LMFDB - Embedded newform 8042.2.a.d.1.8

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:	$101$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.03040 q^{3} +1.00000 q^{4} -4.23996 q^{5} -3.03040 q^{6} +3.85031 q^{7} +1.00000 q^{8} +6.18334 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.03040 q^{3} +1.00000 q^{4} -4.23996 q^{5} -3.03040 q^{6} +3.85031 q^{7} +1.00000 q^{8} +6.18334 q^{9} -4.23996 q^{10} -3.32099 q^{11} -3.03040 q^{12} +5.62072 q^{13} +3.85031 q^{14} +12.8488 q^{15} +1.00000 q^{16} -6.44238 q^{17} +6.18334 q^{18} +0.364283 q^{19} -4.23996 q^{20} -11.6680 q^{21} -3.32099 q^{22} +9.04507 q^{23} -3.03040 q^{24} +12.9773 q^{25} +5.62072 q^{26} -9.64682 q^{27} +3.85031 q^{28} +3.17515 q^{29} +12.8488 q^{30} +8.20528 q^{31} +1.00000 q^{32} +10.0639 q^{33} -6.44238 q^{34} -16.3252 q^{35} +6.18334 q^{36} -10.0221 q^{37} +0.364283 q^{38} -17.0330 q^{39} -4.23996 q^{40} -11.0451 q^{41} -11.6680 q^{42} +6.55246 q^{43} -3.32099 q^{44} -26.2172 q^{45} +9.04507 q^{46} +5.96705 q^{47} -3.03040 q^{48} +7.82492 q^{49} +12.9773 q^{50} +19.5230 q^{51} +5.62072 q^{52} +3.73520 q^{53} -9.64682 q^{54} +14.0809 q^{55} +3.85031 q^{56} -1.10393 q^{57} +3.17515 q^{58} -1.70542 q^{59} +12.8488 q^{60} -8.97226 q^{61} +8.20528 q^{62} +23.8078 q^{63} +1.00000 q^{64} -23.8316 q^{65} +10.0639 q^{66} -3.64697 q^{67} -6.44238 q^{68} -27.4102 q^{69} -16.3252 q^{70} +8.31462 q^{71} +6.18334 q^{72} +10.3297 q^{73} -10.0221 q^{74} -39.3264 q^{75} +0.364283 q^{76} -12.7868 q^{77} -17.0330 q^{78} -3.76618 q^{79} -4.23996 q^{80} +10.6837 q^{81} -11.0451 q^{82} -5.59841 q^{83} -11.6680 q^{84} +27.3155 q^{85} +6.55246 q^{86} -9.62198 q^{87} -3.32099 q^{88} -9.22266 q^{89} -26.2172 q^{90} +21.6415 q^{91} +9.04507 q^{92} -24.8653 q^{93} +5.96705 q^{94} -1.54455 q^{95} -3.03040 q^{96} -1.02328 q^{97} +7.82492 q^{98} -20.5348 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) $ 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9 + 19 * q^10 + 4 * q^11 + 10 * q^12 + 58 * q^13 + 42 * q^14 + 27 * q^15 + 101 * q^16 + 34 * q^17 + 147 * q^18 + 36 * q^19 + 19 * q^20 + 45 * q^21 + 4 * q^22 + 47 * q^23 + 10 * q^24 + 174 * q^25 + 58 * q^26 + 31 * q^27 + 42 * q^28 + 62 * q^29 + 27 * q^30 + 47 * q^31 + 101 * q^32 + 55 * q^33 + 34 * q^34 + 16 * q^35 + 147 * q^36 + 90 * q^37 + 36 * q^38 + 50 * q^39 + 19 * q^40 + 54 * q^41 + 45 * q^42 + 65 * q^43 + 4 * q^44 + 47 * q^45 + 47 * q^46 + 54 * q^47 + 10 * q^48 + 189 * q^49 + 174 * q^50 + 36 * q^51 + 58 * q^52 + 94 * q^53 + 31 * q^54 + 68 * q^55 + 42 * q^56 + 79 * q^57 + 62 * q^58 - 6 * q^59 + 27 * q^60 + 58 * q^61 + 47 * q^62 + 117 * q^63 + 101 * q^64 + 89 * q^65 + 55 * q^66 + 127 * q^67 + 34 * q^68 + 45 * q^69 + 16 * q^70 + 87 * q^71 + 147 * q^72 + 83 * q^73 + 90 * q^74 - 4 * q^75 + 36 * q^76 + 53 * q^77 + 50 * q^78 + 74 * q^79 + 19 * q^80 + 241 * q^81 + 54 * q^82 + 11 * q^83 + 45 * q^84 + 120 * q^85 + 65 * q^86 + 37 * q^87 + 4 * q^88 + 89 * q^89 + 47 * q^90 + 31 * q^91 + 47 * q^92 + 123 * q^93 + 54 * q^94 + 61 * q^95 + 10 * q^96 + 85 * q^97 + 189 * q^98 - 55 * 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
$3$	−3.03040	−1.74960	−0.874802	−	0.484480i	$-0.839009\pi$
$3$	−3.03040	−1.74960	−0.874802	+	0.484480i	$0.839009\pi$
$4$	1.00000	0.500000
$5$	−4.23996	−1.89617	−0.948085	−	0.318018i	$-0.896983\pi$
$5$	−4.23996	−1.89617	−0.948085	+	0.318018i	$0.896983\pi$
$6$	−3.03040	−1.23716
$7$	3.85031	1.45528	0.727641	−	0.685958i	$-0.240617\pi$
$7$	3.85031	1.45528	0.727641	+	0.685958i	$0.240617\pi$
$8$	1.00000	0.353553
$9$	6.18334	2.06111
$10$	−4.23996	−1.34079
$11$	−3.32099	−1.00132	−0.500658	−	0.865645i	$-0.666908\pi$
$11$	−3.32099	−1.00132	−0.500658	+	0.865645i	$0.666908\pi$
$12$	−3.03040	−0.874802
$13$	5.62072	1.55891	0.779453	−	0.626461i	$-0.215497\pi$
$13$	5.62072	1.55891	0.779453	+	0.626461i	$0.215497\pi$
$14$	3.85031	1.02904
$15$	12.8488	3.31755
$16$	1.00000	0.250000
$17$	−6.44238	−1.56251	−0.781253	−	0.624214i	$-0.785419\pi$
$17$	−6.44238	−1.56251	−0.781253	+	0.624214i	$0.785419\pi$
$18$	6.18334	1.45743
$19$	0.364283	0.0835723	0.0417862	−	0.999127i	$-0.486695\pi$
$19$	0.364283	0.0835723	0.0417862	+	0.999127i	$0.486695\pi$
$20$	−4.23996	−0.948085
$21$	−11.6680	−2.54617
$22$	−3.32099	−0.708037
$23$	9.04507	1.88603	0.943014	−	0.332753i	$-0.107977\pi$
$23$	9.04507	1.88603	0.943014	+	0.332753i	$0.107977\pi$
$24$	−3.03040	−0.618578
$25$	12.9773	2.59546
$26$	5.62072	1.10231
$27$	−9.64682	−1.85653
$28$	3.85031	0.727641
$29$	3.17515	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$29$	3.17515	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$30$	12.8488	2.34586
$31$	8.20528	1.47371	0.736856	−	0.676050i	$-0.236310\pi$
$31$	8.20528	1.47371	0.736856	+	0.676050i	$0.236310\pi$
$32$	1.00000	0.176777
$33$	10.0639	1.75191
$34$	−6.44238	−1.10486
$35$	−16.3252	−2.75946
$36$	6.18334	1.03056
$37$	−10.0221	−1.64762	−0.823812	−	0.566862i	$-0.808157\pi$
$37$	−10.0221	−1.64762	−0.823812	+	0.566862i	$0.808157\pi$
$38$	0.364283	0.0590945
$39$	−17.0330	−2.72747
$40$	−4.23996	−0.670397
$41$	−11.0451	−1.72496	−0.862480	−	0.506091i	$-0.831090\pi$
$41$	−11.0451	−1.72496	−0.862480	+	0.506091i	$0.831090\pi$
$42$	−11.6680	−1.80041
$43$	6.55246	0.999241	0.499621	−	0.866244i	$-0.333473\pi$
$43$	6.55246	0.999241	0.499621	+	0.866244i	$0.333473\pi$
$44$	−3.32099	−0.500658
$45$	−26.2172	−3.90822
$46$	9.04507	1.33362
$47$	5.96705	0.870383	0.435192	−	0.900338i	$-0.356681\pi$
$47$	5.96705	0.870383	0.435192	+	0.900338i	$0.356681\pi$
$48$	−3.03040	−0.437401
$49$	7.82492	1.11785
$50$	12.9773	1.83527
$51$	19.5230	2.73377
$52$	5.62072	0.779453
$53$	3.73520	0.513070	0.256535	−	0.966535i	$-0.417419\pi$
$53$	3.73520	0.513070	0.256535	+	0.966535i	$0.417419\pi$
$54$	−9.64682	−1.31277
$55$	14.0809	1.89866
$56$	3.85031	0.514520
$57$	−1.10393	−0.146218
$58$	3.17515	0.416917
$59$	−1.70542	−0.222027	−0.111013	−	0.993819i	$-0.535410\pi$
$59$	−1.70542	−0.222027	−0.111013	+	0.993819i	$0.535410\pi$
$60$	12.8488	1.65877
$61$	−8.97226	−1.14878	−0.574390	−	0.818582i	$-0.694761\pi$
$61$	−8.97226	−1.14878	−0.574390	+	0.818582i	$0.694761\pi$
$62$	8.20528	1.04207
$63$	23.8078	2.99950
$64$	1.00000	0.125000
$65$	−23.8316	−2.95595
$66$	10.0639	1.23878
$67$	−3.64697	−0.445549	−0.222774	−	0.974870i	$-0.571511\pi$
$67$	−3.64697	−0.445549	−0.222774	+	0.974870i	$0.571511\pi$
$68$	−6.44238	−0.781253
$69$	−27.4102	−3.29980
$70$	−16.3252	−1.95123
$71$	8.31462	0.986764	0.493382	−	0.869813i	$-0.335761\pi$
$71$	8.31462	0.986764	0.493382	+	0.869813i	$0.335761\pi$
$72$	6.18334	0.728714
$73$	10.3297	1.20900	0.604498	−	0.796607i	$-0.293374\pi$
$73$	10.3297	1.20900	0.604498	+	0.796607i	$0.293374\pi$
$74$	−10.0221	−1.16505
$75$	−39.3264	−4.54102
$76$	0.364283	0.0417862
$77$	−12.7868	−1.45720
$78$	−17.0330	−1.92861
$79$	−3.76618	−0.423728	−0.211864	−	0.977299i	$-0.567953\pi$
$79$	−3.76618	−0.423728	−0.211864	+	0.977299i	$0.567953\pi$
$80$	−4.23996	−0.474042
$81$	10.6837	1.18708
$82$	−11.0451	−1.21973
$83$	−5.59841	−0.614506	−0.307253	−	0.951628i	$-0.599410\pi$
$83$	−5.59841	−0.614506	−0.307253	+	0.951628i	$0.599410\pi$
$84$	−11.6680	−1.27308
$85$	27.3155	2.96278
$86$	6.55246	0.706570
$87$	−9.62198	−1.03158
$88$	−3.32099	−0.354018
$89$	−9.22266	−0.977600	−0.488800	−	0.872396i	$-0.662565\pi$
$89$	−9.22266	−0.977600	−0.488800	+	0.872396i	$0.662565\pi$
$90$	−26.2172	−2.76353
$91$	21.6415	2.26865
$92$	9.04507	0.943014
$93$	−24.8653	−2.57841
$94$	5.96705	0.615454
$95$	−1.54455	−0.158467
$96$	−3.03040	−0.309289
$97$	−1.02328	−0.103898	−0.0519491	−	0.998650i	$-0.516543\pi$
$97$	−1.02328	−0.103898	−0.0519491	+	0.998650i	$0.516543\pi$
$98$	7.82492	0.790436
$99$	−20.5348	−2.06383
$100$	12.9773	1.29773
$101$	16.1496	1.60694	0.803472	−	0.595343i	$-0.202984\pi$
$101$	16.1496	1.60694	0.803472	+	0.595343i	$0.202984\pi$
$102$	19.5230	1.93307
$103$	−12.8647	−1.26760	−0.633799	−	0.773498i	$-0.718505\pi$
$103$	−12.8647	−1.26760	−0.633799	+	0.773498i	$0.718505\pi$
$104$	5.62072	0.551157
$105$	49.4719	4.82796
$106$	3.73520	0.362795
$107$	−15.6865	−1.51647	−0.758237	−	0.651979i	$-0.773939\pi$
$107$	−15.6865	−1.51647	−0.758237	+	0.651979i	$0.773939\pi$
$108$	−9.64682	−0.928265
$109$	−4.87727	−0.467157	−0.233579	−	0.972338i	$-0.575044\pi$
$109$	−4.87727	−0.467157	−0.233579	+	0.972338i	$0.575044\pi$
$110$	14.0809	1.34256
$111$	30.3710	2.88269
$112$	3.85031	0.363820
$113$	5.81370	0.546907	0.273453	−	0.961885i	$-0.411834\pi$
$113$	5.81370	0.546907	0.273453	+	0.961885i	$0.411834\pi$
$114$	−1.10393	−0.103392
$115$	−38.3508	−3.57623
$116$	3.17515	0.294805
$117$	34.7548	3.21308
$118$	−1.70542	−0.156997
$119$	−24.8052	−2.27389
$120$	12.8488	1.17293
$121$	0.0289569	0.00263244
$122$	−8.97226	−0.812311
$123$	33.4712	3.01800
$124$	8.20528	0.736856
$125$	−33.8234	−3.02526
$126$	23.8078	2.12097
$127$	−10.1000	−0.896233	−0.448117	−	0.893975i	$-0.647905\pi$
$127$	−10.1000	−0.896233	−0.448117	+	0.893975i	$0.647905\pi$
$128$	1.00000	0.0883883
$129$	−19.8566	−1.74828
$130$	−23.8316	−2.09017
$131$	2.18668	0.191051	0.0955254	−	0.995427i	$-0.469547\pi$
$131$	2.18668	0.191051	0.0955254	+	0.995427i	$0.469547\pi$
$132$	10.0639	0.875953
$133$	1.40260	0.121621
$134$	−3.64697	−0.315050
$135$	40.9022	3.52030
$136$	−6.44238	−0.552430
$137$	−2.58867	−0.221165	−0.110582	−	0.993867i	$-0.535272\pi$
$137$	−2.58867	−0.221165	−0.110582	+	0.993867i	$0.535272\pi$
$138$	−27.4102	−2.33331
$139$	15.2704	1.29522	0.647608	−	0.761974i	$-0.275770\pi$
$139$	15.2704	1.29522	0.647608	+	0.761974i	$0.275770\pi$
$140$	−16.3252	−1.37973
$141$	−18.0826	−1.52283
$142$	8.31462	0.697747
$143$	−18.6663	−1.56096
$144$	6.18334	0.515279
$145$	−13.4625	−1.11800
$146$	10.3297	0.854889
$147$	−23.7127	−1.95579
$148$	−10.0221	−0.823812
$149$	7.73549	0.633716	0.316858	−	0.948473i	$-0.397372\pi$
$149$	7.73549	0.633716	0.316858	+	0.948473i	$0.397372\pi$
$150$	−39.3264	−3.21099
$151$	22.7270	1.84949	0.924747	−	0.380583i	$-0.124277\pi$
$151$	22.7270	1.84949	0.924747	+	0.380583i	$0.124277\pi$
$152$	0.364283	0.0295473
$153$	−39.8355	−3.22051
$154$	−12.7868	−1.03039
$155$	−34.7901	−2.79441
$156$	−17.0330	−1.36373
$157$	−22.2835	−1.77842	−0.889209	−	0.457501i	$-0.848745\pi$
$157$	−22.2835	−1.77842	−0.889209	+	0.457501i	$0.848745\pi$
$158$	−3.76618	−0.299621
$159$	−11.3192	−0.897669
$160$	−4.23996	−0.335199
$161$	34.8264	2.74470
$162$	10.6837	0.839392
$163$	−3.20310	−0.250886	−0.125443	−	0.992101i	$-0.540035\pi$
$163$	−3.20310	−0.250886	−0.125443	+	0.992101i	$0.540035\pi$
$164$	−11.0451	−0.862480
$165$	−42.6707	−3.32191
$166$	−5.59841	−0.434521
$167$	−18.0757	−1.39874	−0.699371	−	0.714759i	$-0.746536\pi$
$167$	−18.0757	−1.39874	−0.699371	+	0.714759i	$0.746536\pi$
$168$	−11.6680	−0.900206
$169$	18.5924	1.43019
$170$	27.3155	2.09500
$171$	2.25249	0.172252
$172$	6.55246	0.499621
$173$	4.81958	0.366426	0.183213	−	0.983073i	$-0.441350\pi$
$173$	4.81958	0.366426	0.183213	+	0.983073i	$0.441350\pi$
$174$	−9.62198	−0.729441
$175$	49.9667	3.77712
$176$	−3.32099	−0.250329
$177$	5.16811	0.388459
$178$	−9.22266	−0.691268
$179$	2.01141	0.150340	0.0751701	−	0.997171i	$-0.476050\pi$
$179$	2.01141	0.150340	0.0751701	+	0.997171i	$0.476050\pi$
$180$	−26.2172	−1.95411
$181$	12.2798	0.912749	0.456374	−	0.889788i	$-0.349148\pi$
$181$	12.2798	0.912749	0.456374	+	0.889788i	$0.349148\pi$
$182$	21.6415	1.60418
$183$	27.1896	2.00991
$184$	9.04507	0.666812
$185$	42.4934	3.12418
$186$	−24.8653	−1.82321
$187$	21.3951	1.56456
$188$	5.96705	0.435192
$189$	−37.1433	−2.70178
$190$	−1.54455	−0.112053
$191$	12.3086	0.890621	0.445310	−	0.895376i	$-0.353093\pi$
$191$	12.3086	0.890621	0.445310	+	0.895376i	$0.353093\pi$
$192$	−3.03040	−0.218701
$193$	8.39869	0.604551	0.302276	−	0.953221i	$-0.402254\pi$
$193$	8.39869	0.604551	0.302276	+	0.953221i	$0.402254\pi$
$194$	−1.02328	−0.0734671
$195$	72.2195	5.17174
$196$	7.82492	0.558923
$197$	13.9163	0.991493	0.495746	−	0.868467i	$-0.334895\pi$
$197$	13.9163	0.991493	0.495746	+	0.868467i	$0.334895\pi$
$198$	−20.5348	−1.45935
$199$	3.31855	0.235245	0.117623	−	0.993058i	$-0.462473\pi$
$199$	3.31855	0.235245	0.117623	+	0.993058i	$0.462473\pi$
$200$	12.9773	0.917633
$201$	11.0518	0.779534
$202$	16.1496	1.13628
$203$	12.2253	0.858049
$204$	19.5230	1.36688
$205$	46.8310	3.27082
$206$	−12.8647	−0.896327
$207$	55.9288	3.88732
$208$	5.62072	0.389727
$209$	−1.20978	−0.0836822
$210$	49.4719	3.41389
$211$	−1.82272	−0.125481	−0.0627407	−	0.998030i	$-0.519984\pi$
$211$	−1.82272	−0.125481	−0.0627407	+	0.998030i	$0.519984\pi$
$212$	3.73520	0.256535
$213$	−25.1966	−1.72645
$214$	−15.6865	−1.07231
$215$	−27.7822	−1.89473
$216$	−9.64682	−0.656383
$217$	31.5929	2.14467
$218$	−4.87727	−0.330330
$219$	−31.3030	−2.11526
$220$	14.0809	0.949332
$221$	−36.2108	−2.43580
$222$	30.3710	2.03837
$223$	24.2282	1.62244	0.811221	−	0.584739i	$-0.198803\pi$
$223$	24.2282	1.62244	0.811221	+	0.584739i	$0.198803\pi$
$224$	3.85031	0.257260
$225$	80.2431	5.34954
$226$	5.81370	0.386721
$227$	7.73990	0.513715	0.256858	−	0.966449i	$-0.417313\pi$
$227$	7.73990	0.513715	0.256858	+	0.966449i	$0.417313\pi$
$228$	−1.10393	−0.0731092
$229$	−0.960006	−0.0634390	−0.0317195	−	0.999497i	$-0.510098\pi$
$229$	−0.960006	−0.0634390	−0.0317195	+	0.999497i	$0.510098\pi$
$230$	−38.3508	−2.52878
$231$	38.7493	2.54952
$232$	3.17515	0.208459
$233$	5.98362	0.392000	0.196000	−	0.980604i	$-0.437205\pi$
$233$	5.98362	0.392000	0.196000	+	0.980604i	$0.437205\pi$
$234$	34.7548	2.27199
$235$	−25.3001	−1.65039
$236$	−1.70542	−0.111013
$237$	11.4130	0.741356
$238$	−24.8052	−1.60788
$239$	−12.3127	−0.796445	−0.398223	−	0.917289i	$-0.630373\pi$
$239$	−12.3127	−0.796445	−0.398223	+	0.917289i	$0.630373\pi$
$240$	12.8488	0.829386
$241$	−6.64674	−0.428154	−0.214077	−	0.976817i	$-0.568674\pi$
$241$	−6.64674	−0.428154	−0.214077	+	0.976817i	$0.568674\pi$
$242$	0.0289569	0.00186142
$243$	−3.43551	−0.220388
$244$	−8.97226	−0.574390
$245$	−33.1774	−2.11962
$246$	33.4712	2.13405
$247$	2.04753	0.130281
$248$	8.20528	0.521036
$249$	16.9655	1.07514
$250$	−33.8234	−2.13918
$251$	18.8853	1.19203	0.596015	−	0.802974i	$-0.296750\pi$
$251$	18.8853	1.19203	0.596015	+	0.802974i	$0.296750\pi$
$252$	23.8078	1.49975
$253$	−30.0386	−1.88851
$254$	−10.1000	−0.633733
$255$	−82.7769	−5.18369
$256$	1.00000	0.0625000
$257$	−11.6132	−0.724412	−0.362206	−	0.932098i	$-0.617976\pi$
$257$	−11.6132	−0.724412	−0.362206	+	0.932098i	$0.617976\pi$
$258$	−19.8566	−1.23622
$259$	−38.5883	−2.39776
$260$	−23.8316	−1.47798
$261$	19.6330	1.21525
$262$	2.18668	0.135093
$263$	16.5196	1.01864	0.509322	−	0.860576i	$-0.329896\pi$
$263$	16.5196	1.01864	0.509322	+	0.860576i	$0.329896\pi$
$264$	10.0639	0.619392
$265$	−15.8371	−0.972867
$266$	1.40260	0.0859992
$267$	27.9484	1.71041
$268$	−3.64697	−0.222774
$269$	12.9437	0.789191	0.394595	−	0.918855i	$-0.370885\pi$
$269$	12.9437	0.789191	0.394595	+	0.918855i	$0.370885\pi$
$270$	40.9022	2.48923
$271$	−14.9213	−0.906405	−0.453203	−	0.891408i	$-0.649719\pi$
$271$	−14.9213	−0.906405	−0.453203	+	0.891408i	$0.649719\pi$
$272$	−6.44238	−0.390627
$273$	−65.5825	−3.96924
$274$	−2.58867	−0.156387
$275$	−43.0974	−2.59887
$276$	−27.4102	−1.64990
$277$	15.2701	0.917489	0.458745	−	0.888568i	$-0.348299\pi$
$277$	15.2701	0.917489	0.458745	+	0.888568i	$0.348299\pi$
$278$	15.2704	0.915856
$279$	50.7361	3.03749
$280$	−16.3252	−0.975617
$281$	18.2265	1.08730	0.543649	−	0.839312i	$-0.317042\pi$
$281$	18.2265	1.08730	0.543649	+	0.839312i	$0.317042\pi$
$282$	−18.0826	−1.07680
$283$	−0.912383	−0.0542355	−0.0271178	−	0.999632i	$-0.508633\pi$
$283$	−0.912383	−0.0542355	−0.0271178	+	0.999632i	$0.508633\pi$
$284$	8.31462	0.493382
$285$	4.68060	0.277255
$286$	−18.6663	−1.10376
$287$	−42.5272	−2.51030
$288$	6.18334	0.364357
$289$	24.5043	1.44143
$290$	−13.4625	−0.790546
$291$	3.10095	0.181781
$292$	10.3297	0.604498
$293$	26.4531	1.54541	0.772703	−	0.634768i	$-0.218904\pi$
$293$	26.4531	1.54541	0.772703	+	0.634768i	$0.218904\pi$
$294$	−23.7127	−1.38295
$295$	7.23092	0.421001
$296$	−10.0221	−0.582523
$297$	32.0370	1.85897
$298$	7.73549	0.448105
$299$	50.8398	2.94014
$300$	−39.3264	−2.27051
$301$	25.2290	1.45418
$302$	22.7270	1.30779
$303$	−48.9398	−2.81152
$304$	0.364283	0.0208931
$305$	38.0421	2.17828
$306$	−39.8355	−2.27724
$307$	−16.7402	−0.955413	−0.477707	−	0.878519i	$-0.658532\pi$
$307$	−16.7402	−0.955413	−0.477707	+	0.878519i	$0.658532\pi$
$308$	−12.7868	−0.728598
$309$	38.9853	2.21779
$310$	−34.7901	−1.97594
$311$	−3.83133	−0.217255	−0.108628	−	0.994083i	$-0.534646\pi$
$311$	−3.83133	−0.217255	−0.108628	+	0.994083i	$0.534646\pi$
$312$	−17.0330	−0.964306
$313$	−4.98696	−0.281880	−0.140940	−	0.990018i	$-0.545012\pi$
$313$	−4.98696	−0.281880	−0.140940	+	0.990018i	$0.545012\pi$
$314$	−22.2835	−1.25753
$315$	−100.944	−5.68757
$316$	−3.76618	−0.211864
$317$	−1.43031	−0.0803342	−0.0401671	−	0.999193i	$-0.512789\pi$
$317$	−1.43031	−0.0803342	−0.0401671	+	0.999193i	$0.512789\pi$
$318$	−11.3192	−0.634748
$319$	−10.5446	−0.590386
$320$	−4.23996	−0.237021
$321$	47.5365	2.65323
$322$	34.8264	1.94080
$323$	−2.34685	−0.130582
$324$	10.6837	0.593540
$325$	72.9417	4.04608
$326$	−3.20310	−0.177404
$327$	14.7801	0.817341
$328$	−11.0451	−0.609866
$329$	22.9750	1.26665
$330$	−42.6707	−2.34894
$331$	23.3803	1.28510	0.642550	−	0.766244i	$-0.277877\pi$
$331$	23.3803	1.28510	0.642550	+	0.766244i	$0.277877\pi$
$332$	−5.59841	−0.307253
$333$	−61.9702	−3.39594
$334$	−18.0757	−0.989060
$335$	15.4630	0.844836
$336$	−11.6680	−0.636542
$337$	4.90879	0.267399	0.133699	−	0.991022i	$-0.457314\pi$
$337$	4.90879	0.267399	0.133699	+	0.991022i	$0.457314\pi$
$338$	18.5924	1.01130
$339$	−17.6179	−0.956870
$340$	27.3155	1.48139
$341$	−27.2496	−1.47565
$342$	2.25249	0.121801
$343$	3.17620	0.171498
$344$	6.55246	0.353285
$345$	116.218	6.25699
$346$	4.81958	0.259102
$347$	−28.7056	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$347$	−28.7056	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$348$	−9.62198	−0.515792
$349$	−8.86238	−0.474392	−0.237196	−	0.971462i	$-0.576228\pi$
$349$	−8.86238	−0.474392	−0.237196	+	0.971462i	$0.576228\pi$
$350$	49.9667	2.67083
$351$	−54.2220	−2.89416
$352$	−3.32099	−0.177009
$353$	−5.81121	−0.309300	−0.154650	−	0.987969i	$-0.549425\pi$
$353$	−5.81121	−0.309300	−0.154650	+	0.987969i	$0.549425\pi$
$354$	5.16811	0.274682
$355$	−35.2537	−1.87107
$356$	−9.22266	−0.488800
$357$	75.1697	3.97840
$358$	2.01141	0.106307
$359$	−1.78513	−0.0942153	−0.0471077	−	0.998890i	$-0.515000\pi$
$359$	−1.78513	−0.0942153	−0.0471077	+	0.998890i	$0.515000\pi$
$360$	−26.2172	−1.38177
$361$	−18.8673	−0.993016
$362$	12.2798	0.645411
$363$	−0.0877510	−0.00460573
$364$	21.6415	1.13432
$365$	−43.7974	−2.29246
$366$	27.1896	1.42122
$367$	−4.09505	−0.213760	−0.106880	−	0.994272i	$-0.534086\pi$
$367$	−4.09505	−0.213760	−0.106880	+	0.994272i	$0.534086\pi$
$368$	9.04507	0.471507
$369$	−68.2959	−3.55534
$370$	42.4934	2.20913
$371$	14.3817	0.746661
$372$	−24.8653	−1.28921
$373$	−9.05086	−0.468636	−0.234318	−	0.972160i	$-0.575286\pi$
$373$	−9.05086	−0.468636	−0.234318	+	0.972160i	$0.575286\pi$
$374$	21.3951	1.10631
$375$	102.499	5.29301
$376$	5.96705	0.307727
$377$	17.8466	0.919147
$378$	−37.1433	−1.91044
$379$	27.8350	1.42979	0.714893	−	0.699234i	$-0.246475\pi$
$379$	27.8350	1.42979	0.714893	+	0.699234i	$0.246475\pi$
$380$	−1.54455	−0.0792336
$381$	30.6072	1.56805
$382$	12.3086	0.629764
$383$	31.7484	1.62227	0.811135	−	0.584859i	$-0.198850\pi$
$383$	31.7484	1.62227	0.811135	+	0.584859i	$0.198850\pi$
$384$	−3.03040	−0.154645
$385$	54.2158	2.76309
$386$	8.39869	0.427482
$387$	40.5161	2.05955
$388$	−1.02328	−0.0519491
$389$	21.3369	1.08182	0.540911	−	0.841080i	$-0.318080\pi$
$389$	21.3369	1.08182	0.540911	+	0.841080i	$0.318080\pi$
$390$	72.2195	3.65697
$391$	−58.2718	−2.94693
$392$	7.82492	0.395218
$393$	−6.62651	−0.334263
$394$	13.9163	0.701091
$395$	15.9685	0.803460
$396$	−20.5348	−1.03191
$397$	−7.14884	−0.358790	−0.179395	−	0.983777i	$-0.557414\pi$
$397$	−7.14884	−0.358790	−0.179395	+	0.983777i	$0.557414\pi$
$398$	3.31855	0.166344
$399$	−4.25046	−0.212789
$400$	12.9773	0.648865
$401$	−31.7757	−1.58680	−0.793402	−	0.608698i	$-0.791692\pi$
$401$	−31.7757	−1.58680	−0.793402	+	0.608698i	$0.791692\pi$
$402$	11.0518	0.551214
$403$	46.1195	2.29738
$404$	16.1496	0.803472
$405$	−45.2986	−2.25090
$406$	12.2253	0.606732
$407$	33.2833	1.64979
$408$	19.5230	0.966533
$409$	2.69844	0.133429	0.0667146	−	0.997772i	$-0.478748\pi$
$409$	2.69844	0.133429	0.0667146	+	0.997772i	$0.478748\pi$
$410$	46.8310	2.31282
$411$	7.84471	0.386951
$412$	−12.8647	−0.633799
$413$	−6.56641	−0.323112
$414$	55.9288	2.74875
$415$	23.7371	1.16521
$416$	5.62072	0.275578
$417$	−46.2754	−2.26611
$418$	−1.20978	−0.0591723
$419$	5.26920	0.257417	0.128709	−	0.991682i	$-0.458917\pi$
$419$	5.26920	0.257417	0.128709	+	0.991682i	$0.458917\pi$
$420$	49.4719	2.41398
$421$	31.7474	1.54727	0.773636	−	0.633630i	$-0.218436\pi$
$421$	31.7474	1.54727	0.773636	+	0.633630i	$0.218436\pi$
$422$	−1.82272	−0.0887288
$423$	36.8963	1.79396
$424$	3.73520	0.181398
$425$	−83.6047	−4.05542
$426$	−25.1966	−1.22078
$427$	−34.5460	−1.67180
$428$	−15.6865	−0.758237
$429$	56.5665	2.73106
$430$	−27.7822	−1.33978
$431$	0.425619	0.0205013	0.0102507	−	0.999947i	$-0.496737\pi$
$431$	0.425619	0.0205013	0.0102507	+	0.999947i	$0.496737\pi$
$432$	−9.64682	−0.464133
$433$	23.9771	1.15227	0.576134	−	0.817355i	$-0.304561\pi$
$433$	23.9771	1.15227	0.576134	+	0.817355i	$0.304561\pi$
$434$	31.5929	1.51651
$435$	40.7969	1.95606
$436$	−4.87727	−0.233579
$437$	3.29497	0.157620
$438$	−31.3030	−1.49572
$439$	−5.27599	−0.251809	−0.125905	−	0.992042i	$-0.540183\pi$
$439$	−5.27599	−0.251809	−0.125905	+	0.992042i	$0.540183\pi$
$440$	14.0809	0.671279
$441$	48.3842	2.30401
$442$	−36.2108	−1.72237
$443$	26.0537	1.23785	0.618925	−	0.785450i	$-0.287568\pi$
$443$	26.0537	1.23785	0.618925	+	0.785450i	$0.287568\pi$
$444$	30.3710	1.44135
$445$	39.1038	1.85370
$446$	24.2282	1.14724
$447$	−23.4416	−1.10875
$448$	3.85031	0.181910
$449$	3.82611	0.180565	0.0902827	−	0.995916i	$-0.471223\pi$
$449$	3.82611	0.180565	0.0902827	+	0.995916i	$0.471223\pi$
$450$	80.2431	3.78269
$451$	36.6808	1.72723
$452$	5.81370	0.273453
$453$	−68.8718	−3.23588
$454$	7.73990	0.363251
$455$	−91.7593	−4.30174
$456$	−1.10393	−0.0516960
$457$	36.2399	1.69523	0.847616	−	0.530610i	$-0.178037\pi$
$457$	36.2399	1.69523	0.847616	+	0.530610i	$0.178037\pi$
$458$	−0.960006	−0.0448582
$459$	62.1485	2.90084
$460$	−38.3508	−1.78811
$461$	−29.6960	−1.38308	−0.691540	−	0.722338i	$-0.743067\pi$
$461$	−29.6960	−1.38308	−0.691540	+	0.722338i	$0.743067\pi$
$462$	38.7493	1.80278
$463$	24.1300	1.12142	0.560709	−	0.828013i	$-0.310529\pi$
$463$	24.1300	1.12142	0.560709	+	0.828013i	$0.310529\pi$
$464$	3.17515	0.147403
$465$	105.428	4.88911
$466$	5.98362	0.277186
$467$	7.62506	0.352846	0.176423	−	0.984314i	$-0.443547\pi$
$467$	7.62506	0.352846	0.176423	+	0.984314i	$0.443547\pi$
$468$	34.7548	1.60654
$469$	−14.0420	−0.648399
$470$	−25.3001	−1.16700
$471$	67.5280	3.11153
$472$	−1.70542	−0.0784984
$473$	−21.7606	−1.00056
$474$	11.4130	0.524218
$475$	4.72741	0.216908
$476$	−24.8052	−1.13694
$477$	23.0961	1.05750
$478$	−12.3127	−0.563172
$479$	−1.16234	−0.0531087	−0.0265544	−	0.999647i	$-0.508454\pi$
$479$	−1.16234	−0.0531087	−0.0265544	+	0.999647i	$0.508454\pi$
$480$	12.8488	0.586465
$481$	−56.3314	−2.56849
$482$	−6.64674	−0.302751
$483$	−105.538	−4.80214
$484$	0.0289569	0.00131622
$485$	4.33866	0.197008
$486$	−3.43551	−0.155838
$487$	28.2944	1.28214	0.641070	−	0.767482i	$-0.278491\pi$
$487$	28.2944	1.28214	0.641070	+	0.767482i	$0.278491\pi$
$488$	−8.97226	−0.406155
$489$	9.70670	0.438952
$490$	−33.1774	−1.49880
$491$	−21.0266	−0.948917	−0.474458	−	0.880278i	$-0.657356\pi$
$491$	−21.0266	−0.948917	−0.474458	+	0.880278i	$0.657356\pi$
$492$	33.4712	1.50900
$493$	−20.4555	−0.921270
$494$	2.04753	0.0921228
$495$	87.0668	3.91336
$496$	8.20528	0.368428
$497$	32.0139	1.43602
$498$	16.9655	0.760240
$499$	−0.844870	−0.0378216	−0.0189108	−	0.999821i	$-0.506020\pi$
$499$	−0.844870	−0.0378216	−0.0189108	+	0.999821i	$0.506020\pi$
$500$	−33.8234	−1.51263
$501$	54.7768	2.44725
$502$	18.8853	0.842892
$503$	−9.24615	−0.412265	−0.206133	−	0.978524i	$-0.566088\pi$
$503$	−9.24615	−0.412265	−0.206133	+	0.978524i	$0.566088\pi$
$504$	23.8078	1.06048
$505$	−68.4737	−3.04704
$506$	−30.0386	−1.33538
$507$	−56.3426	−2.50226
$508$	−10.1000	−0.448117
$509$	4.95748	0.219736	0.109868	−	0.993946i	$-0.464957\pi$
$509$	4.95748	0.219736	0.109868	+	0.993946i	$0.464957\pi$
$510$	−82.7769	−3.66542
$511$	39.7724	1.75943
$512$	1.00000	0.0441942
$513$	−3.51417	−0.155155
$514$	−11.6132	−0.512237
$515$	54.5459	2.40358
$516$	−19.8566	−0.874138
$517$	−19.8165	−0.871528
$518$	−38.5883	−1.69547
$519$	−14.6053	−0.641100
$520$	−23.8316	−1.04509
$521$	39.2530	1.71971	0.859853	−	0.510542i	$-0.170555\pi$
$521$	39.2530	1.71971	0.859853	+	0.510542i	$0.170555\pi$
$522$	19.6330	0.859315
$523$	7.82453	0.342143	0.171071	−	0.985259i	$-0.445277\pi$
$523$	7.82453	0.342143	0.171071	+	0.985259i	$0.445277\pi$
$524$	2.18668	0.0955254
$525$	−151.419	−6.60847
$526$	16.5196	0.720289
$527$	−52.8615	−2.30268
$528$	10.0639	0.437976
$529$	58.8134	2.55710
$530$	−15.8371	−0.687921
$531$	−10.5452	−0.457623
$532$	1.40260	0.0608106
$533$	−62.0816	−2.68905
$534$	27.9484	1.20944
$535$	66.5103	2.87549
$536$	−3.64697	−0.157525
$537$	−6.09540	−0.263036
$538$	12.9437	0.558042
$539$	−25.9865	−1.11932
$540$	40.9022	1.76015
$541$	3.25282	0.139850	0.0699248	−	0.997552i	$-0.477724\pi$
$541$	3.25282	0.139850	0.0699248	+	0.997552i	$0.477724\pi$
$542$	−14.9213	−0.640925
$543$	−37.2127	−1.59695
$544$	−6.44238	−0.276215
$545$	20.6794	0.885810
$546$	−65.5825	−2.80667
$547$	4.29841	0.183787	0.0918934	−	0.995769i	$-0.470708\pi$
$547$	4.29841	0.183787	0.0918934	+	0.995769i	$0.470708\pi$
$548$	−2.58867	−0.110582
$549$	−55.4786	−2.36777
$550$	−43.0974	−1.83768
$551$	1.15665	0.0492751
$552$	−27.4102	−1.16666
$553$	−14.5010	−0.616644
$554$	15.2701	0.648763
$555$	−128.772	−5.46607
$556$	15.2704	0.647608
$557$	−7.82462	−0.331540	−0.165770	−	0.986164i	$-0.553011\pi$
$557$	−7.82462	−0.331540	−0.165770	+	0.986164i	$0.553011\pi$
$558$	50.7361	2.14783
$559$	36.8295	1.55772
$560$	−16.3252	−0.689865
$561$	−64.8357	−2.73736
$562$	18.2265	0.768836
$563$	−4.83852	−0.203919	−0.101960	−	0.994789i	$-0.532511\pi$
$563$	−4.83852	−0.203919	−0.101960	+	0.994789i	$0.532511\pi$
$564$	−18.0826	−0.761413
$565$	−24.6499	−1.03703
$566$	−0.912383	−0.0383503
$567$	41.1357	1.72753
$568$	8.31462	0.348874
$569$	9.14247	0.383272	0.191636	−	0.981466i	$-0.438621\pi$
$569$	9.14247	0.383272	0.191636	+	0.981466i	$0.438621\pi$
$570$	4.68060	0.196049
$571$	−6.97744	−0.291997	−0.145998	−	0.989285i	$-0.546639\pi$
$571$	−6.97744	−0.291997	−0.145998	+	0.989285i	$0.546639\pi$
$572$	−18.6663	−0.780478
$573$	−37.3001	−1.55823
$574$	−42.5272	−1.77505
$575$	117.381	4.89511
$576$	6.18334	0.257639
$577$	6.03312	0.251162	0.125581	−	0.992083i	$-0.459920\pi$
$577$	6.03312	0.251162	0.125581	+	0.992083i	$0.459920\pi$
$578$	24.5043	1.01924
$579$	−25.4514	−1.05773
$580$	−13.4625	−0.559001
$581$	−21.5557	−0.894279
$582$	3.10095	0.128538
$583$	−12.4046	−0.513745
$584$	10.3297	0.427444
$585$	−147.359	−6.09255
$586$	26.4531	1.09277
$587$	7.77281	0.320818	0.160409	−	0.987051i	$-0.448719\pi$
$587$	7.77281	0.320818	0.160409	+	0.987051i	$0.448719\pi$
$588$	−23.7127	−0.977894
$589$	2.98905	0.123161
$590$	7.23092	0.297692
$591$	−42.1719	−1.73472
$592$	−10.0221	−0.411906
$593$	27.8470	1.14354	0.571769	−	0.820415i	$-0.306257\pi$
$593$	27.8470	1.14354	0.571769	+	0.820415i	$0.306257\pi$
$594$	32.0370	1.31449
$595$	105.173	4.31168
$596$	7.73549	0.316858
$597$	−10.0565	−0.411587
$598$	50.8398	2.07899
$599$	−0.346379	−0.0141527	−0.00707633	−	0.999975i	$-0.502252\pi$
$599$	−0.346379	−0.0141527	−0.00707633	+	0.999975i	$0.502252\pi$
$600$	−39.3264	−1.60549
$601$	−44.0206	−1.79564	−0.897818	−	0.440367i	$-0.854848\pi$
$601$	−44.0206	−1.79564	−0.897818	+	0.440367i	$0.854848\pi$
$602$	25.2290	1.02826
$603$	−22.5505	−0.918327
$604$	22.7270	0.924747
$605$	−0.122776	−0.00499156
$606$	−48.9398	−1.98804
$607$	1.84255	0.0747867	0.0373933	−	0.999301i	$-0.488095\pi$
$607$	1.84255	0.0747867	0.0373933	+	0.999301i	$0.488095\pi$
$608$	0.364283	0.0147736
$609$	−37.0476	−1.50125
$610$	38.0421	1.54028
$611$	33.5391	1.35685
$612$	−39.8355	−1.61025
$613$	36.4701	1.47301	0.736506	−	0.676431i	$-0.236474\pi$
$613$	36.4701	1.47301	0.736506	+	0.676431i	$0.236474\pi$
$614$	−16.7402	−0.675579
$615$	−141.917	−5.72264
$616$	−12.7868	−0.515197
$617$	−21.1468	−0.851338	−0.425669	−	0.904879i	$-0.639961\pi$
$617$	−21.1468	−0.851338	−0.425669	+	0.904879i	$0.639961\pi$
$618$	38.9853	1.56822
$619$	6.94026	0.278952	0.139476	−	0.990225i	$-0.455458\pi$
$619$	6.94026	0.278952	0.139476	+	0.990225i	$0.455458\pi$
$620$	−34.7901	−1.39720
$621$	−87.2562	−3.50147
$622$	−3.83133	−0.153623
$623$	−35.5101	−1.42268
$624$	−17.0330	−0.681867
$625$	78.5237	3.14095
$626$	−4.98696	−0.199319
$627$	3.66612	0.146411
$628$	−22.2835	−0.889209
$629$	64.5663	2.57443
$630$	−100.944	−4.02172
$631$	8.13644	0.323906	0.161953	−	0.986798i	$-0.448221\pi$
$631$	8.13644	0.323906	0.161953	+	0.986798i	$0.448221\pi$
$632$	−3.76618	−0.149811
$633$	5.52359	0.219543
$634$	−1.43031	−0.0568048
$635$	42.8238	1.69941
$636$	−11.3192	−0.448835
$637$	43.9816	1.74262
$638$	−10.5446	−0.417466
$639$	51.4121	2.03383
$640$	−4.23996	−0.167599
$641$	45.1380	1.78284	0.891422	−	0.453174i	$-0.149708\pi$
$641$	45.1380	1.78284	0.891422	+	0.453174i	$0.149708\pi$
$642$	47.5365	1.87612
$643$	−1.62108	−0.0639290	−0.0319645	−	0.999489i	$-0.510176\pi$
$643$	−1.62108	−0.0639290	−0.0319645	+	0.999489i	$0.510176\pi$
$644$	34.8264	1.37235
$645$	84.1913	3.31503
$646$	−2.34685	−0.0923356
$647$	−3.16529	−0.124440	−0.0622202	−	0.998062i	$-0.519818\pi$
$647$	−3.16529	−0.124440	−0.0622202	+	0.998062i	$0.519818\pi$
$648$	10.6837	0.419696
$649$	5.66368	0.222319
$650$	72.9417	2.86101
$651$	−95.7392	−3.75232
$652$	−3.20310	−0.125443
$653$	−36.0235	−1.40971	−0.704855	−	0.709352i	$-0.748988\pi$
$653$	−36.0235	−1.40971	−0.704855	+	0.709352i	$0.748988\pi$
$654$	14.7801	0.577947
$655$	−9.27143	−0.362265
$656$	−11.0451	−0.431240
$657$	63.8718	2.49188
$658$	22.9750	0.895659
$659$	−23.4383	−0.913025	−0.456513	−	0.889717i	$-0.650902\pi$
$659$	−23.4383	−0.913025	−0.456513	+	0.889717i	$0.650902\pi$
$660$	−42.6707	−1.66095
$661$	−1.64881	−0.0641312	−0.0320656	−	0.999486i	$-0.510209\pi$
$661$	−1.64881	−0.0641312	−0.0320656	+	0.999486i	$0.510209\pi$
$662$	23.3803	0.908703
$663$	109.733	4.26169
$664$	−5.59841	−0.217261
$665$	−5.94699	−0.230615
$666$	−61.9702	−2.40130
$667$	28.7195	1.11202
$668$	−18.0757	−0.699371
$669$	−73.4213	−2.83863
$670$	15.4630	0.597389
$671$	29.7968	1.15029
$672$	−11.6680	−0.450103
$673$	6.87679	0.265081	0.132540	−	0.991178i	$-0.457687\pi$
$673$	6.87679	0.265081	0.132540	+	0.991178i	$0.457687\pi$
$674$	4.90879	0.189080
$675$	−125.190	−4.81855
$676$	18.5924	0.715094
$677$	−33.7222	−1.29605	−0.648024	−	0.761620i	$-0.724405\pi$
$677$	−33.7222	−1.29605	−0.648024	+	0.761620i	$0.724405\pi$
$678$	−17.6179	−0.676609
$679$	−3.93994	−0.151201
$680$	27.3155	1.04750
$681$	−23.4550	−0.898798
$682$	−27.2496	−1.04344
$683$	−35.1649	−1.34555	−0.672774	−	0.739848i	$-0.734897\pi$
$683$	−35.1649	−1.34555	−0.672774	+	0.739848i	$0.734897\pi$
$684$	2.25249	0.0861261
$685$	10.9759	0.419366
$686$	3.17620	0.121268
$687$	2.90921	0.110993
$688$	6.55246	0.249810
$689$	20.9945	0.799828
$690$	116.218	4.42436
$691$	−24.5064	−0.932268	−0.466134	−	0.884714i	$-0.654353\pi$
$691$	−24.5064	−0.932268	−0.466134	+	0.884714i	$0.654353\pi$
$692$	4.81958	0.183213
$693$	−79.0655	−3.00345
$694$	−28.7056	−1.08965
$695$	−64.7458	−2.45595
$696$	−9.62198	−0.364720
$697$	71.1570	2.69526
$698$	−8.86238	−0.335446
$699$	−18.1328	−0.685845
$700$	49.9667	1.88856
$701$	30.1346	1.13817	0.569084	−	0.822279i	$-0.307298\pi$
$701$	30.1346	1.13817	0.569084	+	0.822279i	$0.307298\pi$
$702$	−54.2220	−2.04648
$703$	−3.65089	−0.137696
$704$	−3.32099	−0.125164
$705$	76.6694	2.88754
$706$	−5.81121	−0.218708
$707$	62.1810	2.33856
$708$	5.16811	0.194230
$709$	−16.3715	−0.614846	−0.307423	−	0.951573i	$-0.599467\pi$
$709$	−16.3715	−0.614846	−0.307423	+	0.951573i	$0.599467\pi$
$710$	−35.2537	−1.32305
$711$	−23.2876	−0.873352
$712$	−9.22266	−0.345634
$713$	74.2174	2.77946
$714$	75.1697	2.81316
$715$	79.1445	2.95984
$716$	2.01141	0.0751701
$717$	37.3126	1.39346
$718$	−1.78513	−0.0666203
$719$	−39.4880	−1.47265	−0.736327	−	0.676626i	$-0.763441\pi$
$719$	−39.4880	−1.47265	−0.736327	+	0.676626i	$0.763441\pi$
$720$	−26.2172	−0.977056
$721$	−49.5332	−1.84471
$722$	−18.8673	−0.702168
$723$	20.1423	0.749100
$724$	12.2798	0.456374
$725$	41.2048	1.53031
$726$	−0.0877510	−0.00325675
$727$	26.3112	0.975829	0.487914	−	0.872891i	$-0.337758\pi$
$727$	26.3112	0.975829	0.487914	+	0.872891i	$0.337758\pi$
$728$	21.6415	0.802088
$729$	−21.6402	−0.801487
$730$	−43.7974	−1.62101
$731$	−42.2135	−1.56132
$732$	27.1896	1.00496
$733$	−37.7756	−1.39527	−0.697636	−	0.716452i	$-0.745765\pi$
$733$	−37.7756	−1.39527	−0.697636	+	0.716452i	$0.745765\pi$
$734$	−4.09505	−0.151151
$735$	100.541	3.70850
$736$	9.04507	0.333406
$737$	12.1116	0.446135
$738$	−68.2959	−2.51401
$739$	−31.1269	−1.14502	−0.572511	−	0.819897i	$-0.694031\pi$
$739$	−31.1269	−1.14502	−0.572511	+	0.819897i	$0.694031\pi$
$740$	42.4934	1.56209
$741$	−6.20485	−0.227941
$742$	14.3817	0.527969
$743$	22.9203	0.840866	0.420433	−	0.907324i	$-0.361878\pi$
$743$	22.9203	0.840866	0.420433	+	0.907324i	$0.361878\pi$
$744$	−24.8653	−0.911606
$745$	−32.7982	−1.20163
$746$	−9.05086	−0.331376
$747$	−34.6169	−1.26657
$748$	21.3951	0.782281
$749$	−60.3981	−2.20690
$750$	102.499	3.74272
$751$	−25.3668	−0.925647	−0.462823	−	0.886451i	$-0.653164\pi$
$751$	−25.3668	−0.925647	−0.462823	+	0.886451i	$0.653164\pi$
$752$	5.96705	0.217596
$753$	−57.2301	−2.08558
$754$	17.8466	0.649935
$755$	−96.3615	−3.50695
$756$	−37.1433	−1.35089
$757$	−48.8333	−1.77488	−0.887438	−	0.460926i	$-0.847517\pi$
$757$	−48.8333	−1.77488	−0.887438	+	0.460926i	$0.847517\pi$
$758$	27.8350	1.01101
$759$	91.0290	3.30414
$760$	−1.54455	−0.0560266
$761$	8.73051	0.316481	0.158240	−	0.987401i	$-0.449418\pi$
$761$	8.73051	0.316481	0.158240	+	0.987401i	$0.449418\pi$
$762$	30.6072	1.10878
$763$	−18.7790	−0.679846
$764$	12.3086	0.445310
$765$	168.901	6.10663
$766$	31.7484	1.14712
$767$	−9.58569	−0.346119
$768$	−3.03040	−0.109350
$769$	−41.8605	−1.50953	−0.754764	−	0.655997i	$-0.772249\pi$
$769$	−41.8605	−1.50953	−0.754764	+	0.655997i	$0.772249\pi$
$770$	54.2158	1.95380
$771$	35.1927	1.26743
$772$	8.39869	0.302276
$773$	38.0387	1.36816	0.684079	−	0.729408i	$-0.260204\pi$
$773$	38.0387	1.36816	0.684079	+	0.729408i	$0.260204\pi$
$774$	40.5161	1.45632
$775$	106.482	3.82496
$776$	−1.02328	−0.0367335
$777$	116.938	4.19513
$778$	21.3369	0.764964
$779$	−4.02356	−0.144159
$780$	72.2195	2.58587
$781$	−27.6127	−0.988062
$782$	−58.2718	−2.08380
$783$	−30.6301	−1.09463
$784$	7.82492	0.279461
$785$	94.4813	3.37218
$786$	−6.62651	−0.236360
$787$	38.7842	1.38251	0.691254	−	0.722612i	$-0.257059\pi$
$787$	38.7842	1.38251	0.691254	+	0.722612i	$0.257059\pi$
$788$	13.9163	0.495746
$789$	−50.0611	−1.78222
$790$	15.9685	0.568132
$791$	22.3846	0.795903
$792$	−20.5348	−0.729673
$793$	−50.4305	−1.79084
$794$	−7.14884	−0.253703
$795$	47.9929	1.70213
$796$	3.31855	0.117623
$797$	−2.59734	−0.0920026	−0.0460013	−	0.998941i	$-0.514648\pi$
$797$	−2.59734	−0.0920026	−0.0460013	+	0.998941i	$0.514648\pi$
$798$	−4.25046	−0.150465
$799$	−38.4420	−1.35998
$800$	12.9773	0.458817
$801$	−57.0269	−2.01495
$802$	−31.7757	−1.12204
$803$	−34.3047	−1.21059
$804$	11.0518	0.389767
$805$	−147.663	−5.20442
$806$	46.1195	1.62449
$807$	−39.2246	−1.38077
$808$	16.1496	0.568141
$809$	−7.15197	−0.251450	−0.125725	−	0.992065i	$-0.540126\pi$
$809$	−7.15197	−0.251450	−0.125725	+	0.992065i	$0.540126\pi$
$810$	−45.2986	−1.59163
$811$	40.4129	1.41909	0.709544	−	0.704661i	$-0.248901\pi$
$811$	40.4129	1.41909	0.709544	+	0.704661i	$0.248901\pi$
$812$	12.2253	0.429025
$813$	45.2176	1.58585
$814$	33.2833	1.16658
$815$	13.5810	0.475723
$816$	19.5230	0.683442
$817$	2.38695	0.0835089
$818$	2.69844	0.0943487
$819$	133.817	4.67594
$820$	46.8310	1.63541
$821$	22.9004	0.799228	0.399614	−	0.916683i	$-0.369144\pi$
$821$	22.9004	0.799228	0.399614	+	0.916683i	$0.369144\pi$
$822$	7.84471	0.273616
$823$	−26.7390	−0.932062	−0.466031	−	0.884768i	$-0.654316\pi$
$823$	−26.7390	−0.932062	−0.466031	+	0.884768i	$0.654316\pi$
$824$	−12.8647	−0.448164
$825$	130.603	4.54700
$826$	−6.56641	−0.228474
$827$	−36.6296	−1.27374	−0.636869	−	0.770972i	$-0.719771\pi$
$827$	−36.6296	−1.27374	−0.636869	+	0.770972i	$0.719771\pi$
$828$	55.9288	1.94366
$829$	−21.1680	−0.735194	−0.367597	−	0.929985i	$-0.619819\pi$
$829$	−21.1680	−0.735194	−0.367597	+	0.929985i	$0.619819\pi$
$830$	23.7371	0.823926
$831$	−46.2745	−1.60524
$832$	5.62072	0.194863
$833$	−50.4111	−1.74664
$834$	−46.2754	−1.60238
$835$	76.6404	2.65225
$836$	−1.20978	−0.0418411
$837$	−79.1548	−2.73599
$838$	5.26920	0.182022
$839$	51.2927	1.77082	0.885411	−	0.464809i	$-0.153877\pi$
$839$	51.2927	1.77082	0.885411	+	0.464809i	$0.153877\pi$
$840$	49.4719	1.70694
$841$	−18.9184	−0.652360
$842$	31.7474	1.09409
$843$	−55.2335	−1.90234
$844$	−1.82272	−0.0627407
$845$	−78.8313	−2.71188
$846$	36.8963	1.26852
$847$	0.111493	0.00383095
$848$	3.73520	0.128267
$849$	2.76489	0.0948907
$850$	−83.6047	−2.86762
$851$	−90.6507	−3.10747
$852$	−25.1966	−0.863223
$853$	30.2021	1.03410	0.517050	−	0.855955i	$-0.327030\pi$
$853$	30.2021	1.03410	0.517050	+	0.855955i	$0.327030\pi$
$854$	−34.5460	−1.18214
$855$	−9.55047	−0.326619
$856$	−15.6865	−0.536155
$857$	45.2724	1.54648	0.773238	−	0.634116i	$-0.218636\pi$
$857$	45.2724	1.54648	0.773238	+	0.634116i	$0.218636\pi$
$858$	56.5665	1.93115
$859$	7.74623	0.264298	0.132149	−	0.991230i	$-0.457812\pi$
$859$	7.74623	0.264298	0.132149	+	0.991230i	$0.457812\pi$
$860$	−27.7822	−0.947365
$861$	128.875	4.39204
$862$	0.425619	0.0144966
$863$	−48.8363	−1.66241	−0.831204	−	0.555968i	$-0.812348\pi$
$863$	−48.8363	−1.66241	−0.831204	+	0.555968i	$0.812348\pi$
$864$	−9.64682	−0.328191
$865$	−20.4348	−0.694805
$866$	23.9771	0.814776
$867$	−74.2578	−2.52193
$868$	31.5929	1.07233
$869$	12.5074	0.424285
$870$	40.7969	1.38314
$871$	−20.4986	−0.694569
$872$	−4.87727	−0.165165
$873$	−6.32728	−0.214146
$874$	3.29497	0.111454
$875$	−130.231	−4.40261
$876$	−31.3030	−1.05763
$877$	17.7839	0.600520	0.300260	−	0.953857i	$-0.402926\pi$
$877$	17.7839	0.600520	0.300260	+	0.953857i	$0.402926\pi$
$878$	−5.27599	−0.178056
$879$	−80.1635	−2.70385
$880$	14.0809	0.474666
$881$	−11.3509	−0.382421	−0.191210	−	0.981549i	$-0.561241\pi$
$881$	−11.3509	−0.382421	−0.191210	+	0.981549i	$0.561241\pi$
$882$	48.3842	1.62918
$883$	45.5892	1.53420	0.767100	−	0.641528i	$-0.221699\pi$
$883$	45.5892	1.53420	0.767100	+	0.641528i	$0.221699\pi$
$884$	−36.2108	−1.21790
$885$	−21.9126	−0.736584
$886$	26.0537	0.875292
$887$	15.2506	0.512064	0.256032	−	0.966668i	$-0.417585\pi$
$887$	15.2506	0.512064	0.256032	+	0.966668i	$0.417585\pi$
$888$	30.3710	1.01919
$889$	−38.8883	−1.30427
$890$	39.1038	1.31076
$891$	−35.4805	−1.18864
$892$	24.2282	0.811221
$893$	2.17369	0.0727399
$894$	−23.4416	−0.784006
$895$	−8.52833	−0.285070
$896$	3.85031	0.128630
$897$	−154.065	−5.14408
$898$	3.82611	0.127679
$899$	26.0530	0.868915
$900$	80.2431	2.67477
$901$	−24.0636	−0.801675
$902$	36.6808	1.22134
$903$	−76.4542	−2.54424
$904$	5.81370	0.193361
$905$	−52.0658	−1.73073
$906$	−68.8718	−2.28811
$907$	31.3844	1.04210	0.521051	−	0.853526i	$-0.325540\pi$
$907$	31.3844	1.04210	0.521051	+	0.853526i	$0.325540\pi$
$908$	7.73990	0.256858
$909$	99.8585	3.31210
$910$	−91.7593	−3.04179
$911$	−16.2560	−0.538585	−0.269292	−	0.963058i	$-0.586790\pi$
$911$	−16.2560	−0.538585	−0.269292	+	0.963058i	$0.586790\pi$
$912$	−1.10393	−0.0365546
$913$	18.5923	0.615314
$914$	36.2399	1.19871
$915$	−115.283	−3.81113
$916$	−0.960006	−0.0317195
$917$	8.41939	0.278033
$918$	62.1485	2.05121
$919$	22.0144	0.726188	0.363094	−	0.931752i	$-0.381720\pi$
$919$	22.0144	0.726188	0.363094	+	0.931752i	$0.381720\pi$
$920$	−38.3508	−1.26439
$921$	50.7295	1.67160
$922$	−29.6960	−0.977985
$923$	46.7341	1.53827
$924$	38.7493	1.27476
$925$	−130.060	−4.27634
$926$	24.1300	0.792962
$927$	−79.5470	−2.61267
$928$	3.17515	0.104229
$929$	−8.53506	−0.280026	−0.140013	−	0.990150i	$-0.544714\pi$
$929$	−8.53506	−0.280026	−0.140013	+	0.990150i	$0.544714\pi$
$930$	105.428	3.45712
$931$	2.85049	0.0934209
$932$	5.98362	0.196000
$933$	11.6105	0.380110
$934$	7.62506	0.249500
$935$	−90.7143	−2.96667
$936$	34.7548	1.13600
$937$	29.0665	0.949560	0.474780	−	0.880105i	$-0.342528\pi$
$937$	29.0665	0.949560	0.474780	+	0.880105i	$0.342528\pi$
$938$	−14.0420	−0.458487
$939$	15.1125	0.493178
$940$	−25.3001	−0.825197
$941$	−18.0473	−0.588327	−0.294163	−	0.955755i	$-0.595041\pi$
$941$	−18.0473	−0.588327	−0.294163	+	0.955755i	$0.595041\pi$
$942$	67.5280	2.20018
$943$	−99.9041	−3.25332
$944$	−1.70542	−0.0555067
$945$	157.486	5.12302
$946$	−21.7606	−0.707500
$947$	−3.20030	−0.103996	−0.0519979	−	0.998647i	$-0.516559\pi$
$947$	−3.20030	−0.103996	−0.0519979	+	0.998647i	$0.516559\pi$
$948$	11.4130	0.370678
$949$	58.0601	1.88471
$950$	4.72741	0.153377
$951$	4.33441	0.140553
$952$	−24.8052	−0.803941
$953$	32.1771	1.04232	0.521159	−	0.853460i	$-0.325500\pi$
$953$	32.1771	1.04232	0.521159	+	0.853460i	$0.325500\pi$
$954$	23.0961	0.747762
$955$	−52.1881	−1.68877
$956$	−12.3127	−0.398223
$957$	31.9545	1.03294
$958$	−1.16234	−0.0375536
$959$	−9.96718	−0.321857
$960$	12.8488	0.414693
$961$	36.3266	1.17183
$962$	−56.3314	−1.81620
$963$	−96.9952	−3.12563
$964$	−6.64674	−0.214077
$965$	−35.6102	−1.14633
$966$	−105.538	−3.39563
$967$	−57.6359	−1.85345	−0.926723	−	0.375746i	$-0.877387\pi$
$967$	−57.6359	−1.85345	−0.926723	+	0.375746i	$0.877387\pi$
$968$	0.0289569	0.000930709 0
$969$	7.11191	0.228467
$970$	4.33866	0.139306
$971$	−43.2091	−1.38665	−0.693323	−	0.720627i	$-0.743854\pi$
$971$	−43.2091	−1.38665	−0.693323	+	0.720627i	$0.743854\pi$
$972$	−3.43551	−0.110194
$973$	58.7957	1.88490
$974$	28.2944	0.906610
$975$	−221.043	−7.07903
$976$	−8.97226	−0.287195
$977$	43.2664	1.38421	0.692107	−	0.721795i	$-0.256683\pi$
$977$	43.2664	1.38421	0.692107	+	0.721795i	$0.256683\pi$
$978$	9.70670	0.310386
$979$	30.6283	0.978886
$980$	−33.1774	−1.05981
$981$	−30.1578	−0.962865
$982$	−21.0266	−0.670986
$983$	−44.0293	−1.40432	−0.702158	−	0.712021i	$-0.747780\pi$
$983$	−44.0293	−1.40432	−0.702158	+	0.712021i	$0.747780\pi$
$984$	33.4712	1.06702
$985$	−59.0045	−1.88004
$986$	−20.4555	−0.651436
$987$	−69.6235	−2.21614
$988$	2.04753	0.0651407
$989$	59.2675	1.88460
$990$	87.0668	2.76717
$991$	−19.7582	−0.627639	−0.313819	−	0.949483i	$-0.601609\pi$
$991$	−19.7582	−0.627639	−0.313819	+	0.949483i	$0.601609\pi$
$992$	8.20528	0.260518
$993$	−70.8519	−2.24842
$994$	32.0139	1.01542
$995$	−14.0705	−0.446065
$996$	16.9655	0.537571
$997$	−1.64122	−0.0519779	−0.0259890	−	0.999662i	$-0.508273\pi$
$997$	−1.64122	−0.0519779	−0.0259890	+	0.999662i	$0.508273\pi$
$998$	−0.844870	−0.0267439
$999$	96.6815	3.05887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.d.1.8	✓	101

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.d.1.8	✓	101	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.03040 q^{3} +1.00000 q^{4} -4.23996 q^{5} -3.03040 q^{6} +3.85031 q^{7} +1.00000 q^{8} +6.18334 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.03040 q^{3} +1.00000 q^{4} -4.23996 q^{5} -3.03040 q^{6} +3.85031 q^{7} +1.00000 q^{8} +6.18334 q^{9} -4.23996 q^{10} -3.32099 q^{11} -3.03040 q^{12} +5.62072 q^{13} +3.85031 q^{14} +12.8488 q^{15} +1.00000 q^{16} -6.44238 q^{17} +6.18334 q^{18} +0.364283 q^{19} -4.23996 q^{20} -11.6680 q^{21} -3.32099 q^{22} +9.04507 q^{23} -3.03040 q^{24} +12.9773 q^{25} +5.62072 q^{26} -9.64682 q^{27} +3.85031 q^{28} +3.17515 q^{29} +12.8488 q^{30} +8.20528 q^{31} +1.00000 q^{32} +10.0639 q^{33} -6.44238 q^{34} -16.3252 q^{35} +6.18334 q^{36} -10.0221 q^{37} +0.364283 q^{38} -17.0330 q^{39} -4.23996 q^{40} -11.0451 q^{41} -11.6680 q^{42} +6.55246 q^{43} -3.32099 q^{44} -26.2172 q^{45} +9.04507 q^{46} +5.96705 q^{47} -3.03040 q^{48} +7.82492 q^{49} +12.9773 q^{50} +19.5230 q^{51} +5.62072 q^{52} +3.73520 q^{53} -9.64682 q^{54} +14.0809 q^{55} +3.85031 q^{56} -1.10393 q^{57} +3.17515 q^{58} -1.70542 q^{59} +12.8488 q^{60} -8.97226 q^{61} +8.20528 q^{62} +23.8078 q^{63} +1.00000 q^{64} -23.8316 q^{65} +10.0639 q^{66} -3.64697 q^{67} -6.44238 q^{68} -27.4102 q^{69} -16.3252 q^{70} +8.31462 q^{71} +6.18334 q^{72} +10.3297 q^{73} -10.0221 q^{74} -39.3264 q^{75} +0.364283 q^{76} -12.7868 q^{77} -17.0330 q^{78} -3.76618 q^{79} -4.23996 q^{80} +10.6837 q^{81} -11.0451 q^{82} -5.59841 q^{83} -11.6680 q^{84} +27.3155 q^{85} +6.55246 q^{86} -9.62198 q^{87} -3.32099 q^{88} -9.22266 q^{89} -26.2172 q^{90} +21.6415 q^{91} +9.04507 q^{92} -24.8653 q^{93} +5.96705 q^{94} -1.54455 q^{95} -3.03040 q^{96} -1.02328 q^{97} +7.82492 q^{98} -20.5348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9 + 19 * q^10 + 4 * q^11 + 10 * q^12 + 58 * q^13 + 42 * q^14 + 27 * q^15 + 101 * q^16 + 34 * q^17 + 147 * q^18 + 36 * q^19 + 19 * q^20 + 45 * q^21 + 4 * q^22 + 47 * q^23 + 10 * q^24 + 174 * q^25 + 58 * q^26 + 31 * q^27 + 42 * q^28 + 62 * q^29 + 27 * q^30 + 47 * q^31 + 101 * q^32 + 55 * q^33 + 34 * q^34 + 16 * q^35 + 147 * q^36 + 90 * q^37 + 36 * q^38 + 50 * q^39 + 19 * q^40 + 54 * q^41 + 45 * q^42 + 65 * q^43 + 4 * q^44 + 47 * q^45 + 47 * q^46 + 54 * q^47 + 10 * q^48 + 189 * q^49 + 174 * q^50 + 36 * q^51 + 58 * q^52 + 94 * q^53 + 31 * q^54 + 68 * q^55 + 42 * q^56 + 79 * q^57 + 62 * q^58 - 6 * q^59 + 27 * q^60 + 58 * q^61 + 47 * q^62 + 117 * q^63 + 101 * q^64 + 89 * q^65 + 55 * q^66 + 127 * q^67 + 34 * q^68 + 45 * q^69 + 16 * q^70 + 87 * q^71 + 147 * q^72 + 83 * q^73 + 90 * q^74 - 4 * q^75 + 36 * q^76 + 53 * q^77 + 50 * q^78 + 74 * q^79 + 19 * q^80 + 241 * q^81 + 54 * q^82 + 11 * q^83 + 45 * q^84 + 120 * q^85 + 65 * q^86 + 37 * q^87 + 4 * q^88 + 89 * q^89 + 47 * q^90 + 31 * q^91 + 47 * q^92 + 123 * q^93 + 54 * q^94 + 61 * q^95 + 10 * q^96 + 85 * q^97 + 189 * q^98 - 55 * q^99

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