LMFDB - Embedded newform 6046.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, 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("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.45102 q^{3} +1.00000 q^{4} +0.781132 q^{5} -2.45102 q^{6} -0.0575638 q^{7} +1.00000 q^{8} +3.00749 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.45102 q^{3} +1.00000 q^{4} +0.781132 q^{5} -2.45102 q^{6} -0.0575638 q^{7} +1.00000 q^{8} +3.00749 q^{9} +0.781132 q^{10} -1.75342 q^{11} -2.45102 q^{12} +1.53928 q^{13} -0.0575638 q^{14} -1.91457 q^{15} +1.00000 q^{16} -3.25701 q^{17} +3.00749 q^{18} +6.83443 q^{19} +0.781132 q^{20} +0.141090 q^{21} -1.75342 q^{22} -3.68648 q^{23} -2.45102 q^{24} -4.38983 q^{25} +1.53928 q^{26} -0.0183618 q^{27} -0.0575638 q^{28} -1.95159 q^{29} -1.91457 q^{30} -4.50185 q^{31} +1.00000 q^{32} +4.29768 q^{33} -3.25701 q^{34} -0.0449649 q^{35} +3.00749 q^{36} +6.57508 q^{37} +6.83443 q^{38} -3.77279 q^{39} +0.781132 q^{40} -1.05822 q^{41} +0.141090 q^{42} -1.77728 q^{43} -1.75342 q^{44} +2.34925 q^{45} -3.68648 q^{46} -7.13117 q^{47} -2.45102 q^{48} -6.99669 q^{49} -4.38983 q^{50} +7.98300 q^{51} +1.53928 q^{52} +7.45722 q^{53} -0.0183618 q^{54} -1.36966 q^{55} -0.0575638 q^{56} -16.7513 q^{57} -1.95159 q^{58} -12.8504 q^{59} -1.91457 q^{60} +11.7182 q^{61} -4.50185 q^{62} -0.173123 q^{63} +1.00000 q^{64} +1.20238 q^{65} +4.29768 q^{66} +11.2517 q^{67} -3.25701 q^{68} +9.03563 q^{69} -0.0449649 q^{70} -4.44201 q^{71} +3.00749 q^{72} -1.73802 q^{73} +6.57508 q^{74} +10.7596 q^{75} +6.83443 q^{76} +0.100934 q^{77} -3.77279 q^{78} +4.26198 q^{79} +0.781132 q^{80} -8.97747 q^{81} -1.05822 q^{82} -8.61139 q^{83} +0.141090 q^{84} -2.54416 q^{85} -1.77728 q^{86} +4.78339 q^{87} -1.75342 q^{88} +12.7128 q^{89} +2.34925 q^{90} -0.0886066 q^{91} -3.68648 q^{92} +11.0341 q^{93} -7.13117 q^{94} +5.33859 q^{95} -2.45102 q^{96} -0.351049 q^{97} -6.99669 q^{98} -5.27341 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) $ 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * 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
$3$	−2.45102	−1.41510	−0.707548	−	0.706665i	$-0.750199\pi$
$3$	−2.45102	−1.41510	−0.707548	+	0.706665i	$0.750199\pi$
$4$	1.00000	0.500000
$5$	0.781132	0.349333	0.174666	−	0.984628i	$-0.444115\pi$
$5$	0.781132	0.349333	0.174666	+	0.984628i	$0.444115\pi$
$6$	−2.45102	−1.00062
$7$	−0.0575638	−0.0217571	−0.0108785	−	0.999941i	$-0.503463\pi$
$7$	−0.0575638	−0.0217571	−0.0108785	+	0.999941i	$0.503463\pi$
$8$	1.00000	0.353553
$9$	3.00749	1.00250
$10$	0.781132	0.247016
$11$	−1.75342	−0.528677	−0.264339	−	0.964430i	$-0.585154\pi$
$11$	−1.75342	−0.528677	−0.264339	+	0.964430i	$0.585154\pi$
$12$	−2.45102	−0.707548
$13$	1.53928	0.426918	0.213459	−	0.976952i	$-0.431527\pi$
$13$	1.53928	0.426918	0.213459	+	0.976952i	$0.431527\pi$
$14$	−0.0575638	−0.0153846
$15$	−1.91457	−0.494340
$16$	1.00000	0.250000
$17$	−3.25701	−0.789942	−0.394971	−	0.918694i	$-0.629245\pi$
$17$	−3.25701	−0.789942	−0.394971	+	0.918694i	$0.629245\pi$
$18$	3.00749	0.708873
$19$	6.83443	1.56792	0.783962	−	0.620808i	$-0.213196\pi$
$19$	6.83443	1.56792	0.783962	+	0.620808i	$0.213196\pi$
$20$	0.781132	0.174666
$21$	0.141090	0.0307883
$22$	−1.75342	−0.373831
$23$	−3.68648	−0.768684	−0.384342	−	0.923191i	$-0.625572\pi$
$23$	−3.68648	−0.768684	−0.384342	+	0.923191i	$0.625572\pi$
$24$	−2.45102	−0.500312
$25$	−4.38983	−0.877966
$26$	1.53928	0.301877
$27$	−0.0183618	−0.00353372
$28$	−0.0575638	−0.0108785
$29$	−1.95159	−0.362402	−0.181201	−	0.983446i	$-0.557998\pi$
$29$	−1.95159	−0.362402	−0.181201	+	0.983446i	$0.557998\pi$
$30$	−1.91457	−0.349551
$31$	−4.50185	−0.808556	−0.404278	−	0.914636i	$-0.632477\pi$
$31$	−4.50185	−0.808556	−0.404278	+	0.914636i	$0.632477\pi$
$32$	1.00000	0.176777
$33$	4.29768	0.748129
$34$	−3.25701	−0.558573
$35$	−0.0449649	−0.00760046
$36$	3.00749	0.501249
$37$	6.57508	1.08094	0.540468	−	0.841365i	$-0.318247\pi$
$37$	6.57508	1.08094	0.540468	+	0.841365i	$0.318247\pi$
$38$	6.83443	1.10869
$39$	−3.77279	−0.604130
$40$	0.781132	0.123508
$41$	−1.05822	−0.165267	−0.0826334	−	0.996580i	$-0.526333\pi$
$41$	−1.05822	−0.165267	−0.0826334	+	0.996580i	$0.526333\pi$
$42$	0.141090	0.0217707
$43$	−1.77728	−0.271032	−0.135516	−	0.990775i	$-0.543269\pi$
$43$	−1.77728	−0.271032	−0.135516	+	0.990775i	$0.543269\pi$
$44$	−1.75342	−0.264339
$45$	2.34925	0.350205
$46$	−3.68648	−0.543542
$47$	−7.13117	−1.04019	−0.520094	−	0.854109i	$-0.674103\pi$
$47$	−7.13117	−1.04019	−0.520094	+	0.854109i	$0.674103\pi$
$48$	−2.45102	−0.353774
$49$	−6.99669	−0.999527
$50$	−4.38983	−0.620816
$51$	7.98300	1.11784
$52$	1.53928	0.213459
$53$	7.45722	1.02433	0.512164	−	0.858888i	$-0.328844\pi$
$53$	7.45722	1.02433	0.512164	+	0.858888i	$0.328844\pi$
$54$	−0.0183618	−0.00249872
$55$	−1.36966	−0.184684
$56$	−0.0575638	−0.00769229
$57$	−16.7513	−2.21876
$58$	−1.95159	−0.256257
$59$	−12.8504	−1.67298	−0.836490	−	0.547982i	$-0.815396\pi$
$59$	−12.8504	−1.67298	−0.836490	+	0.547982i	$0.815396\pi$
$60$	−1.91457	−0.247170
$61$	11.7182	1.50036	0.750179	−	0.661234i	$-0.229967\pi$
$61$	11.7182	1.50036	0.750179	+	0.661234i	$0.229967\pi$
$62$	−4.50185	−0.571736
$63$	−0.173123	−0.0218114
$64$	1.00000	0.125000
$65$	1.20238	0.149137
$66$	4.29768	0.529007
$67$	11.2517	1.37461	0.687306	−	0.726368i	$-0.258793\pi$
$67$	11.2517	1.37461	0.687306	+	0.726368i	$0.258793\pi$
$68$	−3.25701	−0.394971
$69$	9.03563	1.08776
$70$	−0.0449649	−0.00537434
$71$	−4.44201	−0.527170	−0.263585	−	0.964636i	$-0.584905\pi$
$71$	−4.44201	−0.527170	−0.263585	+	0.964636i	$0.584905\pi$
$72$	3.00749	0.354436
$73$	−1.73802	−0.203420	−0.101710	−	0.994814i	$-0.532431\pi$
$73$	−1.73802	−0.203420	−0.101710	+	0.994814i	$0.532431\pi$
$74$	6.57508	0.764337
$75$	10.7596	1.24241
$76$	6.83443	0.783962
$77$	0.100934	0.0115025
$78$	−3.77279	−0.427185
$79$	4.26198	0.479510	0.239755	−	0.970833i	$-0.422933\pi$
$79$	4.26198	0.479510	0.239755	+	0.970833i	$0.422933\pi$
$80$	0.781132	0.0873332
$81$	−8.97747	−0.997497
$82$	−1.05822	−0.116861
$83$	−8.61139	−0.945223	−0.472612	−	0.881271i	$-0.656689\pi$
$83$	−8.61139	−0.945223	−0.472612	+	0.881271i	$0.656689\pi$
$84$	0.141090	0.0153942
$85$	−2.54416	−0.275953
$86$	−1.77728	−0.191649
$87$	4.78339	0.512834
$88$	−1.75342	−0.186916
$89$	12.7128	1.34755	0.673775	−	0.738936i	$-0.264672\pi$
$89$	12.7128	1.34755	0.673775	+	0.738936i	$0.264672\pi$
$90$	2.34925	0.247633
$91$	−0.0886066	−0.00928849
$92$	−3.68648	−0.384342
$93$	11.0341	1.14418
$94$	−7.13117	−0.735524
$95$	5.33859	0.547728
$96$	−2.45102	−0.250156
$97$	−0.351049	−0.0356437	−0.0178218	−	0.999841i	$-0.505673\pi$
$97$	−0.351049	−0.0356437	−0.0178218	+	0.999841i	$0.505673\pi$
$98$	−6.99669	−0.706772
$99$	−5.27341	−0.529998
$100$	−4.38983	−0.438983
$101$	4.73321	0.470972	0.235486	−	0.971878i	$-0.424332\pi$
$101$	4.73321	0.470972	0.235486	+	0.971878i	$0.424332\pi$
$102$	7.98300	0.790435
$103$	−3.48850	−0.343732	−0.171866	−	0.985120i	$-0.554980\pi$
$103$	−3.48850	−0.343732	−0.171866	+	0.985120i	$0.554980\pi$
$104$	1.53928	0.150938
$105$	0.110210	0.0107554
$106$	7.45722	0.724309
$107$	−11.2744	−1.08993	−0.544967	−	0.838458i	$-0.683458\pi$
$107$	−11.2744	−1.08993	−0.544967	+	0.838458i	$0.683458\pi$
$108$	−0.0183618	−0.00176686
$109$	−6.72600	−0.644234	−0.322117	−	0.946700i	$-0.604394\pi$
$109$	−6.72600	−0.644234	−0.322117	+	0.946700i	$0.604394\pi$
$110$	−1.36966	−0.130592
$111$	−16.1156	−1.52963
$112$	−0.0575638	−0.00543927
$113$	−10.3856	−0.976992	−0.488496	−	0.872566i	$-0.662454\pi$
$113$	−10.3856	−0.976992	−0.488496	+	0.872566i	$0.662454\pi$
$114$	−16.7513	−1.56890
$115$	−2.87963	−0.268527
$116$	−1.95159	−0.181201
$117$	4.62936	0.427984
$118$	−12.8504	−1.18298
$119$	0.187486	0.0171868
$120$	−1.91457	−0.174775
$121$	−7.92550	−0.720500
$122$	11.7182	1.06091
$123$	2.59373	0.233868
$124$	−4.50185	−0.404278
$125$	−7.33470	−0.656036
$126$	−0.173123	−0.0154230
$127$	14.7640	1.31010	0.655049	−	0.755587i	$-0.272648\pi$
$127$	14.7640	1.31010	0.655049	+	0.755587i	$0.272648\pi$
$128$	1.00000	0.0883883
$129$	4.35614	0.383537
$130$	1.20238	0.105456
$131$	−18.0628	−1.57816	−0.789078	−	0.614293i	$-0.789441\pi$
$131$	−18.0628	−1.57816	−0.789078	+	0.614293i	$0.789441\pi$
$132$	4.29768	0.374065
$133$	−0.393416	−0.0341135
$134$	11.2517	0.971997
$135$	−0.0143430	−0.00123445
$136$	−3.25701	−0.279287
$137$	−12.4326	−1.06219	−0.531096	−	0.847311i	$-0.678220\pi$
$137$	−12.4326	−1.06219	−0.531096	+	0.847311i	$0.678220\pi$
$138$	9.03563	0.769164
$139$	−3.82528	−0.324456	−0.162228	−	0.986753i	$-0.551868\pi$
$139$	−3.82528	−0.324456	−0.162228	+	0.986753i	$0.551868\pi$
$140$	−0.0449649	−0.00380023
$141$	17.4786	1.47197
$142$	−4.44201	−0.372765
$143$	−2.69900	−0.225702
$144$	3.00749	0.250624
$145$	−1.52445	−0.126599
$146$	−1.73802	−0.143840
$147$	17.1490	1.41443
$148$	6.57508	0.540468
$149$	12.8659	1.05401	0.527007	−	0.849861i	$-0.323314\pi$
$149$	12.8659	1.05401	0.527007	+	0.849861i	$0.323314\pi$
$150$	10.7596	0.878514
$151$	7.64168	0.621871	0.310936	−	0.950431i	$-0.399358\pi$
$151$	7.64168	0.621871	0.310936	+	0.950431i	$0.399358\pi$
$152$	6.83443	0.554345
$153$	−9.79544	−0.791914
$154$	0.100934	0.00813348
$155$	−3.51654	−0.282455
$156$	−3.77279	−0.302065
$157$	6.80509	0.543105	0.271553	−	0.962424i	$-0.412463\pi$
$157$	6.80509	0.543105	0.271553	+	0.962424i	$0.412463\pi$
$158$	4.26198	0.339065
$159$	−18.2778	−1.44952
$160$	0.781132	0.0617539
$161$	0.212208	0.0167243
$162$	−8.97747	−0.705337
$163$	−12.6844	−0.993521	−0.496761	−	0.867888i	$-0.665477\pi$
$163$	−12.6844	−0.993521	−0.496761	+	0.867888i	$0.665477\pi$
$164$	−1.05822	−0.0826334
$165$	3.35705	0.261346
$166$	−8.61139	−0.668374
$167$	−15.5080	−1.20005	−0.600024	−	0.799982i	$-0.704842\pi$
$167$	−15.5080	−1.20005	−0.600024	+	0.799982i	$0.704842\pi$
$168$	0.141090	0.0108853
$169$	−10.6306	−0.817741
$170$	−2.54416	−0.195128
$171$	20.5545	1.57184
$172$	−1.77728	−0.135516
$173$	−0.553257	−0.0420633	−0.0210317	−	0.999779i	$-0.506695\pi$
$173$	−0.553257	−0.0420633	−0.0210317	+	0.999779i	$0.506695\pi$
$174$	4.78339	0.362628
$175$	0.252695	0.0191020
$176$	−1.75342	−0.132169
$177$	31.4966	2.36743
$178$	12.7128	0.952862
$179$	7.30569	0.546053	0.273027	−	0.962006i	$-0.411975\pi$
$179$	7.30569	0.546053	0.273027	+	0.962006i	$0.411975\pi$
$180$	2.34925	0.175103
$181$	−16.1658	−1.20159	−0.600797	−	0.799402i	$-0.705150\pi$
$181$	−16.1658	−1.20159	−0.600797	+	0.799402i	$0.705150\pi$
$182$	−0.0886066	−0.00656796
$183$	−28.7215	−2.12315
$184$	−3.68648	−0.271771
$185$	5.13600	0.377606
$186$	11.0341	0.809061
$187$	5.71093	0.417624
$188$	−7.13117	−0.520094
$189$	0.00105697	7.68835e−5 0
$190$	5.33859	0.387302
$191$	15.6847	1.13490	0.567452	−	0.823407i	$-0.307929\pi$
$191$	15.6847	1.13490	0.567452	+	0.823407i	$0.307929\pi$
$192$	−2.45102	−0.176887
$193$	19.3955	1.39612	0.698060	−	0.716039i	$-0.254047\pi$
$193$	19.3955	1.39612	0.698060	+	0.716039i	$0.254047\pi$
$194$	−0.351049	−0.0252039
$195$	−2.94705	−0.211043
$196$	−6.99669	−0.499763
$197$	−8.71870	−0.621181	−0.310591	−	0.950544i	$-0.600527\pi$
$197$	−8.71870	−0.621181	−0.310591	+	0.950544i	$0.600527\pi$
$198$	−5.27341	−0.374765
$199$	−3.79822	−0.269248	−0.134624	−	0.990897i	$-0.542983\pi$
$199$	−3.79822	−0.269248	−0.134624	+	0.990897i	$0.542983\pi$
$200$	−4.38983	−0.310408
$201$	−27.5781	−1.94521
$202$	4.73321	0.333028
$203$	0.112341	0.00788481
$204$	7.98300	0.558922
$205$	−0.826613	−0.0577331
$206$	−3.48850	−0.243055
$207$	−11.0871	−0.770604
$208$	1.53928	0.106730
$209$	−11.9837	−0.828926
$210$	0.110210	0.00760521
$211$	−16.0934	−1.10791	−0.553957	−	0.832546i	$-0.686883\pi$
$211$	−16.0934	−1.10791	−0.553957	+	0.832546i	$0.686883\pi$
$212$	7.45722	0.512164
$213$	10.8874	0.745996
$214$	−11.2744	−0.770700
$215$	−1.38829	−0.0946805
$216$	−0.0183618	−0.00124936
$217$	0.259144	0.0175918
$218$	−6.72600	−0.455542
$219$	4.25992	0.287859
$220$	−1.36966	−0.0923422
$221$	−5.01344	−0.337240
$222$	−16.1156	−1.08161
$223$	−3.60013	−0.241083	−0.120541	−	0.992708i	$-0.538463\pi$
$223$	−3.60013	−0.241083	−0.120541	+	0.992708i	$0.538463\pi$
$224$	−0.0575638	−0.00384614
$225$	−13.2024	−0.880159
$226$	−10.3856	−0.690838
$227$	−1.66634	−0.110599	−0.0552993	−	0.998470i	$-0.517611\pi$
$227$	−1.66634	−0.110599	−0.0552993	+	0.998470i	$0.517611\pi$
$228$	−16.7513	−1.10938
$229$	4.44843	0.293961	0.146980	−	0.989139i	$-0.453045\pi$
$229$	4.44843	0.293961	0.146980	+	0.989139i	$0.453045\pi$
$230$	−2.87963	−0.189877
$231$	−0.247391	−0.0162771
$232$	−1.95159	−0.128128
$233$	−8.18305	−0.536089	−0.268045	−	0.963406i	$-0.586377\pi$
$233$	−8.18305	−0.536089	−0.268045	+	0.963406i	$0.586377\pi$
$234$	4.62936	0.302631
$235$	−5.57039	−0.363372
$236$	−12.8504	−0.836490
$237$	−10.4462	−0.678553
$238$	0.187486	0.0121529
$239$	−12.7237	−0.823027	−0.411514	−	0.911404i	$-0.635000\pi$
$239$	−12.7237	−0.823027	−0.411514	+	0.911404i	$0.635000\pi$
$240$	−1.91457	−0.123585
$241$	−21.2333	−1.36775	−0.683877	−	0.729597i	$-0.739708\pi$
$241$	−21.2333	−1.36775	−0.683877	+	0.729597i	$0.739708\pi$
$242$	−7.92550	−0.509471
$243$	22.0590	1.41509
$244$	11.7182	0.750179
$245$	−5.46534	−0.349168
$246$	2.59373	0.165370
$247$	10.5201	0.669376
$248$	−4.50185	−0.285868
$249$	21.1067	1.33758
$250$	−7.33470	−0.463887
$251$	−28.2428	−1.78267	−0.891334	−	0.453348i	$-0.850230\pi$
$251$	−28.2428	−1.78267	−0.891334	+	0.453348i	$0.850230\pi$
$252$	−0.173123	−0.0109057
$253$	6.46397	0.406386
$254$	14.7640	0.926379
$255$	6.23578	0.390500
$256$	1.00000	0.0625000
$257$	17.9635	1.12053	0.560265	−	0.828313i	$-0.310699\pi$
$257$	17.9635	1.12053	0.560265	+	0.828313i	$0.310699\pi$
$258$	4.35614	0.271201
$259$	−0.378486	−0.0235180
$260$	1.20238	0.0745683
$261$	−5.86940	−0.363307
$262$	−18.0628	−1.11592
$263$	−16.2664	−1.00303	−0.501514	−	0.865149i	$-0.667224\pi$
$263$	−16.2664	−1.00303	−0.501514	+	0.865149i	$0.667224\pi$
$264$	4.29768	0.264504
$265$	5.82508	0.357832
$266$	−0.393416	−0.0241219
$267$	−31.1592	−1.90691
$268$	11.2517	0.687306
$269$	28.7269	1.75151	0.875755	−	0.482756i	$-0.160364\pi$
$269$	28.7269	1.75151	0.875755	+	0.482756i	$0.160364\pi$
$270$	−0.0143430	−0.000872885 0
$271$	−5.78261	−0.351269	−0.175634	−	0.984455i	$-0.556198\pi$
$271$	−5.78261	−0.351269	−0.175634	+	0.984455i	$0.556198\pi$
$272$	−3.25701	−0.197485
$273$	0.217176	0.0131441
$274$	−12.4326	−0.751084
$275$	7.69724	0.464161
$276$	9.03563	0.543881
$277$	1.62820	0.0978292	0.0489146	−	0.998803i	$-0.484424\pi$
$277$	1.62820	0.0978292	0.0489146	+	0.998803i	$0.484424\pi$
$278$	−3.82528	−0.229425
$279$	−13.5393	−0.810575
$280$	−0.0449649	−0.00268717
$281$	−11.2527	−0.671277	−0.335639	−	0.941991i	$-0.608952\pi$
$281$	−11.2527	−0.671277	−0.335639	+	0.941991i	$0.608952\pi$
$282$	17.4786	1.04084
$283$	3.14029	0.186671	0.0933354	−	0.995635i	$-0.470247\pi$
$283$	3.14029	0.186671	0.0933354	+	0.995635i	$0.470247\pi$
$284$	−4.44201	−0.263585
$285$	−13.0850	−0.775087
$286$	−2.69900	−0.159595
$287$	0.0609154	0.00359572
$288$	3.00749	0.177218
$289$	−6.39187	−0.375992
$290$	−1.52445	−0.0895190
$291$	0.860428	0.0504392
$292$	−1.73802	−0.101710
$293$	19.4595	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$293$	19.4595	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$294$	17.1490	1.00015
$295$	−10.0379	−0.584427
$296$	6.57508	0.382169
$297$	0.0321960	0.00186820
$298$	12.8659	0.745300
$299$	−5.67451	−0.328165
$300$	10.7596	0.621204
$301$	0.102307	0.00589687
$302$	7.64168	0.439729
$303$	−11.6012	−0.666471
$304$	6.83443	0.391981
$305$	9.15345	0.524125
$306$	−9.79544	−0.559968
$307$	21.1765	1.20861	0.604305	−	0.796753i	$-0.293451\pi$
$307$	21.1765	1.20861	0.604305	+	0.796753i	$0.293451\pi$
$308$	0.100934	0.00575124
$309$	8.55038	0.486414
$310$	−3.51654	−0.199726
$311$	−30.3912	−1.72333	−0.861663	−	0.507481i	$-0.830577\pi$
$311$	−30.3912	−1.72333	−0.861663	+	0.507481i	$0.830577\pi$
$312$	−3.77279	−0.213592
$313$	−21.8003	−1.23223	−0.616114	−	0.787657i	$-0.711294\pi$
$313$	−21.8003	−1.23223	−0.616114	+	0.787657i	$0.711294\pi$
$314$	6.80509	0.384033
$315$	−0.135232	−0.00761944
$316$	4.26198	0.239755
$317$	−32.6658	−1.83469	−0.917346	−	0.398091i	$-0.869673\pi$
$317$	−32.6658	−1.83469	−0.917346	+	0.398091i	$0.869673\pi$
$318$	−18.2778	−1.02497
$319$	3.42197	0.191594
$320$	0.781132	0.0436666
$321$	27.6337	1.54236
$322$	0.212208	0.0118259
$323$	−22.2598	−1.23857
$324$	−8.97747	−0.498748
$325$	−6.75716	−0.374820
$326$	−12.6844	−0.702526
$327$	16.4856	0.911653
$328$	−1.05822	−0.0584306
$329$	0.410497	0.0226314
$330$	3.35705	0.184800
$331$	−13.1155	−0.720895	−0.360447	−	0.932780i	$-0.617376\pi$
$331$	−13.1155	−0.720895	−0.360447	+	0.932780i	$0.617376\pi$
$332$	−8.61139	−0.472612
$333$	19.7745	1.08364
$334$	−15.5080	−0.848563
$335$	8.78905	0.480197
$336$	0.141090	0.00769709
$337$	−27.2629	−1.48510	−0.742552	−	0.669789i	$-0.766385\pi$
$337$	−27.2629	−1.48510	−0.742552	+	0.669789i	$0.766385\pi$
$338$	−10.6306	−0.578230
$339$	25.4552	1.38254
$340$	−2.54416	−0.137976
$341$	7.89366	0.427465
$342$	20.5545	1.11146
$343$	0.805702	0.0435038
$344$	−1.77728	−0.0958244
$345$	7.05802	0.379991
$346$	−0.553257	−0.0297433
$347$	−2.24845	−0.120703	−0.0603517	−	0.998177i	$-0.519222\pi$
$347$	−2.24845	−0.120703	−0.0603517	+	0.998177i	$0.519222\pi$
$348$	4.78339	0.256417
$349$	1.63381	0.0874557	0.0437278	−	0.999043i	$-0.486077\pi$
$349$	1.63381	0.0874557	0.0437278	+	0.999043i	$0.486077\pi$
$350$	0.252695	0.0135071
$351$	−0.0282638	−0.00150861
$352$	−1.75342	−0.0934579
$353$	5.52490	0.294061	0.147031	−	0.989132i	$-0.453028\pi$
$353$	5.52490	0.294061	0.147031	+	0.989132i	$0.453028\pi$
$354$	31.4966	1.67402
$355$	−3.46980	−0.184158
$356$	12.7128	0.673775
$357$	−0.459532	−0.0243210
$358$	7.30569	0.386118
$359$	−27.4741	−1.45003	−0.725013	−	0.688735i	$-0.758166\pi$
$359$	−27.4741	−1.45003	−0.725013	+	0.688735i	$0.758166\pi$
$360$	2.34925	0.123816
$361$	27.7094	1.45839
$362$	−16.1658	−0.849655
$363$	19.4256	1.01958
$364$	−0.0886066	−0.00464425
$365$	−1.35762	−0.0710613
$366$	−28.7215	−1.50130
$367$	−5.11153	−0.266820	−0.133410	−	0.991061i	$-0.542593\pi$
$367$	−5.11153	−0.266820	−0.133410	+	0.991061i	$0.542593\pi$
$368$	−3.68648	−0.192171
$369$	−3.18260	−0.165679
$370$	5.13600	0.267008
$371$	−0.429266	−0.0222864
$372$	11.0341	0.572092
$373$	26.9146	1.39359	0.696793	−	0.717273i	$-0.254610\pi$
$373$	26.9146	1.39359	0.696793	+	0.717273i	$0.254610\pi$
$374$	5.71093	0.295305
$375$	17.9775	0.928353
$376$	−7.13117	−0.367762
$377$	−3.00404	−0.154716
$378$	0.00105697	5.43648e−5 0
$379$	4.48486	0.230372	0.115186	−	0.993344i	$-0.463254\pi$
$379$	4.48486	0.230372	0.115186	+	0.993344i	$0.463254\pi$
$380$	5.33859	0.273864
$381$	−36.1869	−1.85391
$382$	15.6847	0.802498
$383$	−6.63305	−0.338933	−0.169467	−	0.985536i	$-0.554204\pi$
$383$	−6.63305	−0.338933	−0.169467	+	0.985536i	$0.554204\pi$
$384$	−2.45102	−0.125078
$385$	0.0788426	0.00401819
$386$	19.3955	0.987206
$387$	−5.34515	−0.271709
$388$	−0.351049	−0.0178218
$389$	−22.0735	−1.11917	−0.559584	−	0.828773i	$-0.689039\pi$
$389$	−22.0735	−1.11917	−0.559584	+	0.828773i	$0.689039\pi$
$390$	−2.94705	−0.149230
$391$	12.0069	0.607216
$392$	−6.99669	−0.353386
$393$	44.2723	2.23324
$394$	−8.71870	−0.439242
$395$	3.32917	0.167509
$396$	−5.27341	−0.264999
$397$	−2.41204	−0.121057	−0.0605284	−	0.998166i	$-0.519279\pi$
$397$	−2.41204	−0.121057	−0.0605284	+	0.998166i	$0.519279\pi$
$398$	−3.79822	−0.190387
$399$	0.964269	0.0482738
$400$	−4.38983	−0.219492
$401$	−15.1902	−0.758564	−0.379282	−	0.925281i	$-0.623829\pi$
$401$	−15.1902	−0.758564	−0.379282	+	0.925281i	$0.623829\pi$
$402$	−27.5781	−1.37547
$403$	−6.92959	−0.345187
$404$	4.73321	0.235486
$405$	−7.01259	−0.348458
$406$	0.112341	0.00557540
$407$	−11.5289	−0.571466
$408$	7.98300	0.395217
$409$	−23.9447	−1.18399	−0.591995	−	0.805942i	$-0.701659\pi$
$409$	−23.9447	−1.18399	−0.591995	+	0.805942i	$0.701659\pi$
$410$	−0.826613	−0.0408235
$411$	30.4727	1.50311
$412$	−3.48850	−0.171866
$413$	0.739718	0.0363991
$414$	−11.0871	−0.544899
$415$	−6.72664	−0.330198
$416$	1.53928	0.0754692
$417$	9.37584	0.459137
$418$	−11.9837	−0.586140
$419$	24.0789	1.17633	0.588165	−	0.808741i	$-0.299851\pi$
$419$	24.0789	1.17633	0.588165	+	0.808741i	$0.299851\pi$
$420$	0.110210	0.00537769
$421$	4.44385	0.216580	0.108290	−	0.994119i	$-0.465463\pi$
$421$	4.44385	0.216580	0.108290	+	0.994119i	$0.465463\pi$
$422$	−16.0934	−0.783413
$423$	−21.4469	−1.04279
$424$	7.45722	0.362155
$425$	14.2977	0.693542
$426$	10.8874	0.527499
$427$	−0.674543	−0.0326434
$428$	−11.2744	−0.544967
$429$	6.61531	0.319390
$430$	−1.38829	−0.0669492
$431$	−15.1894	−0.731649	−0.365825	−	0.930684i	$-0.619213\pi$
$431$	−15.1894	−0.731649	−0.365825	+	0.930684i	$0.619213\pi$
$432$	−0.0183618	−0.000883431 0
$433$	16.5244	0.794112	0.397056	−	0.917794i	$-0.370032\pi$
$433$	16.5244	0.794112	0.397056	+	0.917794i	$0.370032\pi$
$434$	0.259144	0.0124393
$435$	3.73646	0.179150
$436$	−6.72600	−0.322117
$437$	−25.1950	−1.20524
$438$	4.25992	0.203547
$439$	−20.0891	−0.958801	−0.479400	−	0.877596i	$-0.659146\pi$
$439$	−20.0891	−0.958801	−0.479400	+	0.877596i	$0.659146\pi$
$440$	−1.36966	−0.0652958
$441$	−21.0425	−1.00202
$442$	−5.01344	−0.238465
$443$	−39.3353	−1.86888	−0.934438	−	0.356125i	$-0.884098\pi$
$443$	−39.3353	−1.86888	−0.934438	+	0.356125i	$0.884098\pi$
$444$	−16.1156	−0.764814
$445$	9.93035	0.470744
$446$	−3.60013	−0.170471
$447$	−31.5345	−1.49153
$448$	−0.0575638	−0.00271963
$449$	1.18741	0.0560373	0.0280187	−	0.999607i	$-0.491080\pi$
$449$	1.18741	0.0560373	0.0280187	+	0.999607i	$0.491080\pi$
$450$	−13.2024	−0.622366
$451$	1.85552	0.0873728
$452$	−10.3856	−0.488496
$453$	−18.7299	−0.880008
$454$	−1.66634	−0.0782050
$455$	−0.0692134	−0.00324478
$456$	−16.7513	−0.784452
$457$	1.47811	0.0691432	0.0345716	−	0.999402i	$-0.488993\pi$
$457$	1.47811	0.0691432	0.0345716	+	0.999402i	$0.488993\pi$
$458$	4.44843	0.207862
$459$	0.0598045	0.00279144
$460$	−2.87963	−0.134263
$461$	4.86416	0.226546	0.113273	−	0.993564i	$-0.463866\pi$
$461$	4.86416	0.226546	0.113273	+	0.993564i	$0.463866\pi$
$462$	−0.247391	−0.0115097
$463$	−22.8349	−1.06123	−0.530613	−	0.847614i	$-0.678038\pi$
$463$	−22.8349	−1.06123	−0.530613	+	0.847614i	$0.678038\pi$
$464$	−1.95159	−0.0906005
$465$	8.61911	0.399701
$466$	−8.18305	−0.379072
$467$	−13.6345	−0.630929	−0.315465	−	0.948937i	$-0.602160\pi$
$467$	−13.6345	−0.630929	−0.315465	+	0.948937i	$0.602160\pi$
$468$	4.62936	0.213992
$469$	−0.647689	−0.0299075
$470$	−5.57039	−0.256943
$471$	−16.6794	−0.768546
$472$	−12.8504	−0.591488
$473$	3.11632	0.143289
$474$	−10.4462	−0.479810
$475$	−30.0020	−1.37659
$476$	0.187486	0.00859341
$477$	22.4275	1.02689
$478$	−12.7237	−0.581968
$479$	−24.2538	−1.10818	−0.554092	−	0.832456i	$-0.686934\pi$
$479$	−24.2538	−1.10818	−0.554092	+	0.832456i	$0.686934\pi$
$480$	−1.91457	−0.0873877
$481$	10.1209	0.461471
$482$	−21.2333	−0.967148
$483$	−0.520125	−0.0236665
$484$	−7.92550	−0.360250
$485$	−0.274216	−0.0124515
$486$	22.0590	1.00062
$487$	18.0981	0.820102	0.410051	−	0.912063i	$-0.365511\pi$
$487$	18.0981	0.820102	0.410051	+	0.912063i	$0.365511\pi$
$488$	11.7182	0.530457
$489$	31.0898	1.40593
$490$	−5.46534	−0.246899
$491$	−21.5514	−0.972602	−0.486301	−	0.873791i	$-0.661654\pi$
$491$	−21.5514	−0.972602	−0.486301	+	0.873791i	$0.661654\pi$
$492$	2.59373	0.116934
$493$	6.35637	0.286276
$494$	10.5201	0.473320
$495$	−4.11923	−0.185146
$496$	−4.50185	−0.202139
$497$	0.255699	0.0114697
$498$	21.1067	0.945813
$499$	1.22402	0.0547947	0.0273974	−	0.999625i	$-0.491278\pi$
$499$	1.22402	0.0547947	0.0273974	+	0.999625i	$0.491278\pi$
$500$	−7.33470	−0.328018
$501$	38.0105	1.69818
$502$	−28.2428	−1.26054
$503$	29.3938	1.31061	0.655303	−	0.755366i	$-0.272541\pi$
$503$	29.3938	1.31061	0.655303	+	0.755366i	$0.272541\pi$
$504$	−0.173123	−0.00771150
$505$	3.69727	0.164526
$506$	6.46397	0.287358
$507$	26.0559	1.15718
$508$	14.7640	0.655049
$509$	−19.1149	−0.847254	−0.423627	−	0.905837i	$-0.639243\pi$
$509$	−19.1149	−0.847254	−0.423627	+	0.905837i	$0.639243\pi$
$510$	6.23578	0.276125
$511$	0.100047	0.00442582
$512$	1.00000	0.0441942
$513$	−0.125492	−0.00554061
$514$	17.9635	0.792334
$515$	−2.72498	−0.120077
$516$	4.35614	0.191768
$517$	12.5040	0.549924
$518$	−0.378486	−0.0166297
$519$	1.35604	0.0595236
$520$	1.20238	0.0527278
$521$	17.5979	0.770979	0.385490	−	0.922712i	$-0.374033\pi$
$521$	17.5979	0.770979	0.385490	+	0.922712i	$0.374033\pi$
$522$	−5.86940	−0.256897
$523$	−34.5730	−1.51177	−0.755887	−	0.654703i	$-0.772794\pi$
$523$	−34.5730	−1.51177	−0.755887	+	0.654703i	$0.772794\pi$
$524$	−18.0628	−0.789078
$525$	−0.619361	−0.0270311
$526$	−16.2664	−0.709248
$527$	14.6626	0.638712
$528$	4.29768	0.187032
$529$	−9.40986	−0.409125
$530$	5.82508	0.253025
$531$	−38.6475	−1.67716
$532$	−0.393416	−0.0170567
$533$	−1.62890	−0.0705554
$534$	−31.1592	−1.34839
$535$	−8.80677	−0.380750
$536$	11.2517	0.485998
$537$	−17.9064	−0.772718
$538$	28.7269	1.23850
$539$	12.2682	0.528427
$540$	−0.0143430	−0.000617223 0
$541$	33.0339	1.42024	0.710119	−	0.704082i	$-0.248641\pi$
$541$	33.0339	1.42024	0.710119	+	0.704082i	$0.248641\pi$
$542$	−5.78261	−0.248384
$543$	39.6226	1.70037
$544$	−3.25701	−0.139643
$545$	−5.25390	−0.225052
$546$	0.217176	0.00929429
$547$	27.3325	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$547$	27.3325	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$548$	−12.4326	−0.531096
$549$	35.2423	1.50411
$550$	7.69724	0.328211
$551$	−13.3380	−0.568219
$552$	9.03563	0.384582
$553$	−0.245336	−0.0104327
$554$	1.62820	0.0691757
$555$	−12.5884	−0.534349
$556$	−3.82528	−0.162228
$557$	16.0869	0.681625	0.340812	−	0.940131i	$-0.389298\pi$
$557$	16.0869	0.681625	0.340812	+	0.940131i	$0.389298\pi$
$558$	−13.5393	−0.573163
$559$	−2.73572	−0.115709
$560$	−0.0449649	−0.00190012
$561$	−13.9976	−0.590979
$562$	−11.2527	−0.474665
$563$	1.64047	0.0691374	0.0345687	−	0.999402i	$-0.488994\pi$
$563$	1.64047	0.0691374	0.0345687	+	0.999402i	$0.488994\pi$
$564$	17.4786	0.735983
$565$	−8.11250	−0.341295
$566$	3.14029	0.131996
$567$	0.516777	0.0217026
$568$	−4.44201	−0.186383
$569$	26.3499	1.10464	0.552322	−	0.833631i	$-0.313742\pi$
$569$	26.3499	1.10464	0.552322	+	0.833631i	$0.313742\pi$
$570$	−13.0850	−0.548070
$571$	−25.8407	−1.08140	−0.540699	−	0.841216i	$-0.681840\pi$
$571$	−25.8407	−1.08140	−0.540699	+	0.841216i	$0.681840\pi$
$572$	−2.69900	−0.112851
$573$	−38.4434	−1.60600
$574$	0.0609154	0.00254256
$575$	16.1830	0.674879
$576$	3.00749	0.125312
$577$	30.5376	1.27130	0.635648	−	0.771979i	$-0.280733\pi$
$577$	30.5376	1.27130	0.635648	+	0.771979i	$0.280733\pi$
$578$	−6.39187	−0.265867
$579$	−47.5388	−1.97564
$580$	−1.52445	−0.0632995
$581$	0.495705	0.0205653
$582$	0.860428	0.0356659
$583$	−13.0757	−0.541539
$584$	−1.73802	−0.0719198
$585$	3.61614	0.149509
$586$	19.4595	0.803864
$587$	−11.4538	−0.472750	−0.236375	−	0.971662i	$-0.575959\pi$
$587$	−11.4538	−0.472750	−0.236375	+	0.971662i	$0.575959\pi$
$588$	17.1490	0.707213
$589$	−30.7676	−1.26776
$590$	−10.0379	−0.413252
$591$	21.3697	0.879031
$592$	6.57508	0.270234
$593$	21.5838	0.886342	0.443171	−	0.896437i	$-0.353853\pi$
$593$	21.5838	0.886342	0.443171	+	0.896437i	$0.353853\pi$
$594$	0.0321960	0.00132102
$595$	0.146451	0.00600392
$596$	12.8659	0.527007
$597$	9.30950	0.381012
$598$	−5.67451	−0.232048
$599$	43.6771	1.78460	0.892298	−	0.451446i	$-0.149092\pi$
$599$	43.6771	1.78460	0.892298	+	0.451446i	$0.149092\pi$
$600$	10.7596	0.439257
$601$	16.8500	0.687324	0.343662	−	0.939093i	$-0.388333\pi$
$601$	16.8500	0.687324	0.343662	+	0.939093i	$0.388333\pi$
$602$	0.102307	0.00416972
$603$	33.8393	1.37804
$604$	7.64168	0.310936
$605$	−6.19086	−0.251694
$606$	−11.6012	−0.471266
$607$	−14.8606	−0.603174	−0.301587	−	0.953439i	$-0.597516\pi$
$607$	−14.8606	−0.603174	−0.301587	+	0.953439i	$0.597516\pi$
$608$	6.83443	0.277173
$609$	−0.275350	−0.0111578
$610$	9.15345	0.370612
$611$	−10.9768	−0.444075
$612$	−9.79544	−0.395957
$613$	−0.474337	−0.0191583	−0.00957915	−	0.999954i	$-0.503049\pi$
$613$	−0.474337	−0.0191583	−0.00957915	+	0.999954i	$0.503049\pi$
$614$	21.1765	0.854616
$615$	2.02604	0.0816979
$616$	0.100934	0.00406674
$617$	−8.97696	−0.361399	−0.180699	−	0.983538i	$-0.557836\pi$
$617$	−8.97696	−0.361399	−0.180699	+	0.983538i	$0.557836\pi$
$618$	8.55038	0.343947
$619$	37.5211	1.50810	0.754051	−	0.656816i	$-0.228097\pi$
$619$	37.5211	1.50810	0.754051	+	0.656816i	$0.228097\pi$
$620$	−3.51654	−0.141228
$621$	0.0676903	0.00271632
$622$	−30.3912	−1.21858
$623$	−0.731795	−0.0293188
$624$	−3.77279	−0.151033
$625$	16.2198	0.648792
$626$	−21.8003	−0.871317
$627$	29.3721	1.17301
$628$	6.80509	0.271553
$629$	−21.4151	−0.853876
$630$	−0.135232	−0.00538776
$631$	−4.99941	−0.199023	−0.0995117	−	0.995036i	$-0.531728\pi$
$631$	−4.99941	−0.199023	−0.0995117	+	0.995036i	$0.531728\pi$
$632$	4.26198	0.169533
$633$	39.4451	1.56780
$634$	−32.6658	−1.29732
$635$	11.5327	0.457660
$636$	−18.2778	−0.724761
$637$	−10.7698	−0.426716
$638$	3.42197	0.135477
$639$	−13.3593	−0.528486
$640$	0.781132	0.0308770
$641$	27.6201	1.09093	0.545465	−	0.838134i	$-0.316353\pi$
$641$	27.6201	1.09093	0.545465	+	0.838134i	$0.316353\pi$
$642$	27.6337	1.09061
$643$	−33.9371	−1.33835	−0.669173	−	0.743106i	$-0.733352\pi$
$643$	−33.9371	−1.33835	−0.669173	+	0.743106i	$0.733352\pi$
$644$	0.212208	0.00836216
$645$	3.40272	0.133982
$646$	−22.2598	−0.875800
$647$	−2.50807	−0.0986023	−0.0493012	−	0.998784i	$-0.515699\pi$
$647$	−2.50807	−0.0986023	−0.0493012	+	0.998784i	$0.515699\pi$
$648$	−8.97747	−0.352668
$649$	22.5322	0.884467
$650$	−6.75716	−0.265038
$651$	−0.635166	−0.0248941
$652$	−12.6844	−0.496761
$653$	50.7798	1.98717	0.993584	−	0.113096i	$-0.0360767\pi$
$653$	50.7798	1.98717	0.993584	+	0.113096i	$0.0360767\pi$
$654$	16.4856	0.644636
$655$	−14.1095	−0.551302
$656$	−1.05822	−0.0413167
$657$	−5.22708	−0.203928
$658$	0.410497	0.0160028
$659$	−22.5059	−0.876705	−0.438352	−	0.898803i	$-0.644438\pi$
$659$	−22.5059	−0.876705	−0.438352	+	0.898803i	$0.644438\pi$
$660$	3.35705	0.130673
$661$	18.9059	0.735355	0.367677	−	0.929953i	$-0.380153\pi$
$661$	18.9059	0.735355	0.367677	+	0.929953i	$0.380153\pi$
$662$	−13.1155	−0.509750
$663$	12.2880	0.477228
$664$	−8.61139	−0.334187
$665$	−0.307310	−0.0119170
$666$	19.7745	0.766246
$667$	7.19451	0.278573
$668$	−15.5080	−0.600024
$669$	8.82399	0.341155
$670$	8.78905	0.339551
$671$	−20.5469	−0.793206
$672$	0.141090	0.00544266
$673$	−8.96978	−0.345760	−0.172880	−	0.984943i	$-0.555307\pi$
$673$	−8.96978	−0.345760	−0.172880	+	0.984943i	$0.555307\pi$
$674$	−27.2629	−1.05013
$675$	0.0806051	0.00310249
$676$	−10.6306	−0.408870
$677$	24.4279	0.938840	0.469420	−	0.882975i	$-0.344463\pi$
$677$	24.4279	0.938840	0.469420	+	0.882975i	$0.344463\pi$
$678$	25.4552	0.977602
$679$	0.0202077	0.000775502 0
$680$	−2.54416	−0.0975640
$681$	4.08422	0.156508
$682$	7.89366	0.302264
$683$	9.15211	0.350196	0.175098	−	0.984551i	$-0.443976\pi$
$683$	9.15211	0.350196	0.175098	+	0.984551i	$0.443976\pi$
$684$	20.5545	0.785920
$685$	−9.71154	−0.371059
$686$	0.805702	0.0307619
$687$	−10.9032	−0.415983
$688$	−1.77728	−0.0677581
$689$	11.4787	0.437304
$690$	7.05802	0.268694
$691$	46.2098	1.75790	0.878951	−	0.476912i	$-0.158244\pi$
$691$	46.2098	1.75790	0.878951	+	0.476912i	$0.158244\pi$
$692$	−0.553257	−0.0210317
$693$	0.303558	0.0115312
$694$	−2.24845	−0.0853501
$695$	−2.98805	−0.113343
$696$	4.78339	0.181314
$697$	3.44665	0.130551
$698$	1.63381	0.0618405
$699$	20.0568	0.758618
$700$	0.252695	0.00955099
$701$	−10.5898	−0.399971	−0.199986	−	0.979799i	$-0.564090\pi$
$701$	−10.5898	−0.399971	−0.199986	+	0.979799i	$0.564090\pi$
$702$	−0.0282638	−0.00106675
$703$	44.9369	1.69483
$704$	−1.75342	−0.0660847
$705$	13.6531	0.514206
$706$	5.52490	0.207933
$707$	−0.272462	−0.0102470
$708$	31.4966	1.18371
$709$	−13.5104	−0.507394	−0.253697	−	0.967284i	$-0.581647\pi$
$709$	−13.5104	−0.507394	−0.253697	+	0.967284i	$0.581647\pi$
$710$	−3.46980	−0.130219
$711$	12.8179	0.480708
$712$	12.7128	0.476431
$713$	16.5960	0.621524
$714$	−0.459532	−0.0171975
$715$	−2.10828	−0.0788452
$716$	7.30569	0.273027
$717$	31.1860	1.16466
$718$	−27.4741	−1.02532
$719$	16.3385	0.609324	0.304662	−	0.952461i	$-0.401456\pi$
$719$	16.3385	0.609324	0.304662	+	0.952461i	$0.401456\pi$
$720$	2.34925	0.0875513
$721$	0.200811	0.00747860
$722$	27.7094	1.03124
$723$	52.0431	1.93550
$724$	−16.1658	−0.600797
$725$	8.56717	0.318177
$726$	19.4256	0.720950
$727$	−22.6819	−0.841226	−0.420613	−	0.907240i	$-0.638185\pi$
$727$	−22.6819	−0.841226	−0.420613	+	0.907240i	$0.638185\pi$
$728$	−0.0886066	−0.00328398
$729$	−27.1347	−1.00499
$730$	−1.35762	−0.0502479
$731$	5.78861	0.214100
$732$	−28.7215	−1.06158
$733$	−24.4311	−0.902383	−0.451191	−	0.892427i	$-0.649001\pi$
$733$	−24.4311	−0.902383	−0.451191	+	0.892427i	$0.649001\pi$
$734$	−5.11153	−0.188670
$735$	13.3956	0.494106
$736$	−3.68648	−0.135885
$737$	−19.7290	−0.726726
$738$	−3.18260	−0.117153
$739$	8.94372	0.329000	0.164500	−	0.986377i	$-0.447399\pi$
$739$	8.94372	0.329000	0.164500	+	0.986377i	$0.447399\pi$
$740$	5.13600	0.188803
$741$	−25.7849	−0.947231
$742$	−0.429266	−0.0157589
$743$	−2.90944	−0.106737	−0.0533684	−	0.998575i	$-0.516996\pi$
$743$	−2.90944	−0.106737	−0.0533684	+	0.998575i	$0.516996\pi$
$744$	11.0341	0.404530
$745$	10.0499	0.368202
$746$	26.9146	0.985414
$747$	−25.8987	−0.947584
$748$	5.71093	0.208812
$749$	0.648995	0.0237138
$750$	17.9775	0.656445
$751$	−23.0818	−0.842267	−0.421134	−	0.906999i	$-0.638368\pi$
$751$	−23.0818	−0.842267	−0.421134	+	0.906999i	$0.638368\pi$
$752$	−7.13117	−0.260047
$753$	69.2235	2.52265
$754$	−3.00404	−0.109401
$755$	5.96916	0.217240
$756$	0.00105697	3.84417e−5 0
$757$	−47.5041	−1.72656	−0.863282	−	0.504721i	$-0.831595\pi$
$757$	−47.5041	−1.72656	−0.863282	+	0.504721i	$0.831595\pi$
$758$	4.48486	0.162898
$759$	−15.8433	−0.575075
$760$	5.33859	0.193651
$761$	49.9564	1.81092	0.905460	−	0.424432i	$-0.139526\pi$
$761$	49.9564	1.81092	0.905460	+	0.424432i	$0.139526\pi$
$762$	−36.1869	−1.31091
$763$	0.387174	0.0140167
$764$	15.6847	0.567452
$765$	−7.65153	−0.276642
$766$	−6.63305	−0.239662
$767$	−19.7803	−0.714226
$768$	−2.45102	−0.0884435
$769$	32.3775	1.16756	0.583781	−	0.811911i	$-0.301573\pi$
$769$	32.3775	1.16756	0.583781	+	0.811911i	$0.301573\pi$
$770$	0.0788426	0.00284129
$771$	−44.0288	−1.58566
$772$	19.3955	0.698060
$773$	46.2978	1.66521	0.832607	−	0.553864i	$-0.186847\pi$
$773$	46.2978	1.66521	0.832607	+	0.553864i	$0.186847\pi$
$774$	−5.34515	−0.192127
$775$	19.7624	0.709885
$776$	−0.351049	−0.0126019
$777$	0.927677	0.0332802
$778$	−22.0735	−0.791372
$779$	−7.23235	−0.259126
$780$	−2.94705	−0.105521
$781$	7.78873	0.278703
$782$	12.0069	0.429366
$783$	0.0358347	0.00128063
$784$	−6.99669	−0.249882
$785$	5.31567	0.189725
$786$	44.2723	1.57914
$787$	−25.5203	−0.909700	−0.454850	−	0.890568i	$-0.650307\pi$
$787$	−25.5203	−0.909700	−0.454850	+	0.890568i	$0.650307\pi$
$788$	−8.71870	−0.310591
$789$	39.8692	1.41938
$790$	3.32917	0.118447
$791$	0.597833	0.0212565
$792$	−5.27341	−0.187382
$793$	18.0375	0.640531
$794$	−2.41204	−0.0856001
$795$	−14.2774	−0.506366
$796$	−3.79822	−0.134624
$797$	42.7129	1.51297	0.756485	−	0.654011i	$-0.226915\pi$
$797$	42.7129	1.51297	0.756485	+	0.654011i	$0.226915\pi$
$798$	0.964269	0.0341347
$799$	23.2263	0.821688
$800$	−4.38983	−0.155204
$801$	38.2335	1.35092
$802$	−15.1902	−0.536386
$803$	3.04749	0.107544
$804$	−27.5781	−0.972604
$805$	0.165762	0.00584236
$806$	−6.92959	−0.244084
$807$	−70.4102	−2.47856
$808$	4.73321	0.166514
$809$	5.95732	0.209448	0.104724	−	0.994501i	$-0.466604\pi$
$809$	5.95732	0.209448	0.104724	+	0.994501i	$0.466604\pi$
$810$	−7.01259	−0.246397
$811$	16.2208	0.569590	0.284795	−	0.958588i	$-0.408074\pi$
$811$	16.2208	0.569590	0.284795	+	0.958588i	$0.408074\pi$
$812$	0.112341	0.00394240
$813$	14.1733	0.497079
$814$	−11.5289	−0.404088
$815$	−9.90822	−0.347070
$816$	7.98300	0.279461
$817$	−12.1467	−0.424958
$818$	−23.9447	−0.837207
$819$	−0.266483	−0.00931169
$820$	−0.826613	−0.0288666
$821$	−41.4444	−1.44642	−0.723210	−	0.690628i	$-0.757334\pi$
$821$	−41.4444	−1.44642	−0.723210	+	0.690628i	$0.757334\pi$
$822$	30.4727	1.06286
$823$	11.0949	0.386742	0.193371	−	0.981126i	$-0.438058\pi$
$823$	11.0949	0.386742	0.193371	+	0.981126i	$0.438058\pi$
$824$	−3.48850	−0.121528
$825$	−18.8661	−0.656833
$826$	0.739718	0.0257381
$827$	−2.26993	−0.0789332	−0.0394666	−	0.999221i	$-0.512566\pi$
$827$	−2.26993	−0.0789332	−0.0394666	+	0.999221i	$0.512566\pi$
$828$	−11.0871	−0.385302
$829$	−11.7772	−0.409038	−0.204519	−	0.978863i	$-0.565563\pi$
$829$	−11.7772	−0.409038	−0.204519	+	0.978863i	$0.565563\pi$
$830$	−6.72664	−0.233485
$831$	−3.99075	−0.138438
$832$	1.53928	0.0533648
$833$	22.7883	0.789568
$834$	9.37584	0.324659
$835$	−12.1138	−0.419217
$836$	−11.9837	−0.414463
$837$	0.0826619	0.00285721
$838$	24.0789	0.831790
$839$	44.0061	1.51926	0.759630	−	0.650355i	$-0.225380\pi$
$839$	44.0061	1.51926	0.759630	+	0.650355i	$0.225380\pi$
$840$	0.110210	0.00380260
$841$	−25.1913	−0.868665
$842$	4.44385	0.153145
$843$	27.5805	0.949922
$844$	−16.0934	−0.553957
$845$	−8.30393	−0.285664
$846$	−21.4469	−0.737361
$847$	0.456222	0.0156760
$848$	7.45722	0.256082
$849$	−7.69690	−0.264157
$850$	14.2977	0.490408
$851$	−24.2389	−0.830898
$852$	10.8874	0.372998
$853$	44.6238	1.52789	0.763946	−	0.645281i	$-0.223260\pi$
$853$	44.6238	1.52789	0.763946	+	0.645281i	$0.223260\pi$
$854$	−0.674543	−0.0230824
$855$	16.0558	0.549096
$856$	−11.2744	−0.385350
$857$	27.2932	0.932319	0.466159	−	0.884701i	$-0.345637\pi$
$857$	27.2932	0.932319	0.466159	+	0.884701i	$0.345637\pi$
$858$	6.61531	0.225843
$859$	−55.8121	−1.90429	−0.952143	−	0.305654i	$-0.901125\pi$
$859$	−55.8121	−1.90429	−0.952143	+	0.305654i	$0.901125\pi$
$860$	−1.38829	−0.0473403
$861$	−0.149305	−0.00508829
$862$	−15.1894	−0.517354
$863$	28.8868	0.983318	0.491659	−	0.870788i	$-0.336391\pi$
$863$	28.8868	0.983318	0.491659	+	0.870788i	$0.336391\pi$
$864$	−0.0183618	−0.000624680 0
$865$	−0.432167	−0.0146941
$866$	16.5244	0.561522
$867$	15.6666	0.532065
$868$	0.259144	0.00879591
$869$	−7.47306	−0.253506
$870$	3.73646	0.126678
$871$	17.3194	0.586847
$872$	−6.72600	−0.227771
$873$	−1.05578	−0.0357327
$874$	−25.1950	−0.852233
$875$	0.422213	0.0142734
$876$	4.25992	0.143929
$877$	−16.8104	−0.567645	−0.283823	−	0.958877i	$-0.591603\pi$
$877$	−16.8104	−0.567645	−0.283823	+	0.958877i	$0.591603\pi$
$878$	−20.0891	−0.677975
$879$	−47.6956	−1.60873
$880$	−1.36966	−0.0461711
$881$	−30.1150	−1.01460	−0.507301	−	0.861769i	$-0.669357\pi$
$881$	−30.1150	−1.01460	−0.507301	+	0.861769i	$0.669357\pi$
$882$	−21.0425	−0.708537
$883$	7.86653	0.264730	0.132365	−	0.991201i	$-0.457743\pi$
$883$	7.86653	0.264730	0.132365	+	0.991201i	$0.457743\pi$
$884$	−5.01344	−0.168620
$885$	24.6030	0.827020
$886$	−39.3353	−1.32150
$887$	33.7066	1.13176	0.565878	−	0.824489i	$-0.308537\pi$
$887$	33.7066	1.13176	0.565878	+	0.824489i	$0.308537\pi$
$888$	−16.1156	−0.540805
$889$	−0.849875	−0.0285039
$890$	9.93035	0.332866
$891$	15.7413	0.527354
$892$	−3.60013	−0.120541
$893$	−48.7374	−1.63094
$894$	−31.5345	−1.05467
$895$	5.70671	0.190754
$896$	−0.0575638	−0.00192307
$897$	13.9083	0.464385
$898$	1.18741	0.0396244
$899$	8.78579	0.293022
$900$	−13.2024	−0.440079
$901$	−24.2883	−0.809159
$902$	1.85552	0.0617819
$903$	−0.250756	−0.00834464
$904$	−10.3856	−0.345419
$905$	−12.6276	−0.419756
$906$	−18.7299	−0.622259
$907$	−21.9693	−0.729477	−0.364739	−	0.931110i	$-0.618842\pi$
$907$	−21.9693	−0.729477	−0.364739	+	0.931110i	$0.618842\pi$
$908$	−1.66634	−0.0552993
$909$	14.2351	0.472148
$910$	−0.0692134	−0.00229440
$911$	27.0581	0.896474	0.448237	−	0.893915i	$-0.352052\pi$
$911$	27.0581	0.896474	0.448237	+	0.893915i	$0.352052\pi$
$912$	−16.7513	−0.554691
$913$	15.0994	0.499718
$914$	1.47811	0.0488916
$915$	−22.4353	−0.741687
$916$	4.44843	0.146980
$917$	1.03976	0.0343361
$918$	0.0598045	0.00197384
$919$	−22.6061	−0.745707	−0.372853	−	0.927890i	$-0.621621\pi$
$919$	−22.6061	−0.745707	−0.372853	+	0.927890i	$0.621621\pi$
$920$	−2.87963	−0.0949385
$921$	−51.9041	−1.71030
$922$	4.86416	0.160193
$923$	−6.83748	−0.225058
$924$	−0.247391	−0.00813855
$925$	−28.8635	−0.949025
$926$	−22.8349	−0.750400
$927$	−10.4916	−0.344590
$928$	−1.95159	−0.0640642
$929$	37.8809	1.24283	0.621416	−	0.783481i	$-0.286558\pi$
$929$	37.8809	1.24283	0.621416	+	0.783481i	$0.286558\pi$
$930$	8.61911	0.282632
$931$	−47.8183	−1.56718
$932$	−8.18305	−0.268045
$933$	74.4893	2.43867
$934$	−13.6345	−0.446134
$935$	4.46099	0.145890
$936$	4.62936	0.151315
$937$	−4.69348	−0.153329	−0.0766646	−	0.997057i	$-0.524427\pi$
$937$	−4.69348	−0.153329	−0.0766646	+	0.997057i	$0.524427\pi$
$938$	−0.647689	−0.0211478
$939$	53.4330	1.74372
$940$	−5.57039	−0.181686
$941$	−22.2469	−0.725229	−0.362614	−	0.931939i	$-0.618116\pi$
$941$	−22.2469	−0.725229	−0.362614	+	0.931939i	$0.618116\pi$
$942$	−16.6794	−0.543444
$943$	3.90112	0.127038
$944$	−12.8504	−0.418245
$945$	0.000825636 0	2.68579e−5 0
$946$	3.11632	0.101320
$947$	−28.3270	−0.920503	−0.460251	−	0.887789i	$-0.652241\pi$
$947$	−28.3270	−0.920503	−0.460251	+	0.887789i	$0.652241\pi$
$948$	−10.4462	−0.339277
$949$	−2.67529	−0.0868437
$950$	−30.0020	−0.973393
$951$	80.0644	2.59627
$952$	0.187486	0.00607646
$953$	22.1388	0.717145	0.358573	−	0.933502i	$-0.383264\pi$
$953$	22.1388	0.717145	0.358573	+	0.933502i	$0.383264\pi$
$954$	22.4275	0.726118
$955$	12.2518	0.396459
$956$	−12.7237	−0.411514
$957$	−8.38732	−0.271124
$958$	−24.2538	−0.783604
$959$	0.715671	0.0231102
$960$	−1.91457	−0.0617925
$961$	−10.7333	−0.346237
$962$	10.1209	0.326309
$963$	−33.9076	−1.09266
$964$	−21.2333	−0.683877
$965$	15.1505	0.487711
$966$	−0.520125	−0.0167348
$967$	51.3225	1.65042	0.825211	−	0.564825i	$-0.191056\pi$
$967$	51.3225	1.65042	0.825211	+	0.564825i	$0.191056\pi$
$968$	−7.92550	−0.254735
$969$	54.5592	1.75269
$970$	−0.274216	−0.00880454
$971$	−36.5930	−1.17432	−0.587162	−	0.809469i	$-0.699755\pi$
$971$	−36.5930	−1.17432	−0.587162	+	0.809469i	$0.699755\pi$
$972$	22.0590	0.707544
$973$	0.220198	0.00705922
$974$	18.0981	0.579900
$975$	16.5619	0.530406
$976$	11.7182	0.375090
$977$	9.51623	0.304451	0.152226	−	0.988346i	$-0.451356\pi$
$977$	9.51623	0.304451	0.152226	+	0.988346i	$0.451356\pi$
$978$	31.0898	0.994141
$979$	−22.2909	−0.712420
$980$	−5.46534	−0.174584
$981$	−20.2284	−0.645843
$982$	−21.5514	−0.687733
$983$	45.3795	1.44738	0.723690	−	0.690125i	$-0.242444\pi$
$983$	45.3795	1.44738	0.723690	+	0.690125i	$0.242444\pi$
$984$	2.59373	0.0826850
$985$	−6.81045	−0.216999
$986$	6.35637	0.202428
$987$	−1.00614	−0.0320257
$988$	10.5201	0.334688
$989$	6.55190	0.208338
$990$	−4.11923	−0.130918
$991$	11.7082	0.371923	0.185961	−	0.982557i	$-0.440460\pi$
$991$	11.7082	0.371923	0.185961	+	0.982557i	$0.440460\pi$
$992$	−4.50185	−0.142934
$993$	32.1464	1.02014
$994$	0.255699	0.00811028
$995$	−2.96691	−0.0940573
$996$	21.1067	0.668791
$997$	7.56438	0.239566	0.119783	−	0.992800i	$-0.461780\pi$
$997$	7.56438	0.239566	0.119783	+	0.992800i	$0.461780\pi$
$998$	1.22402	0.0387457
$999$	−0.120730	−0.00381973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.e.1.8	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.e.1.8	✓	56	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.45102 q^{3} +1.00000 q^{4} +0.781132 q^{5} -2.45102 q^{6} -0.0575638 q^{7} +1.00000 q^{8} +3.00749 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.45102 q^{3} +1.00000 q^{4} +0.781132 q^{5} -2.45102 q^{6} -0.0575638 q^{7} +1.00000 q^{8} +3.00749 q^{9} +0.781132 q^{10} -1.75342 q^{11} -2.45102 q^{12} +1.53928 q^{13} -0.0575638 q^{14} -1.91457 q^{15} +1.00000 q^{16} -3.25701 q^{17} +3.00749 q^{18} +6.83443 q^{19} +0.781132 q^{20} +0.141090 q^{21} -1.75342 q^{22} -3.68648 q^{23} -2.45102 q^{24} -4.38983 q^{25} +1.53928 q^{26} -0.0183618 q^{27} -0.0575638 q^{28} -1.95159 q^{29} -1.91457 q^{30} -4.50185 q^{31} +1.00000 q^{32} +4.29768 q^{33} -3.25701 q^{34} -0.0449649 q^{35} +3.00749 q^{36} +6.57508 q^{37} +6.83443 q^{38} -3.77279 q^{39} +0.781132 q^{40} -1.05822 q^{41} +0.141090 q^{42} -1.77728 q^{43} -1.75342 q^{44} +2.34925 q^{45} -3.68648 q^{46} -7.13117 q^{47} -2.45102 q^{48} -6.99669 q^{49} -4.38983 q^{50} +7.98300 q^{51} +1.53928 q^{52} +7.45722 q^{53} -0.0183618 q^{54} -1.36966 q^{55} -0.0575638 q^{56} -16.7513 q^{57} -1.95159 q^{58} -12.8504 q^{59} -1.91457 q^{60} +11.7182 q^{61} -4.50185 q^{62} -0.173123 q^{63} +1.00000 q^{64} +1.20238 q^{65} +4.29768 q^{66} +11.2517 q^{67} -3.25701 q^{68} +9.03563 q^{69} -0.0449649 q^{70} -4.44201 q^{71} +3.00749 q^{72} -1.73802 q^{73} +6.57508 q^{74} +10.7596 q^{75} +6.83443 q^{76} +0.100934 q^{77} -3.77279 q^{78} +4.26198 q^{79} +0.781132 q^{80} -8.97747 q^{81} -1.05822 q^{82} -8.61139 q^{83} +0.141090 q^{84} -2.54416 q^{85} -1.77728 q^{86} +4.78339 q^{87} -1.75342 q^{88} +12.7128 q^{89} +2.34925 q^{90} -0.0886066 q^{91} -3.68648 q^{92} +11.0341 q^{93} -7.13117 q^{94} +5.33859 q^{95} -2.45102 q^{96} -0.351049 q^{97} -6.99669 q^{98} -5.27341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * q^99

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