LMFDB - Embedded newform 6042.2.a.bf.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + \cdots + 4048 $ x^12 - 5x^11 - 27x^10 + 134x^9 + 294x^8 - 1313x^7 - 1685x^6 + 5910x^5 + 5237x^4 - 12206x^3 - 7876x^2 + 9264*x + 4048
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$4.00527$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00527 q^{5} -1.00000 q^{6} -3.21582 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00527 q^{5} -1.00000 q^{6} -3.21582 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00527 q^{10} +5.12800 q^{11} +1.00000 q^{12} +3.41995 q^{13} +3.21582 q^{14} +3.00527 q^{15} +1.00000 q^{16} -0.940219 q^{17} -1.00000 q^{18} +1.00000 q^{19} +3.00527 q^{20} -3.21582 q^{21} -5.12800 q^{22} +1.92262 q^{23} -1.00000 q^{24} +4.03167 q^{25} -3.41995 q^{26} +1.00000 q^{27} -3.21582 q^{28} +2.19031 q^{29} -3.00527 q^{30} -9.62234 q^{31} -1.00000 q^{32} +5.12800 q^{33} +0.940219 q^{34} -9.66442 q^{35} +1.00000 q^{36} +2.11836 q^{37} -1.00000 q^{38} +3.41995 q^{39} -3.00527 q^{40} +12.0479 q^{41} +3.21582 q^{42} +0.0773802 q^{43} +5.12800 q^{44} +3.00527 q^{45} -1.92262 q^{46} +2.88389 q^{47} +1.00000 q^{48} +3.34150 q^{49} -4.03167 q^{50} -0.940219 q^{51} +3.41995 q^{52} +1.00000 q^{53} -1.00000 q^{54} +15.4110 q^{55} +3.21582 q^{56} +1.00000 q^{57} -2.19031 q^{58} +11.4673 q^{59} +3.00527 q^{60} -9.72441 q^{61} +9.62234 q^{62} -3.21582 q^{63} +1.00000 q^{64} +10.2779 q^{65} -5.12800 q^{66} +14.8803 q^{67} -0.940219 q^{68} +1.92262 q^{69} +9.66442 q^{70} -9.11787 q^{71} -1.00000 q^{72} -7.25794 q^{73} -2.11836 q^{74} +4.03167 q^{75} +1.00000 q^{76} -16.4907 q^{77} -3.41995 q^{78} +2.78210 q^{79} +3.00527 q^{80} +1.00000 q^{81} -12.0479 q^{82} +12.1481 q^{83} -3.21582 q^{84} -2.82561 q^{85} -0.0773802 q^{86} +2.19031 q^{87} -5.12800 q^{88} -12.1649 q^{89} -3.00527 q^{90} -10.9980 q^{91} +1.92262 q^{92} -9.62234 q^{93} -2.88389 q^{94} +3.00527 q^{95} -1.00000 q^{96} -10.3962 q^{97} -3.34150 q^{98} +5.12800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 + 10 * q^13 - 5 * q^14 - 7 * q^15 + 12 * q^16 + 6 * q^17 - 12 * q^18 + 12 * q^19 - 7 * q^20 + 5 * q^21 + 5 * q^22 - 12 * q^24 + 21 * q^25 - 10 * q^26 + 12 * q^27 + 5 * q^28 + 7 * q^30 + 2 * q^31 - 12 * q^32 - 5 * q^33 - 6 * q^34 - 17 * q^35 + 12 * q^36 + 16 * q^37 - 12 * q^38 + 10 * q^39 + 7 * q^40 + 7 * q^41 - 5 * q^42 + 24 * q^43 - 5 * q^44 - 7 * q^45 + 4 * q^47 + 12 * q^48 + 29 * q^49 - 21 * q^50 + 6 * q^51 + 10 * q^52 + 12 * q^53 - 12 * q^54 - 2 * q^55 - 5 * q^56 + 12 * q^57 + 2 * q^59 - 7 * q^60 + 29 * q^61 - 2 * q^62 + 5 * q^63 + 12 * q^64 - 17 * q^65 + 5 * q^66 + 36 * q^67 + 6 * q^68 + 17 * q^70 - 3 * q^71 - 12 * q^72 + 7 * q^73 - 16 * q^74 + 21 * q^75 + 12 * q^76 - 35 * q^77 - 10 * q^78 + 40 * q^79 - 7 * q^80 + 12 * q^81 - 7 * q^82 + 24 * q^83 + 5 * q^84 - 22 * q^85 - 24 * q^86 + 5 * q^88 - 45 * q^89 + 7 * q^90 + 71 * q^91 + 2 * q^93 - 4 * q^94 - 7 * q^95 - 12 * q^96 + q^97 - 29 * q^98 - 5 * 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
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.00527	1.34400	0.672000	−	0.740552i	$-0.265436\pi$
$5$	3.00527	1.34400	0.672000	+	0.740552i	$0.265436\pi$
$6$	−1.00000	−0.408248
$7$	−3.21582	−1.21547	−0.607733	−	0.794141i	$-0.707921\pi$
$7$	−3.21582	−1.21547	−0.607733	+	0.794141i	$0.707921\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00527	−0.950351
$11$	5.12800	1.54615	0.773075	−	0.634315i	$-0.218718\pi$
$11$	5.12800	1.54615	0.773075	+	0.634315i	$0.218718\pi$
$12$	1.00000	0.288675
$13$	3.41995	0.948525	0.474262	−	0.880384i	$-0.342715\pi$
$13$	3.41995	0.948525	0.474262	+	0.880384i	$0.342715\pi$
$14$	3.21582	0.859464
$15$	3.00527	0.775958
$16$	1.00000	0.250000
$17$	−0.940219	−0.228037	−0.114018	−	0.993479i	$-0.536372\pi$
$17$	−0.940219	−0.228037	−0.114018	+	0.993479i	$0.536372\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.00527	0.672000
$21$	−3.21582	−0.701750
$22$	−5.12800	−1.09329
$23$	1.92262	0.400894	0.200447	−	0.979705i	$-0.435761\pi$
$23$	1.92262	0.400894	0.200447	+	0.979705i	$0.435761\pi$
$24$	−1.00000	−0.204124
$25$	4.03167	0.806333
$26$	−3.41995	−0.670708
$27$	1.00000	0.192450
$28$	−3.21582	−0.607733
$29$	2.19031	0.406731	0.203365	−	0.979103i	$-0.434812\pi$
$29$	2.19031	0.406731	0.203365	+	0.979103i	$0.434812\pi$
$30$	−3.00527	−0.548685
$31$	−9.62234	−1.72822	−0.864111	−	0.503301i	$-0.832119\pi$
$31$	−9.62234	−1.72822	−0.864111	+	0.503301i	$0.832119\pi$
$32$	−1.00000	−0.176777
$33$	5.12800	0.892670
$34$	0.940219	0.161246
$35$	−9.66442	−1.63359
$36$	1.00000	0.166667
$37$	2.11836	0.348257	0.174128	−	0.984723i	$-0.444289\pi$
$37$	2.11836	0.348257	0.174128	+	0.984723i	$0.444289\pi$
$38$	−1.00000	−0.162221
$39$	3.41995	0.547631
$40$	−3.00527	−0.475175
$41$	12.0479	1.88156	0.940779	−	0.339020i	$-0.110095\pi$
$41$	12.0479	1.88156	0.940779	+	0.339020i	$0.110095\pi$
$42$	3.21582	0.496212
$43$	0.0773802	0.0118004	0.00590018	−	0.999983i	$-0.498122\pi$
$43$	0.0773802	0.0118004	0.00590018	+	0.999983i	$0.498122\pi$
$44$	5.12800	0.773075
$45$	3.00527	0.448000
$46$	−1.92262	−0.283475
$47$	2.88389	0.420658	0.210329	−	0.977631i	$-0.432546\pi$
$47$	2.88389	0.420658	0.210329	+	0.977631i	$0.432546\pi$
$48$	1.00000	0.144338
$49$	3.34150	0.477358
$50$	−4.03167	−0.570164
$51$	−0.940219	−0.131657
$52$	3.41995	0.474262
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	15.4110	2.07802
$56$	3.21582	0.429732
$57$	1.00000	0.132453
$58$	−2.19031	−0.287602
$59$	11.4673	1.49291	0.746457	−	0.665433i	$-0.231753\pi$
$59$	11.4673	1.49291	0.746457	+	0.665433i	$0.231753\pi$
$60$	3.00527	0.387979
$61$	−9.72441	−1.24508	−0.622542	−	0.782587i	$-0.713900\pi$
$61$	−9.72441	−1.24508	−0.622542	+	0.782587i	$0.713900\pi$
$62$	9.62234	1.22204
$63$	−3.21582	−0.405155
$64$	1.00000	0.125000
$65$	10.2779	1.27482
$66$	−5.12800	−0.631213
$67$	14.8803	1.81792	0.908958	−	0.416888i	$-0.136879\pi$
$67$	14.8803	1.81792	0.908958	+	0.416888i	$0.136879\pi$
$68$	−0.940219	−0.114018
$69$	1.92262	0.231456
$70$	9.66442	1.15512
$71$	−9.11787	−1.08209	−0.541046	−	0.840993i	$-0.681971\pi$
$71$	−9.11787	−1.08209	−0.541046	+	0.840993i	$0.681971\pi$
$72$	−1.00000	−0.117851
$73$	−7.25794	−0.849478	−0.424739	−	0.905316i	$-0.639634\pi$
$73$	−7.25794	−0.849478	−0.424739	+	0.905316i	$0.639634\pi$
$74$	−2.11836	−0.246255
$75$	4.03167	0.465537
$76$	1.00000	0.114708
$77$	−16.4907	−1.87929
$78$	−3.41995	−0.387234
$79$	2.78210	0.313011	0.156505	−	0.987677i	$-0.449977\pi$
$79$	2.78210	0.313011	0.156505	+	0.987677i	$0.449977\pi$
$80$	3.00527	0.336000
$81$	1.00000	0.111111
$82$	−12.0479	−1.33046
$83$	12.1481	1.33342	0.666712	−	0.745315i	$-0.267701\pi$
$83$	12.1481	1.33342	0.666712	+	0.745315i	$0.267701\pi$
$84$	−3.21582	−0.350875
$85$	−2.82561	−0.306481
$86$	−0.0773802	−0.00834412
$87$	2.19031	0.234826
$88$	−5.12800	−0.546647
$89$	−12.1649	−1.28948	−0.644738	−	0.764404i	$-0.723033\pi$
$89$	−12.1649	−1.28948	−0.644738	+	0.764404i	$0.723033\pi$
$90$	−3.00527	−0.316784
$91$	−10.9980	−1.15290
$92$	1.92262	0.200447
$93$	−9.62234	−0.997790
$94$	−2.88389	−0.297450
$95$	3.00527	0.308335
$96$	−1.00000	−0.102062
$97$	−10.3962	−1.05557	−0.527786	−	0.849377i	$-0.676978\pi$
$97$	−10.3962	−1.05557	−0.527786	+	0.849377i	$0.676978\pi$
$98$	−3.34150	−0.337543
$99$	5.12800	0.515383
$100$	4.03167	0.403167
$101$	−6.66768	−0.663459	−0.331730	−	0.943374i	$-0.607632\pi$
$101$	−6.66768	−0.663459	−0.331730	+	0.943374i	$0.607632\pi$
$102$	0.940219	0.0930955
$103$	1.23468	0.121657	0.0608285	−	0.998148i	$-0.480626\pi$
$103$	1.23468	0.121657	0.0608285	+	0.998148i	$0.480626\pi$
$104$	−3.41995	−0.335354
$105$	−9.66442	−0.943151
$106$	−1.00000	−0.0971286
$107$	12.5368	1.21198	0.605990	−	0.795472i	$-0.292777\pi$
$107$	12.5368	1.21198	0.605990	+	0.795472i	$0.292777\pi$
$108$	1.00000	0.0962250
$109$	−19.1952	−1.83857	−0.919283	−	0.393598i	$-0.871230\pi$
$109$	−19.1952	−1.83857	−0.919283	+	0.393598i	$0.871230\pi$
$110$	−15.4110	−1.46939
$111$	2.11836	0.201066
$112$	−3.21582	−0.303867
$113$	−2.66522	−0.250723	−0.125362	−	0.992111i	$-0.540009\pi$
$113$	−2.66522	−0.250723	−0.125362	+	0.992111i	$0.540009\pi$
$114$	−1.00000	−0.0936586
$115$	5.77800	0.538801
$116$	2.19031	0.203365
$117$	3.41995	0.316175
$118$	−11.4673	−1.05565
$119$	3.02357	0.277171
$120$	−3.00527	−0.274343
$121$	15.2964	1.39058
$122$	9.72441	0.880407
$123$	12.0479	1.08632
$124$	−9.62234	−0.864111
$125$	−2.91011	−0.260288
$126$	3.21582	0.286488
$127$	6.78601	0.602161	0.301080	−	0.953599i	$-0.402653\pi$
$127$	6.78601	0.602161	0.301080	+	0.953599i	$0.402653\pi$
$128$	−1.00000	−0.0883883
$129$	0.0773802	0.00681295
$130$	−10.2779	−0.901431
$131$	10.6076	0.926788	0.463394	−	0.886152i	$-0.346631\pi$
$131$	10.6076	0.926788	0.463394	+	0.886152i	$0.346631\pi$
$132$	5.12800	0.446335
$133$	−3.21582	−0.278847
$134$	−14.8803	−1.28546
$135$	3.00527	0.258653
$136$	0.940219	0.0806231
$137$	−9.71691	−0.830172	−0.415086	−	0.909782i	$-0.636248\pi$
$137$	−9.71691	−0.830172	−0.415086	+	0.909782i	$0.636248\pi$
$138$	−1.92262	−0.163664
$139$	23.0088	1.95158	0.975789	−	0.218716i	$-0.0701867\pi$
$139$	23.0088	1.95158	0.975789	+	0.218716i	$0.0701867\pi$
$140$	−9.66442	−0.816793
$141$	2.88389	0.242867
$142$	9.11787	0.765154
$143$	17.5375	1.46656
$144$	1.00000	0.0833333
$145$	6.58249	0.546646
$146$	7.25794	0.600672
$147$	3.34150	0.275603
$148$	2.11836	0.174128
$149$	16.7071	1.36870	0.684348	−	0.729155i	$-0.260087\pi$
$149$	16.7071	1.36870	0.684348	+	0.729155i	$0.260087\pi$
$150$	−4.03167	−0.329184
$151$	−4.01956	−0.327107	−0.163554	−	0.986534i	$-0.552296\pi$
$151$	−4.01956	−0.327107	−0.163554	+	0.986534i	$0.552296\pi$
$152$	−1.00000	−0.0811107
$153$	−0.940219	−0.0760122
$154$	16.4907	1.32886
$155$	−28.9178	−2.32273
$156$	3.41995	0.273815
$157$	4.21613	0.336484	0.168242	−	0.985746i	$-0.446191\pi$
$157$	4.21613	0.336484	0.168242	+	0.985746i	$0.446191\pi$
$158$	−2.78210	−0.221332
$159$	1.00000	0.0793052
$160$	−3.00527	−0.237588
$161$	−6.18280	−0.487273
$162$	−1.00000	−0.0785674
$163$	19.0509	1.49218	0.746089	−	0.665846i	$-0.231929\pi$
$163$	19.0509	1.49218	0.746089	+	0.665846i	$0.231929\pi$
$164$	12.0479	0.940779
$165$	15.4110	1.19975
$166$	−12.1481	−0.942873
$167$	3.76037	0.290986	0.145493	−	0.989359i	$-0.453523\pi$
$167$	3.76037	0.290986	0.145493	+	0.989359i	$0.453523\pi$
$168$	3.21582	0.248106
$169$	−1.30391	−0.100301
$170$	2.82561	0.216715
$171$	1.00000	0.0764719
$172$	0.0773802	0.00590018
$173$	0.854745	0.0649851	0.0324925	−	0.999472i	$-0.489655\pi$
$173$	0.854745	0.0649851	0.0324925	+	0.999472i	$0.489655\pi$
$174$	−2.19031	−0.166047
$175$	−12.9651	−0.980071
$176$	5.12800	0.386538
$177$	11.4673	0.861935
$178$	12.1649	0.911797
$179$	5.76690	0.431038	0.215519	−	0.976500i	$-0.430856\pi$
$179$	5.76690	0.431038	0.215519	+	0.976500i	$0.430856\pi$
$180$	3.00527	0.224000
$181$	−10.8167	−0.804002	−0.402001	−	0.915639i	$-0.631685\pi$
$181$	−10.8167	−0.804002	−0.402001	+	0.915639i	$0.631685\pi$
$182$	10.9980	0.815223
$183$	−9.72441	−0.718849
$184$	−1.92262	−0.141737
$185$	6.36626	0.468057
$186$	9.62234	0.705544
$187$	−4.82144	−0.352579
$188$	2.88389	0.210329
$189$	−3.21582	−0.233917
$190$	−3.00527	−0.218025
$191$	−23.2512	−1.68240	−0.841198	−	0.540728i	$-0.818149\pi$
$191$	−23.2512	−1.68240	−0.841198	+	0.540728i	$0.818149\pi$
$192$	1.00000	0.0721688
$193$	7.91654	0.569845	0.284922	−	0.958551i	$-0.408032\pi$
$193$	7.91654	0.569845	0.284922	+	0.958551i	$0.408032\pi$
$194$	10.3962	0.746402
$195$	10.2779	0.736015
$196$	3.34150	0.238679
$197$	−16.0906	−1.14641	−0.573203	−	0.819413i	$-0.694299\pi$
$197$	−16.0906	−1.14641	−0.573203	+	0.819413i	$0.694299\pi$
$198$	−5.12800	−0.364431
$199$	1.57611	0.111727	0.0558637	−	0.998438i	$-0.482209\pi$
$199$	1.57611	0.111727	0.0558637	+	0.998438i	$0.482209\pi$
$200$	−4.03167	−0.285082
$201$	14.8803	1.04957
$202$	6.66768	0.469137
$203$	−7.04365	−0.494368
$204$	−0.940219	−0.0658285
$205$	36.2071	2.52881
$206$	−1.23468	−0.0860245
$207$	1.92262	0.133631
$208$	3.41995	0.237131
$209$	5.12800	0.354711
$210$	9.66442	0.666908
$211$	−3.81419	−0.262580	−0.131290	−	0.991344i	$-0.541912\pi$
$211$	−3.81419	−0.262580	−0.131290	+	0.991344i	$0.541912\pi$
$212$	1.00000	0.0686803
$213$	−9.11787	−0.624746
$214$	−12.5368	−0.856999
$215$	0.232549	0.0158597
$216$	−1.00000	−0.0680414
$217$	30.9437	2.10060
$218$	19.1952	1.30006
$219$	−7.25794	−0.490446
$220$	15.4110	1.03901
$221$	−3.21550	−0.216298
$222$	−2.11836	−0.142175
$223$	−8.98310	−0.601553	−0.300776	−	0.953695i	$-0.597246\pi$
$223$	−8.98310	−0.601553	−0.300776	+	0.953695i	$0.597246\pi$
$224$	3.21582	0.214866
$225$	4.03167	0.268778
$226$	2.66522	0.177288
$227$	−1.10959	−0.0736463	−0.0368232	−	0.999322i	$-0.511724\pi$
$227$	−1.10959	−0.0736463	−0.0368232	+	0.999322i	$0.511724\pi$
$228$	1.00000	0.0662266
$229$	−2.95869	−0.195515	−0.0977577	−	0.995210i	$-0.531167\pi$
$229$	−2.95869	−0.195515	−0.0977577	+	0.995210i	$0.531167\pi$
$230$	−5.77800	−0.380990
$231$	−16.4907	−1.08501
$232$	−2.19031	−0.143801
$233$	−7.54755	−0.494457	−0.247228	−	0.968957i	$-0.579520\pi$
$233$	−7.54755	−0.494457	−0.247228	+	0.968957i	$0.579520\pi$
$234$	−3.41995	−0.223569
$235$	8.66687	0.565364
$236$	11.4673	0.746457
$237$	2.78210	0.180717
$238$	−3.02357	−0.195989
$239$	22.0369	1.42545	0.712725	−	0.701444i	$-0.247461\pi$
$239$	22.0369	1.42545	0.712725	+	0.701444i	$0.247461\pi$
$240$	3.00527	0.193990
$241$	−21.8449	−1.40715	−0.703576	−	0.710620i	$-0.748414\pi$
$241$	−21.8449	−1.40715	−0.703576	+	0.710620i	$0.748414\pi$
$242$	−15.2964	−0.983289
$243$	1.00000	0.0641500
$244$	−9.72441	−0.622542
$245$	10.0421	0.641568
$246$	−12.0479	−0.768143
$247$	3.41995	0.217606
$248$	9.62234	0.611019
$249$	12.1481	0.769853
$250$	2.91011	0.184051
$251$	−3.74287	−0.236248	−0.118124	−	0.992999i	$-0.537688\pi$
$251$	−3.74287	−0.236248	−0.118124	+	0.992999i	$0.537688\pi$
$252$	−3.21582	−0.202578
$253$	9.85919	0.619842
$254$	−6.78601	−0.425792
$255$	−2.82561	−0.176947
$256$	1.00000	0.0625000
$257$	29.9000	1.86511	0.932555	−	0.361029i	$-0.117574\pi$
$257$	29.9000	1.86511	0.932555	+	0.361029i	$0.117574\pi$
$258$	−0.0773802	−0.00481748
$259$	−6.81228	−0.423294
$260$	10.2779	0.637408
$261$	2.19031	0.135577
$262$	−10.6076	−0.655338
$263$	−15.9329	−0.982464	−0.491232	−	0.871029i	$-0.663453\pi$
$263$	−15.9329	−0.982464	−0.491232	+	0.871029i	$0.663453\pi$
$264$	−5.12800	−0.315607
$265$	3.00527	0.184612
$266$	3.21582	0.197175
$267$	−12.1649	−0.744479
$268$	14.8803	0.908958
$269$	−2.52907	−0.154200	−0.0770999	−	0.997023i	$-0.524566\pi$
$269$	−2.52907	−0.154200	−0.0770999	+	0.997023i	$0.524566\pi$
$270$	−3.00527	−0.182895
$271$	17.4656	1.06096	0.530478	−	0.847698i	$-0.322012\pi$
$271$	17.4656	1.06096	0.530478	+	0.847698i	$0.322012\pi$
$272$	−0.940219	−0.0570091
$273$	−10.9980	−0.665627
$274$	9.71691	0.587020
$275$	20.6744	1.24671
$276$	1.92262	0.115728
$277$	23.5983	1.41789	0.708943	−	0.705266i	$-0.249172\pi$
$277$	23.5983	1.41789	0.708943	+	0.705266i	$0.249172\pi$
$278$	−23.0088	−1.37997
$279$	−9.62234	−0.576074
$280$	9.66442	0.577560
$281$	8.46033	0.504701	0.252351	−	0.967636i	$-0.418796\pi$
$281$	8.46033	0.504701	0.252351	+	0.967636i	$0.418796\pi$
$282$	−2.88389	−0.171733
$283$	−4.05354	−0.240958	−0.120479	−	0.992716i	$-0.538443\pi$
$283$	−4.05354	−0.240958	−0.120479	+	0.992716i	$0.538443\pi$
$284$	−9.11787	−0.541046
$285$	3.00527	0.178017
$286$	−17.5375	−1.03702
$287$	−38.7437	−2.28697
$288$	−1.00000	−0.0589256
$289$	−16.1160	−0.947999
$290$	−6.58249	−0.386537
$291$	−10.3962	−0.609435
$292$	−7.25794	−0.424739
$293$	29.1855	1.70504	0.852518	−	0.522697i	$-0.175074\pi$
$293$	29.1855	1.70504	0.852518	+	0.522697i	$0.175074\pi$
$294$	−3.34150	−0.194880
$295$	34.4623	2.00648
$296$	−2.11836	−0.123127
$297$	5.12800	0.297557
$298$	−16.7071	−0.967815
$299$	6.57527	0.380258
$300$	4.03167	0.232768
$301$	−0.248841	−0.0143429
$302$	4.01956	0.231300
$303$	−6.66768	−0.383048
$304$	1.00000	0.0573539
$305$	−29.2245	−1.67339
$306$	0.940219	0.0537487
$307$	11.8483	0.676218	0.338109	−	0.941107i	$-0.390213\pi$
$307$	11.8483	0.676218	0.338109	+	0.941107i	$0.390213\pi$
$308$	−16.4907	−0.939647
$309$	1.23468	0.0702387
$310$	28.9178	1.64242
$311$	−12.9897	−0.736576	−0.368288	−	0.929712i	$-0.620056\pi$
$311$	−12.9897	−0.736576	−0.368288	+	0.929712i	$0.620056\pi$
$312$	−3.41995	−0.193617
$313$	−28.7246	−1.62361	−0.811806	−	0.583928i	$-0.801515\pi$
$313$	−28.7246	−1.62361	−0.811806	+	0.583928i	$0.801515\pi$
$314$	−4.21613	−0.237930
$315$	−9.66442	−0.544528
$316$	2.78210	0.156505
$317$	32.1285	1.80451	0.902257	−	0.431198i	$-0.141909\pi$
$317$	32.1285	1.80451	0.902257	+	0.431198i	$0.141909\pi$
$318$	−1.00000	−0.0560772
$319$	11.2319	0.628867
$320$	3.00527	0.168000
$321$	12.5368	0.699737
$322$	6.18280	0.344554
$323$	−0.940219	−0.0523152
$324$	1.00000	0.0555556
$325$	13.7881	0.764827
$326$	−19.0509	−1.05513
$327$	−19.1952	−1.06150
$328$	−12.0479	−0.665231
$329$	−9.27407	−0.511296
$330$	−15.4110	−0.848350
$331$	4.86362	0.267329	0.133664	−	0.991027i	$-0.457326\pi$
$331$	4.86362	0.267329	0.133664	+	0.991027i	$0.457326\pi$
$332$	12.1481	0.666712
$333$	2.11836	0.116086
$334$	−3.76037	−0.205758
$335$	44.7193	2.44328
$336$	−3.21582	−0.175437
$337$	9.42897	0.513628	0.256814	−	0.966461i	$-0.417327\pi$
$337$	9.42897	0.513628	0.256814	+	0.966461i	$0.417327\pi$
$338$	1.30391	0.0709235
$339$	−2.66522	−0.144755
$340$	−2.82561	−0.153240
$341$	−49.3433	−2.67209
$342$	−1.00000	−0.0540738
$343$	11.7651	0.635254
$344$	−0.0773802	−0.00417206
$345$	5.77800	0.311077
$346$	−0.854745	−0.0459514
$347$	−34.2341	−1.83778	−0.918892	−	0.394509i	$-0.870915\pi$
$347$	−34.2341	−1.83778	−0.918892	+	0.394509i	$0.870915\pi$
$348$	2.19031	0.117413
$349$	−11.5062	−0.615914	−0.307957	−	0.951400i	$-0.599645\pi$
$349$	−11.5062	−0.615914	−0.307957	+	0.951400i	$0.599645\pi$
$350$	12.9651	0.693015
$351$	3.41995	0.182544
$352$	−5.12800	−0.273323
$353$	−30.2451	−1.60979	−0.804893	−	0.593420i	$-0.797778\pi$
$353$	−30.2451	−1.60979	−0.804893	+	0.593420i	$0.797778\pi$
$354$	−11.4673	−0.609480
$355$	−27.4017	−1.45433
$356$	−12.1649	−0.644738
$357$	3.02357	0.160025
$358$	−5.76690	−0.304790
$359$	−24.7404	−1.30575	−0.652874	−	0.757467i	$-0.726437\pi$
$359$	−24.7404	−1.30575	−0.652874	+	0.757467i	$0.726437\pi$
$360$	−3.00527	−0.158392
$361$	1.00000	0.0526316
$362$	10.8167	0.568515
$363$	15.2964	0.802852
$364$	−10.9980	−0.576450
$365$	−21.8121	−1.14170
$366$	9.72441	0.508303
$367$	32.7819	1.71120	0.855602	−	0.517634i	$-0.173187\pi$
$367$	32.7819	1.71120	0.855602	+	0.517634i	$0.173187\pi$
$368$	1.92262	0.100223
$369$	12.0479	0.627186
$370$	−6.36626	−0.330966
$371$	−3.21582	−0.166957
$372$	−9.62234	−0.498895
$373$	3.34505	0.173200	0.0866000	−	0.996243i	$-0.472400\pi$
$373$	3.34505	0.173200	0.0866000	+	0.996243i	$0.472400\pi$
$374$	4.82144	0.249311
$375$	−2.91011	−0.150277
$376$	−2.88389	−0.148725
$377$	7.49077	0.385794
$378$	3.21582	0.165404
$379$	24.3098	1.24871	0.624357	−	0.781140i	$-0.285361\pi$
$379$	24.3098	1.24871	0.624357	+	0.781140i	$0.285361\pi$
$380$	3.00527	0.154167
$381$	6.78601	0.347658
$382$	23.2512	1.18963
$383$	−10.1870	−0.520532	−0.260266	−	0.965537i	$-0.583810\pi$
$383$	−10.1870	−0.520532	−0.260266	+	0.965537i	$0.583810\pi$
$384$	−1.00000	−0.0510310
$385$	−49.5591	−2.52577
$386$	−7.91654	−0.402941
$387$	0.0773802	0.00393346
$388$	−10.3962	−0.527786
$389$	−7.43376	−0.376906	−0.188453	−	0.982082i	$-0.560347\pi$
$389$	−7.43376	−0.376906	−0.188453	+	0.982082i	$0.560347\pi$
$390$	−10.2779	−0.520442
$391$	−1.80768	−0.0914185
$392$	−3.34150	−0.168771
$393$	10.6076	0.535081
$394$	16.0906	0.810632
$395$	8.36097	0.420686
$396$	5.12800	0.257692
$397$	4.38736	0.220195	0.110098	−	0.993921i	$-0.464884\pi$
$397$	4.38736	0.220195	0.110098	+	0.993921i	$0.464884\pi$
$398$	−1.57611	−0.0790032
$399$	−3.21582	−0.160992
$400$	4.03167	0.201583
$401$	23.3243	1.16476	0.582381	−	0.812916i	$-0.302121\pi$
$401$	23.3243	1.16476	0.582381	+	0.812916i	$0.302121\pi$
$402$	−14.8803	−0.742161
$403$	−32.9080	−1.63926
$404$	−6.66768	−0.331730
$405$	3.00527	0.149333
$406$	7.04365	0.349571
$407$	10.8630	0.538457
$408$	0.940219	0.0465478
$409$	9.18387	0.454113	0.227056	−	0.973882i	$-0.427090\pi$
$409$	9.18387	0.454113	0.227056	+	0.973882i	$0.427090\pi$
$410$	−36.2071	−1.78814
$411$	−9.71691	−0.479300
$412$	1.23468	0.0608285
$413$	−36.8768	−1.81459
$414$	−1.92262	−0.0944916
$415$	36.5083	1.79212
$416$	−3.41995	−0.167677
$417$	23.0088	1.12674
$418$	−5.12800	−0.250819
$419$	−20.2575	−0.989646	−0.494823	−	0.868994i	$-0.664767\pi$
$419$	−20.2575	−0.989646	−0.494823	+	0.868994i	$0.664767\pi$
$420$	−9.66442	−0.471575
$421$	−3.99032	−0.194476	−0.0972381	−	0.995261i	$-0.531001\pi$
$421$	−3.99032	−0.194476	−0.0972381	+	0.995261i	$0.531001\pi$
$422$	3.81419	0.185672
$423$	2.88389	0.140219
$424$	−1.00000	−0.0485643
$425$	−3.79065	−0.183873
$426$	9.11787	0.441762
$427$	31.2720	1.51336
$428$	12.5368	0.605990
$429$	17.5375	0.846720
$430$	−0.232549	−0.0112145
$431$	−15.7467	−0.758491	−0.379245	−	0.925296i	$-0.623816\pi$
$431$	−15.7467	−0.758491	−0.379245	+	0.925296i	$0.623816\pi$
$432$	1.00000	0.0481125
$433$	21.3849	1.02769	0.513846	−	0.857883i	$-0.328220\pi$
$433$	21.3849	1.02769	0.513846	+	0.857883i	$0.328220\pi$
$434$	−30.9437	−1.48535
$435$	6.58249	0.315606
$436$	−19.1952	−0.919283
$437$	1.92262	0.0919714
$438$	7.25794	0.346798
$439$	9.02868	0.430915	0.215458	−	0.976513i	$-0.430876\pi$
$439$	9.02868	0.430915	0.215458	+	0.976513i	$0.430876\pi$
$440$	−15.4110	−0.734693
$441$	3.34150	0.159119
$442$	3.21550	0.152946
$443$	−16.9326	−0.804494	−0.402247	−	0.915531i	$-0.631771\pi$
$443$	−16.9326	−0.804494	−0.402247	+	0.915531i	$0.631771\pi$
$444$	2.11836	0.100533
$445$	−36.5588	−1.73305
$446$	8.98310	0.425362
$447$	16.7071	0.790217
$448$	−3.21582	−0.151933
$449$	−9.83145	−0.463975	−0.231987	−	0.972719i	$-0.574523\pi$
$449$	−9.83145	−0.463975	−0.231987	+	0.972719i	$0.574523\pi$
$450$	−4.03167	−0.190055
$451$	61.7814	2.90917
$452$	−2.66522	−0.125362
$453$	−4.01956	−0.188856
$454$	1.10959	0.0520758
$455$	−33.0519	−1.54950
$456$	−1.00000	−0.0468293
$457$	25.8657	1.20995	0.604973	−	0.796246i	$-0.293184\pi$
$457$	25.8657	1.20995	0.604973	+	0.796246i	$0.293184\pi$
$458$	2.95869	0.138250
$459$	−0.940219	−0.0438857
$460$	5.77800	0.269401
$461$	−4.66666	−0.217348	−0.108674	−	0.994077i	$-0.534660\pi$
$461$	−4.66666	−0.217348	−0.108674	+	0.994077i	$0.534660\pi$
$462$	16.4907	0.767218
$463$	9.94209	0.462048	0.231024	−	0.972948i	$-0.425792\pi$
$463$	9.94209	0.462048	0.231024	+	0.972948i	$0.425792\pi$
$464$	2.19031	0.101683
$465$	−28.9178	−1.34103
$466$	7.54755	0.349634
$467$	−4.60177	−0.212945	−0.106472	−	0.994316i	$-0.533956\pi$
$467$	−4.60177	−0.212945	−0.106472	+	0.994316i	$0.533956\pi$
$468$	3.41995	0.158087
$469$	−47.8523	−2.20961
$470$	−8.66687	−0.399773
$471$	4.21613	0.194269
$472$	−11.4673	−0.527825
$473$	0.396806	0.0182451
$474$	−2.78210	−0.127786
$475$	4.03167	0.184986
$476$	3.02357	0.138585
$477$	1.00000	0.0457869
$478$	−22.0369	−1.00794
$479$	−19.3462	−0.883951	−0.441976	−	0.897027i	$-0.645722\pi$
$479$	−19.3462	−0.883951	−0.441976	+	0.897027i	$0.645722\pi$
$480$	−3.00527	−0.137171
$481$	7.24471	0.330330
$482$	21.8449	0.995006
$483$	−6.18280	−0.281327
$484$	15.2964	0.695290
$485$	−31.2434	−1.41869
$486$	−1.00000	−0.0453609
$487$	−9.18608	−0.416261	−0.208130	−	0.978101i	$-0.566738\pi$
$487$	−9.18608	−0.416261	−0.208130	+	0.978101i	$0.566738\pi$
$488$	9.72441	0.440203
$489$	19.0509	0.861510
$490$	−10.0421	−0.453657
$491$	12.3374	0.556780	0.278390	−	0.960468i	$-0.410199\pi$
$491$	12.3374	0.556780	0.278390	+	0.960468i	$0.410199\pi$
$492$	12.0479	0.543159
$493$	−2.05937	−0.0927495
$494$	−3.41995	−0.153871
$495$	15.4110	0.692675
$496$	−9.62234	−0.432056
$497$	29.3214	1.31525
$498$	−12.1481	−0.544368
$499$	−36.6880	−1.64238	−0.821191	−	0.570653i	$-0.806690\pi$
$499$	−36.6880	−1.64238	−0.821191	+	0.570653i	$0.806690\pi$
$500$	−2.91011	−0.130144
$501$	3.76037	0.168001
$502$	3.74287	0.167052
$503$	35.1088	1.56542	0.782711	−	0.622385i	$-0.213836\pi$
$503$	35.1088	1.56542	0.782711	+	0.622385i	$0.213836\pi$
$504$	3.21582	0.143244
$505$	−20.0382	−0.891689
$506$	−9.85919	−0.438295
$507$	−1.30391	−0.0579088
$508$	6.78601	0.301080
$509$	14.2410	0.631221	0.315610	−	0.948889i	$-0.397791\pi$
$509$	14.2410	0.631221	0.315610	+	0.948889i	$0.397791\pi$
$510$	2.82561	0.125120
$511$	23.3402	1.03251
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	−29.9000	−1.31883
$515$	3.71056	0.163507
$516$	0.0773802	0.00340647
$517$	14.7886	0.650401
$518$	6.81228	0.299314
$519$	0.854745	0.0375191
$520$	−10.2779	−0.450716
$521$	16.4237	0.719536	0.359768	−	0.933042i	$-0.382856\pi$
$521$	16.4237	0.719536	0.359768	+	0.933042i	$0.382856\pi$
$522$	−2.19031	−0.0958674
$523$	36.4396	1.59339	0.796696	−	0.604380i	$-0.206579\pi$
$523$	36.4396	1.59339	0.796696	+	0.604380i	$0.206579\pi$
$524$	10.6076	0.463394
$525$	−12.9651	−0.565844
$526$	15.9329	0.694707
$527$	9.04710	0.394098
$528$	5.12800	0.223168
$529$	−19.3035	−0.839284
$530$	−3.00527	−0.130541
$531$	11.4673	0.497638
$532$	−3.21582	−0.139424
$533$	41.2031	1.78470
$534$	12.1649	0.526426
$535$	37.6766	1.62890
$536$	−14.8803	−0.642730
$537$	5.76690	0.248860
$538$	2.52907	0.109036
$539$	17.1352	0.738067
$540$	3.00527	0.129326
$541$	−1.11213	−0.0478141	−0.0239070	−	0.999714i	$-0.507611\pi$
$541$	−1.11213	−0.0478141	−0.0239070	+	0.999714i	$0.507611\pi$
$542$	−17.4656	−0.750210
$543$	−10.8167	−0.464191
$544$	0.940219	0.0403115
$545$	−57.6868	−2.47103
$546$	10.9980	0.470669
$547$	−38.4923	−1.64581	−0.822906	−	0.568177i	$-0.807649\pi$
$547$	−38.4923	−1.64581	−0.822906	+	0.568177i	$0.807649\pi$
$548$	−9.71691	−0.415086
$549$	−9.72441	−0.415028
$550$	−20.6744	−0.881559
$551$	2.19031	0.0933105
$552$	−1.92262	−0.0818321
$553$	−8.94674	−0.380454
$554$	−23.5983	−1.00260
$555$	6.36626	0.270233
$556$	23.0088	0.975789
$557$	−29.6663	−1.25700	−0.628502	−	0.777808i	$-0.716331\pi$
$557$	−29.6663	−1.25700	−0.628502	+	0.777808i	$0.716331\pi$
$558$	9.62234	0.407346
$559$	0.264637	0.0111929
$560$	−9.66442	−0.408396
$561$	−4.82144	−0.203561
$562$	−8.46033	−0.356878
$563$	−33.8326	−1.42587	−0.712937	−	0.701228i	$-0.752636\pi$
$563$	−33.8326	−1.42587	−0.712937	+	0.701228i	$0.752636\pi$
$564$	2.88389	0.121434
$565$	−8.00972	−0.336972
$566$	4.05354	0.170383
$567$	−3.21582	−0.135052
$568$	9.11787	0.382577
$569$	−5.49246	−0.230256	−0.115128	−	0.993351i	$-0.536728\pi$
$569$	−5.49246	−0.230256	−0.115128	+	0.993351i	$0.536728\pi$
$570$	−3.00527	−0.125877
$571$	25.3738	1.06186	0.530931	−	0.847415i	$-0.321842\pi$
$571$	25.3738	1.06186	0.530931	+	0.847415i	$0.321842\pi$
$572$	17.5375	0.733281
$573$	−23.2512	−0.971331
$574$	38.7437	1.61713
$575$	7.75136	0.323254
$576$	1.00000	0.0416667
$577$	14.0185	0.583596	0.291798	−	0.956480i	$-0.405746\pi$
$577$	14.0185	0.583596	0.291798	+	0.956480i	$0.405746\pi$
$578$	16.1160	0.670337
$579$	7.91654	0.329000
$580$	6.58249	0.273323
$581$	−39.0660	−1.62073
$582$	10.3962	0.430936
$583$	5.12800	0.212380
$584$	7.25794	0.300336
$585$	10.2779	0.424939
$586$	−29.1855	−1.20564
$587$	−27.6328	−1.14053	−0.570264	−	0.821462i	$-0.693159\pi$
$587$	−27.6328	−1.14053	−0.570264	+	0.821462i	$0.693159\pi$
$588$	3.34150	0.137801
$589$	−9.62234	−0.396481
$590$	−34.4623	−1.41879
$591$	−16.0906	−0.661878
$592$	2.11836	0.0870642
$593$	−3.84896	−0.158058	−0.0790290	−	0.996872i	$-0.525182\pi$
$593$	−3.84896	−0.158058	−0.0790290	+	0.996872i	$0.525182\pi$
$594$	−5.12800	−0.210404
$595$	9.08667	0.372517
$596$	16.7071	0.684348
$597$	1.57611	0.0645058
$598$	−6.57527	−0.268883
$599$	17.9692	0.734201	0.367101	−	0.930181i	$-0.380350\pi$
$599$	17.9692	0.734201	0.367101	+	0.930181i	$0.380350\pi$
$600$	−4.03167	−0.164592
$601$	−7.02693	−0.286635	−0.143317	−	0.989677i	$-0.545777\pi$
$601$	−7.02693	−0.286635	−0.143317	+	0.989677i	$0.545777\pi$
$602$	0.248841	0.0101420
$603$	14.8803	0.605972
$604$	−4.01956	−0.163554
$605$	45.9698	1.86894
$606$	6.66768	0.270856
$607$	3.92149	0.159168	0.0795842	−	0.996828i	$-0.474641\pi$
$607$	3.92149	0.159168	0.0795842	+	0.996828i	$0.474641\pi$
$608$	−1.00000	−0.0405554
$609$	−7.04365	−0.285423
$610$	29.2245	1.18327
$611$	9.86277	0.399005
$612$	−0.940219	−0.0380061
$613$	42.6307	1.72184	0.860919	−	0.508742i	$-0.169889\pi$
$613$	42.6307	1.72184	0.860919	+	0.508742i	$0.169889\pi$
$614$	−11.8483	−0.478159
$615$	36.2071	1.46001
$616$	16.4907	0.664430
$617$	1.23042	0.0495349	0.0247675	−	0.999693i	$-0.492115\pi$
$617$	1.23042	0.0495349	0.0247675	+	0.999693i	$0.492115\pi$
$618$	−1.23468	−0.0496663
$619$	26.3056	1.05731	0.528656	−	0.848836i	$-0.322696\pi$
$619$	26.3056	1.05731	0.528656	+	0.848836i	$0.322696\pi$
$620$	−28.9178	−1.16136
$621$	1.92262	0.0771521
$622$	12.9897	0.520838
$623$	39.1201	1.56731
$624$	3.41995	0.136908
$625$	−28.9040	−1.15616
$626$	28.7246	1.14807
$627$	5.12800	0.204793
$628$	4.21613	0.168242
$629$	−1.99172	−0.0794153
$630$	9.66442	0.385040
$631$	−22.3179	−0.888461	−0.444231	−	0.895912i	$-0.646523\pi$
$631$	−22.3179	−0.888461	−0.444231	+	0.895912i	$0.646523\pi$
$632$	−2.78210	−0.110666
$633$	−3.81419	−0.151600
$634$	−32.1285	−1.27598
$635$	20.3938	0.809303
$636$	1.00000	0.0396526
$637$	11.4278	0.452786
$638$	−11.2319	−0.444676
$639$	−9.11787	−0.360697
$640$	−3.00527	−0.118794
$641$	−6.40916	−0.253147	−0.126573	−	0.991957i	$-0.540398\pi$
$641$	−6.40916	−0.253147	−0.126573	+	0.991957i	$0.540398\pi$
$642$	−12.5368	−0.494789
$643$	21.7167	0.856423	0.428211	−	0.903679i	$-0.359144\pi$
$643$	21.7167	0.856423	0.428211	+	0.903679i	$0.359144\pi$
$644$	−6.18280	−0.243636
$645$	0.232549	0.00915659
$646$	0.940219	0.0369924
$647$	−21.7927	−0.856758	−0.428379	−	0.903599i	$-0.640915\pi$
$647$	−21.7927	−0.856758	−0.428379	+	0.903599i	$0.640915\pi$
$648$	−1.00000	−0.0392837
$649$	58.8043	2.30827
$650$	−13.7881	−0.540814
$651$	30.9437	1.21278
$652$	19.0509	0.746089
$653$	10.8601	0.424988	0.212494	−	0.977162i	$-0.431841\pi$
$653$	10.8601	0.424988	0.212494	+	0.977162i	$0.431841\pi$
$654$	19.1952	0.750591
$655$	31.8787	1.24560
$656$	12.0479	0.470390
$657$	−7.25794	−0.283159
$658$	9.27407	0.361541
$659$	−6.80248	−0.264987	−0.132493	−	0.991184i	$-0.542298\pi$
$659$	−6.80248	−0.264987	−0.132493	+	0.991184i	$0.542298\pi$
$660$	15.4110	0.599874
$661$	26.9098	1.04667	0.523335	−	0.852127i	$-0.324688\pi$
$661$	26.9098	1.04667	0.523335	+	0.852127i	$0.324688\pi$
$662$	−4.86362	−0.189030
$663$	−3.21550	−0.124880
$664$	−12.1481	−0.471437
$665$	−9.66442	−0.374770
$666$	−2.11836	−0.0820849
$667$	4.21114	0.163056
$668$	3.76037	0.145493
$669$	−8.98310	−0.347307
$670$	−44.7193	−1.72766
$671$	−49.8668	−1.92509
$672$	3.21582	0.124053
$673$	15.8514	0.611026	0.305513	−	0.952188i	$-0.401172\pi$
$673$	15.8514	0.611026	0.305513	+	0.952188i	$0.401172\pi$
$674$	−9.42897	−0.363190
$675$	4.03167	0.155179
$676$	−1.30391	−0.0501505
$677$	−2.57673	−0.0990318	−0.0495159	−	0.998773i	$-0.515768\pi$
$677$	−2.57673	−0.0990318	−0.0495159	+	0.998773i	$0.515768\pi$
$678$	2.66522	0.102357
$679$	33.4323	1.28301
$680$	2.82561	0.108357
$681$	−1.10959	−0.0425197
$682$	49.3433	1.88945
$683$	1.36058	0.0520613	0.0260306	−	0.999661i	$-0.491713\pi$
$683$	1.36058	0.0520613	0.0260306	+	0.999661i	$0.491713\pi$
$684$	1.00000	0.0382360
$685$	−29.2020	−1.11575
$686$	−11.7651	−0.449192
$687$	−2.95869	−0.112881
$688$	0.0773802	0.00295009
$689$	3.41995	0.130290
$690$	−5.77800	−0.219965
$691$	13.3630	0.508354	0.254177	−	0.967158i	$-0.418195\pi$
$691$	13.3630	0.508354	0.254177	+	0.967158i	$0.418195\pi$
$692$	0.854745	0.0324925
$693$	−16.4907	−0.626431
$694$	34.2341	1.29951
$695$	69.1476	2.62292
$696$	−2.19031	−0.0830236
$697$	−11.3276	−0.429064
$698$	11.5062	0.435517
$699$	−7.54755	−0.285475
$700$	−12.9651	−0.490035
$701$	47.3367	1.78788	0.893940	−	0.448186i	$-0.147930\pi$
$701$	47.3367	1.78788	0.893940	+	0.448186i	$0.147930\pi$
$702$	−3.41995	−0.129078
$703$	2.11836	0.0798956
$704$	5.12800	0.193269
$705$	8.66687	0.326413
$706$	30.2451	1.13829
$707$	21.4421	0.806412
$708$	11.4673	0.430967
$709$	−40.2167	−1.51037	−0.755184	−	0.655512i	$-0.772453\pi$
$709$	−40.2167	−1.51037	−0.755184	+	0.655512i	$0.772453\pi$
$710$	27.4017	1.02837
$711$	2.78210	0.104337
$712$	12.1649	0.455898
$713$	−18.5001	−0.692834
$714$	−3.02357	−0.113154
$715$	52.7051	1.97106
$716$	5.76690	0.215519
$717$	22.0369	0.822984
$718$	24.7404	0.923303
$719$	−38.5792	−1.43876	−0.719381	−	0.694616i	$-0.755574\pi$
$719$	−38.5792	−1.43876	−0.719381	+	0.694616i	$0.755574\pi$
$720$	3.00527	0.112000
$721$	−3.97052	−0.147870
$722$	−1.00000	−0.0372161
$723$	−21.8449	−0.812419
$724$	−10.8167	−0.402001
$725$	8.83061	0.327961
$726$	−15.2964	−0.567702
$727$	50.4658	1.87167	0.935836	−	0.352437i	$-0.114647\pi$
$727$	50.4658	1.87167	0.935836	+	0.352437i	$0.114647\pi$
$728$	10.9980	0.407612
$729$	1.00000	0.0370370
$730$	21.8121	0.807302
$731$	−0.0727543	−0.00269092
$732$	−9.72441	−0.359425
$733$	−40.9083	−1.51098	−0.755491	−	0.655159i	$-0.772601\pi$
$733$	−40.9083	−1.51098	−0.755491	+	0.655159i	$0.772601\pi$
$734$	−32.7819	−1.21000
$735$	10.0421	0.370410
$736$	−1.92262	−0.0708687
$737$	76.3061	2.81077
$738$	−12.0479	−0.443488
$739$	−16.2393	−0.597370	−0.298685	−	0.954352i	$-0.596548\pi$
$739$	−16.2393	−0.597370	−0.298685	+	0.954352i	$0.596548\pi$
$740$	6.36626	0.234028
$741$	3.41995	0.125635
$742$	3.21582	0.118056
$743$	13.8948	0.509752	0.254876	−	0.966974i	$-0.417965\pi$
$743$	13.8948	0.509752	0.254876	+	0.966974i	$0.417965\pi$
$744$	9.62234	0.352772
$745$	50.2093	1.83953
$746$	−3.34505	−0.122471
$747$	12.1481	0.444475
$748$	−4.82144	−0.176289
$749$	−40.3162	−1.47312
$750$	2.91011	0.106262
$751$	11.8570	0.432668	0.216334	−	0.976319i	$-0.430590\pi$
$751$	11.8570	0.432668	0.216334	+	0.976319i	$0.430590\pi$
$752$	2.88389	0.105165
$753$	−3.74287	−0.136398
$754$	−7.49077	−0.272798
$755$	−12.0799	−0.439632
$756$	−3.21582	−0.116958
$757$	36.5667	1.32904	0.664520	−	0.747271i	$-0.268636\pi$
$757$	36.5667	1.32904	0.664520	+	0.747271i	$0.268636\pi$
$758$	−24.3098	−0.882973
$759$	9.85919	0.357866
$760$	−3.00527	−0.109013
$761$	28.8100	1.04436	0.522181	−	0.852835i	$-0.325118\pi$
$761$	28.8100	1.04436	0.522181	+	0.852835i	$0.325118\pi$
$762$	−6.78601	−0.245831
$763$	61.7283	2.23471
$764$	−23.2512	−0.841198
$765$	−2.82561	−0.102160
$766$	10.1870	0.368072
$767$	39.2176	1.41607
$768$	1.00000	0.0360844
$769$	−21.3675	−0.770532	−0.385266	−	0.922806i	$-0.625890\pi$
$769$	−21.3675	−0.770532	−0.385266	+	0.922806i	$0.625890\pi$
$770$	49.5591	1.78599
$771$	29.9000	1.07682
$772$	7.91654	0.284922
$773$	−14.4774	−0.520716	−0.260358	−	0.965512i	$-0.583841\pi$
$773$	−14.4774	−0.520716	−0.260358	+	0.965512i	$0.583841\pi$
$774$	−0.0773802	−0.00278137
$775$	−38.7941	−1.39352
$776$	10.3962	0.373201
$777$	−6.81228	−0.244389
$778$	7.43376	0.266513
$779$	12.0479	0.431659
$780$	10.2779	0.368008
$781$	−46.7564	−1.67308
$782$	1.80768	0.0646426
$783$	2.19031	0.0782754
$784$	3.34150	0.119339
$785$	12.6706	0.452234
$786$	−10.6076	−0.378360
$787$	41.2670	1.47101	0.735506	−	0.677518i	$-0.236945\pi$
$787$	41.2670	1.47101	0.735506	+	0.677518i	$0.236945\pi$
$788$	−16.0906	−0.573203
$789$	−15.9329	−0.567226
$790$	−8.36097	−0.297470
$791$	8.57088	0.304745
$792$	−5.12800	−0.182216
$793$	−33.2570	−1.18099
$794$	−4.38736	−0.155702
$795$	3.00527	0.106586
$796$	1.57611	0.0558637
$797$	−20.9319	−0.741445	−0.370722	−	0.928744i	$-0.620890\pi$
$797$	−20.9319	−0.741445	−0.370722	+	0.928744i	$0.620890\pi$
$798$	3.21582	0.113839
$799$	−2.71149	−0.0959255
$800$	−4.03167	−0.142541
$801$	−12.1649	−0.429825
$802$	−23.3243	−0.823610
$803$	−37.2187	−1.31342
$804$	14.8803	0.524787
$805$	−18.5810	−0.654894
$806$	32.9080	1.15913
$807$	−2.52907	−0.0890273
$808$	6.66768	0.234568
$809$	32.6775	1.14888	0.574441	−	0.818546i	$-0.305220\pi$
$809$	32.6775	1.14888	0.574441	+	0.818546i	$0.305220\pi$
$810$	−3.00527	−0.105595
$811$	−48.2793	−1.69532	−0.847658	−	0.530543i	$-0.821988\pi$
$811$	−48.2793	−1.69532	−0.847658	+	0.530543i	$0.821988\pi$
$812$	−7.04365	−0.247184
$813$	17.4656	0.612544
$814$	−10.8630	−0.380747
$815$	57.2530	2.00549
$816$	−0.940219	−0.0329142
$817$	0.0773802	0.00270719
$818$	−9.18387	−0.321106
$819$	−10.9980	−0.384300
$820$	36.2071	1.26441
$821$	11.2174	0.391490	0.195745	−	0.980655i	$-0.437288\pi$
$821$	11.2174	0.391490	0.195745	+	0.980655i	$0.437288\pi$
$822$	9.71691	0.338916
$823$	10.5550	0.367923	0.183961	−	0.982933i	$-0.441108\pi$
$823$	10.5550	0.367923	0.183961	+	0.982933i	$0.441108\pi$
$824$	−1.23468	−0.0430123
$825$	20.6744	0.719790
$826$	36.8768	1.28311
$827$	−9.74588	−0.338897	−0.169449	−	0.985539i	$-0.554199\pi$
$827$	−9.74588	−0.338897	−0.169449	+	0.985539i	$0.554199\pi$
$828$	1.92262	0.0668157
$829$	−45.6223	−1.58453	−0.792265	−	0.610178i	$-0.791098\pi$
$829$	−45.6223	−1.58453	−0.792265	+	0.610178i	$0.791098\pi$
$830$	−36.5083	−1.26722
$831$	23.5983	0.818617
$832$	3.41995	0.118566
$833$	−3.14174	−0.108855
$834$	−23.0088	−0.796728
$835$	11.3009	0.391085
$836$	5.12800	0.177356
$837$	−9.62234	−0.332597
$838$	20.2575	0.699785
$839$	18.0833	0.624304	0.312152	−	0.950032i	$-0.398950\pi$
$839$	18.0833	0.624304	0.312152	+	0.950032i	$0.398950\pi$
$840$	9.66442	0.333454
$841$	−24.2025	−0.834570
$842$	3.99032	0.137515
$843$	8.46033	0.291389
$844$	−3.81419	−0.131290
$845$	−3.91861	−0.134804
$846$	−2.88389	−0.0991501
$847$	−49.1904	−1.69020
$848$	1.00000	0.0343401
$849$	−4.05354	−0.139117
$850$	3.79065	0.130018
$851$	4.07281	0.139614
$852$	−9.11787	−0.312373
$853$	−12.8919	−0.441410	−0.220705	−	0.975341i	$-0.570836\pi$
$853$	−12.8919	−0.441410	−0.220705	+	0.975341i	$0.570836\pi$
$854$	−31.2720	−1.07010
$855$	3.00527	0.102778
$856$	−12.5368	−0.428500
$857$	−9.84416	−0.336270	−0.168135	−	0.985764i	$-0.553774\pi$
$857$	−9.84416	−0.336270	−0.168135	+	0.985764i	$0.553774\pi$
$858$	−17.5375	−0.598721
$859$	−29.4526	−1.00491	−0.502455	−	0.864603i	$-0.667570\pi$
$859$	−29.4526	−1.00491	−0.502455	+	0.864603i	$0.667570\pi$
$860$	0.232549	0.00792984
$861$	−38.7437	−1.32038
$862$	15.7467	0.536334
$863$	−34.8300	−1.18563	−0.592813	−	0.805340i	$-0.701983\pi$
$863$	−34.8300	−1.18563	−0.592813	+	0.805340i	$0.701983\pi$
$864$	−1.00000	−0.0340207
$865$	2.56874	0.0873399
$866$	−21.3849	−0.726687
$867$	−16.1160	−0.547328
$868$	30.9437	1.05030
$869$	14.2666	0.483962
$870$	−6.58249	−0.223167
$871$	50.8899	1.72434
$872$	19.1952	0.650031
$873$	−10.3962	−0.351857
$874$	−1.92262	−0.0650336
$875$	9.35838	0.316371
$876$	−7.25794	−0.245223
$877$	17.9776	0.607061	0.303531	−	0.952822i	$-0.401835\pi$
$877$	17.9776	0.607061	0.303531	+	0.952822i	$0.401835\pi$
$878$	−9.02868	−0.304703
$879$	29.1855	0.984404
$880$	15.4110	0.519506
$881$	−56.3782	−1.89943	−0.949716	−	0.313114i	$-0.898628\pi$
$881$	−56.3782	−1.89943	−0.949716	+	0.313114i	$0.898628\pi$
$882$	−3.34150	−0.112514
$883$	−50.8475	−1.71116	−0.855578	−	0.517675i	$-0.826798\pi$
$883$	−50.8475	−1.71116	−0.855578	+	0.517675i	$0.826798\pi$
$884$	−3.21550	−0.108149
$885$	34.4623	1.15844
$886$	16.9326	0.568863
$887$	1.29811	0.0435863	0.0217931	−	0.999763i	$-0.493062\pi$
$887$	1.29811	0.0435863	0.0217931	+	0.999763i	$0.493062\pi$
$888$	−2.11836	−0.0710876
$889$	−21.8226	−0.731906
$890$	36.5588	1.22545
$891$	5.12800	0.171794
$892$	−8.98310	−0.300776
$893$	2.88389	0.0965056
$894$	−16.7071	−0.558768
$895$	17.3311	0.579315
$896$	3.21582	0.107433
$897$	6.57527	0.219542
$898$	9.83145	0.328080
$899$	−21.0759	−0.702922
$900$	4.03167	0.134389
$901$	−0.940219	−0.0313232
$902$	−61.7814	−2.05710
$903$	−0.248841	−0.00828091
$904$	2.66522	0.0886440
$905$	−32.5072	−1.08058
$906$	4.01956	0.133541
$907$	−14.5034	−0.481577	−0.240789	−	0.970578i	$-0.577406\pi$
$907$	−14.5034	−0.481577	−0.240789	+	0.970578i	$0.577406\pi$
$908$	−1.10959	−0.0368232
$909$	−6.66768	−0.221153
$910$	33.0519	1.09566
$911$	−17.8622	−0.591801	−0.295900	−	0.955219i	$-0.595620\pi$
$911$	−17.8622	−0.591801	−0.295900	+	0.955219i	$0.595620\pi$
$912$	1.00000	0.0331133
$913$	62.2953	2.06167
$914$	−25.8657	−0.855561
$915$	−29.2245	−0.966133
$916$	−2.95869	−0.0977577
$917$	−34.1121	−1.12648
$918$	0.940219	0.0310318
$919$	23.0713	0.761052	0.380526	−	0.924770i	$-0.375743\pi$
$919$	23.0713	0.761052	0.380526	+	0.924770i	$0.375743\pi$
$920$	−5.77800	−0.190495
$921$	11.8483	0.390415
$922$	4.66666	0.153688
$923$	−31.1827	−1.02639
$924$	−16.4907	−0.542505
$925$	8.54053	0.280811
$926$	−9.94209	−0.326717
$927$	1.23468	0.0405524
$928$	−2.19031	−0.0719005
$929$	26.4077	0.866408	0.433204	−	0.901296i	$-0.357383\pi$
$929$	26.4077	0.866408	0.433204	+	0.901296i	$0.357383\pi$
$930$	28.9178	0.948250
$931$	3.34150	0.109513
$932$	−7.54755	−0.247228
$933$	−12.9897	−0.425262
$934$	4.60177	0.150575
$935$	−14.4897	−0.473865
$936$	−3.41995	−0.111785
$937$	−21.0135	−0.686481	−0.343241	−	0.939247i	$-0.611525\pi$
$937$	−21.0135	−0.686481	−0.343241	+	0.939247i	$0.611525\pi$
$938$	47.8523	1.56243
$939$	−28.7246	−0.937393
$940$	8.66687	0.282682
$941$	−30.2772	−0.987009	−0.493504	−	0.869743i	$-0.664284\pi$
$941$	−30.2772	−0.987009	−0.493504	+	0.869743i	$0.664284\pi$
$942$	−4.21613	−0.137369
$943$	23.1634	0.754305
$944$	11.4673	0.373229
$945$	−9.66442	−0.314384
$946$	−0.396806	−0.0129013
$947$	−31.9675	−1.03880	−0.519402	−	0.854530i	$-0.673845\pi$
$947$	−31.9675	−1.03880	−0.519402	+	0.854530i	$0.673845\pi$
$948$	2.78210	0.0903584
$949$	−24.8218	−0.805751
$950$	−4.03167	−0.130805
$951$	32.1285	1.04184
$952$	−3.02357	−0.0979946
$953$	−32.7470	−1.06078	−0.530390	−	0.847754i	$-0.677954\pi$
$953$	−32.7470	−1.06078	−0.530390	+	0.847754i	$0.677954\pi$
$954$	−1.00000	−0.0323762
$955$	−69.8761	−2.26114
$956$	22.0369	0.712725
$957$	11.2319	0.363077
$958$	19.3462	0.625048
$959$	31.2479	1.00905
$960$	3.00527	0.0969948
$961$	61.5894	1.98675
$962$	−7.24471	−0.233579
$963$	12.5368	0.403993
$964$	−21.8449	−0.703576
$965$	23.7914	0.765871
$966$	6.18280	0.198928
$967$	−28.7771	−0.925410	−0.462705	−	0.886512i	$-0.653121\pi$
$967$	−28.7771	−0.925410	−0.462705	+	0.886512i	$0.653121\pi$
$968$	−15.2964	−0.491644
$969$	−0.940219	−0.0302042
$970$	31.2434	1.00316
$971$	38.7144	1.24240	0.621202	−	0.783650i	$-0.286645\pi$
$971$	38.7144	1.24240	0.621202	+	0.783650i	$0.286645\pi$
$972$	1.00000	0.0320750
$973$	−73.9920	−2.37208
$974$	9.18608	0.294341
$975$	13.7881	0.441573
$976$	−9.72441	−0.311271
$977$	14.5001	0.463898	0.231949	−	0.972728i	$-0.425490\pi$
$977$	14.5001	0.463898	0.231949	+	0.972728i	$0.425490\pi$
$978$	−19.0509	−0.609179
$979$	−62.3815	−1.99372
$980$	10.0421	0.320784
$981$	−19.1952	−0.612855
$982$	−12.3374	−0.393703
$983$	58.3933	1.86246	0.931228	−	0.364436i	$-0.118738\pi$
$983$	58.3933	1.86246	0.931228	+	0.364436i	$0.118738\pi$
$984$	−12.0479	−0.384072
$985$	−48.3566	−1.54077
$986$	2.05937	0.0655838
$987$	−9.27407	−0.295197
$988$	3.41995	0.108803
$989$	0.148773	0.00473070
$990$	−15.4110	−0.489795
$991$	−21.7560	−0.691104	−0.345552	−	0.938400i	$-0.612308\pi$
$991$	−21.7560	−0.691104	−0.345552	+	0.938400i	$0.612308\pi$
$992$	9.62234	0.305510
$993$	4.86362	0.154342
$994$	−29.3214	−0.930019
$995$	4.73664	0.150161
$996$	12.1481	0.384926
$997$	41.9273	1.32785	0.663925	−	0.747799i	$-0.268889\pi$
$997$	41.9273	1.32785	0.663925	+	0.747799i	$0.268889\pi$
$998$	36.6880	1.16134
$999$	2.11836	0.0670220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bf.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bf.1.11	✓	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00527 q^{5} -1.00000 q^{6} -3.21582 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00527 q^{5} -1.00000 q^{6} -3.21582 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00527 q^{10} +5.12800 q^{11} +1.00000 q^{12} +3.41995 q^{13} +3.21582 q^{14} +3.00527 q^{15} +1.00000 q^{16} -0.940219 q^{17} -1.00000 q^{18} +1.00000 q^{19} +3.00527 q^{20} -3.21582 q^{21} -5.12800 q^{22} +1.92262 q^{23} -1.00000 q^{24} +4.03167 q^{25} -3.41995 q^{26} +1.00000 q^{27} -3.21582 q^{28} +2.19031 q^{29} -3.00527 q^{30} -9.62234 q^{31} -1.00000 q^{32} +5.12800 q^{33} +0.940219 q^{34} -9.66442 q^{35} +1.00000 q^{36} +2.11836 q^{37} -1.00000 q^{38} +3.41995 q^{39} -3.00527 q^{40} +12.0479 q^{41} +3.21582 q^{42} +0.0773802 q^{43} +5.12800 q^{44} +3.00527 q^{45} -1.92262 q^{46} +2.88389 q^{47} +1.00000 q^{48} +3.34150 q^{49} -4.03167 q^{50} -0.940219 q^{51} +3.41995 q^{52} +1.00000 q^{53} -1.00000 q^{54} +15.4110 q^{55} +3.21582 q^{56} +1.00000 q^{57} -2.19031 q^{58} +11.4673 q^{59} +3.00527 q^{60} -9.72441 q^{61} +9.62234 q^{62} -3.21582 q^{63} +1.00000 q^{64} +10.2779 q^{65} -5.12800 q^{66} +14.8803 q^{67} -0.940219 q^{68} +1.92262 q^{69} +9.66442 q^{70} -9.11787 q^{71} -1.00000 q^{72} -7.25794 q^{73} -2.11836 q^{74} +4.03167 q^{75} +1.00000 q^{76} -16.4907 q^{77} -3.41995 q^{78} +2.78210 q^{79} +3.00527 q^{80} +1.00000 q^{81} -12.0479 q^{82} +12.1481 q^{83} -3.21582 q^{84} -2.82561 q^{85} -0.0773802 q^{86} +2.19031 q^{87} -5.12800 q^{88} -12.1649 q^{89} -3.00527 q^{90} -10.9980 q^{91} +1.92262 q^{92} -9.62234 q^{93} -2.88389 q^{94} +3.00527 q^{95} -1.00000 q^{96} -10.3962 q^{97} -3.34150 q^{98} +5.12800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 + 10 * q^13 - 5 * q^14 - 7 * q^15 + 12 * q^16 + 6 * q^17 - 12 * q^18 + 12 * q^19 - 7 * q^20 + 5 * q^21 + 5 * q^22 - 12 * q^24 + 21 * q^25 - 10 * q^26 + 12 * q^27 + 5 * q^28 + 7 * q^30 + 2 * q^31 - 12 * q^32 - 5 * q^33 - 6 * q^34 - 17 * q^35 + 12 * q^36 + 16 * q^37 - 12 * q^38 + 10 * q^39 + 7 * q^40 + 7 * q^41 - 5 * q^42 + 24 * q^43 - 5 * q^44 - 7 * q^45 + 4 * q^47 + 12 * q^48 + 29 * q^49 - 21 * q^50 + 6 * q^51 + 10 * q^52 + 12 * q^53 - 12 * q^54 - 2 * q^55 - 5 * q^56 + 12 * q^57 + 2 * q^59 - 7 * q^60 + 29 * q^61 - 2 * q^62 + 5 * q^63 + 12 * q^64 - 17 * q^65 + 5 * q^66 + 36 * q^67 + 6 * q^68 + 17 * q^70 - 3 * q^71 - 12 * q^72 + 7 * q^73 - 16 * q^74 + 21 * q^75 + 12 * q^76 - 35 * q^77 - 10 * q^78 + 40 * q^79 - 7 * q^80 + 12 * q^81 - 7 * q^82 + 24 * q^83 + 5 * q^84 - 22 * q^85 - 24 * q^86 + 5 * q^88 - 45 * q^89 + 7 * q^90 + 71 * q^91 + 2 * q^93 - 4 * q^94 - 7 * q^95 - 12 * q^96 + q^97 - 29 * q^98 - 5 * q^99

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