LMFDB - Embedded newform 4002.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 $ x^7 - x^6 - 13x^5 + 16x^4 + 19x^3 - 8x^2 - 10*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.02256$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.662522 q^{5} +1.00000 q^{6} -0.537195 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.662522 q^{5} +1.00000 q^{6} -0.537195 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.662522 q^{10} +1.91221 q^{11} -1.00000 q^{12} -2.29155 q^{13} +0.537195 q^{14} +0.662522 q^{15} +1.00000 q^{16} -1.24969 q^{17} -1.00000 q^{18} +2.15128 q^{19} -0.662522 q^{20} +0.537195 q^{21} -1.91221 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.56106 q^{25} +2.29155 q^{26} -1.00000 q^{27} -0.537195 q^{28} +1.00000 q^{29} -0.662522 q^{30} +4.51797 q^{31} -1.00000 q^{32} -1.91221 q^{33} +1.24969 q^{34} +0.355903 q^{35} +1.00000 q^{36} -4.68235 q^{37} -2.15128 q^{38} +2.29155 q^{39} +0.662522 q^{40} +3.06169 q^{41} -0.537195 q^{42} +3.78428 q^{43} +1.91221 q^{44} -0.662522 q^{45} -1.00000 q^{46} -3.25727 q^{47} -1.00000 q^{48} -6.71142 q^{49} +4.56106 q^{50} +1.24969 q^{51} -2.29155 q^{52} -1.37436 q^{53} +1.00000 q^{54} -1.26688 q^{55} +0.537195 q^{56} -2.15128 q^{57} -1.00000 q^{58} +10.9987 q^{59} +0.662522 q^{60} +0.986600 q^{61} -4.51797 q^{62} -0.537195 q^{63} +1.00000 q^{64} +1.51820 q^{65} +1.91221 q^{66} -13.5609 q^{67} -1.24969 q^{68} -1.00000 q^{69} -0.355903 q^{70} +4.39652 q^{71} -1.00000 q^{72} +6.04309 q^{73} +4.68235 q^{74} +4.56106 q^{75} +2.15128 q^{76} -1.02723 q^{77} -2.29155 q^{78} +0.602927 q^{79} -0.662522 q^{80} +1.00000 q^{81} -3.06169 q^{82} -5.40097 q^{83} +0.537195 q^{84} +0.827946 q^{85} -3.78428 q^{86} -1.00000 q^{87} -1.91221 q^{88} -1.27788 q^{89} +0.662522 q^{90} +1.23101 q^{91} +1.00000 q^{92} -4.51797 q^{93} +3.25727 q^{94} -1.42527 q^{95} +1.00000 q^{96} +10.7530 q^{97} +6.71142 q^{98} +1.91221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.662522	−0.296289	−0.148144	−	0.988966i	$-0.547330\pi$
$5$	−0.662522	−0.296289	−0.148144	+	0.988966i	$0.547330\pi$
$6$	1.00000	0.408248
$7$	−0.537195	−0.203041	−0.101520	−	0.994833i	$-0.532371\pi$
$7$	−0.537195	−0.203041	−0.101520	+	0.994833i	$0.532371\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.662522	0.209508
$11$	1.91221	0.576553	0.288277	−	0.957547i	$-0.406918\pi$
$11$	1.91221	0.576553	0.288277	+	0.957547i	$0.406918\pi$
$12$	−1.00000	−0.288675
$13$	−2.29155	−0.635562	−0.317781	−	0.948164i	$-0.602938\pi$
$13$	−2.29155	−0.635562	−0.317781	+	0.948164i	$0.602938\pi$
$14$	0.537195	0.143571
$15$	0.662522	0.171062
$16$	1.00000	0.250000
$17$	−1.24969	−0.303094	−0.151547	−	0.988450i	$-0.548426\pi$
$17$	−1.24969	−0.303094	−0.151547	+	0.988450i	$0.548426\pi$
$18$	−1.00000	−0.235702
$19$	2.15128	0.493537	0.246768	−	0.969074i	$-0.420631\pi$
$19$	2.15128	0.493537	0.246768	+	0.969074i	$0.420631\pi$
$20$	−0.662522	−0.148144
$21$	0.537195	0.117226
$22$	−1.91221	−0.407685
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.56106	−0.912213
$26$	2.29155	0.449410
$27$	−1.00000	−0.192450
$28$	−0.537195	−0.101520
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−0.662522	−0.120959
$31$	4.51797	0.811451	0.405725	−	0.913995i	$-0.367019\pi$
$31$	4.51797	0.811451	0.405725	+	0.913995i	$0.367019\pi$
$32$	−1.00000	−0.176777
$33$	−1.91221	−0.332873
$34$	1.24969	0.214320
$35$	0.355903	0.0601586
$36$	1.00000	0.166667
$37$	−4.68235	−0.769773	−0.384887	−	0.922964i	$-0.625759\pi$
$37$	−4.68235	−0.769773	−0.384887	+	0.922964i	$0.625759\pi$
$38$	−2.15128	−0.348983
$39$	2.29155	0.366942
$40$	0.662522	0.104754
$41$	3.06169	0.478155	0.239078	−	0.971000i	$-0.423155\pi$
$41$	3.06169	0.478155	0.239078	+	0.971000i	$0.423155\pi$
$42$	−0.537195	−0.0828910
$43$	3.78428	0.577097	0.288549	−	0.957465i	$-0.406827\pi$
$43$	3.78428	0.577097	0.288549	+	0.957465i	$0.406827\pi$
$44$	1.91221	0.288277
$45$	−0.662522	−0.0987629
$46$	−1.00000	−0.147442
$47$	−3.25727	−0.475122	−0.237561	−	0.971373i	$-0.576348\pi$
$47$	−3.25727	−0.475122	−0.237561	+	0.971373i	$0.576348\pi$
$48$	−1.00000	−0.144338
$49$	−6.71142	−0.958775
$50$	4.56106	0.645032
$51$	1.24969	0.174991
$52$	−2.29155	−0.317781
$53$	−1.37436	−0.188783	−0.0943917	−	0.995535i	$-0.530091\pi$
$53$	−1.37436	−0.188783	−0.0943917	+	0.995535i	$0.530091\pi$
$54$	1.00000	0.136083
$55$	−1.26688	−0.170826
$56$	0.537195	0.0717857
$57$	−2.15128	−0.284944
$58$	−1.00000	−0.131306
$59$	10.9987	1.43190	0.715952	−	0.698150i	$-0.245993\pi$
$59$	10.9987	1.43190	0.715952	+	0.698150i	$0.245993\pi$
$60$	0.662522	0.0855312
$61$	0.986600	0.126321	0.0631606	−	0.998003i	$-0.479882\pi$
$61$	0.986600	0.126321	0.0631606	+	0.998003i	$0.479882\pi$
$62$	−4.51797	−0.573782
$63$	−0.537195	−0.0676802
$64$	1.00000	0.125000
$65$	1.51820	0.188310
$66$	1.91221	0.235377
$67$	−13.5609	−1.65673	−0.828363	−	0.560192i	$-0.810728\pi$
$67$	−13.5609	−1.65673	−0.828363	+	0.560192i	$0.810728\pi$
$68$	−1.24969	−0.151547
$69$	−1.00000	−0.120386
$70$	−0.355903	−0.0425386
$71$	4.39652	0.521771	0.260886	−	0.965370i	$-0.415985\pi$
$71$	4.39652	0.521771	0.260886	+	0.965370i	$0.415985\pi$
$72$	−1.00000	−0.117851
$73$	6.04309	0.707290	0.353645	−	0.935380i	$-0.384942\pi$
$73$	6.04309	0.707290	0.353645	+	0.935380i	$0.384942\pi$
$74$	4.68235	0.544312
$75$	4.56106	0.526666
$76$	2.15128	0.246768
$77$	−1.02723	−0.117064
$78$	−2.29155	−0.259467
$79$	0.602927	0.0678346	0.0339173	−	0.999425i	$-0.489202\pi$
$79$	0.602927	0.0678346	0.0339173	+	0.999425i	$0.489202\pi$
$80$	−0.662522	−0.0740722
$81$	1.00000	0.111111
$82$	−3.06169	−0.338107
$83$	−5.40097	−0.592833	−0.296416	−	0.955059i	$-0.595792\pi$
$83$	−5.40097	−0.592833	−0.296416	+	0.955059i	$0.595792\pi$
$84$	0.537195	0.0586128
$85$	0.827946	0.0898034
$86$	−3.78428	−0.408069
$87$	−1.00000	−0.107211
$88$	−1.91221	−0.203842
$89$	−1.27788	−0.135455	−0.0677277	−	0.997704i	$-0.521575\pi$
$89$	−1.27788	−0.135455	−0.0677277	+	0.997704i	$0.521575\pi$
$90$	0.662522	0.0698359
$91$	1.23101	0.129045
$92$	1.00000	0.104257
$93$	−4.51797	−0.468491
$94$	3.25727	0.335962
$95$	−1.42527	−0.146229
$96$	1.00000	0.102062
$97$	10.7530	1.09180	0.545902	−	0.837849i	$-0.316187\pi$
$97$	10.7530	1.09180	0.545902	+	0.837849i	$0.316187\pi$
$98$	6.71142	0.677956
$99$	1.91221	0.192184
$100$	−4.56106	−0.456106
$101$	−7.87633	−0.783724	−0.391862	−	0.920024i	$-0.628169\pi$
$101$	−7.87633	−0.783724	−0.391862	+	0.920024i	$0.628169\pi$
$102$	−1.24969	−0.123738
$103$	−13.6072	−1.34076	−0.670378	−	0.742019i	$-0.733868\pi$
$103$	−13.6072	−1.34076	−0.670378	+	0.742019i	$0.733868\pi$
$104$	2.29155	0.224705
$105$	−0.355903	−0.0347326
$106$	1.37436	0.133490
$107$	14.6516	1.41642	0.708212	−	0.706000i	$-0.249502\pi$
$107$	14.6516	1.41642	0.708212	+	0.706000i	$0.249502\pi$
$108$	−1.00000	−0.0962250
$109$	−5.86994	−0.562239	−0.281119	−	0.959673i	$-0.590706\pi$
$109$	−5.86994	−0.562239	−0.281119	+	0.959673i	$0.590706\pi$
$110$	1.26688	0.120792
$111$	4.68235	0.444429
$112$	−0.537195	−0.0507601
$113$	12.2930	1.15642	0.578212	−	0.815887i	$-0.303751\pi$
$113$	12.2930	1.15642	0.578212	+	0.815887i	$0.303751\pi$
$114$	2.15128	0.201486
$115$	−0.662522	−0.0617805
$116$	1.00000	0.0928477
$117$	−2.29155	−0.211854
$118$	−10.9987	−1.01251
$119$	0.671326	0.0615404
$120$	−0.662522	−0.0604797
$121$	−7.34345	−0.667586
$122$	−0.986600	−0.0893226
$123$	−3.06169	−0.276063
$124$	4.51797	0.405725
$125$	6.33441	0.566567
$126$	0.537195	0.0478571
$127$	−9.47091	−0.840407	−0.420204	−	0.907430i	$-0.638041\pi$
$127$	−9.47091	−0.840407	−0.420204	+	0.907430i	$0.638041\pi$
$128$	−1.00000	−0.0883883
$129$	−3.78428	−0.333187
$130$	−1.51820	−0.133155
$131$	−15.9426	−1.39291	−0.696455	−	0.717601i	$-0.745240\pi$
$131$	−15.9426	−1.39291	−0.696455	+	0.717601i	$0.745240\pi$
$132$	−1.91221	−0.166437
$133$	−1.15565	−0.100208
$134$	13.5609	1.17148
$135$	0.662522	0.0570208
$136$	1.24969	0.107160
$137$	−4.27806	−0.365500	−0.182750	−	0.983159i	$-0.558500\pi$
$137$	−4.27806	−0.365500	−0.182750	+	0.983159i	$0.558500\pi$
$138$	1.00000	0.0851257
$139$	19.7483	1.67503	0.837515	−	0.546414i	$-0.184007\pi$
$139$	19.7483	1.67503	0.837515	+	0.546414i	$0.184007\pi$
$140$	0.355903	0.0300793
$141$	3.25727	0.274312
$142$	−4.39652	−0.368948
$143$	−4.38193	−0.366435
$144$	1.00000	0.0833333
$145$	−0.662522	−0.0550194
$146$	−6.04309	−0.500129
$147$	6.71142	0.553549
$148$	−4.68235	−0.384887
$149$	11.1893	0.916664	0.458332	−	0.888781i	$-0.348447\pi$
$149$	11.1893	0.916664	0.458332	+	0.888781i	$0.348447\pi$
$150$	−4.56106	−0.372409
$151$	1.02801	0.0836586	0.0418293	−	0.999125i	$-0.486681\pi$
$151$	1.02801	0.0836586	0.0418293	+	0.999125i	$0.486681\pi$
$152$	−2.15128	−0.174492
$153$	−1.24969	−0.101031
$154$	1.02723	0.0827765
$155$	−2.99325	−0.240424
$156$	2.29155	0.183471
$157$	−9.59821	−0.766021	−0.383010	−	0.923744i	$-0.625113\pi$
$157$	−9.59821	−0.766021	−0.383010	+	0.923744i	$0.625113\pi$
$158$	−0.602927	−0.0479663
$159$	1.37436	0.108994
$160$	0.662522	0.0523769
$161$	−0.537195	−0.0423369
$162$	−1.00000	−0.0785674
$163$	−18.3567	−1.43781	−0.718903	−	0.695111i	$-0.755355\pi$
$163$	−18.3567	−1.43781	−0.718903	+	0.695111i	$0.755355\pi$
$164$	3.06169	0.239078
$165$	1.26688	0.0986266
$166$	5.40097	0.419196
$167$	−24.1811	−1.87119	−0.935593	−	0.353079i	$-0.885135\pi$
$167$	−24.1811	−1.87119	−0.935593	+	0.353079i	$0.885135\pi$
$168$	−0.537195	−0.0414455
$169$	−7.74879	−0.596061
$170$	−0.827946	−0.0635006
$171$	2.15128	0.164512
$172$	3.78428	0.288549
$173$	14.9035	1.13309	0.566545	−	0.824031i	$-0.308280\pi$
$173$	14.9035	1.13309	0.566545	+	0.824031i	$0.308280\pi$
$174$	1.00000	0.0758098
$175$	2.45018	0.185216
$176$	1.91221	0.144138
$177$	−10.9987	−0.826710
$178$	1.27788	0.0957814
$179$	−5.77261	−0.431465	−0.215732	−	0.976453i	$-0.569214\pi$
$179$	−5.77261	−0.431465	−0.215732	+	0.976453i	$0.569214\pi$
$180$	−0.662522	−0.0493815
$181$	−6.84199	−0.508561	−0.254281	−	0.967130i	$-0.581839\pi$
$181$	−6.84199	−0.508561	−0.254281	+	0.967130i	$0.581839\pi$
$182$	−1.23101	−0.0912485
$183$	−0.986600	−0.0729316
$184$	−1.00000	−0.0737210
$185$	3.10216	0.228075
$186$	4.51797	0.331273
$187$	−2.38967	−0.174750
$188$	−3.25727	−0.237561
$189$	0.537195	0.0390752
$190$	1.42527	0.103400
$191$	19.2737	1.39460	0.697298	−	0.716782i	$-0.254386\pi$
$191$	19.2737	1.39460	0.697298	+	0.716782i	$0.254386\pi$
$192$	−1.00000	−0.0721688
$193$	0.191419	0.0137786	0.00688932	−	0.999976i	$-0.497807\pi$
$193$	0.191419	0.0137786	0.00688932	+	0.999976i	$0.497807\pi$
$194$	−10.7530	−0.772022
$195$	−1.51820	−0.108721
$196$	−6.71142	−0.479387
$197$	−1.90518	−0.135739	−0.0678693	−	0.997694i	$-0.521620\pi$
$197$	−1.90518	−0.135739	−0.0678693	+	0.997694i	$0.521620\pi$
$198$	−1.91221	−0.135895
$199$	−12.6011	−0.893270	−0.446635	−	0.894716i	$-0.647378\pi$
$199$	−12.6011	−0.893270	−0.446635	+	0.894716i	$0.647378\pi$
$200$	4.56106	0.322516
$201$	13.5609	0.956511
$202$	7.87633	0.554177
$203$	−0.537195	−0.0377037
$204$	1.24969	0.0874957
$205$	−2.02843	−0.141672
$206$	13.6072	0.948058
$207$	1.00000	0.0695048
$208$	−2.29155	−0.158890
$209$	4.11369	0.284550
$210$	0.355903	0.0245597
$211$	−26.0281	−1.79185	−0.895923	−	0.444209i	$-0.853485\pi$
$211$	−26.0281	−1.79185	−0.895923	+	0.444209i	$0.853485\pi$
$212$	−1.37436	−0.0943917
$213$	−4.39652	−0.301245
$214$	−14.6516	−1.00156
$215$	−2.50717	−0.170987
$216$	1.00000	0.0680414
$217$	−2.42703	−0.164757
$218$	5.86994	0.397563
$219$	−6.04309	−0.408354
$220$	−1.26688	−0.0854131
$221$	2.86373	0.192635
$222$	−4.68235	−0.314259
$223$	−17.3745	−1.16348	−0.581740	−	0.813375i	$-0.697628\pi$
$223$	−17.3745	−1.16348	−0.581740	+	0.813375i	$0.697628\pi$
$224$	0.537195	0.0358928
$225$	−4.56106	−0.304071
$226$	−12.2930	−0.817715
$227$	−7.96370	−0.528569	−0.264285	−	0.964445i	$-0.585136\pi$
$227$	−7.96370	−0.528569	−0.264285	+	0.964445i	$0.585136\pi$
$228$	−2.15128	−0.142472
$229$	4.43814	0.293280	0.146640	−	0.989190i	$-0.453154\pi$
$229$	4.43814	0.293280	0.146640	+	0.989190i	$0.453154\pi$
$230$	0.662522	0.0436854
$231$	1.02723	0.0675867
$232$	−1.00000	−0.0656532
$233$	19.2617	1.26188	0.630938	−	0.775833i	$-0.282670\pi$
$233$	19.2617	1.26188	0.630938	+	0.775833i	$0.282670\pi$
$234$	2.29155	0.149803
$235$	2.15801	0.140773
$236$	10.9987	0.715952
$237$	−0.602927	−0.0391643
$238$	−0.671326	−0.0435156
$239$	23.8180	1.54066	0.770331	−	0.637645i	$-0.220091\pi$
$239$	23.8180	1.54066	0.770331	+	0.637645i	$0.220091\pi$
$240$	0.662522	0.0427656
$241$	−5.72420	−0.368728	−0.184364	−	0.982858i	$-0.559023\pi$
$241$	−5.72420	−0.368728	−0.184364	+	0.982858i	$0.559023\pi$
$242$	7.34345	0.472055
$243$	−1.00000	−0.0641500
$244$	0.986600	0.0631606
$245$	4.44646	0.284074
$246$	3.06169	0.195206
$247$	−4.92976	−0.313673
$248$	−4.51797	−0.286891
$249$	5.40097	0.342272
$250$	−6.33441	−0.400623
$251$	−13.2587	−0.836879	−0.418440	−	0.908245i	$-0.637423\pi$
$251$	−13.2587	−0.836879	−0.418440	+	0.908245i	$0.637423\pi$
$252$	−0.537195	−0.0338401
$253$	1.91221	0.120220
$254$	9.47091	0.594258
$255$	−0.827946	−0.0518480
$256$	1.00000	0.0625000
$257$	−15.5021	−0.966994	−0.483497	−	0.875346i	$-0.660634\pi$
$257$	−15.5021	−0.966994	−0.483497	+	0.875346i	$0.660634\pi$
$258$	3.78428	0.235599
$259$	2.51533	0.156295
$260$	1.51820	0.0941549
$261$	1.00000	0.0618984
$262$	15.9426	0.984936
$263$	−31.9201	−1.96828	−0.984138	−	0.177403i	$-0.943230\pi$
$263$	−31.9201	−1.96828	−0.984138	+	0.177403i	$0.943230\pi$
$264$	1.91221	0.117688
$265$	0.910546	0.0559344
$266$	1.15565	0.0708577
$267$	1.27788	0.0782052
$268$	−13.5609	−0.828363
$269$	1.05924	0.0645828	0.0322914	−	0.999478i	$-0.489720\pi$
$269$	1.05924	0.0645828	0.0322914	+	0.999478i	$0.489720\pi$
$270$	−0.662522	−0.0403198
$271$	−7.26353	−0.441228	−0.220614	−	0.975361i	$-0.570806\pi$
$271$	−7.26353	−0.441228	−0.220614	+	0.975361i	$0.570806\pi$
$272$	−1.24969	−0.0757735
$273$	−1.23101	−0.0745041
$274$	4.27806	0.258447
$275$	−8.72172	−0.525939
$276$	−1.00000	−0.0601929
$277$	15.1693	0.911435	0.455718	−	0.890124i	$-0.349383\pi$
$277$	15.1693	0.911435	0.455718	+	0.890124i	$0.349383\pi$
$278$	−19.7483	−1.18443
$279$	4.51797	0.270484
$280$	−0.355903	−0.0212693
$281$	−9.83912	−0.586953	−0.293476	−	0.955966i	$-0.594812\pi$
$281$	−9.83912	−0.586953	−0.293476	+	0.955966i	$0.594812\pi$
$282$	−3.25727	−0.193968
$283$	6.58812	0.391623	0.195811	−	0.980642i	$-0.437266\pi$
$283$	6.58812	0.391623	0.195811	+	0.980642i	$0.437266\pi$
$284$	4.39652	0.260886
$285$	1.42527	0.0844256
$286$	4.38193	0.259109
$287$	−1.64472	−0.0970849
$288$	−1.00000	−0.0589256
$289$	−15.4383	−0.908134
$290$	0.662522	0.0389046
$291$	−10.7530	−0.630353
$292$	6.04309	0.353645
$293$	−26.1212	−1.52602	−0.763008	−	0.646389i	$-0.776278\pi$
$293$	−26.1212	−1.52602	−0.763008	+	0.646389i	$0.776278\pi$
$294$	−6.71142	−0.391418
$295$	−7.28685	−0.424257
$296$	4.68235	0.272156
$297$	−1.91221	−0.110958
$298$	−11.1893	−0.648179
$299$	−2.29155	−0.132524
$300$	4.56106	0.263333
$301$	−2.03290	−0.117174
$302$	−1.02801	−0.0591556
$303$	7.87633	0.452483
$304$	2.15128	0.123384
$305$	−0.653644	−0.0374276
$306$	1.24969	0.0714400
$307$	−18.1507	−1.03591	−0.517957	−	0.855407i	$-0.673307\pi$
$307$	−18.1507	−1.03591	−0.517957	+	0.855407i	$0.673307\pi$
$308$	−1.02723	−0.0585318
$309$	13.6072	0.774086
$310$	2.99325	0.170005
$311$	−1.06964	−0.0606539	−0.0303270	−	0.999540i	$-0.509655\pi$
$311$	−1.06964	−0.0606539	−0.0303270	+	0.999540i	$0.509655\pi$
$312$	−2.29155	−0.129734
$313$	−18.6843	−1.05610	−0.528049	−	0.849214i	$-0.677076\pi$
$313$	−18.6843	−1.05610	−0.528049	+	0.849214i	$0.677076\pi$
$314$	9.59821	0.541658
$315$	0.355903	0.0200529
$316$	0.602927	0.0339173
$317$	12.9906	0.729627	0.364813	−	0.931081i	$-0.381133\pi$
$317$	12.9906	0.729627	0.364813	+	0.931081i	$0.381133\pi$
$318$	−1.37436	−0.0770705
$319$	1.91221	0.107063
$320$	−0.662522	−0.0370361
$321$	−14.6516	−0.817773
$322$	0.537195	0.0299367
$323$	−2.68843	−0.149588
$324$	1.00000	0.0555556
$325$	10.4519	0.579768
$326$	18.3567	1.01668
$327$	5.86994	0.324609
$328$	−3.06169	−0.169053
$329$	1.74979	0.0964691
$330$	−1.26688	−0.0697395
$331$	9.73971	0.535343	0.267671	−	0.963510i	$-0.413746\pi$
$331$	9.73971	0.535343	0.267671	+	0.963510i	$0.413746\pi$
$332$	−5.40097	−0.296416
$333$	−4.68235	−0.256591
$334$	24.1811	1.32313
$335$	8.98438	0.490869
$336$	0.537195	0.0293064
$337$	−7.46365	−0.406571	−0.203286	−	0.979119i	$-0.565162\pi$
$337$	−7.46365	−0.406571	−0.203286	+	0.979119i	$0.565162\pi$
$338$	7.74879	0.421479
$339$	−12.2930	−0.667662
$340$	0.827946	0.0449017
$341$	8.63930	0.467844
$342$	−2.15128	−0.116328
$343$	7.36570	0.397711
$344$	−3.78428	−0.204035
$345$	0.662522	0.0356690
$346$	−14.9035	−0.801215
$347$	−18.1759	−0.975734	−0.487867	−	0.872918i	$-0.662225\pi$
$347$	−18.1759	−0.975734	−0.487867	+	0.872918i	$0.662225\pi$
$348$	−1.00000	−0.0536056
$349$	−21.2660	−1.13834	−0.569170	−	0.822220i	$-0.692735\pi$
$349$	−21.2660	−1.13834	−0.569170	+	0.822220i	$0.692735\pi$
$350$	−2.45018	−0.130968
$351$	2.29155	0.122314
$352$	−1.91221	−0.101921
$353$	26.0964	1.38897	0.694487	−	0.719506i	$-0.255632\pi$
$353$	26.0964	1.38897	0.694487	+	0.719506i	$0.255632\pi$
$354$	10.9987	0.584572
$355$	−2.91279	−0.154595
$356$	−1.27788	−0.0677277
$357$	−0.671326	−0.0355304
$358$	5.77261	0.305092
$359$	−30.9097	−1.63135	−0.815675	−	0.578510i	$-0.803634\pi$
$359$	−30.9097	−1.63135	−0.815675	+	0.578510i	$0.803634\pi$
$360$	0.662522	0.0349180
$361$	−14.3720	−0.756422
$362$	6.84199	0.359607
$363$	7.34345	0.385431
$364$	1.23101	0.0645224
$365$	−4.00368	−0.209562
$366$	0.986600	0.0515704
$367$	−6.59652	−0.344336	−0.172168	−	0.985068i	$-0.555077\pi$
$367$	−6.59652	−0.344336	−0.172168	+	0.985068i	$0.555077\pi$
$368$	1.00000	0.0521286
$369$	3.06169	0.159385
$370$	−3.10216	−0.161273
$371$	0.738301	0.0383307
$372$	−4.51797	−0.234246
$373$	−2.27258	−0.117670	−0.0588349	−	0.998268i	$-0.518739\pi$
$373$	−2.27258	−0.117670	−0.0588349	+	0.998268i	$0.518739\pi$
$374$	2.38967	0.123567
$375$	−6.33441	−0.327108
$376$	3.25727	0.167981
$377$	−2.29155	−0.118021
$378$	−0.537195	−0.0276303
$379$	−12.4181	−0.637872	−0.318936	−	0.947776i	$-0.603326\pi$
$379$	−12.4181	−0.637872	−0.318936	+	0.947776i	$0.603326\pi$
$380$	−1.42527	−0.0731147
$381$	9.47091	0.485209
$382$	−19.2737	−0.986128
$383$	−22.5910	−1.15434	−0.577172	−	0.816623i	$-0.695844\pi$
$383$	−22.5910	−1.15434	−0.577172	+	0.816623i	$0.695844\pi$
$384$	1.00000	0.0510310
$385$	0.680562	0.0346847
$386$	−0.191419	−0.00974297
$387$	3.78428	0.192366
$388$	10.7530	0.545902
$389$	−19.3186	−0.979492	−0.489746	−	0.871865i	$-0.662911\pi$
$389$	−19.3186	−0.979492	−0.489746	+	0.871865i	$0.662911\pi$
$390$	1.51820	0.0768772
$391$	−1.24969	−0.0631995
$392$	6.71142	0.338978
$393$	15.9426	0.804197
$394$	1.90518	0.0959816
$395$	−0.399452	−0.0200986
$396$	1.91221	0.0960922
$397$	−5.49304	−0.275688	−0.137844	−	0.990454i	$-0.544017\pi$
$397$	−5.49304	−0.275688	−0.137844	+	0.990454i	$0.544017\pi$
$398$	12.6011	0.631638
$399$	1.15565	0.0578551
$400$	−4.56106	−0.228053
$401$	6.34075	0.316642	0.158321	−	0.987388i	$-0.449392\pi$
$401$	6.34075	0.316642	0.158321	+	0.987388i	$0.449392\pi$
$402$	−13.5609	−0.676356
$403$	−10.3531	−0.515727
$404$	−7.87633	−0.391862
$405$	−0.662522	−0.0329210
$406$	0.537195	0.0266605
$407$	−8.95363	−0.443815
$408$	−1.24969	−0.0618688
$409$	10.1610	0.502428	0.251214	−	0.967932i	$-0.419170\pi$
$409$	10.1610	0.502428	0.251214	+	0.967932i	$0.419170\pi$
$410$	2.02843	0.100177
$411$	4.27806	0.211021
$412$	−13.6072	−0.670378
$413$	−5.90842	−0.290735
$414$	−1.00000	−0.0491473
$415$	3.57826	0.175650
$416$	2.29155	0.112353
$417$	−19.7483	−0.967079
$418$	−4.11369	−0.201207
$419$	21.4505	1.04793	0.523963	−	0.851741i	$-0.324453\pi$
$419$	21.4505	1.04793	0.523963	+	0.851741i	$0.324453\pi$
$420$	−0.355903	−0.0173663
$421$	−12.7848	−0.623093	−0.311547	−	0.950231i	$-0.600847\pi$
$421$	−12.7848	−0.623093	−0.311547	+	0.950231i	$0.600847\pi$
$422$	26.0281	1.26703
$423$	−3.25727	−0.158374
$424$	1.37436	0.0667450
$425$	5.69991	0.276486
$426$	4.39652	0.213012
$427$	−0.529997	−0.0256483
$428$	14.6516	0.708212
$429$	4.38193	0.211561
$430$	2.50717	0.120906
$431$	35.4828	1.70915	0.854574	−	0.519330i	$-0.173818\pi$
$431$	35.4828	1.70915	0.854574	+	0.519330i	$0.173818\pi$
$432$	−1.00000	−0.0481125
$433$	6.08785	0.292563	0.146282	−	0.989243i	$-0.453269\pi$
$433$	6.08785	0.292563	0.146282	+	0.989243i	$0.453269\pi$
$434$	2.42703	0.116501
$435$	0.662522	0.0317655
$436$	−5.86994	−0.281119
$437$	2.15128	0.102910
$438$	6.04309	0.288750
$439$	−27.4443	−1.30985	−0.654923	−	0.755696i	$-0.727299\pi$
$439$	−27.4443	−1.30985	−0.654923	+	0.755696i	$0.727299\pi$
$440$	1.26688	0.0603962
$441$	−6.71142	−0.319592
$442$	−2.86373	−0.136214
$443$	−25.5492	−1.21388	−0.606940	−	0.794747i	$-0.707603\pi$
$443$	−25.5492	−1.21388	−0.606940	+	0.794747i	$0.707603\pi$
$444$	4.68235	0.222214
$445$	0.846626	0.0401339
$446$	17.3745	0.822704
$447$	−11.1893	−0.529236
$448$	−0.537195	−0.0253801
$449$	−17.0641	−0.805306	−0.402653	−	0.915353i	$-0.631912\pi$
$449$	−17.0641	−0.805306	−0.402653	+	0.915353i	$0.631912\pi$
$450$	4.56106	0.215011
$451$	5.85459	0.275682
$452$	12.2930	0.578212
$453$	−1.02801	−0.0483003
$454$	7.96370	0.373755
$455$	−0.815570	−0.0382345
$456$	2.15128	0.100743
$457$	29.3382	1.37238	0.686192	−	0.727421i	$-0.259281\pi$
$457$	29.3382	1.37238	0.686192	+	0.727421i	$0.259281\pi$
$458$	−4.43814	−0.207381
$459$	1.24969	0.0583305
$460$	−0.662522	−0.0308902
$461$	−4.69336	−0.218591	−0.109296	−	0.994009i	$-0.534860\pi$
$461$	−4.69336	−0.218591	−0.109296	+	0.994009i	$0.534860\pi$
$462$	−1.02723	−0.0477910
$463$	−36.1807	−1.68146	−0.840730	−	0.541454i	$-0.817874\pi$
$463$	−36.1807	−1.68146	−0.840730	+	0.541454i	$0.817874\pi$
$464$	1.00000	0.0464238
$465$	2.99325	0.138809
$466$	−19.2617	−0.892282
$467$	28.4564	1.31681	0.658403	−	0.752666i	$-0.271232\pi$
$467$	28.4564	1.31681	0.658403	+	0.752666i	$0.271232\pi$
$468$	−2.29155	−0.105927
$469$	7.28484	0.336383
$470$	−2.15801	−0.0995418
$471$	9.59821	0.442262
$472$	−10.9987	−0.506255
$473$	7.23634	0.332727
$474$	0.602927	0.0276934
$475$	−9.81211	−0.450211
$476$	0.671326	0.0307702
$477$	−1.37436	−0.0629278
$478$	−23.8180	−1.08941
$479$	−23.5349	−1.07534	−0.537668	−	0.843157i	$-0.680695\pi$
$479$	−23.5349	−1.07534	−0.537668	+	0.843157i	$0.680695\pi$
$480$	−0.662522	−0.0302398
$481$	10.7298	0.489238
$482$	5.72420	0.260730
$483$	0.537195	0.0244432
$484$	−7.34345	−0.333793
$485$	−7.12411	−0.323489
$486$	1.00000	0.0453609
$487$	−4.31147	−0.195371	−0.0976856	−	0.995217i	$-0.531144\pi$
$487$	−4.31147	−0.195371	−0.0976856	+	0.995217i	$0.531144\pi$
$488$	−0.986600	−0.0446613
$489$	18.3567	0.830117
$490$	−4.44646	−0.200871
$491$	−25.5072	−1.15112	−0.575562	−	0.817758i	$-0.695217\pi$
$491$	−25.5072	−1.15112	−0.575562	+	0.817758i	$0.695217\pi$
$492$	−3.06169	−0.138032
$493$	−1.24969	−0.0562832
$494$	4.92976	0.221800
$495$	−1.26688	−0.0569421
$496$	4.51797	0.202863
$497$	−2.36179	−0.105941
$498$	−5.40097	−0.242023
$499$	23.8655	1.06837	0.534184	−	0.845368i	$-0.320619\pi$
$499$	23.8655	1.06837	0.534184	+	0.845368i	$0.320619\pi$
$500$	6.33441	0.283284
$501$	24.1811	1.08033
$502$	13.2587	0.591763
$503$	30.9069	1.37807	0.689036	−	0.724727i	$-0.258034\pi$
$503$	30.9069	1.37807	0.689036	+	0.724727i	$0.258034\pi$
$504$	0.537195	0.0239286
$505$	5.21824	0.232209
$506$	−1.91221	−0.0850081
$507$	7.74879	0.344136
$508$	−9.47091	−0.420204
$509$	−17.4891	−0.775192	−0.387596	−	0.921829i	$-0.626694\pi$
$509$	−17.4891	−0.775192	−0.387596	+	0.921829i	$0.626694\pi$
$510$	0.827946	0.0366621
$511$	−3.24631	−0.143608
$512$	−1.00000	−0.0441942
$513$	−2.15128	−0.0949812
$514$	15.5021	0.683768
$515$	9.01506	0.397251
$516$	−3.78428	−0.166594
$517$	−6.22859	−0.273933
$518$	−2.51533	−0.110517
$519$	−14.9035	−0.654190
$520$	−1.51820	−0.0665776
$521$	−36.4214	−1.59565	−0.797825	−	0.602889i	$-0.794016\pi$
$521$	−36.4214	−1.59565	−0.797825	+	0.602889i	$0.794016\pi$
$522$	−1.00000	−0.0437688
$523$	−23.5458	−1.02959	−0.514793	−	0.857314i	$-0.672131\pi$
$523$	−23.5458	−1.02959	−0.514793	+	0.857314i	$0.672131\pi$
$524$	−15.9426	−0.696455
$525$	−2.45018	−0.106935
$526$	31.9201	1.39178
$527$	−5.64605	−0.245946
$528$	−1.91221	−0.0832183
$529$	1.00000	0.0434783
$530$	−0.910546	−0.0395516
$531$	10.9987	0.477301
$532$	−1.15565	−0.0501040
$533$	−7.01601	−0.303897
$534$	−1.27788	−0.0552994
$535$	−9.70701	−0.419671
$536$	13.5609	0.585741
$537$	5.77261	0.249106
$538$	−1.05924	−0.0456670
$539$	−12.8337	−0.552785
$540$	0.662522	0.0285104
$541$	17.2236	0.740500	0.370250	−	0.928932i	$-0.379272\pi$
$541$	17.2236	0.740500	0.370250	+	0.928932i	$0.379272\pi$
$542$	7.26353	0.311995
$543$	6.84199	0.293618
$544$	1.24969	0.0535800
$545$	3.88897	0.166585
$546$	1.23101	0.0526823
$547$	0.713616	0.0305120	0.0152560	−	0.999884i	$-0.495144\pi$
$547$	0.713616	0.0305120	0.0152560	+	0.999884i	$0.495144\pi$
$548$	−4.27806	−0.182750
$549$	0.986600	0.0421071
$550$	8.72172	0.371895
$551$	2.15128	0.0916475
$552$	1.00000	0.0425628
$553$	−0.323889	−0.0137732
$554$	−15.1693	−0.644482
$555$	−3.10216	−0.131679
$556$	19.7483	0.837515
$557$	10.8643	0.460337	0.230168	−	0.973151i	$-0.426072\pi$
$557$	10.8643	0.460337	0.230168	+	0.973151i	$0.426072\pi$
$558$	−4.51797	−0.191261
$559$	−8.67187	−0.366781
$560$	0.355903	0.0150397
$561$	2.38967	0.100892
$562$	9.83912	0.415038
$563$	−26.2759	−1.10740	−0.553699	−	0.832717i	$-0.686784\pi$
$563$	−26.2759	−1.10740	−0.553699	+	0.832717i	$0.686784\pi$
$564$	3.25727	0.137156
$565$	−8.14435	−0.342635
$566$	−6.58812	−0.276919
$567$	−0.537195	−0.0225601
$568$	−4.39652	−0.184474
$569$	−36.3653	−1.52451	−0.762256	−	0.647276i	$-0.775908\pi$
$569$	−36.3653	−1.52451	−0.762256	+	0.647276i	$0.775908\pi$
$570$	−1.42527	−0.0596979
$571$	−3.95249	−0.165406	−0.0827032	−	0.996574i	$-0.526355\pi$
$571$	−3.95249	−0.165406	−0.0827032	+	0.996574i	$0.526355\pi$
$572$	−4.38193	−0.183218
$573$	−19.2737	−0.805170
$574$	1.64472	0.0686494
$575$	−4.56106	−0.190210
$576$	1.00000	0.0416667
$577$	25.8782	1.07732	0.538662	−	0.842522i	$-0.318930\pi$
$577$	25.8782	1.07732	0.538662	+	0.842522i	$0.318930\pi$
$578$	15.4383	0.642148
$579$	−0.191419	−0.00795510
$580$	−0.662522	−0.0275097
$581$	2.90137	0.120369
$582$	10.7530	0.445727
$583$	−2.62807	−0.108844
$584$	−6.04309	−0.250065
$585$	1.51820	0.0627699
$586$	26.1212	1.07906
$587$	37.4508	1.54576	0.772881	−	0.634551i	$-0.218815\pi$
$587$	37.4508	1.54576	0.772881	+	0.634551i	$0.218815\pi$
$588$	6.71142	0.276774
$589$	9.71939	0.400481
$590$	7.28685	0.299995
$591$	1.90518	0.0783687
$592$	−4.68235	−0.192443
$593$	19.9449	0.819040	0.409520	−	0.912301i	$-0.365696\pi$
$593$	19.9449	0.819040	0.409520	+	0.912301i	$0.365696\pi$
$594$	1.91221	0.0784590
$595$	−0.444768	−0.0182337
$596$	11.1893	0.458332
$597$	12.6011	0.515730
$598$	2.29155	0.0937085
$599$	33.8808	1.38433	0.692165	−	0.721739i	$-0.256657\pi$
$599$	33.8808	1.38433	0.692165	+	0.721739i	$0.256657\pi$
$600$	−4.56106	−0.186205
$601$	16.9018	0.689439	0.344720	−	0.938706i	$-0.387974\pi$
$601$	16.9018	0.689439	0.344720	+	0.938706i	$0.387974\pi$
$602$	2.03290	0.0828546
$603$	−13.5609	−0.552242
$604$	1.02801	0.0418293
$605$	4.86520	0.197798
$606$	−7.87633	−0.319954
$607$	−37.7572	−1.53252	−0.766259	−	0.642532i	$-0.777884\pi$
$607$	−37.7572	−1.53252	−0.766259	+	0.642532i	$0.777884\pi$
$608$	−2.15128	−0.0872458
$609$	0.537195	0.0217682
$610$	0.653644	0.0264653
$611$	7.46421	0.301970
$612$	−1.24969	−0.0505157
$613$	−8.00674	−0.323389	−0.161695	−	0.986841i	$-0.551696\pi$
$613$	−8.00674	−0.323389	−0.161695	+	0.986841i	$0.551696\pi$
$614$	18.1507	0.732502
$615$	2.02843	0.0817944
$616$	1.02723	0.0413883
$617$	−21.4621	−0.864030	−0.432015	−	0.901866i	$-0.642197\pi$
$617$	−21.4621	−0.864030	−0.432015	+	0.901866i	$0.642197\pi$
$618$	−13.6072	−0.547362
$619$	−27.4480	−1.10323	−0.551614	−	0.834099i	$-0.685988\pi$
$619$	−27.4480	−1.10323	−0.551614	+	0.834099i	$0.685988\pi$
$620$	−2.99325	−0.120212
$621$	−1.00000	−0.0401286
$622$	1.06964	0.0428888
$623$	0.686472	0.0275029
$624$	2.29155	0.0917355
$625$	18.6086	0.744346
$626$	18.6843	0.746774
$627$	−4.11369	−0.164285
$628$	−9.59821	−0.383010
$629$	5.85148	0.233314
$630$	−0.355903	−0.0141795
$631$	−39.5511	−1.57450	−0.787252	−	0.616631i	$-0.788497\pi$
$631$	−39.5511	−1.57450	−0.787252	+	0.616631i	$0.788497\pi$
$632$	−0.602927	−0.0239831
$633$	26.0281	1.03452
$634$	−12.9906	−0.515924
$635$	6.27468	0.249003
$636$	1.37436	0.0544971
$637$	15.3796	0.609361
$638$	−1.91221	−0.0757051
$639$	4.39652	0.173924
$640$	0.662522	0.0261885
$641$	−24.7869	−0.979024	−0.489512	−	0.871997i	$-0.662825\pi$
$641$	−24.7869	−0.979024	−0.489512	+	0.871997i	$0.662825\pi$
$642$	14.6516	0.578253
$643$	4.91631	0.193880	0.0969401	−	0.995290i	$-0.469094\pi$
$643$	4.91631	0.193880	0.0969401	+	0.995290i	$0.469094\pi$
$644$	−0.537195	−0.0211684
$645$	2.50717	0.0987196
$646$	2.68843	0.105775
$647$	−3.82063	−0.150205	−0.0751023	−	0.997176i	$-0.523928\pi$
$647$	−3.82063	−0.150205	−0.0751023	+	0.997176i	$0.523928\pi$
$648$	−1.00000	−0.0392837
$649$	21.0318	0.825569
$650$	−10.4519	−0.409958
$651$	2.42703	0.0951227
$652$	−18.3567	−0.718903
$653$	−36.0170	−1.40946	−0.704728	−	0.709478i	$-0.748931\pi$
$653$	−36.0170	−1.40946	−0.704728	+	0.709478i	$0.748931\pi$
$654$	−5.86994	−0.229533
$655$	10.5623	0.412703
$656$	3.06169	0.119539
$657$	6.04309	0.235763
$658$	−1.74979	−0.0682139
$659$	7.32402	0.285303	0.142652	−	0.989773i	$-0.454437\pi$
$659$	7.32402	0.285303	0.142652	+	0.989773i	$0.454437\pi$
$660$	1.26688	0.0493133
$661$	8.92072	0.346976	0.173488	−	0.984836i	$-0.444496\pi$
$661$	8.92072	0.346976	0.173488	+	0.984836i	$0.444496\pi$
$662$	−9.73971	−0.378544
$663$	−2.86373	−0.111218
$664$	5.40097	0.209598
$665$	0.765646	0.0296905
$666$	4.68235	0.181437
$667$	1.00000	0.0387202
$668$	−24.1811	−0.935593
$669$	17.3745	0.671735
$670$	−8.98438	−0.347097
$671$	1.88659	0.0728309
$672$	−0.537195	−0.0207227
$673$	20.4446	0.788082	0.394041	−	0.919093i	$-0.371077\pi$
$673$	20.4446	0.788082	0.394041	+	0.919093i	$0.371077\pi$
$674$	7.46365	0.287489
$675$	4.56106	0.175555
$676$	−7.74879	−0.298031
$677$	19.9088	0.765157	0.382579	−	0.923923i	$-0.375036\pi$
$677$	19.9088	0.765157	0.382579	+	0.923923i	$0.375036\pi$
$678$	12.2930	0.472108
$679$	−5.77647	−0.221680
$680$	−0.827946	−0.0317503
$681$	7.96370	0.305170
$682$	−8.63930	−0.330816
$683$	37.1755	1.42248	0.711241	−	0.702948i	$-0.248134\pi$
$683$	37.1755	1.42248	0.711241	+	0.702948i	$0.248134\pi$
$684$	2.15128	0.0822561
$685$	2.83431	0.108293
$686$	−7.36570	−0.281224
$687$	−4.43814	−0.169326
$688$	3.78428	0.144274
$689$	3.14942	0.119984
$690$	−0.662522	−0.0252218
$691$	−8.31145	−0.316182	−0.158091	−	0.987425i	$-0.550534\pi$
$691$	−8.31145	−0.316182	−0.158091	+	0.987425i	$0.550534\pi$
$692$	14.9035	0.566545
$693$	−1.02723	−0.0390212
$694$	18.1759	0.689948
$695$	−13.0837	−0.496293
$696$	1.00000	0.0379049
$697$	−3.82616	−0.144926
$698$	21.2660	0.804928
$699$	−19.2617	−0.728545
$700$	2.45018	0.0926081
$701$	16.3750	0.618475	0.309238	−	0.950985i	$-0.399926\pi$
$701$	16.3750	0.618475	0.309238	+	0.950985i	$0.399926\pi$
$702$	−2.29155	−0.0864890
$703$	−10.0730	−0.379911
$704$	1.91221	0.0720692
$705$	−2.15801	−0.0812755
$706$	−26.0964	−0.982152
$707$	4.23112	0.159128
$708$	−10.9987	−0.413355
$709$	24.6775	0.926785	0.463392	−	0.886153i	$-0.346632\pi$
$709$	24.6775	0.926785	0.463392	+	0.886153i	$0.346632\pi$
$710$	2.91279	0.109315
$711$	0.602927	0.0226115
$712$	1.27788	0.0478907
$713$	4.51797	0.169199
$714$	0.671326	0.0251238
$715$	2.90312	0.108571
$716$	−5.77261	−0.215732
$717$	−23.8180	−0.889501
$718$	30.9097	1.15354
$719$	26.7816	0.998784	0.499392	−	0.866376i	$-0.333557\pi$
$719$	26.7816	0.998784	0.499392	+	0.866376i	$0.333557\pi$
$720$	−0.662522	−0.0246907
$721$	7.30971	0.272228
$722$	14.3720	0.534871
$723$	5.72420	0.212885
$724$	−6.84199	−0.254281
$725$	−4.56106	−0.169394
$726$	−7.34345	−0.272541
$727$	8.11252	0.300877	0.150438	−	0.988619i	$-0.451932\pi$
$727$	8.11252	0.300877	0.150438	+	0.988619i	$0.451932\pi$
$728$	−1.23101	−0.0456242
$729$	1.00000	0.0370370
$730$	4.00368	0.148183
$731$	−4.72917	−0.174915
$732$	−0.986600	−0.0364658
$733$	26.7905	0.989528	0.494764	−	0.869027i	$-0.335255\pi$
$733$	26.7905	0.989528	0.494764	+	0.869027i	$0.335255\pi$
$734$	6.59652	0.243482
$735$	−4.44646	−0.164010
$736$	−1.00000	−0.0368605
$737$	−25.9313	−0.955191
$738$	−3.06169	−0.112702
$739$	7.60479	0.279747	0.139873	−	0.990169i	$-0.455330\pi$
$739$	7.60479	0.279747	0.139873	+	0.990169i	$0.455330\pi$
$740$	3.10216	0.114038
$741$	4.92976	0.181099
$742$	−0.738301	−0.0271039
$743$	−36.2869	−1.33124	−0.665619	−	0.746292i	$-0.731832\pi$
$743$	−36.2869	−1.33124	−0.665619	+	0.746292i	$0.731832\pi$
$744$	4.51797	0.165637
$745$	−7.41316	−0.271597
$746$	2.27258	0.0832051
$747$	−5.40097	−0.197611
$748$	−2.38967	−0.0873749
$749$	−7.87077	−0.287592
$750$	6.33441	0.231300
$751$	−6.04448	−0.220566	−0.110283	−	0.993900i	$-0.535176\pi$
$751$	−6.04448	−0.220566	−0.110283	+	0.993900i	$0.535176\pi$
$752$	−3.25727	−0.118781
$753$	13.2587	0.483172
$754$	2.29155	0.0834534
$755$	−0.681082	−0.0247871
$756$	0.537195	0.0195376
$757$	−26.7788	−0.973294	−0.486647	−	0.873599i	$-0.661780\pi$
$757$	−26.7788	−0.973294	−0.486647	+	0.873599i	$0.661780\pi$
$758$	12.4181	0.451044
$759$	−1.91221	−0.0694088
$760$	1.42527	0.0516999
$761$	−47.7520	−1.73101	−0.865504	−	0.500901i	$-0.833002\pi$
$761$	−47.7520	−1.73101	−0.865504	+	0.500901i	$0.833002\pi$
$762$	−9.47091	−0.343095
$763$	3.15330	0.114157
$764$	19.2737	0.697298
$765$	0.827946	0.0299345
$766$	22.5910	0.816245
$767$	−25.2040	−0.910064
$768$	−1.00000	−0.0360844
$769$	37.4483	1.35042	0.675211	−	0.737625i	$-0.264053\pi$
$769$	37.4483	1.35042	0.675211	+	0.737625i	$0.264053\pi$
$770$	−0.680562	−0.0245258
$771$	15.5021	0.558294
$772$	0.191419	0.00688932
$773$	5.27035	0.189561	0.0947807	−	0.995498i	$-0.469785\pi$
$773$	5.27035	0.189561	0.0947807	+	0.995498i	$0.469785\pi$
$774$	−3.78428	−0.136023
$775$	−20.6067	−0.740216
$776$	−10.7530	−0.386011
$777$	−2.51533	−0.0902370
$778$	19.3186	0.692606
$779$	6.58654	0.235987
$780$	−1.51820	−0.0543604
$781$	8.40707	0.300829
$782$	1.24969	0.0446888
$783$	−1.00000	−0.0357371
$784$	−6.71142	−0.239694
$785$	6.35902	0.226963
$786$	−15.9426	−0.568653
$787$	16.0621	0.572552	0.286276	−	0.958147i	$-0.407583\pi$
$787$	16.0621	0.572552	0.286276	+	0.958147i	$0.407583\pi$
$788$	−1.90518	−0.0678693
$789$	31.9201	1.13639
$790$	0.399452	0.0142119
$791$	−6.60371	−0.234801
$792$	−1.91221	−0.0679474
$793$	−2.26084	−0.0802850
$794$	5.49304	0.194941
$795$	−0.910546	−0.0322937
$796$	−12.6011	−0.446635
$797$	34.8252	1.23357	0.616787	−	0.787130i	$-0.288434\pi$
$797$	34.8252	1.23357	0.616787	+	0.787130i	$0.288434\pi$
$798$	−1.15565	−0.0409097
$799$	4.07058	0.144007
$800$	4.56106	0.161258
$801$	−1.27788	−0.0451518
$802$	−6.34075	−0.223900
$803$	11.5557	0.407790
$804$	13.5609	0.478256
$805$	0.355903	0.0125439
$806$	10.3531	0.364674
$807$	−1.05924	−0.0372869
$808$	7.87633	0.277088
$809$	45.9202	1.61447	0.807234	−	0.590232i	$-0.200964\pi$
$809$	45.9202	1.61447	0.807234	+	0.590232i	$0.200964\pi$
$810$	0.662522	0.0232786
$811$	44.5143	1.56311	0.781554	−	0.623838i	$-0.214427\pi$
$811$	44.5143	1.56311	0.781554	+	0.623838i	$0.214427\pi$
$812$	−0.537195	−0.0188518
$813$	7.26353	0.254743
$814$	8.95363	0.313825
$815$	12.1617	0.426005
$816$	1.24969	0.0437479
$817$	8.14103	0.284819
$818$	−10.1610	−0.355270
$819$	1.23101	0.0430149
$820$	−2.02843	−0.0708360
$821$	−29.4704	−1.02853	−0.514263	−	0.857633i	$-0.671934\pi$
$821$	−29.4704	−1.02853	−0.514263	+	0.857633i	$0.671934\pi$
$822$	−4.27806	−0.149215
$823$	41.6081	1.45037	0.725183	−	0.688556i	$-0.241755\pi$
$823$	41.6081	1.45037	0.725183	+	0.688556i	$0.241755\pi$
$824$	13.6072	0.474029
$825$	8.72172	0.303651
$826$	5.90842	0.205580
$827$	44.9575	1.56333	0.781663	−	0.623701i	$-0.214372\pi$
$827$	44.9575	1.56333	0.781663	+	0.623701i	$0.214372\pi$
$828$	1.00000	0.0347524
$829$	34.5082	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$829$	34.5082	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$830$	−3.57826	−0.124203
$831$	−15.1693	−0.526217
$832$	−2.29155	−0.0794452
$833$	8.38719	0.290599
$834$	19.7483	0.683828
$835$	16.0205	0.554412
$836$	4.11369	0.142275
$837$	−4.51797	−0.156164
$838$	−21.4505	−0.740995
$839$	56.2374	1.94153	0.970766	−	0.240030i	$-0.0771571\pi$
$839$	56.2374	1.94153	0.970766	+	0.240030i	$0.0771571\pi$
$840$	0.355903	0.0122798
$841$	1.00000	0.0344828
$842$	12.7848	0.440593
$843$	9.83912	0.338877
$844$	−26.0281	−0.895923
$845$	5.13374	0.176606
$846$	3.25727	0.111987
$847$	3.94486	0.135547
$848$	−1.37436	−0.0471958
$849$	−6.58812	−0.226104
$850$	−5.69991	−0.195505
$851$	−4.68235	−0.160509
$852$	−4.39652	−0.150622
$853$	−11.8532	−0.405847	−0.202923	−	0.979195i	$-0.565044\pi$
$853$	−11.8532	−0.405847	−0.202923	+	0.979195i	$0.565044\pi$
$854$	0.529997	0.0181361
$855$	−1.42527	−0.0487431
$856$	−14.6516	−0.500782
$857$	35.4929	1.21241	0.606207	−	0.795307i	$-0.292690\pi$
$857$	35.4929	1.21241	0.606207	+	0.795307i	$0.292690\pi$
$858$	−4.38193	−0.149597
$859$	49.1378	1.67656	0.838281	−	0.545239i	$-0.183561\pi$
$859$	49.1378	1.67656	0.838281	+	0.545239i	$0.183561\pi$
$860$	−2.50717	−0.0854937
$861$	1.64472	0.0560520
$862$	−35.4828	−1.20855
$863$	−17.9972	−0.612633	−0.306317	−	0.951930i	$-0.599097\pi$
$863$	−17.9972	−0.612633	−0.306317	+	0.951930i	$0.599097\pi$
$864$	1.00000	0.0340207
$865$	−9.87387	−0.335722
$866$	−6.08785	−0.206873
$867$	15.4383	0.524311
$868$	−2.42703	−0.0823787
$869$	1.15292	0.0391103
$870$	−0.662522	−0.0224616
$871$	31.0755	1.05295
$872$	5.86994	0.198781
$873$	10.7530	0.363935
$874$	−2.15128	−0.0727680
$875$	−3.40281	−0.115036
$876$	−6.04309	−0.204177
$877$	45.0468	1.52112	0.760561	−	0.649266i	$-0.224924\pi$
$877$	45.0468	1.52112	0.760561	+	0.649266i	$0.224924\pi$
$878$	27.4443	0.926201
$879$	26.1212	0.881046
$880$	−1.26688	−0.0427066
$881$	23.1912	0.781333	0.390666	−	0.920532i	$-0.372245\pi$
$881$	23.1912	0.781333	0.390666	+	0.920532i	$0.372245\pi$
$882$	6.71142	0.225985
$883$	33.2408	1.11864	0.559321	−	0.828951i	$-0.311062\pi$
$883$	33.2408	1.11864	0.559321	+	0.828951i	$0.311062\pi$
$884$	2.86373	0.0963175
$885$	7.28685	0.244945
$886$	25.5492	0.858343
$887$	−50.4077	−1.69252	−0.846262	−	0.532767i	$-0.821152\pi$
$887$	−50.4077	−1.69252	−0.846262	+	0.532767i	$0.821152\pi$
$888$	−4.68235	−0.157129
$889$	5.08772	0.170637
$890$	−0.846626	−0.0283790
$891$	1.91221	0.0640615
$892$	−17.3745	−0.581740
$893$	−7.00730	−0.234490
$894$	11.1893	0.374226
$895$	3.82448	0.127838
$896$	0.537195	0.0179464
$897$	2.29155	0.0765127
$898$	17.0641	0.569438
$899$	4.51797	0.150683
$900$	−4.56106	−0.152035
$901$	1.71753	0.0572191
$902$	−5.85459	−0.194937
$903$	2.03290	0.0676505
$904$	−12.2930	−0.408858
$905$	4.53297	0.150681
$906$	1.02801	0.0341535
$907$	−42.3352	−1.40572	−0.702859	−	0.711330i	$-0.748093\pi$
$907$	−42.3352	−1.40572	−0.702859	+	0.711330i	$0.748093\pi$
$908$	−7.96370	−0.264285
$909$	−7.87633	−0.261241
$910$	0.815570	0.0270359
$911$	−9.78638	−0.324237	−0.162119	−	0.986771i	$-0.551833\pi$
$911$	−9.78638	−0.324237	−0.162119	+	0.986771i	$0.551833\pi$
$912$	−2.15128	−0.0712359
$913$	−10.3278	−0.341800
$914$	−29.3382	−0.970422
$915$	0.653644	0.0216088
$916$	4.43814	0.146640
$917$	8.56427	0.282817
$918$	−1.24969	−0.0412459
$919$	−39.7046	−1.30973	−0.654867	−	0.755744i	$-0.727275\pi$
$919$	−39.7046	−1.30973	−0.654867	+	0.755744i	$0.727275\pi$
$920$	0.662522	0.0218427
$921$	18.1507	0.598085
$922$	4.69336	0.154567
$923$	−10.0749	−0.331618
$924$	1.02723	0.0337934
$925$	21.3565	0.702197
$926$	36.1807	1.18897
$927$	−13.6072	−0.446919
$928$	−1.00000	−0.0328266
$929$	46.1659	1.51465	0.757326	−	0.653037i	$-0.226505\pi$
$929$	46.1659	1.51465	0.757326	+	0.653037i	$0.226505\pi$
$930$	−2.99325	−0.0981525
$931$	−14.4381	−0.473190
$932$	19.2617	0.630938
$933$	1.06964	0.0350186
$934$	−28.4564	−0.931122
$935$	1.58321	0.0517764
$936$	2.29155	0.0749017
$937$	−34.3820	−1.12321	−0.561606	−	0.827405i	$-0.689816\pi$
$937$	−34.3820	−1.12321	−0.561606	+	0.827405i	$0.689816\pi$
$938$	−7.28484	−0.237858
$939$	18.6843	0.609738
$940$	2.15801	0.0703867
$941$	5.80081	0.189101	0.0945505	−	0.995520i	$-0.469859\pi$
$941$	5.80081	0.189101	0.0945505	+	0.995520i	$0.469859\pi$
$942$	−9.59821	−0.312727
$943$	3.06169	0.0997023
$944$	10.9987	0.357976
$945$	−0.355903	−0.0115775
$946$	−7.23634	−0.235274
$947$	−21.4285	−0.696332	−0.348166	−	0.937433i	$-0.613195\pi$
$947$	−21.4285	−0.696332	−0.348166	+	0.937433i	$0.613195\pi$
$948$	−0.602927	−0.0195822
$949$	−13.8480	−0.449526
$950$	9.81211	0.318347
$951$	−12.9906	−0.421250
$952$	−0.671326	−0.0217578
$953$	−20.3925	−0.660579	−0.330289	−	0.943880i	$-0.607146\pi$
$953$	−20.3925	−0.660579	−0.330289	+	0.943880i	$0.607146\pi$
$954$	1.37436	0.0444967
$955$	−12.7692	−0.413203
$956$	23.8180	0.770331
$957$	−1.91221	−0.0618130
$958$	23.5349	0.760378
$959$	2.29815	0.0742113
$960$	0.662522	0.0213828
$961$	−10.5880	−0.341548
$962$	−10.7298	−0.345944
$963$	14.6516	0.472141
$964$	−5.72420	−0.184364
$965$	−0.126819	−0.00408246
$966$	−0.537195	−0.0172840
$967$	0.951791	0.0306075	0.0153038	−	0.999883i	$-0.495128\pi$
$967$	0.951791	0.0306075	0.0153038	+	0.999883i	$0.495128\pi$
$968$	7.34345	0.236027
$969$	2.68843	0.0863647
$970$	7.12411	0.228741
$971$	33.1967	1.06533	0.532666	−	0.846326i	$-0.321190\pi$
$971$	33.1967	1.06533	0.532666	+	0.846326i	$0.321190\pi$
$972$	−1.00000	−0.0320750
$973$	−10.6087	−0.340099
$974$	4.31147	0.138148
$975$	−10.4519	−0.334729
$976$	0.986600	0.0315803
$977$	5.22535	0.167174	0.0835869	−	0.996500i	$-0.473362\pi$
$977$	5.22535	0.167174	0.0835869	+	0.996500i	$0.473362\pi$
$978$	−18.3567	−0.586981
$979$	−2.44358	−0.0780972
$980$	4.44646	0.142037
$981$	−5.86994	−0.187413
$982$	25.5072	0.813968
$983$	−2.55887	−0.0816152	−0.0408076	−	0.999167i	$-0.512993\pi$
$983$	−2.55887	−0.0816152	−0.0408076	+	0.999167i	$0.512993\pi$
$984$	3.06169	0.0976030
$985$	1.26222	0.0402178
$986$	1.24969	0.0397982
$987$	−1.74979	−0.0556965
$988$	−4.92976	−0.156837
$989$	3.78428	0.120333
$990$	1.26688	0.0402641
$991$	−31.4192	−0.998063	−0.499032	−	0.866584i	$-0.666311\pi$
$991$	−31.4192	−0.998063	−0.499032	+	0.866584i	$0.666311\pi$
$992$	−4.51797	−0.143446
$993$	−9.73971	−0.309080
$994$	2.36179	0.0749114
$995$	8.34852	0.264666
$996$	5.40097	0.171136
$997$	−28.9702	−0.917496	−0.458748	−	0.888566i	$-0.651702\pi$
$997$	−28.9702	−0.917496	−0.458748	+	0.888566i	$0.651702\pi$
$998$	−23.8655	−0.755450
$999$	4.68235	0.148143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bg.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bg.1.3	✓	7	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.662522 q^{5} +1.00000 q^{6} -0.537195 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.662522 q^{5} +1.00000 q^{6} -0.537195 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.662522 q^{10} +1.91221 q^{11} -1.00000 q^{12} -2.29155 q^{13} +0.537195 q^{14} +0.662522 q^{15} +1.00000 q^{16} -1.24969 q^{17} -1.00000 q^{18} +2.15128 q^{19} -0.662522 q^{20} +0.537195 q^{21} -1.91221 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.56106 q^{25} +2.29155 q^{26} -1.00000 q^{27} -0.537195 q^{28} +1.00000 q^{29} -0.662522 q^{30} +4.51797 q^{31} -1.00000 q^{32} -1.91221 q^{33} +1.24969 q^{34} +0.355903 q^{35} +1.00000 q^{36} -4.68235 q^{37} -2.15128 q^{38} +2.29155 q^{39} +0.662522 q^{40} +3.06169 q^{41} -0.537195 q^{42} +3.78428 q^{43} +1.91221 q^{44} -0.662522 q^{45} -1.00000 q^{46} -3.25727 q^{47} -1.00000 q^{48} -6.71142 q^{49} +4.56106 q^{50} +1.24969 q^{51} -2.29155 q^{52} -1.37436 q^{53} +1.00000 q^{54} -1.26688 q^{55} +0.537195 q^{56} -2.15128 q^{57} -1.00000 q^{58} +10.9987 q^{59} +0.662522 q^{60} +0.986600 q^{61} -4.51797 q^{62} -0.537195 q^{63} +1.00000 q^{64} +1.51820 q^{65} +1.91221 q^{66} -13.5609 q^{67} -1.24969 q^{68} -1.00000 q^{69} -0.355903 q^{70} +4.39652 q^{71} -1.00000 q^{72} +6.04309 q^{73} +4.68235 q^{74} +4.56106 q^{75} +2.15128 q^{76} -1.02723 q^{77} -2.29155 q^{78} +0.602927 q^{79} -0.662522 q^{80} +1.00000 q^{81} -3.06169 q^{82} -5.40097 q^{83} +0.537195 q^{84} +0.827946 q^{85} -3.78428 q^{86} -1.00000 q^{87} -1.91221 q^{88} -1.27788 q^{89} +0.662522 q^{90} +1.23101 q^{91} +1.00000 q^{92} -4.51797 q^{93} +3.25727 q^{94} -1.42527 q^{95} +1.00000 q^{96} +10.7530 q^{97} +6.71142 q^{98} +1.91221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - q^99

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