LMFDB - Embedded newform 6046.2.a.f.1.4

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			1.4
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.02873 q^{3} +1.00000 q^{4} -2.84681 q^{5} -3.02873 q^{6} -0.956843 q^{7} +1.00000 q^{8} +6.17323 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.02873 q^{3} +1.00000 q^{4} -2.84681 q^{5} -3.02873 q^{6} -0.956843 q^{7} +1.00000 q^{8} +6.17323 q^{9} -2.84681 q^{10} +4.35239 q^{11} -3.02873 q^{12} +4.15301 q^{13} -0.956843 q^{14} +8.62224 q^{15} +1.00000 q^{16} +5.17517 q^{17} +6.17323 q^{18} -5.94820 q^{19} -2.84681 q^{20} +2.89802 q^{21} +4.35239 q^{22} +5.31629 q^{23} -3.02873 q^{24} +3.10434 q^{25} +4.15301 q^{26} -9.61088 q^{27} -0.956843 q^{28} +5.63010 q^{29} +8.62224 q^{30} +3.78076 q^{31} +1.00000 q^{32} -13.1822 q^{33} +5.17517 q^{34} +2.72395 q^{35} +6.17323 q^{36} -2.05065 q^{37} -5.94820 q^{38} -12.5784 q^{39} -2.84681 q^{40} -9.32815 q^{41} +2.89802 q^{42} +10.0923 q^{43} +4.35239 q^{44} -17.5740 q^{45} +5.31629 q^{46} -8.73772 q^{47} -3.02873 q^{48} -6.08445 q^{49} +3.10434 q^{50} -15.6742 q^{51} +4.15301 q^{52} -13.8682 q^{53} -9.61088 q^{54} -12.3904 q^{55} -0.956843 q^{56} +18.0155 q^{57} +5.63010 q^{58} +3.91801 q^{59} +8.62224 q^{60} +8.08533 q^{61} +3.78076 q^{62} -5.90682 q^{63} +1.00000 q^{64} -11.8228 q^{65} -13.1822 q^{66} +7.76343 q^{67} +5.17517 q^{68} -16.1016 q^{69} +2.72395 q^{70} -11.8932 q^{71} +6.17323 q^{72} -6.70709 q^{73} -2.05065 q^{74} -9.40221 q^{75} -5.94820 q^{76} -4.16456 q^{77} -12.5784 q^{78} +15.8623 q^{79} -2.84681 q^{80} +10.5891 q^{81} -9.32815 q^{82} +8.80253 q^{83} +2.89802 q^{84} -14.7327 q^{85} +10.0923 q^{86} -17.0521 q^{87} +4.35239 q^{88} +18.2669 q^{89} -17.5740 q^{90} -3.97378 q^{91} +5.31629 q^{92} -11.4509 q^{93} -8.73772 q^{94} +16.9334 q^{95} -3.02873 q^{96} -2.90814 q^{97} -6.08445 q^{98} +26.8683 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.02873	−1.74864	−0.874320	−	0.485349i	$-0.838692\pi$
$3$	−3.02873	−1.74864	−0.874320	+	0.485349i	$0.838692\pi$
$4$	1.00000	0.500000
$5$	−2.84681	−1.27313	−0.636566	−	0.771222i	$-0.719646\pi$
$5$	−2.84681	−1.27313	−0.636566	+	0.771222i	$0.719646\pi$
$6$	−3.02873	−1.23648
$7$	−0.956843	−0.361653	−0.180826	−	0.983515i	$-0.557877\pi$
$7$	−0.956843	−0.361653	−0.180826	+	0.983515i	$0.557877\pi$
$8$	1.00000	0.353553
$9$	6.17323	2.05774
$10$	−2.84681	−0.900241
$11$	4.35239	1.31230	0.656148	−	0.754633i	$-0.272185\pi$
$11$	4.35239	1.31230	0.656148	+	0.754633i	$0.272185\pi$
$12$	−3.02873	−0.874320
$13$	4.15301	1.15184	0.575919	−	0.817507i	$-0.304644\pi$
$13$	4.15301	1.15184	0.575919	+	0.817507i	$0.304644\pi$
$14$	−0.956843	−0.255727
$15$	8.62224	2.22625
$16$	1.00000	0.250000
$17$	5.17517	1.25516	0.627581	−	0.778551i	$-0.284045\pi$
$17$	5.17517	1.25516	0.627581	+	0.778551i	$0.284045\pi$
$18$	6.17323	1.45504
$19$	−5.94820	−1.36461	−0.682306	−	0.731067i	$-0.739023\pi$
$19$	−5.94820	−1.36461	−0.682306	+	0.731067i	$0.739023\pi$
$20$	−2.84681	−0.636566
$21$	2.89802	0.632401
$22$	4.35239	0.927933
$23$	5.31629	1.10852	0.554262	−	0.832342i	$-0.313001\pi$
$23$	5.31629	1.10852	0.554262	+	0.832342i	$0.313001\pi$
$24$	−3.02873	−0.618238
$25$	3.10434	0.620867
$26$	4.15301	0.814473
$27$	−9.61088	−1.84961
$28$	−0.956843	−0.180826
$29$	5.63010	1.04548	0.522741	−	0.852491i	$-0.324909\pi$
$29$	5.63010	1.04548	0.522741	+	0.852491i	$0.324909\pi$
$30$	8.62224	1.57420
$31$	3.78076	0.679044	0.339522	−	0.940598i	$-0.389735\pi$
$31$	3.78076	0.679044	0.339522	+	0.940598i	$0.389735\pi$
$32$	1.00000	0.176777
$33$	−13.1822	−2.29473
$34$	5.17517	0.887534
$35$	2.72395	0.460432
$36$	6.17323	1.02887
$37$	−2.05065	−0.337124	−0.168562	−	0.985691i	$-0.553912\pi$
$37$	−2.05065	−0.337124	−0.168562	+	0.985691i	$0.553912\pi$
$38$	−5.94820	−0.964926
$39$	−12.5784	−2.01415
$40$	−2.84681	−0.450120
$41$	−9.32815	−1.45681	−0.728406	−	0.685145i	$-0.759739\pi$
$41$	−9.32815	−1.45681	−0.728406	+	0.685145i	$0.759739\pi$
$42$	2.89802	0.447175
$43$	10.0923	1.53905	0.769527	−	0.638614i	$-0.220492\pi$
$43$	10.0923	1.53905	0.769527	+	0.638614i	$0.220492\pi$
$44$	4.35239	0.656148
$45$	−17.5740	−2.61978
$46$	5.31629	0.783845
$47$	−8.73772	−1.27453	−0.637264	−	0.770646i	$-0.719934\pi$
$47$	−8.73772	−1.27453	−0.637264	+	0.770646i	$0.719934\pi$
$48$	−3.02873	−0.437160
$49$	−6.08445	−0.869207
$50$	3.10434	0.439019
$51$	−15.6742	−2.19483
$52$	4.15301	0.575919
$53$	−13.8682	−1.90494	−0.952469	−	0.304634i	$-0.901466\pi$
$53$	−13.8682	−1.90494	−0.952469	+	0.304634i	$0.901466\pi$
$54$	−9.61088	−1.30787
$55$	−12.3904	−1.67073
$56$	−0.956843	−0.127864
$57$	18.0155	2.38621
$58$	5.63010	0.739268
$59$	3.91801	0.510081	0.255040	−	0.966930i	$-0.417911\pi$
$59$	3.91801	0.510081	0.255040	+	0.966930i	$0.417911\pi$
$60$	8.62224	1.11313
$61$	8.08533	1.03522	0.517610	−	0.855617i	$-0.326822\pi$
$61$	8.08533	1.03522	0.517610	+	0.855617i	$0.326822\pi$
$62$	3.78076	0.480157
$63$	−5.90682	−0.744189
$64$	1.00000	0.125000
$65$	−11.8228	−1.46644
$66$	−13.1822	−1.62262
$67$	7.76343	0.948453	0.474227	−	0.880403i	$-0.342728\pi$
$67$	7.76343	0.948453	0.474227	+	0.880403i	$0.342728\pi$
$68$	5.17517	0.627581
$69$	−16.1016	−1.93841
$70$	2.72395	0.325575
$71$	−11.8932	−1.41146	−0.705729	−	0.708482i	$-0.749380\pi$
$71$	−11.8932	−1.41146	−0.705729	+	0.708482i	$0.749380\pi$
$72$	6.17323	0.727522
$73$	−6.70709	−0.785006	−0.392503	−	0.919751i	$-0.628391\pi$
$73$	−6.70709	−0.785006	−0.392503	+	0.919751i	$0.628391\pi$
$74$	−2.05065	−0.238383
$75$	−9.40221	−1.08567
$76$	−5.94820	−0.682306
$77$	−4.16456	−0.474595
$78$	−12.5784	−1.42422
$79$	15.8623	1.78465	0.892325	−	0.451393i	$-0.149073\pi$
$79$	15.8623	1.78465	0.892325	+	0.451393i	$0.149073\pi$
$80$	−2.84681	−0.318283
$81$	10.5891	1.17657
$82$	−9.32815	−1.03012
$83$	8.80253	0.966203	0.483102	−	0.875564i	$-0.339510\pi$
$83$	8.80253	0.966203	0.483102	+	0.875564i	$0.339510\pi$
$84$	2.89802	0.316200
$85$	−14.7327	−1.59799
$86$	10.0923	1.08828
$87$	−17.0521	−1.82817
$88$	4.35239	0.463966
$89$	18.2669	1.93629	0.968143	−	0.250400i	$-0.0805620\pi$
$89$	18.2669	1.93629	0.968143	+	0.250400i	$0.0805620\pi$
$90$	−17.5740	−1.85247
$91$	−3.97378	−0.416565
$92$	5.31629	0.554262
$93$	−11.4509	−1.18740
$94$	−8.73772	−0.901227
$95$	16.9334	1.73733
$96$	−3.02873	−0.309119
$97$	−2.90814	−0.295277	−0.147638	−	0.989041i	$-0.547167\pi$
$97$	−2.90814	−0.295277	−0.147638	+	0.989041i	$0.547167\pi$
$98$	−6.08445	−0.614622
$99$	26.8683	2.70037
$100$	3.10434	0.310434
$101$	−6.47943	−0.644727	−0.322363	−	0.946616i	$-0.604477\pi$
$101$	−6.47943	−0.644727	−0.322363	+	0.946616i	$0.604477\pi$
$102$	−15.6742	−1.55198
$103$	−6.61766	−0.652057	−0.326029	−	0.945360i	$-0.605711\pi$
$103$	−6.61766	−0.652057	−0.326029	+	0.945360i	$0.605711\pi$
$104$	4.15301	0.407236
$105$	−8.25013	−0.805130
$106$	−13.8682	−1.34700
$107$	14.0615	1.35938	0.679690	−	0.733499i	$-0.262114\pi$
$107$	14.0615	1.35938	0.679690	+	0.733499i	$0.262114\pi$
$108$	−9.61088	−0.924807
$109$	−7.43435	−0.712082	−0.356041	−	0.934470i	$-0.615874\pi$
$109$	−7.43435	−0.712082	−0.356041	+	0.934470i	$0.615874\pi$
$110$	−12.3904	−1.18138
$111$	6.21086	0.589509
$112$	−0.956843	−0.0904132
$113$	3.16786	0.298007	0.149004	−	0.988837i	$-0.452393\pi$
$113$	3.16786	0.298007	0.149004	+	0.988837i	$0.452393\pi$
$114$	18.0155	1.68731
$115$	−15.1345	−1.41130
$116$	5.63010	0.522741
$117$	25.6375	2.37019
$118$	3.91801	0.360682
$119$	−4.95182	−0.453933
$120$	8.62224	0.787099
$121$	7.94330	0.722118
$122$	8.08533	0.732011
$123$	28.2525	2.54744
$124$	3.78076	0.339522
$125$	5.39660	0.482686
$126$	−5.90682	−0.526221
$127$	14.1657	1.25700	0.628502	−	0.777808i	$-0.283668\pi$
$127$	14.1657	1.25700	0.628502	+	0.777808i	$0.283668\pi$
$128$	1.00000	0.0883883
$129$	−30.5668	−2.69125
$130$	−11.8228	−1.03693
$131$	−0.439718	−0.0384183	−0.0192092	−	0.999815i	$-0.506115\pi$
$131$	−0.439718	−0.0384183	−0.0192092	+	0.999815i	$0.506115\pi$
$132$	−13.1822	−1.14737
$133$	5.69150	0.493515
$134$	7.76343	0.670658
$135$	27.3604	2.35480
$136$	5.17517	0.443767
$137$	−12.6351	−1.07949	−0.539745	−	0.841829i	$-0.681479\pi$
$137$	−12.6351	−1.07949	−0.539745	+	0.841829i	$0.681479\pi$
$138$	−16.1016	−1.37066
$139$	−17.8716	−1.51585	−0.757927	−	0.652340i	$-0.773788\pi$
$139$	−17.8716	−1.51585	−0.757927	+	0.652340i	$0.773788\pi$
$140$	2.72395	0.230216
$141$	26.4642	2.22869
$142$	−11.8932	−0.998052
$143$	18.0755	1.51155
$144$	6.17323	0.514436
$145$	−16.0278	−1.33104
$146$	−6.70709	−0.555083
$147$	18.4282	1.51993
$148$	−2.05065	−0.168562
$149$	8.75706	0.717406	0.358703	−	0.933452i	$-0.383219\pi$
$149$	8.75706	0.717406	0.358703	+	0.933452i	$0.383219\pi$
$150$	−9.40221	−0.767687
$151$	17.8689	1.45415	0.727073	−	0.686560i	$-0.240880\pi$
$151$	17.8689	1.45415	0.727073	+	0.686560i	$0.240880\pi$
$152$	−5.94820	−0.482463
$153$	31.9475	2.58280
$154$	−4.16456	−0.335589
$155$	−10.7631	−0.864514
$156$	−12.5784	−1.00708
$157$	−5.28336	−0.421658	−0.210829	−	0.977523i	$-0.567616\pi$
$157$	−5.28336	−0.421658	−0.210829	+	0.977523i	$0.567616\pi$
$158$	15.8623	1.26194
$159$	42.0030	3.33105
$160$	−2.84681	−0.225060
$161$	−5.08686	−0.400901
$162$	10.5891	0.831958
$163$	3.06469	0.240045	0.120022	−	0.992771i	$-0.461703\pi$
$163$	3.06469	0.240045	0.120022	+	0.992771i	$0.461703\pi$
$164$	−9.32815	−0.728406
$165$	37.5273	2.92150
$166$	8.80253	0.683209
$167$	−5.66054	−0.438026	−0.219013	−	0.975722i	$-0.570284\pi$
$167$	−5.66054	−0.438026	−0.219013	+	0.975722i	$0.570284\pi$
$168$	2.89802	0.223587
$169$	4.24751	0.326731
$170$	−14.7327	−1.12995
$171$	−36.7196	−2.80802
$172$	10.0923	0.769527
$173$	−7.56463	−0.575128	−0.287564	−	0.957761i	$-0.592845\pi$
$173$	−7.56463	−0.575128	−0.287564	+	0.957761i	$0.592845\pi$
$174$	−17.0521	−1.29271
$175$	−2.97036	−0.224538
$176$	4.35239	0.328074
$177$	−11.8666	−0.891948
$178$	18.2669	1.36916
$179$	3.07304	0.229690	0.114845	−	0.993383i	$-0.463363\pi$
$179$	3.07304	0.229690	0.114845	+	0.993383i	$0.463363\pi$
$180$	−17.5740	−1.30989
$181$	10.7167	0.796566	0.398283	−	0.917262i	$-0.369606\pi$
$181$	10.7167	0.796566	0.398283	+	0.917262i	$0.369606\pi$
$182$	−3.97378	−0.294556
$183$	−24.4883	−1.81023
$184$	5.31629	0.391922
$185$	5.83780	0.429204
$186$	−11.4509	−0.839622
$187$	22.5243	1.64714
$188$	−8.73772	−0.637264
$189$	9.19610	0.668918
$190$	16.9334	1.22848
$191$	−5.34510	−0.386758	−0.193379	−	0.981124i	$-0.561945\pi$
$191$	−5.34510	−0.386758	−0.193379	+	0.981124i	$0.561945\pi$
$192$	−3.02873	−0.218580
$193$	−20.5254	−1.47745	−0.738725	−	0.674007i	$-0.764572\pi$
$193$	−20.5254	−1.47745	−0.738725	+	0.674007i	$0.764572\pi$
$194$	−2.90814	−0.208792
$195$	35.8083	2.56428
$196$	−6.08445	−0.434604
$197$	17.7201	1.26250	0.631252	−	0.775578i	$-0.282541\pi$
$197$	17.7201	1.26250	0.631252	+	0.775578i	$0.282541\pi$
$198$	26.8683	1.90945
$199$	19.2815	1.36683	0.683415	−	0.730030i	$-0.260494\pi$
$199$	19.2815	1.36683	0.683415	+	0.730030i	$0.260494\pi$
$200$	3.10434	0.219510
$201$	−23.5134	−1.65850
$202$	−6.47943	−0.455891
$203$	−5.38712	−0.378102
$204$	−15.6742	−1.09741
$205$	26.5555	1.85472
$206$	−6.61766	−0.461074
$207$	32.8187	2.28106
$208$	4.15301	0.287960
$209$	−25.8889	−1.79077
$210$	−8.25013	−0.569313
$211$	7.67448	0.528333	0.264166	−	0.964477i	$-0.414903\pi$
$211$	7.67448	0.528333	0.264166	+	0.964477i	$0.414903\pi$
$212$	−13.8682	−0.952469
$213$	36.0212	2.46813
$214$	14.0615	0.961227
$215$	−28.7307	−1.95942
$216$	−9.61088	−0.653937
$217$	−3.61759	−0.245578
$218$	−7.43435	−0.503518
$219$	20.3140	1.37269
$220$	−12.3904	−0.835363
$221$	21.4925	1.44574
$222$	6.21086	0.416846
$223$	−28.7792	−1.92720	−0.963599	−	0.267351i	$-0.913852\pi$
$223$	−28.7792	−1.92720	−0.963599	+	0.267351i	$0.913852\pi$
$224$	−0.956843	−0.0639318
$225$	19.1638	1.27759
$226$	3.16786	0.210723
$227$	−11.9565	−0.793579	−0.396790	−	0.917910i	$-0.629876\pi$
$227$	−11.9565	−0.793579	−0.396790	+	0.917910i	$0.629876\pi$
$228$	18.0155	1.19311
$229$	17.6026	1.16321	0.581606	−	0.813470i	$-0.302424\pi$
$229$	17.6026	1.16321	0.581606	+	0.813470i	$0.302424\pi$
$230$	−15.1345	−0.997939
$231$	12.6133	0.829896
$232$	5.63010	0.369634
$233$	−7.73691	−0.506862	−0.253431	−	0.967353i	$-0.581559\pi$
$233$	−7.73691	−0.506862	−0.253431	+	0.967353i	$0.581559\pi$
$234$	25.6375	1.67598
$235$	24.8747	1.62264
$236$	3.91801	0.255040
$237$	−48.0428	−3.12071
$238$	−4.95182	−0.320979
$239$	−12.0376	−0.778650	−0.389325	−	0.921100i	$-0.627292\pi$
$239$	−12.0376	−0.778650	−0.389325	+	0.921100i	$0.627292\pi$
$240$	8.62224	0.556563
$241$	−17.3393	−1.11692	−0.558461	−	0.829531i	$-0.688608\pi$
$241$	−17.3393	−1.11692	−0.558461	+	0.829531i	$0.688608\pi$
$242$	7.94330	0.510615
$243$	−3.23893	−0.207778
$244$	8.08533	0.517610
$245$	17.3213	1.10662
$246$	28.2525	1.80131
$247$	−24.7030	−1.57181
$248$	3.78076	0.240078
$249$	−26.6605	−1.68954
$250$	5.39660	0.341311
$251$	0.282683	0.0178428	0.00892141	−	0.999960i	$-0.497160\pi$
$251$	0.282683	0.0178428	0.00892141	+	0.999960i	$0.497160\pi$
$252$	−5.90682	−0.372094
$253$	23.1386	1.45471
$254$	14.1657	0.888836
$255$	44.6215	2.79431
$256$	1.00000	0.0625000
$257$	21.8179	1.36096	0.680481	−	0.732766i	$-0.261771\pi$
$257$	21.8179	1.36096	0.680481	+	0.732766i	$0.261771\pi$
$258$	−30.5668	−1.90300
$259$	1.96215	0.121922
$260$	−11.8228	−0.733222
$261$	34.7559	2.15134
$262$	−0.439718	−0.0271658
$263$	−0.225707	−0.0139177	−0.00695883	−	0.999976i	$-0.502215\pi$
$263$	−0.225707	−0.0139177	−0.00695883	+	0.999976i	$0.502215\pi$
$264$	−13.1822	−0.811310
$265$	39.4800	2.42524
$266$	5.69150	0.348968
$267$	−55.3255	−3.38587
$268$	7.76343	0.474227
$269$	21.9832	1.34034	0.670169	−	0.742208i	$-0.266221\pi$
$269$	21.9832	1.34034	0.670169	+	0.742208i	$0.266221\pi$
$270$	27.3604	1.66510
$271$	16.0251	0.973457	0.486729	−	0.873553i	$-0.338190\pi$
$271$	16.0251	0.973457	0.486729	+	0.873553i	$0.338190\pi$
$272$	5.17517	0.313791
$273$	12.0355	0.728423
$274$	−12.6351	−0.763315
$275$	13.5113	0.814761
$276$	−16.1016	−0.969205
$277$	6.55158	0.393646	0.196823	−	0.980439i	$-0.436938\pi$
$277$	6.55158	0.393646	0.196823	+	0.980439i	$0.436938\pi$
$278$	−17.8716	−1.07187
$279$	23.3395	1.39730
$280$	2.72395	0.162787
$281$	−2.02312	−0.120689	−0.0603445	−	0.998178i	$-0.519220\pi$
$281$	−2.02312	−0.120689	−0.0603445	+	0.998178i	$0.519220\pi$
$282$	26.4642	1.57592
$283$	29.1277	1.73146	0.865732	−	0.500508i	$-0.166853\pi$
$283$	29.1277	1.73146	0.865732	+	0.500508i	$0.166853\pi$
$284$	−11.8932	−0.705729
$285$	−51.2868	−3.03797
$286$	18.0755	1.06883
$287$	8.92558	0.526860
$288$	6.17323	0.363761
$289$	9.78235	0.575432
$290$	−16.0278	−0.941186
$291$	8.80798	0.516333
$292$	−6.70709	−0.392503
$293$	2.21600	0.129460	0.0647299	−	0.997903i	$-0.479381\pi$
$293$	2.21600	0.129460	0.0647299	+	0.997903i	$0.479381\pi$
$294$	18.4282	1.07475
$295$	−11.1538	−0.649401
$296$	−2.05065	−0.119191
$297$	−41.8303	−2.42724
$298$	8.75706	0.507283
$299$	22.0786	1.27684
$300$	−9.40221	−0.542837
$301$	−9.65670	−0.556603
$302$	17.8689	1.02824
$303$	19.6245	1.12740
$304$	−5.94820	−0.341153
$305$	−23.0174	−1.31797
$306$	31.9475	1.82632
$307$	11.9481	0.681915	0.340957	−	0.940079i	$-0.389249\pi$
$307$	11.9481	0.681915	0.340957	+	0.940079i	$0.389249\pi$
$308$	−4.16456	−0.237298
$309$	20.0431	1.14021
$310$	−10.7631	−0.611303
$311$	9.10913	0.516531	0.258266	−	0.966074i	$-0.416849\pi$
$311$	9.10913	0.516531	0.258266	+	0.966074i	$0.416849\pi$
$312$	−12.5784	−0.712110
$313$	22.2625	1.25835	0.629176	−	0.777263i	$-0.283393\pi$
$313$	22.2625	1.25835	0.629176	+	0.777263i	$0.283393\pi$
$314$	−5.28336	−0.298157
$315$	16.8156	0.947451
$316$	15.8623	0.892325
$317$	−5.75473	−0.323218	−0.161609	−	0.986855i	$-0.551668\pi$
$317$	−5.75473	−0.323218	−0.161609	+	0.986855i	$0.551668\pi$
$318$	42.0030	2.35541
$319$	24.5044	1.37198
$320$	−2.84681	−0.159142
$321$	−42.5887	−2.37707
$322$	−5.08686	−0.283480
$323$	−30.7829	−1.71281
$324$	10.5891	0.588283
$325$	12.8923	0.715139
$326$	3.06469	0.169737
$327$	22.5167	1.24517
$328$	−9.32815	−0.515061
$329$	8.36063	0.460937
$330$	37.5273	2.06581
$331$	26.8263	1.47451	0.737253	−	0.675617i	$-0.236123\pi$
$331$	26.8263	1.47451	0.737253	+	0.675617i	$0.236123\pi$
$332$	8.80253	0.483102
$333$	−12.6591	−0.693715
$334$	−5.66054	−0.309731
$335$	−22.1010	−1.20751
$336$	2.89802	0.158100
$337$	−25.2681	−1.37644	−0.688221	−	0.725502i	$-0.741608\pi$
$337$	−25.2681	−1.37644	−0.688221	+	0.725502i	$0.741608\pi$
$338$	4.24751	0.231034
$339$	−9.59461	−0.521108
$340$	−14.7327	−0.798994
$341$	16.4553	0.891107
$342$	−36.7196	−1.98557
$343$	12.5198	0.676004
$344$	10.0923	0.544138
$345$	45.8383	2.46785
$346$	−7.56463	−0.406677
$347$	8.90906	0.478263	0.239132	−	0.970987i	$-0.423137\pi$
$347$	8.90906	0.478263	0.239132	+	0.970987i	$0.423137\pi$
$348$	−17.0521	−0.914087
$349$	−23.4987	−1.25786	−0.628928	−	0.777464i	$-0.716506\pi$
$349$	−23.4987	−1.25786	−0.628928	+	0.777464i	$0.716506\pi$
$350$	−2.97036	−0.158773
$351$	−39.9141	−2.13046
$352$	4.35239	0.231983
$353$	−6.93841	−0.369294	−0.184647	−	0.982805i	$-0.559114\pi$
$353$	−6.93841	−0.369294	−0.184647	+	0.982805i	$0.559114\pi$
$354$	−11.8666	−0.630703
$355$	33.8576	1.79697
$356$	18.2669	0.968143
$357$	14.9978	0.793765
$358$	3.07304	0.162415
$359$	−2.97480	−0.157004	−0.0785020	−	0.996914i	$-0.525014\pi$
$359$	−2.97480	−0.157004	−0.0785020	+	0.996914i	$0.525014\pi$
$360$	−17.5740	−0.926233
$361$	16.3811	0.862164
$362$	10.7167	0.563258
$363$	−24.0581	−1.26273
$364$	−3.97378	−0.208283
$365$	19.0938	0.999417
$366$	−24.4883	−1.28002
$367$	−11.5928	−0.605141	−0.302571	−	0.953127i	$-0.597845\pi$
$367$	−11.5928	−0.605141	−0.302571	+	0.953127i	$0.597845\pi$
$368$	5.31629	0.277131
$369$	−57.5849	−2.99775
$370$	5.83780	0.303493
$371$	13.2697	0.688926
$372$	−11.4509	−0.593702
$373$	−4.96839	−0.257253	−0.128627	−	0.991693i	$-0.541057\pi$
$373$	−4.96839	−0.257253	−0.128627	+	0.991693i	$0.541057\pi$
$374$	22.5243	1.16471
$375$	−16.3449	−0.844045
$376$	−8.73772	−0.450614
$377$	23.3819	1.20423
$378$	9.19610	0.472996
$379$	−20.3888	−1.04730	−0.523650	−	0.851933i	$-0.675430\pi$
$379$	−20.3888	−1.04730	−0.523650	+	0.851933i	$0.675430\pi$
$380$	16.9334	0.868666
$381$	−42.9042	−2.19805
$382$	−5.34510	−0.273479
$383$	−26.0413	−1.33065	−0.665325	−	0.746554i	$-0.731707\pi$
$383$	−26.0413	−1.33065	−0.665325	+	0.746554i	$0.731707\pi$
$384$	−3.02873	−0.154559
$385$	11.8557	0.604223
$386$	−20.5254	−1.04471
$387$	62.3018	3.16698
$388$	−2.90814	−0.147638
$389$	0.840128	0.0425962	0.0212981	−	0.999773i	$-0.493220\pi$
$389$	0.840128	0.0425962	0.0212981	+	0.999773i	$0.493220\pi$
$390$	35.8083	1.81322
$391$	27.5127	1.39138
$392$	−6.08445	−0.307311
$393$	1.33179	0.0671798
$394$	17.7201	0.892725
$395$	−45.1570	−2.27210
$396$	26.8683	1.35018
$397$	−17.2443	−0.865466	−0.432733	−	0.901522i	$-0.642451\pi$
$397$	−17.2443	−0.865466	−0.432733	+	0.901522i	$0.642451\pi$
$398$	19.2815	0.966494
$399$	−17.2380	−0.862981
$400$	3.10434	0.155217
$401$	11.3503	0.566809	0.283405	−	0.959000i	$-0.408536\pi$
$401$	11.3503	0.566809	0.283405	+	0.959000i	$0.408536\pi$
$402$	−23.5134	−1.17274
$403$	15.7015	0.782149
$404$	−6.47943	−0.322363
$405$	−30.1452	−1.49793
$406$	−5.38712	−0.267358
$407$	−8.92521	−0.442406
$408$	−15.6742	−0.775989
$409$	−21.8386	−1.07985	−0.539924	−	0.841714i	$-0.681547\pi$
$409$	−21.8386	−1.07985	−0.539924	+	0.841714i	$0.681547\pi$
$410$	26.5555	1.31148
$411$	38.2684	1.88764
$412$	−6.61766	−0.326029
$413$	−3.74892	−0.184472
$414$	32.8187	1.61295
$415$	−25.0591	−1.23010
$416$	4.15301	0.203618
$417$	54.1285	2.65068
$418$	−25.8889	−1.26627
$419$	−0.457074	−0.0223295	−0.0111648	−	0.999938i	$-0.503554\pi$
$419$	−0.457074	−0.0223295	−0.0111648	+	0.999938i	$0.503554\pi$
$420$	−8.25013	−0.402565
$421$	2.18031	0.106262	0.0531310	−	0.998588i	$-0.483080\pi$
$421$	2.18031	0.106262	0.0531310	+	0.998588i	$0.483080\pi$
$422$	7.67448	0.373588
$423$	−53.9400	−2.62265
$424$	−13.8682	−0.673498
$425$	16.0655	0.779289
$426$	36.0212	1.74523
$427$	−7.73639	−0.374390
$428$	14.0615	0.679690
$429$	−54.7460	−2.64316
$430$	−28.7307	−1.38552
$431$	28.6513	1.38008	0.690042	−	0.723769i	$-0.257592\pi$
$431$	28.6513	1.38008	0.690042	+	0.723769i	$0.257592\pi$
$432$	−9.61088	−0.462404
$433$	28.3048	1.36024	0.680121	−	0.733100i	$-0.261927\pi$
$433$	28.3048	1.36024	0.680121	+	0.733100i	$0.261927\pi$
$434$	−3.61759	−0.173650
$435$	48.5440	2.32751
$436$	−7.43435	−0.356041
$437$	−31.6224	−1.51270
$438$	20.3140	0.970641
$439$	8.43460	0.402562	0.201281	−	0.979534i	$-0.435490\pi$
$439$	8.43460	0.402562	0.201281	+	0.979534i	$0.435490\pi$
$440$	−12.3904	−0.590691
$441$	−37.5607	−1.78861
$442$	21.4925	1.02230
$443$	24.2070	1.15011	0.575055	−	0.818115i	$-0.304981\pi$
$443$	24.2070	1.15011	0.575055	+	0.818115i	$0.304981\pi$
$444$	6.21086	0.294754
$445$	−52.0024	−2.46515
$446$	−28.7792	−1.36274
$447$	−26.5228	−1.25449
$448$	−0.956843	−0.0452066
$449$	0.267331	0.0126161	0.00630807	−	0.999980i	$-0.497992\pi$
$449$	0.267331	0.0126161	0.00630807	+	0.999980i	$0.497992\pi$
$450$	19.1638	0.903390
$451$	−40.5998	−1.91177
$452$	3.16786	0.149004
$453$	−54.1200	−2.54278
$454$	−11.9565	−0.561145
$455$	11.3126	0.530343
$456$	18.0155	0.843654
$457$	−11.7058	−0.547576	−0.273788	−	0.961790i	$-0.588277\pi$
$457$	−11.7058	−0.547576	−0.273788	+	0.961790i	$0.588277\pi$
$458$	17.6026	0.822516
$459$	−49.7379	−2.32157
$460$	−15.1345	−0.705649
$461$	38.1403	1.77637	0.888186	−	0.459483i	$-0.151965\pi$
$461$	38.1403	1.77637	0.888186	+	0.459483i	$0.151965\pi$
$462$	12.6133	0.586825
$463$	−19.7702	−0.918801	−0.459400	−	0.888229i	$-0.651936\pi$
$463$	−19.7702	−0.918801	−0.459400	+	0.888229i	$0.651936\pi$
$464$	5.63010	0.261371
$465$	32.5986	1.51172
$466$	−7.73691	−0.358405
$467$	22.1508	1.02502	0.512509	−	0.858682i	$-0.328716\pi$
$467$	22.1508	1.02502	0.512509	+	0.858682i	$0.328716\pi$
$468$	25.6375	1.18509
$469$	−7.42838	−0.343011
$470$	24.8747	1.14738
$471$	16.0019	0.737329
$472$	3.91801	0.180341
$473$	43.9254	2.01969
$474$	−48.0428	−2.20668
$475$	−18.4652	−0.847243
$476$	−4.95182	−0.226966
$477$	−85.6114	−3.91988
$478$	−12.0376	−0.550589
$479$	22.6179	1.03344	0.516718	−	0.856155i	$-0.327153\pi$
$479$	22.6179	1.03344	0.516718	+	0.856155i	$0.327153\pi$
$480$	8.62224	0.393549
$481$	−8.51636	−0.388312
$482$	−17.3393	−0.789783
$483$	15.4067	0.701031
$484$	7.94330	0.361059
$485$	8.27892	0.375926
$486$	−3.23893	−0.146921
$487$	19.6769	0.891646	0.445823	−	0.895121i	$-0.352911\pi$
$487$	19.6769	0.891646	0.445823	+	0.895121i	$0.352911\pi$
$488$	8.08533	0.366005
$489$	−9.28212	−0.419752
$490$	17.3213	0.782496
$491$	−33.2162	−1.49903	−0.749513	−	0.661990i	$-0.769712\pi$
$491$	−33.2162	−1.49903	−0.749513	+	0.661990i	$0.769712\pi$
$492$	28.2525	1.27372
$493$	29.1367	1.31225
$494$	−24.7030	−1.11144
$495$	−76.4890	−3.43793
$496$	3.78076	0.169761
$497$	11.3799	0.510458
$498$	−26.6605	−1.19469
$499$	11.8684	0.531301	0.265651	−	0.964069i	$-0.414413\pi$
$499$	11.8684	0.531301	0.265651	+	0.964069i	$0.414413\pi$
$500$	5.39660	0.241343
$501$	17.1443	0.765949
$502$	0.282683	0.0126168
$503$	35.7413	1.59363	0.796813	−	0.604226i	$-0.206518\pi$
$503$	35.7413	1.59363	0.796813	+	0.604226i	$0.206518\pi$
$504$	−5.90682	−0.263110
$505$	18.4457	0.820823
$506$	23.1386	1.02864
$507$	−12.8646	−0.571336
$508$	14.1657	0.628502
$509$	31.3231	1.38837	0.694187	−	0.719795i	$-0.255764\pi$
$509$	31.3231	1.38837	0.694187	+	0.719795i	$0.255764\pi$
$510$	44.6215	1.97587
$511$	6.41764	0.283900
$512$	1.00000	0.0441942
$513$	57.1675	2.52400
$514$	21.8179	0.962345
$515$	18.8392	0.830156
$516$	−30.5668	−1.34563
$517$	−38.0300	−1.67256
$518$	1.96215	0.0862118
$519$	22.9113	1.00569
$520$	−11.8228	−0.518466
$521$	31.0365	1.35973	0.679866	−	0.733336i	$-0.262038\pi$
$521$	31.0365	1.35973	0.679866	+	0.733336i	$0.262038\pi$
$522$	34.7559	1.52122
$523$	2.64744	0.115764	0.0578822	−	0.998323i	$-0.481565\pi$
$523$	2.64744	0.115764	0.0578822	+	0.998323i	$0.481565\pi$
$524$	−0.439718	−0.0192092
$525$	8.99644	0.392637
$526$	−0.225707	−0.00984128
$527$	19.5661	0.852311
$528$	−13.1822	−0.573683
$529$	5.26298	0.228825
$530$	39.4800	1.71490
$531$	24.1868	1.04962
$532$	5.69150	0.246758
$533$	−38.7399	−1.67801
$534$	−55.3255	−2.39417
$535$	−40.0306	−1.73067
$536$	7.76343	0.335329
$537$	−9.30743	−0.401645
$538$	21.9832	0.947763
$539$	−26.4819	−1.14066
$540$	27.3604	1.17740
$541$	−2.19977	−0.0945756	−0.0472878	−	0.998881i	$-0.515058\pi$
$541$	−2.19977	−0.0945756	−0.0472878	+	0.998881i	$0.515058\pi$
$542$	16.0251	0.688338
$543$	−32.4581	−1.39291
$544$	5.17517	0.221883
$545$	21.1642	0.906574
$546$	12.0355	0.515073
$547$	21.9689	0.939321	0.469660	−	0.882847i	$-0.344376\pi$
$547$	21.9689	0.939321	0.469660	+	0.882847i	$0.344376\pi$
$548$	−12.6351	−0.539745
$549$	49.9126	2.13022
$550$	13.5113	0.576123
$551$	−33.4890	−1.42668
$552$	−16.1016	−0.685331
$553$	−15.1778	−0.645424
$554$	6.55158	0.278350
$555$	−17.6812	−0.750523
$556$	−17.8716	−0.757927
$557$	−14.5912	−0.618248	−0.309124	−	0.951022i	$-0.600036\pi$
$557$	−14.5912	−0.618248	−0.309124	+	0.951022i	$0.600036\pi$
$558$	23.3395	0.988040
$559$	41.9132	1.77274
$560$	2.72395	0.115108
$561$	−68.2203	−2.88026
$562$	−2.02312	−0.0853400
$563$	−1.60613	−0.0676904	−0.0338452	−	0.999427i	$-0.510775\pi$
$563$	−1.60613	−0.0676904	−0.0338452	+	0.999427i	$0.510775\pi$
$564$	26.4642	1.11435
$565$	−9.01830	−0.379403
$566$	29.1277	1.22433
$567$	−10.1321	−0.425508
$568$	−11.8932	−0.499026
$569$	19.4476	0.815284	0.407642	−	0.913142i	$-0.366351\pi$
$569$	19.4476	0.815284	0.407642	+	0.913142i	$0.366351\pi$
$570$	−51.2868	−2.14817
$571$	−17.4555	−0.730490	−0.365245	−	0.930911i	$-0.619015\pi$
$571$	−17.4555	−0.730490	−0.365245	+	0.930911i	$0.619015\pi$
$572$	18.0755	0.755776
$573$	16.1889	0.676301
$574$	8.92558	0.372547
$575$	16.5036	0.688246
$576$	6.17323	0.257218
$577$	−36.4145	−1.51596	−0.757978	−	0.652280i	$-0.773813\pi$
$577$	−36.4145	−1.51596	−0.757978	+	0.652280i	$0.773813\pi$
$578$	9.78235	0.406892
$579$	62.1659	2.58353
$580$	−16.0278	−0.665519
$581$	−8.42264	−0.349430
$582$	8.80798	0.365102
$583$	−60.3597	−2.49984
$584$	−6.70709	−0.277541
$585$	−72.9852	−3.01756
$586$	2.21600	0.0915419
$587$	−11.4621	−0.473090	−0.236545	−	0.971621i	$-0.576015\pi$
$587$	−11.4621	−0.473090	−0.236545	+	0.971621i	$0.576015\pi$
$588$	18.4282	0.759966
$589$	−22.4887	−0.926632
$590$	−11.1538	−0.459196
$591$	−53.6694	−2.20766
$592$	−2.05065	−0.0842810
$593$	−7.17279	−0.294551	−0.147276	−	0.989096i	$-0.547050\pi$
$593$	−7.17279	−0.294551	−0.147276	+	0.989096i	$0.547050\pi$
$594$	−41.8303	−1.71632
$595$	14.0969	0.577917
$596$	8.75706	0.358703
$597$	−58.3985	−2.39009
$598$	22.0786	0.902862
$599$	−32.0913	−1.31122	−0.655608	−	0.755102i	$-0.727588\pi$
$599$	−32.0913	−1.31122	−0.655608	+	0.755102i	$0.727588\pi$
$600$	−9.40221	−0.383844
$601$	22.7640	0.928562	0.464281	−	0.885688i	$-0.346313\pi$
$601$	22.7640	0.928562	0.464281	+	0.885688i	$0.346313\pi$
$602$	−9.65670	−0.393578
$603$	47.9254	1.95167
$604$	17.8689	0.727073
$605$	−22.6131	−0.919352
$606$	19.6245	0.797189
$607$	−0.0286533	−0.00116300	−0.000581501	−	1.00000i	$-0.500185\pi$
$607$	−0.0286533	−0.00116300	−0.000581501		1.00000i	$0.500185\pi$
$608$	−5.94820	−0.241231
$609$	16.3162	0.661164
$610$	−23.0174	−0.931947
$611$	−36.2879	−1.46805
$612$	31.9475	1.29140
$613$	31.0394	1.25367	0.626836	−	0.779151i	$-0.284350\pi$
$613$	31.0394	1.25367	0.626836	+	0.779151i	$0.284350\pi$
$614$	11.9481	0.482186
$615$	−80.4295	−3.24323
$616$	−4.16456	−0.167795
$617$	−35.4745	−1.42815	−0.714074	−	0.700070i	$-0.753152\pi$
$617$	−35.4745	−1.42815	−0.714074	+	0.700070i	$0.753152\pi$
$618$	20.0431	0.806253
$619$	−20.1357	−0.809323	−0.404661	−	0.914467i	$-0.632611\pi$
$619$	−20.1357	−0.809323	−0.404661	+	0.914467i	$0.632611\pi$
$620$	−10.7631	−0.432257
$621$	−51.0942	−2.05034
$622$	9.10913	0.365243
$623$	−17.4785	−0.700263
$624$	−12.5784	−0.503538
$625$	−30.8848	−1.23539
$626$	22.2625	0.889789
$627$	78.4106	3.13142
$628$	−5.28336	−0.210829
$629$	−10.6124	−0.423145
$630$	16.8156	0.669949
$631$	−13.1131	−0.522025	−0.261012	−	0.965335i	$-0.584056\pi$
$631$	−13.1131	−0.522025	−0.261012	+	0.965335i	$0.584056\pi$
$632$	15.8623	0.630969
$633$	−23.2439	−0.923864
$634$	−5.75473	−0.228550
$635$	−40.3271	−1.60033
$636$	42.0030	1.66553
$637$	−25.2688	−1.00119
$638$	24.5044	0.970138
$639$	−73.4192	−2.90442
$640$	−2.84681	−0.112530
$641$	6.40716	0.253068	0.126534	−	0.991962i	$-0.459615\pi$
$641$	6.40716	0.253068	0.126534	+	0.991962i	$0.459615\pi$
$642$	−42.5887	−1.68084
$643$	−47.4923	−1.87291	−0.936456	−	0.350785i	$-0.885915\pi$
$643$	−47.4923	−1.87291	−0.936456	+	0.350785i	$0.885915\pi$
$644$	−5.08686	−0.200450
$645$	87.0178	3.42632
$646$	−30.7829	−1.21114
$647$	29.6343	1.16505	0.582523	−	0.812815i	$-0.302066\pi$
$647$	29.6343	1.16505	0.582523	+	0.812815i	$0.302066\pi$
$648$	10.5891	0.415979
$649$	17.0527	0.669377
$650$	12.8923	0.505679
$651$	10.9567	0.429428
$652$	3.06469	0.120022
$653$	−35.1720	−1.37639	−0.688193	−	0.725528i	$-0.741596\pi$
$653$	−35.1720	−1.37639	−0.688193	+	0.725528i	$0.741596\pi$
$654$	22.5167	0.880472
$655$	1.25179	0.0489116
$656$	−9.32815	−0.364203
$657$	−41.4044	−1.61534
$658$	8.36063	0.325931
$659$	−14.9260	−0.581434	−0.290717	−	0.956809i	$-0.593894\pi$
$659$	−14.9260	−0.581434	−0.290717	+	0.956809i	$0.593894\pi$
$660$	37.5273	1.46075
$661$	13.4596	0.523517	0.261758	−	0.965133i	$-0.415698\pi$
$661$	13.4596	0.523517	0.261758	+	0.965133i	$0.415698\pi$
$662$	26.8263	1.04263
$663$	−65.0952	−2.52809
$664$	8.80253	0.341604
$665$	−16.2026	−0.628311
$666$	−12.6591	−0.490531
$667$	29.9313	1.15894
$668$	−5.66054	−0.219013
$669$	87.1646	3.36998
$670$	−22.1010	−0.853837
$671$	35.1905	1.35851
$672$	2.89802	0.111794
$673$	−3.61378	−0.139301	−0.0696505	−	0.997571i	$-0.522188\pi$
$673$	−3.61378	−0.139301	−0.0696505	+	0.997571i	$0.522188\pi$
$674$	−25.2681	−0.973291
$675$	−29.8354	−1.14836
$676$	4.24751	0.163366
$677$	−15.0144	−0.577050	−0.288525	−	0.957472i	$-0.593165\pi$
$677$	−15.0144	−0.577050	−0.288525	+	0.957472i	$0.593165\pi$
$678$	−9.59461	−0.368479
$679$	2.78263	0.106788
$680$	−14.7327	−0.564974
$681$	36.2130	1.38769
$682$	16.4553	0.630107
$683$	38.3026	1.46561	0.732804	−	0.680440i	$-0.238211\pi$
$683$	38.3026	1.46561	0.732804	+	0.680440i	$0.238211\pi$
$684$	−36.7196	−1.40401
$685$	35.9698	1.37433
$686$	12.5198	0.478007
$687$	−53.3136	−2.03404
$688$	10.0923	0.384764
$689$	−57.5946	−2.19418
$690$	45.8383	1.74504
$691$	44.6156	1.69726	0.848628	−	0.528990i	$-0.177429\pi$
$691$	44.6156	1.69726	0.848628	+	0.528990i	$0.177429\pi$
$692$	−7.56463	−0.287564
$693$	−25.7088	−0.976595
$694$	8.90906	0.338183
$695$	50.8772	1.92988
$696$	−17.0521	−0.646357
$697$	−48.2747	−1.82854
$698$	−23.4987	−0.889438
$699$	23.4330	0.886319
$700$	−2.97036	−0.112269
$701$	51.9564	1.96237	0.981184	−	0.193076i	$-0.0618465\pi$
$701$	51.9564	1.96237	0.981184	+	0.193076i	$0.0618465\pi$
$702$	−39.9141	−1.50646
$703$	12.1977	0.460043
$704$	4.35239	0.164037
$705$	−75.3387	−2.83742
$706$	−6.93841	−0.261130
$707$	6.19979	0.233167
$708$	−11.8666	−0.445974
$709$	42.4300	1.59349	0.796747	−	0.604313i	$-0.206552\pi$
$709$	42.4300	1.59349	0.796747	+	0.604313i	$0.206552\pi$
$710$	33.8576	1.27065
$711$	97.9218	3.67235
$712$	18.2669	0.684580
$713$	20.0996	0.752737
$714$	14.9978	0.561277
$715$	−51.4576	−1.92441
$716$	3.07304	0.114845
$717$	36.4588	1.36158
$718$	−2.97480	−0.111019
$719$	11.3755	0.424233	0.212117	−	0.977244i	$-0.431964\pi$
$719$	11.3755	0.424233	0.212117	+	0.977244i	$0.431964\pi$
$720$	−17.5740	−0.654945
$721$	6.33206	0.235818
$722$	16.3811	0.609642
$723$	52.5161	1.95310
$724$	10.7167	0.398283
$725$	17.4777	0.649106
$726$	−24.0581	−0.892882
$727$	−26.1373	−0.969380	−0.484690	−	0.874686i	$-0.661068\pi$
$727$	−26.1373	−0.969380	−0.484690	+	0.874686i	$0.661068\pi$
$728$	−3.97378	−0.147278
$729$	−21.9574	−0.813238
$730$	19.0938	0.706694
$731$	52.2291	1.93176
$732$	−24.4883	−0.905114
$733$	−8.87010	−0.327625	−0.163812	−	0.986492i	$-0.552379\pi$
$733$	−8.87010	−0.327625	−0.163812	+	0.986492i	$0.552379\pi$
$734$	−11.5928	−0.427899
$735$	−52.4616	−1.93507
$736$	5.31629	0.195961
$737$	33.7895	1.24465
$738$	−57.5849	−2.11973
$739$	−20.0101	−0.736084	−0.368042	−	0.929809i	$-0.619972\pi$
$739$	−20.0101	−0.736084	−0.368042	+	0.929809i	$0.619972\pi$
$740$	5.83780	0.214602
$741$	74.8187	2.74853
$742$	13.2697	0.487144
$743$	−14.0748	−0.516354	−0.258177	−	0.966098i	$-0.583122\pi$
$743$	−14.0748	−0.516354	−0.258177	+	0.966098i	$0.583122\pi$
$744$	−11.4509	−0.419811
$745$	−24.9297	−0.913353
$746$	−4.96839	−0.181906
$747$	54.3401	1.98820
$748$	22.5243	0.823572
$749$	−13.4547	−0.491624
$750$	−16.3449	−0.596830
$751$	−14.8255	−0.540988	−0.270494	−	0.962722i	$-0.587187\pi$
$751$	−14.8255	−0.540988	−0.270494	+	0.962722i	$0.587187\pi$
$752$	−8.73772	−0.318632
$753$	−0.856173	−0.0312007
$754$	23.3819	0.851517
$755$	−50.8693	−1.85132
$756$	9.19610	0.334459
$757$	6.09176	0.221409	0.110704	−	0.993853i	$-0.464689\pi$
$757$	6.09176	0.221409	0.110704	+	0.993853i	$0.464689\pi$
$758$	−20.3888	−0.740553
$759$	−70.0806	−2.54377
$760$	16.9334	0.614239
$761$	−25.4018	−0.920815	−0.460407	−	0.887708i	$-0.652297\pi$
$761$	−25.4018	−0.920815	−0.460407	+	0.887708i	$0.652297\pi$
$762$	−42.9042	−1.55426
$763$	7.11351	0.257526
$764$	−5.34510	−0.193379
$765$	−90.9485	−3.28825
$766$	−26.0413	−0.940912
$767$	16.2715	0.587531
$768$	−3.02873	−0.109290
$769$	40.0304	1.44353	0.721766	−	0.692137i	$-0.243331\pi$
$769$	40.0304	1.44353	0.721766	+	0.692137i	$0.243331\pi$
$770$	11.8557	0.427250
$771$	−66.0805	−2.37983
$772$	−20.5254	−0.738725
$773$	−43.3159	−1.55797	−0.778983	−	0.627045i	$-0.784264\pi$
$773$	−43.3159	−1.55797	−0.778983	+	0.627045i	$0.784264\pi$
$774$	62.3018	2.23939
$775$	11.7367	0.421596
$776$	−2.90814	−0.104396
$777$	−5.94282	−0.213198
$778$	0.840128	0.0301200
$779$	55.4858	1.98798
$780$	35.8083	1.28214
$781$	−51.7637	−1.85225
$782$	27.5127	0.983852
$783$	−54.1102	−1.93374
$784$	−6.08445	−0.217302
$785$	15.0407	0.536827
$786$	1.33179	0.0475033
$787$	41.4648	1.47806	0.739031	−	0.673671i	$-0.235284\pi$
$787$	41.4648	1.47806	0.739031	+	0.673671i	$0.235284\pi$
$788$	17.7201	0.631252
$789$	0.683605	0.0243370
$790$	−45.1570	−1.60662
$791$	−3.03115	−0.107775
$792$	26.8683	0.954724
$793$	33.5785	1.19241
$794$	−17.2443	−0.611977
$795$	−119.575	−4.24087
$796$	19.2815	0.683415
$797$	8.75996	0.310294	0.155147	−	0.987891i	$-0.450415\pi$
$797$	8.75996	0.310294	0.155147	+	0.987891i	$0.450415\pi$
$798$	−17.2380	−0.610220
$799$	−45.2192	−1.59974
$800$	3.10434	0.109755
$801$	112.766	3.98438
$802$	11.3503	0.400795
$803$	−29.1919	−1.03016
$804$	−23.5134	−0.829252
$805$	14.4813	0.510400
$806$	15.7015	0.553063
$807$	−66.5813	−2.34377
$808$	−6.47943	−0.227945
$809$	45.4527	1.59803	0.799015	−	0.601311i	$-0.205355\pi$
$809$	45.4527	1.59803	0.799015	+	0.601311i	$0.205355\pi$
$810$	−30.1452	−1.05919
$811$	42.2469	1.48349	0.741745	−	0.670681i	$-0.233998\pi$
$811$	42.2469	1.48349	0.741745	+	0.670681i	$0.233998\pi$
$812$	−5.38712	−0.189051
$813$	−48.5359	−1.70223
$814$	−8.92521	−0.312828
$815$	−8.72458	−0.305609
$816$	−15.6742	−0.548707
$817$	−60.0308	−2.10021
$818$	−21.8386	−0.763568
$819$	−24.5311	−0.857185
$820$	26.5555	0.927358
$821$	−3.97645	−0.138779	−0.0693896	−	0.997590i	$-0.522105\pi$
$821$	−3.97645	−0.138779	−0.0693896	+	0.997590i	$0.522105\pi$
$822$	38.2684	1.33476
$823$	2.91914	0.101755	0.0508775	−	0.998705i	$-0.483798\pi$
$823$	2.91914	0.101755	0.0508775	+	0.998705i	$0.483798\pi$
$824$	−6.61766	−0.230537
$825$	−40.9221	−1.42472
$826$	−3.74892	−0.130442
$827$	36.0510	1.25362	0.626809	−	0.779173i	$-0.284361\pi$
$827$	36.0510	1.25362	0.626809	+	0.779173i	$0.284361\pi$
$828$	32.8187	1.14053
$829$	−6.33054	−0.219869	−0.109934	−	0.993939i	$-0.535064\pi$
$829$	−6.33054	−0.219869	−0.109934	+	0.993939i	$0.535064\pi$
$830$	−25.0591	−0.869816
$831$	−19.8430	−0.688346
$832$	4.15301	0.143980
$833$	−31.4880	−1.09100
$834$	54.1285	1.87432
$835$	16.1145	0.557665
$836$	−25.8889	−0.895386
$837$	−36.3364	−1.25597
$838$	−0.457074	−0.0157894
$839$	−12.5184	−0.432184	−0.216092	−	0.976373i	$-0.569331\pi$
$839$	−12.5184	−0.432184	−0.216092	+	0.976373i	$0.569331\pi$
$840$	−8.25013	−0.284656
$841$	2.69799	0.0930342
$842$	2.18031	0.0751385
$843$	6.12748	0.211042
$844$	7.67448	0.264166
$845$	−12.0919	−0.415973
$846$	−53.9400	−1.85450
$847$	−7.60049	−0.261156
$848$	−13.8682	−0.476235
$849$	−88.2202	−3.02771
$850$	16.0655	0.551041
$851$	−10.9018	−0.373710
$852$	36.0212	1.23407
$853$	−27.5675	−0.943894	−0.471947	−	0.881627i	$-0.656449\pi$
$853$	−27.5675	−0.943894	−0.471947	+	0.881627i	$0.656449\pi$
$854$	−7.73639	−0.264734
$855$	104.534	3.57498
$856$	14.0615	0.480614
$857$	52.1084	1.77999	0.889995	−	0.455970i	$-0.150708\pi$
$857$	52.1084	1.77999	0.889995	+	0.455970i	$0.150708\pi$
$858$	−54.7460	−1.86900
$859$	−6.56243	−0.223907	−0.111954	−	0.993713i	$-0.535711\pi$
$859$	−6.56243	−0.223907	−0.111954	+	0.993713i	$0.535711\pi$
$860$	−28.7307	−0.979710
$861$	−27.0332	−0.921289
$862$	28.6513	0.975867
$863$	40.7185	1.38607	0.693037	−	0.720902i	$-0.256272\pi$
$863$	40.7185	1.38607	0.693037	+	0.720902i	$0.256272\pi$
$864$	−9.61088	−0.326969
$865$	21.5351	0.732215
$866$	28.3048	0.961836
$867$	−29.6281	−1.00622
$868$	−3.61759	−0.122789
$869$	69.0390	2.34199
$870$	48.5440	1.64580
$871$	32.2416	1.09247
$872$	−7.43435	−0.251759
$873$	−17.9526	−0.607604
$874$	−31.6224	−1.06964
$875$	−5.16370	−0.174565
$876$	20.3140	0.686347
$877$	20.5150	0.692741	0.346371	−	0.938098i	$-0.387414\pi$
$877$	20.5150	0.692741	0.346371	+	0.938098i	$0.387414\pi$
$878$	8.43460	0.284654
$879$	−6.71166	−0.226379
$880$	−12.3904	−0.417681
$881$	44.9855	1.51560	0.757800	−	0.652486i	$-0.226274\pi$
$881$	44.9855	1.51560	0.757800	+	0.652486i	$0.226274\pi$
$882$	−37.5607	−1.26474
$883$	−41.2489	−1.38814	−0.694068	−	0.719909i	$-0.744183\pi$
$883$	−41.2489	−1.38814	−0.694068	+	0.719909i	$0.744183\pi$
$884$	21.4925	0.722872
$885$	33.7820	1.13557
$886$	24.2070	0.813250
$887$	42.6465	1.43193	0.715964	−	0.698137i	$-0.245987\pi$
$887$	42.6465	1.43193	0.715964	+	0.698137i	$0.245987\pi$
$888$	6.21086	0.208423
$889$	−13.5544	−0.454599
$890$	−52.0024	−1.74312
$891$	46.0879	1.54400
$892$	−28.7792	−0.963599
$893$	51.9738	1.73924
$894$	−26.5228	−0.887055
$895$	−8.74837	−0.292426
$896$	−0.956843	−0.0319659
$897$	−66.8703	−2.23273
$898$	0.267331	0.00892096
$899$	21.2860	0.709929
$900$	19.1638	0.638793
$901$	−71.7700	−2.39101
$902$	−40.5998	−1.35182
$903$	29.2476	0.973299
$904$	3.16786	0.105361
$905$	−30.5084	−1.01413
$906$	−54.1200	−1.79802
$907$	−42.6471	−1.41607	−0.708037	−	0.706175i	$-0.750419\pi$
$907$	−42.6471	−1.41607	−0.708037	+	0.706175i	$0.750419\pi$
$908$	−11.9565	−0.396790
$909$	−39.9990	−1.32668
$910$	11.3126	0.375009
$911$	44.7402	1.48231	0.741154	−	0.671335i	$-0.234279\pi$
$911$	44.7402	1.48231	0.741154	+	0.671335i	$0.234279\pi$
$912$	18.0155	0.596554
$913$	38.3120	1.26794
$914$	−11.7058	−0.387195
$915$	69.7136	2.30466
$916$	17.6026	0.581606
$917$	0.420741	0.0138941
$918$	−49.7379	−1.64159
$919$	45.4831	1.50035	0.750174	−	0.661240i	$-0.229970\pi$
$919$	45.4831	1.50035	0.750174	+	0.661240i	$0.229970\pi$
$920$	−15.1345	−0.498969
$921$	−36.1876	−1.19242
$922$	38.1403	1.25609
$923$	−49.3924	−1.62577
$924$	12.6133	0.414948
$925$	−6.36589	−0.209309
$926$	−19.7702	−0.649690
$927$	−40.8523	−1.34177
$928$	5.63010	0.184817
$929$	16.1437	0.529658	0.264829	−	0.964295i	$-0.414685\pi$
$929$	16.1437	0.529658	0.264829	+	0.964295i	$0.414685\pi$
$930$	32.5986	1.06895
$931$	36.1916	1.18613
$932$	−7.73691	−0.253431
$933$	−27.5891	−0.903228
$934$	22.1508	0.724797
$935$	−64.1226	−2.09703
$936$	25.6375	0.837988
$937$	−10.3419	−0.337855	−0.168927	−	0.985629i	$-0.554030\pi$
$937$	−10.3419	−0.337855	−0.168927	+	0.985629i	$0.554030\pi$
$938$	−7.42838	−0.242545
$939$	−67.4272	−2.20040
$940$	24.8747	0.811322
$941$	−4.80604	−0.156672	−0.0783361	−	0.996927i	$-0.524961\pi$
$941$	−4.80604	−0.156672	−0.0783361	+	0.996927i	$0.524961\pi$
$942$	16.0019	0.521370
$943$	−49.5912	−1.61491
$944$	3.91801	0.127520
$945$	−26.1796	−0.851622
$946$	43.9254	1.42814
$947$	43.7101	1.42039	0.710193	−	0.704007i	$-0.248608\pi$
$947$	43.7101	1.42039	0.710193	+	0.704007i	$0.248608\pi$
$948$	−48.0428	−1.56036
$949$	−27.8546	−0.904200
$950$	−18.4652	−0.599091
$951$	17.4295	0.565192
$952$	−4.95182	−0.160490
$953$	−0.108963	−0.00352965	−0.00176482	−	0.999998i	$-0.500562\pi$
$953$	−0.108963	−0.00352965	−0.00176482	+	0.999998i	$0.500562\pi$
$954$	−85.6114	−2.77177
$955$	15.2165	0.492395
$956$	−12.0376	−0.389325
$957$	−74.2173	−2.39910
$958$	22.6179	0.730750
$959$	12.0898	0.390400
$960$	8.62224	0.278281
$961$	−16.7059	−0.538899
$962$	−8.51636	−0.274578
$963$	86.8052	2.79726
$964$	−17.3393	−0.558461
$965$	58.4319	1.88099
$966$	15.4067	0.495704
$967$	49.4793	1.59115	0.795574	−	0.605857i	$-0.207169\pi$
$967$	49.4793	1.59115	0.795574	+	0.605857i	$0.207169\pi$
$968$	7.94330	0.255307
$969$	93.2334	2.99509
$970$	8.27892	0.265820
$971$	−44.6391	−1.43254	−0.716268	−	0.697825i	$-0.754151\pi$
$971$	−44.6391	−1.43254	−0.716268	+	0.697825i	$0.754151\pi$
$972$	−3.23893	−0.103889
$973$	17.1004	0.548213
$974$	19.6769	0.630489
$975$	−39.0475	−1.25052
$976$	8.08533	0.258805
$977$	34.9297	1.11750	0.558749	−	0.829337i	$-0.311281\pi$
$977$	34.9297	1.11750	0.558749	+	0.829337i	$0.311281\pi$
$978$	−9.28212	−0.296809
$979$	79.5046	2.54098
$980$	17.3213	0.553308
$981$	−45.8940	−1.46528
$982$	−33.2162	−1.05997
$983$	−14.1797	−0.452262	−0.226131	−	0.974097i	$-0.572608\pi$
$983$	−14.1797	−0.452262	−0.226131	+	0.974097i	$0.572608\pi$
$984$	28.2525	0.900657
$985$	−50.4457	−1.60733
$986$	29.1367	0.927901
$987$	−25.3221	−0.806012
$988$	−24.7030	−0.785906
$989$	53.6534	1.70608
$990$	−76.4890	−2.43098
$991$	2.62484	0.0833806	0.0416903	−	0.999131i	$-0.486726\pi$
$991$	2.62484	0.0833806	0.0416903	+	0.999131i	$0.486726\pi$
$992$	3.78076	0.120039
$993$	−81.2496	−2.57838
$994$	11.3799	0.360948
$995$	−54.8908	−1.74016
$996$	−26.6605	−0.844771
$997$	−48.1845	−1.52602	−0.763009	−	0.646388i	$-0.776279\pi$
$997$	−48.1845	−1.52602	−0.763009	+	0.646388i	$0.776279\pi$
$998$	11.8684	0.375687
$999$	19.7085	0.623550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.4	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.4	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.02873 q^{3} +1.00000 q^{4} -2.84681 q^{5} -3.02873 q^{6} -0.956843 q^{7} +1.00000 q^{8} +6.17323 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.02873 q^{3} +1.00000 q^{4} -2.84681 q^{5} -3.02873 q^{6} -0.956843 q^{7} +1.00000 q^{8} +6.17323 q^{9} -2.84681 q^{10} +4.35239 q^{11} -3.02873 q^{12} +4.15301 q^{13} -0.956843 q^{14} +8.62224 q^{15} +1.00000 q^{16} +5.17517 q^{17} +6.17323 q^{18} -5.94820 q^{19} -2.84681 q^{20} +2.89802 q^{21} +4.35239 q^{22} +5.31629 q^{23} -3.02873 q^{24} +3.10434 q^{25} +4.15301 q^{26} -9.61088 q^{27} -0.956843 q^{28} +5.63010 q^{29} +8.62224 q^{30} +3.78076 q^{31} +1.00000 q^{32} -13.1822 q^{33} +5.17517 q^{34} +2.72395 q^{35} +6.17323 q^{36} -2.05065 q^{37} -5.94820 q^{38} -12.5784 q^{39} -2.84681 q^{40} -9.32815 q^{41} +2.89802 q^{42} +10.0923 q^{43} +4.35239 q^{44} -17.5740 q^{45} +5.31629 q^{46} -8.73772 q^{47} -3.02873 q^{48} -6.08445 q^{49} +3.10434 q^{50} -15.6742 q^{51} +4.15301 q^{52} -13.8682 q^{53} -9.61088 q^{54} -12.3904 q^{55} -0.956843 q^{56} +18.0155 q^{57} +5.63010 q^{58} +3.91801 q^{59} +8.62224 q^{60} +8.08533 q^{61} +3.78076 q^{62} -5.90682 q^{63} +1.00000 q^{64} -11.8228 q^{65} -13.1822 q^{66} +7.76343 q^{67} +5.17517 q^{68} -16.1016 q^{69} +2.72395 q^{70} -11.8932 q^{71} +6.17323 q^{72} -6.70709 q^{73} -2.05065 q^{74} -9.40221 q^{75} -5.94820 q^{76} -4.16456 q^{77} -12.5784 q^{78} +15.8623 q^{79} -2.84681 q^{80} +10.5891 q^{81} -9.32815 q^{82} +8.80253 q^{83} +2.89802 q^{84} -14.7327 q^{85} +10.0923 q^{86} -17.0521 q^{87} +4.35239 q^{88} +18.2669 q^{89} -17.5740 q^{90} -3.97378 q^{91} +5.31629 q^{92} -11.4509 q^{93} -8.73772 q^{94} +16.9334 q^{95} -3.02873 q^{96} -2.90814 q^{97} -6.08445 q^{98} +26.8683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 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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