LMFDB - Embedded newform 4015.2.a.i.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.445505 q^{2} -0.895604 q^{3} -1.80153 q^{4} +1.00000 q^{5} +0.398996 q^{6} -1.53298 q^{7} +1.69360 q^{8} -2.19789 q^{9} +O(q^{10})$	\(q-0.445505 q^{2} -0.895604 q^{3} -1.80153 q^{4} +1.00000 q^{5} +0.398996 q^{6} -1.53298 q^{7} +1.69360 q^{8} -2.19789 q^{9} -0.445505 q^{10} -1.00000 q^{11} +1.61345 q^{12} +0.486692 q^{13} +0.682952 q^{14} -0.895604 q^{15} +2.84854 q^{16} +5.05886 q^{17} +0.979172 q^{18} +6.67052 q^{19} -1.80153 q^{20} +1.37295 q^{21} +0.445505 q^{22} -4.42934 q^{23} -1.51679 q^{24} +1.00000 q^{25} -0.216824 q^{26} +4.65525 q^{27} +2.76171 q^{28} -5.46008 q^{29} +0.398996 q^{30} -7.71305 q^{31} -4.65624 q^{32} +0.895604 q^{33} -2.25375 q^{34} -1.53298 q^{35} +3.95956 q^{36} -11.0650 q^{37} -2.97175 q^{38} -0.435883 q^{39} +1.69360 q^{40} -0.524952 q^{41} -0.611655 q^{42} +0.700201 q^{43} +1.80153 q^{44} -2.19789 q^{45} +1.97329 q^{46} +4.02141 q^{47} -2.55117 q^{48} -4.64996 q^{49} -0.445505 q^{50} -4.53074 q^{51} -0.876788 q^{52} +8.57731 q^{53} -2.07394 q^{54} -1.00000 q^{55} -2.59626 q^{56} -5.97415 q^{57} +2.43249 q^{58} -5.45908 q^{59} +1.61345 q^{60} +13.0898 q^{61} +3.43620 q^{62} +3.36933 q^{63} -3.62271 q^{64} +0.486692 q^{65} -0.398996 q^{66} -12.2452 q^{67} -9.11367 q^{68} +3.96693 q^{69} +0.682952 q^{70} -7.45532 q^{71} -3.72235 q^{72} -1.00000 q^{73} +4.92951 q^{74} -0.895604 q^{75} -12.0171 q^{76} +1.53298 q^{77} +0.194188 q^{78} +1.26837 q^{79} +2.84854 q^{80} +2.42441 q^{81} +0.233869 q^{82} -0.116676 q^{83} -2.47340 q^{84} +5.05886 q^{85} -0.311943 q^{86} +4.89007 q^{87} -1.69360 q^{88} -8.09149 q^{89} +0.979172 q^{90} -0.746091 q^{91} +7.97957 q^{92} +6.90784 q^{93} -1.79156 q^{94} +6.67052 q^{95} +4.17015 q^{96} -3.19162 q^{97} +2.07158 q^{98} +2.19789 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) $ 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * 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 $

$2$	−0.445505	−0.315020	−0.157510	−	0.987517i	$-0.550347\pi$
$2$	−0.445505	−0.315020	−0.157510	+	0.987517i	$0.550347\pi$
$3$	−0.895604	−0.517077	−0.258539	−	0.966001i	$-0.583241\pi$
$3$	−0.895604	−0.517077	−0.258539	+	0.966001i	$0.583241\pi$
$4$	−1.80153	−0.900763
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.398996	0.162889
$7$	−1.53298	−0.579413	−0.289707	−	0.957115i	$-0.593558\pi$
$7$	−1.53298	−0.579413	−0.289707	+	0.957115i	$0.593558\pi$
$8$	1.69360	0.598777
$9$	−2.19789	−0.732631
$10$	−0.445505	−0.140881
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.61345	0.465764
$13$	0.486692	0.134984	0.0674920	−	0.997720i	$-0.478500\pi$
$13$	0.486692	0.134984	0.0674920	+	0.997720i	$0.478500\pi$
$14$	0.682952	0.182527
$15$	−0.895604	−0.231244
$16$	2.84854	0.712136
$17$	5.05886	1.22695	0.613477	−	0.789713i	$-0.289770\pi$
$17$	5.05886	1.22695	0.613477	+	0.789713i	$0.289770\pi$
$18$	0.979172	0.230793
$19$	6.67052	1.53032	0.765161	−	0.643839i	$-0.222659\pi$
$19$	6.67052	1.53032	0.765161	+	0.643839i	$0.222659\pi$
$20$	−1.80153	−0.402833
$21$	1.37295	0.299602
$22$	0.445505	0.0949820
$23$	−4.42934	−0.923581	−0.461790	−	0.886989i	$-0.652793\pi$
$23$	−4.42934	−0.923581	−0.461790	+	0.886989i	$0.652793\pi$
$24$	−1.51679	−0.309614
$25$	1.00000	0.200000
$26$	−0.216824	−0.0425226
$27$	4.65525	0.895904
$28$	2.76171	0.521914
$29$	−5.46008	−1.01391	−0.506956	−	0.861972i	$-0.669229\pi$
$29$	−5.46008	−1.01391	−0.506956	+	0.861972i	$0.669229\pi$
$30$	0.398996	0.0728464
$31$	−7.71305	−1.38530	−0.692652	−	0.721272i	$-0.743558\pi$
$31$	−7.71305	−1.38530	−0.692652	+	0.721272i	$0.743558\pi$
$32$	−4.65624	−0.823114
$33$	0.895604	0.155905
$34$	−2.25375	−0.386514
$35$	−1.53298	−0.259122
$36$	3.95956	0.659927
$37$	−11.0650	−1.81907	−0.909537	−	0.415623i	$-0.863564\pi$
$37$	−11.0650	−1.81907	−0.909537	+	0.415623i	$0.863564\pi$
$38$	−2.97175	−0.482082
$39$	−0.435883	−0.0697972
$40$	1.69360	0.267781
$41$	−0.524952	−0.0819837	−0.0409919	−	0.999159i	$-0.513052\pi$
$41$	−0.524952	−0.0819837	−0.0409919	+	0.999159i	$0.513052\pi$
$42$	−0.611655	−0.0943803
$43$	0.700201	0.106780	0.0533898	−	0.998574i	$-0.482997\pi$
$43$	0.700201	0.106780	0.0533898	+	0.998574i	$0.482997\pi$
$44$	1.80153	0.271590
$45$	−2.19789	−0.327643
$46$	1.97329	0.290946
$47$	4.02141	0.586582	0.293291	−	0.956023i	$-0.405249\pi$
$47$	4.02141	0.586582	0.293291	+	0.956023i	$0.405249\pi$
$48$	−2.55117	−0.368229
$49$	−4.64996	−0.664280
$50$	−0.445505	−0.0630039
$51$	−4.53074	−0.634430
$52$	−0.876788	−0.121589
$53$	8.57731	1.17818	0.589092	−	0.808066i	$-0.299485\pi$
$53$	8.57731	1.17818	0.589092	+	0.808066i	$0.299485\pi$
$54$	−2.07394	−0.282227
$55$	−1.00000	−0.134840
$56$	−2.59626	−0.346940
$57$	−5.97415	−0.791295
$58$	2.43249	0.319402
$59$	−5.45908	−0.710711	−0.355356	−	0.934731i	$-0.615640\pi$
$59$	−5.45908	−0.710711	−0.355356	+	0.934731i	$0.615640\pi$
$60$	1.61345	0.208296
$61$	13.0898	1.67598	0.837991	−	0.545684i	$-0.183730\pi$
$61$	13.0898	1.67598	0.837991	+	0.545684i	$0.183730\pi$
$62$	3.43620	0.436398
$63$	3.36933	0.424496
$64$	−3.62271	−0.452839
$65$	0.486692	0.0603667
$66$	−0.398996	−0.0491130
$67$	−12.2452	−1.49599	−0.747997	−	0.663702i	$-0.768984\pi$
$67$	−12.2452	−1.49599	−0.747997	+	0.663702i	$0.768984\pi$
$68$	−9.11367	−1.10519
$69$	3.96693	0.477563
$70$	0.682952	0.0816284
$71$	−7.45532	−0.884784	−0.442392	−	0.896822i	$-0.645870\pi$
$71$	−7.45532	−0.884784	−0.442392	+	0.896822i	$0.645870\pi$
$72$	−3.72235	−0.438683
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	4.92951	0.573044
$75$	−0.895604	−0.103415
$76$	−12.0171	−1.37846
$77$	1.53298	0.174700
$78$	0.194188	0.0219875
$79$	1.26837	0.142703	0.0713517	−	0.997451i	$-0.477269\pi$
$79$	1.26837	0.142703	0.0713517	+	0.997451i	$0.477269\pi$
$80$	2.84854	0.318477
$81$	2.42441	0.269379
$82$	0.233869	0.0258265
$83$	−0.116676	−0.0128068	−0.00640341	−	0.999979i	$-0.502038\pi$
$83$	−0.116676	−0.0128068	−0.00640341	+	0.999979i	$0.502038\pi$
$84$	−2.47340	−0.269870
$85$	5.05886	0.548711
$86$	−0.311943	−0.0336377
$87$	4.89007	0.524271
$88$	−1.69360	−0.180538
$89$	−8.09149	−0.857696	−0.428848	−	0.903377i	$-0.641080\pi$
$89$	−8.09149	−0.857696	−0.428848	+	0.903377i	$0.641080\pi$
$90$	0.979172	0.103214
$91$	−0.746091	−0.0782115
$92$	7.97957	0.831927
$93$	6.90784	0.716309
$94$	−1.79156	−0.184785
$95$	6.67052	0.684381
$96$	4.17015	0.425614
$97$	−3.19162	−0.324060	−0.162030	−	0.986786i	$-0.551804\pi$
$97$	−3.19162	−0.324060	−0.162030	+	0.986786i	$0.551804\pi$
$98$	2.07158	0.209261
$99$	2.19789	0.220897
$100$	−1.80153	−0.180153
$101$	15.0233	1.49488	0.747439	−	0.664331i	$-0.231283\pi$
$101$	15.0233	1.49488	0.747439	+	0.664331i	$0.231283\pi$
$102$	2.01847	0.199858
$103$	13.2467	1.30524	0.652619	−	0.757686i	$-0.273670\pi$
$103$	13.2467	1.30524	0.652619	+	0.757686i	$0.273670\pi$
$104$	0.824260	0.0808254
$105$	1.37295	0.133986
$106$	−3.82124	−0.371151
$107$	6.99530	0.676262	0.338131	−	0.941099i	$-0.390205\pi$
$107$	6.99530	0.676262	0.338131	+	0.941099i	$0.390205\pi$
$108$	−8.38656	−0.806997
$109$	13.9309	1.33434	0.667171	−	0.744905i	$-0.267505\pi$
$109$	13.9309	1.33434	0.667171	+	0.744905i	$0.267505\pi$
$110$	0.445505	0.0424772
$111$	9.90986	0.940602
$112$	−4.36677	−0.412621
$113$	6.73186	0.633281	0.316640	−	0.948546i	$-0.397445\pi$
$113$	6.73186	0.633281	0.316640	+	0.948546i	$0.397445\pi$
$114$	2.66151	0.249273
$115$	−4.42934	−0.413038
$116$	9.83648	0.913294
$117$	−1.06970	−0.0988935
$118$	2.43205	0.223888
$119$	−7.75515	−0.710914
$120$	−1.51679	−0.138464
$121$	1.00000	0.0909091
$122$	−5.83158	−0.527967
$123$	0.470149	0.0423919
$124$	13.8953	1.24783
$125$	1.00000	0.0894427
$126$	−1.50106	−0.133725
$127$	1.07997	0.0958321	0.0479161	−	0.998851i	$-0.484742\pi$
$127$	1.07997	0.0958321	0.0479161	+	0.998851i	$0.484742\pi$
$128$	10.9264	0.965767
$129$	−0.627103	−0.0552133
$130$	−0.216824	−0.0190167
$131$	7.47701	0.653270	0.326635	−	0.945151i	$-0.394085\pi$
$131$	7.47701	0.653270	0.326635	+	0.945151i	$0.394085\pi$
$132$	−1.61345	−0.140433
$133$	−10.2258	−0.886690
$134$	5.45532	0.471268
$135$	4.65525	0.400661
$136$	8.56768	0.734672
$137$	8.98369	0.767528	0.383764	−	0.923431i	$-0.374628\pi$
$137$	8.98369	0.767528	0.383764	+	0.923431i	$0.374628\pi$
$138$	−1.76729	−0.150442
$139$	7.92166	0.671906	0.335953	−	0.941879i	$-0.390942\pi$
$139$	7.92166	0.671906	0.335953	+	0.941879i	$0.390942\pi$
$140$	2.76171	0.233407
$141$	−3.60159	−0.303308
$142$	3.32138	0.278724
$143$	−0.486692	−0.0406992
$144$	−6.26080	−0.521733
$145$	−5.46008	−0.453435
$146$	0.445505	0.0368702
$147$	4.16452	0.343484
$148$	19.9339	1.63855
$149$	−14.8781	−1.21886	−0.609430	−	0.792840i	$-0.708602\pi$
$149$	−14.8781	−1.21886	−0.609430	+	0.792840i	$0.708602\pi$
$150$	0.398996	0.0325779
$151$	8.32448	0.677436	0.338718	−	0.940888i	$-0.390007\pi$
$151$	8.32448	0.677436	0.338718	+	0.940888i	$0.390007\pi$
$152$	11.2972	0.916323
$153$	−11.1188	−0.898905
$154$	−0.682952	−0.0550338
$155$	−7.71305	−0.619527
$156$	0.785255	0.0628707
$157$	2.84853	0.227337	0.113669	−	0.993519i	$-0.463740\pi$
$157$	2.84853	0.227337	0.113669	+	0.993519i	$0.463740\pi$
$158$	−0.565067	−0.0449543
$159$	−7.68188	−0.609213
$160$	−4.65624	−0.368108
$161$	6.79010	0.535135
$162$	−1.08009	−0.0848597
$163$	6.78916	0.531768	0.265884	−	0.964005i	$-0.414336\pi$
$163$	6.78916	0.531768	0.265884	+	0.964005i	$0.414336\pi$
$164$	0.945714	0.0738479
$165$	0.895604	0.0697227
$166$	0.0519796	0.00403440
$167$	−16.3122	−1.26227	−0.631137	−	0.775671i	$-0.717412\pi$
$167$	−16.3122	−1.26227	−0.631137	+	0.775671i	$0.717412\pi$
$168$	2.32522	0.179395
$169$	−12.7631	−0.981779
$170$	−2.25375	−0.172855
$171$	−14.6611	−1.12116
$172$	−1.26143	−0.0961831
$173$	−10.4023	−0.790873	−0.395437	−	0.918493i	$-0.629407\pi$
$173$	−10.4023	−0.790873	−0.395437	+	0.918493i	$0.629407\pi$
$174$	−2.17855	−0.165156
$175$	−1.53298	−0.115883
$176$	−2.84854	−0.214717
$177$	4.88917	0.367493
$178$	3.60480	0.270191
$179$	3.96976	0.296714	0.148357	−	0.988934i	$-0.452602\pi$
$179$	3.96976	0.296714	0.148357	+	0.988934i	$0.452602\pi$
$180$	3.95956	0.295128
$181$	17.7082	1.31624	0.658119	−	0.752914i	$-0.271352\pi$
$181$	17.7082	1.31624	0.658119	+	0.752914i	$0.271352\pi$
$182$	0.332387	0.0246382
$183$	−11.7233	−0.866612
$184$	−7.50152	−0.553019
$185$	−11.0650	−0.813515
$186$	−3.07748	−0.225651
$187$	−5.05886	−0.369941
$188$	−7.24466	−0.528371
$189$	−7.13643	−0.519099
$190$	−2.97175	−0.215593
$191$	1.88096	0.136102	0.0680508	−	0.997682i	$-0.478322\pi$
$191$	1.88096	0.136102	0.0680508	+	0.997682i	$0.478322\pi$
$192$	3.24452	0.234153
$193$	16.5743	1.19304	0.596520	−	0.802598i	$-0.296549\pi$
$193$	16.5743	1.19304	0.596520	+	0.802598i	$0.296549\pi$
$194$	1.42188	0.102085
$195$	−0.435883	−0.0312142
$196$	8.37702	0.598359
$197$	20.2148	1.44025	0.720123	−	0.693847i	$-0.244086\pi$
$197$	20.2148	1.44025	0.720123	+	0.693847i	$0.244086\pi$
$198$	−0.979172	−0.0695867
$199$	2.84495	0.201673	0.100837	−	0.994903i	$-0.467848\pi$
$199$	2.84495	0.201673	0.100837	+	0.994903i	$0.467848\pi$
$200$	1.69360	0.119755
$201$	10.9669	0.773545
$202$	−6.69297	−0.470916
$203$	8.37022	0.587474
$204$	8.16224	0.571471
$205$	−0.524952	−0.0366642
$206$	−5.90148	−0.411175
$207$	9.73521	0.676644
$208$	1.38636	0.0961270
$209$	−6.67052	−0.461410
$210$	−0.611655	−0.0422082
$211$	14.9366	1.02828	0.514140	−	0.857706i	$-0.328111\pi$
$211$	14.9366	1.02828	0.514140	+	0.857706i	$0.328111\pi$
$212$	−15.4522	−1.06126
$213$	6.67702	0.457502
$214$	−3.11644	−0.213036
$215$	0.700201	0.0477533
$216$	7.88413	0.536447
$217$	11.8240	0.802664
$218$	−6.20630	−0.420344
$219$	0.895604	0.0605193
$220$	1.80153	0.121459
$221$	2.46211	0.165619
$222$	−4.41489	−0.296308
$223$	−27.3802	−1.83351	−0.916757	−	0.399446i	$-0.869203\pi$
$223$	−27.3802	−1.83351	−0.916757	+	0.399446i	$0.869203\pi$
$224$	7.13794	0.476923
$225$	−2.19789	−0.146526
$226$	−2.99908	−0.199496
$227$	8.60977	0.571450	0.285725	−	0.958312i	$-0.407766\pi$
$227$	8.60977	0.571450	0.285725	+	0.958312i	$0.407766\pi$
$228$	10.7626	0.712769
$229$	0.645097	0.0426292	0.0213146	−	0.999773i	$-0.493215\pi$
$229$	0.645097	0.0426292	0.0213146	+	0.999773i	$0.493215\pi$
$230$	1.97329	0.130115
$231$	−1.37295	−0.0903333
$232$	−9.24719	−0.607108
$233$	−3.22258	−0.211118	−0.105559	−	0.994413i	$-0.533663\pi$
$233$	−3.22258	−0.211118	−0.105559	+	0.994413i	$0.533663\pi$
$234$	0.476555	0.0311534
$235$	4.02141	0.262328
$236$	9.83467	0.640182
$237$	−1.13596	−0.0737887
$238$	3.45496	0.223952
$239$	−6.61668	−0.427998	−0.213999	−	0.976834i	$-0.568649\pi$
$239$	−6.61668	−0.427998	−0.213999	+	0.976834i	$0.568649\pi$
$240$	−2.55117	−0.164677
$241$	25.7268	1.65721	0.828604	−	0.559836i	$-0.189136\pi$
$241$	25.7268	1.65721	0.828604	+	0.559836i	$0.189136\pi$
$242$	−0.445505	−0.0286381
$243$	−16.1371	−1.03519
$244$	−23.5817	−1.50966
$245$	−4.64996	−0.297075
$246$	−0.209454	−0.0133543
$247$	3.24649	0.206569
$248$	−13.0628	−0.829489
$249$	0.104495	0.00662211
$250$	−0.445505	−0.0281762
$251$	3.06336	0.193357	0.0966786	−	0.995316i	$-0.469178\pi$
$251$	3.06336	0.193357	0.0966786	+	0.995316i	$0.469178\pi$
$252$	−6.06994	−0.382370
$253$	4.42934	0.278470
$254$	−0.481133	−0.0301890
$255$	−4.53074	−0.283726
$256$	2.37766	0.148604
$257$	−20.6790	−1.28992	−0.644960	−	0.764216i	$-0.723126\pi$
$257$	−20.6790	−1.28992	−0.644960	+	0.764216i	$0.723126\pi$
$258$	0.279377	0.0173933
$259$	16.9625	1.05400
$260$	−0.876788	−0.0543761
$261$	12.0007	0.742823
$262$	−3.33105	−0.205793
$263$	17.3828	1.07187	0.535935	−	0.844259i	$-0.319959\pi$
$263$	17.3828	1.07187	0.535935	+	0.844259i	$0.319959\pi$
$264$	1.51679	0.0933522
$265$	8.57731	0.526900
$266$	4.55565	0.279325
$267$	7.24677	0.443495
$268$	22.0601	1.34754
$269$	−24.7960	−1.51184	−0.755920	−	0.654664i	$-0.772810\pi$
$269$	−24.7960	−1.51184	−0.755920	+	0.654664i	$0.772810\pi$
$270$	−2.07394	−0.126216
$271$	18.0476	1.09631	0.548157	−	0.836376i	$-0.315330\pi$
$271$	18.0476	1.09631	0.548157	+	0.836376i	$0.315330\pi$
$272$	14.4104	0.873758
$273$	0.668202	0.0404414
$274$	−4.00228	−0.241786
$275$	−1.00000	−0.0603023
$276$	−7.14653	−0.430171
$277$	1.00582	0.0604339	0.0302169	−	0.999543i	$-0.490380\pi$
$277$	1.00582	0.0604339	0.0302169	+	0.999543i	$0.490380\pi$
$278$	−3.52914	−0.211664
$279$	16.9525	1.01492
$280$	−2.59626	−0.155156
$281$	27.4307	1.63638	0.818189	−	0.574949i	$-0.194978\pi$
$281$	27.4307	1.63638	0.818189	+	0.574949i	$0.194978\pi$
$282$	1.60452	0.0955481
$283$	−20.2112	−1.20143	−0.600715	−	0.799463i	$-0.705117\pi$
$283$	−20.2112	−1.20143	−0.600715	+	0.799463i	$0.705117\pi$
$284$	13.4309	0.796980
$285$	−5.97415	−0.353878
$286$	0.216824	0.0128210
$287$	0.804743	0.0475025
$288$	10.2339	0.603039
$289$	8.59208	0.505416
$290$	2.43249	0.142841
$291$	2.85843	0.167564
$292$	1.80153	0.105426
$293$	−7.23478	−0.422660	−0.211330	−	0.977415i	$-0.567780\pi$
$293$	−7.23478	−0.422660	−0.211330	+	0.977415i	$0.567780\pi$
$294$	−1.85532	−0.108204
$295$	−5.45908	−0.317840
$296$	−18.7397	−1.08922
$297$	−4.65525	−0.270125
$298$	6.62826	0.383965
$299$	−2.15572	−0.124669
$300$	1.61345	0.0931528
$301$	−1.07340	−0.0618696
$302$	−3.70860	−0.213406
$303$	−13.4550	−0.772967
$304$	19.0013	1.08980
$305$	13.0898	0.749522
$306$	4.95350	0.283172
$307$	26.6941	1.52352	0.761758	−	0.647862i	$-0.224337\pi$
$307$	26.6941	1.52352	0.761758	+	0.647862i	$0.224337\pi$
$308$	−2.76171	−0.157363
$309$	−11.8638	−0.674909
$310$	3.43620	0.195163
$311$	15.4751	0.877514	0.438757	−	0.898606i	$-0.355419\pi$
$311$	15.4751	0.877514	0.438757	+	0.898606i	$0.355419\pi$
$312$	−0.738211	−0.0417930
$313$	−8.74697	−0.494408	−0.247204	−	0.968963i	$-0.579512\pi$
$313$	−8.74697	−0.494408	−0.247204	+	0.968963i	$0.579512\pi$
$314$	−1.26903	−0.0716156
$315$	3.36933	0.189840
$316$	−2.28501	−0.128542
$317$	16.1355	0.906262	0.453131	−	0.891444i	$-0.350307\pi$
$317$	16.1355	0.906262	0.453131	+	0.891444i	$0.350307\pi$
$318$	3.42231	0.191914
$319$	5.46008	0.305706
$320$	−3.62271	−0.202516
$321$	−6.26502	−0.349680
$322$	−3.02502	−0.168578
$323$	33.7452	1.87764
$324$	−4.36764	−0.242647
$325$	0.486692	0.0269968
$326$	−3.02461	−0.167517
$327$	−12.4766	−0.689958
$328$	−0.889058	−0.0490900
$329$	−6.16475	−0.339874
$330$	−0.398996	−0.0219640
$331$	−11.2120	−0.616267	−0.308134	−	0.951343i	$-0.599704\pi$
$331$	−11.2120	−0.616267	−0.308134	+	0.951343i	$0.599704\pi$
$332$	0.210194	0.0115359
$333$	24.3197	1.33271
$334$	7.26716	0.397641
$335$	−12.2452	−0.669029
$336$	3.91090	0.213357
$337$	−18.6645	−1.01672	−0.508361	−	0.861144i	$-0.669748\pi$
$337$	−18.6645	−1.01672	−0.508361	+	0.861144i	$0.669748\pi$
$338$	5.68604	0.309280
$339$	−6.02909	−0.327455
$340$	−9.11367	−0.494258
$341$	7.71305	0.417685
$342$	6.53159	0.353188
$343$	17.8592	0.964306
$344$	1.18586	0.0639372
$345$	3.96693	0.213573
$346$	4.63428	0.249140
$347$	5.44114	0.292095	0.146048	−	0.989278i	$-0.453345\pi$
$347$	5.44114	0.292095	0.146048	+	0.989278i	$0.453345\pi$
$348$	−8.80959	−0.472244
$349$	23.1979	1.24175	0.620877	−	0.783908i	$-0.286777\pi$
$349$	23.1979	1.24175	0.620877	+	0.783908i	$0.286777\pi$
$350$	0.682952	0.0365053
$351$	2.26567	0.120933
$352$	4.65624	0.248178
$353$	−2.98529	−0.158891	−0.0794454	−	0.996839i	$-0.525315\pi$
$353$	−2.98529	−0.158891	−0.0794454	+	0.996839i	$0.525315\pi$
$354$	−2.17815	−0.115767
$355$	−7.45532	−0.395687
$356$	14.5770	0.772581
$357$	6.94555	0.367597
$358$	−1.76855	−0.0934706
$359$	−16.7552	−0.884305	−0.442153	−	0.896940i	$-0.645785\pi$
$359$	−16.7552	−0.884305	−0.442153	+	0.896940i	$0.645785\pi$
$360$	−3.72235	−0.196185
$361$	25.4959	1.34189
$362$	−7.88908	−0.414641
$363$	−0.895604	−0.0470070
$364$	1.34410	0.0704500
$365$	−1.00000	−0.0523424
$366$	5.22279	0.273000
$367$	−9.38385	−0.489833	−0.244916	−	0.969544i	$-0.578761\pi$
$367$	−9.38385	−0.489833	−0.244916	+	0.969544i	$0.578761\pi$
$368$	−12.6172	−0.657715
$369$	1.15379	0.0600638
$370$	4.92951	0.256273
$371$	−13.1489	−0.682656
$372$	−12.4446	−0.645225
$373$	−1.84256	−0.0954044	−0.0477022	−	0.998862i	$-0.515190\pi$
$373$	−1.84256	−0.0954044	−0.0477022	+	0.998862i	$0.515190\pi$
$374$	2.25375	0.116538
$375$	−0.895604	−0.0462488
$376$	6.81064	0.351232
$377$	−2.65738	−0.136862
$378$	3.17931	0.163526
$379$	14.8124	0.760860	0.380430	−	0.924810i	$-0.375776\pi$
$379$	14.8124	0.760860	0.380430	+	0.924810i	$0.375776\pi$
$380$	−12.0171	−0.616465
$381$	−0.967229	−0.0495526
$382$	−0.837978	−0.0428747
$383$	31.2744	1.59805	0.799024	−	0.601300i	$-0.205350\pi$
$383$	31.2744	1.59805	0.799024	+	0.601300i	$0.205350\pi$
$384$	−9.78574	−0.499376
$385$	1.53298	0.0781281
$386$	−7.38391	−0.375831
$387$	−1.53897	−0.0782301
$388$	5.74979	0.291901
$389$	26.1192	1.32429	0.662147	−	0.749374i	$-0.269645\pi$
$389$	26.1192	1.32429	0.662147	+	0.749374i	$0.269645\pi$
$390$	0.194188	0.00983310
$391$	−22.4074	−1.13319
$392$	−7.87516	−0.397756
$393$	−6.69645	−0.337791
$394$	−9.00579	−0.453705
$395$	1.26837	0.0638189
$396$	−3.95956	−0.198975
$397$	31.8860	1.60031	0.800157	−	0.599791i	$-0.204750\pi$
$397$	31.8860	1.60031	0.800157	+	0.599791i	$0.204750\pi$
$398$	−1.26744	−0.0635310
$399$	9.15827	0.458487
$400$	2.84854	0.142427
$401$	22.2721	1.11222	0.556108	−	0.831110i	$-0.312294\pi$
$401$	22.2721	1.11222	0.556108	+	0.831110i	$0.312294\pi$
$402$	−4.88580	−0.243682
$403$	−3.75388	−0.186994
$404$	−27.0649	−1.34653
$405$	2.42441	0.120470
$406$	−3.72897	−0.185066
$407$	11.0650	0.548472
$408$	−7.67325	−0.379882
$409$	34.6860	1.71511	0.857556	−	0.514391i	$-0.171982\pi$
$409$	34.6860	1.71511	0.857556	+	0.514391i	$0.171982\pi$
$410$	0.233869	0.0115499
$411$	−8.04583	−0.396871
$412$	−23.8643	−1.17571
$413$	8.36868	0.411796
$414$	−4.33708	−0.213156
$415$	−0.116676	−0.00572738
$416$	−2.26615	−0.111107
$417$	−7.09467	−0.347427
$418$	2.97175	0.145353
$419$	8.13706	0.397521	0.198761	−	0.980048i	$-0.436308\pi$
$419$	8.13706	0.397521	0.198761	+	0.980048i	$0.436308\pi$
$420$	−2.47340	−0.120689
$421$	24.9533	1.21615	0.608074	−	0.793880i	$-0.291942\pi$
$421$	24.9533	1.21615	0.608074	+	0.793880i	$0.291942\pi$
$422$	−6.65434	−0.323928
$423$	−8.83862	−0.429748
$424$	14.5265	0.705470
$425$	5.05886	0.245391
$426$	−2.97464	−0.144122
$427$	−20.0665	−0.971086
$428$	−12.6022	−0.609151
$429$	0.435883	0.0210446
$430$	−0.311943	−0.0150432
$431$	7.56782	0.364529	0.182265	−	0.983250i	$-0.441657\pi$
$431$	7.56782	0.364529	0.182265	+	0.983250i	$0.441657\pi$
$432$	13.2607	0.638006
$433$	13.0729	0.628241	0.314121	−	0.949383i	$-0.398290\pi$
$433$	13.0729	0.628241	0.314121	+	0.949383i	$0.398290\pi$
$434$	−5.26764	−0.252855
$435$	4.89007	0.234461
$436$	−25.0969	−1.20193
$437$	−29.5460	−1.41338
$438$	−0.398996	−0.0190648
$439$	−19.6035	−0.935625	−0.467813	−	0.883828i	$-0.654958\pi$
$439$	−19.6035	−0.935625	−0.467813	+	0.883828i	$0.654958\pi$
$440$	−1.69360	−0.0807391
$441$	10.2201	0.486672
$442$	−1.09688	−0.0521733
$443$	−12.2675	−0.582846	−0.291423	−	0.956594i	$-0.594129\pi$
$443$	−12.2675	−0.582846	−0.291423	+	0.956594i	$0.594129\pi$
$444$	−17.8529	−0.847259
$445$	−8.09149	−0.383573
$446$	12.1980	0.577593
$447$	13.3249	0.630245
$448$	5.55356	0.262381
$449$	20.2876	0.957431	0.478716	−	0.877970i	$-0.341103\pi$
$449$	20.2876	0.957431	0.478716	+	0.877970i	$0.341103\pi$
$450$	0.979172	0.0461586
$451$	0.524952	0.0247190
$452$	−12.1276	−0.570435
$453$	−7.45544	−0.350287
$454$	−3.83569	−0.180018
$455$	−0.746091	−0.0349773
$456$	−10.1178	−0.473810
$457$	33.9969	1.59031	0.795154	−	0.606408i	$-0.207390\pi$
$457$	33.9969	1.59031	0.795154	+	0.606408i	$0.207390\pi$
$458$	−0.287394	−0.0134290
$459$	23.5503	1.09923
$460$	7.97957	0.372049
$461$	11.1819	0.520794	0.260397	−	0.965502i	$-0.416147\pi$
$461$	11.1819	0.520794	0.260397	+	0.965502i	$0.416147\pi$
$462$	0.611655	0.0284567
$463$	3.21214	0.149281	0.0746404	−	0.997211i	$-0.476219\pi$
$463$	3.21214	0.149281	0.0746404	+	0.997211i	$0.476219\pi$
$464$	−15.5533	−0.722043
$465$	6.90784	0.320343
$466$	1.43568	0.0665064
$467$	−15.7611	−0.729338	−0.364669	−	0.931137i	$-0.618818\pi$
$467$	−15.7611	−0.729338	−0.364669	+	0.931137i	$0.618818\pi$
$468$	1.92709	0.0890795
$469$	18.7718	0.866799
$470$	−1.79156	−0.0826383
$471$	−2.55115	−0.117551
$472$	−9.24549	−0.425558
$473$	−0.700201	−0.0321953
$474$	0.506077	0.0232449
$475$	6.67052	0.306065
$476$	13.9711	0.640364
$477$	−18.8520	−0.863175
$478$	2.94776	0.134828
$479$	−3.12670	−0.142862	−0.0714312	−	0.997446i	$-0.522757\pi$
$479$	−3.12670	−0.142862	−0.0714312	+	0.997446i	$0.522757\pi$
$480$	4.17015	0.190340
$481$	−5.38524	−0.245546
$482$	−11.4614	−0.522053
$483$	−6.08125	−0.276706
$484$	−1.80153	−0.0818875
$485$	−3.19162	−0.144924
$486$	7.18915	0.326106
$487$	−42.8416	−1.94134	−0.970670	−	0.240415i	$-0.922716\pi$
$487$	−42.8416	−1.94134	−0.970670	+	0.240415i	$0.922716\pi$
$488$	22.1689	1.00354
$489$	−6.08040	−0.274965
$490$	2.07158	0.0935845
$491$	−17.7323	−0.800245	−0.400123	−	0.916462i	$-0.631032\pi$
$491$	−17.7323	−0.800245	−0.400123	+	0.916462i	$0.631032\pi$
$492$	−0.846986	−0.0381851
$493$	−27.6218	−1.24402
$494$	−1.44633	−0.0650733
$495$	2.19789	0.0987879
$496$	−21.9710	−0.986525
$497$	11.4289	0.512656
$498$	−0.0465531	−0.00208609
$499$	15.2542	0.682871	0.341435	−	0.939905i	$-0.389087\pi$
$499$	15.2542	0.682871	0.341435	+	0.939905i	$0.389087\pi$
$500$	−1.80153	−0.0805667
$501$	14.6093	0.652693
$502$	−1.36474	−0.0609113
$503$	−38.9003	−1.73448	−0.867240	−	0.497891i	$-0.834108\pi$
$503$	−38.9003	−1.73448	−0.867240	+	0.497891i	$0.834108\pi$
$504$	5.70630	0.254179
$505$	15.0233	0.668529
$506$	−1.97329	−0.0877235
$507$	11.4307	0.507656
$508$	−1.94560	−0.0863220
$509$	13.3650	0.592392	0.296196	−	0.955127i	$-0.404282\pi$
$509$	13.3650	0.592392	0.296196	+	0.955127i	$0.404282\pi$
$510$	2.01847	0.0893792
$511$	1.53298	0.0678152
$512$	−22.9121	−1.01258
$513$	31.0530	1.37102
$514$	9.21259	0.406350
$515$	13.2467	0.583720
$516$	1.12974	0.0497341
$517$	−4.02141	−0.176861
$518$	−7.55686	−0.332029
$519$	9.31635	0.408943
$520$	0.824260	0.0361462
$521$	8.50014	0.372398	0.186199	−	0.982512i	$-0.440383\pi$
$521$	8.50014	0.372398	0.186199	+	0.982512i	$0.440383\pi$
$522$	−5.34636	−0.234004
$523$	−13.5029	−0.590441	−0.295220	−	0.955429i	$-0.595393\pi$
$523$	−13.5029	−0.590441	−0.295220	+	0.955429i	$0.595393\pi$
$524$	−13.4700	−0.588441
$525$	1.37295	0.0599203
$526$	−7.74413	−0.337660
$527$	−39.0192	−1.69970
$528$	2.55117	0.111025
$529$	−3.38096	−0.146998
$530$	−3.82124	−0.165984
$531$	11.9985	0.520689
$532$	18.4220	0.798697
$533$	−0.255490	−0.0110665
$534$	−3.22847	−0.139710
$535$	6.99530	0.302433
$536$	−20.7385	−0.895768
$537$	−3.55533	−0.153424
$538$	11.0467	0.476259
$539$	4.64996	0.200288
$540$	−8.38656	−0.360900
$541$	−0.574599	−0.0247039	−0.0123520	−	0.999924i	$-0.503932\pi$
$541$	−0.574599	−0.0247039	−0.0123520	+	0.999924i	$0.503932\pi$
$542$	−8.04029	−0.345360
$543$	−15.8595	−0.680597
$544$	−23.5553	−1.00992
$545$	13.9309	0.596736
$546$	−0.297687	−0.0127398
$547$	−16.3908	−0.700820	−0.350410	−	0.936596i	$-0.613958\pi$
$547$	−16.3908	−0.700820	−0.350410	+	0.936596i	$0.613958\pi$
$548$	−16.1843	−0.691361
$549$	−28.7701	−1.22788
$550$	0.445505	0.0189964
$551$	−36.4216	−1.55161
$552$	6.71839	0.285954
$553$	−1.94440	−0.0826842
$554$	−0.448098	−0.0190379
$555$	9.90986	0.420650
$556$	−14.2711	−0.605228
$557$	6.09199	0.258126	0.129063	−	0.991636i	$-0.458803\pi$
$557$	6.09199	0.258126	0.129063	+	0.991636i	$0.458803\pi$
$558$	−7.55240	−0.319719
$559$	0.340782	0.0144135
$560$	−4.36677	−0.184530
$561$	4.53074	0.191288
$562$	−12.2205	−0.515491
$563$	10.0884	0.425173	0.212587	−	0.977142i	$-0.431811\pi$
$563$	10.0884	0.425173	0.212587	+	0.977142i	$0.431811\pi$
$564$	6.48835	0.273209
$565$	6.73186	0.283212
$566$	9.00418	0.378474
$567$	−3.71659	−0.156082
$568$	−12.6263	−0.529788
$569$	8.29217	0.347626	0.173813	−	0.984779i	$-0.444391\pi$
$569$	8.29217	0.347626	0.173813	+	0.984779i	$0.444391\pi$
$570$	2.66151	0.111478
$571$	14.0446	0.587747	0.293874	−	0.955844i	$-0.405056\pi$
$571$	14.0446	0.587747	0.293874	+	0.955844i	$0.405056\pi$
$572$	0.876788	0.0366603
$573$	−1.68460	−0.0703751
$574$	−0.358517	−0.0149642
$575$	−4.42934	−0.184716
$576$	7.96234	0.331764
$577$	−4.52282	−0.188288	−0.0941438	−	0.995559i	$-0.530011\pi$
$577$	−4.52282	−0.188288	−0.0941438	+	0.995559i	$0.530011\pi$
$578$	−3.82781	−0.159216
$579$	−14.8440	−0.616894
$580$	9.83648	0.408438
$581$	0.178862	0.00742044
$582$	−1.27345	−0.0527860
$583$	−8.57731	−0.355236
$584$	−1.69360	−0.0700816
$585$	−1.06970	−0.0442265
$586$	3.22313	0.133146
$587$	−17.2015	−0.709981	−0.354991	−	0.934870i	$-0.615516\pi$
$587$	−17.2015	−0.709981	−0.354991	+	0.934870i	$0.615516\pi$
$588$	−7.50250	−0.309398
$589$	−51.4501	−2.11996
$590$	2.43205	0.100126
$591$	−18.1045	−0.744718
$592$	−31.5191	−1.29543
$593$	−16.5096	−0.677969	−0.338984	−	0.940792i	$-0.610083\pi$
$593$	−16.5096	−0.677969	−0.338984	+	0.940792i	$0.610083\pi$
$594$	2.07394	0.0850947
$595$	−7.75515	−0.317930
$596$	26.8032	1.09790
$597$	−2.54795	−0.104281
$598$	0.960385	0.0392731
$599$	16.4076	0.670397	0.335198	−	0.942148i	$-0.391197\pi$
$599$	16.4076	0.670397	0.335198	+	0.942148i	$0.391197\pi$
$600$	−1.51679	−0.0619228
$601$	39.0658	1.59353	0.796763	−	0.604292i	$-0.206544\pi$
$601$	39.0658	1.59353	0.796763	+	0.604292i	$0.206544\pi$
$602$	0.478204	0.0194901
$603$	26.9137	1.09601
$604$	−14.9968	−0.610209
$605$	1.00000	0.0406558
$606$	5.99425	0.243500
$607$	−47.9235	−1.94516	−0.972578	−	0.232579i	$-0.925284\pi$
$607$	−47.9235	−1.94516	−0.972578	+	0.232579i	$0.925284\pi$
$608$	−31.0595	−1.25963
$609$	−7.49640	−0.303770
$610$	−5.83158	−0.236114
$611$	1.95718	0.0791792
$612$	20.0309	0.809700
$613$	−0.0613440	−0.00247766	−0.00123883	−	0.999999i	$-0.500394\pi$
$613$	−0.0613440	−0.00247766	−0.00123883	+	0.999999i	$0.500394\pi$
$614$	−11.8924	−0.479937
$615$	0.470149	0.0189582
$616$	2.59626	0.104606
$617$	−34.3380	−1.38239	−0.691197	−	0.722666i	$-0.742916\pi$
$617$	−34.3380	−1.38239	−0.691197	+	0.722666i	$0.742916\pi$
$618$	5.28539	0.212610
$619$	3.55587	0.142923	0.0714613	−	0.997443i	$-0.477234\pi$
$619$	3.55587	0.142923	0.0714613	+	0.997443i	$0.477234\pi$
$620$	13.8953	0.558047
$621$	−20.6197	−0.827440
$622$	−6.89424	−0.276434
$623$	12.4041	0.496961
$624$	−1.24163	−0.0497051
$625$	1.00000	0.0400000
$626$	3.89682	0.155748
$627$	5.97415	0.238584
$628$	−5.13169	−0.204777
$629$	−55.9763	−2.23192
$630$	−1.50106	−0.0598035
$631$	−43.9602	−1.75003	−0.875014	−	0.484098i	$-0.839148\pi$
$631$	−43.9602	−1.75003	−0.875014	+	0.484098i	$0.839148\pi$
$632$	2.14812	0.0854475
$633$	−13.3773	−0.531700
$634$	−7.18846	−0.285490
$635$	1.07997	0.0428574
$636$	13.8391	0.548756
$637$	−2.26310	−0.0896672
$638$	−2.43249	−0.0963034
$639$	16.3860	0.648220
$640$	10.9264	0.431904
$641$	16.5639	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$641$	16.5639	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$642$	2.79110	0.110156
$643$	21.6130	0.852334	0.426167	−	0.904645i	$-0.359864\pi$
$643$	21.6130	0.852334	0.426167	+	0.904645i	$0.359864\pi$
$644$	−12.2325	−0.482030
$645$	−0.627103	−0.0246922
$646$	−15.0337	−0.591492
$647$	−11.1073	−0.436675	−0.218337	−	0.975873i	$-0.570063\pi$
$647$	−11.1073	−0.436675	−0.218337	+	0.975873i	$0.570063\pi$
$648$	4.10598	0.161298
$649$	5.45908	0.214288
$650$	−0.216824	−0.00850452
$651$	−10.5896	−0.415039
$652$	−12.2309	−0.478997
$653$	35.3268	1.38244	0.691222	−	0.722643i	$-0.257073\pi$
$653$	35.3268	1.38244	0.691222	+	0.722643i	$0.257073\pi$
$654$	5.55839	0.217350
$655$	7.47701	0.292151
$656$	−1.49535	−0.0583836
$657$	2.19789	0.0857480
$658$	2.74643	0.107067
$659$	44.3849	1.72899	0.864495	−	0.502641i	$-0.167638\pi$
$659$	44.3849	1.72899	0.864495	+	0.502641i	$0.167638\pi$
$660$	−1.61345	−0.0628036
$661$	−30.8704	−1.20072	−0.600361	−	0.799729i	$-0.704976\pi$
$661$	−30.8704	−1.20072	−0.600361	+	0.799729i	$0.704976\pi$
$662$	4.99500	0.194136
$663$	−2.20507	−0.0856379
$664$	−0.197602	−0.00766843
$665$	−10.2258	−0.396540
$666$	−10.8345	−0.419830
$667$	24.1846	0.936430
$668$	29.3868	1.13701
$669$	24.5218	0.948068
$670$	5.45532	0.210757
$671$	−13.0898	−0.505327
$672$	−6.39277	−0.246606
$673$	3.39316	0.130797	0.0653984	−	0.997859i	$-0.479168\pi$
$673$	3.39316	0.130797	0.0653984	+	0.997859i	$0.479168\pi$
$674$	8.31514	0.320287
$675$	4.65525	0.179181
$676$	22.9931	0.884350
$677$	17.8210	0.684917	0.342459	−	0.939533i	$-0.388740\pi$
$677$	17.8210	0.684917	0.342459	+	0.939533i	$0.388740\pi$
$678$	2.68599	0.103155
$679$	4.89271	0.187765
$680$	8.56768	0.328555
$681$	−7.71094	−0.295484
$682$	−3.43620	−0.131579
$683$	33.8828	1.29649	0.648244	−	0.761433i	$-0.275504\pi$
$683$	33.8828	1.29649	0.648244	+	0.761433i	$0.275504\pi$
$684$	26.4123	1.00990
$685$	8.98369	0.343249
$686$	−7.95636	−0.303775
$687$	−0.577752	−0.0220426
$688$	1.99455	0.0760417
$689$	4.17451	0.159036
$690$	−1.76729	−0.0672795
$691$	7.83381	0.298012	0.149006	−	0.988836i	$-0.452393\pi$
$691$	7.83381	0.298012	0.149006	+	0.988836i	$0.452393\pi$
$692$	18.7400	0.712389
$693$	−3.36933	−0.127990
$694$	−2.42405	−0.0920158
$695$	7.92166	0.300486
$696$	8.28182	0.313922
$697$	−2.65566	−0.100590
$698$	−10.3348	−0.391177
$699$	2.88616	0.109165
$700$	2.76171	0.104383
$701$	−6.13465	−0.231703	−0.115851	−	0.993267i	$-0.536960\pi$
$701$	−6.13465	−0.231703	−0.115851	+	0.993267i	$0.536960\pi$
$702$	−1.00937	−0.0380962
$703$	−73.8093	−2.78377
$704$	3.62271	0.136536
$705$	−3.60159	−0.135644
$706$	1.32996	0.0500537
$707$	−23.0305	−0.866152
$708$	−8.80797	−0.331024
$709$	−29.9653	−1.12537	−0.562685	−	0.826671i	$-0.690232\pi$
$709$	−29.9653	−1.12537	−0.562685	+	0.826671i	$0.690232\pi$
$710$	3.32138	0.124649
$711$	−2.78775	−0.104549
$712$	−13.7037	−0.513569
$713$	34.1637	1.27944
$714$	−3.09428	−0.115800
$715$	−0.486692	−0.0182012
$716$	−7.15162	−0.267269
$717$	5.92593	0.221308
$718$	7.46452	0.278573
$719$	47.8243	1.78354	0.891772	−	0.452485i	$-0.149462\pi$
$719$	47.8243	1.78354	0.891772	+	0.452485i	$0.149462\pi$
$720$	−6.26080	−0.233326
$721$	−20.3070	−0.756272
$722$	−11.3585	−0.422721
$723$	−23.0410	−0.856904
$724$	−31.9017	−1.18562
$725$	−5.46008	−0.202782
$726$	0.398996	0.0148081
$727$	−47.5988	−1.76534	−0.882671	−	0.469991i	$-0.844257\pi$
$727$	−47.5988	−1.76534	−0.882671	+	0.469991i	$0.844257\pi$
$728$	−1.26358	−0.0468313
$729$	7.17920	0.265896
$730$	0.445505	0.0164889
$731$	3.54222	0.131014
$732$	21.1198	0.780612
$733$	−0.165558	−0.00611502	−0.00305751	−	0.999995i	$-0.500973\pi$
$733$	−0.165558	−0.00611502	−0.00305751	+	0.999995i	$0.500973\pi$
$734$	4.18055	0.154307
$735$	4.16452	0.153611
$736$	20.6240	0.760212
$737$	12.2452	0.451059
$738$	−0.514018	−0.0189213
$739$	20.4153	0.750988	0.375494	−	0.926825i	$-0.377473\pi$
$739$	20.4153	0.750988	0.375494	+	0.926825i	$0.377473\pi$
$740$	19.9339	0.732784
$741$	−2.90757	−0.106812
$742$	5.85789	0.215050
$743$	−14.8499	−0.544792	−0.272396	−	0.962185i	$-0.587816\pi$
$743$	−14.8499	−0.544792	−0.272396	+	0.962185i	$0.587816\pi$
$744$	11.6991	0.428910
$745$	−14.8781	−0.545091
$746$	0.820872	0.0300542
$747$	0.256441	0.00938267
$748$	9.11367	0.333229
$749$	−10.7237	−0.391835
$750$	0.398996	0.0145693
$751$	30.2917	1.10536	0.552680	−	0.833394i	$-0.313605\pi$
$751$	30.2917	1.10536	0.552680	+	0.833394i	$0.313605\pi$
$752$	11.4552	0.417726
$753$	−2.74355	−0.0999807
$754$	1.18387	0.0431142
$755$	8.32448	0.302959
$756$	12.8565	0.467585
$757$	−46.9988	−1.70820	−0.854100	−	0.520109i	$-0.825891\pi$
$757$	−46.9988	−1.70820	−0.854100	+	0.520109i	$0.825891\pi$
$758$	−6.59898	−0.239686
$759$	−3.96693	−0.143991
$760$	11.2972	0.409792
$761$	−27.7381	−1.00550	−0.502752	−	0.864431i	$-0.667679\pi$
$761$	−27.7381	−1.00550	−0.502752	+	0.864431i	$0.667679\pi$
$762$	0.430905	0.0156100
$763$	−21.3559	−0.773136
$764$	−3.38860	−0.122595
$765$	−11.1188	−0.402002
$766$	−13.9329	−0.503416
$767$	−2.65689	−0.0959347
$768$	−2.12944	−0.0768395
$769$	18.9900	0.684796	0.342398	−	0.939555i	$-0.388761\pi$
$769$	18.9900	0.684796	0.342398	+	0.939555i	$0.388761\pi$
$770$	−0.682952	−0.0246119
$771$	18.5202	0.666988
$772$	−29.8589	−1.07465
$773$	10.4128	0.374522	0.187261	−	0.982310i	$-0.440039\pi$
$773$	10.4128	0.374522	0.187261	+	0.982310i	$0.440039\pi$
$774$	0.685617	0.0246440
$775$	−7.71305	−0.277061
$776$	−5.40533	−0.194040
$777$	−15.1917	−0.544997
$778$	−11.6362	−0.417178
$779$	−3.50170	−0.125462
$780$	0.785255	0.0281166
$781$	7.45532	0.266772
$782$	9.98261	0.356977
$783$	−25.4181	−0.908368
$784$	−13.2456	−0.473058
$785$	2.84853	0.101668
$786$	2.98330	0.106411
$787$	−1.21889	−0.0434489	−0.0217244	−	0.999764i	$-0.506916\pi$
$787$	−1.21889	−0.0434489	−0.0217244	+	0.999764i	$0.506916\pi$
$788$	−36.4175	−1.29732
$789$	−15.5681	−0.554240
$790$	−0.565067	−0.0201042
$791$	−10.3198	−0.366931
$792$	3.72235	0.132268
$793$	6.37071	0.226231
$794$	−14.2054	−0.504130
$795$	−7.68188	−0.272448
$796$	−5.12525	−0.181660
$797$	19.7288	0.698830	0.349415	−	0.936968i	$-0.386380\pi$
$797$	19.7288	0.698830	0.349415	+	0.936968i	$0.386380\pi$
$798$	−4.08006	−0.144432
$799$	20.3437	0.719709
$800$	−4.65624	−0.164623
$801$	17.7842	0.628375
$802$	−9.92234	−0.350370
$803$	1.00000	0.0352892
$804$	−19.7571	−0.696780
$805$	6.79010	0.239320
$806$	1.67237	0.0589067
$807$	22.2074	0.781738
$808$	25.4435	0.895099
$809$	16.9493	0.595907	0.297953	−	0.954580i	$-0.403696\pi$
$809$	16.9493	0.595907	0.297953	+	0.954580i	$0.403696\pi$
$810$	−1.08009	−0.0379504
$811$	0.0663287	0.00232911	0.00116456	−	0.999999i	$-0.499629\pi$
$811$	0.0663287	0.00232911	0.00116456	+	0.999999i	$0.499629\pi$
$812$	−15.0792	−0.529175
$813$	−16.1635	−0.566879
$814$	−4.92951	−0.172779
$815$	6.78916	0.237814
$816$	−12.9060	−0.451801
$817$	4.67071	0.163407
$818$	−15.4528	−0.540294
$819$	1.63983	0.0573002
$820$	0.945714	0.0330258
$821$	−36.6408	−1.27877	−0.639387	−	0.768885i	$-0.720812\pi$
$821$	−36.6408	−1.27877	−0.639387	+	0.768885i	$0.720812\pi$
$822$	3.58446	0.125022
$823$	−28.7328	−1.00156	−0.500780	−	0.865574i	$-0.666954\pi$
$823$	−28.7328	−1.00156	−0.500780	+	0.865574i	$0.666954\pi$
$824$	22.4346	0.781547
$825$	0.895604	0.0311809
$826$	−3.72829	−0.129724
$827$	11.4057	0.396614	0.198307	−	0.980140i	$-0.436456\pi$
$827$	11.4057	0.396614	0.198307	+	0.980140i	$0.436456\pi$
$828$	−17.5382	−0.609496
$829$	−34.1731	−1.18688	−0.593440	−	0.804879i	$-0.702230\pi$
$829$	−34.1731	−1.18688	−0.593440	+	0.804879i	$0.702230\pi$
$830$	0.0519796	0.00180424
$831$	−0.900817	−0.0312490
$832$	−1.76314	−0.0611260
$833$	−23.5235	−0.815041
$834$	3.16071	0.109446
$835$	−16.3122	−0.564506
$836$	12.0171	0.415621
$837$	−35.9062	−1.24110
$838$	−3.62510	−0.125227
$839$	11.4499	0.395295	0.197648	−	0.980273i	$-0.436670\pi$
$839$	11.4499	0.395295	0.197648	+	0.980273i	$0.436670\pi$
$840$	2.32522	0.0802277
$841$	0.812516	0.0280178
$842$	−11.1168	−0.383111
$843$	−24.5670	−0.846134
$844$	−26.9087	−0.926236
$845$	−12.7631	−0.439065
$846$	3.93765	0.135379
$847$	−1.53298	−0.0526739
$848$	24.4329	0.839028
$849$	18.1012	0.621232
$850$	−2.25375	−0.0773029
$851$	49.0106	1.68006
$852$	−12.0288	−0.412100
$853$	−25.9466	−0.888396	−0.444198	−	0.895929i	$-0.646511\pi$
$853$	−25.9466	−0.888396	−0.444198	+	0.895929i	$0.646511\pi$
$854$	8.93973	0.305911
$855$	−14.6611	−0.501399
$856$	11.8472	0.404930
$857$	35.7970	1.22280	0.611402	−	0.791320i	$-0.290606\pi$
$857$	35.7970	1.22280	0.611402	+	0.791320i	$0.290606\pi$
$858$	−0.194188	−0.00662947
$859$	−31.4854	−1.07427	−0.537134	−	0.843497i	$-0.680493\pi$
$859$	−31.4854	−1.07427	−0.537134	+	0.843497i	$0.680493\pi$
$860$	−1.26143	−0.0430144
$861$	−0.720731	−0.0245624
$862$	−3.37150	−0.114834
$863$	−9.42228	−0.320738	−0.160369	−	0.987057i	$-0.551268\pi$
$863$	−9.42228	−0.320738	−0.160369	+	0.987057i	$0.551268\pi$
$864$	−21.6760	−0.737431
$865$	−10.4023	−0.353689
$866$	−5.82402	−0.197908
$867$	−7.69510	−0.261339
$868$	−21.3012	−0.723010
$869$	−1.26837	−0.0430267
$870$	−2.17855	−0.0738598
$871$	−5.95966	−0.201935
$872$	23.5934	0.798974
$873$	7.01485	0.237417
$874$	13.1629	0.445241
$875$	−1.53298	−0.0518243
$876$	−1.61345	−0.0545136
$877$	23.1737	0.782520	0.391260	−	0.920280i	$-0.372039\pi$
$877$	23.1737	0.782520	0.391260	+	0.920280i	$0.372039\pi$
$878$	8.73347	0.294740
$879$	6.47950	0.218548
$880$	−2.84854	−0.0960244
$881$	14.8313	0.499680	0.249840	−	0.968287i	$-0.419622\pi$
$881$	14.8313	0.499680	0.249840	+	0.968287i	$0.419622\pi$
$882$	−4.55311	−0.153311
$883$	−45.5007	−1.53122	−0.765610	−	0.643305i	$-0.777563\pi$
$883$	−45.5007	−1.53122	−0.765610	+	0.643305i	$0.777563\pi$
$884$	−4.43555	−0.149184
$885$	4.88917	0.164348
$886$	5.46523	0.183608
$887$	−21.1802	−0.711161	−0.355581	−	0.934646i	$-0.615717\pi$
$887$	−21.1802	−0.711161	−0.355581	+	0.934646i	$0.615717\pi$
$888$	16.7833	0.563211
$889$	−1.65558	−0.0555264
$890$	3.60480	0.120833
$891$	−2.42441	−0.0812209
$892$	49.3261	1.65156
$893$	26.8249	0.897660
$894$	−5.93629	−0.198539
$895$	3.96976	0.132694
$896$	−16.7500	−0.559579
$897$	1.93067	0.0644633
$898$	−9.03823	−0.301610
$899$	42.1139	1.40458
$900$	3.95956	0.131985
$901$	43.3914	1.44558
$902$	−0.233869	−0.00778697
$903$	0.961339	0.0319914
$904$	11.4011	0.379194
$905$	17.7082	0.588640
$906$	3.32143	0.110347
$907$	18.9624	0.629637	0.314819	−	0.949152i	$-0.398056\pi$
$907$	18.9624	0.629637	0.314819	+	0.949152i	$0.398056\pi$
$908$	−15.5107	−0.514741
$909$	−33.0197	−1.09519
$910$	0.332387	0.0110185
$911$	17.4804	0.579151	0.289575	−	0.957155i	$-0.406486\pi$
$911$	17.4804	0.579151	0.289575	+	0.957155i	$0.406486\pi$
$912$	−17.0176	−0.563510
$913$	0.116676	0.00386140
$914$	−15.1458	−0.500978
$915$	−11.7233	−0.387561
$916$	−1.16216	−0.0383988
$917$	−11.4621	−0.378513
$918$	−10.4918	−0.346280
$919$	6.71945	0.221654	0.110827	−	0.993840i	$-0.464650\pi$
$919$	6.71945	0.221654	0.110827	+	0.993840i	$0.464650\pi$
$920$	−7.50152	−0.247318
$921$	−23.9074	−0.787775
$922$	−4.98160	−0.164060
$923$	−3.62844	−0.119432
$924$	2.47340	0.0813688
$925$	−11.0650	−0.363815
$926$	−1.43102	−0.0470264
$927$	−29.1149	−0.956258
$928$	25.4234	0.834565
$929$	−12.7674	−0.418885	−0.209442	−	0.977821i	$-0.567165\pi$
$929$	−12.7674	−0.418885	−0.209442	+	0.977821i	$0.567165\pi$
$930$	−3.07748	−0.100914
$931$	−31.0177	−1.01656
$932$	5.80557	0.190168
$933$	−13.8596	−0.453743
$934$	7.02166	0.229756
$935$	−5.05886	−0.165442
$936$	−1.81164	−0.0592152
$937$	−45.7227	−1.49370	−0.746848	−	0.664995i	$-0.768434\pi$
$937$	−45.7227	−1.49370	−0.746848	+	0.664995i	$0.768434\pi$
$938$	−8.36291	−0.273059
$939$	7.83382	0.255647
$940$	−7.24466	−0.236295
$941$	58.4887	1.90668	0.953338	−	0.301904i	$-0.0976223\pi$
$941$	58.4887	1.90668	0.953338	+	0.301904i	$0.0976223\pi$
$942$	1.13655	0.0370308
$943$	2.32519	0.0757186
$944$	−15.5504	−0.506123
$945$	−7.13643	−0.232148
$946$	0.311943	0.0101421
$947$	1.00051	0.0325121	0.0162561	−	0.999868i	$-0.494825\pi$
$947$	1.00051	0.0325121	0.0162561	+	0.999868i	$0.494825\pi$
$948$	2.04646	0.0664661
$949$	−0.486692	−0.0157987
$950$	−2.97175	−0.0964163
$951$	−14.4511	−0.468608
$952$	−13.1341	−0.425679
$953$	43.2678	1.40158	0.700791	−	0.713367i	$-0.252831\pi$
$953$	43.2678	1.40158	0.700791	+	0.713367i	$0.252831\pi$
$954$	8.39867	0.271917
$955$	1.88096	0.0608665
$956$	11.9201	0.385524
$957$	−4.89007	−0.158074
$958$	1.39296	0.0450045
$959$	−13.7718	−0.444716
$960$	3.24452	0.104716
$961$	28.4911	0.919068
$962$	2.39915	0.0773518
$963$	−15.3749	−0.495450
$964$	−46.3474	−1.49275
$965$	16.5743	0.533544
$966$	2.70922	0.0871679
$967$	0.631600	0.0203109	0.0101554	−	0.999948i	$-0.496767\pi$
$967$	0.631600	0.0203109	0.0101554	+	0.999948i	$0.496767\pi$
$968$	1.69360	0.0544343
$969$	−30.2224	−0.970883
$970$	1.42188	0.0456539
$971$	−46.9169	−1.50563	−0.752817	−	0.658230i	$-0.771306\pi$
$971$	−46.9169	−1.50563	−0.752817	+	0.658230i	$0.771306\pi$
$972$	29.0714	0.932464
$973$	−12.1438	−0.389311
$974$	19.0862	0.611560
$975$	−0.435883	−0.0139594
$976$	37.2870	1.19353
$977$	−30.2167	−0.966718	−0.483359	−	0.875422i	$-0.660583\pi$
$977$	−30.2167	−0.966718	−0.483359	+	0.875422i	$0.660583\pi$
$978$	2.70885	0.0866195
$979$	8.09149	0.258605
$980$	8.37702	0.267594
$981$	−30.6187	−0.977580
$982$	7.89981	0.252093
$983$	4.63624	0.147873	0.0739366	−	0.997263i	$-0.476444\pi$
$983$	4.63624	0.147873	0.0739366	+	0.997263i	$0.476444\pi$
$984$	0.796244	0.0253833
$985$	20.2148	0.644097
$986$	12.3056	0.391892
$987$	5.52118	0.175741
$988$	−5.84863	−0.186070
$989$	−3.10143	−0.0986196
$990$	−0.979172	−0.0311201
$991$	−0.584758	−0.0185754	−0.00928772	−	0.999957i	$-0.502956\pi$
$991$	−0.584758	−0.0185754	−0.00928772	+	0.999957i	$0.502956\pi$
$992$	35.9138	1.14026
$993$	10.0415	0.318658
$994$	−5.09162	−0.161497
$995$	2.84495	0.0901910
$996$	−0.188251	−0.00596495
$997$	37.2033	1.17824	0.589120	−	0.808046i	$-0.299475\pi$
$997$	37.2033	1.17824	0.589120	+	0.808046i	$0.299475\pi$
$998$	−6.79581	−0.215118
$999$	−51.5104	−1.62972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.i.1.16	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.i.1.16	✓	38	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-0.445505 q^{2} -0.895604 q^{3} -1.80153 q^{4} +1.00000 q^{5} +0.398996 q^{6} -1.53298 q^{7} +1.69360 q^{8} -2.19789 q^{9} +O(q^{10})\)	\(q-0.445505 q^{2} -0.895604 q^{3} -1.80153 q^{4} +1.00000 q^{5} +0.398996 q^{6} -1.53298 q^{7} +1.69360 q^{8} -2.19789 q^{9} -0.445505 q^{10} -1.00000 q^{11} +1.61345 q^{12} +0.486692 q^{13} +0.682952 q^{14} -0.895604 q^{15} +2.84854 q^{16} +5.05886 q^{17} +0.979172 q^{18} +6.67052 q^{19} -1.80153 q^{20} +1.37295 q^{21} +0.445505 q^{22} -4.42934 q^{23} -1.51679 q^{24} +1.00000 q^{25} -0.216824 q^{26} +4.65525 q^{27} +2.76171 q^{28} -5.46008 q^{29} +0.398996 q^{30} -7.71305 q^{31} -4.65624 q^{32} +0.895604 q^{33} -2.25375 q^{34} -1.53298 q^{35} +3.95956 q^{36} -11.0650 q^{37} -2.97175 q^{38} -0.435883 q^{39} +1.69360 q^{40} -0.524952 q^{41} -0.611655 q^{42} +0.700201 q^{43} +1.80153 q^{44} -2.19789 q^{45} +1.97329 q^{46} +4.02141 q^{47} -2.55117 q^{48} -4.64996 q^{49} -0.445505 q^{50} -4.53074 q^{51} -0.876788 q^{52} +8.57731 q^{53} -2.07394 q^{54} -1.00000 q^{55} -2.59626 q^{56} -5.97415 q^{57} +2.43249 q^{58} -5.45908 q^{59} +1.61345 q^{60} +13.0898 q^{61} +3.43620 q^{62} +3.36933 q^{63} -3.62271 q^{64} +0.486692 q^{65} -0.398996 q^{66} -12.2452 q^{67} -9.11367 q^{68} +3.96693 q^{69} +0.682952 q^{70} -7.45532 q^{71} -3.72235 q^{72} -1.00000 q^{73} +4.92951 q^{74} -0.895604 q^{75} -12.0171 q^{76} +1.53298 q^{77} +0.194188 q^{78} +1.26837 q^{79} +2.84854 q^{80} +2.42441 q^{81} +0.233869 q^{82} -0.116676 q^{83} -2.47340 q^{84} +5.05886 q^{85} -0.311943 q^{86} +4.89007 q^{87} -1.69360 q^{88} -8.09149 q^{89} +0.979172 q^{90} -0.746091 q^{91} +7.97957 q^{92} +6.90784 q^{93} -1.79156 q^{94} +6.67052 q^{95} +4.17015 q^{96} -3.19162 q^{97} +2.07158 q^{98} +2.19789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * q^99

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