LMFDB - Embedded newform 6042.2.a.be.1.2

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:	$1$
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} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + \cdots + 1496 $ x^12 - 3x^11 - 36x^10 + 105x^9 + 465x^8 - 1287x^7 - 2668x^6 + 6443x^5 + 7140x^4 - 10858x^3 - 10086x^2 + 2072*x + 1496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89261$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.89261 q^{5} +1.00000 q^{6} -4.19999 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.89261 q^{5} +1.00000 q^{6} -4.19999 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.89261 q^{10} -5.07383 q^{11} -1.00000 q^{12} +4.31548 q^{13} +4.19999 q^{14} +2.89261 q^{15} +1.00000 q^{16} -0.164752 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.89261 q^{20} +4.19999 q^{21} +5.07383 q^{22} -6.57103 q^{23} +1.00000 q^{24} +3.36720 q^{25} -4.31548 q^{26} -1.00000 q^{27} -4.19999 q^{28} +2.51473 q^{29} -2.89261 q^{30} +9.86954 q^{31} -1.00000 q^{32} +5.07383 q^{33} +0.164752 q^{34} +12.1489 q^{35} +1.00000 q^{36} -1.06505 q^{37} +1.00000 q^{38} -4.31548 q^{39} +2.89261 q^{40} -9.53136 q^{41} -4.19999 q^{42} +11.9616 q^{43} -5.07383 q^{44} -2.89261 q^{45} +6.57103 q^{46} -2.21548 q^{47} -1.00000 q^{48} +10.6399 q^{49} -3.36720 q^{50} +0.164752 q^{51} +4.31548 q^{52} -1.00000 q^{53} +1.00000 q^{54} +14.6766 q^{55} +4.19999 q^{56} +1.00000 q^{57} -2.51473 q^{58} +7.39486 q^{59} +2.89261 q^{60} -2.35917 q^{61} -9.86954 q^{62} -4.19999 q^{63} +1.00000 q^{64} -12.4830 q^{65} -5.07383 q^{66} -8.01730 q^{67} -0.164752 q^{68} +6.57103 q^{69} -12.1489 q^{70} +12.1481 q^{71} -1.00000 q^{72} +0.402131 q^{73} +1.06505 q^{74} -3.36720 q^{75} -1.00000 q^{76} +21.3100 q^{77} +4.31548 q^{78} -9.61531 q^{79} -2.89261 q^{80} +1.00000 q^{81} +9.53136 q^{82} +14.9921 q^{83} +4.19999 q^{84} +0.476563 q^{85} -11.9616 q^{86} -2.51473 q^{87} +5.07383 q^{88} +9.04557 q^{89} +2.89261 q^{90} -18.1250 q^{91} -6.57103 q^{92} -9.86954 q^{93} +2.21548 q^{94} +2.89261 q^{95} +1.00000 q^{96} +14.2021 q^{97} -10.6399 q^{98} -5.07383 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * 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
$5$	−2.89261	−1.29362	−0.646808	−	0.762653i	$-0.723896\pi$
$5$	−2.89261	−1.29362	−0.646808	+	0.762653i	$0.723896\pi$
$6$	1.00000	0.408248
$7$	−4.19999	−1.58745	−0.793723	−	0.608280i	$-0.791860\pi$
$7$	−4.19999	−1.58745	−0.793723	+	0.608280i	$0.791860\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.89261	0.914724
$11$	−5.07383	−1.52982	−0.764909	−	0.644139i	$-0.777216\pi$
$11$	−5.07383	−1.52982	−0.764909	+	0.644139i	$0.777216\pi$
$12$	−1.00000	−0.288675
$13$	4.31548	1.19690	0.598450	−	0.801160i	$-0.295784\pi$
$13$	4.31548	1.19690	0.598450	+	0.801160i	$0.295784\pi$
$14$	4.19999	1.12249
$15$	2.89261	0.746869
$16$	1.00000	0.250000
$17$	−0.164752	−0.0399582	−0.0199791	−	0.999800i	$-0.506360\pi$
$17$	−0.164752	−0.0399582	−0.0199791	+	0.999800i	$0.506360\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.89261	−0.646808
$21$	4.19999	0.916512
$22$	5.07383	1.08174
$23$	−6.57103	−1.37015	−0.685077	−	0.728471i	$-0.740231\pi$
$23$	−6.57103	−1.37015	−0.685077	+	0.728471i	$0.740231\pi$
$24$	1.00000	0.204124
$25$	3.36720	0.673441
$26$	−4.31548	−0.846336
$27$	−1.00000	−0.192450
$28$	−4.19999	−0.793723
$29$	2.51473	0.466974	0.233487	−	0.972360i	$-0.424986\pi$
$29$	2.51473	0.466974	0.233487	+	0.972360i	$0.424986\pi$
$30$	−2.89261	−0.528116
$31$	9.86954	1.77262	0.886311	−	0.463091i	$-0.153260\pi$
$31$	9.86954	1.77262	0.886311	+	0.463091i	$0.153260\pi$
$32$	−1.00000	−0.176777
$33$	5.07383	0.883240
$34$	0.164752	0.0282547
$35$	12.1489	2.05354
$36$	1.00000	0.166667
$37$	−1.06505	−0.175093	−0.0875464	−	0.996160i	$-0.527903\pi$
$37$	−1.06505	−0.175093	−0.0875464	+	0.996160i	$0.527903\pi$
$38$	1.00000	0.162221
$39$	−4.31548	−0.691030
$40$	2.89261	0.457362
$41$	−9.53136	−1.48855	−0.744274	−	0.667874i	$-0.767204\pi$
$41$	−9.53136	−1.48855	−0.744274	+	0.667874i	$0.767204\pi$
$42$	−4.19999	−0.648072
$43$	11.9616	1.82412	0.912060	−	0.410057i	$-0.134491\pi$
$43$	11.9616	1.82412	0.912060	+	0.410057i	$0.134491\pi$
$44$	−5.07383	−0.764909
$45$	−2.89261	−0.431205
$46$	6.57103	0.968845
$47$	−2.21548	−0.323161	−0.161581	−	0.986860i	$-0.551659\pi$
$47$	−2.21548	−0.323161	−0.161581	+	0.986860i	$0.551659\pi$
$48$	−1.00000	−0.144338
$49$	10.6399	1.51998
$50$	−3.36720	−0.476194
$51$	0.164752	0.0230699
$52$	4.31548	0.598450
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	14.6766	1.97900
$56$	4.19999	0.561247
$57$	1.00000	0.132453
$58$	−2.51473	−0.330201
$59$	7.39486	0.962729	0.481364	−	0.876521i	$-0.340141\pi$
$59$	7.39486	0.962729	0.481364	+	0.876521i	$0.340141\pi$
$60$	2.89261	0.373435
$61$	−2.35917	−0.302061	−0.151031	−	0.988529i	$-0.548259\pi$
$61$	−2.35917	−0.302061	−0.151031	+	0.988529i	$0.548259\pi$
$62$	−9.86954	−1.25343
$63$	−4.19999	−0.529148
$64$	1.00000	0.125000
$65$	−12.4830	−1.54833
$66$	−5.07383	−0.624545
$67$	−8.01730	−0.979469	−0.489735	−	0.871872i	$-0.662906\pi$
$67$	−8.01730	−0.979469	−0.489735	+	0.871872i	$0.662906\pi$
$68$	−0.164752	−0.0199791
$69$	6.57103	0.791058
$70$	−12.1489	−1.45207
$71$	12.1481	1.44171	0.720856	−	0.693085i	$-0.243749\pi$
$71$	12.1481	1.44171	0.720856	+	0.693085i	$0.243749\pi$
$72$	−1.00000	−0.117851
$73$	0.402131	0.0470659	0.0235329	−	0.999723i	$-0.492509\pi$
$73$	0.402131	0.0470659	0.0235329	+	0.999723i	$0.492509\pi$
$74$	1.06505	0.123809
$75$	−3.36720	−0.388811
$76$	−1.00000	−0.114708
$77$	21.3100	2.42850
$78$	4.31548	0.488632
$79$	−9.61531	−1.08181	−0.540903	−	0.841085i	$-0.681918\pi$
$79$	−9.61531	−1.08181	−0.540903	+	0.841085i	$0.681918\pi$
$80$	−2.89261	−0.323404
$81$	1.00000	0.111111
$82$	9.53136	1.05256
$83$	14.9921	1.64560	0.822799	−	0.568333i	$-0.192412\pi$
$83$	14.9921	1.64560	0.822799	+	0.568333i	$0.192412\pi$
$84$	4.19999	0.458256
$85$	0.476563	0.0516906
$86$	−11.9616	−1.28985
$87$	−2.51473	−0.269608
$88$	5.07383	0.540872
$89$	9.04557	0.958828	0.479414	−	0.877589i	$-0.340849\pi$
$89$	9.04557	0.958828	0.479414	+	0.877589i	$0.340849\pi$
$90$	2.89261	0.304908
$91$	−18.1250	−1.90001
$92$	−6.57103	−0.685077
$93$	−9.86954	−1.02342
$94$	2.21548	0.228510
$95$	2.89261	0.296776
$96$	1.00000	0.102062
$97$	14.2021	1.44200	0.721001	−	0.692934i	$-0.243682\pi$
$97$	14.2021	1.44200	0.721001	+	0.692934i	$0.243682\pi$
$98$	−10.6399	−1.07479
$99$	−5.07383	−0.509939
$100$	3.36720	0.336720
$101$	19.4631	1.93665	0.968327	−	0.249686i	$-0.0803274\pi$
$101$	19.4631	1.93665	0.968327	+	0.249686i	$0.0803274\pi$
$102$	−0.164752	−0.0163129
$103$	2.15342	0.212183	0.106092	−	0.994356i	$-0.466166\pi$
$103$	2.15342	0.212183	0.106092	+	0.994356i	$0.466166\pi$
$104$	−4.31548	−0.423168
$105$	−12.1489	−1.18561
$106$	1.00000	0.0971286
$107$	−2.32310	−0.224582	−0.112291	−	0.993675i	$-0.535819\pi$
$107$	−2.32310	−0.224582	−0.112291	+	0.993675i	$0.535819\pi$
$108$	−1.00000	−0.0962250
$109$	−6.97372	−0.667961	−0.333981	−	0.942580i	$-0.608392\pi$
$109$	−6.97372	−0.667961	−0.333981	+	0.942580i	$0.608392\pi$
$110$	−14.6766	−1.39936
$111$	1.06505	0.101090
$112$	−4.19999	−0.396861
$113$	−4.23773	−0.398652	−0.199326	−	0.979933i	$-0.563875\pi$
$113$	−4.23773	−0.398652	−0.199326	+	0.979933i	$0.563875\pi$
$114$	−1.00000	−0.0936586
$115$	19.0074	1.77245
$116$	2.51473	0.233487
$117$	4.31548	0.398966
$118$	−7.39486	−0.680752
$119$	0.691956	0.0634315
$120$	−2.89261	−0.264058
$121$	14.7438	1.34034
$122$	2.35917	0.213590
$123$	9.53136	0.859414
$124$	9.86954	0.886311
$125$	4.72305	0.422442
$126$	4.19999	0.374164
$127$	−13.2802	−1.17843	−0.589216	−	0.807976i	$-0.700563\pi$
$127$	−13.2802	−1.17843	−0.589216	+	0.807976i	$0.700563\pi$
$128$	−1.00000	−0.0883883
$129$	−11.9616	−1.05316
$130$	12.4830	1.09483
$131$	−20.5119	−1.79213	−0.896066	−	0.443921i	$-0.853587\pi$
$131$	−20.5119	−1.79213	−0.896066	+	0.443921i	$0.853587\pi$
$132$	5.07383	0.441620
$133$	4.19999	0.364185
$134$	8.01730	0.692589
$135$	2.89261	0.248956
$136$	0.164752	0.0141274
$137$	21.2098	1.81208	0.906039	−	0.423194i	$-0.139091\pi$
$137$	21.2098	1.81208	0.906039	+	0.423194i	$0.139091\pi$
$138$	−6.57103	−0.559363
$139$	−12.0035	−1.01812	−0.509061	−	0.860730i	$-0.670007\pi$
$139$	−12.0035	−1.01812	−0.509061	+	0.860730i	$0.670007\pi$
$140$	12.1489	1.02677
$141$	2.21548	0.186577
$142$	−12.1481	−1.01944
$143$	−21.8960	−1.83104
$144$	1.00000	0.0833333
$145$	−7.27414	−0.604085
$146$	−0.402131	−0.0332806
$147$	−10.6399	−0.877562
$148$	−1.06505	−0.0875464
$149$	10.1141	0.828578	0.414289	−	0.910145i	$-0.364030\pi$
$149$	10.1141	0.828578	0.414289	+	0.910145i	$0.364030\pi$
$150$	3.36720	0.274931
$151$	22.9799	1.87007	0.935037	−	0.354550i	$-0.115366\pi$
$151$	22.9799	1.87007	0.935037	+	0.354550i	$0.115366\pi$
$152$	1.00000	0.0811107
$153$	−0.164752	−0.0133194
$154$	−21.3100	−1.71721
$155$	−28.5487	−2.29309
$156$	−4.31548	−0.345515
$157$	−18.2952	−1.46012	−0.730059	−	0.683384i	$-0.760507\pi$
$157$	−18.2952	−1.46012	−0.730059	+	0.683384i	$0.760507\pi$
$158$	9.61531	0.764953
$159$	1.00000	0.0793052
$160$	2.89261	0.228681
$161$	27.5982	2.17504
$162$	−1.00000	−0.0785674
$163$	23.7938	1.86368	0.931838	−	0.362875i	$-0.118205\pi$
$163$	23.7938	1.86368	0.931838	+	0.362875i	$0.118205\pi$
$164$	−9.53136	−0.744274
$165$	−14.6766	−1.14257
$166$	−14.9921	−1.16361
$167$	−6.90049	−0.533976	−0.266988	−	0.963700i	$-0.586028\pi$
$167$	−6.90049	−0.533976	−0.266988	+	0.963700i	$0.586028\pi$
$168$	−4.19999	−0.324036
$169$	5.62338	0.432568
$170$	−0.476563	−0.0365507
$171$	−1.00000	−0.0764719
$172$	11.9616	0.912060
$173$	−13.0432	−0.991653	−0.495827	−	0.868422i	$-0.665135\pi$
$173$	−13.0432	−0.991653	−0.495827	+	0.868422i	$0.665135\pi$
$174$	2.51473	0.190641
$175$	−14.1422	−1.06905
$176$	−5.07383	−0.382454
$177$	−7.39486	−0.555832
$178$	−9.04557	−0.677994
$179$	−13.7443	−1.02730	−0.513648	−	0.858001i	$-0.671706\pi$
$179$	−13.7443	−1.02730	−0.513648	+	0.858001i	$0.671706\pi$
$180$	−2.89261	−0.215603
$181$	10.6985	0.795211	0.397606	−	0.917556i	$-0.369841\pi$
$181$	10.6985	0.795211	0.397606	+	0.917556i	$0.369841\pi$
$182$	18.1250	1.34351
$183$	2.35917	0.174395
$184$	6.57103	0.484422
$185$	3.08077	0.226503
$186$	9.86954	0.723670
$187$	0.835923	0.0611288
$188$	−2.21548	−0.161581
$189$	4.19999	0.305504
$190$	−2.89261	−0.209852
$191$	−12.9853	−0.939583	−0.469792	−	0.882777i	$-0.655671\pi$
$191$	−12.9853	−0.939583	−0.469792	+	0.882777i	$0.655671\pi$
$192$	−1.00000	−0.0721688
$193$	−5.55372	−0.399765	−0.199883	−	0.979820i	$-0.564056\pi$
$193$	−5.55372	−0.399765	−0.199883	+	0.979820i	$0.564056\pi$
$194$	−14.2021	−1.01965
$195$	12.4830	0.893927
$196$	10.6399	0.759991
$197$	−15.1348	−1.07831	−0.539156	−	0.842206i	$-0.681257\pi$
$197$	−15.1348	−1.07831	−0.539156	+	0.842206i	$0.681257\pi$
$198$	5.07383	0.360581
$199$	−3.14427	−0.222891	−0.111446	−	0.993771i	$-0.535548\pi$
$199$	−3.14427	−0.222891	−0.111446	+	0.993771i	$0.535548\pi$
$200$	−3.36720	−0.238097
$201$	8.01730	0.565497
$202$	−19.4631	−1.36942
$203$	−10.5618	−0.741296
$204$	0.164752	0.0115349
$205$	27.5705	1.92561
$206$	−2.15342	−0.150036
$207$	−6.57103	−0.456718
$208$	4.31548	0.299225
$209$	5.07383	0.350964
$210$	12.1489	0.838356
$211$	−20.4722	−1.40936	−0.704681	−	0.709524i	$-0.748910\pi$
$211$	−20.4722	−1.40936	−0.704681	+	0.709524i	$0.748910\pi$
$212$	−1.00000	−0.0686803
$213$	−12.1481	−0.832373
$214$	2.32310	0.158804
$215$	−34.6001	−2.35971
$216$	1.00000	0.0680414
$217$	−41.4519	−2.81394
$218$	6.97372	0.472320
$219$	−0.402131	−0.0271735
$220$	14.6766	0.989498
$221$	−0.710984	−0.0478260
$222$	−1.06505	−0.0714813
$223$	1.30931	0.0876777	0.0438389	−	0.999039i	$-0.486041\pi$
$223$	1.30931	0.0876777	0.0438389	+	0.999039i	$0.486041\pi$
$224$	4.19999	0.280623
$225$	3.36720	0.224480
$226$	4.23773	0.281890
$227$	−29.3497	−1.94801	−0.974005	−	0.226528i	$-0.927262\pi$
$227$	−29.3497	−1.94801	−0.974005	+	0.226528i	$0.927262\pi$
$228$	1.00000	0.0662266
$229$	11.8618	0.783852	0.391926	−	0.919997i	$-0.371809\pi$
$229$	11.8618	0.783852	0.391926	+	0.919997i	$0.371809\pi$
$230$	−19.0074	−1.25331
$231$	−21.3100	−1.40210
$232$	−2.51473	−0.165100
$233$	−6.53013	−0.427803	−0.213901	−	0.976855i	$-0.568617\pi$
$233$	−6.53013	−0.427803	−0.213901	+	0.976855i	$0.568617\pi$
$234$	−4.31548	−0.282112
$235$	6.40853	0.418046
$236$	7.39486	0.481364
$237$	9.61531	0.624581
$238$	−0.691956	−0.0448528
$239$	−22.3102	−1.44313	−0.721564	−	0.692347i	$-0.756577\pi$
$239$	−22.3102	−1.44313	−0.721564	+	0.692347i	$0.756577\pi$
$240$	2.89261	0.186717
$241$	8.77862	0.565481	0.282740	−	0.959196i	$-0.408757\pi$
$241$	8.77862	0.565481	0.282740	+	0.959196i	$0.408757\pi$
$242$	−14.7438	−0.947764
$243$	−1.00000	−0.0641500
$244$	−2.35917	−0.151031
$245$	−30.7770	−1.96627
$246$	−9.53136	−0.607697
$247$	−4.31548	−0.274588
$248$	−9.86954	−0.626716
$249$	−14.9921	−0.950086
$250$	−4.72305	−0.298712
$251$	13.4594	0.849552	0.424776	−	0.905299i	$-0.360353\pi$
$251$	13.4594	0.849552	0.424776	+	0.905299i	$0.360353\pi$
$252$	−4.19999	−0.264574
$253$	33.3403	2.09608
$254$	13.2802	0.833277
$255$	−0.476563	−0.0298436
$256$	1.00000	0.0625000
$257$	−7.35628	−0.458872	−0.229436	−	0.973324i	$-0.573688\pi$
$257$	−7.35628	−0.458872	−0.229436	+	0.973324i	$0.573688\pi$
$258$	11.9616	0.744694
$259$	4.47318	0.277950
$260$	−12.4830	−0.774164
$261$	2.51473	0.155658
$262$	20.5119	1.26723
$263$	−22.7323	−1.40173	−0.700867	−	0.713292i	$-0.747203\pi$
$263$	−22.7323	−1.40173	−0.700867	+	0.713292i	$0.747203\pi$
$264$	−5.07383	−0.312273
$265$	2.89261	0.177692
$266$	−4.19999	−0.257518
$267$	−9.04557	−0.553580
$268$	−8.01730	−0.489735
$269$	19.1670	1.16863	0.584316	−	0.811526i	$-0.301363\pi$
$269$	19.1670	1.16863	0.584316	+	0.811526i	$0.301363\pi$
$270$	−2.89261	−0.176039
$271$	−10.0217	−0.608776	−0.304388	−	0.952548i	$-0.598452\pi$
$271$	−10.0217	−0.608776	−0.304388	+	0.952548i	$0.598452\pi$
$272$	−0.164752	−0.00998955
$273$	18.1250	1.09697
$274$	−21.2098	−1.28133
$275$	−17.0846	−1.03024
$276$	6.57103	0.395529
$277$	26.3575	1.58367	0.791836	−	0.610734i	$-0.209126\pi$
$277$	26.3575	1.58367	0.791836	+	0.610734i	$0.209126\pi$
$278$	12.0035	0.719922
$279$	9.86954	0.590874
$280$	−12.1489	−0.726037
$281$	25.1508	1.50037	0.750186	−	0.661227i	$-0.229964\pi$
$281$	25.1508	1.50037	0.750186	+	0.661227i	$0.229964\pi$
$282$	−2.21548	−0.131930
$283$	21.5774	1.28264	0.641322	−	0.767272i	$-0.278386\pi$
$283$	21.5774	1.28264	0.641322	+	0.767272i	$0.278386\pi$
$284$	12.1481	0.720856
$285$	−2.89261	−0.171344
$286$	21.8960	1.29474
$287$	40.0316	2.36299
$288$	−1.00000	−0.0589256
$289$	−16.9729	−0.998403
$290$	7.27414	0.427152
$291$	−14.2021	−0.832541
$292$	0.402131	0.0235329
$293$	4.48113	0.261790	0.130895	−	0.991396i	$-0.458215\pi$
$293$	4.48113	0.261790	0.130895	+	0.991396i	$0.458215\pi$
$294$	10.6399	0.620530
$295$	−21.3905	−1.24540
$296$	1.06505	0.0619046
$297$	5.07383	0.294413
$298$	−10.1141	−0.585893
$299$	−28.3571	−1.63994
$300$	−3.36720	−0.194406
$301$	−50.2383	−2.89569
$302$	−22.9799	−1.32234
$303$	−19.4631	−1.11813
$304$	−1.00000	−0.0573539
$305$	6.82417	0.390751
$306$	0.164752	0.00941824
$307$	−25.9747	−1.48245	−0.741227	−	0.671254i	$-0.765756\pi$
$307$	−25.9747	−1.48245	−0.741227	+	0.671254i	$0.765756\pi$
$308$	21.3100	1.21425
$309$	−2.15342	−0.122504
$310$	28.5487	1.62146
$311$	19.4275	1.10163	0.550817	−	0.834626i	$-0.314316\pi$
$311$	19.4275	1.10163	0.550817	+	0.834626i	$0.314316\pi$
$312$	4.31548	0.244316
$313$	−16.6025	−0.938428	−0.469214	−	0.883085i	$-0.655463\pi$
$313$	−16.6025	−0.938428	−0.469214	+	0.883085i	$0.655463\pi$
$314$	18.2952	1.03246
$315$	12.1489	0.684515
$316$	−9.61531	−0.540903
$317$	−22.4533	−1.26110	−0.630551	−	0.776148i	$-0.717171\pi$
$317$	−22.4533	−1.26110	−0.630551	+	0.776148i	$0.717171\pi$
$318$	−1.00000	−0.0560772
$319$	−12.7593	−0.714385
$320$	−2.89261	−0.161702
$321$	2.32310	0.129663
$322$	−27.5982	−1.53799
$323$	0.164752	0.00916704
$324$	1.00000	0.0555556
$325$	14.5311	0.806040
$326$	−23.7938	−1.31782
$327$	6.97372	0.385648
$328$	9.53136	0.526281
$329$	9.30499	0.513001
$330$	14.6766	0.807921
$331$	2.79548	0.153653	0.0768267	−	0.997044i	$-0.475521\pi$
$331$	2.79548	0.153653	0.0768267	+	0.997044i	$0.475521\pi$
$332$	14.9921	0.822799
$333$	−1.06505	−0.0583642
$334$	6.90049	0.377578
$335$	23.1909	1.26706
$336$	4.19999	0.229128
$337$	−18.2672	−0.995077	−0.497538	−	0.867442i	$-0.665763\pi$
$337$	−18.2672	−0.995077	−0.497538	+	0.867442i	$0.665763\pi$
$338$	−5.62338	−0.305872
$339$	4.23773	0.230162
$340$	0.476563	0.0258453
$341$	−50.0764	−2.71179
$342$	1.00000	0.0540738
$343$	−15.2874	−0.825444
$344$	−11.9616	−0.644924
$345$	−19.0074	−1.02333
$346$	13.0432	0.701205
$347$	−4.25132	−0.228223	−0.114111	−	0.993468i	$-0.536402\pi$
$347$	−4.25132	−0.228223	−0.114111	+	0.993468i	$0.536402\pi$
$348$	−2.51473	−0.134804
$349$	35.4354	1.89681	0.948406	−	0.317057i	$-0.102695\pi$
$349$	35.4354	1.89681	0.948406	+	0.317057i	$0.102695\pi$
$350$	14.1422	0.755933
$351$	−4.31548	−0.230343
$352$	5.07383	0.270436
$353$	19.5956	1.04297	0.521484	−	0.853261i	$-0.325378\pi$
$353$	19.5956	1.04297	0.521484	+	0.853261i	$0.325378\pi$
$354$	7.39486	0.393032
$355$	−35.1397	−1.86502
$356$	9.04557	0.479414
$357$	−0.691956	−0.0366222
$358$	13.7443	0.726408
$359$	−14.0420	−0.741110	−0.370555	−	0.928810i	$-0.620833\pi$
$359$	−14.0420	−0.741110	−0.370555	+	0.928810i	$0.620833\pi$
$360$	2.89261	0.152454
$361$	1.00000	0.0526316
$362$	−10.6985	−0.562299
$363$	−14.7438	−0.773846
$364$	−18.1250	−0.950006
$365$	−1.16321	−0.0608851
$366$	−2.35917	−0.123316
$367$	−2.72418	−0.142201	−0.0711006	−	0.997469i	$-0.522651\pi$
$367$	−2.72418	−0.142201	−0.0711006	+	0.997469i	$0.522651\pi$
$368$	−6.57103	−0.342538
$369$	−9.53136	−0.496183
$370$	−3.08077	−0.160162
$371$	4.19999	0.218052
$372$	−9.86954	−0.511712
$373$	13.4646	0.697173	0.348586	−	0.937277i	$-0.386662\pi$
$373$	13.4646	0.697173	0.348586	+	0.937277i	$0.386662\pi$
$374$	−0.835923	−0.0432246
$375$	−4.72305	−0.243897
$376$	2.21548	0.114255
$377$	10.8523	0.558921
$378$	−4.19999	−0.216024
$379$	−2.12113	−0.108955	−0.0544777	−	0.998515i	$-0.517349\pi$
$379$	−2.12113	−0.108955	−0.0544777	+	0.998515i	$0.517349\pi$
$380$	2.89261	0.148388
$381$	13.2802	0.680367
$382$	12.9853	0.664386
$383$	12.0871	0.617621	0.308811	−	0.951124i	$-0.400069\pi$
$383$	12.0871	0.617621	0.308811	+	0.951124i	$0.400069\pi$
$384$	1.00000	0.0510310
$385$	−61.6416	−3.14155
$386$	5.55372	0.282677
$387$	11.9616	0.608040
$388$	14.2021	0.721001
$389$	18.2322	0.924410	0.462205	−	0.886773i	$-0.347058\pi$
$389$	18.2322	0.924410	0.462205	+	0.886773i	$0.347058\pi$
$390$	−12.4830	−0.632102
$391$	1.08259	0.0547489
$392$	−10.6399	−0.537395
$393$	20.5119	1.03469
$394$	15.1348	0.762482
$395$	27.8133	1.39944
$396$	−5.07383	−0.254970
$397$	0.303597	0.0152371	0.00761854	−	0.999971i	$-0.497575\pi$
$397$	0.303597	0.0152371	0.00761854	+	0.999971i	$0.497575\pi$
$398$	3.14427	0.157608
$399$	−4.19999	−0.210262
$400$	3.36720	0.168360
$401$	6.29918	0.314566	0.157283	−	0.987554i	$-0.449726\pi$
$401$	6.29918	0.314566	0.157283	+	0.987554i	$0.449726\pi$
$402$	−8.01730	−0.399867
$403$	42.5918	2.12165
$404$	19.4631	0.968327
$405$	−2.89261	−0.143735
$406$	10.5618	0.524175
$407$	5.40387	0.267860
$408$	−0.164752	−0.00815644
$409$	35.9705	1.77862	0.889312	−	0.457300i	$-0.151184\pi$
$409$	35.9705	1.77862	0.889312	+	0.457300i	$0.151184\pi$
$410$	−27.5705	−1.36161
$411$	−21.2098	−1.04620
$412$	2.15342	0.106092
$413$	−31.0583	−1.52828
$414$	6.57103	0.322948
$415$	−43.3663	−2.12877
$416$	−4.31548	−0.211584
$417$	12.0035	0.587813
$418$	−5.07383	−0.248169
$419$	−31.9649	−1.56159	−0.780793	−	0.624789i	$-0.785185\pi$
$419$	−31.9649	−1.56159	−0.780793	+	0.624789i	$0.785185\pi$
$420$	−12.1489	−0.592807
$421$	13.1293	0.639885	0.319942	−	0.947437i	$-0.396336\pi$
$421$	13.1293	0.639885	0.319942	+	0.947437i	$0.396336\pi$
$422$	20.4722	0.996570
$423$	−2.21548	−0.107720
$424$	1.00000	0.0485643
$425$	−0.554753	−0.0269095
$426$	12.1481	0.588576
$427$	9.90850	0.479506
$428$	−2.32310	−0.112291
$429$	21.8960	1.05715
$430$	34.6001	1.66857
$431$	18.8667	0.908779	0.454389	−	0.890803i	$-0.349857\pi$
$431$	18.8667	0.908779	0.454389	+	0.890803i	$0.349857\pi$
$432$	−1.00000	−0.0481125
$433$	−15.9897	−0.768418	−0.384209	−	0.923246i	$-0.625526\pi$
$433$	−15.9897	−0.768418	−0.384209	+	0.923246i	$0.625526\pi$
$434$	41.4519	1.98976
$435$	7.27414	0.348769
$436$	−6.97372	−0.333981
$437$	6.57103	0.314335
$438$	0.402131	0.0192146
$439$	36.3616	1.73545	0.867723	−	0.497048i	$-0.165583\pi$
$439$	36.3616	1.73545	0.867723	+	0.497048i	$0.165583\pi$
$440$	−14.6766	−0.699680
$441$	10.6399	0.506661
$442$	0.710984	0.0338181
$443$	9.83169	0.467118	0.233559	−	0.972343i	$-0.424963\pi$
$443$	9.83169	0.467118	0.233559	+	0.972343i	$0.424963\pi$
$444$	1.06505	0.0505449
$445$	−26.1653	−1.24035
$446$	−1.30931	−0.0619975
$447$	−10.1141	−0.478380
$448$	−4.19999	−0.198431
$449$	2.21385	0.104478	0.0522391	−	0.998635i	$-0.483364\pi$
$449$	2.21385	0.104478	0.0522391	+	0.998635i	$0.483364\pi$
$450$	−3.36720	−0.158731
$451$	48.3605	2.27721
$452$	−4.23773	−0.199326
$453$	−22.9799	−1.07969
$454$	29.3497	1.37745
$455$	52.4285	2.45788
$456$	−1.00000	−0.0468293
$457$	−33.7502	−1.57877	−0.789384	−	0.613900i	$-0.789600\pi$
$457$	−33.7502	−1.57877	−0.789384	+	0.613900i	$0.789600\pi$
$458$	−11.8618	−0.554267
$459$	0.164752	0.00768996
$460$	19.0074	0.886226
$461$	27.9582	1.30214	0.651071	−	0.759017i	$-0.274320\pi$
$461$	27.9582	1.30214	0.651071	+	0.759017i	$0.274320\pi$
$462$	21.3100	0.991432
$463$	−34.4387	−1.60050	−0.800251	−	0.599666i	$-0.795300\pi$
$463$	−34.4387	−1.60050	−0.800251	+	0.599666i	$0.795300\pi$
$464$	2.51473	0.116744
$465$	28.5487	1.32392
$466$	6.53013	0.302502
$467$	−21.8458	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$467$	−21.8458	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$468$	4.31548	0.199483
$469$	33.6726	1.55485
$470$	−6.40853	−0.295603
$471$	18.2952	0.842999
$472$	−7.39486	−0.340376
$473$	−60.6909	−2.79057
$474$	−9.61531	−0.441646
$475$	−3.36720	−0.154498
$476$	0.691956	0.0317157
$477$	−1.00000	−0.0457869
$478$	22.3102	1.02045
$479$	−12.2579	−0.560076	−0.280038	−	0.959989i	$-0.590347\pi$
$479$	−12.2579	−0.560076	−0.280038	+	0.959989i	$0.590347\pi$
$480$	−2.89261	−0.132029
$481$	−4.59619	−0.209568
$482$	−8.77862	−0.399855
$483$	−27.5982	−1.25576
$484$	14.7438	0.670171
$485$	−41.0811	−1.86540
$486$	1.00000	0.0453609
$487$	7.38220	0.334519	0.167260	−	0.985913i	$-0.446508\pi$
$487$	7.38220	0.334519	0.167260	+	0.985913i	$0.446508\pi$
$488$	2.35917	0.106795
$489$	−23.7938	−1.07599
$490$	30.7770	1.39036
$491$	18.0802	0.815947	0.407973	−	0.912994i	$-0.366236\pi$
$491$	18.0802	0.815947	0.407973	+	0.912994i	$0.366236\pi$
$492$	9.53136	0.429707
$493$	−0.414307	−0.0186594
$494$	4.31548	0.194163
$495$	14.6766	0.659665
$496$	9.86954	0.443155
$497$	−51.0218	−2.28864
$498$	14.9921	0.671812
$499$	7.19500	0.322093	0.161046	−	0.986947i	$-0.448513\pi$
$499$	7.19500	0.322093	0.161046	+	0.986947i	$0.448513\pi$
$500$	4.72305	0.211221
$501$	6.90049	0.308291
$502$	−13.4594	−0.600724
$503$	−7.01264	−0.312678	−0.156339	−	0.987703i	$-0.549969\pi$
$503$	−7.01264	−0.312678	−0.156339	+	0.987703i	$0.549969\pi$
$504$	4.19999	0.187082
$505$	−56.2993	−2.50528
$506$	−33.3403	−1.48216
$507$	−5.62338	−0.249743
$508$	−13.2802	−0.589216
$509$	−35.5288	−1.57479	−0.787393	−	0.616452i	$-0.788570\pi$
$509$	−35.5288	−1.57479	−0.787393	+	0.616452i	$0.788570\pi$
$510$	0.476563	0.0211026
$511$	−1.68894	−0.0747145
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	7.35628	0.324471
$515$	−6.22902	−0.274483
$516$	−11.9616	−0.526578
$517$	11.2410	0.494378
$518$	−4.47318	−0.196540
$519$	13.0432	0.572531
$520$	12.4830	0.547416
$521$	−19.9271	−0.873022	−0.436511	−	0.899699i	$-0.643786\pi$
$521$	−19.9271	−0.873022	−0.436511	+	0.899699i	$0.643786\pi$
$522$	−2.51473	−0.110067
$523$	−28.6054	−1.25083	−0.625414	−	0.780293i	$-0.715070\pi$
$523$	−28.6054	−1.25083	−0.625414	+	0.780293i	$0.715070\pi$
$524$	−20.5119	−0.896066
$525$	14.1422	0.617216
$526$	22.7323	0.991175
$527$	−1.62603	−0.0708308
$528$	5.07383	0.220810
$529$	20.1784	0.877321
$530$	−2.89261	−0.125647
$531$	7.39486	0.320910
$532$	4.19999	0.182092
$533$	−41.1324	−1.78164
$534$	9.04557	0.391440
$535$	6.71982	0.290523
$536$	8.01730	0.346295
$537$	13.7443	0.593109
$538$	−19.1670	−0.826348
$539$	−53.9849	−2.32530
$540$	2.89261	0.124478
$541$	−16.5564	−0.711815	−0.355908	−	0.934521i	$-0.615828\pi$
$541$	−16.5564	−0.711815	−0.355908	+	0.934521i	$0.615828\pi$
$542$	10.0217	0.430469
$543$	−10.6985	−0.459115
$544$	0.164752	0.00706368
$545$	20.1723	0.864085
$546$	−18.1250	−0.775677
$547$	−14.5329	−0.621381	−0.310691	−	0.950511i	$-0.600560\pi$
$547$	−14.5329	−0.621381	−0.310691	+	0.950511i	$0.600560\pi$
$548$	21.2098	0.906039
$549$	−2.35917	−0.100687
$550$	17.0846	0.728490
$551$	−2.51473	−0.107131
$552$	−6.57103	−0.279681
$553$	40.3842	1.71731
$554$	−26.3575	−1.11982
$555$	−3.08077	−0.130771
$556$	−12.0035	−0.509061
$557$	−36.3376	−1.53967	−0.769836	−	0.638241i	$-0.779662\pi$
$557$	−36.3376	−1.53967	−0.769836	+	0.638241i	$0.779662\pi$
$558$	−9.86954	−0.417811
$559$	51.6199	2.18329
$560$	12.1489	0.513386
$561$	−0.835923	−0.0352927
$562$	−25.1508	−1.06092
$563$	−25.6272	−1.08006	−0.540029	−	0.841646i	$-0.681587\pi$
$563$	−25.6272	−1.08006	−0.540029	+	0.841646i	$0.681587\pi$
$564$	2.21548	0.0932886
$565$	12.2581	0.515702
$566$	−21.5774	−0.906966
$567$	−4.19999	−0.176383
$568$	−12.1481	−0.509722
$569$	−13.8891	−0.582259	−0.291130	−	0.956684i	$-0.594031\pi$
$569$	−13.8891	−0.582259	−0.291130	+	0.956684i	$0.594031\pi$
$570$	2.89261	0.121158
$571$	−13.8942	−0.581456	−0.290728	−	0.956806i	$-0.593897\pi$
$571$	−13.8942	−0.581456	−0.290728	+	0.956806i	$0.593897\pi$
$572$	−21.8960	−0.915519
$573$	12.9853	0.542469
$574$	−40.0316	−1.67089
$575$	−22.1260	−0.922717
$576$	1.00000	0.0416667
$577$	−9.46687	−0.394111	−0.197056	−	0.980392i	$-0.563138\pi$
$577$	−9.46687	−0.394111	−0.197056	+	0.980392i	$0.563138\pi$
$578$	16.9729	0.705978
$579$	5.55372	0.230805
$580$	−7.27414	−0.302042
$581$	−62.9666	−2.61230
$582$	14.2021	0.588695
$583$	5.07383	0.210137
$584$	−0.402131	−0.0166403
$585$	−12.4830	−0.516109
$586$	−4.48113	−0.185114
$587$	7.96978	0.328948	0.164474	−	0.986381i	$-0.447407\pi$
$587$	7.96978	0.328948	0.164474	+	0.986381i	$0.447407\pi$
$588$	−10.6399	−0.438781
$589$	−9.86954	−0.406667
$590$	21.3905	0.880631
$591$	15.1348	0.622564
$592$	−1.06505	−0.0437732
$593$	28.5880	1.17397	0.586984	−	0.809598i	$-0.300315\pi$
$593$	28.5880	1.17397	0.586984	+	0.809598i	$0.300315\pi$
$594$	−5.07383	−0.208182
$595$	−2.00156	−0.0820559
$596$	10.1141	0.414289
$597$	3.14427	0.128686
$598$	28.3571	1.15961
$599$	−46.9205	−1.91712	−0.958560	−	0.284892i	$-0.908042\pi$
$599$	−46.9205	−1.91712	−0.958560	+	0.284892i	$0.908042\pi$
$600$	3.36720	0.137465
$601$	20.2150	0.824587	0.412293	−	0.911051i	$-0.364728\pi$
$601$	20.2150	0.824587	0.412293	+	0.911051i	$0.364728\pi$
$602$	50.2383	2.04756
$603$	−8.01730	−0.326490
$604$	22.9799	0.935037
$605$	−42.6480	−1.73389
$606$	19.4631	0.790636
$607$	−26.0644	−1.05792	−0.528962	−	0.848646i	$-0.677418\pi$
$607$	−26.0644	−1.05792	−0.528962	+	0.848646i	$0.677418\pi$
$608$	1.00000	0.0405554
$609$	10.5618	0.427987
$610$	−6.82417	−0.276303
$611$	−9.56087	−0.386792
$612$	−0.164752	−0.00665970
$613$	−40.0784	−1.61875	−0.809375	−	0.587292i	$-0.800194\pi$
$613$	−40.0784	−1.61875	−0.809375	+	0.587292i	$0.800194\pi$
$614$	25.9747	1.04825
$615$	−27.5705	−1.11175
$616$	−21.3100	−0.858605
$617$	−3.89038	−0.156621	−0.0783103	−	0.996929i	$-0.524953\pi$
$617$	−3.89038	−0.156621	−0.0783103	+	0.996929i	$0.524953\pi$
$618$	2.15342	0.0866234
$619$	−39.9122	−1.60421	−0.802104	−	0.597184i	$-0.796286\pi$
$619$	−39.9122	−1.60421	−0.802104	+	0.597184i	$0.796286\pi$
$620$	−28.5487	−1.14655
$621$	6.57103	0.263686
$622$	−19.4275	−0.778973
$623$	−37.9912	−1.52209
$624$	−4.31548	−0.172758
$625$	−30.4980	−1.21992
$626$	16.6025	0.663569
$627$	−5.07383	−0.202629
$628$	−18.2952	−0.730059
$629$	0.175469	0.00699639
$630$	−12.1489	−0.484025
$631$	12.8924	0.513239	0.256619	−	0.966513i	$-0.417391\pi$
$631$	12.8924	0.513239	0.256619	+	0.966513i	$0.417391\pi$
$632$	9.61531	0.382476
$633$	20.4722	0.813696
$634$	22.4533	0.891733
$635$	38.4146	1.52444
$636$	1.00000	0.0396526
$637$	45.9162	1.81927
$638$	12.7593	0.505146
$639$	12.1481	0.480571
$640$	2.89261	0.114341
$641$	14.9520	0.590569	0.295284	−	0.955409i	$-0.404586\pi$
$641$	14.9520	0.590569	0.295284	+	0.955409i	$0.404586\pi$
$642$	−2.32310	−0.0916854
$643$	26.4447	1.04288	0.521438	−	0.853289i	$-0.325396\pi$
$643$	26.4447	1.04288	0.521438	+	0.853289i	$0.325396\pi$
$644$	27.5982	1.08752
$645$	34.6001	1.36238
$646$	−0.164752	−0.00648208
$647$	29.4064	1.15608	0.578042	−	0.816007i	$-0.303817\pi$
$647$	29.4064	1.15608	0.578042	+	0.816007i	$0.303817\pi$
$648$	−1.00000	−0.0392837
$649$	−37.5203	−1.47280
$650$	−14.5311	−0.569957
$651$	41.4519	1.62463
$652$	23.7938	0.931838
$653$	44.8056	1.75338	0.876690	−	0.481056i	$-0.159747\pi$
$653$	44.8056	1.75338	0.876690	+	0.481056i	$0.159747\pi$
$654$	−6.97372	−0.272694
$655$	59.3329	2.31833
$656$	−9.53136	−0.372137
$657$	0.402131	0.0156886
$658$	−9.30499	−0.362746
$659$	11.0523	0.430536	0.215268	−	0.976555i	$-0.430937\pi$
$659$	11.0523	0.430536	0.215268	+	0.976555i	$0.430937\pi$
$660$	−14.6766	−0.571287
$661$	−32.1873	−1.25194	−0.625971	−	0.779846i	$-0.715297\pi$
$661$	−32.1873	−1.25194	−0.625971	+	0.779846i	$0.715297\pi$
$662$	−2.79548	−0.108649
$663$	0.710984	0.0276123
$664$	−14.9921	−0.581806
$665$	−12.1489	−0.471115
$666$	1.06505	0.0412697
$667$	−16.5244	−0.639826
$668$	−6.90049	−0.266988
$669$	−1.30931	−0.0506208
$670$	−23.1909	−0.895944
$671$	11.9700	0.462099
$672$	−4.19999	−0.162018
$673$	8.10843	0.312557	0.156279	−	0.987713i	$-0.450050\pi$
$673$	8.10843	0.312557	0.156279	+	0.987713i	$0.450050\pi$
$674$	18.2672	0.703625
$675$	−3.36720	−0.129604
$676$	5.62338	0.216284
$677$	29.3481	1.12794	0.563971	−	0.825795i	$-0.309273\pi$
$677$	29.3481	1.12794	0.563971	+	0.825795i	$0.309273\pi$
$678$	−4.23773	−0.162749
$679$	−59.6485	−2.28910
$680$	−0.476563	−0.0182754
$681$	29.3497	1.12468
$682$	50.0764	1.91752
$683$	30.4604	1.16554	0.582768	−	0.812639i	$-0.301970\pi$
$683$	30.4604	1.16554	0.582768	+	0.812639i	$0.301970\pi$
$684$	−1.00000	−0.0382360
$685$	−61.3518	−2.34413
$686$	15.2874	0.583677
$687$	−11.8618	−0.452557
$688$	11.9616	0.456030
$689$	−4.31548	−0.164407
$690$	19.0074	0.723600
$691$	18.2783	0.695340	0.347670	−	0.937617i	$-0.386973\pi$
$691$	18.2783	0.695340	0.347670	+	0.937617i	$0.386973\pi$
$692$	−13.0432	−0.495827
$693$	21.3100	0.809500
$694$	4.25132	0.161378
$695$	34.7214	1.31706
$696$	2.51473	0.0953207
$697$	1.57031	0.0594797
$698$	−35.4354	−1.34125
$699$	6.53013	0.246992
$700$	−14.1422	−0.534525
$701$	−29.1262	−1.10008	−0.550041	−	0.835137i	$-0.685388\pi$
$701$	−29.1262	−1.10008	−0.550041	+	0.835137i	$0.685388\pi$
$702$	4.31548	0.162877
$703$	1.06505	0.0401690
$704$	−5.07383	−0.191227
$705$	−6.40853	−0.241359
$706$	−19.5956	−0.737490
$707$	−81.7449	−3.07433
$708$	−7.39486	−0.277916
$709$	−7.13353	−0.267905	−0.133953	−	0.990988i	$-0.542767\pi$
$709$	−7.13353	−0.267905	−0.133953	+	0.990988i	$0.542767\pi$
$710$	35.1397	1.31877
$711$	−9.61531	−0.360602
$712$	−9.04557	−0.338997
$713$	−64.8530	−2.42876
$714$	0.691956	0.0258958
$715$	63.3367	2.36866
$716$	−13.7443	−0.513648
$717$	22.3102	0.833191
$718$	14.0420	0.524044
$719$	2.23576	0.0833796	0.0416898	−	0.999131i	$-0.486726\pi$
$719$	2.23576	0.0833796	0.0416898	+	0.999131i	$0.486726\pi$
$720$	−2.89261	−0.107801
$721$	−9.04435	−0.336829
$722$	−1.00000	−0.0372161
$723$	−8.77862	−0.326480
$724$	10.6985	0.397606
$725$	8.46761	0.314479
$726$	14.7438	0.547192
$727$	27.1373	1.00647	0.503233	−	0.864151i	$-0.332144\pi$
$727$	27.1373	1.00647	0.503233	+	0.864151i	$0.332144\pi$
$728$	18.1250	0.671756
$729$	1.00000	0.0370370
$730$	1.16321	0.0430523
$731$	−1.97069	−0.0728886
$732$	2.35917	0.0871976
$733$	29.5464	1.09132	0.545661	−	0.838006i	$-0.316279\pi$
$733$	29.5464	1.09132	0.545661	+	0.838006i	$0.316279\pi$
$734$	2.72418	0.100551
$735$	30.7770	1.13523
$736$	6.57103	0.242211
$737$	40.6784	1.49841
$738$	9.53136	0.350854
$739$	−6.27892	−0.230974	−0.115487	−	0.993309i	$-0.536843\pi$
$739$	−6.27892	−0.230974	−0.115487	+	0.993309i	$0.536843\pi$
$740$	3.08077	0.113251
$741$	4.31548	0.158533
$742$	−4.19999	−0.154186
$743$	39.2227	1.43894	0.719470	−	0.694523i	$-0.244385\pi$
$743$	39.2227	1.43894	0.719470	+	0.694523i	$0.244385\pi$
$744$	9.86954	0.361835
$745$	−29.2561	−1.07186
$746$	−13.4646	−0.492975
$747$	14.9921	0.548532
$748$	0.835923	0.0305644
$749$	9.75698	0.356512
$750$	4.72305	0.172461
$751$	34.3063	1.25185	0.625927	−	0.779882i	$-0.284721\pi$
$751$	34.3063	1.25185	0.625927	+	0.779882i	$0.284721\pi$
$752$	−2.21548	−0.0807903
$753$	−13.4594	−0.490489
$754$	−10.8523	−0.395217
$755$	−66.4718	−2.41916
$756$	4.19999	0.152752
$757$	−6.27235	−0.227972	−0.113986	−	0.993482i	$-0.536362\pi$
$757$	−6.27235	−0.227972	−0.113986	+	0.993482i	$0.536362\pi$
$758$	2.12113	0.0770431
$759$	−33.3403	−1.21017
$760$	−2.89261	−0.104926
$761$	15.0822	0.546731	0.273365	−	0.961910i	$-0.411863\pi$
$761$	15.0822	0.546731	0.273365	+	0.961910i	$0.411863\pi$
$762$	−13.2802	−0.481092
$763$	29.2895	1.06035
$764$	−12.9853	−0.469792
$765$	0.476563	0.0172302
$766$	−12.0871	−0.436724
$767$	31.9124	1.15229
$768$	−1.00000	−0.0360844
$769$	−5.17273	−0.186533	−0.0932667	−	0.995641i	$-0.529731\pi$
$769$	−5.17273	−0.186533	−0.0932667	+	0.995641i	$0.529731\pi$
$770$	61.6416	2.22141
$771$	7.35628	0.264930
$772$	−5.55372	−0.199883
$773$	−15.2176	−0.547339	−0.273669	−	0.961824i	$-0.588237\pi$
$773$	−15.2176	−0.547339	−0.273669	+	0.961824i	$0.588237\pi$
$774$	−11.9616	−0.429949
$775$	33.2327	1.19376
$776$	−14.2021	−0.509825
$777$	−4.47318	−0.160475
$778$	−18.2322	−0.653657
$779$	9.53136	0.341496
$780$	12.4830	0.446964
$781$	−61.6373	−2.20556
$782$	−1.08259	−0.0387133
$783$	−2.51473	−0.0898692
$784$	10.6399	0.379996
$785$	52.9210	1.88883
$786$	−20.5119	−0.731635
$787$	13.3623	0.476314	0.238157	−	0.971227i	$-0.423457\pi$
$787$	13.3623	0.476314	0.238157	+	0.971227i	$0.423457\pi$
$788$	−15.1348	−0.539156
$789$	22.7323	0.809291
$790$	−27.8133	−0.989555
$791$	17.7984	0.632838
$792$	5.07383	0.180291
$793$	−10.1810	−0.361537
$794$	−0.303597	−0.0107742
$795$	−2.89261	−0.102590
$796$	−3.14427	−0.111446
$797$	−36.0017	−1.27524	−0.637622	−	0.770349i	$-0.720082\pi$
$797$	−36.0017	−1.27524	−0.637622	+	0.770349i	$0.720082\pi$
$798$	4.19999	0.148678
$799$	0.365005	0.0129129
$800$	−3.36720	−0.119049
$801$	9.04557	0.319609
$802$	−6.29918	−0.222432
$803$	−2.04034	−0.0720022
$804$	8.01730	0.282748
$805$	−79.8309	−2.81367
$806$	−42.5918	−1.50023
$807$	−19.1670	−0.674710
$808$	−19.4631	−0.684710
$809$	33.4734	1.17686	0.588432	−	0.808547i	$-0.299746\pi$
$809$	33.4734	1.17686	0.588432	+	0.808547i	$0.299746\pi$
$810$	2.89261	0.101636
$811$	−17.0307	−0.598030	−0.299015	−	0.954248i	$-0.596658\pi$
$811$	−17.0307	−0.598030	−0.299015	+	0.954248i	$0.596658\pi$
$812$	−10.5618	−0.370648
$813$	10.0217	0.351477
$814$	−5.40387	−0.189406
$815$	−68.8263	−2.41088
$816$	0.164752	0.00576747
$817$	−11.9616	−0.418482
$818$	−35.9705	−1.25768
$819$	−18.1250	−0.633337
$820$	27.5705	0.962804
$821$	28.3759	0.990324	0.495162	−	0.868801i	$-0.335109\pi$
$821$	28.3759	0.990324	0.495162	+	0.868801i	$0.335109\pi$
$822$	21.2098	0.739778
$823$	40.6574	1.41723	0.708613	−	0.705597i	$-0.249321\pi$
$823$	40.6574	1.41723	0.708613	+	0.705597i	$0.249321\pi$
$824$	−2.15342	−0.0750181
$825$	17.0846	0.594810
$826$	31.0583	1.08066
$827$	−49.3362	−1.71559	−0.857793	−	0.513995i	$-0.828165\pi$
$827$	−49.3362	−1.71559	−0.857793	+	0.513995i	$0.828165\pi$
$828$	−6.57103	−0.228359
$829$	0.520715	0.0180852	0.00904258	−	0.999959i	$-0.497122\pi$
$829$	0.520715	0.0180852	0.00904258	+	0.999959i	$0.497122\pi$
$830$	43.3663	1.50527
$831$	−26.3575	−0.914333
$832$	4.31548	0.149612
$833$	−1.75294	−0.0607358
$834$	−12.0035	−0.415647
$835$	19.9604	0.690759
$836$	5.07383	0.175482
$837$	−9.86954	−0.341141
$838$	31.9649	1.10421
$839$	−44.9576	−1.55211	−0.776055	−	0.630665i	$-0.782782\pi$
$839$	−44.9576	−1.55211	−0.776055	+	0.630665i	$0.782782\pi$
$840$	12.1489	0.419178
$841$	−22.6761	−0.781935
$842$	−13.1293	−0.452467
$843$	−25.1508	−0.866240
$844$	−20.4722	−0.704681
$845$	−16.2663	−0.559576
$846$	2.21548	0.0761699
$847$	−61.9235	−2.12772
$848$	−1.00000	−0.0343401
$849$	−21.5774	−0.740535
$850$	0.554753	0.0190279
$851$	6.99845	0.239904
$852$	−12.1481	−0.416186
$853$	31.7249	1.08624	0.543121	−	0.839655i	$-0.317243\pi$
$853$	31.7249	1.08624	0.543121	+	0.839655i	$0.317243\pi$
$854$	−9.90850	−0.339062
$855$	2.89261	0.0989252
$856$	2.32310	0.0794019
$857$	41.9051	1.43145	0.715725	−	0.698383i	$-0.246097\pi$
$857$	41.9051	1.43145	0.715725	+	0.698383i	$0.246097\pi$
$858$	−21.8960	−0.747518
$859$	−36.6132	−1.24923	−0.624613	−	0.780934i	$-0.714743\pi$
$859$	−36.6132	−1.24923	−0.624613	+	0.780934i	$0.714743\pi$
$860$	−34.6001	−1.17985
$861$	−40.0316	−1.36427
$862$	−18.8667	−0.642604
$863$	30.8006	1.04847	0.524233	−	0.851575i	$-0.324352\pi$
$863$	30.8006	1.04847	0.524233	+	0.851575i	$0.324352\pi$
$864$	1.00000	0.0340207
$865$	37.7288	1.28282
$866$	15.9897	0.543354
$867$	16.9729	0.576428
$868$	−41.4519	−1.40697
$869$	48.7864	1.65497
$870$	−7.27414	−0.246617
$871$	−34.5985	−1.17233
$872$	6.97372	0.236160
$873$	14.2021	0.480668
$874$	−6.57103	−0.222268
$875$	−19.8367	−0.670604
$876$	−0.402131	−0.0135867
$877$	−30.8781	−1.04268	−0.521339	−	0.853350i	$-0.674567\pi$
$877$	−30.8781	−1.04268	−0.521339	+	0.853350i	$0.674567\pi$
$878$	−36.3616	−1.22715
$879$	−4.48113	−0.151145
$880$	14.6766	0.494749
$881$	29.4034	0.990626	0.495313	−	0.868715i	$-0.335053\pi$
$881$	29.4034	0.990626	0.495313	+	0.868715i	$0.335053\pi$
$882$	−10.6399	−0.358263
$883$	−27.0905	−0.911667	−0.455833	−	0.890065i	$-0.650659\pi$
$883$	−27.0905	−0.911667	−0.455833	+	0.890065i	$0.650659\pi$
$884$	−0.710984	−0.0239130
$885$	21.3905	0.719033
$886$	−9.83169	−0.330302
$887$	16.8178	0.564686	0.282343	−	0.959313i	$-0.408888\pi$
$887$	16.8178	0.564686	0.282343	+	0.959313i	$0.408888\pi$
$888$	−1.06505	−0.0357407
$889$	55.7768	1.87069
$890$	26.1653	0.877063
$891$	−5.07383	−0.169980
$892$	1.30931	0.0438389
$893$	2.21548	0.0741383
$894$	10.1141	0.338265
$895$	39.7569	1.32893
$896$	4.19999	0.140312
$897$	28.3571	0.946817
$898$	−2.21385	−0.0738772
$899$	24.8193	0.827768
$900$	3.36720	0.112240
$901$	0.164752	0.00548868
$902$	−48.3605	−1.61023
$903$	50.2383	1.67183
$904$	4.23773	0.140945
$905$	−30.9465	−1.02870
$906$	22.9799	0.763454
$907$	−12.1895	−0.404747	−0.202373	−	0.979308i	$-0.564865\pi$
$907$	−12.1895	−0.404747	−0.202373	+	0.979308i	$0.564865\pi$
$908$	−29.3497	−0.974005
$909$	19.4631	0.645551
$910$	−52.4285	−1.73799
$911$	9.37909	0.310743	0.155372	−	0.987856i	$-0.450343\pi$
$911$	9.37909	0.310743	0.155372	+	0.987856i	$0.450343\pi$
$912$	1.00000	0.0331133
$913$	−76.0674	−2.51746
$914$	33.7502	1.11636
$915$	−6.82417	−0.225600
$916$	11.8618	0.391926
$917$	86.1496	2.84491
$918$	−0.164752	−0.00543762
$919$	37.3252	1.23125	0.615623	−	0.788041i	$-0.288904\pi$
$919$	37.3252	1.23125	0.615623	+	0.788041i	$0.288904\pi$
$920$	−19.0074	−0.626656
$921$	25.9747	0.855896
$922$	−27.9582	−0.920754
$923$	52.4248	1.72558
$924$	−21.3100	−0.701048
$925$	−3.58623	−0.117915
$926$	34.4387	1.13173
$927$	2.15342	0.0707277
$928$	−2.51473	−0.0825501
$929$	−37.0078	−1.21418	−0.607092	−	0.794631i	$-0.707664\pi$
$929$	−37.0078	−1.21418	−0.607092	+	0.794631i	$0.707664\pi$
$930$	−28.5487	−0.936150
$931$	−10.6399	−0.348708
$932$	−6.53013	−0.213901
$933$	−19.4275	−0.636029
$934$	21.8458	0.714817
$935$	−2.41800	−0.0790771
$936$	−4.31548	−0.141056
$937$	1.39487	0.0455683	0.0227842	−	0.999740i	$-0.492747\pi$
$937$	1.39487	0.0455683	0.0227842	+	0.999740i	$0.492747\pi$
$938$	−33.6726	−1.09945
$939$	16.6025	0.541801
$940$	6.40853	0.209023
$941$	18.6777	0.608874	0.304437	−	0.952532i	$-0.401532\pi$
$941$	18.6777	0.608874	0.304437	+	0.952532i	$0.401532\pi$
$942$	−18.2952	−0.596090
$943$	62.6308	2.03954
$944$	7.39486	0.240682
$945$	−12.1489	−0.395205
$946$	60.6909	1.97323
$947$	5.60661	0.182190	0.0910951	−	0.995842i	$-0.470963\pi$
$947$	5.60661	0.182190	0.0910951	+	0.995842i	$0.470963\pi$
$948$	9.61531	0.312291
$949$	1.73539	0.0563331
$950$	3.36720	0.109246
$951$	22.4533	0.728097
$952$	−0.691956	−0.0224264
$953$	12.3673	0.400616	0.200308	−	0.979733i	$-0.435806\pi$
$953$	12.3673	0.400616	0.200308	+	0.979733i	$0.435806\pi$
$954$	1.00000	0.0323762
$955$	37.5614	1.21546
$956$	−22.3102	−0.721564
$957$	12.7593	0.412450
$958$	12.2579	0.396033
$959$	−89.0810	−2.87658
$960$	2.89261	0.0933586
$961$	66.4078	2.14219
$962$	4.59619	0.148187
$963$	−2.32310	−0.0748608
$964$	8.77862	0.282740
$965$	16.0647	0.517143
$966$	27.5982	0.887958
$967$	−54.8024	−1.76233	−0.881163	−	0.472812i	$-0.843239\pi$
$967$	−54.8024	−1.76233	−0.881163	+	0.472812i	$0.843239\pi$
$968$	−14.7438	−0.473882
$969$	−0.164752	−0.00529259
$970$	41.0811	1.31903
$971$	−5.23079	−0.167864	−0.0839321	−	0.996471i	$-0.526748\pi$
$971$	−5.23079	−0.167864	−0.0839321	+	0.996471i	$0.526748\pi$
$972$	−1.00000	−0.0320750
$973$	50.4145	1.61621
$974$	−7.38220	−0.236541
$975$	−14.5311	−0.465368
$976$	−2.35917	−0.0755153
$977$	42.6129	1.36331	0.681653	−	0.731675i	$-0.261261\pi$
$977$	42.6129	1.36331	0.681653	+	0.731675i	$0.261261\pi$
$978$	23.7938	0.760842
$979$	−45.8957	−1.46683
$980$	−30.7770	−0.983136
$981$	−6.97372	−0.222654
$982$	−18.0802	−0.576962
$983$	−46.7968	−1.49259	−0.746293	−	0.665617i	$-0.768168\pi$
$983$	−46.7968	−1.49259	−0.746293	+	0.665617i	$0.768168\pi$
$984$	−9.53136	−0.303849
$985$	43.7792	1.39492
$986$	0.414307	0.0131942
$987$	−9.30499	−0.296181
$988$	−4.31548	−0.137294
$989$	−78.5997	−2.49932
$990$	−14.6766	−0.466454
$991$	38.4547	1.22156	0.610778	−	0.791802i	$-0.290857\pi$
$991$	38.4547	1.22156	0.610778	+	0.791802i	$0.290857\pi$
$992$	−9.86954	−0.313358
$993$	−2.79548	−0.0887118
$994$	51.0218	1.61831
$995$	9.09515	0.288336
$996$	−14.9921	−0.475043
$997$	9.74444	0.308609	0.154305	−	0.988023i	$-0.450686\pi$
$997$	9.74444	0.308609	0.154305	+	0.988023i	$0.450686\pi$
$998$	−7.19500	−0.227754
$999$	1.06505	0.0336966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.be.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.be.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.89261 q^{5} +1.00000 q^{6} -4.19999 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.89261 q^{5} +1.00000 q^{6} -4.19999 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.89261 q^{10} -5.07383 q^{11} -1.00000 q^{12} +4.31548 q^{13} +4.19999 q^{14} +2.89261 q^{15} +1.00000 q^{16} -0.164752 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.89261 q^{20} +4.19999 q^{21} +5.07383 q^{22} -6.57103 q^{23} +1.00000 q^{24} +3.36720 q^{25} -4.31548 q^{26} -1.00000 q^{27} -4.19999 q^{28} +2.51473 q^{29} -2.89261 q^{30} +9.86954 q^{31} -1.00000 q^{32} +5.07383 q^{33} +0.164752 q^{34} +12.1489 q^{35} +1.00000 q^{36} -1.06505 q^{37} +1.00000 q^{38} -4.31548 q^{39} +2.89261 q^{40} -9.53136 q^{41} -4.19999 q^{42} +11.9616 q^{43} -5.07383 q^{44} -2.89261 q^{45} +6.57103 q^{46} -2.21548 q^{47} -1.00000 q^{48} +10.6399 q^{49} -3.36720 q^{50} +0.164752 q^{51} +4.31548 q^{52} -1.00000 q^{53} +1.00000 q^{54} +14.6766 q^{55} +4.19999 q^{56} +1.00000 q^{57} -2.51473 q^{58} +7.39486 q^{59} +2.89261 q^{60} -2.35917 q^{61} -9.86954 q^{62} -4.19999 q^{63} +1.00000 q^{64} -12.4830 q^{65} -5.07383 q^{66} -8.01730 q^{67} -0.164752 q^{68} +6.57103 q^{69} -12.1489 q^{70} +12.1481 q^{71} -1.00000 q^{72} +0.402131 q^{73} +1.06505 q^{74} -3.36720 q^{75} -1.00000 q^{76} +21.3100 q^{77} +4.31548 q^{78} -9.61531 q^{79} -2.89261 q^{80} +1.00000 q^{81} +9.53136 q^{82} +14.9921 q^{83} +4.19999 q^{84} +0.476563 q^{85} -11.9616 q^{86} -2.51473 q^{87} +5.07383 q^{88} +9.04557 q^{89} +2.89261 q^{90} -18.1250 q^{91} -6.57103 q^{92} -9.86954 q^{93} +2.21548 q^{94} +2.89261 q^{95} +1.00000 q^{96} +14.2021 q^{97} -10.6399 q^{98} -5.07383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 12 * q^3 + 12 * q^4 - 3 * q^5 + 12 * q^6 - q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 7 * q^11 - 12 * q^12 + 4 * q^13 + q^14 + 3 * q^15 + 12 * q^16 - 4 * q^17 - 12 * q^18 - 12 * q^19 - 3 * q^20 + q^21 + 7 * q^22 - 18 * q^23 + 12 * q^24 + 21 * q^25 - 4 * q^26 - 12 * q^27 - q^28 + 10 * q^29 - 3 * q^30 + 14 * q^31 - 12 * q^32 + 7 * q^33 + 4 * q^34 - 19 * q^35 + 12 * q^36 + 14 * q^37 + 12 * q^38 - 4 * q^39 + 3 * q^40 - 9 * q^41 - q^42 - 4 * q^43 - 7 * q^44 - 3 * q^45 + 18 * q^46 - 14 * q^47 - 12 * q^48 + 37 * q^49 - 21 * q^50 + 4 * q^51 + 4 * q^52 - 12 * q^53 + 12 * q^54 - 4 * q^55 + q^56 + 12 * q^57 - 10 * q^58 - 18 * q^59 + 3 * q^60 - 7 * q^61 - 14 * q^62 - q^63 + 12 * q^64 + 3 * q^65 - 7 * q^66 + 8 * q^67 - 4 * q^68 + 18 * q^69 + 19 * q^70 - q^71 - 12 * q^72 - 31 * q^73 - 14 * q^74 - 21 * q^75 - 12 * q^76 - 25 * q^77 + 4 * q^78 - 3 * q^80 + 12 * q^81 + 9 * q^82 - 48 * q^83 + q^84 + 4 * q^85 + 4 * q^86 - 10 * q^87 + 7 * q^88 + 13 * q^89 + 3 * q^90 + 9 * q^91 - 18 * q^92 - 14 * q^93 + 14 * q^94 + 3 * q^95 + 12 * q^96 + 25 * q^97 - 37 * q^98 - 7 * q^99

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