LMFDB - Embedded newform 8042.2.a.c.1.11

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			1.11
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.49912 q^{3} +1.00000 q^{4} -3.77840 q^{5} +2.49912 q^{6} -0.508744 q^{7} -1.00000 q^{8} +3.24558 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.49912 q^{3} +1.00000 q^{4} -3.77840 q^{5} +2.49912 q^{6} -0.508744 q^{7} -1.00000 q^{8} +3.24558 q^{9} +3.77840 q^{10} +4.72877 q^{11} -2.49912 q^{12} -1.56616 q^{13} +0.508744 q^{14} +9.44266 q^{15} +1.00000 q^{16} -1.70941 q^{17} -3.24558 q^{18} -7.33030 q^{19} -3.77840 q^{20} +1.27141 q^{21} -4.72877 q^{22} +6.82278 q^{23} +2.49912 q^{24} +9.27631 q^{25} +1.56616 q^{26} -0.613729 q^{27} -0.508744 q^{28} +6.46138 q^{29} -9.44266 q^{30} -0.0712862 q^{31} -1.00000 q^{32} -11.8177 q^{33} +1.70941 q^{34} +1.92224 q^{35} +3.24558 q^{36} +9.42635 q^{37} +7.33030 q^{38} +3.91402 q^{39} +3.77840 q^{40} -5.14690 q^{41} -1.27141 q^{42} +1.98166 q^{43} +4.72877 q^{44} -12.2631 q^{45} -6.82278 q^{46} +5.36150 q^{47} -2.49912 q^{48} -6.74118 q^{49} -9.27631 q^{50} +4.27200 q^{51} -1.56616 q^{52} +0.630655 q^{53} +0.613729 q^{54} -17.8672 q^{55} +0.508744 q^{56} +18.3193 q^{57} -6.46138 q^{58} -3.96183 q^{59} +9.44266 q^{60} +10.9418 q^{61} +0.0712862 q^{62} -1.65117 q^{63} +1.00000 q^{64} +5.91758 q^{65} +11.8177 q^{66} -0.192213 q^{67} -1.70941 q^{68} -17.0509 q^{69} -1.92224 q^{70} +4.53092 q^{71} -3.24558 q^{72} +0.588797 q^{73} -9.42635 q^{74} -23.1826 q^{75} -7.33030 q^{76} -2.40573 q^{77} -3.91402 q^{78} -16.1449 q^{79} -3.77840 q^{80} -8.20296 q^{81} +5.14690 q^{82} +6.65631 q^{83} +1.27141 q^{84} +6.45882 q^{85} -1.98166 q^{86} -16.1477 q^{87} -4.72877 q^{88} -0.667289 q^{89} +12.2631 q^{90} +0.796775 q^{91} +6.82278 q^{92} +0.178152 q^{93} -5.36150 q^{94} +27.6968 q^{95} +2.49912 q^{96} +2.10060 q^{97} +6.74118 q^{98} +15.3476 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.49912	−1.44287	−0.721433	−	0.692485i	$-0.756516\pi$
$3$	−2.49912	−1.44287	−0.721433	+	0.692485i	$0.756516\pi$
$4$	1.00000	0.500000
$5$	−3.77840	−1.68975	−0.844876	−	0.534962i	$-0.820326\pi$
$5$	−3.77840	−1.68975	−0.844876	+	0.534962i	$0.820326\pi$
$6$	2.49912	1.02026
$7$	−0.508744	−0.192287	−0.0961436	−	0.995367i	$-0.530651\pi$
$7$	−0.508744	−0.192287	−0.0961436	+	0.995367i	$0.530651\pi$
$8$	−1.00000	−0.353553
$9$	3.24558	1.08186
$10$	3.77840	1.19483
$11$	4.72877	1.42578	0.712889	−	0.701277i	$-0.247386\pi$
$11$	4.72877	1.42578	0.712889	+	0.701277i	$0.247386\pi$
$12$	−2.49912	−0.721433
$13$	−1.56616	−0.434375	−0.217187	−	0.976130i	$-0.569688\pi$
$13$	−1.56616	−0.434375	−0.217187	+	0.976130i	$0.569688\pi$
$14$	0.508744	0.135968
$15$	9.44266	2.43808
$16$	1.00000	0.250000
$17$	−1.70941	−0.414592	−0.207296	−	0.978278i	$-0.566466\pi$
$17$	−1.70941	−0.414592	−0.207296	+	0.978278i	$0.566466\pi$
$18$	−3.24558	−0.764990
$19$	−7.33030	−1.68169	−0.840843	−	0.541278i	$-0.817941\pi$
$19$	−7.33030	−1.68169	−0.840843	+	0.541278i	$0.817941\pi$
$20$	−3.77840	−0.844876
$21$	1.27141	0.277444
$22$	−4.72877	−1.00818
$23$	6.82278	1.42265	0.711324	−	0.702865i	$-0.248096\pi$
$23$	6.82278	1.42265	0.711324	+	0.702865i	$0.248096\pi$
$24$	2.49912	0.510130
$25$	9.27631	1.85526
$26$	1.56616	0.307149
$27$	−0.613729	−0.118112
$28$	−0.508744	−0.0961436
$29$	6.46138	1.19985	0.599924	−	0.800057i	$-0.295197\pi$
$29$	6.46138	1.19985	0.599924	+	0.800057i	$0.295197\pi$
$30$	−9.44266	−1.72399
$31$	−0.0712862	−0.0128034	−0.00640169	−	0.999980i	$-0.502038\pi$
$31$	−0.0712862	−0.0128034	−0.00640169	+	0.999980i	$0.502038\pi$
$32$	−1.00000	−0.176777
$33$	−11.8177	−2.05720
$34$	1.70941	0.293161
$35$	1.92224	0.324917
$36$	3.24558	0.540930
$37$	9.42635	1.54968	0.774841	−	0.632156i	$-0.217830\pi$
$37$	9.42635	1.54968	0.774841	+	0.632156i	$0.217830\pi$
$38$	7.33030	1.18913
$39$	3.91402	0.626744
$40$	3.77840	0.597417
$41$	−5.14690	−0.803810	−0.401905	−	0.915681i	$-0.631652\pi$
$41$	−5.14690	−0.803810	−0.401905	+	0.915681i	$0.631652\pi$
$42$	−1.27141	−0.196183
$43$	1.98166	0.302200	0.151100	−	0.988518i	$-0.451718\pi$
$43$	1.98166	0.302200	0.151100	+	0.988518i	$0.451718\pi$
$44$	4.72877	0.712889
$45$	−12.2631	−1.82807
$46$	−6.82278	−1.00596
$47$	5.36150	0.782055	0.391028	−	0.920379i	$-0.372120\pi$
$47$	5.36150	0.782055	0.391028	+	0.920379i	$0.372120\pi$
$48$	−2.49912	−0.360716
$49$	−6.74118	−0.963026
$50$	−9.27631	−1.31187
$51$	4.27200	0.598200
$52$	−1.56616	−0.217187
$53$	0.630655	0.0866271	0.0433136	−	0.999062i	$-0.486209\pi$
$53$	0.630655	0.0866271	0.0433136	+	0.999062i	$0.486209\pi$
$54$	0.613729	0.0835179
$55$	−17.8672	−2.40921
$56$	0.508744	0.0679838
$57$	18.3193	2.42645
$58$	−6.46138	−0.848421
$59$	−3.96183	−0.515787	−0.257893	−	0.966173i	$-0.583028\pi$
$59$	−3.96183	−0.515787	−0.257893	+	0.966173i	$0.583028\pi$
$60$	9.44266	1.21904
$61$	10.9418	1.40095	0.700477	−	0.713675i	$-0.252971\pi$
$61$	10.9418	1.40095	0.700477	+	0.713675i	$0.252971\pi$
$62$	0.0712862	0.00905335
$63$	−1.65117	−0.208028
$64$	1.00000	0.125000
$65$	5.91758	0.733986
$66$	11.8177	1.45466
$67$	−0.192213	−0.0234825	−0.0117412	−	0.999931i	$-0.503737\pi$
$67$	−0.192213	−0.0234825	−0.0117412	+	0.999931i	$0.503737\pi$
$68$	−1.70941	−0.207296
$69$	−17.0509	−2.05269
$70$	−1.92224	−0.229751
$71$	4.53092	0.537721	0.268861	−	0.963179i	$-0.413353\pi$
$71$	4.53092	0.537721	0.268861	+	0.963179i	$0.413353\pi$
$72$	−3.24558	−0.382495
$73$	0.588797	0.0689134	0.0344567	−	0.999406i	$-0.489030\pi$
$73$	0.588797	0.0689134	0.0344567	+	0.999406i	$0.489030\pi$
$74$	−9.42635	−1.09579
$75$	−23.1826	−2.67689
$76$	−7.33030	−0.840843
$77$	−2.40573	−0.274159
$78$	−3.91402	−0.443175
$79$	−16.1449	−1.81644	−0.908221	−	0.418492i	$-0.862559\pi$
$79$	−16.1449	−1.81644	−0.908221	+	0.418492i	$0.862559\pi$
$80$	−3.77840	−0.422438
$81$	−8.20296	−0.911440
$82$	5.14690	0.568380
$83$	6.65631	0.730625	0.365312	−	0.930885i	$-0.380962\pi$
$83$	6.65631	0.730625	0.365312	+	0.930885i	$0.380962\pi$
$84$	1.27141	0.138722
$85$	6.45882	0.700558
$86$	−1.98166	−0.213688
$87$	−16.1477	−1.73122
$88$	−4.72877	−0.504089
$89$	−0.667289	−0.0707325	−0.0353662	−	0.999374i	$-0.511260\pi$
$89$	−0.667289	−0.0707325	−0.0353662	+	0.999374i	$0.511260\pi$
$90$	12.2631	1.29264
$91$	0.796775	0.0835247
$92$	6.82278	0.711324
$93$	0.178152	0.0184735
$94$	−5.36150	−0.552997
$95$	27.6968	2.84163
$96$	2.49912	0.255065
$97$	2.10060	0.213284	0.106642	−	0.994297i	$-0.465990\pi$
$97$	2.10060	0.213284	0.106642	+	0.994297i	$0.465990\pi$
$98$	6.74118	0.680962
$99$	15.3476	1.54249
$100$	9.27631	0.927631
$101$	2.07226	0.206198	0.103099	−	0.994671i	$-0.467124\pi$
$101$	2.07226	0.206198	0.103099	+	0.994671i	$0.467124\pi$
$102$	−4.27200	−0.422992
$103$	14.2849	1.40753	0.703764	−	0.710434i	$-0.251501\pi$
$103$	14.2849	1.40753	0.703764	+	0.710434i	$0.251501\pi$
$104$	1.56616	0.153575
$105$	−4.80389	−0.468812
$106$	−0.630655	−0.0612546
$107$	−10.1314	−0.979437	−0.489718	−	0.871881i	$-0.662900\pi$
$107$	−10.1314	−0.979437	−0.489718	+	0.871881i	$0.662900\pi$
$108$	−0.613729	−0.0590561
$109$	13.6923	1.31148	0.655742	−	0.754985i	$-0.272356\pi$
$109$	13.6923	1.31148	0.655742	+	0.754985i	$0.272356\pi$
$110$	17.8672	1.70357
$111$	−23.5575	−2.23598
$112$	−0.508744	−0.0480718
$113$	−8.47807	−0.797550	−0.398775	−	0.917049i	$-0.630565\pi$
$113$	−8.47807	−0.797550	−0.398775	+	0.917049i	$0.630565\pi$
$114$	−18.3193	−1.71576
$115$	−25.7792	−2.40392
$116$	6.46138	0.599924
$117$	−5.08310	−0.469933
$118$	3.96183	0.364716
$119$	0.869650	0.0797207
$120$	−9.44266	−0.861993
$121$	11.3613	1.03284
$122$	−10.9418	−0.990624
$123$	12.8627	1.15979
$124$	−0.0712862	−0.00640169
$125$	−16.1576	−1.44518
$126$	1.65117	0.147098
$127$	−14.9791	−1.32918	−0.664590	−	0.747208i	$-0.731394\pi$
$127$	−14.9791	−1.32918	−0.664590	+	0.747208i	$0.731394\pi$
$128$	−1.00000	−0.0883883
$129$	−4.95240	−0.436034
$130$	−5.91758	−0.519006
$131$	−10.5654	−0.923106	−0.461553	−	0.887113i	$-0.652708\pi$
$131$	−10.5654	−0.923106	−0.461553	+	0.887113i	$0.652708\pi$
$132$	−11.8177	−1.02860
$133$	3.72925	0.323367
$134$	0.192213	0.0166046
$135$	2.31891	0.199580
$136$	1.70941	0.146580
$137$	−11.8114	−1.00912	−0.504559	−	0.863377i	$-0.668345\pi$
$137$	−11.8114	−1.00912	−0.504559	+	0.863377i	$0.668345\pi$
$138$	17.0509	1.45147
$139$	−2.76617	−0.234624	−0.117312	−	0.993095i	$-0.537428\pi$
$139$	−2.76617	−0.234624	−0.117312	+	0.993095i	$0.537428\pi$
$140$	1.92224	0.162459
$141$	−13.3990	−1.12840
$142$	−4.53092	−0.380226
$143$	−7.40602	−0.619322
$144$	3.24558	0.270465
$145$	−24.4137	−2.02745
$146$	−0.588797	−0.0487291
$147$	16.8470	1.38952
$148$	9.42635	0.774841
$149$	0.0224216	0.00183685	0.000918426	−	1.00000i	$-0.499708\pi$
$149$	0.0224216	0.00183685	0.000918426		1.00000i	$0.499708\pi$
$150$	23.1826	1.89285
$151$	15.8603	1.29070	0.645348	−	0.763888i	$-0.276712\pi$
$151$	15.8603	1.29070	0.645348	+	0.763888i	$0.276712\pi$
$152$	7.33030	0.594566
$153$	−5.54801	−0.448530
$154$	2.40573	0.193859
$155$	0.269348	0.0216345
$156$	3.91402	0.313372
$157$	5.68936	0.454061	0.227030	−	0.973888i	$-0.427098\pi$
$157$	5.68936	0.454061	0.227030	+	0.973888i	$0.427098\pi$
$158$	16.1449	1.28442
$159$	−1.57608	−0.124991
$160$	3.77840	0.298709
$161$	−3.47105	−0.273557
$162$	8.20296	0.644485
$163$	4.12095	0.322777	0.161389	−	0.986891i	$-0.448403\pi$
$163$	4.12095	0.322777	0.161389	+	0.986891i	$0.448403\pi$
$164$	−5.14690	−0.401905
$165$	44.6522	3.47617
$166$	−6.65631	−0.516630
$167$	2.34577	0.181521	0.0907607	−	0.995873i	$-0.471070\pi$
$167$	2.34577	0.181521	0.0907607	+	0.995873i	$0.471070\pi$
$168$	−1.27141	−0.0980914
$169$	−10.5471	−0.811318
$170$	−6.45882	−0.495369
$171$	−23.7911	−1.81935
$172$	1.98166	0.151100
$173$	−16.9073	−1.28544	−0.642718	−	0.766103i	$-0.722193\pi$
$173$	−16.9073	−1.28544	−0.642718	+	0.766103i	$0.722193\pi$
$174$	16.1477	1.22416
$175$	−4.71926	−0.356743
$176$	4.72877	0.356444
$177$	9.90108	0.744211
$178$	0.667289	0.0500154
$179$	−25.4604	−1.90300	−0.951498	−	0.307654i	$-0.900456\pi$
$179$	−25.4604	−1.90300	−0.951498	+	0.307654i	$0.900456\pi$
$180$	−12.2631	−0.914037
$181$	−1.99371	−0.148191	−0.0740957	−	0.997251i	$-0.523607\pi$
$181$	−1.99371	−0.148191	−0.0740957	+	0.997251i	$0.523607\pi$
$182$	−0.796775	−0.0590609
$183$	−27.3448	−2.02139
$184$	−6.82278	−0.502982
$185$	−35.6165	−2.61858
$186$	−0.178152	−0.0130628
$187$	−8.08339	−0.591116
$188$	5.36150	0.391028
$189$	0.312231	0.0227114
$190$	−27.6968	−2.00934
$191$	−14.4859	−1.04816	−0.524081	−	0.851669i	$-0.675591\pi$
$191$	−14.4859	−1.04816	−0.524081	+	0.851669i	$0.675591\pi$
$192$	−2.49912	−0.180358
$193$	21.0958	1.51851	0.759254	−	0.650794i	$-0.225564\pi$
$193$	21.0958	1.51851	0.759254	+	0.650794i	$0.225564\pi$
$194$	−2.10060	−0.150815
$195$	−14.7887	−1.05904
$196$	−6.74118	−0.481513
$197$	−24.1967	−1.72394	−0.861970	−	0.506959i	$-0.830770\pi$
$197$	−24.1967	−1.72394	−0.861970	+	0.506959i	$0.830770\pi$
$198$	−15.3476	−1.09071
$199$	16.5051	1.17002	0.585008	−	0.811028i	$-0.301091\pi$
$199$	16.5051	1.17002	0.585008	+	0.811028i	$0.301091\pi$
$200$	−9.27631	−0.655934
$201$	0.480361	0.0338821
$202$	−2.07226	−0.145804
$203$	−3.28719	−0.230715
$204$	4.27200	0.299100
$205$	19.4470	1.35824
$206$	−14.2849	−0.995273
$207$	22.1439	1.53910
$208$	−1.56616	−0.108594
$209$	−34.6633	−2.39771
$210$	4.80389	0.331500
$211$	12.4250	0.855375	0.427688	−	0.903927i	$-0.359328\pi$
$211$	12.4250	0.855375	0.427688	+	0.903927i	$0.359328\pi$
$212$	0.630655	0.0433136
$213$	−11.3233	−0.775859
$214$	10.1314	0.692566
$215$	−7.48751	−0.510644
$216$	0.613729	0.0417589
$217$	0.0362664	0.00246192
$218$	−13.6923	−0.927360
$219$	−1.47147	−0.0994328
$220$	−17.8672	−1.20461
$221$	2.67721	0.180088
$222$	23.5575	1.58108
$223$	−13.1348	−0.879569	−0.439784	−	0.898103i	$-0.644945\pi$
$223$	−13.1348	−0.879569	−0.439784	+	0.898103i	$0.644945\pi$
$224$	0.508744	0.0339919
$225$	30.1070	2.00713
$226$	8.47807	0.563953
$227$	−8.32519	−0.552562	−0.276281	−	0.961077i	$-0.589102\pi$
$227$	−8.32519	−0.552562	−0.276281	+	0.961077i	$0.589102\pi$
$228$	18.3193	1.21322
$229$	−10.2884	−0.679875	−0.339937	−	0.940448i	$-0.610406\pi$
$229$	−10.2884	−0.679875	−0.339937	+	0.940448i	$0.610406\pi$
$230$	25.7792	1.69983
$231$	6.01220	0.395574
$232$	−6.46138	−0.424210
$233$	29.6620	1.94322	0.971611	−	0.236582i	$-0.0760273\pi$
$233$	29.6620	1.94322	0.971611	+	0.236582i	$0.0760273\pi$
$234$	5.08310	0.332293
$235$	−20.2579	−1.32148
$236$	−3.96183	−0.257893
$237$	40.3479	2.62088
$238$	−0.869650	−0.0563710
$239$	−27.9429	−1.80748	−0.903738	−	0.428086i	$-0.859188\pi$
$239$	−27.9429	−1.80748	−0.903738	+	0.428086i	$0.859188\pi$
$240$	9.44266	0.609521
$241$	8.10538	0.522113	0.261057	−	0.965323i	$-0.415929\pi$
$241$	8.10538	0.522113	0.261057	+	0.965323i	$0.415929\pi$
$242$	−11.3613	−0.730330
$243$	22.3413	1.43320
$244$	10.9418	0.700477
$245$	25.4709	1.62727
$246$	−12.8627	−0.820095
$247$	11.4804	0.730483
$248$	0.0712862	0.00452668
$249$	−16.6349	−1.05419
$250$	16.1576	1.02190
$251$	22.2154	1.40223	0.701113	−	0.713050i	$-0.252687\pi$
$251$	22.2154	1.40223	0.701113	+	0.713050i	$0.252687\pi$
$252$	−1.65117	−0.104014
$253$	32.2633	2.02838
$254$	14.9791	0.939873
$255$	−16.1413	−1.01081
$256$	1.00000	0.0625000
$257$	−11.3368	−0.707168	−0.353584	−	0.935403i	$-0.615037\pi$
$257$	−11.3368	−0.707168	−0.353584	+	0.935403i	$0.615037\pi$
$258$	4.95240	0.308323
$259$	−4.79560	−0.297984
$260$	5.91758	0.366993
$261$	20.9709	1.29807
$262$	10.5654	0.652735
$263$	−14.7510	−0.909589	−0.454794	−	0.890596i	$-0.650287\pi$
$263$	−14.7510	−0.909589	−0.454794	+	0.890596i	$0.650287\pi$
$264$	11.8177	0.727332
$265$	−2.38287	−0.146378
$266$	−3.72925	−0.228655
$267$	1.66763	0.102057
$268$	−0.192213	−0.0117412
$269$	20.7231	1.26351	0.631753	−	0.775169i	$-0.282336\pi$
$269$	20.7231	1.26351	0.631753	+	0.775169i	$0.282336\pi$
$270$	−2.31891	−0.141124
$271$	−28.7047	−1.74369	−0.871844	−	0.489784i	$-0.837076\pi$
$271$	−28.7047	−1.74369	−0.871844	+	0.489784i	$0.837076\pi$
$272$	−1.70941	−0.103648
$273$	−1.99123	−0.120515
$274$	11.8114	0.713554
$275$	43.8655	2.64519
$276$	−17.0509	−1.02634
$277$	−18.6874	−1.12282	−0.561409	−	0.827539i	$-0.689740\pi$
$277$	−18.6874	−1.12282	−0.561409	+	0.827539i	$0.689740\pi$
$278$	2.76617	0.165904
$279$	−0.231365	−0.0138515
$280$	−1.92224	−0.114876
$281$	−21.3536	−1.27385	−0.636924	−	0.770926i	$-0.719794\pi$
$281$	−21.3536	−1.27385	−0.636924	+	0.770926i	$0.719794\pi$
$282$	13.3990	0.797899
$283$	22.4804	1.33632	0.668160	−	0.744018i	$-0.267082\pi$
$283$	22.4804	1.33632	0.668160	+	0.744018i	$0.267082\pi$
$284$	4.53092	0.268861
$285$	−69.2175	−4.10009
$286$	7.40602	0.437927
$287$	2.61845	0.154562
$288$	−3.24558	−0.191248
$289$	−14.0779	−0.828113
$290$	24.4137	1.43362
$291$	−5.24965	−0.307740
$292$	0.588797	0.0344567
$293$	14.4598	0.844752	0.422376	−	0.906421i	$-0.361196\pi$
$293$	14.4598	0.844752	0.422376	+	0.906421i	$0.361196\pi$
$294$	−16.8470	−0.982536
$295$	14.9694	0.871552
$296$	−9.42635	−0.547896
$297$	−2.90218	−0.168402
$298$	−0.0224216	−0.00129885
$299$	−10.6856	−0.617962
$300$	−23.1826	−1.33845
$301$	−1.00816	−0.0581093
$302$	−15.8603	−0.912660
$303$	−5.17882	−0.297516
$304$	−7.33030	−0.420422
$305$	−41.3425	−2.36726
$306$	5.54801	0.317159
$307$	−8.08162	−0.461242	−0.230621	−	0.973044i	$-0.574076\pi$
$307$	−8.08162	−0.461242	−0.230621	+	0.973044i	$0.574076\pi$
$308$	−2.40573	−0.137079
$309$	−35.6995	−2.03087
$310$	−0.269348	−0.0152979
$311$	−21.1984	−1.20205	−0.601027	−	0.799229i	$-0.705242\pi$
$311$	−21.1984	−1.20205	−0.601027	+	0.799229i	$0.705242\pi$
$312$	−3.91402	−0.221588
$313$	16.5817	0.937252	0.468626	−	0.883397i	$-0.344749\pi$
$313$	16.5817	0.937252	0.468626	+	0.883397i	$0.344749\pi$
$314$	−5.68936	−0.321069
$315$	6.23877	0.351515
$316$	−16.1449	−0.908221
$317$	20.8454	1.17079	0.585397	−	0.810747i	$-0.300939\pi$
$317$	20.8454	1.17079	0.585397	+	0.810747i	$0.300939\pi$
$318$	1.57608	0.0883822
$319$	30.5544	1.71072
$320$	−3.77840	−0.211219
$321$	25.3195	1.41320
$322$	3.47105	0.193434
$323$	12.5305	0.697214
$324$	−8.20296	−0.455720
$325$	−14.5282	−0.805879
$326$	−4.12095	−0.228238
$327$	−34.2186	−1.89229
$328$	5.14690	0.284190
$329$	−2.72763	−0.150379
$330$	−44.6522	−2.45802
$331$	−3.40899	−0.187375	−0.0936875	−	0.995602i	$-0.529865\pi$
$331$	−3.40899	−0.187375	−0.0936875	+	0.995602i	$0.529865\pi$
$332$	6.65631	0.365312
$333$	30.5940	1.67654
$334$	−2.34577	−0.128355
$335$	0.726256	0.0396796
$336$	1.27141	0.0693611
$337$	13.2438	0.721434	0.360717	−	0.932675i	$-0.382532\pi$
$337$	13.2438	0.721434	0.360717	+	0.932675i	$0.382532\pi$
$338$	10.5471	0.573689
$339$	21.1877	1.15076
$340$	6.45882	0.350279
$341$	−0.337096	−0.0182548
$342$	23.7911	1.28647
$343$	6.99074	0.377465
$344$	−1.98166	−0.106844
$345$	64.4252	3.46853
$346$	16.9073	0.908940
$347$	16.4422	0.882664	0.441332	−	0.897344i	$-0.354506\pi$
$347$	16.4422	0.882664	0.441332	+	0.897344i	$0.354506\pi$
$348$	−16.1477	−0.865610
$349$	4.96654	0.265853	0.132926	−	0.991126i	$-0.457563\pi$
$349$	4.96654	0.265853	0.132926	+	0.991126i	$0.457563\pi$
$350$	4.71926	0.252255
$351$	0.961198	0.0513049
$352$	−4.72877	−0.252044
$353$	−12.8929	−0.686217	−0.343109	−	0.939296i	$-0.611480\pi$
$353$	−12.8929	−0.686217	−0.343109	+	0.939296i	$0.611480\pi$
$354$	−9.90108	−0.526236
$355$	−17.1196	−0.908616
$356$	−0.667289	−0.0353662
$357$	−2.17336	−0.115026
$358$	25.4604	1.34562
$359$	−0.764622	−0.0403552	−0.0201776	−	0.999796i	$-0.506423\pi$
$359$	−0.764622	−0.0403552	−0.0201776	+	0.999796i	$0.506423\pi$
$360$	12.2631	0.646322
$361$	34.7333	1.82807
$362$	1.99371	0.104787
$363$	−28.3931	−1.49025
$364$	0.796775	0.0417623
$365$	−2.22471	−0.116447
$366$	27.3448	1.42934
$367$	−20.4308	−1.06648	−0.533239	−	0.845964i	$-0.679025\pi$
$367$	−20.4308	−1.06648	−0.533239	+	0.845964i	$0.679025\pi$
$368$	6.82278	0.355662
$369$	−16.7047	−0.869610
$370$	35.6165	1.85162
$371$	−0.320842	−0.0166573
$372$	0.178152	0.00923677
$373$	12.5231	0.648422	0.324211	−	0.945985i	$-0.394901\pi$
$373$	12.5231	0.648422	0.324211	+	0.945985i	$0.394901\pi$
$374$	8.08339	0.417982
$375$	40.3797	2.08520
$376$	−5.36150	−0.276498
$377$	−10.1196	−0.521184
$378$	−0.312231	−0.0160594
$379$	32.1780	1.65287	0.826436	−	0.563031i	$-0.190365\pi$
$379$	32.1780	1.65287	0.826436	+	0.563031i	$0.190365\pi$
$380$	27.6968	1.42082
$381$	37.4345	1.91783
$382$	14.4859	0.741162
$383$	37.0698	1.89418	0.947089	−	0.320971i	$-0.104009\pi$
$383$	37.0698	1.89418	0.947089	+	0.320971i	$0.104009\pi$
$384$	2.49912	0.127532
$385$	9.08982	0.463260
$386$	−21.0958	−1.07375
$387$	6.43164	0.326938
$388$	2.10060	0.106642
$389$	−10.0922	−0.511697	−0.255849	−	0.966717i	$-0.582355\pi$
$389$	−10.0922	−0.511697	−0.255849	+	0.966717i	$0.582355\pi$
$390$	14.7887	0.748856
$391$	−11.6629	−0.589818
$392$	6.74118	0.340481
$393$	26.4042	1.33192
$394$	24.1967	1.21901
$395$	61.0018	3.06933
$396$	15.3476	0.771246
$397$	19.5206	0.979713	0.489856	−	0.871803i	$-0.337049\pi$
$397$	19.5206	0.979713	0.489856	+	0.871803i	$0.337049\pi$
$398$	−16.5051	−0.827326
$399$	−9.31982	−0.466575
$400$	9.27631	0.463815
$401$	23.5426	1.17566	0.587829	−	0.808985i	$-0.299983\pi$
$401$	23.5426	1.17566	0.587829	+	0.808985i	$0.299983\pi$
$402$	−0.480361	−0.0239582
$403$	0.111646	0.00556147
$404$	2.07226	0.103099
$405$	30.9940	1.54011
$406$	3.28719	0.163140
$407$	44.5751	2.20950
$408$	−4.27200	−0.211496
$409$	−36.4807	−1.80385	−0.901927	−	0.431889i	$-0.857847\pi$
$409$	−36.4807	−1.80385	−0.901927	+	0.431889i	$0.857847\pi$
$410$	−19.4470	−0.960421
$411$	29.5181	1.45602
$412$	14.2849	0.703764
$413$	2.01556	0.0991792
$414$	−22.1439	−1.08831
$415$	−25.1502	−1.23457
$416$	1.56616	0.0767874
$417$	6.91299	0.338531
$418$	34.6633	1.69544
$419$	5.85624	0.286096	0.143048	−	0.989716i	$-0.454310\pi$
$419$	5.85624	0.286096	0.143048	+	0.989716i	$0.454310\pi$
$420$	−4.80389	−0.234406
$421$	14.3591	0.699819	0.349909	−	0.936784i	$-0.386212\pi$
$421$	14.3591	0.699819	0.349909	+	0.936784i	$0.386212\pi$
$422$	−12.4250	−0.604842
$423$	17.4012	0.846074
$424$	−0.630655	−0.0306273
$425$	−15.8570	−0.769177
$426$	11.3233	0.548615
$427$	−5.56657	−0.269385
$428$	−10.1314	−0.489718
$429$	18.5085	0.893598
$430$	7.48751	0.361080
$431$	−10.4899	−0.505280	−0.252640	−	0.967560i	$-0.581299\pi$
$431$	−10.4899	−0.505280	−0.252640	+	0.967560i	$0.581299\pi$
$432$	−0.613729	−0.0295280
$433$	−36.7641	−1.76677	−0.883384	−	0.468650i	$-0.844741\pi$
$433$	−36.7641	−1.76677	−0.883384	+	0.468650i	$0.844741\pi$
$434$	−0.0362664	−0.00174084
$435$	61.0126	2.92533
$436$	13.6923	0.655742
$437$	−50.0130	−2.39245
$438$	1.47147	0.0703096
$439$	38.0471	1.81589	0.907943	−	0.419093i	$-0.137652\pi$
$439$	38.0471	1.81589	0.907943	+	0.419093i	$0.137652\pi$
$440$	17.8672	0.851785
$441$	−21.8790	−1.04186
$442$	−2.67721	−0.127342
$443$	38.1362	1.81190	0.905952	−	0.423381i	$-0.139157\pi$
$443$	38.1362	1.81190	0.905952	+	0.423381i	$0.139157\pi$
$444$	−23.5575	−1.11799
$445$	2.52128	0.119520
$446$	13.1348	0.621949
$447$	−0.0560342	−0.00265033
$448$	−0.508744	−0.0240359
$449$	9.26607	0.437293	0.218646	−	0.975804i	$-0.429836\pi$
$449$	9.26607	0.437293	0.218646	+	0.975804i	$0.429836\pi$
$450$	−30.1070	−1.41926
$451$	−24.3385	−1.14605
$452$	−8.47807	−0.398775
$453$	−39.6368	−1.86230
$454$	8.32519	0.390720
$455$	−3.01053	−0.141136
$456$	−18.3193	−0.857879
$457$	40.9023	1.91333	0.956665	−	0.291189i	$-0.0940509\pi$
$457$	40.9023	1.91333	0.956665	+	0.291189i	$0.0940509\pi$
$458$	10.2884	0.480744
$459$	1.04911	0.0489683
$460$	−25.7792	−1.20196
$461$	4.97150	0.231546	0.115773	−	0.993276i	$-0.463065\pi$
$461$	4.97150	0.231546	0.115773	+	0.993276i	$0.463065\pi$
$462$	−6.01220	−0.279713
$463$	26.8750	1.24899	0.624494	−	0.781030i	$-0.285305\pi$
$463$	26.8750	1.24899	0.624494	+	0.781030i	$0.285305\pi$
$464$	6.46138	0.299962
$465$	−0.673131	−0.0312157
$466$	−29.6620	−1.37407
$467$	3.24381	0.150105	0.0750527	−	0.997180i	$-0.476087\pi$
$467$	3.24381	0.150105	0.0750527	+	0.997180i	$0.476087\pi$
$468$	−5.08310	−0.234966
$469$	0.0977869	0.00451538
$470$	20.2579	0.934427
$471$	−14.2184	−0.655148
$472$	3.96183	0.182358
$473$	9.37082	0.430871
$474$	−40.3479	−1.85324
$475$	−67.9981	−3.11997
$476$	0.869650	0.0398604
$477$	2.04684	0.0937184
$478$	27.9429	1.27808
$479$	28.9709	1.32371	0.661856	−	0.749631i	$-0.269769\pi$
$479$	28.9709	1.32371	0.661856	+	0.749631i	$0.269769\pi$
$480$	−9.44266	−0.430996
$481$	−14.7632	−0.673143
$482$	−8.10538	−0.369190
$483$	8.67455	0.394705
$484$	11.3613	0.516421
$485$	−7.93692	−0.360397
$486$	−22.3413	−1.01342
$487$	−5.33853	−0.241912	−0.120956	−	0.992658i	$-0.538596\pi$
$487$	−5.33853	−0.241912	−0.120956	+	0.992658i	$0.538596\pi$
$488$	−10.9418	−0.495312
$489$	−10.2987	−0.465724
$490$	−25.4709	−1.15066
$491$	26.8776	1.21297	0.606484	−	0.795096i	$-0.292580\pi$
$491$	26.8776	1.21297	0.606484	+	0.795096i	$0.292580\pi$
$492$	12.8627	0.579895
$493$	−11.0451	−0.497448
$494$	−11.4804	−0.516529
$495$	−57.9893	−2.60643
$496$	−0.0712862	−0.00320084
$497$	−2.30508	−0.103397
$498$	16.6349	0.745427
$499$	25.8270	1.15618	0.578088	−	0.815975i	$-0.303799\pi$
$499$	25.8270	1.15618	0.578088	+	0.815975i	$0.303799\pi$
$500$	−16.1576	−0.722590
$501$	−5.86236	−0.261911
$502$	−22.2154	−0.991524
$503$	27.8491	1.24173	0.620866	−	0.783917i	$-0.286781\pi$
$503$	27.8491	1.24173	0.620866	+	0.783917i	$0.286781\pi$
$504$	1.65117	0.0735489
$505$	−7.82983	−0.348423
$506$	−32.2633	−1.43428
$507$	26.3585	1.17062
$508$	−14.9791	−0.664590
$509$	7.29862	0.323505	0.161753	−	0.986831i	$-0.448285\pi$
$509$	7.29862	0.323505	0.161753	+	0.986831i	$0.448285\pi$
$510$	16.1413	0.714751
$511$	−0.299547	−0.0132512
$512$	−1.00000	−0.0441942
$513$	4.49882	0.198628
$514$	11.3368	0.500043
$515$	−53.9739	−2.37837
$516$	−4.95240	−0.218017
$517$	25.3533	1.11504
$518$	4.79560	0.210707
$519$	42.2532	1.85471
$520$	−5.91758	−0.259503
$521$	−29.9555	−1.31237	−0.656187	−	0.754598i	$-0.727832\pi$
$521$	−29.9555	−1.31237	−0.656187	+	0.754598i	$0.727832\pi$
$522$	−20.9709	−0.917872
$523$	29.8986	1.30737	0.653686	−	0.756766i	$-0.273222\pi$
$523$	29.8986	1.30737	0.653686	+	0.756766i	$0.273222\pi$
$524$	−10.5654	−0.461553
$525$	11.7940	0.514732
$526$	14.7510	0.643176
$527$	0.121857	0.00530818
$528$	−11.8177	−0.514301
$529$	23.5503	1.02393
$530$	2.38287	0.103505
$531$	−12.8584	−0.558009
$532$	3.72925	0.161683
$533$	8.06087	0.349155
$534$	−1.66763	−0.0721655
$535$	38.2804	1.65501
$536$	0.192213	0.00830232
$537$	63.6284	2.74577
$538$	−20.7231	−0.893434
$539$	−31.8775	−1.37306
$540$	2.31891	0.0997901
$541$	−11.9125	−0.512158	−0.256079	−	0.966656i	$-0.582431\pi$
$541$	−11.9125	−0.512158	−0.256079	+	0.966656i	$0.582431\pi$
$542$	28.7047	1.23297
$543$	4.98251	0.213820
$544$	1.70941	0.0732902
$545$	−51.7350	−2.21608
$546$	1.99123	0.0852169
$547$	−20.9557	−0.896001	−0.448001	−	0.894033i	$-0.647864\pi$
$547$	−20.9557	−0.896001	−0.448001	+	0.894033i	$0.647864\pi$
$548$	−11.8114	−0.504559
$549$	35.5125	1.51563
$550$	−43.8655	−1.87043
$551$	−47.3639	−2.01777
$552$	17.0509	0.725735
$553$	8.21361	0.349278
$554$	18.6874	0.793952
$555$	89.0098	3.77826
$556$	−2.76617	−0.117312
$557$	−40.3513	−1.70974	−0.854869	−	0.518843i	$-0.826363\pi$
$557$	−40.3513	−1.70974	−0.854869	+	0.518843i	$0.826363\pi$
$558$	0.231365	0.00979446
$559$	−3.10360	−0.131268
$560$	1.92224	0.0812294
$561$	20.2013	0.852901
$562$	21.3536	0.900747
$563$	−30.5264	−1.28653	−0.643266	−	0.765643i	$-0.722421\pi$
$563$	−30.5264	−1.28653	−0.643266	+	0.765643i	$0.722421\pi$
$564$	−13.3990	−0.564200
$565$	32.0335	1.34766
$566$	−22.4804	−0.944920
$567$	4.17320	0.175258
$568$	−4.53092	−0.190113
$569$	−5.77079	−0.241924	−0.120962	−	0.992657i	$-0.538598\pi$
$569$	−5.77079	−0.241924	−0.120962	+	0.992657i	$0.538598\pi$
$570$	69.2175	2.89920
$571$	1.36888	0.0572859	0.0286429	−	0.999590i	$-0.490881\pi$
$571$	1.36888	0.0572859	0.0286429	+	0.999590i	$0.490881\pi$
$572$	−7.40602	−0.309661
$573$	36.2019	1.51235
$574$	−2.61845	−0.109292
$575$	63.2902	2.63938
$576$	3.24558	0.135232
$577$	8.91467	0.371123	0.185561	−	0.982633i	$-0.440590\pi$
$577$	8.91467	0.371123	0.185561	+	0.982633i	$0.440590\pi$
$578$	14.0779	0.585565
$579$	−52.7208	−2.19100
$580$	−24.4137	−1.01372
$581$	−3.38636	−0.140490
$582$	5.24965	0.217605
$583$	2.98222	0.123511
$584$	−0.588797	−0.0243646
$585$	19.2060	0.794069
$586$	−14.4598	−0.597330
$587$	−40.2858	−1.66277	−0.831386	−	0.555695i	$-0.812452\pi$
$587$	−40.2858	−1.66277	−0.831386	+	0.555695i	$0.812452\pi$
$588$	16.8470	0.694758
$589$	0.522549	0.0215313
$590$	−14.9694	−0.616280
$591$	60.4702	2.48741
$592$	9.42635	0.387421
$593$	−11.7372	−0.481990	−0.240995	−	0.970526i	$-0.577474\pi$
$593$	−11.7372	−0.481990	−0.240995	+	0.970526i	$0.577474\pi$
$594$	2.90218	0.119078
$595$	−3.28589	−0.134708
$596$	0.0224216	0.000918426 0
$597$	−41.2482	−1.68817
$598$	10.6856	0.436965
$599$	−24.5752	−1.00411	−0.502057	−	0.864835i	$-0.667423\pi$
$599$	−24.5752	−1.00411	−0.502057	+	0.864835i	$0.667423\pi$
$600$	23.1826	0.946424
$601$	−1.53950	−0.0627973	−0.0313987	−	0.999507i	$-0.509996\pi$
$601$	−1.53950	−0.0627973	−0.0313987	+	0.999507i	$0.509996\pi$
$602$	1.00816	0.0410894
$603$	−0.623841	−0.0254048
$604$	15.8603	0.645348
$605$	−42.9274	−1.74525
$606$	5.17882	0.210375
$607$	16.9070	0.686232	0.343116	−	0.939293i	$-0.388518\pi$
$607$	16.9070	0.686232	0.343116	+	0.939293i	$0.388518\pi$
$608$	7.33030	0.297283
$609$	8.21506	0.332891
$610$	41.3425	1.67391
$611$	−8.39697	−0.339705
$612$	−5.54801	−0.224265
$613$	29.9590	1.21003	0.605017	−	0.796212i	$-0.293166\pi$
$613$	29.9590	1.21003	0.605017	+	0.796212i	$0.293166\pi$
$614$	8.08162	0.326148
$615$	−48.6004	−1.95976
$616$	2.40573	0.0969297
$617$	−23.8586	−0.960510	−0.480255	−	0.877129i	$-0.659456\pi$
$617$	−23.8586	−0.960510	−0.480255	+	0.877129i	$0.659456\pi$
$618$	35.6995	1.43604
$619$	−22.9469	−0.922316	−0.461158	−	0.887318i	$-0.652566\pi$
$619$	−22.9469	−0.922316	−0.461158	+	0.887318i	$0.652566\pi$
$620$	0.269348	0.0108173
$621$	−4.18733	−0.168032
$622$	21.1984	0.849980
$623$	0.339479	0.0136009
$624$	3.91402	0.156686
$625$	14.6683	0.586733
$626$	−16.5817	−0.662737
$627$	86.6276	3.45957
$628$	5.68936	0.227030
$629$	−16.1135	−0.642486
$630$	−6.23877	−0.248559
$631$	−30.4685	−1.21293	−0.606467	−	0.795109i	$-0.707414\pi$
$631$	−30.4685	−1.21293	−0.606467	+	0.795109i	$0.707414\pi$
$632$	16.1449	0.642209
$633$	−31.0516	−1.23419
$634$	−20.8454	−0.827876
$635$	56.5971	2.24599
$636$	−1.57608	−0.0624956
$637$	10.5578	0.418314
$638$	−30.5544	−1.20966
$639$	14.7055	0.581739
$640$	3.77840	0.149354
$641$	−26.6974	−1.05448	−0.527241	−	0.849716i	$-0.676774\pi$
$641$	−26.6974	−1.05448	−0.527241	+	0.849716i	$0.676774\pi$
$642$	−25.3195	−0.999280
$643$	22.7392	0.896748	0.448374	−	0.893846i	$-0.352003\pi$
$643$	22.7392	0.896748	0.448374	+	0.893846i	$0.352003\pi$
$644$	−3.47105	−0.136778
$645$	18.7121	0.736790
$646$	−12.5305	−0.493005
$647$	7.31694	0.287659	0.143829	−	0.989603i	$-0.454058\pi$
$647$	7.31694	0.287659	0.143829	+	0.989603i	$0.454058\pi$
$648$	8.20296	0.322243
$649$	−18.7346	−0.735397
$650$	14.5282	0.569842
$651$	−0.0906339	−0.00355222
$652$	4.12095	0.161389
$653$	−17.8937	−0.700236	−0.350118	−	0.936706i	$-0.613858\pi$
$653$	−17.8937	−0.700236	−0.350118	+	0.936706i	$0.613858\pi$
$654$	34.2186	1.33805
$655$	39.9204	1.55982
$656$	−5.14690	−0.200953
$657$	1.91099	0.0745546
$658$	2.72763	0.106334
$659$	23.3582	0.909905	0.454952	−	0.890516i	$-0.349656\pi$
$659$	23.3582	0.909905	0.454952	+	0.890516i	$0.349656\pi$
$660$	44.6522	1.73808
$661$	22.6576	0.881280	0.440640	−	0.897684i	$-0.354752\pi$
$661$	22.6576	0.881280	0.440640	+	0.897684i	$0.354752\pi$
$662$	3.40899	0.132494
$663$	−6.69065	−0.259843
$664$	−6.65631	−0.258315
$665$	−14.0906	−0.546409
$666$	−30.5940	−1.18549
$667$	44.0846	1.70696
$668$	2.34577	0.0907607
$669$	32.8253	1.26910
$670$	−0.726256	−0.0280577
$671$	51.7412	1.99745
$672$	−1.27141	−0.0490457
$673$	−7.86813	−0.303294	−0.151647	−	0.988435i	$-0.548458\pi$
$673$	−7.86813	−0.303294	−0.151647	+	0.988435i	$0.548458\pi$
$674$	−13.2438	−0.510131
$675$	−5.69313	−0.219129
$676$	−10.5471	−0.405659
$677$	36.0153	1.38418	0.692090	−	0.721811i	$-0.256690\pi$
$677$	36.0153	1.38418	0.692090	+	0.721811i	$0.256690\pi$
$678$	−21.1877	−0.813708
$679$	−1.06867	−0.0410118
$680$	−6.45882	−0.247685
$681$	20.8056	0.797273
$682$	0.337096	0.0129081
$683$	−26.3791	−1.00937	−0.504685	−	0.863304i	$-0.668391\pi$
$683$	−26.3791	−1.00937	−0.504685	+	0.863304i	$0.668391\pi$
$684$	−23.7911	−0.909674
$685$	44.6283	1.70516
$686$	−6.99074	−0.266908
$687$	25.7118	0.980968
$688$	1.98166	0.0755501
$689$	−0.987707	−0.0376287
$690$	−64.4252	−2.45262
$691$	−1.26320	−0.0480543	−0.0240271	−	0.999711i	$-0.507649\pi$
$691$	−1.26320	−0.0480543	−0.0240271	+	0.999711i	$0.507649\pi$
$692$	−16.9073	−0.642718
$693$	−7.80799	−0.296601
$694$	−16.4422	−0.624138
$695$	10.4517	0.396456
$696$	16.1477	0.612078
$697$	8.79814	0.333253
$698$	−4.96654	−0.187986
$699$	−74.1288	−2.80381
$700$	−4.71926	−0.178371
$701$	−21.1907	−0.800362	−0.400181	−	0.916436i	$-0.631053\pi$
$701$	−21.1907	−0.800362	−0.400181	+	0.916436i	$0.631053\pi$
$702$	−0.961198	−0.0362781
$703$	−69.0980	−2.60608
$704$	4.72877	0.178222
$705$	50.6268	1.90672
$706$	12.8929	0.485229
$707$	−1.05425	−0.0396492
$708$	9.90108	0.372105
$709$	−35.9556	−1.35034	−0.675171	−	0.737661i	$-0.735930\pi$
$709$	−35.9556	−1.35034	−0.675171	+	0.737661i	$0.735930\pi$
$710$	17.1196	0.642488
$711$	−52.3995	−1.96513
$712$	0.667289	0.0250077
$713$	−0.486370	−0.0182147
$714$	2.17336	0.0813358
$715$	27.9829	1.04650
$716$	−25.4604	−0.951498
$717$	69.8325	2.60794
$718$	0.764622	0.0285354
$719$	−24.7782	−0.924070	−0.462035	−	0.886862i	$-0.652881\pi$
$719$	−24.7782	−0.924070	−0.462035	+	0.886862i	$0.652881\pi$
$720$	−12.2631	−0.457018
$721$	−7.26733	−0.270650
$722$	−34.7333	−1.29264
$723$	−20.2563	−0.753339
$724$	−1.99371	−0.0740957
$725$	59.9378	2.22603
$726$	28.3931	1.05377
$727$	10.8869	0.403775	0.201887	−	0.979409i	$-0.435293\pi$
$727$	10.8869	0.403775	0.201887	+	0.979409i	$0.435293\pi$
$728$	−0.796775	−0.0295304
$729$	−31.2247	−1.15647
$730$	2.22471	0.0823402
$731$	−3.38746	−0.125290
$732$	−27.3448	−1.01069
$733$	32.3175	1.19368	0.596838	−	0.802362i	$-0.296424\pi$
$733$	32.3175	1.19368	0.596838	+	0.802362i	$0.296424\pi$
$734$	20.4308	0.754114
$735$	−63.6547	−2.34794
$736$	−6.82278	−0.251491
$737$	−0.908929	−0.0334808
$738$	16.7047	0.614907
$739$	−20.6425	−0.759346	−0.379673	−	0.925121i	$-0.623963\pi$
$739$	−20.6425	−0.759346	−0.379673	+	0.925121i	$0.623963\pi$
$740$	−35.6165	−1.30929
$741$	−28.6909	−1.05399
$742$	0.320842	0.0117785
$743$	32.4736	1.19134	0.595670	−	0.803230i	$-0.296887\pi$
$743$	32.4736	1.19134	0.595670	+	0.803230i	$0.296887\pi$
$744$	−0.178152	−0.00653138
$745$	−0.0847179	−0.00310382
$746$	−12.5231	−0.458503
$747$	21.6036	0.790434
$748$	−8.08339	−0.295558
$749$	5.15427	0.188333
$750$	−40.3797	−1.47446
$751$	1.17709	0.0429525	0.0214763	−	0.999769i	$-0.493163\pi$
$751$	1.17709	0.0429525	0.0214763	+	0.999769i	$0.493163\pi$
$752$	5.36150	0.195514
$753$	−55.5189	−2.02322
$754$	10.1196	0.368533
$755$	−59.9267	−2.18096
$756$	0.312231	0.0113557
$757$	−6.43499	−0.233884	−0.116942	−	0.993139i	$-0.537309\pi$
$757$	−6.43499	−0.233884	−0.116942	+	0.993139i	$0.537309\pi$
$758$	−32.1780	−1.16876
$759$	−80.6298	−2.92668
$760$	−27.6968	−1.00467
$761$	21.5453	0.781015	0.390508	−	0.920600i	$-0.372300\pi$
$761$	21.5453	0.781015	0.390508	+	0.920600i	$0.372300\pi$
$762$	−37.4345	−1.35611
$763$	−6.96587	−0.252182
$764$	−14.4859	−0.524081
$765$	20.9626	0.757905
$766$	−37.0698	−1.33939
$767$	6.20487	0.224045
$768$	−2.49912	−0.0901791
$769$	−23.3646	−0.842550	−0.421275	−	0.906933i	$-0.638417\pi$
$769$	−23.3646	−0.842550	−0.421275	+	0.906933i	$0.638417\pi$
$770$	−9.08982	−0.327574
$771$	28.3319	1.02035
$772$	21.0958	0.759254
$773$	12.0401	0.433052	0.216526	−	0.976277i	$-0.430527\pi$
$773$	12.0401	0.433052	0.216526	+	0.976277i	$0.430527\pi$
$774$	−6.43164	−0.231180
$775$	−0.661272	−0.0237536
$776$	−2.10060	−0.0754073
$777$	11.9848	0.429951
$778$	10.0922	0.361825
$779$	37.7283	1.35176
$780$	−14.7887	−0.529521
$781$	21.4257	0.766671
$782$	11.6629	0.417065
$783$	−3.96553	−0.141717
$784$	−6.74118	−0.240756
$785$	−21.4967	−0.767250
$786$	−26.4042	−0.941808
$787$	23.8812	0.851272	0.425636	−	0.904894i	$-0.360050\pi$
$787$	23.8812	0.851272	0.425636	+	0.904894i	$0.360050\pi$
$788$	−24.1967	−0.861970
$789$	36.8646	1.31241
$790$	−61.0018	−2.17035
$791$	4.31317	0.153359
$792$	−15.3476	−0.545353
$793$	−17.1366	−0.608539
$794$	−19.5206	−0.692761
$795$	5.95506	0.211204
$796$	16.5051	0.585008
$797$	22.4521	0.795295	0.397648	−	0.917538i	$-0.369827\pi$
$797$	22.4521	0.795295	0.397648	+	0.917538i	$0.369827\pi$
$798$	9.31982	0.329918
$799$	−9.16498	−0.324234
$800$	−9.27631	−0.327967
$801$	−2.16574	−0.0765226
$802$	−23.5426	−0.831316
$803$	2.78428	0.0982552
$804$	0.480361	0.0169410
$805$	13.1150	0.462243
$806$	−0.111646	−0.00393255
$807$	−51.7893	−1.82307
$808$	−2.07226	−0.0729019
$809$	−19.8751	−0.698772	−0.349386	−	0.936979i	$-0.613610\pi$
$809$	−19.8751	−0.698772	−0.349386	+	0.936979i	$0.613610\pi$
$810$	−30.9940	−1.08902
$811$	6.78435	0.238231	0.119115	−	0.992880i	$-0.461994\pi$
$811$	6.78435	0.238231	0.119115	+	0.992880i	$0.461994\pi$
$812$	−3.28719	−0.115358
$813$	71.7364	2.51591
$814$	−44.5751	−1.56235
$815$	−15.5706	−0.545414
$816$	4.27200	0.149550
$817$	−14.5262	−0.508207
$818$	36.4807	1.27552
$819$	2.58600	0.0903620
$820$	19.4470	0.679120
$821$	−31.2628	−1.09108	−0.545540	−	0.838085i	$-0.683675\pi$
$821$	−31.2628	−1.09108	−0.545540	+	0.838085i	$0.683675\pi$
$822$	−29.5181	−1.02956
$823$	42.3528	1.47632	0.738162	−	0.674623i	$-0.235694\pi$
$823$	42.3528	1.47632	0.738162	+	0.674623i	$0.235694\pi$
$824$	−14.2849	−0.497636
$825$	−109.625	−3.81665
$826$	−2.01556	−0.0701303
$827$	35.5451	1.23602	0.618011	−	0.786169i	$-0.287939\pi$
$827$	35.5451	1.23602	0.618011	+	0.786169i	$0.287939\pi$
$828$	22.1439	0.769552
$829$	4.59014	0.159422	0.0797111	−	0.996818i	$-0.474600\pi$
$829$	4.59014	0.159422	0.0797111	+	0.996818i	$0.474600\pi$
$830$	25.1502	0.872976
$831$	46.7020	1.62007
$832$	−1.56616	−0.0542969
$833$	11.5234	0.399263
$834$	−6.91299	−0.239377
$835$	−8.86327	−0.306726
$836$	−34.6633	−1.19886
$837$	0.0437504	0.00151223
$838$	−5.85624	−0.202300
$839$	−1.67928	−0.0579753	−0.0289877	−	0.999580i	$-0.509228\pi$
$839$	−1.67928	−0.0579753	−0.0289877	+	0.999580i	$0.509228\pi$
$840$	4.80389	0.165750
$841$	12.7495	0.439636
$842$	−14.3591	−0.494847
$843$	53.3651	1.83799
$844$	12.4250	0.427688
$845$	39.8513	1.37093
$846$	−17.4012	−0.598265
$847$	−5.77997	−0.198602
$848$	0.630655	0.0216568
$849$	−56.1810	−1.92813
$850$	15.8570	0.543890
$851$	64.3139	2.20465
$852$	−11.3233	−0.387930
$853$	−20.2165	−0.692201	−0.346101	−	0.938197i	$-0.612494\pi$
$853$	−20.2165	−0.692201	−0.346101	+	0.938197i	$0.612494\pi$
$854$	5.56657	0.190484
$855$	89.8922	3.07425
$856$	10.1314	0.346283
$857$	−7.18968	−0.245595	−0.122797	−	0.992432i	$-0.539187\pi$
$857$	−7.18968	−0.245595	−0.122797	+	0.992432i	$0.539187\pi$
$858$	−18.5085	−0.631869
$859$	−31.0062	−1.05792	−0.528959	−	0.848647i	$-0.677417\pi$
$859$	−31.0062	−1.05792	−0.528959	+	0.848647i	$0.677417\pi$
$860$	−7.48751	−0.255322
$861$	−6.54381	−0.223013
$862$	10.4899	0.357287
$863$	19.8583	0.675986	0.337993	−	0.941149i	$-0.390252\pi$
$863$	19.8583	0.675986	0.337993	+	0.941149i	$0.390252\pi$
$864$	0.613729	0.0208795
$865$	63.8824	2.17207
$866$	36.7641	1.24929
$867$	35.1824	1.19486
$868$	0.0362664	0.00123096
$869$	−76.3454	−2.58984
$870$	−61.0126	−2.06852
$871$	0.301036	0.0102002
$872$	−13.6923	−0.463680
$873$	6.81767	0.230743
$874$	50.0130	1.69172
$875$	8.22008	0.277889
$876$	−1.47147	−0.0497164
$877$	23.8428	0.805114	0.402557	−	0.915395i	$-0.368122\pi$
$877$	23.8428	0.805114	0.402557	+	0.915395i	$0.368122\pi$
$878$	−38.0471	−1.28403
$879$	−36.1368	−1.21886
$880$	−17.8672	−0.602303
$881$	−21.4631	−0.723110	−0.361555	−	0.932351i	$-0.617754\pi$
$881$	−21.4631	−0.723110	−0.361555	+	0.932351i	$0.617754\pi$
$882$	21.8790	0.736705
$883$	−9.49777	−0.319626	−0.159813	−	0.987147i	$-0.551089\pi$
$883$	−9.49777	−0.319626	−0.159813	+	0.987147i	$0.551089\pi$
$884$	2.67721	0.0900442
$885$	−37.4102	−1.25753
$886$	−38.1362	−1.28121
$887$	16.9315	0.568504	0.284252	−	0.958750i	$-0.408255\pi$
$887$	16.9315	0.568504	0.284252	+	0.958750i	$0.408255\pi$
$888$	23.5575	0.790539
$889$	7.62053	0.255584
$890$	−2.52128	−0.0845136
$891$	−38.7899	−1.29951
$892$	−13.1348	−0.439784
$893$	−39.3014	−1.31517
$894$	0.0560342	0.00187407
$895$	96.1994	3.21559
$896$	0.508744	0.0169959
$897$	26.7045	0.891636
$898$	−9.26607	−0.309213
$899$	−0.460607	−0.0153621
$900$	30.1070	1.00357
$901$	−1.07805	−0.0359149
$902$	24.3385	0.810383
$903$	2.51950	0.0838438
$904$	8.47807	0.281976
$905$	7.53304	0.250407
$906$	39.6368	1.31685
$907$	45.8683	1.52303	0.761516	−	0.648146i	$-0.224456\pi$
$907$	45.8683	1.52303	0.761516	+	0.648146i	$0.224456\pi$
$908$	−8.32519	−0.276281
$909$	6.72569	0.223077
$910$	3.01053	0.0997982
$911$	49.6128	1.64375	0.821873	−	0.569671i	$-0.192929\pi$
$911$	49.6128	1.64375	0.821873	+	0.569671i	$0.192929\pi$
$912$	18.3193	0.606612
$913$	31.4762	1.04171
$914$	−40.9023	−1.35293
$915$	103.320	3.41564
$916$	−10.2884	−0.339937
$917$	5.37510	0.177501
$918$	−1.04911	−0.0346258
$919$	44.4103	1.46496	0.732481	−	0.680787i	$-0.238362\pi$
$919$	44.4103	1.46496	0.732481	+	0.680787i	$0.238362\pi$
$920$	25.7792	0.849914
$921$	20.1969	0.665510
$922$	−4.97150	−0.163728
$923$	−7.09615	−0.233573
$924$	6.01220	0.197787
$925$	87.4417	2.87507
$926$	−26.8750	−0.883167
$927$	46.3626	1.52275
$928$	−6.46138	−0.212105
$929$	11.7818	0.386549	0.193274	−	0.981145i	$-0.438089\pi$
$929$	11.7818	0.386549	0.193274	+	0.981145i	$0.438089\pi$
$930$	0.673131	0.0220728
$931$	49.4149	1.61951
$932$	29.6620	0.971611
$933$	52.9774	1.73440
$934$	−3.24381	−0.106141
$935$	30.5423	0.998840
$936$	5.08310	0.166146
$937$	1.07946	0.0352644	0.0176322	−	0.999845i	$-0.494387\pi$
$937$	1.07946	0.0352644	0.0176322	+	0.999845i	$0.494387\pi$
$938$	−0.0977869	−0.00319286
$939$	−41.4395	−1.35233
$940$	−20.2579	−0.660740
$941$	42.9535	1.40024	0.700121	−	0.714024i	$-0.253129\pi$
$941$	42.9535	1.40024	0.700121	+	0.714024i	$0.253129\pi$
$942$	14.2184	0.463260
$943$	−35.1161	−1.14354
$944$	−3.96183	−0.128947
$945$	−1.17973	−0.0383767
$946$	−9.37082	−0.304672
$947$	9.97975	0.324298	0.162149	−	0.986766i	$-0.448157\pi$
$947$	9.97975	0.324298	0.162149	+	0.986766i	$0.448157\pi$
$948$	40.3479	1.31044
$949$	−0.922150	−0.0299343
$950$	67.9981	2.20615
$951$	−52.0950	−1.68930
$952$	−0.869650	−0.0281855
$953$	−17.2418	−0.558516	−0.279258	−	0.960216i	$-0.590089\pi$
$953$	−17.2418	−0.558516	−0.279258	+	0.960216i	$0.590089\pi$
$954$	−2.04684	−0.0662689
$955$	54.7334	1.77113
$956$	−27.9429	−0.903738
$957$	−76.3589	−2.46833
$958$	−28.9709	−0.936006
$959$	6.00899	0.194040
$960$	9.44266	0.304760
$961$	−30.9949	−0.999836
$962$	14.7632	0.475984
$963$	−32.8822	−1.05961
$964$	8.10538	0.261057
$965$	−79.7083	−2.56590
$966$	−8.67455	−0.279099
$967$	−2.59124	−0.0833286	−0.0416643	−	0.999132i	$-0.513266\pi$
$967$	−2.59124	−0.0833286	−0.0416643	+	0.999132i	$0.513266\pi$
$968$	−11.3613	−0.365165
$969$	−31.3151	−1.00599
$970$	7.93692	0.254839
$971$	−56.9588	−1.82790	−0.913948	−	0.405832i	$-0.866982\pi$
$971$	−56.9588	−1.82790	−0.913948	+	0.405832i	$0.866982\pi$
$972$	22.3413	0.716598
$973$	1.40727	0.0451151
$974$	5.33853	0.171058
$975$	36.3076	1.16277
$976$	10.9418	0.350238
$977$	14.5138	0.464338	0.232169	−	0.972676i	$-0.425418\pi$
$977$	14.5138	0.464338	0.232169	+	0.972676i	$0.425418\pi$
$978$	10.2987	0.329317
$979$	−3.15546	−0.100849
$980$	25.4709	0.813637
$981$	44.4394	1.41884
$982$	−26.8776	−0.857697
$983$	−28.6998	−0.915381	−0.457691	−	0.889112i	$-0.651323\pi$
$983$	−28.6998	−0.915381	−0.457691	+	0.889112i	$0.651323\pi$
$984$	−12.8627	−0.410048
$985$	91.4246	2.91303
$986$	11.0451	0.351749
$987$	6.81666	0.216977
$988$	11.4804	0.365241
$989$	13.5204	0.429925
$990$	57.9893	1.84302
$991$	−32.0748	−1.01889	−0.509444	−	0.860504i	$-0.670149\pi$
$991$	−32.0748	−1.01889	−0.509444	+	0.860504i	$0.670149\pi$
$992$	0.0712862	0.00226334
$993$	8.51946	0.270357
$994$	2.30508	0.0731126
$995$	−62.3629	−1.97704
$996$	−16.6349	−0.527097
$997$	17.8809	0.566295	0.283147	−	0.959076i	$-0.408621\pi$
$997$	17.8809	0.566295	0.283147	+	0.959076i	$0.408621\pi$
$998$	−25.8270	−0.817539
$999$	−5.78522	−0.183036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.11	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.49912 q^{3} +1.00000 q^{4} -3.77840 q^{5} +2.49912 q^{6} -0.508744 q^{7} -1.00000 q^{8} +3.24558 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.49912 q^{3} +1.00000 q^{4} -3.77840 q^{5} +2.49912 q^{6} -0.508744 q^{7} -1.00000 q^{8} +3.24558 q^{9} +3.77840 q^{10} +4.72877 q^{11} -2.49912 q^{12} -1.56616 q^{13} +0.508744 q^{14} +9.44266 q^{15} +1.00000 q^{16} -1.70941 q^{17} -3.24558 q^{18} -7.33030 q^{19} -3.77840 q^{20} +1.27141 q^{21} -4.72877 q^{22} +6.82278 q^{23} +2.49912 q^{24} +9.27631 q^{25} +1.56616 q^{26} -0.613729 q^{27} -0.508744 q^{28} +6.46138 q^{29} -9.44266 q^{30} -0.0712862 q^{31} -1.00000 q^{32} -11.8177 q^{33} +1.70941 q^{34} +1.92224 q^{35} +3.24558 q^{36} +9.42635 q^{37} +7.33030 q^{38} +3.91402 q^{39} +3.77840 q^{40} -5.14690 q^{41} -1.27141 q^{42} +1.98166 q^{43} +4.72877 q^{44} -12.2631 q^{45} -6.82278 q^{46} +5.36150 q^{47} -2.49912 q^{48} -6.74118 q^{49} -9.27631 q^{50} +4.27200 q^{51} -1.56616 q^{52} +0.630655 q^{53} +0.613729 q^{54} -17.8672 q^{55} +0.508744 q^{56} +18.3193 q^{57} -6.46138 q^{58} -3.96183 q^{59} +9.44266 q^{60} +10.9418 q^{61} +0.0712862 q^{62} -1.65117 q^{63} +1.00000 q^{64} +5.91758 q^{65} +11.8177 q^{66} -0.192213 q^{67} -1.70941 q^{68} -17.0509 q^{69} -1.92224 q^{70} +4.53092 q^{71} -3.24558 q^{72} +0.588797 q^{73} -9.42635 q^{74} -23.1826 q^{75} -7.33030 q^{76} -2.40573 q^{77} -3.91402 q^{78} -16.1449 q^{79} -3.77840 q^{80} -8.20296 q^{81} +5.14690 q^{82} +6.65631 q^{83} +1.27141 q^{84} +6.45882 q^{85} -1.98166 q^{86} -16.1477 q^{87} -4.72877 q^{88} -0.667289 q^{89} +12.2631 q^{90} +0.796775 q^{91} +6.82278 q^{92} +0.178152 q^{93} -5.36150 q^{94} +27.6968 q^{95} +2.49912 q^{96} +2.10060 q^{97} +6.74118 q^{98} +15.3476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 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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