LMFDB - Embedded newform 8031.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.01958 q^{2} -1.00000 q^{3} +2.07869 q^{4} -3.91431 q^{5} +2.01958 q^{6} +1.55136 q^{7} -0.158921 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.01958 q^{2} -1.00000 q^{3} +2.07869 q^{4} -3.91431 q^{5} +2.01958 q^{6} +1.55136 q^{7} -0.158921 q^{8} +1.00000 q^{9} +7.90525 q^{10} +2.49461 q^{11} -2.07869 q^{12} -4.97450 q^{13} -3.13309 q^{14} +3.91431 q^{15} -3.83643 q^{16} +1.21523 q^{17} -2.01958 q^{18} -1.44450 q^{19} -8.13664 q^{20} -1.55136 q^{21} -5.03806 q^{22} +0.000901793 q^{23} +0.158921 q^{24} +10.3218 q^{25} +10.0464 q^{26} -1.00000 q^{27} +3.22479 q^{28} -2.86225 q^{29} -7.90525 q^{30} +6.66781 q^{31} +8.06580 q^{32} -2.49461 q^{33} -2.45425 q^{34} -6.07249 q^{35} +2.07869 q^{36} +1.14698 q^{37} +2.91728 q^{38} +4.97450 q^{39} +0.622066 q^{40} -10.7711 q^{41} +3.13309 q^{42} +8.92688 q^{43} +5.18553 q^{44} -3.91431 q^{45} -0.00182124 q^{46} -6.91126 q^{47} +3.83643 q^{48} -4.59329 q^{49} -20.8457 q^{50} -1.21523 q^{51} -10.3405 q^{52} +0.762037 q^{53} +2.01958 q^{54} -9.76468 q^{55} -0.246544 q^{56} +1.44450 q^{57} +5.78054 q^{58} +4.41002 q^{59} +8.13664 q^{60} +6.80128 q^{61} -13.4662 q^{62} +1.55136 q^{63} -8.61665 q^{64} +19.4717 q^{65} +5.03806 q^{66} -3.44974 q^{67} +2.52608 q^{68} -0.000901793 q^{69} +12.2639 q^{70} -14.1111 q^{71} -0.158921 q^{72} +7.81730 q^{73} -2.31641 q^{74} -10.3218 q^{75} -3.00267 q^{76} +3.87004 q^{77} -10.0464 q^{78} +11.3893 q^{79} +15.0170 q^{80} +1.00000 q^{81} +21.7532 q^{82} -2.44424 q^{83} -3.22479 q^{84} -4.75678 q^{85} -18.0285 q^{86} +2.86225 q^{87} -0.396446 q^{88} -10.8863 q^{89} +7.90525 q^{90} -7.71723 q^{91} +0.00187455 q^{92} -6.66781 q^{93} +13.9578 q^{94} +5.65423 q^{95} -8.06580 q^{96} -9.60470 q^{97} +9.27650 q^{98} +2.49461 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.01958	−1.42806	−0.714028	−	0.700117i	$-0.753131\pi$
$2$	−2.01958	−1.42806	−0.714028	+	0.700117i	$0.753131\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.07869	1.03935
$5$	−3.91431	−1.75053	−0.875266	−	0.483642i	$-0.839314\pi$
$5$	−3.91431	−1.75053	−0.875266	+	0.483642i	$0.839314\pi$
$6$	2.01958	0.824489
$7$	1.55136	0.586358	0.293179	−	0.956058i	$-0.405287\pi$
$7$	1.55136	0.586358	0.293179	+	0.956058i	$0.405287\pi$
$8$	−0.158921	−0.0561871
$9$	1.00000	0.333333
$10$	7.90525	2.49986
$11$	2.49461	0.752154	0.376077	−	0.926588i	$-0.377273\pi$
$11$	2.49461	0.752154	0.376077	+	0.926588i	$0.377273\pi$
$12$	−2.07869	−0.600066
$13$	−4.97450	−1.37968	−0.689839	−	0.723962i	$-0.742319\pi$
$13$	−4.97450	−1.37968	−0.689839	+	0.723962i	$0.742319\pi$
$14$	−3.13309	−0.837353
$15$	3.91431	1.01067
$16$	−3.83643	−0.959107
$17$	1.21523	0.294736	0.147368	−	0.989082i	$-0.452920\pi$
$17$	1.21523	0.294736	0.147368	+	0.989082i	$0.452920\pi$
$18$	−2.01958	−0.476019
$19$	−1.44450	−0.331391	−0.165696	−	0.986177i	$-0.552987\pi$
$19$	−1.44450	−0.331391	−0.165696	+	0.986177i	$0.552987\pi$
$20$	−8.13664	−1.81941
$21$	−1.55136	−0.338534
$22$	−5.03806	−1.07412
$23$	0.000901793 0	0.000188037 0	9.40184e−5	−	1.00000i	$-0.499970\pi$
$23$	0.000901793 0	0.000188037 0	9.40184e−5		1.00000i	$0.499970\pi$
$24$	0.158921	0.0324396
$25$	10.3218	2.06436
$26$	10.0464	1.97026
$27$	−1.00000	−0.192450
$28$	3.22479	0.609429
$29$	−2.86225	−0.531507	−0.265754	−	0.964041i	$-0.585621\pi$
$29$	−2.86225	−0.531507	−0.265754	+	0.964041i	$0.585621\pi$
$30$	−7.90525	−1.44329
$31$	6.66781	1.19757	0.598787	−	0.800908i	$-0.295650\pi$
$31$	6.66781	1.19757	0.598787	+	0.800908i	$0.295650\pi$
$32$	8.06580	1.42585
$33$	−2.49461	−0.434256
$34$	−2.45425	−0.420900
$35$	−6.07249	−1.02644
$36$	2.07869	0.346448
$37$	1.14698	0.188562	0.0942812	−	0.995546i	$-0.469945\pi$
$37$	1.14698	0.188562	0.0942812	+	0.995546i	$0.469945\pi$
$38$	2.91728	0.473246
$39$	4.97450	0.796558
$40$	0.622066	0.0983573
$41$	−10.7711	−1.68217	−0.841085	−	0.540903i	$-0.818083\pi$
$41$	−10.7711	−1.68217	−0.841085	+	0.540903i	$0.818083\pi$
$42$	3.13309	0.483446
$43$	8.92688	1.36134	0.680668	−	0.732592i	$-0.261689\pi$
$43$	8.92688	1.36134	0.680668	+	0.732592i	$0.261689\pi$
$44$	5.18553	0.781747
$45$	−3.91431	−0.583511
$46$	−0.00182124	−0.000268527 0
$47$	−6.91126	−1.00811	−0.504056	−	0.863671i	$-0.668159\pi$
$47$	−6.91126	−1.00811	−0.504056	+	0.863671i	$0.668159\pi$
$48$	3.83643	0.553741
$49$	−4.59329	−0.656184
$50$	−20.8457	−2.94803
$51$	−1.21523	−0.170166
$52$	−10.3405	−1.43396
$53$	0.762037	0.104674	0.0523369	−	0.998629i	$-0.483333\pi$
$53$	0.762037	0.104674	0.0523369	+	0.998629i	$0.483333\pi$
$54$	2.01958	0.274830
$55$	−9.76468	−1.31667
$56$	−0.246544	−0.0329458
$57$	1.44450	0.191329
$58$	5.78054	0.759022
$59$	4.41002	0.574135	0.287068	−	0.957910i	$-0.407320\pi$
$59$	4.41002	0.574135	0.287068	+	0.957910i	$0.407320\pi$
$60$	8.13664	1.05044
$61$	6.80128	0.870814	0.435407	−	0.900234i	$-0.356604\pi$
$61$	6.80128	0.870814	0.435407	+	0.900234i	$0.356604\pi$
$62$	−13.4662	−1.71020
$63$	1.55136	0.195453
$64$	−8.61665	−1.07708
$65$	19.4717	2.41517
$66$	5.03806	0.620142
$67$	−3.44974	−0.421452	−0.210726	−	0.977545i	$-0.567583\pi$
$67$	−3.44974	−0.421452	−0.210726	+	0.977545i	$0.567583\pi$
$68$	2.52608	0.306332
$69$	−0.000901793 0	−0.000108563 0
$70$	12.2639	1.46581
$71$	−14.1111	−1.67468	−0.837338	−	0.546686i	$-0.815889\pi$
$71$	−14.1111	−1.67468	−0.837338	+	0.546686i	$0.815889\pi$
$72$	−0.158921	−0.0187290
$73$	7.81730	0.914946	0.457473	−	0.889224i	$-0.348755\pi$
$73$	7.81730	0.914946	0.457473	+	0.889224i	$0.348755\pi$
$74$	−2.31641	−0.269278
$75$	−10.3218	−1.19186
$76$	−3.00267	−0.344430
$77$	3.87004	0.441032
$78$	−10.0464	−1.13753
$79$	11.3893	1.28140	0.640699	−	0.767792i	$-0.278645\pi$
$79$	11.3893	1.28140	0.640699	+	0.767792i	$0.278645\pi$
$80$	15.0170	1.67895
$81$	1.00000	0.111111
$82$	21.7532	2.40223
$83$	−2.44424	−0.268291	−0.134145	−	0.990962i	$-0.542829\pi$
$83$	−2.44424	−0.268291	−0.134145	+	0.990962i	$0.542829\pi$
$84$	−3.22479	−0.351854
$85$	−4.75678	−0.515945
$86$	−18.0285	−1.94407
$87$	2.86225	0.306866
$88$	−0.396446	−0.0422613
$89$	−10.8863	−1.15394	−0.576971	−	0.816764i	$-0.695766\pi$
$89$	−10.8863	−1.15394	−0.576971	+	0.816764i	$0.695766\pi$
$90$	7.90525	0.833286
$91$	−7.71723	−0.808986
$92$	0.00187455	0.000195435 0
$93$	−6.66781	−0.691420
$94$	13.9578	1.43964
$95$	5.65423	0.580111
$96$	−8.06580	−0.823212
$97$	−9.60470	−0.975210	−0.487605	−	0.873064i	$-0.662129\pi$
$97$	−9.60470	−0.975210	−0.487605	+	0.873064i	$0.662129\pi$
$98$	9.27650	0.937068
$99$	2.49461	0.250718
$100$	21.4559	2.14559
$101$	−8.50964	−0.846741	−0.423370	−	0.905957i	$-0.639153\pi$
$101$	−8.50964	−0.846741	−0.423370	+	0.905957i	$0.639153\pi$
$102$	2.45425	0.243006
$103$	−9.15337	−0.901908	−0.450954	−	0.892547i	$-0.648916\pi$
$103$	−9.15337	−0.901908	−0.450954	+	0.892547i	$0.648916\pi$
$104$	0.790553	0.0775201
$105$	6.07249	0.592615
$106$	−1.53899	−0.149480
$107$	0.568615	0.0549701	0.0274851	−	0.999622i	$-0.491250\pi$
$107$	0.568615	0.0549701	0.0274851	+	0.999622i	$0.491250\pi$
$108$	−2.07869	−0.200022
$109$	17.6014	1.68591	0.842954	−	0.537985i	$-0.180814\pi$
$109$	17.6014	1.68591	0.842954	+	0.537985i	$0.180814\pi$
$110$	19.7205	1.88028
$111$	−1.14698	−0.108867
$112$	−5.95167	−0.562380
$113$	3.03073	0.285107	0.142554	−	0.989787i	$-0.454469\pi$
$113$	3.03073	0.285107	0.142554	+	0.989787i	$0.454469\pi$
$114$	−2.91728	−0.273229
$115$	−0.00352990	−0.000329165 0
$116$	−5.94974	−0.552419
$117$	−4.97450	−0.459893
$118$	−8.90637	−0.819898
$119$	1.88525	0.172821
$120$	−0.622066	−0.0567866
$121$	−4.77691	−0.434265
$122$	−13.7357	−1.24357
$123$	10.7711	0.971201
$124$	13.8603	1.24469
$125$	−20.8312	−1.86320
$126$	−3.13309	−0.279118
$127$	−18.0156	−1.59863	−0.799314	−	0.600913i	$-0.794804\pi$
$127$	−18.0156	−1.59863	−0.799314	+	0.600913i	$0.794804\pi$
$128$	1.27038	0.112287
$129$	−8.92688	−0.785968
$130$	−39.3247	−3.44900
$131$	16.8904	1.47572	0.737862	−	0.674952i	$-0.235836\pi$
$131$	16.8904	1.47572	0.737862	+	0.674952i	$0.235836\pi$
$132$	−5.18553	−0.451342
$133$	−2.24094	−0.194314
$134$	6.96701	0.601858
$135$	3.91431	0.336890
$136$	−0.193125	−0.0165604
$137$	1.77437	0.151595	0.0757973	−	0.997123i	$-0.475850\pi$
$137$	1.77437	0.151595	0.0757973	+	0.997123i	$0.475850\pi$
$138$	0.00182124	0.000155034 0
$139$	8.63991	0.732827	0.366414	−	0.930452i	$-0.380586\pi$
$139$	8.63991	0.732827	0.366414	+	0.930452i	$0.380586\pi$
$140$	−12.6228	−1.06682
$141$	6.91126	0.582033
$142$	28.4984	2.39153
$143$	−12.4095	−1.03773
$144$	−3.83643	−0.319702
$145$	11.2037	0.930420
$146$	−15.7876	−1.30659
$147$	4.59329	0.378848
$148$	2.38422	0.195981
$149$	3.37415	0.276421	0.138210	−	0.990403i	$-0.455865\pi$
$149$	3.37415	0.276421	0.138210	+	0.990403i	$0.455865\pi$
$150$	20.8457	1.70204
$151$	13.0701	1.06363	0.531817	−	0.846859i	$-0.321510\pi$
$151$	13.0701	1.06363	0.531817	+	0.846859i	$0.321510\pi$
$152$	0.229562	0.0186199
$153$	1.21523	0.0982453
$154$	−7.81583	−0.629818
$155$	−26.0999	−2.09639
$156$	10.3405	0.827899
$157$	−0.347707	−0.0277500	−0.0138750	−	0.999904i	$-0.504417\pi$
$157$	−0.347707	−0.0277500	−0.0138750	+	0.999904i	$0.504417\pi$
$158$	−23.0016	−1.82991
$159$	−0.762037	−0.0604334
$160$	−31.5720	−2.49599
$161$	0.00139900	0.000110257 0
$162$	−2.01958	−0.158673
$163$	8.76663	0.686655	0.343328	−	0.939216i	$-0.388446\pi$
$163$	8.76663	0.686655	0.343328	+	0.939216i	$0.388446\pi$
$164$	−22.3899	−1.74836
$165$	9.76468	0.760179
$166$	4.93634	0.383134
$167$	8.93423	0.691351	0.345676	−	0.938354i	$-0.387650\pi$
$167$	8.93423	0.691351	0.345676	+	0.938354i	$0.387650\pi$
$168$	0.246544	0.0190212
$169$	11.7457	0.903514
$170$	9.60667	0.736798
$171$	−1.44450	−0.110464
$172$	18.5562	1.41490
$173$	24.0776	1.83058	0.915292	−	0.402790i	$-0.131960\pi$
$173$	24.0776	1.83058	0.915292	+	0.402790i	$0.131960\pi$
$174$	−5.78054	−0.438222
$175$	16.0128	1.21046
$176$	−9.57040	−0.721396
$177$	−4.41002	−0.331477
$178$	21.9857	1.64790
$179$	−10.0336	−0.749949	−0.374975	−	0.927035i	$-0.622349\pi$
$179$	−10.0336	−0.749949	−0.374975	+	0.927035i	$0.622349\pi$
$180$	−8.13664	−0.606469
$181$	5.63606	0.418925	0.209463	−	0.977817i	$-0.432829\pi$
$181$	5.63606	0.418925	0.209463	+	0.977817i	$0.432829\pi$
$182$	15.5855	1.15528
$183$	−6.80128	−0.502765
$184$	−0.000143314 0	−1.05652e−5 0
$185$	−4.48963	−0.330084
$186$	13.4662	0.987386
$187$	3.03152	0.221687
$188$	−14.3664	−1.04778
$189$	−1.55136	−0.112845
$190$	−11.4191	−0.828432
$191$	7.30965	0.528907	0.264454	−	0.964398i	$-0.414808\pi$
$191$	7.30965	0.528907	0.264454	+	0.964398i	$0.414808\pi$
$192$	8.61665	0.621853
$193$	20.1856	1.45299	0.726497	−	0.687169i	$-0.241147\pi$
$193$	20.1856	1.45299	0.726497	+	0.687169i	$0.241147\pi$
$194$	19.3974	1.39265
$195$	−19.4717	−1.39440
$196$	−9.54802	−0.682002
$197$	18.9783	1.35215	0.676075	−	0.736833i	$-0.263680\pi$
$197$	18.9783	1.35215	0.676075	+	0.736833i	$0.263680\pi$
$198$	−5.03806	−0.358039
$199$	7.31139	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$199$	7.31139	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$200$	−1.64035	−0.115991
$201$	3.44974	0.243326
$202$	17.1859	1.20919
$203$	−4.44038	−0.311654
$204$	−2.52608	−0.176861
$205$	42.1616	2.94469
$206$	18.4859	1.28798
$207$	0.000901793 0	6.26790e−5 0
$208$	19.0843	1.32326
$209$	−3.60347	−0.249257
$210$	−12.2639	−0.846287
$211$	−17.1362	−1.17970	−0.589851	−	0.807512i	$-0.700814\pi$
$211$	−17.1362	−1.17970	−0.589851	+	0.807512i	$0.700814\pi$
$212$	1.58404	0.108792
$213$	14.1111	0.966874
$214$	−1.14836	−0.0785004
$215$	−34.9426	−2.38306
$216$	0.158921	0.0108132
$217$	10.3442	0.702207
$218$	−35.5474	−2.40757
$219$	−7.81730	−0.528244
$220$	−20.2977	−1.36847
$221$	−6.04515	−0.406641
$222$	2.31641	0.155468
$223$	24.6099	1.64800	0.823999	−	0.566591i	$-0.191738\pi$
$223$	24.6099	1.64800	0.823999	+	0.566591i	$0.191738\pi$
$224$	12.5129	0.836056
$225$	10.3218	0.688121
$226$	−6.12080	−0.407149
$227$	15.9175	1.05648	0.528239	−	0.849095i	$-0.322852\pi$
$227$	15.9175	1.05648	0.528239	+	0.849095i	$0.322852\pi$
$228$	3.00267	0.198857
$229$	16.9688	1.12133	0.560665	−	0.828043i	$-0.310546\pi$
$229$	16.9688	1.12133	0.560665	+	0.828043i	$0.310546\pi$
$230$	0.00712890	0.000470066 0
$231$	−3.87004	−0.254630
$232$	0.454872	0.0298638
$233$	−7.12581	−0.466827	−0.233414	−	0.972378i	$-0.574990\pi$
$233$	−7.12581	−0.466827	−0.233414	+	0.972378i	$0.574990\pi$
$234$	10.0464	0.656753
$235$	27.0528	1.76473
$236$	9.16706	0.596725
$237$	−11.3893	−0.739815
$238$	−3.80741	−0.246798
$239$	−14.7012	−0.950943	−0.475472	−	0.879731i	$-0.657723\pi$
$239$	−14.7012	−0.950943	−0.475472	+	0.879731i	$0.657723\pi$
$240$	−15.0170	−0.969341
$241$	−6.15661	−0.396582	−0.198291	−	0.980143i	$-0.563539\pi$
$241$	−6.15661	−0.396582	−0.198291	+	0.980143i	$0.563539\pi$
$242$	9.64734	0.620155
$243$	−1.00000	−0.0641500
$244$	14.1377	0.905076
$245$	17.9795	1.14867
$246$	−21.7532	−1.38693
$247$	7.18568	0.457214
$248$	−1.05966	−0.0672882
$249$	2.44424	0.154898
$250$	42.0703	2.66076
$251$	−24.6175	−1.55384	−0.776921	−	0.629598i	$-0.783220\pi$
$251$	−24.6175	−1.55384	−0.776921	+	0.629598i	$0.783220\pi$
$252$	3.22479	0.203143
$253$	0.00224962	0.000141433 0
$254$	36.3840	2.28293
$255$	4.75678	0.297881
$256$	14.6677	0.916729
$257$	−7.82689	−0.488228	−0.244114	−	0.969747i	$-0.578497\pi$
$257$	−7.82689	−0.488228	−0.244114	+	0.969747i	$0.578497\pi$
$258$	18.0285	1.12241
$259$	1.77938	0.110565
$260$	40.4757	2.51020
$261$	−2.86225	−0.177169
$262$	−34.1115	−2.10742
$263$	4.09864	0.252733	0.126367	−	0.991984i	$-0.459668\pi$
$263$	4.09864	0.252733	0.126367	+	0.991984i	$0.459668\pi$
$264$	0.396446	0.0243996
$265$	−2.98285	−0.183235
$266$	4.52575	0.277491
$267$	10.8863	0.666229
$268$	−7.17093	−0.438034
$269$	−2.83228	−0.172687	−0.0863436	−	0.996265i	$-0.527518\pi$
$269$	−2.83228	−0.172687	−0.0863436	+	0.996265i	$0.527518\pi$
$270$	−7.90525	−0.481098
$271$	0.247146	0.0150130	0.00750651	−	0.999972i	$-0.497611\pi$
$271$	0.247146	0.0150130	0.00750651	+	0.999972i	$0.497611\pi$
$272$	−4.66213	−0.282683
$273$	7.71723	0.467068
$274$	−3.58348	−0.216486
$275$	25.7489	1.55272
$276$	−0.00187455	−0.000112835 0
$277$	28.3684	1.70449	0.852247	−	0.523139i	$-0.175239\pi$
$277$	28.3684	1.70449	0.852247	+	0.523139i	$0.175239\pi$
$278$	−17.4490	−1.04652
$279$	6.66781	0.399191
$280$	0.965047	0.0576726
$281$	30.9062	1.84371	0.921853	−	0.387539i	$-0.126675\pi$
$281$	30.9062	1.84371	0.921853	+	0.387539i	$0.126675\pi$
$282$	−13.9578	−0.831176
$283$	8.21793	0.488505	0.244253	−	0.969712i	$-0.421457\pi$
$283$	8.21793	0.488505	0.244253	+	0.969712i	$0.421457\pi$
$284$	−29.3325	−1.74057
$285$	−5.65423	−0.334927
$286$	25.0618	1.48194
$287$	−16.7099	−0.986354
$288$	8.06580	0.475282
$289$	−15.5232	−0.913131
$290$	−22.6268	−1.32869
$291$	9.60470	0.563038
$292$	16.2497	0.950944
$293$	−12.8488	−0.750632	−0.375316	−	0.926897i	$-0.622466\pi$
$293$	−12.8488	−0.750632	−0.375316	+	0.926897i	$0.622466\pi$
$294$	−9.27650	−0.541016
$295$	−17.2622	−1.00504
$296$	−0.182279	−0.0105948
$297$	−2.49461	−0.144752
$298$	−6.81435	−0.394744
$299$	−0.00448597	−0.000259430 0
$300$	−21.4559	−1.23875
$301$	13.8488	0.798231
$302$	−26.3962	−1.51893
$303$	8.50964	0.488866
$304$	5.54173	0.317840
$305$	−26.6223	−1.52439
$306$	−2.45425	−0.140300
$307$	29.4364	1.68002	0.840012	−	0.542568i	$-0.182548\pi$
$307$	29.4364	1.68002	0.840012	+	0.542568i	$0.182548\pi$
$308$	8.04461	0.458384
$309$	9.15337	0.520717
$310$	52.7107	2.99377
$311$	8.42068	0.477493	0.238746	−	0.971082i	$-0.423264\pi$
$311$	8.42068	0.477493	0.238746	+	0.971082i	$0.423264\pi$
$312$	−0.790553	−0.0447563
$313$	−30.1506	−1.70421	−0.852107	−	0.523368i	$-0.824675\pi$
$313$	−30.1506	−1.70421	−0.852107	+	0.523368i	$0.824675\pi$
$314$	0.702221	0.0396286
$315$	−6.07249	−0.342146
$316$	23.6749	1.33181
$317$	2.37564	0.133429	0.0667147	−	0.997772i	$-0.478748\pi$
$317$	2.37564	0.133429	0.0667147	+	0.997772i	$0.478748\pi$
$318$	1.53899	0.0863024
$319$	−7.14021	−0.399775
$320$	33.7282	1.88547
$321$	−0.568615	−0.0317370
$322$	−0.00282540	−0.000157453 0
$323$	−1.75540	−0.0976730
$324$	2.07869	0.115483
$325$	−51.3459	−2.84816
$326$	−17.7049	−0.980583
$327$	−17.6014	−0.973360
$328$	1.71176	0.0945162
$329$	−10.7218	−0.591114
$330$	−19.7205	−1.08558
$331$	−26.4915	−1.45610	−0.728052	−	0.685522i	$-0.759574\pi$
$331$	−26.4915	−1.45610	−0.728052	+	0.685522i	$0.759574\pi$
$332$	−5.08082	−0.278846
$333$	1.14698	0.0628541
$334$	−18.0434	−0.987289
$335$	13.5033	0.737766
$336$	5.95167	0.324690
$337$	17.9639	0.978554	0.489277	−	0.872128i	$-0.337261\pi$
$337$	17.9639	0.978554	0.489277	+	0.872128i	$0.337261\pi$
$338$	−23.7213	−1.29027
$339$	−3.03073	−0.164607
$340$	−9.88786	−0.536245
$341$	16.6336	0.900760
$342$	2.91728	0.157749
$343$	−17.9853	−0.971117
$344$	−1.41867	−0.0764896
$345$	0.00352990	0.000190043 0
$346$	−48.6265	−2.61418
$347$	5.56403	0.298693	0.149346	−	0.988785i	$-0.452283\pi$
$347$	5.56403	0.298693	0.149346	+	0.988785i	$0.452283\pi$
$348$	5.94974	0.318939
$349$	−26.3964	−1.41297	−0.706484	−	0.707729i	$-0.749720\pi$
$349$	−26.3964	−1.41297	−0.706484	+	0.707729i	$0.749720\pi$
$350$	−32.3391	−1.72860
$351$	4.97450	0.265519
$352$	20.1210	1.07246
$353$	−28.4042	−1.51180	−0.755902	−	0.654685i	$-0.772801\pi$
$353$	−28.4042	−1.51180	−0.755902	+	0.654685i	$0.772801\pi$
$354$	8.90637	0.473368
$355$	55.2351	2.93157
$356$	−22.6292	−1.19934
$357$	−1.88525	−0.0997782
$358$	20.2637	1.07097
$359$	−33.3063	−1.75784	−0.878919	−	0.476971i	$-0.841735\pi$
$359$	−33.3063	−1.75784	−0.878919	+	0.476971i	$0.841735\pi$
$360$	0.622066	0.0327858
$361$	−16.9134	−0.890180
$362$	−11.3825	−0.598249
$363$	4.77691	0.250723
$364$	−16.0417	−0.840816
$365$	−30.5993	−1.60164
$366$	13.7357	0.717976
$367$	−19.2426	−1.00446	−0.502228	−	0.864735i	$-0.667486\pi$
$367$	−19.2426	−1.00446	−0.502228	+	0.864735i	$0.667486\pi$
$368$	−0.00345966	−0.000180347 0
$369$	−10.7711	−0.560723
$370$	9.06716	0.471379
$371$	1.18219	0.0613763
$372$	−13.8603	−0.718624
$373$	20.0623	1.03879	0.519394	−	0.854535i	$-0.326158\pi$
$373$	20.0623	1.03879	0.519394	+	0.854535i	$0.326158\pi$
$374$	−6.12239	−0.316581
$375$	20.8312	1.07572
$376$	1.09835	0.0566428
$377$	14.2383	0.733309
$378$	3.13309	0.161149
$379$	−0.276075	−0.0141810	−0.00709052	−	0.999975i	$-0.502257\pi$
$379$	−0.276075	−0.0141810	−0.00709052	+	0.999975i	$0.502257\pi$
$380$	11.7534	0.602936
$381$	18.0156	0.922969
$382$	−14.7624	−0.755310
$383$	4.03578	0.206219	0.103109	−	0.994670i	$-0.467121\pi$
$383$	4.03578	0.206219	0.103109	+	0.994670i	$0.467121\pi$
$384$	−1.27038	−0.0648290
$385$	−15.1485	−0.772040
$386$	−40.7665	−2.07496
$387$	8.92688	0.453779
$388$	−19.9652	−1.01358
$389$	−2.26318	−0.114748	−0.0573739	−	0.998353i	$-0.518273\pi$
$389$	−2.26318	−0.114748	−0.0573739	+	0.998353i	$0.518273\pi$
$390$	39.3247	1.99128
$391$	0.00109588	5.54212e−5 0
$392$	0.729970	0.0368691
$393$	−16.8904	−0.852010
$394$	−38.3282	−1.93095
$395$	−44.5813	−2.24313
$396$	5.18553	0.260582
$397$	34.2274	1.71782	0.858912	−	0.512123i	$-0.171141\pi$
$397$	34.2274	1.71782	0.858912	+	0.512123i	$0.171141\pi$
$398$	−14.7659	−0.740149
$399$	2.24094	0.112187
$400$	−39.5989	−1.97994
$401$	16.1305	0.805518	0.402759	−	0.915306i	$-0.368051\pi$
$401$	16.1305	0.805518	0.402759	+	0.915306i	$0.368051\pi$
$402$	−6.96701	−0.347483
$403$	−33.1690	−1.65227
$404$	−17.6889	−0.880056
$405$	−3.91431	−0.194504
$406$	8.96769	0.445059
$407$	2.86127	0.141828
$408$	0.193125	0.00956113
$409$	6.19327	0.306238	0.153119	−	0.988208i	$-0.451068\pi$
$409$	6.19327	0.306238	0.153119	+	0.988208i	$0.451068\pi$
$410$	−85.1485	−4.20519
$411$	−1.77437	−0.0875232
$412$	−19.0270	−0.937394
$413$	6.84152	0.336649
$414$	−0.00182124	−8.95091e−5 0
$415$	9.56752	0.469651
$416$	−40.1234	−1.96721
$417$	−8.63991	−0.423098
$418$	7.27749	0.355953
$419$	5.06244	0.247316	0.123658	−	0.992325i	$-0.460537\pi$
$419$	5.06244	0.247316	0.123658	+	0.992325i	$0.460537\pi$
$420$	12.6228	0.615931
$421$	16.5975	0.808914	0.404457	−	0.914557i	$-0.367460\pi$
$421$	16.5975	0.808914	0.404457	+	0.914557i	$0.367460\pi$
$422$	34.6078	1.68468
$423$	−6.91126	−0.336037
$424$	−0.121104	−0.00588132
$425$	12.5434	0.608442
$426$	−28.4984	−1.38075
$427$	10.5512	0.510609
$428$	1.18197	0.0571329
$429$	12.4095	0.599134
$430$	70.5692	3.40315
$431$	−26.5184	−1.27735	−0.638674	−	0.769478i	$-0.720517\pi$
$431$	−26.5184	−1.27735	−0.638674	+	0.769478i	$0.720517\pi$
$432$	3.83643	0.184580
$433$	−27.1649	−1.30546	−0.652730	−	0.757590i	$-0.726377\pi$
$433$	−27.1649	−1.30546	−0.652730	+	0.757590i	$0.726377\pi$
$434$	−20.8908	−1.00279
$435$	−11.2037	−0.537178
$436$	36.5879	1.75224
$437$	−0.00130264	−6.23138e−5 0
$438$	15.7876	0.754363
$439$	−9.95828	−0.475283	−0.237641	−	0.971353i	$-0.576374\pi$
$439$	−9.95828	−0.475283	−0.237641	+	0.971353i	$0.576374\pi$
$440$	1.55181	0.0739798
$441$	−4.59329	−0.218728
$442$	12.2086	0.580706
$443$	−37.1638	−1.76570	−0.882852	−	0.469651i	$-0.844380\pi$
$443$	−37.1638	−1.76570	−0.882852	+	0.469651i	$0.844380\pi$
$444$	−2.38422	−0.113150
$445$	42.6122	2.02001
$446$	−49.7015	−2.35343
$447$	−3.37415	−0.159592
$448$	−13.3675	−0.631555
$449$	−1.86333	−0.0879361	−0.0439680	−	0.999033i	$-0.514000\pi$
$449$	−1.86333	−0.0879361	−0.0439680	+	0.999033i	$0.514000\pi$
$450$	−20.8457	−0.982675
$451$	−26.8698	−1.26525
$452$	6.29995	0.296325
$453$	−13.0701	−0.614089
$454$	−32.1465	−1.50871
$455$	30.2076	1.41616
$456$	−0.229562	−0.0107502
$457$	−5.52912	−0.258641	−0.129321	−	0.991603i	$-0.541280\pi$
$457$	−5.52912	−0.258641	−0.129321	+	0.991603i	$0.541280\pi$
$458$	−34.2698	−1.60132
$459$	−1.21523	−0.0567220
$460$	−0.00733756	−0.000342116 0
$461$	7.96400	0.370920	0.185460	−	0.982652i	$-0.440622\pi$
$461$	7.96400	0.370920	0.185460	+	0.982652i	$0.440622\pi$
$462$	7.81583	0.363626
$463$	−11.9967	−0.557535	−0.278768	−	0.960359i	$-0.589926\pi$
$463$	−11.9967	−0.557535	−0.278768	+	0.960359i	$0.589926\pi$
$464$	10.9808	0.509772
$465$	26.0999	1.21035
$466$	14.3911	0.666655
$467$	−27.7292	−1.28316	−0.641578	−	0.767058i	$-0.721720\pi$
$467$	−27.7292	−1.28316	−0.641578	+	0.767058i	$0.721720\pi$
$468$	−10.3405	−0.477988
$469$	−5.35178	−0.247122
$470$	−54.6352	−2.52014
$471$	0.347707	0.0160215
$472$	−0.700845	−0.0322590
$473$	22.2691	1.02393
$474$	23.0016	1.05650
$475$	−14.9099	−0.684112
$476$	3.91886	0.179621
$477$	0.762037	0.0348913
$478$	29.6903	1.35800
$479$	29.6115	1.35298	0.676492	−	0.736450i	$-0.263499\pi$
$479$	29.6115	1.35298	0.676492	+	0.736450i	$0.263499\pi$
$480$	31.5720	1.44106
$481$	−5.70566	−0.260155
$482$	12.4337	0.566341
$483$	−0.00139900	−6.36569e−5 0
$484$	−9.92972	−0.451351
$485$	37.5958	1.70714
$486$	2.01958	0.0916099
$487$	29.8422	1.35228	0.676139	−	0.736774i	$-0.263652\pi$
$487$	29.8422	1.35228	0.676139	+	0.736774i	$0.263652\pi$
$488$	−1.08087	−0.0489285
$489$	−8.76663	−0.396441
$490$	−36.3111	−1.64037
$491$	39.9891	1.80468	0.902342	−	0.431021i	$-0.141847\pi$
$491$	39.9891	1.80468	0.902342	+	0.431021i	$0.141847\pi$
$492$	22.3899	1.00941
$493$	−3.47829	−0.156654
$494$	−14.5120	−0.652927
$495$	−9.76468	−0.438890
$496$	−25.5806	−1.14860
$497$	−21.8913	−0.981960
$498$	−4.93634	−0.221203
$499$	−37.3125	−1.67034	−0.835169	−	0.549994i	$-0.814630\pi$
$499$	−37.3125	−1.67034	−0.835169	+	0.549994i	$0.814630\pi$
$500$	−43.3017	−1.93651
$501$	−8.93423	−0.399152
$502$	49.7169	2.21897
$503$	3.92233	0.174888	0.0874440	−	0.996169i	$-0.472130\pi$
$503$	3.92233	0.174888	0.0874440	+	0.996169i	$0.472130\pi$
$504$	−0.246544	−0.0109819
$505$	33.3094	1.48225
$506$	−0.00454329	−0.000201974 0
$507$	−11.7457	−0.521644
$508$	−37.4489	−1.66153
$509$	36.0026	1.59579	0.797894	−	0.602798i	$-0.205947\pi$
$509$	36.0026	1.59579	0.797894	+	0.602798i	$0.205947\pi$
$510$	−9.60667	−0.425391
$511$	12.1274	0.536486
$512$	−32.1632	−1.42143
$513$	1.44450	0.0637763
$514$	15.8070	0.697217
$515$	35.8291	1.57882
$516$	−18.5562	−0.816892
$517$	−17.2409	−0.758255
$518$	−3.59359	−0.157893
$519$	−24.0776	−1.05689
$520$	−3.09447	−0.135702
$521$	9.66617	0.423483	0.211741	−	0.977326i	$-0.432087\pi$
$521$	9.66617	0.423483	0.211741	+	0.977326i	$0.432087\pi$
$522$	5.78054	0.253007
$523$	3.30733	0.144619	0.0723097	−	0.997382i	$-0.476963\pi$
$523$	3.30733	0.144619	0.0723097	+	0.997382i	$0.476963\pi$
$524$	35.1100	1.53379
$525$	−16.0128	−0.698857
$526$	−8.27752	−0.360917
$527$	8.10290	0.352968
$528$	9.57040	0.416498
$529$	−23.0000	−1.00000
$530$	6.02409	0.261670
$531$	4.41002	0.191378
$532$	−4.65822	−0.201959
$533$	53.5811	2.32085
$534$	−21.9857	−0.951413
$535$	−2.22574	−0.0962269
$536$	0.548236	0.0236802
$537$	10.0336	0.432983
$538$	5.72001	0.246607
$539$	−11.4585	−0.493551
$540$	8.13664	0.350145
$541$	−4.33458	−0.186358	−0.0931790	−	0.995649i	$-0.529703\pi$
$541$	−4.33458	−0.186358	−0.0931790	+	0.995649i	$0.529703\pi$
$542$	−0.499130	−0.0214395
$543$	−5.63606	−0.241866
$544$	9.80178	0.420248
$545$	−68.8973	−2.95124
$546$	−15.5855	−0.667000
$547$	31.7352	1.35690	0.678449	−	0.734648i	$-0.262652\pi$
$547$	31.7352	1.35690	0.678449	+	0.734648i	$0.262652\pi$
$548$	3.68836	0.157559
$549$	6.80128	0.290271
$550$	−52.0019	−2.21737
$551$	4.13453	0.176137
$552$	0.000143314 0	6.09985e−6 0
$553$	17.6689	0.751358
$554$	−57.2922	−2.43411
$555$	4.48963	0.190574
$556$	17.9597	0.761660
$557$	28.6538	1.21410	0.607050	−	0.794664i	$-0.292353\pi$
$557$	28.6538	1.21410	0.607050	+	0.794664i	$0.292353\pi$
$558$	−13.4662	−0.570068
$559$	−44.4068	−1.87821
$560$	23.2967	0.984465
$561$	−3.03152	−0.127991
$562$	−62.4173	−2.63292
$563$	−41.9835	−1.76939	−0.884697	−	0.466166i	$-0.845635\pi$
$563$	−41.9835	−1.76939	−0.884697	+	0.466166i	$0.845635\pi$
$564$	14.3664	0.604933
$565$	−11.8632	−0.499089
$566$	−16.5967	−0.697613
$567$	1.55136	0.0651509
$568$	2.24255	0.0940952
$569$	−39.3725	−1.65058	−0.825290	−	0.564710i	$-0.808988\pi$
$569$	−39.3725	−1.65058	−0.825290	+	0.564710i	$0.808988\pi$
$570$	11.4191	0.478295
$571$	−31.2113	−1.30615	−0.653077	−	0.757292i	$-0.726522\pi$
$571$	−31.2113	−1.30615	−0.653077	+	0.757292i	$0.726522\pi$
$572$	−25.7954	−1.07856
$573$	−7.30965	−0.305365
$574$	33.7469	1.40857
$575$	0.00930814	0.000388176 0
$576$	−8.61665	−0.359027
$577$	−18.9722	−0.789823	−0.394911	−	0.918719i	$-0.629225\pi$
$577$	−18.9722	−0.789823	−0.394911	+	0.918719i	$0.629225\pi$
$578$	31.3503	1.30400
$579$	−20.1856	−0.838887
$580$	23.2891	0.967028
$581$	−3.79190	−0.157314
$582$	−19.3974	−0.804050
$583$	1.90099	0.0787308
$584$	−1.24233	−0.0514081
$585$	19.4717	0.805057
$586$	25.9491	1.07195
$587$	−23.8873	−0.985933	−0.492967	−	0.870048i	$-0.664087\pi$
$587$	−23.8873	−0.985933	−0.492967	+	0.870048i	$0.664087\pi$
$588$	9.54802	0.393754
$589$	−9.63166	−0.396866
$590$	34.8623	1.43526
$591$	−18.9783	−0.780664
$592$	−4.40031	−0.180851
$593$	−12.4157	−0.509850	−0.254925	−	0.966961i	$-0.582051\pi$
$593$	−12.4157	−0.509850	−0.254925	+	0.966961i	$0.582051\pi$
$594$	5.03806	0.206714
$595$	−7.37946	−0.302528
$596$	7.01380	0.287297
$597$	−7.31139	−0.299235
$598$	0.00905977	0.000370481 0
$599$	−33.6031	−1.37299	−0.686494	−	0.727136i	$-0.740851\pi$
$599$	−33.6031	−1.37299	−0.686494	+	0.727136i	$0.740851\pi$
$600$	1.64035	0.0669672
$601$	1.90217	0.0775912	0.0387956	−	0.999247i	$-0.487648\pi$
$601$	1.90217	0.0775912	0.0387956	+	0.999247i	$0.487648\pi$
$602$	−27.9687	−1.13992
$603$	−3.44974	−0.140484
$604$	27.1688	1.10548
$605$	18.6983	0.760194
$606$	−17.1859	−0.698128
$607$	−14.7727	−0.599605	−0.299803	−	0.954001i	$-0.596921\pi$
$607$	−14.7727	−0.599605	−0.299803	+	0.954001i	$0.596921\pi$
$608$	−11.6511	−0.472513
$609$	4.44038	0.179933
$610$	53.7658	2.17691
$611$	34.3801	1.39087
$612$	2.52608	0.102111
$613$	−14.9644	−0.604408	−0.302204	−	0.953243i	$-0.597722\pi$
$613$	−14.9644	−0.604408	−0.302204	+	0.953243i	$0.597722\pi$
$614$	−59.4491	−2.39917
$615$	−42.1616	−1.70012
$616$	−0.615030	−0.0247803
$617$	19.7849	0.796511	0.398256	−	0.917275i	$-0.369616\pi$
$617$	19.7849	0.796511	0.398256	+	0.917275i	$0.369616\pi$
$618$	−18.4859	−0.743613
$619$	−23.1653	−0.931091	−0.465545	−	0.885024i	$-0.654142\pi$
$619$	−23.1653	−0.931091	−0.465545	+	0.885024i	$0.654142\pi$
$620$	−54.2535	−2.17887
$621$	−0.000901793 0	−3.61877e−5 0
$622$	−17.0062	−0.681887
$623$	−16.8885	−0.676624
$624$	−19.0843	−0.763984
$625$	29.9308	1.19723
$626$	60.8915	2.43371
$627$	3.60347	0.143909
$628$	−0.722775	−0.0288419
$629$	1.39384	0.0555761
$630$	12.2639	0.488604
$631$	−29.4381	−1.17191	−0.585957	−	0.810342i	$-0.699281\pi$
$631$	−29.4381	−1.17191	−0.585957	+	0.810342i	$0.699281\pi$
$632$	−1.81000	−0.0719980
$633$	17.1362	0.681101
$634$	−4.79780	−0.190545
$635$	70.5187	2.79845
$636$	−1.58404	−0.0628112
$637$	22.8493	0.905323
$638$	14.4202	0.570901
$639$	−14.1111	−0.558225
$640$	−4.97268	−0.196562
$641$	32.8902	1.29909	0.649543	−	0.760325i	$-0.274960\pi$
$641$	32.8902	1.29909	0.649543	+	0.760325i	$0.274960\pi$
$642$	1.14836	0.0453222
$643$	−18.4830	−0.728896	−0.364448	−	0.931224i	$-0.618742\pi$
$643$	−18.4830	−0.728896	−0.364448	+	0.931224i	$0.618742\pi$
$644$	0.00290810	0.000114595 0
$645$	34.9426	1.37586
$646$	3.54516	0.139483
$647$	−15.4006	−0.605459	−0.302729	−	0.953077i	$-0.597898\pi$
$647$	−15.4006	−0.605459	−0.302729	+	0.953077i	$0.597898\pi$
$648$	−0.158921	−0.00624301
$649$	11.0013	0.431838
$650$	103.697	4.06733
$651$	−10.3442	−0.405420
$652$	18.2231	0.713672
$653$	−28.9390	−1.13247	−0.566235	−	0.824244i	$-0.691601\pi$
$653$	−28.9390	−1.13247	−0.566235	+	0.824244i	$0.691601\pi$
$654$	35.5474	1.39001
$655$	−66.1144	−2.58330
$656$	41.3227	1.61338
$657$	7.81730	0.304982
$658$	21.6536	0.844144
$659$	36.0751	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$659$	36.0751	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$660$	20.2977	0.790089
$661$	−20.7773	−0.808142	−0.404071	−	0.914728i	$-0.632405\pi$
$661$	−20.7773	−0.808142	−0.404071	+	0.914728i	$0.632405\pi$
$662$	53.5016	2.07940
$663$	6.04515	0.234774
$664$	0.388442	0.0150745
$665$	8.77173	0.340153
$666$	−2.31641	−0.0897592
$667$	−0.00258116	−9.99429e−5 0
$668$	18.5715	0.718553
$669$	−24.6099	−0.951472
$670$	−27.2710	−1.05357
$671$	16.9665	0.654986
$672$	−12.5129	−0.482697
$673$	2.14373	0.0826347	0.0413174	−	0.999146i	$-0.486845\pi$
$673$	2.14373	0.0826347	0.0413174	+	0.999146i	$0.486845\pi$
$674$	−36.2794	−1.39743
$675$	−10.3218	−0.397287
$676$	24.4156	0.939063
$677$	26.9811	1.03697	0.518485	−	0.855087i	$-0.326496\pi$
$677$	26.9811	1.03697	0.518485	+	0.855087i	$0.326496\pi$
$678$	6.12080	0.235068
$679$	−14.9003	−0.571822
$680$	0.755952	0.0289894
$681$	−15.9175	−0.609958
$682$	−33.5928	−1.28634
$683$	15.6127	0.597404	0.298702	−	0.954346i	$-0.403446\pi$
$683$	15.6127	0.597404	0.298702	+	0.954346i	$0.403446\pi$
$684$	−3.00267	−0.114810
$685$	−6.94543	−0.265371
$686$	36.3228	1.38681
$687$	−16.9688	−0.647400
$688$	−34.2473	−1.30567
$689$	−3.79075	−0.144416
$690$	−0.00712890	−0.000271392 0
$691$	−18.0405	−0.686291	−0.343146	−	0.939282i	$-0.611492\pi$
$691$	−18.0405	−0.686291	−0.343146	+	0.939282i	$0.611492\pi$
$692$	50.0498	1.90261
$693$	3.87004	0.147011
$694$	−11.2370	−0.426550
$695$	−33.8193	−1.28284
$696$	−0.454872	−0.0172419
$697$	−13.0894	−0.495796
$698$	53.3096	2.01780
$699$	7.12581	0.269523
$700$	33.2857	1.25808
$701$	−2.60800	−0.0985029	−0.0492515	−	0.998786i	$-0.515684\pi$
$701$	−2.60800	−0.0985029	−0.0492515	+	0.998786i	$0.515684\pi$
$702$	−10.0464	−0.379177
$703$	−1.65681	−0.0624879
$704$	−21.4952	−0.810131
$705$	−27.0528	−1.01887
$706$	57.3645	2.15894
$707$	−13.2015	−0.496493
$708$	−9.16706	−0.344519
$709$	46.1807	1.73435	0.867177	−	0.498000i	$-0.165932\pi$
$709$	46.1807	1.73435	0.867177	+	0.498000i	$0.165932\pi$
$710$	−111.551	−4.18645
$711$	11.3893	0.427133
$712$	1.73006	0.0648367
$713$	0.00601298	0.000225188 0
$714$	3.80741	0.142489
$715$	48.5744	1.81658
$716$	−20.8568	−0.779456
$717$	14.7012	0.549027
$718$	67.2646	2.51029
$719$	−25.3830	−0.946628	−0.473314	−	0.880894i	$-0.656942\pi$
$719$	−25.3830	−0.946628	−0.473314	+	0.880894i	$0.656942\pi$
$720$	15.0170	0.559649
$721$	−14.2002	−0.528841
$722$	34.1579	1.27123
$723$	6.15661	0.228967
$724$	11.7156	0.435408
$725$	−29.5436	−1.09722
$726$	−9.64734	−0.358046
$727$	44.0689	1.63443	0.817213	−	0.576336i	$-0.195518\pi$
$727$	44.0689	1.63443	0.817213	+	0.576336i	$0.195518\pi$
$728$	1.22643	0.0454546
$729$	1.00000	0.0370370
$730$	61.7977	2.28724
$731$	10.8482	0.401235
$732$	−14.1377	−0.522546
$733$	−21.7600	−0.803723	−0.401861	−	0.915701i	$-0.631637\pi$
$733$	−21.7600	−0.803723	−0.401861	+	0.915701i	$0.631637\pi$
$734$	38.8619	1.43442
$735$	−17.9795	−0.663186
$736$	0.00727368	0.000268112 0
$737$	−8.60575	−0.316997
$738$	21.7532	0.800745
$739$	−47.9948	−1.76552	−0.882758	−	0.469827i	$-0.844316\pi$
$739$	−47.9948	−1.76552	−0.882758	+	0.469827i	$0.844316\pi$
$740$	−9.33256	−0.343072
$741$	−7.18568	−0.263972
$742$	−2.38753	−0.0876489
$743$	5.50233	0.201861	0.100931	−	0.994893i	$-0.467818\pi$
$743$	5.50233	0.201861	0.100931	+	0.994893i	$0.467818\pi$
$744$	1.05966	0.0388489
$745$	−13.2074	−0.483883
$746$	−40.5174	−1.48345
$747$	−2.44424	−0.0894302
$748$	6.30159	0.230409
$749$	0.882126	0.0322322
$750$	−42.0703	−1.53619
$751$	−4.66971	−0.170400	−0.0852001	−	0.996364i	$-0.527153\pi$
$751$	−4.66971	−0.170400	−0.0852001	+	0.996364i	$0.527153\pi$
$752$	26.5146	0.966886
$753$	24.6175	0.897111
$754$	−28.7553	−1.04721
$755$	−51.1606	−1.86192
$756$	−3.22479	−0.117285
$757$	43.3173	1.57439	0.787197	−	0.616702i	$-0.211532\pi$
$757$	43.3173	1.57439	0.787197	+	0.616702i	$0.211532\pi$
$758$	0.557556	0.0202513
$759$	−0.00224962	−8.16562e−5 0
$760$	−0.898576	−0.0325948
$761$	0.814696	0.0295327	0.0147663	−	0.999891i	$-0.495300\pi$
$761$	0.814696	0.0295327	0.0147663	+	0.999891i	$0.495300\pi$
$762$	−36.3840	−1.31805
$763$	27.3061	0.988546
$764$	15.1945	0.549717
$765$	−4.75678	−0.171982
$766$	−8.15056	−0.294492
$767$	−21.9376	−0.792122
$768$	−14.6677	−0.529274
$769$	−5.81646	−0.209747	−0.104873	−	0.994486i	$-0.533444\pi$
$769$	−5.81646	−0.209747	−0.104873	+	0.994486i	$0.533444\pi$
$770$	30.5936	1.10252
$771$	7.82689	0.281879
$772$	41.9597	1.51016
$773$	−19.9457	−0.717398	−0.358699	−	0.933453i	$-0.616780\pi$
$773$	−19.9457	−0.717398	−0.358699	+	0.933453i	$0.616780\pi$
$774$	−18.0285	−0.648022
$775$	68.8239	2.47223
$776$	1.52639	0.0547942
$777$	−1.77938	−0.0638348
$778$	4.57067	0.163866
$779$	15.5589	0.557457
$780$	−40.4757	−1.44926
$781$	−35.2016	−1.25961
$782$	−0.00221322	−7.91446e−5 0
$783$	2.86225	0.102289
$784$	17.6218	0.629351
$785$	1.36103	0.0485773
$786$	34.1115	1.21672
$787$	−4.07080	−0.145108	−0.0725541	−	0.997364i	$-0.523115\pi$
$787$	−4.07080	−0.145108	−0.0725541	+	0.997364i	$0.523115\pi$
$788$	39.4501	1.40535
$789$	−4.09864	−0.145916
$790$	90.0353	3.20331
$791$	4.70175	0.167175
$792$	−0.396446	−0.0140871
$793$	−33.8330	−1.20144
$794$	−69.1249	−2.45315
$795$	2.98285	0.105791
$796$	15.1981	0.538683
$797$	−47.7612	−1.69179	−0.845893	−	0.533352i	$-0.820932\pi$
$797$	−47.7612	−1.69179	−0.845893	+	0.533352i	$0.820932\pi$
$798$	−4.52575	−0.160210
$799$	−8.39875	−0.297127
$800$	83.2537	2.94346
$801$	−10.8863	−0.384648
$802$	−32.5768	−1.15033
$803$	19.5011	0.688180
$804$	7.17093	0.252899
$805$	−0.00547613	−0.000193008 0
$806$	66.9874	2.35953
$807$	2.83228	0.0997011
$808$	1.35236	0.0475759
$809$	45.5831	1.60262	0.801309	−	0.598251i	$-0.204137\pi$
$809$	45.5831	1.60262	0.801309	+	0.598251i	$0.204137\pi$
$810$	7.90525	0.277762
$811$	−56.2640	−1.97569	−0.987847	−	0.155427i	$-0.950325\pi$
$811$	−56.2640	−1.97569	−0.987847	+	0.155427i	$0.950325\pi$
$812$	−9.23017	−0.323916
$813$	−0.247146	−0.00866778
$814$	−5.77855	−0.202538
$815$	−34.3153	−1.20201
$816$	4.66213	0.163207
$817$	−12.8949	−0.451135
$818$	−12.5078	−0.437324
$819$	−7.71723	−0.269662
$820$	87.6409	3.06055
$821$	−31.4169	−1.09646	−0.548228	−	0.836329i	$-0.684698\pi$
$821$	−31.4169	−1.09646	−0.548228	+	0.836329i	$0.684698\pi$
$822$	3.58348	0.124988
$823$	10.6761	0.372144	0.186072	−	0.982536i	$-0.440424\pi$
$823$	10.6761	0.372144	0.186072	+	0.982536i	$0.440424\pi$
$824$	1.45466	0.0506756
$825$	−25.7489	−0.896462
$826$	−13.8170	−0.480754
$827$	−3.20869	−0.111577	−0.0557885	−	0.998443i	$-0.517767\pi$
$827$	−3.20869	−0.111577	−0.0557885	+	0.998443i	$0.517767\pi$
$828$	0.00187455	6.51451e−5 0
$829$	−53.4175	−1.85527	−0.927633	−	0.373493i	$-0.878160\pi$
$829$	−53.4175	−1.85527	−0.927633	+	0.373493i	$0.878160\pi$
$830$	−19.3223	−0.670688
$831$	−28.3684	−0.984090
$832$	42.8636	1.48603
$833$	−5.58189	−0.193401
$834$	17.4490	0.604208
$835$	−34.9713	−1.21023
$836$	−7.49050	−0.259064
$837$	−6.66781	−0.230473
$838$	−10.2240	−0.353182
$839$	37.9321	1.30956	0.654781	−	0.755819i	$-0.272761\pi$
$839$	37.9321	1.30956	0.654781	+	0.755819i	$0.272761\pi$
$840$	−0.965047	−0.0332973
$841$	−20.8075	−0.717500
$842$	−33.5200	−1.15518
$843$	−30.9062	−1.06446
$844$	−35.6208	−1.22612
$845$	−45.9762	−1.58163
$846$	13.9578	0.479880
$847$	−7.41070	−0.254635
$848$	−2.92350	−0.100393
$849$	−8.21793	−0.282039
$850$	−25.3323	−0.868889
$851$	0.00103434	3.54567e−5 0
$852$	29.3325	1.00492
$853$	6.14513	0.210405	0.105203	−	0.994451i	$-0.466451\pi$
$853$	6.14513	0.210405	0.105203	+	0.994451i	$0.466451\pi$
$854$	−21.3090	−0.729178
$855$	5.65423	0.193370
$856$	−0.0903650	−0.00308861
$857$	6.26084	0.213866	0.106933	−	0.994266i	$-0.465897\pi$
$857$	6.26084	0.213866	0.106933	+	0.994266i	$0.465897\pi$
$858$	−25.0618	−0.855597
$859$	24.6229	0.840122	0.420061	−	0.907496i	$-0.362009\pi$
$859$	24.6229	0.840122	0.420061	+	0.907496i	$0.362009\pi$
$860$	−72.6348	−2.47683
$861$	16.7099	0.569472
$862$	53.5560	1.82412
$863$	23.1377	0.787616	0.393808	−	0.919193i	$-0.371157\pi$
$863$	23.1377	0.787616	0.393808	+	0.919193i	$0.371157\pi$
$864$	−8.06580	−0.274404
$865$	−94.2471	−3.20450
$866$	54.8616	1.86427
$867$	15.5232	0.527196
$868$	21.5023	0.729836
$869$	28.4119	0.963808
$870$	22.6268	0.767121
$871$	17.1607	0.581469
$872$	−2.79723	−0.0947263
$873$	−9.60470	−0.325070
$874$	0.00263079	8.89876e−5 0
$875$	−32.3167	−1.09250
$876$	−16.2497	−0.549028
$877$	26.6624	0.900325	0.450163	−	0.892947i	$-0.351366\pi$
$877$	26.6624	0.900325	0.450163	+	0.892947i	$0.351366\pi$
$878$	20.1115	0.678730
$879$	12.8488	0.433378
$880$	37.4615	1.26283
$881$	−19.5234	−0.657760	−0.328880	−	0.944372i	$-0.606671\pi$
$881$	−19.5234	−0.657760	−0.328880	+	0.944372i	$0.606671\pi$
$882$	9.27650	0.312356
$883$	−9.12625	−0.307123	−0.153561	−	0.988139i	$-0.549074\pi$
$883$	−9.12625	−0.307123	−0.153561	+	0.988139i	$0.549074\pi$
$884$	−12.5660	−0.422640
$885$	17.2622	0.580261
$886$	75.0551	2.52153
$887$	−0.462641	−0.0155340	−0.00776699	−	0.999970i	$-0.502472\pi$
$887$	−0.462641	−0.0155340	−0.00776699	+	0.999970i	$0.502472\pi$
$888$	0.182279	0.00611689
$889$	−27.9487	−0.937369
$890$	−86.0587	−2.88469
$891$	2.49461	0.0835726
$892$	51.1563	1.71284
$893$	9.98333	0.334079
$894$	6.81435	0.227906
$895$	39.2748	1.31281
$896$	1.97082	0.0658405
$897$	0.00448597	0.000149782 0
$898$	3.76314	0.125578
$899$	−19.0850	−0.636519
$900$	21.4559	0.715195
$901$	0.926048	0.0308511
$902$	54.2657	1.80685
$903$	−13.8488	−0.460859
$904$	−0.481647	−0.0160193
$905$	−22.0613	−0.733342
$906$	26.3962	0.876954
$907$	24.7656	0.822328	0.411164	−	0.911561i	$-0.365122\pi$
$907$	24.7656	0.822328	0.411164	+	0.911561i	$0.365122\pi$
$908$	33.0875	1.09805
$909$	−8.50964	−0.282247
$910$	−61.0066	−2.02235
$911$	−20.0687	−0.664907	−0.332454	−	0.943120i	$-0.607876\pi$
$911$	−20.0687	−0.664907	−0.332454	+	0.943120i	$0.607876\pi$
$912$	−5.54173	−0.183505
$913$	−6.09744	−0.201796
$914$	11.1665	0.369355
$915$	26.6223	0.880106
$916$	35.2729	1.16545
$917$	26.2031	0.865303
$918$	2.45425	0.0810022
$919$	−24.0164	−0.792228	−0.396114	−	0.918201i	$-0.629641\pi$
$919$	−24.0164	−0.792228	−0.396114	+	0.918201i	$0.629641\pi$
$920$	0.000560975 0	1.84948e−5 0
$921$	−29.4364	−0.969962
$922$	−16.0839	−0.529695
$923$	70.1955	2.31051
$924$	−8.04461	−0.264648
$925$	11.8389	0.389261
$926$	24.2283	0.796191
$927$	−9.15337	−0.300636
$928$	−23.0864	−0.757847
$929$	−46.0397	−1.51051	−0.755256	−	0.655430i	$-0.772487\pi$
$929$	−46.0397	−1.51051	−0.755256	+	0.655430i	$0.772487\pi$
$930$	−52.7107	−1.72845
$931$	6.63501	0.217454
$932$	−14.8123	−0.485194
$933$	−8.42068	−0.275681
$934$	56.0013	1.83242
$935$	−11.8663	−0.388070
$936$	0.790553	0.0258400
$937$	−47.7584	−1.56020	−0.780100	−	0.625655i	$-0.784832\pi$
$937$	−47.7584	−1.56020	−0.780100	+	0.625655i	$0.784832\pi$
$938$	10.8083	0.352904
$939$	30.1506	0.983928
$940$	56.2344	1.83416
$941$	−32.1587	−1.04834	−0.524172	−	0.851612i	$-0.675625\pi$
$941$	−32.1587	−1.04834	−0.524172	+	0.851612i	$0.675625\pi$
$942$	−0.702221	−0.0228796
$943$	−0.00971334	−0.000316310 0
$944$	−16.9187	−0.550657
$945$	6.07249	0.197538
$946$	−44.9742	−1.46224
$947$	7.55311	0.245443	0.122721	−	0.992441i	$-0.460838\pi$
$947$	7.55311	0.245443	0.122721	+	0.992441i	$0.460838\pi$
$948$	−23.6749	−0.768924
$949$	−38.8872	−1.26233
$950$	30.1116	0.976951
$951$	−2.37564	−0.0770355
$952$	−0.299606	−0.00971030
$953$	54.4854	1.76495	0.882477	−	0.470356i	$-0.155874\pi$
$953$	54.4854	1.76495	0.882477	+	0.470356i	$0.155874\pi$
$954$	−1.53899	−0.0498267
$955$	−28.6122	−0.925869
$956$	−30.5593	−0.988358
$957$	7.14021	0.230810
$958$	−59.8027	−1.93214
$959$	2.75268	0.0888887
$960$	−33.7282	−1.08857
$961$	13.4597	0.434183
$962$	11.5230	0.371517
$963$	0.568615	0.0183234
$964$	−12.7977	−0.412185
$965$	−79.0129	−2.54351
$966$	0.00282540	9.09056e−5 0
$967$	−45.8157	−1.47333	−0.736667	−	0.676255i	$-0.763602\pi$
$967$	−45.8157	−1.47333	−0.736667	+	0.676255i	$0.763602\pi$
$968$	0.759152	0.0244001
$969$	1.75540	0.0563915
$970$	−75.9276	−2.43789
$971$	9.92508	0.318511	0.159255	−	0.987237i	$-0.449091\pi$
$971$	9.92508	0.318511	0.159255	+	0.987237i	$0.449091\pi$
$972$	−2.07869	−0.0666740
$973$	13.4036	0.429699
$974$	−60.2685	−1.93113
$975$	51.3459	1.64438
$976$	−26.0926	−0.835204
$977$	16.1734	0.517431	0.258716	−	0.965954i	$-0.416701\pi$
$977$	16.1734	0.517431	0.258716	+	0.965954i	$0.416701\pi$
$978$	17.7049	0.566140
$979$	−27.1570	−0.867942
$980$	37.3739	1.19387
$981$	17.6014	0.561970
$982$	−80.7611	−2.57719
$983$	−1.52206	−0.0485462	−0.0242731	−	0.999705i	$-0.507727\pi$
$983$	−1.52206	−0.0485462	−0.0242731	+	0.999705i	$0.507727\pi$
$984$	−1.71176	−0.0545690
$985$	−74.2870	−2.36698
$986$	7.02467	0.223711
$987$	10.7218	0.341280
$988$	14.9368	0.475203
$989$	0.00805020	0.000255981 0
$990$	19.7205	0.626759
$991$	−46.2553	−1.46935	−0.734675	−	0.678420i	$-0.762665\pi$
$991$	−46.2553	−1.46935	−0.734675	+	0.678420i	$0.762665\pi$
$992$	53.7812	1.70756
$993$	26.4915	0.840682
$994$	44.2112	1.40229
$995$	−28.6190	−0.907285
$996$	5.08082	0.160992
$997$	−13.3117	−0.421586	−0.210793	−	0.977531i	$-0.567605\pi$
$997$	−13.3117	−0.421586	−0.210793	+	0.977531i	$0.567605\pi$
$998$	75.3555	2.38534
$999$	−1.14698	−0.0362888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.19	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.19	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.01958 q^{2} -1.00000 q^{3} +2.07869 q^{4} -3.91431 q^{5} +2.01958 q^{6} +1.55136 q^{7} -0.158921 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.01958 q^{2} -1.00000 q^{3} +2.07869 q^{4} -3.91431 q^{5} +2.01958 q^{6} +1.55136 q^{7} -0.158921 q^{8} +1.00000 q^{9} +7.90525 q^{10} +2.49461 q^{11} -2.07869 q^{12} -4.97450 q^{13} -3.13309 q^{14} +3.91431 q^{15} -3.83643 q^{16} +1.21523 q^{17} -2.01958 q^{18} -1.44450 q^{19} -8.13664 q^{20} -1.55136 q^{21} -5.03806 q^{22} +0.000901793 q^{23} +0.158921 q^{24} +10.3218 q^{25} +10.0464 q^{26} -1.00000 q^{27} +3.22479 q^{28} -2.86225 q^{29} -7.90525 q^{30} +6.66781 q^{31} +8.06580 q^{32} -2.49461 q^{33} -2.45425 q^{34} -6.07249 q^{35} +2.07869 q^{36} +1.14698 q^{37} +2.91728 q^{38} +4.97450 q^{39} +0.622066 q^{40} -10.7711 q^{41} +3.13309 q^{42} +8.92688 q^{43} +5.18553 q^{44} -3.91431 q^{45} -0.00182124 q^{46} -6.91126 q^{47} +3.83643 q^{48} -4.59329 q^{49} -20.8457 q^{50} -1.21523 q^{51} -10.3405 q^{52} +0.762037 q^{53} +2.01958 q^{54} -9.76468 q^{55} -0.246544 q^{56} +1.44450 q^{57} +5.78054 q^{58} +4.41002 q^{59} +8.13664 q^{60} +6.80128 q^{61} -13.4662 q^{62} +1.55136 q^{63} -8.61665 q^{64} +19.4717 q^{65} +5.03806 q^{66} -3.44974 q^{67} +2.52608 q^{68} -0.000901793 q^{69} +12.2639 q^{70} -14.1111 q^{71} -0.158921 q^{72} +7.81730 q^{73} -2.31641 q^{74} -10.3218 q^{75} -3.00267 q^{76} +3.87004 q^{77} -10.0464 q^{78} +11.3893 q^{79} +15.0170 q^{80} +1.00000 q^{81} +21.7532 q^{82} -2.44424 q^{83} -3.22479 q^{84} -4.75678 q^{85} -18.0285 q^{86} +2.86225 q^{87} -0.396446 q^{88} -10.8863 q^{89} +7.90525 q^{90} -7.71723 q^{91} +0.00187455 q^{92} -6.66781 q^{93} +13.9578 q^{94} +5.65423 q^{95} -8.06580 q^{96} -9.60470 q^{97} +9.27650 q^{98} +2.49461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

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