LMFDB - Embedded newform 6038.2.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$70$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.44970 q^{3} +1.00000 q^{4} -2.67633 q^{5} -2.44970 q^{6} +2.27602 q^{7} +1.00000 q^{8} +3.00102 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.44970 q^{3} +1.00000 q^{4} -2.67633 q^{5} -2.44970 q^{6} +2.27602 q^{7} +1.00000 q^{8} +3.00102 q^{9} -2.67633 q^{10} -6.52281 q^{11} -2.44970 q^{12} -4.75174 q^{13} +2.27602 q^{14} +6.55620 q^{15} +1.00000 q^{16} +3.90437 q^{17} +3.00102 q^{18} -1.19221 q^{19} -2.67633 q^{20} -5.57556 q^{21} -6.52281 q^{22} +1.82865 q^{23} -2.44970 q^{24} +2.16273 q^{25} -4.75174 q^{26} -0.00251085 q^{27} +2.27602 q^{28} -1.82306 q^{29} +6.55620 q^{30} -6.49186 q^{31} +1.00000 q^{32} +15.9789 q^{33} +3.90437 q^{34} -6.09137 q^{35} +3.00102 q^{36} -3.85316 q^{37} -1.19221 q^{38} +11.6403 q^{39} -2.67633 q^{40} -6.32690 q^{41} -5.57556 q^{42} -6.71695 q^{43} -6.52281 q^{44} -8.03173 q^{45} +1.82865 q^{46} -3.83195 q^{47} -2.44970 q^{48} -1.81974 q^{49} +2.16273 q^{50} -9.56454 q^{51} -4.75174 q^{52} -3.92791 q^{53} -0.00251085 q^{54} +17.4572 q^{55} +2.27602 q^{56} +2.92056 q^{57} -1.82306 q^{58} +6.38413 q^{59} +6.55620 q^{60} +7.22153 q^{61} -6.49186 q^{62} +6.83039 q^{63} +1.00000 q^{64} +12.7172 q^{65} +15.9789 q^{66} -2.66923 q^{67} +3.90437 q^{68} -4.47965 q^{69} -6.09137 q^{70} +9.20807 q^{71} +3.00102 q^{72} +13.7243 q^{73} -3.85316 q^{74} -5.29805 q^{75} -1.19221 q^{76} -14.8460 q^{77} +11.6403 q^{78} +5.97476 q^{79} -2.67633 q^{80} -8.99692 q^{81} -6.32690 q^{82} -5.47309 q^{83} -5.57556 q^{84} -10.4494 q^{85} -6.71695 q^{86} +4.46595 q^{87} -6.52281 q^{88} -4.06384 q^{89} -8.03173 q^{90} -10.8150 q^{91} +1.82865 q^{92} +15.9031 q^{93} -3.83195 q^{94} +3.19075 q^{95} -2.44970 q^{96} -6.75097 q^{97} -1.81974 q^{98} -19.5751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) $ 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * 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
$3$	−2.44970	−1.41433	−0.707167	−	0.707046i	$-0.750027\pi$
$3$	−2.44970	−1.41433	−0.707167	+	0.707046i	$0.750027\pi$
$4$	1.00000	0.500000
$5$	−2.67633	−1.19689	−0.598445	−	0.801164i	$-0.704215\pi$
$5$	−2.67633	−1.19689	−0.598445	+	0.801164i	$0.704215\pi$
$6$	−2.44970	−1.00009
$7$	2.27602	0.860254	0.430127	−	0.902768i	$-0.358469\pi$
$7$	2.27602	0.860254	0.430127	+	0.902768i	$0.358469\pi$
$8$	1.00000	0.353553
$9$	3.00102	1.00034
$10$	−2.67633	−0.846329
$11$	−6.52281	−1.96670	−0.983351	−	0.181718i	$-0.941834\pi$
$11$	−6.52281	−1.96670	−0.983351	+	0.181718i	$0.941834\pi$
$12$	−2.44970	−0.707167
$13$	−4.75174	−1.31790	−0.658948	−	0.752189i	$-0.728998\pi$
$13$	−4.75174	−1.31790	−0.658948	+	0.752189i	$0.728998\pi$
$14$	2.27602	0.608291
$15$	6.55620	1.69280
$16$	1.00000	0.250000
$17$	3.90437	0.946950	0.473475	−	0.880807i	$-0.342999\pi$
$17$	3.90437	0.946950	0.473475	+	0.880807i	$0.342999\pi$
$18$	3.00102	0.707348
$19$	−1.19221	−0.273512	−0.136756	−	0.990605i	$-0.543668\pi$
$19$	−1.19221	−0.273512	−0.136756	+	0.990605i	$0.543668\pi$
$20$	−2.67633	−0.598445
$21$	−5.57556	−1.21669
$22$	−6.52281	−1.39067
$23$	1.82865	0.381300	0.190650	−	0.981658i	$-0.438940\pi$
$23$	1.82865	0.381300	0.190650	+	0.981658i	$0.438940\pi$
$24$	−2.44970	−0.500043
$25$	2.16273	0.432547
$26$	−4.75174	−0.931893
$27$	−0.00251085	−0.000483214 0
$28$	2.27602	0.430127
$29$	−1.82306	−0.338534	−0.169267	−	0.985570i	$-0.554140\pi$
$29$	−1.82306	−0.338534	−0.169267	+	0.985570i	$0.554140\pi$
$30$	6.55620	1.19699
$31$	−6.49186	−1.16597	−0.582986	−	0.812482i	$-0.698116\pi$
$31$	−6.49186	−1.16597	−0.582986	+	0.812482i	$0.698116\pi$
$32$	1.00000	0.176777
$33$	15.9789	2.78157
$34$	3.90437	0.669595
$35$	−6.09137	−1.02963
$36$	3.00102	0.500171
$37$	−3.85316	−0.633455	−0.316727	−	0.948517i	$-0.602584\pi$
$37$	−3.85316	−0.633455	−0.316727	+	0.948517i	$0.602584\pi$
$38$	−1.19221	−0.193402
$39$	11.6403	1.86395
$40$	−2.67633	−0.423165
$41$	−6.32690	−0.988096	−0.494048	−	0.869435i	$-0.664483\pi$
$41$	−6.32690	−0.988096	−0.494048	+	0.869435i	$0.664483\pi$
$42$	−5.57556	−0.860327
$43$	−6.71695	−1.02433	−0.512163	−	0.858888i	$-0.671156\pi$
$43$	−6.71695	−1.02433	−0.512163	+	0.858888i	$0.671156\pi$
$44$	−6.52281	−0.983351
$45$	−8.03173	−1.19730
$46$	1.82865	0.269620
$47$	−3.83195	−0.558947	−0.279474	−	0.960153i	$-0.590160\pi$
$47$	−3.83195	−0.558947	−0.279474	+	0.960153i	$0.590160\pi$
$48$	−2.44970	−0.353584
$49$	−1.81974	−0.259963
$50$	2.16273	0.305857
$51$	−9.56454	−1.33930
$52$	−4.75174	−0.658948
$53$	−3.92791	−0.539540	−0.269770	−	0.962925i	$-0.586948\pi$
$53$	−3.92791	−0.539540	−0.269770	+	0.962925i	$0.586948\pi$
$54$	−0.00251085	−0.000341684 0
$55$	17.4572	2.35393
$56$	2.27602	0.304146
$57$	2.92056	0.386838
$58$	−1.82306	−0.239380
$59$	6.38413	0.831143	0.415572	−	0.909560i	$-0.363582\pi$
$59$	6.38413	0.831143	0.415572	+	0.909560i	$0.363582\pi$
$60$	6.55620	0.846402
$61$	7.22153	0.924622	0.462311	−	0.886718i	$-0.347020\pi$
$61$	7.22153	0.924622	0.462311	+	0.886718i	$0.347020\pi$
$62$	−6.49186	−0.824467
$63$	6.83039	0.860548
$64$	1.00000	0.125000
$65$	12.7172	1.57738
$66$	15.9789	1.96687
$67$	−2.66923	−0.326098	−0.163049	−	0.986618i	$-0.552133\pi$
$67$	−2.66923	−0.326098	−0.163049	+	0.986618i	$0.552133\pi$
$68$	3.90437	0.473475
$69$	−4.47965	−0.539286
$70$	−6.09137	−0.728058
$71$	9.20807	1.09280	0.546399	−	0.837525i	$-0.315998\pi$
$71$	9.20807	1.09280	0.546399	+	0.837525i	$0.315998\pi$
$72$	3.00102	0.353674
$73$	13.7243	1.60631	0.803156	−	0.595768i	$-0.203152\pi$
$73$	13.7243	1.60631	0.803156	+	0.595768i	$0.203152\pi$
$74$	−3.85316	−0.447920
$75$	−5.29805	−0.611766
$76$	−1.19221	−0.136756
$77$	−14.8460	−1.69186
$78$	11.6403	1.31801
$79$	5.97476	0.672213	0.336106	−	0.941824i	$-0.390890\pi$
$79$	5.97476	0.672213	0.336106	+	0.941824i	$0.390890\pi$
$80$	−2.67633	−0.299223
$81$	−8.99692	−0.999658
$82$	−6.32690	−0.698689
$83$	−5.47309	−0.600750	−0.300375	−	0.953821i	$-0.597112\pi$
$83$	−5.47309	−0.600750	−0.300375	+	0.953821i	$0.597112\pi$
$84$	−5.57556	−0.608343
$85$	−10.4494	−1.13340
$86$	−6.71695	−0.724308
$87$	4.46595	0.478800
$88$	−6.52281	−0.695334
$89$	−4.06384	−0.430766	−0.215383	−	0.976530i	$-0.569100\pi$
$89$	−4.06384	−0.430766	−0.215383	+	0.976530i	$0.569100\pi$
$90$	−8.03173	−0.846619
$91$	−10.8150	−1.13372
$92$	1.82865	0.190650
$93$	15.9031	1.64908
$94$	−3.83195	−0.395235
$95$	3.19075	0.327364
$96$	−2.44970	−0.250021
$97$	−6.75097	−0.685457	−0.342728	−	0.939435i	$-0.611351\pi$
$97$	−6.75097	−0.685457	−0.342728	+	0.939435i	$0.611351\pi$
$98$	−1.81974	−0.183822
$99$	−19.5751	−1.96737
$100$	2.16273	0.216273
$101$	−0.327972	−0.0326344	−0.0163172	−	0.999867i	$-0.505194\pi$
$101$	−0.327972	−0.0326344	−0.0163172	+	0.999867i	$0.505194\pi$
$102$	−9.56454	−0.947031
$103$	7.82920	0.771434	0.385717	−	0.922617i	$-0.373954\pi$
$103$	7.82920	0.771434	0.385717	+	0.922617i	$0.373954\pi$
$104$	−4.75174	−0.465946
$105$	14.9220	1.45624
$106$	−3.92791	−0.381512
$107$	−10.5781	−1.02262	−0.511311	−	0.859396i	$-0.670840\pi$
$107$	−10.5781	−1.02262	−0.511311	+	0.859396i	$0.670840\pi$
$108$	−0.00251085	−0.000241607 0
$109$	−2.36734	−0.226751	−0.113375	−	0.993552i	$-0.536166\pi$
$109$	−2.36734	−0.226751	−0.113375	+	0.993552i	$0.536166\pi$
$110$	17.4572	1.66448
$111$	9.43907	0.895917
$112$	2.27602	0.215063
$113$	−15.2418	−1.43383	−0.716913	−	0.697162i	$-0.754446\pi$
$113$	−15.2418	−1.43383	−0.716913	+	0.697162i	$0.754446\pi$
$114$	2.92056	0.273536
$115$	−4.89408	−0.456375
$116$	−1.82306	−0.169267
$117$	−14.2601	−1.31835
$118$	6.38413	0.587707
$119$	8.88643	0.814617
$120$	6.55620	0.598496
$121$	31.5471	2.86791
$122$	7.22153	0.653807
$123$	15.4990	1.39750
$124$	−6.49186	−0.582986
$125$	7.59345	0.679179
$126$	6.83039	0.608499
$127$	11.7267	1.04057	0.520287	−	0.853991i	$-0.325825\pi$
$127$	11.7267	1.04057	0.520287	+	0.853991i	$0.325825\pi$
$128$	1.00000	0.0883883
$129$	16.4545	1.44874
$130$	12.7172	1.11537
$131$	−4.38596	−0.383203	−0.191602	−	0.981473i	$-0.561368\pi$
$131$	−4.38596	−0.383203	−0.191602	+	0.981473i	$0.561368\pi$
$132$	15.9789	1.39079
$133$	−2.71350	−0.235290
$134$	−2.66923	−0.230586
$135$	0.00671986	0.000578354 0
$136$	3.90437	0.334797
$137$	−14.6198	−1.24906	−0.624528	−	0.781002i	$-0.714709\pi$
$137$	−14.6198	−1.24906	−0.624528	+	0.781002i	$0.714709\pi$
$138$	−4.47965	−0.381333
$139$	−0.682611	−0.0578983	−0.0289492	−	0.999581i	$-0.509216\pi$
$139$	−0.682611	−0.0578983	−0.0289492	+	0.999581i	$0.509216\pi$
$140$	−6.09137	−0.514815
$141$	9.38712	0.790538
$142$	9.20807	0.772724
$143$	30.9947	2.59191
$144$	3.00102	0.250085
$145$	4.87911	0.405188
$146$	13.7243	1.13583
$147$	4.45782	0.367675
$148$	−3.85316	−0.316727
$149$	14.1359	1.15806	0.579030	−	0.815307i	$-0.303432\pi$
$149$	14.1359	1.15806	0.579030	+	0.815307i	$0.303432\pi$
$150$	−5.29805	−0.432584
$151$	−21.6396	−1.76100	−0.880502	−	0.474042i	$-0.842794\pi$
$151$	−21.6396	−1.76100	−0.880502	+	0.474042i	$0.842794\pi$
$152$	−1.19221	−0.0967012
$153$	11.7171	0.947273
$154$	−14.8460	−1.19633
$155$	17.3744	1.39554
$156$	11.6403	0.931973
$157$	0.327618	0.0261468	0.0130734	−	0.999915i	$-0.495838\pi$
$157$	0.327618	0.0261468	0.0130734	+	0.999915i	$0.495838\pi$
$158$	5.97476	0.475326
$159$	9.62219	0.763090
$160$	−2.67633	−0.211582
$161$	4.16205	0.328015
$162$	−8.99692	−0.706865
$163$	12.2765	0.961572	0.480786	−	0.876838i	$-0.340351\pi$
$163$	12.2765	0.961572	0.480786	+	0.876838i	$0.340351\pi$
$164$	−6.32690	−0.494048
$165$	−42.7648	−3.32924
$166$	−5.47309	−0.424794
$167$	−1.16486	−0.0901395	−0.0450698	−	0.998984i	$-0.514351\pi$
$167$	−1.16486	−0.0901395	−0.0450698	+	0.998984i	$0.514351\pi$
$168$	−5.57556	−0.430164
$169$	9.57904	0.736849
$170$	−10.4494	−0.801432
$171$	−3.57786	−0.273606
$172$	−6.71695	−0.512163
$173$	2.45218	0.186436	0.0932179	−	0.995646i	$-0.470285\pi$
$173$	2.45218	0.186436	0.0932179	+	0.995646i	$0.470285\pi$
$174$	4.46595	0.338563
$175$	4.92242	0.372100
$176$	−6.52281	−0.491675
$177$	−15.6392	−1.17551
$178$	−4.06384	−0.304598
$179$	7.90177	0.590606	0.295303	−	0.955404i	$-0.404579\pi$
$179$	7.90177	0.590606	0.295303	+	0.955404i	$0.404579\pi$
$180$	−8.03173	−0.598650
$181$	22.9301	1.70438	0.852192	−	0.523229i	$-0.175273\pi$
$181$	22.9301	1.70438	0.852192	+	0.523229i	$0.175273\pi$
$182$	−10.8150	−0.801665
$183$	−17.6906	−1.30773
$184$	1.82865	0.134810
$185$	10.3123	0.758176
$186$	15.9031	1.16607
$187$	−25.4675	−1.86237
$188$	−3.83195	−0.279474
$189$	−0.00571474	−0.000415686 0
$190$	3.19075	0.231481
$191$	15.5730	1.12682	0.563410	−	0.826178i	$-0.309489\pi$
$191$	15.5730	1.12682	0.563410	+	0.826178i	$0.309489\pi$
$192$	−2.44970	−0.176792
$193$	0.264860	0.0190650	0.00953251	−	0.999955i	$-0.496966\pi$
$193$	0.264860	0.0190650	0.00953251	+	0.999955i	$0.496966\pi$
$194$	−6.75097	−0.484691
$195$	−31.1534	−2.23094
$196$	−1.81974	−0.129982
$197$	−12.1841	−0.868081	−0.434040	−	0.900893i	$-0.642912\pi$
$197$	−12.1841	−0.868081	−0.434040	+	0.900893i	$0.642912\pi$
$198$	−19.5751	−1.39114
$199$	16.8013	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$199$	16.8013	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$200$	2.16273	0.152928
$201$	6.53880	0.461211
$202$	−0.327972	−0.0230760
$203$	−4.14932	−0.291225
$204$	−9.56454	−0.669652
$205$	16.9329	1.18264
$206$	7.82920	0.545486
$207$	5.48783	0.381431
$208$	−4.75174	−0.329474
$209$	7.77657	0.537917
$210$	14.9220	1.02972
$211$	−1.44362	−0.0993829	−0.0496914	−	0.998765i	$-0.515824\pi$
$211$	−1.44362	−0.0993829	−0.0496914	+	0.998765i	$0.515824\pi$
$212$	−3.92791	−0.269770
$213$	−22.5570	−1.54558
$214$	−10.5781	−0.723102
$215$	17.9768	1.22601
$216$	−0.00251085	−0.000170842 0
$217$	−14.7756	−1.00303
$218$	−2.36734	−0.160337
$219$	−33.6205	−2.27186
$220$	17.4572	1.17696
$221$	−18.5526	−1.24798
$222$	9.43907	0.633509
$223$	1.43504	0.0960977	0.0480488	−	0.998845i	$-0.484700\pi$
$223$	1.43504	0.0960977	0.0480488	+	0.998845i	$0.484700\pi$
$224$	2.27602	0.152073
$225$	6.49042	0.432695
$226$	−15.2418	−1.01387
$227$	13.2451	0.879107	0.439554	−	0.898216i	$-0.355137\pi$
$227$	13.2451	0.879107	0.439554	+	0.898216i	$0.355137\pi$
$228$	2.92056	0.193419
$229$	0.964996	0.0637687	0.0318844	−	0.999492i	$-0.489849\pi$
$229$	0.964996	0.0637687	0.0318844	+	0.999492i	$0.489849\pi$
$230$	−4.89408	−0.322706
$231$	36.3683	2.39286
$232$	−1.82306	−0.119690
$233$	−3.26796	−0.214091	−0.107046	−	0.994254i	$-0.534139\pi$
$233$	−3.26796	−0.214091	−0.107046	+	0.994254i	$0.534139\pi$
$234$	−14.2601	−0.932211
$235$	10.2556	0.668999
$236$	6.38413	0.415572
$237$	−14.6364	−0.950733
$238$	8.88643	0.576022
$239$	3.58436	0.231853	0.115926	−	0.993258i	$-0.463016\pi$
$239$	3.58436	0.231853	0.115926	+	0.993258i	$0.463016\pi$
$240$	6.55620	0.423201
$241$	6.23100	0.401374	0.200687	−	0.979655i	$-0.435683\pi$
$241$	6.23100	0.401374	0.200687	+	0.979655i	$0.435683\pi$
$242$	31.5471	2.02792
$243$	22.0473	1.41433
$244$	7.22153	0.462311
$245$	4.87023	0.311147
$246$	15.4990	0.988180
$247$	5.66508	0.360460
$248$	−6.49186	−0.412234
$249$	13.4074	0.849661
$250$	7.59345	0.480252
$251$	9.87296	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$251$	9.87296	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$252$	6.83039	0.430274
$253$	−11.9280	−0.749904
$254$	11.7267	0.735797
$255$	25.5979	1.60300
$256$	1.00000	0.0625000
$257$	14.0851	0.878606	0.439303	−	0.898339i	$-0.355225\pi$
$257$	14.0851	0.878606	0.439303	+	0.898339i	$0.355225\pi$
$258$	16.4545	1.02441
$259$	−8.76985	−0.544932
$260$	12.7172	0.788688
$261$	−5.47105	−0.338650
$262$	−4.38596	−0.270966
$263$	−28.3471	−1.74795	−0.873977	−	0.485967i	$-0.838467\pi$
$263$	−28.3471	−1.74795	−0.873977	+	0.485967i	$0.838467\pi$
$264$	15.9789	0.983435
$265$	10.5124	0.645770
$266$	−2.71350	−0.166375
$267$	9.95518	0.609247
$268$	−2.66923	−0.163049
$269$	−23.2309	−1.41641	−0.708206	−	0.706006i	$-0.750495\pi$
$269$	−23.2309	−1.41641	−0.708206	+	0.706006i	$0.750495\pi$
$270$	0.00671986	0.000408958 0
$271$	21.9264	1.33193	0.665967	−	0.745981i	$-0.268019\pi$
$271$	21.9264	1.33193	0.665967	+	0.745981i	$0.268019\pi$
$272$	3.90437	0.236737
$273$	26.4936	1.60347
$274$	−14.6198	−0.883216
$275$	−14.1071	−0.850690
$276$	−4.47965	−0.269643
$277$	27.7747	1.66882	0.834411	−	0.551142i	$-0.185808\pi$
$277$	27.7747	1.66882	0.834411	+	0.551142i	$0.185808\pi$
$278$	−0.682611	−0.0409403
$279$	−19.4822	−1.16637
$280$	−6.09137	−0.364029
$281$	16.9077	1.00863	0.504315	−	0.863520i	$-0.331745\pi$
$281$	16.9077	1.00863	0.504315	+	0.863520i	$0.331745\pi$
$282$	9.38712	0.558995
$283$	22.1355	1.31582	0.657910	−	0.753096i	$-0.271441\pi$
$283$	22.1355	1.31582	0.657910	+	0.753096i	$0.271441\pi$
$284$	9.20807	0.546399
$285$	−7.81638	−0.463002
$286$	30.9947	1.83275
$287$	−14.4001	−0.850013
$288$	3.00102	0.176837
$289$	−1.75586	−0.103286
$290$	4.87911	0.286511
$291$	16.5378	0.969465
$292$	13.7243	0.803156
$293$	23.1266	1.35107	0.675534	−	0.737329i	$-0.263913\pi$
$293$	23.1266	1.35107	0.675534	+	0.737329i	$0.263913\pi$
$294$	4.45782	0.259985
$295$	−17.0860	−0.994787
$296$	−3.85316	−0.223960
$297$	0.0163778	0.000950337 0
$298$	14.1359	0.818871
$299$	−8.68928	−0.502514
$300$	−5.29805	−0.305883
$301$	−15.2879	−0.881181
$302$	−21.6396	−1.24522
$303$	0.803432	0.0461559
$304$	−1.19221	−0.0683780
$305$	−19.3272	−1.10667
$306$	11.7171	0.669824
$307$	10.1586	0.579783	0.289892	−	0.957059i	$-0.406381\pi$
$307$	10.1586	0.579783	0.289892	+	0.957059i	$0.406381\pi$
$308$	−14.8460	−0.845931
$309$	−19.1792	−1.09107
$310$	17.3744	0.986797
$311$	−5.71818	−0.324248	−0.162124	−	0.986770i	$-0.551834\pi$
$311$	−5.71818	−0.324248	−0.162124	+	0.986770i	$0.551834\pi$
$312$	11.6403	0.659004
$313$	20.8998	1.18132	0.590662	−	0.806919i	$-0.298867\pi$
$313$	20.8998	1.18132	0.590662	+	0.806919i	$0.298867\pi$
$314$	0.327618	0.0184885
$315$	−18.2804	−1.02998
$316$	5.97476	0.336106
$317$	−4.12986	−0.231956	−0.115978	−	0.993252i	$-0.537000\pi$
$317$	−4.12986	−0.231956	−0.115978	+	0.993252i	$0.537000\pi$
$318$	9.62219	0.539586
$319$	11.8915	0.665795
$320$	−2.67633	−0.149611
$321$	25.9131	1.44633
$322$	4.16205	0.231942
$323$	−4.65484	−0.259002
$324$	−8.99692	−0.499829
$325$	−10.2768	−0.570052
$326$	12.2765	0.679934
$327$	5.79928	0.320701
$328$	−6.32690	−0.349345
$329$	−8.72159	−0.480837
$330$	−42.7648	−2.35413
$331$	11.0862	0.609354	0.304677	−	0.952456i	$-0.401451\pi$
$331$	11.0862	0.609354	0.304677	+	0.952456i	$0.401451\pi$
$332$	−5.47309	−0.300375
$333$	−11.5634	−0.633671
$334$	−1.16486	−0.0637383
$335$	7.14372	0.390303
$336$	−5.57556	−0.304172
$337$	1.50386	0.0819205	0.0409603	−	0.999161i	$-0.486958\pi$
$337$	1.50386	0.0819205	0.0409603	+	0.999161i	$0.486958\pi$
$338$	9.57904	0.521031
$339$	37.3378	2.02791
$340$	−10.4494	−0.566698
$341$	42.3452	2.29312
$342$	−3.57786	−0.193468
$343$	−20.0739	−1.08389
$344$	−6.71695	−0.362154
$345$	11.9890	0.645467
$346$	2.45218	0.131830
$347$	−10.7260	−0.575803	−0.287901	−	0.957660i	$-0.592958\pi$
$347$	−10.7260	−0.575803	−0.287901	+	0.957660i	$0.592958\pi$
$348$	4.46595	0.239400
$349$	30.5424	1.63489	0.817447	−	0.576003i	$-0.195389\pi$
$349$	30.5424	1.63489	0.817447	+	0.576003i	$0.195389\pi$
$350$	4.92242	0.263115
$351$	0.0119309	0.000636825 0
$352$	−6.52281	−0.347667
$353$	29.8069	1.58646	0.793232	−	0.608920i	$-0.208397\pi$
$353$	29.8069	1.58646	0.793232	+	0.608920i	$0.208397\pi$
$354$	−15.6392	−0.831214
$355$	−24.6438	−1.30796
$356$	−4.06384	−0.215383
$357$	−21.7691	−1.15214
$358$	7.90177	0.417621
$359$	8.77055	0.462892	0.231446	−	0.972848i	$-0.425654\pi$
$359$	8.77055	0.462892	0.231446	+	0.972848i	$0.425654\pi$
$360$	−8.03173	−0.423309
$361$	−17.5786	−0.925191
$362$	22.9301	1.20518
$363$	−77.2808	−4.05619
$364$	−10.8150	−0.566862
$365$	−36.7309	−1.92258
$366$	−17.6906	−0.924701
$367$	−11.2179	−0.585568	−0.292784	−	0.956179i	$-0.594582\pi$
$367$	−11.2179	−0.585568	−0.292784	+	0.956179i	$0.594582\pi$
$368$	1.82865	0.0953251
$369$	−18.9872	−0.988433
$370$	10.3123	0.536112
$371$	−8.93999	−0.464141
$372$	15.9031	0.824538
$373$	−28.5222	−1.47682	−0.738411	−	0.674351i	$-0.764423\pi$
$373$	−28.5222	−1.47682	−0.738411	+	0.674351i	$0.764423\pi$
$374$	−25.4675	−1.31689
$375$	−18.6017	−0.960587
$376$	−3.83195	−0.197618
$377$	8.66271	0.446152
$378$	−0.00571474	−0.000293935 0
$379$	1.81207	0.0930798	0.0465399	−	0.998916i	$-0.485181\pi$
$379$	1.81207	0.0930798	0.0465399	+	0.998916i	$0.485181\pi$
$380$	3.19075	0.163682
$381$	−28.7268	−1.47172
$382$	15.5730	0.796782
$383$	0.236314	0.0120751	0.00603754	−	0.999982i	$-0.498078\pi$
$383$	0.236314	0.0120751	0.00603754	+	0.999982i	$0.498078\pi$
$384$	−2.44970	−0.125011
$385$	39.7329	2.02497
$386$	0.264860	0.0134810
$387$	−20.1577	−1.02468
$388$	−6.75097	−0.342728
$389$	−8.46353	−0.429118	−0.214559	−	0.976711i	$-0.568831\pi$
$389$	−8.46353	−0.429118	−0.214559	+	0.976711i	$0.568831\pi$
$390$	−31.1534	−1.57751
$391$	7.13975	0.361072
$392$	−1.81974	−0.0919108
$393$	10.7443	0.541977
$394$	−12.1841	−0.613826
$395$	−15.9904	−0.804565
$396$	−19.5751	−0.983687
$397$	−19.3698	−0.972145	−0.486072	−	0.873919i	$-0.661571\pi$
$397$	−19.3698	−0.972145	−0.486072	+	0.873919i	$0.661571\pi$
$398$	16.8013	0.842172
$399$	6.64725	0.332779
$400$	2.16273	0.108137
$401$	−2.87853	−0.143747	−0.0718734	−	0.997414i	$-0.522898\pi$
$401$	−2.87853	−0.143747	−0.0718734	+	0.997414i	$0.522898\pi$
$402$	6.53880	0.326126
$403$	30.8476	1.53663
$404$	−0.327972	−0.0163172
$405$	24.0787	1.19648
$406$	−4.14932	−0.205927
$407$	25.1334	1.24582
$408$	−9.56454	−0.473515
$409$	−16.3002	−0.805991	−0.402996	−	0.915202i	$-0.632031\pi$
$409$	−16.3002	−0.805991	−0.402996	+	0.915202i	$0.632031\pi$
$410$	16.9329	0.836254
$411$	35.8142	1.76658
$412$	7.82920	0.385717
$413$	14.5304	0.714994
$414$	5.48783	0.269712
$415$	14.6478	0.719032
$416$	−4.75174	−0.232973
$417$	1.67219	0.0818876
$418$	7.77657	0.380365
$419$	18.1398	0.886185	0.443093	−	0.896476i	$-0.353881\pi$
$419$	18.1398	0.886185	0.443093	+	0.896476i	$0.353881\pi$
$420$	14.9220	0.728120
$421$	−13.3419	−0.650244	−0.325122	−	0.945672i	$-0.605405\pi$
$421$	−13.3419	−0.650244	−0.325122	+	0.945672i	$0.605405\pi$
$422$	−1.44362	−0.0702743
$423$	−11.4998	−0.559138
$424$	−3.92791	−0.190756
$425$	8.44413	0.409600
$426$	−22.5570	−1.09289
$427$	16.4363	0.795410
$428$	−10.5781	−0.511311
$429$	−75.9277	−3.66582
$430$	17.9768	0.866917
$431$	26.3994	1.27162	0.635808	−	0.771847i	$-0.280667\pi$
$431$	26.3994	1.27162	0.635808	+	0.771847i	$0.280667\pi$
$432$	−0.00251085	−0.000120803 0
$433$	3.79957	0.182596	0.0912978	−	0.995824i	$-0.470898\pi$
$433$	3.79957	0.182596	0.0912978	+	0.995824i	$0.470898\pi$
$434$	−14.7756	−0.709251
$435$	−11.9524	−0.573071
$436$	−2.36734	−0.113375
$437$	−2.18014	−0.104290
$438$	−33.6205	−1.60645
$439$	−14.3589	−0.685314	−0.342657	−	0.939461i	$-0.611327\pi$
$439$	−14.3589	−0.685314	−0.342657	+	0.939461i	$0.611327\pi$
$440$	17.4572	0.832239
$441$	−5.46109	−0.260052
$442$	−18.5526	−0.882456
$443$	1.42478	0.0676933	0.0338466	−	0.999427i	$-0.489224\pi$
$443$	1.42478	0.0676933	0.0338466	+	0.999427i	$0.489224\pi$
$444$	9.43907	0.447959
$445$	10.8762	0.515580
$446$	1.43504	0.0679513
$447$	−34.6287	−1.63788
$448$	2.27602	0.107532
$449$	−4.76063	−0.224668	−0.112334	−	0.993670i	$-0.535833\pi$
$449$	−4.76063	−0.224668	−0.112334	+	0.993670i	$0.535833\pi$
$450$	6.49042	0.305961
$451$	41.2692	1.94329
$452$	−15.2418	−0.716913
$453$	53.0105	2.49065
$454$	13.2451	0.621623
$455$	28.9446	1.35694
$456$	2.92056	0.136768
$457$	24.6893	1.15491	0.577457	−	0.816421i	$-0.304045\pi$
$457$	24.6893	1.15491	0.577457	+	0.816421i	$0.304045\pi$
$458$	0.964996	0.0450913
$459$	−0.00980331	−0.000457579 0
$460$	−4.89408	−0.228187
$461$	−9.58534	−0.446434	−0.223217	−	0.974769i	$-0.571656\pi$
$461$	−9.58534	−0.446434	−0.223217	+	0.974769i	$0.571656\pi$
$462$	36.3683	1.69201
$463$	32.1030	1.49195	0.745976	−	0.665973i	$-0.231984\pi$
$463$	32.1030	1.49195	0.745976	+	0.665973i	$0.231984\pi$
$464$	−1.82306	−0.0846335
$465$	−42.5619	−1.97376
$466$	−3.26796	−0.151385
$467$	12.5338	0.579996	0.289998	−	0.957027i	$-0.406345\pi$
$467$	12.5338	0.579996	0.289998	+	0.957027i	$0.406345\pi$
$468$	−14.2601	−0.659173
$469$	−6.07521	−0.280527
$470$	10.2556	0.473054
$471$	−0.802565	−0.0369803
$472$	6.38413	0.293854
$473$	43.8134	2.01454
$474$	−14.6364	−0.672270
$475$	−2.57844	−0.118307
$476$	8.88643	0.407309
$477$	−11.7878	−0.539724
$478$	3.58436	0.163945
$479$	−0.365441	−0.0166974	−0.00834871	−	0.999965i	$-0.502658\pi$
$479$	−0.365441	−0.0166974	−0.00834871	+	0.999965i	$0.502658\pi$
$480$	6.55620	0.299248
$481$	18.3092	0.834828
$482$	6.23100	0.283814
$483$	−10.1958	−0.463923
$484$	31.5471	1.43396
$485$	18.0678	0.820417
$486$	22.0473	1.00009
$487$	27.5887	1.25016	0.625082	−	0.780559i	$-0.285065\pi$
$487$	27.5887	1.25016	0.625082	+	0.780559i	$0.285065\pi$
$488$	7.22153	0.326903
$489$	−30.0738	−1.35998
$490$	4.87023	0.220014
$491$	−17.6219	−0.795267	−0.397634	−	0.917544i	$-0.630169\pi$
$491$	−17.6219	−0.795267	−0.397634	+	0.917544i	$0.630169\pi$
$492$	15.4990	0.698749
$493$	−7.11791	−0.320575
$494$	5.66508	0.254884
$495$	52.3894	2.35473
$496$	−6.49186	−0.291493
$497$	20.9577	0.940083
$498$	13.4074	0.600801
$499$	21.3209	0.954454	0.477227	−	0.878780i	$-0.341642\pi$
$499$	21.3209	0.954454	0.477227	+	0.878780i	$0.341642\pi$
$500$	7.59345	0.339590
$501$	2.85355	0.127487
$502$	9.87296	0.440652
$503$	−38.9298	−1.73579	−0.867897	−	0.496744i	$-0.834529\pi$
$503$	−38.9298	−1.73579	−0.867897	+	0.496744i	$0.834529\pi$
$504$	6.83039	0.304250
$505$	0.877760	0.0390598
$506$	−11.9280	−0.530262
$507$	−23.4658	−1.04215
$508$	11.7267	0.520287
$509$	−29.4122	−1.30367	−0.651836	−	0.758360i	$-0.726001\pi$
$509$	−29.4122	−1.30367	−0.651836	+	0.758360i	$0.726001\pi$
$510$	25.5979	1.13349
$511$	31.2369	1.38184
$512$	1.00000	0.0441942
$513$	0.00299347	0.000132165 0
$514$	14.0851	0.621268
$515$	−20.9535	−0.923322
$516$	16.4545	0.724370
$517$	24.9951	1.09928
$518$	−8.76985	−0.385325
$519$	−6.00710	−0.263682
$520$	12.7172	0.557687
$521$	10.0562	0.440571	0.220285	−	0.975435i	$-0.429301\pi$
$521$	10.0562	0.440571	0.220285	+	0.975435i	$0.429301\pi$
$522$	−5.47105	−0.239461
$523$	−0.365783	−0.0159946	−0.00799729	−	0.999968i	$-0.502546\pi$
$523$	−0.365783	−0.0159946	−0.00799729	+	0.999968i	$0.502546\pi$
$524$	−4.38596	−0.191602
$525$	−12.0585	−0.526274
$526$	−28.3471	−1.23599
$527$	−25.3467	−1.10412
$528$	15.9789	0.695393
$529$	−19.6560	−0.854610
$530$	10.5124	0.456628
$531$	19.1589	0.831427
$532$	−2.71350	−0.117645
$533$	30.0638	1.30221
$534$	9.95518	0.430803
$535$	28.3104	1.22397
$536$	−2.66923	−0.115293
$537$	−19.3569	−0.835314
$538$	−23.2309	−1.00155
$539$	11.8698	0.511270
$540$	0.00671986	0.000289177 0
$541$	26.9994	1.16079	0.580397	−	0.814333i	$-0.302897\pi$
$541$	26.9994	1.16079	0.580397	+	0.814333i	$0.302897\pi$
$542$	21.9264	0.941820
$543$	−56.1720	−2.41057
$544$	3.90437	0.167399
$545$	6.33579	0.271396
$546$	26.4936	1.13382
$547$	−32.9400	−1.40841	−0.704207	−	0.709995i	$-0.748697\pi$
$547$	−32.9400	−1.40841	−0.704207	+	0.709995i	$0.748697\pi$
$548$	−14.6198	−0.624528
$549$	21.6720	0.924938
$550$	−14.1071	−0.601529
$551$	2.17347	0.0925931
$552$	−4.47965	−0.190667
$553$	13.5987	0.578274
$554$	27.7747	1.18004
$555$	−25.2621	−1.07231
$556$	−0.682611	−0.0289492
$557$	38.1917	1.61823	0.809116	−	0.587648i	$-0.199946\pi$
$557$	38.1917	1.61823	0.809116	+	0.587648i	$0.199946\pi$
$558$	−19.4822	−0.824749
$559$	31.9172	1.34995
$560$	−6.09137	−0.257407
$561$	62.3877	2.63401
$562$	16.9077	0.713210
$563$	−14.5855	−0.614707	−0.307353	−	0.951595i	$-0.599443\pi$
$563$	−14.5855	−0.614707	−0.307353	+	0.951595i	$0.599443\pi$
$564$	9.38712	0.395269
$565$	40.7920	1.71613
$566$	22.1355	0.930425
$567$	−20.4772	−0.859960
$568$	9.20807	0.386362
$569$	−30.0842	−1.26119	−0.630597	−	0.776110i	$-0.717190\pi$
$569$	−30.0842	−1.26119	−0.630597	+	0.776110i	$0.717190\pi$
$570$	−7.81638	−0.327392
$571$	32.6313	1.36558	0.682790	−	0.730615i	$-0.260767\pi$
$571$	32.6313	1.36558	0.682790	+	0.730615i	$0.260767\pi$
$572$	30.9947	1.29595
$573$	−38.1491	−1.59370
$574$	−14.4001	−0.601050
$575$	3.95489	0.164930
$576$	3.00102	0.125043
$577$	−23.0609	−0.960038	−0.480019	−	0.877258i	$-0.659370\pi$
$577$	−23.0609	−0.960038	−0.480019	+	0.877258i	$0.659370\pi$
$578$	−1.75586	−0.0730341
$579$	−0.648826	−0.0269643
$580$	4.87911	0.202594
$581$	−12.4569	−0.516797
$582$	16.5378	0.685515
$583$	25.6210	1.06111
$584$	13.7243	0.567917
$585$	38.1647	1.57792
$586$	23.1266	0.955349
$587$	−22.7734	−0.939960	−0.469980	−	0.882677i	$-0.655739\pi$
$587$	−22.7734	−0.939960	−0.469980	+	0.882677i	$0.655739\pi$
$588$	4.45782	0.183837
$589$	7.73967	0.318908
$590$	−17.0860	−0.703421
$591$	29.8474	1.22776
$592$	−3.85316	−0.158364
$593$	9.20280	0.377914	0.188957	−	0.981985i	$-0.439489\pi$
$593$	9.20280	0.377914	0.188957	+	0.981985i	$0.439489\pi$
$594$	0.0163778	0.000671990 0
$595$	−23.7830	−0.975008
$596$	14.1359	0.579030
$597$	−41.1581	−1.68449
$598$	−8.68928	−0.355331
$599$	−28.6410	−1.17024	−0.585119	−	0.810948i	$-0.698952\pi$
$599$	−28.6410	−1.17024	−0.585119	+	0.810948i	$0.698952\pi$
$600$	−5.29805	−0.216292
$601$	−1.42507	−0.0581296	−0.0290648	−	0.999578i	$-0.509253\pi$
$601$	−1.42507	−0.0581296	−0.0290648	+	0.999578i	$0.509253\pi$
$602$	−15.2879	−0.623089
$603$	−8.01041	−0.326209
$604$	−21.6396	−0.880502
$605$	−84.4303	−3.43258
$606$	0.803432	0.0326372
$607$	20.7564	0.842477	0.421239	−	0.906950i	$-0.361596\pi$
$607$	20.7564	0.842477	0.421239	+	0.906950i	$0.361596\pi$
$608$	−1.19221	−0.0483506
$609$	10.1646	0.411890
$610$	−19.3272	−0.782535
$611$	18.2084	0.736634
$612$	11.7171	0.473637
$613$	9.37206	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$613$	9.37206	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$614$	10.1586	0.409969
$615$	−41.4804	−1.67265
$616$	−14.8460	−0.598164
$617$	−41.2046	−1.65883	−0.829417	−	0.558629i	$-0.811327\pi$
$617$	−41.2046	−1.65883	−0.829417	+	0.558629i	$0.811327\pi$
$618$	−19.1792	−0.771500
$619$	−20.0468	−0.805750	−0.402875	−	0.915255i	$-0.631989\pi$
$619$	−20.0468	−0.805750	−0.402875	+	0.915255i	$0.631989\pi$
$620$	17.3744	0.697771
$621$	−0.00459148	−0.000184250 0
$622$	−5.71818	−0.229278
$623$	−9.24937	−0.370568
$624$	11.6403	0.465986
$625$	−31.1363	−1.24545
$626$	20.8998	0.835323
$627$	−19.0503	−0.760794
$628$	0.327618	0.0130734
$629$	−15.0442	−0.599850
$630$	−18.2804	−0.728307
$631$	13.3280	0.530579	0.265289	−	0.964169i	$-0.414533\pi$
$631$	13.3280	0.530579	0.265289	+	0.964169i	$0.414533\pi$
$632$	5.97476	0.237663
$633$	3.53643	0.140561
$634$	−4.12986	−0.164018
$635$	−31.3844	−1.24545
$636$	9.62219	0.381545
$637$	8.64694	0.342604
$638$	11.8915	0.470788
$639$	27.6337	1.09317
$640$	−2.67633	−0.105791
$641$	−38.8894	−1.53604	−0.768019	−	0.640428i	$-0.778757\pi$
$641$	−38.8894	−1.53604	−0.768019	+	0.640428i	$0.778757\pi$
$642$	25.9131	1.02271
$643$	36.8098	1.45164	0.725819	−	0.687886i	$-0.241461\pi$
$643$	36.8098	1.45164	0.725819	+	0.687886i	$0.241461\pi$
$644$	4.16205	0.164008
$645$	−44.0377	−1.73398
$646$	−4.65484	−0.183142
$647$	−9.03605	−0.355244	−0.177622	−	0.984099i	$-0.556840\pi$
$647$	−9.03605	−0.355244	−0.177622	+	0.984099i	$0.556840\pi$
$648$	−8.99692	−0.353433
$649$	−41.6425	−1.63461
$650$	−10.2768	−0.403087
$651$	36.1958	1.41862
$652$	12.2765	0.480786
$653$	−32.2030	−1.26020	−0.630100	−	0.776514i	$-0.716986\pi$
$653$	−32.2030	−1.26020	−0.630100	+	0.776514i	$0.716986\pi$
$654$	5.79928	0.226770
$655$	11.7383	0.458652
$656$	−6.32690	−0.247024
$657$	41.1871	1.60686
$658$	−8.72159	−0.340003
$659$	−13.1612	−0.512689	−0.256344	−	0.966586i	$-0.582518\pi$
$659$	−13.1612	−0.512689	−0.256344	+	0.966586i	$0.582518\pi$
$660$	−42.7648	−1.66462
$661$	−27.1435	−1.05576	−0.527881	−	0.849319i	$-0.677013\pi$
$661$	−27.1435	−1.05576	−0.527881	+	0.849319i	$0.677013\pi$
$662$	11.0862	0.430878
$663$	45.4482	1.76506
$664$	−5.47309	−0.212397
$665$	7.26221	0.281616
$666$	−11.5634	−0.448073
$667$	−3.33375	−0.129083
$668$	−1.16486	−0.0450698
$669$	−3.51543	−0.135914
$670$	7.14372	0.275986
$671$	−47.1047	−1.81846
$672$	−5.57556	−0.215082
$673$	1.17445	0.0452716	0.0226358	−	0.999744i	$-0.492794\pi$
$673$	1.17445	0.0452716	0.0226358	+	0.999744i	$0.492794\pi$
$674$	1.50386	0.0579266
$675$	−0.00543031	−0.000209013 0
$676$	9.57904	0.368424
$677$	−45.8341	−1.76155	−0.880773	−	0.473538i	$-0.842977\pi$
$677$	−45.8341	−1.76155	−0.880773	+	0.473538i	$0.842977\pi$
$678$	37.3378	1.43395
$679$	−15.3653	−0.589667
$680$	−10.4494	−0.400716
$681$	−32.4465	−1.24335
$682$	42.3452	1.62148
$683$	44.3992	1.69889	0.849443	−	0.527680i	$-0.176938\pi$
$683$	44.3992	1.69889	0.849443	+	0.527680i	$0.176938\pi$
$684$	−3.57786	−0.136803
$685$	39.1275	1.49498
$686$	−20.0739	−0.766425
$687$	−2.36395	−0.0901903
$688$	−6.71695	−0.256081
$689$	18.6644	0.711057
$690$	11.9890	0.456414
$691$	25.4638	0.968687	0.484344	−	0.874878i	$-0.339058\pi$
$691$	25.4638	0.968687	0.484344	+	0.874878i	$0.339058\pi$
$692$	2.45218	0.0932179
$693$	−44.5533	−1.69244
$694$	−10.7260	−0.407154
$695$	1.82689	0.0692980
$696$	4.46595	0.169281
$697$	−24.7026	−0.935677
$698$	30.5424	1.15605
$699$	8.00552	0.302797
$700$	4.92242	0.186050
$701$	33.4467	1.26326	0.631632	−	0.775269i	$-0.282386\pi$
$701$	33.4467	1.26326	0.631632	+	0.775269i	$0.282386\pi$
$702$	0.0119309	0.000450303 0
$703$	4.59378	0.173258
$704$	−6.52281	−0.245838
$705$	−25.1230	−0.946188
$706$	29.8069	1.12180
$707$	−0.746469	−0.0280739
$708$	−15.6392	−0.587757
$709$	−15.5535	−0.584122	−0.292061	−	0.956400i	$-0.594341\pi$
$709$	−15.5535	−0.584122	−0.292061	+	0.956400i	$0.594341\pi$
$710$	−24.6438	−0.924866
$711$	17.9304	0.672442
$712$	−4.06384	−0.152299
$713$	−11.8714	−0.444586
$714$	−21.7691	−0.814687
$715$	−82.9520	−3.10223
$716$	7.90177	0.295303
$717$	−8.78060	−0.327917
$718$	8.77055	0.327314
$719$	−10.3709	−0.386770	−0.193385	−	0.981123i	$-0.561947\pi$
$719$	−10.3709	−0.386770	−0.193385	+	0.981123i	$0.561947\pi$
$720$	−8.03173	−0.299325
$721$	17.8194	0.663629
$722$	−17.5786	−0.654209
$723$	−15.2641	−0.567677
$724$	22.9301	0.852192
$725$	−3.94280	−0.146432
$726$	−77.2808	−2.86816
$727$	−19.4143	−0.720035	−0.360018	−	0.932945i	$-0.617229\pi$
$727$	−19.4143	−0.720035	−0.360018	+	0.932945i	$0.617229\pi$
$728$	−10.8150	−0.400832
$729$	−27.0184	−1.00068
$730$	−36.7309	−1.35947
$731$	−26.2255	−0.969985
$732$	−17.6906	−0.653863
$733$	−13.1053	−0.484056	−0.242028	−	0.970269i	$-0.577813\pi$
$733$	−13.1053	−0.484056	−0.242028	+	0.970269i	$0.577813\pi$
$734$	−11.2179	−0.414059
$735$	−11.9306	−0.440066
$736$	1.82865	0.0674050
$737$	17.4109	0.641337
$738$	−18.9872	−0.698928
$739$	11.7900	0.433703	0.216851	−	0.976205i	$-0.430421\pi$
$739$	11.7900	0.433703	0.216851	+	0.976205i	$0.430421\pi$
$740$	10.3123	0.379088
$741$	−13.8777	−0.509812
$742$	−8.93999	−0.328197
$743$	5.47382	0.200815	0.100407	−	0.994946i	$-0.467985\pi$
$743$	5.47382	0.200815	0.100407	+	0.994946i	$0.467985\pi$
$744$	15.9031	0.583036
$745$	−37.8323	−1.38607
$746$	−28.5222	−1.04427
$747$	−16.4249	−0.600955
$748$	−25.4675	−0.931184
$749$	−24.0759	−0.879714
$750$	−18.6017	−0.679237
$751$	19.7618	0.721119	0.360559	−	0.932736i	$-0.382586\pi$
$751$	19.7618	0.721119	0.360559	+	0.932736i	$0.382586\pi$
$752$	−3.83195	−0.139737
$753$	−24.1858	−0.881379
$754$	8.66271	0.315477
$755$	57.9146	2.10773
$756$	−0.00571474	−0.000207843 0
$757$	−25.0728	−0.911286	−0.455643	−	0.890163i	$-0.650591\pi$
$757$	−25.0728	−0.911286	−0.455643	+	0.890163i	$0.650591\pi$
$758$	1.81207	0.0658174
$759$	29.2199	1.06062
$760$	3.19075	0.115741
$761$	−7.95425	−0.288341	−0.144171	−	0.989553i	$-0.546051\pi$
$761$	−7.95425	−0.288341	−0.144171	+	0.989553i	$0.546051\pi$
$762$	−28.7268	−1.04066
$763$	−5.38812	−0.195063
$764$	15.5730	0.563410
$765$	−31.3589	−1.13378
$766$	0.236314	0.00853838
$767$	−30.3357	−1.09536
$768$	−2.44970	−0.0883959
$769$	−4.20179	−0.151520	−0.0757602	−	0.997126i	$-0.524138\pi$
$769$	−4.20179	−0.151520	−0.0757602	+	0.997126i	$0.524138\pi$
$770$	39.7329	1.43187
$771$	−34.5043	−1.24264
$772$	0.264860	0.00953251
$773$	1.30104	0.0467951	0.0233975	−	0.999726i	$-0.492552\pi$
$773$	1.30104	0.0467951	0.0233975	+	0.999726i	$0.492552\pi$
$774$	−20.1577	−0.724555
$775$	−14.0402	−0.504338
$776$	−6.75097	−0.242346
$777$	21.4835	0.770716
$778$	−8.46353	−0.303432
$779$	7.54300	0.270256
$780$	−31.1534	−1.11547
$781$	−60.0625	−2.14921
$782$	7.13975	0.255317
$783$	0.00457744	0.000163584 0
$784$	−1.81974	−0.0649908
$785$	−0.876813	−0.0312948
$786$	10.7443	0.383236
$787$	53.7896	1.91739	0.958696	−	0.284432i	$-0.0918051\pi$
$787$	53.7896	1.91739	0.958696	+	0.284432i	$0.0918051\pi$
$788$	−12.1841	−0.434040
$789$	69.4417	2.47219
$790$	−15.9904	−0.568913
$791$	−34.6906	−1.23345
$792$	−19.5751	−0.695571
$793$	−34.3148	−1.21856
$794$	−19.3698	−0.687410
$795$	−25.7522	−0.913335
$796$	16.8013	0.595506
$797$	−29.4052	−1.04159	−0.520793	−	0.853683i	$-0.674364\pi$
$797$	−29.4052	−1.04159	−0.520793	+	0.853683i	$0.674364\pi$
$798$	6.64725	0.235310
$799$	−14.9614	−0.529295
$800$	2.16273	0.0764642
$801$	−12.1957	−0.430913
$802$	−2.87853	−0.101644
$803$	−89.5213	−3.15914
$804$	6.53880	0.230606
$805$	−11.1390	−0.392598
$806$	30.8476	1.08656
$807$	56.9087	2.00328
$808$	−0.327972	−0.0115380
$809$	35.4574	1.24662	0.623308	−	0.781976i	$-0.285788\pi$
$809$	35.4574	1.24662	0.623308	+	0.781976i	$0.285788\pi$
$810$	24.0787	0.846040
$811$	42.3222	1.48613	0.743066	−	0.669218i	$-0.233371\pi$
$811$	42.3222	1.48613	0.743066	+	0.669218i	$0.233371\pi$
$812$	−4.14932	−0.145613
$813$	−53.7131	−1.88380
$814$	25.1334	0.880925
$815$	−32.8560	−1.15090
$816$	−9.56454	−0.334826
$817$	8.00803	0.280166
$818$	−16.3002	−0.569922
$819$	−32.4562	−1.13411
$820$	16.9329	0.591321
$821$	−35.5605	−1.24107	−0.620536	−	0.784178i	$-0.713085\pi$
$821$	−35.5605	−1.24107	−0.620536	+	0.784178i	$0.713085\pi$
$822$	35.8142	1.24916
$823$	19.4345	0.677444	0.338722	−	0.940886i	$-0.390005\pi$
$823$	19.4345	0.677444	0.338722	+	0.940886i	$0.390005\pi$
$824$	7.82920	0.272743
$825$	34.5582	1.20316
$826$	14.5304	0.505577
$827$	−8.75877	−0.304572	−0.152286	−	0.988336i	$-0.548664\pi$
$827$	−8.75877	−0.304572	−0.152286	+	0.988336i	$0.548664\pi$
$828$	5.48783	0.190715
$829$	−6.46731	−0.224619	−0.112310	−	0.993673i	$-0.535825\pi$
$829$	−6.46731	−0.224619	−0.112310	+	0.993673i	$0.535825\pi$
$830$	14.6478	0.508432
$831$	−68.0398	−2.36027
$832$	−4.75174	−0.164737
$833$	−7.10495	−0.246172
$834$	1.67219	0.0579033
$835$	3.11755	0.107887
$836$	7.77657	0.268958
$837$	0.0163001	0.000563414 0
$838$	18.1398	0.626628
$839$	−1.53718	−0.0530694	−0.0265347	−	0.999648i	$-0.508447\pi$
$839$	−1.53718	−0.0530694	−0.0265347	+	0.999648i	$0.508447\pi$
$840$	14.9220	0.514859
$841$	−25.6764	−0.885395
$842$	−13.3419	−0.459792
$843$	−41.4188	−1.42654
$844$	−1.44362	−0.0496914
$845$	−25.6366	−0.881927
$846$	−11.4998	−0.395370
$847$	71.8017	2.46713
$848$	−3.92791	−0.134885
$849$	−54.2254	−1.86101
$850$	8.44413	0.289631
$851$	−7.04608	−0.241537
$852$	−22.5570	−0.772790
$853$	13.4851	0.461722	0.230861	−	0.972987i	$-0.425846\pi$
$853$	13.4851	0.461722	0.230861	+	0.972987i	$0.425846\pi$
$854$	16.4363	0.562440
$855$	9.57552	0.327476
$856$	−10.5781	−0.361551
$857$	−16.0303	−0.547585	−0.273793	−	0.961789i	$-0.588278\pi$
$857$	−16.0303	−0.547585	−0.273793	+	0.961789i	$0.588278\pi$
$858$	−75.9277	−2.59213
$859$	−11.5460	−0.393944	−0.196972	−	0.980409i	$-0.563111\pi$
$859$	−11.5460	−0.393944	−0.196972	+	0.980409i	$0.563111\pi$
$860$	17.9768	0.613003
$861$	35.2760	1.20220
$862$	26.3994	0.899169
$863$	−21.9320	−0.746573	−0.373286	−	0.927716i	$-0.621769\pi$
$863$	−21.9320	−0.746573	−0.373286	+	0.927716i	$0.621769\pi$
$864$	−0.00251085	−8.54209e−5 0
$865$	−6.56284	−0.223143
$866$	3.79957	0.129115
$867$	4.30132	0.146081
$868$	−14.7756	−0.501516
$869$	−38.9722	−1.32204
$870$	−11.9524	−0.405223
$871$	12.6835	0.429763
$872$	−2.36734	−0.0801684
$873$	−20.2598	−0.685691
$874$	−2.18014	−0.0737444
$875$	17.2828	0.584267
$876$	−33.6205	−1.13593
$877$	−12.1993	−0.411940	−0.205970	−	0.978558i	$-0.566035\pi$
$877$	−12.1993	−0.411940	−0.205970	+	0.978558i	$0.566035\pi$
$878$	−14.3589	−0.484590
$879$	−56.6531	−1.91086
$880$	17.4572	0.588482
$881$	1.76232	0.0593740	0.0296870	−	0.999559i	$-0.490549\pi$
$881$	1.76232	0.0593740	0.0296870	+	0.999559i	$0.490549\pi$
$882$	−5.46109	−0.183884
$883$	−13.0982	−0.440791	−0.220395	−	0.975411i	$-0.570735\pi$
$883$	−13.0982	−0.440791	−0.220395	+	0.975411i	$0.570735\pi$
$884$	−18.5526	−0.623991
$885$	41.8556	1.40696
$886$	1.42478	0.0478664
$887$	36.5090	1.22585	0.612926	−	0.790140i	$-0.289992\pi$
$887$	36.5090	1.22585	0.612926	+	0.790140i	$0.289992\pi$
$888$	9.43907	0.316755
$889$	26.6901	0.895158
$890$	10.8762	0.364570
$891$	58.6852	1.96603
$892$	1.43504	0.0480488
$893$	4.56850	0.152879
$894$	−34.6287	−1.15816
$895$	−21.1477	−0.706890
$896$	2.27602	0.0760364
$897$	21.2861	0.710723
$898$	−4.76063	−0.158864
$899$	11.8351	0.394721
$900$	6.49042	0.216347
$901$	−15.3360	−0.510917
$902$	41.2692	1.37411
$903$	37.4508	1.24628
$904$	−15.2418	−0.506934
$905$	−61.3686	−2.03996
$906$	53.0105	1.76115
$907$	16.8730	0.560260	0.280130	−	0.959962i	$-0.409622\pi$
$907$	16.8730	0.560260	0.280130	+	0.959962i	$0.409622\pi$
$908$	13.2451	0.439554
$909$	−0.984251	−0.0326455
$910$	28.9446	0.959505
$911$	−24.9358	−0.826161	−0.413081	−	0.910694i	$-0.635547\pi$
$911$	−24.9358	−0.826161	−0.413081	+	0.910694i	$0.635547\pi$
$912$	2.92056	0.0967094
$913$	35.6999	1.18150
$914$	24.6893	0.816648
$915$	47.3458	1.56520
$916$	0.964996	0.0318844
$917$	−9.98253	−0.329652
$918$	−0.00980331	−0.000323557 0
$919$	−32.6782	−1.07795	−0.538977	−	0.842320i	$-0.681189\pi$
$919$	−32.6782	−1.07795	−0.538977	+	0.842320i	$0.681189\pi$
$920$	−4.89408	−0.161353
$921$	−24.8856	−0.820008
$922$	−9.58534	−0.315676
$923$	−43.7544	−1.44019
$924$	36.3683	1.19643
$925$	−8.33335	−0.273999
$926$	32.1030	1.05497
$927$	23.4956	0.771698
$928$	−1.82306	−0.0598449
$929$	−4.26095	−0.139797	−0.0698987	−	0.997554i	$-0.522268\pi$
$929$	−4.26095	−0.139797	−0.0698987	+	0.997554i	$0.522268\pi$
$930$	−42.5619	−1.39566
$931$	2.16952	0.0711031
$932$	−3.26796	−0.107046
$933$	14.0078	0.458595
$934$	12.5338	0.410119
$935$	68.1594	2.22905
$936$	−14.2601	−0.466106
$937$	−2.86973	−0.0937501	−0.0468750	−	0.998901i	$-0.514926\pi$
$937$	−2.86973	−0.0937501	−0.0468750	+	0.998901i	$0.514926\pi$
$938$	−6.07521	−0.198362
$939$	−51.1981	−1.67079
$940$	10.2556	0.334499
$941$	−31.7839	−1.03613	−0.518063	−	0.855342i	$-0.673347\pi$
$941$	−31.7839	−1.03613	−0.518063	+	0.855342i	$0.673347\pi$
$942$	−0.802565	−0.0261490
$943$	−11.5697	−0.376761
$944$	6.38413	0.207786
$945$	0.0152945	0.000497531 0
$946$	43.8134	1.42450
$947$	−21.5901	−0.701584	−0.350792	−	0.936453i	$-0.614088\pi$
$947$	−21.5901	−0.701584	−0.350792	+	0.936453i	$0.614088\pi$
$948$	−14.6364	−0.475367
$949$	−65.2145	−2.11695
$950$	−2.57844	−0.0836556
$951$	10.1169	0.328063
$952$	8.88643	0.288011
$953$	−33.3533	−1.08042	−0.540210	−	0.841530i	$-0.681655\pi$
$953$	−33.3533	−1.08042	−0.540210	+	0.841530i	$0.681655\pi$
$954$	−11.7878	−0.381643
$955$	−41.6784	−1.34868
$956$	3.58436	0.115926
$957$	−29.1305	−0.941657
$958$	−0.365441	−0.0118069
$959$	−33.2750	−1.07451
$960$	6.55620	0.211600
$961$	11.1443	0.359493
$962$	18.3092	0.590312
$963$	−31.7451	−1.02297
$964$	6.23100	0.200687
$965$	−0.708851	−0.0228187
$966$	−10.1958	−0.328043
$967$	−18.6083	−0.598403	−0.299202	−	0.954190i	$-0.596720\pi$
$967$	−18.6083	−0.598403	−0.299202	+	0.954190i	$0.596720\pi$
$968$	31.5471	1.01396
$969$	11.4030	0.366316
$970$	18.0678	0.580122
$971$	−20.8883	−0.670339	−0.335170	−	0.942158i	$-0.608794\pi$
$971$	−20.8883	−0.670339	−0.335170	+	0.942158i	$0.608794\pi$
$972$	22.0473	0.707167
$973$	−1.55364	−0.0498073
$974$	27.5887	0.884000
$975$	25.1749	0.806244
$976$	7.22153	0.231156
$977$	−3.44158	−0.110106	−0.0550530	−	0.998483i	$-0.517533\pi$
$977$	−3.44158	−0.110106	−0.0550530	+	0.998483i	$0.517533\pi$
$978$	−30.0738	−0.961654
$979$	26.5076	0.847188
$980$	4.87023	0.155574
$981$	−7.10446	−0.226828
$982$	−17.6219	−0.562339
$983$	−4.83926	−0.154348	−0.0771742	−	0.997018i	$-0.524590\pi$
$983$	−4.83926	−0.154348	−0.0771742	+	0.997018i	$0.524590\pi$
$984$	15.4990	0.494090
$985$	32.6086	1.03900
$986$	−7.11791	−0.226681
$987$	21.3653	0.680064
$988$	5.66508	0.180230
$989$	−12.2830	−0.390576
$990$	52.3894	1.66505
$991$	−24.4676	−0.777240	−0.388620	−	0.921398i	$-0.627048\pi$
$991$	−24.4676	−0.777240	−0.388620	+	0.921398i	$0.627048\pi$
$992$	−6.49186	−0.206117
$993$	−27.1579	−0.861830
$994$	20.9577	0.664739
$995$	−44.9657	−1.42551
$996$	13.4074	0.424831
$997$	−13.3195	−0.421832	−0.210916	−	0.977504i	$-0.567645\pi$
$997$	−13.3195	−0.421832	−0.210916	+	0.977504i	$0.567645\pi$
$998$	21.3209	0.674901
$999$	0.00967470	0.000306094 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.e.1.9	✓	70

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.e.1.9	✓	70	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.44970 q^{3} +1.00000 q^{4} -2.67633 q^{5} -2.44970 q^{6} +2.27602 q^{7} +1.00000 q^{8} +3.00102 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.44970 q^{3} +1.00000 q^{4} -2.67633 q^{5} -2.44970 q^{6} +2.27602 q^{7} +1.00000 q^{8} +3.00102 q^{9} -2.67633 q^{10} -6.52281 q^{11} -2.44970 q^{12} -4.75174 q^{13} +2.27602 q^{14} +6.55620 q^{15} +1.00000 q^{16} +3.90437 q^{17} +3.00102 q^{18} -1.19221 q^{19} -2.67633 q^{20} -5.57556 q^{21} -6.52281 q^{22} +1.82865 q^{23} -2.44970 q^{24} +2.16273 q^{25} -4.75174 q^{26} -0.00251085 q^{27} +2.27602 q^{28} -1.82306 q^{29} +6.55620 q^{30} -6.49186 q^{31} +1.00000 q^{32} +15.9789 q^{33} +3.90437 q^{34} -6.09137 q^{35} +3.00102 q^{36} -3.85316 q^{37} -1.19221 q^{38} +11.6403 q^{39} -2.67633 q^{40} -6.32690 q^{41} -5.57556 q^{42} -6.71695 q^{43} -6.52281 q^{44} -8.03173 q^{45} +1.82865 q^{46} -3.83195 q^{47} -2.44970 q^{48} -1.81974 q^{49} +2.16273 q^{50} -9.56454 q^{51} -4.75174 q^{52} -3.92791 q^{53} -0.00251085 q^{54} +17.4572 q^{55} +2.27602 q^{56} +2.92056 q^{57} -1.82306 q^{58} +6.38413 q^{59} +6.55620 q^{60} +7.22153 q^{61} -6.49186 q^{62} +6.83039 q^{63} +1.00000 q^{64} +12.7172 q^{65} +15.9789 q^{66} -2.66923 q^{67} +3.90437 q^{68} -4.47965 q^{69} -6.09137 q^{70} +9.20807 q^{71} +3.00102 q^{72} +13.7243 q^{73} -3.85316 q^{74} -5.29805 q^{75} -1.19221 q^{76} -14.8460 q^{77} +11.6403 q^{78} +5.97476 q^{79} -2.67633 q^{80} -8.99692 q^{81} -6.32690 q^{82} -5.47309 q^{83} -5.57556 q^{84} -10.4494 q^{85} -6.71695 q^{86} +4.46595 q^{87} -6.52281 q^{88} -4.06384 q^{89} -8.03173 q^{90} -10.8150 q^{91} +1.82865 q^{92} +15.9031 q^{93} -3.83195 q^{94} +3.19075 q^{95} -2.44970 q^{96} -6.75097 q^{97} -1.81974 q^{98} -19.5751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * q^99

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