LMFDB - Embedded newform 6046.2.a.f.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, 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("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.39039 q^{3} +1.00000 q^{4} +0.762400 q^{5} -2.39039 q^{6} -3.68654 q^{7} +1.00000 q^{8} +2.71398 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.39039 q^{3} +1.00000 q^{4} +0.762400 q^{5} -2.39039 q^{6} -3.68654 q^{7} +1.00000 q^{8} +2.71398 q^{9} +0.762400 q^{10} +1.57765 q^{11} -2.39039 q^{12} +4.47685 q^{13} -3.68654 q^{14} -1.82244 q^{15} +1.00000 q^{16} -6.09503 q^{17} +2.71398 q^{18} +5.66985 q^{19} +0.762400 q^{20} +8.81229 q^{21} +1.57765 q^{22} -1.53390 q^{23} -2.39039 q^{24} -4.41875 q^{25} +4.47685 q^{26} +0.683707 q^{27} -3.68654 q^{28} +1.51082 q^{29} -1.82244 q^{30} +0.811037 q^{31} +1.00000 q^{32} -3.77120 q^{33} -6.09503 q^{34} -2.81062 q^{35} +2.71398 q^{36} -6.68159 q^{37} +5.66985 q^{38} -10.7014 q^{39} +0.762400 q^{40} +3.92454 q^{41} +8.81229 q^{42} +8.16724 q^{43} +1.57765 q^{44} +2.06914 q^{45} -1.53390 q^{46} -2.30434 q^{47} -2.39039 q^{48} +6.59061 q^{49} -4.41875 q^{50} +14.5695 q^{51} +4.47685 q^{52} -13.9354 q^{53} +0.683707 q^{54} +1.20280 q^{55} -3.68654 q^{56} -13.5532 q^{57} +1.51082 q^{58} -1.30051 q^{59} -1.82244 q^{60} -7.03368 q^{61} +0.811037 q^{62} -10.0052 q^{63} +1.00000 q^{64} +3.41315 q^{65} -3.77120 q^{66} -7.80466 q^{67} -6.09503 q^{68} +3.66661 q^{69} -2.81062 q^{70} +13.0109 q^{71} +2.71398 q^{72} +10.7556 q^{73} -6.68159 q^{74} +10.5625 q^{75} +5.66985 q^{76} -5.81608 q^{77} -10.7014 q^{78} -0.904168 q^{79} +0.762400 q^{80} -9.77626 q^{81} +3.92454 q^{82} +7.25579 q^{83} +8.81229 q^{84} -4.64685 q^{85} +8.16724 q^{86} -3.61145 q^{87} +1.57765 q^{88} +8.04225 q^{89} +2.06914 q^{90} -16.5041 q^{91} -1.53390 q^{92} -1.93870 q^{93} -2.30434 q^{94} +4.32269 q^{95} -2.39039 q^{96} -0.761190 q^{97} +6.59061 q^{98} +4.28171 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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
$3$	−2.39039	−1.38009	−0.690047	−	0.723765i	$-0.742410\pi$
$3$	−2.39039	−1.38009	−0.690047	+	0.723765i	$0.742410\pi$
$4$	1.00000	0.500000
$5$	0.762400	0.340956	0.170478	−	0.985362i	$-0.445469\pi$
$5$	0.762400	0.340956	0.170478	+	0.985362i	$0.445469\pi$
$6$	−2.39039	−0.975874
$7$	−3.68654	−1.39338	−0.696691	−	0.717371i	$-0.745345\pi$
$7$	−3.68654	−1.39338	−0.696691	+	0.717371i	$0.745345\pi$
$8$	1.00000	0.353553
$9$	2.71398	0.904659
$10$	0.762400	0.241092
$11$	1.57765	0.475679	0.237840	−	0.971304i	$-0.423561\pi$
$11$	1.57765	0.475679	0.237840	+	0.971304i	$0.423561\pi$
$12$	−2.39039	−0.690047
$13$	4.47685	1.24165	0.620827	−	0.783948i	$-0.286797\pi$
$13$	4.47685	1.24165	0.620827	+	0.783948i	$0.286797\pi$
$14$	−3.68654	−0.985270
$15$	−1.82244	−0.470551
$16$	1.00000	0.250000
$17$	−6.09503	−1.47826	−0.739131	−	0.673562i	$-0.764763\pi$
$17$	−6.09503	−1.47826	−0.739131	+	0.673562i	$0.764763\pi$
$18$	2.71398	0.639691
$19$	5.66985	1.30075	0.650376	−	0.759612i	$-0.274611\pi$
$19$	5.66985	1.30075	0.650376	+	0.759612i	$0.274611\pi$
$20$	0.762400	0.170478
$21$	8.81229	1.92300
$22$	1.57765	0.336356
$23$	−1.53390	−0.319839	−0.159920	−	0.987130i	$-0.551124\pi$
$23$	−1.53390	−0.319839	−0.159920	+	0.987130i	$0.551124\pi$
$24$	−2.39039	−0.487937
$25$	−4.41875	−0.883749
$26$	4.47685	0.877982
$27$	0.683707	0.131580
$28$	−3.68654	−0.696691
$29$	1.51082	0.280552	0.140276	−	0.990112i	$-0.455201\pi$
$29$	1.51082	0.280552	0.140276	+	0.990112i	$0.455201\pi$
$30$	−1.82244	−0.332730
$31$	0.811037	0.145667	0.0728333	−	0.997344i	$-0.476796\pi$
$31$	0.811037	0.145667	0.0728333	+	0.997344i	$0.476796\pi$
$32$	1.00000	0.176777
$33$	−3.77120	−0.656482
$34$	−6.09503	−1.04529
$35$	−2.81062	−0.475082
$36$	2.71398	0.452330
$37$	−6.68159	−1.09845	−0.549223	−	0.835676i	$-0.685076\pi$
$37$	−6.68159	−1.09845	−0.549223	+	0.835676i	$0.685076\pi$
$38$	5.66985	0.919771
$39$	−10.7014	−1.71360
$40$	0.762400	0.120546
$41$	3.92454	0.612910	0.306455	−	0.951885i	$-0.400857\pi$
$41$	3.92454	0.612910	0.306455	+	0.951885i	$0.400857\pi$
$42$	8.81229	1.35977
$43$	8.16724	1.24549	0.622746	−	0.782424i	$-0.286017\pi$
$43$	8.16724	1.24549	0.622746	+	0.782424i	$0.286017\pi$
$44$	1.57765	0.237840
$45$	2.06914	0.308449
$46$	−1.53390	−0.226161
$47$	−2.30434	−0.336123	−0.168061	−	0.985777i	$-0.553751\pi$
$47$	−2.30434	−0.336123	−0.168061	+	0.985777i	$0.553751\pi$
$48$	−2.39039	−0.345023
$49$	6.59061	0.941516
$50$	−4.41875	−0.624905
$51$	14.5695	2.04014
$52$	4.47685	0.620827
$53$	−13.9354	−1.91417	−0.957087	−	0.289802i	$-0.906410\pi$
$53$	−13.9354	−1.91417	−0.957087	+	0.289802i	$0.906410\pi$
$54$	0.683707	0.0930408
$55$	1.20280	0.162186
$56$	−3.68654	−0.492635
$57$	−13.5532	−1.79516
$58$	1.51082	0.198380
$59$	−1.30051	−0.169313	−0.0846563	−	0.996410i	$-0.526979\pi$
$59$	−1.30051	−0.169313	−0.0846563	+	0.996410i	$0.526979\pi$
$60$	−1.82244	−0.235275
$61$	−7.03368	−0.900571	−0.450285	−	0.892885i	$-0.648678\pi$
$61$	−7.03368	−0.900571	−0.450285	+	0.892885i	$0.648678\pi$
$62$	0.811037	0.103002
$63$	−10.0052	−1.26054
$64$	1.00000	0.125000
$65$	3.41315	0.423349
$66$	−3.77120	−0.464203
$67$	−7.80466	−0.953491	−0.476746	−	0.879041i	$-0.658184\pi$
$67$	−7.80466	−0.953491	−0.476746	+	0.879041i	$0.658184\pi$
$68$	−6.09503	−0.739131
$69$	3.66661	0.441408
$70$	−2.81062	−0.335934
$71$	13.0109	1.54410	0.772052	−	0.635559i	$-0.219230\pi$
$71$	13.0109	1.54410	0.772052	+	0.635559i	$0.219230\pi$
$72$	2.71398	0.319845
$73$	10.7556	1.25885	0.629423	−	0.777063i	$-0.283291\pi$
$73$	10.7556	1.25885	0.629423	+	0.777063i	$0.283291\pi$
$74$	−6.68159	−0.776719
$75$	10.5625	1.21966
$76$	5.66985	0.650376
$77$	−5.81608	−0.662804
$78$	−10.7014	−1.21170
$79$	−0.904168	−0.101727	−0.0508634	−	0.998706i	$-0.516197\pi$
$79$	−0.904168	−0.101727	−0.0508634	+	0.998706i	$0.516197\pi$
$80$	0.762400	0.0852389
$81$	−9.77626	−1.08625
$82$	3.92454	0.433392
$83$	7.25579	0.796426	0.398213	−	0.917293i	$-0.369630\pi$
$83$	7.25579	0.796426	0.398213	+	0.917293i	$0.369630\pi$
$84$	8.81229	0.961500
$85$	−4.64685	−0.504022
$86$	8.16724	0.880696
$87$	−3.61145	−0.387189
$88$	1.57765	0.168178
$89$	8.04225	0.852477	0.426238	−	0.904611i	$-0.359838\pi$
$89$	8.04225	0.852477	0.426238	+	0.904611i	$0.359838\pi$
$90$	2.06914	0.218106
$91$	−16.5041	−1.73010
$92$	−1.53390	−0.159920
$93$	−1.93870	−0.201034
$94$	−2.30434	−0.237675
$95$	4.32269	0.443499
$96$	−2.39039	−0.243968
$97$	−0.761190	−0.0772871	−0.0386436	−	0.999253i	$-0.512304\pi$
$97$	−0.761190	−0.0772871	−0.0386436	+	0.999253i	$0.512304\pi$
$98$	6.59061	0.665752
$99$	4.28171	0.430328
$100$	−4.41875	−0.441875
$101$	−7.32480	−0.728845	−0.364422	−	0.931234i	$-0.618734\pi$
$101$	−7.32480	−0.728845	−0.364422	+	0.931234i	$0.618734\pi$
$102$	14.5695	1.44260
$103$	1.56624	0.154326	0.0771630	−	0.997018i	$-0.475414\pi$
$103$	1.56624	0.154326	0.0771630	+	0.997018i	$0.475414\pi$
$104$	4.47685	0.438991
$105$	6.71849	0.655657
$106$	−13.9354	−1.35352
$107$	5.67301	0.548431	0.274216	−	0.961668i	$-0.411582\pi$
$107$	5.67301	0.548431	0.274216	+	0.961668i	$0.411582\pi$
$108$	0.683707	0.0657898
$109$	13.4265	1.28602	0.643010	−	0.765857i	$-0.277685\pi$
$109$	13.4265	1.28602	0.643010	+	0.765857i	$0.277685\pi$
$110$	1.20280	0.114683
$111$	15.9716	1.51596
$112$	−3.68654	−0.348346
$113$	19.6129	1.84503	0.922515	−	0.385962i	$-0.126130\pi$
$113$	19.6129	1.84503	0.922515	+	0.385962i	$0.126130\pi$
$114$	−13.5532	−1.26937
$115$	−1.16944	−0.109051
$116$	1.51082	0.140276
$117$	12.1501	1.12327
$118$	−1.30051	−0.119722
$119$	22.4696	2.05978
$120$	−1.82244	−0.166365
$121$	−8.51102	−0.773729
$122$	−7.03368	−0.636800
$123$	−9.38118	−0.845873
$124$	0.811037	0.0728333
$125$	−7.18085	−0.642275
$126$	−10.0052	−0.891334
$127$	−12.5365	−1.11243	−0.556217	−	0.831037i	$-0.687748\pi$
$127$	−12.5365	−1.11243	−0.556217	+	0.831037i	$0.687748\pi$
$128$	1.00000	0.0883883
$129$	−19.5229	−1.71890
$130$	3.41315	0.299353
$131$	−3.34526	−0.292277	−0.146138	−	0.989264i	$-0.546685\pi$
$131$	−3.34526	−0.292277	−0.146138	+	0.989264i	$0.546685\pi$
$132$	−3.77120	−0.328241
$133$	−20.9021	−1.81245
$134$	−7.80466	−0.674220
$135$	0.521258	0.0448628
$136$	−6.09503	−0.522644
$137$	−1.91050	−0.163225	−0.0816127	−	0.996664i	$-0.526007\pi$
$137$	−1.91050	−0.163225	−0.0816127	+	0.996664i	$0.526007\pi$
$138$	3.66661	0.312123
$139$	5.03838	0.427350	0.213675	−	0.976905i	$-0.431457\pi$
$139$	5.03838	0.427350	0.213675	+	0.976905i	$0.431457\pi$
$140$	−2.81062	−0.237541
$141$	5.50828	0.463881
$142$	13.0109	1.09185
$143$	7.06290	0.590629
$144$	2.71398	0.226165
$145$	1.15185	0.0956559
$146$	10.7556	0.890139
$147$	−15.7541	−1.29938
$148$	−6.68159	−0.549223
$149$	12.3890	1.01495	0.507473	−	0.861668i	$-0.330580\pi$
$149$	12.3890	1.01495	0.507473	+	0.861668i	$0.330580\pi$
$150$	10.5625	0.862428
$151$	18.8108	1.53080	0.765400	−	0.643555i	$-0.222541\pi$
$151$	18.8108	1.53080	0.765400	+	0.643555i	$0.222541\pi$
$152$	5.66985	0.459885
$153$	−16.5418	−1.33732
$154$	−5.81608	−0.468673
$155$	0.618335	0.0496659
$156$	−10.7014	−0.856799
$157$	−2.74649	−0.219194	−0.109597	−	0.993976i	$-0.534956\pi$
$157$	−2.74649	−0.219194	−0.109597	+	0.993976i	$0.534956\pi$
$158$	−0.904168	−0.0719317
$159$	33.3111	2.64174
$160$	0.762400	0.0602730
$161$	5.65478	0.445659
$162$	−9.77626	−0.768096
$163$	1.41018	0.110453	0.0552267	−	0.998474i	$-0.482412\pi$
$163$	1.41018	0.110453	0.0552267	+	0.998474i	$0.482412\pi$
$164$	3.92454	0.306455
$165$	−2.87517	−0.223831
$166$	7.25579	0.563158
$167$	2.30657	0.178487	0.0892437	−	0.996010i	$-0.471555\pi$
$167$	2.30657	0.178487	0.0892437	+	0.996010i	$0.471555\pi$
$168$	8.81229	0.679883
$169$	7.04216	0.541704
$170$	−4.64685	−0.356397
$171$	15.3878	1.17674
$172$	8.16724	0.622746
$173$	20.4254	1.55291	0.776456	−	0.630171i	$-0.217015\pi$
$173$	20.4254	1.55291	0.776456	+	0.630171i	$0.217015\pi$
$174$	−3.61145	−0.273784
$175$	16.2899	1.23140
$176$	1.57765	0.118920
$177$	3.10874	0.233667
$178$	8.04225	0.602792
$179$	−4.77720	−0.357065	−0.178532	−	0.983934i	$-0.557135\pi$
$179$	−4.77720	−0.357065	−0.178532	+	0.983934i	$0.557135\pi$
$180$	2.06914	0.154224
$181$	7.81999	0.581255	0.290628	−	0.956836i	$-0.406136\pi$
$181$	7.81999	0.581255	0.290628	+	0.956836i	$0.406136\pi$
$182$	−16.5041	−1.22336
$183$	16.8133	1.24287
$184$	−1.53390	−0.113080
$185$	−5.09404	−0.374522
$186$	−1.93870	−0.142152
$187$	−9.61582	−0.703179
$188$	−2.30434	−0.168061
$189$	−2.52052	−0.183341
$190$	4.32269	0.313601
$191$	19.5984	1.41809	0.709046	−	0.705162i	$-0.249126\pi$
$191$	19.5984	1.41809	0.709046	+	0.705162i	$0.249126\pi$
$192$	−2.39039	−0.172512
$193$	0.0591449	0.00425734	0.00212867	−	0.999998i	$-0.499322\pi$
$193$	0.0591449	0.00425734	0.00212867	+	0.999998i	$0.499322\pi$
$194$	−0.761190	−0.0546503
$195$	−8.15876	−0.584261
$196$	6.59061	0.470758
$197$	4.51017	0.321336	0.160668	−	0.987008i	$-0.448635\pi$
$197$	4.51017	0.321336	0.160668	+	0.987008i	$0.448635\pi$
$198$	4.28171	0.304288
$199$	21.9251	1.55423	0.777113	−	0.629361i	$-0.216683\pi$
$199$	21.9251	1.55423	0.777113	+	0.629361i	$0.216683\pi$
$200$	−4.41875	−0.312453
$201$	18.6562	1.31591
$202$	−7.32480	−0.515371
$203$	−5.56971	−0.390917
$204$	14.5695	1.02007
$205$	2.99207	0.208975
$206$	1.56624	0.109125
$207$	−4.16296	−0.289346
$208$	4.47685	0.310413
$209$	8.94504	0.618741
$210$	6.71849	0.463620
$211$	13.4452	0.925604	0.462802	−	0.886462i	$-0.346844\pi$
$211$	13.4452	0.925604	0.462802	+	0.886462i	$0.346844\pi$
$212$	−13.9354	−0.957087
$213$	−31.1011	−2.13101
$214$	5.67301	0.387799
$215$	6.22671	0.424658
$216$	0.683707	0.0465204
$217$	−2.98993	−0.202969
$218$	13.4265	0.909354
$219$	−25.7101	−1.73733
$220$	1.20280	0.0810928
$221$	−27.2865	−1.83549
$222$	15.9716	1.07195
$223$	12.4427	0.833222	0.416611	−	0.909085i	$-0.363218\pi$
$223$	12.4427	0.833222	0.416611	+	0.909085i	$0.363218\pi$
$224$	−3.68654	−0.246318
$225$	−11.9924	−0.799492
$226$	19.6129	1.30463
$227$	−5.98097	−0.396971	−0.198485	−	0.980104i	$-0.563602\pi$
$227$	−5.98097	−0.396971	−0.198485	+	0.980104i	$0.563602\pi$
$228$	−13.5532	−0.897580
$229$	−10.9331	−0.722480	−0.361240	−	0.932473i	$-0.617646\pi$
$229$	−10.9331	−0.722480	−0.361240	+	0.932473i	$0.617646\pi$
$230$	−1.16944	−0.0771108
$231$	13.9027	0.914731
$232$	1.51082	0.0991902
$233$	18.3966	1.20520	0.602602	−	0.798042i	$-0.294131\pi$
$233$	18.3966	1.20520	0.602602	+	0.798042i	$0.294131\pi$
$234$	12.1501	0.794274
$235$	−1.75683	−0.114603
$236$	−1.30051	−0.0846563
$237$	2.16132	0.140393
$238$	22.4696	1.45649
$239$	−0.0152298	−0.000985136 0	−0.000492568	−	1.00000i	$-0.500157\pi$
$239$	−0.0152298	−0.000985136 0	−0.000492568		1.00000i	$0.500157\pi$
$240$	−1.82244	−0.117638
$241$	−16.1634	−1.04117	−0.520587	−	0.853809i	$-0.674287\pi$
$241$	−16.1634	−1.04117	−0.520587	+	0.853809i	$0.674287\pi$
$242$	−8.51102	−0.547109
$243$	21.3180	1.36755
$244$	−7.03368	−0.450285
$245$	5.02468	0.321015
$246$	−9.38118	−0.598122
$247$	25.3830	1.61508
$248$	0.811037	0.0515009
$249$	−17.3442	−1.09914
$250$	−7.18085	−0.454157
$251$	0.996978	0.0629287	0.0314644	−	0.999505i	$-0.489983\pi$
$251$	0.996978	0.0629287	0.0314644	+	0.999505i	$0.489983\pi$
$252$	−10.0052	−0.630268
$253$	−2.41995	−0.152141
$254$	−12.5365	−0.786609
$255$	11.1078	0.695597
$256$	1.00000	0.0625000
$257$	−15.8029	−0.985757	−0.492878	−	0.870098i	$-0.664055\pi$
$257$	−15.8029	−0.985757	−0.492878	+	0.870098i	$0.664055\pi$
$258$	−19.5229	−1.21544
$259$	24.6320	1.53056
$260$	3.41315	0.211674
$261$	4.10033	0.253804
$262$	−3.34526	−0.206671
$263$	25.9754	1.60171	0.800854	−	0.598859i	$-0.204379\pi$
$263$	25.9754	1.60171	0.800854	+	0.598859i	$0.204379\pi$
$264$	−3.77120	−0.232102
$265$	−10.6243	−0.652648
$266$	−20.9021	−1.28159
$267$	−19.2241	−1.17650
$268$	−7.80466	−0.476746
$269$	10.9520	0.667758	0.333879	−	0.942616i	$-0.391642\pi$
$269$	10.9520	0.667758	0.333879	+	0.942616i	$0.391642\pi$
$270$	0.521258	0.0317228
$271$	23.1454	1.40599	0.702993	−	0.711197i	$-0.251847\pi$
$271$	23.1454	1.40599	0.702993	+	0.711197i	$0.251847\pi$
$272$	−6.09503	−0.369565
$273$	39.4513	2.38770
$274$	−1.91050	−0.115418
$275$	−6.97124	−0.420381
$276$	3.66661	0.220704
$277$	5.97177	0.358809	0.179404	−	0.983775i	$-0.442583\pi$
$277$	5.97177	0.358809	0.179404	+	0.983775i	$0.442583\pi$
$278$	5.03838	0.302182
$279$	2.20114	0.131779
$280$	−2.81062	−0.167967
$281$	−26.8667	−1.60273	−0.801366	−	0.598174i	$-0.795893\pi$
$281$	−26.8667	−1.60273	−0.801366	+	0.598174i	$0.795893\pi$
$282$	5.50828	0.328013
$283$	−12.2294	−0.726960	−0.363480	−	0.931602i	$-0.618412\pi$
$283$	−12.2294	−0.726960	−0.363480	+	0.931602i	$0.618412\pi$
$284$	13.0109	0.772052
$285$	−10.3329	−0.612070
$286$	7.06290	0.417638
$287$	−14.4680	−0.854018
$288$	2.71398	0.159923
$289$	20.1494	1.18526
$290$	1.15185	0.0676389
$291$	1.81954	0.106663
$292$	10.7556	0.629423
$293$	25.8506	1.51021	0.755105	−	0.655604i	$-0.227586\pi$
$293$	25.8506	1.51021	0.755105	+	0.655604i	$0.227586\pi$
$294$	−15.7541	−0.918801
$295$	−0.991512	−0.0577281
$296$	−6.68159	−0.388360
$297$	1.07865	0.0625897
$298$	12.3890	0.717675
$299$	−6.86702	−0.397130
$300$	10.5625	0.609828
$301$	−30.1089	−1.73545
$302$	18.8108	1.08244
$303$	17.5091	1.00587
$304$	5.66985	0.325188
$305$	−5.36248	−0.307055
$306$	−16.5418	−0.945630
$307$	−14.1372	−0.806852	−0.403426	−	0.915012i	$-0.632181\pi$
$307$	−14.1372	−0.806852	−0.403426	+	0.915012i	$0.632181\pi$
$308$	−5.81608	−0.331402
$309$	−3.74392	−0.212984
$310$	0.618335	0.0351191
$311$	21.3720	1.21190	0.605949	−	0.795504i	$-0.292794\pi$
$311$	21.3720	1.21190	0.605949	+	0.795504i	$0.292794\pi$
$312$	−10.7014	−0.605849
$313$	−26.0314	−1.47138	−0.735691	−	0.677317i	$-0.763142\pi$
$313$	−26.0314	−1.47138	−0.735691	+	0.677317i	$0.763142\pi$
$314$	−2.74649	−0.154993
$315$	−7.62796	−0.429787
$316$	−0.904168	−0.0508634
$317$	4.72941	0.265630	0.132815	−	0.991141i	$-0.457598\pi$
$317$	4.72941	0.265630	0.132815	+	0.991141i	$0.457598\pi$
$318$	33.3111	1.86799
$319$	2.38355	0.133453
$320$	0.762400	0.0426195
$321$	−13.5607	−0.756886
$322$	5.65478	0.315128
$323$	−34.5579	−1.92285
$324$	−9.77626	−0.543126
$325$	−19.7820	−1.09731
$326$	1.41018	0.0781024
$327$	−32.0945	−1.77483
$328$	3.92454	0.216696
$329$	8.49506	0.468348
$330$	−2.87517	−0.158273
$331$	4.17730	0.229605	0.114803	−	0.993388i	$-0.463376\pi$
$331$	4.17730	0.229605	0.114803	+	0.993388i	$0.463376\pi$
$332$	7.25579	0.398213
$333$	−18.1337	−0.993720
$334$	2.30657	0.126210
$335$	−5.95027	−0.325098
$336$	8.81229	0.480750
$337$	11.3999	0.620992	0.310496	−	0.950575i	$-0.399505\pi$
$337$	11.3999	0.620992	0.310496	+	0.950575i	$0.399505\pi$
$338$	7.04216	0.383043
$339$	−46.8826	−2.54631
$340$	−4.64685	−0.252011
$341$	1.27953	0.0692906
$342$	15.3878	0.832079
$343$	1.50923	0.0814909
$344$	8.16724	0.440348
$345$	2.79543	0.150501
$346$	20.4254	1.09807
$347$	8.89103	0.477296	0.238648	−	0.971106i	$-0.423296\pi$
$347$	8.89103	0.477296	0.238648	+	0.971106i	$0.423296\pi$
$348$	−3.61145	−0.193594
$349$	17.6602	0.945331	0.472666	−	0.881242i	$-0.343292\pi$
$349$	17.6602	0.945331	0.472666	+	0.881242i	$0.343292\pi$
$350$	16.2899	0.870732
$351$	3.06085	0.163376
$352$	1.57765	0.0840890
$353$	−4.47475	−0.238167	−0.119083	−	0.992884i	$-0.537996\pi$
$353$	−4.47475	−0.238167	−0.119083	+	0.992884i	$0.537996\pi$
$354$	3.10874	0.165228
$355$	9.91948	0.526471
$356$	8.04225	0.426238
$357$	−53.7111	−2.84270
$358$	−4.77720	−0.252483
$359$	23.6996	1.25081	0.625407	−	0.780298i	$-0.284933\pi$
$359$	23.6996	1.25081	0.625407	+	0.780298i	$0.284933\pi$
$360$	2.06914	0.109053
$361$	13.1472	0.691957
$362$	7.81999	0.411009
$363$	20.3447	1.06782
$364$	−16.5041	−0.865050
$365$	8.20006	0.429211
$366$	16.8133	0.878844
$367$	−16.7817	−0.875996	−0.437998	−	0.898976i	$-0.644312\pi$
$367$	−16.7817	−0.875996	−0.437998	+	0.898976i	$0.644312\pi$
$368$	−1.53390	−0.0799599
$369$	10.6511	0.554474
$370$	−5.09404	−0.264827
$371$	51.3734	2.66718
$372$	−1.93870	−0.100517
$373$	−0.891755	−0.0461733	−0.0230867	−	0.999733i	$-0.507349\pi$
$373$	−0.891755	−0.0461733	−0.0230867	+	0.999733i	$0.507349\pi$
$374$	−9.61582	−0.497222
$375$	17.1651	0.886400
$376$	−2.30434	−0.118837
$377$	6.76371	0.348349
$378$	−2.52052	−0.129641
$379$	2.62415	0.134793	0.0673967	−	0.997726i	$-0.478531\pi$
$379$	2.62415	0.134793	0.0673967	+	0.997726i	$0.478531\pi$
$380$	4.32269	0.221749
$381$	29.9671	1.53526
$382$	19.5984	1.00274
$383$	−12.5602	−0.641795	−0.320898	−	0.947114i	$-0.603985\pi$
$383$	−12.5602	−0.641795	−0.320898	+	0.947114i	$0.603985\pi$
$384$	−2.39039	−0.121984
$385$	−4.43418	−0.225987
$386$	0.0591449	0.00301040
$387$	22.1657	1.12675
$388$	−0.761190	−0.0386436
$389$	0.676024	0.0342758	0.0171379	−	0.999853i	$-0.494545\pi$
$389$	0.676024	0.0342758	0.0171379	+	0.999853i	$0.494545\pi$
$390$	−8.15876	−0.413135
$391$	9.34914	0.472806
$392$	6.59061	0.332876
$393$	7.99649	0.403370
$394$	4.51017	0.227219
$395$	−0.689338	−0.0346843
$396$	4.28171	0.215164
$397$	−20.4484	−1.02628	−0.513139	−	0.858306i	$-0.671517\pi$
$397$	−20.4484	−1.02628	−0.513139	+	0.858306i	$0.671517\pi$
$398$	21.9251	1.09900
$399$	49.9643	2.50135
$400$	−4.41875	−0.220937
$401$	16.0855	0.803272	0.401636	−	0.915799i	$-0.368442\pi$
$401$	16.0855	0.803272	0.401636	+	0.915799i	$0.368442\pi$
$402$	18.6562	0.930487
$403$	3.63089	0.180868
$404$	−7.32480	−0.364422
$405$	−7.45342	−0.370363
$406$	−5.56971	−0.276420
$407$	−10.5412	−0.522509
$408$	14.5695	0.721298
$409$	16.5640	0.819039	0.409520	−	0.912301i	$-0.365696\pi$
$409$	16.5640	0.819039	0.409520	+	0.912301i	$0.365696\pi$
$410$	2.99207	0.147768
$411$	4.56686	0.225266
$412$	1.56624	0.0771630
$413$	4.79440	0.235917
$414$	−4.16296	−0.204598
$415$	5.53181	0.271546
$416$	4.47685	0.219495
$417$	−12.0437	−0.589783
$418$	8.94504	0.437516
$419$	37.5032	1.83215	0.916076	−	0.401006i	$-0.131339\pi$
$419$	37.5032	1.83215	0.916076	+	0.401006i	$0.131339\pi$
$420$	6.71849	0.327829
$421$	11.8066	0.575421	0.287710	−	0.957717i	$-0.407106\pi$
$421$	11.8066	0.575421	0.287710	+	0.957717i	$0.407106\pi$
$422$	13.4452	0.654501
$423$	−6.25393	−0.304077
$424$	−13.9354	−0.676762
$425$	26.9324	1.30641
$426$	−31.1011	−1.50685
$427$	25.9300	1.25484
$428$	5.67301	0.274216
$429$	−16.8831	−0.815124
$430$	6.22671	0.300278
$431$	12.4027	0.597418	0.298709	−	0.954344i	$-0.403444\pi$
$431$	12.4027	0.597418	0.298709	+	0.954344i	$0.403444\pi$
$432$	0.683707	0.0328949
$433$	23.8173	1.14459	0.572294	−	0.820049i	$-0.306054\pi$
$433$	23.8173	1.14459	0.572294	+	0.820049i	$0.306054\pi$
$434$	−2.98993	−0.143521
$435$	−2.75337	−0.132014
$436$	13.4265	0.643010
$437$	−8.69696	−0.416032
$438$	−25.7101	−1.22848
$439$	−19.6432	−0.937520	−0.468760	−	0.883326i	$-0.655299\pi$
$439$	−19.6432	−0.937520	−0.468760	+	0.883326i	$0.655299\pi$
$440$	1.20280	0.0573413
$441$	17.8868	0.851751
$442$	−27.2865	−1.29789
$443$	13.0205	0.618623	0.309311	−	0.950961i	$-0.399901\pi$
$443$	13.0205	0.618623	0.309311	+	0.950961i	$0.399901\pi$
$444$	15.9716	0.757980
$445$	6.13141	0.290657
$446$	12.4427	0.589177
$447$	−29.6146	−1.40072
$448$	−3.68654	−0.174173
$449$	3.62308	0.170984	0.0854918	−	0.996339i	$-0.472754\pi$
$449$	3.62308	0.170984	0.0854918	+	0.996339i	$0.472754\pi$
$450$	−11.9924	−0.565326
$451$	6.19155	0.291548
$452$	19.6129	0.922515
$453$	−44.9652	−2.11265
$454$	−5.98097	−0.280701
$455$	−12.5827	−0.589887
$456$	−13.5532	−0.634685
$457$	−14.3878	−0.673031	−0.336516	−	0.941678i	$-0.609248\pi$
$457$	−14.3878	−0.673031	−0.336516	+	0.941678i	$0.609248\pi$
$458$	−10.9331	−0.510870
$459$	−4.16721	−0.194509
$460$	−1.16944	−0.0545255
$461$	−23.7968	−1.10833	−0.554164	−	0.832407i	$-0.686962\pi$
$461$	−23.7968	−1.10833	−0.554164	+	0.832407i	$0.686962\pi$
$462$	13.9027	0.646813
$463$	20.2483	0.941017	0.470508	−	0.882396i	$-0.344071\pi$
$463$	20.2483	0.941017	0.470508	+	0.882396i	$0.344071\pi$
$464$	1.51082	0.0701381
$465$	−1.47806	−0.0685436
$466$	18.3966	0.852208
$467$	−22.5327	−1.04269	−0.521343	−	0.853347i	$-0.674569\pi$
$467$	−22.5327	−1.04269	−0.521343	+	0.853347i	$0.674569\pi$
$468$	12.1501	0.561637
$469$	28.7722	1.32858
$470$	−1.75683	−0.0810365
$471$	6.56518	0.302508
$472$	−1.30051	−0.0598610
$473$	12.8851	0.592455
$474$	2.16132	0.0992725
$475$	−25.0536	−1.14954
$476$	22.4696	1.02989
$477$	−37.8203	−1.73167
$478$	−0.0152298	−0.000696596 0
$479$	−36.3982	−1.66308	−0.831538	−	0.555468i	$-0.812539\pi$
$479$	−36.3982	−1.66308	−0.831538	+	0.555468i	$0.812539\pi$
$480$	−1.82244	−0.0831824
$481$	−29.9125	−1.36389
$482$	−16.1634	−0.736221
$483$	−13.5171	−0.615051
$484$	−8.51102	−0.386865
$485$	−0.580331	−0.0263515
$486$	21.3180	0.967003
$487$	10.5691	0.478930	0.239465	−	0.970905i	$-0.423028\pi$
$487$	10.5691	0.478930	0.239465	+	0.970905i	$0.423028\pi$
$488$	−7.03368	−0.318400
$489$	−3.37087	−0.152436
$490$	5.02468	0.226992
$491$	7.02955	0.317239	0.158620	−	0.987340i	$-0.449296\pi$
$491$	7.02955	0.317239	0.158620	+	0.987340i	$0.449296\pi$
$492$	−9.38118	−0.422936
$493$	−9.20849	−0.414730
$494$	25.3830	1.14204
$495$	3.26437	0.146723
$496$	0.811037	0.0364167
$497$	−47.9651	−2.15153
$498$	−17.3442	−0.777211
$499$	29.6479	1.32722	0.663611	−	0.748078i	$-0.269023\pi$
$499$	29.6479	1.32722	0.663611	+	0.748078i	$0.269023\pi$
$500$	−7.18085	−0.321137
$501$	−5.51360	−0.246329
$502$	0.996978	0.0444973
$503$	−22.1655	−0.988309	−0.494154	−	0.869374i	$-0.664522\pi$
$503$	−22.1655	−0.988309	−0.494154	+	0.869374i	$0.664522\pi$
$504$	−10.0052	−0.445667
$505$	−5.58443	−0.248504
$506$	−2.41995	−0.107580
$507$	−16.8335	−0.747603
$508$	−12.5365	−0.556217
$509$	−17.3511	−0.769073	−0.384536	−	0.923110i	$-0.625639\pi$
$509$	−17.3511	−0.769073	−0.384536	+	0.923110i	$0.625639\pi$
$510$	11.1078	0.491861
$511$	−39.6510	−1.75406
$512$	1.00000	0.0441942
$513$	3.87652	0.171152
$514$	−15.8029	−0.697035
$515$	1.19410	0.0526183
$516$	−19.5229	−0.859448
$517$	−3.63545	−0.159887
$518$	24.6320	1.08227
$519$	−48.8247	−2.14316
$520$	3.41315	0.149676
$521$	−9.37099	−0.410551	−0.205275	−	0.978704i	$-0.565809\pi$
$521$	−9.37099	−0.410551	−0.205275	+	0.978704i	$0.565809\pi$
$522$	4.10033	0.179467
$523$	15.5880	0.681618	0.340809	−	0.940133i	$-0.389299\pi$
$523$	15.5880	0.681618	0.340809	+	0.940133i	$0.389299\pi$
$524$	−3.34526	−0.146138
$525$	−38.9393	−1.69945
$526$	25.9754	1.13258
$527$	−4.94330	−0.215333
$528$	−3.77120	−0.164121
$529$	−20.6472	−0.897703
$530$	−10.6243	−0.461492
$531$	−3.52957	−0.153170
$532$	−20.9021	−0.906223
$533$	17.5695	0.761022
$534$	−19.2241	−0.831910
$535$	4.32511	0.186991
$536$	−7.80466	−0.337110
$537$	11.4194	0.492783
$538$	10.9520	0.472176
$539$	10.3977	0.447860
$540$	0.521258	0.0224314
$541$	−26.3062	−1.13099	−0.565495	−	0.824752i	$-0.691315\pi$
$541$	−26.3062	−1.13099	−0.565495	+	0.824752i	$0.691315\pi$
$542$	23.1454	0.994182
$543$	−18.6928	−0.802187
$544$	−6.09503	−0.261322
$545$	10.2363	0.438476
$546$	39.4513	1.68836
$547$	−11.1780	−0.477937	−0.238969	−	0.971027i	$-0.576809\pi$
$547$	−11.1780	−0.477937	−0.238969	+	0.971027i	$0.576809\pi$
$548$	−1.91050	−0.0816127
$549$	−19.0893	−0.814710
$550$	−6.97124	−0.297255
$551$	8.56612	0.364929
$552$	3.66661	0.156061
$553$	3.33326	0.141744
$554$	5.97177	0.253716
$555$	12.1768	0.516875
$556$	5.03838	0.213675
$557$	32.4234	1.37382	0.686912	−	0.726740i	$-0.258966\pi$
$557$	32.4234	1.37382	0.686912	+	0.726740i	$0.258966\pi$
$558$	2.20114	0.0931816
$559$	36.5635	1.54647
$560$	−2.81062	−0.118770
$561$	22.9856	0.970453
$562$	−26.8667	−1.13330
$563$	−8.44783	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$563$	−8.44783	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$564$	5.50828	0.231941
$565$	14.9529	0.629073
$566$	−12.2294	−0.514038
$567$	36.0406	1.51356
$568$	13.0109	0.545923
$569$	33.5068	1.40468	0.702339	−	0.711842i	$-0.252139\pi$
$569$	33.5068	1.40468	0.702339	+	0.711842i	$0.252139\pi$
$570$	−10.3329	−0.432799
$571$	−8.35183	−0.349513	−0.174757	−	0.984612i	$-0.555914\pi$
$571$	−8.35183	−0.349513	−0.174757	+	0.984612i	$0.555914\pi$
$572$	7.06290	0.295315
$573$	−46.8479	−1.95710
$574$	−14.4680	−0.603882
$575$	6.77790	0.282658
$576$	2.71398	0.113082
$577$	4.38337	0.182482	0.0912410	−	0.995829i	$-0.470917\pi$
$577$	4.38337	0.182482	0.0912410	+	0.995829i	$0.470917\pi$
$578$	20.1494	0.838103
$579$	−0.141380	−0.00587553
$580$	1.15185	0.0478280
$581$	−26.7488	−1.10973
$582$	1.81954	0.0754225
$583$	−21.9852	−0.910533
$584$	10.7556	0.445069
$585$	9.26321	0.382986
$586$	25.8506	1.06788
$587$	12.4430	0.513577	0.256788	−	0.966468i	$-0.417336\pi$
$587$	12.4430	0.513577	0.256788	+	0.966468i	$0.417336\pi$
$588$	−15.7541	−0.649690
$589$	4.59846	0.189476
$590$	−0.991512	−0.0408199
$591$	−10.7811	−0.443474
$592$	−6.68159	−0.274612
$593$	10.0085	0.410999	0.205499	−	0.978657i	$-0.434118\pi$
$593$	10.0085	0.410999	0.205499	+	0.978657i	$0.434118\pi$
$594$	1.07865	0.0442576
$595$	17.1308	0.702295
$596$	12.3890	0.507473
$597$	−52.4095	−2.14498
$598$	−6.86702	−0.280813
$599$	−31.7536	−1.29742	−0.648709	−	0.761037i	$-0.724691\pi$
$599$	−31.7536	−1.29742	−0.648709	+	0.761037i	$0.724691\pi$
$600$	10.5625	0.431214
$601$	−16.6503	−0.679180	−0.339590	−	0.940574i	$-0.610288\pi$
$601$	−16.6503	−0.679180	−0.339590	+	0.940574i	$0.610288\pi$
$602$	−30.1089	−1.22715
$603$	−21.1817	−0.862584
$604$	18.8108	0.765400
$605$	−6.48880	−0.263807
$606$	17.5091	0.711260
$607$	31.3882	1.27401	0.637005	−	0.770860i	$-0.280173\pi$
$607$	31.3882	1.27401	0.637005	+	0.770860i	$0.280173\pi$
$608$	5.66985	0.229943
$609$	13.3138	0.539502
$610$	−5.36248	−0.217120
$611$	−10.3162	−0.417348
$612$	−16.5418	−0.668661
$613$	5.05727	0.204261	0.102131	−	0.994771i	$-0.467434\pi$
$613$	5.05727	0.204261	0.102131	+	0.994771i	$0.467434\pi$
$614$	−14.1372	−0.570530
$615$	−7.15221	−0.288405
$616$	−5.81608	−0.234336
$617$	−16.5442	−0.666043	−0.333022	−	0.942919i	$-0.608068\pi$
$617$	−16.5442	−0.666043	−0.333022	+	0.942919i	$0.608068\pi$
$618$	−3.74392	−0.150603
$619$	−3.67853	−0.147853	−0.0739263	−	0.997264i	$-0.523553\pi$
$619$	−3.67853	−0.147853	−0.0739263	+	0.997264i	$0.523553\pi$
$620$	0.618335	0.0248329
$621$	−1.04874	−0.0420843
$622$	21.3720	0.856941
$623$	−29.6481	−1.18783
$624$	−10.7014	−0.428400
$625$	16.6190	0.664762
$626$	−26.0314	−1.04042
$627$	−21.3822	−0.853921
$628$	−2.74649	−0.109597
$629$	40.7245	1.62379
$630$	−7.62796	−0.303905
$631$	28.9782	1.15360	0.576801	−	0.816885i	$-0.304301\pi$
$631$	28.9782	1.15360	0.576801	+	0.816885i	$0.304301\pi$
$632$	−0.904168	−0.0359659
$633$	−32.1393	−1.27742
$634$	4.72941	0.187829
$635$	−9.55782	−0.379291
$636$	33.3111	1.32087
$637$	29.5052	1.16904
$638$	2.38355	0.0943655
$639$	35.3112	1.39689
$640$	0.762400	0.0301365
$641$	−28.8735	−1.14043	−0.570217	−	0.821494i	$-0.693141\pi$
$641$	−28.8735	−1.14043	−0.570217	+	0.821494i	$0.693141\pi$
$642$	−13.5607	−0.535199
$643$	−45.7825	−1.80549	−0.902743	−	0.430181i	$-0.858450\pi$
$643$	−45.7825	−1.80549	−0.902743	+	0.430181i	$0.858450\pi$
$644$	5.65478	0.222829
$645$	−14.8843	−0.586068
$646$	−34.5579	−1.35966
$647$	11.1956	0.440145	0.220072	−	0.975484i	$-0.429371\pi$
$647$	11.1956	0.440145	0.220072	+	0.975484i	$0.429371\pi$
$648$	−9.77626	−0.384048
$649$	−2.05176	−0.0805385
$650$	−19.7820	−0.775916
$651$	7.14710	0.280117
$652$	1.41018	0.0552267
$653$	−17.3974	−0.680813	−0.340406	−	0.940278i	$-0.610565\pi$
$653$	−17.3974	−0.680813	−0.340406	+	0.940278i	$0.610565\pi$
$654$	−32.0945	−1.25499
$655$	−2.55043	−0.0996535
$656$	3.92454	0.153227
$657$	29.1904	1.13883
$658$	8.49506	0.331172
$659$	12.0408	0.469045	0.234522	−	0.972111i	$-0.424647\pi$
$659$	12.0408	0.469045	0.234522	+	0.972111i	$0.424647\pi$
$660$	−2.87517	−0.111916
$661$	34.4764	1.34098	0.670488	−	0.741920i	$-0.266085\pi$
$661$	34.4764	1.34098	0.670488	+	0.741920i	$0.266085\pi$
$662$	4.17730	0.162355
$663$	65.2255	2.53315
$664$	7.25579	0.281579
$665$	−15.9358	−0.617964
$666$	−18.1337	−0.702666
$667$	−2.31744	−0.0897317
$668$	2.30657	0.0892437
$669$	−29.7429	−1.14993
$670$	−5.95027	−0.229879
$671$	−11.0967	−0.428383
$672$	8.81229	0.339941
$673$	10.8196	0.417064	0.208532	−	0.978016i	$-0.433132\pi$
$673$	10.8196	0.417064	0.208532	+	0.978016i	$0.433132\pi$
$674$	11.3999	0.439107
$675$	−3.02113	−0.116283
$676$	7.04216	0.270852
$677$	0.711773	0.0273557	0.0136778	−	0.999906i	$-0.495646\pi$
$677$	0.711773	0.0273557	0.0136778	+	0.999906i	$0.495646\pi$
$678$	−46.8826	−1.80052
$679$	2.80616	0.107691
$680$	−4.64685	−0.178199
$681$	14.2969	0.547857
$682$	1.27953	0.0489959
$683$	−22.1217	−0.846463	−0.423231	−	0.906022i	$-0.639104\pi$
$683$	−22.1217	−0.846463	−0.423231	+	0.906022i	$0.639104\pi$
$684$	15.3878	0.588369
$685$	−1.45657	−0.0556526
$686$	1.50923	0.0576228
$687$	26.1344	0.997090
$688$	8.16724	0.311373
$689$	−62.3866	−2.37674
$690$	2.79543	0.106420
$691$	13.6698	0.520023	0.260012	−	0.965606i	$-0.416274\pi$
$691$	13.6698	0.520023	0.260012	+	0.965606i	$0.416274\pi$
$692$	20.4254	0.776456
$693$	−15.7847	−0.599611
$694$	8.89103	0.337499
$695$	3.84126	0.145707
$696$	−3.61145	−0.136892
$697$	−23.9202	−0.906041
$698$	17.6602	0.668450
$699$	−43.9752	−1.66330
$700$	16.2899	0.615701
$701$	15.5041	0.585581	0.292790	−	0.956177i	$-0.405416\pi$
$701$	15.5041	0.585581	0.292790	+	0.956177i	$0.405416\pi$
$702$	3.06085	0.115524
$703$	−37.8836	−1.42881
$704$	1.57765	0.0594599
$705$	4.19951	0.158163
$706$	−4.47475	−0.168409
$707$	27.0032	1.01556
$708$	3.10874	0.116834
$709$	−20.6610	−0.775939	−0.387969	−	0.921672i	$-0.626823\pi$
$709$	−20.6610	−0.775939	−0.387969	+	0.921672i	$0.626823\pi$
$710$	9.91948	0.372271
$711$	−2.45389	−0.0920281
$712$	8.04225	0.301396
$713$	−1.24405	−0.0465899
$714$	−53.7111	−2.01009
$715$	5.38475	0.201378
$716$	−4.77720	−0.178532
$717$	0.0364053	0.00135958
$718$	23.6996	0.884460
$719$	20.1067	0.749854	0.374927	−	0.927054i	$-0.377668\pi$
$719$	20.1067	0.749854	0.374927	+	0.927054i	$0.377668\pi$
$720$	2.06914	0.0771122
$721$	−5.77400	−0.215035
$722$	13.1472	0.489287
$723$	38.6368	1.43692
$724$	7.81999	0.290628
$725$	−6.67593	−0.247938
$726$	20.3447	0.755062
$727$	12.8652	0.477143	0.238572	−	0.971125i	$-0.423321\pi$
$727$	12.8652	0.477143	0.238572	+	0.971125i	$0.423321\pi$
$728$	−16.5041	−0.611682
$729$	−21.6296	−0.801095
$730$	8.20006	0.303498
$731$	−49.7796	−1.84116
$732$	16.8133	0.621436
$733$	10.5577	0.389957	0.194978	−	0.980808i	$-0.437536\pi$
$733$	10.5577	0.389957	0.194978	+	0.980808i	$0.437536\pi$
$734$	−16.7817	−0.619423
$735$	−12.0110	−0.443031
$736$	−1.53390	−0.0565402
$737$	−12.3130	−0.453556
$738$	10.6511	0.392072
$739$	14.5799	0.536331	0.268166	−	0.963373i	$-0.413583\pi$
$739$	14.5799	0.536331	0.268166	+	0.963373i	$0.413583\pi$
$740$	−5.09404	−0.187261
$741$	−60.6754	−2.22897
$742$	51.3734	1.88598
$743$	−10.0383	−0.368271	−0.184135	−	0.982901i	$-0.558948\pi$
$743$	−10.0383	−0.368271	−0.184135	+	0.982901i	$0.558948\pi$
$744$	−1.93870	−0.0710761
$745$	9.44537	0.346052
$746$	−0.891755	−0.0326495
$747$	19.6920	0.720494
$748$	−9.61582	−0.351589
$749$	−20.9138	−0.764174
$750$	17.1651	0.626779
$751$	30.9025	1.12765	0.563825	−	0.825894i	$-0.309329\pi$
$751$	30.9025	1.12765	0.563825	+	0.825894i	$0.309329\pi$
$752$	−2.30434	−0.0840307
$753$	−2.38317	−0.0868475
$754$	6.76371	0.246320
$755$	14.3413	0.521935
$756$	−2.52052	−0.0916703
$757$	27.3107	0.992624	0.496312	−	0.868144i	$-0.334687\pi$
$757$	27.3107	0.992624	0.496312	+	0.868144i	$0.334687\pi$
$758$	2.62415	0.0953133
$759$	5.78464	0.209969
$760$	4.32269	0.156801
$761$	32.4367	1.17583	0.587916	−	0.808922i	$-0.299949\pi$
$761$	32.4367	1.17583	0.587916	+	0.808922i	$0.299949\pi$
$762$	29.9671	1.08559
$763$	−49.4972	−1.79192
$764$	19.5984	0.709046
$765$	−12.6114	−0.455968
$766$	−12.5602	−0.453818
$767$	−5.82220	−0.210228
$768$	−2.39039	−0.0862559
$769$	1.20431	0.0434285	0.0217143	−	0.999764i	$-0.493088\pi$
$769$	1.20431	0.0434285	0.0217143	+	0.999764i	$0.493088\pi$
$770$	−4.43418	−0.159797
$771$	37.7751	1.36044
$772$	0.0591449	0.00212867
$773$	−25.4707	−0.916118	−0.458059	−	0.888922i	$-0.651455\pi$
$773$	−25.4707	−0.916118	−0.458059	+	0.888922i	$0.651455\pi$
$774$	22.1657	0.796730
$775$	−3.58377	−0.128733
$776$	−0.761190	−0.0273251
$777$	−58.8801	−2.11231
$778$	0.676024	0.0242366
$779$	22.2515	0.797244
$780$	−8.15876	−0.292131
$781$	20.5266	0.734499
$782$	9.34914	0.334325
$783$	1.03296	0.0369149
$784$	6.59061	0.235379
$785$	−2.09392	−0.0747353
$786$	7.99649	0.285225
$787$	46.6967	1.66456	0.832279	−	0.554357i	$-0.187036\pi$
$787$	46.6967	1.66456	0.832279	+	0.554357i	$0.187036\pi$
$788$	4.51017	0.160668
$789$	−62.0913	−2.21051
$790$	−0.689338	−0.0245255
$791$	−72.3039	−2.57083
$792$	4.28171	0.152144
$793$	−31.4887	−1.11820
$794$	−20.4484	−0.725688
$795$	25.3963	0.900716
$796$	21.9251	0.777113
$797$	−37.5880	−1.33144	−0.665719	−	0.746203i	$-0.731875\pi$
$797$	−37.5880	−1.33144	−0.665719	+	0.746203i	$0.731875\pi$
$798$	49.9643	1.76872
$799$	14.0450	0.496877
$800$	−4.41875	−0.156226
$801$	21.8265	0.771201
$802$	16.0855	0.567999
$803$	16.9686	0.598807
$804$	18.6562	0.657954
$805$	4.31120	0.151950
$806$	3.63089	0.127893
$807$	−26.1797	−0.921568
$808$	−7.32480	−0.257686
$809$	−9.65071	−0.339301	−0.169650	−	0.985504i	$-0.554264\pi$
$809$	−9.65071	−0.339301	−0.169650	+	0.985504i	$0.554264\pi$
$810$	−7.45342	−0.261887
$811$	36.3042	1.27481	0.637407	−	0.770528i	$-0.280007\pi$
$811$	36.3042	1.27481	0.637407	+	0.770528i	$0.280007\pi$
$812$	−5.56971	−0.195458
$813$	−55.3267	−1.94039
$814$	−10.5412	−0.369469
$815$	1.07512	0.0376597
$816$	14.5695	0.510035
$817$	46.3070	1.62008
$818$	16.5640	0.579148
$819$	−44.7917	−1.56515
$820$	2.99207	0.104487
$821$	3.54173	0.123607	0.0618036	−	0.998088i	$-0.480315\pi$
$821$	3.54173	0.123607	0.0618036	+	0.998088i	$0.480315\pi$
$822$	4.56686	0.159287
$823$	−1.73273	−0.0603992	−0.0301996	−	0.999544i	$-0.509614\pi$
$823$	−1.73273	−0.0603992	−0.0301996	+	0.999544i	$0.509614\pi$
$824$	1.56624	0.0545625
$825$	16.6640	0.580166
$826$	4.79440	0.166819
$827$	4.88000	0.169694	0.0848471	−	0.996394i	$-0.472960\pi$
$827$	4.88000	0.169694	0.0848471	+	0.996394i	$0.472960\pi$
$828$	−4.16296	−0.144673
$829$	−36.4691	−1.26662	−0.633311	−	0.773897i	$-0.718305\pi$
$829$	−36.4691	−1.26662	−0.633311	+	0.773897i	$0.718305\pi$
$830$	5.53181	0.192012
$831$	−14.2749	−0.495190
$832$	4.47685	0.155207
$833$	−40.1700	−1.39181
$834$	−12.0437	−0.417040
$835$	1.75853	0.0608563
$836$	8.94504	0.309371
$837$	0.554512	0.0191667
$838$	37.5032	1.29553
$839$	−29.2926	−1.01129	−0.505647	−	0.862740i	$-0.668746\pi$
$839$	−29.2926	−1.01129	−0.505647	+	0.862740i	$0.668746\pi$
$840$	6.71849	0.231810
$841$	−26.7174	−0.921290
$842$	11.8066	0.406884
$843$	64.2219	2.21192
$844$	13.4452	0.462802
$845$	5.36894	0.184697
$846$	−6.25393	−0.215015
$847$	31.3763	1.07810
$848$	−13.9354	−0.478543
$849$	29.2330	1.00327
$850$	26.9324	0.923773
$851$	10.2489	0.351327
$852$	−31.1011	−1.06550
$853$	−53.3985	−1.82833	−0.914165	−	0.405341i	$-0.867153\pi$
$853$	−53.3985	−1.82833	−0.914165	+	0.405341i	$0.867153\pi$
$854$	25.9300	0.887306
$855$	11.7317	0.401215
$856$	5.67301	0.193900
$857$	−5.00701	−0.171036	−0.0855181	−	0.996337i	$-0.527255\pi$
$857$	−5.00701	−0.171036	−0.0855181	+	0.996337i	$0.527255\pi$
$858$	−16.8831	−0.576380
$859$	−54.4377	−1.85739	−0.928695	−	0.370843i	$-0.879069\pi$
$859$	−54.4377	−1.85739	−0.928695	+	0.370843i	$0.879069\pi$
$860$	6.22671	0.212329
$861$	34.5841	1.17862
$862$	12.4027	0.422438
$863$	−21.1398	−0.719608	−0.359804	−	0.933028i	$-0.617156\pi$
$863$	−21.1398	−0.719608	−0.359804	+	0.933028i	$0.617156\pi$
$864$	0.683707	0.0232602
$865$	15.5723	0.529474
$866$	23.8173	0.809346
$867$	−48.1649	−1.63577
$868$	−2.98993	−0.101485
$869$	−1.42646	−0.0483894
$870$	−2.75337	−0.0933481
$871$	−34.9403	−1.18391
$872$	13.4265	0.454677
$873$	−2.06585	−0.0699185
$874$	−8.69696	−0.294179
$875$	26.4725	0.894935
$876$	−25.7101	−0.868663
$877$	7.27763	0.245748	0.122874	−	0.992422i	$-0.460789\pi$
$877$	7.27763	0.245748	0.122874	+	0.992422i	$0.460789\pi$
$878$	−19.6432	−0.662927
$879$	−61.7931	−2.08423
$880$	1.20280	0.0405464
$881$	−1.85264	−0.0624170	−0.0312085	−	0.999513i	$-0.509936\pi$
$881$	−1.85264	−0.0624170	−0.0312085	+	0.999513i	$0.509936\pi$
$882$	17.8868	0.602279
$883$	−19.3328	−0.650601	−0.325300	−	0.945611i	$-0.605465\pi$
$883$	−19.3328	−0.650601	−0.325300	+	0.945611i	$0.605465\pi$
$884$	−27.2865	−0.917745
$885$	2.37010	0.0796702
$886$	13.0205	0.437432
$887$	−5.74240	−0.192811	−0.0964055	−	0.995342i	$-0.530735\pi$
$887$	−5.74240	−0.192811	−0.0964055	+	0.995342i	$0.530735\pi$
$888$	15.9716	0.535973
$889$	46.2163	1.55005
$890$	6.13141	0.205525
$891$	−15.4235	−0.516707
$892$	12.4427	0.416611
$893$	−13.0653	−0.437213
$894$	−29.6146	−0.990459
$895$	−3.64214	−0.121743
$896$	−3.68654	−0.123159
$897$	16.4149	0.548077
$898$	3.62308	0.120904
$899$	1.22533	0.0408671
$900$	−11.9924	−0.399746
$901$	84.9366	2.82965
$902$	6.19155	0.206156
$903$	71.9721	2.39508
$904$	19.6129	0.652316
$905$	5.96196	0.198182
$906$	−44.9652	−1.49387
$907$	0.604932	0.0200864	0.0100432	−	0.999950i	$-0.496803\pi$
$907$	0.604932	0.0200864	0.0100432	+	0.999950i	$0.496803\pi$
$908$	−5.98097	−0.198485
$909$	−19.8793	−0.659356
$910$	−12.5827	−0.417113
$911$	20.0672	0.664857	0.332429	−	0.943128i	$-0.392132\pi$
$911$	20.0672	0.664857	0.332429	+	0.943128i	$0.392132\pi$
$912$	−13.5532	−0.448790
$913$	11.4471	0.378844
$914$	−14.3878	−0.475905
$915$	12.8184	0.423764
$916$	−10.9331	−0.361240
$917$	12.3325	0.407254
$918$	−4.16721	−0.137539
$919$	−52.8574	−1.74360	−0.871802	−	0.489858i	$-0.837049\pi$
$919$	−52.8574	−1.74360	−0.871802	+	0.489858i	$0.837049\pi$
$920$	−1.16944	−0.0385554
$921$	33.7934	1.11353
$922$	−23.7968	−0.783707
$923$	58.2476	1.91724
$924$	13.9027	0.457366
$925$	29.5242	0.970751
$926$	20.2483	0.665399
$927$	4.25073	0.139612
$928$	1.51082	0.0495951
$929$	−16.4744	−0.540508	−0.270254	−	0.962789i	$-0.587108\pi$
$929$	−16.4744	−0.540508	−0.270254	+	0.962789i	$0.587108\pi$
$930$	−1.47806	−0.0484676
$931$	37.3678	1.22468
$932$	18.3966	0.602602
$933$	−51.0876	−1.67253
$934$	−22.5327	−0.737291
$935$	−7.33110	−0.239753
$936$	12.1501	0.397137
$937$	−2.77510	−0.0906587	−0.0453294	−	0.998972i	$-0.514434\pi$
$937$	−2.77510	−0.0906587	−0.0453294	+	0.998972i	$0.514434\pi$
$938$	28.7722	0.939447
$939$	62.2253	2.03065
$940$	−1.75683	−0.0573015
$941$	−10.2840	−0.335249	−0.167625	−	0.985851i	$-0.553610\pi$
$941$	−10.2840	−0.335249	−0.167625	+	0.985851i	$0.553610\pi$
$942$	6.56518	0.213905
$943$	−6.01983	−0.196033
$944$	−1.30051	−0.0423281
$945$	−1.92164	−0.0625110
$946$	12.8851	0.418929
$947$	16.0836	0.522645	0.261323	−	0.965251i	$-0.415841\pi$
$947$	16.0836	0.522645	0.261323	+	0.965251i	$0.415841\pi$
$948$	2.16132	0.0701963
$949$	48.1511	1.56305
$950$	−25.0536	−0.812847
$951$	−11.3051	−0.366594
$952$	22.4696	0.728244
$953$	35.1512	1.13866	0.569330	−	0.822109i	$-0.307203\pi$
$953$	35.1512	1.13866	0.569330	+	0.822109i	$0.307203\pi$
$954$	−37.8203	−1.22448
$955$	14.9418	0.483507
$956$	−0.0152298	−0.000492568 0
$957$	−5.69761	−0.184178
$958$	−36.3982	−1.17597
$959$	7.04316	0.227435
$960$	−1.82244	−0.0588188
$961$	−30.3422	−0.978781
$962$	−29.9125	−0.964416
$963$	15.3964	0.496143
$964$	−16.1634	−0.520587
$965$	0.0450921	0.00145157
$966$	−13.5171	−0.434907
$967$	41.8521	1.34587	0.672937	−	0.739700i	$-0.265033\pi$
$967$	41.8521	1.34587	0.672937	+	0.739700i	$0.265033\pi$
$968$	−8.51102	−0.273555
$969$	82.6069	2.65372
$970$	−0.580331	−0.0186333
$971$	−40.9868	−1.31533	−0.657665	−	0.753311i	$-0.728456\pi$
$971$	−40.9868	−1.31533	−0.657665	+	0.753311i	$0.728456\pi$
$972$	21.3180	0.683774
$973$	−18.5742	−0.595462
$974$	10.5691	0.338655
$975$	47.2869	1.51439
$976$	−7.03368	−0.225143
$977$	27.1376	0.868208	0.434104	−	0.900863i	$-0.357065\pi$
$977$	27.1376	0.868208	0.434104	+	0.900863i	$0.357065\pi$
$978$	−3.37087	−0.107789
$979$	12.6879	0.405506
$980$	5.02468	0.160508
$981$	36.4391	1.16341
$982$	7.02955	0.224322
$983$	8.69458	0.277314	0.138657	−	0.990340i	$-0.455721\pi$
$983$	8.69458	0.277314	0.138657	+	0.990340i	$0.455721\pi$
$984$	−9.38118	−0.299061
$985$	3.43855	0.109561
$986$	−9.20849	−0.293258
$987$	−20.3065	−0.646364
$988$	25.3830	0.807542
$989$	−12.5277	−0.398358
$990$	3.26437	0.103749
$991$	17.5660	0.558003	0.279001	−	0.960291i	$-0.409997\pi$
$991$	17.5660	0.558003	0.279001	+	0.960291i	$0.409997\pi$
$992$	0.811037	0.0257505
$993$	−9.98538	−0.316877
$994$	−47.9651	−1.52136
$995$	16.7157	0.529922
$996$	−17.3442	−0.549571
$997$	51.0803	1.61773	0.808864	−	0.587995i	$-0.200083\pi$
$997$	51.0803	1.61773	0.808864	+	0.587995i	$0.200083\pi$
$998$	29.6479	0.938487
$999$	−4.56825	−0.144533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.10	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.10	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.39039 q^{3} +1.00000 q^{4} +0.762400 q^{5} -2.39039 q^{6} -3.68654 q^{7} +1.00000 q^{8} +2.71398 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.39039 q^{3} +1.00000 q^{4} +0.762400 q^{5} -2.39039 q^{6} -3.68654 q^{7} +1.00000 q^{8} +2.71398 q^{9} +0.762400 q^{10} +1.57765 q^{11} -2.39039 q^{12} +4.47685 q^{13} -3.68654 q^{14} -1.82244 q^{15} +1.00000 q^{16} -6.09503 q^{17} +2.71398 q^{18} +5.66985 q^{19} +0.762400 q^{20} +8.81229 q^{21} +1.57765 q^{22} -1.53390 q^{23} -2.39039 q^{24} -4.41875 q^{25} +4.47685 q^{26} +0.683707 q^{27} -3.68654 q^{28} +1.51082 q^{29} -1.82244 q^{30} +0.811037 q^{31} +1.00000 q^{32} -3.77120 q^{33} -6.09503 q^{34} -2.81062 q^{35} +2.71398 q^{36} -6.68159 q^{37} +5.66985 q^{38} -10.7014 q^{39} +0.762400 q^{40} +3.92454 q^{41} +8.81229 q^{42} +8.16724 q^{43} +1.57765 q^{44} +2.06914 q^{45} -1.53390 q^{46} -2.30434 q^{47} -2.39039 q^{48} +6.59061 q^{49} -4.41875 q^{50} +14.5695 q^{51} +4.47685 q^{52} -13.9354 q^{53} +0.683707 q^{54} +1.20280 q^{55} -3.68654 q^{56} -13.5532 q^{57} +1.51082 q^{58} -1.30051 q^{59} -1.82244 q^{60} -7.03368 q^{61} +0.811037 q^{62} -10.0052 q^{63} +1.00000 q^{64} +3.41315 q^{65} -3.77120 q^{66} -7.80466 q^{67} -6.09503 q^{68} +3.66661 q^{69} -2.81062 q^{70} +13.0109 q^{71} +2.71398 q^{72} +10.7556 q^{73} -6.68159 q^{74} +10.5625 q^{75} +5.66985 q^{76} -5.81608 q^{77} -10.7014 q^{78} -0.904168 q^{79} +0.762400 q^{80} -9.77626 q^{81} +3.92454 q^{82} +7.25579 q^{83} +8.81229 q^{84} -4.64685 q^{85} +8.16724 q^{86} -3.61145 q^{87} +1.57765 q^{88} +8.04225 q^{89} +2.06914 q^{90} -16.5041 q^{91} -1.53390 q^{92} -1.93870 q^{93} -2.30434 q^{94} +4.32269 q^{95} -2.39039 q^{96} -0.761190 q^{97} +6.59061 q^{98} +4.28171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

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