LMFDB - Embedded newform 465.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [465,2,Mod(1,465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.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:	$3.71304369399$
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	$\chi$	$=$	465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -4.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -4.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} -1.46410 q^{11} -1.00000 q^{12} -1.26795 q^{13} -8.19615 q^{14} +1.00000 q^{15} -5.00000 q^{16} -1.46410 q^{17} +1.73205 q^{18} +5.46410 q^{19} -1.00000 q^{20} +4.73205 q^{21} -2.53590 q^{22} -0.535898 q^{23} +1.73205 q^{24} +1.00000 q^{25} -2.19615 q^{26} -1.00000 q^{27} -4.73205 q^{28} +0.732051 q^{29} +1.73205 q^{30} -1.00000 q^{31} -5.19615 q^{32} +1.46410 q^{33} -2.53590 q^{34} +4.73205 q^{35} +1.00000 q^{36} -6.73205 q^{37} +9.46410 q^{38} +1.26795 q^{39} +1.73205 q^{40} +3.46410 q^{41} +8.19615 q^{42} -4.00000 q^{43} -1.46410 q^{44} -1.00000 q^{45} -0.928203 q^{46} -6.00000 q^{47} +5.00000 q^{48} +15.3923 q^{49} +1.73205 q^{50} +1.46410 q^{51} -1.26795 q^{52} +12.3923 q^{53} -1.73205 q^{54} +1.46410 q^{55} +8.19615 q^{56} -5.46410 q^{57} +1.26795 q^{58} -12.1962 q^{59} +1.00000 q^{60} -10.3923 q^{61} -1.73205 q^{62} -4.73205 q^{63} +1.00000 q^{64} +1.26795 q^{65} +2.53590 q^{66} -10.1962 q^{67} -1.46410 q^{68} +0.535898 q^{69} +8.19615 q^{70} -3.80385 q^{71} -1.73205 q^{72} +5.66025 q^{73} -11.6603 q^{74} -1.00000 q^{75} +5.46410 q^{76} +6.92820 q^{77} +2.19615 q^{78} -10.0000 q^{79} +5.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +15.4641 q^{83} +4.73205 q^{84} +1.46410 q^{85} -6.92820 q^{86} -0.732051 q^{87} +2.53590 q^{88} +7.26795 q^{89} -1.73205 q^{90} +6.00000 q^{91} -0.535898 q^{92} +1.00000 q^{93} -10.3923 q^{94} -5.46410 q^{95} +5.19615 q^{96} -2.00000 q^{97} +26.6603 q^{98} -1.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{12} - 6 q^{13} - 6 q^{14} + 2 q^{15} - 10 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} + 6 q^{21} - 12 q^{22} - 8 q^{23} + 2 q^{25} + 6 q^{26} - 2 q^{27} - 6 q^{28} - 2 q^{29} - 2 q^{31} - 4 q^{33} - 12 q^{34} + 6 q^{35} + 2 q^{36} - 10 q^{37} + 12 q^{38} + 6 q^{39} + 6 q^{42} - 8 q^{43} + 4 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} + 10 q^{48} + 10 q^{49} - 4 q^{51} - 6 q^{52} + 4 q^{53} - 4 q^{55} + 6 q^{56} - 4 q^{57} + 6 q^{58} - 14 q^{59} + 2 q^{60} - 6 q^{63} + 2 q^{64} + 6 q^{65} + 12 q^{66} - 10 q^{67} + 4 q^{68} + 8 q^{69} + 6 q^{70} - 18 q^{71} - 6 q^{73} - 6 q^{74} - 2 q^{75} + 4 q^{76} - 6 q^{78} - 20 q^{79} + 10 q^{80} + 2 q^{81} + 12 q^{82} + 24 q^{83} + 6 q^{84} - 4 q^{85} + 2 q^{87} + 12 q^{88} + 18 q^{89} + 12 q^{91} - 8 q^{92} + 2 q^{93} - 4 q^{95} - 4 q^{97} + 36 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^12 - 6 * q^13 - 6 * q^14 + 2 * q^15 - 10 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 + 6 * q^21 - 12 * q^22 - 8 * q^23 + 2 * q^25 + 6 * q^26 - 2 * q^27 - 6 * q^28 - 2 * q^29 - 2 * q^31 - 4 * q^33 - 12 * q^34 + 6 * q^35 + 2 * q^36 - 10 * q^37 + 12 * q^38 + 6 * q^39 + 6 * q^42 - 8 * q^43 + 4 * q^44 - 2 * q^45 + 12 * q^46 - 12 * q^47 + 10 * q^48 + 10 * q^49 - 4 * q^51 - 6 * q^52 + 4 * q^53 - 4 * q^55 + 6 * q^56 - 4 * q^57 + 6 * q^58 - 14 * q^59 + 2 * q^60 - 6 * q^63 + 2 * q^64 + 6 * q^65 + 12 * q^66 - 10 * q^67 + 4 * q^68 + 8 * q^69 + 6 * q^70 - 18 * q^71 - 6 * q^73 - 6 * q^74 - 2 * q^75 + 4 * q^76 - 6 * q^78 - 20 * q^79 + 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 + 6 * q^84 - 4 * q^85 + 2 * q^87 + 12 * q^88 + 18 * q^89 + 12 * q^91 - 8 * q^92 + 2 * q^93 - 4 * q^95 - 4 * q^97 + 36 * q^98 + 4 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.73205	−0.707107
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	−1.73205	−0.547723
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	−1.00000	−0.288675
$13$	−1.26795	−0.351666	−0.175833	−	0.984420i	$-0.556262\pi$
$13$	−1.26795	−0.351666	−0.175833	+	0.984420i	$0.556262\pi$
$14$	−8.19615	−2.19051
$15$	1.00000	0.258199
$16$	−5.00000	−1.25000
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	1.73205	0.408248
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	−1.00000	−0.223607
$21$	4.73205	1.03262
$22$	−2.53590	−0.540655
$23$	−0.535898	−0.111743	−0.0558713	−	0.998438i	$-0.517794\pi$
$23$	−0.535898	−0.111743	−0.0558713	+	0.998438i	$0.517794\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	−2.19615	−0.430701
$27$	−1.00000	−0.192450
$28$	−4.73205	−0.894274
$29$	0.732051	0.135938	0.0679692	−	0.997687i	$-0.478348\pi$
$29$	0.732051	0.135938	0.0679692	+	0.997687i	$0.478348\pi$
$30$	1.73205	0.316228
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.19615	−0.918559
$33$	1.46410	0.254867
$34$	−2.53590	−0.434903
$35$	4.73205	0.799863
$36$	1.00000	0.166667
$37$	−6.73205	−1.10674	−0.553371	−	0.832935i	$-0.686659\pi$
$37$	−6.73205	−1.10674	−0.553371	+	0.832935i	$0.686659\pi$
$38$	9.46410	1.53528
$39$	1.26795	0.203034
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	8.19615	1.26469
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.46410	−0.220722
$45$	−1.00000	−0.149071
$46$	−0.928203	−0.136856
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	5.00000	0.721688
$49$	15.3923	2.19890
$50$	1.73205	0.244949
$51$	1.46410	0.205015
$52$	−1.26795	−0.175833
$53$	12.3923	1.70221	0.851107	−	0.524992i	$-0.175932\pi$
$53$	12.3923	1.70221	0.851107	+	0.524992i	$0.175932\pi$
$54$	−1.73205	−0.235702
$55$	1.46410	0.197419
$56$	8.19615	1.09526
$57$	−5.46410	−0.723738
$58$	1.26795	0.166490
$59$	−12.1962	−1.58780	−0.793902	−	0.608046i	$-0.791954\pi$
$59$	−12.1962	−1.58780	−0.793902	+	0.608046i	$0.791954\pi$
$60$	1.00000	0.129099
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	−1.73205	−0.219971
$63$	−4.73205	−0.596182
$64$	1.00000	0.125000
$65$	1.26795	0.157270
$66$	2.53590	0.312148
$67$	−10.1962	−1.24566	−0.622829	−	0.782358i	$-0.714017\pi$
$67$	−10.1962	−1.24566	−0.622829	+	0.782358i	$0.714017\pi$
$68$	−1.46410	−0.177548
$69$	0.535898	0.0645146
$70$	8.19615	0.979628
$71$	−3.80385	−0.451434	−0.225717	−	0.974193i	$-0.572472\pi$
$71$	−3.80385	−0.451434	−0.225717	+	0.974193i	$0.572472\pi$
$72$	−1.73205	−0.204124
$73$	5.66025	0.662483	0.331241	−	0.943546i	$-0.392533\pi$
$73$	5.66025	0.662483	0.331241	+	0.943546i	$0.392533\pi$
$74$	−11.6603	−1.35548
$75$	−1.00000	−0.115470
$76$	5.46410	0.626775
$77$	6.92820	0.789542
$78$	2.19615	0.248665
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	5.00000	0.559017
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	4.73205	0.516309
$85$	1.46410	0.158804
$86$	−6.92820	−0.747087
$87$	−0.732051	−0.0784841
$88$	2.53590	0.270328
$89$	7.26795	0.770401	0.385201	−	0.922833i	$-0.374132\pi$
$89$	7.26795	0.770401	0.385201	+	0.922833i	$0.374132\pi$
$90$	−1.73205	−0.182574
$91$	6.00000	0.628971
$92$	−0.535898	−0.0558713
$93$	1.00000	0.103695
$94$	−10.3923	−1.07188
$95$	−5.46410	−0.560605
$96$	5.19615	0.530330
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	26.6603	2.69309
$99$	−1.46410	−0.147148
$100$	1.00000	0.100000
$101$	4.53590	0.451339	0.225669	−	0.974204i	$-0.427543\pi$
$101$	4.53590	0.451339	0.225669	+	0.974204i	$0.427543\pi$
$102$	2.53590	0.251091
$103$	3.26795	0.322001	0.161000	−	0.986954i	$-0.448528\pi$
$103$	3.26795	0.322001	0.161000	+	0.986954i	$0.448528\pi$
$104$	2.19615	0.215350
$105$	−4.73205	−0.461801
$106$	21.4641	2.08478
$107$	−15.8564	−1.53290	−0.766448	−	0.642306i	$-0.777978\pi$
$107$	−15.8564	−1.53290	−0.766448	+	0.642306i	$0.777978\pi$
$108$	−1.00000	−0.0962250
$109$	−2.53590	−0.242895	−0.121448	−	0.992598i	$-0.538754\pi$
$109$	−2.53590	−0.242895	−0.121448	+	0.992598i	$0.538754\pi$
$110$	2.53590	0.241788
$111$	6.73205	0.638978
$112$	23.6603	2.23568
$113$	−4.92820	−0.463606	−0.231803	−	0.972763i	$-0.574463\pi$
$113$	−4.92820	−0.463606	−0.231803	+	0.972763i	$0.574463\pi$
$114$	−9.46410	−0.886394
$115$	0.535898	0.0499728
$116$	0.732051	0.0679692
$117$	−1.26795	−0.117222
$118$	−21.1244	−1.94465
$119$	6.92820	0.635107
$120$	−1.73205	−0.158114
$121$	−8.85641	−0.805128
$122$	−18.0000	−1.62964
$123$	−3.46410	−0.312348
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−8.19615	−0.730171
$127$	−10.5359	−0.934910	−0.467455	−	0.884017i	$-0.654829\pi$
$127$	−10.5359	−0.934910	−0.467455	+	0.884017i	$0.654829\pi$
$128$	12.1244	1.07165
$129$	4.00000	0.352180
$130$	2.19615	0.192615
$131$	6.73205	0.588182	0.294091	−	0.955777i	$-0.404983\pi$
$131$	6.73205	0.588182	0.294091	+	0.955777i	$0.404983\pi$
$132$	1.46410	0.127434
$133$	−25.8564	−2.24203
$134$	−17.6603	−1.52561
$135$	1.00000	0.0860663
$136$	2.53590	0.217451
$137$	18.9282	1.61715	0.808573	−	0.588396i	$-0.200240\pi$
$137$	18.9282	1.61715	0.808573	+	0.588396i	$0.200240\pi$
$138$	0.928203	0.0790139
$139$	9.85641	0.836009	0.418005	−	0.908445i	$-0.362730\pi$
$139$	9.85641	0.836009	0.418005	+	0.908445i	$0.362730\pi$
$140$	4.73205	0.399931
$141$	6.00000	0.505291
$142$	−6.58846	−0.552891
$143$	1.85641	0.155241
$144$	−5.00000	−0.416667
$145$	−0.732051	−0.0607935
$146$	9.80385	0.811372
$147$	−15.3923	−1.26954
$148$	−6.73205	−0.553371
$149$	21.3205	1.74664	0.873322	−	0.487143i	$-0.161961\pi$
$149$	21.3205	1.74664	0.873322	+	0.487143i	$0.161961\pi$
$150$	−1.73205	−0.141421
$151$	−20.9282	−1.70311	−0.851557	−	0.524263i	$-0.824341\pi$
$151$	−20.9282	−1.70311	−0.851557	+	0.524263i	$0.824341\pi$
$152$	−9.46410	−0.767640
$153$	−1.46410	−0.118366
$154$	12.0000	0.966988
$155$	1.00000	0.0803219
$156$	1.26795	0.101517
$157$	−12.9282	−1.03178	−0.515891	−	0.856654i	$-0.672539\pi$
$157$	−12.9282	−1.03178	−0.515891	+	0.856654i	$0.672539\pi$
$158$	−17.3205	−1.37795
$159$	−12.3923	−0.982774
$160$	5.19615	0.410792
$161$	2.53590	0.199857
$162$	1.73205	0.136083
$163$	−14.1962	−1.11193	−0.555964	−	0.831206i	$-0.687651\pi$
$163$	−14.1962	−1.11193	−0.555964	+	0.831206i	$0.687651\pi$
$164$	3.46410	0.270501
$165$	−1.46410	−0.113980
$166$	26.7846	2.07889
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	−8.19615	−0.632347
$169$	−11.3923	−0.876331
$170$	2.53590	0.194495
$171$	5.46410	0.417850
$172$	−4.00000	−0.304997
$173$	−22.7846	−1.73228	−0.866141	−	0.499800i	$-0.833407\pi$
$173$	−22.7846	−1.73228	−0.866141	+	0.499800i	$0.833407\pi$
$174$	−1.26795	−0.0961230
$175$	−4.73205	−0.357709
$176$	7.32051	0.551804
$177$	12.1962	0.916719
$178$	12.5885	0.943545
$179$	−9.07180	−0.678058	−0.339029	−	0.940776i	$-0.610098\pi$
$179$	−9.07180	−0.678058	−0.339029	+	0.940776i	$0.610098\pi$
$180$	−1.00000	−0.0745356
$181$	−10.3923	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$181$	−10.3923	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$182$	10.3923	0.770329
$183$	10.3923	0.768221
$184$	0.928203	0.0684280
$185$	6.73205	0.494950
$186$	1.73205	0.127000
$187$	2.14359	0.156755
$188$	−6.00000	−0.437595
$189$	4.73205	0.344206
$190$	−9.46410	−0.686598
$191$	−11.8038	−0.854096	−0.427048	−	0.904229i	$-0.640447\pi$
$191$	−11.8038	−0.854096	−0.427048	+	0.904229i	$0.640447\pi$
$192$	−1.00000	−0.0721688
$193$	−23.8564	−1.71722	−0.858611	−	0.512628i	$-0.828672\pi$
$193$	−23.8564	−1.71722	−0.858611	+	0.512628i	$0.828672\pi$
$194$	−3.46410	−0.248708
$195$	−1.26795	−0.0907997
$196$	15.3923	1.09945
$197$	13.0718	0.931327	0.465663	−	0.884962i	$-0.345816\pi$
$197$	13.0718	0.931327	0.465663	+	0.884962i	$0.345816\pi$
$198$	−2.53590	−0.180218
$199$	16.9282	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$199$	16.9282	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$200$	−1.73205	−0.122474
$201$	10.1962	0.719181
$202$	7.85641	0.552775
$203$	−3.46410	−0.243132
$204$	1.46410	0.102508
$205$	−3.46410	−0.241943
$206$	5.66025	0.394369
$207$	−0.535898	−0.0372475
$208$	6.33975	0.439582
$209$	−8.00000	−0.553372
$210$	−8.19615	−0.565588
$211$	7.32051	0.503965	0.251982	−	0.967732i	$-0.418918\pi$
$211$	7.32051	0.503965	0.251982	+	0.967732i	$0.418918\pi$
$212$	12.3923	0.851107
$213$	3.80385	0.260635
$214$	−27.4641	−1.87741
$215$	4.00000	0.272798
$216$	1.73205	0.117851
$217$	4.73205	0.321233
$218$	−4.39230	−0.297484
$219$	−5.66025	−0.382485
$220$	1.46410	0.0987097
$221$	1.85641	0.124875
$222$	11.6603	0.782585
$223$	−0.392305	−0.0262707	−0.0131353	−	0.999914i	$-0.504181\pi$
$223$	−0.392305	−0.0262707	−0.0131353	+	0.999914i	$0.504181\pi$
$224$	24.5885	1.64289
$225$	1.00000	0.0666667
$226$	−8.53590	−0.567800
$227$	13.3205	0.884113	0.442057	−	0.896987i	$-0.354249\pi$
$227$	13.3205	0.884113	0.442057	+	0.896987i	$0.354249\pi$
$228$	−5.46410	−0.361869
$229$	18.7846	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$229$	18.7846	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$230$	0.928203	0.0612039
$231$	−6.92820	−0.455842
$232$	−1.26795	−0.0832449
$233$	10.9282	0.715930	0.357965	−	0.933735i	$-0.383471\pi$
$233$	10.9282	0.715930	0.357965	+	0.933735i	$0.383471\pi$
$234$	−2.19615	−0.143567
$235$	6.00000	0.391397
$236$	−12.1962	−0.793902
$237$	10.0000	0.649570
$238$	12.0000	0.777844
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	−5.00000	−0.322749
$241$	23.8564	1.53673	0.768363	−	0.640014i	$-0.221072\pi$
$241$	23.8564	1.53673	0.768363	+	0.640014i	$0.221072\pi$
$242$	−15.3397	−0.986076
$243$	−1.00000	−0.0641500
$244$	−10.3923	−0.665299
$245$	−15.3923	−0.983378
$246$	−6.00000	−0.382546
$247$	−6.92820	−0.440831
$248$	1.73205	0.109985
$249$	−15.4641	−0.979998
$250$	−1.73205	−0.109545
$251$	2.92820	0.184827	0.0924133	−	0.995721i	$-0.470542\pi$
$251$	2.92820	0.184827	0.0924133	+	0.995721i	$0.470542\pi$
$252$	−4.73205	−0.298091
$253$	0.784610	0.0493280
$254$	−18.2487	−1.14503
$255$	−1.46410	−0.0916856
$256$	19.0000	1.18750
$257$	−9.85641	−0.614826	−0.307413	−	0.951576i	$-0.599463\pi$
$257$	−9.85641	−0.614826	−0.307413	+	0.951576i	$0.599463\pi$
$258$	6.92820	0.431331
$259$	31.8564	1.97946
$260$	1.26795	0.0786349
$261$	0.732051	0.0453128
$262$	11.6603	0.720373
$263$	−17.0718	−1.05269	−0.526346	−	0.850270i	$-0.676438\pi$
$263$	−17.0718	−1.05269	−0.526346	+	0.850270i	$0.676438\pi$
$264$	−2.53590	−0.156074
$265$	−12.3923	−0.761253
$266$	−44.7846	−2.74592
$267$	−7.26795	−0.444791
$268$	−10.1962	−0.622829
$269$	9.80385	0.597751	0.298876	−	0.954292i	$-0.403388\pi$
$269$	9.80385	0.597751	0.298876	+	0.954292i	$0.403388\pi$
$270$	1.73205	0.105409
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	7.32051	0.443871
$273$	−6.00000	−0.363137
$274$	32.7846	1.98059
$275$	−1.46410	−0.0882886
$276$	0.535898	0.0322573
$277$	−15.1244	−0.908734	−0.454367	−	0.890814i	$-0.650135\pi$
$277$	−15.1244	−0.908734	−0.454367	+	0.890814i	$0.650135\pi$
$278$	17.0718	1.02390
$279$	−1.00000	−0.0598684
$280$	−8.19615	−0.489814
$281$	−16.5359	−0.986449	−0.493224	−	0.869902i	$-0.664182\pi$
$281$	−16.5359	−0.986449	−0.493224	+	0.869902i	$0.664182\pi$
$282$	10.3923	0.618853
$283$	7.26795	0.432035	0.216017	−	0.976390i	$-0.430693\pi$
$283$	7.26795	0.432035	0.216017	+	0.976390i	$0.430693\pi$
$284$	−3.80385	−0.225717
$285$	5.46410	0.323665
$286$	3.21539	0.190130
$287$	−16.3923	−0.967607
$288$	−5.19615	−0.306186
$289$	−14.8564	−0.873906
$290$	−1.26795	−0.0744565
$291$	2.00000	0.117242
$292$	5.66025	0.331241
$293$	5.07180	0.296298	0.148149	−	0.988965i	$-0.452669\pi$
$293$	5.07180	0.296298	0.148149	+	0.988965i	$0.452669\pi$
$294$	−26.6603	−1.55486
$295$	12.1962	0.710087
$296$	11.6603	0.677738
$297$	1.46410	0.0849558
$298$	36.9282	2.13919
$299$	0.679492	0.0392960
$300$	−1.00000	−0.0577350
$301$	18.9282	1.09100
$302$	−36.2487	−2.08588
$303$	−4.53590	−0.260581
$304$	−27.3205	−1.56694
$305$	10.3923	0.595062
$306$	−2.53590	−0.144968
$307$	26.5885	1.51748	0.758742	−	0.651392i	$-0.225814\pi$
$307$	26.5885	1.51748	0.758742	+	0.651392i	$0.225814\pi$
$308$	6.92820	0.394771
$309$	−3.26795	−0.185907
$310$	1.73205	0.0983739
$311$	13.6603	0.774602	0.387301	−	0.921953i	$-0.373407\pi$
$311$	13.6603	0.774602	0.387301	+	0.921953i	$0.373407\pi$
$312$	−2.19615	−0.124333
$313$	−21.6603	−1.22431	−0.612155	−	0.790738i	$-0.709697\pi$
$313$	−21.6603	−1.22431	−0.612155	+	0.790738i	$0.709697\pi$
$314$	−22.3923	−1.26367
$315$	4.73205	0.266621
$316$	−10.0000	−0.562544
$317$	−22.9282	−1.28778	−0.643888	−	0.765120i	$-0.722680\pi$
$317$	−22.9282	−1.28778	−0.643888	+	0.765120i	$0.722680\pi$
$318$	−21.4641	−1.20365
$319$	−1.07180	−0.0600091
$320$	−1.00000	−0.0559017
$321$	15.8564	0.885018
$322$	4.39230	0.244774
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	−1.26795	−0.0703332
$326$	−24.5885	−1.36183
$327$	2.53590	0.140236
$328$	−6.00000	−0.331295
$329$	28.3923	1.56532
$330$	−2.53590	−0.139597
$331$	24.9282	1.37018	0.685089	−	0.728459i	$-0.259763\pi$
$331$	24.9282	1.37018	0.685089	+	0.728459i	$0.259763\pi$
$332$	15.4641	0.848703
$333$	−6.73205	−0.368914
$334$	24.0000	1.31322
$335$	10.1962	0.557075
$336$	−23.6603	−1.29077
$337$	18.0526	0.983386	0.491693	−	0.870769i	$-0.336378\pi$
$337$	18.0526	0.983386	0.491693	+	0.870769i	$0.336378\pi$
$338$	−19.7321	−1.07328
$339$	4.92820	0.267663
$340$	1.46410	0.0794021
$341$	1.46410	0.0792855
$342$	9.46410	0.511760
$343$	−39.7128	−2.14429
$344$	6.92820	0.373544
$345$	−0.535898	−0.0288518
$346$	−39.4641	−2.12160
$347$	−12.7846	−0.686314	−0.343157	−	0.939278i	$-0.611496\pi$
$347$	−12.7846	−0.686314	−0.343157	+	0.939278i	$0.611496\pi$
$348$	−0.732051	−0.0392420
$349$	6.53590	0.349859	0.174929	−	0.984581i	$-0.444030\pi$
$349$	6.53590	0.349859	0.174929	+	0.984581i	$0.444030\pi$
$350$	−8.19615	−0.438103
$351$	1.26795	0.0676781
$352$	7.60770	0.405492
$353$	−4.39230	−0.233779	−0.116889	−	0.993145i	$-0.537292\pi$
$353$	−4.39230	−0.233779	−0.116889	+	0.993145i	$0.537292\pi$
$354$	21.1244	1.12275
$355$	3.80385	0.201887
$356$	7.26795	0.385201
$357$	−6.92820	−0.366679
$358$	−15.7128	−0.830448
$359$	−1.66025	−0.0876249	−0.0438124	−	0.999040i	$-0.513950\pi$
$359$	−1.66025	−0.0876249	−0.0438124	+	0.999040i	$0.513950\pi$
$360$	1.73205	0.0912871
$361$	10.8564	0.571390
$362$	−18.0000	−0.946059
$363$	8.85641	0.464841
$364$	6.00000	0.314485
$365$	−5.66025	−0.296271
$366$	18.0000	0.940875
$367$	−21.8564	−1.14090	−0.570448	−	0.821334i	$-0.693230\pi$
$367$	−21.8564	−1.14090	−0.570448	+	0.821334i	$0.693230\pi$
$368$	2.67949	0.139678
$369$	3.46410	0.180334
$370$	11.6603	0.606188
$371$	−58.6410	−3.04449
$372$	1.00000	0.0518476
$373$	26.3923	1.36654	0.683271	−	0.730165i	$-0.260557\pi$
$373$	26.3923	1.36654	0.683271	+	0.730165i	$0.260557\pi$
$374$	3.71281	0.191985
$375$	1.00000	0.0516398
$376$	10.3923	0.535942
$377$	−0.928203	−0.0478049
$378$	8.19615	0.421565
$379$	32.3923	1.66388	0.831940	−	0.554865i	$-0.187230\pi$
$379$	32.3923	1.66388	0.831940	+	0.554865i	$0.187230\pi$
$380$	−5.46410	−0.280302
$381$	10.5359	0.539770
$382$	−20.4449	−1.04605
$383$	−13.3205	−0.680646	−0.340323	−	0.940309i	$-0.610536\pi$
$383$	−13.3205	−0.680646	−0.340323	+	0.940309i	$0.610536\pi$
$384$	−12.1244	−0.618718
$385$	−6.92820	−0.353094
$386$	−41.3205	−2.10316
$387$	−4.00000	−0.203331
$388$	−2.00000	−0.101535
$389$	12.3397	0.625650	0.312825	−	0.949811i	$-0.398725\pi$
$389$	12.3397	0.625650	0.312825	+	0.949811i	$0.398725\pi$
$390$	−2.19615	−0.111207
$391$	0.784610	0.0396794
$392$	−26.6603	−1.34655
$393$	−6.73205	−0.339587
$394$	22.6410	1.14064
$395$	10.0000	0.503155
$396$	−1.46410	−0.0735739
$397$	23.8564	1.19732	0.598659	−	0.801004i	$-0.295700\pi$
$397$	23.8564	1.19732	0.598659	+	0.801004i	$0.295700\pi$
$398$	29.3205	1.46970
$399$	25.8564	1.29444
$400$	−5.00000	−0.250000
$401$	9.51666	0.475239	0.237620	−	0.971358i	$-0.423633\pi$
$401$	9.51666	0.475239	0.237620	+	0.971358i	$0.423633\pi$
$402$	17.6603	0.880813
$403$	1.26795	0.0631610
$404$	4.53590	0.225669
$405$	−1.00000	−0.0496904
$406$	−6.00000	−0.297775
$407$	9.85641	0.488564
$408$	−2.53590	−0.125546
$409$	−17.3205	−0.856444	−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205	−0.856444	−0.428222	+	0.903674i	$0.640860\pi$
$410$	−6.00000	−0.296319
$411$	−18.9282	−0.933659
$412$	3.26795	0.161000
$413$	57.7128	2.83986
$414$	−0.928203	−0.0456187
$415$	−15.4641	−0.759103
$416$	6.58846	0.323026
$417$	−9.85641	−0.482670
$418$	−13.8564	−0.677739
$419$	−27.1244	−1.32511	−0.662556	−	0.749013i	$-0.730528\pi$
$419$	−27.1244	−1.32511	−0.662556	+	0.749013i	$0.730528\pi$
$420$	−4.73205	−0.230900
$421$	−15.3205	−0.746676	−0.373338	−	0.927695i	$-0.621787\pi$
$421$	−15.3205	−0.746676	−0.373338	+	0.927695i	$0.621787\pi$
$422$	12.6795	0.617228
$423$	−6.00000	−0.291730
$424$	−21.4641	−1.04239
$425$	−1.46410	−0.0710194
$426$	6.58846	0.319212
$427$	49.1769	2.37984
$428$	−15.8564	−0.766448
$429$	−1.85641	−0.0896281
$430$	6.92820	0.334108
$431$	8.87564	0.427525	0.213762	−	0.976886i	$-0.431428\pi$
$431$	8.87564	0.427525	0.213762	+	0.976886i	$0.431428\pi$
$432$	5.00000	0.240563
$433$	15.5167	0.745683	0.372842	−	0.927895i	$-0.378383\pi$
$433$	15.5167	0.745683	0.372842	+	0.927895i	$0.378383\pi$
$434$	8.19615	0.393428
$435$	0.732051	0.0350991
$436$	−2.53590	−0.121448
$437$	−2.92820	−0.140075
$438$	−9.80385	−0.468446
$439$	−35.7128	−1.70448	−0.852240	−	0.523151i	$-0.824756\pi$
$439$	−35.7128	−1.70448	−0.852240	+	0.523151i	$0.824756\pi$
$440$	−2.53590	−0.120894
$441$	15.3923	0.732967
$442$	3.21539	0.152941
$443$	30.3923	1.44398	0.721991	−	0.691902i	$-0.243227\pi$
$443$	30.3923	1.44398	0.721991	+	0.691902i	$0.243227\pi$
$444$	6.73205	0.319489
$445$	−7.26795	−0.344534
$446$	−0.679492	−0.0321749
$447$	−21.3205	−1.00843
$448$	−4.73205	−0.223568
$449$	40.7321	1.92226	0.961132	−	0.276089i	$-0.0890384\pi$
$449$	40.7321	1.92226	0.961132	+	0.276089i	$0.0890384\pi$
$450$	1.73205	0.0816497
$451$	−5.07180	−0.238822
$452$	−4.92820	−0.231803
$453$	20.9282	0.983293
$454$	23.0718	1.08281
$455$	−6.00000	−0.281284
$456$	9.46410	0.443197
$457$	32.5885	1.52442	0.762212	−	0.647328i	$-0.224113\pi$
$457$	32.5885	1.52442	0.762212	+	0.647328i	$0.224113\pi$
$458$	32.5359	1.52030
$459$	1.46410	0.0683384
$460$	0.535898	0.0249864
$461$	−0.732051	−0.0340950	−0.0170475	−	0.999855i	$-0.505427\pi$
$461$	−0.732051	−0.0340950	−0.0170475	+	0.999855i	$0.505427\pi$
$462$	−12.0000	−0.558291
$463$	39.7128	1.84561	0.922805	−	0.385266i	$-0.125890\pi$
$463$	39.7128	1.84561	0.922805	+	0.385266i	$0.125890\pi$
$464$	−3.66025	−0.169923
$465$	−1.00000	−0.0463739
$466$	18.9282	0.876832
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	−1.26795	−0.0586110
$469$	48.2487	2.22792
$470$	10.3923	0.479361
$471$	12.9282	0.595700
$472$	21.1244	0.972327
$473$	5.85641	0.269278
$474$	17.3205	0.795557
$475$	5.46410	0.250710
$476$	6.92820	0.317554
$477$	12.3923	0.567405
$478$	−24.0000	−1.09773
$479$	−27.8038	−1.27039	−0.635195	−	0.772352i	$-0.719080\pi$
$479$	−27.8038	−1.27039	−0.635195	+	0.772352i	$0.719080\pi$
$480$	−5.19615	−0.237171
$481$	8.53590	0.389203
$482$	41.3205	1.88210
$483$	−2.53590	−0.115387
$484$	−8.85641	−0.402564
$485$	2.00000	0.0908153
$486$	−1.73205	−0.0785674
$487$	10.5359	0.477427	0.238714	−	0.971090i	$-0.423274\pi$
$487$	10.5359	0.477427	0.238714	+	0.971090i	$0.423274\pi$
$488$	18.0000	0.814822
$489$	14.1962	0.641972
$490$	−26.6603	−1.20439
$491$	24.3923	1.10081	0.550405	−	0.834898i	$-0.314473\pi$
$491$	24.3923	1.10081	0.550405	+	0.834898i	$0.314473\pi$
$492$	−3.46410	−0.156174
$493$	−1.07180	−0.0482713
$494$	−12.0000	−0.539906
$495$	1.46410	0.0658065
$496$	5.00000	0.224507
$497$	18.0000	0.807410
$498$	−26.7846	−1.20025
$499$	−14.9282	−0.668278	−0.334139	−	0.942524i	$-0.608446\pi$
$499$	−14.9282	−0.668278	−0.334139	+	0.942524i	$0.608446\pi$
$500$	−1.00000	−0.0447214
$501$	−13.8564	−0.619059
$502$	5.07180	0.226365
$503$	−11.8564	−0.528651	−0.264326	−	0.964434i	$-0.585149\pi$
$503$	−11.8564	−0.528651	−0.264326	+	0.964434i	$0.585149\pi$
$504$	8.19615	0.365086
$505$	−4.53590	−0.201845
$506$	1.35898	0.0604142
$507$	11.3923	0.505950
$508$	−10.5359	−0.467455
$509$	16.7321	0.741635	0.370818	−	0.928706i	$-0.379078\pi$
$509$	16.7321	0.741635	0.370818	+	0.928706i	$0.379078\pi$
$510$	−2.53590	−0.112291
$511$	−26.7846	−1.18488
$512$	8.66025	0.382733
$513$	−5.46410	−0.241246
$514$	−17.0718	−0.753005
$515$	−3.26795	−0.144003
$516$	4.00000	0.176090
$517$	8.78461	0.386347
$518$	55.1769	2.42433
$519$	22.7846	1.00013
$520$	−2.19615	−0.0963077
$521$	21.3205	0.934068	0.467034	−	0.884239i	$-0.345322\pi$
$521$	21.3205	0.934068	0.467034	+	0.884239i	$0.345322\pi$
$522$	1.26795	0.0554966
$523$	−11.6077	−0.507569	−0.253785	−	0.967261i	$-0.581675\pi$
$523$	−11.6077	−0.507569	−0.253785	+	0.967261i	$0.581675\pi$
$524$	6.73205	0.294091
$525$	4.73205	0.206524
$526$	−29.5692	−1.28928
$527$	1.46410	0.0637773
$528$	−7.32051	−0.318584
$529$	−22.7128	−0.987514
$530$	−21.4641	−0.932341
$531$	−12.1962	−0.529268
$532$	−25.8564	−1.12102
$533$	−4.39230	−0.190252
$534$	−12.5885	−0.544756
$535$	15.8564	0.685532
$536$	17.6603	0.762807
$537$	9.07180	0.391477
$538$	16.9808	0.732093
$539$	−22.5359	−0.970690
$540$	1.00000	0.0430331
$541$	−6.53590	−0.281000	−0.140500	−	0.990081i	$-0.544871\pi$
$541$	−6.53590	−0.281000	−0.140500	+	0.990081i	$0.544871\pi$
$542$	−29.0718	−1.24874
$543$	10.3923	0.445976
$544$	7.60770	0.326177
$545$	2.53590	0.108626
$546$	−10.3923	−0.444750
$547$	−34.1962	−1.46212	−0.731061	−	0.682312i	$-0.760975\pi$
$547$	−34.1962	−1.46212	−0.731061	+	0.682312i	$0.760975\pi$
$548$	18.9282	0.808573
$549$	−10.3923	−0.443533
$550$	−2.53590	−0.108131
$551$	4.00000	0.170406
$552$	−0.928203	−0.0395070
$553$	47.3205	2.01227
$554$	−26.1962	−1.11297
$555$	−6.73205	−0.285760
$556$	9.85641	0.418005
$557$	−9.46410	−0.401007	−0.200503	−	0.979693i	$-0.564258\pi$
$557$	−9.46410	−0.401007	−0.200503	+	0.979693i	$0.564258\pi$
$558$	−1.73205	−0.0733236
$559$	5.07180	0.214514
$560$	−23.6603	−0.999828
$561$	−2.14359	−0.0905026
$562$	−28.6410	−1.20815
$563$	40.6410	1.71281	0.856407	−	0.516301i	$-0.172691\pi$
$563$	40.6410	1.71281	0.856407	+	0.516301i	$0.172691\pi$
$564$	6.00000	0.252646
$565$	4.92820	0.207331
$566$	12.5885	0.529132
$567$	−4.73205	−0.198727
$568$	6.58846	0.276446
$569$	−28.7321	−1.20451	−0.602255	−	0.798304i	$-0.705731\pi$
$569$	−28.7321	−1.20451	−0.602255	+	0.798304i	$0.705731\pi$
$570$	9.46410	0.396408
$571$	−23.7128	−0.992350	−0.496175	−	0.868222i	$-0.665263\pi$
$571$	−23.7128	−0.992350	−0.496175	+	0.868222i	$0.665263\pi$
$572$	1.85641	0.0776203
$573$	11.8038	0.493113
$574$	−28.3923	−1.18507
$575$	−0.535898	−0.0223485
$576$	1.00000	0.0416667
$577$	5.32051	0.221496	0.110748	−	0.993849i	$-0.464675\pi$
$577$	5.32051	0.221496	0.110748	+	0.993849i	$0.464675\pi$
$578$	−25.7321	−1.07031
$579$	23.8564	0.991438
$580$	−0.732051	−0.0303968
$581$	−73.1769	−3.03589
$582$	3.46410	0.143592
$583$	−18.1436	−0.751431
$584$	−9.80385	−0.405686
$585$	1.26795	0.0524232
$586$	8.78461	0.362889
$587$	10.9282	0.451055	0.225528	−	0.974237i	$-0.427589\pi$
$587$	10.9282	0.451055	0.225528	+	0.974237i	$0.427589\pi$
$588$	−15.3923	−0.634768
$589$	−5.46410	−0.225144
$590$	21.1244	0.869676
$591$	−13.0718	−0.537702
$592$	33.6603	1.38343
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	2.53590	0.104049
$595$	−6.92820	−0.284029
$596$	21.3205	0.873322
$597$	−16.9282	−0.692825
$598$	1.17691	0.0481276
$599$	41.3731	1.69046	0.845229	−	0.534405i	$-0.179464\pi$
$599$	41.3731	1.69046	0.845229	+	0.534405i	$0.179464\pi$
$600$	1.73205	0.0707107
$601$	−37.3205	−1.52234	−0.761168	−	0.648555i	$-0.775374\pi$
$601$	−37.3205	−1.52234	−0.761168	+	0.648555i	$0.775374\pi$
$602$	32.7846	1.33620
$603$	−10.1962	−0.415219
$604$	−20.9282	−0.851557
$605$	8.85641	0.360064
$606$	−7.85641	−0.319145
$607$	−14.8756	−0.603784	−0.301892	−	0.953342i	$-0.597618\pi$
$607$	−14.8756	−0.603784	−0.301892	+	0.953342i	$0.597618\pi$
$608$	−28.3923	−1.15146
$609$	3.46410	0.140372
$610$	18.0000	0.728799
$611$	7.60770	0.307774
$612$	−1.46410	−0.0591828
$613$	−10.4449	−0.421864	−0.210932	−	0.977501i	$-0.567650\pi$
$613$	−10.4449	−0.421864	−0.210932	+	0.977501i	$0.567650\pi$
$614$	46.0526	1.85853
$615$	3.46410	0.139686
$616$	−12.0000	−0.483494
$617$	−34.7846	−1.40038	−0.700188	−	0.713959i	$-0.746900\pi$
$617$	−34.7846	−1.40038	−0.700188	+	0.713959i	$0.746900\pi$
$618$	−5.66025	−0.227689
$619$	15.8564	0.637323	0.318661	−	0.947869i	$-0.396767\pi$
$619$	15.8564	0.637323	0.318661	+	0.947869i	$0.396767\pi$
$620$	1.00000	0.0401610
$621$	0.535898	0.0215049
$622$	23.6603	0.948690
$623$	−34.3923	−1.37790
$624$	−6.33975	−0.253793
$625$	1.00000	0.0400000
$626$	−37.5167	−1.49947
$627$	8.00000	0.319489
$628$	−12.9282	−0.515891
$629$	9.85641	0.393001
$630$	8.19615	0.326543
$631$	2.92820	0.116570	0.0582850	−	0.998300i	$-0.481437\pi$
$631$	2.92820	0.116570	0.0582850	+	0.998300i	$0.481437\pi$
$632$	17.3205	0.688973
$633$	−7.32051	−0.290964
$634$	−39.7128	−1.57720
$635$	10.5359	0.418104
$636$	−12.3923	−0.491387
$637$	−19.5167	−0.773278
$638$	−1.85641	−0.0734958
$639$	−3.80385	−0.150478
$640$	−12.1244	−0.479257
$641$	−49.5167	−1.95579	−0.977895	−	0.209095i	$-0.932948\pi$
$641$	−49.5167	−1.95579	−0.977895	+	0.209095i	$0.932948\pi$
$642$	27.4641	1.08392
$643$	46.2487	1.82387	0.911936	−	0.410333i	$-0.134588\pi$
$643$	46.2487	1.82387	0.911936	+	0.410333i	$0.134588\pi$
$644$	2.53590	0.0999284
$645$	−4.00000	−0.157500
$646$	−13.8564	−0.545173
$647$	3.21539	0.126410	0.0632050	−	0.998001i	$-0.479868\pi$
$647$	3.21539	0.126410	0.0632050	+	0.998001i	$0.479868\pi$
$648$	−1.73205	−0.0680414
$649$	17.8564	0.700925
$650$	−2.19615	−0.0861402
$651$	−4.73205	−0.185464
$652$	−14.1962	−0.555964
$653$	−25.8564	−1.01184	−0.505920	−	0.862581i	$-0.668847\pi$
$653$	−25.8564	−1.01184	−0.505920	+	0.862581i	$0.668847\pi$
$654$	4.39230	0.171753
$655$	−6.73205	−0.263043
$656$	−17.3205	−0.676252
$657$	5.66025	0.220828
$658$	49.1769	1.91712
$659$	−39.1244	−1.52407	−0.762034	−	0.647537i	$-0.775799\pi$
$659$	−39.1244	−1.52407	−0.762034	+	0.647537i	$0.775799\pi$
$660$	−1.46410	−0.0569901
$661$	−44.6410	−1.73633	−0.868167	−	0.496272i	$-0.834702\pi$
$661$	−44.6410	−1.73633	−0.868167	+	0.496272i	$0.834702\pi$
$662$	43.1769	1.67812
$663$	−1.85641	−0.0720969
$664$	−26.7846	−1.03944
$665$	25.8564	1.00267
$666$	−11.6603	−0.451826
$667$	−0.392305	−0.0151901
$668$	13.8564	0.536120
$669$	0.392305	0.0151674
$670$	17.6603	0.682275
$671$	15.2154	0.587384
$672$	−24.5885	−0.948520
$673$	−9.26795	−0.357253	−0.178627	−	0.983917i	$-0.557165\pi$
$673$	−9.26795	−0.357253	−0.178627	+	0.983917i	$0.557165\pi$
$674$	31.2679	1.20440
$675$	−1.00000	−0.0384900
$676$	−11.3923	−0.438166
$677$	28.3923	1.09120	0.545602	−	0.838044i	$-0.316301\pi$
$677$	28.3923	1.09120	0.545602	+	0.838044i	$0.316301\pi$
$678$	8.53590	0.327819
$679$	9.46410	0.363199
$680$	−2.53590	−0.0972473
$681$	−13.3205	−0.510443
$682$	2.53590	0.0971046
$683$	−19.1769	−0.733784	−0.366892	−	0.930263i	$-0.619578\pi$
$683$	−19.1769	−0.733784	−0.366892	+	0.930263i	$0.619578\pi$
$684$	5.46410	0.208925
$685$	−18.9282	−0.723209
$686$	−68.7846	−2.62621
$687$	−18.7846	−0.716678
$688$	20.0000	0.762493
$689$	−15.7128	−0.598610
$690$	−0.928203	−0.0353361
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−22.7846	−0.866141
$693$	6.92820	0.263181
$694$	−22.1436	−0.840559
$695$	−9.85641	−0.373875
$696$	1.26795	0.0480615
$697$	−5.07180	−0.192108
$698$	11.3205	0.428488
$699$	−10.9282	−0.413343
$700$	−4.73205	−0.178855
$701$	6.78461	0.256251	0.128126	−	0.991758i	$-0.459104\pi$
$701$	6.78461	0.256251	0.128126	+	0.991758i	$0.459104\pi$
$702$	2.19615	0.0828884
$703$	−36.7846	−1.38736
$704$	−1.46410	−0.0551804
$705$	−6.00000	−0.225973
$706$	−7.60770	−0.286319
$707$	−21.4641	−0.807241
$708$	12.1962	0.458359
$709$	32.9282	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$709$	32.9282	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$710$	6.58846	0.247260
$711$	−10.0000	−0.375029
$712$	−12.5885	−0.471772
$713$	0.535898	0.0200696
$714$	−12.0000	−0.449089
$715$	−1.85641	−0.0694257
$716$	−9.07180	−0.339029
$717$	13.8564	0.517477
$718$	−2.87564	−0.107318
$719$	−9.46410	−0.352951	−0.176476	−	0.984305i	$-0.556470\pi$
$719$	−9.46410	−0.352951	−0.176476	+	0.984305i	$0.556470\pi$
$720$	5.00000	0.186339
$721$	−15.4641	−0.575913
$722$	18.8038	0.699807
$723$	−23.8564	−0.887229
$724$	−10.3923	−0.386227
$725$	0.732051	0.0271877
$726$	15.3397	0.569311
$727$	17.5167	0.649657	0.324828	−	0.945773i	$-0.394693\pi$
$727$	17.5167	0.649657	0.324828	+	0.945773i	$0.394693\pi$
$728$	−10.3923	−0.385164
$729$	1.00000	0.0370370
$730$	−9.80385	−0.362857
$731$	5.85641	0.216607
$732$	10.3923	0.384111
$733$	8.92820	0.329771	0.164885	−	0.986313i	$-0.447275\pi$
$733$	8.92820	0.329771	0.164885	+	0.986313i	$0.447275\pi$
$734$	−37.8564	−1.39731
$735$	15.3923	0.567754
$736$	2.78461	0.102642
$737$	14.9282	0.549887
$738$	6.00000	0.220863
$739$	−39.5692	−1.45558	−0.727789	−	0.685802i	$-0.759452\pi$
$739$	−39.5692	−1.45558	−0.727789	+	0.685802i	$0.759452\pi$
$740$	6.73205	0.247475
$741$	6.92820	0.254514
$742$	−101.569	−3.72872
$743$	48.4974	1.77920	0.889599	−	0.456743i	$-0.150984\pi$
$743$	48.4974	1.77920	0.889599	+	0.456743i	$0.150984\pi$
$744$	−1.73205	−0.0635001
$745$	−21.3205	−0.781123
$746$	45.7128	1.67366
$747$	15.4641	0.565802
$748$	2.14359	0.0783775
$749$	75.0333	2.74166
$750$	1.73205	0.0632456
$751$	−8.78461	−0.320555	−0.160277	−	0.987072i	$-0.551239\pi$
$751$	−8.78461	−0.320555	−0.160277	+	0.987072i	$0.551239\pi$
$752$	30.0000	1.09399
$753$	−2.92820	−0.106710
$754$	−1.60770	−0.0585488
$755$	20.9282	0.761655
$756$	4.73205	0.172103
$757$	22.7321	0.826210	0.413105	−	0.910683i	$-0.364444\pi$
$757$	22.7321	0.826210	0.413105	+	0.910683i	$0.364444\pi$
$758$	56.1051	2.03783
$759$	−0.784610	−0.0284795
$760$	9.46410	0.343299
$761$	−31.6603	−1.14768	−0.573842	−	0.818966i	$-0.694548\pi$
$761$	−31.6603	−1.14768	−0.573842	+	0.818966i	$0.694548\pi$
$762$	18.2487	0.661081
$763$	12.0000	0.434429
$764$	−11.8038	−0.427048
$765$	1.46410	0.0529347
$766$	−23.0718	−0.833618
$767$	15.4641	0.558376
$768$	−19.0000	−0.685603
$769$	−36.3923	−1.31234	−0.656170	−	0.754613i	$-0.727825\pi$
$769$	−36.3923	−1.31234	−0.656170	+	0.754613i	$0.727825\pi$
$770$	−12.0000	−0.432450
$771$	9.85641	0.354970
$772$	−23.8564	−0.858611
$773$	−17.1769	−0.617811	−0.308905	−	0.951093i	$-0.599963\pi$
$773$	−17.1769	−0.617811	−0.308905	+	0.951093i	$0.599963\pi$
$774$	−6.92820	−0.249029
$775$	−1.00000	−0.0359211
$776$	3.46410	0.124354
$777$	−31.8564	−1.14284
$778$	21.3731	0.766262
$779$	18.9282	0.678173
$780$	−1.26795	−0.0453999
$781$	5.56922	0.199282
$782$	1.35898	0.0485972
$783$	−0.732051	−0.0261614
$784$	−76.9615	−2.74863
$785$	12.9282	0.461427
$786$	−11.6603	−0.415907
$787$	−53.4641	−1.90579	−0.952895	−	0.303301i	$-0.901911\pi$
$787$	−53.4641	−1.90579	−0.952895	+	0.303301i	$0.901911\pi$
$788$	13.0718	0.465663
$789$	17.0718	0.607772
$790$	17.3205	0.616236
$791$	23.3205	0.829182
$792$	2.53590	0.0901092
$793$	13.1769	0.467926
$794$	41.3205	1.46641
$795$	12.3923	0.439510
$796$	16.9282	0.600004
$797$	−1.85641	−0.0657573	−0.0328786	−	0.999459i	$-0.510467\pi$
$797$	−1.85641	−0.0657573	−0.0328786	+	0.999459i	$0.510467\pi$
$798$	44.7846	1.58536
$799$	8.78461	0.310777
$800$	−5.19615	−0.183712
$801$	7.26795	0.256800
$802$	16.4833	0.582047
$803$	−8.28719	−0.292448
$804$	10.1962	0.359591
$805$	−2.53590	−0.0893787
$806$	2.19615	0.0773562
$807$	−9.80385	−0.345112
$808$	−7.85641	−0.276387
$809$	−28.8372	−1.01386	−0.506930	−	0.861987i	$-0.669220\pi$
$809$	−28.8372	−1.01386	−0.506930	+	0.861987i	$0.669220\pi$
$810$	−1.73205	−0.0608581
$811$	−30.2487	−1.06218	−0.531088	−	0.847317i	$-0.678217\pi$
$811$	−30.2487	−1.06218	−0.531088	+	0.847317i	$0.678217\pi$
$812$	−3.46410	−0.121566
$813$	16.7846	0.588662
$814$	17.0718	0.598366
$815$	14.1962	0.497270
$816$	−7.32051	−0.256269
$817$	−21.8564	−0.764659
$818$	−30.0000	−1.04893
$819$	6.00000	0.209657
$820$	−3.46410	−0.120972
$821$	17.1244	0.597644	0.298822	−	0.954309i	$-0.403406\pi$
$821$	17.1244	0.597644	0.298822	+	0.954309i	$0.403406\pi$
$822$	−32.7846	−1.14349
$823$	−23.7128	−0.826577	−0.413288	−	0.910600i	$-0.635620\pi$
$823$	−23.7128	−0.826577	−0.413288	+	0.910600i	$0.635620\pi$
$824$	−5.66025	−0.197184
$825$	1.46410	0.0509735
$826$	99.9615	3.47811
$827$	−34.1051	−1.18595	−0.592976	−	0.805220i	$-0.702047\pi$
$827$	−34.1051	−1.18595	−0.592976	+	0.805220i	$0.702047\pi$
$828$	−0.535898	−0.0186238
$829$	−54.1051	−1.87915	−0.939574	−	0.342345i	$-0.888779\pi$
$829$	−54.1051	−1.87915	−0.939574	+	0.342345i	$0.888779\pi$
$830$	−26.7846	−0.929707
$831$	15.1244	0.524658
$832$	−1.26795	−0.0439582
$833$	−22.5359	−0.780823
$834$	−17.0718	−0.591148
$835$	−13.8564	−0.479521
$836$	−8.00000	−0.276686
$837$	1.00000	0.0345651
$838$	−46.9808	−1.62292
$839$	−44.5885	−1.53936	−0.769682	−	0.638427i	$-0.779585\pi$
$839$	−44.5885	−1.53936	−0.769682	+	0.638427i	$0.779585\pi$
$840$	8.19615	0.282794
$841$	−28.4641	−0.981521
$842$	−26.5359	−0.914487
$843$	16.5359	0.569527
$844$	7.32051	0.251982
$845$	11.3923	0.391907
$846$	−10.3923	−0.357295
$847$	41.9090	1.44001
$848$	−61.9615	−2.12777
$849$	−7.26795	−0.249435
$850$	−2.53590	−0.0869806
$851$	3.60770	0.123670
$852$	3.80385	0.130318
$853$	52.2487	1.78896	0.894481	−	0.447106i	$-0.147545\pi$
$853$	52.2487	1.78896	0.894481	+	0.447106i	$0.147545\pi$
$854$	85.1769	2.91469
$855$	−5.46410	−0.186868
$856$	27.4641	0.938704
$857$	9.71281	0.331783	0.165892	−	0.986144i	$-0.446950\pi$
$857$	9.71281	0.331783	0.165892	+	0.986144i	$0.446950\pi$
$858$	−3.21539	−0.109772
$859$	39.7128	1.35498	0.677492	−	0.735530i	$-0.263067\pi$
$859$	39.7128	1.35498	0.677492	+	0.735530i	$0.263067\pi$
$860$	4.00000	0.136399
$861$	16.3923	0.558648
$862$	15.3731	0.523609
$863$	27.7128	0.943355	0.471678	−	0.881771i	$-0.343649\pi$
$863$	27.7128	0.943355	0.471678	+	0.881771i	$0.343649\pi$
$864$	5.19615	0.176777
$865$	22.7846	0.774700
$866$	26.8756	0.913272
$867$	14.8564	0.504550
$868$	4.73205	0.160616
$869$	14.6410	0.496662
$870$	1.26795	0.0429875
$871$	12.9282	0.438055
$872$	4.39230	0.148742
$873$	−2.00000	−0.0676897
$874$	−5.07180	−0.171556
$875$	4.73205	0.159973
$876$	−5.66025	−0.191242
$877$	−34.1051	−1.15165	−0.575824	−	0.817574i	$-0.695319\pi$
$877$	−34.1051	−1.15165	−0.575824	+	0.817574i	$0.695319\pi$
$878$	−61.8564	−2.08755
$879$	−5.07180	−0.171067
$880$	−7.32051	−0.246774
$881$	48.8372	1.64537	0.822683	−	0.568500i	$-0.192476\pi$
$881$	48.8372	1.64537	0.822683	+	0.568500i	$0.192476\pi$
$882$	26.6603	0.897697
$883$	−19.3205	−0.650187	−0.325093	−	0.945682i	$-0.605396\pi$
$883$	−19.3205	−0.650187	−0.325093	+	0.945682i	$0.605396\pi$
$884$	1.85641	0.0624377
$885$	−12.1962	−0.409969
$886$	52.6410	1.76851
$887$	−15.4641	−0.519234	−0.259617	−	0.965712i	$-0.583596\pi$
$887$	−15.4641	−0.519234	−0.259617	+	0.965712i	$0.583596\pi$
$888$	−11.6603	−0.391293
$889$	49.8564	1.67213
$890$	−12.5885	−0.421966
$891$	−1.46410	−0.0490492
$892$	−0.392305	−0.0131353
$893$	−32.7846	−1.09710
$894$	−36.9282	−1.23506
$895$	9.07180	0.303237
$896$	−57.3731	−1.91670
$897$	−0.679492	−0.0226876
$898$	70.5500	2.35428
$899$	−0.732051	−0.0244153
$900$	1.00000	0.0333333
$901$	−18.1436	−0.604451
$902$	−8.78461	−0.292496
$903$	−18.9282	−0.629891
$904$	8.53590	0.283900
$905$	10.3923	0.345452
$906$	36.2487	1.20428
$907$	20.7321	0.688396	0.344198	−	0.938897i	$-0.388151\pi$
$907$	20.7321	0.688396	0.344198	+	0.938897i	$0.388151\pi$
$908$	13.3205	0.442057
$909$	4.53590	0.150446
$910$	−10.3923	−0.344502
$911$	−9.46410	−0.313560	−0.156780	−	0.987634i	$-0.550111\pi$
$911$	−9.46410	−0.313560	−0.156780	+	0.987634i	$0.550111\pi$
$912$	27.3205	0.904672
$913$	−22.6410	−0.749308
$914$	56.4449	1.86703
$915$	−10.3923	−0.343559
$916$	18.7846	0.620661
$917$	−31.8564	−1.05199
$918$	2.53590	0.0836971
$919$	−10.5359	−0.347547	−0.173774	−	0.984786i	$-0.555596\pi$
$919$	−10.5359	−0.347547	−0.173774	+	0.984786i	$0.555596\pi$
$920$	−0.928203	−0.0306020
$921$	−26.5885	−0.876119
$922$	−1.26795	−0.0417577
$923$	4.82309	0.158754
$924$	−6.92820	−0.227921
$925$	−6.73205	−0.221348
$926$	68.7846	2.26040
$927$	3.26795	0.107334
$928$	−3.80385	−0.124867
$929$	−50.1962	−1.64688	−0.823441	−	0.567402i	$-0.807949\pi$
$929$	−50.1962	−1.64688	−0.823441	+	0.567402i	$0.807949\pi$
$930$	−1.73205	−0.0567962
$931$	84.1051	2.75643
$932$	10.9282	0.357965
$933$	−13.6603	−0.447217
$934$	3.46410	0.113349
$935$	−2.14359	−0.0701030
$936$	2.19615	0.0717835
$937$	−0.143594	−0.00469100	−0.00234550	−	0.999997i	$-0.500747\pi$
$937$	−0.143594	−0.00469100	−0.00234550	+	0.999997i	$0.500747\pi$
$938$	83.5692	2.72863
$939$	21.6603	0.706856
$940$	6.00000	0.195698
$941$	−6.58846	−0.214778	−0.107389	−	0.994217i	$-0.534249\pi$
$941$	−6.58846	−0.214778	−0.107389	+	0.994217i	$0.534249\pi$
$942$	22.3923	0.729581
$943$	−1.85641	−0.0604529
$944$	60.9808	1.98475
$945$	−4.73205	−0.153934
$946$	10.1436	0.329797
$947$	23.1769	0.753149	0.376574	−	0.926386i	$-0.377102\pi$
$947$	23.1769	0.753149	0.376574	+	0.926386i	$0.377102\pi$
$948$	10.0000	0.324785
$949$	−7.17691	−0.232973
$950$	9.46410	0.307056
$951$	22.9282	0.743498
$952$	−12.0000	−0.388922
$953$	−44.7846	−1.45072	−0.725358	−	0.688372i	$-0.758326\pi$
$953$	−44.7846	−1.45072	−0.725358	+	0.688372i	$0.758326\pi$
$954$	21.4641	0.694926
$955$	11.8038	0.381964
$956$	−13.8564	−0.448148
$957$	1.07180	0.0346463
$958$	−48.1577	−1.55590
$959$	−89.5692	−2.89234
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	14.7846	0.476675
$963$	−15.8564	−0.510966
$964$	23.8564	0.768363
$965$	23.8564	0.767965
$966$	−4.39230	−0.141320
$967$	−20.6795	−0.665008	−0.332504	−	0.943102i	$-0.607893\pi$
$967$	−20.6795	−0.665008	−0.332504	+	0.943102i	$0.607893\pi$
$968$	15.3397	0.493038
$969$	8.00000	0.256997
$970$	3.46410	0.111226
$971$	27.8038	0.892268	0.446134	−	0.894966i	$-0.352800\pi$
$971$	27.8038	0.892268	0.446134	+	0.894966i	$0.352800\pi$
$972$	−1.00000	−0.0320750
$973$	−46.6410	−1.49524
$974$	18.2487	0.584726
$975$	1.26795	0.0406069
$976$	51.9615	1.66325
$977$	−20.9282	−0.669553	−0.334776	−	0.942298i	$-0.608661\pi$
$977$	−20.9282	−0.669553	−0.334776	+	0.942298i	$0.608661\pi$
$978$	24.5885	0.786252
$979$	−10.6410	−0.340088
$980$	−15.3923	−0.491689
$981$	−2.53590	−0.0809650
$982$	42.2487	1.34821
$983$	−45.5692	−1.45343	−0.726716	−	0.686938i	$-0.758954\pi$
$983$	−45.5692	−1.45343	−0.726716	+	0.686938i	$0.758954\pi$
$984$	6.00000	0.191273
$985$	−13.0718	−0.416502
$986$	−1.85641	−0.0591200
$987$	−28.3923	−0.903737
$988$	−6.92820	−0.220416
$989$	2.14359	0.0681623
$990$	2.53590	0.0805961
$991$	18.1436	0.576350	0.288175	−	0.957578i	$-0.406951\pi$
$991$	18.1436	0.576350	0.288175	+	0.957578i	$0.406951\pi$
$992$	5.19615	0.164978
$993$	−24.9282	−0.791073
$994$	31.1769	0.988872
$995$	−16.9282	−0.536660
$996$	−15.4641	−0.489999
$997$	−29.6077	−0.937685	−0.468843	−	0.883282i	$-0.655329\pi$
$997$	−29.6077	−0.937685	−0.468843	+	0.883282i	$0.655329\pi$
$998$	−25.8564	−0.818470
$999$	6.73205	0.212993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.d.1.2	✓	2
3.2	odd	2		1395.2.a.f.1.1		2
4.3	odd	2		7440.2.a.bk.1.2		2
5.2	odd	4		2325.2.c.j.1024.4		4
5.3	odd	4		2325.2.c.j.1024.1		4
5.4	even	2		2325.2.a.m.1.1		2
15.14	odd	2		6975.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.d.1.2	✓	2	1.1	even	1	trivial
1395.2.a.f.1.1		2	3.2	odd	2
2325.2.a.m.1.1		2	5.4	even	2
2325.2.c.j.1024.1		4	5.3	odd	4
2325.2.c.j.1024.4		4	5.2	odd	4
6975.2.a.v.1.2		2	15.14	odd	2
7440.2.a.bk.1.2		2	4.3	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:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -4.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} -4.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} -1.46410 q^{11} -1.00000 q^{12} -1.26795 q^{13} -8.19615 q^{14} +1.00000 q^{15} -5.00000 q^{16} -1.46410 q^{17} +1.73205 q^{18} +5.46410 q^{19} -1.00000 q^{20} +4.73205 q^{21} -2.53590 q^{22} -0.535898 q^{23} +1.73205 q^{24} +1.00000 q^{25} -2.19615 q^{26} -1.00000 q^{27} -4.73205 q^{28} +0.732051 q^{29} +1.73205 q^{30} -1.00000 q^{31} -5.19615 q^{32} +1.46410 q^{33} -2.53590 q^{34} +4.73205 q^{35} +1.00000 q^{36} -6.73205 q^{37} +9.46410 q^{38} +1.26795 q^{39} +1.73205 q^{40} +3.46410 q^{41} +8.19615 q^{42} -4.00000 q^{43} -1.46410 q^{44} -1.00000 q^{45} -0.928203 q^{46} -6.00000 q^{47} +5.00000 q^{48} +15.3923 q^{49} +1.73205 q^{50} +1.46410 q^{51} -1.26795 q^{52} +12.3923 q^{53} -1.73205 q^{54} +1.46410 q^{55} +8.19615 q^{56} -5.46410 q^{57} +1.26795 q^{58} -12.1962 q^{59} +1.00000 q^{60} -10.3923 q^{61} -1.73205 q^{62} -4.73205 q^{63} +1.00000 q^{64} +1.26795 q^{65} +2.53590 q^{66} -10.1962 q^{67} -1.46410 q^{68} +0.535898 q^{69} +8.19615 q^{70} -3.80385 q^{71} -1.73205 q^{72} +5.66025 q^{73} -11.6603 q^{74} -1.00000 q^{75} +5.46410 q^{76} +6.92820 q^{77} +2.19615 q^{78} -10.0000 q^{79} +5.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +15.4641 q^{83} +4.73205 q^{84} +1.46410 q^{85} -6.92820 q^{86} -0.732051 q^{87} +2.53590 q^{88} +7.26795 q^{89} -1.73205 q^{90} +6.00000 q^{91} -0.535898 q^{92} +1.00000 q^{93} -10.3923 q^{94} -5.46410 q^{95} +5.19615 q^{96} -2.00000 q^{97} +26.6603 q^{98} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{12} - 6 q^{13} - 6 q^{14} + 2 q^{15} - 10 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} + 6 q^{21} - 12 q^{22} - 8 q^{23} + 2 q^{25} + 6 q^{26} - 2 q^{27} - 6 q^{28} - 2 q^{29} - 2 q^{31} - 4 q^{33} - 12 q^{34} + 6 q^{35} + 2 q^{36} - 10 q^{37} + 12 q^{38} + 6 q^{39} + 6 q^{42} - 8 q^{43} + 4 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} + 10 q^{48} + 10 q^{49} - 4 q^{51} - 6 q^{52} + 4 q^{53} - 4 q^{55} + 6 q^{56} - 4 q^{57} + 6 q^{58} - 14 q^{59} + 2 q^{60} - 6 q^{63} + 2 q^{64} + 6 q^{65} + 12 q^{66} - 10 q^{67} + 4 q^{68} + 8 q^{69} + 6 q^{70} - 18 q^{71} - 6 q^{73} - 6 q^{74} - 2 q^{75} + 4 q^{76} - 6 q^{78} - 20 q^{79} + 10 q^{80} + 2 q^{81} + 12 q^{82} + 24 q^{83} + 6 q^{84} - 4 q^{85} + 2 q^{87} + 12 q^{88} + 18 q^{89} + 12 q^{91} - 8 q^{92} + 2 q^{93} - 4 q^{95} - 4 q^{97} + 36 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^12 - 6 * q^13 - 6 * q^14 + 2 * q^15 - 10 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 + 6 * q^21 - 12 * q^22 - 8 * q^23 + 2 * q^25 + 6 * q^26 - 2 * q^27 - 6 * q^28 - 2 * q^29 - 2 * q^31 - 4 * q^33 - 12 * q^34 + 6 * q^35 + 2 * q^36 - 10 * q^37 + 12 * q^38 + 6 * q^39 + 6 * q^42 - 8 * q^43 + 4 * q^44 - 2 * q^45 + 12 * q^46 - 12 * q^47 + 10 * q^48 + 10 * q^49 - 4 * q^51 - 6 * q^52 + 4 * q^53 - 4 * q^55 + 6 * q^56 - 4 * q^57 + 6 * q^58 - 14 * q^59 + 2 * q^60 - 6 * q^63 + 2 * q^64 + 6 * q^65 + 12 * q^66 - 10 * q^67 + 4 * q^68 + 8 * q^69 + 6 * q^70 - 18 * q^71 - 6 * q^73 - 6 * q^74 - 2 * q^75 + 4 * q^76 - 6 * q^78 - 20 * q^79 + 10 * q^80 + 2 * q^81 + 12 * q^82 + 24 * q^83 + 6 * q^84 - 4 * q^85 + 2 * q^87 + 12 * q^88 + 18 * q^89 + 12 * q^91 - 8 * q^92 + 2 * q^93 - 4 * q^95 - 4 * q^97 + 36 * q^98 + 4 * q^99

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