3013	3.13861
$592$	0	0
$593$	17.7846	0.730326	0.365163	−	0.930944i	$-0.381013\pi$
$593$	17.7846	0.730326	0.365163	+	0.930944i	$0.381013\pi$
$594$	18.9282	0.776634
$595$	−3.46410	−0.142014
$596$	−5.53590	−0.226759
$597$	30.2487	1.23800
$598$	2.53590	0.103701
$599$	7.94744	0.324724	0.162362	−	0.986731i	$-0.448089\pi$
$599$	7.94744	0.324724	0.162362	+	0.986731i	$0.448089\pi$
$600$	40.3923	1.64901
$601$	31.3923	1.28052	0.640259	−	0.768159i	$-0.278827\pi$
$601$	31.3923	1.28052	0.640259	+	0.768159i	$0.278827\pi$
$602$	12.0000	0.489083
$603$	32.4449	1.32126
$604$	16.5885	0.674975
$605$	−3.05256	−0.124104
$606$	24.5885	0.998838
$607$	−5.66025	−0.229743	−0.114871	−	0.993380i	$-0.536646\pi$
$607$	−5.66025	−0.229743	−0.114871	+	0.993380i	$0.536646\pi$
$608$	−6.33975	−0.257111
$609$	−2.53590	−0.102760
$610$	−1.14359	−0.0463027
$611$	7.60770	0.307774
$612$	−16.6603	−0.673451
$613$	−3.39230	−0.137014	−0.0685070	−	0.997651i	$-0.521824\pi$
$613$	−3.39230	−0.137014	−0.0685070	+	0.997651i	$0.521824\pi$
$614$	14.3923	0.580826
$615$	2.19615	0.0885574
$616$	−49.1769	−1.98139
$617$	25.1769	1.01358	0.506792	−	0.862068i	$-0.330831\pi$
$617$	25.1769	1.01358	0.506792	+	0.862068i	$0.330831\pi$
$618$	35.3205	1.42080
$619$	−6.78461	−0.272696	−0.136348	−	0.990661i	$-0.543537\pi$
$619$	−6.78461	−0.272696	−0.136348	+	0.990661i	$0.543537\pi$
$620$	2.19615	0.0881996
$621$	−2.92820	−0.117505
$622$	30.3923	1.21862
$623$	−31.8564	−1.27630
$624$	9.46410	0.378867
$625$	23.9282	0.957128
$626$	−15.5885	−0.623040
$627$	−16.3923	−0.654646
$628$	14.3205	0.571450
$629$	0	0
$630$	−4.14359	−0.165085
$631$	−19.2679	−0.767045	−0.383522	−	0.923532i	$-0.625289\pi$
$631$	−19.2679	−0.767045	−0.383522	+	0.923532i	$0.625289\pi$
$632$	11.4115	0.453927
$633$	−27.3205	−1.08589
$634$	15.9282	0.632590
$635$	3.26795	0.129685
$636$	6.92820	0.274721
$637$	−17.3205	−0.686264
$638$	−1.26795	−0.0501986
$639$	26.7846	1.05958
$640$	0.803848	0.0317749
$641$	−17.7846	−0.702450	−0.351225	−	0.936291i	$-0.614235\pi$
$641$	−17.7846	−0.702450	−0.351225	+	0.936291i	$0.614235\pi$
$642$	−35.3205	−1.39399
$643$	−5.66025	−0.223219	−0.111609	−	0.993752i	$-0.535601\pi$
$643$	−5.66025	−0.223219	−0.111609	+	0.993752i	$0.535601\pi$
$644$	2.53590	0.0999284
$645$	−2.53590	−0.0998509
$646$	−4.73205	−0.186180
$647$	43.4641	1.70875	0.854375	−	0.519657i	$-0.173940\pi$
$647$	43.4641	1.70875	0.854375	+	0.519657i	$0.173940\pi$
$648$	7.39230	0.290397
$649$	−63.7128	−2.50095
$650$	17.0718	0.669611
$651$	77.5692	3.04018
$652$	22.3923	0.876950
$653$	−2.80385	−0.109723	−0.0548615	−	0.998494i	$-0.517472\pi$
$653$	−2.80385	−0.109723	−0.0548615	+	0.998494i	$0.517472\pi$
$654$	−18.5885	−0.726866
$655$	−2.83717	−0.110857
$656$	3.00000	0.117130
$657$	0	0
$658$	−7.60770	−0.296579
$659$	−9.46410	−0.368669	−0.184335	−	0.982864i	$-0.559013\pi$
$659$	−9.46410	−0.368669	−0.184335	+	0.982864i	$0.559013\pi$
$660$	3.46410	0.134840
$661$	−12.8038	−0.498012	−0.249006	−	0.968502i	$-0.580104\pi$
$661$	−12.8038	−0.498012	−0.249006	+	0.968502i	$0.580104\pi$
$662$	32.7846	1.27421
$663$	−35.3205	−1.37173
$664$	24.5885	0.954217
$665$	1.17691	0.0456388
$666$	0	0
$667$	0.196152	0.00759505
$668$	10.9282	0.422825
$669$	12.5359	0.484666
$670$	−1.94744	−0.0752362
$671$	20.1962	0.779664
$672$	47.3205	1.82543
$673$	15.7128	0.605684	0.302842	−	0.953041i	$-0.402064\pi$
$673$	15.7128	0.605684	0.302842	+	0.953041i	$0.402064\pi$
$674$	−29.3923	−1.13215
$675$	−19.7128	−0.758747
$676$	1.00000	0.0384615
$677$	44.3205	1.70338	0.851688	−	0.524050i	$-0.175579\pi$
$677$	44.3205	1.70338	0.851688	+	0.524050i	$0.175579\pi$
$678$	12.3923	0.475924
$679$	−14.7846	−0.567381
$680$	3.00000	0.115045
$681$	−63.1769	−2.42094
$682$	38.7846	1.48514
$683$	31.0718	1.18893	0.594465	−	0.804122i	$-0.297364\pi$
$683$	31.0718	1.18893	0.594465	+	0.804122i	$0.297364\pi$
$684$	5.66025	0.216425
$685$	1.73205	0.0661783
$686$	−6.92820	−0.264520
$687$	19.1244	0.729640
$688$	−3.46410	−0.132068
$689$	8.78461	0.334667
$690$	0.535898	0.0204013
$691$	−16.0526	−0.610668	−0.305334	−	0.952245i	$-0.598768\pi$
$691$	−16.0526	−0.610668	−0.305334	+	0.952245i	$0.598768\pi$
$692$	9.92820	0.377414
$693$	73.1769	2.77976
$694$	−12.7321	−0.483302
$695$	3.37307	0.127948
$696$	2.19615	0.0832449
$697$	−11.1962	−0.424085
$698$	8.07180	0.305522
$699$	29.6603	1.12185
$700$	17.0718	0.645253
$701$	−5.85641	−0.221193	−0.110597	−	0.993865i	$-0.535276\pi$
$701$	−5.85641	−0.221193	−0.110597	+	0.993865i	$0.535276\pi$
$702$	−13.8564	−0.522976
$703$	0	0
$704$	33.1244	1.24842
$705$	1.60770	0.0605493
$706$	−12.2679	−0.461710
$707$	31.1769	1.17253
$708$	36.7846	1.38245
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−1.60770	−0.0603357
$711$	−16.9808	−0.636828
$712$	27.5885	1.03392
$713$	−6.00000	−0.224702
$714$	35.3205	1.32184
$715$	4.39230	0.164263
$716$	1.46410	0.0547160
$717$	46.2487	1.72719
$718$	16.3923	0.611755
$719$	26.4449	0.986227	0.493114	−	0.869965i	$-0.335859\pi$
$719$	26.4449	0.986227	0.493114	+	0.869965i	$0.335859\pi$
$720$	1.19615	0.0445780
$721$	44.7846	1.66787
$722$	−17.3923	−0.647275
$723$	42.2487	1.57125
$724$	−9.92820	−0.368979
$725$	1.32051	0.0490424
$726$	31.1244	1.15513
$727$	−1.85641	−0.0688503	−0.0344252	−	0.999407i	$-0.510960\pi$
$727$	−1.85641	−0.0688503	−0.0344252	+	0.999407i	$0.510960\pi$
$728$	36.0000	1.33425
$729$	−43.7846	−1.62165
$730$	0	0
$731$	12.9282	0.478167
$732$	−11.6603	−0.430975
$733$	−28.7846	−1.06318	−0.531592	−	0.847001i	$-0.678406\pi$
$733$	−28.7846	−1.06318	−0.531592	+	0.847001i	$0.678406\pi$
$734$	−20.3923	−0.752694
$735$	−3.66025	−0.135011
$736$	−3.66025	−0.134919
$737$	34.3923	1.26686
$738$	−13.3923	−0.492978
$739$	23.8038	0.875639	0.437819	−	0.899063i	$-0.355751\pi$
$739$	23.8038	0.875639	0.437819	+	0.899063i	$0.355751\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	−8.78461	−0.322493
$743$	43.2679	1.58735	0.793674	−	0.608344i	$-0.208166\pi$
$743$	43.2679	1.58735	0.793674	+	0.608344i	$0.208166\pi$
$744$	−67.1769	−2.46283
$745$	−1.48334	−0.0543454
$746$	−17.5359	−0.642035
$747$	−36.5885	−1.33870
$748$	−17.6603	−0.645723
$749$	−44.7846	−1.63639
$750$	7.26795	0.265388
$751$	9.21539	0.336274	0.168137	−	0.985764i	$-0.446225\pi$
$751$	9.21539	0.336274	0.168137	+	0.985764i	$0.446225\pi$
$752$	2.19615	0.0800854
$753$	−70.2487	−2.56001
$754$	0.928203	0.0338032
$755$	4.44486	0.161765
$756$	−13.8564	−0.503953
$757$	42.3731	1.54008	0.770038	−	0.637998i	$-0.220237\pi$
$757$	42.3731	1.54008	0.770038	+	0.637998i	$0.220237\pi$
$758$	8.53590	0.310038
$759$	−9.46410	−0.343525
$760$	−1.01924	−0.0369716
$761$	3.67949	0.133381	0.0666907	−	0.997774i	$-0.478756\pi$
$761$	3.67949	0.133381	0.0666907	+	0.997774i	$0.478756\pi$
$762$	−33.3205	−1.20707
$763$	−23.5692	−0.853263
$764$	6.19615	0.224169
$765$	−4.46410	−0.161400
$766$	15.8564	0.572915
$767$	46.6410	1.68411
$768$	−46.4449	−1.67593
$769$	−20.5359	−0.740543	−0.370272	−	0.928923i	$-0.620735\pi$
$769$	−20.5359	−0.740543	−0.370272	+	0.928923i	$0.620735\pi$
$770$	−4.39230	−0.158288
$771$	53.5167	1.92736
$772$	14.6603	0.527634
$773$	−48.0333	−1.72764	−0.863819	−	0.503802i	$-0.831934\pi$
$773$	−48.0333	−1.72764	−0.863819	+	0.503802i	$0.831934\pi$
$774$	15.4641	0.555846
$775$	−40.3923	−1.45093
$776$	12.8038	0.459631
$777$	0	0
$778$	−31.1962	−1.11844
$779$	3.80385	0.136287
$780$	−2.53590	−0.0907997
$781$	28.3923	1.01596
$782$	−2.73205	−0.0976979
$783$	−1.07180	−0.0383029
$784$	−5.00000	−0.178571
$785$	3.83717	0.136954
$786$	28.9282	1.03183
$787$	0.392305	0.0139842	0.00699208	−	0.999976i	$-0.497774\pi$
$787$	0.392305	0.0139842	0.00699208	+	0.999976i	$0.497774\pi$
$788$	−27.9282	−0.994901
$789$	20.7846	0.739952
$790$	1.01924	0.0362629
$791$	15.7128	0.558683
$792$	−63.3731	−2.25186
$793$	−14.7846	−0.525017
$794$	21.2487	0.754089
$795$	1.85641	0.0658400
$796$	−11.0718	−0.392429
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	−12.0000	−0.424795
$799$	−8.19615	−0.289959
$800$	−24.6410	−0.871191
$801$	−41.0526	−1.45052
$802$	5.60770	0.198015
$803$	0	0
$804$	−19.8564	−0.700281
$805$	0.679492	0.0239489
$806$	−28.3923	−1.00008
$807$	39.7128	1.39796
$808$	−27.0000	−0.949857
$809$	33.8564	1.19033	0.595164	−	0.803604i	$-0.297087\pi$
$809$	33.8564	1.19033	0.595164	+	0.803604i	$0.297087\pi$
$810$	0.660254	0.0231990
$811$	18.2487	0.640799	0.320399	−	0.947283i	$-0.396183\pi$
$811$	18.2487	0.640799	0.320399	+	0.947283i	$0.396183\pi$
$812$	0.928203	0.0325735
$813$	−31.8564	−1.11725
$814$	0	0
$815$	6.00000	0.210171
$816$	−10.1962	−0.356937
$817$	−4.39230	−0.153667
$818$	−8.66025	−0.302799
$819$	−53.5692	−1.87186
$820$	−0.803848	−0.0280716
$821$	49.1769	1.71629	0.858143	−	0.513411i	$-0.171618\pi$
$821$	49.1769	1.71629	0.858143	+	0.513411i	$0.171618\pi$
$822$	−17.6603	−0.615972
$823$	−3.01924	−0.105244	−0.0526220	−	0.998615i	$-0.516758\pi$
$823$	−3.01924	−0.105244	−0.0526220	+	0.998615i	$0.516758\pi$
$824$	−38.7846	−1.35113
$825$	−63.7128	−2.21820
$826$	−46.6410	−1.62285
$827$	22.9282	0.797292	0.398646	−	0.917105i	$-0.369480\pi$
$827$	22.9282	0.797292	0.398646	+	0.917105i	$0.369480\pi$
$828$	3.26795	0.113569
$829$	44.5359	1.54680	0.773398	−	0.633921i	$-0.218556\pi$
$829$	44.5359	1.54680	0.773398	+	0.633921i	$0.218556\pi$
$830$	2.19615	0.0762296
$831$	−28.0526	−0.973132
$832$	−24.2487	−0.840673
$833$	18.6603	0.646539
$834$	−34.3923	−1.19091
$835$	2.92820	0.101335
$836$	6.00000	0.207514
$837$	32.7846	1.13320
$838$	17.3205	0.598327
$839$	29.6603	1.02399	0.511993	−	0.858990i	$-0.328907\pi$
$839$	29.6603	1.02399	0.511993	+	0.858990i	$0.328907\pi$
$840$	7.60770	0.262490
$841$	−28.9282	−0.997524
$842$	−16.2679	−0.560631
$843$	90.6936	3.12365
$844$	10.0000	0.344214
$845$	0.267949	0.00921773
$846$	−9.80385	−0.337063
$847$	39.4641	1.35600
$848$	2.53590	0.0870831
$849$	28.3923	0.974421
$850$	−18.3923	−0.630851
$851$	0	0
$852$	−16.3923	−0.561591
$853$	−42.1244	−1.44231	−0.721155	−	0.692774i	$-0.756389\pi$
$853$	−42.1244	−1.44231	−0.721155	+	0.692774i	$0.756389\pi$
$854$	14.7846	0.505919
$855$	1.51666	0.0518687
$856$	38.7846	1.32563
$857$	23.7321	0.810671	0.405336	−	0.914168i	$-0.367155\pi$
$857$	23.7321	0.810671	0.405336	+	0.914168i	$0.367155\pi$
$858$	−44.7846	−1.52892
$859$	−8.53590	−0.291241	−0.145621	−	0.989341i	$-0.546518\pi$
$859$	−8.53590	−0.291241	−0.145621	+	0.989341i	$0.546518\pi$
$860$	0.928203	0.0316515
$861$	−28.3923	−0.967607
$862$	−11.5167	−0.392259
$863$	38.5359	1.31178	0.655889	−	0.754858i	$-0.272294\pi$
$863$	38.5359	1.31178	0.655889	+	0.754858i	$0.272294\pi$
$864$	20.0000	0.680414
$865$	2.66025	0.0904514
$866$	27.0000	0.917497
$867$	−8.39230	−0.285018
$868$	−28.3923	−0.963698
$869$	−18.0000	−0.610608
$870$	0.196152	0.00665019
$871$	−25.1769	−0.853087
$872$	20.4115	0.691222
$873$	−19.0526	−0.644831
$874$	0.928203	0.0313969
$875$	9.21539	0.311537
$876$	0	0
$877$	5.78461	0.195332	0.0976662	−	0.995219i	$-0.468862\pi$
$877$	5.78461	0.195332	0.0976662	+	0.995219i	$0.468862\pi$
$878$	14.5359	0.490563
$879$	78.8372	2.65911
$880$	1.26795	0.0427426
$881$	−32.3205	−1.08891	−0.544453	−	0.838791i	$-0.683263\pi$
$881$	−32.3205	−1.08891	−0.544453	+	0.838791i	$0.683263\pi$
$882$	22.3205	0.751571
$883$	−35.6603	−1.20006	−0.600032	−	0.799976i	$-0.704845\pi$
$883$	−35.6603	−1.20006	−0.600032	+	0.799976i	$0.704845\pi$
$884$	12.9282	0.434823
$885$	9.85641	0.331319
$886$	−33.4641	−1.12425
$887$	−28.9808	−0.973079	−0.486539	−	0.873659i	$-0.661741\pi$
$887$	−28.9808	−0.973079	−0.486539	+	0.873659i	$0.661741\pi$
$888$	0	0
$889$	−42.2487	−1.41698
$890$	2.46410	0.0825969
$891$	−11.6603	−0.390633
$892$	−4.58846	−0.153633
$893$	2.78461	0.0931834
$894$	15.1244	0.505834
$895$	0.392305	0.0131133
$896$	−10.3923	−0.347183
$897$	6.92820	0.231326
$898$	32.2487	1.07615
$899$	−2.19615	−0.0732458
$900$	22.0000	0.733333
$901$	−9.46410	−0.315295
$902$	−14.1962	−0.472680
$903$	32.7846	1.09100
$904$	−13.6077	−0.452585
$905$	−2.66025	−0.0884298
$906$	−45.3205	−1.50567
$907$	−1.26795	−0.0421016	−0.0210508	−	0.999778i	$-0.506701\pi$
$907$	−1.26795	−0.0421016	−0.0210508	+	0.999778i	$0.506701\pi$
$908$	23.1244	0.767409
$909$	40.1769	1.33258
$910$	3.21539	0.106589
$911$	−53.5167	−1.77309	−0.886543	−	0.462646i	$-0.846900\pi$
$911$	−53.5167	−1.77309	−0.886543	+	0.462646i	$0.846900\pi$
$912$	3.46410	0.114708
$913$	−38.7846	−1.28358
$914$	36.1244	1.19489
$915$	−3.12436	−0.103288
$916$	−7.00000	−0.231287
$917$	36.6795	1.21126
$918$	14.9282	0.492704
$919$	−4.05256	−0.133682	−0.0668408	−	0.997764i	$-0.521292\pi$
$919$	−4.05256	−0.133682	−0.0668408	+	0.997764i	$0.521292\pi$
$920$	−0.588457	−0.0194009
$921$	39.3205	1.29565
$922$	−16.5359	−0.544581
$923$	−20.7846	−0.684134
$924$	−44.7846	−1.47331
$925$	0	0
$926$	15.4641	0.508182
$927$	57.7128	1.89554
$928$	−1.33975	−0.0439793
$929$	30.4641	0.999495	0.499747	−	0.866171i	$-0.333426\pi$
$929$	30.4641	0.999495	0.499747	+	0.866171i	$0.333426\pi$
$930$	−6.00000	−0.196748
$931$	−6.33975	−0.207777
$932$	−10.8564	−0.355613
$933$	83.0333	2.71839
$934$	−3.32051	−0.108650
$935$	−4.73205	−0.154755
$936$	46.3923	1.51638
$937$	−12.6077	−0.411875	−0.205938	−	0.978565i	$-0.566024\pi$
$937$	−12.6077	−0.411875	−0.205938	+	0.978565i	$0.566024\pi$
$938$	25.1769	0.822055
$939$	−42.5885	−1.38982
$940$	−0.588457	−0.0191934
$941$	−4.17691	−0.136164	−0.0680818	−	0.997680i	$-0.521688\pi$
$941$	−4.17691	−0.136164	−0.0680818	+	0.997680i	$0.521688\pi$
$942$	−39.1244	−1.27474
$943$	2.19615	0.0715166
$944$	13.4641	0.438219
$945$	−3.71281	−0.120778
$946$	16.3923	0.532960
$947$	−27.6077	−0.897130	−0.448565	−	0.893750i	$-0.648065\pi$
$947$	−27.6077	−0.897130	−0.448565	+	0.893750i	$0.648065\pi$
$948$	10.3923	0.337526
$949$	0	0
$950$	6.24871	0.202735
$951$	43.5167	1.41112
$952$	−38.7846	−1.25702
$953$	4.14359	0.134224	0.0671121	−	0.997745i	$-0.478621\pi$
$953$	4.14359	0.134224	0.0671121	+	0.997745i	$0.478621\pi$
$954$	−11.3205	−0.366515
$955$	1.66025	0.0537246
$956$	−16.9282	−0.547497
$957$	−3.46410	−0.111979
$958$	−39.3205	−1.27039
$959$	−22.3923	−0.723085
$960$	−5.12436	−0.165388
$961$	36.1769	1.16700
$962$	0	0
$963$	−57.7128	−1.85977
$964$	−15.4641	−0.498065
$965$	3.92820	0.126453
$966$	−6.92820	−0.222911
$967$	−32.7846	−1.05428	−0.527141	−	0.849778i	$-0.676736\pi$
$967$	−32.7846	−1.05428	−0.527141	+	0.849778i	$0.676736\pi$
$968$	−34.1769	−1.09849
$969$	−12.9282	−0.415314
$970$	1.14359	0.0367186
$971$	−8.19615	−0.263027	−0.131514	−	0.991314i	$-0.541984\pi$
$971$	−8.19615	−0.263027	−0.131514	+	0.991314i	$0.541984\pi$
$972$	18.7321	0.600831
$973$	−43.6077	−1.39800
$974$	32.4449	1.03960
$975$	46.6410	1.49371
$976$	−4.26795	−0.136614
$977$	16.2487	0.519842	0.259921	−	0.965630i	$-0.416303\pi$
$977$	16.2487	0.519842	0.259921	+	0.965630i	$0.416303\pi$
$978$	−61.1769	−1.95622
$979$	−43.5167	−1.39080
$980$	1.33975	0.0427966
$981$	−30.3731	−0.969737
$982$	31.2679	0.997800
$983$	−57.1244	−1.82198	−0.910992	−	0.412424i	$-0.864682\pi$
$983$	−57.1244	−1.82198	−0.910992	+	0.412424i	$0.864682\pi$
$984$	24.5885	0.783851
$985$	−7.48334	−0.238439
$986$	−1.00000	−0.0318465
$987$	−20.7846	−0.661581
$988$	−4.39230	−0.139738
$989$	−2.53590	−0.0806369
$990$	−5.66025	−0.179895
$991$	46.3923	1.47370	0.736850	−	0.676056i	$-0.236312\pi$
$991$	46.3923	1.47370	0.736850	+	0.676056i	$0.236312\pi$
$992$	40.9808	1.30114
$993$	89.5692	2.84239
$994$	20.7846	0.659248
$995$	−2.96668	−0.0940500
$996$	22.3923	0.709527
$997$	20.5359	0.650378	0.325189	−	0.945649i	$-0.394572\pi$
$997$	20.5359	0.650378	0.325189	+	0.945649i	$0.394572\pi$
$998$	12.5885	0.398481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.a.h.1.2		2
37.6	odd	4		1369.2.b.d.1368.1		4
37.23	odd	12		37.2.e.a.11.1	✓	4
37.29	odd	12		37.2.e.a.27.1	yes	4
37.31	odd	4		1369.2.b.d.1368.3		4
37.36	even	2		1369.2.a.g.1.2		2
111.23	even	12		333.2.s.b.307.2		4
111.29	even	12		333.2.s.b.64.2		4
148.23	even	12		592.2.w.c.529.1		4
148.103	even	12		592.2.w.c.545.1		4
185.23	even	12		925.2.m.b.899.2		4
185.29	odd	12		925.2.n.a.101.2		4
185.97	even	12		925.2.m.a.899.1		4
185.103	even	12		925.2.m.a.249.1		4
185.134	odd	12		925.2.n.a.751.2		4
185.177	even	12		925.2.m.b.249.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.e.a.11.1	✓	4	37.23	odd	12
37.2.e.a.27.1	yes	4	37.29	odd	12
333.2.s.b.64.2		4	111.29	even	12
333.2.s.b.307.2		4	111.23	even	12
592.2.w.c.529.1		4	148.23	even	12
592.2.w.c.545.1		4	148.103	even	12
925.2.m.a.249.1		4	185.103	even	12
925.2.m.a.899.1		4	185.97	even	12
925.2.m.b.249.2		4	185.177	even	12
925.2.m.b.899.2		4	185.23	even	12
925.2.n.a.101.2		4	185.29	odd	12
925.2.n.a.751.2		4	185.134	odd	12
1369.2.a.g.1.2		2	37.36	even	2
1369.2.a.h.1.2		2	1.1	even	1	trivial
1369.2.b.d.1368.1		4	37.6	odd	4
1369.2.b.d.1368.3		4	37.31	odd	4

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:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.73205 q^{3} -1.00000 q^{4} -0.267949 q^{5} +2.73205 q^{6} +3.46410 q^{7} -3.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.73205 q^{3} -1.00000 q^{4} -0.267949 q^{5} +2.73205 q^{6} +3.46410 q^{7} -3.00000 q^{8} +4.46410 q^{9} -0.267949 q^{10} +4.73205 q^{11} -2.73205 q^{12} -3.46410 q^{13} +3.46410 q^{14} -0.732051 q^{15} -1.00000 q^{16} +3.73205 q^{17} +4.46410 q^{18} -1.26795 q^{19} +0.267949 q^{20} +9.46410 q^{21} +4.73205 q^{22} -0.732051 q^{23} -8.19615 q^{24} -4.92820 q^{25} -3.46410 q^{26} +4.00000 q^{27} -3.46410 q^{28} -0.267949 q^{29} -0.732051 q^{30} +8.19615 q^{31} +5.00000 q^{32} +12.9282 q^{33} +3.73205 q^{34} -0.928203 q^{35} -4.46410 q^{36} -1.26795 q^{38} -9.46410 q^{39} +0.803848 q^{40} -3.00000 q^{41} +9.46410 q^{42} +3.46410 q^{43} -4.73205 q^{44} -1.19615 q^{45} -0.732051 q^{46} -2.19615 q^{47} -2.73205 q^{48} +5.00000 q^{49} -4.92820 q^{50} +10.1962 q^{51} +3.46410 q^{52} -2.53590 q^{53} +4.00000 q^{54} -1.26795 q^{55} -10.3923 q^{56} -3.46410 q^{57} -0.267949 q^{58} -13.4641 q^{59} +0.732051 q^{60} +4.26795 q^{61} +8.19615 q^{62} +15.4641 q^{63} +7.00000 q^{64} +0.928203 q^{65} +12.9282 q^{66} +7.26795 q^{67} -3.73205 q^{68} -2.00000 q^{69} -0.928203 q^{70} +6.00000 q^{71} -13.3923 q^{72} -13.4641 q^{75} +1.26795 q^{76} +16.3923 q^{77} -9.46410 q^{78} -3.80385 q^{79} +0.267949 q^{80} -2.46410 q^{81} -3.00000 q^{82} -8.19615 q^{83} -9.46410 q^{84} -1.00000 q^{85} +3.46410 q^{86} -0.732051 q^{87} -14.1962 q^{88} -9.19615 q^{89} -1.19615 q^{90} -12.0000 q^{91} +0.732051 q^{92} +22.3923 q^{93} -2.19615 q^{94} +0.339746 q^{95} +13.6603 q^{96} -4.26795 q^{97} +5.00000 q^{98} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} - 4 q^{10} + 6 q^{11} - 2 q^{12} + 2 q^{15} - 2 q^{16} + 4 q^{17} + 2 q^{18} - 6 q^{19} + 4 q^{20} + 12 q^{21} + 6 q^{22} + 2 q^{23} - 6 q^{24} + 4 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{30} + 6 q^{31} + 10 q^{32} + 12 q^{33} + 4 q^{34} + 12 q^{35} - 2 q^{36} - 6 q^{38} - 12 q^{39} + 12 q^{40} - 6 q^{41} + 12 q^{42} - 6 q^{44} + 8 q^{45} + 2 q^{46} + 6 q^{47} - 2 q^{48} + 10 q^{49} + 4 q^{50} + 10 q^{51} - 12 q^{53} + 8 q^{54} - 6 q^{55} - 4 q^{58} - 20 q^{59} - 2 q^{60} + 12 q^{61} + 6 q^{62} + 24 q^{63} + 14 q^{64} - 12 q^{65} + 12 q^{66} + 18 q^{67} - 4 q^{68} - 4 q^{69} + 12 q^{70} + 12 q^{71} - 6 q^{72} - 20 q^{75} + 6 q^{76} + 12 q^{77} - 12 q^{78} - 18 q^{79} + 4 q^{80} + 2 q^{81} - 6 q^{82} - 6 q^{83} - 12 q^{84} - 2 q^{85} + 2 q^{87} - 18 q^{88} - 8 q^{89} + 8 q^{90} - 24 q^{91} - 2 q^{92} + 24 q^{93} + 6 q^{94} + 18 q^{95} + 10 q^{96} - 12 q^{97} + 10 q^{98} + 18 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 + 2 * q^9 - 4 * q^10 + 6 * q^11 - 2 * q^12 + 2 * q^15 - 2 * q^16 + 4 * q^17 + 2 * q^18 - 6 * q^19 + 4 * q^20 + 12 * q^21 + 6 * q^22 + 2 * q^23 - 6 * q^24 + 4 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^30 + 6 * q^31 + 10 * q^32 + 12 * q^33 + 4 * q^34 + 12 * q^35 - 2 * q^36 - 6 * q^38 - 12 * q^39 + 12 * q^40 - 6 * q^41 + 12 * q^42 - 6 * q^44 + 8 * q^45 + 2 * q^46 + 6 * q^47 - 2 * q^48 + 10 * q^49 + 4 * q^50 + 10 * q^51 - 12 * q^53 + 8 * q^54 - 6 * q^55 - 4 * q^58 - 20 * q^59 - 2 * q^60 + 12 * q^61 + 6 * q^62 + 24 * q^63 + 14 * q^64 - 12 * q^65 + 12 * q^66 + 18 * q^67 - 4 * q^68 - 4 * q^69 + 12 * q^70 + 12 * q^71 - 6 * q^72 - 20 * q^75 + 6 * q^76 + 12 * q^77 - 12 * q^78 - 18 * q^79 + 4 * q^80 + 2 * q^81 - 6 * q^82 - 6 * q^83 - 12 * q^84 - 2 * q^85 + 2 * q^87 - 18 * q^88 - 8 * q^89 + 8 * q^90 - 24 * q^91 - 2 * q^92 + 24 * q^93 + 6 * q^94 + 18 * q^95 + 10 * q^96 - 12 * q^97 + 10 * q^98 + 18 * q^99

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