LMFDB - Embedded newform 8034.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
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} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} + \cdots + 768 $ x^12 - 6x^11 - 8x^10 + 94x^9 - 7x^8 - 580x^7 + 180x^6 + 1787x^5 - 308x^4 - 2790x^3 - 352x^2 + 1768*x + 768
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.78941$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.74714 q^{10} -0.860848 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.39290 q^{14} -2.74714 q^{15} +1.00000 q^{16} -4.14293 q^{17} -1.00000 q^{18} -2.70287 q^{19} -2.74714 q^{20} +3.39290 q^{21} +0.860848 q^{22} +0.937524 q^{23} -1.00000 q^{24} +2.54676 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.39290 q^{28} +0.521486 q^{29} +2.74714 q^{30} +7.56769 q^{31} -1.00000 q^{32} -0.860848 q^{33} +4.14293 q^{34} -9.32077 q^{35} +1.00000 q^{36} -0.373450 q^{37} +2.70287 q^{38} +1.00000 q^{39} +2.74714 q^{40} -5.16599 q^{41} -3.39290 q^{42} -9.15820 q^{43} -0.860848 q^{44} -2.74714 q^{45} -0.937524 q^{46} +3.73475 q^{47} +1.00000 q^{48} +4.51179 q^{49} -2.54676 q^{50} -4.14293 q^{51} +1.00000 q^{52} +6.82448 q^{53} -1.00000 q^{54} +2.36487 q^{55} -3.39290 q^{56} -2.70287 q^{57} -0.521486 q^{58} -10.7140 q^{59} -2.74714 q^{60} +4.99993 q^{61} -7.56769 q^{62} +3.39290 q^{63} +1.00000 q^{64} -2.74714 q^{65} +0.860848 q^{66} +7.00412 q^{67} -4.14293 q^{68} +0.937524 q^{69} +9.32077 q^{70} -9.02958 q^{71} -1.00000 q^{72} -8.63216 q^{73} +0.373450 q^{74} +2.54676 q^{75} -2.70287 q^{76} -2.92077 q^{77} -1.00000 q^{78} +1.21230 q^{79} -2.74714 q^{80} +1.00000 q^{81} +5.16599 q^{82} +15.5907 q^{83} +3.39290 q^{84} +11.3812 q^{85} +9.15820 q^{86} +0.521486 q^{87} +0.860848 q^{88} -4.60576 q^{89} +2.74714 q^{90} +3.39290 q^{91} +0.937524 q^{92} +7.56769 q^{93} -3.73475 q^{94} +7.42515 q^{95} -1.00000 q^{96} -8.14027 q^{97} -4.51179 q^{98} -0.860848 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * 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
$5$	−2.74714	−1.22856	−0.614278	−	0.789089i	$-0.710553\pi$
$5$	−2.74714	−1.22856	−0.614278	+	0.789089i	$0.710553\pi$
$6$	−1.00000	−0.408248
$7$	3.39290	1.28240	0.641198	−	0.767375i	$-0.278438\pi$
$7$	3.39290	1.28240	0.641198	+	0.767375i	$0.278438\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.74714	0.868721
$11$	−0.860848	−0.259556	−0.129778	−	0.991543i	$-0.541426\pi$
$11$	−0.860848	−0.259556	−0.129778	+	0.991543i	$0.541426\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.39290	−0.906791
$15$	−2.74714	−0.709308
$16$	1.00000	0.250000
$17$	−4.14293	−1.00481	−0.502404	−	0.864633i	$-0.667551\pi$
$17$	−4.14293	−1.00481	−0.502404	+	0.864633i	$0.667551\pi$
$18$	−1.00000	−0.235702
$19$	−2.70287	−0.620081	−0.310040	−	0.950723i	$-0.600343\pi$
$19$	−2.70287	−0.620081	−0.310040	+	0.950723i	$0.600343\pi$
$20$	−2.74714	−0.614278
$21$	3.39290	0.740392
$22$	0.860848	0.183533
$23$	0.937524	0.195487	0.0977436	−	0.995212i	$-0.468837\pi$
$23$	0.937524	0.195487	0.0977436	+	0.995212i	$0.468837\pi$
$24$	−1.00000	−0.204124
$25$	2.54676	0.509352
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	3.39290	0.641198
$29$	0.521486	0.0968375	0.0484187	−	0.998827i	$-0.484582\pi$
$29$	0.521486	0.0968375	0.0484187	+	0.998827i	$0.484582\pi$
$30$	2.74714	0.501556
$31$	7.56769	1.35920	0.679598	−	0.733584i	$-0.262154\pi$
$31$	7.56769	1.35920	0.679598	+	0.733584i	$0.262154\pi$
$32$	−1.00000	−0.176777
$33$	−0.860848	−0.149854
$34$	4.14293	0.710507
$35$	−9.32077	−1.57550
$36$	1.00000	0.166667
$37$	−0.373450	−0.0613947	−0.0306974	−	0.999529i	$-0.509773\pi$
$37$	−0.373450	−0.0613947	−0.0306974	+	0.999529i	$0.509773\pi$
$38$	2.70287	0.438463
$39$	1.00000	0.160128
$40$	2.74714	0.434360
$41$	−5.16599	−0.806792	−0.403396	−	0.915025i	$-0.632170\pi$
$41$	−5.16599	−0.806792	−0.403396	+	0.915025i	$0.632170\pi$
$42$	−3.39290	−0.523536
$43$	−9.15820	−1.39661	−0.698306	−	0.715799i	$-0.746063\pi$
$43$	−9.15820	−1.39661	−0.698306	+	0.715799i	$0.746063\pi$
$44$	−0.860848	−0.129778
$45$	−2.74714	−0.409519
$46$	−0.937524	−0.138230
$47$	3.73475	0.544770	0.272385	−	0.962188i	$-0.412188\pi$
$47$	3.73475	0.544770	0.272385	+	0.962188i	$0.412188\pi$
$48$	1.00000	0.144338
$49$	4.51179	0.644541
$50$	−2.54676	−0.360166
$51$	−4.14293	−0.580126
$52$	1.00000	0.138675
$53$	6.82448	0.937414	0.468707	−	0.883354i	$-0.344720\pi$
$53$	6.82448	0.937414	0.468707	+	0.883354i	$0.344720\pi$
$54$	−1.00000	−0.136083
$55$	2.36487	0.318879
$56$	−3.39290	−0.453396
$57$	−2.70287	−0.358004
$58$	−0.521486	−0.0684744
$59$	−10.7140	−1.39484	−0.697422	−	0.716661i	$-0.745670\pi$
$59$	−10.7140	−1.39484	−0.697422	+	0.716661i	$0.745670\pi$
$60$	−2.74714	−0.354654
$61$	4.99993	0.640175	0.320087	−	0.947388i	$-0.396288\pi$
$61$	4.99993	0.640175	0.320087	+	0.947388i	$0.396288\pi$
$62$	−7.56769	−0.961097
$63$	3.39290	0.427465
$64$	1.00000	0.125000
$65$	−2.74714	−0.340740
$66$	0.860848	0.105963
$67$	7.00412	0.855689	0.427845	−	0.903852i	$-0.359273\pi$
$67$	7.00412	0.855689	0.427845	+	0.903852i	$0.359273\pi$
$68$	−4.14293	−0.502404
$69$	0.937524	0.112865
$70$	9.32077	1.11404
$71$	−9.02958	−1.07161	−0.535807	−	0.844341i	$-0.679992\pi$
$71$	−9.02958	−1.07161	−0.535807	+	0.844341i	$0.679992\pi$
$72$	−1.00000	−0.117851
$73$	−8.63216	−1.01032	−0.505159	−	0.863026i	$-0.668566\pi$
$73$	−8.63216	−1.01032	−0.505159	+	0.863026i	$0.668566\pi$
$74$	0.373450	0.0434126
$75$	2.54676	0.294075
$76$	−2.70287	−0.310040
$77$	−2.92077	−0.332853
$78$	−1.00000	−0.113228
$79$	1.21230	0.136394	0.0681970	−	0.997672i	$-0.478275\pi$
$79$	1.21230	0.136394	0.0681970	+	0.997672i	$0.478275\pi$
$80$	−2.74714	−0.307139
$81$	1.00000	0.111111
$82$	5.16599	0.570488
$83$	15.5907	1.71130	0.855649	−	0.517556i	$-0.173158\pi$
$83$	15.5907	1.71130	0.855649	+	0.517556i	$0.173158\pi$
$84$	3.39290	0.370196
$85$	11.3812	1.23446
$86$	9.15820	0.987554
$87$	0.521486	0.0559092
$88$	0.860848	0.0917667
$89$	−4.60576	−0.488209	−0.244105	−	0.969749i	$-0.578494\pi$
$89$	−4.60576	−0.488209	−0.244105	+	0.969749i	$0.578494\pi$
$90$	2.74714	0.289574
$91$	3.39290	0.355673
$92$	0.937524	0.0977436
$93$	7.56769	0.784732
$94$	−3.73475	−0.385210
$95$	7.42515	0.761805
$96$	−1.00000	−0.102062
$97$	−8.14027	−0.826519	−0.413260	−	0.910613i	$-0.635610\pi$
$97$	−8.14027	−0.826519	−0.413260	+	0.910613i	$0.635610\pi$
$98$	−4.51179	−0.455759
$99$	−0.860848	−0.0865185
$100$	2.54676	0.254676
$101$	−7.07845	−0.704332	−0.352166	−	0.935938i	$-0.614555\pi$
$101$	−7.07845	−0.704332	−0.352166	+	0.935938i	$0.614555\pi$
$102$	4.14293	0.410211
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−9.32077	−0.909614
$106$	−6.82448	−0.662852
$107$	−17.1864	−1.66148	−0.830738	−	0.556664i	$-0.812081\pi$
$107$	−17.1864	−1.66148	−0.830738	+	0.556664i	$0.812081\pi$
$108$	1.00000	0.0962250
$109$	4.88430	0.467831	0.233916	−	0.972257i	$-0.424846\pi$
$109$	4.88430	0.467831	0.233916	+	0.972257i	$0.424846\pi$
$110$	−2.36487	−0.225481
$111$	−0.373450	−0.0354463
$112$	3.39290	0.320599
$113$	−14.8209	−1.39423	−0.697115	−	0.716959i	$-0.745533\pi$
$113$	−14.8209	−1.39423	−0.697115	+	0.716959i	$0.745533\pi$
$114$	2.70287	0.253147
$115$	−2.57551	−0.240167
$116$	0.521486	0.0484187
$117$	1.00000	0.0924500
$118$	10.7140	0.986303
$119$	−14.0566	−1.28856
$120$	2.74714	0.250778
$121$	−10.2589	−0.932631
$122$	−4.99993	−0.452672
$123$	−5.16599	−0.465802
$124$	7.56769	0.679598
$125$	6.73938	0.602789
$126$	−3.39290	−0.302264
$127$	−11.1401	−0.988524	−0.494262	−	0.869313i	$-0.664562\pi$
$127$	−11.1401	−0.988524	−0.494262	+	0.869313i	$0.664562\pi$
$128$	−1.00000	−0.0883883
$129$	−9.15820	−0.806335
$130$	2.74714	0.240940
$131$	3.97766	0.347529	0.173765	−	0.984787i	$-0.444407\pi$
$131$	3.97766	0.347529	0.173765	+	0.984787i	$0.444407\pi$
$132$	−0.860848	−0.0749272
$133$	−9.17057	−0.795190
$134$	−7.00412	−0.605064
$135$	−2.74714	−0.236436
$136$	4.14293	0.355253
$137$	−3.45070	−0.294813	−0.147406	−	0.989076i	$-0.547093\pi$
$137$	−3.45070	−0.294813	−0.147406	+	0.989076i	$0.547093\pi$
$138$	−0.937524	−0.0798073
$139$	1.88476	0.159863	0.0799317	−	0.996800i	$-0.474530\pi$
$139$	1.88476	0.159863	0.0799317	+	0.996800i	$0.474530\pi$
$140$	−9.32077	−0.787749
$141$	3.73475	0.314523
$142$	9.02958	0.757745
$143$	−0.860848	−0.0719878
$144$	1.00000	0.0833333
$145$	−1.43259	−0.118970
$146$	8.63216	0.714402
$147$	4.51179	0.372126
$148$	−0.373450	−0.0306974
$149$	3.18177	0.260661	0.130330	−	0.991471i	$-0.458396\pi$
$149$	3.18177	0.260661	0.130330	+	0.991471i	$0.458396\pi$
$150$	−2.54676	−0.207942
$151$	21.2767	1.73147	0.865736	−	0.500501i	$-0.166851\pi$
$151$	21.2767	1.73147	0.865736	+	0.500501i	$0.166851\pi$
$152$	2.70287	0.219232
$153$	−4.14293	−0.334936
$154$	2.92077	0.235363
$155$	−20.7895	−1.66985
$156$	1.00000	0.0800641
$157$	8.89329	0.709762	0.354881	−	0.934912i	$-0.384521\pi$
$157$	8.89329	0.709762	0.354881	+	0.934912i	$0.384521\pi$
$158$	−1.21230	−0.0964452
$159$	6.82448	0.541216
$160$	2.74714	0.217180
$161$	3.18093	0.250692
$162$	−1.00000	−0.0785674
$163$	−18.6770	−1.46289	−0.731447	−	0.681898i	$-0.761155\pi$
$163$	−18.6770	−1.46289	−0.731447	+	0.681898i	$0.761155\pi$
$164$	−5.16599	−0.403396
$165$	2.36487	0.184105
$166$	−15.5907	−1.21007
$167$	−12.3917	−0.958896	−0.479448	−	0.877570i	$-0.659163\pi$
$167$	−12.3917	−0.958896	−0.479448	+	0.877570i	$0.659163\pi$
$168$	−3.39290	−0.261768
$169$	1.00000	0.0769231
$170$	−11.3812	−0.872898
$171$	−2.70287	−0.206694
$172$	−9.15820	−0.698306
$173$	−10.7413	−0.816644	−0.408322	−	0.912838i	$-0.633886\pi$
$173$	−10.7413	−0.816644	−0.408322	+	0.912838i	$0.633886\pi$
$174$	−0.521486	−0.0395337
$175$	8.64091	0.653191
$176$	−0.860848	−0.0648889
$177$	−10.7140	−0.805313
$178$	4.60576	0.345216
$179$	−5.05121	−0.377545	−0.188772	−	0.982021i	$-0.560451\pi$
$179$	−5.05121	−0.377545	−0.188772	+	0.982021i	$0.560451\pi$
$180$	−2.74714	−0.204759
$181$	−17.9386	−1.33337	−0.666684	−	0.745341i	$-0.732287\pi$
$181$	−17.9386	−1.33337	−0.666684	+	0.745341i	$0.732287\pi$
$182$	−3.39290	−0.251499
$183$	4.99993	0.369605
$184$	−0.937524	−0.0691152
$185$	1.02592	0.0754269
$186$	−7.56769	−0.554890
$187$	3.56644	0.260804
$188$	3.73475	0.272385
$189$	3.39290	0.246797
$190$	−7.42515	−0.538677
$191$	16.2968	1.17919	0.589597	−	0.807697i	$-0.299286\pi$
$191$	16.2968	1.17919	0.589597	+	0.807697i	$0.299286\pi$
$192$	1.00000	0.0721688
$193$	25.9923	1.87097	0.935485	−	0.353366i	$-0.114963\pi$
$193$	25.9923	1.87097	0.935485	+	0.353366i	$0.114963\pi$
$194$	8.14027	0.584437
$195$	−2.74714	−0.196727
$196$	4.51179	0.322270
$197$	26.0591	1.85663	0.928315	−	0.371794i	$-0.121257\pi$
$197$	26.0591	1.85663	0.928315	+	0.371794i	$0.121257\pi$
$198$	0.860848	0.0611778
$199$	−12.4115	−0.879829	−0.439915	−	0.898040i	$-0.644991\pi$
$199$	−12.4115	−0.879829	−0.439915	+	0.898040i	$0.644991\pi$
$200$	−2.54676	−0.180083
$201$	7.00412	0.494032
$202$	7.07845	0.498038
$203$	1.76935	0.124184
$204$	−4.14293	−0.290063
$205$	14.1917	0.991190
$206$	1.00000	0.0696733
$207$	0.937524	0.0651624
$208$	1.00000	0.0693375
$209$	2.32676	0.160945
$210$	9.32077	0.643194
$211$	−15.7658	−1.08537	−0.542683	−	0.839938i	$-0.682591\pi$
$211$	−15.7658	−1.08537	−0.542683	+	0.839938i	$0.682591\pi$
$212$	6.82448	0.468707
$213$	−9.02958	−0.618696
$214$	17.1864	1.17484
$215$	25.1588	1.71582
$216$	−1.00000	−0.0680414
$217$	25.6764	1.74303
$218$	−4.88430	−0.330807
$219$	−8.63216	−0.583307
$220$	2.36487	0.159439
$221$	−4.14293	−0.278684
$222$	0.373450	0.0250643
$223$	26.3575	1.76503	0.882514	−	0.470286i	$-0.155849\pi$
$223$	26.3575	1.76503	0.882514	+	0.470286i	$0.155849\pi$
$224$	−3.39290	−0.226698
$225$	2.54676	0.169784
$226$	14.8209	0.985870
$227$	−6.92514	−0.459638	−0.229819	−	0.973233i	$-0.573813\pi$
$227$	−6.92514	−0.459638	−0.229819	+	0.973233i	$0.573813\pi$
$228$	−2.70287	−0.179002
$229$	24.9537	1.64899	0.824495	−	0.565870i	$-0.191459\pi$
$229$	24.9537	1.64899	0.824495	+	0.565870i	$0.191459\pi$
$230$	2.57551	0.169824
$231$	−2.92077	−0.192173
$232$	−0.521486	−0.0342372
$233$	−8.28474	−0.542751	−0.271376	−	0.962474i	$-0.587479\pi$
$233$	−8.28474	−0.542751	−0.271376	+	0.962474i	$0.587479\pi$
$234$	−1.00000	−0.0653720
$235$	−10.2599	−0.669281
$236$	−10.7140	−0.697422
$237$	1.21230	0.0787471
$238$	14.0566	0.911152
$239$	−26.6667	−1.72493	−0.862464	−	0.506118i	$-0.831080\pi$
$239$	−26.6667	−1.72493	−0.862464	+	0.506118i	$0.831080\pi$
$240$	−2.74714	−0.177327
$241$	−27.2863	−1.75767	−0.878834	−	0.477128i	$-0.841678\pi$
$241$	−27.2863	−1.75767	−0.878834	+	0.477128i	$0.841678\pi$
$242$	10.2589	0.659470
$243$	1.00000	0.0641500
$244$	4.99993	0.320087
$245$	−12.3945	−0.791855
$246$	5.16599	0.329372
$247$	−2.70287	−0.171980
$248$	−7.56769	−0.480549
$249$	15.5907	0.988019
$250$	−6.73938	−0.426236
$251$	19.0037	1.19950	0.599750	−	0.800187i	$-0.295267\pi$
$251$	19.0037	1.19950	0.599750	+	0.800187i	$0.295267\pi$
$252$	3.39290	0.213733
$253$	−0.807066	−0.0507398
$254$	11.1401	0.698992
$255$	11.3812	0.712718
$256$	1.00000	0.0625000
$257$	−18.7819	−1.17158	−0.585791	−	0.810462i	$-0.699216\pi$
$257$	−18.7819	−1.17158	−0.585791	+	0.810462i	$0.699216\pi$
$258$	9.15820	0.570165
$259$	−1.26708	−0.0787324
$260$	−2.74714	−0.170370
$261$	0.521486	0.0322792
$262$	−3.97766	−0.245740
$263$	7.84505	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$263$	7.84505	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$264$	0.860848	0.0529816
$265$	−18.7478	−1.15167
$266$	9.17057	0.562284
$267$	−4.60576	−0.281868
$268$	7.00412	0.427845
$269$	−7.66512	−0.467351	−0.233675	−	0.972315i	$-0.575075\pi$
$269$	−7.66512	−0.467351	−0.233675	+	0.972315i	$0.575075\pi$
$270$	2.74714	0.167185
$271$	0.226570	0.0137631	0.00688157	−	0.999976i	$-0.497810\pi$
$271$	0.226570	0.0137631	0.00688157	+	0.999976i	$0.497810\pi$
$272$	−4.14293	−0.251202
$273$	3.39290	0.205348
$274$	3.45070	0.208464
$275$	−2.19237	−0.132205
$276$	0.937524	0.0564323
$277$	−31.6272	−1.90029	−0.950147	−	0.311802i	$-0.899067\pi$
$277$	−31.6272	−1.90029	−0.950147	+	0.311802i	$0.899067\pi$
$278$	−1.88476	−0.113040
$279$	7.56769	0.453066
$280$	9.32077	0.557022
$281$	−5.79701	−0.345820	−0.172910	−	0.984938i	$-0.555317\pi$
$281$	−5.79701	−0.345820	−0.172910	+	0.984938i	$0.555317\pi$
$282$	−3.73475	−0.222401
$283$	31.1609	1.85232	0.926161	−	0.377129i	$-0.123089\pi$
$283$	31.1609	1.85232	0.926161	+	0.377129i	$0.123089\pi$
$284$	−9.02958	−0.535807
$285$	7.42515	0.439828
$286$	0.860848	0.0509030
$287$	−17.5277	−1.03463
$288$	−1.00000	−0.0589256
$289$	0.163881	0.00964004
$290$	1.43259	0.0841248
$291$	−8.14027	−0.477191
$292$	−8.63216	−0.505159
$293$	−29.9482	−1.74959	−0.874795	−	0.484493i	$-0.839004\pi$
$293$	−29.9482	−1.74959	−0.874795	+	0.484493i	$0.839004\pi$
$294$	−4.51179	−0.263133
$295$	29.4328	1.71364
$296$	0.373450	0.0217063
$297$	−0.860848	−0.0499515
$298$	−3.18177	−0.184315
$299$	0.937524	0.0542184
$300$	2.54676	0.147037
$301$	−31.0729	−1.79101
$302$	−21.2767	−1.22434
$303$	−7.07845	−0.406646
$304$	−2.70287	−0.155020
$305$	−13.7355	−0.786491
$306$	4.14293	0.236836
$307$	−23.7820	−1.35731	−0.678657	−	0.734456i	$-0.737438\pi$
$307$	−23.7820	−1.35731	−0.678657	+	0.734456i	$0.737438\pi$
$308$	−2.92077	−0.166427
$309$	−1.00000	−0.0568880
$310$	20.7895	1.18076
$311$	29.4598	1.67051	0.835256	−	0.549861i	$-0.185319\pi$
$311$	29.4598	1.67051	0.835256	+	0.549861i	$0.185319\pi$
$312$	−1.00000	−0.0566139
$313$	−5.15810	−0.291553	−0.145776	−	0.989318i	$-0.546568\pi$
$313$	−5.15810	−0.291553	−0.145776	+	0.989318i	$0.546568\pi$
$314$	−8.89329	−0.501877
$315$	−9.32077	−0.525166
$316$	1.21230	0.0681970
$317$	16.2036	0.910085	0.455043	−	0.890470i	$-0.349624\pi$
$317$	16.2036	0.910085	0.455043	+	0.890470i	$0.349624\pi$
$318$	−6.82448	−0.382698
$319$	−0.448920	−0.0251347
$320$	−2.74714	−0.153570
$321$	−17.1864	−0.959254
$322$	−3.18093	−0.177266
$323$	11.1978	0.623063
$324$	1.00000	0.0555556
$325$	2.54676	0.141269
$326$	18.6770	1.03442
$327$	4.88430	0.270103
$328$	5.16599	0.285244
$329$	12.6717	0.698611
$330$	−2.36487	−0.130182
$331$	32.8982	1.80825	0.904125	−	0.427269i	$-0.140524\pi$
$331$	32.8982	1.80825	0.904125	+	0.427269i	$0.140524\pi$
$332$	15.5907	0.855649
$333$	−0.373450	−0.0204649
$334$	12.3917	0.678042
$335$	−19.2413	−1.05126
$336$	3.39290	0.185098
$337$	−11.5902	−0.631357	−0.315678	−	0.948866i	$-0.602232\pi$
$337$	−11.5902	−0.631357	−0.315678	+	0.948866i	$0.602232\pi$
$338$	−1.00000	−0.0543928
$339$	−14.8209	−0.804959
$340$	11.3812	0.617232
$341$	−6.51463	−0.352787
$342$	2.70287	0.146154
$343$	−8.44227	−0.455840
$344$	9.15820	0.493777
$345$	−2.57551	−0.138661
$346$	10.7413	0.577455
$347$	14.5311	0.780072	0.390036	−	0.920800i	$-0.372463\pi$
$347$	14.5311	0.780072	0.390036	+	0.920800i	$0.372463\pi$
$348$	0.521486	0.0279546
$349$	2.57425	0.137796	0.0688981	−	0.997624i	$-0.478052\pi$
$349$	2.57425	0.137796	0.0688981	+	0.997624i	$0.478052\pi$
$350$	−8.64091	−0.461876
$351$	1.00000	0.0533761
$352$	0.860848	0.0458834
$353$	4.27233	0.227393	0.113697	−	0.993516i	$-0.463731\pi$
$353$	4.27233	0.227393	0.113697	+	0.993516i	$0.463731\pi$
$354$	10.7140	0.569442
$355$	24.8055	1.31654
$356$	−4.60576	−0.244105
$357$	−14.0566	−0.743952
$358$	5.05121	0.266965
$359$	−12.0179	−0.634279	−0.317139	−	0.948379i	$-0.602722\pi$
$359$	−12.0179	−0.634279	−0.317139	+	0.948379i	$0.602722\pi$
$360$	2.74714	0.144787
$361$	−11.6945	−0.615500
$362$	17.9386	0.942833
$363$	−10.2589	−0.538455
$364$	3.39290	0.177836
$365$	23.7137	1.24123
$366$	−4.99993	−0.261350
$367$	3.95835	0.206624	0.103312	−	0.994649i	$-0.467056\pi$
$367$	3.95835	0.206624	0.103312	+	0.994649i	$0.467056\pi$
$368$	0.937524	0.0488718
$369$	−5.16599	−0.268931
$370$	−1.02592	−0.0533349
$371$	23.1548	1.20214
$372$	7.56769	0.392366
$373$	6.22306	0.322218	0.161109	−	0.986937i	$-0.448493\pi$
$373$	6.22306	0.322218	0.161109	+	0.986937i	$0.448493\pi$
$374$	−3.56644	−0.184416
$375$	6.73938	0.348020
$376$	−3.73475	−0.192605
$377$	0.521486	0.0268579
$378$	−3.39290	−0.174512
$379$	−10.0976	−0.518678	−0.259339	−	0.965786i	$-0.583505\pi$
$379$	−10.0976	−0.518678	−0.259339	+	0.965786i	$0.583505\pi$
$380$	7.42515	0.380902
$381$	−11.1401	−0.570725
$382$	−16.2968	−0.833816
$383$	−38.4445	−1.96442	−0.982211	−	0.187779i	$-0.939871\pi$
$383$	−38.4445	−1.96442	−0.982211	+	0.187779i	$0.939871\pi$
$384$	−1.00000	−0.0510310
$385$	8.02377	0.408929
$386$	−25.9923	−1.32298
$387$	−9.15820	−0.465537
$388$	−8.14027	−0.413260
$389$	−23.4430	−1.18861	−0.594304	−	0.804240i	$-0.702573\pi$
$389$	−23.4430	−1.18861	−0.594304	+	0.804240i	$0.702573\pi$
$390$	2.74714	0.139107
$391$	−3.88410	−0.196427
$392$	−4.51179	−0.227880
$393$	3.97766	0.200646
$394$	−26.0591	−1.31284
$395$	−3.33035	−0.167568
$396$	−0.860848	−0.0432593
$397$	−16.4848	−0.827350	−0.413675	−	0.910425i	$-0.635755\pi$
$397$	−16.4848	−0.827350	−0.413675	+	0.910425i	$0.635755\pi$
$398$	12.4115	0.622133
$399$	−9.17057	−0.459103
$400$	2.54676	0.127338
$401$	0.0156239	0.000780218 0	0.000390109	−	1.00000i	$-0.499876\pi$
$401$	0.0156239	0.000780218 0	0.000390109		1.00000i	$0.499876\pi$
$402$	−7.00412	−0.349334
$403$	7.56769	0.376973
$404$	−7.07845	−0.352166
$405$	−2.74714	−0.136506
$406$	−1.76935	−0.0878114
$407$	0.321483	0.0159353
$408$	4.14293	0.205106
$409$	−38.0942	−1.88364	−0.941818	−	0.336124i	$-0.890884\pi$
$409$	−38.0942	−1.88364	−0.941818	+	0.336124i	$0.890884\pi$
$410$	−14.1917	−0.700877
$411$	−3.45070	−0.170210
$412$	−1.00000	−0.0492665
$413$	−36.3515	−1.78874
$414$	−0.937524	−0.0460768
$415$	−42.8297	−2.10243
$416$	−1.00000	−0.0490290
$417$	1.88476	0.0922972
$418$	−2.32676	−0.113806
$419$	31.0704	1.51789	0.758943	−	0.651157i	$-0.225716\pi$
$419$	31.0704	1.51789	0.758943	+	0.651157i	$0.225716\pi$
$420$	−9.32077	−0.454807
$421$	−15.5883	−0.759729	−0.379864	−	0.925042i	$-0.624029\pi$
$421$	−15.5883	−0.759729	−0.379864	+	0.925042i	$0.624029\pi$
$422$	15.7658	0.767470
$423$	3.73475	0.181590
$424$	−6.82448	−0.331426
$425$	−10.5511	−0.511801
$426$	9.02958	0.437484
$427$	16.9643	0.820958
$428$	−17.1864	−0.830738
$429$	−0.860848	−0.0415621
$430$	−25.1588	−1.21327
$431$	−17.0616	−0.821830	−0.410915	−	0.911674i	$-0.634791\pi$
$431$	−17.0616	−0.821830	−0.410915	+	0.911674i	$0.634791\pi$
$432$	1.00000	0.0481125
$433$	5.18891	0.249363	0.124682	−	0.992197i	$-0.460209\pi$
$433$	5.18891	0.249363	0.124682	+	0.992197i	$0.460209\pi$
$434$	−25.6764	−1.23251
$435$	−1.43259	−0.0686876
$436$	4.88430	0.233916
$437$	−2.53401	−0.121218
$438$	8.63216	0.412460
$439$	−41.6551	−1.98809	−0.994045	−	0.108975i	$-0.965243\pi$
$439$	−41.6551	−1.98809	−0.994045	+	0.108975i	$0.965243\pi$
$440$	−2.36487	−0.112741
$441$	4.51179	0.214847
$442$	4.14293	0.197059
$443$	−23.0790	−1.09652	−0.548258	−	0.836309i	$-0.684709\pi$
$443$	−23.0790	−1.09652	−0.548258	+	0.836309i	$0.684709\pi$
$444$	−0.373450	−0.0177231
$445$	12.6526	0.599793
$446$	−26.3575	−1.24806
$447$	3.18177	0.150492
$448$	3.39290	0.160300
$449$	7.77027	0.366702	0.183351	−	0.983048i	$-0.441306\pi$
$449$	7.77027	0.366702	0.183351	+	0.983048i	$0.441306\pi$
$450$	−2.54676	−0.120055
$451$	4.44713	0.209407
$452$	−14.8209	−0.697115
$453$	21.2767	0.999665
$454$	6.92514	0.325013
$455$	−9.32077	−0.436964
$456$	2.70287	0.126574
$457$	−24.1142	−1.12801	−0.564007	−	0.825770i	$-0.690741\pi$
$457$	−24.1142	−1.12801	−0.564007	+	0.825770i	$0.690741\pi$
$458$	−24.9537	−1.16601
$459$	−4.14293	−0.193375
$460$	−2.57551	−0.120084
$461$	−4.96952	−0.231453	−0.115727	−	0.993281i	$-0.536920\pi$
$461$	−4.96952	−0.231453	−0.115727	+	0.993281i	$0.536920\pi$
$462$	2.92077	0.135887
$463$	−0.774424	−0.0359905	−0.0179953	−	0.999838i	$-0.505728\pi$
$463$	−0.774424	−0.0359905	−0.0179953	+	0.999838i	$0.505728\pi$
$464$	0.521486	0.0242094
$465$	−20.7895	−0.964089
$466$	8.28474	0.383783
$467$	−18.2679	−0.845336	−0.422668	−	0.906285i	$-0.638906\pi$
$467$	−18.2679	−0.845336	−0.422668	+	0.906285i	$0.638906\pi$
$468$	1.00000	0.0462250
$469$	23.7643	1.09733
$470$	10.2599	0.473253
$471$	8.89329	0.409781
$472$	10.7140	0.493152
$473$	7.88382	0.362498
$474$	−1.21230	−0.0556826
$475$	−6.88357	−0.315840
$476$	−14.0566	−0.644281
$477$	6.82448	0.312471
$478$	26.6667	1.21971
$479$	5.82877	0.266323	0.133162	−	0.991094i	$-0.457487\pi$
$479$	5.82877	0.266323	0.133162	+	0.991094i	$0.457487\pi$
$480$	2.74714	0.125389
$481$	−0.373450	−0.0170278
$482$	27.2863	1.24286
$483$	3.18093	0.144737
$484$	−10.2589	−0.466315
$485$	22.3624	1.01543
$486$	−1.00000	−0.0453609
$487$	−7.63754	−0.346090	−0.173045	−	0.984914i	$-0.555361\pi$
$487$	−7.63754	−0.346090	−0.173045	+	0.984914i	$0.555361\pi$
$488$	−4.99993	−0.226336
$489$	−18.6770	−0.844602
$490$	12.3945	0.559926
$491$	10.2639	0.463203	0.231602	−	0.972811i	$-0.425603\pi$
$491$	10.2639	0.463203	0.231602	+	0.972811i	$0.425603\pi$
$492$	−5.16599	−0.232901
$493$	−2.16048	−0.0973031
$494$	2.70287	0.121608
$495$	2.36487	0.106293
$496$	7.56769	0.339799
$497$	−30.6365	−1.37423
$498$	−15.5907	−0.698635
$499$	18.1451	0.812288	0.406144	−	0.913809i	$-0.366873\pi$
$499$	18.1451	0.812288	0.406144	+	0.913809i	$0.366873\pi$
$500$	6.73938	0.301394
$501$	−12.3917	−0.553619
$502$	−19.0037	−0.848175
$503$	−3.82760	−0.170664	−0.0853321	−	0.996353i	$-0.527195\pi$
$503$	−3.82760	−0.170664	−0.0853321	+	0.996353i	$0.527195\pi$
$504$	−3.39290	−0.151132
$505$	19.4455	0.865312
$506$	0.807066	0.0358785
$507$	1.00000	0.0444116
$508$	−11.1401	−0.494262
$509$	23.7607	1.05318	0.526588	−	0.850121i	$-0.323471\pi$
$509$	23.7607	1.05318	0.526588	+	0.850121i	$0.323471\pi$
$510$	−11.3812	−0.503968
$511$	−29.2881	−1.29563
$512$	−1.00000	−0.0441942
$513$	−2.70287	−0.119335
$514$	18.7819	0.828434
$515$	2.74714	0.121053
$516$	−9.15820	−0.403167
$517$	−3.21506	−0.141398
$518$	1.26708	0.0556722
$519$	−10.7413	−0.471490
$520$	2.74714	0.120470
$521$	−17.9485	−0.786338	−0.393169	−	0.919466i	$-0.628621\pi$
$521$	−17.9485	−0.786338	−0.393169	+	0.919466i	$0.628621\pi$
$522$	−0.521486	−0.0228248
$523$	31.3989	1.37298	0.686490	−	0.727139i	$-0.259151\pi$
$523$	31.3989	1.37298	0.686490	+	0.727139i	$0.259151\pi$
$524$	3.97766	0.173765
$525$	8.64091	0.377120
$526$	−7.84505	−0.342060
$527$	−31.3524	−1.36573
$528$	−0.860848	−0.0374636
$529$	−22.1210	−0.961785
$530$	18.7478	0.814351
$531$	−10.7140	−0.464948
$532$	−9.17057	−0.397595
$533$	−5.16599	−0.223764
$534$	4.60576	0.199311
$535$	47.2135	2.04122
$536$	−7.00412	−0.302532
$537$	−5.05121	−0.217976
$538$	7.66512	0.330467
$539$	−3.88396	−0.167294
$540$	−2.74714	−0.118218
$541$	−26.4259	−1.13614	−0.568070	−	0.822980i	$-0.692310\pi$
$541$	−26.4259	−1.13614	−0.568070	+	0.822980i	$0.692310\pi$
$542$	−0.226570	−0.00973201
$543$	−17.9386	−0.769820
$544$	4.14293	0.177627
$545$	−13.4178	−0.574757
$546$	−3.39290	−0.145203
$547$	−1.01814	−0.0435325	−0.0217663	−	0.999763i	$-0.506929\pi$
$547$	−1.01814	−0.0435325	−0.0217663	+	0.999763i	$0.506929\pi$
$548$	−3.45070	−0.147406
$549$	4.99993	0.213392
$550$	2.19237	0.0934832
$551$	−1.40951	−0.0600471
$552$	−0.937524	−0.0399037
$553$	4.11320	0.174911
$554$	31.6272	1.34371
$555$	1.02592	0.0435478
$556$	1.88476	0.0799317
$557$	18.7478	0.794371	0.397185	−	0.917738i	$-0.369987\pi$
$557$	18.7478	0.794371	0.397185	+	0.917738i	$0.369987\pi$
$558$	−7.56769	−0.320366
$559$	−9.15820	−0.387351
$560$	−9.32077	−0.393874
$561$	3.56644	0.150575
$562$	5.79701	0.244532
$563$	−29.7944	−1.25568	−0.627841	−	0.778341i	$-0.716061\pi$
$563$	−29.7944	−1.25568	−0.627841	+	0.778341i	$0.716061\pi$
$564$	3.73475	0.157261
$565$	40.7150	1.71289
$566$	−31.1609	−1.30979
$567$	3.39290	0.142488
$568$	9.02958	0.378873
$569$	−29.3063	−1.22858	−0.614292	−	0.789079i	$-0.710558\pi$
$569$	−29.3063	−1.22858	−0.614292	+	0.789079i	$0.710558\pi$
$570$	−7.42515	−0.311006
$571$	−34.2507	−1.43335	−0.716673	−	0.697410i	$-0.754336\pi$
$571$	−34.2507	−1.43335	−0.716673	+	0.697410i	$0.754336\pi$
$572$	−0.860848	−0.0359939
$573$	16.2968	0.680808
$574$	17.5277	0.731592
$575$	2.38765	0.0995719
$576$	1.00000	0.0416667
$577$	20.9906	0.873848	0.436924	−	0.899498i	$-0.356068\pi$
$577$	20.9906	0.873848	0.436924	+	0.899498i	$0.356068\pi$
$578$	−0.163881	−0.00681654
$579$	25.9923	1.08021
$580$	−1.43259	−0.0594852
$581$	52.8976	2.19456
$582$	8.14027	0.337425
$583$	−5.87484	−0.243311
$584$	8.63216	0.357201
$585$	−2.74714	−0.113580
$586$	29.9482	1.23715
$587$	4.97107	0.205178	0.102589	−	0.994724i	$-0.467287\pi$
$587$	4.97107	0.205178	0.102589	+	0.994724i	$0.467287\pi$
$588$	4.51179	0.186063
$589$	−20.4545	−0.842812
$590$	−29.4328	−1.21173
$591$	26.0591	1.07193
$592$	−0.373450	−0.0153487
$593$	−18.4742	−0.758643	−0.379321	−	0.925265i	$-0.623842\pi$
$593$	−18.4742	−0.758643	−0.379321	+	0.925265i	$0.623842\pi$
$594$	0.860848	0.0353210
$595$	38.6153	1.58307
$596$	3.18177	0.130330
$597$	−12.4115	−0.507970
$598$	−0.937524	−0.0383382
$599$	−10.0889	−0.412222	−0.206111	−	0.978529i	$-0.566081\pi$
$599$	−10.0889	−0.412222	−0.206111	+	0.978529i	$0.566081\pi$
$600$	−2.54676	−0.103971
$601$	−17.7439	−0.723788	−0.361894	−	0.932219i	$-0.617870\pi$
$601$	−17.7439	−0.723788	−0.361894	+	0.932219i	$0.617870\pi$
$602$	31.0729	1.26644
$603$	7.00412	0.285230
$604$	21.2767	0.865736
$605$	28.1827	1.14579
$606$	7.07845	0.287542
$607$	10.5752	0.429233	0.214616	−	0.976698i	$-0.431150\pi$
$607$	10.5752	0.429233	0.214616	+	0.976698i	$0.431150\pi$
$608$	2.70287	0.109616
$609$	1.76935	0.0716977
$610$	13.7355	0.556133
$611$	3.73475	0.151092
$612$	−4.14293	−0.167468
$613$	−3.09381	−0.124958	−0.0624789	−	0.998046i	$-0.519901\pi$
$613$	−3.09381	−0.124958	−0.0624789	+	0.998046i	$0.519901\pi$
$614$	23.7820	0.959765
$615$	14.1917	0.572264
$616$	2.92077	0.117681
$617$	0.671803	0.0270458	0.0135229	−	0.999909i	$-0.495695\pi$
$617$	0.671803	0.0270458	0.0135229	+	0.999909i	$0.495695\pi$
$618$	1.00000	0.0402259
$619$	−0.425707	−0.0171106	−0.00855531	−	0.999963i	$-0.502723\pi$
$619$	−0.425707	−0.0171106	−0.00855531	+	0.999963i	$0.502723\pi$
$620$	−20.7895	−0.834925
$621$	0.937524	0.0376215
$622$	−29.4598	−1.18123
$623$	−15.6269	−0.626078
$624$	1.00000	0.0400320
$625$	−31.2478	−1.24991
$626$	5.15810	0.206159
$627$	2.32676	0.0929219
$628$	8.89329	0.354881
$629$	1.54718	0.0616900
$630$	9.32077	0.371348
$631$	36.4937	1.45279	0.726395	−	0.687278i	$-0.241194\pi$
$631$	36.4937	1.45279	0.726395	+	0.687278i	$0.241194\pi$
$632$	−1.21230	−0.0482226
$633$	−15.7658	−0.626636
$634$	−16.2036	−0.643528
$635$	30.6034	1.21446
$636$	6.82448	0.270608
$637$	4.51179	0.178763
$638$	0.448920	0.0177729
$639$	−9.02958	−0.357204
$640$	2.74714	0.108590
$641$	−12.5755	−0.496702	−0.248351	−	0.968670i	$-0.579889\pi$
$641$	−12.5755	−0.496702	−0.248351	+	0.968670i	$0.579889\pi$
$642$	17.1864	0.678295
$643$	−7.38509	−0.291239	−0.145620	−	0.989341i	$-0.546518\pi$
$643$	−7.38509	−0.291239	−0.145620	+	0.989341i	$0.546518\pi$
$644$	3.18093	0.125346
$645$	25.1588	0.990628
$646$	−11.1978	−0.440572
$647$	42.7103	1.67911	0.839557	−	0.543271i	$-0.182814\pi$
$647$	42.7103	1.67911	0.839557	+	0.543271i	$0.182814\pi$
$648$	−1.00000	−0.0392837
$649$	9.22313	0.362039
$650$	−2.54676	−0.0998922
$651$	25.6764	1.00634
$652$	−18.6770	−0.731447
$653$	−2.17138	−0.0849726	−0.0424863	−	0.999097i	$-0.513528\pi$
$653$	−2.17138	−0.0849726	−0.0424863	+	0.999097i	$0.513528\pi$
$654$	−4.88430	−0.190991
$655$	−10.9272	−0.426960
$656$	−5.16599	−0.201698
$657$	−8.63216	−0.336773
$658$	−12.6717	−0.493992
$659$	28.6191	1.11484	0.557421	−	0.830230i	$-0.311791\pi$
$659$	28.6191	1.11484	0.557421	+	0.830230i	$0.311791\pi$
$660$	2.36487	0.0920524
$661$	−23.2652	−0.904912	−0.452456	−	0.891787i	$-0.649452\pi$
$661$	−23.2652	−0.904912	−0.452456	+	0.891787i	$0.649452\pi$
$662$	−32.8982	−1.27863
$663$	−4.14293	−0.160898
$664$	−15.5907	−0.605035
$665$	25.1928	0.976936
$666$	0.373450	0.0144709
$667$	0.488905	0.0189305
$668$	−12.3917	−0.479448
$669$	26.3575	1.01904
$670$	19.2413	0.743355
$671$	−4.30418	−0.166161
$672$	−3.39290	−0.130884
$673$	−15.9535	−0.614961	−0.307480	−	0.951554i	$-0.599486\pi$
$673$	−15.9535	−0.614961	−0.307480	+	0.951554i	$0.599486\pi$
$674$	11.5902	0.446437
$675$	2.54676	0.0980249
$676$	1.00000	0.0384615
$677$	39.7105	1.52620	0.763098	−	0.646282i	$-0.223677\pi$
$677$	39.7105	1.52620	0.763098	+	0.646282i	$0.223677\pi$
$678$	14.8209	0.569192
$679$	−27.6191	−1.05993
$680$	−11.3812	−0.436449
$681$	−6.92514	−0.265372
$682$	6.51463	0.249458
$683$	−4.31372	−0.165060	−0.0825299	−	0.996589i	$-0.526300\pi$
$683$	−4.31372	−0.165060	−0.0825299	+	0.996589i	$0.526300\pi$
$684$	−2.70287	−0.103347
$685$	9.47953	0.362194
$686$	8.44227	0.322327
$687$	24.9537	0.952044
$688$	−9.15820	−0.349153
$689$	6.82448	0.259992
$690$	2.57551	0.0980479
$691$	−16.3101	−0.620464	−0.310232	−	0.950661i	$-0.600407\pi$
$691$	−16.3101	−0.620464	−0.310232	+	0.950661i	$0.600407\pi$
$692$	−10.7413	−0.408322
$693$	−2.92077	−0.110951
$694$	−14.5311	−0.551594
$695$	−5.17770	−0.196401
$696$	−0.521486	−0.0197669
$697$	21.4023	0.810672
$698$	−2.57425	−0.0974367
$699$	−8.28474	−0.313358
$700$	8.64091	0.326596
$701$	−46.1044	−1.74134	−0.870670	−	0.491868i	$-0.836314\pi$
$701$	−46.1044	−1.74134	−0.870670	+	0.491868i	$0.836314\pi$
$702$	−1.00000	−0.0377426
$703$	1.00939	0.0380697
$704$	−0.860848	−0.0324444
$705$	−10.2599	−0.386409
$706$	−4.27233	−0.160791
$707$	−24.0165	−0.903233
$708$	−10.7140	−0.402657
$709$	29.9630	1.12529	0.562643	−	0.826700i	$-0.309785\pi$
$709$	29.9630	1.12529	0.562643	+	0.826700i	$0.309785\pi$
$710$	−24.8055	−0.930933
$711$	1.21230	0.0454647
$712$	4.60576	0.172608
$713$	7.09489	0.265706
$714$	14.0566	0.526054
$715$	2.36487	0.0884411
$716$	−5.05121	−0.188772
$717$	−26.6667	−0.995888
$718$	12.0179	0.448503
$719$	28.1540	1.04997	0.524984	−	0.851112i	$-0.324071\pi$
$719$	28.1540	1.04997	0.524984	+	0.851112i	$0.324071\pi$
$720$	−2.74714	−0.102380
$721$	−3.39290	−0.126358
$722$	11.6945	0.435224
$723$	−27.2863	−1.01479
$724$	−17.9386	−0.666684
$725$	1.32810	0.0493244
$726$	10.2589	0.380745
$727$	−41.1786	−1.52723	−0.763614	−	0.645673i	$-0.776577\pi$
$727$	−41.1786	−1.52723	−0.763614	+	0.645673i	$0.776577\pi$
$728$	−3.39290	−0.125749
$729$	1.00000	0.0370370
$730$	−23.7137	−0.877684
$731$	37.9418	1.40333
$732$	4.99993	0.184803
$733$	29.4372	1.08729	0.543645	−	0.839315i	$-0.317044\pi$
$733$	29.4372	1.08729	0.543645	+	0.839315i	$0.317044\pi$
$734$	−3.95835	−0.146106
$735$	−12.3945	−0.457178
$736$	−0.937524	−0.0345576
$737$	−6.02948	−0.222099
$738$	5.16599	0.190163
$739$	39.2150	1.44255	0.721274	−	0.692650i	$-0.243557\pi$
$739$	39.2150	1.44255	0.721274	+	0.692650i	$0.243557\pi$
$740$	1.02592	0.0377135
$741$	−2.70287	−0.0992924
$742$	−23.1548	−0.850039
$743$	39.5114	1.44953	0.724766	−	0.688995i	$-0.241948\pi$
$743$	39.5114	1.44953	0.724766	+	0.688995i	$0.241948\pi$
$744$	−7.56769	−0.277445
$745$	−8.74075	−0.320236
$746$	−6.22306	−0.227843
$747$	15.5907	0.570433
$748$	3.56644	0.130402
$749$	−58.3119	−2.13067
$750$	−6.73938	−0.246087
$751$	2.11914	0.0773284	0.0386642	−	0.999252i	$-0.487690\pi$
$751$	2.11914	0.0773284	0.0386642	+	0.999252i	$0.487690\pi$
$752$	3.73475	0.136192
$753$	19.0037	0.692532
$754$	−0.521486	−0.0189914
$755$	−58.4499	−2.12721
$756$	3.39290	0.123399
$757$	−51.6443	−1.87704	−0.938521	−	0.345221i	$-0.887804\pi$
$757$	−51.6443	−1.87704	−0.938521	+	0.345221i	$0.887804\pi$
$758$	10.0976	0.366761
$759$	−0.807066	−0.0292946
$760$	−7.42515	−0.269339
$761$	14.3106	0.518758	0.259379	−	0.965776i	$-0.416482\pi$
$761$	14.3106	0.518758	0.259379	+	0.965776i	$0.416482\pi$
$762$	11.1401	0.403563
$763$	16.5720	0.599945
$764$	16.2968	0.589597
$765$	11.3812	0.411488
$766$	38.4445	1.38906
$767$	−10.7140	−0.386860
$768$	1.00000	0.0360844
$769$	−23.6577	−0.853118	−0.426559	−	0.904460i	$-0.640274\pi$
$769$	−23.6577	−0.853118	−0.426559	+	0.904460i	$0.640274\pi$
$770$	−8.02377	−0.289156
$771$	−18.7819	−0.676414
$772$	25.9923	0.935485
$773$	42.3109	1.52182	0.760908	−	0.648859i	$-0.224754\pi$
$773$	42.3109	1.52182	0.760908	+	0.648859i	$0.224754\pi$
$774$	9.15820	0.329185
$775$	19.2731	0.692310
$776$	8.14027	0.292219
$777$	−1.26708	−0.0454562
$778$	23.4430	0.840473
$779$	13.9630	0.500277
$780$	−2.74714	−0.0983633
$781$	7.77309	0.278143
$782$	3.88410	0.138895
$783$	0.521486	0.0186364
$784$	4.51179	0.161135
$785$	−24.4311	−0.871983
$786$	−3.97766	−0.141878
$787$	−15.9337	−0.567976	−0.283988	−	0.958828i	$-0.591658\pi$
$787$	−15.9337	−0.567976	−0.283988	+	0.958828i	$0.591658\pi$
$788$	26.0591	0.928315
$789$	7.84505	0.279291
$790$	3.33035	0.118488
$791$	−50.2858	−1.78796
$792$	0.860848	0.0305889
$793$	4.99993	0.177553
$794$	16.4848	0.585025
$795$	−18.7478	−0.664915
$796$	−12.4115	−0.439915
$797$	−25.3438	−0.897724	−0.448862	−	0.893601i	$-0.648171\pi$
$797$	−25.3438	−0.897724	−0.448862	+	0.893601i	$0.648171\pi$
$798$	9.17057	0.324635
$799$	−15.4728	−0.547389
$800$	−2.54676	−0.0900416
$801$	−4.60576	−0.162736
$802$	−0.0156239	−0.000551698 0
$803$	7.43098	0.262234
$804$	7.00412	0.247016
$805$	−8.73844	−0.307990
$806$	−7.56769	−0.266560
$807$	−7.66512	−0.269825
$808$	7.07845	0.249019
$809$	26.9385	0.947108	0.473554	−	0.880765i	$-0.342971\pi$
$809$	26.9385	0.947108	0.473554	+	0.880765i	$0.342971\pi$
$810$	2.74714	0.0965246
$811$	−37.2117	−1.30668	−0.653341	−	0.757064i	$-0.726633\pi$
$811$	−37.2117	−1.30668	−0.653341	+	0.757064i	$0.726633\pi$
$812$	1.76935	0.0620920
$813$	0.226570	0.00794615
$814$	−0.321483	−0.0112680
$815$	51.3082	1.79725
$816$	−4.14293	−0.145032
$817$	24.7534	0.866013
$818$	38.0942	1.33193
$819$	3.39290	0.118558
$820$	14.1917	0.495595
$821$	22.9658	0.801511	0.400756	−	0.916185i	$-0.368748\pi$
$821$	22.9658	0.801511	0.400756	+	0.916185i	$0.368748\pi$
$822$	3.45070	0.120357
$823$	2.33619	0.0814343	0.0407172	−	0.999171i	$-0.487036\pi$
$823$	2.33619	0.0814343	0.0407172	+	0.999171i	$0.487036\pi$
$824$	1.00000	0.0348367
$825$	−2.19237	−0.0763287
$826$	36.3515	1.26483
$827$	11.5453	0.401468	0.200734	−	0.979646i	$-0.435667\pi$
$827$	11.5453	0.401468	0.200734	+	0.979646i	$0.435667\pi$
$828$	0.937524	0.0325812
$829$	−30.4372	−1.05713	−0.528563	−	0.848894i	$-0.677269\pi$
$829$	−30.4372	−1.05713	−0.528563	+	0.848894i	$0.677269\pi$
$830$	42.8297	1.48664
$831$	−31.6272	−1.09714
$832$	1.00000	0.0346688
$833$	−18.6920	−0.647640
$834$	−1.88476	−0.0652639
$835$	34.0416	1.17806
$836$	2.32676	0.0804727
$837$	7.56769	0.261577
$838$	−31.0704	−1.07331
$839$	45.1025	1.55711	0.778556	−	0.627575i	$-0.215952\pi$
$839$	45.1025	1.55711	0.778556	+	0.627575i	$0.215952\pi$
$840$	9.32077	0.321597
$841$	−28.7281	−0.990622
$842$	15.5883	0.537210
$843$	−5.79701	−0.199659
$844$	−15.7658	−0.542683
$845$	−2.74714	−0.0945044
$846$	−3.73475	−0.128403
$847$	−34.8076	−1.19600
$848$	6.82448	0.234354
$849$	31.1609	1.06944
$850$	10.5511	0.361898
$851$	−0.350118	−0.0120019
$852$	−9.02958	−0.309348
$853$	−44.8775	−1.53658	−0.768289	−	0.640103i	$-0.778891\pi$
$853$	−44.8775	−1.53658	−0.768289	+	0.640103i	$0.778891\pi$
$854$	−16.9643	−0.580505
$855$	7.42515	0.253935
$856$	17.1864	0.587420
$857$	−37.1959	−1.27059	−0.635294	−	0.772271i	$-0.719121\pi$
$857$	−37.1959	−1.27059	−0.635294	+	0.772271i	$0.719121\pi$
$858$	0.860848	0.0293889
$859$	−19.7851	−0.675060	−0.337530	−	0.941315i	$-0.609591\pi$
$859$	−19.7851	−0.675060	−0.337530	+	0.941315i	$0.609591\pi$
$860$	25.1588	0.857909
$861$	−17.5277	−0.597342
$862$	17.0616	0.581122
$863$	−14.5853	−0.496490	−0.248245	−	0.968697i	$-0.579854\pi$
$863$	−14.5853	−0.496490	−0.248245	+	0.968697i	$0.579854\pi$
$864$	−1.00000	−0.0340207
$865$	29.5078	1.00329
$866$	−5.18891	−0.176326
$867$	0.163881	0.00556568
$868$	25.6764	0.871514
$869$	−1.04360	−0.0354018
$870$	1.43259	0.0485695
$871$	7.00412	0.237325
$872$	−4.88430	−0.165403
$873$	−8.14027	−0.275506
$874$	2.53401	0.0857140
$875$	22.8661	0.773014
$876$	−8.63216	−0.291654
$877$	34.7902	1.17478	0.587391	−	0.809303i	$-0.300155\pi$
$877$	34.7902	1.17478	0.587391	+	0.809303i	$0.300155\pi$
$878$	41.6551	1.40579
$879$	−29.9482	−1.01013
$880$	2.36487	0.0797197
$881$	25.2054	0.849192	0.424596	−	0.905383i	$-0.360416\pi$
$881$	25.2054	0.849192	0.424596	+	0.905383i	$0.360416\pi$
$882$	−4.51179	−0.151920
$883$	34.9651	1.17667	0.588335	−	0.808617i	$-0.299784\pi$
$883$	34.9651	1.17667	0.588335	+	0.808617i	$0.299784\pi$
$884$	−4.14293	−0.139342
$885$	29.4328	0.989373
$886$	23.0790	0.775354
$887$	−41.5902	−1.39646	−0.698231	−	0.715873i	$-0.746029\pi$
$887$	−41.5902	−1.39646	−0.698231	+	0.715873i	$0.746029\pi$
$888$	0.373450	0.0125321
$889$	−37.7973	−1.26768
$890$	−12.6526	−0.424117
$891$	−0.860848	−0.0288395
$892$	26.3575	0.882514
$893$	−10.0946	−0.337801
$894$	−3.18177	−0.106414
$895$	13.8764	0.463836
$896$	−3.39290	−0.113349
$897$	0.937524	0.0313030
$898$	−7.77027	−0.259297
$899$	3.94644	0.131621
$900$	2.54676	0.0848920
$901$	−28.2733	−0.941922
$902$	−4.44713	−0.148073
$903$	−31.0729	−1.03404
$904$	14.8209	0.492935
$905$	49.2799	1.63812
$906$	−21.2767	−0.706870
$907$	−35.7939	−1.18852	−0.594259	−	0.804274i	$-0.702555\pi$
$907$	−35.7939	−1.18852	−0.594259	+	0.804274i	$0.702555\pi$
$908$	−6.92514	−0.229819
$909$	−7.07845	−0.234777
$910$	9.32077	0.308980
$911$	−49.4881	−1.63961	−0.819806	−	0.572641i	$-0.805919\pi$
$911$	−49.4881	−1.63961	−0.819806	+	0.572641i	$0.805919\pi$
$912$	−2.70287	−0.0895010
$913$	−13.4212	−0.444177
$914$	24.1142	0.797626
$915$	−13.7355	−0.454081
$916$	24.9537	0.824495
$917$	13.4958	0.445670
$918$	4.14293	0.136737
$919$	−4.19055	−0.138233	−0.0691167	−	0.997609i	$-0.522018\pi$
$919$	−4.19055	−0.138233	−0.0691167	+	0.997609i	$0.522018\pi$
$920$	2.57551	0.0849119
$921$	−23.7820	−0.783645
$922$	4.96952	0.163662
$923$	−9.02958	−0.297212
$924$	−2.92077	−0.0960864
$925$	−0.951087	−0.0312715
$926$	0.774424	0.0254491
$927$	−1.00000	−0.0328443
$928$	−0.521486	−0.0171186
$929$	−12.6464	−0.414916	−0.207458	−	0.978244i	$-0.566519\pi$
$929$	−12.6464	−0.414916	−0.207458	+	0.978244i	$0.566519\pi$
$930$	20.7895	0.681714
$931$	−12.1948	−0.399667
$932$	−8.28474	−0.271376
$933$	29.4598	0.964471
$934$	18.2679	0.597743
$935$	−9.79749	−0.320412
$936$	−1.00000	−0.0326860
$937$	−8.42532	−0.275243	−0.137622	−	0.990485i	$-0.543946\pi$
$937$	−8.42532	−0.275243	−0.137622	+	0.990485i	$0.543946\pi$
$938$	−23.7643	−0.775931
$939$	−5.15810	−0.168328
$940$	−10.2599	−0.334640
$941$	10.7156	0.349319	0.174659	−	0.984629i	$-0.444118\pi$
$941$	10.7156	0.349319	0.174659	+	0.984629i	$0.444118\pi$
$942$	−8.89329	−0.289759
$943$	−4.84324	−0.157718
$944$	−10.7140	−0.348711
$945$	−9.32077	−0.303205
$946$	−7.88382	−0.256325
$947$	36.7982	1.19578	0.597891	−	0.801578i	$-0.296006\pi$
$947$	36.7982	1.19578	0.597891	+	0.801578i	$0.296006\pi$
$948$	1.21230	0.0393736
$949$	−8.63216	−0.280212
$950$	6.88357	0.223332
$951$	16.2036	0.525438
$952$	14.0566	0.455576
$953$	12.7429	0.412783	0.206392	−	0.978469i	$-0.433828\pi$
$953$	12.7429	0.412783	0.206392	+	0.978469i	$0.433828\pi$
$954$	−6.82448	−0.220951
$955$	−44.7695	−1.44871
$956$	−26.6667	−0.862464
$957$	−0.448920	−0.0145115
$958$	−5.82877	−0.188319
$959$	−11.7079	−0.378067
$960$	−2.74714	−0.0886635
$961$	26.2699	0.847415
$962$	0.373450	0.0120405
$963$	−17.1864	−0.553825
$964$	−27.2863	−0.878834
$965$	−71.4045	−2.29859
$966$	−3.18093	−0.102345
$967$	19.1336	0.615293	0.307647	−	0.951501i	$-0.400459\pi$
$967$	19.1336	0.615293	0.307647	+	0.951501i	$0.400459\pi$
$968$	10.2589	0.329735
$969$	11.1978	0.359725
$970$	−22.3624	−0.718014
$971$	13.9506	0.447697	0.223849	−	0.974624i	$-0.428138\pi$
$971$	13.9506	0.447697	0.223849	+	0.974624i	$0.428138\pi$
$972$	1.00000	0.0320750
$973$	6.39481	0.205008
$974$	7.63754	0.244722
$975$	2.54676	0.0815616
$976$	4.99993	0.160044
$977$	20.4548	0.654406	0.327203	−	0.944954i	$-0.393894\pi$
$977$	20.4548	0.654406	0.327203	+	0.944954i	$0.393894\pi$
$978$	18.6770	0.597224
$979$	3.96486	0.126717
$980$	−12.3945	−0.395928
$981$	4.88430	0.155944
$982$	−10.2639	−0.327534
$983$	−38.5517	−1.22961	−0.614805	−	0.788679i	$-0.710765\pi$
$983$	−38.5517	−1.22961	−0.614805	+	0.788679i	$0.710765\pi$
$984$	5.16599	0.164686
$985$	−71.5878	−2.28098
$986$	2.16048	0.0688037
$987$	12.6717	0.403343
$988$	−2.70287	−0.0859898
$989$	−8.58603	−0.273020
$990$	−2.36487	−0.0751604
$991$	−31.7584	−1.00884	−0.504419	−	0.863459i	$-0.668293\pi$
$991$	−31.7584	−1.00884	−0.504419	+	0.863459i	$0.668293\pi$
$992$	−7.56769	−0.240274
$993$	32.8982	1.04399
$994$	30.6365	0.971730
$995$	34.0961	1.08092
$996$	15.5907	0.494009
$997$	−15.8176	−0.500947	−0.250474	−	0.968123i	$-0.580586\pi$
$997$	−15.8176	−0.500947	−0.250474	+	0.968123i	$0.580586\pi$
$998$	−18.1451	−0.574374
$999$	−0.373450	−0.0118154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.w.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.w.1.3	✓	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.74714 q^{10} -0.860848 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.39290 q^{14} -2.74714 q^{15} +1.00000 q^{16} -4.14293 q^{17} -1.00000 q^{18} -2.70287 q^{19} -2.74714 q^{20} +3.39290 q^{21} +0.860848 q^{22} +0.937524 q^{23} -1.00000 q^{24} +2.54676 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.39290 q^{28} +0.521486 q^{29} +2.74714 q^{30} +7.56769 q^{31} -1.00000 q^{32} -0.860848 q^{33} +4.14293 q^{34} -9.32077 q^{35} +1.00000 q^{36} -0.373450 q^{37} +2.70287 q^{38} +1.00000 q^{39} +2.74714 q^{40} -5.16599 q^{41} -3.39290 q^{42} -9.15820 q^{43} -0.860848 q^{44} -2.74714 q^{45} -0.937524 q^{46} +3.73475 q^{47} +1.00000 q^{48} +4.51179 q^{49} -2.54676 q^{50} -4.14293 q^{51} +1.00000 q^{52} +6.82448 q^{53} -1.00000 q^{54} +2.36487 q^{55} -3.39290 q^{56} -2.70287 q^{57} -0.521486 q^{58} -10.7140 q^{59} -2.74714 q^{60} +4.99993 q^{61} -7.56769 q^{62} +3.39290 q^{63} +1.00000 q^{64} -2.74714 q^{65} +0.860848 q^{66} +7.00412 q^{67} -4.14293 q^{68} +0.937524 q^{69} +9.32077 q^{70} -9.02958 q^{71} -1.00000 q^{72} -8.63216 q^{73} +0.373450 q^{74} +2.54676 q^{75} -2.70287 q^{76} -2.92077 q^{77} -1.00000 q^{78} +1.21230 q^{79} -2.74714 q^{80} +1.00000 q^{81} +5.16599 q^{82} +15.5907 q^{83} +3.39290 q^{84} +11.3812 q^{85} +9.15820 q^{86} +0.521486 q^{87} +0.860848 q^{88} -4.60576 q^{89} +2.74714 q^{90} +3.39290 q^{91} +0.937524 q^{92} +7.56769 q^{93} -3.73475 q^{94} +7.42515 q^{95} -1.00000 q^{96} -8.14027 q^{97} -4.51179 q^{98} -0.860848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * q^99

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