LMFDB - Embedded newform 4027.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$0$
Dimension:	$174$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54479 q^{2} +0.596838 q^{3} +4.47598 q^{4} +4.06364 q^{5} -1.51883 q^{6} -0.751735 q^{7} -6.30086 q^{8} -2.64378 q^{9} +O(q^{10})$	\(q-2.54479 q^{2} +0.596838 q^{3} +4.47598 q^{4} +4.06364 q^{5} -1.51883 q^{6} -0.751735 q^{7} -6.30086 q^{8} -2.64378 q^{9} -10.3411 q^{10} +1.59169 q^{11} +2.67143 q^{12} +0.889782 q^{13} +1.91301 q^{14} +2.42533 q^{15} +7.08244 q^{16} +6.61510 q^{17} +6.72789 q^{18} +4.89661 q^{19} +18.1888 q^{20} -0.448664 q^{21} -4.05052 q^{22} +3.07346 q^{23} -3.76059 q^{24} +11.5132 q^{25} -2.26431 q^{26} -3.36842 q^{27} -3.36475 q^{28} +4.87557 q^{29} -6.17198 q^{30} +8.96241 q^{31} -5.42164 q^{32} +0.949981 q^{33} -16.8341 q^{34} -3.05478 q^{35} -11.8335 q^{36} +8.64979 q^{37} -12.4609 q^{38} +0.531056 q^{39} -25.6044 q^{40} -7.20078 q^{41} +1.14176 q^{42} -2.22059 q^{43} +7.12437 q^{44} -10.7434 q^{45} -7.82132 q^{46} -2.58487 q^{47} +4.22707 q^{48} -6.43489 q^{49} -29.2986 q^{50} +3.94814 q^{51} +3.98265 q^{52} -0.647148 q^{53} +8.57195 q^{54} +6.46805 q^{55} +4.73658 q^{56} +2.92248 q^{57} -12.4073 q^{58} -10.6319 q^{59} +10.8557 q^{60} +9.23648 q^{61} -22.8075 q^{62} +1.98743 q^{63} -0.367927 q^{64} +3.61575 q^{65} -2.41751 q^{66} -6.56328 q^{67} +29.6090 q^{68} +1.83436 q^{69} +7.77379 q^{70} -14.7856 q^{71} +16.6581 q^{72} -7.61205 q^{73} -22.0119 q^{74} +6.87149 q^{75} +21.9171 q^{76} -1.19653 q^{77} -1.35143 q^{78} +1.24194 q^{79} +28.7805 q^{80} +5.92095 q^{81} +18.3245 q^{82} -0.207324 q^{83} -2.00821 q^{84} +26.8814 q^{85} +5.65095 q^{86} +2.90993 q^{87} -10.0290 q^{88} +16.3645 q^{89} +27.3397 q^{90} -0.668881 q^{91} +13.7567 q^{92} +5.34911 q^{93} +6.57797 q^{94} +19.8981 q^{95} -3.23584 q^{96} -8.12779 q^{97} +16.3755 q^{98} -4.20809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) $ 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * 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 $

$2$	−2.54479	−1.79944	−0.899721	−	0.436466i	$-0.856230\pi$
$2$	−2.54479	−1.79944	−0.899721	+	0.436466i	$0.856230\pi$
$3$	0.596838	0.344584	0.172292	−	0.985046i	$-0.444883\pi$
$3$	0.596838	0.344584	0.172292	+	0.985046i	$0.444883\pi$
$4$	4.47598	2.23799
$5$	4.06364	1.81731	0.908657	−	0.417543i	$-0.137109\pi$
$5$	4.06364	1.81731	0.908657	+	0.417543i	$0.137109\pi$
$6$	−1.51883	−0.620060
$7$	−0.751735	−0.284129	−0.142065	−	0.989857i	$-0.545374\pi$
$7$	−0.751735	−0.284129	−0.142065	+	0.989857i	$0.545374\pi$
$8$	−6.30086	−2.22769
$9$	−2.64378	−0.881262
$10$	−10.3411	−3.27015
$11$	1.59169	0.479913	0.239956	−	0.970784i	$-0.422867\pi$
$11$	1.59169	0.479913	0.239956	+	0.970784i	$0.422867\pi$
$12$	2.67143	0.771177
$13$	0.889782	0.246781	0.123391	−	0.992358i	$-0.460623\pi$
$13$	0.889782	0.246781	0.123391	+	0.992358i	$0.460623\pi$
$14$	1.91301	0.511274
$15$	2.42533	0.626219
$16$	7.08244	1.77061
$17$	6.61510	1.60440	0.802198	−	0.597058i	$-0.203664\pi$
$17$	6.61510	1.60440	0.802198	+	0.597058i	$0.203664\pi$
$18$	6.72789	1.58578
$19$	4.89661	1.12336	0.561680	−	0.827355i	$-0.310155\pi$
$19$	4.89661	1.12336	0.561680	+	0.827355i	$0.310155\pi$
$20$	18.1888	4.06713
$21$	−0.448664	−0.0979065
$22$	−4.05052	−0.863575
$23$	3.07346	0.640860	0.320430	−	0.947272i	$-0.396173\pi$
$23$	3.07346	0.640860	0.320430	+	0.947272i	$0.396173\pi$
$24$	−3.76059	−0.767628
$25$	11.5132	2.30263
$26$	−2.26431	−0.444068
$27$	−3.36842	−0.648254
$28$	−3.36475	−0.635879
$29$	4.87557	0.905371	0.452686	−	0.891670i	$-0.350466\pi$
$29$	4.87557	0.905371	0.452686	+	0.891670i	$0.350466\pi$
$30$	−6.17198	−1.12684
$31$	8.96241	1.60970	0.804848	−	0.593481i	$-0.202247\pi$
$31$	8.96241	1.60970	0.804848	+	0.593481i	$0.202247\pi$
$32$	−5.42164	−0.958420
$33$	0.949981	0.165370
$34$	−16.8341	−2.88702
$35$	−3.05478	−0.516352
$36$	−11.8335	−1.97225
$37$	8.64979	1.42202	0.711008	−	0.703184i	$-0.248239\pi$
$37$	8.64979	1.42202	0.711008	+	0.703184i	$0.248239\pi$
$38$	−12.4609	−2.02142
$39$	0.531056	0.0850370
$40$	−25.6044	−4.04842
$41$	−7.20078	−1.12457	−0.562287	−	0.826942i	$-0.690078\pi$
$41$	−7.20078	−1.12457	−0.562287	+	0.826942i	$0.690078\pi$
$42$	1.14176	0.176177
$43$	−2.22059	−0.338637	−0.169318	−	0.985561i	$-0.554157\pi$
$43$	−2.22059	−0.338637	−0.169318	+	0.985561i	$0.554157\pi$
$44$	7.12437	1.07404
$45$	−10.7434	−1.60153
$46$	−7.82132	−1.15319
$47$	−2.58487	−0.377043	−0.188521	−	0.982069i	$-0.560369\pi$
$47$	−2.58487	−0.377043	−0.188521	+	0.982069i	$0.560369\pi$
$48$	4.22707	0.610125
$49$	−6.43489	−0.919271
$50$	−29.2986	−4.14345
$51$	3.94814	0.552850
$52$	3.98265	0.552294
$53$	−0.647148	−0.0888926	−0.0444463	−	0.999012i	$-0.514152\pi$
$53$	−0.647148	−0.0888926	−0.0444463	+	0.999012i	$0.514152\pi$
$54$	8.57195	1.16649
$55$	6.46805	0.872152
$56$	4.73658	0.632952
$57$	2.92248	0.387092
$58$	−12.4073	−1.62916
$59$	−10.6319	−1.38416	−0.692078	−	0.721823i	$-0.743305\pi$
$59$	−10.6319	−1.38416	−0.692078	+	0.721823i	$0.743305\pi$
$60$	10.8557	1.40147
$61$	9.23648	1.18261	0.591305	−	0.806448i	$-0.298613\pi$
$61$	9.23648	1.18261	0.591305	+	0.806448i	$0.298613\pi$
$62$	−22.8075	−2.89655
$63$	1.98743	0.250392
$64$	−0.367927	−0.0459908
$65$	3.61575	0.448479
$66$	−2.41751	−0.297574
$67$	−6.56328	−0.801832	−0.400916	−	0.916115i	$-0.631308\pi$
$67$	−6.56328	−0.801832	−0.400916	+	0.916115i	$0.631308\pi$
$68$	29.6090	3.59062
$69$	1.83436	0.220830
$70$	7.77379	0.929146
$71$	−14.7856	−1.75473	−0.877364	−	0.479825i	$-0.840700\pi$
$71$	−14.7856	−1.75473	−0.877364	+	0.479825i	$0.840700\pi$
$72$	16.6581	1.96318
$73$	−7.61205	−0.890923	−0.445462	−	0.895301i	$-0.646960\pi$
$73$	−7.61205	−0.890923	−0.445462	+	0.895301i	$0.646960\pi$
$74$	−22.0119	−2.55884
$75$	6.87149	0.793452
$76$	21.9171	2.51407
$77$	−1.19653	−0.136357
$78$	−1.35143	−0.153019
$79$	1.24194	0.139730	0.0698648	−	0.997556i	$-0.477743\pi$
$79$	1.24194	0.139730	0.0698648	+	0.997556i	$0.477743\pi$
$80$	28.7805	3.21776
$81$	5.92095	0.657883
$82$	18.3245	2.02360
$83$	−0.207324	−0.0227568	−0.0113784	−	0.999935i	$-0.503622\pi$
$83$	−0.207324	−0.0227568	−0.0113784	+	0.999935i	$0.503622\pi$
$84$	−2.00821	−0.219114
$85$	26.8814	2.91569
$86$	5.65095	0.609357
$87$	2.90993	0.311977
$88$	−10.0290	−1.06910
$89$	16.3645	1.73463	0.867317	−	0.497755i	$-0.165842\pi$
$89$	16.3645	1.73463	0.867317	+	0.497755i	$0.165842\pi$
$90$	27.3397	2.88186
$91$	−0.668881	−0.0701177
$92$	13.7567	1.43424
$93$	5.34911	0.554676
$94$	6.57797	0.678466
$95$	19.8981	2.04150
$96$	−3.23584	−0.330257
$97$	−8.12779	−0.825252	−0.412626	−	0.910901i	$-0.635388\pi$
$97$	−8.12779	−0.825252	−0.412626	+	0.910901i	$0.635388\pi$
$98$	16.3755	1.65417
$99$	−4.20809	−0.422928
$100$	51.5327	5.15327
$101$	−16.5090	−1.64271	−0.821355	−	0.570417i	$-0.806782\pi$
$101$	−16.5090	−1.64271	−0.821355	+	0.570417i	$0.806782\pi$
$102$	−10.0472	−0.994822
$103$	−12.6195	−1.24343	−0.621716	−	0.783242i	$-0.713564\pi$
$103$	−12.6195	−1.24343	−0.621716	+	0.783242i	$0.713564\pi$
$104$	−5.60640	−0.549752
$105$	−1.82321	−0.177927
$106$	1.64686	0.159957
$107$	−5.88071	−0.568509	−0.284255	−	0.958749i	$-0.591746\pi$
$107$	−5.88071	−0.568509	−0.284255	+	0.958749i	$0.591746\pi$
$108$	−15.0770	−1.45079
$109$	−10.5666	−1.01210	−0.506049	−	0.862505i	$-0.668894\pi$
$109$	−10.5666	−1.01210	−0.506049	+	0.862505i	$0.668894\pi$
$110$	−16.4599	−1.56939
$111$	5.16252	0.490005
$112$	−5.32412	−0.503082
$113$	15.3333	1.44243	0.721216	−	0.692710i	$-0.243584\pi$
$113$	15.3333	1.44243	0.721216	+	0.692710i	$0.243584\pi$
$114$	−7.43712	−0.696550
$115$	12.4894	1.16464
$116$	21.8230	2.02621
$117$	−2.35239	−0.217479
$118$	27.0560	2.49071
$119$	−4.97280	−0.455856
$120$	−15.2817	−1.39502
$121$	−8.46652	−0.769684
$122$	−23.5049	−2.12804
$123$	−4.29770	−0.387511
$124$	40.1156	3.60249
$125$	26.4672	2.36729
$126$	−5.05759	−0.450566
$127$	5.59628	0.496590	0.248295	−	0.968685i	$-0.420130\pi$
$127$	5.59628	0.496590	0.248295	+	0.968685i	$0.420130\pi$
$128$	11.7796	1.04118
$129$	−1.32533	−0.116689
$130$	−9.20135	−0.807012
$131$	19.3051	1.68669	0.843347	−	0.537369i	$-0.180582\pi$
$131$	19.3051	1.68669	0.843347	+	0.537369i	$0.180582\pi$
$132$	4.25210	0.370097
$133$	−3.68096	−0.319179
$134$	16.7022	1.44285
$135$	−13.6881	−1.17808
$136$	−41.6808	−3.57410
$137$	3.80008	0.324662	0.162331	−	0.986736i	$-0.448099\pi$
$137$	3.80008	0.324662	0.162331	+	0.986736i	$0.448099\pi$
$138$	−4.66806	−0.397371
$139$	9.54205	0.809346	0.404673	−	0.914461i	$-0.367385\pi$
$139$	9.54205	0.809346	0.404673	+	0.914461i	$0.367385\pi$
$140$	−13.6731	−1.15559
$141$	−1.54275	−0.129923
$142$	37.6263	3.15753
$143$	1.41626	0.118433
$144$	−18.7245	−1.56037
$145$	19.8126	1.64534
$146$	19.3711	1.60316
$147$	−3.84059	−0.316766
$148$	38.7163	3.18246
$149$	−0.583891	−0.0478342	−0.0239171	−	0.999714i	$-0.507614\pi$
$149$	−0.583891	−0.0478342	−0.0239171	+	0.999714i	$0.507614\pi$
$150$	−17.4865	−1.42777
$151$	3.52011	0.286463	0.143231	−	0.989689i	$-0.454251\pi$
$151$	3.52011	0.286463	0.143231	+	0.989689i	$0.454251\pi$
$152$	−30.8529	−2.50250
$153$	−17.4889	−1.41389
$154$	3.04492	0.245367
$155$	36.4200	2.92533
$156$	2.37700	0.190312
$157$	−7.64180	−0.609882	−0.304941	−	0.952371i	$-0.598637\pi$
$157$	−7.64180	−0.609882	−0.304941	+	0.952371i	$0.598637\pi$
$158$	−3.16049	−0.251435
$159$	−0.386242	−0.0306310
$160$	−22.0316	−1.74175
$161$	−2.31043	−0.182087
$162$	−15.0676	−1.18382
$163$	−9.88999	−0.774644	−0.387322	−	0.921945i	$-0.626600\pi$
$163$	−9.88999	−0.774644	−0.387322	+	0.921945i	$0.626600\pi$
$164$	−32.2306	−2.51678
$165$	3.86038	0.300530
$166$	0.527597	0.0409494
$167$	3.14610	0.243452	0.121726	−	0.992564i	$-0.461157\pi$
$167$	3.14610	0.243452	0.121726	+	0.992564i	$0.461157\pi$
$168$	2.82697	0.218106
$169$	−12.2083	−0.939099
$170$	−68.4076	−5.24662
$171$	−12.9456	−0.989974
$172$	−9.93932	−0.757866
$173$	−1.05704	−0.0803649	−0.0401824	−	0.999192i	$-0.512794\pi$
$173$	−1.05704	−0.0803649	−0.0401824	+	0.999192i	$0.512794\pi$
$174$	−7.40517	−0.561384
$175$	−8.65485	−0.654245
$176$	11.2731	0.849739
$177$	−6.34552	−0.476959
$178$	−41.6443	−3.12137
$179$	−5.90938	−0.441688	−0.220844	−	0.975309i	$-0.570881\pi$
$179$	−5.90938	−0.441688	−0.220844	+	0.975309i	$0.570881\pi$
$180$	−48.0872	−3.58421
$181$	−15.6026	−1.15973	−0.579865	−	0.814712i	$-0.696895\pi$
$181$	−15.6026	−1.15973	−0.579865	+	0.814712i	$0.696895\pi$
$182$	1.70216	0.126173
$183$	5.51268	0.407509
$184$	−19.3654	−1.42764
$185$	35.1496	2.58425
$186$	−13.6124	−0.998108
$187$	10.5292	0.769970
$188$	−11.5698	−0.843818
$189$	2.53216	0.184188
$190$	−50.6365	−3.67356
$191$	2.15350	0.155822	0.0779109	−	0.996960i	$-0.475175\pi$
$191$	2.15350	0.155822	0.0779109	+	0.996960i	$0.475175\pi$
$192$	−0.219592	−0.0158477
$193$	25.4873	1.83461	0.917307	−	0.398181i	$-0.130358\pi$
$193$	25.4873	1.83461	0.917307	+	0.398181i	$0.130358\pi$
$194$	20.6836	1.48499
$195$	2.15802	0.154539
$196$	−28.8025	−2.05732
$197$	10.0064	0.712924	0.356462	−	0.934310i	$-0.383983\pi$
$197$	10.0064	0.712924	0.356462	+	0.934310i	$0.383983\pi$
$198$	10.7087	0.761035
$199$	−18.0972	−1.28288	−0.641438	−	0.767175i	$-0.721662\pi$
$199$	−18.0972	−1.28288	−0.641438	+	0.767175i	$0.721662\pi$
$200$	−72.5429	−5.12956
$201$	−3.91721	−0.276299
$202$	42.0121	2.95596
$203$	−3.66514	−0.257242
$204$	17.6718	1.23727
$205$	−29.2614	−2.04370
$206$	32.1140	2.23748
$207$	−8.12556	−0.564765
$208$	6.30183	0.436953
$209$	7.79389	0.539114
$210$	4.63969	0.320169
$211$	−14.6750	−1.01027	−0.505135	−	0.863040i	$-0.668557\pi$
$211$	−14.6750	−1.01027	−0.505135	+	0.863040i	$0.668557\pi$
$212$	−2.89662	−0.198941
$213$	−8.82461	−0.604652
$214$	14.9652	1.02300
$215$	−9.02368	−0.615410
$216$	21.2240	1.44411
$217$	−6.73736	−0.457362
$218$	26.8898	1.82121
$219$	−4.54316	−0.306998
$220$	28.9509	1.95187
$221$	5.88599	0.395935
$222$	−13.1376	−0.881735
$223$	−17.2178	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$223$	−17.2178	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$224$	4.07564	0.272315
$225$	−30.4383	−2.02922
$226$	−39.0200	−2.59557
$227$	14.2320	0.944614	0.472307	−	0.881434i	$-0.343421\pi$
$227$	14.2320	0.944614	0.472307	+	0.881434i	$0.343421\pi$
$228$	13.0810	0.866309
$229$	26.9662	1.78197	0.890987	−	0.454029i	$-0.150014\pi$
$229$	26.9662	1.78197	0.890987	+	0.454029i	$0.150014\pi$
$230$	−31.7830	−2.09571
$231$	−0.714134	−0.0469866
$232$	−30.7203	−2.01689
$233$	−8.57031	−0.561460	−0.280730	−	0.959787i	$-0.590576\pi$
$233$	−8.57031	−0.561460	−0.280730	+	0.959787i	$0.590576\pi$
$234$	5.98636	0.391340
$235$	−10.5040	−0.685205
$236$	−47.5882	−3.09773
$237$	0.741239	0.0481486
$238$	12.6548	0.820286
$239$	−11.0017	−0.711643	−0.355822	−	0.934554i	$-0.615799\pi$
$239$	−11.0017	−0.711643	−0.355822	+	0.934554i	$0.615799\pi$
$240$	17.1773	1.10879
$241$	−0.868425	−0.0559402	−0.0279701	−	0.999609i	$-0.508904\pi$
$241$	−0.868425	−0.0559402	−0.0279701	+	0.999609i	$0.508904\pi$
$242$	21.5456	1.38500
$243$	13.6391	0.874950
$244$	41.3423	2.64667
$245$	−26.1491	−1.67060
$246$	10.9368	0.697303
$247$	4.35692	0.277224
$248$	−56.4709	−3.58591
$249$	−0.123739	−0.00784162
$250$	−67.3535	−4.25981
$251$	25.5662	1.61373	0.806863	−	0.590738i	$-0.201163\pi$
$251$	25.5662	1.61373	0.806863	+	0.590738i	$0.201163\pi$
$252$	8.89568	0.560375
$253$	4.89199	0.307557
$254$	−14.2414	−0.893584
$255$	16.0438	1.00470
$256$	−29.2408	−1.82755
$257$	15.7501	0.982466	0.491233	−	0.871028i	$-0.336546\pi$
$257$	15.7501	0.982466	0.491233	+	0.871028i	$0.336546\pi$
$258$	3.37270	0.209975
$259$	−6.50235	−0.404036
$260$	16.1840	1.00369
$261$	−12.8900	−0.797869
$262$	−49.1275	−3.03511
$263$	19.8859	1.22622	0.613110	−	0.789998i	$-0.289918\pi$
$263$	19.8859	1.22622	0.613110	+	0.789998i	$0.289918\pi$
$264$	−5.98570	−0.368394
$265$	−2.62977	−0.161546
$266$	9.36728	0.574345
$267$	9.76696	0.597728
$268$	−29.3771	−1.79449
$269$	−16.8977	−1.03027	−0.515134	−	0.857109i	$-0.672258\pi$
$269$	−16.8977	−1.03027	−0.515134	+	0.857109i	$0.672258\pi$
$270$	34.8333	2.11989
$271$	−16.1903	−0.983490	−0.491745	−	0.870739i	$-0.663641\pi$
$271$	−16.1903	−0.983490	−0.491745	+	0.870739i	$0.663641\pi$
$272$	46.8510	2.84076
$273$	−0.399213	−0.0241615
$274$	−9.67041	−0.584211
$275$	18.3254	1.10506
$276$	8.21054	0.494216
$277$	21.1958	1.27353	0.636765	−	0.771058i	$-0.280272\pi$
$277$	21.1958	1.27353	0.636765	+	0.771058i	$0.280272\pi$
$278$	−24.2826	−1.45637
$279$	−23.6947	−1.41856
$280$	19.2478	1.15027
$281$	25.8399	1.54148	0.770741	−	0.637149i	$-0.219886\pi$
$281$	25.8399	1.54148	0.770741	+	0.637149i	$0.219886\pi$
$282$	3.92598	0.233789
$283$	30.3561	1.80448	0.902241	−	0.431233i	$-0.141921\pi$
$283$	30.3561	1.80448	0.902241	+	0.431233i	$0.141921\pi$
$284$	−66.1801	−3.92707
$285$	11.8759	0.703469
$286$	−3.60408	−0.213114
$287$	5.41308	0.319524
$288$	14.3336	0.844618
$289$	26.7595	1.57409
$290$	−50.4189	−2.96070
$291$	−4.85097	−0.284369
$292$	−34.0714	−1.99388
$293$	13.3191	0.778109	0.389054	−	0.921215i	$-0.372802\pi$
$293$	13.3191	0.778109	0.389054	+	0.921215i	$0.372802\pi$
$294$	9.77351	0.570003
$295$	−43.2042	−2.51545
$296$	−54.5011	−3.16781
$297$	−5.36149	−0.311105
$298$	1.48588	0.0860748
$299$	2.73471	0.158152
$300$	30.7567	1.77574
$301$	1.66930	0.0962166
$302$	−8.95796	−0.515473
$303$	−9.85322	−0.566052
$304$	34.6800	1.98903
$305$	37.5337	2.14917
$306$	44.5056	2.54422
$307$	11.3473	0.647625	0.323812	−	0.946121i	$-0.395035\pi$
$307$	11.3473	0.647625	0.323812	+	0.946121i	$0.395035\pi$
$308$	−5.35564	−0.305166
$309$	−7.53177	−0.428468
$310$	−92.6814	−5.26395
$311$	28.8444	1.63562	0.817809	−	0.575490i	$-0.195188\pi$
$311$	28.8444	1.63562	0.817809	+	0.575490i	$0.195188\pi$
$312$	−3.34611	−0.189436
$313$	−7.27211	−0.411044	−0.205522	−	0.978652i	$-0.565889\pi$
$313$	−7.27211	−0.411044	−0.205522	+	0.978652i	$0.565889\pi$
$314$	19.4468	1.09745
$315$	8.07618	0.455041
$316$	5.55892	0.312713
$317$	2.30266	0.129330	0.0646651	−	0.997907i	$-0.479402\pi$
$317$	2.30266	0.129330	0.0646651	+	0.997907i	$0.479402\pi$
$318$	0.982907	0.0551187
$319$	7.76040	0.434499
$320$	−1.49512	−0.0835798
$321$	−3.50983	−0.195899
$322$	5.87956	0.327655
$323$	32.3915	1.80231
$324$	26.5021	1.47234
$325$	10.2442	0.568246
$326$	25.1680	1.39393
$327$	−6.30655	−0.348753
$328$	45.3712	2.50520
$329$	1.94314	0.107129
$330$	−9.82387	−0.540786
$331$	−13.9717	−0.767952	−0.383976	−	0.923343i	$-0.625446\pi$
$331$	−13.9717	−0.767952	−0.383976	+	0.923343i	$0.625446\pi$
$332$	−0.927978	−0.0509294
$333$	−22.8682	−1.25317
$334$	−8.00617	−0.438078
$335$	−26.6708	−1.45718
$336$	−3.17764	−0.173354
$337$	−7.20428	−0.392442	−0.196221	−	0.980560i	$-0.562867\pi$
$337$	−7.20428	−0.392442	−0.196221	+	0.980560i	$0.562867\pi$
$338$	31.0676	1.68985
$339$	9.15147	0.497040
$340$	120.320	6.52529
$341$	14.2654	0.772514
$342$	32.9439	1.78140
$343$	10.0995	0.545321
$344$	13.9916	0.754379
$345$	7.45416	0.401318
$346$	2.68994	0.144612
$347$	−11.8618	−0.636774	−0.318387	−	0.947961i	$-0.603141\pi$
$347$	−11.8618	−0.636774	−0.318387	+	0.947961i	$0.603141\pi$
$348$	13.0248	0.698201
$349$	−9.42552	−0.504536	−0.252268	−	0.967657i	$-0.581177\pi$
$349$	−9.42552	−0.504536	−0.252268	+	0.967657i	$0.581177\pi$
$350$	22.0248	1.17728
$351$	−2.99716	−0.159977
$352$	−8.62957	−0.459958
$353$	−21.1243	−1.12433	−0.562167	−	0.827024i	$-0.690032\pi$
$353$	−21.1243	−1.12433	−0.562167	+	0.827024i	$0.690032\pi$
$354$	16.1481	0.858259
$355$	−60.0834	−3.18889
$356$	73.2472	3.88210
$357$	−2.96796	−0.157081
$358$	15.0382	0.794791
$359$	26.8636	1.41780	0.708902	−	0.705307i	$-0.249191\pi$
$359$	26.8636	1.41780	0.708902	+	0.705307i	$0.249191\pi$
$360$	67.6926	3.56771
$361$	4.97680	0.261937
$362$	39.7054	2.08687
$363$	−5.05314	−0.265221
$364$	−2.99390	−0.156923
$365$	−30.9326	−1.61909
$366$	−14.0286	−0.733289
$367$	−0.0496759	−0.00259306	−0.00129653	−	0.999999i	$-0.500413\pi$
$367$	−0.0496759	−0.00259306	−0.00129653	+	0.999999i	$0.500413\pi$
$368$	21.7676	1.13471
$369$	19.0373	0.991043
$370$	−89.4486	−4.65021
$371$	0.486484	0.0252570
$372$	23.9425	1.24136
$373$	29.7032	1.53797	0.768986	−	0.639265i	$-0.220761\pi$
$373$	29.7032	1.53797	0.768986	+	0.639265i	$0.220761\pi$
$374$	−26.7946	−1.38552
$375$	15.7966	0.815733
$376$	16.2869	0.839935
$377$	4.33820	0.223429
$378$	−6.44384	−0.331435
$379$	7.27050	0.373461	0.186730	−	0.982411i	$-0.440211\pi$
$379$	7.27050	0.373461	0.186730	+	0.982411i	$0.440211\pi$
$380$	89.0633	4.56885
$381$	3.34007	0.171117
$382$	−5.48022	−0.280392
$383$	14.1152	0.721252	0.360626	−	0.932711i	$-0.382563\pi$
$383$	14.1152	0.721252	0.360626	+	0.932711i	$0.382563\pi$
$384$	7.03050	0.358774
$385$	−4.86226	−0.247804
$386$	−64.8599	−3.30128
$387$	5.87076	0.298428
$388$	−36.3798	−1.84691
$389$	28.4979	1.44490	0.722451	−	0.691423i	$-0.243016\pi$
$389$	28.4979	1.44490	0.722451	+	0.691423i	$0.243016\pi$
$390$	−5.49172	−0.278084
$391$	20.3312	1.02819
$392$	40.5454	2.04785
$393$	11.5220	0.581209
$394$	−25.4642	−1.28287
$395$	5.04681	0.253933
$396$	−18.8353	−0.946510
$397$	11.0392	0.554041	0.277021	−	0.960864i	$-0.410653\pi$
$397$	11.0392	0.554041	0.277021	+	0.960864i	$0.410653\pi$
$398$	46.0536	2.30846
$399$	−2.19693	−0.109984
$400$	81.5413	4.07707
$401$	−31.4484	−1.57046	−0.785230	−	0.619204i	$-0.787455\pi$
$401$	−31.4484	−1.57046	−0.785230	+	0.619204i	$0.787455\pi$
$402$	9.96850	0.497184
$403$	7.97459	0.397243
$404$	−73.8941	−3.67637
$405$	24.0606	1.19558
$406$	9.32703	0.462893
$407$	13.7678	0.682443
$408$	−24.8767	−1.23158
$409$	−23.4429	−1.15917	−0.579587	−	0.814910i	$-0.696786\pi$
$409$	−23.4429	−1.15917	−0.579587	+	0.814910i	$0.696786\pi$
$410$	74.4642	3.67753
$411$	2.26803	0.111874
$412$	−56.4845	−2.78279
$413$	7.99238	0.393279
$414$	20.6779	1.01626
$415$	−0.842489	−0.0413562
$416$	−4.82408	−0.236520
$417$	5.69506	0.278888
$418$	−19.8338	−0.970105
$419$	17.4406	0.852030	0.426015	−	0.904716i	$-0.359917\pi$
$419$	17.4406	0.852030	0.426015	+	0.904716i	$0.359917\pi$
$420$	−8.16065	−0.398199
$421$	−29.7259	−1.44875	−0.724375	−	0.689406i	$-0.757872\pi$
$421$	−29.7259	−1.44875	−0.724375	+	0.689406i	$0.757872\pi$
$422$	37.3449	1.81792
$423$	6.83385	0.332273
$424$	4.07759	0.198025
$425$	76.1607	3.69434
$426$	22.4568	1.08804
$427$	−6.94339	−0.336014
$428$	−26.3219	−1.27232
$429$	0.845276	0.0408103
$430$	22.9634	1.10739
$431$	−27.5179	−1.32549	−0.662745	−	0.748845i	$-0.730609\pi$
$431$	−27.5179	−1.32549	−0.662745	+	0.748845i	$0.730609\pi$
$432$	−23.8567	−1.14780
$433$	−10.0841	−0.484613	−0.242306	−	0.970200i	$-0.577904\pi$
$433$	−10.0841	−0.484613	−0.242306	+	0.970200i	$0.577904\pi$
$434$	17.1452	0.822996
$435$	11.8249	0.566960
$436$	−47.2959	−2.26506
$437$	15.0495	0.719916
$438$	11.5614	0.552426
$439$	−27.8354	−1.32851	−0.664256	−	0.747505i	$-0.731252\pi$
$439$	−27.8354	−1.32851	−0.664256	+	0.747505i	$0.731252\pi$
$440$	−40.7543	−1.94289
$441$	17.0125	0.810118
$442$	−14.9786	−0.712462
$443$	−37.2220	−1.76847	−0.884235	−	0.467043i	$-0.845319\pi$
$443$	−37.2220	−1.76847	−0.884235	+	0.467043i	$0.845319\pi$
$444$	23.1073	1.09663
$445$	66.4995	3.15238
$446$	43.8157	2.07473
$447$	−0.348488	−0.0164829
$448$	0.276583	0.0130673
$449$	11.0794	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$449$	11.0794	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$450$	77.4593	3.65147
$451$	−11.4614	−0.539697
$452$	68.6314	3.22815
$453$	2.10094	0.0987106
$454$	−36.2176	−1.69978
$455$	−2.71809	−0.127426
$456$	−18.4142	−0.862322
$457$	15.9552	0.746352	0.373176	−	0.927761i	$-0.378269\pi$
$457$	15.9552	0.746352	0.373176	+	0.927761i	$0.378269\pi$
$458$	−68.6233	−3.20656
$459$	−22.2824	−1.04006
$460$	55.9024	2.60646
$461$	−3.50016	−0.163019	−0.0815093	−	0.996673i	$-0.525974\pi$
$461$	−3.50016	−0.163019	−0.0815093	+	0.996673i	$0.525974\pi$
$462$	1.81733	0.0845496
$463$	33.6980	1.56608	0.783040	−	0.621971i	$-0.213668\pi$
$463$	33.6980	1.56608	0.783040	+	0.621971i	$0.213668\pi$
$464$	34.5310	1.60306
$465$	21.7368	1.00802
$466$	21.8097	1.01031
$467$	4.40699	0.203931	0.101966	−	0.994788i	$-0.467487\pi$
$467$	4.40699	0.203931	0.101966	+	0.994788i	$0.467487\pi$
$468$	−10.5293	−0.486715
$469$	4.93385	0.227824
$470$	26.7305	1.23299
$471$	−4.56092	−0.210156
$472$	66.9902	3.08347
$473$	−3.53449	−0.162516
$474$	−1.88630	−0.0866407
$475$	56.3755	2.58668
$476$	−22.2582	−1.02020
$477$	1.71092	0.0783376
$478$	27.9972	1.28056
$479$	−17.4431	−0.796997	−0.398499	−	0.917169i	$-0.630469\pi$
$479$	−17.4431	−0.796997	−0.398499	+	0.917169i	$0.630469\pi$
$480$	−13.1493	−0.600180
$481$	7.69643	0.350927
$482$	2.20996	0.100661
$483$	−1.37895	−0.0627444
$484$	−37.8960	−1.72255
$485$	−33.0284	−1.49974
$486$	−34.7088	−1.57442
$487$	−16.6629	−0.755068	−0.377534	−	0.925996i	$-0.623228\pi$
$487$	−16.6629	−0.755068	−0.377534	+	0.925996i	$0.623228\pi$
$488$	−58.1978	−2.63449
$489$	−5.90272	−0.266930
$490$	66.5441	3.00615
$491$	−6.65911	−0.300522	−0.150261	−	0.988646i	$-0.548011\pi$
$491$	−6.65911	−0.300522	−0.150261	+	0.988646i	$0.548011\pi$
$492$	−19.2364	−0.867245
$493$	32.2524	1.45257
$494$	−11.0875	−0.498848
$495$	−17.1001	−0.768594
$496$	63.4758	2.85015
$497$	11.1149	0.498570
$498$	0.314890	0.0141105
$499$	0.409460	0.0183300	0.00916498	−	0.999958i	$-0.497083\pi$
$499$	0.409460	0.0183300	0.00916498	+	0.999958i	$0.497083\pi$
$500$	118.466	5.29798
$501$	1.87771	0.0838899
$502$	−65.0608	−2.90381
$503$	−32.9937	−1.47112	−0.735558	−	0.677462i	$-0.763080\pi$
$503$	−32.9937	−1.47112	−0.735558	+	0.677462i	$0.763080\pi$
$504$	−12.5225	−0.557797
$505$	−67.0868	−2.98532
$506$	−12.4491	−0.553430
$507$	−7.28637	−0.323599
$508$	25.0488	1.11136
$509$	3.71307	0.164579	0.0822895	−	0.996608i	$-0.473777\pi$
$509$	3.71307	0.164579	0.0822895	+	0.996608i	$0.473777\pi$
$510$	−40.8282	−1.80790
$511$	5.72225	0.253137
$512$	50.8526	2.24739
$513$	−16.4939	−0.728222
$514$	−40.0808	−1.76789
$515$	−51.2810	−2.25971
$516$	−5.93216	−0.261149
$517$	−4.11432	−0.180947
$518$	16.5471	0.727040
$519$	−0.630879	−0.0276925
$520$	−22.7824	−0.999073
$521$	15.1139	0.662154	0.331077	−	0.943604i	$-0.392588\pi$
$521$	15.1139	0.662154	0.331077	+	0.943604i	$0.392588\pi$
$522$	32.8023	1.43572
$523$	−24.4498	−1.06912	−0.534558	−	0.845132i	$-0.679522\pi$
$523$	−24.4498	−1.06912	−0.534558	+	0.845132i	$0.679522\pi$
$524$	86.4093	3.77481
$525$	−5.16554	−0.225443
$526$	−50.6056	−2.20651
$527$	59.2872	2.58259
$528$	6.72819	0.292807
$529$	−13.5539	−0.589299
$530$	6.69224	0.290692
$531$	28.1085	1.21980
$532$	−16.4759	−0.714320
$533$	−6.40713	−0.277524
$534$	−24.8549	−1.07558
$535$	−23.8971	−1.03316
$536$	41.3543	1.78623
$537$	−3.52694	−0.152199
$538$	43.0011	1.85391
$539$	−10.2424	−0.441170
$540$	−61.2675	−2.63653
$541$	−25.5179	−1.09710	−0.548550	−	0.836118i	$-0.684820\pi$
$541$	−25.5179	−1.09710	−0.548550	+	0.836118i	$0.684820\pi$
$542$	41.2010	1.76973
$543$	−9.31221	−0.399625
$544$	−35.8647	−1.53768
$545$	−42.9389	−1.83930
$546$	1.01592	0.0434772
$547$	−30.7848	−1.31626	−0.658132	−	0.752903i	$-0.728653\pi$
$547$	−30.7848	−1.31626	−0.658132	+	0.752903i	$0.728653\pi$
$548$	17.0091	0.726591
$549$	−24.4193	−1.04219
$550$	−46.6344	−1.98850
$551$	23.8738	1.01706
$552$	−11.5580	−0.491942
$553$	−0.933613	−0.0397013
$554$	−53.9388	−2.29164
$555$	20.9786	0.890493
$556$	42.7100	1.81131
$557$	−18.8415	−0.798341	−0.399171	−	0.916877i	$-0.630702\pi$
$557$	−18.8415	−0.798341	−0.399171	+	0.916877i	$0.630702\pi$
$558$	60.2981	2.55262
$559$	−1.97584	−0.0835692
$560$	−21.6353	−0.914259
$561$	6.28421	0.265320
$562$	−65.7573	−2.77381
$563$	30.5558	1.28777	0.643886	−	0.765121i	$-0.277321\pi$
$563$	30.5558	1.28777	0.643886	+	0.765121i	$0.277321\pi$
$564$	−6.90532	−0.290766
$565$	62.3088	2.62135
$566$	−77.2500	−3.24706
$567$	−4.45099	−0.186924
$568$	93.1621	3.90900
$569$	−36.8104	−1.54317	−0.771586	−	0.636125i	$-0.780536\pi$
$569$	−36.8104	−1.54317	−0.771586	+	0.636125i	$0.780536\pi$
$570$	−30.2218	−1.26585
$571$	−10.2175	−0.427591	−0.213795	−	0.976878i	$-0.568583\pi$
$571$	−10.2175	−0.427591	−0.213795	+	0.976878i	$0.568583\pi$
$572$	6.33914	0.265053
$573$	1.28529	0.0536938
$574$	−13.7752	−0.574965
$575$	35.3852	1.47567
$576$	0.972719	0.0405299
$577$	14.4440	0.601314	0.300657	−	0.953732i	$-0.402794\pi$
$577$	14.4440	0.601314	0.300657	+	0.953732i	$0.402794\pi$
$578$	−68.0974	−2.83248
$579$	15.2118	0.632180
$580$	88.6807	3.68226
$581$	0.155853	0.00646586
$582$	12.3447	0.511706
$583$	−1.03006	−0.0426607
$584$	47.9625	1.98470
$585$	−9.55927	−0.395227
$586$	−33.8943	−1.40016
$587$	38.3102	1.58123	0.790616	−	0.612312i	$-0.209760\pi$
$587$	38.3102	1.58123	0.790616	+	0.612312i	$0.209760\pi$
$588$	−17.1904	−0.708920
$589$	43.8854	1.80827
$590$	109.946	4.52640
$591$	5.97218	0.245663
$592$	61.2616	2.51784
$593$	37.2269	1.52873	0.764363	−	0.644787i	$-0.223054\pi$
$593$	37.2269	1.52873	0.764363	+	0.644787i	$0.223054\pi$
$594$	13.6439	0.559815
$595$	−20.2077	−0.828434
$596$	−2.61348	−0.107052
$597$	−10.8011	−0.442059
$598$	−6.95927	−0.284586
$599$	−27.4364	−1.12102	−0.560511	−	0.828147i	$-0.689395\pi$
$599$	−27.4364	−1.12102	−0.560511	+	0.828147i	$0.689395\pi$
$600$	−43.2963	−1.76757
$601$	17.9505	0.732216	0.366108	−	0.930572i	$-0.380690\pi$
$601$	17.9505	0.732216	0.366108	+	0.930572i	$0.380690\pi$
$602$	−4.24802	−0.173136
$603$	17.3519	0.706624
$604$	15.7560	0.641101
$605$	−34.4049	−1.39876
$606$	25.0744	1.01858
$607$	−15.7500	−0.639272	−0.319636	−	0.947540i	$-0.603561\pi$
$607$	−15.7500	−0.639272	−0.319636	+	0.947540i	$0.603561\pi$
$608$	−26.5477	−1.07665
$609$	−2.18749	−0.0886417
$610$	−95.5156	−3.86731
$611$	−2.29997	−0.0930470
$612$	−78.2799	−3.16428
$613$	−1.84663	−0.0745845	−0.0372923	−	0.999304i	$-0.511873\pi$
$613$	−1.84663	−0.0745845	−0.0372923	+	0.999304i	$0.511873\pi$
$614$	−28.8766	−1.16536
$615$	−17.4643	−0.704229
$616$	7.53917	0.303762
$617$	−14.1576	−0.569962	−0.284981	−	0.958533i	$-0.591987\pi$
$617$	−14.1576	−0.569962	−0.284981	+	0.958533i	$0.591987\pi$
$618$	19.1668	0.771003
$619$	5.43999	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$619$	5.43999	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$620$	163.015	6.54685
$621$	−10.3527	−0.415440
$622$	−73.4032	−2.94320
$623$	−12.3018	−0.492860
$624$	3.76117	0.150567
$625$	49.9871	1.99949
$626$	18.5060	0.739650
$627$	4.65169	0.185770
$628$	−34.2046	−1.36491
$629$	57.2192	2.28148
$630$	−20.5522	−0.818821
$631$	−25.5739	−1.01808	−0.509039	−	0.860743i	$-0.669999\pi$
$631$	−25.5739	−1.01808	−0.509039	+	0.860743i	$0.669999\pi$
$632$	−7.82532	−0.311274
$633$	−8.75860	−0.348123
$634$	−5.85980	−0.232722
$635$	22.7413	0.902460
$636$	−1.72881	−0.0685519
$637$	−5.72565	−0.226859
$638$	−19.7486	−0.781856
$639$	39.0900	1.54638
$640$	47.8680	1.89215
$641$	−17.4812	−0.690467	−0.345234	−	0.938517i	$-0.612200\pi$
$641$	−17.4812	−0.690467	−0.345234	+	0.938517i	$0.612200\pi$
$642$	8.93179	0.352510
$643$	5.57208	0.219741	0.109871	−	0.993946i	$-0.464956\pi$
$643$	5.57208	0.219741	0.109871	+	0.993946i	$0.464956\pi$
$644$	−10.3414	−0.407509
$645$	−5.38567	−0.212061
$646$	−82.4299	−3.24316
$647$	−39.3096	−1.54542	−0.772709	−	0.634760i	$-0.781099\pi$
$647$	−39.3096	−1.54542	−0.772709	+	0.634760i	$0.781099\pi$
$648$	−37.3071	−1.46556
$649$	−16.9227	−0.664274
$650$	−26.0694	−1.02253
$651$	−4.02111	−0.157600
$652$	−44.2674	−1.73365
$653$	−24.1760	−0.946081	−0.473040	−	0.881041i	$-0.656843\pi$
$653$	−24.1760	−0.946081	−0.473040	+	0.881041i	$0.656843\pi$
$654$	16.0489	0.627561
$655$	78.4490	3.06525
$656$	−50.9991	−1.99118
$657$	20.1246	0.785136
$658$	−4.94489	−0.192772
$659$	−7.82605	−0.304859	−0.152430	−	0.988314i	$-0.548710\pi$
$659$	−7.82605	−0.304859	−0.152430	+	0.988314i	$0.548710\pi$
$660$	17.2790	0.672584
$661$	−24.2306	−0.942462	−0.471231	−	0.882010i	$-0.656190\pi$
$661$	−24.2306	−0.942462	−0.471231	+	0.882010i	$0.656190\pi$
$662$	35.5550	1.38189
$663$	3.51298	0.136433
$664$	1.30632	0.0506950
$665$	−14.9581	−0.580049
$666$	58.1948	2.25500
$667$	14.9849	0.580216
$668$	14.0819	0.544844
$669$	−10.2762	−0.397301
$670$	67.8717	2.62211
$671$	14.7016	0.567549
$672$	2.43250	0.0938355
$673$	32.4551	1.25105	0.625526	−	0.780204i	$-0.284885\pi$
$673$	32.4551	1.25105	0.625526	+	0.780204i	$0.284885\pi$
$674$	18.3334	0.706177
$675$	−38.7812	−1.49269
$676$	−54.6441	−2.10169
$677$	37.8082	1.45309	0.726543	−	0.687121i	$-0.241126\pi$
$677$	37.8082	1.45309	0.726543	+	0.687121i	$0.241126\pi$
$678$	−23.2886	−0.894394
$679$	6.10995	0.234478
$680$	−169.376	−6.49527
$681$	8.49422	0.325499
$682$	−36.3025	−1.39009
$683$	−36.8369	−1.40952	−0.704762	−	0.709444i	$-0.748946\pi$
$683$	−36.8369	−1.40952	−0.704762	+	0.709444i	$0.748946\pi$
$684$	−57.9442	−2.21555
$685$	15.4421	0.590014
$686$	−25.7011	−0.981273
$687$	16.0944	0.614041
$688$	−15.7272	−0.599594
$689$	−0.575820	−0.0219370
$690$	−18.9693	−0.722149
$691$	−47.2915	−1.79905	−0.899527	−	0.436864i	$-0.856089\pi$
$691$	−47.2915	−1.79905	−0.899527	+	0.436864i	$0.856089\pi$
$692$	−4.73127	−0.179856
$693$	3.16337	0.120166
$694$	30.1858	1.14584
$695$	38.7754	1.47084
$696$	−18.3350	−0.694988
$697$	−47.6339	−1.80426
$698$	23.9860	0.907884
$699$	−5.11509	−0.193470
$700$	−38.7390	−1.46420
$701$	6.17292	0.233148	0.116574	−	0.993182i	$-0.462809\pi$
$701$	6.17292	0.233148	0.116574	+	0.993182i	$0.462809\pi$
$702$	7.62717	0.287869
$703$	42.3546	1.59744
$704$	−0.585625	−0.0220716
$705$	−6.26918	−0.236111
$706$	53.7570	2.02317
$707$	12.4104	0.466742
$708$	−28.4024	−1.06743
$709$	48.6056	1.82542	0.912711	−	0.408607i	$-0.133985\pi$
$709$	48.6056	1.82542	0.912711	+	0.408607i	$0.133985\pi$
$710$	152.900	5.73823
$711$	−3.28343	−0.123138
$712$	−103.111	−3.86423
$713$	27.5456	1.03159
$714$	7.55284	0.282658
$715$	5.75516	0.215231
$716$	−26.4503	−0.988493
$717$	−6.56626	−0.245221
$718$	−68.3623	−2.55126
$719$	−52.0520	−1.94121	−0.970605	−	0.240676i	$-0.922631\pi$
$719$	−52.0520	−1.94121	−0.970605	+	0.240676i	$0.922631\pi$
$720$	−76.0894	−2.83569
$721$	9.48650	0.353296
$722$	−12.6649	−0.471340
$723$	−0.518309	−0.0192761
$724$	−69.8369	−2.59547
$725$	56.1333	2.08474
$726$	12.8592	0.477250
$727$	47.1599	1.74906	0.874531	−	0.484969i	$-0.161169\pi$
$727$	47.1599	1.74906	0.874531	+	0.484969i	$0.161169\pi$
$728$	4.21453	0.156201
$729$	−9.62251	−0.356389
$730$	78.7172	2.91345
$731$	−14.6894	−0.543308
$732$	24.6747	0.912001
$733$	−14.4317	−0.533047	−0.266523	−	0.963828i	$-0.585875\pi$
$733$	−14.4317	−0.533047	−0.266523	+	0.963828i	$0.585875\pi$
$734$	0.126415	0.00466606
$735$	−15.6068	−0.575664
$736$	−16.6632	−0.614213
$737$	−10.4467	−0.384809
$738$	−48.4461	−1.78332
$739$	15.0399	0.553251	0.276626	−	0.960978i	$-0.410784\pi$
$739$	15.0399	0.553251	0.276626	+	0.960978i	$0.410784\pi$
$740$	157.329	5.78353
$741$	2.60037	0.0955271
$742$	−1.23800	−0.0454485
$743$	30.6441	1.12422	0.562112	−	0.827061i	$-0.309989\pi$
$743$	30.6441	1.12422	0.562112	+	0.827061i	$0.309989\pi$
$744$	−33.7040	−1.23565
$745$	−2.37272	−0.0869298
$746$	−75.5885	−2.76749
$747$	0.548120	0.0200546
$748$	47.1284	1.72319
$749$	4.42073	0.161530
$750$	−40.1991	−1.46786
$751$	−6.00695	−0.219197	−0.109598	−	0.993976i	$-0.534956\pi$
$751$	−6.00695	−0.219197	−0.109598	+	0.993976i	$0.534956\pi$
$752$	−18.3072	−0.667596
$753$	15.2589	0.556065
$754$	−11.0398	−0.402047
$755$	14.3045	0.520593
$756$	11.3339	0.412211
$757$	−5.08382	−0.184774	−0.0923872	−	0.995723i	$-0.529450\pi$
$757$	−5.08382	−0.184774	−0.0923872	+	0.995723i	$0.529450\pi$
$758$	−18.5019	−0.672021
$759$	2.91972	0.105979
$760$	−125.375	−4.54783
$761$	3.62296	0.131332	0.0656660	−	0.997842i	$-0.479083\pi$
$761$	3.62296	0.131332	0.0656660	+	0.997842i	$0.479083\pi$
$762$	−8.49980	−0.307915
$763$	7.94329	0.287566
$764$	9.63903	0.348728
$765$	−71.0685	−2.56949
$766$	−35.9202	−1.29785
$767$	−9.46008	−0.341584
$768$	−17.4520	−0.629744
$769$	13.2246	0.476891	0.238446	−	0.971156i	$-0.423362\pi$
$769$	13.2246	0.476891	0.238446	+	0.971156i	$0.423362\pi$
$770$	12.3735	0.445909
$771$	9.40027	0.338543
$772$	114.081	4.10585
$773$	19.8487	0.713908	0.356954	−	0.934122i	$-0.383815\pi$
$773$	19.8487	0.713908	0.356954	+	0.934122i	$0.383815\pi$
$774$	−14.9399	−0.537003
$775$	103.186	3.70654
$776$	51.2121	1.83841
$777$	−3.88085	−0.139225
$778$	−72.5213	−2.60002
$779$	−35.2594	−1.26330
$780$	9.65925	0.345857
$781$	−23.5341	−0.842116
$782$	−51.7388	−1.85017
$783$	−16.4230	−0.586910
$784$	−45.5748	−1.62767
$785$	−31.0535	−1.10835
$786$	−29.3212	−1.04585
$787$	−42.7482	−1.52381	−0.761904	−	0.647690i	$-0.775735\pi$
$787$	−42.7482	−1.52381	−0.761904	+	0.647690i	$0.775735\pi$
$788$	44.7883	1.59552
$789$	11.8687	0.422536
$790$	−12.8431	−0.456937
$791$	−11.5266	−0.409837
$792$	26.5146	0.942154
$793$	8.21845	0.291846
$794$	−28.0925	−0.996965
$795$	−1.56955	−0.0556662
$796$	−81.0027	−2.87106
$797$	−26.2040	−0.928192	−0.464096	−	0.885785i	$-0.653621\pi$
$797$	−26.2040	−0.928192	−0.464096	+	0.885785i	$0.653621\pi$
$798$	5.59075	0.197910
$799$	−17.0992	−0.604926
$800$	−62.4202	−2.20689
$801$	−43.2642	−1.52867
$802$	80.0298	2.82595
$803$	−12.1160	−0.427565
$804$	−17.5334	−0.618354
$805$	−9.38874	−0.330909
$806$	−20.2937	−0.714815
$807$	−10.0852	−0.355015
$808$	104.021	3.65945
$809$	38.1138	1.34001	0.670005	−	0.742357i	$-0.266292\pi$
$809$	38.1138	1.34001	0.670005	+	0.742357i	$0.266292\pi$
$810$	−61.2293	−2.15138
$811$	−10.3856	−0.364689	−0.182345	−	0.983235i	$-0.558369\pi$
$811$	−10.3856	−0.364689	−0.182345	+	0.983235i	$0.558369\pi$
$812$	−16.4051	−0.575706
$813$	−9.66298	−0.338895
$814$	−35.0362	−1.22802
$815$	−40.1894	−1.40777
$816$	27.9625	0.978883
$817$	−10.8734	−0.380411
$818$	59.6573	2.08587
$819$	1.76838	0.0617921
$820$	−130.973	−4.57379
$821$	0.0281231	0.000981503 0	0.000490751	−	1.00000i	$-0.499844\pi$
$821$	0.0281231	0.000981503 0	0.000490751		1.00000i	$0.499844\pi$
$822$	−5.77167	−0.201310
$823$	−47.7297	−1.66375	−0.831876	−	0.554962i	$-0.812733\pi$
$823$	−47.7297	−1.66375	−0.831876	+	0.554962i	$0.812733\pi$
$824$	79.5135	2.76999
$825$	10.9373	0.380787
$826$	−20.3390	−0.707683
$827$	−19.1070	−0.664414	−0.332207	−	0.943206i	$-0.607793\pi$
$827$	−19.1070	−0.664414	−0.332207	+	0.943206i	$0.607793\pi$
$828$	−36.3698	−1.26394
$829$	−36.9455	−1.28317	−0.641585	−	0.767052i	$-0.721723\pi$
$829$	−36.9455	−1.28317	−0.641585	+	0.767052i	$0.721723\pi$
$830$	2.14396	0.0744180
$831$	12.6504	0.438838
$832$	−0.327374	−0.0113497
$833$	−42.5674	−1.47487
$834$	−14.4927	−0.501843
$835$	12.7846	0.442429
$836$	34.8853	1.20653
$837$	−30.1892	−1.04349
$838$	−44.3828	−1.53318
$839$	−10.7769	−0.372059	−0.186029	−	0.982544i	$-0.559562\pi$
$839$	−10.7769	−0.372059	−0.186029	+	0.982544i	$0.559562\pi$
$840$	11.4878	0.396367
$841$	−5.22879	−0.180303
$842$	75.6462	2.60694
$843$	15.4223	0.531171
$844$	−65.6851	−2.26097
$845$	−49.6101	−1.70664
$846$	−17.3907	−0.597906
$847$	6.36458	0.218690
$848$	−4.58339	−0.157394
$849$	18.1177	0.621796
$850$	−193.813	−6.64774
$851$	26.5847	0.911313
$852$	−39.4988	−1.35321
$853$	−13.4397	−0.460167	−0.230084	−	0.973171i	$-0.573900\pi$
$853$	−13.4397	−0.460167	−0.230084	+	0.973171i	$0.573900\pi$
$854$	17.6695	0.604638
$855$	−52.6062	−1.79909
$856$	37.0535	1.26646
$857$	34.9765	1.19478	0.597388	−	0.801952i	$-0.296205\pi$
$857$	34.9765	1.19478	0.597388	+	0.801952i	$0.296205\pi$
$858$	−2.15105	−0.0734358
$859$	49.8304	1.70019	0.850096	−	0.526627i	$-0.176544\pi$
$859$	49.8304	1.70019	0.850096	+	0.526627i	$0.176544\pi$
$860$	−40.3898	−1.37728
$861$	3.23073	0.110103
$862$	70.0274	2.38514
$863$	40.0338	1.36277	0.681383	−	0.731927i	$-0.261379\pi$
$863$	40.0338	1.36277	0.681383	+	0.731927i	$0.261379\pi$
$864$	18.2624	0.621299
$865$	−4.29541	−0.146048
$866$	25.6621	0.872033
$867$	15.9711	0.542406
$868$	−30.1563	−1.02357
$869$	1.97679	0.0670580
$870$	−30.0919	−1.02021
$871$	−5.83989	−0.197877
$872$	66.5787	2.25464
$873$	21.4881	0.727263
$874$	−38.2979	−1.29545
$875$	−19.8963	−0.672618
$876$	−20.3351	−0.687059
$877$	30.1705	1.01879	0.509393	−	0.860534i	$-0.329870\pi$
$877$	30.1705	1.01879	0.509393	+	0.860534i	$0.329870\pi$
$878$	70.8355	2.39058
$879$	7.94933	0.268124
$880$	45.8096	1.54424
$881$	46.5443	1.56812	0.784058	−	0.620687i	$-0.213146\pi$
$881$	46.5443	1.56812	0.784058	+	0.620687i	$0.213146\pi$
$882$	−43.2933	−1.45776
$883$	−7.16157	−0.241006	−0.120503	−	0.992713i	$-0.538451\pi$
$883$	−7.16157	−0.241006	−0.120503	+	0.992713i	$0.538451\pi$
$884$	26.3456	0.886098
$885$	−25.7859	−0.866784
$886$	94.7223	3.18226
$887$	40.3578	1.35508	0.677541	−	0.735485i	$-0.263046\pi$
$887$	40.3578	1.35508	0.677541	+	0.735485i	$0.263046\pi$
$888$	−32.5283	−1.09158
$889$	−4.20692	−0.141096
$890$	−169.228	−5.67252
$891$	9.42432	0.315727
$892$	−77.0664	−2.58037
$893$	−12.6571	−0.423554
$894$	0.886831	0.0296601
$895$	−24.0136	−0.802686
$896$	−8.85512	−0.295829
$897$	1.63218	0.0544968
$898$	−28.1947	−0.940869
$899$	43.6969	1.45737
$900$	−136.241	−4.54138
$901$	−4.28094	−0.142619
$902$	29.1669	0.971153
$903$	0.996299	0.0331548
$904$	−96.6128	−3.21329
$905$	−63.4033	−2.10760
$906$	−5.34645	−0.177624
$907$	6.22170	0.206588	0.103294	−	0.994651i	$-0.467062\pi$
$907$	6.22170	0.206588	0.103294	+	0.994651i	$0.467062\pi$
$908$	63.7023	2.11404
$909$	43.6463	1.44766
$910$	6.91698	0.229296
$911$	−15.9935	−0.529890	−0.264945	−	0.964264i	$-0.585354\pi$
$911$	−15.9935	−0.529890	−0.264945	+	0.964264i	$0.585354\pi$
$912$	20.6983	0.685390
$913$	−0.329995	−0.0109213
$914$	−40.6026	−1.34302
$915$	22.4015	0.740572
$916$	120.700	3.98804
$917$	−14.5123	−0.479239
$918$	56.7043	1.87152
$919$	6.37731	0.210368	0.105184	−	0.994453i	$-0.466457\pi$
$919$	6.37731	0.210368	0.105184	+	0.994453i	$0.466457\pi$
$920$	−78.6941	−2.59447
$921$	6.77250	0.223161
$922$	8.90718	0.293342
$923$	−13.1560	−0.433034
$924$	−3.19645	−0.105156
$925$	99.5864	3.27438
$926$	−85.7546	−2.81807
$927$	33.3631	1.09579
$928$	−26.4336	−0.867725
$929$	36.3215	1.19167	0.595835	−	0.803107i	$-0.296821\pi$
$929$	36.3215	1.19167	0.595835	+	0.803107i	$0.296821\pi$
$930$	−55.3158	−1.81388
$931$	−31.5092	−1.03267
$932$	−38.3605	−1.25654
$933$	17.2155	0.563609
$934$	−11.2149	−0.366963
$935$	42.7868	1.39928
$936$	14.8221	0.484476
$937$	−12.6650	−0.413748	−0.206874	−	0.978368i	$-0.566329\pi$
$937$	−12.6650	−0.413748	−0.206874	+	0.978368i	$0.566329\pi$
$938$	−12.5556	−0.409956
$939$	−4.34027	−0.141639
$940$	−47.0157	−1.53348
$941$	15.8726	0.517431	0.258715	−	0.965954i	$-0.416701\pi$
$941$	15.8726	0.517431	0.258715	+	0.965954i	$0.416701\pi$
$942$	11.6066	0.378164
$943$	−22.1313	−0.720694
$944$	−75.2999	−2.45080
$945$	10.2898	0.334727
$946$	8.99456	0.292438
$947$	24.0174	0.780462	0.390231	−	0.920717i	$-0.372395\pi$
$947$	24.0174	0.780462	0.390231	+	0.920717i	$0.372395\pi$
$948$	3.31777	0.107756
$949$	−6.77307	−0.219863
$950$	−143.464	−4.65459
$951$	1.37431	0.0445652
$952$	31.3329	1.01551
$953$	−47.1370	−1.52692	−0.763459	−	0.645856i	$-0.776500\pi$
$953$	−47.1370	−1.52692	−0.763459	+	0.645856i	$0.776500\pi$
$954$	−4.35394	−0.140964
$955$	8.75105	0.283177
$956$	−49.2436	−1.59265
$957$	4.63170	0.149722
$958$	44.3892	1.43415
$959$	−2.85665	−0.0922461
$960$	−0.892345	−0.0288003
$961$	49.3248	1.59112
$962$	−19.5858	−0.631472
$963$	15.5473	0.501005
$964$	−3.88705	−0.125194
$965$	103.571	3.33407
$966$	3.50914	0.112905
$967$	−39.0783	−1.25667	−0.628336	−	0.777942i	$-0.716264\pi$
$967$	−39.0783	−1.25667	−0.628336	+	0.777942i	$0.716264\pi$
$968$	53.3464	1.71462
$969$	19.3325	0.621049
$970$	84.0505	2.69870
$971$	−39.9791	−1.28299	−0.641496	−	0.767127i	$-0.721686\pi$
$971$	−39.9791	−1.28299	−0.641496	+	0.767127i	$0.721686\pi$
$972$	61.0484	1.95813
$973$	−7.17310	−0.229959
$974$	42.4037	1.35870
$975$	6.11413	0.195809
$976$	65.4168	2.09394
$977$	19.1459	0.612532	0.306266	−	0.951946i	$-0.400920\pi$
$977$	19.1459	0.612532	0.306266	+	0.951946i	$0.400920\pi$
$978$	15.0212	0.480325
$979$	26.0472	0.832473
$980$	−117.043	−3.73880
$981$	27.9358	0.891922
$982$	16.9461	0.540771
$983$	0.0268941	0.000857788 0	0.000428894	−	1.00000i	$-0.499863\pi$
$983$	0.0268941	0.000857788 0	0.000428894		1.00000i	$0.499863\pi$
$984$	27.0792	0.863254
$985$	40.6623	1.29561
$986$	−82.0757	−2.61382
$987$	1.15974	0.0369149
$988$	19.5015	0.620425
$989$	−6.82489	−0.217019
$990$	43.5164	1.38304
$991$	5.07781	0.161302	0.0806510	−	0.996742i	$-0.474300\pi$
$991$	5.07781	0.161302	0.0806510	+	0.996742i	$0.474300\pi$
$992$	−48.5910	−1.54276
$993$	−8.33882	−0.264624
$994$	−28.2851	−0.897147
$995$	−73.5404	−2.33139
$996$	−0.553852	−0.0175495
$997$	12.7721	0.404495	0.202247	−	0.979334i	$-0.435175\pi$
$997$	12.7721	0.404495	0.202247	+	0.979334i	$0.435175\pi$
$998$	−1.04199	−0.0329837
$999$	−29.1362	−0.921827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.c.1.8	✓	174

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.c.1.8	✓	174	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54479 q^{2} +0.596838 q^{3} +4.47598 q^{4} +4.06364 q^{5} -1.51883 q^{6} -0.751735 q^{7} -6.30086 q^{8} -2.64378 q^{9} +O(q^{10})\)	\(q-2.54479 q^{2} +0.596838 q^{3} +4.47598 q^{4} +4.06364 q^{5} -1.51883 q^{6} -0.751735 q^{7} -6.30086 q^{8} -2.64378 q^{9} -10.3411 q^{10} +1.59169 q^{11} +2.67143 q^{12} +0.889782 q^{13} +1.91301 q^{14} +2.42533 q^{15} +7.08244 q^{16} +6.61510 q^{17} +6.72789 q^{18} +4.89661 q^{19} +18.1888 q^{20} -0.448664 q^{21} -4.05052 q^{22} +3.07346 q^{23} -3.76059 q^{24} +11.5132 q^{25} -2.26431 q^{26} -3.36842 q^{27} -3.36475 q^{28} +4.87557 q^{29} -6.17198 q^{30} +8.96241 q^{31} -5.42164 q^{32} +0.949981 q^{33} -16.8341 q^{34} -3.05478 q^{35} -11.8335 q^{36} +8.64979 q^{37} -12.4609 q^{38} +0.531056 q^{39} -25.6044 q^{40} -7.20078 q^{41} +1.14176 q^{42} -2.22059 q^{43} +7.12437 q^{44} -10.7434 q^{45} -7.82132 q^{46} -2.58487 q^{47} +4.22707 q^{48} -6.43489 q^{49} -29.2986 q^{50} +3.94814 q^{51} +3.98265 q^{52} -0.647148 q^{53} +8.57195 q^{54} +6.46805 q^{55} +4.73658 q^{56} +2.92248 q^{57} -12.4073 q^{58} -10.6319 q^{59} +10.8557 q^{60} +9.23648 q^{61} -22.8075 q^{62} +1.98743 q^{63} -0.367927 q^{64} +3.61575 q^{65} -2.41751 q^{66} -6.56328 q^{67} +29.6090 q^{68} +1.83436 q^{69} +7.77379 q^{70} -14.7856 q^{71} +16.6581 q^{72} -7.61205 q^{73} -22.0119 q^{74} +6.87149 q^{75} +21.9171 q^{76} -1.19653 q^{77} -1.35143 q^{78} +1.24194 q^{79} +28.7805 q^{80} +5.92095 q^{81} +18.3245 q^{82} -0.207324 q^{83} -2.00821 q^{84} +26.8814 q^{85} +5.65095 q^{86} +2.90993 q^{87} -10.0290 q^{88} +16.3645 q^{89} +27.3397 q^{90} -0.668881 q^{91} +13.7567 q^{92} +5.34911 q^{93} +6.57797 q^{94} +19.8981 q^{95} -3.23584 q^{96} -8.12779 q^{97} +16.3755 q^{98} -4.20809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * q^99

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