LMFDB - Embedded newform 4002.2.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.652958$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.652958 q^{5} -1.00000 q^{6} -3.89658 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.652958 q^{5} -1.00000 q^{6} -3.89658 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.652958 q^{10} +5.43246 q^{11} -1.00000 q^{12} +3.14335 q^{13} -3.89658 q^{14} +0.652958 q^{15} +1.00000 q^{16} +4.77950 q^{17} +1.00000 q^{18} -7.24090 q^{19} -0.652958 q^{20} +3.89658 q^{21} +5.43246 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.57365 q^{25} +3.14335 q^{26} -1.00000 q^{27} -3.89658 q^{28} -1.00000 q^{29} +0.652958 q^{30} +5.19796 q^{31} +1.00000 q^{32} -5.43246 q^{33} +4.77950 q^{34} +2.54430 q^{35} +1.00000 q^{36} -1.28454 q^{37} -7.24090 q^{38} -3.14335 q^{39} -0.652958 q^{40} -5.34588 q^{41} +3.89658 q^{42} -2.02783 q^{43} +5.43246 q^{44} -0.652958 q^{45} -1.00000 q^{46} +4.83411 q^{47} -1.00000 q^{48} +8.18332 q^{49} -4.57365 q^{50} -4.77950 q^{51} +3.14335 q^{52} +9.97902 q^{53} -1.00000 q^{54} -3.54717 q^{55} -3.89658 q^{56} +7.24090 q^{57} -1.00000 q^{58} +11.0844 q^{59} +0.652958 q^{60} -3.43246 q^{61} +5.19796 q^{62} -3.89658 q^{63} +1.00000 q^{64} -2.05247 q^{65} -5.43246 q^{66} -4.99831 q^{67} +4.77950 q^{68} +1.00000 q^{69} +2.54430 q^{70} -0.305382 q^{71} +1.00000 q^{72} -4.98203 q^{73} -1.28454 q^{74} +4.57365 q^{75} -7.24090 q^{76} -21.1680 q^{77} -3.14335 q^{78} +3.65080 q^{79} -0.652958 q^{80} +1.00000 q^{81} -5.34588 q^{82} +8.46140 q^{83} +3.89658 q^{84} -3.12081 q^{85} -2.02783 q^{86} +1.00000 q^{87} +5.43246 q^{88} +7.44828 q^{89} -0.652958 q^{90} -12.2483 q^{91} -1.00000 q^{92} -5.19796 q^{93} +4.83411 q^{94} +4.72800 q^{95} -1.00000 q^{96} +16.6682 q^{97} +8.18332 q^{98} +5.43246 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * 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
$5$	−0.652958	−0.292012	−0.146006	−	0.989284i	$-0.546642\pi$
$5$	−0.652958	−0.292012	−0.146006	+	0.989284i	$0.546642\pi$
$6$	−1.00000	−0.408248
$7$	−3.89658	−1.47277	−0.736384	−	0.676564i	$-0.763468\pi$
$7$	−3.89658	−1.47277	−0.736384	+	0.676564i	$0.763468\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.652958	−0.206483
$11$	5.43246	1.63795	0.818974	−	0.573831i	$-0.194543\pi$
$11$	5.43246	1.63795	0.818974	+	0.573831i	$0.194543\pi$
$12$	−1.00000	−0.288675
$13$	3.14335	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$13$	3.14335	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$14$	−3.89658	−1.04140
$15$	0.652958	0.168593
$16$	1.00000	0.250000
$17$	4.77950	1.15920	0.579600	−	0.814901i	$-0.303209\pi$
$17$	4.77950	1.15920	0.579600	+	0.814901i	$0.303209\pi$
$18$	1.00000	0.235702
$19$	−7.24090	−1.66118	−0.830589	−	0.556886i	$-0.811996\pi$
$19$	−7.24090	−1.66118	−0.830589	+	0.556886i	$0.811996\pi$
$20$	−0.652958	−0.146006
$21$	3.89658	0.850303
$22$	5.43246	1.15820
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−4.57365	−0.914729
$26$	3.14335	0.616462
$27$	−1.00000	−0.192450
$28$	−3.89658	−0.736384
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.652958	0.119213
$31$	5.19796	0.933581	0.466791	−	0.884368i	$-0.345410\pi$
$31$	5.19796	0.933581	0.466791	+	0.884368i	$0.345410\pi$
$32$	1.00000	0.176777
$33$	−5.43246	−0.945670
$34$	4.77950	0.819678
$35$	2.54430	0.430065
$36$	1.00000	0.166667
$37$	−1.28454	−0.211177	−0.105588	−	0.994410i	$-0.533673\pi$
$37$	−1.28454	−0.211177	−0.105588	+	0.994410i	$0.533673\pi$
$38$	−7.24090	−1.17463
$39$	−3.14335	−0.503339
$40$	−0.652958	−0.103242
$41$	−5.34588	−0.834887	−0.417443	−	0.908703i	$-0.637074\pi$
$41$	−5.34588	−0.834887	−0.417443	+	0.908703i	$0.637074\pi$
$42$	3.89658	0.601255
$43$	−2.02783	−0.309241	−0.154621	−	0.987974i	$-0.549416\pi$
$43$	−2.02783	−0.309241	−0.154621	+	0.987974i	$0.549416\pi$
$44$	5.43246	0.818974
$45$	−0.652958	−0.0973372
$46$	−1.00000	−0.147442
$47$	4.83411	0.705128	0.352564	−	0.935788i	$-0.385310\pi$
$47$	4.83411	0.705128	0.352564	+	0.935788i	$0.385310\pi$
$48$	−1.00000	−0.144338
$49$	8.18332	1.16905
$50$	−4.57365	−0.646811
$51$	−4.77950	−0.669264
$52$	3.14335	0.435904
$53$	9.97902	1.37072	0.685362	−	0.728203i	$-0.259644\pi$
$53$	9.97902	1.37072	0.685362	+	0.728203i	$0.259644\pi$
$54$	−1.00000	−0.136083
$55$	−3.54717	−0.478300
$56$	−3.89658	−0.520702
$57$	7.24090	0.959081
$58$	−1.00000	−0.131306
$59$	11.0844	1.44307	0.721535	−	0.692378i	$-0.243437\pi$
$59$	11.0844	1.44307	0.721535	+	0.692378i	$0.243437\pi$
$60$	0.652958	0.0842965
$61$	−3.43246	−0.439481	−0.219741	−	0.975558i	$-0.570521\pi$
$61$	−3.43246	−0.439481	−0.219741	+	0.975558i	$0.570521\pi$
$62$	5.19796	0.660142
$63$	−3.89658	−0.490923
$64$	1.00000	0.125000
$65$	−2.05247	−0.254578
$66$	−5.43246	−0.668689
$67$	−4.99831	−0.610640	−0.305320	−	0.952250i	$-0.598763\pi$
$67$	−4.99831	−0.610640	−0.305320	+	0.952250i	$0.598763\pi$
$68$	4.77950	0.579600
$69$	1.00000	0.120386
$70$	2.54430	0.304102
$71$	−0.305382	−0.0362422	−0.0181211	−	0.999836i	$-0.505768\pi$
$71$	−0.305382	−0.0362422	−0.0181211	+	0.999836i	$0.505768\pi$
$72$	1.00000	0.117851
$73$	−4.98203	−0.583103	−0.291552	−	0.956555i	$-0.594171\pi$
$73$	−4.98203	−0.583103	−0.291552	+	0.956555i	$0.594171\pi$
$74$	−1.28454	−0.149324
$75$	4.57365	0.528119
$76$	−7.24090	−0.830589
$77$	−21.1680	−2.41232
$78$	−3.14335	−0.355914
$79$	3.65080	0.410747	0.205373	−	0.978684i	$-0.434159\pi$
$79$	3.65080	0.410747	0.205373	+	0.978684i	$0.434159\pi$
$80$	−0.652958	−0.0730029
$81$	1.00000	0.111111
$82$	−5.34588	−0.590354
$83$	8.46140	0.928760	0.464380	−	0.885636i	$-0.346277\pi$
$83$	8.46140	0.928760	0.464380	+	0.885636i	$0.346277\pi$
$84$	3.89658	0.425151
$85$	−3.12081	−0.338500
$86$	−2.02783	−0.218667
$87$	1.00000	0.107211
$88$	5.43246	0.579102
$89$	7.44828	0.789516	0.394758	−	0.918785i	$-0.370829\pi$
$89$	7.44828	0.789516	0.394758	+	0.918785i	$0.370829\pi$
$90$	−0.652958	−0.0688278
$91$	−12.2483	−1.28397
$92$	−1.00000	−0.104257
$93$	−5.19796	−0.539003
$94$	4.83411	0.498601
$95$	4.72800	0.485083
$96$	−1.00000	−0.102062
$97$	16.6682	1.69240	0.846199	−	0.532867i	$-0.178885\pi$
$97$	16.6682	1.69240	0.846199	+	0.532867i	$0.178885\pi$
$98$	8.18332	0.826640
$99$	5.43246	0.545983
$100$	−4.57365	−0.457365
$101$	14.1175	1.40474	0.702371	−	0.711811i	$-0.252125\pi$
$101$	14.1175	1.40474	0.702371	+	0.711811i	$0.252125\pi$
$102$	−4.77950	−0.473241
$103$	0.187986	0.0185228	0.00926139	−	0.999957i	$-0.497052\pi$
$103$	0.187986	0.0185228	0.00926139	+	0.999957i	$0.497052\pi$
$104$	3.14335	0.308231
$105$	−2.54430	−0.248298
$106$	9.97902	0.969248
$107$	10.1145	0.977809	0.488904	−	0.872337i	$-0.337397\pi$
$107$	10.1145	0.977809	0.488904	+	0.872337i	$0.337397\pi$
$108$	−1.00000	−0.0962250
$109$	16.9130	1.61997	0.809985	−	0.586451i	$-0.199475\pi$
$109$	16.9130	1.61997	0.809985	+	0.586451i	$0.199475\pi$
$110$	−3.54717	−0.338209
$111$	1.28454	0.121923
$112$	−3.89658	−0.368192
$113$	19.0614	1.79315	0.896574	−	0.442894i	$-0.146048\pi$
$113$	19.0614	1.79315	0.896574	+	0.442894i	$0.146048\pi$
$114$	7.24090	0.678173
$115$	0.652958	0.0608886
$116$	−1.00000	−0.0928477
$117$	3.14335	0.290603
$118$	11.0844	1.02040
$119$	−18.6237	−1.70723
$120$	0.652958	0.0596066
$121$	18.5116	1.68287
$122$	−3.43246	−0.310760
$123$	5.34588	0.482022
$124$	5.19796	0.466791
$125$	6.25119	0.559123
$126$	−3.89658	−0.347135
$127$	−17.0070	−1.50912	−0.754562	−	0.656229i	$-0.772150\pi$
$127$	−17.0070	−1.50912	−0.754562	+	0.656229i	$0.772150\pi$
$128$	1.00000	0.0883883
$129$	2.02783	0.178540
$130$	−2.05247	−0.180014
$131$	−14.8267	−1.29541	−0.647707	−	0.761890i	$-0.724272\pi$
$131$	−14.8267	−1.29541	−0.647707	+	0.761890i	$0.724272\pi$
$132$	−5.43246	−0.472835
$133$	28.2147	2.44653
$134$	−4.99831	−0.431788
$135$	0.652958	0.0561976
$136$	4.77950	0.409839
$137$	5.65080	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$137$	5.65080	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$138$	1.00000	0.0851257
$139$	−10.3334	−0.876469	−0.438234	−	0.898861i	$-0.644396\pi$
$139$	−10.3334	−0.876469	−0.438234	+	0.898861i	$0.644396\pi$
$140$	2.54430	0.215033
$141$	−4.83411	−0.407106
$142$	−0.305382	−0.0256271
$143$	17.0761	1.42798
$144$	1.00000	0.0833333
$145$	0.652958	0.0542252
$146$	−4.98203	−0.412316
$147$	−8.18332	−0.674949
$148$	−1.28454	−0.105588
$149$	15.9143	1.30375	0.651873	−	0.758328i	$-0.273983\pi$
$149$	15.9143	1.30375	0.651873	+	0.758328i	$0.273983\pi$
$150$	4.57365	0.373437
$151$	0.926954	0.0754344	0.0377172	−	0.999288i	$-0.487991\pi$
$151$	0.926954	0.0754344	0.0377172	+	0.999288i	$0.487991\pi$
$152$	−7.24090	−0.587315
$153$	4.77950	0.386400
$154$	−21.1680	−1.70577
$155$	−3.39405	−0.272616
$156$	−3.14335	−0.251669
$157$	8.37925	0.668737	0.334369	−	0.942442i	$-0.391477\pi$
$157$	8.37925	0.668737	0.334369	+	0.942442i	$0.391477\pi$
$158$	3.65080	0.290442
$159$	−9.97902	−0.791388
$160$	−0.652958	−0.0516208
$161$	3.89658	0.307093
$162$	1.00000	0.0785674
$163$	15.1905	1.18981	0.594907	−	0.803794i	$-0.297189\pi$
$163$	15.1905	1.18981	0.594907	+	0.803794i	$0.297189\pi$
$164$	−5.34588	−0.417443
$165$	3.54717	0.276146
$166$	8.46140	0.656732
$167$	14.5570	1.12645	0.563226	−	0.826303i	$-0.309560\pi$
$167$	14.5570	1.12645	0.563226	+	0.826303i	$0.309560\pi$
$168$	3.89658	0.300627
$169$	−3.11935	−0.239950
$170$	−3.12081	−0.239355
$171$	−7.24090	−0.553726
$172$	−2.02783	−0.154621
$173$	−10.8097	−0.821848	−0.410924	−	0.911670i	$-0.634794\pi$
$173$	−10.8097	−0.821848	−0.410924	+	0.911670i	$0.634794\pi$
$174$	1.00000	0.0758098
$175$	17.8216	1.34718
$176$	5.43246	0.409487
$177$	−11.0844	−0.833157
$178$	7.44828	0.558272
$179$	−8.20054	−0.612937	−0.306469	−	0.951881i	$-0.599147\pi$
$179$	−8.20054	−0.612937	−0.306469	+	0.951881i	$0.599147\pi$
$180$	−0.652958	−0.0486686
$181$	−4.40352	−0.327311	−0.163655	−	0.986518i	$-0.552329\pi$
$181$	−4.40352	−0.327311	−0.163655	+	0.986518i	$0.552329\pi$
$182$	−12.2483	−0.907905
$183$	3.43246	0.253735
$184$	−1.00000	−0.0737210
$185$	0.838748	0.0616660
$186$	−5.19796	−0.381133
$187$	25.9644	1.89871
$188$	4.83411	0.352564
$189$	3.89658	0.283434
$190$	4.72800	0.343005
$191$	3.17330	0.229612	0.114806	−	0.993388i	$-0.463375\pi$
$191$	3.17330	0.229612	0.114806	+	0.993388i	$0.463375\pi$
$192$	−1.00000	−0.0721688
$193$	24.6898	1.77721	0.888606	−	0.458671i	$-0.151674\pi$
$193$	24.6898	1.77721	0.888606	+	0.458671i	$0.151674\pi$
$194$	16.6682	1.19671
$195$	2.05247	0.146981
$196$	8.18332	0.584523
$197$	−1.61793	−0.115272	−0.0576362	−	0.998338i	$-0.518356\pi$
$197$	−1.61793	−0.115272	−0.0576362	+	0.998338i	$0.518356\pi$
$198$	5.43246	0.386068
$199$	1.43101	0.101441	0.0507207	−	0.998713i	$-0.483848\pi$
$199$	1.43101	0.101441	0.0507207	+	0.998713i	$0.483848\pi$
$200$	−4.57365	−0.323406
$201$	4.99831	0.352553
$202$	14.1175	0.993303
$203$	3.89658	0.273486
$204$	−4.77950	−0.334632
$205$	3.49063	0.243797
$206$	0.187986	0.0130976
$207$	−1.00000	−0.0695048
$208$	3.14335	0.217952
$209$	−39.3359	−2.72092
$210$	−2.54430	−0.175573
$211$	4.87050	0.335299	0.167649	−	0.985847i	$-0.446382\pi$
$211$	4.87050	0.335299	0.167649	+	0.985847i	$0.446382\pi$
$212$	9.97902	0.685362
$213$	0.305382	0.0209244
$214$	10.1145	0.691415
$215$	1.32409	0.0903020
$216$	−1.00000	−0.0680414
$217$	−20.2543	−1.37495
$218$	16.9130	1.14549
$219$	4.98203	0.336655
$220$	−3.54717	−0.239150
$221$	15.0236	1.01060
$222$	1.28454	0.0862125
$223$	−26.7563	−1.79173	−0.895867	−	0.444323i	$-0.853444\pi$
$223$	−26.7563	−1.79173	−0.895867	+	0.444323i	$0.853444\pi$
$224$	−3.89658	−0.260351
$225$	−4.57365	−0.304910
$226$	19.0614	1.26795
$227$	−22.8435	−1.51618	−0.758089	−	0.652152i	$-0.773867\pi$
$227$	−22.8435	−1.51618	−0.758089	+	0.652152i	$0.773867\pi$
$228$	7.24090	0.479541
$229$	−12.2649	−0.810485	−0.405243	−	0.914209i	$-0.632813\pi$
$229$	−12.2649	−0.810485	−0.405243	+	0.914209i	$0.632813\pi$
$230$	0.652958	0.0430548
$231$	21.1680	1.39275
$232$	−1.00000	−0.0656532
$233$	26.6506	1.74594	0.872971	−	0.487773i	$-0.162191\pi$
$233$	26.6506	1.74594	0.872971	+	0.487773i	$0.162191\pi$
$234$	3.14335	0.205487
$235$	−3.15647	−0.205905
$236$	11.0844	0.721535
$237$	−3.65080	−0.237145
$238$	−18.6237	−1.20720
$239$	−13.4123	−0.867573	−0.433786	−	0.901016i	$-0.642823\pi$
$239$	−13.4123	−0.867573	−0.433786	+	0.901016i	$0.642823\pi$
$240$	0.652958	0.0421482
$241$	11.8596	0.763945	0.381973	−	0.924174i	$-0.375245\pi$
$241$	11.8596	0.763945	0.381973	+	0.924174i	$0.375245\pi$
$242$	18.5116	1.18997
$243$	−1.00000	−0.0641500
$244$	−3.43246	−0.219741
$245$	−5.34336	−0.341375
$246$	5.34588	0.340841
$247$	−22.7607	−1.44823
$248$	5.19796	0.330071
$249$	−8.46140	−0.536220
$250$	6.25119	0.395360
$251$	−3.49169	−0.220393	−0.110197	−	0.993910i	$-0.535148\pi$
$251$	−3.49169	−0.220393	−0.110197	+	0.993910i	$0.535148\pi$
$252$	−3.89658	−0.245461
$253$	−5.43246	−0.341536
$254$	−17.0070	−1.06711
$255$	3.12081	0.195433
$256$	1.00000	0.0625000
$257$	−19.4752	−1.21483	−0.607416	−	0.794384i	$-0.707794\pi$
$257$	−19.4752	−1.21483	−0.607416	+	0.794384i	$0.707794\pi$
$258$	2.02783	0.126247
$259$	5.00530	0.311014
$260$	−2.05247	−0.127289
$261$	−1.00000	−0.0618984
$262$	−14.8267	−0.915996
$263$	5.45886	0.336607	0.168304	−	0.985735i	$-0.446171\pi$
$263$	5.45886	0.336607	0.168304	+	0.985735i	$0.446171\pi$
$264$	−5.43246	−0.334345
$265$	−6.51588	−0.400267
$266$	28.2147	1.72996
$267$	−7.44828	−0.455827
$268$	−4.99831	−0.305320
$269$	−6.24599	−0.380825	−0.190412	−	0.981704i	$-0.560982\pi$
$269$	−6.24599	−0.380825	−0.190412	+	0.981704i	$0.560982\pi$
$270$	0.652958	0.0397377
$271$	−1.67221	−0.101580	−0.0507898	−	0.998709i	$-0.516174\pi$
$271$	−1.67221	−0.101580	−0.0507898	+	0.998709i	$0.516174\pi$
$272$	4.77950	0.289800
$273$	12.2483	0.741301
$274$	5.65080	0.341377
$275$	−24.8461	−1.49828
$276$	1.00000	0.0601929
$277$	0.419039	0.0251776	0.0125888	−	0.999921i	$-0.495993\pi$
$277$	0.419039	0.0251776	0.0125888	+	0.999921i	$0.495993\pi$
$278$	−10.3334	−0.619757
$279$	5.19796	0.311194
$280$	2.54430	0.152051
$281$	−13.6630	−0.815067	−0.407534	−	0.913190i	$-0.633611\pi$
$281$	−13.6630	−0.815067	−0.407534	+	0.913190i	$0.633611\pi$
$282$	−4.83411	−0.287867
$283$	21.4837	1.27707	0.638536	−	0.769592i	$-0.279540\pi$
$283$	21.4837	1.27707	0.638536	+	0.769592i	$0.279540\pi$
$284$	−0.305382	−0.0181211
$285$	−4.72800	−0.280063
$286$	17.0761	1.00973
$287$	20.8306	1.22959
$288$	1.00000	0.0589256
$289$	5.84363	0.343743
$290$	0.652958	0.0383430
$291$	−16.6682	−0.977106
$292$	−4.98203	−0.291552
$293$	−26.4618	−1.54591	−0.772956	−	0.634459i	$-0.781223\pi$
$293$	−26.4618	−1.54591	−0.772956	+	0.634459i	$0.781223\pi$
$294$	−8.18332	−0.477261
$295$	−7.23766	−0.421393
$296$	−1.28454	−0.0746622
$297$	−5.43246	−0.315223
$298$	15.9143	0.921888
$299$	−3.14335	−0.181785
$300$	4.57365	0.264060
$301$	7.90160	0.455440
$302$	0.926954	0.0533402
$303$	−14.1175	−0.811029
$304$	−7.24090	−0.415294
$305$	2.24125	0.128334
$306$	4.77950	0.273226
$307$	26.1141	1.49041	0.745205	−	0.666835i	$-0.232351\pi$
$307$	26.1141	1.49041	0.745205	+	0.666835i	$0.232351\pi$
$308$	−21.1680	−1.20616
$309$	−0.187986	−0.0106941
$310$	−3.39405	−0.192769
$311$	27.9298	1.58375	0.791877	−	0.610681i	$-0.209104\pi$
$311$	27.9298	1.58375	0.791877	+	0.610681i	$0.209104\pi$
$312$	−3.14335	−0.177957
$313$	0.185103	0.0104627	0.00523133	−	0.999986i	$-0.498335\pi$
$313$	0.185103	0.0104627	0.00523133	+	0.999986i	$0.498335\pi$
$314$	8.37925	0.472869
$315$	2.54430	0.143355
$316$	3.65080	0.205373
$317$	−21.8614	−1.22786	−0.613928	−	0.789362i	$-0.710412\pi$
$317$	−21.8614	−1.22786	−0.613928	+	0.789362i	$0.710412\pi$
$318$	−9.97902	−0.559596
$319$	−5.43246	−0.304159
$320$	−0.652958	−0.0365014
$321$	−10.1145	−0.564538
$322$	3.89658	0.217148
$323$	−34.6079	−1.92564
$324$	1.00000	0.0555556
$325$	−14.3766	−0.797469
$326$	15.1905	0.841326
$327$	−16.9130	−0.935290
$328$	−5.34588	−0.295177
$329$	−18.8365	−1.03849
$330$	3.54717	0.195265
$331$	−7.34947	−0.403963	−0.201982	−	0.979389i	$-0.564738\pi$
$331$	−7.34947	−0.403963	−0.201982	+	0.979389i	$0.564738\pi$
$332$	8.46140	0.464380
$333$	−1.28454	−0.0703922
$334$	14.5570	0.796522
$335$	3.26368	0.178314
$336$	3.89658	0.212576
$337$	20.0701	1.09329	0.546645	−	0.837365i	$-0.315905\pi$
$337$	20.0701	1.09329	0.546645	+	0.837365i	$0.315905\pi$
$338$	−3.11935	−0.169670
$339$	−19.0614	−1.03527
$340$	−3.12081	−0.169250
$341$	28.2377	1.52916
$342$	−7.24090	−0.391543
$343$	−4.61089	−0.248964
$344$	−2.02783	−0.109333
$345$	−0.652958	−0.0351541
$346$	−10.8097	−0.581135
$347$	−15.6156	−0.838290	−0.419145	−	0.907919i	$-0.637670\pi$
$347$	−15.6156	−0.838290	−0.419145	+	0.907919i	$0.637670\pi$
$348$	1.00000	0.0536056
$349$	−13.1335	−0.703019	−0.351509	−	0.936184i	$-0.614331\pi$
$349$	−13.1335	−0.703019	−0.351509	+	0.936184i	$0.614331\pi$
$350$	17.8216	0.952603
$351$	−3.14335	−0.167780
$352$	5.43246	0.289551
$353$	4.91671	0.261690	0.130845	−	0.991403i	$-0.458231\pi$
$353$	4.91671	0.261690	0.130845	+	0.991403i	$0.458231\pi$
$354$	−11.0844	−0.589131
$355$	0.199402	0.0105831
$356$	7.44828	0.394758
$357$	18.6237	0.985671
$358$	−8.20054	−0.433412
$359$	12.3197	0.650210	0.325105	−	0.945678i	$-0.394600\pi$
$359$	12.3197	0.650210	0.325105	+	0.945678i	$0.394600\pi$
$360$	−0.652958	−0.0344139
$361$	33.4307	1.75951
$362$	−4.40352	−0.231444
$363$	−18.5116	−0.971608
$364$	−12.2483	−0.641986
$365$	3.25306	0.170273
$366$	3.43246	0.179418
$367$	−33.3993	−1.74343	−0.871715	−	0.490012i	$-0.836992\pi$
$367$	−33.3993	−1.74343	−0.871715	+	0.490012i	$0.836992\pi$
$368$	−1.00000	−0.0521286
$369$	−5.34588	−0.278296
$370$	0.838748	0.0436044
$371$	−38.8840	−2.01876
$372$	−5.19796	−0.269502
$373$	11.0525	0.572278	0.286139	−	0.958188i	$-0.407628\pi$
$373$	11.0525	0.572278	0.286139	+	0.958188i	$0.407628\pi$
$374$	25.9644	1.34259
$375$	−6.25119	−0.322810
$376$	4.83411	0.249300
$377$	−3.14335	−0.161891
$378$	3.89658	0.200418
$379$	−22.4468	−1.15302	−0.576509	−	0.817091i	$-0.695585\pi$
$379$	−22.4468	−1.15302	−0.576509	+	0.817091i	$0.695585\pi$
$380$	4.72800	0.242541
$381$	17.0070	0.871293
$382$	3.17330	0.162360
$383$	15.0211	0.767543	0.383771	−	0.923428i	$-0.374625\pi$
$383$	15.0211	0.767543	0.383771	+	0.923428i	$0.374625\pi$
$384$	−1.00000	−0.0510310
$385$	13.8218	0.704424
$386$	24.6898	1.25668
$387$	−2.02783	−0.103080
$388$	16.6682	0.846199
$389$	0.578161	0.0293139	0.0146570	−	0.999893i	$-0.495334\pi$
$389$	0.578161	0.0293139	0.0146570	+	0.999893i	$0.495334\pi$
$390$	2.05247	0.103931
$391$	−4.77950	−0.241710
$392$	8.18332	0.413320
$393$	14.8267	0.747907
$394$	−1.61793	−0.0815099
$395$	−2.38381	−0.119943
$396$	5.43246	0.272991
$397$	−17.2881	−0.867667	−0.433834	−	0.900993i	$-0.642839\pi$
$397$	−17.2881	−0.867667	−0.433834	+	0.900993i	$0.642839\pi$
$398$	1.43101	0.0717299
$399$	−28.2147	−1.41250
$400$	−4.57365	−0.228682
$401$	−24.7150	−1.23421	−0.617103	−	0.786882i	$-0.711694\pi$
$401$	−24.7150	−1.23421	−0.617103	+	0.786882i	$0.711694\pi$
$402$	4.99831	0.249293
$403$	16.3390	0.813904
$404$	14.1175	0.702371
$405$	−0.652958	−0.0324457
$406$	3.89658	0.193384
$407$	−6.97820	−0.345896
$408$	−4.77950	−0.236621
$409$	−21.3137	−1.05389	−0.526947	−	0.849898i	$-0.676663\pi$
$409$	−21.3137	−1.05389	−0.526947	+	0.849898i	$0.676663\pi$
$410$	3.49063	0.172390
$411$	−5.65080	−0.278733
$412$	0.187986	0.00926139
$413$	−43.1913	−2.12531
$414$	−1.00000	−0.0491473
$415$	−5.52494	−0.271208
$416$	3.14335	0.154115
$417$	10.3334	0.506030
$418$	−39.3359	−1.92398
$419$	−17.1381	−0.837253	−0.418627	−	0.908158i	$-0.637488\pi$
$419$	−17.1381	−0.837253	−0.418627	+	0.908158i	$0.637488\pi$
$420$	−2.54430	−0.124149
$421$	34.8943	1.70064	0.850322	−	0.526263i	$-0.176407\pi$
$421$	34.8943	1.70064	0.850322	+	0.526263i	$0.176407\pi$
$422$	4.87050	0.237092
$423$	4.83411	0.235043
$424$	9.97902	0.484624
$425$	−21.8597	−1.06035
$426$	0.305382	0.0147958
$427$	13.3748	0.647254
$428$	10.1145	0.488904
$429$	−17.0761	−0.824443
$430$	1.32409	0.0638531
$431$	−23.7793	−1.14541	−0.572705	−	0.819762i	$-0.694106\pi$
$431$	−23.7793	−1.14541	−0.572705	+	0.819762i	$0.694106\pi$
$432$	−1.00000	−0.0481125
$433$	7.11867	0.342101	0.171051	−	0.985262i	$-0.445284\pi$
$433$	7.11867	0.342101	0.171051	+	0.985262i	$0.445284\pi$
$434$	−20.2543	−0.972235
$435$	−0.652958	−0.0313069
$436$	16.9130	0.809985
$437$	7.24090	0.346379
$438$	4.98203	0.238051
$439$	22.6902	1.08294	0.541471	−	0.840719i	$-0.317868\pi$
$439$	22.6902	1.08294	0.541471	+	0.840719i	$0.317868\pi$
$440$	−3.54717	−0.169104
$441$	8.18332	0.389682
$442$	15.0236	0.714602
$443$	22.0150	1.04597	0.522983	−	0.852343i	$-0.324819\pi$
$443$	22.0150	1.04597	0.522983	+	0.852343i	$0.324819\pi$
$444$	1.28454	0.0609614
$445$	−4.86341	−0.230548
$446$	−26.7563	−1.26695
$447$	−15.9143	−0.752718
$448$	−3.89658	−0.184096
$449$	−0.292116	−0.0137858	−0.00689290	−	0.999976i	$-0.502194\pi$
$449$	−0.292116	−0.0137858	−0.00689290	+	0.999976i	$0.502194\pi$
$450$	−4.57365	−0.215604
$451$	−29.0413	−1.36750
$452$	19.0614	0.896574
$453$	−0.926954	−0.0435521
$454$	−22.8435	−1.07210
$455$	7.99762	0.374934
$456$	7.24090	0.339086
$457$	25.2951	1.18326	0.591628	−	0.806211i	$-0.298485\pi$
$457$	25.2951	1.18326	0.591628	+	0.806211i	$0.298485\pi$
$458$	−12.2649	−0.573099
$459$	−4.77950	−0.223088
$460$	0.652958	0.0304443
$461$	−13.6709	−0.636717	−0.318358	−	0.947970i	$-0.603131\pi$
$461$	−13.6709	−0.636717	−0.318358	+	0.947970i	$0.603131\pi$
$462$	21.1680	0.984824
$463$	12.7660	0.593288	0.296644	−	0.954988i	$-0.404132\pi$
$463$	12.7660	0.593288	0.296644	+	0.954988i	$0.404132\pi$
$464$	−1.00000	−0.0464238
$465$	3.39405	0.157395
$466$	26.6506	1.23457
$467$	−32.6378	−1.51030	−0.755148	−	0.655555i	$-0.772435\pi$
$467$	−32.6378	−1.51030	−0.755148	+	0.655555i	$0.772435\pi$
$468$	3.14335	0.145301
$469$	19.4763	0.899332
$470$	−3.15647	−0.145597
$471$	−8.37925	−0.386096
$472$	11.0844	0.510202
$473$	−11.0161	−0.506521
$474$	−3.65080	−0.167687
$475$	33.1173	1.51953
$476$	−18.6237	−0.853616
$477$	9.97902	0.456908
$478$	−13.4123	−0.613466
$479$	10.5246	0.480882	0.240441	−	0.970664i	$-0.422708\pi$
$479$	10.5246	0.480882	0.240441	+	0.970664i	$0.422708\pi$
$480$	0.652958	0.0298033
$481$	−4.03775	−0.184106
$482$	11.8596	0.540191
$483$	−3.89658	−0.177300
$484$	18.5116	0.841437
$485$	−10.8836	−0.494200
$486$	−1.00000	−0.0453609
$487$	−14.4635	−0.655402	−0.327701	−	0.944782i	$-0.606274\pi$
$487$	−14.4635	−0.655402	−0.327701	+	0.944782i	$0.606274\pi$
$488$	−3.43246	−0.155380
$489$	−15.1905	−0.686940
$490$	−5.34336	−0.241388
$491$	32.7684	1.47882	0.739410	−	0.673256i	$-0.235105\pi$
$491$	32.7684	1.47882	0.739410	+	0.673256i	$0.235105\pi$
$492$	5.34588	0.241011
$493$	−4.77950	−0.215258
$494$	−22.7607	−1.02405
$495$	−3.54717	−0.159433
$496$	5.19796	0.233395
$497$	1.18995	0.0533763
$498$	−8.46140	−0.379165
$499$	−2.09729	−0.0938875	−0.0469437	−	0.998898i	$-0.514948\pi$
$499$	−2.09729	−0.0938875	−0.0469437	+	0.998898i	$0.514948\pi$
$500$	6.25119	0.279562
$501$	−14.5570	−0.650357
$502$	−3.49169	−0.155842
$503$	9.30673	0.414967	0.207483	−	0.978239i	$-0.433473\pi$
$503$	9.30673	0.414967	0.207483	+	0.978239i	$0.433473\pi$
$504$	−3.89658	−0.173567
$505$	−9.21812	−0.410201
$506$	−5.43246	−0.241502
$507$	3.11935	0.138535
$508$	−17.0070	−0.754562
$509$	−27.7983	−1.23214	−0.616068	−	0.787693i	$-0.711275\pi$
$509$	−27.7983	−1.23214	−0.616068	+	0.787693i	$0.711275\pi$
$510$	3.12081	0.138192
$511$	19.4129	0.858775
$512$	1.00000	0.0441942
$513$	7.24090	0.319694
$514$	−19.4752	−0.859015
$515$	−0.122747	−0.00540886
$516$	2.02783	0.0892702
$517$	26.2611	1.15496
$518$	5.00530	0.219920
$519$	10.8097	0.474494
$520$	−2.05247	−0.0900069
$521$	−27.6140	−1.20979	−0.604895	−	0.796305i	$-0.706785\pi$
$521$	−27.6140	−1.20979	−0.604895	+	0.796305i	$0.706785\pi$
$522$	−1.00000	−0.0437688
$523$	11.6070	0.507540	0.253770	−	0.967265i	$-0.418329\pi$
$523$	11.6070	0.507540	0.253770	+	0.967265i	$0.418329\pi$
$524$	−14.8267	−0.647707
$525$	−17.8216	−0.777797
$526$	5.45886	0.238017
$527$	24.8437	1.08221
$528$	−5.43246	−0.236417
$529$	1.00000	0.0434783
$530$	−6.51588	−0.283032
$531$	11.0844	0.481023
$532$	28.2147	1.22326
$533$	−16.8040	−0.727861
$534$	−7.44828	−0.322318
$535$	−6.60436	−0.285531
$536$	−4.99831	−0.215894
$537$	8.20054	0.353879
$538$	−6.24599	−0.269284
$539$	44.4555	1.91484
$540$	0.652958	0.0280988
$541$	25.6713	1.10369	0.551847	−	0.833945i	$-0.313923\pi$
$541$	25.6713	1.10369	0.551847	+	0.833945i	$0.313923\pi$
$542$	−1.67221	−0.0718277
$543$	4.40352	0.188973
$544$	4.77950	0.204919
$545$	−11.0435	−0.473050
$546$	12.2483	0.524179
$547$	−8.82855	−0.377482	−0.188741	−	0.982027i	$-0.560441\pi$
$547$	−8.82855	−0.377482	−0.188741	+	0.982027i	$0.560441\pi$
$548$	5.65080	0.241390
$549$	−3.43246	−0.146494
$550$	−24.8461	−1.05944
$551$	7.24090	0.308473
$552$	1.00000	0.0425628
$553$	−14.2256	−0.604934
$554$	0.419039	0.0178032
$555$	−0.838748	−0.0356029
$556$	−10.3334	−0.438234
$557$	23.2592	0.985523	0.492761	−	0.870164i	$-0.335988\pi$
$557$	23.2592	0.985523	0.492761	+	0.870164i	$0.335988\pi$
$558$	5.19796	0.220047
$559$	−6.37418	−0.269599
$560$	2.54430	0.107516
$561$	−25.9644	−1.09622
$562$	−13.6630	−0.576340
$563$	46.2484	1.94914	0.974568	−	0.224090i	$-0.0719411\pi$
$563$	46.2484	1.94914	0.974568	+	0.224090i	$0.0719411\pi$
$564$	−4.83411	−0.203553
$565$	−12.4463	−0.523620
$566$	21.4837	0.903027
$567$	−3.89658	−0.163641
$568$	−0.305382	−0.0128136
$569$	33.3629	1.39864	0.699322	−	0.714807i	$-0.253485\pi$
$569$	33.3629	1.39864	0.699322	+	0.714807i	$0.253485\pi$
$570$	−4.72800	−0.198034
$571$	28.9022	1.20952	0.604760	−	0.796408i	$-0.293269\pi$
$571$	28.9022	1.20952	0.604760	+	0.796408i	$0.293269\pi$
$572$	17.0761	0.713988
$573$	−3.17330	−0.132566
$574$	20.8306	0.869454
$575$	4.57365	0.190734
$576$	1.00000	0.0416667
$577$	−25.6706	−1.06868	−0.534341	−	0.845269i	$-0.679440\pi$
$577$	−25.6706	−1.06868	−0.534341	+	0.845269i	$0.679440\pi$
$578$	5.84363	0.243063
$579$	−24.6898	−1.02607
$580$	0.652958	0.0271126
$581$	−32.9705	−1.36785
$582$	−16.6682	−0.690918
$583$	54.2106	2.24517
$584$	−4.98203	−0.206158
$585$	−2.05247	−0.0848594
$586$	−26.4618	−1.09313
$587$	9.88963	0.408189	0.204094	−	0.978951i	$-0.434575\pi$
$587$	9.88963	0.408189	0.204094	+	0.978951i	$0.434575\pi$
$588$	−8.18332	−0.337474
$589$	−37.6379	−1.55084
$590$	−7.23766	−0.297970
$591$	1.61793	0.0665526
$592$	−1.28454	−0.0527941
$593$	−11.6160	−0.477011	−0.238506	−	0.971141i	$-0.576658\pi$
$593$	−11.6160	−0.477011	−0.238506	+	0.971141i	$0.576658\pi$
$594$	−5.43246	−0.222896
$595$	12.1605	0.498531
$596$	15.9143	0.651873
$597$	−1.43101	−0.0585672
$598$	−3.14335	−0.128541
$599$	12.0689	0.493123	0.246562	−	0.969127i	$-0.420699\pi$
$599$	12.0689	0.493123	0.246562	+	0.969127i	$0.420699\pi$
$600$	4.57365	0.186718
$601$	20.5093	0.836593	0.418296	−	0.908311i	$-0.362627\pi$
$601$	20.5093	0.836593	0.418296	+	0.908311i	$0.362627\pi$
$602$	7.90160	0.322045
$603$	−4.99831	−0.203547
$604$	0.926954	0.0377172
$605$	−12.0873	−0.491419
$606$	−14.1175	−0.573484
$607$	9.81332	0.398310	0.199155	−	0.979968i	$-0.436180\pi$
$607$	9.81332	0.398310	0.199155	+	0.979968i	$0.436180\pi$
$608$	−7.24090	−0.293657
$609$	−3.89658	−0.157897
$610$	2.24125	0.0907456
$611$	15.1953	0.614736
$612$	4.77950	0.193200
$613$	1.34070	0.0541505	0.0270752	−	0.999633i	$-0.491381\pi$
$613$	1.34070	0.0541505	0.0270752	+	0.999633i	$0.491381\pi$
$614$	26.1141	1.05388
$615$	−3.49063	−0.140756
$616$	−21.1680	−0.852883
$617$	33.6334	1.35403	0.677015	−	0.735970i	$-0.263273\pi$
$617$	33.6334	1.35403	0.677015	+	0.735970i	$0.263273\pi$
$618$	−0.187986	−0.00756189
$619$	−26.8643	−1.07977	−0.539884	−	0.841739i	$-0.681532\pi$
$619$	−26.8643	−1.07977	−0.539884	+	0.841739i	$0.681532\pi$
$620$	−3.39405	−0.136308
$621$	1.00000	0.0401286
$622$	27.9298	1.11988
$623$	−29.0228	−1.16277
$624$	−3.14335	−0.125835
$625$	18.7865	0.751459
$626$	0.185103	0.00739822
$627$	39.3359	1.57093
$628$	8.37925	0.334369
$629$	−6.13945	−0.244796
$630$	2.54430	0.101367
$631$	−44.5346	−1.77290	−0.886448	−	0.462828i	$-0.846835\pi$
$631$	−44.5346	−1.77290	−0.886448	+	0.462828i	$0.846835\pi$
$632$	3.65080	0.145221
$633$	−4.87050	−0.193585
$634$	−21.8614	−0.868225
$635$	11.1048	0.440681
$636$	−9.97902	−0.395694
$637$	25.7230	1.01918
$638$	−5.43246	−0.215073
$639$	−0.305382	−0.0120807
$640$	−0.652958	−0.0258104
$641$	35.3226	1.39516	0.697579	−	0.716508i	$-0.254261\pi$
$641$	35.3226	1.39516	0.697579	+	0.716508i	$0.254261\pi$
$642$	−10.1145	−0.399189
$643$	−19.0175	−0.749977	−0.374989	−	0.927029i	$-0.622353\pi$
$643$	−19.0175	−0.749977	−0.374989	+	0.927029i	$0.622353\pi$
$644$	3.89658	0.153547
$645$	−1.32409	−0.0521359
$646$	−34.6079	−1.36163
$647$	−38.0528	−1.49601	−0.748004	−	0.663694i	$-0.768988\pi$
$647$	−38.0528	−1.49601	−0.748004	+	0.663694i	$0.768988\pi$
$648$	1.00000	0.0392837
$649$	60.2157	2.36367
$650$	−14.3766	−0.563895
$651$	20.2543	0.793827
$652$	15.1905	0.594907
$653$	31.7439	1.24223	0.621117	−	0.783718i	$-0.286679\pi$
$653$	31.7439	1.24223	0.621117	+	0.783718i	$0.286679\pi$
$654$	−16.9130	−0.661350
$655$	9.68120	0.378276
$656$	−5.34588	−0.208722
$657$	−4.98203	−0.194368
$658$	−18.8365	−0.734323
$659$	10.9149	0.425182	0.212591	−	0.977141i	$-0.431810\pi$
$659$	10.9149	0.425182	0.212591	+	0.977141i	$0.431810\pi$
$660$	3.54717	0.138073
$661$	8.37866	0.325892	0.162946	−	0.986635i	$-0.447900\pi$
$661$	8.37866	0.325892	0.162946	+	0.986635i	$0.447900\pi$
$662$	−7.34947	−0.285645
$663$	−15.0236	−0.583470
$664$	8.46140	0.328366
$665$	−18.4230	−0.714415
$666$	−1.28454	−0.0497748
$667$	1.00000	0.0387202
$668$	14.5570	0.563226
$669$	26.7563	1.03446
$670$	3.26368	0.126087
$671$	−18.6467	−0.719848
$672$	3.89658	0.150314
$673$	16.0566	0.618937	0.309468	−	0.950910i	$-0.399849\pi$
$673$	16.0566	0.618937	0.309468	+	0.950910i	$0.399849\pi$
$674$	20.0701	0.773072
$675$	4.57365	0.176040
$676$	−3.11935	−0.119975
$677$	−5.05183	−0.194158	−0.0970788	−	0.995277i	$-0.530950\pi$
$677$	−5.05183	−0.194158	−0.0970788	+	0.995277i	$0.530950\pi$
$678$	−19.0614	−0.732049
$679$	−64.9489	−2.49251
$680$	−3.12081	−0.119678
$681$	22.8435	0.875365
$682$	28.2377	1.08128
$683$	51.1418	1.95689	0.978444	−	0.206511i	$-0.0662110\pi$
$683$	51.1418	1.95689	0.978444	+	0.206511i	$0.0662110\pi$
$684$	−7.24090	−0.276863
$685$	−3.68973	−0.140977
$686$	−4.61089	−0.176044
$687$	12.2649	0.467934
$688$	−2.02783	−0.0773103
$689$	31.3675	1.19501
$690$	−0.652958	−0.0248577
$691$	−29.9142	−1.13799	−0.568994	−	0.822341i	$-0.692667\pi$
$691$	−29.9142	−1.13799	−0.568994	+	0.822341i	$0.692667\pi$
$692$	−10.8097	−0.410924
$693$	−21.1680	−0.804106
$694$	−15.6156	−0.592760
$695$	6.74728	0.255939
$696$	1.00000	0.0379049
$697$	−25.5507	−0.967800
$698$	−13.1335	−0.497109
$699$	−26.6506	−1.00802
$700$	17.8216	0.673592
$701$	−40.3529	−1.52411	−0.762054	−	0.647513i	$-0.775809\pi$
$701$	−40.3529	−1.52411	−0.762054	+	0.647513i	$0.775809\pi$
$702$	−3.14335	−0.118638
$703$	9.30121	0.350802
$704$	5.43246	0.204744
$705$	3.15647	0.118880
$706$	4.91671	0.185043
$707$	−55.0099	−2.06886
$708$	−11.0844	−0.416578
$709$	20.5292	0.770988	0.385494	−	0.922710i	$-0.374031\pi$
$709$	20.5292	0.770988	0.385494	+	0.922710i	$0.374031\pi$
$710$	0.199402	0.00748341
$711$	3.65080	0.136916
$712$	7.44828	0.279136
$713$	−5.19796	−0.194665
$714$	18.6237	0.696974
$715$	−11.1500	−0.416986
$716$	−8.20054	−0.306469
$717$	13.4123	0.500893
$718$	12.3197	0.459768
$719$	−36.0665	−1.34505	−0.672526	−	0.740073i	$-0.734791\pi$
$719$	−36.0665	−1.34505	−0.672526	+	0.740073i	$0.734791\pi$
$720$	−0.652958	−0.0243343
$721$	−0.732501	−0.0272797
$722$	33.4307	1.24416
$723$	−11.8596	−0.441064
$724$	−4.40352	−0.163655
$725$	4.57365	0.169861
$726$	−18.5116	−0.687030
$727$	−43.4068	−1.60987	−0.804934	−	0.593364i	$-0.797800\pi$
$727$	−43.4068	−1.60987	−0.804934	+	0.593364i	$0.797800\pi$
$728$	−12.2483	−0.453952
$729$	1.00000	0.0370370
$730$	3.25306	0.120401
$731$	−9.69202	−0.358472
$732$	3.43246	0.126867
$733$	−45.9201	−1.69610	−0.848048	−	0.529919i	$-0.822222\pi$
$733$	−45.9201	−1.69610	−0.848048	+	0.529919i	$0.822222\pi$
$734$	−33.3993	−1.23279
$735$	5.34336	0.197093
$736$	−1.00000	−0.0368605
$737$	−27.1531	−1.00020
$738$	−5.34588	−0.196785
$739$	−19.2125	−0.706742	−0.353371	−	0.935483i	$-0.614965\pi$
$739$	−19.2125	−0.706742	−0.353371	+	0.935483i	$0.614965\pi$
$740$	0.838748	0.0308330
$741$	22.7607	0.836135
$742$	−38.8840	−1.42748
$743$	−51.9343	−1.90529	−0.952643	−	0.304092i	$-0.901647\pi$
$743$	−51.9343	−1.90529	−0.952643	+	0.304092i	$0.901647\pi$
$744$	−5.19796	−0.190566
$745$	−10.3913	−0.380709
$746$	11.0525	0.404662
$747$	8.46140	0.309587
$748$	25.9644	0.949354
$749$	−39.4121	−1.44009
$750$	−6.25119	−0.228261
$751$	−26.3797	−0.962610	−0.481305	−	0.876553i	$-0.659837\pi$
$751$	−26.3797	−0.962610	−0.481305	+	0.876553i	$0.659837\pi$
$752$	4.83411	0.176282
$753$	3.49169	0.127244
$754$	−3.14335	−0.114474
$755$	−0.605262	−0.0220277
$756$	3.89658	0.141717
$757$	−1.11874	−0.0406614	−0.0203307	−	0.999793i	$-0.506472\pi$
$757$	−1.11874	−0.0406614	−0.0203307	+	0.999793i	$0.506472\pi$
$758$	−22.4468	−0.815306
$759$	5.43246	0.197186
$760$	4.72800	0.171503
$761$	−20.8316	−0.755147	−0.377573	−	0.925980i	$-0.623241\pi$
$761$	−20.8316	−0.755147	−0.377573	+	0.925980i	$0.623241\pi$
$762$	17.0070	0.616097
$763$	−65.9028	−2.38584
$764$	3.17330	0.114806
$765$	−3.12081	−0.112833
$766$	15.0211	0.542735
$767$	34.8422	1.25808
$768$	−1.00000	−0.0360844
$769$	20.3376	0.733394	0.366697	−	0.930340i	$-0.380489\pi$
$769$	20.3376	0.733394	0.366697	+	0.930340i	$0.380489\pi$
$770$	13.8218	0.498103
$771$	19.4752	0.701383
$772$	24.6898	0.888606
$773$	47.2163	1.69825	0.849126	−	0.528190i	$-0.177129\pi$
$773$	47.2163	1.69825	0.849126	+	0.528190i	$0.177129\pi$
$774$	−2.02783	−0.0728888
$775$	−23.7736	−0.853974
$776$	16.6682	0.598353
$777$	−5.00530	−0.179564
$778$	0.578161	0.0207281
$779$	38.7090	1.38689
$780$	2.05247	0.0734904
$781$	−1.65898	−0.0593628
$782$	−4.77950	−0.170915
$783$	1.00000	0.0357371
$784$	8.18332	0.292261
$785$	−5.47130	−0.195279
$786$	14.8267	0.528850
$787$	31.8505	1.13535	0.567675	−	0.823253i	$-0.307843\pi$
$787$	31.8505	1.13535	0.567675	+	0.823253i	$0.307843\pi$
$788$	−1.61793	−0.0576362
$789$	−5.45886	−0.194340
$790$	−2.38381	−0.0848123
$791$	−74.2743	−2.64089
$792$	5.43246	0.193034
$793$	−10.7894	−0.383144
$794$	−17.2881	−0.613533
$795$	6.51588	0.231094
$796$	1.43101	0.0507207
$797$	−14.9421	−0.529278	−0.264639	−	0.964348i	$-0.585253\pi$
$797$	−14.9421	−0.529278	−0.264639	+	0.964348i	$0.585253\pi$
$798$	−28.2147	−0.998791
$799$	23.1046	0.817384
$800$	−4.57365	−0.161703
$801$	7.44828	0.263172
$802$	−24.7150	−0.872716
$803$	−27.0647	−0.955092
$804$	4.99831	0.176277
$805$	−2.54430	−0.0896748
$806$	16.3390	0.575517
$807$	6.24599	0.219869
$808$	14.1175	0.496652
$809$	37.2618	1.31005	0.655027	−	0.755606i	$-0.272657\pi$
$809$	37.2618	1.31005	0.655027	+	0.755606i	$0.272657\pi$
$810$	−0.652958	−0.0229426
$811$	11.5548	0.405744	0.202872	−	0.979205i	$-0.434972\pi$
$811$	11.5548	0.405744	0.202872	+	0.979205i	$0.434972\pi$
$812$	3.89658	0.136743
$813$	1.67221	0.0586471
$814$	−6.97820	−0.244586
$815$	−9.91878	−0.347440
$816$	−4.77950	−0.167316
$817$	14.6833	0.513704
$818$	−21.3137	−0.745216
$819$	−12.2483	−0.427990
$820$	3.49063	0.121898
$821$	−48.0919	−1.67842	−0.839209	−	0.543810i	$-0.816981\pi$
$821$	−48.0919	−1.67842	−0.839209	+	0.543810i	$0.816981\pi$
$822$	−5.65080	−0.197094
$823$	−16.1071	−0.561457	−0.280728	−	0.959787i	$-0.590576\pi$
$823$	−16.1071	−0.561457	−0.280728	+	0.959787i	$0.590576\pi$
$824$	0.187986	0.00654879
$825$	24.8461	0.865032
$826$	−43.1913	−1.50282
$827$	−19.8361	−0.689769	−0.344884	−	0.938645i	$-0.612082\pi$
$827$	−19.8361	−0.689769	−0.344884	+	0.938645i	$0.612082\pi$
$828$	−1.00000	−0.0347524
$829$	−24.9727	−0.867339	−0.433670	−	0.901072i	$-0.642781\pi$
$829$	−24.9727	−0.867339	−0.433670	+	0.901072i	$0.642781\pi$
$830$	−5.52494	−0.191773
$831$	−0.419039	−0.0145363
$832$	3.14335	0.108976
$833$	39.1122	1.35516
$834$	10.3334	0.357817
$835$	−9.50508	−0.328937
$836$	−39.3359	−1.36046
$837$	−5.19796	−0.179668
$838$	−17.1381	−0.592027
$839$	28.9294	0.998753	0.499376	−	0.866385i	$-0.333562\pi$
$839$	28.9294	0.998753	0.499376	+	0.866385i	$0.333562\pi$
$840$	−2.54430	−0.0877867
$841$	1.00000	0.0344828
$842$	34.8943	1.20254
$843$	13.6630	0.470579
$844$	4.87050	0.167649
$845$	2.03681	0.0700682
$846$	4.83411	0.166200
$847$	−72.1319	−2.47848
$848$	9.97902	0.342681
$849$	−21.4837	−0.737318
$850$	−21.8597	−0.749783
$851$	1.28454	0.0440334
$852$	0.305382	0.0104622
$853$	−42.1465	−1.44307	−0.721535	−	0.692378i	$-0.756563\pi$
$853$	−42.1465	−1.44307	−0.721535	+	0.692378i	$0.756563\pi$
$854$	13.3748	0.457678
$855$	4.72800	0.161694
$856$	10.1145	0.345708
$857$	41.3067	1.41101	0.705506	−	0.708704i	$-0.250720\pi$
$857$	41.3067	1.41101	0.705506	+	0.708704i	$0.250720\pi$
$858$	−17.0761	−0.582969
$859$	−19.9708	−0.681396	−0.340698	−	0.940173i	$-0.610663\pi$
$859$	−19.9708	−0.681396	−0.340698	+	0.940173i	$0.610663\pi$
$860$	1.32409	0.0451510
$861$	−20.8306	−0.709907
$862$	−23.7793	−0.809927
$863$	−37.1083	−1.26318	−0.631590	−	0.775302i	$-0.717597\pi$
$863$	−37.1083	−1.26318	−0.631590	+	0.775302i	$0.717597\pi$
$864$	−1.00000	−0.0340207
$865$	7.05829	0.239989
$866$	7.11867	0.241902
$867$	−5.84363	−0.198460
$868$	−20.2543	−0.687474
$869$	19.8328	0.672782
$870$	−0.652958	−0.0221373
$871$	−15.7114	−0.532361
$872$	16.9130	0.572746
$873$	16.6682	0.564133
$874$	7.24090	0.244927
$875$	−24.3582	−0.823458
$876$	4.98203	0.168327
$877$	39.2808	1.32642	0.663210	−	0.748433i	$-0.269194\pi$
$877$	39.2808	1.32642	0.663210	+	0.748433i	$0.269194\pi$
$878$	22.6902	0.765756
$879$	26.4618	0.892533
$880$	−3.54717	−0.119575
$881$	−53.5324	−1.80355	−0.901776	−	0.432203i	$-0.857736\pi$
$881$	−53.5324	−1.80355	−0.901776	+	0.432203i	$0.857736\pi$
$882$	8.18332	0.275547
$883$	51.9053	1.74675	0.873377	−	0.487045i	$-0.161925\pi$
$883$	51.9053	1.74675	0.873377	+	0.487045i	$0.161925\pi$
$884$	15.0236	0.505300
$885$	7.23766	0.243291
$886$	22.0150	0.739609
$887$	−3.83770	−0.128857	−0.0644286	−	0.997922i	$-0.520522\pi$
$887$	−3.83770	−0.128857	−0.0644286	+	0.997922i	$0.520522\pi$
$888$	1.28454	0.0431062
$889$	66.2689	2.22259
$890$	−4.86341	−0.163022
$891$	5.43246	0.181994
$892$	−26.7563	−0.895867
$893$	−35.0033	−1.17134
$894$	−15.9143	−0.532252
$895$	5.35461	0.178985
$896$	−3.89658	−0.130176
$897$	3.14335	0.104953
$898$	−0.292116	−0.00974803
$899$	−5.19796	−0.173362
$900$	−4.57365	−0.152455
$901$	47.6947	1.58894
$902$	−29.0413	−0.966969
$903$	−7.90160	−0.262949
$904$	19.0614	0.633973
$905$	2.87531	0.0955785
$906$	−0.926954	−0.0307960
$907$	−37.8368	−1.25635	−0.628176	−	0.778071i	$-0.716198\pi$
$907$	−37.8368	−1.25635	−0.628176	+	0.778071i	$0.716198\pi$
$908$	−22.8435	−0.758089
$909$	14.1175	0.468248
$910$	7.99762	0.265119
$911$	−24.0810	−0.797838	−0.398919	−	0.916986i	$-0.630615\pi$
$911$	−24.0810	−0.797838	−0.398919	+	0.916986i	$0.630615\pi$
$912$	7.24090	0.239770
$913$	45.9662	1.52126
$914$	25.2951	0.836689
$915$	−2.24125	−0.0740934
$916$	−12.2649	−0.405243
$917$	57.7733	1.90784
$918$	−4.77950	−0.157747
$919$	−48.0124	−1.58378	−0.791891	−	0.610663i	$-0.790903\pi$
$919$	−48.0124	−1.58378	−0.791891	+	0.610663i	$0.790903\pi$
$920$	0.652958	0.0215274
$921$	−26.1141	−0.860489
$922$	−13.6709	−0.450227
$923$	−0.959923	−0.0315962
$924$	21.1680	0.696376
$925$	5.87502	0.193169
$926$	12.7660	0.419518
$927$	0.187986	0.00617426
$928$	−1.00000	−0.0328266
$929$	−32.3276	−1.06064	−0.530318	−	0.847799i	$-0.677927\pi$
$929$	−32.3276	−1.06064	−0.530318	+	0.847799i	$0.677927\pi$
$930$	3.39405	0.111295
$931$	−59.2546	−1.94199
$932$	26.6506	0.872971
$933$	−27.9298	−0.914381
$934$	−32.6378	−1.06794
$935$	−16.9537	−0.554445
$936$	3.14335	0.102744
$937$	31.2124	1.01966	0.509832	−	0.860274i	$-0.329708\pi$
$937$	31.2124	1.01966	0.509832	+	0.860274i	$0.329708\pi$
$938$	19.4763	0.635923
$939$	−0.185103	−0.00604062
$940$	−3.15647	−0.102953
$941$	24.6942	0.805009	0.402504	−	0.915418i	$-0.368140\pi$
$941$	24.6942	0.805009	0.402504	+	0.915418i	$0.368140\pi$
$942$	−8.37925	−0.273011
$943$	5.34588	0.174086
$944$	11.0844	0.360767
$945$	−2.54430	−0.0827661
$946$	−11.0161	−0.358164
$947$	−28.0075	−0.910120	−0.455060	−	0.890461i	$-0.650382\pi$
$947$	−28.0075	−0.910120	−0.455060	+	0.890461i	$0.650382\pi$
$948$	−3.65080	−0.118572
$949$	−15.6603	−0.508354
$950$	33.1173	1.07447
$951$	21.8614	0.708903
$952$	−18.6237	−0.603598
$953$	37.6733	1.22036	0.610179	−	0.792264i	$-0.291098\pi$
$953$	37.6733	1.22036	0.610179	+	0.792264i	$0.291098\pi$
$954$	9.97902	0.323083
$955$	−2.07203	−0.0670492
$956$	−13.4123	−0.433786
$957$	5.43246	0.175606
$958$	10.5246	0.340035
$959$	−22.0188	−0.711023
$960$	0.652958	0.0210741
$961$	−3.98120	−0.128426
$962$	−4.03775	−0.130182
$963$	10.1145	0.325936
$964$	11.8596	0.381973
$965$	−16.1214	−0.518966
$966$	−3.89658	−0.125370
$967$	−34.7085	−1.11615	−0.558075	−	0.829790i	$-0.688460\pi$
$967$	−34.7085	−1.11615	−0.558075	+	0.829790i	$0.688460\pi$
$968$	18.5116	0.594986
$969$	34.6079	1.11177
$970$	−10.8836	−0.349452
$971$	7.32986	0.235226	0.117613	−	0.993059i	$-0.462476\pi$
$971$	7.32986	0.235226	0.117613	+	0.993059i	$0.462476\pi$
$972$	−1.00000	−0.0320750
$973$	40.2650	1.29084
$974$	−14.4635	−0.463439
$975$	14.3766	0.460419
$976$	−3.43246	−0.109870
$977$	−8.22242	−0.263059	−0.131529	−	0.991312i	$-0.541989\pi$
$977$	−8.22242	−0.263059	−0.131529	+	0.991312i	$0.541989\pi$
$978$	−15.1905	−0.485740
$979$	40.4624	1.29319
$980$	−5.34336	−0.170687
$981$	16.9130	0.539990
$982$	32.7684	1.04568
$983$	40.7529	1.29982	0.649908	−	0.760013i	$-0.274807\pi$
$983$	40.7529	1.29982	0.649908	+	0.760013i	$0.274807\pi$
$984$	5.34588	0.170421
$985$	1.05644	0.0336609
$986$	−4.77950	−0.152210
$987$	18.8365	0.599572
$988$	−22.7607	−0.724114
$989$	2.02783	0.0644812
$990$	−3.54717	−0.112736
$991$	−51.3663	−1.63171	−0.815853	−	0.578260i	$-0.803732\pi$
$991$	−51.3663	−1.63171	−0.815853	+	0.578260i	$0.803732\pi$
$992$	5.19796	0.165035
$993$	7.34947	0.233228
$994$	1.18995	0.0377428
$995$	−0.934386	−0.0296220
$996$	−8.46140	−0.268110
$997$	−19.9052	−0.630405	−0.315202	−	0.949025i	$-0.602072\pi$
$997$	−19.9052	−0.630405	−0.315202	+	0.949025i	$0.602072\pi$
$998$	−2.09729	−0.0663885
$999$	1.28454	0.0406410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.4	✓	8	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.652958 q^{5} -1.00000 q^{6} -3.89658 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.652958 q^{5} -1.00000 q^{6} -3.89658 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.652958 q^{10} +5.43246 q^{11} -1.00000 q^{12} +3.14335 q^{13} -3.89658 q^{14} +0.652958 q^{15} +1.00000 q^{16} +4.77950 q^{17} +1.00000 q^{18} -7.24090 q^{19} -0.652958 q^{20} +3.89658 q^{21} +5.43246 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.57365 q^{25} +3.14335 q^{26} -1.00000 q^{27} -3.89658 q^{28} -1.00000 q^{29} +0.652958 q^{30} +5.19796 q^{31} +1.00000 q^{32} -5.43246 q^{33} +4.77950 q^{34} +2.54430 q^{35} +1.00000 q^{36} -1.28454 q^{37} -7.24090 q^{38} -3.14335 q^{39} -0.652958 q^{40} -5.34588 q^{41} +3.89658 q^{42} -2.02783 q^{43} +5.43246 q^{44} -0.652958 q^{45} -1.00000 q^{46} +4.83411 q^{47} -1.00000 q^{48} +8.18332 q^{49} -4.57365 q^{50} -4.77950 q^{51} +3.14335 q^{52} +9.97902 q^{53} -1.00000 q^{54} -3.54717 q^{55} -3.89658 q^{56} +7.24090 q^{57} -1.00000 q^{58} +11.0844 q^{59} +0.652958 q^{60} -3.43246 q^{61} +5.19796 q^{62} -3.89658 q^{63} +1.00000 q^{64} -2.05247 q^{65} -5.43246 q^{66} -4.99831 q^{67} +4.77950 q^{68} +1.00000 q^{69} +2.54430 q^{70} -0.305382 q^{71} +1.00000 q^{72} -4.98203 q^{73} -1.28454 q^{74} +4.57365 q^{75} -7.24090 q^{76} -21.1680 q^{77} -3.14335 q^{78} +3.65080 q^{79} -0.652958 q^{80} +1.00000 q^{81} -5.34588 q^{82} +8.46140 q^{83} +3.89658 q^{84} -3.12081 q^{85} -2.02783 q^{86} +1.00000 q^{87} +5.43246 q^{88} +7.44828 q^{89} -0.652958 q^{90} -12.2483 q^{91} -1.00000 q^{92} -5.19796 q^{93} +4.83411 q^{94} +4.72800 q^{95} -1.00000 q^{96} +16.6682 q^{97} +8.18332 q^{98} +5.43246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

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