LMFDB - Embedded newform 8034.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 45x^{6} + 7x^{5} - 123x^{4} + 37x^{3} + 87x^{2} - 54x + 8 $ x^9 - 4x^8 - 9x^7 + 45x^6 + 7x^5 - 123x^4 + 37x^3 + 87x^2 - 54x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.20969$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.20969 q^{5} -1.00000 q^{6} -2.02210 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.20969 q^{5} -1.00000 q^{6} -2.02210 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.20969 q^{10} -1.72557 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.02210 q^{14} +3.20969 q^{15} +1.00000 q^{16} -0.669533 q^{17} +1.00000 q^{18} +6.70293 q^{19} -3.20969 q^{20} +2.02210 q^{21} -1.72557 q^{22} +2.38462 q^{23} -1.00000 q^{24} +5.30211 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.02210 q^{28} -2.29980 q^{29} +3.20969 q^{30} -4.65872 q^{31} +1.00000 q^{32} +1.72557 q^{33} -0.669533 q^{34} +6.49033 q^{35} +1.00000 q^{36} -2.89073 q^{37} +6.70293 q^{38} -1.00000 q^{39} -3.20969 q^{40} -1.52823 q^{41} +2.02210 q^{42} +3.74925 q^{43} -1.72557 q^{44} -3.20969 q^{45} +2.38462 q^{46} +8.51407 q^{47} -1.00000 q^{48} -2.91109 q^{49} +5.30211 q^{50} +0.669533 q^{51} +1.00000 q^{52} +11.6372 q^{53} -1.00000 q^{54} +5.53854 q^{55} -2.02210 q^{56} -6.70293 q^{57} -2.29980 q^{58} -3.76784 q^{59} +3.20969 q^{60} -3.46636 q^{61} -4.65872 q^{62} -2.02210 q^{63} +1.00000 q^{64} -3.20969 q^{65} +1.72557 q^{66} -13.5209 q^{67} -0.669533 q^{68} -2.38462 q^{69} +6.49033 q^{70} +6.69404 q^{71} +1.00000 q^{72} +11.7410 q^{73} -2.89073 q^{74} -5.30211 q^{75} +6.70293 q^{76} +3.48928 q^{77} -1.00000 q^{78} +0.941600 q^{79} -3.20969 q^{80} +1.00000 q^{81} -1.52823 q^{82} +0.0679630 q^{83} +2.02210 q^{84} +2.14899 q^{85} +3.74925 q^{86} +2.29980 q^{87} -1.72557 q^{88} +15.8437 q^{89} -3.20969 q^{90} -2.02210 q^{91} +2.38462 q^{92} +4.65872 q^{93} +8.51407 q^{94} -21.5143 q^{95} -1.00000 q^{96} -14.9124 q^{97} -2.91109 q^{98} -1.72557 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9 - 4 * q^10 - 5 * q^11 - 9 * q^12 + 9 * q^13 - 4 * q^14 + 4 * q^15 + 9 * q^16 - 6 * q^17 + 9 * q^18 - 4 * q^19 - 4 * q^20 + 4 * q^21 - 5 * q^22 - 6 * q^23 - 9 * q^24 - 11 * q^25 + 9 * q^26 - 9 * q^27 - 4 * q^28 - 19 * q^29 + 4 * q^30 - 6 * q^31 + 9 * q^32 + 5 * q^33 - 6 * q^34 + 10 * q^35 + 9 * q^36 - 13 * q^37 - 4 * q^38 - 9 * q^39 - 4 * q^40 - 18 * q^41 + 4 * q^42 - 20 * q^43 - 5 * q^44 - 4 * q^45 - 6 * q^46 + 14 * q^47 - 9 * q^48 - 3 * q^49 - 11 * q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 - 9 * q^54 - 4 * q^55 - 4 * q^56 + 4 * q^57 - 19 * q^58 - 9 * q^59 + 4 * q^60 - 24 * q^61 - 6 * q^62 - 4 * q^63 + 9 * q^64 - 4 * q^65 + 5 * q^66 - 4 * q^67 - 6 * q^68 + 6 * q^69 + 10 * q^70 - 9 * q^71 + 9 * q^72 - 24 * q^73 - 13 * q^74 + 11 * q^75 - 4 * q^76 + 3 * q^77 - 9 * q^78 - 15 * q^79 - 4 * q^80 + 9 * q^81 - 18 * q^82 + 20 * q^83 + 4 * q^84 - 31 * q^85 - 20 * q^86 + 19 * q^87 - 5 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^91 - 6 * q^92 + 6 * q^93 + 14 * q^94 - 4 * q^95 - 9 * q^96 - 19 * q^97 - 3 * q^98 - 5 * 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
$5$	−3.20969	−1.43542	−0.717709	−	0.696344i	$-0.754809\pi$
$5$	−3.20969	−1.43542	−0.717709	+	0.696344i	$0.754809\pi$
$6$	−1.00000	−0.408248
$7$	−2.02210	−0.764284	−0.382142	−	0.924104i	$-0.624813\pi$
$7$	−2.02210	−0.764284	−0.382142	+	0.924104i	$0.624813\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.20969	−1.01499
$11$	−1.72557	−0.520279	−0.260139	−	0.965571i	$-0.583769\pi$
$11$	−1.72557	−0.520279	−0.260139	+	0.965571i	$0.583769\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.02210	−0.540430
$15$	3.20969	0.828739
$16$	1.00000	0.250000
$17$	−0.669533	−0.162386	−0.0811928	−	0.996698i	$-0.525873\pi$
$17$	−0.669533	−0.162386	−0.0811928	+	0.996698i	$0.525873\pi$
$18$	1.00000	0.235702
$19$	6.70293	1.53776	0.768879	−	0.639395i	$-0.220815\pi$
$19$	6.70293	1.53776	0.768879	+	0.639395i	$0.220815\pi$
$20$	−3.20969	−0.717709
$21$	2.02210	0.441259
$22$	−1.72557	−0.367893
$23$	2.38462	0.497227	0.248613	−	0.968603i	$-0.420025\pi$
$23$	2.38462	0.497227	0.248613	+	0.968603i	$0.420025\pi$
$24$	−1.00000	−0.204124
$25$	5.30211	1.06042
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	−2.02210	−0.382142
$29$	−2.29980	−0.427063	−0.213531	−	0.976936i	$-0.568497\pi$
$29$	−2.29980	−0.427063	−0.213531	+	0.976936i	$0.568497\pi$
$30$	3.20969	0.586007
$31$	−4.65872	−0.836731	−0.418365	−	0.908279i	$-0.637397\pi$
$31$	−4.65872	−0.836731	−0.418365	+	0.908279i	$0.637397\pi$
$32$	1.00000	0.176777
$33$	1.72557	0.300383
$34$	−0.669533	−0.114824
$35$	6.49033	1.09707
$36$	1.00000	0.166667
$37$	−2.89073	−0.475233	−0.237617	−	0.971359i	$-0.576366\pi$
$37$	−2.89073	−0.475233	−0.237617	+	0.971359i	$0.576366\pi$
$38$	6.70293	1.08736
$39$	−1.00000	−0.160128
$40$	−3.20969	−0.507497
$41$	−1.52823	−0.238670	−0.119335	−	0.992854i	$-0.538076\pi$
$41$	−1.52823	−0.238670	−0.119335	+	0.992854i	$0.538076\pi$
$42$	2.02210	0.312018
$43$	3.74925	0.571755	0.285878	−	0.958266i	$-0.407715\pi$
$43$	3.74925	0.571755	0.285878	+	0.958266i	$0.407715\pi$
$44$	−1.72557	−0.260139
$45$	−3.20969	−0.478472
$46$	2.38462	0.351593
$47$	8.51407	1.24190	0.620952	−	0.783848i	$-0.286746\pi$
$47$	8.51407	1.24190	0.620952	+	0.783848i	$0.286746\pi$
$48$	−1.00000	−0.144338
$49$	−2.91109	−0.415870
$50$	5.30211	0.749832
$51$	0.669533	0.0937534
$52$	1.00000	0.138675
$53$	11.6372	1.59849	0.799247	−	0.601003i	$-0.205232\pi$
$53$	11.6372	1.59849	0.799247	+	0.601003i	$0.205232\pi$
$54$	−1.00000	−0.136083
$55$	5.53854	0.746817
$56$	−2.02210	−0.270215
$57$	−6.70293	−0.887825
$58$	−2.29980	−0.301979
$59$	−3.76784	−0.490530	−0.245265	−	0.969456i	$-0.578875\pi$
$59$	−3.76784	−0.490530	−0.245265	+	0.969456i	$0.578875\pi$
$60$	3.20969	0.414369
$61$	−3.46636	−0.443822	−0.221911	−	0.975067i	$-0.571229\pi$
$61$	−3.46636	−0.443822	−0.221911	+	0.975067i	$0.571229\pi$
$62$	−4.65872	−0.591658
$63$	−2.02210	−0.254761
$64$	1.00000	0.125000
$65$	−3.20969	−0.398113
$66$	1.72557	0.212403
$67$	−13.5209	−1.65184	−0.825919	−	0.563789i	$-0.809343\pi$
$67$	−13.5209	−1.65184	−0.825919	+	0.563789i	$0.809343\pi$
$68$	−0.669533	−0.0811928
$69$	−2.38462	−0.287074
$70$	6.49033	0.775743
$71$	6.69404	0.794436	0.397218	−	0.917724i	$-0.369976\pi$
$71$	6.69404	0.794436	0.397218	+	0.917724i	$0.369976\pi$
$72$	1.00000	0.117851
$73$	11.7410	1.37418	0.687091	−	0.726571i	$-0.258887\pi$
$73$	11.7410	1.37418	0.687091	+	0.726571i	$0.258887\pi$
$74$	−2.89073	−0.336041
$75$	−5.30211	−0.612235
$76$	6.70293	0.768879
$77$	3.48928	0.397641
$78$	−1.00000	−0.113228
$79$	0.941600	0.105938	0.0529692	−	0.998596i	$-0.483132\pi$
$79$	0.941600	0.105938	0.0529692	+	0.998596i	$0.483132\pi$
$80$	−3.20969	−0.358854
$81$	1.00000	0.111111
$82$	−1.52823	−0.168765
$83$	0.0679630	0.00745991	0.00372996	−	0.999993i	$-0.498813\pi$
$83$	0.0679630	0.00745991	0.00372996	+	0.999993i	$0.498813\pi$
$84$	2.02210	0.220630
$85$	2.14899	0.233091
$86$	3.74925	0.404292
$87$	2.29980	0.246565
$88$	−1.72557	−0.183946
$89$	15.8437	1.67943	0.839716	−	0.543026i	$-0.182721\pi$
$89$	15.8437	1.67943	0.839716	+	0.543026i	$0.182721\pi$
$90$	−3.20969	−0.338331
$91$	−2.02210	−0.211974
$92$	2.38462	0.248613
$93$	4.65872	0.483087
$94$	8.51407	0.878159
$95$	−21.5143	−2.20732
$96$	−1.00000	−0.102062
$97$	−14.9124	−1.51413	−0.757064	−	0.653341i	$-0.773367\pi$
$97$	−14.9124	−1.51413	−0.757064	+	0.653341i	$0.773367\pi$
$98$	−2.91109	−0.294065
$99$	−1.72557	−0.173426
$100$	5.30211	0.530211
$101$	18.7134	1.86205	0.931027	−	0.364950i	$-0.118914\pi$
$101$	18.7134	1.86205	0.931027	+	0.364950i	$0.118914\pi$
$102$	0.669533	0.0662937
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	1.00000	0.0980581
$105$	−6.49033	−0.633391
$106$	11.6372	1.13031
$107$	3.67967	0.355727	0.177863	−	0.984055i	$-0.443081\pi$
$107$	3.67967	0.355727	0.177863	+	0.984055i	$0.443081\pi$
$108$	−1.00000	−0.0962250
$109$	−0.643981	−0.0616822	−0.0308411	−	0.999524i	$-0.509819\pi$
$109$	−0.643981	−0.0616822	−0.0308411	+	0.999524i	$0.509819\pi$
$110$	5.53854	0.528079
$111$	2.89073	0.274376
$112$	−2.02210	−0.191071
$113$	−2.61607	−0.246099	−0.123049	−	0.992401i	$-0.539267\pi$
$113$	−2.61607	−0.246099	−0.123049	+	0.992401i	$0.539267\pi$
$114$	−6.70293	−0.627787
$115$	−7.65388	−0.713728
$116$	−2.29980	−0.213531
$117$	1.00000	0.0924500
$118$	−3.76784	−0.346857
$119$	1.35387	0.124109
$120$	3.20969	0.293003
$121$	−8.02241	−0.729310
$122$	−3.46636	−0.313829
$123$	1.52823	0.137796
$124$	−4.65872	−0.418365
$125$	−0.969688	−0.0867315
$126$	−2.02210	−0.180143
$127$	−17.1214	−1.51928	−0.759639	−	0.650345i	$-0.774624\pi$
$127$	−17.1214	−1.51928	−0.759639	+	0.650345i	$0.774624\pi$
$128$	1.00000	0.0883883
$129$	−3.74925	−0.330103
$130$	−3.20969	−0.281508
$131$	−11.9320	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$131$	−11.9320	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$132$	1.72557	0.150192
$133$	−13.5540	−1.17528
$134$	−13.5209	−1.16803
$135$	3.20969	0.276246
$136$	−0.669533	−0.0574120
$137$	−6.86144	−0.586212	−0.293106	−	0.956080i	$-0.594689\pi$
$137$	−6.86144	−0.586212	−0.293106	+	0.956080i	$0.594689\pi$
$138$	−2.38462	−0.202992
$139$	−13.0202	−1.10436	−0.552181	−	0.833724i	$-0.686204\pi$
$139$	−13.0202	−1.10436	−0.552181	+	0.833724i	$0.686204\pi$
$140$	6.49033	0.548533
$141$	−8.51407	−0.717014
$142$	6.69404	0.561751
$143$	−1.72557	−0.144299
$144$	1.00000	0.0833333
$145$	7.38166	0.613013
$146$	11.7410	0.971694
$147$	2.91109	0.240103
$148$	−2.89073	−0.237617
$149$	11.3872	0.932873	0.466437	−	0.884555i	$-0.345538\pi$
$149$	11.3872	0.932873	0.466437	+	0.884555i	$0.345538\pi$
$150$	−5.30211	−0.432916
$151$	6.07828	0.494643	0.247322	−	0.968933i	$-0.420450\pi$
$151$	6.07828	0.494643	0.247322	+	0.968933i	$0.420450\pi$
$152$	6.70293	0.543679
$153$	−0.669533	−0.0541285
$154$	3.48928	0.281174
$155$	14.9530	1.20106
$156$	−1.00000	−0.0800641
$157$	−10.3162	−0.823326	−0.411663	−	0.911336i	$-0.635052\pi$
$157$	−10.3162	−0.823326	−0.411663	+	0.911336i	$0.635052\pi$
$158$	0.941600	0.0749097
$159$	−11.6372	−0.922891
$160$	−3.20969	−0.253748
$161$	−4.82194	−0.380022
$162$	1.00000	0.0785674
$163$	−15.6784	−1.22803	−0.614013	−	0.789296i	$-0.710446\pi$
$163$	−15.6784	−1.22803	−0.614013	+	0.789296i	$0.710446\pi$
$164$	−1.52823	−0.119335
$165$	−5.53854	−0.431175
$166$	0.0679630	0.00527495
$167$	1.59761	0.123627	0.0618134	−	0.998088i	$-0.480312\pi$
$167$	1.59761	0.123627	0.0618134	+	0.998088i	$0.480312\pi$
$168$	2.02210	0.156009
$169$	1.00000	0.0769231
$170$	2.14899	0.164820
$171$	6.70293	0.512586
$172$	3.74925	0.285878
$173$	−1.82588	−0.138819	−0.0694096	−	0.997588i	$-0.522112\pi$
$173$	−1.82588	−0.138819	−0.0694096	+	0.997588i	$0.522112\pi$
$174$	2.29980	0.174348
$175$	−10.7214	−0.810464
$176$	−1.72557	−0.130070
$177$	3.76784	0.283208
$178$	15.8437	1.18754
$179$	−6.67663	−0.499035	−0.249517	−	0.968370i	$-0.580272\pi$
$179$	−6.67663	−0.499035	−0.249517	+	0.968370i	$0.580272\pi$
$180$	−3.20969	−0.239236
$181$	−21.0273	−1.56295	−0.781475	−	0.623936i	$-0.785532\pi$
$181$	−21.0273	−1.56295	−0.781475	+	0.623936i	$0.785532\pi$
$182$	−2.02210	−0.149888
$183$	3.46636	0.256241
$184$	2.38462	0.175796
$185$	9.27836	0.682158
$186$	4.65872	0.341594
$187$	1.15533	0.0844858
$188$	8.51407	0.620952
$189$	2.02210	0.147086
$190$	−21.5143	−1.56081
$191$	−20.1793	−1.46013	−0.730063	−	0.683380i	$-0.760509\pi$
$191$	−20.1793	−1.46013	−0.730063	+	0.683380i	$0.760509\pi$
$192$	−1.00000	−0.0721688
$193$	−6.28353	−0.452298	−0.226149	−	0.974093i	$-0.572614\pi$
$193$	−6.28353	−0.452298	−0.226149	+	0.974093i	$0.572614\pi$
$194$	−14.9124	−1.07065
$195$	3.20969	0.229851
$196$	−2.91109	−0.207935
$197$	−17.1542	−1.22219	−0.611094	−	0.791558i	$-0.709270\pi$
$197$	−17.1542	−1.22219	−0.611094	+	0.791558i	$0.709270\pi$
$198$	−1.72557	−0.122631
$199$	−25.3168	−1.79466	−0.897330	−	0.441360i	$-0.854496\pi$
$199$	−25.3168	−1.79466	−0.897330	+	0.441360i	$0.854496\pi$
$200$	5.30211	0.374916
$201$	13.5209	0.953689
$202$	18.7134	1.31667
$203$	4.65044	0.326397
$204$	0.669533	0.0468767
$205$	4.90515	0.342591
$206$	−1.00000	−0.0696733
$207$	2.38462	0.165742
$208$	1.00000	0.0693375
$209$	−11.5664	−0.800063
$210$	−6.49033	−0.447875
$211$	2.77247	0.190865	0.0954323	−	0.995436i	$-0.469577\pi$
$211$	2.77247	0.190865	0.0954323	+	0.995436i	$0.469577\pi$
$212$	11.6372	0.799247
$213$	−6.69404	−0.458668
$214$	3.67967	0.251537
$215$	−12.0339	−0.820708
$216$	−1.00000	−0.0680414
$217$	9.42042	0.639500
$218$	−0.643981	−0.0436159
$219$	−11.7410	−0.793385
$220$	5.53854	0.373409
$221$	−0.669533	−0.0450377
$222$	2.89073	0.194013
$223$	29.2727	1.96025	0.980123	−	0.198392i	$-0.0635718\pi$
$223$	29.2727	1.96025	0.980123	+	0.198392i	$0.0635718\pi$
$224$	−2.02210	−0.135108
$225$	5.30211	0.353474
$226$	−2.61607	−0.174018
$227$	17.6781	1.17334	0.586669	−	0.809827i	$-0.300439\pi$
$227$	17.6781	1.17334	0.586669	+	0.809827i	$0.300439\pi$
$228$	−6.70293	−0.443912
$229$	−22.4144	−1.48119	−0.740593	−	0.671953i	$-0.765455\pi$
$229$	−22.4144	−1.48119	−0.740593	+	0.671953i	$0.765455\pi$
$230$	−7.65388	−0.504682
$231$	−3.48928	−0.229578
$232$	−2.29980	−0.150989
$233$	−20.3329	−1.33205	−0.666026	−	0.745929i	$-0.732006\pi$
$233$	−20.3329	−1.33205	−0.666026	+	0.745929i	$0.732006\pi$
$234$	1.00000	0.0653720
$235$	−27.3275	−1.78265
$236$	−3.76784	−0.245265
$237$	−0.941600	−0.0611635
$238$	1.35387	0.0877581
$239$	−18.4706	−1.19477	−0.597383	−	0.801956i	$-0.703793\pi$
$239$	−18.4706	−1.19477	−0.597383	+	0.801956i	$0.703793\pi$
$240$	3.20969	0.207185
$241$	15.6356	1.00718	0.503588	−	0.863944i	$-0.332013\pi$
$241$	15.6356	1.00718	0.503588	+	0.863944i	$0.332013\pi$
$242$	−8.02241	−0.515700
$243$	−1.00000	−0.0641500
$244$	−3.46636	−0.221911
$245$	9.34370	0.596947
$246$	1.52823	0.0974365
$247$	6.70293	0.426497
$248$	−4.65872	−0.295829
$249$	−0.0679630	−0.00430698
$250$	−0.969688	−0.0613284
$251$	−12.3105	−0.777033	−0.388517	−	0.921442i	$-0.627012\pi$
$251$	−12.3105	−0.777033	−0.388517	+	0.921442i	$0.627012\pi$
$252$	−2.02210	−0.127381
$253$	−4.11482	−0.258697
$254$	−17.1214	−1.07429
$255$	−2.14899	−0.134575
$256$	1.00000	0.0625000
$257$	−25.6930	−1.60269	−0.801344	−	0.598204i	$-0.795881\pi$
$257$	−25.6930	−1.60269	−0.801344	+	0.598204i	$0.795881\pi$
$258$	−3.74925	−0.233418
$259$	5.84536	0.363213
$260$	−3.20969	−0.199057
$261$	−2.29980	−0.142354
$262$	−11.9320	−0.737163
$263$	−12.2397	−0.754734	−0.377367	−	0.926064i	$-0.623171\pi$
$263$	−12.2397	−0.754734	−0.377367	+	0.926064i	$0.623171\pi$
$264$	1.72557	0.106201
$265$	−37.3518	−2.29451
$266$	−13.5540	−0.831051
$267$	−15.8437	−0.969620
$268$	−13.5209	−0.825919
$269$	−18.2413	−1.11219	−0.556097	−	0.831117i	$-0.687702\pi$
$269$	−18.2413	−1.11219	−0.556097	+	0.831117i	$0.687702\pi$
$270$	3.20969	0.195336
$271$	3.96166	0.240654	0.120327	−	0.992734i	$-0.461606\pi$
$271$	3.96166	0.240654	0.120327	+	0.992734i	$0.461606\pi$
$272$	−0.669533	−0.0405964
$273$	2.02210	0.122383
$274$	−6.86144	−0.414515
$275$	−9.14916	−0.551715
$276$	−2.38462	−0.143537
$277$	−18.5109	−1.11221	−0.556106	−	0.831112i	$-0.687705\pi$
$277$	−18.5109	−1.11221	−0.556106	+	0.831112i	$0.687705\pi$
$278$	−13.0202	−0.780903
$279$	−4.65872	−0.278910
$280$	6.49033	0.387871
$281$	18.6586	1.11308	0.556541	−	0.830820i	$-0.312128\pi$
$281$	18.6586	1.11308	0.556541	+	0.830820i	$0.312128\pi$
$282$	−8.51407	−0.507005
$283$	26.3081	1.56386	0.781928	−	0.623368i	$-0.214236\pi$
$283$	26.3081	1.56386	0.781928	+	0.623368i	$0.214236\pi$
$284$	6.69404	0.397218
$285$	21.5143	1.27440
$286$	−1.72557	−0.102035
$287$	3.09025	0.182411
$288$	1.00000	0.0589256
$289$	−16.5517	−0.973631
$290$	7.38166	0.433466
$291$	14.9124	0.874182
$292$	11.7410	0.687091
$293$	−8.53157	−0.498420	−0.249210	−	0.968450i	$-0.580171\pi$
$293$	−8.53157	−0.498420	−0.249210	+	0.968450i	$0.580171\pi$
$294$	2.91109	0.169778
$295$	12.0936	0.704116
$296$	−2.89073	−0.168020
$297$	1.72557	0.100128
$298$	11.3872	0.659641
$299$	2.38462	0.137906
$300$	−5.30211	−0.306118
$301$	−7.58138	−0.436983
$302$	6.07828	0.349766
$303$	−18.7134	−1.07506
$304$	6.70293	0.384439
$305$	11.1259	0.637069
$306$	−0.669533	−0.0382747
$307$	−17.3226	−0.988651	−0.494325	−	0.869277i	$-0.664585\pi$
$307$	−17.3226	−0.988651	−0.494325	+	0.869277i	$0.664585\pi$
$308$	3.48928	0.198820
$309$	1.00000	0.0568880
$310$	14.9530	0.849276
$311$	−1.53808	−0.0872165	−0.0436083	−	0.999049i	$-0.513885\pi$
$311$	−1.53808	−0.0872165	−0.0436083	+	0.999049i	$0.513885\pi$
$312$	−1.00000	−0.0566139
$313$	−19.6423	−1.11025	−0.555125	−	0.831767i	$-0.687330\pi$
$313$	−19.6423	−1.11025	−0.555125	+	0.831767i	$0.687330\pi$
$314$	−10.3162	−0.582179
$315$	6.49033	0.365689
$316$	0.941600	0.0529692
$317$	11.6707	0.655489	0.327745	−	0.944766i	$-0.393711\pi$
$317$	11.6707	0.655489	0.327745	+	0.944766i	$0.393711\pi$
$318$	−11.6372	−0.652582
$319$	3.96847	0.222192
$320$	−3.20969	−0.179427
$321$	−3.67967	−0.205379
$322$	−4.82194	−0.268716
$323$	−4.48783	−0.249710
$324$	1.00000	0.0555556
$325$	5.30211	0.294108
$326$	−15.6784	−0.868345
$327$	0.643981	0.0356122
$328$	−1.52823	−0.0843825
$329$	−17.2163	−0.949168
$330$	−5.53854	−0.304887
$331$	28.6487	1.57467	0.787337	−	0.616522i	$-0.211459\pi$
$331$	28.6487	1.57467	0.787337	+	0.616522i	$0.211459\pi$
$332$	0.0679630	0.00372996
$333$	−2.89073	−0.158411
$334$	1.59761	0.0874173
$335$	43.3978	2.37108
$336$	2.02210	0.110315
$337$	−4.12431	−0.224666	−0.112333	−	0.993671i	$-0.535832\pi$
$337$	−4.12431	−0.224666	−0.112333	+	0.993671i	$0.535832\pi$
$338$	1.00000	0.0543928
$339$	2.61607	0.142085
$340$	2.14899	0.116546
$341$	8.03894	0.435333
$342$	6.70293	0.362453
$343$	20.0413	1.08213
$344$	3.74925	0.202146
$345$	7.65388	0.412071
$346$	−1.82588	−0.0981600
$347$	−0.273562	−0.0146856	−0.00734278	−	0.999973i	$-0.502337\pi$
$347$	−0.273562	−0.0146856	−0.00734278	+	0.999973i	$0.502337\pi$
$348$	2.29980	0.123282
$349$	−12.3688	−0.662086	−0.331043	−	0.943616i	$-0.607400\pi$
$349$	−12.3688	−0.662086	−0.331043	+	0.943616i	$0.607400\pi$
$350$	−10.7214	−0.573084
$351$	−1.00000	−0.0533761
$352$	−1.72557	−0.0919732
$353$	18.6874	0.994631	0.497315	−	0.867570i	$-0.334319\pi$
$353$	18.6874	0.994631	0.497315	+	0.867570i	$0.334319\pi$
$354$	3.76784	0.200258
$355$	−21.4858	−1.14035
$356$	15.8437	0.839716
$357$	−1.35387	−0.0716542
$358$	−6.67663	−0.352871
$359$	−10.1397	−0.535151	−0.267576	−	0.963537i	$-0.586223\pi$
$359$	−10.1397	−0.535151	−0.267576	+	0.963537i	$0.586223\pi$
$360$	−3.20969	−0.169166
$361$	25.9293	1.36470
$362$	−21.0273	−1.10517
$363$	8.02241	0.421067
$364$	−2.02210	−0.105987
$365$	−37.6850	−1.97253
$366$	3.46636	0.181189
$367$	−16.8044	−0.877185	−0.438592	−	0.898686i	$-0.644523\pi$
$367$	−16.8044	−0.877185	−0.438592	+	0.898686i	$0.644523\pi$
$368$	2.38462	0.124307
$369$	−1.52823	−0.0795566
$370$	9.27836	0.482359
$371$	−23.5317	−1.22170
$372$	4.65872	0.241543
$373$	1.24141	0.0642777	0.0321389	−	0.999483i	$-0.489768\pi$
$373$	1.24141	0.0642777	0.0321389	+	0.999483i	$0.489768\pi$
$374$	1.15533	0.0597405
$375$	0.969688	0.0500745
$376$	8.51407	0.439080
$377$	−2.29980	−0.118446
$378$	2.02210	0.104006
$379$	2.51633	0.129255	0.0646277	−	0.997909i	$-0.479414\pi$
$379$	2.51633	0.129255	0.0646277	+	0.997909i	$0.479414\pi$
$380$	−21.5143	−1.10366
$381$	17.1214	0.877155
$382$	−20.1793	−1.03246
$383$	16.4674	0.841447	0.420724	−	0.907189i	$-0.361776\pi$
$383$	16.4674	0.841447	0.420724	+	0.907189i	$0.361776\pi$
$384$	−1.00000	−0.0510310
$385$	−11.1995	−0.570780
$386$	−6.28353	−0.319823
$387$	3.74925	0.190585
$388$	−14.9124	−0.757064
$389$	4.75752	0.241216	0.120608	−	0.992700i	$-0.461516\pi$
$389$	4.75752	0.241216	0.120608	+	0.992700i	$0.461516\pi$
$390$	3.20969	0.162529
$391$	−1.59658	−0.0807425
$392$	−2.91109	−0.147032
$393$	11.9320	0.601891
$394$	−17.1542	−0.864217
$395$	−3.02225	−0.152066
$396$	−1.72557	−0.0867131
$397$	−28.4890	−1.42982	−0.714912	−	0.699215i	$-0.753533\pi$
$397$	−28.4890	−1.42982	−0.714912	+	0.699215i	$0.753533\pi$
$398$	−25.3168	−1.26902
$399$	13.5540	0.678550
$400$	5.30211	0.265106
$401$	−33.9746	−1.69661	−0.848306	−	0.529507i	$-0.822377\pi$
$401$	−33.9746	−1.69661	−0.848306	+	0.529507i	$0.822377\pi$
$402$	13.5209	0.674360
$403$	−4.65872	−0.232067
$404$	18.7134	0.931027
$405$	−3.20969	−0.159491
$406$	4.65044	0.230798
$407$	4.98816	0.247254
$408$	0.669533	0.0331468
$409$	27.7269	1.37101	0.685504	−	0.728069i	$-0.259582\pi$
$409$	27.7269	1.37101	0.685504	+	0.728069i	$0.259582\pi$
$410$	4.90515	0.242248
$411$	6.86144	0.338450
$412$	−1.00000	−0.0492665
$413$	7.61896	0.374904
$414$	2.38462	0.117198
$415$	−0.218140	−0.0107081
$416$	1.00000	0.0490290
$417$	13.0202	0.637604
$418$	−11.5664	−0.565730
$419$	14.6390	0.715161	0.357581	−	0.933882i	$-0.383602\pi$
$419$	14.6390	0.715161	0.357581	+	0.933882i	$0.383602\pi$
$420$	−6.49033	−0.316696
$421$	7.33989	0.357724	0.178862	−	0.983874i	$-0.442758\pi$
$421$	7.33989	0.357724	0.178862	+	0.983874i	$0.442758\pi$
$422$	2.77247	0.134962
$423$	8.51407	0.413968
$424$	11.6372	0.565153
$425$	−3.54994	−0.172197
$426$	−6.69404	−0.324327
$427$	7.00934	0.339206
$428$	3.67967	0.177863
$429$	1.72557	0.0833113
$430$	−12.0339	−0.580328
$431$	25.2104	1.21434	0.607171	−	0.794571i	$-0.292304\pi$
$431$	25.2104	1.21434	0.607171	+	0.794571i	$0.292304\pi$
$432$	−1.00000	−0.0481125
$433$	15.4861	0.744213	0.372107	−	0.928190i	$-0.378636\pi$
$433$	15.4861	0.744213	0.372107	+	0.928190i	$0.378636\pi$
$434$	9.42042	0.452195
$435$	−7.38166	−0.353923
$436$	−0.643981	−0.0308411
$437$	15.9839	0.764614
$438$	−11.7410	−0.561008
$439$	31.5746	1.50697	0.753486	−	0.657464i	$-0.228371\pi$
$439$	31.5746	1.50697	0.753486	+	0.657464i	$0.228371\pi$
$440$	5.53854	0.264040
$441$	−2.91109	−0.138623
$442$	−0.669533	−0.0318464
$443$	−6.71458	−0.319019	−0.159510	−	0.987196i	$-0.550991\pi$
$443$	−6.71458	−0.319019	−0.159510	+	0.987196i	$0.550991\pi$
$444$	2.89073	0.137188
$445$	−50.8535	−2.41069
$446$	29.2727	1.38610
$447$	−11.3872	−0.538595
$448$	−2.02210	−0.0955355
$449$	16.4048	0.774190	0.387095	−	0.922040i	$-0.373479\pi$
$449$	16.4048	0.774190	0.387095	+	0.922040i	$0.373479\pi$
$450$	5.30211	0.249944
$451$	2.63707	0.124175
$452$	−2.61607	−0.123049
$453$	−6.07828	−0.285582
$454$	17.6781	0.829675
$455$	6.49033	0.304271
$456$	−6.70293	−0.313893
$457$	−22.4703	−1.05112	−0.525559	−	0.850757i	$-0.676144\pi$
$457$	−22.4703	−1.05112	−0.525559	+	0.850757i	$0.676144\pi$
$458$	−22.4144	−1.04736
$459$	0.669533	0.0312511
$460$	−7.65388	−0.356864
$461$	−18.0083	−0.838729	−0.419365	−	0.907818i	$-0.637747\pi$
$461$	−18.0083	−0.838729	−0.419365	+	0.907818i	$0.637747\pi$
$462$	−3.48928	−0.162336
$463$	−5.02690	−0.233620	−0.116810	−	0.993154i	$-0.537267\pi$
$463$	−5.02690	−0.233620	−0.116810	+	0.993154i	$0.537267\pi$
$464$	−2.29980	−0.106766
$465$	−14.9530	−0.693431
$466$	−20.3329	−0.941903
$467$	−8.50870	−0.393736	−0.196868	−	0.980430i	$-0.563077\pi$
$467$	−8.50870	−0.393736	−0.196868	+	0.980430i	$0.563077\pi$
$468$	1.00000	0.0462250
$469$	27.3406	1.26247
$470$	−27.3275	−1.26052
$471$	10.3162	0.475347
$472$	−3.76784	−0.173429
$473$	−6.46959	−0.297472
$474$	−0.941600	−0.0432491
$475$	35.5397	1.63067
$476$	1.35387	0.0620544
$477$	11.6372	0.532831
$478$	−18.4706	−0.844827
$479$	6.33157	0.289297	0.144648	−	0.989483i	$-0.453795\pi$
$479$	6.33157	0.289297	0.144648	+	0.989483i	$0.453795\pi$
$480$	3.20969	0.146502
$481$	−2.89073	−0.131806
$482$	15.6356	0.712182
$483$	4.82194	0.219406
$484$	−8.02241	−0.364655
$485$	47.8643	2.17340
$486$	−1.00000	−0.0453609
$487$	28.0705	1.27199	0.635997	−	0.771692i	$-0.280589\pi$
$487$	28.0705	1.27199	0.635997	+	0.771692i	$0.280589\pi$
$488$	−3.46636	−0.156915
$489$	15.6784	0.709001
$490$	9.34370	0.422105
$491$	−25.3670	−1.14480	−0.572399	−	0.819975i	$-0.693987\pi$
$491$	−25.3670	−1.14480	−0.572399	+	0.819975i	$0.693987\pi$
$492$	1.52823	0.0688980
$493$	1.53979	0.0693489
$494$	6.70293	0.301579
$495$	5.53854	0.248939
$496$	−4.65872	−0.209183
$497$	−13.5360	−0.607175
$498$	−0.0679630	−0.00304550
$499$	−9.91564	−0.443885	−0.221942	−	0.975060i	$-0.571240\pi$
$499$	−9.91564	−0.443885	−0.221942	+	0.975060i	$0.571240\pi$
$500$	−0.969688	−0.0433658
$501$	−1.59761	−0.0713759
$502$	−12.3105	−0.549445
$503$	33.6282	1.49941	0.749704	−	0.661773i	$-0.230196\pi$
$503$	33.6282	1.49941	0.749704	+	0.661773i	$0.230196\pi$
$504$	−2.02210	−0.0900717
$505$	−60.0643	−2.67283
$506$	−4.11482	−0.182926
$507$	−1.00000	−0.0444116
$508$	−17.1214	−0.759639
$509$	5.30327	0.235063	0.117532	−	0.993069i	$-0.462502\pi$
$509$	5.30327	0.235063	0.117532	+	0.993069i	$0.462502\pi$
$510$	−2.14899	−0.0951591
$511$	−23.7416	−1.05027
$512$	1.00000	0.0441942
$513$	−6.70293	−0.295942
$514$	−25.6930	−1.13327
$515$	3.20969	0.141436
$516$	−3.74925	−0.165052
$517$	−14.6916	−0.646136
$518$	5.84536	0.256831
$519$	1.82588	0.0801473
$520$	−3.20969	−0.140754
$521$	12.0147	0.526376	0.263188	−	0.964745i	$-0.415226\pi$
$521$	12.0147	0.526376	0.263188	+	0.964745i	$0.415226\pi$
$522$	−2.29980	−0.100660
$523$	−17.9439	−0.784631	−0.392316	−	0.919831i	$-0.628326\pi$
$523$	−17.9439	−0.784631	−0.392316	+	0.919831i	$0.628326\pi$
$524$	−11.9320	−0.521253
$525$	10.7214	0.467921
$526$	−12.2397	−0.533678
$527$	3.11917	0.135873
$528$	1.72557	0.0750958
$529$	−17.3136	−0.752765
$530$	−37.3518	−1.62246
$531$	−3.76784	−0.163510
$532$	−13.5540	−0.587642
$533$	−1.52823	−0.0661951
$534$	−15.8437	−0.685625
$535$	−11.8106	−0.510616
$536$	−13.5209	−0.584013
$537$	6.67663	0.288118
$538$	−18.2413	−0.786440
$539$	5.02329	0.216368
$540$	3.20969	0.138123
$541$	32.7794	1.40930	0.704649	−	0.709556i	$-0.251104\pi$
$541$	32.7794	1.40930	0.704649	+	0.709556i	$0.251104\pi$
$542$	3.96166	0.170168
$543$	21.0273	0.902370
$544$	−0.669533	−0.0287060
$545$	2.06698	0.0885396
$546$	2.02210	0.0865381
$547$	8.57173	0.366501	0.183250	−	0.983066i	$-0.441338\pi$
$547$	8.57173	0.366501	0.183250	+	0.983066i	$0.441338\pi$
$548$	−6.86144	−0.293106
$549$	−3.46636	−0.147941
$550$	−9.14916	−0.390122
$551$	−15.4154	−0.656719
$552$	−2.38462	−0.101496
$553$	−1.90401	−0.0809669
$554$	−18.5109	−0.786452
$555$	−9.27836	−0.393844
$556$	−13.0202	−0.552181
$557$	0.0836573	0.00354467	0.00177234	−	0.999998i	$-0.499436\pi$
$557$	0.0836573	0.00354467	0.00177234	+	0.999998i	$0.499436\pi$
$558$	−4.65872	−0.197219
$559$	3.74925	0.158576
$560$	6.49033	0.274267
$561$	−1.15533	−0.0487779
$562$	18.6586	0.787067
$563$	4.82841	0.203493	0.101747	−	0.994810i	$-0.467557\pi$
$563$	4.82841	0.203493	0.101747	+	0.994810i	$0.467557\pi$
$564$	−8.51407	−0.358507
$565$	8.39677	0.353255
$566$	26.3081	1.10581
$567$	−2.02210	−0.0849204
$568$	6.69404	0.280876
$569$	41.6502	1.74607	0.873033	−	0.487661i	$-0.162150\pi$
$569$	41.6502	1.74607	0.873033	+	0.487661i	$0.162150\pi$
$570$	21.5143	0.901136
$571$	30.5388	1.27801	0.639004	−	0.769203i	$-0.279347\pi$
$571$	30.5388	1.27801	0.639004	+	0.769203i	$0.279347\pi$
$572$	−1.72557	−0.0721497
$573$	20.1793	0.843004
$574$	3.09025	0.128984
$575$	12.6435	0.527271
$576$	1.00000	0.0416667
$577$	−25.1388	−1.04654	−0.523272	−	0.852166i	$-0.675289\pi$
$577$	−25.1388	−1.04654	−0.523272	+	0.852166i	$0.675289\pi$
$578$	−16.5517	−0.688461
$579$	6.28353	0.261134
$580$	7.38166	0.306507
$581$	−0.137428	−0.00570149
$582$	14.9124	0.618140
$583$	−20.0808	−0.831662
$584$	11.7410	0.485847
$585$	−3.20969	−0.132704
$586$	−8.53157	−0.352436
$587$	−21.8038	−0.899938	−0.449969	−	0.893044i	$-0.648565\pi$
$587$	−21.8038	−0.899938	−0.449969	+	0.893044i	$0.648565\pi$
$588$	2.91109	0.120051
$589$	−31.2271	−1.28669
$590$	12.0936	0.497885
$591$	17.1542	0.705630
$592$	−2.89073	−0.118808
$593$	24.1980	0.993691	0.496846	−	0.867839i	$-0.334492\pi$
$593$	24.1980	0.993691	0.496846	+	0.867839i	$0.334492\pi$
$594$	1.72557	0.0708010
$595$	−4.34549	−0.178148
$596$	11.3872	0.466437
$597$	25.3168	1.03615
$598$	2.38462	0.0975142
$599$	−7.97659	−0.325915	−0.162957	−	0.986633i	$-0.552103\pi$
$599$	−7.97659	−0.325915	−0.162957	+	0.986633i	$0.552103\pi$
$600$	−5.30211	−0.216458
$601$	45.4210	1.85276	0.926381	−	0.376588i	$-0.122903\pi$
$601$	45.4210	1.85276	0.926381	+	0.376588i	$0.122903\pi$
$602$	−7.58138	−0.308994
$603$	−13.5209	−0.550612
$604$	6.07828	0.247322
$605$	25.7495	1.04686
$606$	−18.7134	−0.760181
$607$	−2.64554	−0.107379	−0.0536897	−	0.998558i	$-0.517098\pi$
$607$	−2.64554	−0.107379	−0.0536897	+	0.998558i	$0.517098\pi$
$608$	6.70293	0.271840
$609$	−4.65044	−0.188445
$610$	11.1259	0.450476
$611$	8.51407	0.344442
$612$	−0.669533	−0.0270643
$613$	−33.0710	−1.33573	−0.667863	−	0.744284i	$-0.732791\pi$
$613$	−33.0710	−1.33573	−0.667863	+	0.744284i	$0.732791\pi$
$614$	−17.3226	−0.699082
$615$	−4.90515	−0.197795
$616$	3.48928	0.140587
$617$	−23.4625	−0.944564	−0.472282	−	0.881447i	$-0.656570\pi$
$617$	−23.4625	−0.944564	−0.472282	+	0.881447i	$0.656570\pi$
$618$	1.00000	0.0402259
$619$	−24.7794	−0.995968	−0.497984	−	0.867186i	$-0.665926\pi$
$619$	−24.7794	−0.995968	−0.497984	+	0.867186i	$0.665926\pi$
$620$	14.9530	0.600529
$621$	−2.38462	−0.0956914
$622$	−1.53808	−0.0616714
$623$	−32.0377	−1.28356
$624$	−1.00000	−0.0400320
$625$	−23.3982	−0.935927
$626$	−19.6423	−0.785065
$627$	11.5664	0.461916
$628$	−10.3162	−0.411663
$629$	1.93544	0.0771711
$630$	6.49033	0.258581
$631$	9.08319	0.361596	0.180798	−	0.983520i	$-0.442132\pi$
$631$	9.08319	0.361596	0.180798	+	0.983520i	$0.442132\pi$
$632$	0.941600	0.0374548
$633$	−2.77247	−0.110196
$634$	11.6707	0.463501
$635$	54.9544	2.18080
$636$	−11.6372	−0.461445
$637$	−2.91109	−0.115342
$638$	3.96847	0.157113
$639$	6.69404	0.264812
$640$	−3.20969	−0.126874
$641$	−44.2213	−1.74664	−0.873318	−	0.487150i	$-0.838036\pi$
$641$	−44.2213	−1.74664	−0.873318	+	0.487150i	$0.838036\pi$
$642$	−3.67967	−0.145225
$643$	−24.8201	−0.978810	−0.489405	−	0.872057i	$-0.662786\pi$
$643$	−24.8201	−0.978810	−0.489405	+	0.872057i	$0.662786\pi$
$644$	−4.82194	−0.190011
$645$	12.0339	0.473836
$646$	−4.48783	−0.176571
$647$	21.8686	0.859744	0.429872	−	0.902890i	$-0.358559\pi$
$647$	21.8686	0.859744	0.429872	+	0.902890i	$0.358559\pi$
$648$	1.00000	0.0392837
$649$	6.50166	0.255213
$650$	5.30211	0.207966
$651$	−9.42042	−0.369215
$652$	−15.6784	−0.614013
$653$	33.3162	1.30376	0.651882	−	0.758320i	$-0.273980\pi$
$653$	33.3162	1.30376	0.651882	+	0.758320i	$0.273980\pi$
$654$	0.643981	0.0251816
$655$	38.2981	1.49643
$656$	−1.52823	−0.0596675
$657$	11.7410	0.458061
$658$	−17.2163	−0.671163
$659$	−25.0845	−0.977153	−0.488577	−	0.872521i	$-0.662484\pi$
$659$	−25.0845	−0.977153	−0.488577	+	0.872521i	$0.662484\pi$
$660$	−5.53854	−0.215588
$661$	33.1111	1.28787	0.643936	−	0.765079i	$-0.277300\pi$
$661$	33.1111	1.28787	0.643936	+	0.765079i	$0.277300\pi$
$662$	28.6487	1.11346
$663$	0.669533	0.0260025
$664$	0.0679630	0.00263748
$665$	43.5042	1.68702
$666$	−2.89073	−0.112014
$667$	−5.48415	−0.212347
$668$	1.59761	0.0618134
$669$	−29.2727	−1.13175
$670$	43.3978	1.67660
$671$	5.98144	0.230911
$672$	2.02210	0.0780044
$673$	−19.3297	−0.745105	−0.372553	−	0.928011i	$-0.621517\pi$
$673$	−19.3297	−0.745105	−0.372553	+	0.928011i	$0.621517\pi$
$674$	−4.12431	−0.158863
$675$	−5.30211	−0.204078
$676$	1.00000	0.0384615
$677$	42.8994	1.64876	0.824380	−	0.566037i	$-0.191524\pi$
$677$	42.8994	1.64876	0.824380	+	0.566037i	$0.191524\pi$
$678$	2.61607	0.100469
$679$	30.1545	1.15722
$680$	2.14899	0.0824102
$681$	−17.6781	−0.677427
$682$	8.03894	0.307827
$683$	−26.5193	−1.01473	−0.507365	−	0.861731i	$-0.669381\pi$
$683$	−26.5193	−1.01473	−0.507365	+	0.861731i	$0.669381\pi$
$684$	6.70293	0.256293
$685$	22.0231	0.841459
$686$	20.0413	0.765179
$687$	22.4144	0.855164
$688$	3.74925	0.142939
$689$	11.6372	0.443342
$690$	7.65388	0.291378
$691$	−30.4249	−1.15742	−0.578709	−	0.815534i	$-0.696443\pi$
$691$	−30.4249	−1.15742	−0.578709	+	0.815534i	$0.696443\pi$
$692$	−1.82588	−0.0694096
$693$	3.48928	0.132547
$694$	−0.273562	−0.0103843
$695$	41.7910	1.58522
$696$	2.29980	0.0871738
$697$	1.02320	0.0387566
$698$	−12.3688	−0.468165
$699$	20.3329	0.769060
$700$	−10.7214	−0.405232
$701$	−6.24257	−0.235778	−0.117889	−	0.993027i	$-0.537613\pi$
$701$	−6.24257	−0.235778	−0.117889	+	0.993027i	$0.537613\pi$
$702$	−1.00000	−0.0377426
$703$	−19.3764	−0.730794
$704$	−1.72557	−0.0650348
$705$	27.3275	1.02921
$706$	18.6874	0.703310
$707$	−37.8405	−1.42314
$708$	3.76784	0.141604
$709$	1.68983	0.0634627	0.0317314	−	0.999496i	$-0.489898\pi$
$709$	1.68983	0.0634627	0.0317314	+	0.999496i	$0.489898\pi$
$710$	−21.4858	−0.806347
$711$	0.941600	0.0353128
$712$	15.8437	0.593769
$713$	−11.1093	−0.416045
$714$	−1.35387	−0.0506672
$715$	5.53854	0.207130
$716$	−6.67663	−0.249517
$717$	18.4706	0.689798
$718$	−10.1397	−0.378409
$719$	−46.1759	−1.72207	−0.861034	−	0.508547i	$-0.830183\pi$
$719$	−46.1759	−1.72207	−0.861034	+	0.508547i	$0.830183\pi$
$720$	−3.20969	−0.119618
$721$	2.02210	0.0753071
$722$	25.9293	0.964988
$723$	−15.6356	−0.581494
$724$	−21.0273	−0.781475
$725$	−12.1938	−0.452867
$726$	8.02241	0.297740
$727$	8.06916	0.299268	0.149634	−	0.988741i	$-0.452190\pi$
$727$	8.06916	0.299268	0.149634	+	0.988741i	$0.452190\pi$
$728$	−2.02210	−0.0749442
$729$	1.00000	0.0370370
$730$	−37.6850	−1.39479
$731$	−2.51025	−0.0928449
$732$	3.46636	0.128120
$733$	7.59474	0.280518	0.140259	−	0.990115i	$-0.455206\pi$
$733$	7.59474	0.280518	0.140259	+	0.990115i	$0.455206\pi$
$734$	−16.8044	−0.620263
$735$	−9.34370	−0.344648
$736$	2.38462	0.0878981
$737$	23.3312	0.859416
$738$	−1.52823	−0.0562550
$739$	−18.5847	−0.683648	−0.341824	−	0.939764i	$-0.611045\pi$
$739$	−18.5847	−0.683648	−0.341824	+	0.939764i	$0.611045\pi$
$740$	9.27836	0.341079
$741$	−6.70293	−0.246238
$742$	−23.5317	−0.863874
$743$	−16.2897	−0.597613	−0.298806	−	0.954314i	$-0.596588\pi$
$743$	−16.2897	−0.597613	−0.298806	+	0.954314i	$0.596588\pi$
$744$	4.65872	0.170797
$745$	−36.5493	−1.33906
$746$	1.24141	0.0454512
$747$	0.0679630	0.00248664
$748$	1.15533	0.0422429
$749$	−7.44067	−0.271876
$750$	0.969688	0.0354080
$751$	−40.2701	−1.46948	−0.734739	−	0.678350i	$-0.762695\pi$
$751$	−40.2701	−1.46948	−0.734739	+	0.678350i	$0.762695\pi$
$752$	8.51407	0.310476
$753$	12.3105	0.448620
$754$	−2.29980	−0.0837539
$755$	−19.5094	−0.710020
$756$	2.02210	0.0735432
$757$	−36.6762	−1.33302	−0.666510	−	0.745496i	$-0.732213\pi$
$757$	−36.6762	−1.33302	−0.666510	+	0.745496i	$0.732213\pi$
$758$	2.51633	0.0913973
$759$	4.11482	0.149359
$760$	−21.5143	−0.780407
$761$	−0.506560	−0.0183628	−0.00918140	−	0.999958i	$-0.502923\pi$
$761$	−0.506560	−0.0183628	−0.00918140	+	0.999958i	$0.502923\pi$
$762$	17.1214	0.620243
$763$	1.30220	0.0471427
$764$	−20.1793	−0.730063
$765$	2.14899	0.0776970
$766$	16.4674	0.594993
$767$	−3.76784	−0.136049
$768$	−1.00000	−0.0360844
$769$	−19.2648	−0.694705	−0.347353	−	0.937735i	$-0.612919\pi$
$769$	−19.2648	−0.694705	−0.347353	+	0.937735i	$0.612919\pi$
$770$	−11.1995	−0.403603
$771$	25.6930	0.925312
$772$	−6.28353	−0.226149
$773$	−19.7072	−0.708818	−0.354409	−	0.935091i	$-0.615318\pi$
$773$	−19.7072	−0.708818	−0.354409	+	0.935091i	$0.615318\pi$
$774$	3.74925	0.134764
$775$	−24.7011	−0.887288
$776$	−14.9124	−0.535325
$777$	−5.84536	−0.209701
$778$	4.75752	0.170565
$779$	−10.2436	−0.367016
$780$	3.20969	0.114925
$781$	−11.5510	−0.413328
$782$	−1.59658	−0.0570936
$783$	2.29980	0.0821883
$784$	−2.91109	−0.103968
$785$	33.1120	1.18182
$786$	11.9320	0.425601
$787$	41.3177	1.47282	0.736409	−	0.676536i	$-0.236520\pi$
$787$	41.3177	1.47282	0.736409	+	0.676536i	$0.236520\pi$
$788$	−17.1542	−0.611094
$789$	12.2397	0.435746
$790$	−3.02225	−0.107527
$791$	5.28996	0.188089
$792$	−1.72557	−0.0613154
$793$	−3.46636	−0.123094
$794$	−28.4890	−1.01104
$795$	37.3518	1.32473
$796$	−25.3168	−0.897330
$797$	43.7552	1.54989	0.774944	−	0.632030i	$-0.217778\pi$
$797$	43.7552	1.54989	0.774944	+	0.632030i	$0.217778\pi$
$798$	13.5540	0.479807
$799$	−5.70045	−0.201667
$800$	5.30211	0.187458
$801$	15.8437	0.559811
$802$	−33.9746	−1.19969
$803$	−20.2599	−0.714958
$804$	13.5209	0.476844
$805$	15.4770	0.545491
$806$	−4.65872	−0.164096
$807$	18.2413	0.642126
$808$	18.7134	0.658336
$809$	−32.3788	−1.13838	−0.569189	−	0.822207i	$-0.692743\pi$
$809$	−32.3788	−1.13838	−0.569189	+	0.822207i	$0.692743\pi$
$810$	−3.20969	−0.112777
$811$	−7.11397	−0.249805	−0.124903	−	0.992169i	$-0.539862\pi$
$811$	−7.11397	−0.249805	−0.124903	+	0.992169i	$0.539862\pi$
$812$	4.65044	0.163199
$813$	−3.96166	−0.138942
$814$	4.98816	0.174835
$815$	50.3227	1.76273
$816$	0.669533	0.0234383
$817$	25.1310	0.879221
$818$	27.7269	0.969450
$819$	−2.02210	−0.0706581
$820$	4.90515	0.171295
$821$	−40.6722	−1.41947	−0.709735	−	0.704469i	$-0.751185\pi$
$821$	−40.6722	−1.41947	−0.709735	+	0.704469i	$0.751185\pi$
$822$	6.86144	0.239320
$823$	23.9583	0.835132	0.417566	−	0.908647i	$-0.362883\pi$
$823$	23.9583	0.835132	0.417566	+	0.908647i	$0.362883\pi$
$824$	−1.00000	−0.0348367
$825$	9.14916	0.318533
$826$	7.61896	0.265097
$827$	−8.40864	−0.292397	−0.146198	−	0.989255i	$-0.546704\pi$
$827$	−8.40864	−0.292397	−0.146198	+	0.989255i	$0.546704\pi$
$828$	2.38462	0.0828712
$829$	44.9658	1.56173	0.780864	−	0.624701i	$-0.214779\pi$
$829$	44.9658	1.56173	0.780864	+	0.624701i	$0.214779\pi$
$830$	−0.218140	−0.00757176
$831$	18.5109	0.642136
$832$	1.00000	0.0346688
$833$	1.94907	0.0675314
$834$	13.0202	0.450854
$835$	−5.12783	−0.177456
$836$	−11.5664	−0.400031
$837$	4.65872	0.161029
$838$	14.6390	0.505695
$839$	7.81663	0.269860	0.134930	−	0.990855i	$-0.456919\pi$
$839$	7.81663	0.269860	0.134930	+	0.990855i	$0.456919\pi$
$840$	−6.49033	−0.223938
$841$	−23.7109	−0.817617
$842$	7.33989	0.252949
$843$	−18.6586	−0.642638
$844$	2.77247	0.0954323
$845$	−3.20969	−0.110417
$846$	8.51407	0.292720
$847$	16.2222	0.557400
$848$	11.6372	0.399623
$849$	−26.3081	−0.902893
$850$	−3.54994	−0.121762
$851$	−6.89329	−0.236299
$852$	−6.69404	−0.229334
$853$	24.0738	0.824270	0.412135	−	0.911123i	$-0.364783\pi$
$853$	24.0738	0.824270	0.412135	+	0.911123i	$0.364783\pi$
$854$	7.00934	0.239855
$855$	−21.5143	−0.735775
$856$	3.67967	0.125768
$857$	−25.2461	−0.862389	−0.431194	−	0.902259i	$-0.641908\pi$
$857$	−25.2461	−0.862389	−0.431194	+	0.902259i	$0.641908\pi$
$858$	1.72557	0.0589100
$859$	−30.8226	−1.05165	−0.525826	−	0.850592i	$-0.676244\pi$
$859$	−30.8226	−1.05165	−0.525826	+	0.850592i	$0.676244\pi$
$860$	−12.0339	−0.410354
$861$	−3.09025	−0.105315
$862$	25.2104	0.858670
$863$	47.8889	1.63016	0.815079	−	0.579350i	$-0.196694\pi$
$863$	47.8889	1.63016	0.815079	+	0.579350i	$0.196694\pi$
$864$	−1.00000	−0.0340207
$865$	5.86051	0.199263
$866$	15.4861	0.526238
$867$	16.5517	0.562126
$868$	9.42042	0.319750
$869$	−1.62480	−0.0551174
$870$	−7.38166	−0.250262
$871$	−13.5209	−0.458137
$872$	−0.643981	−0.0218079
$873$	−14.9124	−0.504709
$874$	15.9839	0.540664
$875$	1.96081	0.0662875
$876$	−11.7410	−0.396692
$877$	−38.3110	−1.29367	−0.646835	−	0.762630i	$-0.723908\pi$
$877$	−38.3110	−1.29367	−0.646835	+	0.762630i	$0.723908\pi$
$878$	31.5746	1.06559
$879$	8.53157	0.287763
$880$	5.53854	0.186704
$881$	−26.6819	−0.898936	−0.449468	−	0.893296i	$-0.648386\pi$
$881$	−26.6819	−0.898936	−0.449468	+	0.893296i	$0.648386\pi$
$882$	−2.91109	−0.0980215
$883$	28.2227	0.949769	0.474884	−	0.880048i	$-0.342490\pi$
$883$	28.2227	0.949769	0.474884	+	0.880048i	$0.342490\pi$
$884$	−0.669533	−0.0225188
$885$	−12.0936	−0.406521
$886$	−6.71458	−0.225581
$887$	40.3094	1.35346	0.676729	−	0.736232i	$-0.263397\pi$
$887$	40.3094	1.35346	0.676729	+	0.736232i	$0.263397\pi$
$888$	2.89073	0.0970066
$889$	34.6212	1.16116
$890$	−50.8535	−1.70461
$891$	−1.72557	−0.0578087
$892$	29.2727	0.980123
$893$	57.0692	1.90975
$894$	−11.3872	−0.380844
$895$	21.4299	0.716323
$896$	−2.02210	−0.0675538
$897$	−2.38462	−0.0796200
$898$	16.4048	0.547435
$899$	10.7141	0.357337
$900$	5.30211	0.176737
$901$	−7.79150	−0.259572
$902$	2.63707	0.0878049
$903$	7.58138	0.252293
$904$	−2.61607	−0.0870091
$905$	67.4913	2.24349
$906$	−6.07828	−0.201937
$907$	38.1235	1.26587	0.632935	−	0.774205i	$-0.281850\pi$
$907$	38.1235	1.26587	0.632935	+	0.774205i	$0.281850\pi$
$908$	17.6781	0.586669
$909$	18.7134	0.620685
$910$	6.49033	0.215152
$911$	26.6119	0.881691	0.440845	−	0.897583i	$-0.354679\pi$
$911$	26.6119	0.881691	0.440845	+	0.897583i	$0.354679\pi$
$912$	−6.70293	−0.221956
$913$	−0.117275	−0.00388123
$914$	−22.4703	−0.743253
$915$	−11.1259	−0.367812
$916$	−22.4144	−0.740593
$917$	24.1278	0.796770
$918$	0.669533	0.0220979
$919$	8.12414	0.267991	0.133995	−	0.990982i	$-0.457219\pi$
$919$	8.12414	0.267991	0.133995	+	0.990982i	$0.457219\pi$
$920$	−7.65388	−0.252341
$921$	17.3226	0.570798
$922$	−18.0083	−0.593071
$923$	6.69404	0.220337
$924$	−3.48928	−0.114789
$925$	−15.3270	−0.503948
$926$	−5.02690	−0.165194
$927$	−1.00000	−0.0328443
$928$	−2.29980	−0.0754947
$929$	1.09825	0.0360323	0.0180162	−	0.999838i	$-0.494265\pi$
$929$	1.09825	0.0360323	0.0180162	+	0.999838i	$0.494265\pi$
$930$	−14.9530	−0.490330
$931$	−19.5128	−0.639508
$932$	−20.3329	−0.666026
$933$	1.53808	0.0503545
$934$	−8.50870	−0.278413
$935$	−3.70824	−0.121272
$936$	1.00000	0.0326860
$937$	15.9949	0.522530	0.261265	−	0.965267i	$-0.415860\pi$
$937$	15.9949	0.522530	0.261265	+	0.965267i	$0.415860\pi$
$938$	27.3406	0.892703
$939$	19.6423	0.641003
$940$	−27.3275	−0.891326
$941$	13.6922	0.446354	0.223177	−	0.974778i	$-0.428357\pi$
$941$	13.6922	0.446354	0.223177	+	0.974778i	$0.428357\pi$
$942$	10.3162	0.336121
$943$	−3.64425	−0.118673
$944$	−3.76784	−0.122633
$945$	−6.49033	−0.211130
$946$	−6.46959	−0.210345
$947$	−6.82474	−0.221774	−0.110887	−	0.993833i	$-0.535369\pi$
$947$	−6.82474	−0.221774	−0.110887	+	0.993833i	$0.535369\pi$
$948$	−0.941600	−0.0305818
$949$	11.7410	0.381130
$950$	35.5397	1.15306
$951$	−11.6707	−0.378447
$952$	1.35387	0.0438791
$953$	−54.0833	−1.75193	−0.875965	−	0.482374i	$-0.839775\pi$
$953$	−54.0833	−1.75193	−0.875965	+	0.482374i	$0.839775\pi$
$954$	11.6372	0.376769
$955$	64.7694	2.09589
$956$	−18.4706	−0.597383
$957$	−3.96847	−0.128282
$958$	6.33157	0.204564
$959$	13.8746	0.448033
$960$	3.20969	0.103592
$961$	−9.29633	−0.299882
$962$	−2.89073	−0.0932010
$963$	3.67967	0.118576
$964$	15.6356	0.503588
$965$	20.1682	0.649237
$966$	4.82194	0.155144
$967$	−27.6290	−0.888489	−0.444244	−	0.895906i	$-0.646528\pi$
$967$	−27.6290	−0.888489	−0.444244	+	0.895906i	$0.646528\pi$
$968$	−8.02241	−0.257850
$969$	4.48783	0.144170
$970$	47.8643	1.53683
$971$	−53.1314	−1.70507	−0.852535	−	0.522671i	$-0.824936\pi$
$971$	−53.1314	−1.70507	−0.852535	+	0.522671i	$0.824936\pi$
$972$	−1.00000	−0.0320750
$973$	26.3283	0.844047
$974$	28.0705	0.899435
$975$	−5.30211	−0.169804
$976$	−3.46636	−0.110955
$977$	−6.80034	−0.217562	−0.108781	−	0.994066i	$-0.534695\pi$
$977$	−6.80034	−0.217562	−0.108781	+	0.994066i	$0.534695\pi$
$978$	15.6784	0.501339
$979$	−27.3395	−0.873773
$980$	9.34370	0.298474
$981$	−0.643981	−0.0205607
$982$	−25.3670	−0.809494
$983$	9.75903	0.311265	0.155632	−	0.987815i	$-0.450258\pi$
$983$	9.75903	0.311265	0.155632	+	0.987815i	$0.450258\pi$
$984$	1.52823	0.0487183
$985$	55.0597	1.75435
$986$	1.53979	0.0490371
$987$	17.2163	0.548002
$988$	6.70293	0.213249
$989$	8.94053	0.284292
$990$	5.53854	0.176026
$991$	25.2699	0.802724	0.401362	−	0.915919i	$-0.368537\pi$
$991$	25.2699	0.802724	0.401362	+	0.915919i	$0.368537\pi$
$992$	−4.65872	−0.147915
$993$	−28.6487	−0.909139
$994$	−13.5360	−0.429337
$995$	81.2591	2.57609
$996$	−0.0679630	−0.00215349
$997$	−19.5110	−0.617920	−0.308960	−	0.951075i	$-0.599981\pi$
$997$	−19.5110	−0.617920	−0.308960	+	0.951075i	$0.599981\pi$
$998$	−9.91564	−0.313874
$999$	2.89073	0.0914587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.r.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.r.1.1	✓	9	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.20969 q^{5} -1.00000 q^{6} -2.02210 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.20969 q^{5} -1.00000 q^{6} -2.02210 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.20969 q^{10} -1.72557 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.02210 q^{14} +3.20969 q^{15} +1.00000 q^{16} -0.669533 q^{17} +1.00000 q^{18} +6.70293 q^{19} -3.20969 q^{20} +2.02210 q^{21} -1.72557 q^{22} +2.38462 q^{23} -1.00000 q^{24} +5.30211 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.02210 q^{28} -2.29980 q^{29} +3.20969 q^{30} -4.65872 q^{31} +1.00000 q^{32} +1.72557 q^{33} -0.669533 q^{34} +6.49033 q^{35} +1.00000 q^{36} -2.89073 q^{37} +6.70293 q^{38} -1.00000 q^{39} -3.20969 q^{40} -1.52823 q^{41} +2.02210 q^{42} +3.74925 q^{43} -1.72557 q^{44} -3.20969 q^{45} +2.38462 q^{46} +8.51407 q^{47} -1.00000 q^{48} -2.91109 q^{49} +5.30211 q^{50} +0.669533 q^{51} +1.00000 q^{52} +11.6372 q^{53} -1.00000 q^{54} +5.53854 q^{55} -2.02210 q^{56} -6.70293 q^{57} -2.29980 q^{58} -3.76784 q^{59} +3.20969 q^{60} -3.46636 q^{61} -4.65872 q^{62} -2.02210 q^{63} +1.00000 q^{64} -3.20969 q^{65} +1.72557 q^{66} -13.5209 q^{67} -0.669533 q^{68} -2.38462 q^{69} +6.49033 q^{70} +6.69404 q^{71} +1.00000 q^{72} +11.7410 q^{73} -2.89073 q^{74} -5.30211 q^{75} +6.70293 q^{76} +3.48928 q^{77} -1.00000 q^{78} +0.941600 q^{79} -3.20969 q^{80} +1.00000 q^{81} -1.52823 q^{82} +0.0679630 q^{83} +2.02210 q^{84} +2.14899 q^{85} +3.74925 q^{86} +2.29980 q^{87} -1.72557 q^{88} +15.8437 q^{89} -3.20969 q^{90} -2.02210 q^{91} +2.38462 q^{92} +4.65872 q^{93} +8.51407 q^{94} -21.5143 q^{95} -1.00000 q^{96} -14.9124 q^{97} -2.91109 q^{98} -1.72557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9 - 4 * q^10 - 5 * q^11 - 9 * q^12 + 9 * q^13 - 4 * q^14 + 4 * q^15 + 9 * q^16 - 6 * q^17 + 9 * q^18 - 4 * q^19 - 4 * q^20 + 4 * q^21 - 5 * q^22 - 6 * q^23 - 9 * q^24 - 11 * q^25 + 9 * q^26 - 9 * q^27 - 4 * q^28 - 19 * q^29 + 4 * q^30 - 6 * q^31 + 9 * q^32 + 5 * q^33 - 6 * q^34 + 10 * q^35 + 9 * q^36 - 13 * q^37 - 4 * q^38 - 9 * q^39 - 4 * q^40 - 18 * q^41 + 4 * q^42 - 20 * q^43 - 5 * q^44 - 4 * q^45 - 6 * q^46 + 14 * q^47 - 9 * q^48 - 3 * q^49 - 11 * q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 - 9 * q^54 - 4 * q^55 - 4 * q^56 + 4 * q^57 - 19 * q^58 - 9 * q^59 + 4 * q^60 - 24 * q^61 - 6 * q^62 - 4 * q^63 + 9 * q^64 - 4 * q^65 + 5 * q^66 - 4 * q^67 - 6 * q^68 + 6 * q^69 + 10 * q^70 - 9 * q^71 + 9 * q^72 - 24 * q^73 - 13 * q^74 + 11 * q^75 - 4 * q^76 + 3 * q^77 - 9 * q^78 - 15 * q^79 - 4 * q^80 + 9 * q^81 - 18 * q^82 + 20 * q^83 + 4 * q^84 - 31 * q^85 - 20 * q^86 + 19 * q^87 - 5 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^91 - 6 * q^92 + 6 * q^93 + 14 * q^94 - 4 * q^95 - 9 * q^96 - 19 * q^97 - 3 * q^98 - 5 * q^99

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