LMFDB - Embedded newform 8045.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46500 q^{2} +2.31843 q^{3} +4.07621 q^{4} +1.00000 q^{5} -5.71491 q^{6} -0.590235 q^{7} -5.11785 q^{8} +2.37510 q^{9} +O(q^{10})$	\(q-2.46500 q^{2} +2.31843 q^{3} +4.07621 q^{4} +1.00000 q^{5} -5.71491 q^{6} -0.590235 q^{7} -5.11785 q^{8} +2.37510 q^{9} -2.46500 q^{10} -2.69051 q^{11} +9.45039 q^{12} -1.40343 q^{13} +1.45493 q^{14} +2.31843 q^{15} +4.46306 q^{16} -7.17556 q^{17} -5.85461 q^{18} +1.15356 q^{19} +4.07621 q^{20} -1.36842 q^{21} +6.63211 q^{22} +4.82918 q^{23} -11.8654 q^{24} +1.00000 q^{25} +3.45944 q^{26} -1.44879 q^{27} -2.40592 q^{28} +9.11423 q^{29} -5.71491 q^{30} -5.08505 q^{31} -0.765737 q^{32} -6.23775 q^{33} +17.6877 q^{34} -0.590235 q^{35} +9.68140 q^{36} -0.266993 q^{37} -2.84352 q^{38} -3.25374 q^{39} -5.11785 q^{40} +8.27632 q^{41} +3.37314 q^{42} +2.72460 q^{43} -10.9671 q^{44} +2.37510 q^{45} -11.9039 q^{46} +10.3841 q^{47} +10.3473 q^{48} -6.65162 q^{49} -2.46500 q^{50} -16.6360 q^{51} -5.72066 q^{52} +12.6442 q^{53} +3.57126 q^{54} -2.69051 q^{55} +3.02073 q^{56} +2.67445 q^{57} -22.4665 q^{58} -9.10146 q^{59} +9.45039 q^{60} +1.88782 q^{61} +12.5346 q^{62} -1.40187 q^{63} -7.03859 q^{64} -1.40343 q^{65} +15.3760 q^{66} +3.27272 q^{67} -29.2491 q^{68} +11.1961 q^{69} +1.45493 q^{70} -11.9740 q^{71} -12.1554 q^{72} +9.33169 q^{73} +0.658138 q^{74} +2.31843 q^{75} +4.70216 q^{76} +1.58803 q^{77} +8.02045 q^{78} +14.9604 q^{79} +4.46306 q^{80} -10.4842 q^{81} -20.4011 q^{82} -5.43223 q^{83} -5.57795 q^{84} -7.17556 q^{85} -6.71612 q^{86} +21.1307 q^{87} +13.7696 q^{88} -1.91746 q^{89} -5.85461 q^{90} +0.828350 q^{91} +19.6847 q^{92} -11.7893 q^{93} -25.5967 q^{94} +1.15356 q^{95} -1.77530 q^{96} +11.5717 q^{97} +16.3962 q^{98} -6.39023 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) $ 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * 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 $

$2$	−2.46500	−1.74302	−0.871508	−	0.490381i	$-0.836858\pi$
$2$	−2.46500	−1.74302	−0.871508	+	0.490381i	$0.836858\pi$
$3$	2.31843	1.33854	0.669272	−	0.743018i	$-0.266606\pi$
$3$	2.31843	1.33854	0.669272	+	0.743018i	$0.266606\pi$
$4$	4.07621	2.03810
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−5.71491	−2.33310
$7$	−0.590235	−0.223088	−0.111544	−	0.993760i	$-0.535580\pi$
$7$	−0.590235	−0.223088	−0.111544	+	0.993760i	$0.535580\pi$
$8$	−5.11785	−1.80943
$9$	2.37510	0.791699
$10$	−2.46500	−0.779500
$11$	−2.69051	−0.811220	−0.405610	−	0.914046i	$-0.632941\pi$
$11$	−2.69051	−0.811220	−0.405610	+	0.914046i	$0.632941\pi$
$12$	9.45039	2.72809
$13$	−1.40343	−0.389240	−0.194620	−	0.980879i	$-0.562347\pi$
$13$	−1.40343	−0.389240	−0.194620	+	0.980879i	$0.562347\pi$
$14$	1.45493	0.388845
$15$	2.31843	0.598615
$16$	4.46306	1.11577
$17$	−7.17556	−1.74033	−0.870164	−	0.492762i	$-0.835987\pi$
$17$	−7.17556	−1.74033	−0.870164	+	0.492762i	$0.835987\pi$
$18$	−5.85461	−1.37994
$19$	1.15356	0.264645	0.132323	−	0.991207i	$-0.457757\pi$
$19$	1.15356	0.264645	0.132323	+	0.991207i	$0.457757\pi$
$20$	4.07621	0.911468
$21$	−1.36842	−0.298613
$22$	6.63211	1.41397
$23$	4.82918	1.00695	0.503477	−	0.864009i	$-0.332054\pi$
$23$	4.82918	1.00695	0.503477	+	0.864009i	$0.332054\pi$
$24$	−11.8654	−2.42200
$25$	1.00000	0.200000
$26$	3.45944	0.678452
$27$	−1.44879	−0.278820
$28$	−2.40592	−0.454676
$29$	9.11423	1.69247	0.846235	−	0.532810i	$-0.178864\pi$
$29$	9.11423	1.69247	0.846235	+	0.532810i	$0.178864\pi$
$30$	−5.71491	−1.04340
$31$	−5.08505	−0.913302	−0.456651	−	0.889646i	$-0.650951\pi$
$31$	−5.08505	−0.913302	−0.456651	+	0.889646i	$0.650951\pi$
$32$	−0.765737	−0.135364
$33$	−6.23775	−1.08585
$34$	17.6877	3.03342
$35$	−0.590235	−0.0997679
$36$	9.68140	1.61357
$37$	−0.266993	−0.0438934	−0.0219467	−	0.999759i	$-0.506986\pi$
$37$	−0.266993	−0.0438934	−0.0219467	+	0.999759i	$0.506986\pi$
$38$	−2.84352	−0.461281
$39$	−3.25374	−0.521015
$40$	−5.11785	−0.809203
$41$	8.27632	1.29254	0.646272	−	0.763107i	$-0.276327\pi$
$41$	8.27632	1.29254	0.646272	+	0.763107i	$0.276327\pi$
$42$	3.37314	0.520487
$43$	2.72460	0.415497	0.207749	−	0.978182i	$-0.433386\pi$
$43$	2.72460	0.415497	0.207749	+	0.978182i	$0.433386\pi$
$44$	−10.9671	−1.65335
$45$	2.37510	0.354059
$46$	−11.9039	−1.75514
$47$	10.3841	1.51467	0.757336	−	0.653026i	$-0.226501\pi$
$47$	10.3841	1.51467	0.757336	+	0.653026i	$0.226501\pi$
$48$	10.3473	1.49350
$49$	−6.65162	−0.950232
$50$	−2.46500	−0.348603
$51$	−16.6360	−2.32951
$52$	−5.72066	−0.793312
$53$	12.6442	1.73681	0.868406	−	0.495854i	$-0.165145\pi$
$53$	12.6442	1.73681	0.868406	+	0.495854i	$0.165145\pi$
$54$	3.57126	0.485987
$55$	−2.69051	−0.362789
$56$	3.02073	0.403662
$57$	2.67445	0.354239
$58$	−22.4665	−2.95000
$59$	−9.10146	−1.18491	−0.592454	−	0.805604i	$-0.701841\pi$
$59$	−9.10146	−1.18491	−0.592454	+	0.805604i	$0.701841\pi$
$60$	9.45039	1.22004
$61$	1.88782	0.241710	0.120855	−	0.992670i	$-0.461436\pi$
$61$	1.88782	0.241710	0.120855	+	0.992670i	$0.461436\pi$
$62$	12.5346	1.59190
$63$	−1.40187	−0.176618
$64$	−7.03859	−0.879823
$65$	−1.40343	−0.174074
$66$	15.3760	1.89266
$67$	3.27272	0.399826	0.199913	−	0.979814i	$-0.435934\pi$
$67$	3.27272	0.399826	0.199913	+	0.979814i	$0.435934\pi$
$68$	−29.2491	−3.54697
$69$	11.1961	1.34785
$70$	1.45493	0.173897
$71$	−11.9740	−1.42105	−0.710527	−	0.703670i	$-0.751544\pi$
$71$	−11.9740	−1.42105	−0.710527	+	0.703670i	$0.751544\pi$
$72$	−12.1554	−1.43253
$73$	9.33169	1.09219	0.546096	−	0.837723i	$-0.316113\pi$
$73$	9.33169	1.09219	0.546096	+	0.837723i	$0.316113\pi$
$74$	0.658138	0.0765069
$75$	2.31843	0.267709
$76$	4.70216	0.539374
$77$	1.58803	0.180973
$78$	8.02045	0.908138
$79$	14.9604	1.68317	0.841587	−	0.540121i	$-0.181621\pi$
$79$	14.9604	1.68317	0.841587	+	0.540121i	$0.181621\pi$
$80$	4.46306	0.498986
$81$	−10.4842	−1.16491
$82$	−20.4011	−2.25292
$83$	−5.43223	−0.596264	−0.298132	−	0.954525i	$-0.596364\pi$
$83$	−5.43223	−0.596264	−0.298132	+	0.954525i	$0.596364\pi$
$84$	−5.57795	−0.608604
$85$	−7.17556	−0.778298
$86$	−6.71612	−0.724218
$87$	21.1307	2.26545
$88$	13.7696	1.46785
$89$	−1.91746	−0.203251	−0.101625	−	0.994823i	$-0.532404\pi$
$89$	−1.91746	−0.203251	−0.101625	+	0.994823i	$0.532404\pi$
$90$	−5.85461	−0.617130
$91$	0.828350	0.0868347
$92$	19.6847	2.05228
$93$	−11.7893	−1.22249
$94$	−25.5967	−2.64010
$95$	1.15356	0.118353
$96$	−1.77530	−0.181191
$97$	11.5717	1.17493	0.587466	−	0.809249i	$-0.300125\pi$
$97$	11.5717	1.17493	0.587466	+	0.809249i	$0.300125\pi$
$98$	16.3962	1.65627
$99$	−6.39023	−0.642243
$100$	4.07621	0.407621
$101$	−12.6389	−1.25762	−0.628810	−	0.777559i	$-0.716458\pi$
$101$	−12.6389	−1.25762	−0.628810	+	0.777559i	$0.716458\pi$
$102$	41.0077	4.06037
$103$	4.30295	0.423982	0.211991	−	0.977272i	$-0.432005\pi$
$103$	4.30295	0.423982	0.211991	+	0.977272i	$0.432005\pi$
$104$	7.18252	0.704304
$105$	−1.36842	−0.133544
$106$	−31.1679	−3.02729
$107$	−11.4864	−1.11044	−0.555218	−	0.831705i	$-0.687365\pi$
$107$	−11.4864	−1.11044	−0.555218	+	0.831705i	$0.687365\pi$
$108$	−5.90557	−0.568263
$109$	15.0294	1.43955	0.719777	−	0.694206i	$-0.244244\pi$
$109$	15.0294	1.43955	0.719777	+	0.694206i	$0.244244\pi$
$110$	6.63211	0.632346
$111$	−0.619004	−0.0587533
$112$	−2.63425	−0.248914
$113$	−17.6544	−1.66078	−0.830391	−	0.557181i	$-0.811883\pi$
$113$	−17.6544	−1.66078	−0.830391	+	0.557181i	$0.811883\pi$
$114$	−6.59250	−0.617444
$115$	4.82918	0.450323
$116$	37.1515	3.44943
$117$	−3.33327	−0.308161
$118$	22.4351	2.06532
$119$	4.23526	0.388246
$120$	−11.8654	−1.08315
$121$	−3.76114	−0.341922
$122$	−4.65346	−0.421304
$123$	19.1880	1.73013
$124$	−20.7277	−1.86141
$125$	1.00000	0.0894427
$126$	3.45559	0.307849
$127$	6.70772	0.595214	0.297607	−	0.954688i	$-0.403812\pi$
$127$	6.70772	0.595214	0.297607	+	0.954688i	$0.403812\pi$
$128$	18.8816	1.66891
$129$	6.31678	0.556161
$130$	3.45944	0.303413
$131$	−8.75634	−0.765045	−0.382522	−	0.923946i	$-0.624945\pi$
$131$	−8.75634	−0.765045	−0.382522	+	0.923946i	$0.624945\pi$
$132$	−25.4264	−2.21308
$133$	−0.680872	−0.0590391
$134$	−8.06725	−0.696904
$135$	−1.44879	−0.124692
$136$	36.7234	3.14901
$137$	−8.15637	−0.696846	−0.348423	−	0.937337i	$-0.613283\pi$
$137$	−8.15637	−0.696846	−0.348423	+	0.937337i	$0.613283\pi$
$138$	−27.5983	−2.34933
$139$	3.83065	0.324911	0.162456	−	0.986716i	$-0.448059\pi$
$139$	3.83065	0.324911	0.162456	+	0.986716i	$0.448059\pi$
$140$	−2.40592	−0.203337
$141$	24.0747	2.02745
$142$	29.5159	2.47692
$143$	3.77594	0.315760
$144$	10.6002	0.883351
$145$	9.11423	0.756896
$146$	−23.0026	−1.90371
$147$	−15.4213	−1.27193
$148$	−1.08832	−0.0894594
$149$	−4.88918	−0.400538	−0.200269	−	0.979741i	$-0.564182\pi$
$149$	−4.88918	−0.400538	−0.200269	+	0.979741i	$0.564182\pi$
$150$	−5.71491	−0.466621
$151$	18.1820	1.47963	0.739816	−	0.672809i	$-0.234912\pi$
$151$	18.1820	1.47963	0.739816	+	0.672809i	$0.234912\pi$
$152$	−5.90375	−0.478857
$153$	−17.0427	−1.37782
$154$	−3.91450	−0.315439
$155$	−5.08505	−0.408441
$156$	−13.2629	−1.06188
$157$	−4.32031	−0.344798	−0.172399	−	0.985027i	$-0.555152\pi$
$157$	−4.32031	−0.344798	−0.172399	+	0.985027i	$0.555152\pi$
$158$	−36.8773	−2.93380
$159$	29.3146	2.32480
$160$	−0.765737	−0.0605368
$161$	−2.85035	−0.224639
$162$	25.8435	2.03046
$163$	4.44193	0.347919	0.173959	−	0.984753i	$-0.444344\pi$
$163$	4.44193	0.347919	0.173959	+	0.984753i	$0.444344\pi$
$164$	33.7360	2.63434
$165$	−6.23775	−0.485609
$166$	13.3904	1.03930
$167$	3.95589	0.306116	0.153058	−	0.988217i	$-0.451088\pi$
$167$	3.95589	0.306116	0.153058	+	0.988217i	$0.451088\pi$
$168$	7.00334	0.540320
$169$	−11.0304	−0.848492
$170$	17.6877	1.35659
$171$	2.73982	0.209519
$172$	11.1060	0.846827
$173$	−0.597238	−0.0454072	−0.0227036	−	0.999742i	$-0.507227\pi$
$173$	−0.597238	−0.0454072	−0.0227036	+	0.999742i	$0.507227\pi$
$174$	−52.0870	−3.94871
$175$	−0.590235	−0.0446175
$176$	−12.0079	−0.905132
$177$	−21.1011	−1.58605
$178$	4.72654	0.354269
$179$	18.4707	1.38057	0.690283	−	0.723540i	$-0.257486\pi$
$179$	18.4707	1.38057	0.690283	+	0.723540i	$0.257486\pi$
$180$	9.68140	0.721609
$181$	−2.61828	−0.194615	−0.0973076	−	0.995254i	$-0.531023\pi$
$181$	−2.61828	−0.194615	−0.0973076	+	0.995254i	$0.531023\pi$
$182$	−2.04188	−0.151354
$183$	4.37676	0.323539
$184$	−24.7150	−1.82201
$185$	−0.266993	−0.0196297
$186$	29.0606	2.13083
$187$	19.3059	1.41179
$188$	42.3276	3.08706
$189$	0.855125	0.0622012
$190$	−2.84352	−0.206291
$191$	−12.0222	−0.869893	−0.434946	−	0.900456i	$-0.643233\pi$
$191$	−12.0222	−0.869893	−0.434946	+	0.900456i	$0.643233\pi$
$192$	−16.3184	−1.17768
$193$	7.47499	0.538062	0.269031	−	0.963132i	$-0.413297\pi$
$193$	7.47499	0.538062	0.269031	+	0.963132i	$0.413297\pi$
$194$	−28.5243	−2.04793
$195$	−3.25374	−0.233005
$196$	−27.1134	−1.93667
$197$	8.08509	0.576039	0.288020	−	0.957625i	$-0.407003\pi$
$197$	8.08509	0.576039	0.288020	+	0.957625i	$0.407003\pi$
$198$	15.7519	1.11944
$199$	19.2069	1.36154	0.680771	−	0.732496i	$-0.261645\pi$
$199$	19.2069	1.36154	0.680771	+	0.732496i	$0.261645\pi$
$200$	−5.11785	−0.361887
$201$	7.58756	0.535185
$202$	31.1549	2.19205
$203$	−5.37953	−0.377569
$204$	−67.8118	−4.74778
$205$	8.27632	0.578043
$206$	−10.6068	−0.739008
$207$	11.4698	0.797204
$208$	−6.26358	−0.434301
$209$	−3.10367	−0.214685
$210$	3.37314	0.232769
$211$	12.6002	0.867433	0.433717	−	0.901049i	$-0.357202\pi$
$211$	12.6002	0.867433	0.433717	+	0.901049i	$0.357202\pi$
$212$	51.5403	3.53980
$213$	−27.7609	−1.90214
$214$	28.3140	1.93551
$215$	2.72460	0.185816
$216$	7.41468	0.504505
$217$	3.00137	0.203746
$218$	−37.0474	−2.50916
$219$	21.6348	1.46195
$220$	−10.9671	−0.739401
$221$	10.0704	0.677406
$222$	1.52584	0.102408
$223$	3.55032	0.237747	0.118873	−	0.992909i	$-0.462072\pi$
$223$	3.55032	0.237747	0.118873	+	0.992909i	$0.462072\pi$
$224$	0.451964	0.0301981
$225$	2.37510	0.158340
$226$	43.5179	2.89477
$227$	16.9883	1.12755	0.563777	−	0.825927i	$-0.309348\pi$
$227$	16.9883	1.12755	0.563777	+	0.825927i	$0.309348\pi$
$228$	10.9016	0.721976
$229$	−13.2438	−0.875176	−0.437588	−	0.899176i	$-0.644167\pi$
$229$	−13.2438	−0.875176	−0.437588	+	0.899176i	$0.644167\pi$
$230$	−11.9039	−0.784920
$231$	3.68174	0.242241
$232$	−46.6452	−3.06241
$233$	−20.5750	−1.34791	−0.673957	−	0.738770i	$-0.735407\pi$
$233$	−20.5750	−1.34791	−0.673957	+	0.738770i	$0.735407\pi$
$234$	8.21651	0.537130
$235$	10.3841	0.677382
$236$	−37.0994	−2.41497
$237$	34.6845	2.25300
$238$	−10.4399	−0.676719
$239$	17.9237	1.15939	0.579694	−	0.814834i	$-0.303172\pi$
$239$	17.9237	1.15939	0.579694	+	0.814834i	$0.303172\pi$
$240$	10.3473	0.667914
$241$	26.4714	1.70517	0.852586	−	0.522587i	$-0.175033\pi$
$241$	26.4714	1.70517	0.852586	+	0.522587i	$0.175033\pi$
$242$	9.27120	0.595975
$243$	−19.9605	−1.28047
$244$	7.69513	0.492630
$245$	−6.65162	−0.424957
$246$	−47.2984	−3.01564
$247$	−1.61894	−0.103011
$248$	26.0245	1.65256
$249$	−12.5942	−0.798126
$250$	−2.46500	−0.155900
$251$	6.97566	0.440300	0.220150	−	0.975466i	$-0.429345\pi$
$251$	6.97566	0.440300	0.220150	+	0.975466i	$0.429345\pi$
$252$	−5.71430	−0.359967
$253$	−12.9930	−0.816861
$254$	−16.5345	−1.03747
$255$	−16.6360	−1.04179
$256$	−32.4658	−2.02911
$257$	3.58498	0.223625	0.111812	−	0.993729i	$-0.464334\pi$
$257$	3.58498	0.223625	0.111812	+	0.993729i	$0.464334\pi$
$258$	−15.5708	−0.969398
$259$	0.157589	0.00979209
$260$	−5.72066	−0.354780
$261$	21.6472	1.33993
$262$	21.5843	1.33349
$263$	9.44517	0.582414	0.291207	−	0.956660i	$-0.405943\pi$
$263$	9.44517	0.582414	0.291207	+	0.956660i	$0.405943\pi$
$264$	31.9239	1.96478
$265$	12.6442	0.776726
$266$	1.67835	0.102906
$267$	−4.44550	−0.272060
$268$	13.3403	0.814888
$269$	27.9813	1.70605	0.853025	−	0.521869i	$-0.174765\pi$
$269$	27.9813	1.70605	0.853025	+	0.521869i	$0.174765\pi$
$270$	3.57126	0.217340
$271$	0.509792	0.0309677	0.0154838	−	0.999880i	$-0.495071\pi$
$271$	0.509792	0.0309677	0.0154838	+	0.999880i	$0.495071\pi$
$272$	−32.0250	−1.94180
$273$	1.92047	0.116232
$274$	20.1054	1.21461
$275$	−2.69051	−0.162244
$276$	45.6376	2.74706
$277$	22.3890	1.34523	0.672613	−	0.739995i	$-0.265172\pi$
$277$	22.3890	1.34523	0.672613	+	0.739995i	$0.265172\pi$
$278$	−9.44253	−0.566326
$279$	−12.0775	−0.723061
$280$	3.02073	0.180523
$281$	4.36230	0.260233	0.130116	−	0.991499i	$-0.458465\pi$
$281$	4.36230	0.260233	0.130116	+	0.991499i	$0.458465\pi$
$282$	−59.3440	−3.53388
$283$	12.0052	0.713634	0.356817	−	0.934174i	$-0.383862\pi$
$283$	12.0052	0.713634	0.356817	+	0.934174i	$0.383862\pi$
$284$	−48.8086	−2.89626
$285$	2.67445	0.158420
$286$	−9.30767	−0.550374
$287$	−4.88497	−0.288351
$288$	−1.81870	−0.107168
$289$	34.4886	2.02874
$290$	−22.4665	−1.31928
$291$	26.8282	1.57270
$292$	38.0379	2.22600
$293$	9.84924	0.575399	0.287699	−	0.957721i	$-0.407110\pi$
$293$	9.84924	0.575399	0.287699	+	0.957721i	$0.407110\pi$
$294$	38.0134	2.21699
$295$	−9.10146	−0.529907
$296$	1.36643	0.0794222
$297$	3.89798	0.226184
$298$	12.0518	0.698143
$299$	−6.77739	−0.391947
$300$	9.45039	0.545618
$301$	−1.60815	−0.0926923
$302$	−44.8186	−2.57902
$303$	−29.3024	−1.68338
$304$	5.14841	0.295282
$305$	1.88782	0.108096
$306$	42.0101	2.40156
$307$	10.8312	0.618167	0.309083	−	0.951035i	$-0.399978\pi$
$307$	10.8312	0.618167	0.309083	+	0.951035i	$0.399978\pi$
$308$	6.47316	0.368842
$309$	9.97607	0.567519
$310$	12.5346	0.711919
$311$	4.88857	0.277205	0.138603	−	0.990348i	$-0.455739\pi$
$311$	4.88857	0.277205	0.138603	+	0.990348i	$0.455739\pi$
$312$	16.6521	0.942742
$313$	0.215223	0.0121651	0.00608256	−	0.999982i	$-0.498064\pi$
$313$	0.215223	0.0121651	0.00608256	+	0.999982i	$0.498064\pi$
$314$	10.6495	0.600989
$315$	−1.40187	−0.0789862
$316$	60.9817	3.43049
$317$	16.8546	0.946647	0.473324	−	0.880889i	$-0.343054\pi$
$317$	16.8546	0.946647	0.473324	+	0.880889i	$0.343054\pi$
$318$	−72.2604	−4.05216
$319$	−24.5220	−1.37297
$320$	−7.03859	−0.393469
$321$	−26.6305	−1.48637
$322$	7.02610	0.391549
$323$	−8.27744	−0.460569
$324$	−42.7358	−2.37421
$325$	−1.40343	−0.0778480
$326$	−10.9493	−0.606428
$327$	34.8445	1.92691
$328$	−42.3569	−2.33877
$329$	−6.12903	−0.337905
$330$	15.3760	0.846423
$331$	25.2101	1.38567	0.692836	−	0.721095i	$-0.256361\pi$
$331$	25.2101	1.38567	0.692836	+	0.721095i	$0.256361\pi$
$332$	−22.1429	−1.21525
$333$	−0.634135	−0.0347504
$334$	−9.75125	−0.533565
$335$	3.27272	0.178808
$336$	−6.10732	−0.333182
$337$	20.7489	1.13026	0.565132	−	0.825000i	$-0.308825\pi$
$337$	20.7489	1.13026	0.565132	+	0.825000i	$0.308825\pi$
$338$	27.1899	1.47894
$339$	−40.9303	−2.22303
$340$	−29.2491	−1.58625
$341$	13.6814	0.740889
$342$	−6.75365	−0.365196
$343$	8.05766	0.435073
$344$	−13.9441	−0.751814
$345$	11.1961	0.602777
$346$	1.47219	0.0791454
$347$	21.4055	1.14911	0.574553	−	0.818468i	$-0.305176\pi$
$347$	21.4055	1.14911	0.574553	+	0.818468i	$0.305176\pi$
$348$	86.1330	4.61721
$349$	16.8243	0.900583	0.450291	−	0.892882i	$-0.351320\pi$
$349$	16.8243	0.900583	0.450291	+	0.892882i	$0.351320\pi$
$350$	1.45493	0.0777691
$351$	2.03327	0.108528
$352$	2.06022	0.109810
$353$	−16.2815	−0.866576	−0.433288	−	0.901255i	$-0.642647\pi$
$353$	−16.2815	−0.866576	−0.433288	+	0.901255i	$0.642647\pi$
$354$	52.0140	2.76451
$355$	−11.9740	−0.635515
$356$	−7.81598	−0.414246
$357$	9.81914	0.519684
$358$	−45.5302	−2.40635
$359$	9.04224	0.477231	0.238616	−	0.971114i	$-0.423306\pi$
$359$	9.04224	0.477231	0.238616	+	0.971114i	$0.423306\pi$
$360$	−12.1554	−0.640645
$361$	−17.6693	−0.929963
$362$	6.45405	0.339218
$363$	−8.71992	−0.457677
$364$	3.37653	0.176978
$365$	9.33169	0.488443
$366$	−10.7887	−0.563934
$367$	13.8114	0.720947	0.360474	−	0.932769i	$-0.382615\pi$
$367$	13.8114	0.720947	0.360474	+	0.932769i	$0.382615\pi$
$368$	21.5529	1.12352
$369$	19.6571	1.02331
$370$	0.658138	0.0342149
$371$	−7.46303	−0.387461
$372$	−48.0557	−2.49157
$373$	28.5165	1.47653	0.738263	−	0.674513i	$-0.235646\pi$
$373$	28.5165	1.47653	0.738263	+	0.674513i	$0.235646\pi$
$374$	−47.5891	−2.46077
$375$	2.31843	0.119723
$376$	−53.1441	−2.74070
$377$	−12.7911	−0.658777
$378$	−2.10788	−0.108418
$379$	−22.0897	−1.13467	−0.567335	−	0.823487i	$-0.692025\pi$
$379$	−22.0897	−1.13467	−0.567335	+	0.823487i	$0.692025\pi$
$380$	4.70216	0.241216
$381$	15.5514	0.796720
$382$	29.6346	1.51624
$383$	16.1439	0.824913	0.412456	−	0.910977i	$-0.364671\pi$
$383$	16.1439	0.824913	0.412456	+	0.910977i	$0.364671\pi$
$384$	43.7755	2.23391
$385$	1.58803	0.0809337
$386$	−18.4258	−0.937850
$387$	6.47119	0.328949
$388$	47.1689	2.39464
$389$	−5.30284	−0.268865	−0.134432	−	0.990923i	$-0.542921\pi$
$389$	−5.30284	−0.268865	−0.134432	+	0.990923i	$0.542921\pi$
$390$	8.02045	0.406131
$391$	−34.6520	−1.75243
$392$	34.0420	1.71938
$393$	−20.3009	−1.02405
$394$	−19.9297	−1.00405
$395$	14.9604	0.752739
$396$	−26.0479	−1.30896
$397$	33.1212	1.66231	0.831153	−	0.556044i	$-0.187682\pi$
$397$	33.1212	1.66231	0.831153	+	0.556044i	$0.187682\pi$
$398$	−47.3450	−2.37319
$399$	−1.57855	−0.0790264
$400$	4.46306	0.223153
$401$	11.1128	0.554946	0.277473	−	0.960733i	$-0.410503\pi$
$401$	11.1128	0.554946	0.277473	+	0.960733i	$0.410503\pi$
$402$	−18.7033	−0.932836
$403$	7.13649	0.355494
$404$	−51.5189	−2.56316
$405$	−10.4842	−0.520964
$406$	13.2605	0.658109
$407$	0.718349	0.0356072
$408$	85.1405	4.21508
$409$	15.7732	0.779936	0.389968	−	0.920828i	$-0.372486\pi$
$409$	15.7732	0.779936	0.389968	+	0.920828i	$0.372486\pi$
$410$	−20.4011	−1.00754
$411$	−18.9099	−0.932759
$412$	17.5397	0.864120
$413$	5.37200	0.264339
$414$	−28.2730	−1.38954
$415$	−5.43223	−0.266658
$416$	1.07465	0.0526893
$417$	8.88107	0.434908
$418$	7.65054	0.374200
$419$	−38.4160	−1.87674	−0.938371	−	0.345629i	$-0.887665\pi$
$419$	−38.4160	−1.87674	−0.938371	+	0.345629i	$0.887665\pi$
$420$	−5.57795	−0.272176
$421$	6.14365	0.299423	0.149712	−	0.988730i	$-0.452165\pi$
$421$	6.14365	0.299423	0.149712	+	0.988730i	$0.452165\pi$
$422$	−31.0594	−1.51195
$423$	24.6632	1.19916
$424$	−64.7110	−3.14264
$425$	−7.17556	−0.348066
$426$	68.4305	3.31547
$427$	−1.11425	−0.0539225
$428$	−46.8211	−2.26319
$429$	8.75423	0.422658
$430$	−6.71612	−0.323880
$431$	−14.2893	−0.688293	−0.344147	−	0.938916i	$-0.611832\pi$
$431$	−14.2893	−0.688293	−0.344147	+	0.938916i	$0.611832\pi$
$432$	−6.46603	−0.311097
$433$	13.6066	0.653890	0.326945	−	0.945043i	$-0.393981\pi$
$433$	13.6066	0.653890	0.326945	+	0.945043i	$0.393981\pi$
$434$	−7.39838	−0.355133
$435$	21.1307	1.01314
$436$	61.2629	2.93396
$437$	5.57075	0.266485
$438$	−53.3298	−2.54820
$439$	−37.8100	−1.80457	−0.902286	−	0.431138i	$-0.858112\pi$
$439$	−37.8100	−1.80457	−0.902286	+	0.431138i	$0.858112\pi$
$440$	13.7696	0.656442
$441$	−15.7983	−0.752298
$442$	−24.8234	−1.18073
$443$	17.8352	0.847375	0.423688	−	0.905808i	$-0.360735\pi$
$443$	17.8352	0.847375	0.423688	+	0.905808i	$0.360735\pi$
$444$	−2.52319	−0.119745
$445$	−1.91746	−0.0908965
$446$	−8.75152	−0.414397
$447$	−11.3352	−0.536137
$448$	4.15442	0.196278
$449$	−7.35484	−0.347096	−0.173548	−	0.984825i	$-0.555523\pi$
$449$	−7.35484	−0.347096	−0.173548	+	0.984825i	$0.555523\pi$
$450$	−5.85461	−0.275989
$451$	−22.2675	−1.04854
$452$	−71.9629	−3.38485
$453$	42.1537	1.98055
$454$	−41.8761	−1.96534
$455$	0.828350	0.0388337
$456$	−13.6874	−0.640972
$457$	−7.75468	−0.362749	−0.181374	−	0.983414i	$-0.558055\pi$
$457$	−7.75468	−0.362749	−0.181374	+	0.983414i	$0.558055\pi$
$458$	32.6460	1.52545
$459$	10.3959	0.485237
$460$	19.6847	0.917806
$461$	−18.8146	−0.876282	−0.438141	−	0.898906i	$-0.644363\pi$
$461$	−18.8146	−0.876282	−0.438141	+	0.898906i	$0.644363\pi$
$462$	−9.07548	−0.422229
$463$	7.68154	0.356992	0.178496	−	0.983941i	$-0.442877\pi$
$463$	7.68154	0.356992	0.178496	+	0.983941i	$0.442877\pi$
$464$	40.6774	1.88840
$465$	−11.7893	−0.546716
$466$	50.7174	2.34944
$467$	−19.6216	−0.907979	−0.453990	−	0.891007i	$-0.650000\pi$
$467$	−19.6216	−0.907979	−0.453990	+	0.891007i	$0.650000\pi$
$468$	−13.5871	−0.628065
$469$	−1.93167	−0.0891964
$470$	−25.5967	−1.18069
$471$	−10.0163	−0.461527
$472$	46.5799	2.14401
$473$	−7.33057	−0.337060
$474$	−85.4973	−3.92702
$475$	1.15356	0.0529290
$476$	17.2638	0.791286
$477$	30.0312	1.37503
$478$	−44.1819	−2.02083
$479$	−10.8043	−0.493659	−0.246829	−	0.969059i	$-0.579389\pi$
$479$	−10.8043	−0.493659	−0.246829	+	0.969059i	$0.579389\pi$
$480$	−1.77530	−0.0810312
$481$	0.374705	0.0170851
$482$	−65.2519	−2.97214
$483$	−6.60832	−0.300689
$484$	−15.3312	−0.696872
$485$	11.5717	0.525446
$486$	49.2025	2.23187
$487$	−1.63413	−0.0740493	−0.0370246	−	0.999314i	$-0.511788\pi$
$487$	−1.63413	−0.0740493	−0.0370246	+	0.999314i	$0.511788\pi$
$488$	−9.66155	−0.437358
$489$	10.2983	0.465705
$490$	16.3962	0.740706
$491$	−33.5077	−1.51218	−0.756090	−	0.654468i	$-0.772893\pi$
$491$	−33.5077	−1.51218	−0.756090	+	0.654468i	$0.772893\pi$
$492$	78.2144	3.52618
$493$	−65.3997	−2.94545
$494$	3.99067	0.179549
$495$	−6.39023	−0.287220
$496$	−22.6949	−1.01903
$497$	7.06748	0.317020
$498$	31.0447	1.39115
$499$	−27.4950	−1.23084	−0.615422	−	0.788198i	$-0.711014\pi$
$499$	−27.4950	−1.23084	−0.615422	+	0.788198i	$0.711014\pi$
$500$	4.07621	0.182294
$501$	9.17144	0.409750
$502$	−17.1950	−0.767449
$503$	36.9087	1.64568	0.822840	−	0.568274i	$-0.192388\pi$
$503$	36.9087	1.64568	0.822840	+	0.568274i	$0.192388\pi$
$504$	7.17453	0.319579
$505$	−12.6389	−0.562425
$506$	32.0276	1.42380
$507$	−25.5732	−1.13574
$508$	27.3421	1.21311
$509$	−16.3931	−0.726611	−0.363305	−	0.931670i	$-0.618352\pi$
$509$	−16.3931	−0.726611	−0.363305	+	0.931670i	$0.618352\pi$
$510$	41.0077	1.81585
$511$	−5.50789	−0.243654
$512$	42.2650	1.86787
$513$	−1.67127	−0.0737882
$514$	−8.83696	−0.389782
$515$	4.30295	0.189611
$516$	25.7485	1.13351
$517$	−27.9385	−1.22873
$518$	−0.388456	−0.0170678
$519$	−1.38465	−0.0607795
$520$	7.18252	0.314974
$521$	24.3309	1.06596	0.532978	−	0.846129i	$-0.321073\pi$
$521$	24.3309	1.06596	0.532978	+	0.846129i	$0.321073\pi$
$522$	−53.3603	−2.33551
$523$	−38.4690	−1.68213	−0.841067	−	0.540931i	$-0.818072\pi$
$523$	−38.4690	−1.68213	−0.841067	+	0.540931i	$0.818072\pi$
$524$	−35.6927	−1.55924
$525$	−1.36842	−0.0597225
$526$	−23.2823	−1.01516
$527$	36.4881	1.58945
$528$	−27.8395	−1.21156
$529$	0.320962	0.0139549
$530$	−31.1679	−1.35385
$531$	−21.6169	−0.938092
$532$	−2.77538	−0.120328
$533$	−11.6152	−0.503110
$534$	10.9581	0.474205
$535$	−11.4864	−0.496602
$536$	−16.7493	−0.723459
$537$	42.8230	1.84795
$538$	−68.9738	−2.97367
$539$	17.8963	0.770847
$540$	−5.90557	−0.254135
$541$	33.9010	1.45752	0.728759	−	0.684771i	$-0.240098\pi$
$541$	33.9010	1.45752	0.728759	+	0.684771i	$0.240098\pi$
$542$	−1.25664	−0.0539771
$543$	−6.07029	−0.260501
$544$	5.49459	0.235579
$545$	15.0294	0.643788
$546$	−4.73395	−0.202594
$547$	25.6887	1.09837	0.549184	−	0.835701i	$-0.314939\pi$
$547$	25.6887	1.09837	0.549184	+	0.835701i	$0.314939\pi$
$548$	−33.2471	−1.42024
$549$	4.48375	0.191362
$550$	6.63211	0.282794
$551$	10.5138	0.447904
$552$	−57.2999	−2.43885
$553$	−8.83014	−0.375496
$554$	−55.1888	−2.34475
$555$	−0.619004	−0.0262753
$556$	15.6145	0.662203
$557$	−42.7436	−1.81111	−0.905553	−	0.424233i	$-0.860544\pi$
$557$	−42.7436	−1.81111	−0.905553	+	0.424233i	$0.860544\pi$
$558$	29.7710	1.26031
$559$	−3.82377	−0.161728
$560$	−2.63425	−0.111318
$561$	44.7594	1.88974
$562$	−10.7531	−0.453590
$563$	40.3154	1.69909	0.849546	−	0.527515i	$-0.176876\pi$
$563$	40.3154	1.69909	0.849546	+	0.527515i	$0.176876\pi$
$564$	98.1334	4.13216
$565$	−17.6544	−0.742725
$566$	−29.5927	−1.24388
$567$	6.18814	0.259877
$568$	61.2812	2.57130
$569$	−30.2635	−1.26871	−0.634356	−	0.773041i	$-0.718735\pi$
$569$	−30.2635	−1.26871	−0.634356	+	0.773041i	$0.718735\pi$
$570$	−6.59250	−0.276129
$571$	−33.0384	−1.38261	−0.691307	−	0.722561i	$-0.742965\pi$
$571$	−33.0384	−1.38261	−0.691307	+	0.722561i	$0.742965\pi$
$572$	15.3915	0.643551
$573$	−27.8725	−1.16439
$574$	12.0414	0.502600
$575$	4.82918	0.201391
$576$	−16.7173	−0.696556
$577$	−32.9546	−1.37192	−0.685960	−	0.727640i	$-0.740617\pi$
$577$	−32.9546	−1.37192	−0.685960	+	0.727640i	$0.740617\pi$
$578$	−85.0144	−3.53613
$579$	17.3302	0.720219
$580$	37.1515	1.54263
$581$	3.20629	0.133019
$582$	−66.1315	−2.74124
$583$	−34.0193	−1.40894
$584$	−47.7582	−1.97625
$585$	−3.33327	−0.137814
$586$	−24.2783	−1.00293
$587$	−5.51207	−0.227507	−0.113754	−	0.993509i	$-0.536287\pi$
$587$	−5.51207	−0.227507	−0.113754	+	0.993509i	$0.536287\pi$
$588$	−62.8604	−2.59232
$589$	−5.86592	−0.241701
$590$	22.4351	0.923637
$591$	18.7447	0.771054
$592$	−1.19161	−0.0489748
$593$	27.6945	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$593$	27.6945	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$594$	−9.60852	−0.394242
$595$	4.23526	0.173629
$596$	−19.9293	−0.816337
$597$	44.5298	1.82248
$598$	16.7063	0.683169
$599$	38.1460	1.55860	0.779302	−	0.626649i	$-0.215574\pi$
$599$	38.1460	1.55860	0.779302	+	0.626649i	$0.215574\pi$
$600$	−11.8654	−0.484401
$601$	−10.7473	−0.438390	−0.219195	−	0.975681i	$-0.570343\pi$
$601$	−10.7473	−0.438390	−0.219195	+	0.975681i	$0.570343\pi$
$602$	3.96409	0.161564
$603$	7.77303	0.316542
$604$	74.1138	3.01565
$605$	−3.76114	−0.152912
$606$	72.2304	2.93416
$607$	44.5904	1.80987	0.904933	−	0.425553i	$-0.139920\pi$
$607$	44.5904	1.80987	0.904933	+	0.425553i	$0.139920\pi$
$608$	−0.883324	−0.0358235
$609$	−12.4721	−0.505393
$610$	−4.65346	−0.188413
$611$	−14.5733	−0.589571
$612$	−69.4694	−2.80813
$613$	9.82393	0.396785	0.198392	−	0.980123i	$-0.436428\pi$
$613$	9.82393	0.396785	0.198392	+	0.980123i	$0.436428\pi$
$614$	−26.6988	−1.07747
$615$	19.1880	0.773736
$616$	−8.12732	−0.327459
$617$	−20.8745	−0.840375	−0.420187	−	0.907437i	$-0.638036\pi$
$617$	−20.8745	−0.840375	−0.420187	+	0.907437i	$0.638036\pi$
$618$	−24.5910	−0.989194
$619$	20.1097	0.808278	0.404139	−	0.914698i	$-0.367571\pi$
$619$	20.1097	0.808278	0.404139	+	0.914698i	$0.367571\pi$
$620$	−20.7277	−0.832446
$621$	−6.99646	−0.280758
$622$	−12.0503	−0.483173
$623$	1.13175	0.0453427
$624$	−14.5216	−0.581331
$625$	1.00000	0.0400000
$626$	−0.530524	−0.0212040
$627$	−7.19563	−0.287366
$628$	−17.6105	−0.702735
$629$	1.91583	0.0763890
$630$	3.45559	0.137674
$631$	1.46997	0.0585185	0.0292592	−	0.999572i	$-0.490685\pi$
$631$	1.46997	0.0585185	0.0292592	+	0.999572i	$0.490685\pi$
$632$	−76.5650	−3.04559
$633$	29.2126	1.16110
$634$	−41.5465	−1.65002
$635$	6.70772	0.266188
$636$	119.492	4.73818
$637$	9.33506	0.369868
$638$	60.4465	2.39310
$639$	−28.4395	−1.12505
$640$	18.8816	0.746359
$641$	−45.1720	−1.78419	−0.892094	−	0.451849i	$-0.850764\pi$
$641$	−45.1720	−1.78419	−0.892094	+	0.451849i	$0.850764\pi$
$642$	65.6440	2.59076
$643$	−17.4923	−0.689829	−0.344914	−	0.938634i	$-0.612092\pi$
$643$	−17.4923	−0.689829	−0.344914	+	0.938634i	$0.612092\pi$
$644$	−11.6186	−0.457838
$645$	6.31678	0.248723
$646$	20.4039	0.802780
$647$	−20.3546	−0.800221	−0.400111	−	0.916467i	$-0.631028\pi$
$647$	−20.3546	−0.800221	−0.400111	+	0.916467i	$0.631028\pi$
$648$	53.6566	2.10783
$649$	24.4876	0.961222
$650$	3.45944	0.135690
$651$	6.95846	0.272724
$652$	18.1062	0.709095
$653$	30.7763	1.20437	0.602185	−	0.798357i	$-0.294297\pi$
$653$	30.7763	1.20437	0.602185	+	0.798357i	$0.294297\pi$
$654$	−85.8916	−3.35863
$655$	−8.75634	−0.342138
$656$	36.9377	1.44218
$657$	22.1637	0.864687
$658$	15.1080	0.588973
$659$	−7.07947	−0.275777	−0.137888	−	0.990448i	$-0.544032\pi$
$659$	−7.07947	−0.275777	−0.137888	+	0.990448i	$0.544032\pi$
$660$	−25.4264	−0.989721
$661$	2.31363	0.0899898	0.0449949	−	0.998987i	$-0.485673\pi$
$661$	2.31363	0.0899898	0.0449949	+	0.998987i	$0.485673\pi$
$662$	−62.1428	−2.41525
$663$	23.3474	0.906737
$664$	27.8013	1.07890
$665$	−0.680872	−0.0264031
$666$	1.56314	0.0605705
$667$	44.0142	1.70424
$668$	16.1250	0.623896
$669$	8.23115	0.318235
$670$	−8.06725	−0.311665
$671$	−5.07919	−0.196080
$672$	1.04785	0.0404215
$673$	−11.0695	−0.426698	−0.213349	−	0.976976i	$-0.568437\pi$
$673$	−11.0695	−0.426698	−0.213349	+	0.976976i	$0.568437\pi$
$674$	−51.1460	−1.97007
$675$	−1.44879	−0.0557639
$676$	−44.9622	−1.72932
$677$	3.67430	0.141215	0.0706074	−	0.997504i	$-0.477506\pi$
$677$	3.67430	0.141215	0.0706074	+	0.997504i	$0.477506\pi$
$678$	100.893	3.87478
$679$	−6.83005	−0.262113
$680$	36.7234	1.40828
$681$	39.3861	1.50928
$682$	−33.7246	−1.29138
$683$	22.5385	0.862413	0.431206	−	0.902253i	$-0.358088\pi$
$683$	22.5385	0.862413	0.431206	+	0.902253i	$0.358088\pi$
$684$	11.1681	0.427022
$685$	−8.15637	−0.311639
$686$	−19.8621	−0.758339
$687$	−30.7048	−1.17146
$688$	12.1600	0.463598
$689$	−17.7452	−0.676037
$690$	−27.5983	−1.05065
$691$	−29.2831	−1.11398	−0.556990	−	0.830519i	$-0.688044\pi$
$691$	−29.2831	−1.11398	−0.556990	+	0.830519i	$0.688044\pi$
$692$	−2.43447	−0.0925445
$693$	3.77174	0.143276
$694$	−52.7644	−2.00291
$695$	3.83065	0.145305
$696$	−108.144	−4.09917
$697$	−59.3872	−2.24945
$698$	−41.4718	−1.56973
$699$	−47.7017	−1.80424
$700$	−2.40592	−0.0909352
$701$	31.9784	1.20781	0.603904	−	0.797057i	$-0.293611\pi$
$701$	31.9784	1.20781	0.603904	+	0.797057i	$0.293611\pi$
$702$	−5.01200	−0.189166
$703$	−0.307993	−0.0116162
$704$	18.9374	0.713730
$705$	24.0747	0.906705
$706$	40.1338	1.51046
$707$	7.45993	0.280560
$708$	−86.0123	−3.23254
$709$	47.1293	1.76998	0.884989	−	0.465613i	$-0.154166\pi$
$709$	47.1293	1.76998	0.884989	+	0.465613i	$0.154166\pi$
$710$	29.5159	1.10771
$711$	35.5324	1.33257
$712$	9.81328	0.367768
$713$	−24.5566	−0.919652
$714$	−24.2042	−0.905818
$715$	3.77594	0.141212
$716$	75.2905	2.81374
$717$	41.5548	1.55189
$718$	−22.2891	−0.831821
$719$	−36.3590	−1.35596	−0.677981	−	0.735080i	$-0.737145\pi$
$719$	−36.3590	−1.35596	−0.677981	+	0.735080i	$0.737145\pi$
$720$	10.6002	0.395047
$721$	−2.53975	−0.0945853
$722$	43.5548	1.62094
$723$	61.3720	2.28245
$724$	−10.6727	−0.396646
$725$	9.11423	0.338494
$726$	21.4946	0.797739
$727$	14.2109	0.527054	0.263527	−	0.964652i	$-0.415114\pi$
$727$	14.2109	0.527054	0.263527	+	0.964652i	$0.415114\pi$
$728$	−4.23937	−0.157122
$729$	−14.8243	−0.549048
$730$	−23.0026	−0.851364
$731$	−19.5505	−0.723102
$732$	17.8406	0.659407
$733$	8.10828	0.299486	0.149743	−	0.988725i	$-0.452155\pi$
$733$	8.10828	0.299486	0.149743	+	0.988725i	$0.452155\pi$
$734$	−34.0450	−1.25662
$735$	−15.4213	−0.568823
$736$	−3.69788	−0.136306
$737$	−8.80530	−0.324347
$738$	−48.4546	−1.78364
$739$	−27.0936	−0.996656	−0.498328	−	0.866989i	$-0.666052\pi$
$739$	−27.0936	−0.996656	−0.498328	+	0.866989i	$0.666052\pi$
$740$	−1.08832	−0.0400075
$741$	−3.75339	−0.137884
$742$	18.3964	0.675351
$743$	32.4299	1.18974	0.594868	−	0.803823i	$-0.297204\pi$
$743$	32.4299	1.18974	0.594868	+	0.803823i	$0.297204\pi$
$744$	60.3359	2.21202
$745$	−4.88918	−0.179126
$746$	−70.2930	−2.57361
$747$	−12.9021	−0.472062
$748$	78.6950	2.87737
$749$	6.77970	0.247725
$750$	−5.71491	−0.208679
$751$	13.9822	0.510217	0.255108	−	0.966912i	$-0.417889\pi$
$751$	13.9822	0.510217	0.255108	+	0.966912i	$0.417889\pi$
$752$	46.3447	1.69002
$753$	16.1725	0.589360
$754$	31.5301	1.14826
$755$	18.1820	0.661712
$756$	3.48567	0.126773
$757$	2.80401	0.101913	0.0509567	−	0.998701i	$-0.483773\pi$
$757$	2.80401	0.101913	0.0509567	+	0.998701i	$0.483773\pi$
$758$	54.4509	1.97775
$759$	−30.1232	−1.09340
$760$	−5.90375	−0.214152
$761$	−1.72427	−0.0625048	−0.0312524	−	0.999512i	$-0.509950\pi$
$761$	−1.72427	−0.0625048	−0.0312524	+	0.999512i	$0.509950\pi$
$762$	−38.3340	−1.38870
$763$	−8.87086	−0.321147
$764$	−49.0048	−1.77293
$765$	−17.0427	−0.616178
$766$	−39.7946	−1.43784
$767$	12.7732	0.461214
$768$	−75.2696	−2.71606
$769$	−7.86191	−0.283508	−0.141754	−	0.989902i	$-0.545274\pi$
$769$	−7.86191	−0.283508	−0.141754	+	0.989902i	$0.545274\pi$
$770$	−3.91450	−0.141069
$771$	8.31151	0.299332
$772$	30.4696	1.09663
$773$	−37.7301	−1.35706	−0.678529	−	0.734574i	$-0.737382\pi$
$773$	−37.7301	−1.35706	−0.678529	+	0.734574i	$0.737382\pi$
$774$	−15.9515	−0.573363
$775$	−5.08505	−0.182660
$776$	−59.2224	−2.12596
$777$	0.365358	0.0131071
$778$	13.0715	0.468635
$779$	9.54724	0.342065
$780$	−13.2629	−0.474889
$781$	32.2163	1.15279
$782$	85.4172	3.05451
$783$	−13.2046	−0.471894
$784$	−29.6866	−1.06024
$785$	−4.32031	−0.154198
$786$	50.0417	1.78493
$787$	23.0236	0.820704	0.410352	−	0.911927i	$-0.365406\pi$
$787$	23.0236	0.820704	0.410352	+	0.911927i	$0.365406\pi$
$788$	32.9565	1.17403
$789$	21.8979	0.779587
$790$	−36.8773	−1.31204
$791$	10.4202	0.370500
$792$	32.7042	1.16209
$793$	−2.64941	−0.0940833
$794$	−81.6436	−2.89742
$795$	29.3146	1.03968
$796$	78.2914	2.77497
$797$	39.0653	1.38376	0.691881	−	0.722011i	$-0.256782\pi$
$797$	39.0653	1.38376	0.691881	+	0.722011i	$0.256782\pi$
$798$	3.89112	0.137744
$799$	−74.5114	−2.63603
$800$	−0.765737	−0.0270729
$801$	−4.55416	−0.160913
$802$	−27.3930	−0.967279
$803$	−25.1070	−0.886008
$804$	30.9285	1.09076
$805$	−2.85035	−0.100462
$806$	−17.5914	−0.619632
$807$	64.8726	2.28362
$808$	64.6841	2.27558
$809$	−1.09525	−0.0385070	−0.0192535	−	0.999815i	$-0.506129\pi$
$809$	−1.09525	−0.0385070	−0.0192535	+	0.999815i	$0.506129\pi$
$810$	25.8435	0.908049
$811$	10.1543	0.356567	0.178284	−	0.983979i	$-0.442946\pi$
$811$	10.1543	0.356567	0.178284	+	0.983979i	$0.442946\pi$
$812$	−21.9281	−0.769526
$813$	1.18191	0.0414516
$814$	−1.77073	−0.0620640
$815$	4.44193	0.155594
$816$	−74.2475	−2.59918
$817$	3.14299	0.109959
$818$	−38.8810	−1.35944
$819$	1.96741	0.0687470
$820$	33.7360	1.17811
$821$	−35.4909	−1.23864	−0.619320	−	0.785139i	$-0.712592\pi$
$821$	−35.4909	−1.23864	−0.619320	+	0.785139i	$0.712592\pi$
$822$	46.6129	1.62581
$823$	4.45656	0.155346	0.0776730	−	0.996979i	$-0.475251\pi$
$823$	4.45656	0.155346	0.0776730	+	0.996979i	$0.475251\pi$
$824$	−22.0219	−0.767168
$825$	−6.23775	−0.217171
$826$	−13.2420	−0.460747
$827$	−29.0916	−1.01161	−0.505806	−	0.862647i	$-0.668805\pi$
$827$	−29.0916	−1.01161	−0.505806	+	0.862647i	$0.668805\pi$
$828$	46.7532	1.62479
$829$	−30.7588	−1.06830	−0.534149	−	0.845390i	$-0.679368\pi$
$829$	−30.7588	−1.06830	−0.534149	+	0.845390i	$0.679368\pi$
$830$	13.3904	0.464788
$831$	51.9073	1.80064
$832$	9.87813	0.342463
$833$	47.7291	1.65372
$834$	−21.8918	−0.758052
$835$	3.95589	0.136899
$836$	−12.6512	−0.437551
$837$	7.36716	0.254646
$838$	94.6952	3.27119
$839$	−13.3625	−0.461325	−0.230662	−	0.973034i	$-0.574089\pi$
$839$	−13.3625	−0.461325	−0.230662	+	0.973034i	$0.574089\pi$
$840$	7.00334	0.241638
$841$	54.0692	1.86445
$842$	−15.1441	−0.521899
$843$	10.1137	0.348333
$844$	51.3610	1.76792
$845$	−11.0304	−0.379457
$846$	−60.7946	−2.09016
$847$	2.21995	0.0762786
$848$	56.4318	1.93787
$849$	27.8331	0.955231
$850$	17.6877	0.606684
$851$	−1.28936	−0.0441986
$852$	−113.159	−3.87677
$853$	−16.6798	−0.571105	−0.285553	−	0.958363i	$-0.592177\pi$
$853$	−16.6798	−0.571105	−0.285553	+	0.958363i	$0.592177\pi$
$854$	2.74663	0.0939878
$855$	2.73982	0.0936999
$856$	58.7859	2.00926
$857$	−52.6163	−1.79734	−0.898670	−	0.438625i	$-0.855466\pi$
$857$	−52.6163	−1.79734	−0.898670	+	0.438625i	$0.855466\pi$
$858$	−21.5791	−0.736700
$859$	−3.77216	−0.128704	−0.0643522	−	0.997927i	$-0.520498\pi$
$859$	−3.77216	−0.128704	−0.0643522	+	0.997927i	$0.520498\pi$
$860$	11.1060	0.378712
$861$	−11.3254	−0.385970
$862$	35.2232	1.19971
$863$	38.2862	1.30328	0.651639	−	0.758529i	$-0.274082\pi$
$863$	38.2862	1.30328	0.651639	+	0.758529i	$0.274082\pi$
$864$	1.10939	0.0377422
$865$	−0.597238	−0.0203067
$866$	−33.5402	−1.13974
$867$	79.9593	2.71556
$868$	12.2342	0.415257
$869$	−40.2511	−1.36543
$870$	−52.0870	−1.76592
$871$	−4.59302	−0.155629
$872$	−76.9181	−2.60478
$873$	27.4840	0.930194
$874$	−13.7319	−0.464488
$875$	−0.590235	−0.0199536
$876$	88.1881	2.97960
$877$	−2.58574	−0.0873142	−0.0436571	−	0.999047i	$-0.513901\pi$
$877$	−2.58574	−0.0873142	−0.0436571	+	0.999047i	$0.513901\pi$
$878$	93.2015	3.14540
$879$	22.8347	0.770197
$880$	−12.0079	−0.404787
$881$	36.1171	1.21682	0.608408	−	0.793624i	$-0.291808\pi$
$881$	36.1171	1.21682	0.608408	+	0.793624i	$0.291808\pi$
$882$	38.9427	1.31127
$883$	0.707931	0.0238238	0.0119119	−	0.999929i	$-0.496208\pi$
$883$	0.707931	0.0238238	0.0119119	+	0.999929i	$0.496208\pi$
$884$	41.0489	1.38062
$885$	−21.1011	−0.709304
$886$	−43.9637	−1.47699
$887$	0.340871	0.0114453	0.00572266	−	0.999984i	$-0.498178\pi$
$887$	0.340871	0.0114453	0.00572266	+	0.999984i	$0.498178\pi$
$888$	3.16797	0.106310
$889$	−3.95913	−0.132785
$890$	4.72654	0.158434
$891$	28.2079	0.945000
$892$	14.4718	0.484553
$893$	11.9787	0.400850
$894$	27.9413	0.934495
$895$	18.4707	0.617408
$896$	−11.1446	−0.372313
$897$	−15.7129	−0.524638
$898$	18.1297	0.604995
$899$	−46.3463	−1.54574
$900$	9.68140	0.322713
$901$	−90.7291	−3.02262
$902$	54.8894	1.82762
$903$	−3.72838	−0.124073
$904$	90.3523	3.00507
$905$	−2.61828	−0.0870346
$906$	−103.909	−3.45214
$907$	−33.7757	−1.12150	−0.560752	−	0.827984i	$-0.689488\pi$
$907$	−33.7757	−1.12150	−0.560752	+	0.827984i	$0.689488\pi$
$908$	69.2479	2.29807
$909$	−30.0187	−0.995657
$910$	−2.04188	−0.0676877
$911$	−30.9348	−1.02491	−0.512457	−	0.858713i	$-0.671265\pi$
$911$	−30.9348	−1.02491	−0.512457	+	0.858713i	$0.671265\pi$
$912$	11.9362	0.395248
$913$	14.6155	0.483702
$914$	19.1153	0.632277
$915$	4.37676	0.144691
$916$	−53.9845	−1.78370
$917$	5.16829	0.170672
$918$	−25.6258	−0.845777
$919$	−57.7380	−1.90460	−0.952300	−	0.305163i	$-0.901289\pi$
$919$	−57.7380	−1.90460	−0.952300	+	0.305163i	$0.901289\pi$
$920$	−24.7150	−0.814830
$921$	25.1112	0.827443
$922$	46.3779	1.52737
$923$	16.8046	0.553132
$924$	15.0075	0.493712
$925$	−0.266993	−0.00877869
$926$	−18.9350	−0.622242
$927$	10.2199	0.335667
$928$	−6.97910	−0.229100
$929$	46.5149	1.52611	0.763053	−	0.646336i	$-0.223700\pi$
$929$	46.5149	1.52611	0.763053	+	0.646336i	$0.223700\pi$
$930$	29.0606	0.952935
$931$	−7.67305	−0.251474
$932$	−83.8681	−2.74719
$933$	11.3338	0.371051
$934$	48.3672	1.58262
$935$	19.3059	0.631371
$936$	17.0592	0.557597
$937$	3.68212	0.120289	0.0601447	−	0.998190i	$-0.480844\pi$
$937$	3.68212	0.120289	0.0601447	+	0.998190i	$0.480844\pi$
$938$	4.76157	0.155471
$939$	0.498979	0.0162836
$940$	42.3276	1.38057
$941$	−29.7163	−0.968724	−0.484362	−	0.874868i	$-0.660948\pi$
$941$	−29.7163	−0.968724	−0.484362	+	0.874868i	$0.660948\pi$
$942$	24.6902	0.804450
$943$	39.9678	1.30153
$944$	−40.6204	−1.32208
$945$	0.855125	0.0278172
$946$	18.0698	0.587501
$947$	43.8018	1.42337	0.711684	−	0.702500i	$-0.247933\pi$
$947$	43.8018	1.42337	0.711684	+	0.702500i	$0.247933\pi$
$948$	141.381	4.59186
$949$	−13.0963	−0.425125
$950$	−2.84352	−0.0922561
$951$	39.0761	1.26713
$952$	−21.6754	−0.702505
$953$	32.9716	1.06806	0.534028	−	0.845467i	$-0.320678\pi$
$953$	32.9716	1.06806	0.534028	+	0.845467i	$0.320678\pi$
$954$	−74.0267	−2.39670
$955$	−12.0222	−0.389028
$956$	73.0608	2.36295
$957$	−56.8523	−1.83777
$958$	26.6324	0.860455
$959$	4.81417	0.155458
$960$	−16.3184	−0.526675
$961$	−5.14226	−0.165879
$962$	−0.923647	−0.0297796
$963$	−27.2814	−0.879132
$964$	107.903	3.47532
$965$	7.47499	0.240629
$966$	16.2895	0.524106
$967$	−42.9757	−1.38201	−0.691003	−	0.722852i	$-0.742831\pi$
$967$	−42.9757	−1.38201	−0.691003	+	0.722852i	$0.742831\pi$
$968$	19.2489	0.618684
$969$	−19.1906	−0.616492
$970$	−28.5243	−0.915861
$971$	−48.3438	−1.55143	−0.775713	−	0.631086i	$-0.782610\pi$
$971$	−48.3438	−1.55143	−0.775713	+	0.631086i	$0.782610\pi$
$972$	−81.3631	−2.60972
$973$	−2.26098	−0.0724837
$974$	4.02811	0.129069
$975$	−3.25374	−0.104203
$976$	8.42544	0.269692
$977$	51.8608	1.65917	0.829587	−	0.558378i	$-0.188576\pi$
$977$	51.8608	1.65917	0.829587	+	0.558378i	$0.188576\pi$
$978$	−25.3852	−0.811731
$979$	5.15896	0.164881
$980$	−27.1134	−0.866106
$981$	35.6963	1.13969
$982$	82.5963	2.63575
$983$	29.2350	0.932452	0.466226	−	0.884666i	$-0.345613\pi$
$983$	29.2350	0.932452	0.466226	+	0.884666i	$0.345613\pi$
$984$	−98.2014	−3.13055
$985$	8.08509	0.257613
$986$	161.210	5.13397
$987$	−14.2097	−0.452300
$988$	−6.59913	−0.209946
$989$	13.1576	0.418386
$990$	15.7519	0.500628
$991$	−50.5185	−1.60477	−0.802386	−	0.596806i	$-0.796436\pi$
$991$	−50.5185	−1.60477	−0.802386	+	0.596806i	$0.796436\pi$
$992$	3.89381	0.123629
$993$	58.4477	1.85478
$994$	−17.4213	−0.552571
$995$	19.2069	0.608900
$996$	−51.3367	−1.62666
$997$	−21.0808	−0.667635	−0.333818	−	0.942638i	$-0.608337\pi$
$997$	−21.0808	−0.667635	−0.333818	+	0.942638i	$0.608337\pi$
$998$	67.7750	2.14538
$999$	0.386817	0.0122383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.e.1.11	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.e.1.11	✓	142	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46500 q^{2} +2.31843 q^{3} +4.07621 q^{4} +1.00000 q^{5} -5.71491 q^{6} -0.590235 q^{7} -5.11785 q^{8} +2.37510 q^{9} +O(q^{10})\)	\(q-2.46500 q^{2} +2.31843 q^{3} +4.07621 q^{4} +1.00000 q^{5} -5.71491 q^{6} -0.590235 q^{7} -5.11785 q^{8} +2.37510 q^{9} -2.46500 q^{10} -2.69051 q^{11} +9.45039 q^{12} -1.40343 q^{13} +1.45493 q^{14} +2.31843 q^{15} +4.46306 q^{16} -7.17556 q^{17} -5.85461 q^{18} +1.15356 q^{19} +4.07621 q^{20} -1.36842 q^{21} +6.63211 q^{22} +4.82918 q^{23} -11.8654 q^{24} +1.00000 q^{25} +3.45944 q^{26} -1.44879 q^{27} -2.40592 q^{28} +9.11423 q^{29} -5.71491 q^{30} -5.08505 q^{31} -0.765737 q^{32} -6.23775 q^{33} +17.6877 q^{34} -0.590235 q^{35} +9.68140 q^{36} -0.266993 q^{37} -2.84352 q^{38} -3.25374 q^{39} -5.11785 q^{40} +8.27632 q^{41} +3.37314 q^{42} +2.72460 q^{43} -10.9671 q^{44} +2.37510 q^{45} -11.9039 q^{46} +10.3841 q^{47} +10.3473 q^{48} -6.65162 q^{49} -2.46500 q^{50} -16.6360 q^{51} -5.72066 q^{52} +12.6442 q^{53} +3.57126 q^{54} -2.69051 q^{55} +3.02073 q^{56} +2.67445 q^{57} -22.4665 q^{58} -9.10146 q^{59} +9.45039 q^{60} +1.88782 q^{61} +12.5346 q^{62} -1.40187 q^{63} -7.03859 q^{64} -1.40343 q^{65} +15.3760 q^{66} +3.27272 q^{67} -29.2491 q^{68} +11.1961 q^{69} +1.45493 q^{70} -11.9740 q^{71} -12.1554 q^{72} +9.33169 q^{73} +0.658138 q^{74} +2.31843 q^{75} +4.70216 q^{76} +1.58803 q^{77} +8.02045 q^{78} +14.9604 q^{79} +4.46306 q^{80} -10.4842 q^{81} -20.4011 q^{82} -5.43223 q^{83} -5.57795 q^{84} -7.17556 q^{85} -6.71612 q^{86} +21.1307 q^{87} +13.7696 q^{88} -1.91746 q^{89} -5.85461 q^{90} +0.828350 q^{91} +19.6847 q^{92} -11.7893 q^{93} -25.5967 q^{94} +1.15356 q^{95} -1.77530 q^{96} +11.5717 q^{97} +16.3962 q^{98} -6.39023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9	\( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 142 * q + 21 * q^2 + 33 * q^3 + 157 * q^4 + 142 * q^5 + 15 * q^6 + 63 * q^7 + 60 * q^8 + 157 * q^9 + 21 * q^10 + 36 * q^11 + 55 * q^12 + 57 * q^13 + 2 * q^14 + 33 * q^15 + 179 * q^16 + 55 * q^17 + 65 * q^18 + 130 * q^19 + 157 * q^20 + 28 * q^21 + 30 * q^22 + 117 * q^23 + 21 * q^24 + 142 * q^25 + 21 * q^26 + 120 * q^27 + 135 * q^28 + 12 * q^29 + 15 * q^30 + 74 * q^31 + 126 * q^32 + 55 * q^33 + 35 * q^34 + 63 * q^35 + 186 * q^36 + 75 * q^37 + 65 * q^38 + 23 * q^39 + 60 * q^40 + 22 * q^41 + 10 * q^42 + 190 * q^43 + 22 * q^44 + 157 * q^45 + 56 * q^46 + 102 * q^47 + 78 * q^48 + 197 * q^49 + 21 * q^50 + 30 * q^51 + 120 * q^52 + 56 * q^53 - 6 * q^54 + 36 * q^55 + 3 * q^56 + 68 * q^57 + 31 * q^58 + 55 * q^59 + 55 * q^60 + 90 * q^61 + 68 * q^62 + 167 * q^63 + 180 * q^64 + 57 * q^65 + 17 * q^66 + 151 * q^67 + 119 * q^68 + 21 * q^69 + 2 * q^70 + 4 * q^71 + 130 * q^72 + 143 * q^73 - 46 * q^74 + 33 * q^75 + 213 * q^76 + 75 * q^77 - 24 * q^78 + 47 * q^79 + 179 * q^80 + 150 * q^81 + 69 * q^82 + 201 * q^83 - 31 * q^84 + 55 * q^85 - 4 * q^86 + 153 * q^87 + 37 * q^88 + 25 * q^89 + 65 * q^90 + 132 * q^91 + 194 * q^92 + 52 * q^93 + 18 * q^94 + 130 * q^95 + 13 * q^96 + 80 * q^97 + 58 * q^98 + 103 * q^99

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