LMFDB - Embedded newform 8042.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.12962 q^{3} +1.00000 q^{4} +1.55366 q^{5} +3.12962 q^{6} +0.219811 q^{7} -1.00000 q^{8} +6.79451 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.12962 q^{3} +1.00000 q^{4} +1.55366 q^{5} +3.12962 q^{6} +0.219811 q^{7} -1.00000 q^{8} +6.79451 q^{9} -1.55366 q^{10} -3.38226 q^{11} -3.12962 q^{12} -1.24197 q^{13} -0.219811 q^{14} -4.86235 q^{15} +1.00000 q^{16} -6.07371 q^{17} -6.79451 q^{18} -4.37899 q^{19} +1.55366 q^{20} -0.687924 q^{21} +3.38226 q^{22} -0.273819 q^{23} +3.12962 q^{24} -2.58615 q^{25} +1.24197 q^{26} -11.8754 q^{27} +0.219811 q^{28} -9.14771 q^{29} +4.86235 q^{30} -0.0762978 q^{31} -1.00000 q^{32} +10.5852 q^{33} +6.07371 q^{34} +0.341511 q^{35} +6.79451 q^{36} -10.1804 q^{37} +4.37899 q^{38} +3.88690 q^{39} -1.55366 q^{40} -10.4153 q^{41} +0.687924 q^{42} -2.44954 q^{43} -3.38226 q^{44} +10.5563 q^{45} +0.273819 q^{46} -4.26477 q^{47} -3.12962 q^{48} -6.95168 q^{49} +2.58615 q^{50} +19.0084 q^{51} -1.24197 q^{52} +0.795398 q^{53} +11.8754 q^{54} -5.25487 q^{55} -0.219811 q^{56} +13.7046 q^{57} +9.14771 q^{58} -2.78406 q^{59} -4.86235 q^{60} -9.70482 q^{61} +0.0762978 q^{62} +1.49351 q^{63} +1.00000 q^{64} -1.92960 q^{65} -10.5852 q^{66} +5.80502 q^{67} -6.07371 q^{68} +0.856950 q^{69} -0.341511 q^{70} +4.38757 q^{71} -6.79451 q^{72} +1.82521 q^{73} +10.1804 q^{74} +8.09367 q^{75} -4.37899 q^{76} -0.743458 q^{77} -3.88690 q^{78} -2.72630 q^{79} +1.55366 q^{80} +16.7818 q^{81} +10.4153 q^{82} +9.27875 q^{83} -0.687924 q^{84} -9.43647 q^{85} +2.44954 q^{86} +28.6288 q^{87} +3.38226 q^{88} -12.0825 q^{89} -10.5563 q^{90} -0.272999 q^{91} -0.273819 q^{92} +0.238783 q^{93} +4.26477 q^{94} -6.80344 q^{95} +3.12962 q^{96} +10.0952 q^{97} +6.95168 q^{98} -22.9808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.12962	−1.80689	−0.903443	−	0.428708i	$-0.858969\pi$
$3$	−3.12962	−1.80689	−0.903443	+	0.428708i	$0.858969\pi$
$4$	1.00000	0.500000
$5$	1.55366	0.694816	0.347408	−	0.937714i	$-0.387062\pi$
$5$	1.55366	0.694816	0.347408	+	0.937714i	$0.387062\pi$
$6$	3.12962	1.27766
$7$	0.219811	0.0830807	0.0415403	−	0.999137i	$-0.486773\pi$
$7$	0.219811	0.0830807	0.0415403	+	0.999137i	$0.486773\pi$
$8$	−1.00000	−0.353553
$9$	6.79451	2.26484
$10$	−1.55366	−0.491309
$11$	−3.38226	−1.01979	−0.509895	−	0.860237i	$-0.670316\pi$
$11$	−3.38226	−1.01979	−0.509895	+	0.860237i	$0.670316\pi$
$12$	−3.12962	−0.903443
$13$	−1.24197	−0.344461	−0.172231	−	0.985057i	$-0.555097\pi$
$13$	−1.24197	−0.344461	−0.172231	+	0.985057i	$0.555097\pi$
$14$	−0.219811	−0.0587469
$15$	−4.86235	−1.25545
$16$	1.00000	0.250000
$17$	−6.07371	−1.47309	−0.736546	−	0.676387i	$-0.763545\pi$
$17$	−6.07371	−1.47309	−0.736546	+	0.676387i	$0.763545\pi$
$18$	−6.79451	−1.60148
$19$	−4.37899	−1.00461	−0.502304	−	0.864691i	$-0.667514\pi$
$19$	−4.37899	−1.00461	−0.502304	+	0.864691i	$0.667514\pi$
$20$	1.55366	0.347408
$21$	−0.687924	−0.150117
$22$	3.38226	0.721101
$23$	−0.273819	−0.0570953	−0.0285477	−	0.999592i	$-0.509088\pi$
$23$	−0.273819	−0.0570953	−0.0285477	+	0.999592i	$0.509088\pi$
$24$	3.12962	0.638831
$25$	−2.58615	−0.517230
$26$	1.24197	0.243571
$27$	−11.8754	−2.28541
$28$	0.219811	0.0415403
$29$	−9.14771	−1.69869	−0.849344	−	0.527840i	$-0.823002\pi$
$29$	−9.14771	−1.69869	−0.849344	+	0.527840i	$0.823002\pi$
$30$	4.86235	0.887740
$31$	−0.0762978	−0.0137035	−0.00685175	−	0.999977i	$-0.502181\pi$
$31$	−0.0762978	−0.0137035	−0.00685175	+	0.999977i	$0.502181\pi$
$32$	−1.00000	−0.176777
$33$	10.5852	1.84264
$34$	6.07371	1.04163
$35$	0.341511	0.0577258
$36$	6.79451	1.13242
$37$	−10.1804	−1.67365	−0.836823	−	0.547473i	$-0.815590\pi$
$37$	−10.1804	−1.67365	−0.836823	+	0.547473i	$0.815590\pi$
$38$	4.37899	0.710365
$39$	3.88690	0.622402
$40$	−1.55366	−0.245655
$41$	−10.4153	−1.62660	−0.813300	−	0.581845i	$-0.802331\pi$
$41$	−10.4153	−1.62660	−0.813300	+	0.581845i	$0.802331\pi$
$42$	0.687924	0.106149
$43$	−2.44954	−0.373551	−0.186776	−	0.982403i	$-0.559804\pi$
$43$	−2.44954	−0.373551	−0.186776	+	0.982403i	$0.559804\pi$
$44$	−3.38226	−0.509895
$45$	10.5563	1.57364
$46$	0.273819	0.0403725
$47$	−4.26477	−0.622081	−0.311040	−	0.950397i	$-0.600677\pi$
$47$	−4.26477	−0.622081	−0.311040	+	0.950397i	$0.600677\pi$
$48$	−3.12962	−0.451721
$49$	−6.95168	−0.993098
$50$	2.58615	0.365737
$51$	19.0084	2.66171
$52$	−1.24197	−0.172231
$53$	0.795398	0.109256	0.0546282	−	0.998507i	$-0.482603\pi$
$53$	0.795398	0.109256	0.0546282	+	0.998507i	$0.482603\pi$
$54$	11.8754	1.61603
$55$	−5.25487	−0.708567
$56$	−0.219811	−0.0293735
$57$	13.7046	1.81521
$58$	9.14771	1.20115
$59$	−2.78406	−0.362453	−0.181227	−	0.983441i	$-0.558007\pi$
$59$	−2.78406	−0.362453	−0.181227	+	0.983441i	$0.558007\pi$
$60$	−4.86235	−0.627727
$61$	−9.70482	−1.24258	−0.621288	−	0.783583i	$-0.713390\pi$
$61$	−9.70482	−1.24258	−0.621288	+	0.783583i	$0.713390\pi$
$62$	0.0762978	0.00968983
$63$	1.49351	0.188164
$64$	1.00000	0.125000
$65$	−1.92960	−0.239337
$66$	−10.5852	−1.30295
$67$	5.80502	0.709196	0.354598	−	0.935019i	$-0.384618\pi$
$67$	5.80502	0.709196	0.354598	+	0.935019i	$0.384618\pi$
$68$	−6.07371	−0.736546
$69$	0.856950	0.103165
$70$	−0.341511	−0.0408183
$71$	4.38757	0.520709	0.260355	−	0.965513i	$-0.416161\pi$
$71$	4.38757	0.520709	0.260355	+	0.965513i	$0.416161\pi$
$72$	−6.79451	−0.800741
$73$	1.82521	0.213625	0.106812	−	0.994279i	$-0.465936\pi$
$73$	1.82521	0.213625	0.106812	+	0.994279i	$0.465936\pi$
$74$	10.1804	1.18345
$75$	8.09367	0.934576
$76$	−4.37899	−0.502304
$77$	−0.743458	−0.0847249
$78$	−3.88690	−0.440105
$79$	−2.72630	−0.306733	−0.153366	−	0.988169i	$-0.549011\pi$
$79$	−2.72630	−0.306733	−0.153366	+	0.988169i	$0.549011\pi$
$80$	1.55366	0.173704
$81$	16.7818	1.86465
$82$	10.4153	1.15018
$83$	9.27875	1.01847	0.509237	−	0.860626i	$-0.329927\pi$
$83$	9.27875	1.01847	0.509237	+	0.860626i	$0.329927\pi$
$84$	−0.687924	−0.0750587
$85$	−9.43647	−1.02353
$86$	2.44954	0.264141
$87$	28.6288	3.06933
$88$	3.38226	0.360550
$89$	−12.0825	−1.28074	−0.640370	−	0.768067i	$-0.721219\pi$
$89$	−12.0825	−1.28074	−0.640370	+	0.768067i	$0.721219\pi$
$90$	−10.5563	−1.11274
$91$	−0.272999	−0.0286181
$92$	−0.273819	−0.0285477
$93$	0.238783	0.0247606
$94$	4.26477	0.439877
$95$	−6.80344	−0.698018
$96$	3.12962	0.319415
$97$	10.0952	1.02501	0.512505	−	0.858684i	$-0.328718\pi$
$97$	10.0952	1.02501	0.512505	+	0.858684i	$0.328718\pi$
$98$	6.95168	0.702226
$99$	−22.9808	−2.30966
$100$	−2.58615	−0.258615
$101$	17.8857	1.77970	0.889849	−	0.456255i	$-0.150810\pi$
$101$	17.8857	1.77970	0.889849	+	0.456255i	$0.150810\pi$
$102$	−19.0084	−1.88211
$103$	9.55210	0.941196	0.470598	−	0.882348i	$-0.344038\pi$
$103$	9.55210	0.941196	0.470598	+	0.882348i	$0.344038\pi$
$104$	1.24197	0.121785
$105$	−1.06880	−0.104304
$106$	−0.795398	−0.0772559
$107$	14.1746	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$107$	14.1746	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$108$	−11.8754	−1.14271
$109$	9.42284	0.902544	0.451272	−	0.892386i	$-0.350970\pi$
$109$	9.42284	0.902544	0.451272	+	0.892386i	$0.350970\pi$
$110$	5.25487	0.501032
$111$	31.8607	3.02409
$112$	0.219811	0.0207702
$113$	−10.9128	−1.02659	−0.513293	−	0.858214i	$-0.671574\pi$
$113$	−10.9128	−1.02659	−0.513293	+	0.858214i	$0.671574\pi$
$114$	−13.7046	−1.28355
$115$	−0.425421	−0.0396707
$116$	−9.14771	−0.849344
$117$	−8.43859	−0.780148
$118$	2.78406	0.256293
$119$	−1.33507	−0.122386
$120$	4.86235	0.443870
$121$	0.439694	0.0399722
$122$	9.70482	0.878633
$123$	32.5960	2.93908
$124$	−0.0762978	−0.00685175
$125$	−11.7863	−1.05420
$126$	−1.49351	−0.133052
$127$	7.44426	0.660571	0.330286	−	0.943881i	$-0.392855\pi$
$127$	7.44426	0.660571	0.330286	+	0.943881i	$0.392855\pi$
$128$	−1.00000	−0.0883883
$129$	7.66612	0.674965
$130$	1.92960	0.169237
$131$	8.11615	0.709111	0.354556	−	0.935035i	$-0.384632\pi$
$131$	8.11615	0.709111	0.354556	+	0.935035i	$0.384632\pi$
$132$	10.5852	0.921322
$133$	−0.962549	−0.0834636
$134$	−5.80502	−0.501477
$135$	−18.4502	−1.58794
$136$	6.07371	0.520817
$137$	−17.5206	−1.49689	−0.748444	−	0.663198i	$-0.769199\pi$
$137$	−17.5206	−1.49689	−0.748444	+	0.663198i	$0.769199\pi$
$138$	−0.856950	−0.0729484
$139$	−3.70299	−0.314084	−0.157042	−	0.987592i	$-0.550196\pi$
$139$	−3.70299	−0.314084	−0.157042	+	0.987592i	$0.550196\pi$
$140$	0.341511	0.0288629
$141$	13.3471	1.12403
$142$	−4.38757	−0.368197
$143$	4.20067	0.351278
$144$	6.79451	0.566209
$145$	−14.2124	−1.18028
$146$	−1.82521	−0.151055
$147$	21.7561	1.79441
$148$	−10.1804	−0.836823
$149$	−5.88730	−0.482306	−0.241153	−	0.970487i	$-0.577526\pi$
$149$	−5.88730	−0.482306	−0.241153	+	0.970487i	$0.577526\pi$
$150$	−8.09367	−0.660845
$151$	−4.04743	−0.329375	−0.164688	−	0.986346i	$-0.552662\pi$
$151$	−4.04743	−0.329375	−0.164688	+	0.986346i	$0.552662\pi$
$152$	4.37899	0.355183
$153$	−41.2679	−3.33631
$154$	0.743458	0.0599095
$155$	−0.118541	−0.00952141
$156$	3.88690	0.311201
$157$	10.9058	0.870378	0.435189	−	0.900339i	$-0.356682\pi$
$157$	10.9058	0.870378	0.435189	+	0.900339i	$0.356682\pi$
$158$	2.72630	0.216893
$159$	−2.48929	−0.197414
$160$	−1.55366	−0.122827
$161$	−0.0601885	−0.00474352
$162$	−16.7818	−1.31850
$163$	12.8049	1.00295	0.501477	−	0.865171i	$-0.332790\pi$
$163$	12.8049	1.00295	0.501477	+	0.865171i	$0.332790\pi$
$164$	−10.4153	−0.813300
$165$	16.4457	1.28030
$166$	−9.27875	−0.720171
$167$	−7.34115	−0.568076	−0.284038	−	0.958813i	$-0.591674\pi$
$167$	−7.34115	−0.568076	−0.284038	+	0.958813i	$0.591674\pi$
$168$	0.687924	0.0530745
$169$	−11.4575	−0.881347
$170$	9.43647	0.723744
$171$	−29.7531	−2.27527
$172$	−2.44954	−0.186776
$173$	−17.3213	−1.31691	−0.658456	−	0.752619i	$-0.728790\pi$
$173$	−17.3213	−1.31691	−0.658456	+	0.752619i	$0.728790\pi$
$174$	−28.6288	−2.17035
$175$	−0.568464	−0.0429719
$176$	−3.38226	−0.254948
$177$	8.71303	0.654912
$178$	12.0825	0.905620
$179$	6.26424	0.468211	0.234106	−	0.972211i	$-0.424784\pi$
$179$	6.26424	0.468211	0.234106	+	0.972211i	$0.424784\pi$
$180$	10.5563	0.786822
$181$	9.17332	0.681847	0.340924	−	0.940091i	$-0.389260\pi$
$181$	9.17332	0.681847	0.340924	+	0.940091i	$0.389260\pi$
$182$	0.272999	0.0202360
$183$	30.3724	2.24519
$184$	0.273819	0.0201862
$185$	−15.8168	−1.16288
$186$	−0.238783	−0.0175084
$187$	20.5429	1.50225
$188$	−4.26477	−0.311040
$189$	−2.61033	−0.189874
$190$	6.80344	0.493573
$191$	−22.1924	−1.60579	−0.802894	−	0.596122i	$-0.796707\pi$
$191$	−22.1924	−1.60579	−0.802894	+	0.596122i	$0.796707\pi$
$192$	−3.12962	−0.225861
$193$	1.84237	0.132616	0.0663082	−	0.997799i	$-0.478878\pi$
$193$	1.84237	0.132616	0.0663082	+	0.997799i	$0.478878\pi$
$194$	−10.0952	−0.724791
$195$	6.03890	0.432455
$196$	−6.95168	−0.496549
$197$	3.30779	0.235670	0.117835	−	0.993033i	$-0.462405\pi$
$197$	3.30779	0.235670	0.117835	+	0.993033i	$0.462405\pi$
$198$	22.9808	1.63317
$199$	−4.01799	−0.284828	−0.142414	−	0.989807i	$-0.545486\pi$
$199$	−4.01799	−0.284828	−0.142414	+	0.989807i	$0.545486\pi$
$200$	2.58615	0.182869
$201$	−18.1675	−1.28144
$202$	−17.8857	−1.25844
$203$	−2.01077	−0.141128
$204$	19.0084	1.33085
$205$	−16.1818	−1.13019
$206$	−9.55210	−0.665526
$207$	−1.86047	−0.129312
$208$	−1.24197	−0.0861153
$209$	14.8109	1.02449
$210$	1.06880	0.0737540
$211$	5.26154	0.362219	0.181110	−	0.983463i	$-0.442031\pi$
$211$	5.26154	0.362219	0.181110	+	0.983463i	$0.442031\pi$
$212$	0.795398	0.0546282
$213$	−13.7314	−0.940862
$214$	−14.1746	−0.968959
$215$	−3.80574	−0.259550
$216$	11.8754	0.808016
$217$	−0.0167711	−0.00113850
$218$	−9.42284	−0.638195
$219$	−5.71221	−0.385995
$220$	−5.25487	−0.354283
$221$	7.54338	0.507423
$222$	−31.8607	−2.13835
$223$	−8.36690	−0.560289	−0.280144	−	0.959958i	$-0.590382\pi$
$223$	−8.36690	−0.560289	−0.280144	+	0.959958i	$0.590382\pi$
$224$	−0.219811	−0.0146867
$225$	−17.5716	−1.17144
$226$	10.9128	0.725905
$227$	8.48387	0.563094	0.281547	−	0.959547i	$-0.409152\pi$
$227$	8.48387	0.563094	0.281547	+	0.959547i	$0.409152\pi$
$228$	13.7046	0.907606
$229$	5.27650	0.348681	0.174340	−	0.984685i	$-0.444221\pi$
$229$	5.27650	0.348681	0.174340	+	0.984685i	$0.444221\pi$
$230$	0.425421	0.0280515
$231$	2.32674	0.153088
$232$	9.14771	0.600577
$233$	4.75489	0.311503	0.155752	−	0.987796i	$-0.450220\pi$
$233$	4.75489	0.311503	0.155752	+	0.987796i	$0.450220\pi$
$234$	8.43859	0.551648
$235$	−6.62599	−0.432232
$236$	−2.78406	−0.181227
$237$	8.53227	0.554231
$238$	1.33507	0.0865396
$239$	−17.2172	−1.11369	−0.556844	−	0.830617i	$-0.687988\pi$
$239$	−17.2172	−1.11369	−0.556844	+	0.830617i	$0.687988\pi$
$240$	−4.86235	−0.313863
$241$	−7.68743	−0.495191	−0.247595	−	0.968864i	$-0.579640\pi$
$241$	−7.68743	−0.495191	−0.247595	+	0.968864i	$0.579640\pi$
$242$	−0.439694	−0.0282646
$243$	−16.8946	−1.08379
$244$	−9.70482	−0.621288
$245$	−10.8005	−0.690020
$246$	−32.5960	−2.07824
$247$	5.43858	0.346049
$248$	0.0762978	0.00484492
$249$	−29.0389	−1.84027
$250$	11.7863	0.745429
$251$	−11.0951	−0.700316	−0.350158	−	0.936691i	$-0.613872\pi$
$251$	−11.0951	−0.700316	−0.350158	+	0.936691i	$0.613872\pi$
$252$	1.49351	0.0940821
$253$	0.926129	0.0582252
$254$	−7.44426	−0.467094
$255$	29.5325	1.84940
$256$	1.00000	0.0625000
$257$	−6.88536	−0.429497	−0.214749	−	0.976669i	$-0.568893\pi$
$257$	−6.88536	−0.429497	−0.214749	+	0.976669i	$0.568893\pi$
$258$	−7.66612	−0.477272
$259$	−2.23776	−0.139048
$260$	−1.92960	−0.119669
$261$	−62.1542	−3.84725
$262$	−8.11615	−0.501417
$263$	1.08777	0.0670747	0.0335374	−	0.999437i	$-0.489323\pi$
$263$	1.08777	0.0670747	0.0335374	+	0.999437i	$0.489323\pi$
$264$	−10.5852	−0.651473
$265$	1.23578	0.0759131
$266$	0.962549	0.0590177
$267$	37.8135	2.31415
$268$	5.80502	0.354598
$269$	−19.0127	−1.15922	−0.579611	−	0.814893i	$-0.696796\pi$
$269$	−19.0127	−1.15922	−0.579611	+	0.814893i	$0.696796\pi$
$270$	18.4502	1.12285
$271$	−26.4945	−1.60943	−0.804714	−	0.593663i	$-0.797681\pi$
$271$	−26.4945	−1.60943	−0.804714	+	0.593663i	$0.797681\pi$
$272$	−6.07371	−0.368273
$273$	0.854382	0.0517096
$274$	17.5206	1.05846
$275$	8.74704	0.527467
$276$	0.856950	0.0515823
$277$	24.1553	1.45135	0.725677	−	0.688036i	$-0.241527\pi$
$277$	24.1553	1.45135	0.725677	+	0.688036i	$0.241527\pi$
$278$	3.70299	0.222091
$279$	−0.518406	−0.0310362
$280$	−0.341511	−0.0204092
$281$	−1.76711	−0.105417	−0.0527083	−	0.998610i	$-0.516785\pi$
$281$	−1.76711	−0.105417	−0.0527083	+	0.998610i	$0.516785\pi$
$282$	−13.3471	−0.794808
$283$	−4.06682	−0.241747	−0.120874	−	0.992668i	$-0.538570\pi$
$283$	−4.06682	−0.241747	−0.120874	+	0.992668i	$0.538570\pi$
$284$	4.38757	0.260355
$285$	21.2922	1.26124
$286$	−4.20067	−0.248391
$287$	−2.28940	−0.135139
$288$	−6.79451	−0.400370
$289$	19.8900	1.17000
$290$	14.2124	0.834581
$291$	−31.5940	−1.85208
$292$	1.82521	0.106812
$293$	29.4996	1.72338	0.861691	−	0.507433i	$-0.169405\pi$
$293$	29.4996	1.72338	0.861691	+	0.507433i	$0.169405\pi$
$294$	−21.7561	−1.26884
$295$	−4.32547	−0.251838
$296$	10.1804	0.591723
$297$	40.1656	2.33064
$298$	5.88730	0.341042
$299$	0.340076	0.0196671
$300$	8.09367	0.467288
$301$	−0.538436	−0.0310349
$302$	4.04743	0.232903
$303$	−55.9755	−3.21571
$304$	−4.37899	−0.251152
$305$	−15.0780	−0.863361
$306$	41.2679	2.35913
$307$	1.49105	0.0850989	0.0425495	−	0.999094i	$-0.486452\pi$
$307$	1.49105	0.0850989	0.0425495	+	0.999094i	$0.486452\pi$
$308$	−0.743458	−0.0423624
$309$	−29.8944	−1.70063
$310$	0.118541	0.00673265
$311$	−8.31280	−0.471376	−0.235688	−	0.971829i	$-0.575734\pi$
$311$	−8.31280	−0.471376	−0.235688	+	0.971829i	$0.575734\pi$
$312$	−3.88690	−0.220052
$313$	−5.24047	−0.296209	−0.148104	−	0.988972i	$-0.547317\pi$
$313$	−5.24047	−0.296209	−0.148104	+	0.988972i	$0.547317\pi$
$314$	−10.9058	−0.615450
$315$	2.32040	0.130740
$316$	−2.72630	−0.153366
$317$	−0.491069	−0.0275812	−0.0137906	−	0.999905i	$-0.504390\pi$
$317$	−0.491069	−0.0275812	−0.0137906	+	0.999905i	$0.504390\pi$
$318$	2.48929	0.139593
$319$	30.9400	1.73231
$320$	1.55366	0.0868520
$321$	−44.3612	−2.47600
$322$	0.0601885	0.00335417
$323$	26.5967	1.47988
$324$	16.7818	0.932323
$325$	3.21193	0.178166
$326$	−12.8049	−0.709195
$327$	−29.4899	−1.63079
$328$	10.4153	0.575090
$329$	−0.937443	−0.0516829
$330$	−16.4457	−0.905308
$331$	22.2297	1.22185	0.610926	−	0.791688i	$-0.290797\pi$
$331$	22.2297	1.22185	0.610926	+	0.791688i	$0.290797\pi$
$332$	9.27875	0.509237
$333$	−69.1708	−3.79054
$334$	7.34115	0.401690
$335$	9.01900	0.492761
$336$	−0.687924	−0.0375293
$337$	−3.14974	−0.171577	−0.0857886	−	0.996313i	$-0.527341\pi$
$337$	−3.14974	−0.171577	−0.0857886	+	0.996313i	$0.527341\pi$
$338$	11.4575	0.623206
$339$	34.1527	1.85492
$340$	−9.43647	−0.511764
$341$	0.258059	0.0139747
$342$	29.7531	1.60886
$343$	−3.06673	−0.165588
$344$	2.44954	0.132070
$345$	1.33141	0.0716805
$346$	17.3213	0.931198
$347$	10.2642	0.551012	0.275506	−	0.961299i	$-0.411155\pi$
$347$	10.2642	0.551012	0.275506	+	0.961299i	$0.411155\pi$
$348$	28.6288	1.53467
$349$	−7.88385	−0.422013	−0.211007	−	0.977485i	$-0.567674\pi$
$349$	−7.88385	−0.422013	−0.211007	+	0.977485i	$0.567674\pi$
$350$	0.568464	0.0303857
$351$	14.7489	0.787236
$352$	3.38226	0.180275
$353$	−4.02178	−0.214058	−0.107029	−	0.994256i	$-0.534134\pi$
$353$	−4.02178	−0.214058	−0.107029	+	0.994256i	$0.534134\pi$
$354$	−8.71303	−0.463093
$355$	6.81678	0.361797
$356$	−12.0825	−0.640370
$357$	4.17825	0.221137
$358$	−6.26424	−0.331075
$359$	10.7412	0.566901	0.283450	−	0.958987i	$-0.408521\pi$
$359$	10.7412	0.566901	0.283450	+	0.958987i	$0.408521\pi$
$360$	−10.5563	−0.556368
$361$	0.175525	0.00923817
$362$	−9.17332	−0.482139
$363$	−1.37607	−0.0722252
$364$	−0.272999	−0.0143090
$365$	2.83575	0.148430
$366$	−30.3724	−1.58759
$367$	−11.0269	−0.575598	−0.287799	−	0.957691i	$-0.592924\pi$
$367$	−11.0269	−0.575598	−0.287799	+	0.957691i	$0.592924\pi$
$368$	−0.273819	−0.0142738
$369$	−70.7670	−3.68398
$370$	15.8168	0.822278
$371$	0.174837	0.00907709
$372$	0.238783	0.0123803
$373$	19.9346	1.03217	0.516086	−	0.856537i	$-0.327388\pi$
$373$	19.9346	1.03217	0.516086	+	0.856537i	$0.327388\pi$
$374$	−20.5429	−1.06225
$375$	36.8865	1.90481
$376$	4.26477	0.219939
$377$	11.3612	0.585132
$378$	2.61033	0.134261
$379$	−12.6839	−0.651530	−0.325765	−	0.945451i	$-0.605622\pi$
$379$	−12.6839	−0.651530	−0.325765	+	0.945451i	$0.605622\pi$
$380$	−6.80344	−0.349009
$381$	−23.2977	−1.19358
$382$	22.1924	1.13546
$383$	−4.56730	−0.233378	−0.116689	−	0.993169i	$-0.537228\pi$
$383$	−4.56730	−0.233378	−0.116689	+	0.993169i	$0.537228\pi$
$384$	3.12962	0.159708
$385$	−1.15508	−0.0588682
$386$	−1.84237	−0.0937740
$387$	−16.6434	−0.846033
$388$	10.0952	0.512505
$389$	2.20896	0.111999	0.0559993	−	0.998431i	$-0.482166\pi$
$389$	2.20896	0.111999	0.0559993	+	0.998431i	$0.482166\pi$
$390$	−6.03890	−0.305792
$391$	1.66310	0.0841066
$392$	6.95168	0.351113
$393$	−25.4005	−1.28128
$394$	−3.30779	−0.166644
$395$	−4.23573	−0.213123
$396$	−22.9808	−1.15483
$397$	−23.3040	−1.16960	−0.584798	−	0.811179i	$-0.698826\pi$
$397$	−23.3040	−1.16960	−0.584798	+	0.811179i	$0.698826\pi$
$398$	4.01799	0.201404
$399$	3.01241	0.150809
$400$	−2.58615	−0.129308
$401$	15.6411	0.781077	0.390539	−	0.920587i	$-0.372289\pi$
$401$	15.6411	0.781077	0.390539	+	0.920587i	$0.372289\pi$
$402$	18.1675	0.906112
$403$	0.0947597	0.00472032
$404$	17.8857	0.889849
$405$	26.0732	1.29559
$406$	2.01077	0.0997927
$407$	34.4328	1.70677
$408$	−19.0084	−0.941056
$409$	21.1945	1.04800	0.524000	−	0.851718i	$-0.324439\pi$
$409$	21.1945	1.04800	0.524000	+	0.851718i	$0.324439\pi$
$410$	16.1818	0.799163
$411$	54.8329	2.70471
$412$	9.55210	0.470598
$413$	−0.611966	−0.0301129
$414$	1.86047	0.0914370
$415$	14.4160	0.707653
$416$	1.24197	0.0608927
$417$	11.5889	0.567513
$418$	−14.8109	−0.724424
$419$	1.48447	0.0725212	0.0362606	−	0.999342i	$-0.488455\pi$
$419$	1.48447	0.0725212	0.0362606	+	0.999342i	$0.488455\pi$
$420$	−1.06880	−0.0521520
$421$	27.3072	1.33087	0.665435	−	0.746456i	$-0.268246\pi$
$421$	27.3072	1.33087	0.665435	+	0.746456i	$0.268246\pi$
$422$	−5.26154	−0.256128
$423$	−28.9770	−1.40891
$424$	−0.795398	−0.0386279
$425$	15.7075	0.761928
$426$	13.7314	0.665290
$427$	−2.13323	−0.103234
$428$	14.1746	0.685158
$429$	−13.1465	−0.634719
$430$	3.80574	0.183529
$431$	−20.5582	−0.990253	−0.495126	−	0.868821i	$-0.664878\pi$
$431$	−20.5582	−0.990253	−0.495126	+	0.868821i	$0.664878\pi$
$432$	−11.8754	−0.571354
$433$	−11.5076	−0.553021	−0.276510	−	0.961011i	$-0.589178\pi$
$433$	−11.5076	−0.553021	−0.276510	+	0.961011i	$0.589178\pi$
$434$	0.0167711	0.000805038 0
$435$	44.4794	2.13262
$436$	9.42284	0.451272
$437$	1.19905	0.0573584
$438$	5.71221	0.272940
$439$	13.6627	0.652085	0.326043	−	0.945355i	$-0.394285\pi$
$439$	13.6627	0.652085	0.326043	+	0.945355i	$0.394285\pi$
$440$	5.25487	0.250516
$441$	−47.2333	−2.24920
$442$	−7.54338	−0.358802
$443$	−0.761260	−0.0361686	−0.0180843	−	0.999836i	$-0.505757\pi$
$443$	−0.761260	−0.0361686	−0.0180843	+	0.999836i	$0.505757\pi$
$444$	31.8607	1.51204
$445$	−18.7720	−0.889879
$446$	8.36690	0.396184
$447$	18.4250	0.871472
$448$	0.219811	0.0103851
$449$	39.8287	1.87963	0.939817	−	0.341677i	$-0.110995\pi$
$449$	39.8287	1.87963	0.939817	+	0.341677i	$0.110995\pi$
$450$	17.5716	0.828335
$451$	35.2273	1.65879
$452$	−10.9128	−0.513293
$453$	12.6669	0.595143
$454$	−8.48387	−0.398168
$455$	−0.424147	−0.0198843
$456$	−13.7046	−0.641775
$457$	−5.39496	−0.252366	−0.126183	−	0.992007i	$-0.540273\pi$
$457$	−5.39496	−0.252366	−0.126183	+	0.992007i	$0.540273\pi$
$458$	−5.27650	−0.246555
$459$	72.1276	3.36663
$460$	−0.425421	−0.0198354
$461$	−0.921782	−0.0429317	−0.0214658	−	0.999770i	$-0.506833\pi$
$461$	−0.921782	−0.0429317	−0.0214658	+	0.999770i	$0.506833\pi$
$462$	−2.32674	−0.108250
$463$	−27.7349	−1.28895	−0.644475	−	0.764626i	$-0.722924\pi$
$463$	−27.7349	−1.28895	−0.644475	+	0.764626i	$0.722924\pi$
$464$	−9.14771	−0.424672
$465$	0.370987	0.0172041
$466$	−4.75489	−0.220266
$467$	12.2482	0.566779	0.283389	−	0.959005i	$-0.408541\pi$
$467$	12.2482	0.566779	0.283389	+	0.959005i	$0.408541\pi$
$468$	−8.43859	−0.390074
$469$	1.27601	0.0589205
$470$	6.62599	0.305634
$471$	−34.1310	−1.57267
$472$	2.78406	0.128147
$473$	8.28499	0.380944
$474$	−8.53227	−0.391900
$475$	11.3247	0.519614
$476$	−1.33507	−0.0611928
$477$	5.40434	0.247448
$478$	17.2172	0.787496
$479$	30.3247	1.38557	0.692785	−	0.721145i	$-0.256384\pi$
$479$	30.3247	1.38557	0.692785	+	0.721145i	$0.256384\pi$
$480$	4.86235	0.221935
$481$	12.6438	0.576506
$482$	7.68743	0.350153
$483$	0.188367	0.00857099
$484$	0.439694	0.0199861
$485$	15.6844	0.712194
$486$	16.8946	0.766355
$487$	7.93744	0.359680	0.179840	−	0.983696i	$-0.442442\pi$
$487$	7.93744	0.359680	0.179840	+	0.983696i	$0.442442\pi$
$488$	9.70482	0.439317
$489$	−40.0743	−1.81222
$490$	10.8005	0.487918
$491$	−35.2377	−1.59026	−0.795128	−	0.606441i	$-0.792596\pi$
$491$	−35.2377	−1.59026	−0.795128	+	0.606441i	$0.792596\pi$
$492$	32.5960	1.46954
$493$	55.5606	2.50232
$494$	−5.43858	−0.244693
$495$	−35.7043	−1.60479
$496$	−0.0762978	−0.00342587
$497$	0.964436	0.0432609
$498$	29.0389	1.30127
$499$	−16.3186	−0.730521	−0.365261	−	0.930905i	$-0.619020\pi$
$499$	−16.3186	−0.730521	−0.365261	+	0.930905i	$0.619020\pi$
$500$	−11.7863	−0.527098
$501$	22.9750	1.02645
$502$	11.0951	0.495198
$503$	−14.4790	−0.645585	−0.322792	−	0.946470i	$-0.604622\pi$
$503$	−14.4790	−0.645585	−0.322792	+	0.946470i	$0.604622\pi$
$504$	−1.49351	−0.0665261
$505$	27.7883	1.23656
$506$	−0.926129	−0.0411715
$507$	35.8576	1.59249
$508$	7.44426	0.330286
$509$	1.18958	0.0527272	0.0263636	−	0.999652i	$-0.491607\pi$
$509$	1.18958	0.0527272	0.0263636	+	0.999652i	$0.491607\pi$
$510$	−29.5325	−1.30772
$511$	0.401201	0.0177481
$512$	−1.00000	−0.0441942
$513$	52.0021	2.29595
$514$	6.88536	0.303700
$515$	14.8407	0.653958
$516$	7.66612	0.337482
$517$	14.4246	0.634392
$518$	2.23776	0.0983216
$519$	54.2090	2.37951
$520$	1.92960	0.0846185
$521$	−30.8116	−1.34988	−0.674939	−	0.737873i	$-0.735830\pi$
$521$	−30.8116	−1.34988	−0.674939	+	0.737873i	$0.735830\pi$
$522$	62.1542	2.72042
$523$	−27.7969	−1.21547	−0.607736	−	0.794139i	$-0.707922\pi$
$523$	−27.7969	−1.21547	−0.607736	+	0.794139i	$0.707922\pi$
$524$	8.11615	0.354556
$525$	1.77908	0.0776452
$526$	−1.08777	−0.0474290
$527$	0.463411	0.0201865
$528$	10.5852	0.460661
$529$	−22.9250	−0.996740
$530$	−1.23578	−0.0536786
$531$	−18.9163	−0.820897
$532$	−0.962549	−0.0417318
$533$	12.9355	0.560300
$534$	−37.8135	−1.63635
$535$	22.0225	0.952117
$536$	−5.80502	−0.250738
$537$	−19.6047	−0.846004
$538$	19.0127	0.819694
$539$	23.5124	1.01275
$540$	−18.4502	−0.793972
$541$	−18.2859	−0.786171	−0.393085	−	0.919502i	$-0.628592\pi$
$541$	−18.2859	−0.786171	−0.393085	+	0.919502i	$0.628592\pi$
$542$	26.4945	1.13804
$543$	−28.7090	−1.23202
$544$	6.07371	0.260408
$545$	14.6399	0.627103
$546$	−0.854382	−0.0365642
$547$	10.7652	0.460288	0.230144	−	0.973157i	$-0.426080\pi$
$547$	10.7652	0.460288	0.230144	+	0.973157i	$0.426080\pi$
$548$	−17.5206	−0.748444
$549$	−65.9395	−2.81423
$550$	−8.74704	−0.372975
$551$	40.0577	1.70652
$552$	−0.856950	−0.0364742
$553$	−0.599270	−0.0254836
$554$	−24.1553	−1.02626
$555$	49.5006	2.10119
$556$	−3.70299	−0.157042
$557$	−8.56630	−0.362966	−0.181483	−	0.983394i	$-0.558090\pi$
$557$	−8.56630	−0.362966	−0.181483	+	0.983394i	$0.558090\pi$
$558$	0.518406	0.0219459
$559$	3.04226	0.128674
$560$	0.341511	0.0144315
$561$	−64.2914	−2.71439
$562$	1.76711	0.0745409
$563$	−10.9206	−0.460250	−0.230125	−	0.973161i	$-0.573913\pi$
$563$	−10.9206	−0.460250	−0.230125	+	0.973161i	$0.573913\pi$
$564$	13.3471	0.562014
$565$	−16.9547	−0.713288
$566$	4.06682	0.170941
$567$	3.68883	0.154916
$568$	−4.38757	−0.184099
$569$	−7.06327	−0.296108	−0.148054	−	0.988979i	$-0.547301\pi$
$569$	−7.06327	−0.296108	−0.148054	+	0.988979i	$0.547301\pi$
$570$	−21.2922	−0.891831
$571$	11.0907	0.464132	0.232066	−	0.972700i	$-0.425451\pi$
$571$	11.0907	0.464132	0.232066	+	0.972700i	$0.425451\pi$
$572$	4.20067	0.175639
$573$	69.4538	2.90147
$574$	2.28940	0.0955577
$575$	0.708139	0.0295314
$576$	6.79451	0.283105
$577$	1.35623	0.0564606	0.0282303	−	0.999601i	$-0.491013\pi$
$577$	1.35623	0.0564606	0.0282303	+	0.999601i	$0.491013\pi$
$578$	−19.8900	−0.827315
$579$	−5.76590	−0.239623
$580$	−14.2124	−0.590138
$581$	2.03957	0.0846156
$582$	31.5940	1.30962
$583$	−2.69024	−0.111419
$584$	−1.82521	−0.0755277
$585$	−13.1107	−0.542059
$586$	−29.4996	−1.21862
$587$	7.40091	0.305468	0.152734	−	0.988267i	$-0.451192\pi$
$587$	7.40091	0.305468	0.152734	+	0.988267i	$0.451192\pi$
$588$	21.7561	0.897207
$589$	0.334107	0.0137666
$590$	4.32547	0.178077
$591$	−10.3521	−0.425829
$592$	−10.1804	−0.418412
$593$	−6.06585	−0.249095	−0.124547	−	0.992214i	$-0.539748\pi$
$593$	−6.06585	−0.249095	−0.124547	+	0.992214i	$0.539748\pi$
$594$	−40.1656	−1.64801
$595$	−2.07424	−0.0850355
$596$	−5.88730	−0.241153
$597$	12.5748	0.514652
$598$	−0.340076	−0.0139067
$599$	−11.0821	−0.452802	−0.226401	−	0.974034i	$-0.572696\pi$
$599$	−11.0821	−0.452802	−0.226401	+	0.974034i	$0.572696\pi$
$600$	−8.09367	−0.330423
$601$	24.8649	1.01426	0.507129	−	0.861870i	$-0.330707\pi$
$601$	24.8649	1.01426	0.507129	+	0.861870i	$0.330707\pi$
$602$	0.538436	0.0219450
$603$	39.4422	1.60621
$604$	−4.04743	−0.164688
$605$	0.683133	0.0277733
$606$	55.9755	2.27385
$607$	−37.8681	−1.53702	−0.768510	−	0.639838i	$-0.779001\pi$
$607$	−37.8681	−1.53702	−0.768510	+	0.639838i	$0.779001\pi$
$608$	4.37899	0.177591
$609$	6.29293	0.255002
$610$	15.0780	0.610489
$611$	5.29672	0.214283
$612$	−41.2679	−1.66816
$613$	3.14628	0.127077	0.0635385	−	0.997979i	$-0.479761\pi$
$613$	3.14628	0.127077	0.0635385	+	0.997979i	$0.479761\pi$
$614$	−1.49105	−0.0601740
$615$	50.6429	2.04212
$616$	0.743458	0.0299548
$617$	7.40240	0.298010	0.149005	−	0.988836i	$-0.452393\pi$
$617$	7.40240	0.298010	0.149005	+	0.988836i	$0.452393\pi$
$618$	29.8944	1.20253
$619$	−11.1699	−0.448955	−0.224478	−	0.974479i	$-0.572068\pi$
$619$	−11.1699	−0.448955	−0.224478	+	0.974479i	$0.572068\pi$
$620$	−0.118541	−0.00476070
$621$	3.25170	0.130486
$622$	8.31280	0.333313
$623$	−2.65586	−0.106405
$624$	3.88690	0.155600
$625$	−5.38106	−0.215242
$626$	5.24047	0.209451
$627$	−46.3524	−1.85114
$628$	10.9058	0.435189
$629$	61.8328	2.46544
$630$	−2.32040	−0.0924468
$631$	31.4977	1.25390	0.626951	−	0.779059i	$-0.284303\pi$
$631$	31.4977	1.25390	0.626951	+	0.779059i	$0.284303\pi$
$632$	2.72630	0.108446
$633$	−16.4666	−0.654489
$634$	0.491069	0.0195028
$635$	11.5658	0.458975
$636$	−2.48929	−0.0987069
$637$	8.63380	0.342083
$638$	−30.9400	−1.22492
$639$	29.8114	1.17932
$640$	−1.55366	−0.0614137
$641$	3.60696	0.142466	0.0712332	−	0.997460i	$-0.477307\pi$
$641$	3.60696	0.142466	0.0712332	+	0.997460i	$0.477307\pi$
$642$	44.3612	1.75080
$643$	20.4998	0.808433	0.404217	−	0.914663i	$-0.367544\pi$
$643$	20.4998	0.808433	0.404217	+	0.914663i	$0.367544\pi$
$644$	−0.0601885	−0.00237176
$645$	11.9105	0.468976
$646$	−26.5967	−1.04643
$647$	39.0587	1.53556	0.767779	−	0.640715i	$-0.221362\pi$
$647$	39.0587	1.53556	0.767779	+	0.640715i	$0.221362\pi$
$648$	−16.7818	−0.659252
$649$	9.41641	0.369626
$650$	−3.21193	−0.125982
$651$	0.0524871	0.00205713
$652$	12.8049	0.501477
$653$	−6.28188	−0.245829	−0.122915	−	0.992417i	$-0.539224\pi$
$653$	−6.28188	−0.245829	−0.122915	+	0.992417i	$0.539224\pi$
$654$	29.4899	1.15315
$655$	12.6097	0.492702
$656$	−10.4153	−0.406650
$657$	12.4014	0.483825
$658$	0.937443	0.0365453
$659$	8.49788	0.331030	0.165515	−	0.986207i	$-0.447071\pi$
$659$	8.49788	0.331030	0.165515	+	0.986207i	$0.447071\pi$
$660$	16.4457	0.640150
$661$	−6.18180	−0.240444	−0.120222	−	0.992747i	$-0.538361\pi$
$661$	−6.18180	−0.240444	−0.120222	+	0.992747i	$0.538361\pi$
$662$	−22.2297	−0.863980
$663$	−23.6079	−0.916855
$664$	−9.27875	−0.360085
$665$	−1.49547	−0.0579918
$666$	69.1708	2.68031
$667$	2.50482	0.0969871
$668$	−7.34115	−0.284038
$669$	26.1852	1.01238
$670$	−9.01900	−0.348434
$671$	32.8243	1.26717
$672$	0.687924	0.0265372
$673$	34.3862	1.32549	0.662746	−	0.748845i	$-0.269391\pi$
$673$	34.3862	1.32549	0.662746	+	0.748845i	$0.269391\pi$
$674$	3.14974	0.121323
$675$	30.7115	1.18209
$676$	−11.4575	−0.440673
$677$	−2.44515	−0.0939748	−0.0469874	−	0.998895i	$-0.514962\pi$
$677$	−2.44515	−0.0939748	−0.0469874	+	0.998895i	$0.514962\pi$
$678$	−34.1527	−1.31163
$679$	2.21903	0.0851585
$680$	9.43647	0.361872
$681$	−26.5513	−1.01745
$682$	−0.258059	−0.00988160
$683$	−38.7481	−1.48266	−0.741328	−	0.671143i	$-0.765804\pi$
$683$	−38.7481	−1.48266	−0.741328	+	0.671143i	$0.765804\pi$
$684$	−29.7531	−1.13764
$685$	−27.2210	−1.04006
$686$	3.06673	0.117088
$687$	−16.5134	−0.630027
$688$	−2.44954	−0.0933878
$689$	−0.987862	−0.0376346
$690$	−1.33141	−0.0506858
$691$	−5.20610	−0.198049	−0.0990247	−	0.995085i	$-0.531572\pi$
$691$	−5.20610	−0.198049	−0.0990247	+	0.995085i	$0.531572\pi$
$692$	−17.3213	−0.658456
$693$	−5.05143	−0.191888
$694$	−10.2642	−0.389624
$695$	−5.75318	−0.218230
$696$	−28.6288	−1.08517
$697$	63.2597	2.39613
$698$	7.88385	0.298408
$699$	−14.8810	−0.562851
$700$	−0.568464	−0.0214859
$701$	−12.5641	−0.474539	−0.237270	−	0.971444i	$-0.576252\pi$
$701$	−12.5641	−0.474539	−0.237270	+	0.971444i	$0.576252\pi$
$702$	−14.7489	−0.556660
$703$	44.5798	1.68136
$704$	−3.38226	−0.127474
$705$	20.7368	0.780993
$706$	4.02178	0.151362
$707$	3.93148	0.147859
$708$	8.71303	0.327456
$709$	−29.9877	−1.12621	−0.563106	−	0.826385i	$-0.690394\pi$
$709$	−29.9877	−1.12621	−0.563106	+	0.826385i	$0.690394\pi$
$710$	−6.81678	−0.255829
$711$	−18.5239	−0.694699
$712$	12.0825	0.452810
$713$	0.0208918	0.000782405 0
$714$	−4.17825	−0.156367
$715$	6.52640	0.244074
$716$	6.26424	0.234106
$717$	53.8832	2.01231
$718$	−10.7412	−0.400859
$719$	−20.6031	−0.768364	−0.384182	−	0.923257i	$-0.625517\pi$
$719$	−20.6031	−0.768364	−0.384182	+	0.923257i	$0.625517\pi$
$720$	10.5563	0.393411
$721$	2.09965	0.0781952
$722$	−0.175525	−0.00653237
$723$	24.0587	0.894753
$724$	9.17332	0.340924
$725$	23.6574	0.878613
$726$	1.37607	0.0510709
$727$	27.1998	1.00879	0.504393	−	0.863474i	$-0.331716\pi$
$727$	27.1998	1.00879	0.504393	+	0.863474i	$0.331716\pi$
$728$	0.272999	0.0101180
$729$	2.52818	0.0936364
$730$	−2.83575	−0.104956
$731$	14.8778	0.550276
$732$	30.3724	1.12260
$733$	21.6061	0.798041	0.399021	−	0.916942i	$-0.369350\pi$
$733$	21.6061	0.798041	0.399021	+	0.916942i	$0.369350\pi$
$734$	11.0269	0.407009
$735$	33.8015	1.24679
$736$	0.273819	0.0100931
$737$	−19.6341	−0.723231
$738$	70.7670	2.60497
$739$	−19.2258	−0.707232	−0.353616	−	0.935391i	$-0.615048\pi$
$739$	−19.2258	−0.707232	−0.353616	+	0.935391i	$0.615048\pi$
$740$	−15.8168	−0.581438
$741$	−17.0207	−0.625270
$742$	−0.174837	−0.00641847
$743$	−2.22384	−0.0815846	−0.0407923	−	0.999168i	$-0.512988\pi$
$743$	−2.22384	−0.0815846	−0.0407923	+	0.999168i	$0.512988\pi$
$744$	−0.238783	−0.00875421
$745$	−9.14684	−0.335114
$746$	−19.9346	−0.729856
$747$	63.0445	2.30668
$748$	20.5429	0.751123
$749$	3.11574	0.113847
$750$	−36.8865	−1.34691
$751$	47.4396	1.73110	0.865548	−	0.500825i	$-0.166970\pi$
$751$	47.4396	1.73110	0.865548	+	0.500825i	$0.166970\pi$
$752$	−4.26477	−0.155520
$753$	34.7234	1.26539
$754$	−11.3612	−0.413751
$755$	−6.28831	−0.228855
$756$	−2.61033	−0.0949369
$757$	19.0557	0.692592	0.346296	−	0.938125i	$-0.387439\pi$
$757$	19.0557	0.692592	0.346296	+	0.938125i	$0.387439\pi$
$758$	12.6839	0.460701
$759$	−2.89843	−0.105206
$760$	6.80344	0.246787
$761$	−25.8493	−0.937035	−0.468518	−	0.883454i	$-0.655212\pi$
$761$	−25.8493	−0.937035	−0.468518	+	0.883454i	$0.655212\pi$
$762$	23.2977	0.843986
$763$	2.07124	0.0749840
$764$	−22.1924	−0.802894
$765$	−64.1161	−2.31812
$766$	4.56730	0.165023
$767$	3.45772	0.124851
$768$	−3.12962	−0.112930
$769$	−13.3045	−0.479774	−0.239887	−	0.970801i	$-0.577110\pi$
$769$	−13.3045	−0.479774	−0.239887	+	0.970801i	$0.577110\pi$
$770$	1.15508	0.0416261
$771$	21.5486	0.776052
$772$	1.84237	0.0663082
$773$	23.4232	0.842474	0.421237	−	0.906951i	$-0.361596\pi$
$773$	23.4232	0.842474	0.421237	+	0.906951i	$0.361596\pi$
$774$	16.6434	0.598235
$775$	0.197318	0.00708786
$776$	−10.0952	−0.362396
$777$	7.00334	0.251243
$778$	−2.20896	−0.0791950
$779$	45.6085	1.63410
$780$	6.03890	0.216227
$781$	−14.8399	−0.531014
$782$	−1.66310	−0.0594724
$783$	108.632	3.88221
$784$	−6.95168	−0.248274
$785$	16.9439	0.604753
$786$	25.4005	0.906004
$787$	−39.9705	−1.42480	−0.712398	−	0.701776i	$-0.752391\pi$
$787$	−39.9705	−1.42480	−0.712398	+	0.701776i	$0.752391\pi$
$788$	3.30779	0.117835
$789$	−3.40430	−0.121196
$790$	4.23573	0.150701
$791$	−2.39874	−0.0852894
$792$	22.9808	0.816587
$793$	12.0531	0.428019
$794$	23.3040	0.827029
$795$	−3.86750	−0.137166
$796$	−4.01799	−0.142414
$797$	−15.3193	−0.542638	−0.271319	−	0.962490i	$-0.587460\pi$
$797$	−15.3193	−0.542638	−0.271319	+	0.962490i	$0.587460\pi$
$798$	−3.01241	−0.106638
$799$	25.9030	0.916382
$800$	2.58615	0.0914343
$801$	−82.0945	−2.90067
$802$	−15.6411	−0.552305
$803$	−6.17334	−0.217852
$804$	−18.1675	−0.640718
$805$	−0.0935122	−0.00329587
$806$	−0.0947597	−0.00333777
$807$	59.5024	2.09458
$808$	−17.8857	−0.629218
$809$	−24.5586	−0.863435	−0.431718	−	0.902009i	$-0.642092\pi$
$809$	−24.5586	−0.863435	−0.431718	+	0.902009i	$0.642092\pi$
$810$	−26.0732	−0.916118
$811$	1.59901	0.0561487	0.0280743	−	0.999606i	$-0.491062\pi$
$811$	1.59901	0.0561487	0.0280743	+	0.999606i	$0.491062\pi$
$812$	−2.01077	−0.0705641
$813$	82.9177	2.90805
$814$	−34.4328	−1.20687
$815$	19.8943	0.696868
$816$	19.0084	0.665427
$817$	10.7265	0.375273
$818$	−21.1945	−0.741048
$819$	−1.85489	−0.0648152
$820$	−16.1818	−0.565094
$821$	−26.2787	−0.917134	−0.458567	−	0.888660i	$-0.651637\pi$
$821$	−26.2787	−0.917134	−0.458567	+	0.888660i	$0.651637\pi$
$822$	−54.8329	−1.91252
$823$	−7.95634	−0.277341	−0.138670	−	0.990339i	$-0.544283\pi$
$823$	−7.95634	−0.277341	−0.138670	+	0.990339i	$0.544283\pi$
$824$	−9.55210	−0.332763
$825$	−27.3749	−0.953072
$826$	0.611966	0.0212930
$827$	−54.3440	−1.88973	−0.944863	−	0.327465i	$-0.893806\pi$
$827$	−54.3440	−1.88973	−0.944863	+	0.327465i	$0.893806\pi$
$828$	−1.86047	−0.0646558
$829$	−44.8886	−1.55905	−0.779523	−	0.626374i	$-0.784538\pi$
$829$	−44.8886	−1.55905	−0.779523	+	0.626374i	$0.784538\pi$
$830$	−14.4160	−0.500386
$831$	−75.5970	−2.62243
$832$	−1.24197	−0.0430576
$833$	42.2225	1.46292
$834$	−11.5889	−0.401292
$835$	−11.4056	−0.394708
$836$	14.8109	0.512245
$837$	0.906064	0.0313182
$838$	−1.48447	−0.0512802
$839$	−32.8818	−1.13521	−0.567603	−	0.823302i	$-0.692129\pi$
$839$	−32.8818	−1.13521	−0.567603	+	0.823302i	$0.692129\pi$
$840$	1.06880	0.0368770
$841$	54.6807	1.88554
$842$	−27.3072	−0.941067
$843$	5.53037	0.190476
$844$	5.26154	0.181110
$845$	−17.8010	−0.612374
$846$	28.9770	0.996250
$847$	0.0966495	0.00332092
$848$	0.795398	0.0273141
$849$	12.7276	0.436810
$850$	−15.7075	−0.538765
$851$	2.78759	0.0955573
$852$	−13.7314	−0.470431
$853$	23.6695	0.810429	0.405214	−	0.914222i	$-0.367197\pi$
$853$	23.6695	0.810429	0.405214	+	0.914222i	$0.367197\pi$
$854$	2.13323	0.0729975
$855$	−46.2260	−1.58090
$856$	−14.1746	−0.484480
$857$	17.6919	0.604343	0.302171	−	0.953254i	$-0.402288\pi$
$857$	17.6919	0.604343	0.302171	+	0.953254i	$0.402288\pi$
$858$	13.1465	0.448814
$859$	−43.1607	−1.47262	−0.736312	−	0.676642i	$-0.763435\pi$
$859$	−43.1607	−1.47262	−0.736312	+	0.676642i	$0.763435\pi$
$860$	−3.80574	−0.129775
$861$	7.16495	0.244181
$862$	20.5582	0.700215
$863$	−1.23931	−0.0421867	−0.0210933	−	0.999778i	$-0.506715\pi$
$863$	−1.23931	−0.0421867	−0.0210933	+	0.999778i	$0.506715\pi$
$864$	11.8754	0.404008
$865$	−26.9113	−0.915012
$866$	11.5076	0.391045
$867$	−62.2481	−2.11406
$868$	−0.0167711	−0.000569248 0
$869$	9.22106	0.312803
$870$	−44.4794	−1.50799
$871$	−7.20967	−0.244290
$872$	−9.42284	−0.319098
$873$	68.5918	2.32148
$874$	−1.19905	−0.0405585
$875$	−2.59075	−0.0875834
$876$	−5.71221	−0.192998
$877$	−32.2716	−1.08973	−0.544867	−	0.838522i	$-0.683420\pi$
$877$	−32.2716	−1.08973	−0.544867	+	0.838522i	$0.683420\pi$
$878$	−13.6627	−0.461094
$879$	−92.3224	−3.11396
$880$	−5.25487	−0.177142
$881$	−12.6428	−0.425945	−0.212973	−	0.977058i	$-0.568315\pi$
$881$	−12.6428	−0.425945	−0.212973	+	0.977058i	$0.568315\pi$
$882$	47.2333	1.59043
$883$	−15.6762	−0.527546	−0.263773	−	0.964585i	$-0.584967\pi$
$883$	−15.6762	−0.527546	−0.263773	+	0.964585i	$0.584967\pi$
$884$	7.54338	0.253711
$885$	13.5371	0.455043
$886$	0.761260	0.0255750
$887$	−42.6791	−1.43302	−0.716512	−	0.697575i	$-0.754263\pi$
$887$	−42.6791	−1.43302	−0.716512	+	0.697575i	$0.754263\pi$
$888$	−31.8607	−1.06918
$889$	1.63633	0.0548807
$890$	18.7720	0.629239
$891$	−56.7605	−1.90155
$892$	−8.36690	−0.280144
$893$	18.6754	0.624947
$894$	−18.4250	−0.616224
$895$	9.73247	0.325321
$896$	−0.219811	−0.00734337
$897$	−1.06431	−0.0355362
$898$	−39.8287	−1.32910
$899$	0.697951	0.0232780
$900$	−17.5716	−0.585721
$901$	−4.83102	−0.160945
$902$	−35.2273	−1.17294
$903$	1.68510	0.0560765
$904$	10.9128	0.362953
$905$	14.2522	0.473758
$906$	−12.6669	−0.420830
$907$	−55.9234	−1.85691	−0.928453	−	0.371450i	$-0.878861\pi$
$907$	−55.9234	−1.85691	−0.928453	+	0.371450i	$0.878861\pi$
$908$	8.48387	0.281547
$909$	121.525	4.03072
$910$	0.424147	0.0140603
$911$	0.460704	0.0152638	0.00763191	−	0.999971i	$-0.497571\pi$
$911$	0.460704	0.0152638	0.00763191	+	0.999971i	$0.497571\pi$
$912$	13.7046	0.453803
$913$	−31.3832	−1.03863
$914$	5.39496	0.178450
$915$	47.1883	1.56000
$916$	5.27650	0.174340
$917$	1.78402	0.0589135
$918$	−72.1276	−2.38056
$919$	−11.4287	−0.376998	−0.188499	−	0.982073i	$-0.560362\pi$
$919$	−11.4287	−0.376998	−0.188499	+	0.982073i	$0.560362\pi$
$920$	0.425421	0.0140257
$921$	−4.66643	−0.153764
$922$	0.921782	0.0303573
$923$	−5.44924	−0.179364
$924$	2.32674	0.0765441
$925$	26.3280	0.865661
$926$	27.7349	0.911425
$927$	64.9018	2.13166
$928$	9.14771	0.300288
$929$	−27.4956	−0.902102	−0.451051	−	0.892498i	$-0.648951\pi$
$929$	−27.4956	−0.902102	−0.451051	+	0.892498i	$0.648951\pi$
$930$	−0.370987	−0.0121651
$931$	30.4413	0.997674
$932$	4.75489	0.155752
$933$	26.0159	0.851722
$934$	−12.2482	−0.400773
$935$	31.9166	1.04378
$936$	8.43859	0.275824
$937$	1.71085	0.0558911	0.0279455	−	0.999609i	$-0.491104\pi$
$937$	1.71085	0.0558911	0.0279455	+	0.999609i	$0.491104\pi$
$938$	−1.27601	−0.0416631
$939$	16.4007	0.535216
$940$	−6.62599	−0.216116
$941$	−59.5502	−1.94128	−0.970640	−	0.240537i	$-0.922677\pi$
$941$	−59.5502	−1.94128	−0.970640	+	0.240537i	$0.922677\pi$
$942$	34.1310	1.11205
$943$	2.85192	0.0928712
$944$	−2.78406	−0.0906133
$945$	−4.05556	−0.131927
$946$	−8.28499	−0.269368
$947$	10.7417	0.349057	0.174529	−	0.984652i	$-0.444160\pi$
$947$	10.7417	0.349057	0.174529	+	0.984652i	$0.444160\pi$
$948$	8.53227	0.277115
$949$	−2.26686	−0.0735854
$950$	−11.3247	−0.367423
$951$	1.53686	0.0498360
$952$	1.33507	0.0432698
$953$	49.8715	1.61550	0.807748	−	0.589528i	$-0.200686\pi$
$953$	49.8715	1.61550	0.807748	+	0.589528i	$0.200686\pi$
$954$	−5.40434	−0.174972
$955$	−34.4794	−1.11573
$956$	−17.2172	−0.556844
$957$	−96.8303	−3.13008
$958$	−30.3247	−0.979745
$959$	−3.85123	−0.124363
$960$	−4.86235	−0.156932
$961$	−30.9942	−0.999812
$962$	−12.6438	−0.407651
$963$	96.3098	3.10354
$964$	−7.68743	−0.247595
$965$	2.86240	0.0921441
$966$	−0.188367	−0.00606061
$967$	38.6912	1.24423	0.622113	−	0.782927i	$-0.286274\pi$
$967$	38.6912	1.24423	0.622113	+	0.782927i	$0.286274\pi$
$968$	−0.439694	−0.0141323
$969$	−83.2376	−2.67398
$970$	−15.6844	−0.503597
$971$	−0.377391	−0.0121110	−0.00605552	−	0.999982i	$-0.501928\pi$
$971$	−0.377391	−0.0121110	−0.00605552	+	0.999982i	$0.501928\pi$
$972$	−16.8946	−0.541895
$973$	−0.813958	−0.0260943
$974$	−7.93744	−0.254332
$975$	−10.0521	−0.321925
$976$	−9.70482	−0.310644
$977$	−59.4031	−1.90047	−0.950237	−	0.311527i	$-0.899160\pi$
$977$	−59.4031	−1.90047	−0.950237	+	0.311527i	$0.899160\pi$
$978$	40.0743	1.28143
$979$	40.8661	1.30609
$980$	−10.8005	−0.345010
$981$	64.0236	2.04412
$982$	35.2377	1.12448
$983$	−12.9560	−0.413234	−0.206617	−	0.978422i	$-0.566245\pi$
$983$	−12.9560	−0.413234	−0.206617	+	0.978422i	$0.566245\pi$
$984$	−32.5960	−1.03912
$985$	5.13917	0.163747
$986$	−55.5606	−1.76941
$987$	2.93384	0.0933851
$988$	5.43858	0.173024
$989$	0.670732	0.0213280
$990$	35.7043	1.13476
$991$	36.2446	1.15135	0.575673	−	0.817680i	$-0.304740\pi$
$991$	36.2446	1.15135	0.575673	+	0.817680i	$0.304740\pi$
$992$	0.0762978	0.00242246
$993$	−69.5703	−2.20775
$994$	−0.964436	−0.0305901
$995$	−6.24258	−0.197903
$996$	−29.0389	−0.920134
$997$	−28.8741	−0.914451	−0.457226	−	0.889351i	$-0.651157\pi$
$997$	−28.8741	−0.914451	−0.457226	+	0.889351i	$0.651157\pi$
$998$	16.3186	0.516557
$999$	120.896	3.82498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.2	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.2	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.12962 q^{3} +1.00000 q^{4} +1.55366 q^{5} +3.12962 q^{6} +0.219811 q^{7} -1.00000 q^{8} +6.79451 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.12962 q^{3} +1.00000 q^{4} +1.55366 q^{5} +3.12962 q^{6} +0.219811 q^{7} -1.00000 q^{8} +6.79451 q^{9} -1.55366 q^{10} -3.38226 q^{11} -3.12962 q^{12} -1.24197 q^{13} -0.219811 q^{14} -4.86235 q^{15} +1.00000 q^{16} -6.07371 q^{17} -6.79451 q^{18} -4.37899 q^{19} +1.55366 q^{20} -0.687924 q^{21} +3.38226 q^{22} -0.273819 q^{23} +3.12962 q^{24} -2.58615 q^{25} +1.24197 q^{26} -11.8754 q^{27} +0.219811 q^{28} -9.14771 q^{29} +4.86235 q^{30} -0.0762978 q^{31} -1.00000 q^{32} +10.5852 q^{33} +6.07371 q^{34} +0.341511 q^{35} +6.79451 q^{36} -10.1804 q^{37} +4.37899 q^{38} +3.88690 q^{39} -1.55366 q^{40} -10.4153 q^{41} +0.687924 q^{42} -2.44954 q^{43} -3.38226 q^{44} +10.5563 q^{45} +0.273819 q^{46} -4.26477 q^{47} -3.12962 q^{48} -6.95168 q^{49} +2.58615 q^{50} +19.0084 q^{51} -1.24197 q^{52} +0.795398 q^{53} +11.8754 q^{54} -5.25487 q^{55} -0.219811 q^{56} +13.7046 q^{57} +9.14771 q^{58} -2.78406 q^{59} -4.86235 q^{60} -9.70482 q^{61} +0.0762978 q^{62} +1.49351 q^{63} +1.00000 q^{64} -1.92960 q^{65} -10.5852 q^{66} +5.80502 q^{67} -6.07371 q^{68} +0.856950 q^{69} -0.341511 q^{70} +4.38757 q^{71} -6.79451 q^{72} +1.82521 q^{73} +10.1804 q^{74} +8.09367 q^{75} -4.37899 q^{76} -0.743458 q^{77} -3.88690 q^{78} -2.72630 q^{79} +1.55366 q^{80} +16.7818 q^{81} +10.4153 q^{82} +9.27875 q^{83} -0.687924 q^{84} -9.43647 q^{85} +2.44954 q^{86} +28.6288 q^{87} +3.38226 q^{88} -12.0825 q^{89} -10.5563 q^{90} -0.272999 q^{91} -0.273819 q^{92} +0.238783 q^{93} +4.26477 q^{94} -6.80344 q^{95} +3.12962 q^{96} +10.0952 q^{97} +6.95168 q^{98} -22.9808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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