LMFDB - Embedded newform 8038.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.90498 q^{3} +1.00000 q^{4} -2.22164 q^{5} +2.90498 q^{6} -2.98182 q^{7} -1.00000 q^{8} +5.43889 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.90498 q^{3} +1.00000 q^{4} -2.22164 q^{5} +2.90498 q^{6} -2.98182 q^{7} -1.00000 q^{8} +5.43889 q^{9} +2.22164 q^{10} +4.22828 q^{11} -2.90498 q^{12} +4.16523 q^{13} +2.98182 q^{14} +6.45381 q^{15} +1.00000 q^{16} -1.74613 q^{17} -5.43889 q^{18} +3.08078 q^{19} -2.22164 q^{20} +8.66212 q^{21} -4.22828 q^{22} -8.61325 q^{23} +2.90498 q^{24} -0.0643144 q^{25} -4.16523 q^{26} -7.08493 q^{27} -2.98182 q^{28} +4.24751 q^{29} -6.45381 q^{30} -3.62697 q^{31} -1.00000 q^{32} -12.2831 q^{33} +1.74613 q^{34} +6.62453 q^{35} +5.43889 q^{36} -7.43425 q^{37} -3.08078 q^{38} -12.0999 q^{39} +2.22164 q^{40} +0.156049 q^{41} -8.66212 q^{42} +7.01163 q^{43} +4.22828 q^{44} -12.0833 q^{45} +8.61325 q^{46} -7.08595 q^{47} -2.90498 q^{48} +1.89125 q^{49} +0.0643144 q^{50} +5.07246 q^{51} +4.16523 q^{52} -7.29358 q^{53} +7.08493 q^{54} -9.39372 q^{55} +2.98182 q^{56} -8.94960 q^{57} -4.24751 q^{58} +8.81234 q^{59} +6.45381 q^{60} +2.37193 q^{61} +3.62697 q^{62} -16.2178 q^{63} +1.00000 q^{64} -9.25365 q^{65} +12.2831 q^{66} +4.36626 q^{67} -1.74613 q^{68} +25.0213 q^{69} -6.62453 q^{70} -6.53149 q^{71} -5.43889 q^{72} -14.3645 q^{73} +7.43425 q^{74} +0.186832 q^{75} +3.08078 q^{76} -12.6080 q^{77} +12.0999 q^{78} +15.4884 q^{79} -2.22164 q^{80} +4.26487 q^{81} -0.156049 q^{82} +12.1839 q^{83} +8.66212 q^{84} +3.87927 q^{85} -7.01163 q^{86} -12.3389 q^{87} -4.22828 q^{88} -16.8165 q^{89} +12.0833 q^{90} -12.4200 q^{91} -8.61325 q^{92} +10.5363 q^{93} +7.08595 q^{94} -6.84439 q^{95} +2.90498 q^{96} +10.9548 q^{97} -1.89125 q^{98} +22.9972 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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
$3$	−2.90498	−1.67719	−0.838595	−	0.544756i	$-0.816622\pi$
$3$	−2.90498	−1.67719	−0.838595	+	0.544756i	$0.816622\pi$
$4$	1.00000	0.500000
$5$	−2.22164	−0.993548	−0.496774	−	0.867880i	$-0.665482\pi$
$5$	−2.22164	−0.993548	−0.496774	+	0.867880i	$0.665482\pi$
$6$	2.90498	1.18595
$7$	−2.98182	−1.12702	−0.563511	−	0.826109i	$-0.690550\pi$
$7$	−2.98182	−1.12702	−0.563511	+	0.826109i	$0.690550\pi$
$8$	−1.00000	−0.353553
$9$	5.43889	1.81296
$10$	2.22164	0.702544
$11$	4.22828	1.27487	0.637437	−	0.770502i	$-0.279995\pi$
$11$	4.22828	1.27487	0.637437	+	0.770502i	$0.279995\pi$
$12$	−2.90498	−0.838595
$13$	4.16523	1.15523	0.577614	−	0.816310i	$-0.303984\pi$
$13$	4.16523	1.15523	0.577614	+	0.816310i	$0.303984\pi$
$14$	2.98182	0.796925
$15$	6.45381	1.66637
$16$	1.00000	0.250000
$17$	−1.74613	−0.423498	−0.211749	−	0.977324i	$-0.567916\pi$
$17$	−1.74613	−0.423498	−0.211749	+	0.977324i	$0.567916\pi$
$18$	−5.43889	−1.28196
$19$	3.08078	0.706780	0.353390	−	0.935476i	$-0.385029\pi$
$19$	3.08078	0.706780	0.353390	+	0.935476i	$0.385029\pi$
$20$	−2.22164	−0.496774
$21$	8.66212	1.89023
$22$	−4.22828	−0.901472
$23$	−8.61325	−1.79599	−0.897994	−	0.440008i	$-0.854976\pi$
$23$	−8.61325	−1.79599	−0.897994	+	0.440008i	$0.854976\pi$
$24$	2.90498	0.592976
$25$	−0.0643144	−0.0128629
$26$	−4.16523	−0.816869
$27$	−7.08493	−1.36349
$28$	−2.98182	−0.563511
$29$	4.24751	0.788742	0.394371	−	0.918951i	$-0.370962\pi$
$29$	4.24751	0.788742	0.394371	+	0.918951i	$0.370962\pi$
$30$	−6.45381	−1.17830
$31$	−3.62697	−0.651422	−0.325711	−	0.945469i	$-0.605604\pi$
$31$	−3.62697	−0.651422	−0.325711	+	0.945469i	$0.605604\pi$
$32$	−1.00000	−0.176777
$33$	−12.2831	−2.13821
$34$	1.74613	0.299458
$35$	6.62453	1.11975
$36$	5.43889	0.906482
$37$	−7.43425	−1.22218	−0.611092	−	0.791560i	$-0.709269\pi$
$37$	−7.43425	−1.22218	−0.611092	+	0.791560i	$0.709269\pi$
$38$	−3.08078	−0.499769
$39$	−12.0999	−1.93754
$40$	2.22164	0.351272
$41$	0.156049	0.0243708	0.0121854	−	0.999926i	$-0.496121\pi$
$41$	0.156049	0.0243708	0.0121854	+	0.999926i	$0.496121\pi$
$42$	−8.66212	−1.33659
$43$	7.01163	1.06926	0.534632	−	0.845085i	$-0.320450\pi$
$43$	7.01163	1.06926	0.534632	+	0.845085i	$0.320450\pi$
$44$	4.22828	0.637437
$45$	−12.0833	−1.80127
$46$	8.61325	1.26995
$47$	−7.08595	−1.03359	−0.516796	−	0.856108i	$-0.672876\pi$
$47$	−7.08595	−1.03359	−0.516796	+	0.856108i	$0.672876\pi$
$48$	−2.90498	−0.419297
$49$	1.89125	0.270178
$50$	0.0643144	0.00909542
$51$	5.07246	0.710287
$52$	4.16523	0.577614
$53$	−7.29358	−1.00185	−0.500925	−	0.865491i	$-0.667007\pi$
$53$	−7.29358	−1.00185	−0.500925	+	0.865491i	$0.667007\pi$
$54$	7.08493	0.964136
$55$	−9.39372	−1.26665
$56$	2.98182	0.398462
$57$	−8.94960	−1.18540
$58$	−4.24751	−0.557725
$59$	8.81234	1.14727	0.573634	−	0.819112i	$-0.305533\pi$
$59$	8.81234	1.14727	0.573634	+	0.819112i	$0.305533\pi$
$60$	6.45381	0.833184
$61$	2.37193	0.303695	0.151847	−	0.988404i	$-0.451478\pi$
$61$	2.37193	0.303695	0.151847	+	0.988404i	$0.451478\pi$
$62$	3.62697	0.460625
$63$	−16.2178	−2.04325
$64$	1.00000	0.125000
$65$	−9.25365	−1.14777
$66$	12.2831	1.51194
$67$	4.36626	0.533424	0.266712	−	0.963776i	$-0.414063\pi$
$67$	4.36626	0.533424	0.266712	+	0.963776i	$0.414063\pi$
$68$	−1.74613	−0.211749
$69$	25.0213	3.01221
$70$	−6.62453	−0.791783
$71$	−6.53149	−0.775145	−0.387572	−	0.921839i	$-0.626686\pi$
$71$	−6.53149	−0.775145	−0.387572	+	0.921839i	$0.626686\pi$
$72$	−5.43889	−0.640980
$73$	−14.3645	−1.68124	−0.840621	−	0.541624i	$-0.817810\pi$
$73$	−14.3645	−1.68124	−0.840621	+	0.541624i	$0.817810\pi$
$74$	7.43425	0.864214
$75$	0.186832	0.0215735
$76$	3.08078	0.353390
$77$	−12.6080	−1.43681
$78$	12.0999	1.37004
$79$	15.4884	1.74259	0.871293	−	0.490764i	$-0.163282\pi$
$79$	15.4884	1.74259	0.871293	+	0.490764i	$0.163282\pi$
$80$	−2.22164	−0.248387
$81$	4.26487	0.473874
$82$	−0.156049	−0.0172328
$83$	12.1839	1.33736	0.668680	−	0.743551i	$-0.266860\pi$
$83$	12.1839	1.33736	0.668680	+	0.743551i	$0.266860\pi$
$84$	8.66212	0.945114
$85$	3.87927	0.420766
$86$	−7.01163	−0.756083
$87$	−12.3389	−1.32287
$88$	−4.22828	−0.450736
$89$	−16.8165	−1.78255	−0.891273	−	0.453466i	$-0.850187\pi$
$89$	−16.8165	−1.78255	−0.891273	+	0.453466i	$0.850187\pi$
$90$	12.0833	1.27369
$91$	−12.4200	−1.30197
$92$	−8.61325	−0.897994
$93$	10.5363	1.09256
$94$	7.08595	0.730860
$95$	−6.84439	−0.702220
$96$	2.90498	0.296488
$97$	10.9548	1.11229	0.556146	−	0.831084i	$-0.312279\pi$
$97$	10.9548	1.11229	0.556146	+	0.831084i	$0.312279\pi$
$98$	−1.89125	−0.191045
$99$	22.9972	2.31130
$100$	−0.0643144	−0.00643144
$101$	6.51027	0.647796	0.323898	−	0.946092i	$-0.395007\pi$
$101$	6.51027	0.647796	0.323898	+	0.946092i	$0.395007\pi$
$102$	−5.07246	−0.502248
$103$	−0.772479	−0.0761147	−0.0380573	−	0.999276i	$-0.512117\pi$
$103$	−0.772479	−0.0761147	−0.0380573	+	0.999276i	$0.512117\pi$
$104$	−4.16523	−0.408435
$105$	−19.2441	−1.87803
$106$	7.29358	0.708415
$107$	5.68912	0.549988	0.274994	−	0.961446i	$-0.411324\pi$
$107$	5.68912	0.549988	0.274994	+	0.961446i	$0.411324\pi$
$108$	−7.08493	−0.681747
$109$	10.8355	1.03785	0.518926	−	0.854819i	$-0.326332\pi$
$109$	10.8355	1.03785	0.518926	+	0.854819i	$0.326332\pi$
$110$	9.39372	0.895656
$111$	21.5963	2.04983
$112$	−2.98182	−0.281755
$113$	1.46427	0.137747	0.0688736	−	0.997625i	$-0.478059\pi$
$113$	1.46427	0.137747	0.0688736	+	0.997625i	$0.478059\pi$
$114$	8.94960	0.838207
$115$	19.1356	1.78440
$116$	4.24751	0.394371
$117$	22.6542	2.09439
$118$	−8.81234	−0.811241
$119$	5.20664	0.477292
$120$	−6.45381	−0.589150
$121$	6.87835	0.625305
$122$	−2.37193	−0.214745
$123$	−0.453320	−0.0408745
$124$	−3.62697	−0.325711
$125$	11.2511	1.00633
$126$	16.2178	1.44480
$127$	20.7453	1.84085	0.920424	−	0.390923i	$-0.127844\pi$
$127$	20.7453	1.84085	0.920424	+	0.390923i	$0.127844\pi$
$128$	−1.00000	−0.0883883
$129$	−20.3686	−1.79336
$130$	9.25365	0.811599
$131$	5.55416	0.485269	0.242634	−	0.970118i	$-0.421988\pi$
$131$	5.55416	0.485269	0.242634	+	0.970118i	$0.421988\pi$
$132$	−12.2831	−1.06910
$133$	−9.18633	−0.796556
$134$	−4.36626	−0.377188
$135$	15.7402	1.35470
$136$	1.74613	0.149729
$137$	−0.208084	−0.0177778	−0.00888890	−	0.999960i	$-0.502829\pi$
$137$	−0.208084	−0.0177778	−0.00888890	+	0.999960i	$0.502829\pi$
$138$	−25.0213	−2.12995
$139$	6.13225	0.520130	0.260065	−	0.965591i	$-0.416256\pi$
$139$	6.13225	0.520130	0.260065	+	0.965591i	$0.416256\pi$
$140$	6.62453	0.559875
$141$	20.5845	1.73353
$142$	6.53149	0.548110
$143$	17.6118	1.47277
$144$	5.43889	0.453241
$145$	−9.43644	−0.783653
$146$	14.3645	1.18882
$147$	−5.49403	−0.453140
$148$	−7.43425	−0.611092
$149$	2.55628	0.209418	0.104709	−	0.994503i	$-0.466609\pi$
$149$	2.55628	0.209418	0.104709	+	0.994503i	$0.466609\pi$
$150$	−0.186832	−0.0152547
$151$	−17.1035	−1.39186	−0.695932	−	0.718107i	$-0.745009\pi$
$151$	−17.1035	−1.39186	−0.695932	+	0.718107i	$0.745009\pi$
$152$	−3.08078	−0.249884
$153$	−9.49700	−0.767787
$154$	12.6080	1.01598
$155$	8.05782	0.647219
$156$	−12.0999	−0.968768
$157$	−8.20628	−0.654932	−0.327466	−	0.944863i	$-0.606195\pi$
$157$	−8.20628	−0.654932	−0.327466	+	0.944863i	$0.606195\pi$
$158$	−15.4884	−1.23219
$159$	21.1877	1.68029
$160$	2.22164	0.175636
$161$	25.6832	2.02412
$162$	−4.26487	−0.335080
$163$	13.0551	1.02256	0.511278	−	0.859415i	$-0.329172\pi$
$163$	13.0551	1.02256	0.511278	+	0.859415i	$0.329172\pi$
$164$	0.156049	0.0121854
$165$	27.2885	2.12441
$166$	−12.1839	−0.945656
$167$	14.8981	1.15285	0.576423	−	0.817151i	$-0.304448\pi$
$167$	14.8981	1.15285	0.576423	+	0.817151i	$0.304448\pi$
$168$	−8.66212	−0.668297
$169$	4.34916	0.334551
$170$	−3.87927	−0.297526
$171$	16.7560	1.28137
$172$	7.01163	0.534632
$173$	−11.2622	−0.856249	−0.428124	−	0.903720i	$-0.640825\pi$
$173$	−11.2622	−0.856249	−0.428124	+	0.903720i	$0.640825\pi$
$174$	12.3389	0.935411
$175$	0.191774	0.0144967
$176$	4.22828	0.318719
$177$	−25.5996	−1.92419
$178$	16.8165	1.26045
$179$	−1.48501	−0.110995	−0.0554973	−	0.998459i	$-0.517674\pi$
$179$	−1.48501	−0.110995	−0.0554973	+	0.998459i	$0.517674\pi$
$180$	−12.0833	−0.900633
$181$	9.69309	0.720482	0.360241	−	0.932859i	$-0.382694\pi$
$181$	9.69309	0.720482	0.360241	+	0.932859i	$0.382694\pi$
$182$	12.4200	0.920629
$183$	−6.89040	−0.509353
$184$	8.61325	0.634977
$185$	16.5162	1.21430
$186$	−10.5363	−0.772556
$187$	−7.38312	−0.539907
$188$	−7.08595	−0.516796
$189$	21.1260	1.53669
$190$	6.84439	0.496544
$191$	4.59560	0.332526	0.166263	−	0.986081i	$-0.446830\pi$
$191$	4.59560	0.332526	0.166263	+	0.986081i	$0.446830\pi$
$192$	−2.90498	−0.209649
$193$	−7.09125	−0.510439	−0.255219	−	0.966883i	$-0.582148\pi$
$193$	−7.09125	−0.510439	−0.255219	+	0.966883i	$0.582148\pi$
$194$	−10.9548	−0.786510
$195$	26.8816	1.92503
$196$	1.89125	0.135089
$197$	−14.1422	−1.00759	−0.503795	−	0.863823i	$-0.668063\pi$
$197$	−14.1422	−1.00759	−0.503795	+	0.863823i	$0.668063\pi$
$198$	−22.9972	−1.63434
$199$	−24.0763	−1.70673	−0.853363	−	0.521317i	$-0.825441\pi$
$199$	−24.0763	−1.70673	−0.853363	+	0.521317i	$0.825441\pi$
$200$	0.0643144	0.00454771
$201$	−12.6839	−0.894653
$202$	−6.51027	−0.458061
$203$	−12.6653	−0.888930
$204$	5.07246	0.355143
$205$	−0.346686	−0.0242136
$206$	0.772479	0.0538212
$207$	−46.8466	−3.25606
$208$	4.16523	0.288807
$209$	13.0264	0.901056
$210$	19.2441	1.32797
$211$	−9.05552	−0.623407	−0.311704	−	0.950179i	$-0.600900\pi$
$211$	−9.05552	−0.623407	−0.311704	+	0.950179i	$0.600900\pi$
$212$	−7.29358	−0.500925
$213$	18.9738	1.30006
$214$	−5.68912	−0.388900
$215$	−15.5773	−1.06236
$216$	7.08493	0.482068
$217$	10.8150	0.734167
$218$	−10.8355	−0.733872
$219$	41.7286	2.81976
$220$	−9.39372	−0.633324
$221$	−7.27303	−0.489237
$222$	−21.5963	−1.44945
$223$	4.71896	0.316005	0.158003	−	0.987439i	$-0.449495\pi$
$223$	4.71896	0.316005	0.158003	+	0.987439i	$0.449495\pi$
$224$	2.98182	0.199231
$225$	−0.349799	−0.0233199
$226$	−1.46427	−0.0974020
$227$	−4.59127	−0.304733	−0.152367	−	0.988324i	$-0.548689\pi$
$227$	−4.59127	−0.304733	−0.152367	+	0.988324i	$0.548689\pi$
$228$	−8.94960	−0.592702
$229$	25.9849	1.71713	0.858567	−	0.512702i	$-0.171355\pi$
$229$	25.9849	1.71713	0.858567	+	0.512702i	$0.171355\pi$
$230$	−19.1356	−1.26176
$231$	36.6259	2.40980
$232$	−4.24751	−0.278863
$233$	−15.8008	−1.03514	−0.517572	−	0.855640i	$-0.673164\pi$
$233$	−15.8008	−1.03514	−0.517572	+	0.855640i	$0.673164\pi$
$234$	−22.6542	−1.48095
$235$	15.7424	1.02692
$236$	8.81234	0.573634
$237$	−44.9936	−2.92265
$238$	−5.20664	−0.337496
$239$	11.3809	0.736168	0.368084	−	0.929793i	$-0.380014\pi$
$239$	11.3809	0.736168	0.368084	+	0.929793i	$0.380014\pi$
$240$	6.45381	0.416592
$241$	−25.6313	−1.65106	−0.825530	−	0.564359i	$-0.809124\pi$
$241$	−25.6313	−1.65106	−0.825530	+	0.564359i	$0.809124\pi$
$242$	−6.87835	−0.442157
$243$	8.86543	0.568717
$244$	2.37193	0.151847
$245$	−4.20167	−0.268435
$246$	0.453320	0.0289026
$247$	12.8322	0.816492
$248$	3.62697	0.230313
$249$	−35.3940	−2.24300
$250$	−11.2511	−0.711581
$251$	−27.1197	−1.71178	−0.855891	−	0.517157i	$-0.826990\pi$
$251$	−27.1197	−1.71178	−0.855891	+	0.517157i	$0.826990\pi$
$252$	−16.2178	−1.02162
$253$	−36.4192	−2.28966
$254$	−20.7453	−1.30168
$255$	−11.2692	−0.705704
$256$	1.00000	0.0625000
$257$	13.6984	0.854483	0.427241	−	0.904138i	$-0.359485\pi$
$257$	13.6984	0.854483	0.427241	+	0.904138i	$0.359485\pi$
$258$	20.3686	1.26809
$259$	22.1676	1.37743
$260$	−9.25365	−0.573887
$261$	23.1017	1.42996
$262$	−5.55416	−0.343137
$263$	14.5403	0.896592	0.448296	−	0.893885i	$-0.352031\pi$
$263$	14.5403	0.896592	0.448296	+	0.893885i	$0.352031\pi$
$264$	12.2831	0.755970
$265$	16.2037	0.995386
$266$	9.18633	0.563250
$267$	48.8516	2.98967
$268$	4.36626	0.266712
$269$	−4.37316	−0.266636	−0.133318	−	0.991073i	$-0.542563\pi$
$269$	−4.37316	−0.266636	−0.133318	+	0.991073i	$0.542563\pi$
$270$	−15.7402	−0.957915
$271$	7.66118	0.465383	0.232692	−	0.972551i	$-0.425247\pi$
$271$	7.66118	0.465383	0.232692	+	0.972551i	$0.425247\pi$
$272$	−1.74613	−0.105875
$273$	36.0797	2.18364
$274$	0.208084	0.0125708
$275$	−0.271939	−0.0163985
$276$	25.0213	1.50611
$277$	3.41987	0.205480	0.102740	−	0.994708i	$-0.467239\pi$
$277$	3.41987	0.205480	0.102740	+	0.994708i	$0.467239\pi$
$278$	−6.13225	−0.367788
$279$	−19.7267	−1.18101
$280$	−6.62453	−0.395891
$281$	24.5440	1.46417	0.732086	−	0.681212i	$-0.238547\pi$
$281$	24.5440	1.46417	0.732086	+	0.681212i	$0.238547\pi$
$282$	−20.5845	−1.22579
$283$	−33.3354	−1.98158	−0.990792	−	0.135395i	$-0.956770\pi$
$283$	−33.3354	−1.98158	−0.990792	+	0.135395i	$0.956770\pi$
$284$	−6.53149	−0.387572
$285$	19.8828	1.17776
$286$	−17.6118	−1.04141
$287$	−0.465311	−0.0274664
$288$	−5.43889	−0.320490
$289$	−13.9510	−0.820649
$290$	9.43644	0.554127
$291$	−31.8235	−1.86553
$292$	−14.3645	−0.840621
$293$	−9.37984	−0.547976	−0.273988	−	0.961733i	$-0.588343\pi$
$293$	−9.37984	−0.547976	−0.273988	+	0.961733i	$0.588343\pi$
$294$	5.49403	0.320418
$295$	−19.5778	−1.13987
$296$	7.43425	0.432107
$297$	−29.9570	−1.73828
$298$	−2.55628	−0.148081
$299$	−35.8762	−2.07477
$300$	0.186832	0.0107867
$301$	−20.9074	−1.20508
$302$	17.1035	0.984197
$303$	−18.9122	−1.08648
$304$	3.08078	0.176695
$305$	−5.26958	−0.301735
$306$	9.49700	0.542907
$307$	28.1814	1.60840	0.804199	−	0.594360i	$-0.202595\pi$
$307$	28.1814	1.60840	0.804199	+	0.594360i	$0.202595\pi$
$308$	−12.6080	−0.718406
$309$	2.24403	0.127659
$310$	−8.05782	−0.457653
$311$	−11.8992	−0.674741	−0.337370	−	0.941372i	$-0.609537\pi$
$311$	−11.8992	−0.674741	−0.337370	+	0.941372i	$0.609537\pi$
$312$	12.0999	0.685022
$313$	24.9438	1.40990	0.704952	−	0.709255i	$-0.250968\pi$
$313$	24.9438	1.40990	0.704952	+	0.709255i	$0.250968\pi$
$314$	8.20628	0.463107
$315$	36.0301	2.03007
$316$	15.4884	0.871293
$317$	−11.0242	−0.619183	−0.309592	−	0.950870i	$-0.600192\pi$
$317$	−11.0242	−0.619183	−0.309592	+	0.950870i	$0.600192\pi$
$318$	−21.1877	−1.18815
$319$	17.9597	1.00555
$320$	−2.22164	−0.124193
$321$	−16.5268	−0.922434
$322$	−25.6832	−1.43127
$323$	−5.37944	−0.299320
$324$	4.26487	0.236937
$325$	−0.267884	−0.0148595
$326$	−13.0551	−0.723056
$327$	−31.4769	−1.74067
$328$	−0.156049	−0.00861638
$329$	21.1290	1.16488
$330$	−27.2885	−1.50218
$331$	31.3549	1.72342	0.861711	−	0.507400i	$-0.169393\pi$
$331$	31.3549	1.72342	0.861711	+	0.507400i	$0.169393\pi$
$332$	12.1839	0.668680
$333$	−40.4341	−2.21578
$334$	−14.8981	−0.815186
$335$	−9.70027	−0.529982
$336$	8.66212	0.472557
$337$	−2.35703	−0.128396	−0.0641979	−	0.997937i	$-0.520449\pi$
$337$	−2.35703	−0.128396	−0.0641979	+	0.997937i	$0.520449\pi$
$338$	−4.34916	−0.236563
$339$	−4.25368	−0.231028
$340$	3.87927	0.210383
$341$	−15.3358	−0.830482
$342$	−16.7560	−0.906063
$343$	15.2334	0.822525
$344$	−7.01163	−0.378042
$345$	−55.5883	−2.99278
$346$	11.2622	0.605459
$347$	21.2560	1.14108	0.570540	−	0.821270i	$-0.306734\pi$
$347$	21.2560	1.14108	0.570540	+	0.821270i	$0.306734\pi$
$348$	−12.3389	−0.661435
$349$	−30.2578	−1.61966	−0.809830	−	0.586664i	$-0.800441\pi$
$349$	−30.2578	−1.61966	−0.809830	+	0.586664i	$0.800441\pi$
$350$	−0.191774	−0.0102507
$351$	−29.5104	−1.57515
$352$	−4.22828	−0.225368
$353$	33.0283	1.75792	0.878961	−	0.476894i	$-0.158237\pi$
$353$	33.0283	1.75792	0.878961	+	0.476894i	$0.158237\pi$
$354$	25.5996	1.36061
$355$	14.5106	0.770143
$356$	−16.8165	−0.891273
$357$	−15.1252	−0.800508
$358$	1.48501	0.0784850
$359$	−19.9543	−1.05315	−0.526575	−	0.850129i	$-0.676524\pi$
$359$	−19.9543	−1.05315	−0.526575	+	0.850129i	$0.676524\pi$
$360$	12.0833	0.636844
$361$	−9.50878	−0.500462
$362$	−9.69309	−0.509458
$363$	−19.9815	−1.04875
$364$	−12.4200	−0.650983
$365$	31.9128	1.67039
$366$	6.89040	0.360167
$367$	10.5997	0.553299	0.276650	−	0.960971i	$-0.410776\pi$
$367$	10.5997	0.553299	0.276650	+	0.960971i	$0.410776\pi$
$368$	−8.61325	−0.448997
$369$	0.848736	0.0441834
$370$	−16.5162	−0.858638
$371$	21.7481	1.12911
$372$	10.5363	0.546279
$373$	2.32809	0.120544	0.0602721	−	0.998182i	$-0.480803\pi$
$373$	2.32809	0.120544	0.0602721	+	0.998182i	$0.480803\pi$
$374$	7.38312	0.381772
$375$	−32.6841	−1.68780
$376$	7.08595	0.365430
$377$	17.6919	0.911177
$378$	−21.1260	−1.08660
$379$	−11.5106	−0.591261	−0.295630	−	0.955302i	$-0.595530\pi$
$379$	−11.5106	−0.591261	−0.295630	+	0.955302i	$0.595530\pi$
$380$	−6.84439	−0.351110
$381$	−60.2646	−3.08745
$382$	−4.59560	−0.235131
$383$	−23.9955	−1.22611	−0.613055	−	0.790040i	$-0.710060\pi$
$383$	−23.9955	−1.22611	−0.613055	+	0.790040i	$0.710060\pi$
$384$	2.90498	0.148244
$385$	28.0104	1.42754
$386$	7.09125	0.360935
$387$	38.1355	1.93854
$388$	10.9548	0.556146
$389$	−12.7748	−0.647710	−0.323855	−	0.946107i	$-0.604979\pi$
$389$	−12.7748	−0.647710	−0.323855	+	0.946107i	$0.604979\pi$
$390$	−26.8816	−1.36120
$391$	15.0398	0.760597
$392$	−1.89125	−0.0955223
$393$	−16.1347	−0.813888
$394$	14.1422	0.712473
$395$	−34.4097	−1.73134
$396$	22.9972	1.15565
$397$	33.0658	1.65953	0.829763	−	0.558115i	$-0.188475\pi$
$397$	33.0658	1.65953	0.829763	+	0.558115i	$0.188475\pi$
$398$	24.0763	1.20684
$399$	26.6861	1.33598
$400$	−0.0643144	−0.00321572
$401$	15.7180	0.784922	0.392461	−	0.919769i	$-0.371624\pi$
$401$	15.7180	0.784922	0.392461	+	0.919769i	$0.371624\pi$
$402$	12.6839	0.632615
$403$	−15.1072	−0.752541
$404$	6.51027	0.323898
$405$	−9.47501	−0.470817
$406$	12.6653	0.628568
$407$	−31.4341	−1.55813
$408$	−5.07246	−0.251124
$409$	9.58737	0.474065	0.237033	−	0.971502i	$-0.423825\pi$
$409$	9.58737	0.474065	0.237033	+	0.971502i	$0.423825\pi$
$410$	0.346686	0.0171216
$411$	0.604478	0.0298167
$412$	−0.772479	−0.0380573
$413$	−26.2768	−1.29300
$414$	46.8466	2.30238
$415$	−27.0683	−1.32873
$416$	−4.16523	−0.204217
$417$	−17.8140	−0.872357
$418$	−13.0264	−0.637142
$419$	8.06688	0.394093	0.197046	−	0.980394i	$-0.436865\pi$
$419$	8.06688	0.394093	0.197046	+	0.980394i	$0.436865\pi$
$420$	−19.2441	−0.939016
$421$	27.5373	1.34208	0.671042	−	0.741420i	$-0.265847\pi$
$421$	27.5373	1.34208	0.671042	+	0.741420i	$0.265847\pi$
$422$	9.05552	0.440816
$423$	−38.5397	−1.87387
$424$	7.29358	0.354207
$425$	0.112301	0.00544740
$426$	−18.9738	−0.919284
$427$	−7.07267	−0.342270
$428$	5.68912	0.274994
$429$	−51.1618	−2.47011
$430$	15.5773	0.751205
$431$	32.7333	1.57671	0.788353	−	0.615223i	$-0.210934\pi$
$431$	32.7333	1.57671	0.788353	+	0.615223i	$0.210934\pi$
$432$	−7.08493	−0.340874
$433$	18.3532	0.881999	0.441000	−	0.897507i	$-0.354624\pi$
$433$	18.3532	0.881999	0.441000	+	0.897507i	$0.354624\pi$
$434$	−10.8150	−0.519135
$435$	27.4126	1.31433
$436$	10.8355	0.518926
$437$	−26.5356	−1.26937
$438$	−41.7286	−1.99387
$439$	−6.14427	−0.293250	−0.146625	−	0.989192i	$-0.546841\pi$
$439$	−6.14427	−0.293250	−0.146625	+	0.989192i	$0.546841\pi$
$440$	9.39372	0.447828
$441$	10.2863	0.489823
$442$	7.27303	0.345943
$443$	11.6745	0.554671	0.277335	−	0.960773i	$-0.410549\pi$
$443$	11.6745	0.554671	0.277335	+	0.960773i	$0.410549\pi$
$444$	21.5963	1.02492
$445$	37.3602	1.77105
$446$	−4.71896	−0.223449
$447$	−7.42593	−0.351234
$448$	−2.98182	−0.140878
$449$	−25.6521	−1.21060	−0.605299	−	0.795998i	$-0.706947\pi$
$449$	−25.6521	−1.21060	−0.605299	+	0.795998i	$0.706947\pi$
$450$	0.349799	0.0164897
$451$	0.659820	0.0310697
$452$	1.46427	0.0688736
$453$	49.6853	2.33442
$454$	4.59127	0.215479
$455$	27.5927	1.29357
$456$	8.94960	0.419103
$457$	−9.86781	−0.461597	−0.230798	−	0.973002i	$-0.574134\pi$
$457$	−9.86781	−0.461597	−0.230798	+	0.973002i	$0.574134\pi$
$458$	−25.9849	−1.21420
$459$	12.3712	0.577437
$460$	19.1356	0.892200
$461$	−12.2072	−0.568544	−0.284272	−	0.958744i	$-0.591752\pi$
$461$	−12.2072	−0.568544	−0.284272	+	0.958744i	$0.591752\pi$
$462$	−36.6259	−1.70399
$463$	19.6093	0.911323	0.455661	−	0.890153i	$-0.349403\pi$
$463$	19.6093	0.911323	0.455661	+	0.890153i	$0.349403\pi$
$464$	4.24751	0.197186
$465$	−23.4078	−1.08551
$466$	15.8008	0.731957
$467$	−34.2488	−1.58485	−0.792423	−	0.609972i	$-0.791181\pi$
$467$	−34.2488	−1.58485	−0.792423	+	0.609972i	$0.791181\pi$
$468$	22.6542	1.04719
$469$	−13.0194	−0.601180
$470$	−15.7424	−0.726144
$471$	23.8391	1.09845
$472$	−8.81234	−0.405621
$473$	29.6471	1.36318
$474$	44.9936	2.06662
$475$	−0.198139	−0.00909122
$476$	5.20664	0.238646
$477$	−39.6690	−1.81632
$478$	−11.3809	−0.520549
$479$	−40.9632	−1.87166	−0.935829	−	0.352454i	$-0.885347\pi$
$479$	−40.9632	−1.87166	−0.935829	+	0.352454i	$0.885347\pi$
$480$	−6.45381	−0.294575
$481$	−30.9654	−1.41190
$482$	25.6313	1.16748
$483$	−74.6090	−3.39483
$484$	6.87835	0.312652
$485$	−24.3377	−1.10512
$486$	−8.86543	−0.402144
$487$	26.4807	1.19995	0.599977	−	0.800018i	$-0.295177\pi$
$487$	26.4807	1.19995	0.599977	+	0.800018i	$0.295177\pi$
$488$	−2.37193	−0.107372
$489$	−37.9248	−1.71502
$490$	4.20167	0.189812
$491$	40.8604	1.84401	0.922003	−	0.387184i	$-0.126552\pi$
$491$	40.8604	1.84401	0.922003	+	0.387184i	$0.126552\pi$
$492$	−0.453320	−0.0204372
$493$	−7.41669	−0.334031
$494$	−12.8322	−0.577347
$495$	−51.0914	−2.29639
$496$	−3.62697	−0.162856
$497$	19.4757	0.873605
$498$	35.3940	1.58604
$499$	0.124691	0.00558192	0.00279096	−	0.999996i	$-0.499112\pi$
$499$	0.124691	0.00558192	0.00279096	+	0.999996i	$0.499112\pi$
$500$	11.2511	0.503164
$501$	−43.2785	−1.93354
$502$	27.1197	1.21041
$503$	−40.2763	−1.79583	−0.897916	−	0.440167i	$-0.854919\pi$
$503$	−40.2763	−1.79583	−0.897916	+	0.440167i	$0.854919\pi$
$504$	16.2178	0.722398
$505$	−14.4635	−0.643616
$506$	36.4192	1.61903
$507$	−12.6342	−0.561105
$508$	20.7453	0.920424
$509$	−31.0106	−1.37452	−0.687260	−	0.726411i	$-0.741187\pi$
$509$	−31.0106	−1.37452	−0.687260	+	0.726411i	$0.741187\pi$
$510$	11.2692	0.499008
$511$	42.8324	1.89480
$512$	−1.00000	−0.0441942
$513$	−21.8271	−0.963690
$514$	−13.6984	−0.604210
$515$	1.71617	0.0756235
$516$	−20.3686	−0.896678
$517$	−29.9614	−1.31770
$518$	−22.1676	−0.973988
$519$	32.7164	1.43609
$520$	9.25365	0.405799
$521$	−2.12258	−0.0929917	−0.0464959	−	0.998918i	$-0.514805\pi$
$521$	−2.12258	−0.0929917	−0.0464959	+	0.998918i	$0.514805\pi$
$522$	−23.1017	−1.01114
$523$	21.0472	0.920331	0.460165	−	0.887833i	$-0.347790\pi$
$523$	21.0472	0.920331	0.460165	+	0.887833i	$0.347790\pi$
$524$	5.55416	0.242634
$525$	−0.557098	−0.0243138
$526$	−14.5403	−0.633986
$527$	6.33315	0.275876
$528$	−12.2831	−0.534551
$529$	51.1881	2.22557
$530$	−16.2037	−0.703844
$531$	47.9294	2.07996
$532$	−9.18633	−0.398278
$533$	0.649982	0.0281538
$534$	−48.8516	−2.11402
$535$	−12.6392	−0.546440
$536$	−4.36626	−0.188594
$537$	4.31391	0.186159
$538$	4.37316	0.188540
$539$	7.99672	0.344443
$540$	15.7402	0.677348
$541$	−21.2615	−0.914104	−0.457052	−	0.889440i	$-0.651095\pi$
$541$	−21.2615	−0.914104	−0.457052	+	0.889440i	$0.651095\pi$
$542$	−7.66118	−0.329076
$543$	−28.1582	−1.20838
$544$	1.74613	0.0748646
$545$	−24.0726	−1.03116
$546$	−36.0797	−1.54407
$547$	21.0490	0.899992	0.449996	−	0.893031i	$-0.351425\pi$
$547$	21.0490	0.899992	0.449996	+	0.893031i	$0.351425\pi$
$548$	−0.208084	−0.00888890
$549$	12.9007	0.550587
$550$	0.271939	0.0115955
$551$	13.0856	0.557467
$552$	−25.0213	−1.06498
$553$	−46.1837	−1.96393
$554$	−3.41987	−0.145296
$555$	−47.9793	−2.03661
$556$	6.13225	0.260065
$557$	−10.5194	−0.445721	−0.222860	−	0.974850i	$-0.571539\pi$
$557$	−10.5194	−0.445721	−0.222860	+	0.974850i	$0.571539\pi$
$558$	19.7267	0.835097
$559$	29.2051	1.23524
$560$	6.62453	0.279937
$561$	21.4478	0.905526
$562$	−24.5440	−1.03533
$563$	−36.3521	−1.53206	−0.766030	−	0.642805i	$-0.777771\pi$
$563$	−36.3521	−1.53206	−0.766030	+	0.642805i	$0.777771\pi$
$564$	20.5845	0.866765
$565$	−3.25309	−0.136858
$566$	33.3354	1.40119
$567$	−12.7171	−0.534067
$568$	6.53149	0.274055
$569$	37.1806	1.55869	0.779346	−	0.626594i	$-0.215551\pi$
$569$	37.1806	1.55869	0.779346	+	0.626594i	$0.215551\pi$
$570$	−19.8828	−0.832799
$571$	37.0425	1.55018	0.775089	−	0.631852i	$-0.217705\pi$
$571$	37.0425	1.55018	0.775089	+	0.631852i	$0.217705\pi$
$572$	17.6118	0.736385
$573$	−13.3501	−0.557709
$574$	0.465311	0.0194217
$575$	0.553956	0.0231016
$576$	5.43889	0.226620
$577$	−1.42174	−0.0591878	−0.0295939	−	0.999562i	$-0.509421\pi$
$577$	−1.42174	−0.0591878	−0.0295939	+	0.999562i	$0.509421\pi$
$578$	13.9510	0.580287
$579$	20.5999	0.856103
$580$	−9.43644	−0.391827
$581$	−36.3303	−1.50723
$582$	31.8235	1.31913
$583$	−30.8393	−1.27723
$584$	14.3645	0.594409
$585$	−50.3296	−2.08087
$586$	9.37984	0.387478
$587$	−8.92929	−0.368551	−0.184276	−	0.982875i	$-0.558994\pi$
$587$	−8.92929	−0.368551	−0.184276	+	0.982875i	$0.558994\pi$
$588$	−5.49403	−0.226570
$589$	−11.1739	−0.460412
$590$	19.5778	0.806007
$591$	41.0827	1.68992
$592$	−7.43425	−0.305546
$593$	16.1147	0.661750	0.330875	−	0.943675i	$-0.392656\pi$
$593$	16.1147	0.661750	0.330875	+	0.943675i	$0.392656\pi$
$594$	29.9570	1.22915
$595$	−11.5673	−0.474212
$596$	2.55628	0.104709
$597$	69.9412	2.86250
$598$	35.8762	1.46709
$599$	−5.93054	−0.242315	−0.121158	−	0.992633i	$-0.538661\pi$
$599$	−5.93054	−0.242315	−0.121158	+	0.992633i	$0.538661\pi$
$600$	−0.186832	−0.00762737
$601$	9.80133	0.399805	0.199902	−	0.979816i	$-0.435938\pi$
$601$	9.80133	0.399805	0.199902	+	0.979816i	$0.435938\pi$
$602$	20.9074	0.852122
$603$	23.7476	0.967078
$604$	−17.1035	−0.695932
$605$	−15.2812	−0.621270
$606$	18.9122	0.768255
$607$	−9.10195	−0.369437	−0.184718	−	0.982791i	$-0.559137\pi$
$607$	−9.10195	−0.369437	−0.184718	+	0.982791i	$0.559137\pi$
$608$	−3.08078	−0.124942
$609$	36.7924	1.49090
$610$	5.26958	0.213359
$611$	−29.5146	−1.19403
$612$	−9.49700	−0.383893
$613$	−47.0337	−1.89967	−0.949836	−	0.312748i	$-0.898750\pi$
$613$	−47.0337	−1.89967	−0.949836	+	0.312748i	$0.898750\pi$
$614$	−28.1814	−1.13731
$615$	1.00711	0.0406107
$616$	12.6080	0.507989
$617$	4.81967	0.194032	0.0970162	−	0.995283i	$-0.469070\pi$
$617$	4.81967	0.194032	0.0970162	+	0.995283i	$0.469070\pi$
$618$	−2.24403	−0.0902683
$619$	−7.14748	−0.287281	−0.143641	−	0.989630i	$-0.545881\pi$
$619$	−7.14748	−0.287281	−0.143641	+	0.989630i	$0.545881\pi$
$620$	8.05782	0.323610
$621$	61.0243	2.44882
$622$	11.8992	0.477114
$623$	50.1438	2.00897
$624$	−12.0999	−0.484384
$625$	−24.6743	−0.986972
$626$	−24.9438	−0.996953
$627$	−37.8414	−1.51124
$628$	−8.20628	−0.327466
$629$	12.9812	0.517593
$630$	−36.0301	−1.43547
$631$	−35.8168	−1.42584	−0.712922	−	0.701244i	$-0.752629\pi$
$631$	−35.8168	−1.42584	−0.712922	+	0.701244i	$0.752629\pi$
$632$	−15.4884	−0.616097
$633$	26.3061	1.04557
$634$	11.0242	0.437829
$635$	−46.0886	−1.82897
$636$	21.1877	0.840146
$637$	7.87748	0.312117
$638$	−17.9597	−0.711030
$639$	−35.5240	−1.40531
$640$	2.22164	0.0878180
$641$	−49.3164	−1.94788	−0.973941	−	0.226802i	$-0.927173\pi$
$641$	−49.3164	−1.94788	−0.973941	+	0.226802i	$0.927173\pi$
$642$	16.5268	0.652260
$643$	−27.4410	−1.08217	−0.541083	−	0.840969i	$-0.681986\pi$
$643$	−27.4410	−1.08217	−0.541083	+	0.840969i	$0.681986\pi$
$644$	25.6832	1.01206
$645$	45.2517	1.78179
$646$	5.37944	0.211651
$647$	4.49293	0.176635	0.0883176	−	0.996092i	$-0.471851\pi$
$647$	4.49293	0.176635	0.0883176	+	0.996092i	$0.471851\pi$
$648$	−4.26487	−0.167540
$649$	37.2610	1.46262
$650$	0.267884	0.0105073
$651$	−31.4172	−1.23134
$652$	13.0551	0.511278
$653$	24.1545	0.945238	0.472619	−	0.881267i	$-0.343309\pi$
$653$	24.1545	0.945238	0.472619	+	0.881267i	$0.343309\pi$
$654$	31.4769	1.23084
$655$	−12.3393	−0.482138
$656$	0.156049	0.00609270
$657$	−78.1271	−3.04803
$658$	−21.1290	−0.823695
$659$	−29.9730	−1.16758	−0.583791	−	0.811904i	$-0.698431\pi$
$659$	−29.9730	−1.16758	−0.583791	+	0.811904i	$0.698431\pi$
$660$	27.2885	1.06220
$661$	24.0135	0.934017	0.467008	−	0.884253i	$-0.345332\pi$
$661$	24.0135	0.934017	0.467008	+	0.884253i	$0.345332\pi$
$662$	−31.3549	−1.21864
$663$	21.1280	0.820543
$664$	−12.1839	−0.472828
$665$	20.4087	0.791417
$666$	40.4341	1.56679
$667$	−36.5849	−1.41657
$668$	14.8981	0.576423
$669$	−13.7085	−0.530001
$670$	9.70027	0.374754
$671$	10.0292	0.387172
$672$	−8.66212	−0.334148
$673$	−5.17308	−0.199408	−0.0997038	−	0.995017i	$-0.531790\pi$
$673$	−5.17308	−0.199408	−0.0997038	+	0.995017i	$0.531790\pi$
$674$	2.35703	0.0907895
$675$	0.455662	0.0175385
$676$	4.34916	0.167275
$677$	6.15055	0.236385	0.118192	−	0.992991i	$-0.462290\pi$
$677$	6.15055	0.236385	0.118192	+	0.992991i	$0.462290\pi$
$678$	4.25368	0.163362
$679$	−32.6653	−1.25358
$680$	−3.87927	−0.148763
$681$	13.3375	0.511096
$682$	15.3358	0.587239
$683$	15.2282	0.582693	0.291346	−	0.956618i	$-0.405897\pi$
$683$	15.2282	0.582693	0.291346	+	0.956618i	$0.405897\pi$
$684$	16.7560	0.640683
$685$	0.462287	0.0176631
$686$	−15.2334	−0.581613
$687$	−75.4857	−2.87996
$688$	7.01163	0.267316
$689$	−30.3794	−1.15736
$690$	55.5883	2.11621
$691$	−31.9631	−1.21593	−0.607966	−	0.793963i	$-0.708014\pi$
$691$	−31.9631	−1.21593	−0.607966	+	0.793963i	$0.708014\pi$
$692$	−11.2622	−0.428124
$693$	−68.5734	−2.60489
$694$	−21.2560	−0.806865
$695$	−13.6236	−0.516774
$696$	12.3389	0.467705
$697$	−0.272482	−0.0103210
$698$	30.2578	1.14527
$699$	45.9009	1.73613
$700$	0.191774	0.00724837
$701$	−18.0689	−0.682452	−0.341226	−	0.939981i	$-0.610842\pi$
$701$	−18.0689	−0.682452	−0.341226	+	0.939981i	$0.610842\pi$
$702$	29.5104	1.11380
$703$	−22.9033	−0.863815
$704$	4.22828	0.159359
$705$	−45.7314	−1.72234
$706$	−33.0283	−1.24304
$707$	−19.4124	−0.730080
$708$	−25.5996	−0.962093
$709$	28.9897	1.08873	0.544365	−	0.838849i	$-0.316771\pi$
$709$	28.9897	1.08873	0.544365	+	0.838849i	$0.316771\pi$
$710$	−14.5106	−0.544574
$711$	84.2399	3.15924
$712$	16.8165	0.630226
$713$	31.2400	1.16995
$714$	15.1252	0.566045
$715$	−39.1270	−1.46327
$716$	−1.48501	−0.0554973
$717$	−33.0612	−1.23469
$718$	19.9543	0.744690
$719$	−45.4316	−1.69431	−0.847157	−	0.531343i	$-0.821688\pi$
$719$	−45.4316	−1.69431	−0.847157	+	0.531343i	$0.821688\pi$
$720$	−12.0833	−0.450317
$721$	2.30339	0.0857829
$722$	9.50878	0.353880
$723$	74.4584	2.76914
$724$	9.69309	0.360241
$725$	−0.273176	−0.0101455
$726$	19.9815	0.741581
$727$	−35.0270	−1.29908	−0.649540	−	0.760328i	$-0.725038\pi$
$727$	−35.0270	−1.29908	−0.649540	+	0.760328i	$0.725038\pi$
$728$	12.4200	0.460315
$729$	−38.5485	−1.42772
$730$	−31.9128	−1.18115
$731$	−12.2432	−0.452831
$732$	−6.89040	−0.254677
$733$	−35.3337	−1.30508	−0.652540	−	0.757754i	$-0.726297\pi$
$733$	−35.3337	−1.30508	−0.652540	+	0.757754i	$0.726297\pi$
$734$	−10.5997	−0.391242
$735$	12.2057	0.450216
$736$	8.61325	0.317489
$737$	18.4618	0.680049
$738$	−0.848736	−0.0312424
$739$	25.0265	0.920616	0.460308	−	0.887759i	$-0.347739\pi$
$739$	25.0265	0.920616	0.460308	+	0.887759i	$0.347739\pi$
$740$	16.5162	0.607149
$741$	−37.2772	−1.36941
$742$	−21.7481	−0.798399
$743$	7.76972	0.285043	0.142522	−	0.989792i	$-0.454479\pi$
$743$	7.76972	0.285043	0.142522	+	0.989792i	$0.454479\pi$
$744$	−10.5363	−0.386278
$745$	−5.67913	−0.208067
$746$	−2.32809	−0.0852376
$747$	66.2670	2.42458
$748$	−7.38312	−0.269953
$749$	−16.9639	−0.619849
$750$	32.6841	1.19346
$751$	36.6582	1.33768	0.668838	−	0.743408i	$-0.266792\pi$
$751$	36.6582	1.33768	0.668838	+	0.743408i	$0.266792\pi$
$752$	−7.08595	−0.258398
$753$	78.7821	2.87098
$754$	−17.6919	−0.644299
$755$	37.9979	1.38288
$756$	21.1260	0.768344
$757$	2.70400	0.0982786	0.0491393	−	0.998792i	$-0.484352\pi$
$757$	2.70400	0.0982786	0.0491393	+	0.998792i	$0.484352\pi$
$758$	11.5106	0.418084
$759$	105.797	3.84019
$760$	6.84439	0.248272
$761$	5.98631	0.217004	0.108502	−	0.994096i	$-0.465395\pi$
$761$	5.98631	0.217004	0.108502	+	0.994096i	$0.465395\pi$
$762$	60.2646	2.18316
$763$	−32.3095	−1.16968
$764$	4.59560	0.166263
$765$	21.0989	0.762833
$766$	23.9955	0.866991
$767$	36.7054	1.32536
$768$	−2.90498	−0.104824
$769$	−18.2094	−0.656648	−0.328324	−	0.944565i	$-0.606484\pi$
$769$	−18.2094	−0.656648	−0.328324	+	0.944565i	$0.606484\pi$
$770$	−28.0104	−1.00942
$771$	−39.7935	−1.43313
$772$	−7.09125	−0.255219
$773$	50.4674	1.81519	0.907594	−	0.419850i	$-0.137917\pi$
$773$	50.4674	1.81519	0.907594	+	0.419850i	$0.137917\pi$
$774$	−38.1355	−1.37075
$775$	0.233266	0.00837916
$776$	−10.9548	−0.393255
$777$	−64.3964	−2.31021
$778$	12.7748	0.458000
$779$	0.480754	0.0172248
$780$	26.8816	0.962517
$781$	−27.6170	−0.988212
$782$	−15.0398	−0.537824
$783$	−30.0933	−1.07545
$784$	1.89125	0.0675445
$785$	18.2314	0.650707
$786$	16.1347	0.575505
$787$	−36.5869	−1.30418	−0.652091	−	0.758140i	$-0.726108\pi$
$787$	−36.5869	−1.30418	−0.652091	+	0.758140i	$0.726108\pi$
$788$	−14.1422	−0.503795
$789$	−42.2392	−1.50375
$790$	34.4097	1.22424
$791$	−4.36620	−0.155244
$792$	−22.9972	−0.817168
$793$	9.87964	0.350836
$794$	−33.0658	−1.17346
$795$	−47.0714	−1.66945
$796$	−24.0763	−0.853363
$797$	1.79396	0.0635454	0.0317727	−	0.999495i	$-0.489885\pi$
$797$	1.79396	0.0635454	0.0317727	+	0.999495i	$0.489885\pi$
$798$	−26.6861	−0.944677
$799$	12.3730	0.437724
$800$	0.0643144	0.00227386
$801$	−91.4632	−3.23169
$802$	−15.7180	−0.555024
$803$	−60.7373	−2.14337
$804$	−12.6839	−0.447327
$805$	−57.0588	−2.01106
$806$	15.1072	0.532127
$807$	12.7039	0.447199
$808$	−6.51027	−0.229030
$809$	−43.9256	−1.54434	−0.772170	−	0.635416i	$-0.780829\pi$
$809$	−43.9256	−1.54434	−0.772170	+	0.635416i	$0.780829\pi$
$810$	9.47501	0.332918
$811$	6.39978	0.224727	0.112363	−	0.993667i	$-0.464158\pi$
$811$	6.39978	0.224727	0.112363	+	0.993667i	$0.464158\pi$
$812$	−12.6653	−0.444465
$813$	−22.2555	−0.780536
$814$	31.4341	1.10176
$815$	−29.0038	−1.01596
$816$	5.07246	0.177572
$817$	21.6013	0.755734
$818$	−9.58737	−0.335215
$819$	−67.5509	−2.36042
$820$	−0.346686	−0.0121068
$821$	12.0153	0.419336	0.209668	−	0.977773i	$-0.432762\pi$
$821$	12.0153	0.419336	0.209668	+	0.977773i	$0.432762\pi$
$822$	−0.604478	−0.0210836
$823$	45.1188	1.57274	0.786372	−	0.617754i	$-0.211957\pi$
$823$	45.1188	1.57274	0.786372	+	0.617754i	$0.211957\pi$
$824$	0.772479	0.0269106
$825$	0.789977	0.0275035
$826$	26.2768	0.914287
$827$	−3.43612	−0.119486	−0.0597428	−	0.998214i	$-0.519028\pi$
$827$	−3.43612	−0.119486	−0.0597428	+	0.998214i	$0.519028\pi$
$828$	−46.8466	−1.62803
$829$	19.7907	0.687361	0.343680	−	0.939087i	$-0.388326\pi$
$829$	19.7907	0.687361	0.343680	+	0.939087i	$0.388326\pi$
$830$	27.0683	0.939554
$831$	−9.93464	−0.344629
$832$	4.16523	0.144403
$833$	−3.30236	−0.114420
$834$	17.8140	0.616850
$835$	−33.0981	−1.14541
$836$	13.0264	0.450528
$837$	25.6968	0.888211
$838$	−8.06688	−0.278666
$839$	−25.8269	−0.891645	−0.445823	−	0.895121i	$-0.647089\pi$
$839$	−25.8269	−0.891645	−0.445823	+	0.895121i	$0.647089\pi$
$840$	19.2441	0.663985
$841$	−10.9587	−0.377885
$842$	−27.5373	−0.948996
$843$	−71.2998	−2.45569
$844$	−9.05552	−0.311704
$845$	−9.66227	−0.332392
$846$	38.5397	1.32502
$847$	−20.5100	−0.704732
$848$	−7.29358	−0.250462
$849$	96.8386	3.32349
$850$	−0.112301	−0.00385190
$851$	64.0331	2.19503
$852$	18.9738	0.650032
$853$	−3.35061	−0.114723	−0.0573613	−	0.998353i	$-0.518269\pi$
$853$	−3.35061	−0.114723	−0.0573613	+	0.998353i	$0.518269\pi$
$854$	7.07267	0.242022
$855$	−37.2259	−1.27310
$856$	−5.68912	−0.194450
$857$	16.3248	0.557643	0.278822	−	0.960343i	$-0.410056\pi$
$857$	16.3248	0.557643	0.278822	+	0.960343i	$0.410056\pi$
$858$	51.1618	1.74663
$859$	25.5045	0.870201	0.435101	−	0.900382i	$-0.356713\pi$
$859$	25.5045	0.870201	0.435101	+	0.900382i	$0.356713\pi$
$860$	−15.5773	−0.531182
$861$	1.35172	0.0460664
$862$	−32.7333	−1.11490
$863$	30.0601	1.02326	0.511628	−	0.859207i	$-0.329042\pi$
$863$	30.0601	1.02326	0.511628	+	0.859207i	$0.329042\pi$
$864$	7.08493	0.241034
$865$	25.0205	0.850724
$866$	−18.3532	−0.623667
$867$	40.5274	1.37638
$868$	10.8150	0.367084
$869$	65.4895	2.22158
$870$	−27.4126	−0.929375
$871$	18.1865	0.616226
$872$	−10.8355	−0.366936
$873$	59.5820	2.01655
$874$	26.5356	0.897579
$875$	−33.5487	−1.13415
$876$	41.7286	1.40988
$877$	−6.02200	−0.203348	−0.101674	−	0.994818i	$-0.532420\pi$
$877$	−6.02200	−0.203348	−0.101674	+	0.994818i	$0.532420\pi$
$878$	6.14427	0.207359
$879$	27.2482	0.919060
$880$	−9.39372	−0.316662
$881$	8.46317	0.285132	0.142566	−	0.989785i	$-0.454465\pi$
$881$	8.46317	0.285132	0.142566	+	0.989785i	$0.454465\pi$
$882$	−10.2863	−0.346357
$883$	−7.26351	−0.244436	−0.122218	−	0.992503i	$-0.539001\pi$
$883$	−7.26351	−0.244436	−0.122218	+	0.992503i	$0.539001\pi$
$884$	−7.27303	−0.244618
$885$	56.8732	1.91177
$886$	−11.6745	−0.392212
$887$	−50.7311	−1.70338	−0.851692	−	0.524042i	$-0.824423\pi$
$887$	−50.7311	−1.70338	−0.851692	+	0.524042i	$0.824423\pi$
$888$	−21.5963	−0.724726
$889$	−61.8587	−2.07467
$890$	−37.3602	−1.25232
$891$	18.0331	0.604130
$892$	4.71896	0.158003
$893$	−21.8303	−0.730522
$894$	7.42593	0.248360
$895$	3.29915	0.110278
$896$	2.98182	0.0996156
$897$	104.220	3.47979
$898$	25.6521	0.856023
$899$	−15.4056	−0.513805
$900$	−0.349799	−0.0116600
$901$	12.7355	0.424282
$902$	−0.659820	−0.0219696
$903$	60.7355	2.02115
$904$	−1.46427	−0.0487010
$905$	−21.5346	−0.715833
$906$	−49.6853	−1.65068
$907$	24.0171	0.797475	0.398737	−	0.917065i	$-0.369449\pi$
$907$	24.0171	0.797475	0.398737	+	0.917065i	$0.369449\pi$
$908$	−4.59127	−0.152367
$909$	35.4086	1.17443
$910$	−27.5927	−0.914689
$911$	−19.7259	−0.653549	−0.326774	−	0.945102i	$-0.605962\pi$
$911$	−19.7259	−0.653549	−0.326774	+	0.945102i	$0.605962\pi$
$912$	−8.94960	−0.296351
$913$	51.5170	1.70496
$914$	9.86781	0.326398
$915$	15.3080	0.506067
$916$	25.9849	0.858567
$917$	−16.5615	−0.546908
$918$	−12.3712	−0.408310
$919$	24.0270	0.792576	0.396288	−	0.918126i	$-0.370298\pi$
$919$	24.0270	0.792576	0.396288	+	0.918126i	$0.370298\pi$
$920$	−19.1356	−0.630880
$921$	−81.8663	−2.69759
$922$	12.2072	0.402021
$923$	−27.2052	−0.895469
$924$	36.6259	1.20490
$925$	0.478129	0.0157208
$926$	−19.6093	−0.644402
$927$	−4.20143	−0.137993
$928$	−4.24751	−0.139431
$929$	8.15255	0.267476	0.133738	−	0.991017i	$-0.457302\pi$
$929$	8.15255	0.267476	0.133738	+	0.991017i	$0.457302\pi$
$930$	23.4078	0.767571
$931$	5.82652	0.190956
$932$	−15.8008	−0.517572
$933$	34.5669	1.13167
$934$	34.2488	1.12066
$935$	16.4026	0.536423
$936$	−22.6542	−0.740477
$937$	31.2993	1.02250	0.511252	−	0.859431i	$-0.329182\pi$
$937$	31.2993	1.02250	0.511252	+	0.859431i	$0.329182\pi$
$938$	13.0194	0.425099
$939$	−72.4611	−2.36468
$940$	15.7424	0.513462
$941$	−47.7817	−1.55764	−0.778819	−	0.627248i	$-0.784181\pi$
$941$	−47.7817	−1.55764	−0.778819	+	0.627248i	$0.784181\pi$
$942$	−23.8391	−0.776718
$943$	−1.34409	−0.0437697
$944$	8.81234	0.286817
$945$	−46.9343	−1.52677
$946$	−29.6471	−0.963911
$947$	38.7437	1.25900	0.629500	−	0.777000i	$-0.283260\pi$
$947$	38.7437	1.25900	0.629500	+	0.777000i	$0.283260\pi$
$948$	−44.9936	−1.46132
$949$	−59.8316	−1.94222
$950$	0.198139	0.00642846
$951$	32.0252	1.03849
$952$	−5.20664	−0.168748
$953$	10.6717	0.345689	0.172845	−	0.984949i	$-0.444704\pi$
$953$	10.6717	0.345689	0.172845	+	0.984949i	$0.444704\pi$
$954$	39.6690	1.28433
$955$	−10.2098	−0.330380
$956$	11.3809	0.368084
$957$	−52.1724	−1.68649
$958$	40.9632	1.32346
$959$	0.620468	0.0200360
$960$	6.45381	0.208296
$961$	−17.8451	−0.575649
$962$	30.9654	0.998364
$963$	30.9425	0.997109
$964$	−25.6313	−0.825530
$965$	15.7542	0.507146
$966$	74.6090	2.40051
$967$	−41.4809	−1.33394	−0.666968	−	0.745086i	$-0.732408\pi$
$967$	−41.4809	−1.33394	−0.666968	+	0.745086i	$0.732408\pi$
$968$	−6.87835	−0.221079
$969$	15.6271	0.502016
$970$	24.3377	0.781435
$971$	1.50345	0.0482481	0.0241240	−	0.999709i	$-0.492320\pi$
$971$	1.50345	0.0482481	0.0241240	+	0.999709i	$0.492320\pi$
$972$	8.86543	0.284359
$973$	−18.2853	−0.586198
$974$	−26.4807	−0.848495
$975$	0.778198	0.0249223
$976$	2.37193	0.0759237
$977$	−40.3994	−1.29249	−0.646246	−	0.763129i	$-0.723662\pi$
$977$	−40.3994	−1.29249	−0.646246	+	0.763129i	$0.723662\pi$
$978$	37.9248	1.21270
$979$	−71.1049	−2.27252
$980$	−4.20167	−0.134217
$981$	58.9331	1.88159
$982$	−40.8604	−1.30391
$983$	−8.66913	−0.276502	−0.138251	−	0.990397i	$-0.544148\pi$
$983$	−8.66913	−0.276502	−0.138251	+	0.990397i	$0.544148\pi$
$984$	0.453320	0.0144513
$985$	31.4189	1.00109
$986$	7.41669	0.236196
$987$	−61.3793	−1.95373
$988$	12.8322	0.408246
$989$	−60.3929	−1.92038
$990$	51.0914	1.62379
$991$	−34.6322	−1.10013	−0.550064	−	0.835123i	$-0.685397\pi$
$991$	−34.6322	−1.10013	−0.550064	+	0.835123i	$0.685397\pi$
$992$	3.62697	0.115156
$993$	−91.0853	−2.89050
$994$	−19.4757	−0.617732
$995$	53.4890	1.69571
$996$	−35.3940	−1.12150
$997$	−61.4414	−1.94587	−0.972935	−	0.231079i	$-0.925774\pi$
$997$	−61.4414	−1.94587	−0.972935	+	0.231079i	$0.925774\pi$
$998$	−0.124691	−0.00394701
$999$	52.6711	1.66644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.8	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.8	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.90498 q^{3} +1.00000 q^{4} -2.22164 q^{5} +2.90498 q^{6} -2.98182 q^{7} -1.00000 q^{8} +5.43889 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.90498 q^{3} +1.00000 q^{4} -2.22164 q^{5} +2.90498 q^{6} -2.98182 q^{7} -1.00000 q^{8} +5.43889 q^{9} +2.22164 q^{10} +4.22828 q^{11} -2.90498 q^{12} +4.16523 q^{13} +2.98182 q^{14} +6.45381 q^{15} +1.00000 q^{16} -1.74613 q^{17} -5.43889 q^{18} +3.08078 q^{19} -2.22164 q^{20} +8.66212 q^{21} -4.22828 q^{22} -8.61325 q^{23} +2.90498 q^{24} -0.0643144 q^{25} -4.16523 q^{26} -7.08493 q^{27} -2.98182 q^{28} +4.24751 q^{29} -6.45381 q^{30} -3.62697 q^{31} -1.00000 q^{32} -12.2831 q^{33} +1.74613 q^{34} +6.62453 q^{35} +5.43889 q^{36} -7.43425 q^{37} -3.08078 q^{38} -12.0999 q^{39} +2.22164 q^{40} +0.156049 q^{41} -8.66212 q^{42} +7.01163 q^{43} +4.22828 q^{44} -12.0833 q^{45} +8.61325 q^{46} -7.08595 q^{47} -2.90498 q^{48} +1.89125 q^{49} +0.0643144 q^{50} +5.07246 q^{51} +4.16523 q^{52} -7.29358 q^{53} +7.08493 q^{54} -9.39372 q^{55} +2.98182 q^{56} -8.94960 q^{57} -4.24751 q^{58} +8.81234 q^{59} +6.45381 q^{60} +2.37193 q^{61} +3.62697 q^{62} -16.2178 q^{63} +1.00000 q^{64} -9.25365 q^{65} +12.2831 q^{66} +4.36626 q^{67} -1.74613 q^{68} +25.0213 q^{69} -6.62453 q^{70} -6.53149 q^{71} -5.43889 q^{72} -14.3645 q^{73} +7.43425 q^{74} +0.186832 q^{75} +3.08078 q^{76} -12.6080 q^{77} +12.0999 q^{78} +15.4884 q^{79} -2.22164 q^{80} +4.26487 q^{81} -0.156049 q^{82} +12.1839 q^{83} +8.66212 q^{84} +3.87927 q^{85} -7.01163 q^{86} -12.3389 q^{87} -4.22828 q^{88} -16.8165 q^{89} +12.0833 q^{90} -12.4200 q^{91} -8.61325 q^{92} +10.5363 q^{93} +7.08595 q^{94} -6.84439 q^{95} +2.90498 q^{96} +10.9548 q^{97} -1.89125 q^{98} +22.9972 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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