LMFDB - Embedded newform 4022.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.91271 q^{3} +1.00000 q^{4} +3.14140 q^{5} -2.91271 q^{6} -3.38046 q^{7} +1.00000 q^{8} +5.48388 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.91271 q^{3} +1.00000 q^{4} +3.14140 q^{5} -2.91271 q^{6} -3.38046 q^{7} +1.00000 q^{8} +5.48388 q^{9} +3.14140 q^{10} -3.13583 q^{11} -2.91271 q^{12} -2.60951 q^{13} -3.38046 q^{14} -9.14998 q^{15} +1.00000 q^{16} +0.366508 q^{17} +5.48388 q^{18} +6.14906 q^{19} +3.14140 q^{20} +9.84631 q^{21} -3.13583 q^{22} +1.69553 q^{23} -2.91271 q^{24} +4.86839 q^{25} -2.60951 q^{26} -7.23481 q^{27} -3.38046 q^{28} +0.436032 q^{29} -9.14998 q^{30} -0.0906844 q^{31} +1.00000 q^{32} +9.13376 q^{33} +0.366508 q^{34} -10.6194 q^{35} +5.48388 q^{36} -0.999538 q^{37} +6.14906 q^{38} +7.60074 q^{39} +3.14140 q^{40} -4.35490 q^{41} +9.84631 q^{42} -12.9033 q^{43} -3.13583 q^{44} +17.2270 q^{45} +1.69553 q^{46} -1.75832 q^{47} -2.91271 q^{48} +4.42753 q^{49} +4.86839 q^{50} -1.06753 q^{51} -2.60951 q^{52} +13.3950 q^{53} -7.23481 q^{54} -9.85089 q^{55} -3.38046 q^{56} -17.9104 q^{57} +0.436032 q^{58} -4.83880 q^{59} -9.14998 q^{60} -5.12806 q^{61} -0.0906844 q^{62} -18.5380 q^{63} +1.00000 q^{64} -8.19751 q^{65} +9.13376 q^{66} -2.23264 q^{67} +0.366508 q^{68} -4.93858 q^{69} -10.6194 q^{70} -14.2186 q^{71} +5.48388 q^{72} +4.50336 q^{73} -0.999538 q^{74} -14.1802 q^{75} +6.14906 q^{76} +10.6006 q^{77} +7.60074 q^{78} +2.60491 q^{79} +3.14140 q^{80} +4.62127 q^{81} -4.35490 q^{82} +1.12799 q^{83} +9.84631 q^{84} +1.15135 q^{85} -12.9033 q^{86} -1.27004 q^{87} -3.13583 q^{88} +8.46529 q^{89} +17.2270 q^{90} +8.82135 q^{91} +1.69553 q^{92} +0.264137 q^{93} -1.75832 q^{94} +19.3166 q^{95} -2.91271 q^{96} +6.44708 q^{97} +4.42753 q^{98} -17.1965 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - 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
$3$	−2.91271	−1.68165	−0.840827	−	0.541304i	$-0.817931\pi$
$3$	−2.91271	−1.68165	−0.840827	+	0.541304i	$0.817931\pi$
$4$	1.00000	0.500000
$5$	3.14140	1.40488	0.702438	−	0.711745i	$-0.252095\pi$
$5$	3.14140	1.40488	0.702438	+	0.711745i	$0.252095\pi$
$6$	−2.91271	−1.18911
$7$	−3.38046	−1.27769	−0.638847	−	0.769333i	$-0.720589\pi$
$7$	−3.38046	−1.27769	−0.638847	+	0.769333i	$0.720589\pi$
$8$	1.00000	0.353553
$9$	5.48388	1.82796
$10$	3.14140	0.993398
$11$	−3.13583	−0.945488	−0.472744	−	0.881200i	$-0.656736\pi$
$11$	−3.13583	−0.945488	−0.472744	+	0.881200i	$0.656736\pi$
$12$	−2.91271	−0.840827
$13$	−2.60951	−0.723748	−0.361874	−	0.932227i	$-0.617863\pi$
$13$	−2.60951	−0.723748	−0.361874	+	0.932227i	$0.617863\pi$
$14$	−3.38046	−0.903467
$15$	−9.14998	−2.36252
$16$	1.00000	0.250000
$17$	0.366508	0.0888911	0.0444456	−	0.999012i	$-0.485848\pi$
$17$	0.366508	0.0888911	0.0444456	+	0.999012i	$0.485848\pi$
$18$	5.48388	1.29256
$19$	6.14906	1.41069	0.705345	−	0.708864i	$-0.250792\pi$
$19$	6.14906	1.41069	0.705345	+	0.708864i	$0.250792\pi$
$20$	3.14140	0.702438
$21$	9.84631	2.14864
$22$	−3.13583	−0.668561
$23$	1.69553	0.353542	0.176771	−	0.984252i	$-0.443435\pi$
$23$	1.69553	0.353542	0.176771	+	0.984252i	$0.443435\pi$
$24$	−2.91271	−0.594554
$25$	4.86839	0.973678
$26$	−2.60951	−0.511767
$27$	−7.23481	−1.39234
$28$	−3.38046	−0.638847
$29$	0.436032	0.0809692	0.0404846	−	0.999180i	$-0.487110\pi$
$29$	0.436032	0.0809692	0.0404846	+	0.999180i	$0.487110\pi$
$30$	−9.14998	−1.67055
$31$	−0.0906844	−0.0162874	−0.00814370	−	0.999967i	$-0.502592\pi$
$31$	−0.0906844	−0.0162874	−0.00814370	+	0.999967i	$0.502592\pi$
$32$	1.00000	0.176777
$33$	9.13376	1.58998
$34$	0.366508	0.0628555
$35$	−10.6194	−1.79500
$36$	5.48388	0.913979
$37$	−0.999538	−0.164323	−0.0821615	−	0.996619i	$-0.526182\pi$
$37$	−0.999538	−0.164323	−0.0821615	+	0.996619i	$0.526182\pi$
$38$	6.14906	0.997509
$39$	7.60074	1.21709
$40$	3.14140	0.496699
$41$	−4.35490	−0.680121	−0.340060	−	0.940404i	$-0.610447\pi$
$41$	−4.35490	−0.680121	−0.340060	+	0.940404i	$0.610447\pi$
$42$	9.84631	1.51932
$43$	−12.9033	−1.96773	−0.983866	−	0.178908i	$-0.942744\pi$
$43$	−12.9033	−1.96773	−0.983866	+	0.178908i	$0.942744\pi$
$44$	−3.13583	−0.472744
$45$	17.2270	2.56806
$46$	1.69553	0.249992
$47$	−1.75832	−0.256478	−0.128239	−	0.991743i	$-0.540932\pi$
$47$	−1.75832	−0.256478	−0.128239	+	0.991743i	$0.540932\pi$
$48$	−2.91271	−0.420413
$49$	4.42753	0.632504
$50$	4.86839	0.688494
$51$	−1.06753	−0.149484
$52$	−2.60951	−0.361874
$53$	13.3950	1.83994	0.919970	−	0.391988i	$-0.128213\pi$
$53$	13.3950	1.83994	0.919970	+	0.391988i	$0.128213\pi$
$54$	−7.23481	−0.984533
$55$	−9.85089	−1.32829
$56$	−3.38046	−0.451733
$57$	−17.9104	−2.37229
$58$	0.436032	0.0572539
$59$	−4.83880	−0.629958	−0.314979	−	0.949099i	$-0.601997\pi$
$59$	−4.83880	−0.629958	−0.314979	+	0.949099i	$0.601997\pi$
$60$	−9.14998	−1.18126
$61$	−5.12806	−0.656580	−0.328290	−	0.944577i	$-0.606472\pi$
$61$	−5.12806	−0.656580	−0.328290	+	0.944577i	$0.606472\pi$
$62$	−0.0906844	−0.0115169
$63$	−18.5380	−2.33557
$64$	1.00000	0.125000
$65$	−8.19751	−1.01678
$66$	9.13376	1.12429
$67$	−2.23264	−0.272761	−0.136380	−	0.990657i	$-0.543547\pi$
$67$	−2.23264	−0.272761	−0.136380	+	0.990657i	$0.543547\pi$
$68$	0.366508	0.0444456
$69$	−4.93858	−0.594535
$70$	−10.6194	−1.26926
$71$	−14.2186	−1.68743	−0.843717	−	0.536789i	$-0.819637\pi$
$71$	−14.2186	−1.68743	−0.843717	+	0.536789i	$0.819637\pi$
$72$	5.48388	0.646281
$73$	4.50336	0.527078	0.263539	−	0.964649i	$-0.415110\pi$
$73$	4.50336	0.527078	0.263539	+	0.964649i	$0.415110\pi$
$74$	−0.999538	−0.116194
$75$	−14.1802	−1.63739
$76$	6.14906	0.705345
$77$	10.6006	1.20805
$78$	7.60074	0.860615
$79$	2.60491	0.293075	0.146538	−	0.989205i	$-0.453187\pi$
$79$	2.60491	0.293075	0.146538	+	0.989205i	$0.453187\pi$
$80$	3.14140	0.351219
$81$	4.62127	0.513475
$82$	−4.35490	−0.480918
$83$	1.12799	0.123813	0.0619065	−	0.998082i	$-0.480282\pi$
$83$	1.12799	0.123813	0.0619065	+	0.998082i	$0.480282\pi$
$84$	9.84631	1.07432
$85$	1.15135	0.124881
$86$	−12.9033	−1.39140
$87$	−1.27004	−0.136162
$88$	−3.13583	−0.334280
$89$	8.46529	0.897319	0.448660	−	0.893703i	$-0.351902\pi$
$89$	8.46529	0.897319	0.448660	+	0.893703i	$0.351902\pi$
$90$	17.2270	1.81589
$91$	8.82135	0.924729
$92$	1.69553	0.176771
$93$	0.264137	0.0273898
$94$	−1.75832	−0.181357
$95$	19.3166	1.98185
$96$	−2.91271	−0.297277
$97$	6.44708	0.654602	0.327301	−	0.944920i	$-0.393861\pi$
$97$	6.44708	0.654602	0.327301	+	0.944920i	$0.393861\pi$
$98$	4.42753	0.447248
$99$	−17.1965	−1.72831
$100$	4.86839	0.486839
$101$	−6.39827	−0.636652	−0.318326	−	0.947981i	$-0.603121\pi$
$101$	−6.39827	−0.636652	−0.318326	+	0.947981i	$0.603121\pi$
$102$	−1.06753	−0.105701
$103$	−5.39959	−0.532037	−0.266019	−	0.963968i	$-0.585708\pi$
$103$	−5.39959	−0.532037	−0.266019	+	0.963968i	$0.585708\pi$
$104$	−2.60951	−0.255883
$105$	30.9312	3.01857
$106$	13.3950	1.30103
$107$	−5.57151	−0.538618	−0.269309	−	0.963054i	$-0.586795\pi$
$107$	−5.57151	−0.538618	−0.269309	+	0.963054i	$0.586795\pi$
$108$	−7.23481	−0.696170
$109$	−9.43150	−0.903374	−0.451687	−	0.892176i	$-0.649178\pi$
$109$	−9.43150	−0.903374	−0.451687	+	0.892176i	$0.649178\pi$
$110$	−9.85089	−0.939246
$111$	2.91136	0.276334
$112$	−3.38046	−0.319424
$113$	−14.8205	−1.39419	−0.697096	−	0.716977i	$-0.745525\pi$
$113$	−14.8205	−1.39419	−0.697096	+	0.716977i	$0.745525\pi$
$114$	−17.9104	−1.67746
$115$	5.32633	0.496683
$116$	0.436032	0.0404846
$117$	−14.3102	−1.32298
$118$	−4.83880	−0.445447
$119$	−1.23897	−0.113576
$120$	−9.14998	−0.835275
$121$	−1.16658	−0.106053
$122$	−5.12806	−0.464272
$123$	12.6845	1.14373
$124$	−0.0906844	−0.00814370
$125$	−0.413439	−0.0369791
$126$	−18.5380	−1.65150
$127$	−9.79058	−0.868774	−0.434387	−	0.900726i	$-0.643035\pi$
$127$	−9.79058	−0.868774	−0.434387	+	0.900726i	$0.643035\pi$
$128$	1.00000	0.0883883
$129$	37.5835	3.30904
$130$	−8.19751	−0.718969
$131$	0.848480	0.0741320	0.0370660	−	0.999313i	$-0.488199\pi$
$131$	0.848480	0.0741320	0.0370660	+	0.999313i	$0.488199\pi$
$132$	9.13376	0.794992
$133$	−20.7867	−1.80243
$134$	−2.23264	−0.192871
$135$	−22.7274	−1.95607
$136$	0.366508	0.0314278
$137$	−14.1738	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$137$	−14.1738	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$138$	−4.93858	−0.420400
$139$	−4.34021	−0.368132	−0.184066	−	0.982914i	$-0.558926\pi$
$139$	−4.34021	−0.368132	−0.184066	+	0.982914i	$0.558926\pi$
$140$	−10.6194	−0.897502
$141$	5.12149	0.431307
$142$	−14.2186	−1.19320
$143$	8.18298	0.684295
$144$	5.48388	0.456990
$145$	1.36975	0.113752
$146$	4.50336	0.372701
$147$	−12.8961	−1.06365
$148$	−0.999538	−0.0821615
$149$	13.9324	1.14139	0.570693	−	0.821164i	$-0.306675\pi$
$149$	13.9324	1.14139	0.570693	+	0.821164i	$0.306675\pi$
$150$	−14.1802	−1.15781
$151$	7.80718	0.635340	0.317670	−	0.948201i	$-0.397100\pi$
$151$	7.80718	0.635340	0.317670	+	0.948201i	$0.397100\pi$
$152$	6.14906	0.498754
$153$	2.00988	0.162489
$154$	10.6006	0.854217
$155$	−0.284876	−0.0228818
$156$	7.60074	0.608547
$157$	−8.51395	−0.679487	−0.339743	−	0.940518i	$-0.610340\pi$
$157$	−8.51395	−0.679487	−0.339743	+	0.940518i	$0.610340\pi$
$158$	2.60491	0.207235
$159$	−39.0156	−3.09414
$160$	3.14140	0.248349
$161$	−5.73167	−0.451719
$162$	4.62127	0.363081
$163$	9.17904	0.718958	0.359479	−	0.933153i	$-0.382954\pi$
$163$	9.17904	0.718958	0.359479	+	0.933153i	$0.382954\pi$
$164$	−4.35490	−0.340060
$165$	28.6928	2.23373
$166$	1.12799	0.0875490
$167$	−6.57274	−0.508614	−0.254307	−	0.967124i	$-0.581847\pi$
$167$	−6.57274	−0.508614	−0.254307	+	0.967124i	$0.581847\pi$
$168$	9.84631	0.759659
$169$	−6.19046	−0.476189
$170$	1.15135	0.0883043
$171$	33.7207	2.57868
$172$	−12.9033	−0.983866
$173$	−17.1138	−1.30113	−0.650567	−	0.759449i	$-0.725469\pi$
$173$	−17.1138	−1.30113	−0.650567	+	0.759449i	$0.725469\pi$
$174$	−1.27004	−0.0962812
$175$	−16.4574	−1.24406
$176$	−3.13583	−0.236372
$177$	14.0940	1.05937
$178$	8.46529	0.634501
$179$	17.8257	1.33236	0.666178	−	0.745793i	$-0.267929\pi$
$179$	17.8257	1.33236	0.666178	+	0.745793i	$0.267929\pi$
$180$	17.2270	1.28403
$181$	−4.36222	−0.324242	−0.162121	−	0.986771i	$-0.551833\pi$
$181$	−4.36222	−0.324242	−0.162121	+	0.986771i	$0.551833\pi$
$182$	8.82135	0.653882
$183$	14.9365	1.10414
$184$	1.69553	0.124996
$185$	−3.13995	−0.230854
$186$	0.264137	0.0193675
$187$	−1.14930	−0.0840455
$188$	−1.75832	−0.128239
$189$	24.4570	1.77899
$190$	19.3166	1.40138
$191$	−9.47458	−0.685557	−0.342778	−	0.939416i	$-0.611368\pi$
$191$	−9.47458	−0.685557	−0.342778	+	0.939416i	$0.611368\pi$
$192$	−2.91271	−0.210207
$193$	21.7127	1.56291	0.781456	−	0.623961i	$-0.214477\pi$
$193$	21.7127	1.56291	0.781456	+	0.623961i	$0.214477\pi$
$194$	6.44708	0.462874
$195$	23.8770	1.70987
$196$	4.42753	0.316252
$197$	−0.0884114	−0.00629905	−0.00314953	−	0.999995i	$-0.501003\pi$
$197$	−0.0884114	−0.00629905	−0.00314953	+	0.999995i	$0.501003\pi$
$198$	−17.1965	−1.22210
$199$	8.34831	0.591796	0.295898	−	0.955219i	$-0.404381\pi$
$199$	8.34831	0.591796	0.295898	+	0.955219i	$0.404381\pi$
$200$	4.86839	0.344247
$201$	6.50304	0.458689
$202$	−6.39827	−0.450181
$203$	−1.47399	−0.103454
$204$	−1.06753	−0.0747420
$205$	−13.6805	−0.955485
$206$	−5.39959	−0.376207
$207$	9.29806	0.646260
$208$	−2.60951	−0.180937
$209$	−19.2824	−1.33379
$210$	30.9312	2.13445
$211$	−8.66197	−0.596315	−0.298157	−	0.954517i	$-0.596372\pi$
$211$	−8.66197	−0.596315	−0.298157	+	0.954517i	$0.596372\pi$
$212$	13.3950	0.919970
$213$	41.4146	2.83768
$214$	−5.57151	−0.380861
$215$	−40.5344	−2.76442
$216$	−7.23481	−0.492266
$217$	0.306555	0.0208103
$218$	−9.43150	−0.638782
$219$	−13.1170	−0.886363
$220$	−9.85089	−0.664147
$221$	−0.956405	−0.0643348
$222$	2.91136	0.195398
$223$	−28.0460	−1.87810	−0.939050	−	0.343780i	$-0.888292\pi$
$223$	−28.0460	−1.87810	−0.939050	+	0.343780i	$0.888292\pi$
$224$	−3.38046	−0.225867
$225$	26.6976	1.77984
$226$	−14.8205	−0.985843
$227$	−18.0222	−1.19618	−0.598088	−	0.801431i	$-0.704073\pi$
$227$	−18.0222	−1.19618	−0.598088	+	0.801431i	$0.704073\pi$
$228$	−17.9104	−1.18615
$229$	−12.9895	−0.858369	−0.429184	−	0.903217i	$-0.641199\pi$
$229$	−12.9895	−0.858369	−0.429184	+	0.903217i	$0.641199\pi$
$230$	5.32633	0.351208
$231$	−30.8763	−2.03151
$232$	0.436032	0.0286269
$233$	3.69359	0.241975	0.120987	−	0.992654i	$-0.461394\pi$
$233$	3.69359	0.241975	0.120987	+	0.992654i	$0.461394\pi$
$234$	−14.3102	−0.935489
$235$	−5.52360	−0.360320
$236$	−4.83880	−0.314979
$237$	−7.58734	−0.492851
$238$	−1.23897	−0.0803102
$239$	26.1741	1.69306	0.846529	−	0.532342i	$-0.178688\pi$
$239$	26.1741	1.69306	0.846529	+	0.532342i	$0.178688\pi$
$240$	−9.14998	−0.590629
$241$	10.2496	0.660234	0.330117	−	0.943940i	$-0.392912\pi$
$241$	10.2496	0.660234	0.330117	+	0.943940i	$0.392912\pi$
$242$	−1.16658	−0.0749905
$243$	8.24401	0.528854
$244$	−5.12806	−0.328290
$245$	13.9086	0.888590
$246$	12.6845	0.808737
$247$	−16.0460	−1.02098
$248$	−0.0906844	−0.00575846
$249$	−3.28551	−0.208211
$250$	−0.413439	−0.0261482
$251$	3.62024	0.228507	0.114254	−	0.993452i	$-0.463552\pi$
$251$	3.62024	0.228507	0.114254	+	0.993452i	$0.463552\pi$
$252$	−18.5380	−1.16779
$253$	−5.31688	−0.334270
$254$	−9.79058	−0.614316
$255$	−3.35354	−0.210007
$256$	1.00000	0.0625000
$257$	−29.7972	−1.85870	−0.929348	−	0.369206i	$-0.879630\pi$
$257$	−29.7972	−1.85870	−0.929348	+	0.369206i	$0.879630\pi$
$258$	37.5835	2.33985
$259$	3.37890	0.209955
$260$	−8.19751	−0.508388
$261$	2.39115	0.148008
$262$	0.848480	0.0524193
$263$	−17.0765	−1.05298	−0.526490	−	0.850181i	$-0.676492\pi$
$263$	−17.0765	−1.05298	−0.526490	+	0.850181i	$0.676492\pi$
$264$	9.13376	0.562144
$265$	42.0789	2.58489
$266$	−20.7867	−1.27451
$267$	−24.6569	−1.50898
$268$	−2.23264	−0.136380
$269$	−4.94157	−0.301293	−0.150647	−	0.988588i	$-0.548136\pi$
$269$	−4.94157	−0.301293	−0.150647	+	0.988588i	$0.548136\pi$
$270$	−22.7274	−1.38315
$271$	9.74561	0.592003	0.296002	−	0.955187i	$-0.404347\pi$
$271$	9.74561	0.592003	0.296002	+	0.955187i	$0.404347\pi$
$272$	0.366508	0.0222228
$273$	−25.6940	−1.55507
$274$	−14.1738	−0.856270
$275$	−15.2664	−0.920601
$276$	−4.93858	−0.297268
$277$	−22.3545	−1.34315	−0.671576	−	0.740936i	$-0.734382\pi$
$277$	−22.3545	−1.34315	−0.671576	+	0.740936i	$0.734382\pi$
$278$	−4.34021	−0.260309
$279$	−0.497302	−0.0297727
$280$	−10.6194	−0.634630
$281$	−6.39814	−0.381681	−0.190841	−	0.981621i	$-0.561121\pi$
$281$	−6.39814	−0.381681	−0.190841	+	0.981621i	$0.561121\pi$
$282$	5.12149	0.304980
$283$	−25.4747	−1.51431	−0.757157	−	0.653233i	$-0.773412\pi$
$283$	−25.4747	−1.51431	−0.757157	+	0.653233i	$0.773412\pi$
$284$	−14.2186	−0.843717
$285$	−56.2638	−3.33278
$286$	8.18298	0.483869
$287$	14.7216	0.868987
$288$	5.48388	0.323141
$289$	−16.8657	−0.992098
$290$	1.36975	0.0804346
$291$	−18.7785	−1.10081
$292$	4.50336	0.263539
$293$	−6.78388	−0.396318	−0.198159	−	0.980170i	$-0.563496\pi$
$293$	−6.78388	−0.396318	−0.198159	+	0.980170i	$0.563496\pi$
$294$	−12.8961	−0.752116
$295$	−15.2006	−0.885013
$296$	−0.999538	−0.0580970
$297$	22.6871	1.31644
$298$	13.9324	0.807081
$299$	−4.42450	−0.255875
$300$	−14.1802	−0.818695
$301$	43.6191	2.51416
$302$	7.80718	0.449253
$303$	18.6363	1.07063
$304$	6.14906	0.352673
$305$	−16.1093	−0.922414
$306$	2.00988	0.114897
$307$	−26.9925	−1.54054	−0.770271	−	0.637717i	$-0.779879\pi$
$307$	−26.9925	−1.54054	−0.770271	+	0.637717i	$0.779879\pi$
$308$	10.6006	0.604023
$309$	15.7274	0.894702
$310$	−0.284876	−0.0161799
$311$	−19.5024	−1.10588	−0.552941	−	0.833221i	$-0.686494\pi$
$311$	−19.5024	−1.10588	−0.552941	+	0.833221i	$0.686494\pi$
$312$	7.60074	0.430307
$313$	12.3748	0.699463	0.349732	−	0.936850i	$-0.386273\pi$
$313$	12.3748	0.699463	0.349732	+	0.936850i	$0.386273\pi$
$314$	−8.51395	−0.480470
$315$	−58.2354	−3.28119
$316$	2.60491	0.146538
$317$	−10.6382	−0.597499	−0.298749	−	0.954332i	$-0.596569\pi$
$317$	−10.6382	−0.597499	−0.298749	+	0.954332i	$0.596569\pi$
$318$	−39.0156	−2.18789
$319$	−1.36732	−0.0765554
$320$	3.14140	0.175610
$321$	16.2282	0.905770
$322$	−5.73167	−0.319413
$323$	2.25368	0.125398
$324$	4.62127	0.256737
$325$	−12.7041	−0.704697
$326$	9.17904	0.508380
$327$	27.4712	1.51916
$328$	−4.35490	−0.240459
$329$	5.94395	0.327700
$330$	28.6928	1.57949
$331$	19.5927	1.07691	0.538455	−	0.842654i	$-0.319008\pi$
$331$	19.5927	1.07691	0.538455	+	0.842654i	$0.319008\pi$
$332$	1.12799	0.0619065
$333$	−5.48134	−0.300376
$334$	−6.57274	−0.359644
$335$	−7.01362	−0.383195
$336$	9.84631	0.537160
$337$	−27.7501	−1.51164	−0.755821	−	0.654778i	$-0.772762\pi$
$337$	−27.7501	−1.51164	−0.755821	+	0.654778i	$0.772762\pi$
$338$	−6.19046	−0.336717
$339$	43.1677	2.34455
$340$	1.15135	0.0624405
$341$	0.284371	0.0153995
$342$	33.7207	1.82340
$343$	8.69614	0.469547
$344$	−12.9033	−0.695698
$345$	−15.5140	−0.835248
$346$	−17.1138	−0.920041
$347$	−17.8851	−0.960120	−0.480060	−	0.877236i	$-0.659385\pi$
$347$	−17.8851	−0.960120	−0.480060	+	0.877236i	$0.659385\pi$
$348$	−1.27004	−0.0680811
$349$	23.4965	1.25774	0.628868	−	0.777512i	$-0.283518\pi$
$349$	23.4965	1.25774	0.628868	+	0.777512i	$0.283518\pi$
$350$	−16.4574	−0.879686
$351$	18.8793	1.00770
$352$	−3.13583	−0.167140
$353$	−23.8324	−1.26847	−0.634234	−	0.773141i	$-0.718684\pi$
$353$	−23.8324	−1.26847	−0.634234	+	0.773141i	$0.718684\pi$
$354$	14.0940	0.749088
$355$	−44.6662	−2.37064
$356$	8.46529	0.448660
$357$	3.60875	0.190995
$358$	17.8257	0.942118
$359$	−3.74493	−0.197650	−0.0988248	−	0.995105i	$-0.531508\pi$
$359$	−3.74493	−0.197650	−0.0988248	+	0.995105i	$0.531508\pi$
$360$	17.2270	0.907945
$361$	18.8109	0.990048
$362$	−4.36222	−0.229273
$363$	3.39790	0.178344
$364$	8.82135	0.462364
$365$	14.1468	0.740480
$366$	14.9365	0.780745
$367$	2.27771	0.118896	0.0594478	−	0.998231i	$-0.481066\pi$
$367$	2.27771	0.118896	0.0594478	+	0.998231i	$0.481066\pi$
$368$	1.69553	0.0883855
$369$	−23.8817	−1.24323
$370$	−3.13995	−0.163238
$371$	−45.2812	−2.35088
$372$	0.264137	0.0136949
$373$	18.4091	0.953185	0.476593	−	0.879124i	$-0.341872\pi$
$373$	18.4091	0.953185	0.476593	+	0.879124i	$0.341872\pi$
$374$	−1.14930	−0.0594291
$375$	1.20423	0.0621861
$376$	−1.75832	−0.0906786
$377$	−1.13783	−0.0586013
$378$	24.4570	1.25793
$379$	0.187951	0.00965440	0.00482720	−	0.999988i	$-0.498463\pi$
$379$	0.187951	0.00965440	0.00482720	+	0.999988i	$0.498463\pi$
$380$	19.3166	0.990923
$381$	28.5171	1.46098
$382$	−9.47458	−0.484762
$383$	−2.33557	−0.119342	−0.0596710	−	0.998218i	$-0.519005\pi$
$383$	−2.33557	−0.119342	−0.0596710	+	0.998218i	$0.519005\pi$
$384$	−2.91271	−0.148639
$385$	33.3006	1.69715
$386$	21.7127	1.10515
$387$	−70.7600	−3.59693
$388$	6.44708	0.327301
$389$	13.2974	0.674206	0.337103	−	0.941468i	$-0.390553\pi$
$389$	13.2974	0.674206	0.337103	+	0.941468i	$0.390553\pi$
$390$	23.8770	1.20906
$391$	0.621424	0.0314267
$392$	4.42753	0.223624
$393$	−2.47138	−0.124664
$394$	−0.0884114	−0.00445410
$395$	8.18306	0.411734
$396$	−17.1965	−0.864156
$397$	−13.5550	−0.680306	−0.340153	−	0.940370i	$-0.610479\pi$
$397$	−13.5550	−0.680306	−0.340153	+	0.940370i	$0.610479\pi$
$398$	8.34831	0.418463
$399$	60.5455	3.03107
$400$	4.86839	0.243420
$401$	−1.47507	−0.0736615	−0.0368307	−	0.999322i	$-0.511726\pi$
$401$	−1.47507	−0.0736615	−0.0368307	+	0.999322i	$0.511726\pi$
$402$	6.50304	0.324342
$403$	0.236642	0.0117880
$404$	−6.39827	−0.318326
$405$	14.5173	0.721368
$406$	−1.47399	−0.0731530
$407$	3.13438	0.155365
$408$	−1.06753	−0.0528506
$409$	38.3426	1.89592	0.947961	−	0.318388i	$-0.103141\pi$
$409$	38.3426	1.89592	0.947961	+	0.318388i	$0.103141\pi$
$410$	−13.6805	−0.675630
$411$	41.2841	2.03640
$412$	−5.39959	−0.266019
$413$	16.3574	0.804894
$414$	9.29806	0.456975
$415$	3.54347	0.173942
$416$	−2.60951	−0.127942
$417$	12.6418	0.619070
$418$	−19.2824	−0.943133
$419$	6.74454	0.329493	0.164746	−	0.986336i	$-0.447319\pi$
$419$	6.74454	0.329493	0.164746	+	0.986336i	$0.447319\pi$
$420$	30.9312	1.50929
$421$	21.6910	1.05715	0.528577	−	0.848885i	$-0.322726\pi$
$421$	21.6910	1.05715	0.528577	+	0.848885i	$0.322726\pi$
$422$	−8.66197	−0.421658
$423$	−9.64243	−0.468831
$424$	13.3950	0.650517
$425$	1.78430	0.0865513
$426$	41.4146	2.00654
$427$	17.3352	0.838909
$428$	−5.57151	−0.269309
$429$	−23.8346	−1.15075
$430$	−40.5344	−1.95474
$431$	4.90220	0.236131	0.118065	−	0.993006i	$-0.462331\pi$
$431$	4.90220	0.236131	0.118065	+	0.993006i	$0.462331\pi$
$432$	−7.23481	−0.348085
$433$	−17.5439	−0.843106	−0.421553	−	0.906804i	$-0.638515\pi$
$433$	−17.5439	−0.843106	−0.421553	+	0.906804i	$0.638515\pi$
$434$	0.306555	0.0147151
$435$	−3.98969	−0.191291
$436$	−9.43150	−0.451687
$437$	10.4259	0.498738
$438$	−13.1170	−0.626753
$439$	37.8919	1.80848	0.904242	−	0.427021i	$-0.140437\pi$
$439$	37.8919	1.80848	0.904242	+	0.427021i	$0.140437\pi$
$440$	−9.85089	−0.469623
$441$	24.2800	1.15619
$442$	−0.956405	−0.0454915
$443$	4.34345	0.206364	0.103182	−	0.994663i	$-0.467098\pi$
$443$	4.34345	0.206364	0.103182	+	0.994663i	$0.467098\pi$
$444$	2.91136	0.138167
$445$	26.5929	1.26062
$446$	−28.0460	−1.32802
$447$	−40.5810	−1.91941
$448$	−3.38046	−0.159712
$449$	3.62944	0.171284	0.0856420	−	0.996326i	$-0.472706\pi$
$449$	3.62944	0.171284	0.0856420	+	0.996326i	$0.472706\pi$
$450$	26.6976	1.25854
$451$	13.6562	0.643046
$452$	−14.8205	−0.697096
$453$	−22.7401	−1.06842
$454$	−18.0222	−0.845824
$455$	27.7114	1.29913
$456$	−17.9104	−0.838732
$457$	31.1170	1.45559	0.727796	−	0.685793i	$-0.240545\pi$
$457$	31.1170	1.45559	0.727796	+	0.685793i	$0.240545\pi$
$458$	−12.9895	−0.606958
$459$	−2.65161	−0.123767
$460$	5.32633	0.248341
$461$	17.6275	0.820995	0.410497	−	0.911862i	$-0.365355\pi$
$461$	17.6275	0.820995	0.410497	+	0.911862i	$0.365355\pi$
$462$	−30.8763	−1.43650
$463$	29.5658	1.37404	0.687020	−	0.726638i	$-0.258918\pi$
$463$	29.5658	1.37404	0.687020	+	0.726638i	$0.258918\pi$
$464$	0.436032	0.0202423
$465$	0.829760	0.0384792
$466$	3.69359	0.171102
$467$	9.75296	0.451313	0.225657	−	0.974207i	$-0.427547\pi$
$467$	9.75296	0.451313	0.225657	+	0.974207i	$0.427547\pi$
$468$	−14.3102	−0.661491
$469$	7.54737	0.348505
$470$	−5.52360	−0.254785
$471$	24.7987	1.14266
$472$	−4.83880	−0.222724
$473$	40.4625	1.86047
$474$	−7.58734	−0.348498
$475$	29.9360	1.37356
$476$	−1.23897	−0.0567879
$477$	73.4563	3.36334
$478$	26.1741	1.19717
$479$	19.3596	0.884561	0.442281	−	0.896877i	$-0.354170\pi$
$479$	19.3596	0.884561	0.442281	+	0.896877i	$0.354170\pi$
$480$	−9.14998	−0.417638
$481$	2.60830	0.118928
$482$	10.2496	0.466856
$483$	16.6947	0.759634
$484$	−1.16658	−0.0530263
$485$	20.2529	0.919635
$486$	8.24401	0.373956
$487$	−10.0379	−0.454862	−0.227431	−	0.973794i	$-0.573033\pi$
$487$	−10.0379	−0.454862	−0.227431	+	0.973794i	$0.573033\pi$
$488$	−5.12806	−0.232136
$489$	−26.7359	−1.20904
$490$	13.9086	0.628328
$491$	1.11320	0.0502380	0.0251190	−	0.999684i	$-0.492004\pi$
$491$	1.11320	0.0502380	0.0251190	+	0.999684i	$0.492004\pi$
$492$	12.6845	0.571864
$493$	0.159809	0.00719744
$494$	−16.0460	−0.721945
$495$	−54.0211	−2.42807
$496$	−0.0906844	−0.00407185
$497$	48.0653	2.15602
$498$	−3.28551	−0.147227
$499$	14.4178	0.645428	0.322714	−	0.946497i	$-0.395405\pi$
$499$	14.4178	0.645428	0.322714	+	0.946497i	$0.395405\pi$
$500$	−0.413439	−0.0184896
$501$	19.1445	0.855312
$502$	3.62024	0.161579
$503$	−31.2931	−1.39529	−0.697645	−	0.716444i	$-0.745769\pi$
$503$	−31.2931	−1.39529	−0.697645	+	0.716444i	$0.745769\pi$
$504$	−18.5380	−0.825750
$505$	−20.0995	−0.894417
$506$	−5.31688	−0.236364
$507$	18.0310	0.800785
$508$	−9.79058	−0.434387
$509$	−17.9500	−0.795622	−0.397811	−	0.917467i	$-0.630230\pi$
$509$	−17.9500	−0.795622	−0.397811	+	0.917467i	$0.630230\pi$
$510$	−3.35354	−0.148497
$511$	−15.2234	−0.673445
$512$	1.00000	0.0441942
$513$	−44.4873	−1.96416
$514$	−29.7972	−1.31430
$515$	−16.9623	−0.747446
$516$	37.5835	1.65452
$517$	5.51380	0.242497
$518$	3.37890	0.148460
$519$	49.8474	2.18806
$520$	−8.19751	−0.359485
$521$	−8.55558	−0.374827	−0.187413	−	0.982281i	$-0.560010\pi$
$521$	−8.55558	−0.374827	−0.187413	+	0.982281i	$0.560010\pi$
$522$	2.39115	0.104658
$523$	26.6700	1.16620	0.583098	−	0.812402i	$-0.301840\pi$
$523$	26.6700	1.16620	0.583098	+	0.812402i	$0.301840\pi$
$524$	0.848480	0.0370660
$525$	47.9357	2.09208
$526$	−17.0765	−0.744569
$527$	−0.0332365	−0.00144780
$528$	9.13376	0.397496
$529$	−20.1252	−0.875008
$530$	42.0789	1.82779
$531$	−26.5354	−1.15154
$532$	−20.7867	−0.901216
$533$	11.3641	0.492236
$534$	−24.6569	−1.06701
$535$	−17.5023	−0.756692
$536$	−2.23264	−0.0964355
$537$	−51.9211	−2.24056
$538$	−4.94157	−0.213046
$539$	−13.8840	−0.598025
$540$	−22.7274	−0.978033
$541$	39.4034	1.69408	0.847042	−	0.531525i	$-0.178381\pi$
$541$	39.4034	1.69408	0.847042	+	0.531525i	$0.178381\pi$
$542$	9.74561	0.418610
$543$	12.7059	0.545262
$544$	0.366508	0.0157139
$545$	−29.6281	−1.26913
$546$	−25.6940	−1.09960
$547$	24.7568	1.05852	0.529261	−	0.848459i	$-0.322469\pi$
$547$	24.7568	1.05852	0.529261	+	0.848459i	$0.322469\pi$
$548$	−14.1738	−0.605474
$549$	−28.1216	−1.20020
$550$	−15.2664	−0.650963
$551$	2.68119	0.114222
$552$	−4.93858	−0.210200
$553$	−8.80580	−0.374460
$554$	−22.3545	−0.949752
$555$	9.14576	0.388216
$556$	−4.34021	−0.184066
$557$	39.8024	1.68648	0.843241	−	0.537535i	$-0.180644\pi$
$557$	39.8024	1.68648	0.843241	+	0.537535i	$0.180644\pi$
$558$	−0.497302	−0.0210525
$559$	33.6712	1.42414
$560$	−10.6194	−0.448751
$561$	3.34759	0.141335
$562$	−6.39814	−0.269889
$563$	36.2981	1.52978	0.764891	−	0.644159i	$-0.222793\pi$
$563$	36.2981	1.52978	0.764891	+	0.644159i	$0.222793\pi$
$564$	5.12149	0.215653
$565$	−46.5570	−1.95867
$566$	−25.4747	−1.07078
$567$	−15.6220	−0.656064
$568$	−14.2186	−0.596598
$569$	−19.0314	−0.797839	−0.398920	−	0.916986i	$-0.630615\pi$
$569$	−19.0314	−0.797839	−0.398920	+	0.916986i	$0.630615\pi$
$570$	−56.2638	−2.35663
$571$	−8.29239	−0.347026	−0.173513	−	0.984832i	$-0.555512\pi$
$571$	−8.29239	−0.347026	−0.173513	+	0.984832i	$0.555512\pi$
$572$	8.18298	0.342147
$573$	27.5967	1.15287
$574$	14.7216	0.614466
$575$	8.25449	0.344236
$576$	5.48388	0.228495
$577$	24.2176	1.00819	0.504095	−	0.863648i	$-0.331826\pi$
$577$	24.2176	1.00819	0.504095	+	0.863648i	$0.331826\pi$
$578$	−16.8657	−0.701519
$579$	−63.2427	−2.62828
$580$	1.36975	0.0568759
$581$	−3.81313	−0.158195
$582$	−18.7785	−0.778393
$583$	−42.0043	−1.73964
$584$	4.50336	0.186350
$585$	−44.9541	−1.85863
$586$	−6.78388	−0.280239
$587$	−29.3637	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$587$	−29.3637	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$588$	−12.8961	−0.531827
$589$	−0.557623	−0.0229765
$590$	−15.2006	−0.625799
$591$	0.257517	0.0105928
$592$	−0.999538	−0.0410808
$593$	3.81216	0.156547	0.0782733	−	0.996932i	$-0.475059\pi$
$593$	3.81216	0.156547	0.0782733	+	0.996932i	$0.475059\pi$
$594$	22.6871	0.930864
$595$	−3.89208	−0.159560
$596$	13.9324	0.570693
$597$	−24.3162	−0.995196
$598$	−4.42450	−0.180931
$599$	5.07049	0.207174	0.103587	−	0.994620i	$-0.466968\pi$
$599$	5.07049	0.207174	0.103587	+	0.994620i	$0.466968\pi$
$600$	−14.1802	−0.578904
$601$	33.2800	1.35752	0.678759	−	0.734361i	$-0.262518\pi$
$601$	33.2800	1.35752	0.678759	+	0.734361i	$0.262518\pi$
$602$	43.6191	1.77778
$603$	−12.2435	−0.498595
$604$	7.80718	0.317670
$605$	−3.66469	−0.148991
$606$	18.6363	0.757048
$607$	−42.3901	−1.72056	−0.860281	−	0.509819i	$-0.829712\pi$
$607$	−42.3901	−1.72056	−0.860281	+	0.509819i	$0.829712\pi$
$608$	6.14906	0.249377
$609$	4.29331	0.173974
$610$	−16.1093	−0.652246
$611$	4.58836	0.185625
$612$	2.00988	0.0812447
$613$	−41.9327	−1.69365	−0.846823	−	0.531875i	$-0.821488\pi$
$613$	−41.9327	−1.69365	−0.846823	+	0.531875i	$0.821488\pi$
$614$	−26.9925	−1.08933
$615$	39.8472	1.60680
$616$	10.6006	0.427108
$617$	−26.6810	−1.07413	−0.537067	−	0.843539i	$-0.680468\pi$
$617$	−26.6810	−1.07413	−0.537067	+	0.843539i	$0.680468\pi$
$618$	15.7274	0.632650
$619$	20.6104	0.828403	0.414202	−	0.910185i	$-0.364061\pi$
$619$	20.6104	0.828403	0.414202	+	0.910185i	$0.364061\pi$
$620$	−0.284876	−0.0114409
$621$	−12.2668	−0.492250
$622$	−19.5024	−0.781976
$623$	−28.6166	−1.14650
$624$	7.60074	0.304273
$625$	−25.6407	−1.02563
$626$	12.3748	0.494595
$627$	56.1640	2.24297
$628$	−8.51395	−0.339743
$629$	−0.366338	−0.0146069
$630$	−58.2354	−2.32015
$631$	3.14342	0.125138	0.0625689	−	0.998041i	$-0.480071\pi$
$631$	3.14342	0.125138	0.0625689	+	0.998041i	$0.480071\pi$
$632$	2.60491	0.103618
$633$	25.2298	1.00279
$634$	−10.6382	−0.422495
$635$	−30.7561	−1.22052
$636$	−39.0156	−1.54707
$637$	−11.5537	−0.457774
$638$	−1.36732	−0.0541328
$639$	−77.9729	−3.08456
$640$	3.14140	0.124175
$641$	−6.84336	−0.270297	−0.135148	−	0.990825i	$-0.543151\pi$
$641$	−6.84336	−0.270297	−0.135148	+	0.990825i	$0.543151\pi$
$642$	16.2282	0.640476
$643$	−13.6996	−0.540259	−0.270130	−	0.962824i	$-0.587067\pi$
$643$	−13.6996	−0.540259	−0.270130	+	0.962824i	$0.587067\pi$
$644$	−5.73167	−0.225859
$645$	118.065	4.64880
$646$	2.25368	0.0886697
$647$	22.7183	0.893148	0.446574	−	0.894747i	$-0.352644\pi$
$647$	22.7183	0.893148	0.446574	+	0.894747i	$0.352644\pi$
$648$	4.62127	0.181541
$649$	15.1736	0.595618
$650$	−12.7041	−0.498296
$651$	−0.892906	−0.0349957
$652$	9.17904	0.359479
$653$	4.13841	0.161949	0.0809743	−	0.996716i	$-0.474197\pi$
$653$	4.13841	0.161949	0.0809743	+	0.996716i	$0.474197\pi$
$654$	27.4712	1.07421
$655$	2.66541	0.104146
$656$	−4.35490	−0.170030
$657$	24.6959	0.963477
$658$	5.94395	0.231719
$659$	17.8843	0.696673	0.348337	−	0.937370i	$-0.386747\pi$
$659$	17.8843	0.696673	0.348337	+	0.937370i	$0.386747\pi$
$660$	28.6928	1.11686
$661$	−19.9676	−0.776648	−0.388324	−	0.921523i	$-0.626946\pi$
$661$	−19.9676	−0.776648	−0.388324	+	0.921523i	$0.626946\pi$
$662$	19.5927	0.761490
$663$	2.78573	0.108189
$664$	1.12799	0.0437745
$665$	−65.2992	−2.53219
$666$	−5.48134	−0.212398
$667$	0.739305	0.0286260
$668$	−6.57274	−0.254307
$669$	81.6899	3.15831
$670$	−7.01362	−0.270960
$671$	16.0807	0.620789
$672$	9.84631	0.379830
$673$	−5.14952	−0.198499	−0.0992496	−	0.995063i	$-0.531644\pi$
$673$	−5.14952	−0.198499	−0.0992496	+	0.995063i	$0.531644\pi$
$674$	−27.7501	−1.06889
$675$	−35.2219	−1.35569
$676$	−6.19046	−0.238095
$677$	2.93149	0.112666	0.0563331	−	0.998412i	$-0.482059\pi$
$677$	2.93149	0.112666	0.0563331	+	0.998412i	$0.482059\pi$
$678$	43.1677	1.65785
$679$	−21.7941	−0.836382
$680$	1.15135	0.0441521
$681$	52.4934	2.01155
$682$	0.284371	0.0108891
$683$	29.0550	1.11176	0.555880	−	0.831263i	$-0.312381\pi$
$683$	29.0550	1.11176	0.555880	+	0.831263i	$0.312381\pi$
$684$	33.7207	1.28934
$685$	−44.5255	−1.70123
$686$	8.69614	0.332020
$687$	37.8346	1.44348
$688$	−12.9033	−0.491933
$689$	−34.9543	−1.33165
$690$	−15.5140	−0.590610
$691$	−14.1943	−0.539976	−0.269988	−	0.962864i	$-0.587020\pi$
$691$	−14.1943	−0.539976	−0.269988	+	0.962864i	$0.587020\pi$
$692$	−17.1138	−0.650567
$693$	58.1321	2.20826
$694$	−17.8851	−0.678907
$695$	−13.6343	−0.517180
$696$	−1.27004	−0.0481406
$697$	−1.59610	−0.0604567
$698$	23.4965	0.889354
$699$	−10.7583	−0.406918
$700$	−16.4574	−0.622032
$701$	−0.746384	−0.0281905	−0.0140953	−	0.999901i	$-0.504487\pi$
$701$	−0.746384	−0.0281905	−0.0140953	+	0.999901i	$0.504487\pi$
$702$	18.8793	0.712554
$703$	−6.14622	−0.231809
$704$	−3.13583	−0.118186
$705$	16.0886	0.605933
$706$	−23.8324	−0.896942
$707$	21.6291	0.813446
$708$	14.0940	0.529685
$709$	24.5997	0.923860	0.461930	−	0.886916i	$-0.347157\pi$
$709$	24.5997	0.923860	0.461930	+	0.886916i	$0.347157\pi$
$710$	−44.6662	−1.67629
$711$	14.2850	0.535729
$712$	8.46529	0.317250
$713$	−0.153758	−0.00575828
$714$	3.60875	0.135054
$715$	25.7060	0.961350
$716$	17.8257	0.666178
$717$	−76.2374	−2.84714
$718$	−3.74493	−0.139759
$719$	−16.0114	−0.597123	−0.298561	−	0.954390i	$-0.596507\pi$
$719$	−16.0114	−0.597123	−0.298561	+	0.954390i	$0.596507\pi$
$720$	17.2270	0.642014
$721$	18.2531	0.679781
$722$	18.8109	0.700069
$723$	−29.8541	−1.11028
$724$	−4.36222	−0.162121
$725$	2.12278	0.0788379
$726$	3.39790	0.126108
$727$	18.0237	0.668463	0.334232	−	0.942491i	$-0.391523\pi$
$727$	18.0237	0.668463	0.334232	+	0.942491i	$0.391523\pi$
$728$	8.82135	0.326941
$729$	−37.8762	−1.40282
$730$	14.1468	0.523598
$731$	−4.72915	−0.174914
$732$	14.9365	0.552070
$733$	26.4852	0.978255	0.489127	−	0.872212i	$-0.337315\pi$
$733$	26.4852	0.978255	0.489127	+	0.872212i	$0.337315\pi$
$734$	2.27771	0.0840719
$735$	−40.5118	−1.49430
$736$	1.69553	0.0624980
$737$	7.00119	0.257892
$738$	−23.8817	−0.879098
$739$	48.8775	1.79799	0.898993	−	0.437962i	$-0.144300\pi$
$739$	48.8775	1.79799	0.898993	+	0.437962i	$0.144300\pi$
$740$	−3.13995	−0.115427
$741$	46.7374	1.71694
$742$	−45.2812	−1.66232
$743$	16.2789	0.597214	0.298607	−	0.954376i	$-0.403478\pi$
$743$	16.2789	0.597214	0.298607	+	0.954376i	$0.403478\pi$
$744$	0.264137	0.00968374
$745$	43.7672	1.60351
$746$	18.4091	0.674004
$747$	6.18576	0.226325
$748$	−1.14930	−0.0420227
$749$	18.8343	0.688190
$750$	1.20423	0.0439722
$751$	28.2556	1.03106	0.515530	−	0.856871i	$-0.327595\pi$
$751$	28.2556	1.03106	0.515530	+	0.856871i	$0.327595\pi$
$752$	−1.75832	−0.0641195
$753$	−10.5447	−0.384270
$754$	−1.13783	−0.0414374
$755$	24.5255	0.892574
$756$	24.4570	0.889493
$757$	2.74658	0.0998261	0.0499130	−	0.998754i	$-0.484106\pi$
$757$	2.74658	0.0998261	0.0499130	+	0.998754i	$0.484106\pi$
$758$	0.187951	0.00682669
$759$	15.4865	0.562126
$760$	19.3166	0.700688
$761$	12.3862	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$761$	12.3862	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$762$	28.5171	1.03307
$763$	31.8828	1.15424
$764$	−9.47458	−0.342778
$765$	6.31384	0.228277
$766$	−2.33557	−0.0843876
$767$	12.6269	0.455931
$768$	−2.91271	−0.105103
$769$	19.1025	0.688852	0.344426	−	0.938813i	$-0.388073\pi$
$769$	19.1025	0.688852	0.344426	+	0.938813i	$0.388073\pi$
$770$	33.3006	1.20007
$771$	86.7905	3.12568
$772$	21.7127	0.781456
$773$	−15.8590	−0.570408	−0.285204	−	0.958467i	$-0.592061\pi$
$773$	−15.8590	−0.570408	−0.285204	+	0.958467i	$0.592061\pi$
$774$	−70.7600	−2.54342
$775$	−0.441487	−0.0158587
$776$	6.44708	0.231437
$777$	−9.84176	−0.353071
$778$	13.2974	0.476736
$779$	−26.7785	−0.959440
$780$	23.8770	0.854933
$781$	44.5870	1.59545
$782$	0.621424	0.0222221
$783$	−3.15461	−0.112737
$784$	4.42753	0.158126
$785$	−26.7457	−0.954595
$786$	−2.47138	−0.0881511
$787$	38.2525	1.36356	0.681778	−	0.731559i	$-0.261207\pi$
$787$	38.2525	1.36356	0.681778	+	0.731559i	$0.261207\pi$
$788$	−0.0884114	−0.00314953
$789$	49.7388	1.77075
$790$	8.18306	0.291140
$791$	50.1001	1.78135
$792$	−17.1965	−0.611051
$793$	13.3817	0.475199
$794$	−13.5550	−0.481049
$795$	−122.564	−4.34689
$796$	8.34831	0.295898
$797$	35.8075	1.26837	0.634184	−	0.773182i	$-0.281336\pi$
$797$	35.8075	1.26837	0.634184	+	0.773182i	$0.281336\pi$
$798$	60.5455	2.14329
$799$	−0.644439	−0.0227986
$800$	4.86839	0.172124
$801$	46.4226	1.64026
$802$	−1.47507	−0.0520865
$803$	−14.1218	−0.498346
$804$	6.50304	0.229345
$805$	−18.0055	−0.634609
$806$	0.236642	0.00833535
$807$	14.3934	0.506670
$808$	−6.39827	−0.225090
$809$	−1.23816	−0.0435316	−0.0217658	−	0.999763i	$-0.506929\pi$
$809$	−1.23816	−0.0435316	−0.0217658	+	0.999763i	$0.506929\pi$
$810$	14.5173	0.510084
$811$	7.05344	0.247680	0.123840	−	0.992302i	$-0.460479\pi$
$811$	7.05344	0.247680	0.123840	+	0.992302i	$0.460479\pi$
$812$	−1.47399	−0.0517270
$813$	−28.3861	−0.995545
$814$	3.13438	0.109860
$815$	28.8350	1.01005
$816$	−1.06753	−0.0373710
$817$	−79.3430	−2.77586
$818$	38.3426	1.34062
$819$	48.3752	1.69037
$820$	−13.6805	−0.477743
$821$	3.50488	0.122321	0.0611605	−	0.998128i	$-0.480520\pi$
$821$	3.50488	0.122321	0.0611605	+	0.998128i	$0.480520\pi$
$822$	41.2841	1.43995
$823$	−20.6806	−0.720879	−0.360439	−	0.932783i	$-0.617373\pi$
$823$	−20.6806	−0.720879	−0.360439	+	0.932783i	$0.617373\pi$
$824$	−5.39959	−0.188103
$825$	44.4667	1.54813
$826$	16.3574	0.569146
$827$	−31.9818	−1.11212	−0.556059	−	0.831143i	$-0.687687\pi$
$827$	−31.9818	−1.11212	−0.556059	+	0.831143i	$0.687687\pi$
$828$	9.29806	0.323130
$829$	−39.5658	−1.37418	−0.687089	−	0.726574i	$-0.741112\pi$
$829$	−39.5658	−1.37418	−0.687089	+	0.726574i	$0.741112\pi$
$830$	3.54347	0.122996
$831$	65.1122	2.25872
$832$	−2.60951	−0.0904685
$833$	1.62272	0.0562240
$834$	12.6418	0.437749
$835$	−20.6476	−0.714539
$836$	−19.2824	−0.666895
$837$	0.656084	0.0226776
$838$	6.74454	0.232986
$839$	−21.2172	−0.732499	−0.366249	−	0.930517i	$-0.619358\pi$
$839$	−21.2172	−0.732499	−0.366249	+	0.930517i	$0.619358\pi$
$840$	30.9312	1.06723
$841$	−28.8099	−0.993444
$842$	21.6910	0.747521
$843$	18.6359	0.641856
$844$	−8.66197	−0.298157
$845$	−19.4467	−0.668987
$846$	−9.64243	−0.331514
$847$	3.94358	0.135503
$848$	13.3950	0.459985
$849$	74.2004	2.54655
$850$	1.78430	0.0612010
$851$	−1.69474	−0.0580951
$852$	41.4146	1.41884
$853$	8.41989	0.288292	0.144146	−	0.989556i	$-0.453957\pi$
$853$	8.41989	0.288292	0.144146	+	0.989556i	$0.453957\pi$
$854$	17.3352	0.593199
$855$	105.930	3.62273
$856$	−5.57151	−0.190430
$857$	−39.9293	−1.36396	−0.681980	−	0.731371i	$-0.738881\pi$
$857$	−39.9293	−1.36396	−0.681980	+	0.731371i	$0.738881\pi$
$858$	−23.8346	−0.813701
$859$	−18.0059	−0.614355	−0.307177	−	0.951652i	$-0.599384\pi$
$859$	−18.0059	−0.614355	−0.307177	+	0.951652i	$0.599384\pi$
$860$	−40.5344	−1.38221
$861$	−42.8796	−1.46133
$862$	4.90220	0.166970
$863$	21.1342	0.719416	0.359708	−	0.933065i	$-0.382876\pi$
$863$	21.1342	0.719416	0.359708	+	0.933065i	$0.382876\pi$
$864$	−7.23481	−0.246133
$865$	−53.7611	−1.82793
$866$	−17.5439	−0.596166
$867$	49.1248	1.66837
$868$	0.306555	0.0104052
$869$	−8.16855	−0.277099
$870$	−3.98969	−0.135263
$871$	5.82610	0.197410
$872$	−9.43150	−0.319391
$873$	35.3550	1.19659
$874$	10.4259	0.352661
$875$	1.39762	0.0472480
$876$	−13.1170	−0.443182
$877$	−1.46740	−0.0495505	−0.0247753	−	0.999693i	$-0.507887\pi$
$877$	−1.46740	−0.0495505	−0.0247753	+	0.999693i	$0.507887\pi$
$878$	37.8919	1.27879
$879$	19.7595	0.666470
$880$	−9.85089	−0.332073
$881$	−43.1519	−1.45382	−0.726912	−	0.686731i	$-0.759045\pi$
$881$	−43.1519	−1.45382	−0.726912	+	0.686731i	$0.759045\pi$
$882$	24.2800	0.817551
$883$	−12.8371	−0.432002	−0.216001	−	0.976393i	$-0.569301\pi$
$883$	−12.8371	−0.432002	−0.216001	+	0.976393i	$0.569301\pi$
$884$	−0.956405	−0.0321674
$885$	44.2749	1.48829
$886$	4.34345	0.145921
$887$	54.5982	1.83323	0.916614	−	0.399774i	$-0.130911\pi$
$887$	54.5982	1.83323	0.916614	+	0.399774i	$0.130911\pi$
$888$	2.91136	0.0976990
$889$	33.0967	1.11003
$890$	26.5929	0.891395
$891$	−14.4915	−0.485484
$892$	−28.0460	−0.939050
$893$	−10.8120	−0.361811
$894$	−40.5810	−1.35723
$895$	55.9977	1.87180
$896$	−3.38046	−0.112933
$897$	12.8873	0.430293
$898$	3.62944	0.121116
$899$	−0.0395413	−0.00131878
$900$	26.6976	0.889922
$901$	4.90936	0.163554
$902$	13.6562	0.454702
$903$	−127.050	−4.22795
$904$	−14.8205	−0.492922
$905$	−13.7035	−0.455519
$906$	−22.7401	−0.755488
$907$	−28.5389	−0.947617	−0.473809	−	0.880628i	$-0.657121\pi$
$907$	−28.5389	−0.947617	−0.473809	+	0.880628i	$0.657121\pi$
$908$	−18.0222	−0.598088
$909$	−35.0873	−1.16377
$910$	27.7114	0.918624
$911$	4.44421	0.147243	0.0736216	−	0.997286i	$-0.476544\pi$
$911$	4.44421	0.147243	0.0736216	+	0.997286i	$0.476544\pi$
$912$	−17.9104	−0.593073
$913$	−3.53718	−0.117064
$914$	31.1170	1.02926
$915$	46.9216	1.55118
$916$	−12.9895	−0.429184
$917$	−2.86826	−0.0947181
$918$	−2.65161	−0.0875163
$919$	1.48667	0.0490407	0.0245204	−	0.999699i	$-0.492194\pi$
$919$	1.48667	0.0490407	0.0245204	+	0.999699i	$0.492194\pi$
$920$	5.32633	0.175604
$921$	78.6212	2.59066
$922$	17.6275	0.580531
$923$	37.1035	1.22128
$924$	−30.8763	−1.01576
$925$	−4.86614	−0.159998
$926$	29.5658	0.971593
$927$	−29.6107	−0.972542
$928$	0.436032	0.0143135
$929$	−32.0819	−1.05257	−0.526287	−	0.850307i	$-0.676416\pi$
$929$	−32.0819	−1.05257	−0.526287	+	0.850307i	$0.676416\pi$
$930$	0.829760	0.0272089
$931$	27.2251	0.892268
$932$	3.69359	0.120987
$933$	56.8049	1.85971
$934$	9.75296	0.319127
$935$	−3.61043	−0.118074
$936$	−14.3102	−0.467744
$937$	35.7946	1.16936	0.584679	−	0.811265i	$-0.301221\pi$
$937$	35.7946	1.16936	0.584679	+	0.811265i	$0.301221\pi$
$938$	7.54737	0.246430
$939$	−36.0441	−1.17625
$940$	−5.52360	−0.180160
$941$	−49.9425	−1.62808	−0.814039	−	0.580810i	$-0.802736\pi$
$941$	−49.9425	−1.62808	−0.814039	+	0.580810i	$0.802736\pi$
$942$	24.7987	0.807984
$943$	−7.38385	−0.240451
$944$	−4.83880	−0.157489
$945$	76.8292	2.49925
$946$	40.4625	1.31555
$947$	14.9287	0.485117	0.242559	−	0.970137i	$-0.422013\pi$
$947$	14.9287	0.485117	0.242559	+	0.970137i	$0.422013\pi$
$948$	−7.58734	−0.246425
$949$	−11.7516	−0.381472
$950$	29.9360	0.971252
$951$	30.9859	1.00479
$952$	−1.23897	−0.0401551
$953$	−0.0492779	−0.00159627	−0.000798134	−	1.00000i	$-0.500254\pi$
$953$	−0.0492779	−0.00159627	−0.000798134		1.00000i	$0.500254\pi$
$954$	73.4563	2.37824
$955$	−29.7634	−0.963123
$956$	26.1741	0.846529
$957$	3.98261	0.128740
$958$	19.3596	0.625479
$959$	47.9140	1.54722
$960$	−9.14998	−0.295314
$961$	−30.9918	−0.999735
$962$	2.60830	0.0840951
$963$	−30.5535	−0.984572
$964$	10.2496	0.330117
$965$	68.2081	2.19570
$966$	16.6947	0.537143
$967$	−32.1631	−1.03430	−0.517148	−	0.855896i	$-0.673006\pi$
$967$	−32.1631	−1.03430	−0.517148	+	0.855896i	$0.673006\pi$
$968$	−1.16658	−0.0374953
$969$	−6.56430	−0.210876
$970$	20.2529	0.650280
$971$	−4.26100	−0.136742	−0.0683710	−	0.997660i	$-0.521780\pi$
$971$	−4.26100	−0.136742	−0.0683710	+	0.997660i	$0.521780\pi$
$972$	8.24401	0.264427
$973$	14.6719	0.470360
$974$	−10.0379	−0.321636
$975$	37.0034	1.18506
$976$	−5.12806	−0.164145
$977$	59.5141	1.90402	0.952012	−	0.306062i	$-0.0990114\pi$
$977$	59.5141	1.90402	0.952012	+	0.306062i	$0.0990114\pi$
$978$	−26.7359	−0.854919
$979$	−26.5457	−0.848405
$980$	13.9086	0.444295
$981$	−51.7212	−1.65133
$982$	1.11320	0.0355236
$983$	−61.9502	−1.97591	−0.987953	−	0.154755i	$-0.950541\pi$
$983$	−61.9502	−1.97591	−0.987953	+	0.154755i	$0.950541\pi$
$984$	12.6845	0.404369
$985$	−0.277736	−0.00884939
$986$	0.159809	0.00508936
$987$	−17.3130	−0.551079
$988$	−16.0460	−0.510492
$989$	−21.8779	−0.695676
$990$	−54.0211	−1.71690
$991$	37.0869	1.17811	0.589053	−	0.808095i	$-0.299501\pi$
$991$	37.0869	1.17811	0.589053	+	0.808095i	$0.299501\pi$
$992$	−0.0906844	−0.00287923
$993$	−57.0677	−1.81099
$994$	48.0653	1.52454
$995$	26.2254	0.831401
$996$	−3.28551	−0.104105
$997$	−38.0548	−1.20521	−0.602603	−	0.798041i	$-0.705870\pi$
$997$	−38.0548	−1.20521	−0.602603	+	0.798041i	$0.705870\pi$
$998$	14.4178	0.456386
$999$	7.23147	0.228793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.c.1.4	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.c.1.4	✓	31	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.91271 q^{3} +1.00000 q^{4} +3.14140 q^{5} -2.91271 q^{6} -3.38046 q^{7} +1.00000 q^{8} +5.48388 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.91271 q^{3} +1.00000 q^{4} +3.14140 q^{5} -2.91271 q^{6} -3.38046 q^{7} +1.00000 q^{8} +5.48388 q^{9} +3.14140 q^{10} -3.13583 q^{11} -2.91271 q^{12} -2.60951 q^{13} -3.38046 q^{14} -9.14998 q^{15} +1.00000 q^{16} +0.366508 q^{17} +5.48388 q^{18} +6.14906 q^{19} +3.14140 q^{20} +9.84631 q^{21} -3.13583 q^{22} +1.69553 q^{23} -2.91271 q^{24} +4.86839 q^{25} -2.60951 q^{26} -7.23481 q^{27} -3.38046 q^{28} +0.436032 q^{29} -9.14998 q^{30} -0.0906844 q^{31} +1.00000 q^{32} +9.13376 q^{33} +0.366508 q^{34} -10.6194 q^{35} +5.48388 q^{36} -0.999538 q^{37} +6.14906 q^{38} +7.60074 q^{39} +3.14140 q^{40} -4.35490 q^{41} +9.84631 q^{42} -12.9033 q^{43} -3.13583 q^{44} +17.2270 q^{45} +1.69553 q^{46} -1.75832 q^{47} -2.91271 q^{48} +4.42753 q^{49} +4.86839 q^{50} -1.06753 q^{51} -2.60951 q^{52} +13.3950 q^{53} -7.23481 q^{54} -9.85089 q^{55} -3.38046 q^{56} -17.9104 q^{57} +0.436032 q^{58} -4.83880 q^{59} -9.14998 q^{60} -5.12806 q^{61} -0.0906844 q^{62} -18.5380 q^{63} +1.00000 q^{64} -8.19751 q^{65} +9.13376 q^{66} -2.23264 q^{67} +0.366508 q^{68} -4.93858 q^{69} -10.6194 q^{70} -14.2186 q^{71} +5.48388 q^{72} +4.50336 q^{73} -0.999538 q^{74} -14.1802 q^{75} +6.14906 q^{76} +10.6006 q^{77} +7.60074 q^{78} +2.60491 q^{79} +3.14140 q^{80} +4.62127 q^{81} -4.35490 q^{82} +1.12799 q^{83} +9.84631 q^{84} +1.15135 q^{85} -12.9033 q^{86} -1.27004 q^{87} -3.13583 q^{88} +8.46529 q^{89} +17.2270 q^{90} +8.82135 q^{91} +1.69553 q^{92} +0.264137 q^{93} -1.75832 q^{94} +19.3166 q^{95} -2.91271 q^{96} +6.44708 q^{97} +4.42753 q^{98} -17.1965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - q^99

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