LMFDB - Embedded newform 4019.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, 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("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.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:	$32.0918765724$
Analytic rank:	$0$
Dimension:	$186$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69011 q^{2} +0.353602 q^{3} +5.23670 q^{4} -0.999886 q^{5} -0.951229 q^{6} +3.25121 q^{7} -8.70707 q^{8} -2.87497 q^{9} +O(q^{10})$	\(q-2.69011 q^{2} +0.353602 q^{3} +5.23670 q^{4} -0.999886 q^{5} -0.951229 q^{6} +3.25121 q^{7} -8.70707 q^{8} -2.87497 q^{9} +2.68981 q^{10} -3.92179 q^{11} +1.85171 q^{12} +2.55128 q^{13} -8.74612 q^{14} -0.353562 q^{15} +12.9496 q^{16} -3.24551 q^{17} +7.73398 q^{18} -6.18143 q^{19} -5.23610 q^{20} +1.14964 q^{21} +10.5500 q^{22} -6.21887 q^{23} -3.07884 q^{24} -4.00023 q^{25} -6.86322 q^{26} -2.07740 q^{27} +17.0256 q^{28} +2.78073 q^{29} +0.951121 q^{30} -2.88281 q^{31} -17.4217 q^{32} -1.38675 q^{33} +8.73079 q^{34} -3.25084 q^{35} -15.0553 q^{36} +7.41193 q^{37} +16.6287 q^{38} +0.902137 q^{39} +8.70609 q^{40} -2.56356 q^{41} -3.09265 q^{42} -7.30959 q^{43} -20.5372 q^{44} +2.87464 q^{45} +16.7295 q^{46} -0.0357877 q^{47} +4.57901 q^{48} +3.57036 q^{49} +10.7611 q^{50} -1.14762 q^{51} +13.3603 q^{52} +12.9978 q^{53} +5.58844 q^{54} +3.92134 q^{55} -28.3085 q^{56} -2.18577 q^{57} -7.48046 q^{58} -7.42610 q^{59} -1.85150 q^{60} +1.59266 q^{61} +7.75507 q^{62} -9.34712 q^{63} +20.9671 q^{64} -2.55099 q^{65} +3.73052 q^{66} -5.79715 q^{67} -16.9958 q^{68} -2.19901 q^{69} +8.74512 q^{70} -0.814408 q^{71} +25.0325 q^{72} +13.1201 q^{73} -19.9389 q^{74} -1.41449 q^{75} -32.3703 q^{76} -12.7506 q^{77} -2.42685 q^{78} +9.45733 q^{79} -12.9481 q^{80} +7.89032 q^{81} +6.89627 q^{82} -1.90763 q^{83} +6.02029 q^{84} +3.24514 q^{85} +19.6636 q^{86} +0.983271 q^{87} +34.1473 q^{88} +18.3507 q^{89} -7.73310 q^{90} +8.29473 q^{91} -32.5663 q^{92} -1.01937 q^{93} +0.0962729 q^{94} +6.18073 q^{95} -6.16036 q^{96} +3.98815 q^{97} -9.60468 q^{98} +11.2750 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) $ 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * 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 $

$2$	−2.69011	−1.90220	−0.951098	−	0.308890i	$-0.900043\pi$
$2$	−2.69011	−1.90220	−0.951098	+	0.308890i	$0.900043\pi$
$3$	0.353602	0.204152	0.102076	−	0.994777i	$-0.467451\pi$
$3$	0.353602	0.204152	0.102076	+	0.994777i	$0.467451\pi$
$4$	5.23670	2.61835
$5$	−0.999886	−0.447163	−0.223581	−	0.974685i	$-0.571775\pi$
$5$	−0.999886	−0.447163	−0.223581	+	0.974685i	$0.571775\pi$
$6$	−0.951229	−0.388338
$7$	3.25121	1.22884	0.614421	−	0.788978i	$-0.289390\pi$
$7$	3.25121	1.22884	0.614421	+	0.788978i	$0.289390\pi$
$8$	−8.70707	−3.07842
$9$	−2.87497	−0.958322
$10$	2.68981	0.850591
$11$	−3.92179	−1.18246	−0.591232	−	0.806502i	$-0.701358\pi$
$11$	−3.92179	−1.18246	−0.591232	+	0.806502i	$0.701358\pi$
$12$	1.85171	0.534542
$13$	2.55128	0.707597	0.353798	−	0.935322i	$-0.384890\pi$
$13$	2.55128	0.707597	0.353798	+	0.935322i	$0.384890\pi$
$14$	−8.74612	−2.33750
$15$	−0.353562	−0.0912894
$16$	12.9496	3.23740
$17$	−3.24551	−0.787153	−0.393576	−	0.919292i	$-0.628762\pi$
$17$	−3.24551	−0.787153	−0.393576	+	0.919292i	$0.628762\pi$
$18$	7.73398	1.82292
$19$	−6.18143	−1.41812	−0.709059	−	0.705149i	$-0.750880\pi$
$19$	−6.18143	−1.41812	−0.709059	+	0.705149i	$0.750880\pi$
$20$	−5.23610	−1.17083
$21$	1.14964	0.250871
$22$	10.5500	2.24928
$23$	−6.21887	−1.29672	−0.648362	−	0.761332i	$-0.724546\pi$
$23$	−6.21887	−1.29672	−0.648362	+	0.761332i	$0.724546\pi$
$24$	−3.07884	−0.628466
$25$	−4.00023	−0.800045
$26$	−6.86322	−1.34599
$27$	−2.07740	−0.399796
$28$	17.0256	3.21754
$29$	2.78073	0.516368	0.258184	−	0.966096i	$-0.416876\pi$
$29$	2.78073	0.516368	0.258184	+	0.966096i	$0.416876\pi$
$30$	0.951121	0.173650
$31$	−2.88281	−0.517767	−0.258884	−	0.965909i	$-0.583355\pi$
$31$	−2.88281	−0.517767	−0.258884	+	0.965909i	$0.583355\pi$
$32$	−17.4217	−3.07975
$33$	−1.38675	−0.241403
$34$	8.73079	1.49732
$35$	−3.25084	−0.549492
$36$	−15.0553	−2.50922
$37$	7.41193	1.21851	0.609257	−	0.792973i	$-0.291468\pi$
$37$	7.41193	1.21851	0.609257	+	0.792973i	$0.291468\pi$
$38$	16.6287	2.69754
$39$	0.902137	0.144458
$40$	8.70609	1.37655
$41$	−2.56356	−0.400361	−0.200181	−	0.979759i	$-0.564153\pi$
$41$	−2.56356	−0.400361	−0.200181	+	0.979759i	$0.564153\pi$
$42$	−3.09265	−0.477206
$43$	−7.30959	−1.11470	−0.557351	−	0.830277i	$-0.688182\pi$
$43$	−7.30959	−1.11470	−0.557351	+	0.830277i	$0.688182\pi$
$44$	−20.5372	−3.09610
$45$	2.87464	0.428526
$46$	16.7295	2.46662
$47$	−0.0357877	−0.00522017	−0.00261009	−	0.999997i	$-0.500831\pi$
$47$	−0.0357877	−0.00522017	−0.00261009	+	0.999997i	$0.500831\pi$
$48$	4.57901	0.660923
$49$	3.57036	0.510052
$50$	10.7611	1.52184
$51$	−1.14762	−0.160699
$52$	13.3603	1.85273
$53$	12.9978	1.78539	0.892695	−	0.450662i	$-0.148812\pi$
$53$	12.9978	1.78539	0.892695	+	0.450662i	$0.148812\pi$
$54$	5.58844	0.760490
$55$	3.92134	0.528754
$56$	−28.3085	−3.78289
$57$	−2.18577	−0.289512
$58$	−7.48046	−0.982233
$59$	−7.42610	−0.966796	−0.483398	−	0.875401i	$-0.660598\pi$
$59$	−7.42610	−0.966796	−0.483398	+	0.875401i	$0.660598\pi$
$60$	−1.85150	−0.239027
$61$	1.59266	0.203920	0.101960	−	0.994789i	$-0.467489\pi$
$61$	1.59266	0.203920	0.101960	+	0.994789i	$0.467489\pi$
$62$	7.75507	0.984895
$63$	−9.34712	−1.17763
$64$	20.9671	2.62089
$65$	−2.55099	−0.316411
$66$	3.73052	0.459195
$67$	−5.79715	−0.708234	−0.354117	−	0.935201i	$-0.615219\pi$
$67$	−5.79715	−0.708234	−0.354117	+	0.935201i	$0.615219\pi$
$68$	−16.9958	−2.06104
$69$	−2.19901	−0.264729
$70$	8.74512	1.04524
$71$	−0.814408	−0.0966524	−0.0483262	−	0.998832i	$-0.515389\pi$
$71$	−0.814408	−0.0966524	−0.0483262	+	0.998832i	$0.515389\pi$
$72$	25.0325	2.95011
$73$	13.1201	1.53559	0.767794	−	0.640697i	$-0.221355\pi$
$73$	13.1201	1.53559	0.767794	+	0.640697i	$0.221355\pi$
$74$	−19.9389	−2.31785
$75$	−1.41449	−0.163331
$76$	−32.3703	−3.71313
$77$	−12.7506	−1.45306
$78$	−2.42685	−0.274787
$79$	9.45733	1.06403	0.532016	−	0.846734i	$-0.321434\pi$
$79$	9.45733	1.06403	0.532016	+	0.846734i	$0.321434\pi$
$80$	−12.9481	−1.44765
$81$	7.89032	0.876702
$82$	6.89627	0.761565
$83$	−1.90763	−0.209390	−0.104695	−	0.994504i	$-0.533387\pi$
$83$	−1.90763	−0.209390	−0.104695	+	0.994504i	$0.533387\pi$
$84$	6.02029	0.656868
$85$	3.24514	0.351985
$86$	19.6636	2.12038
$87$	0.983271	0.105418
$88$	34.1473	3.64012
$89$	18.3507	1.94517	0.972587	−	0.232541i	$-0.0747039\pi$
$89$	18.3507	1.94517	0.972587	+	0.232541i	$0.0747039\pi$
$90$	−7.73310	−0.815140
$91$	8.29473	0.869524
$92$	−32.5663	−3.39528
$93$	−1.01937	−0.105703
$94$	0.0962729	0.00992979
$95$	6.18073	0.634129
$96$	−6.16036	−0.628739
$97$	3.98815	0.404935	0.202468	−	0.979289i	$-0.435104\pi$
$97$	3.98815	0.404935	0.202468	+	0.979289i	$0.435104\pi$
$98$	−9.60468	−0.970219
$99$	11.2750	1.13318
$100$	−20.9480	−2.09480
$101$	9.07095	0.902594	0.451297	−	0.892374i	$-0.350962\pi$
$101$	9.07095	0.902594	0.451297	+	0.892374i	$0.350962\pi$
$102$	3.08723	0.305681
$103$	9.80257	0.965876	0.482938	−	0.875655i	$-0.339570\pi$
$103$	9.80257	0.965876	0.482938	+	0.875655i	$0.339570\pi$
$104$	−22.2142	−2.17828
$105$	−1.14950	−0.112180
$106$	−34.9656	−3.39616
$107$	14.2465	1.37726	0.688628	−	0.725115i	$-0.258213\pi$
$107$	14.2465	1.37726	0.688628	+	0.725115i	$0.258213\pi$
$108$	−10.8787	−1.04681
$109$	10.3058	0.987112	0.493556	−	0.869714i	$-0.335697\pi$
$109$	10.3058	0.987112	0.493556	+	0.869714i	$0.335697\pi$
$110$	−10.5488	−1.00579
$111$	2.62088	0.248762
$112$	42.1019	3.97825
$113$	12.1223	1.14037	0.570183	−	0.821518i	$-0.306872\pi$
$113$	12.1223	1.14037	0.570183	+	0.821518i	$0.306872\pi$
$114$	5.87996	0.550709
$115$	6.21816	0.579847
$116$	14.5618	1.35203
$117$	−7.33483	−0.678105
$118$	19.9770	1.83903
$119$	−10.5518	−0.967286
$120$	3.07849	0.281027
$121$	4.38043	0.398221
$122$	−4.28444	−0.387895
$123$	−0.906482	−0.0817347
$124$	−15.0964	−1.35570
$125$	8.99921	0.804913
$126$	25.1448	2.24007
$127$	−8.05780	−0.715014	−0.357507	−	0.933910i	$-0.616373\pi$
$127$	−8.05780	−0.715014	−0.357507	+	0.933910i	$0.616373\pi$
$128$	−21.5605	−1.90570
$129$	−2.58469	−0.227569
$130$	6.86244	0.601876
$131$	−8.09546	−0.707304	−0.353652	−	0.935377i	$-0.615060\pi$
$131$	−8.09546	−0.707304	−0.353652	+	0.935377i	$0.615060\pi$
$132$	−7.26201	−0.632077
$133$	−20.0971	−1.74264
$134$	15.5950	1.34720
$135$	2.07717	0.178774
$136$	28.2589	2.42318
$137$	22.1702	1.89413	0.947066	−	0.321039i	$-0.104032\pi$
$137$	22.1702	1.89413	0.947066	+	0.321039i	$0.104032\pi$
$138$	5.91557	0.503567
$139$	5.71832	0.485022	0.242511	−	0.970149i	$-0.422029\pi$
$139$	5.71832	0.485022	0.242511	+	0.970149i	$0.422029\pi$
$140$	−17.0237	−1.43876
$141$	−0.0126546	−0.00106571
$142$	2.19085	0.183852
$143$	−10.0056	−0.836708
$144$	−37.2297	−3.10247
$145$	−2.78041	−0.230901
$146$	−35.2944	−2.92099
$147$	1.26249	0.104128
$148$	38.8140	3.19049
$149$	−11.6974	−0.958284	−0.479142	−	0.877737i	$-0.659052\pi$
$149$	−11.6974	−0.958284	−0.479142	+	0.877737i	$0.659052\pi$
$150$	3.80513	0.310688
$151$	5.06610	0.412273	0.206137	−	0.978523i	$-0.433911\pi$
$151$	5.06610	0.412273	0.206137	+	0.978523i	$0.433911\pi$
$152$	53.8222	4.36556
$153$	9.33074	0.754345
$154$	34.3004	2.76401
$155$	2.88248	0.231526
$156$	4.72422	0.378240
$157$	4.41311	0.352205	0.176102	−	0.984372i	$-0.443651\pi$
$157$	4.41311	0.352205	0.176102	+	0.984372i	$0.443651\pi$
$158$	−25.4413	−2.02400
$159$	4.59606	0.364491
$160$	17.4197	1.37715
$161$	−20.2189	−1.59347
$162$	−21.2258	−1.66766
$163$	−6.15144	−0.481818	−0.240909	−	0.970548i	$-0.577446\pi$
$163$	−6.15144	−0.481818	−0.240909	+	0.970548i	$0.577446\pi$
$164$	−13.4246	−1.04829
$165$	1.38660	0.107946
$166$	5.13174	0.398300
$167$	−2.18316	−0.168938	−0.0844690	−	0.996426i	$-0.526919\pi$
$167$	−2.18316	−0.168938	−0.0844690	+	0.996426i	$0.526919\pi$
$168$	−10.0100	−0.772285
$169$	−6.49099	−0.499307
$170$	−8.72980	−0.669545
$171$	17.7714	1.35901
$172$	−38.2781	−2.91868
$173$	−7.39065	−0.561901	−0.280950	−	0.959722i	$-0.590650\pi$
$173$	−7.39065	−0.561901	−0.280950	+	0.959722i	$0.590650\pi$
$174$	−2.64511	−0.200525
$175$	−13.0056	−0.983129
$176$	−50.7856	−3.82811
$177$	−2.62589	−0.197374
$178$	−49.3655	−3.70010
$179$	−4.52767	−0.338414	−0.169207	−	0.985581i	$-0.554121\pi$
$179$	−4.52767	−0.338414	−0.169207	+	0.985581i	$0.554121\pi$
$180$	15.0536	1.12203
$181$	2.35496	0.175043	0.0875216	−	0.996163i	$-0.472105\pi$
$181$	2.35496	0.175043	0.0875216	+	0.996163i	$0.472105\pi$
$182$	−22.3138	−1.65401
$183$	0.563169	0.0416307
$184$	54.1482	3.99186
$185$	−7.41109	−0.544874
$186$	2.74221	0.201069
$187$	12.7282	0.930779
$188$	−0.187409	−0.0136682
$189$	−6.75407	−0.491286
$190$	−16.6268	−1.20624
$191$	−15.2688	−1.10481	−0.552406	−	0.833575i	$-0.686290\pi$
$191$	−15.2688	−1.10481	−0.552406	+	0.833575i	$0.686290\pi$
$192$	7.41403	0.535062
$193$	6.24930	0.449835	0.224917	−	0.974378i	$-0.427789\pi$
$193$	6.24930	0.449835	0.224917	+	0.974378i	$0.427789\pi$
$194$	−10.7286	−0.770266
$195$	−0.902035	−0.0645961
$196$	18.6969	1.33549
$197$	−7.42210	−0.528803	−0.264401	−	0.964413i	$-0.585174\pi$
$197$	−7.42210	−0.528803	−0.264401	+	0.964413i	$0.585174\pi$
$198$	−30.3310	−2.15553
$199$	−8.90657	−0.631370	−0.315685	−	0.948864i	$-0.602234\pi$
$199$	−8.90657	−0.631370	−0.315685	+	0.948864i	$0.602234\pi$
$200$	34.8303	2.46287
$201$	−2.04988	−0.144588
$202$	−24.4019	−1.71691
$203$	9.04073	0.634535
$204$	−6.00974	−0.420766
$205$	2.56327	0.179027
$206$	−26.3700	−1.83728
$207$	17.8790	1.24268
$208$	33.0380	2.29077
$209$	24.2423	1.67687
$210$	3.09230	0.213389
$211$	−0.682757	−0.0470029	−0.0235015	−	0.999724i	$-0.507481\pi$
$211$	−0.682757	−0.0470029	−0.0235015	+	0.999724i	$0.507481\pi$
$212$	68.0657	4.67477
$213$	−0.287976	−0.0197318
$214$	−38.3245	−2.61981
$215$	7.30876	0.498453
$216$	18.0881	1.23074
$217$	−9.37261	−0.636254
$218$	−27.7236	−1.87768
$219$	4.63928	0.313494
$220$	20.5349	1.38446
$221$	−8.28020	−0.556987
$222$	−7.05044	−0.473195
$223$	−19.2400	−1.28841	−0.644204	−	0.764854i	$-0.722811\pi$
$223$	−19.2400	−1.28841	−0.644204	+	0.764854i	$0.722811\pi$
$224$	−56.6417	−3.78453
$225$	11.5005	0.766701
$226$	−32.6102	−2.16920
$227$	22.9038	1.52018	0.760088	−	0.649820i	$-0.225156\pi$
$227$	22.9038	1.52018	0.760088	+	0.649820i	$0.225156\pi$
$228$	−11.4462	−0.758044
$229$	1.05414	0.0696598	0.0348299	−	0.999393i	$-0.488911\pi$
$229$	1.05414	0.0696598	0.0348299	+	0.999393i	$0.488911\pi$
$230$	−16.7276	−1.10298
$231$	−4.50863	−0.296646
$232$	−24.2120	−1.58960
$233$	−15.9142	−1.04258	−0.521288	−	0.853381i	$-0.674548\pi$
$233$	−15.9142	−1.04258	−0.521288	+	0.853381i	$0.674548\pi$
$234$	19.7315	1.28989
$235$	0.0357836	0.00233427
$236$	−38.8882	−2.53141
$237$	3.34413	0.217225
$238$	28.3856	1.83997
$239$	4.44642	0.287615	0.143808	−	0.989606i	$-0.454065\pi$
$239$	4.44642	0.287615	0.143808	+	0.989606i	$0.454065\pi$
$240$	−4.57849	−0.295540
$241$	4.66671	0.300609	0.150304	−	0.988640i	$-0.451975\pi$
$241$	4.66671	0.300609	0.150304	+	0.988640i	$0.451975\pi$
$242$	−11.7838	−0.757494
$243$	9.02224	0.578777
$244$	8.34029	0.533933
$245$	−3.56996	−0.228076
$246$	2.43854	0.155475
$247$	−15.7705	−1.00346
$248$	25.1008	1.59390
$249$	−0.674542	−0.0427474
$250$	−24.2089	−1.53110
$251$	18.6306	1.17595	0.587976	−	0.808879i	$-0.299925\pi$
$251$	18.6306	1.17595	0.587976	+	0.808879i	$0.299925\pi$
$252$	−48.9480	−3.08343
$253$	24.3891	1.53333
$254$	21.6764	1.36010
$255$	1.14749	0.0718587
$256$	16.0659	1.00412
$257$	5.18730	0.323575	0.161787	−	0.986826i	$-0.448274\pi$
$257$	5.18730	0.323575	0.161787	+	0.986826i	$0.448274\pi$
$258$	6.95309	0.432881
$259$	24.0977	1.49736
$260$	−13.3587	−0.828474
$261$	−7.99449	−0.494847
$262$	21.7777	1.34543
$263$	0.929089	0.0572901	0.0286450	−	0.999590i	$-0.490881\pi$
$263$	0.929089	0.0572901	0.0286450	+	0.999590i	$0.490881\pi$
$264$	12.0746	0.743138
$265$	−12.9964	−0.798360
$266$	54.0635	3.31485
$267$	6.48886	0.397112
$268$	−30.3579	−1.85440
$269$	7.96326	0.485528	0.242764	−	0.970085i	$-0.421946\pi$
$269$	7.96326	0.485528	0.242764	+	0.970085i	$0.421946\pi$
$270$	−5.58781	−0.340063
$271$	1.94501	0.118151	0.0590754	−	0.998254i	$-0.481185\pi$
$271$	1.94501	0.118151	0.0590754	+	0.998254i	$0.481185\pi$
$272$	−42.0281	−2.54833
$273$	2.93304	0.177515
$274$	−59.6404	−3.60301
$275$	15.6880	0.946025
$276$	−11.5155	−0.693154
$277$	12.0364	0.723195	0.361597	−	0.932334i	$-0.382232\pi$
$277$	12.0364	0.723195	0.361597	+	0.932334i	$0.382232\pi$
$278$	−15.3829	−0.922606
$279$	8.28797	0.496188
$280$	28.3053	1.69157
$281$	−10.4581	−0.623878	−0.311939	−	0.950102i	$-0.600979\pi$
$281$	−10.4581	−0.623878	−0.311939	+	0.950102i	$0.600979\pi$
$282$	0.0340423	0.00202719
$283$	−1.06203	−0.0631310	−0.0315655	−	0.999502i	$-0.510049\pi$
$283$	−1.06203	−0.0631310	−0.0315655	+	0.999502i	$0.510049\pi$
$284$	−4.26481	−0.253070
$285$	2.18552	0.129459
$286$	26.9161	1.59158
$287$	−8.33468	−0.491981
$288$	50.0868	2.95139
$289$	−6.46664	−0.380391
$290$	7.47961	0.439218
$291$	1.41022	0.0826685
$292$	68.7058	4.02070
$293$	8.35925	0.488352	0.244176	−	0.969731i	$-0.421482\pi$
$293$	8.35925	0.488352	0.244176	+	0.969731i	$0.421482\pi$
$294$	−3.39624	−0.198073
$295$	7.42525	0.432315
$296$	−64.5362	−3.75109
$297$	8.14713	0.472744
$298$	31.4672	1.82284
$299$	−15.8661	−0.917558
$300$	−7.40725	−0.427658
$301$	−23.7650	−1.36979
$302$	−13.6284	−0.784224
$303$	3.20751	0.184267
$304$	−80.0471	−4.59101
$305$	−1.59248	−0.0911853
$306$	−25.1007	−1.43491
$307$	−22.3856	−1.27762	−0.638808	−	0.769366i	$-0.720572\pi$
$307$	−22.3856	−1.27762	−0.638808	+	0.769366i	$0.720572\pi$
$308$	−66.7708	−3.80462
$309$	3.46621	0.197186
$310$	−7.75419	−0.440408
$311$	−15.4306	−0.874989	−0.437494	−	0.899221i	$-0.644134\pi$
$311$	−15.4306	−0.874989	−0.437494	+	0.899221i	$0.644134\pi$
$312$	−7.85498	−0.444700
$313$	−12.7769	−0.722195	−0.361097	−	0.932528i	$-0.617598\pi$
$313$	−12.7769	−0.722195	−0.361097	+	0.932528i	$0.617598\pi$
$314$	−11.8718	−0.669962
$315$	9.34605	0.526591
$316$	49.5252	2.78601
$317$	−6.96753	−0.391336	−0.195668	−	0.980670i	$-0.562687\pi$
$317$	−6.96753	−0.391336	−0.195668	+	0.980670i	$0.562687\pi$
$318$	−12.3639	−0.693334
$319$	−10.9054	−0.610586
$320$	−20.9648	−1.17197
$321$	5.03758	0.281170
$322$	54.3910	3.03109
$323$	20.0619	1.11627
$324$	41.3192	2.29551
$325$	−10.2057	−0.566110
$326$	16.5481	0.916513
$327$	3.64414	0.201521
$328$	22.3211	1.23248
$329$	−0.116353	−0.00641477
$330$	−3.73010	−0.205335
$331$	35.3489	1.94295	0.971476	−	0.237139i	$-0.0762098\pi$
$331$	35.3489	1.94295	0.971476	+	0.237139i	$0.0762098\pi$
$332$	−9.98968	−0.548255
$333$	−21.3090	−1.16773
$334$	5.87294	0.321353
$335$	5.79649	0.316696
$336$	14.8873	0.812170
$337$	−3.83616	−0.208969	−0.104484	−	0.994527i	$-0.533319\pi$
$337$	−3.83616	−0.208969	−0.104484	+	0.994527i	$0.533319\pi$
$338$	17.4615	0.949779
$339$	4.28646	0.232808
$340$	16.9938	0.921620
$341$	11.3058	0.612241
$342$	−47.8070	−2.58511
$343$	−11.1505	−0.602068
$344$	63.6451	3.43151
$345$	2.19876	0.118377
$346$	19.8817	1.06884
$347$	−21.8511	−1.17303	−0.586515	−	0.809938i	$-0.699500\pi$
$347$	−21.8511	−1.17303	−0.586515	+	0.809938i	$0.699500\pi$
$348$	5.14909	0.276020
$349$	6.87369	0.367940	0.183970	−	0.982932i	$-0.441105\pi$
$349$	6.87369	0.367940	0.183970	+	0.982932i	$0.441105\pi$
$350$	34.9864	1.87010
$351$	−5.30002	−0.282894
$352$	68.3243	3.64170
$353$	−22.6485	−1.20546	−0.602728	−	0.797946i	$-0.705920\pi$
$353$	−22.6485	−1.20546	−0.602728	+	0.797946i	$0.705920\pi$
$354$	7.06392	0.375443
$355$	0.814315	0.0432194
$356$	96.0972	5.09314
$357$	−3.73116	−0.197474
$358$	12.1799	0.643729
$359$	−12.9264	−0.682228	−0.341114	−	0.940022i	$-0.610804\pi$
$359$	−12.9264	−0.682228	−0.341114	+	0.940022i	$0.610804\pi$
$360$	−25.0297	−1.31918
$361$	19.2101	1.01106
$362$	−6.33512	−0.332966
$363$	1.54893	0.0812977
$364$	43.4370	2.27672
$365$	−13.1186	−0.686658
$366$	−1.51499	−0.0791897
$367$	−2.24222	−0.117043	−0.0585215	−	0.998286i	$-0.518639\pi$
$367$	−2.24222	−0.117043	−0.0585215	+	0.998286i	$0.518639\pi$
$368$	−80.5319	−4.19802
$369$	7.37015	0.383675
$370$	19.9366	1.03646
$371$	42.2587	2.19396
$372$	−5.33812	−0.276768
$373$	−35.5116	−1.83872	−0.919361	−	0.393414i	$-0.871294\pi$
$373$	−35.5116	−1.83872	−0.919361	+	0.393414i	$0.871294\pi$
$374$	−34.2403	−1.77052
$375$	3.18214	0.164325
$376$	0.311606	0.0160699
$377$	7.09440	0.365380
$378$	18.1692	0.934522
$379$	−11.5009	−0.590762	−0.295381	−	0.955380i	$-0.595447\pi$
$379$	−11.5009	−0.590762	−0.295381	+	0.955380i	$0.595447\pi$
$380$	32.3666	1.66037
$381$	−2.84926	−0.145972
$382$	41.0748	2.10157
$383$	19.5259	0.997729	0.498864	−	0.866680i	$-0.333751\pi$
$383$	19.5259	0.997729	0.498864	+	0.866680i	$0.333751\pi$
$384$	−7.62385	−0.389053
$385$	12.7491	0.649755
$386$	−16.8113	−0.855674
$387$	21.0148	1.06824
$388$	20.8847	1.06026
$389$	13.3406	0.676395	0.338198	−	0.941075i	$-0.390183\pi$
$389$	13.3406	0.676395	0.338198	+	0.941075i	$0.390183\pi$
$390$	2.42657	0.122874
$391$	20.1834	1.02072
$392$	−31.0874	−1.57015
$393$	−2.86257	−0.144398
$394$	19.9663	1.00589
$395$	−9.45626	−0.475796
$396$	59.0438	2.96706
$397$	−36.1041	−1.81202	−0.906008	−	0.423262i	$-0.860885\pi$
$397$	−36.1041	−1.81202	−0.906008	+	0.423262i	$0.860885\pi$
$398$	23.9597	1.20099
$399$	−7.10639	−0.355765
$400$	−51.8013	−2.59007
$401$	20.3990	1.01868	0.509340	−	0.860566i	$-0.329890\pi$
$401$	20.3990	1.01868	0.509340	+	0.860566i	$0.329890\pi$
$402$	5.51442	0.275034
$403$	−7.35484	−0.366370
$404$	47.5018	2.36330
$405$	−7.88943	−0.392029
$406$	−24.3206	−1.20701
$407$	−29.0680	−1.44085
$408$	9.99242	0.494699
$409$	25.8625	1.27882	0.639409	−	0.768867i	$-0.279179\pi$
$409$	25.8625	1.27882	0.639409	+	0.768867i	$0.279179\pi$
$410$	−6.89548	−0.340544
$411$	7.83945	0.386692
$412$	51.3331	2.52900
$413$	−24.1438	−1.18804
$414$	−48.0966	−2.36382
$415$	1.90741	0.0936313
$416$	−44.4476	−2.17922
$417$	2.02201	0.0990184
$418$	−65.2144	−3.18974
$419$	−2.01758	−0.0985652	−0.0492826	−	0.998785i	$-0.515693\pi$
$419$	−2.01758	−0.0985652	−0.0492826	+	0.998785i	$0.515693\pi$
$420$	−6.01961	−0.293727
$421$	20.4455	0.996454	0.498227	−	0.867047i	$-0.333985\pi$
$421$	20.4455	0.996454	0.498227	+	0.867047i	$0.333985\pi$
$422$	1.83669	0.0894088
$423$	0.102888	0.00500260
$424$	−113.173	−5.49617
$425$	12.9828	0.629758
$426$	0.774689	0.0375338
$427$	5.17808	0.250585
$428$	74.6043	3.60614
$429$	−3.53799	−0.170816
$430$	−19.6614	−0.948155
$431$	26.3069	1.26716	0.633580	−	0.773677i	$-0.281585\pi$
$431$	26.3069	1.26716	0.633580	+	0.773677i	$0.281585\pi$
$432$	−26.9015	−1.29430
$433$	−4.77867	−0.229648	−0.114824	−	0.993386i	$-0.536630\pi$
$433$	−4.77867	−0.229648	−0.114824	+	0.993386i	$0.536630\pi$
$434$	25.2134	1.21028
$435$	−0.983160	−0.0471389
$436$	53.9681	2.58460
$437$	38.4415	1.83891
$438$	−12.4802	−0.596327
$439$	36.1670	1.72616	0.863078	−	0.505071i	$-0.168534\pi$
$439$	36.1670	1.72616	0.863078	+	0.505071i	$0.168534\pi$
$440$	−34.1434	−1.62772
$441$	−10.2647	−0.488794
$442$	22.2747	1.05950
$443$	−11.3463	−0.539080	−0.269540	−	0.962989i	$-0.586872\pi$
$443$	−11.3463	−0.539080	−0.269540	+	0.962989i	$0.586872\pi$
$444$	13.7247	0.651347
$445$	−18.3486	−0.869809
$446$	51.7578	2.45080
$447$	−4.13621	−0.195636
$448$	68.1686	3.22066
$449$	40.6597	1.91885	0.959425	−	0.281965i	$-0.0909862\pi$
$449$	40.6597	1.91885	0.959425	+	0.281965i	$0.0909862\pi$
$450$	−30.9377	−1.45842
$451$	10.0538	0.473413
$452$	63.4806	2.98587
$453$	1.79138	0.0841665
$454$	−61.6137	−2.89167
$455$	−8.29379	−0.388819
$456$	19.0316	0.891239
$457$	36.6821	1.71592	0.857958	−	0.513720i	$-0.171733\pi$
$457$	36.6821	1.71592	0.857958	+	0.513720i	$0.171733\pi$
$458$	−2.83576	−0.132507
$459$	6.74223	0.314701
$460$	32.5626	1.51824
$461$	14.3424	0.667990	0.333995	−	0.942575i	$-0.391603\pi$
$461$	14.3424	0.667990	0.333995	+	0.942575i	$0.391603\pi$
$462$	12.1287	0.564278
$463$	20.5777	0.956326	0.478163	−	0.878271i	$-0.341303\pi$
$463$	20.5777	0.956326	0.478163	+	0.878271i	$0.341303\pi$
$464$	36.0093	1.67169
$465$	1.01925	0.0472666
$466$	42.8110	1.98318
$467$	10.8678	0.502902	0.251451	−	0.967870i	$-0.419092\pi$
$467$	10.8678	0.502902	0.251451	+	0.967870i	$0.419092\pi$
$468$	−38.4103	−1.77552
$469$	−18.8477	−0.870308
$470$	−0.0962619	−0.00444023
$471$	1.56049	0.0719035
$472$	64.6596	2.97620
$473$	28.6667	1.31809
$474$	−8.99609	−0.413204
$475$	24.7271	1.13456
$476$	−55.2568	−2.53269
$477$	−37.3683	−1.71098
$478$	−11.9614	−0.547100
$479$	14.5044	0.662723	0.331362	−	0.943504i	$-0.392492\pi$
$479$	14.5044	0.662723	0.331362	+	0.943504i	$0.392492\pi$
$480$	6.15966	0.281149
$481$	18.9099	0.862216
$482$	−12.5540	−0.571817
$483$	−7.14943	−0.325310
$484$	22.9390	1.04268
$485$	−3.98770	−0.181072
$486$	−24.2708	−1.10095
$487$	−22.3017	−1.01059	−0.505294	−	0.862947i	$-0.668616\pi$
$487$	−22.3017	−1.01059	−0.505294	+	0.862947i	$0.668616\pi$
$488$	−13.8674	−0.627749
$489$	−2.17516	−0.0983644
$490$	9.60359	0.433846
$491$	7.30407	0.329628	0.164814	−	0.986325i	$-0.447298\pi$
$491$	7.30407	0.329628	0.164814	+	0.986325i	$0.447298\pi$
$492$	−4.74697	−0.214010
$493$	−9.02489	−0.406460
$494$	42.4245	1.90877
$495$	−11.2737	−0.506716
$496$	−37.3312	−1.67622
$497$	−2.64781	−0.118771
$498$	1.81459	0.0813139
$499$	28.9663	1.29671	0.648355	−	0.761338i	$-0.275457\pi$
$499$	28.9663	1.29671	0.648355	+	0.761338i	$0.275457\pi$
$500$	47.1261	2.10754
$501$	−0.771971	−0.0344891
$502$	−50.1183	−2.23689
$503$	17.6833	0.788459	0.394230	−	0.919012i	$-0.371011\pi$
$503$	17.6833	0.788459	0.394230	+	0.919012i	$0.371011\pi$
$504$	81.3860	3.62522
$505$	−9.06992	−0.403606
$506$	−65.6094	−2.91669
$507$	−2.29523	−0.101935
$508$	−42.1963	−1.87216
$509$	−22.5151	−0.997966	−0.498983	−	0.866612i	$-0.666293\pi$
$509$	−22.5151	−0.997966	−0.498983	+	0.866612i	$0.666293\pi$
$510$	−3.08688	−0.136689
$511$	42.6561	1.88699
$512$	−0.0980122	−0.00433157
$513$	12.8413	0.566958
$514$	−13.9544	−0.615502
$515$	−9.80145	−0.431904
$516$	−13.5352	−0.595855
$517$	0.140352	0.00617267
$518$	−64.8256	−2.84827
$519$	−2.61335	−0.114713
$520$	22.2116	0.974044
$521$	29.1819	1.27848	0.639240	−	0.769007i	$-0.279249\pi$
$521$	29.1819	1.27848	0.639240	+	0.769007i	$0.279249\pi$
$522$	21.5061	0.941295
$523$	−23.3445	−1.02078	−0.510392	−	0.859942i	$-0.670500\pi$
$523$	−23.3445	−1.02078	−0.510392	+	0.859942i	$0.670500\pi$
$524$	−42.3935	−1.85197
$525$	−4.59880	−0.200708
$526$	−2.49935	−0.108977
$527$	9.35619	0.407562
$528$	−17.9579	−0.781518
$529$	15.6744	0.681494
$530$	34.9616	1.51864
$531$	21.3498	0.926501
$532$	−105.243	−4.56284
$533$	−6.54036	−0.283294
$534$	−17.4558	−0.755384
$535$	−14.2448	−0.615858
$536$	50.4762	2.18024
$537$	−1.60099	−0.0690879
$538$	−21.4220	−0.923570
$539$	−14.0022	−0.603118
$540$	10.8775	0.468092
$541$	−9.84022	−0.423064	−0.211532	−	0.977371i	$-0.567845\pi$
$541$	−9.84022	−0.423064	−0.211532	+	0.977371i	$0.567845\pi$
$542$	−5.23229	−0.224746
$543$	0.832721	0.0357355
$544$	56.5424	2.42424
$545$	−10.3046	−0.441400
$546$	−7.89020	−0.337669
$547$	12.5086	0.534828	0.267414	−	0.963582i	$-0.413831\pi$
$547$	12.5086	0.534828	0.267414	+	0.963582i	$0.413831\pi$
$548$	116.099	4.95950
$549$	−4.57885	−0.195421
$550$	−42.2026	−1.79952
$551$	−17.1889	−0.732271
$552$	19.1469	0.814947
$553$	30.7478	1.30753
$554$	−32.3791	−1.37566
$555$	−2.62058	−0.111237
$556$	29.9451	1.26996
$557$	−13.1604	−0.557625	−0.278813	−	0.960346i	$-0.589941\pi$
$557$	−13.1604	−0.557625	−0.278813	+	0.960346i	$0.589941\pi$
$558$	−22.2956	−0.943846
$559$	−18.6488	−0.788759
$560$	−42.0971	−1.77893
$561$	4.50073	0.190021
$562$	28.1335	1.18674
$563$	10.4178	0.439059	0.219529	−	0.975606i	$-0.429548\pi$
$563$	10.4178	0.439059	0.219529	+	0.975606i	$0.429548\pi$
$564$	−0.0662684	−0.00279040
$565$	−12.1209	−0.509929
$566$	2.85697	0.120087
$567$	25.6531	1.07733
$568$	7.09111	0.297536
$569$	33.3673	1.39883	0.699415	−	0.714716i	$-0.253444\pi$
$569$	33.3673	1.39883	0.699415	+	0.714716i	$0.253444\pi$
$570$	−5.87929	−0.246256
$571$	4.00863	0.167756	0.0838779	−	0.996476i	$-0.473269\pi$
$571$	4.00863	0.167756	0.0838779	+	0.996476i	$0.473269\pi$
$572$	−52.3961	−2.19079
$573$	−5.39909	−0.225550
$574$	22.4212	0.935843
$575$	24.8769	1.03744
$576$	−60.2798	−2.51166
$577$	−12.4877	−0.519868	−0.259934	−	0.965626i	$-0.583701\pi$
$577$	−12.4877	−0.519868	−0.259934	+	0.965626i	$0.583701\pi$
$578$	17.3960	0.723578
$579$	2.20977	0.0918348
$580$	−14.5602	−0.604578
$581$	−6.20211	−0.257307
$582$	−3.79364	−0.157252
$583$	−50.9747	−2.11116
$584$	−114.237	−4.72718
$585$	7.33400	0.303224
$586$	−22.4873	−0.928942
$587$	−36.3147	−1.49887	−0.749434	−	0.662079i	$-0.769674\pi$
$587$	−36.3147	−1.49887	−0.749434	+	0.662079i	$0.769674\pi$
$588$	6.61127	0.272644
$589$	17.8199	0.734255
$590$	−19.9748	−0.822348
$591$	−2.62447	−0.107956
$592$	95.9815	3.94482
$593$	−21.4713	−0.881719	−0.440859	−	0.897576i	$-0.645326\pi$
$593$	−21.4713	−0.881719	−0.440859	+	0.897576i	$0.645326\pi$
$594$	−21.9167	−0.899252
$595$	10.5506	0.432534
$596$	−61.2555	−2.50912
$597$	−3.14938	−0.128896
$598$	42.6815	1.74537
$599$	41.4723	1.69451	0.847255	−	0.531186i	$-0.178253\pi$
$599$	41.4723	1.69451	0.847255	+	0.531186i	$0.178253\pi$
$600$	12.3161	0.502801
$601$	−21.7104	−0.885586	−0.442793	−	0.896624i	$-0.646012\pi$
$601$	−21.7104	−0.885586	−0.442793	+	0.896624i	$0.646012\pi$
$602$	63.9305	2.60561
$603$	16.6666	0.678716
$604$	26.5296	1.07947
$605$	−4.37993	−0.178070
$606$	−8.62856	−0.350511
$607$	−16.2310	−0.658795	−0.329397	−	0.944191i	$-0.606846\pi$
$607$	−16.2310	−0.658795	−0.329397	+	0.944191i	$0.606846\pi$
$608$	107.691	4.36745
$609$	3.19682	0.129542
$610$	4.28395	0.173452
$611$	−0.0913043	−0.00369378
$612$	48.8622	1.97514
$613$	−9.52547	−0.384730	−0.192365	−	0.981323i	$-0.561616\pi$
$613$	−9.52547	−0.384730	−0.192365	+	0.981323i	$0.561616\pi$
$614$	60.2199	2.43028
$615$	0.906379	0.0365487
$616$	111.020	4.47313
$617$	−22.0116	−0.886154	−0.443077	−	0.896484i	$-0.646113\pi$
$617$	−22.0116	−0.886154	−0.443077	+	0.896484i	$0.646113\pi$
$618$	−9.32449	−0.375086
$619$	2.39082	0.0960952	0.0480476	−	0.998845i	$-0.484700\pi$
$619$	2.39082	0.0960952	0.0480476	+	0.998845i	$0.484700\pi$
$620$	15.0947	0.606217
$621$	12.9191	0.518425
$622$	41.5100	1.66440
$623$	59.6621	2.39031
$624$	11.6823	0.467667
$625$	11.0030	0.440118
$626$	34.3714	1.37376
$627$	8.57212	0.342338
$628$	23.1101	0.922195
$629$	−24.0555	−0.959156
$630$	−25.1419	−1.00168
$631$	−7.37154	−0.293456	−0.146728	−	0.989177i	$-0.546874\pi$
$631$	−7.37154	−0.293456	−0.146728	+	0.989177i	$0.546874\pi$
$632$	−82.3457	−3.27554
$633$	−0.241425	−0.00959576
$634$	18.7434	0.744397
$635$	8.05689	0.319728
$636$	24.0682	0.954366
$637$	9.10899	0.360911
$638$	29.3368	1.16145
$639$	2.34139	0.0926241
$640$	21.5581	0.852158
$641$	14.4403	0.570358	0.285179	−	0.958474i	$-0.407947\pi$
$641$	14.4403	0.570358	0.285179	+	0.958474i	$0.407947\pi$
$642$	−13.5516	−0.534841
$643$	−33.7128	−1.32950	−0.664751	−	0.747065i	$-0.731462\pi$
$643$	−33.7128	−1.32950	−0.664751	+	0.747065i	$0.731462\pi$
$644$	−105.880	−4.17226
$645$	2.58439	0.101760
$646$	−53.9688	−2.12337
$647$	2.48216	0.0975839	0.0487920	−	0.998809i	$-0.484463\pi$
$647$	2.48216	0.0975839	0.0487920	+	0.998809i	$0.484463\pi$
$648$	−68.7016	−2.69885
$649$	29.1236	1.14320
$650$	27.4544	1.07685
$651$	−3.31418	−0.129893
$652$	−32.2133	−1.26157
$653$	16.5022	0.645782	0.322891	−	0.946436i	$-0.395345\pi$
$653$	16.5022	0.645782	0.322891	+	0.946436i	$0.395345\pi$
$654$	−9.80314	−0.383333
$655$	8.09454	0.316280
$656$	−33.1971	−1.29613
$657$	−37.7197	−1.47159
$658$	0.313003	0.0122021
$659$	−46.8300	−1.82424	−0.912119	−	0.409925i	$-0.865555\pi$
$659$	−46.8300	−1.82424	−0.912119	+	0.409925i	$0.865555\pi$
$660$	7.26118	0.282641
$661$	27.1955	1.05778	0.528891	−	0.848689i	$-0.322608\pi$
$661$	27.1955	1.05778	0.528891	+	0.848689i	$0.322608\pi$
$662$	−95.0925	−3.69587
$663$	−2.92790	−0.113710
$664$	16.6099	0.644588
$665$	20.0948	0.779245
$666$	57.3237	2.22125
$667$	−17.2930	−0.669587
$668$	−11.4326	−0.442339
$669$	−6.80332	−0.263031
$670$	−15.5932	−0.602418
$671$	−6.24609	−0.241128
$672$	−20.0286	−0.772621
$673$	−16.8344	−0.648917	−0.324458	−	0.945900i	$-0.605182\pi$
$673$	−16.8344	−0.648917	−0.324458	+	0.945900i	$0.605182\pi$
$674$	10.3197	0.397500
$675$	8.31008	0.319855
$676$	−33.9913	−1.30736
$677$	41.9534	1.61240	0.806201	−	0.591642i	$-0.201520\pi$
$677$	41.9534	1.61240	0.806201	+	0.591642i	$0.201520\pi$
$678$	−11.5310	−0.442847
$679$	12.9663	0.497601
$680$	−28.2557	−1.08356
$681$	8.09883	0.310348
$682$	−30.4137	−1.16460
$683$	29.2647	1.11978	0.559892	−	0.828566i	$-0.310843\pi$
$683$	29.2647	1.11978	0.559892	+	0.828566i	$0.310843\pi$
$684$	93.0634	3.55837
$685$	−22.1677	−0.846985
$686$	29.9960	1.14525
$687$	0.372748	0.0142212
$688$	−94.6562	−3.60873
$689$	33.1611	1.26334
$690$	−5.91490	−0.225176
$691$	24.0618	0.915354	0.457677	−	0.889119i	$-0.348682\pi$
$691$	24.0618	0.915354	0.457677	+	0.889119i	$0.348682\pi$
$692$	−38.7026	−1.47125
$693$	36.6574	1.39250
$694$	58.7819	2.23133
$695$	−5.71767	−0.216884
$696$	−8.56142	−0.324520
$697$	8.32008	0.315145
$698$	−18.4910	−0.699894
$699$	−5.62731	−0.212844
$700$	−68.1063	−2.57417
$701$	−45.2373	−1.70859	−0.854294	−	0.519790i	$-0.826010\pi$
$701$	−45.2373	−1.70859	−0.854294	+	0.519790i	$0.826010\pi$
$702$	14.2577	0.538121
$703$	−45.8163	−1.72800
$704$	−82.2287	−3.09911
$705$	0.0126532	0.000476546 0
$706$	60.9269	2.29301
$707$	29.4916	1.10914
$708$	−13.7510	−0.516793
$709$	−34.1604	−1.28292	−0.641461	−	0.767156i	$-0.721671\pi$
$709$	−34.1604	−1.28292	−0.641461	+	0.767156i	$0.721671\pi$
$710$	−2.19060	−0.0822117
$711$	−27.1895	−1.01969
$712$	−159.781	−5.98805
$713$	17.9278	0.671401
$714$	10.0372	0.375634
$715$	10.0044	0.374145
$716$	−23.7100	−0.886085
$717$	1.57227	0.0587173
$718$	34.7734	1.29773
$719$	−41.1197	−1.53350	−0.766752	−	0.641943i	$-0.778129\pi$
$719$	−41.1197	−1.53350	−0.766752	+	0.641943i	$0.778129\pi$
$720$	37.2254	1.38731
$721$	31.8702	1.18691
$722$	−51.6773	−1.92323
$723$	1.65016	0.0613700
$724$	12.3322	0.458324
$725$	−11.1235	−0.413118
$726$	−4.16679	−0.154644
$727$	27.0131	1.00186	0.500931	−	0.865487i	$-0.332991\pi$
$727$	27.0131	1.00186	0.500931	+	0.865487i	$0.332991\pi$
$728$	−72.2229	−2.67676
$729$	−20.4807	−0.758544
$730$	35.2904	1.30616
$731$	23.7234	0.877440
$732$	2.94915	0.109004
$733$	−42.6350	−1.57476	−0.787380	−	0.616468i	$-0.788563\pi$
$733$	−42.6350	−1.57476	−0.787380	+	0.616468i	$0.788563\pi$
$734$	6.03182	0.222639
$735$	−1.26235	−0.0465623
$736$	108.343	3.99359
$737$	22.7352	0.837461
$738$	−19.8265	−0.729825
$739$	−2.49327	−0.0917163	−0.0458582	−	0.998948i	$-0.514602\pi$
$739$	−2.49327	−0.0917163	−0.0458582	+	0.998948i	$0.514602\pi$
$740$	−38.8096	−1.42667
$741$	−5.57650	−0.204858
$742$	−113.681	−4.17334
$743$	1.90224	0.0697865	0.0348932	−	0.999391i	$-0.488891\pi$
$743$	1.90224	0.0697865	0.0348932	+	0.999391i	$0.488891\pi$
$744$	8.87570	0.325399
$745$	11.6960	0.428509
$746$	95.5302	3.49761
$747$	5.48437	0.200663
$748$	66.6538	2.43711
$749$	46.3182	1.69243
$750$	−8.56031	−0.312578
$751$	−26.0973	−0.952303	−0.476152	−	0.879363i	$-0.657969\pi$
$751$	−26.0973	−0.952303	−0.476152	+	0.879363i	$0.657969\pi$
$752$	−0.463436	−0.0168998
$753$	6.58782	0.240073
$754$	−19.0847	−0.695025
$755$	−5.06552	−0.184353
$756$	−35.3690	−1.28636
$757$	18.2526	0.663403	0.331701	−	0.943384i	$-0.392377\pi$
$757$	18.2526	0.663403	0.331701	+	0.943384i	$0.392377\pi$
$758$	30.9387	1.12374
$759$	8.62404	0.313033
$760$	−53.8161	−1.95211
$761$	−33.4256	−1.21168	−0.605838	−	0.795588i	$-0.707162\pi$
$761$	−33.4256	−1.21168	−0.605838	+	0.795588i	$0.707162\pi$
$762$	7.66482	0.277667
$763$	33.5062	1.21300
$764$	−79.9582	−2.89278
$765$	−9.32968	−0.337315
$766$	−52.5269	−1.89788
$767$	−18.9460	−0.684101
$768$	5.68094	0.204993
$769$	15.3937	0.555112	0.277556	−	0.960709i	$-0.410476\pi$
$769$	15.3937	0.555112	0.277556	+	0.960709i	$0.410476\pi$
$770$	−34.2965	−1.23596
$771$	1.83424	0.0660585
$772$	32.7257	1.17782
$773$	20.3970	0.733631	0.366815	−	0.930294i	$-0.380448\pi$
$773$	20.3970	0.733631	0.366815	+	0.930294i	$0.380448\pi$
$774$	−56.5322	−2.03201
$775$	11.5319	0.414237
$776$	−34.7251	−1.24656
$777$	8.52101	0.305690
$778$	−35.8877	−1.28664
$779$	15.8465	0.567759
$780$	−4.72368	−0.169135
$781$	3.19394	0.114288
$782$	−54.2957	−1.94161
$783$	−5.77669	−0.206442
$784$	46.2348	1.65124
$785$	−4.41261	−0.157493
$786$	7.70064	0.274673
$787$	9.49213	0.338358	0.169179	−	0.985585i	$-0.445888\pi$
$787$	9.49213	0.338358	0.169179	+	0.985585i	$0.445888\pi$
$788$	−38.8673	−1.38459
$789$	0.328528	0.0116959
$790$	25.4384	0.905057
$791$	39.4120	1.40133
$792$	−98.1723	−3.48840
$793$	4.06332	0.144293
$794$	97.1242	3.44681
$795$	−4.59554	−0.162987
$796$	−46.6410	−1.65315
$797$	10.2928	0.364590	0.182295	−	0.983244i	$-0.441647\pi$
$797$	10.2928	0.364590	0.182295	+	0.983244i	$0.441647\pi$
$798$	19.1170	0.676734
$799$	0.116149	0.00410907
$800$	69.6908	2.46394
$801$	−52.7577	−1.86410
$802$	−54.8757	−1.93773
$803$	−51.4541	−1.81578
$804$	−10.7346	−0.378581
$805$	20.2166	0.712540
$806$	19.7853	0.696908
$807$	2.81583	0.0991218
$808$	−78.9815	−2.77856
$809$	−22.8025	−0.801693	−0.400847	−	0.916145i	$-0.631284\pi$
$809$	−22.8025	−0.801693	−0.400847	+	0.916145i	$0.631284\pi$
$810$	21.2234	0.745715
$811$	50.9046	1.78750	0.893751	−	0.448564i	$-0.148065\pi$
$811$	50.9046	1.78750	0.893751	+	0.448564i	$0.148065\pi$
$812$	47.3435	1.66143
$813$	0.687759	0.0241208
$814$	78.1962	2.74078
$815$	6.15075	0.215451
$816$	−14.8612	−0.520247
$817$	45.1837	1.58078
$818$	−69.5730	−2.43256
$819$	−23.8471	−0.833284
$820$	13.4231	0.468754
$821$	48.5255	1.69355	0.846776	−	0.531949i	$-0.178540\pi$
$821$	48.5255	1.69355	0.846776	+	0.531949i	$0.178540\pi$
$822$	−21.0890	−0.735563
$823$	37.8842	1.32056	0.660280	−	0.751020i	$-0.270438\pi$
$823$	37.8842	1.32056	0.660280	+	0.751020i	$0.270438\pi$
$824$	−85.3517	−2.97337
$825$	5.54733	0.193133
$826$	64.9495	2.25988
$827$	−0.662653	−0.0230427	−0.0115214	−	0.999934i	$-0.503667\pi$
$827$	−0.662653	−0.0230427	−0.0115214	+	0.999934i	$0.503667\pi$
$828$	93.6271	3.25377
$829$	55.5636	1.92981	0.964903	−	0.262608i	$-0.0845826\pi$
$829$	55.5636	1.92981	0.964903	+	0.262608i	$0.0845826\pi$
$830$	−5.13115	−0.178105
$831$	4.25608	0.147642
$832$	53.4930	1.85454
$833$	−11.5877	−0.401489
$834$	−5.43944	−0.188352
$835$	2.18291	0.0755428
$836$	126.949	4.39064
$837$	5.98875	0.207001
$838$	5.42751	0.187490
$839$	9.46898	0.326906	0.163453	−	0.986551i	$-0.447737\pi$
$839$	9.46898	0.326906	0.163453	+	0.986551i	$0.447737\pi$
$840$	10.0088	0.345337
$841$	−21.2676	−0.733364
$842$	−55.0008	−1.89545
$843$	−3.69801	−0.127366
$844$	−3.57539	−0.123070
$845$	6.49025	0.223271
$846$	−0.276781	−0.00951593
$847$	14.2417	0.489350
$848$	168.317	5.78002
$849$	−0.375535	−0.0128883
$850$	−34.9251	−1.19792
$851$	−46.0938	−1.58008
$852$	−1.50805	−0.0516648
$853$	13.3248	0.456231	0.228116	−	0.973634i	$-0.426744\pi$
$853$	13.3248	0.456231	0.228116	+	0.973634i	$0.426744\pi$
$854$	−13.9296	−0.476662
$855$	−17.7694	−0.607700
$856$	−124.045	−4.23977
$857$	−23.0047	−0.785826	−0.392913	−	0.919576i	$-0.628533\pi$
$857$	−23.0047	−0.785826	−0.392913	+	0.919576i	$0.628533\pi$
$858$	9.51759	0.324925
$859$	40.7816	1.39145	0.695725	−	0.718309i	$-0.255083\pi$
$859$	40.7816	1.39145	0.695725	+	0.718309i	$0.255083\pi$
$860$	38.2737	1.30512
$861$	−2.94716	−0.100439
$862$	−70.7686	−2.41039
$863$	−14.3339	−0.487933	−0.243966	−	0.969784i	$-0.578449\pi$
$863$	−14.3339	−0.487933	−0.243966	+	0.969784i	$0.578449\pi$
$864$	36.1919	1.23127
$865$	7.38981	0.251261
$866$	12.8551	0.436835
$867$	−2.28662	−0.0776577
$868$	−49.0815	−1.66594
$869$	−37.0897	−1.25818
$870$	2.64481	0.0896674
$871$	−14.7901	−0.501144
$872$	−89.7330	−3.03874
$873$	−11.4658	−0.388058
$874$	−103.412	−3.49796
$875$	29.2583	0.989111
$876$	24.2945	0.820836
$877$	−12.2334	−0.413094	−0.206547	−	0.978437i	$-0.566223\pi$
$877$	−12.2334	−0.413094	−0.206547	+	0.978437i	$0.566223\pi$
$878$	−97.2932	−3.28349
$879$	2.95585	0.0996983
$880$	50.7798	1.71179
$881$	−40.0849	−1.35049	−0.675247	−	0.737592i	$-0.735963\pi$
$881$	−40.0849	−1.35049	−0.675247	+	0.737592i	$0.735963\pi$
$882$	27.6131	0.929782
$883$	30.9326	1.04096	0.520482	−	0.853873i	$-0.325752\pi$
$883$	30.9326	1.04096	0.520482	+	0.853873i	$0.325752\pi$
$884$	−43.3609	−1.45839
$885$	2.62559	0.0882581
$886$	30.5228	1.02543
$887$	−56.7326	−1.90490	−0.952448	−	0.304702i	$-0.901443\pi$
$887$	−56.7326	−1.90490	−0.952448	+	0.304702i	$0.901443\pi$
$888$	−22.8202	−0.765794
$889$	−26.1976	−0.878639
$890$	49.3599	1.65455
$891$	−30.9442	−1.03667
$892$	−100.754	−3.37350
$893$	0.221219	0.00740282
$894$	11.1269	0.372138
$895$	4.52715	0.151326
$896$	−70.0978	−2.34180
$897$	−5.61027	−0.187322
$898$	−109.379	−3.65003
$899$	−8.01630	−0.267358
$900$	60.2247	2.00749
$901$	−42.1846	−1.40537
$902$	−27.0457	−0.900524
$903$	−8.40336	−0.279646
$904$	−105.549	−3.51052
$905$	−2.35470	−0.0782728
$906$	−4.81902	−0.160101
$907$	−30.8420	−1.02409	−0.512047	−	0.858958i	$-0.671112\pi$
$907$	−30.8420	−1.02409	−0.512047	+	0.858958i	$0.671112\pi$
$908$	119.940	3.98035
$909$	−26.0787	−0.864975
$910$	22.3112	0.739610
$911$	−16.2193	−0.537370	−0.268685	−	0.963228i	$-0.586589\pi$
$911$	−16.2193	−0.537370	−0.268685	+	0.963228i	$0.586589\pi$
$912$	−28.3048	−0.937267
$913$	7.48132	0.247596
$914$	−98.6789	−3.26401
$915$	−0.563105	−0.0186157
$916$	5.52023	0.182394
$917$	−26.3200	−0.869164
$918$	−18.1374	−0.598622
$919$	−4.34727	−0.143403	−0.0717015	−	0.997426i	$-0.522843\pi$
$919$	−4.34727	−0.143403	−0.0717015	+	0.997426i	$0.522843\pi$
$920$	−54.1420	−1.78501
$921$	−7.91561	−0.260828
$922$	−38.5825	−1.27065
$923$	−2.07778	−0.0683909
$924$	−23.6103	−0.776722
$925$	−29.6494	−0.974866
$926$	−55.3563	−1.81912
$927$	−28.1820	−0.925620
$928$	−48.4450	−1.59029
$929$	−2.22372	−0.0729580	−0.0364790	−	0.999334i	$-0.511614\pi$
$929$	−2.22372	−0.0729580	−0.0364790	+	0.999334i	$0.511614\pi$
$930$	−2.74190	−0.0899104
$931$	−22.0700	−0.723314
$932$	−83.3380	−2.72983
$933$	−5.45629	−0.178631
$934$	−29.2356	−0.956618
$935$	−12.7268	−0.416210
$936$	63.8649	2.08749
$937$	−27.6279	−0.902564	−0.451282	−	0.892381i	$-0.649033\pi$
$937$	−27.6279	−0.902564	−0.451282	+	0.892381i	$0.649033\pi$
$938$	50.7025	1.65550
$939$	−4.51795	−0.147438
$940$	0.187388	0.00611192
$941$	26.9990	0.880144	0.440072	−	0.897963i	$-0.354953\pi$
$941$	26.9990	0.880144	0.440072	+	0.897963i	$0.354953\pi$
$942$	−4.19788	−0.136774
$943$	15.9425	0.519158
$944$	−96.1650	−3.12990
$945$	6.75330	0.219685
$946$	−77.1165	−2.50727
$947$	15.4751	0.502872	0.251436	−	0.967874i	$-0.419097\pi$
$947$	15.4751	0.502872	0.251436	+	0.967874i	$0.419097\pi$
$948$	17.5122	0.568770
$949$	33.4729	1.08658
$950$	−66.5187	−2.15815
$951$	−2.46374	−0.0798921
$952$	91.8757	2.97771
$953$	2.03284	0.0658501	0.0329250	−	0.999458i	$-0.489518\pi$
$953$	2.03284	0.0658501	0.0329250	+	0.999458i	$0.489518\pi$
$954$	100.525	3.25461
$955$	15.2671	0.494031
$956$	23.2846	0.753077
$957$	−3.85618	−0.124653
$958$	−39.0185	−1.26063
$959$	72.0801	2.32759
$960$	−7.41319	−0.239260
$961$	−22.6894	−0.731917
$962$	−50.8697	−1.64010
$963$	−40.9581	−1.31985
$964$	24.4381	0.787099
$965$	−6.24859	−0.201149
$966$	19.2328	0.618804
$967$	−48.7916	−1.56903	−0.784516	−	0.620109i	$-0.787088\pi$
$967$	−48.7916	−1.56903	−0.784516	+	0.620109i	$0.787088\pi$
$968$	−38.1407	−1.22589
$969$	7.09394	0.227890
$970$	10.7273	0.344434
$971$	39.4950	1.26746	0.633728	−	0.773556i	$-0.281524\pi$
$971$	39.4950	1.26746	0.633728	+	0.773556i	$0.281524\pi$
$972$	47.2467	1.51544
$973$	18.5915	0.596015
$974$	59.9941	1.92234
$975$	−3.60875	−0.115573
$976$	20.6243	0.660169
$977$	−22.4125	−0.717039	−0.358519	−	0.933522i	$-0.616718\pi$
$977$	−22.4125	−0.717039	−0.358519	+	0.933522i	$0.616718\pi$
$978$	5.85144	0.187108
$979$	−71.9677	−2.30010
$980$	−18.6948	−0.597183
$981$	−29.6287	−0.945971
$982$	−19.6488	−0.627017
$983$	37.5986	1.19921	0.599605	−	0.800296i	$-0.295324\pi$
$983$	37.5986	1.19921	0.599605	+	0.800296i	$0.295324\pi$
$984$	7.89280	0.251613
$985$	7.42126	0.236461
$986$	24.2779	0.773167
$987$	−0.0411428	−0.00130959
$988$	−82.5855	−2.62740
$989$	45.4574	1.44546
$990$	30.3276	0.963874
$991$	−25.3130	−0.804094	−0.402047	−	0.915619i	$-0.631701\pi$
$991$	−25.3130	−0.804094	−0.402047	+	0.915619i	$0.631701\pi$
$992$	50.2234	1.59460
$993$	12.4995	0.396658
$994$	7.12290	0.225925
$995$	8.90556	0.282325
$996$	−3.53237	−0.111928
$997$	40.1089	1.27026	0.635130	−	0.772405i	$-0.280946\pi$
$997$	40.1089	1.27026	0.635130	+	0.772405i	$0.280946\pi$
$998$	−77.9226	−2.46660
$999$	−15.3976	−0.487157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.b.1.7	✓	186

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.b.1.7	✓	186	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69011 q^{2} +0.353602 q^{3} +5.23670 q^{4} -0.999886 q^{5} -0.951229 q^{6} +3.25121 q^{7} -8.70707 q^{8} -2.87497 q^{9} +O(q^{10})\)	\(q-2.69011 q^{2} +0.353602 q^{3} +5.23670 q^{4} -0.999886 q^{5} -0.951229 q^{6} +3.25121 q^{7} -8.70707 q^{8} -2.87497 q^{9} +2.68981 q^{10} -3.92179 q^{11} +1.85171 q^{12} +2.55128 q^{13} -8.74612 q^{14} -0.353562 q^{15} +12.9496 q^{16} -3.24551 q^{17} +7.73398 q^{18} -6.18143 q^{19} -5.23610 q^{20} +1.14964 q^{21} +10.5500 q^{22} -6.21887 q^{23} -3.07884 q^{24} -4.00023 q^{25} -6.86322 q^{26} -2.07740 q^{27} +17.0256 q^{28} +2.78073 q^{29} +0.951121 q^{30} -2.88281 q^{31} -17.4217 q^{32} -1.38675 q^{33} +8.73079 q^{34} -3.25084 q^{35} -15.0553 q^{36} +7.41193 q^{37} +16.6287 q^{38} +0.902137 q^{39} +8.70609 q^{40} -2.56356 q^{41} -3.09265 q^{42} -7.30959 q^{43} -20.5372 q^{44} +2.87464 q^{45} +16.7295 q^{46} -0.0357877 q^{47} +4.57901 q^{48} +3.57036 q^{49} +10.7611 q^{50} -1.14762 q^{51} +13.3603 q^{52} +12.9978 q^{53} +5.58844 q^{54} +3.92134 q^{55} -28.3085 q^{56} -2.18577 q^{57} -7.48046 q^{58} -7.42610 q^{59} -1.85150 q^{60} +1.59266 q^{61} +7.75507 q^{62} -9.34712 q^{63} +20.9671 q^{64} -2.55099 q^{65} +3.73052 q^{66} -5.79715 q^{67} -16.9958 q^{68} -2.19901 q^{69} +8.74512 q^{70} -0.814408 q^{71} +25.0325 q^{72} +13.1201 q^{73} -19.9389 q^{74} -1.41449 q^{75} -32.3703 q^{76} -12.7506 q^{77} -2.42685 q^{78} +9.45733 q^{79} -12.9481 q^{80} +7.89032 q^{81} +6.89627 q^{82} -1.90763 q^{83} +6.02029 q^{84} +3.24514 q^{85} +19.6636 q^{86} +0.983271 q^{87} +34.1473 q^{88} +18.3507 q^{89} -7.73310 q^{90} +8.29473 q^{91} -32.5663 q^{92} -1.01937 q^{93} +0.0962729 q^{94} +6.18073 q^{95} -6.16036 q^{96} +3.98815 q^{97} -9.60468 q^{98} +11.2750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9}+O(q^{10}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9	\( 186 q + 6 q^{2} + 10 q^{3} + 212 q^{4} + 38 q^{5} + 47 q^{6} + 32 q^{7} + 15 q^{8} + 216 q^{9} + 50 q^{10} + 25 q^{11} + 17 q^{12} + 113 q^{13} + 12 q^{14} + 12 q^{15} + 252 q^{16} + 35 q^{17} + 13 q^{18} + 97 q^{19} + 55 q^{20} + 115 q^{21} + 14 q^{22} + 27 q^{23} + 122 q^{24} + 244 q^{25} + 39 q^{26} + 34 q^{27} + 66 q^{28} + 91 q^{29} + 4 q^{30} + 135 q^{31} + 21 q^{32} + 32 q^{33} + 58 q^{34} + 17 q^{35} + 273 q^{36} + 133 q^{37} - 3 q^{38} + 55 q^{39} + 142 q^{40} + 97 q^{41} - 8 q^{42} + 67 q^{43} + 44 q^{44} + 154 q^{45} + 101 q^{46} + 20 q^{47} - 7 q^{48} + 312 q^{49} + 21 q^{50} + 23 q^{51} + 193 q^{52} + 22 q^{53} + 141 q^{54} + 88 q^{55} + 28 q^{56} + 65 q^{57} + 62 q^{58} + 41 q^{59} + q^{60} + 377 q^{61} + 29 q^{62} + 39 q^{63} + 311 q^{64} + 21 q^{65} + 35 q^{66} + 42 q^{67} + 24 q^{68} + 137 q^{69} + 35 q^{70} + 17 q^{71} - 8 q^{72} + 213 q^{73} - 9 q^{74} + 2 q^{75} + 242 q^{76} + 60 q^{77} + 103 q^{79} + 80 q^{80} + 270 q^{81} + 84 q^{82} + 42 q^{83} + 137 q^{84} + 294 q^{85} - 9 q^{86} + 22 q^{87} - 13 q^{88} + 78 q^{89} + 69 q^{90} + 118 q^{91} + 49 q^{92} + 51 q^{93} + 93 q^{94} + 10 q^{95} + 260 q^{96} + 142 q^{97} - 31 q^{98} + 78 q^{99}+O(q^{100}) \) 186 * q + 6 * q^2 + 10 * q^3 + 212 * q^4 + 38 * q^5 + 47 * q^6 + 32 * q^7 + 15 * q^8 + 216 * q^9 + 50 * q^10 + 25 * q^11 + 17 * q^12 + 113 * q^13 + 12 * q^14 + 12 * q^15 + 252 * q^16 + 35 * q^17 + 13 * q^18 + 97 * q^19 + 55 * q^20 + 115 * q^21 + 14 * q^22 + 27 * q^23 + 122 * q^24 + 244 * q^25 + 39 * q^26 + 34 * q^27 + 66 * q^28 + 91 * q^29 + 4 * q^30 + 135 * q^31 + 21 * q^32 + 32 * q^33 + 58 * q^34 + 17 * q^35 + 273 * q^36 + 133 * q^37 - 3 * q^38 + 55 * q^39 + 142 * q^40 + 97 * q^41 - 8 * q^42 + 67 * q^43 + 44 * q^44 + 154 * q^45 + 101 * q^46 + 20 * q^47 - 7 * q^48 + 312 * q^49 + 21 * q^50 + 23 * q^51 + 193 * q^52 + 22 * q^53 + 141 * q^54 + 88 * q^55 + 28 * q^56 + 65 * q^57 + 62 * q^58 + 41 * q^59 + q^60 + 377 * q^61 + 29 * q^62 + 39 * q^63 + 311 * q^64 + 21 * q^65 + 35 * q^66 + 42 * q^67 + 24 * q^68 + 137 * q^69 + 35 * q^70 + 17 * q^71 - 8 * q^72 + 213 * q^73 - 9 * q^74 + 2 * q^75 + 242 * q^76 + 60 * q^77 + 103 * q^79 + 80 * q^80 + 270 * q^81 + 84 * q^82 + 42 * q^83 + 137 * q^84 + 294 * q^85 - 9 * q^86 + 22 * q^87 - 13 * q^88 + 78 * q^89 + 69 * q^90 + 118 * q^91 + 49 * q^92 + 51 * q^93 + 93 * q^94 + 10 * q^95 + 260 * q^96 + 142 * q^97 - 31 * q^98 + 78 * q^99

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