LMFDB - Embedded newform 1155.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} -2.73205 q^{10} -1.00000 q^{11} -5.46410 q^{12} +6.19615 q^{13} +2.73205 q^{14} +1.00000 q^{15} +14.9282 q^{16} +0.464102 q^{17} +2.73205 q^{18} -8.46410 q^{19} -5.46410 q^{20} -1.00000 q^{21} -2.73205 q^{22} +1.53590 q^{23} -9.46410 q^{24} +1.00000 q^{25} +16.9282 q^{26} -1.00000 q^{27} +5.46410 q^{28} +2.26795 q^{29} +2.73205 q^{30} -8.73205 q^{31} +21.8564 q^{32} +1.00000 q^{33} +1.26795 q^{34} -1.00000 q^{35} +5.46410 q^{36} +6.19615 q^{37} -23.1244 q^{38} -6.19615 q^{39} -9.46410 q^{40} +9.66025 q^{41} -2.73205 q^{42} -1.73205 q^{43} -5.46410 q^{44} -1.00000 q^{45} +4.19615 q^{46} +4.73205 q^{47} -14.9282 q^{48} +1.00000 q^{49} +2.73205 q^{50} -0.464102 q^{51} +33.8564 q^{52} -2.46410 q^{53} -2.73205 q^{54} +1.00000 q^{55} +9.46410 q^{56} +8.46410 q^{57} +6.19615 q^{58} +10.2679 q^{59} +5.46410 q^{60} -9.39230 q^{61} -23.8564 q^{62} +1.00000 q^{63} +29.8564 q^{64} -6.19615 q^{65} +2.73205 q^{66} -8.92820 q^{67} +2.53590 q^{68} -1.53590 q^{69} -2.73205 q^{70} -9.66025 q^{71} +9.46410 q^{72} -12.3923 q^{73} +16.9282 q^{74} -1.00000 q^{75} -46.2487 q^{76} -1.00000 q^{77} -16.9282 q^{78} -13.6603 q^{79} -14.9282 q^{80} +1.00000 q^{81} +26.3923 q^{82} -4.46410 q^{83} -5.46410 q^{84} -0.464102 q^{85} -4.73205 q^{86} -2.26795 q^{87} -9.46410 q^{88} -2.66025 q^{89} -2.73205 q^{90} +6.19615 q^{91} +8.39230 q^{92} +8.73205 q^{93} +12.9282 q^{94} +8.46410 q^{95} -21.8564 q^{96} -5.73205 q^{97} +2.73205 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 16 q^{16} - 6 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} - 12 q^{24} + 2 q^{25} + 20 q^{26} - 2 q^{27} + 4 q^{28} + 8 q^{29} + 2 q^{30} - 14 q^{31} + 16 q^{32} + 2 q^{33} + 6 q^{34} - 2 q^{35} + 4 q^{36} + 2 q^{37} - 22 q^{38} - 2 q^{39} - 12 q^{40} + 2 q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{45} - 2 q^{46} + 6 q^{47} - 16 q^{48} + 2 q^{49} + 2 q^{50} + 6 q^{51} + 40 q^{52} + 2 q^{53} - 2 q^{54} + 2 q^{55} + 12 q^{56} + 10 q^{57} + 2 q^{58} + 24 q^{59} + 4 q^{60} + 2 q^{61} - 20 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} + 2 q^{66} - 4 q^{67} + 12 q^{68} - 10 q^{69} - 2 q^{70} - 2 q^{71} + 12 q^{72} - 4 q^{73} + 20 q^{74} - 2 q^{75} - 44 q^{76} - 2 q^{77} - 20 q^{78} - 10 q^{79} - 16 q^{80} + 2 q^{81} + 32 q^{82} - 2 q^{83} - 4 q^{84} + 6 q^{85} - 6 q^{86} - 8 q^{87} - 12 q^{88} + 12 q^{89} - 2 q^{90} + 2 q^{91} - 4 q^{92} + 14 q^{93} + 12 q^{94} + 10 q^{95} - 16 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 16 * q^16 - 6 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 - 12 * q^24 + 2 * q^25 + 20 * q^26 - 2 * q^27 + 4 * q^28 + 8 * q^29 + 2 * q^30 - 14 * q^31 + 16 * q^32 + 2 * q^33 + 6 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 22 * q^38 - 2 * q^39 - 12 * q^40 + 2 * q^41 - 2 * q^42 - 4 * q^44 - 2 * q^45 - 2 * q^46 + 6 * q^47 - 16 * q^48 + 2 * q^49 + 2 * q^50 + 6 * q^51 + 40 * q^52 + 2 * q^53 - 2 * q^54 + 2 * q^55 + 12 * q^56 + 10 * q^57 + 2 * q^58 + 24 * q^59 + 4 * q^60 + 2 * q^61 - 20 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 + 2 * q^66 - 4 * q^67 + 12 * q^68 - 10 * q^69 - 2 * q^70 - 2 * q^71 + 12 * q^72 - 4 * q^73 + 20 * q^74 - 2 * q^75 - 44 * q^76 - 2 * q^77 - 20 * q^78 - 10 * q^79 - 16 * q^80 + 2 * q^81 + 32 * q^82 - 2 * q^83 - 4 * q^84 + 6 * q^85 - 6 * q^86 - 8 * q^87 - 12 * q^88 + 12 * q^89 - 2 * q^90 + 2 * q^91 - 4 * q^92 + 14 * q^93 + 12 * q^94 + 10 * q^95 - 16 * q^96 - 8 * q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.73205	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$2$	2.73205	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.46410	2.73205
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.73205	−1.11536
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	9.46410	3.34607
$9$	1.00000	0.333333
$10$	−2.73205	−0.863950
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−5.46410	−1.57735
$13$	6.19615	1.71850	0.859252	−	0.511553i	$-0.170930\pi$
$13$	6.19615	1.71850	0.859252	+	0.511553i	$0.170930\pi$
$14$	2.73205	0.730171
$15$	1.00000	0.258199
$16$	14.9282	3.73205
$17$	0.464102	0.112561	0.0562806	−	0.998415i	$-0.482076\pi$
$17$	0.464102	0.112561	0.0562806	+	0.998415i	$0.482076\pi$
$18$	2.73205	0.643951
$19$	−8.46410	−1.94180	−0.970899	−	0.239489i	$-0.923020\pi$
$19$	−8.46410	−1.94180	−0.970899	+	0.239489i	$0.923020\pi$
$20$	−5.46410	−1.22181
$21$	−1.00000	−0.218218
$22$	−2.73205	−0.582475
$23$	1.53590	0.320257	0.160128	−	0.987096i	$-0.448809\pi$
$23$	1.53590	0.320257	0.160128	+	0.987096i	$0.448809\pi$
$24$	−9.46410	−1.93185
$25$	1.00000	0.200000
$26$	16.9282	3.31989
$27$	−1.00000	−0.192450
$28$	5.46410	1.03262
$29$	2.26795	0.421148	0.210574	−	0.977578i	$-0.432467\pi$
$29$	2.26795	0.421148	0.210574	+	0.977578i	$0.432467\pi$
$30$	2.73205	0.498802
$31$	−8.73205	−1.56832	−0.784161	−	0.620557i	$-0.786907\pi$
$31$	−8.73205	−1.56832	−0.784161	+	0.620557i	$0.786907\pi$
$32$	21.8564	3.86370
$33$	1.00000	0.174078
$34$	1.26795	0.217451
$35$	−1.00000	−0.169031
$36$	5.46410	0.910684
$37$	6.19615	1.01864	0.509321	−	0.860577i	$-0.329897\pi$
$37$	6.19615	1.01864	0.509321	+	0.860577i	$0.329897\pi$
$38$	−23.1244	−3.75127
$39$	−6.19615	−0.992178
$40$	−9.46410	−1.49641
$41$	9.66025	1.50868	0.754339	−	0.656485i	$-0.227957\pi$
$41$	9.66025	1.50868	0.754339	+	0.656485i	$0.227957\pi$
$42$	−2.73205	−0.421565
$43$	−1.73205	−0.264135	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	−1.73205	−0.264135	−0.132068	+	0.991241i	$0.542162\pi$
$44$	−5.46410	−0.823744
$45$	−1.00000	−0.149071
$46$	4.19615	0.618689
$47$	4.73205	0.690241	0.345120	−	0.938558i	$-0.387838\pi$
$47$	4.73205	0.690241	0.345120	+	0.938558i	$0.387838\pi$
$48$	−14.9282	−2.15470
$49$	1.00000	0.142857
$50$	2.73205	0.386370
$51$	−0.464102	−0.0649872
$52$	33.8564	4.69504
$53$	−2.46410	−0.338470	−0.169235	−	0.985576i	$-0.554130\pi$
$53$	−2.46410	−0.338470	−0.169235	+	0.985576i	$0.554130\pi$
$54$	−2.73205	−0.371785
$55$	1.00000	0.134840
$56$	9.46410	1.26469
$57$	8.46410	1.12110
$58$	6.19615	0.813595
$59$	10.2679	1.33677	0.668387	−	0.743814i	$-0.266985\pi$
$59$	10.2679	1.33677	0.668387	+	0.743814i	$0.266985\pi$
$60$	5.46410	0.705412
$61$	−9.39230	−1.20256	−0.601281	−	0.799038i	$-0.705343\pi$
$61$	−9.39230	−1.20256	−0.601281	+	0.799038i	$0.705343\pi$
$62$	−23.8564	−3.02977
$63$	1.00000	0.125988
$64$	29.8564	3.73205
$65$	−6.19615	−0.768538
$66$	2.73205	0.336292
$67$	−8.92820	−1.09075	−0.545377	−	0.838191i	$-0.683613\pi$
$67$	−8.92820	−1.09075	−0.545377	+	0.838191i	$0.683613\pi$
$68$	2.53590	0.307523
$69$	−1.53590	−0.184900
$70$	−2.73205	−0.326543
$71$	−9.66025	−1.14646	−0.573231	−	0.819394i	$-0.694310\pi$
$71$	−9.66025	−1.14646	−0.573231	+	0.819394i	$0.694310\pi$
$72$	9.46410	1.11536
$73$	−12.3923	−1.45041	−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923	−1.45041	−0.725205	+	0.688533i	$0.758255\pi$
$74$	16.9282	1.96786
$75$	−1.00000	−0.115470
$76$	−46.2487	−5.30509
$77$	−1.00000	−0.113961
$78$	−16.9282	−1.91674
$79$	−13.6603	−1.53690	−0.768449	−	0.639911i	$-0.778971\pi$
$79$	−13.6603	−1.53690	−0.768449	+	0.639911i	$0.778971\pi$
$80$	−14.9282	−1.66902
$81$	1.00000	0.111111
$82$	26.3923	2.91454
$83$	−4.46410	−0.489999	−0.244999	−	0.969523i	$-0.578788\pi$
$83$	−4.46410	−0.489999	−0.244999	+	0.969523i	$0.578788\pi$
$84$	−5.46410	−0.596182
$85$	−0.464102	−0.0503389
$86$	−4.73205	−0.510270
$87$	−2.26795	−0.243150
$88$	−9.46410	−1.00888
$89$	−2.66025	−0.281986	−0.140993	−	0.990011i	$-0.545030\pi$
$89$	−2.66025	−0.281986	−0.140993	+	0.990011i	$0.545030\pi$
$90$	−2.73205	−0.287983
$91$	6.19615	0.649533
$92$	8.39230	0.874958
$93$	8.73205	0.905471
$94$	12.9282	1.33344
$95$	8.46410	0.868399
$96$	−21.8564	−2.23071
$97$	−5.73205	−0.582002	−0.291001	−	0.956723i	$-0.593988\pi$
$97$	−5.73205	−0.582002	−0.291001	+	0.956723i	$0.593988\pi$
$98$	2.73205	0.275979
$99$	−1.00000	−0.100504
$100$	5.46410	0.546410
$101$	3.07180	0.305655	0.152828	−	0.988253i	$-0.451162\pi$
$101$	3.07180	0.305655	0.152828	+	0.988253i	$0.451162\pi$
$102$	−1.26795	−0.125546
$103$	−17.7321	−1.74719	−0.873595	−	0.486653i	$-0.838218\pi$
$103$	−17.7321	−1.74719	−0.873595	+	0.486653i	$0.838218\pi$
$104$	58.6410	5.75022
$105$	1.00000	0.0975900
$106$	−6.73205	−0.653875
$107$	8.73205	0.844159	0.422080	−	0.906559i	$-0.361300\pi$
$107$	8.73205	0.844159	0.422080	+	0.906559i	$0.361300\pi$
$108$	−5.46410	−0.525783
$109$	−8.73205	−0.836379	−0.418189	−	0.908360i	$-0.637335\pi$
$109$	−8.73205	−0.836379	−0.418189	+	0.908360i	$0.637335\pi$
$110$	2.73205	0.260491
$111$	−6.19615	−0.588113
$112$	14.9282	1.41058
$113$	−16.8564	−1.58572	−0.792859	−	0.609406i	$-0.791408\pi$
$113$	−16.8564	−1.58572	−0.792859	+	0.609406i	$0.791408\pi$
$114$	23.1244	2.16579
$115$	−1.53590	−0.143223
$116$	12.3923	1.15060
$117$	6.19615	0.572834
$118$	28.0526	2.58245
$119$	0.464102	0.0425441
$120$	9.46410	0.863950
$121$	1.00000	0.0909091
$122$	−25.6603	−2.32317
$123$	−9.66025	−0.871036
$124$	−47.7128	−4.28474
$125$	−1.00000	−0.0894427
$126$	2.73205	0.243390
$127$	2.12436	0.188506	0.0942530	−	0.995548i	$-0.469954\pi$
$127$	2.12436	0.188506	0.0942530	+	0.995548i	$0.469954\pi$
$128$	37.8564	3.34607
$129$	1.73205	0.152499
$130$	−16.9282	−1.48470
$131$	5.80385	0.507085	0.253542	−	0.967324i	$-0.418404\pi$
$131$	5.80385	0.507085	0.253542	+	0.967324i	$0.418404\pi$
$132$	5.46410	0.475589
$133$	−8.46410	−0.733931
$134$	−24.3923	−2.10717
$135$	1.00000	0.0860663
$136$	4.39230	0.376637
$137$	−13.4641	−1.15032	−0.575158	−	0.818042i	$-0.695059\pi$
$137$	−13.4641	−1.15032	−0.575158	+	0.818042i	$0.695059\pi$
$138$	−4.19615	−0.357200
$139$	4.53590	0.384730	0.192365	−	0.981323i	$-0.438384\pi$
$139$	4.53590	0.384730	0.192365	+	0.981323i	$0.438384\pi$
$140$	−5.46410	−0.461801
$141$	−4.73205	−0.398511
$142$	−26.3923	−2.21479
$143$	−6.19615	−0.518148
$144$	14.9282	1.24402
$145$	−2.26795	−0.188343
$146$	−33.8564	−2.80198
$147$	−1.00000	−0.0824786
$148$	33.8564	2.78298
$149$	8.53590	0.699288	0.349644	−	0.936883i	$-0.386303\pi$
$149$	8.53590	0.699288	0.349644	+	0.936883i	$0.386303\pi$
$150$	−2.73205	−0.223071
$151$	15.8564	1.29038	0.645188	−	0.764024i	$-0.276779\pi$
$151$	15.8564	1.29038	0.645188	+	0.764024i	$0.276779\pi$
$152$	−80.1051	−6.49738
$153$	0.464102	0.0375204
$154$	−2.73205	−0.220155
$155$	8.73205	0.701375
$156$	−33.8564	−2.71068
$157$	10.1244	0.808012	0.404006	−	0.914756i	$-0.367618\pi$
$157$	10.1244	0.808012	0.404006	+	0.914756i	$0.367618\pi$
$158$	−37.3205	−2.96906
$159$	2.46410	0.195416
$160$	−21.8564	−1.72790
$161$	1.53590	0.121046
$162$	2.73205	0.214650
$163$	−5.12436	−0.401371	−0.200685	−	0.979656i	$-0.564317\pi$
$163$	−5.12436	−0.401371	−0.200685	+	0.979656i	$0.564317\pi$
$164$	52.7846	4.12179
$165$	−1.00000	−0.0778499
$166$	−12.1962	−0.946605
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	−9.46410	−0.730171
$169$	25.3923	1.95325
$170$	−1.26795	−0.0972473
$171$	−8.46410	−0.647266
$172$	−9.46410	−0.721631
$173$	−18.5359	−1.40926	−0.704629	−	0.709576i	$-0.748887\pi$
$173$	−18.5359	−1.40926	−0.704629	+	0.709576i	$0.748887\pi$
$174$	−6.19615	−0.469729
$175$	1.00000	0.0755929
$176$	−14.9282	−1.12526
$177$	−10.2679	−0.771786
$178$	−7.26795	−0.544756
$179$	18.0526	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$179$	18.0526	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$180$	−5.46410	−0.407270
$181$	20.9282	1.55558	0.777791	−	0.628524i	$-0.216340\pi$
$181$	20.9282	1.55558	0.777791	+	0.628524i	$0.216340\pi$
$182$	16.9282	1.25480
$183$	9.39230	0.694299
$184$	14.5359	1.07160
$185$	−6.19615	−0.455550
$186$	23.8564	1.74924
$187$	−0.464102	−0.0339385
$188$	25.8564	1.88577
$189$	−1.00000	−0.0727393
$190$	23.1244	1.67762
$191$	−14.9282	−1.08017	−0.540083	−	0.841611i	$-0.681607\pi$
$191$	−14.9282	−1.08017	−0.540083	+	0.841611i	$0.681607\pi$
$192$	−29.8564	−2.15470
$193$	−0.535898	−0.0385748	−0.0192874	−	0.999814i	$-0.506140\pi$
$193$	−0.535898	−0.0385748	−0.0192874	+	0.999814i	$0.506140\pi$
$194$	−15.6603	−1.12434
$195$	6.19615	0.443716
$196$	5.46410	0.390293
$197$	−1.46410	−0.104313	−0.0521565	−	0.998639i	$-0.516609\pi$
$197$	−1.46410	−0.104313	−0.0521565	+	0.998639i	$0.516609\pi$
$198$	−2.73205	−0.194158
$199$	−1.07180	−0.0759777	−0.0379888	−	0.999278i	$-0.512095\pi$
$199$	−1.07180	−0.0759777	−0.0379888	+	0.999278i	$0.512095\pi$
$200$	9.46410	0.669213
$201$	8.92820	0.629747
$202$	8.39230	0.590481
$203$	2.26795	0.159179
$204$	−2.53590	−0.177548
$205$	−9.66025	−0.674701
$206$	−48.4449	−3.37531
$207$	1.53590	0.106752
$208$	92.4974	6.41354
$209$	8.46410	0.585474
$210$	2.73205	0.188529
$211$	11.4641	0.789221	0.394611	−	0.918848i	$-0.370879\pi$
$211$	11.4641	0.789221	0.394611	+	0.918848i	$0.370879\pi$
$212$	−13.4641	−0.924718
$213$	9.66025	0.661910
$214$	23.8564	1.63079
$215$	1.73205	0.118125
$216$	−9.46410	−0.643951
$217$	−8.73205	−0.592770
$218$	−23.8564	−1.61576
$219$	12.3923	0.837394
$220$	5.46410	0.368390
$221$	2.87564	0.193437
$222$	−16.9282	−1.13615
$223$	11.1962	0.749750	0.374875	−	0.927075i	$-0.377686\pi$
$223$	11.1962	0.749750	0.374875	+	0.927075i	$0.377686\pi$
$224$	21.8564	1.46034
$225$	1.00000	0.0666667
$226$	−46.0526	−3.06337
$227$	15.3923	1.02162	0.510812	−	0.859693i	$-0.329345\pi$
$227$	15.3923	1.02162	0.510812	+	0.859693i	$0.329345\pi$
$228$	46.2487	3.06290
$229$	17.8038	1.17651	0.588256	−	0.808675i	$-0.299815\pi$
$229$	17.8038	1.17651	0.588256	+	0.808675i	$0.299815\pi$
$230$	−4.19615	−0.276686
$231$	1.00000	0.0657952
$232$	21.4641	1.40919
$233$	0.196152	0.0128504	0.00642519	−	0.999979i	$-0.497955\pi$
$233$	0.196152	0.0128504	0.00642519	+	0.999979i	$0.497955\pi$
$234$	16.9282	1.10663
$235$	−4.73205	−0.308685
$236$	56.1051	3.65213
$237$	13.6603	0.887329
$238$	1.26795	0.0821889
$239$	−6.12436	−0.396152	−0.198076	−	0.980187i	$-0.563469\pi$
$239$	−6.12436	−0.396152	−0.198076	+	0.980187i	$0.563469\pi$
$240$	14.9282	0.963611
$241$	13.0718	0.842028	0.421014	−	0.907054i	$-0.361674\pi$
$241$	13.0718	0.842028	0.421014	+	0.907054i	$0.361674\pi$
$242$	2.73205	0.175623
$243$	−1.00000	−0.0641500
$244$	−51.3205	−3.28546
$245$	−1.00000	−0.0638877
$246$	−26.3923	−1.68271
$247$	−52.4449	−3.33699
$248$	−82.6410	−5.24771
$249$	4.46410	0.282901
$250$	−2.73205	−0.172790
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	5.46410	0.344206
$253$	−1.53590	−0.0965611
$254$	5.80385	0.364166
$255$	0.464102	0.0290632
$256$	43.7128	2.73205
$257$	24.1962	1.50931	0.754657	−	0.656119i	$-0.227803\pi$
$257$	24.1962	1.50931	0.754657	+	0.656119i	$0.227803\pi$
$258$	4.73205	0.294605
$259$	6.19615	0.385010
$260$	−33.8564	−2.09969
$261$	2.26795	0.140383
$262$	15.8564	0.979612
$263$	−23.3205	−1.43800	−0.719002	−	0.695008i	$-0.755401\pi$
$263$	−23.3205	−1.43800	−0.719002	+	0.695008i	$0.755401\pi$
$264$	9.46410	0.582475
$265$	2.46410	0.151369
$266$	−23.1244	−1.41785
$267$	2.66025	0.162805
$268$	−48.7846	−2.97999
$269$	−23.5885	−1.43821	−0.719107	−	0.694900i	$-0.755449\pi$
$269$	−23.5885	−1.43821	−0.719107	+	0.694900i	$0.755449\pi$
$270$	2.73205	0.166267
$271$	−4.07180	−0.247344	−0.123672	−	0.992323i	$-0.539467\pi$
$271$	−4.07180	−0.247344	−0.123672	+	0.992323i	$0.539467\pi$
$272$	6.92820	0.420084
$273$	−6.19615	−0.375008
$274$	−36.7846	−2.22224
$275$	−1.00000	−0.0603023
$276$	−8.39230	−0.505157
$277$	−22.9282	−1.37762	−0.688811	−	0.724941i	$-0.741867\pi$
$277$	−22.9282	−1.37762	−0.688811	+	0.724941i	$0.741867\pi$
$278$	12.3923	0.743241
$279$	−8.73205	−0.522774
$280$	−9.46410	−0.565588
$281$	30.9282	1.84502	0.922511	−	0.385971i	$-0.126133\pi$
$281$	30.9282	1.84502	0.922511	+	0.385971i	$0.126133\pi$
$282$	−12.9282	−0.769863
$283$	10.1962	0.606098	0.303049	−	0.952975i	$-0.401995\pi$
$283$	10.1962	0.606098	0.303049	+	0.952975i	$0.401995\pi$
$284$	−52.7846	−3.13219
$285$	−8.46410	−0.501370
$286$	−16.9282	−1.00099
$287$	9.66025	0.570227
$288$	21.8564	1.28790
$289$	−16.7846	−0.987330
$290$	−6.19615	−0.363851
$291$	5.73205	0.336019
$292$	−67.7128	−3.96259
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−2.73205	−0.159336
$295$	−10.2679	−0.597823
$296$	58.6410	3.40844
$297$	1.00000	0.0580259
$298$	23.3205	1.35092
$299$	9.51666	0.550363
$300$	−5.46410	−0.315470
$301$	−1.73205	−0.0998337
$302$	43.3205	2.49282
$303$	−3.07180	−0.176470
$304$	−126.354	−7.24689
$305$	9.39230	0.537802
$306$	1.26795	0.0724838
$307$	10.3923	0.593120	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593120	0.296560	+	0.955014i	$0.404160\pi$
$308$	−5.46410	−0.311346
$309$	17.7321	1.00874
$310$	23.8564	1.35495
$311$	−27.3205	−1.54920	−0.774602	−	0.632449i	$-0.782050\pi$
$311$	−27.3205	−1.54920	−0.774602	+	0.632449i	$0.782050\pi$
$312$	−58.6410	−3.31989
$313$	−1.73205	−0.0979013	−0.0489506	−	0.998801i	$-0.515588\pi$
$313$	−1.73205	−0.0979013	−0.0489506	+	0.998801i	$0.515588\pi$
$314$	27.6603	1.56096
$315$	−1.00000	−0.0563436
$316$	−74.6410	−4.19889
$317$	5.07180	0.284860	0.142430	−	0.989805i	$-0.454508\pi$
$317$	5.07180	0.284860	0.142430	+	0.989805i	$0.454508\pi$
$318$	6.73205	0.377515
$319$	−2.26795	−0.126981
$320$	−29.8564	−1.66902
$321$	−8.73205	−0.487376
$322$	4.19615	0.233842
$323$	−3.92820	−0.218571
$324$	5.46410	0.303561
$325$	6.19615	0.343701
$326$	−14.0000	−0.775388
$327$	8.73205	0.482884
$328$	91.4256	5.04814
$329$	4.73205	0.260886
$330$	−2.73205	−0.150394
$331$	−26.3205	−1.44671	−0.723353	−	0.690478i	$-0.757400\pi$
$331$	−26.3205	−1.44671	−0.723353	+	0.690478i	$0.757400\pi$
$332$	−24.3923	−1.33870
$333$	6.19615	0.339547
$334$	28.3923	1.55356
$335$	8.92820	0.487800
$336$	−14.9282	−0.814400
$337$	6.66025	0.362807	0.181404	−	0.983409i	$-0.441936\pi$
$337$	6.66025	0.362807	0.181404	+	0.983409i	$0.441936\pi$
$338$	69.3731	3.77340
$339$	16.8564	0.915514
$340$	−2.53590	−0.137528
$341$	8.73205	0.472867
$342$	−23.1244	−1.25042
$343$	1.00000	0.0539949
$344$	−16.3923	−0.883814
$345$	1.53590	0.0826900
$346$	−50.6410	−2.72248
$347$	5.46410	0.293328	0.146664	−	0.989186i	$-0.453146\pi$
$347$	5.46410	0.293328	0.146664	+	0.989186i	$0.453146\pi$
$348$	−12.3923	−0.664297
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	2.73205	0.146034
$351$	−6.19615	−0.330726
$352$	−21.8564	−1.16495
$353$	8.53590	0.454320	0.227160	−	0.973857i	$-0.427056\pi$
$353$	8.53590	0.454320	0.227160	+	0.973857i	$0.427056\pi$
$354$	−28.0526	−1.49098
$355$	9.66025	0.512713
$356$	−14.5359	−0.770401
$357$	−0.464102	−0.0245629
$358$	49.3205	2.60667
$359$	10.5167	0.555048	0.277524	−	0.960719i	$-0.410486\pi$
$359$	10.5167	0.555048	0.277524	+	0.960719i	$0.410486\pi$
$360$	−9.46410	−0.498802
$361$	52.6410	2.77058
$362$	57.1769	3.00515
$363$	−1.00000	−0.0524864
$364$	33.8564	1.77456
$365$	12.3923	0.648643
$366$	25.6603	1.34128
$367$	27.1962	1.41963	0.709814	−	0.704389i	$-0.248779\pi$
$367$	27.1962	1.41963	0.709814	+	0.704389i	$0.248779\pi$
$368$	22.9282	1.19522
$369$	9.66025	0.502893
$370$	−16.9282	−0.880055
$371$	−2.46410	−0.127930
$372$	47.7128	2.47379
$373$	18.8038	0.973626	0.486813	−	0.873506i	$-0.338159\pi$
$373$	18.8038	0.973626	0.486813	+	0.873506i	$0.338159\pi$
$374$	−1.26795	−0.0655641
$375$	1.00000	0.0516398
$376$	44.7846	2.30959
$377$	14.0526	0.723744
$378$	−2.73205	−0.140522
$379$	−6.85641	−0.352190	−0.176095	−	0.984373i	$-0.556347\pi$
$379$	−6.85641	−0.352190	−0.176095	+	0.984373i	$0.556347\pi$
$380$	46.2487	2.37251
$381$	−2.12436	−0.108834
$382$	−40.7846	−2.08672
$383$	29.2679	1.49552	0.747761	−	0.663968i	$-0.231129\pi$
$383$	29.2679	1.49552	0.747761	+	0.663968i	$0.231129\pi$
$384$	−37.8564	−1.93185
$385$	1.00000	0.0509647
$386$	−1.46410	−0.0745208
$387$	−1.73205	−0.0880451
$388$	−31.3205	−1.59006
$389$	29.5167	1.49655	0.748277	−	0.663386i	$-0.230881\pi$
$389$	29.5167	1.49655	0.748277	+	0.663386i	$0.230881\pi$
$390$	16.9282	0.857193
$391$	0.712813	0.0360485
$392$	9.46410	0.478009
$393$	−5.80385	−0.292765
$394$	−4.00000	−0.201517
$395$	13.6603	0.687322
$396$	−5.46410	−0.274581
$397$	10.1436	0.509092	0.254546	−	0.967061i	$-0.418074\pi$
$397$	10.1436	0.509092	0.254546	+	0.967061i	$0.418074\pi$
$398$	−2.92820	−0.146778
$399$	8.46410	0.423735
$400$	14.9282	0.746410
$401$	−8.87564	−0.443229	−0.221614	−	0.975134i	$-0.571133\pi$
$401$	−8.87564	−0.443229	−0.221614	+	0.975134i	$0.571133\pi$
$402$	24.3923	1.21658
$403$	−54.1051	−2.69517
$404$	16.7846	0.835066
$405$	−1.00000	−0.0496904
$406$	6.19615	0.307510
$407$	−6.19615	−0.307132
$408$	−4.39230	−0.217451
$409$	−22.9282	−1.13373	−0.566863	−	0.823812i	$-0.691843\pi$
$409$	−22.9282	−1.13373	−0.566863	+	0.823812i	$0.691843\pi$
$410$	−26.3923	−1.30342
$411$	13.4641	0.664135
$412$	−96.8897	−4.77341
$413$	10.2679	0.505253
$414$	4.19615	0.206230
$415$	4.46410	0.219134
$416$	135.426	6.63979
$417$	−4.53590	−0.222124
$418$	23.1244	1.13105
$419$	26.2679	1.28327	0.641637	−	0.767009i	$-0.278256\pi$
$419$	26.2679	1.28327	0.641637	+	0.767009i	$0.278256\pi$
$420$	5.46410	0.266621
$421$	−30.7128	−1.49685	−0.748425	−	0.663219i	$-0.769190\pi$
$421$	−30.7128	−1.49685	−0.748425	+	0.663219i	$0.769190\pi$
$422$	31.3205	1.52466
$423$	4.73205	0.230080
$424$	−23.3205	−1.13254
$425$	0.464102	0.0225122
$426$	26.3923	1.27871
$427$	−9.39230	−0.454525
$428$	47.7128	2.30629
$429$	6.19615	0.299153
$430$	4.73205	0.228200
$431$	−3.60770	−0.173777	−0.0868883	−	0.996218i	$-0.527692\pi$
$431$	−3.60770	−0.173777	−0.0868883	+	0.996218i	$0.527692\pi$
$432$	−14.9282	−0.718234
$433$	−25.3205	−1.21683	−0.608413	−	0.793621i	$-0.708194\pi$
$433$	−25.3205	−1.21683	−0.608413	+	0.793621i	$0.708194\pi$
$434$	−23.8564	−1.14514
$435$	2.26795	0.108740
$436$	−47.7128	−2.28503
$437$	−13.0000	−0.621874
$438$	33.8564	1.61772
$439$	28.7128	1.37039	0.685194	−	0.728361i	$-0.259717\pi$
$439$	28.7128	1.37039	0.685194	+	0.728361i	$0.259717\pi$
$440$	9.46410	0.451183
$441$	1.00000	0.0476190
$442$	7.85641	0.373691
$443$	−16.5359	−0.785644	−0.392822	−	0.919614i	$-0.628501\pi$
$443$	−16.5359	−0.785644	−0.392822	+	0.919614i	$0.628501\pi$
$444$	−33.8564	−1.60675
$445$	2.66025	0.126108
$446$	30.5885	1.44841
$447$	−8.53590	−0.403734
$448$	29.8564	1.41058
$449$	−22.0526	−1.04072	−0.520362	−	0.853946i	$-0.674203\pi$
$449$	−22.0526	−1.04072	−0.520362	+	0.853946i	$0.674203\pi$
$450$	2.73205	0.128790
$451$	−9.66025	−0.454884
$452$	−92.1051	−4.33226
$453$	−15.8564	−0.744999
$454$	42.0526	1.97362
$455$	−6.19615	−0.290480
$456$	80.1051	3.75127
$457$	14.6603	0.685778	0.342889	−	0.939376i	$-0.388595\pi$
$457$	14.6603	0.685778	0.342889	+	0.939376i	$0.388595\pi$
$458$	48.6410	2.27285
$459$	−0.464102	−0.0216624
$460$	−8.39230	−0.391293
$461$	−5.32051	−0.247801	−0.123900	−	0.992295i	$-0.539540\pi$
$461$	−5.32051	−0.247801	−0.123900	+	0.992295i	$0.539540\pi$
$462$	2.73205	0.127107
$463$	19.4641	0.904574	0.452287	−	0.891873i	$-0.350608\pi$
$463$	19.4641	0.904574	0.452287	+	0.891873i	$0.350608\pi$
$464$	33.8564	1.57174
$465$	−8.73205	−0.404939
$466$	0.535898	0.0248250
$467$	7.32051	0.338753	0.169376	−	0.985551i	$-0.445825\pi$
$467$	7.32051	0.338753	0.169376	+	0.985551i	$0.445825\pi$
$468$	33.8564	1.56501
$469$	−8.92820	−0.412266
$470$	−12.9282	−0.596334
$471$	−10.1244	−0.466506
$472$	97.1769	4.47293
$473$	1.73205	0.0796398
$474$	37.3205	1.71419
$475$	−8.46410	−0.388360
$476$	2.53590	0.116233
$477$	−2.46410	−0.112823
$478$	−16.7321	−0.765306
$479$	−3.26795	−0.149316	−0.0746582	−	0.997209i	$-0.523787\pi$
$479$	−3.26795	−0.149316	−0.0746582	+	0.997209i	$0.523787\pi$
$480$	21.8564	0.997604
$481$	38.3923	1.75054
$482$	35.7128	1.62667
$483$	−1.53590	−0.0698858
$484$	5.46410	0.248368
$485$	5.73205	0.260279
$486$	−2.73205	−0.123928
$487$	7.46410	0.338231	0.169115	−	0.985596i	$-0.445909\pi$
$487$	7.46410	0.338231	0.169115	+	0.985596i	$0.445909\pi$
$488$	−88.8897	−4.02385
$489$	5.12436	0.231731
$490$	−2.73205	−0.123421
$491$	20.6603	0.932384	0.466192	−	0.884684i	$-0.345626\pi$
$491$	20.6603	0.932384	0.466192	+	0.884684i	$0.345626\pi$
$492$	−52.7846	−2.37971
$493$	1.05256	0.0474049
$494$	−143.282	−6.44656
$495$	1.00000	0.0449467
$496$	−130.354	−5.85306
$497$	−9.66025	−0.433322
$498$	12.1962	0.546523
$499$	8.60770	0.385333	0.192667	−	0.981264i	$-0.438286\pi$
$499$	8.60770	0.385333	0.192667	+	0.981264i	$0.438286\pi$
$500$	−5.46410	−0.244362
$501$	−10.3923	−0.464294
$502$	18.9282	0.844807
$503$	−23.0000	−1.02552	−0.512760	−	0.858532i	$-0.671377\pi$
$503$	−23.0000	−1.02552	−0.512760	+	0.858532i	$0.671377\pi$
$504$	9.46410	0.421565
$505$	−3.07180	−0.136693
$506$	−4.19615	−0.186542
$507$	−25.3923	−1.12771
$508$	11.6077	0.515008
$509$	−6.80385	−0.301575	−0.150788	−	0.988566i	$-0.548181\pi$
$509$	−6.80385	−0.301575	−0.150788	+	0.988566i	$0.548181\pi$
$510$	1.26795	0.0561457
$511$	−12.3923	−0.548203
$512$	43.7128	1.93185
$513$	8.46410	0.373699
$514$	66.1051	2.91577
$515$	17.7321	0.781368
$516$	9.46410	0.416634
$517$	−4.73205	−0.208115
$518$	16.9282	0.743783
$519$	18.5359	0.813636
$520$	−58.6410	−2.57158
$521$	29.0526	1.27282	0.636408	−	0.771353i	$-0.280420\pi$
$521$	29.0526	1.27282	0.636408	+	0.771353i	$0.280420\pi$
$522$	6.19615	0.271198
$523$	7.32051	0.320103	0.160052	−	0.987109i	$-0.448834\pi$
$523$	7.32051	0.320103	0.160052	+	0.987109i	$0.448834\pi$
$524$	31.7128	1.38538
$525$	−1.00000	−0.0436436
$526$	−63.7128	−2.77801
$527$	−4.05256	−0.176532
$528$	14.9282	0.649667
$529$	−20.6410	−0.897435
$530$	6.73205	0.292422
$531$	10.2679	0.445591
$532$	−46.2487	−2.00514
$533$	59.8564	2.59267
$534$	7.26795	0.314515
$535$	−8.73205	−0.377519
$536$	−84.4974	−3.64973
$537$	−18.0526	−0.779025
$538$	−64.4449	−2.77842
$539$	−1.00000	−0.0430730
$540$	5.46410	0.235137
$541$	4.05256	0.174233	0.0871166	−	0.996198i	$-0.472235\pi$
$541$	4.05256	0.174233	0.0871166	+	0.996198i	$0.472235\pi$
$542$	−11.1244	−0.477832
$543$	−20.9282	−0.898115
$544$	10.1436	0.434903
$545$	8.73205	0.374040
$546$	−16.9282	−0.724460
$547$	−7.58846	−0.324459	−0.162230	−	0.986753i	$-0.551868\pi$
$547$	−7.58846	−0.324459	−0.162230	+	0.986753i	$0.551868\pi$
$548$	−73.5692	−3.14272
$549$	−9.39230	−0.400854
$550$	−2.73205	−0.116495
$551$	−19.1962	−0.817784
$552$	−14.5359	−0.618689
$553$	−13.6603	−0.580893
$554$	−62.6410	−2.66136
$555$	6.19615	0.263012
$556$	24.7846	1.05110
$557$	15.4641	0.655235	0.327618	−	0.944810i	$-0.393754\pi$
$557$	15.4641	0.655235	0.327618	+	0.944810i	$0.393754\pi$
$558$	−23.8564	−1.00992
$559$	−10.7321	−0.453917
$560$	−14.9282	−0.630832
$561$	0.464102	0.0195944
$562$	84.4974	3.56431
$563$	−11.0718	−0.466621	−0.233310	−	0.972402i	$-0.574956\pi$
$563$	−11.0718	−0.466621	−0.233310	+	0.972402i	$0.574956\pi$
$564$	−25.8564	−1.08875
$565$	16.8564	0.709154
$566$	27.8564	1.17089
$567$	1.00000	0.0419961
$568$	−91.4256	−3.83613
$569$	12.6603	0.530745	0.265373	−	0.964146i	$-0.414505\pi$
$569$	12.6603	0.530745	0.265373	+	0.964146i	$0.414505\pi$
$570$	−23.1244	−0.968573
$571$	−7.12436	−0.298145	−0.149073	−	0.988826i	$-0.547629\pi$
$571$	−7.12436	−0.298145	−0.149073	+	0.988826i	$0.547629\pi$
$572$	−33.8564	−1.41561
$573$	14.9282	0.623635
$574$	26.3923	1.10159
$575$	1.53590	0.0640514
$576$	29.8564	1.24402
$577$	−20.9282	−0.871253	−0.435626	−	0.900128i	$-0.643473\pi$
$577$	−20.9282	−0.871253	−0.435626	+	0.900128i	$0.643473\pi$
$578$	−45.8564	−1.90738
$579$	0.535898	0.0222712
$580$	−12.3923	−0.514562
$581$	−4.46410	−0.185202
$582$	15.6603	0.649138
$583$	2.46410	0.102053
$584$	−117.282	−4.85317
$585$	−6.19615	−0.256179
$586$	24.5885	1.01574
$587$	12.5885	0.519581	0.259791	−	0.965665i	$-0.416346\pi$
$587$	12.5885	0.519581	0.259791	+	0.965665i	$0.416346\pi$
$588$	−5.46410	−0.225336
$589$	73.9090	3.04537
$590$	−28.0526	−1.15491
$591$	1.46410	0.0602251
$592$	92.4974	3.80162
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	2.73205	0.112097
$595$	−0.464102	−0.0190263
$596$	46.6410	1.91049
$597$	1.07180	0.0438657
$598$	26.0000	1.06322
$599$	−20.2487	−0.827340	−0.413670	−	0.910427i	$-0.635753\pi$
$599$	−20.2487	−0.827340	−0.413670	+	0.910427i	$0.635753\pi$
$600$	−9.46410	−0.386370
$601$	−23.7846	−0.970194	−0.485097	−	0.874460i	$-0.661216\pi$
$601$	−23.7846	−0.970194	−0.485097	+	0.874460i	$0.661216\pi$
$602$	−4.73205	−0.192864
$603$	−8.92820	−0.363585
$604$	86.6410	3.52537
$605$	−1.00000	−0.0406558
$606$	−8.39230	−0.340914
$607$	−39.1769	−1.59014	−0.795071	−	0.606516i	$-0.792566\pi$
$607$	−39.1769	−1.59014	−0.795071	+	0.606516i	$0.792566\pi$
$608$	−184.995	−7.50253
$609$	−2.26795	−0.0919019
$610$	25.6603	1.03895
$611$	29.3205	1.18618
$612$	2.53590	0.102508
$613$	−6.78461	−0.274028	−0.137014	−	0.990569i	$-0.543751\pi$
$613$	−6.78461	−0.274028	−0.137014	+	0.990569i	$0.543751\pi$
$614$	28.3923	1.14582
$615$	9.66025	0.389539
$616$	−9.46410	−0.381320
$617$	22.6410	0.911493	0.455746	−	0.890110i	$-0.349372\pi$
$617$	22.6410	0.911493	0.455746	+	0.890110i	$0.349372\pi$
$618$	48.4449	1.94874
$619$	−11.0718	−0.445013	−0.222507	−	0.974931i	$-0.571424\pi$
$619$	−11.0718	−0.445013	−0.222507	+	0.974931i	$0.571424\pi$
$620$	47.7128	1.91619
$621$	−1.53590	−0.0616335
$622$	−74.6410	−2.99283
$623$	−2.66025	−0.106581
$624$	−92.4974	−3.70286
$625$	1.00000	0.0400000
$626$	−4.73205	−0.189131
$627$	−8.46410	−0.338024
$628$	55.3205	2.20753
$629$	2.87564	0.114659
$630$	−2.73205	−0.108848
$631$	38.7128	1.54113	0.770566	−	0.637360i	$-0.219973\pi$
$631$	38.7128	1.54113	0.770566	+	0.637360i	$0.219973\pi$
$632$	−129.282	−5.14256
$633$	−11.4641	−0.455657
$634$	13.8564	0.550308
$635$	−2.12436	−0.0843025
$636$	13.4641	0.533886
$637$	6.19615	0.245500
$638$	−6.19615	−0.245308
$639$	−9.66025	−0.382154
$640$	−37.8564	−1.49641
$641$	40.6410	1.60522	0.802612	−	0.596502i	$-0.203443\pi$
$641$	40.6410	1.60522	0.802612	+	0.596502i	$0.203443\pi$
$642$	−23.8564	−0.941537
$643$	−17.1962	−0.678150	−0.339075	−	0.940759i	$-0.610114\pi$
$643$	−17.1962	−0.678150	−0.339075	+	0.940759i	$0.610114\pi$
$644$	8.39230	0.330703
$645$	−1.73205	−0.0681994
$646$	−10.7321	−0.422247
$647$	−32.1051	−1.26218	−0.631091	−	0.775709i	$-0.717393\pi$
$647$	−32.1051	−1.26218	−0.631091	+	0.775709i	$0.717393\pi$
$648$	9.46410	0.371785
$649$	−10.2679	−0.403052
$650$	16.9282	0.663979
$651$	8.73205	0.342236
$652$	−28.0000	−1.09656
$653$	−36.4641	−1.42695	−0.713475	−	0.700680i	$-0.752880\pi$
$653$	−36.4641	−1.42695	−0.713475	+	0.700680i	$0.752880\pi$
$654$	23.8564	0.932859
$655$	−5.80385	−0.226775
$656$	144.210	5.63046
$657$	−12.3923	−0.483470
$658$	12.9282	0.503994
$659$	−5.87564	−0.228883	−0.114441	−	0.993430i	$-0.536508\pi$
$659$	−5.87564	−0.228883	−0.114441	+	0.993430i	$0.536508\pi$
$660$	−5.46410	−0.212690
$661$	0.732051	0.0284735	0.0142367	−	0.999899i	$-0.495468\pi$
$661$	0.732051	0.0284735	0.0142367	+	0.999899i	$0.495468\pi$
$662$	−71.9090	−2.79482
$663$	−2.87564	−0.111681
$664$	−42.2487	−1.63957
$665$	8.46410	0.328224
$666$	16.9282	0.655955
$667$	3.48334	0.134875
$668$	56.7846	2.19706
$669$	−11.1962	−0.432868
$670$	24.3923	0.942357
$671$	9.39230	0.362586
$672$	−21.8564	−0.843129
$673$	13.5885	0.523797	0.261898	−	0.965095i	$-0.415652\pi$
$673$	13.5885	0.523797	0.261898	+	0.965095i	$0.415652\pi$
$674$	18.1962	0.700890
$675$	−1.00000	−0.0384900
$676$	138.746	5.33639
$677$	9.24871	0.355457	0.177728	−	0.984080i	$-0.443125\pi$
$677$	9.24871	0.355457	0.177728	+	0.984080i	$0.443125\pi$
$678$	46.0526	1.76864
$679$	−5.73205	−0.219976
$680$	−4.39230	−0.168437
$681$	−15.3923	−0.589834
$682$	23.8564	0.913509
$683$	−33.0718	−1.26546	−0.632729	−	0.774374i	$-0.718065\pi$
$683$	−33.0718	−1.26546	−0.632729	+	0.774374i	$0.718065\pi$
$684$	−46.2487	−1.76836
$685$	13.4641	0.514437
$686$	2.73205	0.104310
$687$	−17.8038	−0.679259
$688$	−25.8564	−0.985766
$689$	−15.2679	−0.581663
$690$	4.19615	0.159745
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	−101.282	−3.85017
$693$	−1.00000	−0.0379869
$694$	14.9282	0.566667
$695$	−4.53590	−0.172056
$696$	−21.4641	−0.813595
$697$	4.48334	0.169819
$698$	−40.9808	−1.55114
$699$	−0.196152	−0.00741917
$700$	5.46410	0.206524
$701$	22.8038	0.861289	0.430645	−	0.902522i	$-0.358286\pi$
$701$	22.8038	0.861289	0.430645	+	0.902522i	$0.358286\pi$
$702$	−16.9282	−0.638914
$703$	−52.4449	−1.97800
$704$	−29.8564	−1.12526
$705$	4.73205	0.178219
$706$	23.3205	0.877679
$707$	3.07180	0.115527
$708$	−56.1051	−2.10856
$709$	29.5359	1.10924	0.554622	−	0.832102i	$-0.312863\pi$
$709$	29.5359	1.10924	0.554622	+	0.832102i	$0.312863\pi$
$710$	26.3923	0.990486
$711$	−13.6603	−0.512300
$712$	−25.1769	−0.943545
$713$	−13.4115	−0.502266
$714$	−1.26795	−0.0474518
$715$	6.19615	0.231723
$716$	98.6410	3.68639
$717$	6.12436	0.228718
$718$	28.7321	1.07227
$719$	15.3397	0.572076	0.286038	−	0.958218i	$-0.407662\pi$
$719$	15.3397	0.572076	0.286038	+	0.958218i	$0.407662\pi$
$720$	−14.9282	−0.556341
$721$	−17.7321	−0.660376
$722$	143.818	5.35235
$723$	−13.0718	−0.486145
$724$	114.354	4.24993
$725$	2.26795	0.0842295
$726$	−2.73205	−0.101396
$727$	−22.9090	−0.849646	−0.424823	−	0.905276i	$-0.639664\pi$
$727$	−22.9090	−0.849646	−0.424823	+	0.905276i	$0.639664\pi$
$728$	58.6410	2.17338
$729$	1.00000	0.0370370
$730$	33.8564	1.25308
$731$	−0.803848	−0.0297314
$732$	51.3205	1.89686
$733$	−11.1769	−0.412829	−0.206414	−	0.978465i	$-0.566179\pi$
$733$	−11.1769	−0.412829	−0.206414	+	0.978465i	$0.566179\pi$
$734$	74.3013	2.74251
$735$	1.00000	0.0368856
$736$	33.5692	1.23738
$737$	8.92820	0.328875
$738$	26.3923	0.971514
$739$	15.4641	0.568856	0.284428	−	0.958697i	$-0.408196\pi$
$739$	15.4641	0.568856	0.284428	+	0.958697i	$0.408196\pi$
$740$	−33.8564	−1.24459
$741$	52.4449	1.92661
$742$	−6.73205	−0.247141
$743$	−43.1769	−1.58401	−0.792004	−	0.610516i	$-0.790962\pi$
$743$	−43.1769	−1.58401	−0.792004	+	0.610516i	$0.790962\pi$
$744$	82.6410	3.02977
$745$	−8.53590	−0.312731
$746$	51.3731	1.88090
$747$	−4.46410	−0.163333
$748$	−2.53590	−0.0927216
$749$	8.73205	0.319062
$750$	2.73205	0.0997604
$751$	−18.3205	−0.668525	−0.334262	−	0.942480i	$-0.608487\pi$
$751$	−18.3205	−0.668525	−0.334262	+	0.942480i	$0.608487\pi$
$752$	70.6410	2.57601
$753$	−6.92820	−0.252478
$754$	38.3923	1.39817
$755$	−15.8564	−0.577074
$756$	−5.46410	−0.198727
$757$	15.4641	0.562052	0.281026	−	0.959700i	$-0.409325\pi$
$757$	15.4641	0.562052	0.281026	+	0.959700i	$0.409325\pi$
$758$	−18.7321	−0.680379
$759$	1.53590	0.0557496
$760$	80.1051	2.90572
$761$	−24.9282	−0.903647	−0.451823	−	0.892107i	$-0.649226\pi$
$761$	−24.9282	−0.903647	−0.451823	+	0.892107i	$0.649226\pi$
$762$	−5.80385	−0.210251
$763$	−8.73205	−0.316121
$764$	−81.5692	−2.95107
$765$	−0.464102	−0.0167796
$766$	79.9615	2.88913
$767$	63.6218	2.29725
$768$	−43.7128	−1.57735
$769$	17.7846	0.641329	0.320665	−	0.947193i	$-0.396094\pi$
$769$	17.7846	0.641329	0.320665	+	0.947193i	$0.396094\pi$
$770$	2.73205	0.0984563
$771$	−24.1962	−0.871403
$772$	−2.92820	−0.105388
$773$	−12.2487	−0.440556	−0.220278	−	0.975437i	$-0.570696\pi$
$773$	−12.2487	−0.440556	−0.220278	+	0.975437i	$0.570696\pi$
$774$	−4.73205	−0.170090
$775$	−8.73205	−0.313665
$776$	−54.2487	−1.94742
$777$	−6.19615	−0.222286
$778$	80.6410	2.89112
$779$	−81.7654	−2.92955
$780$	33.8564	1.21225
$781$	9.66025	0.345671
$782$	1.94744	0.0696404
$783$	−2.26795	−0.0810499
$784$	14.9282	0.533150
$785$	−10.1244	−0.361354
$786$	−15.8564	−0.565579
$787$	26.6410	0.949650	0.474825	−	0.880080i	$-0.342511\pi$
$787$	26.6410	0.949650	0.474825	+	0.880080i	$0.342511\pi$
$788$	−8.00000	−0.284988
$789$	23.3205	0.830232
$790$	37.3205	1.32780
$791$	−16.8564	−0.599345
$792$	−9.46410	−0.336292
$793$	−58.1962	−2.06661
$794$	27.7128	0.983491
$795$	−2.46410	−0.0873927
$796$	−5.85641	−0.207575
$797$	51.1769	1.81278	0.906390	−	0.422443i	$-0.138827\pi$
$797$	51.1769	1.81278	0.906390	+	0.422443i	$0.138827\pi$
$798$	23.1244	0.818593
$799$	2.19615	0.0776943
$800$	21.8564	0.772741
$801$	−2.66025	−0.0939955
$802$	−24.2487	−0.856252
$803$	12.3923	0.437315
$804$	48.7846	1.72050
$805$	−1.53590	−0.0541333
$806$	−147.818	−5.20666
$807$	23.5885	0.830353
$808$	29.0718	1.02274
$809$	15.4641	0.543689	0.271844	−	0.962341i	$-0.412366\pi$
$809$	15.4641	0.543689	0.271844	+	0.962341i	$0.412366\pi$
$810$	−2.73205	−0.0959945
$811$	−36.7846	−1.29168	−0.645841	−	0.763472i	$-0.723493\pi$
$811$	−36.7846	−1.29168	−0.645841	+	0.763472i	$0.723493\pi$
$812$	12.3923	0.434885
$813$	4.07180	0.142804
$814$	−16.9282	−0.593333
$815$	5.12436	0.179498
$816$	−6.92820	−0.242536
$817$	14.6603	0.512897
$818$	−62.6410	−2.19019
$819$	6.19615	0.216511
$820$	−52.7846	−1.84332
$821$	41.9808	1.46514	0.732569	−	0.680692i	$-0.238321\pi$
$821$	41.9808	1.46514	0.732569	+	0.680692i	$0.238321\pi$
$822$	36.7846	1.28301
$823$	−40.4449	−1.40982	−0.704910	−	0.709297i	$-0.749012\pi$
$823$	−40.4449	−1.40982	−0.704910	+	0.709297i	$0.749012\pi$
$824$	−167.818	−5.84621
$825$	1.00000	0.0348155
$826$	28.0526	0.976073
$827$	6.53590	0.227275	0.113638	−	0.993522i	$-0.463750\pi$
$827$	6.53590	0.227275	0.113638	+	0.993522i	$0.463750\pi$
$828$	8.39230	0.291653
$829$	27.1769	0.943893	0.471947	−	0.881627i	$-0.343551\pi$
$829$	27.1769	0.943893	0.471947	+	0.881627i	$0.343551\pi$
$830$	12.1962	0.423335
$831$	22.9282	0.795371
$832$	184.995	6.41354
$833$	0.464102	0.0160802
$834$	−12.3923	−0.429110
$835$	−10.3923	−0.359641
$836$	46.2487	1.59955
$837$	8.73205	0.301824
$838$	71.7654	2.47909
$839$	32.6603	1.12756	0.563779	−	0.825926i	$-0.309347\pi$
$839$	32.6603	1.12756	0.563779	+	0.825926i	$0.309347\pi$
$840$	9.46410	0.326543
$841$	−23.8564	−0.822635
$842$	−83.9090	−2.89169
$843$	−30.9282	−1.06522
$844$	62.6410	2.15619
$845$	−25.3923	−0.873522
$846$	12.9282	0.444481
$847$	1.00000	0.0343604
$848$	−36.7846	−1.26319
$849$	−10.1962	−0.349931
$850$	1.26795	0.0434903
$851$	9.51666	0.326227
$852$	52.7846	1.80837
$853$	35.1244	1.20264	0.601318	−	0.799010i	$-0.294643\pi$
$853$	35.1244	1.20264	0.601318	+	0.799010i	$0.294643\pi$
$854$	−25.6603	−0.878076
$855$	8.46410	0.289466
$856$	82.6410	2.82461
$857$	−5.71281	−0.195146	−0.0975730	−	0.995228i	$-0.531108\pi$
$857$	−5.71281	−0.195146	−0.0975730	+	0.995228i	$0.531108\pi$
$858$	16.9282	0.577919
$859$	−17.4115	−0.594074	−0.297037	−	0.954866i	$-0.595999\pi$
$859$	−17.4115	−0.594074	−0.297037	+	0.954866i	$0.595999\pi$
$860$	9.46410	0.322723
$861$	−9.66025	−0.329221
$862$	−9.85641	−0.335711
$863$	−32.4641	−1.10509	−0.552545	−	0.833483i	$-0.686343\pi$
$863$	−32.4641	−1.10509	−0.552545	+	0.833483i	$0.686343\pi$
$864$	−21.8564	−0.743570
$865$	18.5359	0.630239
$866$	−69.1769	−2.35073
$867$	16.7846	0.570035
$868$	−47.7128	−1.61948
$869$	13.6603	0.463392
$870$	6.19615	0.210069
$871$	−55.3205	−1.87446
$872$	−82.6410	−2.79858
$873$	−5.73205	−0.194001
$874$	−35.5167	−1.20137
$875$	−1.00000	−0.0338062
$876$	67.7128	2.28780
$877$	−54.3731	−1.83605	−0.918024	−	0.396525i	$-0.870216\pi$
$877$	−54.3731	−1.83605	−0.918024	+	0.396525i	$0.870216\pi$
$878$	78.4449	2.64739
$879$	−9.00000	−0.303562
$880$	14.9282	0.503230
$881$	−48.9090	−1.64778	−0.823892	−	0.566746i	$-0.808202\pi$
$881$	−48.9090	−1.64778	−0.823892	+	0.566746i	$0.808202\pi$
$882$	2.73205	0.0919929
$883$	25.1769	0.847271	0.423635	−	0.905833i	$-0.360754\pi$
$883$	25.1769	0.847271	0.423635	+	0.905833i	$0.360754\pi$
$884$	15.7128	0.528479
$885$	10.2679	0.345153
$886$	−45.1769	−1.51775
$887$	−0.607695	−0.0204044	−0.0102022	−	0.999948i	$-0.503248\pi$
$887$	−0.607695	−0.0204044	−0.0102022	+	0.999948i	$0.503248\pi$
$888$	−58.6410	−1.96786
$889$	2.12436	0.0712486
$890$	7.26795	0.243622
$891$	−1.00000	−0.0335013
$892$	61.1769	2.04835
$893$	−40.0526	−1.34031
$894$	−23.3205	−0.779954
$895$	−18.0526	−0.603430
$896$	37.8564	1.26469
$897$	−9.51666	−0.317752
$898$	−60.2487	−2.01053
$899$	−19.8038	−0.660495
$900$	5.46410	0.182137
$901$	−1.14359	−0.0380986
$902$	−26.3923	−0.878768
$903$	1.73205	0.0576390
$904$	−159.531	−5.30591
$905$	−20.9282	−0.695677
$906$	−43.3205	−1.43923
$907$	35.6603	1.18408	0.592040	−	0.805909i	$-0.298323\pi$
$907$	35.6603	1.18408	0.592040	+	0.805909i	$0.298323\pi$
$908$	84.1051	2.79113
$909$	3.07180	0.101885
$910$	−16.9282	−0.561164
$911$	52.3923	1.73583	0.867917	−	0.496709i	$-0.165458\pi$
$911$	52.3923	1.73583	0.867917	+	0.496709i	$0.165458\pi$
$912$	126.354	4.18399
$913$	4.46410	0.147740
$914$	40.0526	1.32482
$915$	−9.39230	−0.310500
$916$	97.2820	3.21429
$917$	5.80385	0.191660
$918$	−1.26795	−0.0418486
$919$	50.4974	1.66576	0.832878	−	0.553456i	$-0.186691\pi$
$919$	50.4974	1.66576	0.832878	+	0.553456i	$0.186691\pi$
$920$	−14.5359	−0.479234
$921$	−10.3923	−0.342438
$922$	−14.5359	−0.478714
$923$	−59.8564	−1.97020
$924$	5.46410	0.179756
$925$	6.19615	0.203728
$926$	53.1769	1.74750
$927$	−17.7321	−0.582397
$928$	49.5692	1.62719
$929$	39.7128	1.30294	0.651468	−	0.758676i	$-0.274154\pi$
$929$	39.7128	1.30294	0.651468	+	0.758676i	$0.274154\pi$
$930$	−23.8564	−0.782282
$931$	−8.46410	−0.277400
$932$	1.07180	0.0351079
$933$	27.3205	0.894433
$934$	20.0000	0.654420
$935$	0.464102	0.0151777
$936$	58.6410	1.91674
$937$	−24.9808	−0.816086	−0.408043	−	0.912963i	$-0.633789\pi$
$937$	−24.9808	−0.816086	−0.408043	+	0.912963i	$0.633789\pi$
$938$	−24.3923	−0.796437
$939$	1.73205	0.0565233
$940$	−25.8564	−0.843343
$941$	−52.8372	−1.72244	−0.861221	−	0.508230i	$-0.830300\pi$
$941$	−52.8372	−1.72244	−0.861221	+	0.508230i	$0.830300\pi$
$942$	−27.6603	−0.901220
$943$	14.8372	0.483165
$944$	153.282	4.98891
$945$	1.00000	0.0325300
$946$	4.73205	0.153852
$947$	42.7128	1.38798	0.693990	−	0.719985i	$-0.255851\pi$
$947$	42.7128	1.38798	0.693990	+	0.719985i	$0.255851\pi$
$948$	74.6410	2.42423
$949$	−76.7846	−2.49253
$950$	−23.1244	−0.750253
$951$	−5.07180	−0.164464
$952$	4.39230	0.142355
$953$	−4.92820	−0.159640	−0.0798201	−	0.996809i	$-0.525435\pi$
$953$	−4.92820	−0.159640	−0.0798201	+	0.996809i	$0.525435\pi$
$954$	−6.73205	−0.217958
$955$	14.9282	0.483065
$956$	−33.4641	−1.08231
$957$	2.26795	0.0733124
$958$	−8.92820	−0.288457
$959$	−13.4641	−0.434779
$960$	29.8564	0.963611
$961$	45.2487	1.45964
$962$	104.890	3.38178
$963$	8.73205	0.281386
$964$	71.4256	2.30046
$965$	0.535898	0.0172512
$966$	−4.19615	−0.135009
$967$	−14.5167	−0.466824	−0.233412	−	0.972378i	$-0.574989\pi$
$967$	−14.5167	−0.466824	−0.233412	+	0.972378i	$0.574989\pi$
$968$	9.46410	0.304188
$969$	3.92820	0.126192
$970$	15.6603	0.502820
$971$	25.3397	0.813191	0.406596	−	0.913608i	$-0.366716\pi$
$971$	25.3397	0.813191	0.406596	+	0.913608i	$0.366716\pi$
$972$	−5.46410	−0.175261
$973$	4.53590	0.145414
$974$	20.3923	0.653412
$975$	−6.19615	−0.198436
$976$	−140.210	−4.48802
$977$	−9.39230	−0.300486	−0.150243	−	0.988649i	$-0.548006\pi$
$977$	−9.39230	−0.300486	−0.150243	+	0.988649i	$0.548006\pi$
$978$	14.0000	0.447671
$979$	2.66025	0.0850221
$980$	−5.46410	−0.174544
$981$	−8.73205	−0.278793
$982$	56.4449	1.80123
$983$	45.3205	1.44550	0.722750	−	0.691110i	$-0.242878\pi$
$983$	45.3205	1.44550	0.722750	+	0.691110i	$0.242878\pi$
$984$	−91.4256	−2.91454
$985$	1.46410	0.0466502
$986$	2.87564	0.0915792
$987$	−4.73205	−0.150623
$988$	−286.564	−9.11682
$989$	−2.66025	−0.0845912
$990$	2.73205	0.0868303
$991$	52.5692	1.66992	0.834958	−	0.550313i	$-0.185492\pi$
$991$	52.5692	1.66992	0.834958	+	0.550313i	$0.185492\pi$
$992$	−190.851	−6.05953
$993$	26.3205	0.835256
$994$	−26.3923	−0.837113
$995$	1.07180	0.0339782
$996$	24.3923	0.772900
$997$	25.7128	0.814333	0.407166	−	0.913354i	$-0.366517\pi$
$997$	25.7128	0.814333	0.407166	+	0.913354i	$0.366517\pi$
$998$	23.5167	0.744407
$999$	−6.19615	−0.196038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.r.1.2	✓	2
3.2	odd	2		3465.2.a.v.1.1		2
5.4	even	2		5775.2.a.bc.1.1		2
7.6	odd	2		8085.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.r.1.2	✓	2	1.1	even	1	trivial
3465.2.a.v.1.1		2	3.2	odd	2
5775.2.a.bc.1.1		2	5.4	even	2
8085.2.a.bh.1.2		2	7.6	odd	2

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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} -2.73205 q^{10} -1.00000 q^{11} -5.46410 q^{12} +6.19615 q^{13} +2.73205 q^{14} +1.00000 q^{15} +14.9282 q^{16} +0.464102 q^{17} +2.73205 q^{18} -8.46410 q^{19} -5.46410 q^{20} -1.00000 q^{21} -2.73205 q^{22} +1.53590 q^{23} -9.46410 q^{24} +1.00000 q^{25} +16.9282 q^{26} -1.00000 q^{27} +5.46410 q^{28} +2.26795 q^{29} +2.73205 q^{30} -8.73205 q^{31} +21.8564 q^{32} +1.00000 q^{33} +1.26795 q^{34} -1.00000 q^{35} +5.46410 q^{36} +6.19615 q^{37} -23.1244 q^{38} -6.19615 q^{39} -9.46410 q^{40} +9.66025 q^{41} -2.73205 q^{42} -1.73205 q^{43} -5.46410 q^{44} -1.00000 q^{45} +4.19615 q^{46} +4.73205 q^{47} -14.9282 q^{48} +1.00000 q^{49} +2.73205 q^{50} -0.464102 q^{51} +33.8564 q^{52} -2.46410 q^{53} -2.73205 q^{54} +1.00000 q^{55} +9.46410 q^{56} +8.46410 q^{57} +6.19615 q^{58} +10.2679 q^{59} +5.46410 q^{60} -9.39230 q^{61} -23.8564 q^{62} +1.00000 q^{63} +29.8564 q^{64} -6.19615 q^{65} +2.73205 q^{66} -8.92820 q^{67} +2.53590 q^{68} -1.53590 q^{69} -2.73205 q^{70} -9.66025 q^{71} +9.46410 q^{72} -12.3923 q^{73} +16.9282 q^{74} -1.00000 q^{75} -46.2487 q^{76} -1.00000 q^{77} -16.9282 q^{78} -13.6603 q^{79} -14.9282 q^{80} +1.00000 q^{81} +26.3923 q^{82} -4.46410 q^{83} -5.46410 q^{84} -0.464102 q^{85} -4.73205 q^{86} -2.26795 q^{87} -9.46410 q^{88} -2.66025 q^{89} -2.73205 q^{90} +6.19615 q^{91} +8.39230 q^{92} +8.73205 q^{93} +12.9282 q^{94} +8.46410 q^{95} -21.8564 q^{96} -5.73205 q^{97} +2.73205 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 16 q^{16} - 6 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} - 12 q^{24} + 2 q^{25} + 20 q^{26} - 2 q^{27} + 4 q^{28} + 8 q^{29} + 2 q^{30} - 14 q^{31} + 16 q^{32} + 2 q^{33} + 6 q^{34} - 2 q^{35} + 4 q^{36} + 2 q^{37} - 22 q^{38} - 2 q^{39} - 12 q^{40} + 2 q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{45} - 2 q^{46} + 6 q^{47} - 16 q^{48} + 2 q^{49} + 2 q^{50} + 6 q^{51} + 40 q^{52} + 2 q^{53} - 2 q^{54} + 2 q^{55} + 12 q^{56} + 10 q^{57} + 2 q^{58} + 24 q^{59} + 4 q^{60} + 2 q^{61} - 20 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} + 2 q^{66} - 4 q^{67} + 12 q^{68} - 10 q^{69} - 2 q^{70} - 2 q^{71} + 12 q^{72} - 4 q^{73} + 20 q^{74} - 2 q^{75} - 44 q^{76} - 2 q^{77} - 20 q^{78} - 10 q^{79} - 16 q^{80} + 2 q^{81} + 32 q^{82} - 2 q^{83} - 4 q^{84} + 6 q^{85} - 6 q^{86} - 8 q^{87} - 12 q^{88} + 12 q^{89} - 2 q^{90} + 2 q^{91} - 4 q^{92} + 14 q^{93} + 12 q^{94} + 10 q^{95} - 16 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 16 * q^16 - 6 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 - 12 * q^24 + 2 * q^25 + 20 * q^26 - 2 * q^27 + 4 * q^28 + 8 * q^29 + 2 * q^30 - 14 * q^31 + 16 * q^32 + 2 * q^33 + 6 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 22 * q^38 - 2 * q^39 - 12 * q^40 + 2 * q^41 - 2 * q^42 - 4 * q^44 - 2 * q^45 - 2 * q^46 + 6 * q^47 - 16 * q^48 + 2 * q^49 + 2 * q^50 + 6 * q^51 + 40 * q^52 + 2 * q^53 - 2 * q^54 + 2 * q^55 + 12 * q^56 + 10 * q^57 + 2 * q^58 + 24 * q^59 + 4 * q^60 + 2 * q^61 - 20 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 + 2 * q^66 - 4 * q^67 + 12 * q^68 - 10 * q^69 - 2 * q^70 - 2 * q^71 + 12 * q^72 - 4 * q^73 + 20 * q^74 - 2 * q^75 - 44 * q^76 - 2 * q^77 - 20 * q^78 - 10 * q^79 - 16 * q^80 + 2 * q^81 + 32 * q^82 - 2 * q^83 - 4 * q^84 + 6 * q^85 - 6 * q^86 - 8 * q^87 - 12 * q^88 + 12 * q^89 - 2 * q^90 + 2 * q^91 - 4 * q^92 + 14 * q^93 + 12 * q^94 + 10 * q^95 - 16 * q^96 - 8 * q^97 + 2 * q^98 - 2 * q^99

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