LMFDB - Embedded newform 8014.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8014,2,Mod(1,8014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8014 = 2 \cdot 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8014.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:	$63.9921121799$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.23607 q^{6} -3.85410 q^{7} +1.00000 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.23607 q^{6} -3.85410 q^{7} +1.00000 q^{8} -1.47214 q^{9} -0.618034 q^{10} -3.85410 q^{11} +1.23607 q^{12} -3.23607 q^{13} -3.85410 q^{14} -0.763932 q^{15} +1.00000 q^{16} +5.70820 q^{17} -1.47214 q^{18} +4.85410 q^{19} -0.618034 q^{20} -4.76393 q^{21} -3.85410 q^{22} -5.09017 q^{23} +1.23607 q^{24} -4.61803 q^{25} -3.23607 q^{26} -5.52786 q^{27} -3.85410 q^{28} +2.00000 q^{29} -0.763932 q^{30} -1.23607 q^{31} +1.00000 q^{32} -4.76393 q^{33} +5.70820 q^{34} +2.38197 q^{35} -1.47214 q^{36} +9.23607 q^{37} +4.85410 q^{38} -4.00000 q^{39} -0.618034 q^{40} +6.76393 q^{41} -4.76393 q^{42} +4.47214 q^{43} -3.85410 q^{44} +0.909830 q^{45} -5.09017 q^{46} +0.763932 q^{47} +1.23607 q^{48} +7.85410 q^{49} -4.61803 q^{50} +7.05573 q^{51} -3.23607 q^{52} +0.909830 q^{53} -5.52786 q^{54} +2.38197 q^{55} -3.85410 q^{56} +6.00000 q^{57} +2.00000 q^{58} +13.7082 q^{59} -0.763932 q^{60} +6.00000 q^{61} -1.23607 q^{62} +5.67376 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.76393 q^{66} +1.61803 q^{67} +5.70820 q^{68} -6.29180 q^{69} +2.38197 q^{70} +12.7984 q^{71} -1.47214 q^{72} +4.85410 q^{73} +9.23607 q^{74} -5.70820 q^{75} +4.85410 q^{76} +14.8541 q^{77} -4.00000 q^{78} +4.00000 q^{79} -0.618034 q^{80} -2.41641 q^{81} +6.76393 q^{82} -0.145898 q^{83} -4.76393 q^{84} -3.52786 q^{85} +4.47214 q^{86} +2.47214 q^{87} -3.85410 q^{88} -3.38197 q^{89} +0.909830 q^{90} +12.4721 q^{91} -5.09017 q^{92} -1.52786 q^{93} +0.763932 q^{94} -3.00000 q^{95} +1.23607 q^{96} -17.2361 q^{97} +7.85410 q^{98} +5.67376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9} + q^{10} - q^{11} - 2 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{17} + 6 q^{18} + 3 q^{19} + q^{20} - 14 q^{21} - q^{22} + q^{23} - 2 q^{24} - 7 q^{25} - 2 q^{26} - 20 q^{27} - q^{28} + 4 q^{29} - 6 q^{30} + 2 q^{31} + 2 q^{32} - 14 q^{33} - 2 q^{34} + 7 q^{35} + 6 q^{36} + 14 q^{37} + 3 q^{38} - 8 q^{39} + q^{40} + 18 q^{41} - 14 q^{42} - q^{44} + 13 q^{45} + q^{46} + 6 q^{47} - 2 q^{48} + 9 q^{49} - 7 q^{50} + 32 q^{51} - 2 q^{52} + 13 q^{53} - 20 q^{54} + 7 q^{55} - q^{56} + 12 q^{57} + 4 q^{58} + 14 q^{59} - 6 q^{60} + 12 q^{61} + 2 q^{62} + 27 q^{63} + 2 q^{64} + 4 q^{65} - 14 q^{66} + q^{67} - 2 q^{68} - 26 q^{69} + 7 q^{70} + q^{71} + 6 q^{72} + 3 q^{73} + 14 q^{74} + 2 q^{75} + 3 q^{76} + 23 q^{77} - 8 q^{78} + 8 q^{79} + q^{80} + 22 q^{81} + 18 q^{82} - 7 q^{83} - 14 q^{84} - 16 q^{85} - 4 q^{87} - q^{88} - 9 q^{89} + 13 q^{90} + 16 q^{91} + q^{92} - 12 q^{93} + 6 q^{94} - 6 q^{95} - 2 q^{96} - 30 q^{97} + 9 q^{98} + 27 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 6 * q^9 + q^10 - q^11 - 2 * q^12 - 2 * q^13 - q^14 - 6 * q^15 + 2 * q^16 - 2 * q^17 + 6 * q^18 + 3 * q^19 + q^20 - 14 * q^21 - q^22 + q^23 - 2 * q^24 - 7 * q^25 - 2 * q^26 - 20 * q^27 - q^28 + 4 * q^29 - 6 * q^30 + 2 * q^31 + 2 * q^32 - 14 * q^33 - 2 * q^34 + 7 * q^35 + 6 * q^36 + 14 * q^37 + 3 * q^38 - 8 * q^39 + q^40 + 18 * q^41 - 14 * q^42 - q^44 + 13 * q^45 + q^46 + 6 * q^47 - 2 * q^48 + 9 * q^49 - 7 * q^50 + 32 * q^51 - 2 * q^52 + 13 * q^53 - 20 * q^54 + 7 * q^55 - q^56 + 12 * q^57 + 4 * q^58 + 14 * q^59 - 6 * q^60 + 12 * q^61 + 2 * q^62 + 27 * q^63 + 2 * q^64 + 4 * q^65 - 14 * q^66 + q^67 - 2 * q^68 - 26 * q^69 + 7 * q^70 + q^71 + 6 * q^72 + 3 * q^73 + 14 * q^74 + 2 * q^75 + 3 * q^76 + 23 * q^77 - 8 * q^78 + 8 * q^79 + q^80 + 22 * q^81 + 18 * q^82 - 7 * q^83 - 14 * q^84 - 16 * q^85 - 4 * q^87 - q^88 - 9 * q^89 + 13 * q^90 + 16 * q^91 + q^92 - 12 * q^93 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 30 * q^97 + 9 * q^98 + 27 * 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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	1.00000	0.500000
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	1.23607	0.504623
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	1.00000	0.353553
$9$	−1.47214	−0.490712
$10$	−0.618034	−0.195440
$11$	−3.85410	−1.16206	−0.581028	−	0.813884i	$-0.697349\pi$
$11$	−3.85410	−1.16206	−0.581028	+	0.813884i	$0.697349\pi$
$12$	1.23607	0.356822
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	−3.85410	−1.03005
$15$	−0.763932	−0.197246
$16$	1.00000	0.250000
$17$	5.70820	1.38444	0.692221	−	0.721685i	$-0.256632\pi$
$17$	5.70820	1.38444	0.692221	+	0.721685i	$0.256632\pi$
$18$	−1.47214	−0.346986
$19$	4.85410	1.11361	0.556804	−	0.830644i	$-0.312028\pi$
$19$	4.85410	1.11361	0.556804	+	0.830644i	$0.312028\pi$
$20$	−0.618034	−0.138197
$21$	−4.76393	−1.03958
$22$	−3.85410	−0.821697
$23$	−5.09017	−1.06137	−0.530687	−	0.847568i	$-0.678066\pi$
$23$	−5.09017	−1.06137	−0.530687	+	0.847568i	$0.678066\pi$
$24$	1.23607	0.252311
$25$	−4.61803	−0.923607
$26$	−3.23607	−0.634645
$27$	−5.52786	−1.06384
$28$	−3.85410	−0.728357
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−0.763932	−0.139474
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	1.00000	0.176777
$33$	−4.76393	−0.829294
$34$	5.70820	0.978949
$35$	2.38197	0.402626
$36$	−1.47214	−0.245356
$37$	9.23607	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$37$	9.23607	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$38$	4.85410	0.787439
$39$	−4.00000	−0.640513
$40$	−0.618034	−0.0977198
$41$	6.76393	1.05635	0.528174	−	0.849136i	$-0.322877\pi$
$41$	6.76393	1.05635	0.528174	+	0.849136i	$0.322877\pi$
$42$	−4.76393	−0.735091
$43$	4.47214	0.681994	0.340997	−	0.940064i	$-0.389235\pi$
$43$	4.47214	0.681994	0.340997	+	0.940064i	$0.389235\pi$
$44$	−3.85410	−0.581028
$45$	0.909830	0.135629
$46$	−5.09017	−0.750505
$47$	0.763932	0.111431	0.0557155	−	0.998447i	$-0.482256\pi$
$47$	0.763932	0.111431	0.0557155	+	0.998447i	$0.482256\pi$
$48$	1.23607	0.178411
$49$	7.85410	1.12201
$50$	−4.61803	−0.653089
$51$	7.05573	0.988000
$52$	−3.23607	−0.448762
$53$	0.909830	0.124975	0.0624874	−	0.998046i	$-0.480097\pi$
$53$	0.909830	0.124975	0.0624874	+	0.998046i	$0.480097\pi$
$54$	−5.52786	−0.752247
$55$	2.38197	0.321184
$56$	−3.85410	−0.515026
$57$	6.00000	0.794719
$58$	2.00000	0.262613
$59$	13.7082	1.78466	0.892328	−	0.451387i	$-0.149071\pi$
$59$	13.7082	1.78466	0.892328	+	0.451387i	$0.149071\pi$
$60$	−0.763932	−0.0986232
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.23607	−0.156981
$63$	5.67376	0.714827
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.76393	−0.586399
$67$	1.61803	0.197674	0.0988372	−	0.995104i	$-0.468488\pi$
$67$	1.61803	0.197674	0.0988372	+	0.995104i	$0.468488\pi$
$68$	5.70820	0.692221
$69$	−6.29180	−0.757443
$70$	2.38197	0.284699
$71$	12.7984	1.51889	0.759444	−	0.650573i	$-0.225471\pi$
$71$	12.7984	1.51889	0.759444	+	0.650573i	$0.225471\pi$
$72$	−1.47214	−0.173493
$73$	4.85410	0.568130	0.284065	−	0.958805i	$-0.408317\pi$
$73$	4.85410	0.568130	0.284065	+	0.958805i	$0.408317\pi$
$74$	9.23607	1.07367
$75$	−5.70820	−0.659127
$76$	4.85410	0.556804
$77$	14.8541	1.69278
$78$	−4.00000	−0.452911
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−0.618034	−0.0690983
$81$	−2.41641	−0.268490
$82$	6.76393	0.746951
$83$	−0.145898	−0.0160144	−0.00800719	−	0.999968i	$-0.502549\pi$
$83$	−0.145898	−0.0160144	−0.00800719	+	0.999968i	$0.502549\pi$
$84$	−4.76393	−0.519788
$85$	−3.52786	−0.382651
$86$	4.47214	0.482243
$87$	2.47214	0.265041
$88$	−3.85410	−0.410849
$89$	−3.38197	−0.358488	−0.179244	−	0.983805i	$-0.557365\pi$
$89$	−3.38197	−0.358488	−0.179244	+	0.983805i	$0.557365\pi$
$90$	0.909830	0.0959045
$91$	12.4721	1.30744
$92$	−5.09017	−0.530687
$93$	−1.52786	−0.158432
$94$	0.763932	0.0787936
$95$	−3.00000	−0.307794
$96$	1.23607	0.126156
$97$	−17.2361	−1.75006	−0.875029	−	0.484071i	$-0.839158\pi$
$97$	−17.2361	−1.75006	−0.875029	+	0.484071i	$0.839158\pi$
$98$	7.85410	0.793384
$99$	5.67376	0.570235
$100$	−4.61803	−0.461803
$101$	5.23607	0.521008	0.260504	−	0.965473i	$-0.416111\pi$
$101$	5.23607	0.521008	0.260504	+	0.965473i	$0.416111\pi$
$102$	7.05573	0.698621
$103$	19.2361	1.89539	0.947693	−	0.319183i	$-0.103409\pi$
$103$	19.2361	1.89539	0.947693	+	0.319183i	$0.103409\pi$
$104$	−3.23607	−0.317323
$105$	2.94427	0.287332
$106$	0.909830	0.0883705
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−5.52786	−0.531919
$109$	1.52786	0.146343	0.0731714	−	0.997319i	$-0.476688\pi$
$109$	1.52786	0.146343	0.0731714	+	0.997319i	$0.476688\pi$
$110$	2.38197	0.227112
$111$	11.4164	1.08360
$112$	−3.85410	−0.364178
$113$	−4.61803	−0.434428	−0.217214	−	0.976124i	$-0.569697\pi$
$113$	−4.61803	−0.434428	−0.217214	+	0.976124i	$0.569697\pi$
$114$	6.00000	0.561951
$115$	3.14590	0.293357
$116$	2.00000	0.185695
$117$	4.76393	0.440426
$118$	13.7082	1.26194
$119$	−22.0000	−2.01674
$120$	−0.763932	−0.0697371
$121$	3.85410	0.350373
$122$	6.00000	0.543214
$123$	8.36068	0.753857
$124$	−1.23607	−0.111002
$125$	5.94427	0.531672
$126$	5.67376	0.505459
$127$	6.76393	0.600202	0.300101	−	0.953907i	$-0.402980\pi$
$127$	6.76393	0.600202	0.300101	+	0.953907i	$0.402980\pi$
$128$	1.00000	0.0883883
$129$	5.52786	0.486701
$130$	2.00000	0.175412
$131$	−18.5623	−1.62180	−0.810898	−	0.585187i	$-0.801021\pi$
$131$	−18.5623	−1.62180	−0.810898	+	0.585187i	$0.801021\pi$
$132$	−4.76393	−0.414647
$133$	−18.7082	−1.62221
$134$	1.61803	0.139777
$135$	3.41641	0.294038
$136$	5.70820	0.489474
$137$	−20.4721	−1.74905	−0.874526	−	0.484978i	$-0.838828\pi$
$137$	−20.4721	−1.74905	−0.874526	+	0.484978i	$0.838828\pi$
$138$	−6.29180	−0.535593
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	2.38197	0.201313
$141$	0.944272	0.0795220
$142$	12.7984	1.07402
$143$	12.4721	1.04297
$144$	−1.47214	−0.122678
$145$	−1.23607	−0.102650
$146$	4.85410	0.401728
$147$	9.70820	0.800719
$148$	9.23607	0.759200
$149$	−2.76393	−0.226430	−0.113215	−	0.993571i	$-0.536115\pi$
$149$	−2.76393	−0.226430	−0.113215	+	0.993571i	$0.536115\pi$
$150$	−5.70820	−0.466073
$151$	−5.70820	−0.464527	−0.232264	−	0.972653i	$-0.574613\pi$
$151$	−5.70820	−0.464527	−0.232264	+	0.972653i	$0.574613\pi$
$152$	4.85410	0.393720
$153$	−8.40325	−0.679363
$154$	14.8541	1.19698
$155$	0.763932	0.0613605
$156$	−4.00000	−0.320256
$157$	−10.6525	−0.850160	−0.425080	−	0.905156i	$-0.639754\pi$
$157$	−10.6525	−0.850160	−0.425080	+	0.905156i	$0.639754\pi$
$158$	4.00000	0.318223
$159$	1.12461	0.0891875
$160$	−0.618034	−0.0488599
$161$	19.6180	1.54612
$162$	−2.41641	−0.189851
$163$	−6.32624	−0.495509	−0.247755	−	0.968823i	$-0.579693\pi$
$163$	−6.32624	−0.495509	−0.247755	+	0.968823i	$0.579693\pi$
$164$	6.76393	0.528174
$165$	2.94427	0.229211
$166$	−0.145898	−0.0113239
$167$	7.32624	0.566921	0.283461	−	0.958984i	$-0.408517\pi$
$167$	7.32624	0.566921	0.283461	+	0.958984i	$0.408517\pi$
$168$	−4.76393	−0.367545
$169$	−2.52786	−0.194451
$170$	−3.52786	−0.270575
$171$	−7.14590	−0.546460
$172$	4.47214	0.340997
$173$	−5.05573	−0.384380	−0.192190	−	0.981358i	$-0.561559\pi$
$173$	−5.05573	−0.384380	−0.192190	+	0.981358i	$0.561559\pi$
$174$	2.47214	0.187412
$175$	17.7984	1.34543
$176$	−3.85410	−0.290514
$177$	16.9443	1.27361
$178$	−3.38197	−0.253489
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0.909830	0.0678147
$181$	7.23607	0.537853	0.268926	−	0.963161i	$-0.413331\pi$
$181$	7.23607	0.537853	0.268926	+	0.963161i	$0.413331\pi$
$182$	12.4721	0.924496
$183$	7.41641	0.548237
$184$	−5.09017	−0.375252
$185$	−5.70820	−0.419675
$186$	−1.52786	−0.112028
$187$	−22.0000	−1.60880
$188$	0.763932	0.0557155
$189$	21.3050	1.54971
$190$	−3.00000	−0.217643
$191$	5.43769	0.393458	0.196729	−	0.980458i	$-0.436968\pi$
$191$	5.43769	0.393458	0.196729	+	0.980458i	$0.436968\pi$
$192$	1.23607	0.0892055
$193$	−2.38197	−0.171458	−0.0857288	−	0.996319i	$-0.527322\pi$
$193$	−2.38197	−0.171458	−0.0857288	+	0.996319i	$0.527322\pi$
$194$	−17.2361	−1.23748
$195$	2.47214	0.177033
$196$	7.85410	0.561007
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	5.67376	0.403217
$199$	10.3820	0.735958	0.367979	−	0.929834i	$-0.380050\pi$
$199$	10.3820	0.735958	0.367979	+	0.929834i	$0.380050\pi$
$200$	−4.61803	−0.326544
$201$	2.00000	0.141069
$202$	5.23607	0.368408
$203$	−7.70820	−0.541010
$204$	7.05573	0.494000
$205$	−4.18034	−0.291968
$206$	19.2361	1.34024
$207$	7.49342	0.520829
$208$	−3.23607	−0.224381
$209$	−18.7082	−1.29407
$210$	2.94427	0.203174
$211$	−11.3262	−0.779730	−0.389865	−	0.920872i	$-0.627478\pi$
$211$	−11.3262	−0.779730	−0.389865	+	0.920872i	$0.627478\pi$
$212$	0.909830	0.0624874
$213$	15.8197	1.08395
$214$	0	0
$215$	−2.76393	−0.188499
$216$	−5.52786	−0.376124
$217$	4.76393	0.323397
$218$	1.52786	0.103480
$219$	6.00000	0.405442
$220$	2.38197	0.160592
$221$	−18.4721	−1.24257
$222$	11.4164	0.766219
$223$	−5.61803	−0.376211	−0.188106	−	0.982149i	$-0.560235\pi$
$223$	−5.61803	−0.376211	−0.188106	+	0.982149i	$0.560235\pi$
$224$	−3.85410	−0.257513
$225$	6.79837	0.453225
$226$	−4.61803	−0.307187
$227$	−0.763932	−0.0507039	−0.0253520	−	0.999679i	$-0.508071\pi$
$227$	−0.763932	−0.0507039	−0.0253520	+	0.999679i	$0.508071\pi$
$228$	6.00000	0.397360
$229$	−2.38197	−0.157405	−0.0787024	−	0.996898i	$-0.525078\pi$
$229$	−2.38197	−0.157405	−0.0787024	+	0.996898i	$0.525078\pi$
$230$	3.14590	0.207434
$231$	18.3607	1.20804
$232$	2.00000	0.131306
$233$	3.09017	0.202444	0.101222	−	0.994864i	$-0.467725\pi$
$233$	3.09017	0.202444	0.101222	+	0.994864i	$0.467725\pi$
$234$	4.76393	0.311428
$235$	−0.472136	−0.0307988
$236$	13.7082	0.892328
$237$	4.94427	0.321165
$238$	−22.0000	−1.42605
$239$	29.7082	1.92166	0.960832	−	0.277132i	$-0.0893838\pi$
$239$	29.7082	1.92166	0.960832	+	0.277132i	$0.0893838\pi$
$240$	−0.763932	−0.0493116
$241$	22.9443	1.47797	0.738985	−	0.673722i	$-0.235305\pi$
$241$	22.9443	1.47797	0.738985	+	0.673722i	$0.235305\pi$
$242$	3.85410	0.247751
$243$	13.5967	0.872232
$244$	6.00000	0.384111
$245$	−4.85410	−0.310117
$246$	8.36068	0.533057
$247$	−15.7082	−0.999489
$248$	−1.23607	−0.0784904
$249$	−0.180340	−0.0114286
$250$	5.94427	0.375949
$251$	−5.70820	−0.360299	−0.180149	−	0.983639i	$-0.557658\pi$
$251$	−5.70820	−0.360299	−0.180149	+	0.983639i	$0.557658\pi$
$252$	5.67376	0.357413
$253$	19.6180	1.23338
$254$	6.76393	0.424407
$255$	−4.36068	−0.273076
$256$	1.00000	0.0625000
$257$	9.52786	0.594332	0.297166	−	0.954826i	$-0.403959\pi$
$257$	9.52786	0.594332	0.297166	+	0.954826i	$0.403959\pi$
$258$	5.52786	0.344150
$259$	−35.5967	−2.21187
$260$	2.00000	0.124035
$261$	−2.94427	−0.182246
$262$	−18.5623	−1.14678
$263$	−5.41641	−0.333990	−0.166995	−	0.985958i	$-0.553406\pi$
$263$	−5.41641	−0.333990	−0.166995	+	0.985958i	$0.553406\pi$
$264$	−4.76393	−0.293200
$265$	−0.562306	−0.0345422
$266$	−18.7082	−1.14707
$267$	−4.18034	−0.255833
$268$	1.61803	0.0988372
$269$	−4.09017	−0.249382	−0.124691	−	0.992196i	$-0.539794\pi$
$269$	−4.09017	−0.249382	−0.124691	+	0.992196i	$0.539794\pi$
$270$	3.41641	0.207916
$271$	9.23607	0.561051	0.280526	−	0.959847i	$-0.409491\pi$
$271$	9.23607	0.561051	0.280526	+	0.959847i	$0.409491\pi$
$272$	5.70820	0.346111
$273$	15.4164	0.933043
$274$	−20.4721	−1.23677
$275$	17.7984	1.07328
$276$	−6.29180	−0.378722
$277$	−2.58359	−0.155233	−0.0776165	−	0.996983i	$-0.524731\pi$
$277$	−2.58359	−0.155233	−0.0776165	+	0.996983i	$0.524731\pi$
$278$	6.47214	0.388173
$279$	1.81966	0.108940
$280$	2.38197	0.142350
$281$	−8.29180	−0.494647	−0.247324	−	0.968933i	$-0.579551\pi$
$281$	−8.29180	−0.494647	−0.247324	+	0.968933i	$0.579551\pi$
$282$	0.944272	0.0562306
$283$	24.9443	1.48278	0.741392	−	0.671073i	$-0.234166\pi$
$283$	24.9443	1.48278	0.741392	+	0.671073i	$0.234166\pi$
$284$	12.7984	0.759444
$285$	−3.70820	−0.219655
$286$	12.4721	0.737493
$287$	−26.0689	−1.53880
$288$	−1.47214	−0.0867464
$289$	15.5836	0.916682
$290$	−1.23607	−0.0725844
$291$	−21.3050	−1.24892
$292$	4.85410	0.284065
$293$	−2.29180	−0.133888	−0.0669441	−	0.997757i	$-0.521325\pi$
$293$	−2.29180	−0.133888	−0.0669441	+	0.997757i	$0.521325\pi$
$294$	9.70820	0.566194
$295$	−8.47214	−0.493267
$296$	9.23607	0.536836
$297$	21.3050	1.23624
$298$	−2.76393	−0.160110
$299$	16.4721	0.952608
$300$	−5.70820	−0.329563
$301$	−17.2361	−0.993470
$302$	−5.70820	−0.328470
$303$	6.47214	0.371814
$304$	4.85410	0.278402
$305$	−3.70820	−0.212331
$306$	−8.40325	−0.480382
$307$	−18.3820	−1.04911	−0.524557	−	0.851375i	$-0.675769\pi$
$307$	−18.3820	−1.04911	−0.524557	+	0.851375i	$0.675769\pi$
$308$	14.8541	0.846391
$309$	23.7771	1.35263
$310$	0.763932	0.0433884
$311$	27.5066	1.55975	0.779877	−	0.625932i	$-0.215281\pi$
$311$	27.5066	1.55975	0.779877	+	0.625932i	$0.215281\pi$
$312$	−4.00000	−0.226455
$313$	11.5279	0.651593	0.325797	−	0.945440i	$-0.394368\pi$
$313$	11.5279	0.651593	0.325797	+	0.945440i	$0.394368\pi$
$314$	−10.6525	−0.601154
$315$	−3.50658	−0.197573
$316$	4.00000	0.225018
$317$	−7.50658	−0.421611	−0.210806	−	0.977528i	$-0.567609\pi$
$317$	−7.50658	−0.421611	−0.210806	+	0.977528i	$0.567609\pi$
$318$	1.12461	0.0630651
$319$	−7.70820	−0.431577
$320$	−0.618034	−0.0345492
$321$	0	0
$322$	19.6180	1.09327
$323$	27.7082	1.54173
$324$	−2.41641	−0.134245
$325$	14.9443	0.828959
$326$	−6.32624	−0.350378
$327$	1.88854	0.104437
$328$	6.76393	0.373476
$329$	−2.94427	−0.162323
$330$	2.94427	0.162077
$331$	−26.8328	−1.47486	−0.737432	−	0.675421i	$-0.763962\pi$
$331$	−26.8328	−1.47486	−0.737432	+	0.675421i	$0.763962\pi$
$332$	−0.145898	−0.00800719
$333$	−13.5967	−0.745097
$334$	7.32624	0.400874
$335$	−1.00000	−0.0546358
$336$	−4.76393	−0.259894
$337$	−3.50658	−0.191015	−0.0955077	−	0.995429i	$-0.530447\pi$
$337$	−3.50658	−0.191015	−0.0955077	+	0.995429i	$0.530447\pi$
$338$	−2.52786	−0.137498
$339$	−5.70820	−0.310027
$340$	−3.52786	−0.191325
$341$	4.76393	0.257981
$342$	−7.14590	−0.386406
$343$	−3.29180	−0.177740
$344$	4.47214	0.241121
$345$	3.88854	0.209352
$346$	−5.05573	−0.271798
$347$	2.94427	0.158057	0.0790284	−	0.996872i	$-0.474818\pi$
$347$	2.94427	0.158057	0.0790284	+	0.996872i	$0.474818\pi$
$348$	2.47214	0.132520
$349$	18.3607	0.982825	0.491412	−	0.870927i	$-0.336481\pi$
$349$	18.3607	0.982825	0.491412	+	0.870927i	$0.336481\pi$
$350$	17.7984	0.951363
$351$	17.8885	0.954820
$352$	−3.85410	−0.205424
$353$	−8.00000	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$354$	16.9443	0.900578
$355$	−7.90983	−0.419810
$356$	−3.38197	−0.179244
$357$	−27.1935	−1.43923
$358$	10.0000	0.528516
$359$	23.1459	1.22159	0.610797	−	0.791787i	$-0.290849\pi$
$359$	23.1459	1.22159	0.610797	+	0.791787i	$0.290849\pi$
$360$	0.909830	0.0479523
$361$	4.56231	0.240121
$362$	7.23607	0.380319
$363$	4.76393	0.250042
$364$	12.4721	0.653718
$365$	−3.00000	−0.157027
$366$	7.41641	0.387662
$367$	13.7426	0.717360	0.358680	−	0.933461i	$-0.383227\pi$
$367$	13.7426	0.717360	0.358680	+	0.933461i	$0.383227\pi$
$368$	−5.09017	−0.265343
$369$	−9.95743	−0.518363
$370$	−5.70820	−0.296755
$371$	−3.50658	−0.182052
$372$	−1.52786	−0.0792161
$373$	21.0344	1.08912	0.544561	−	0.838721i	$-0.316696\pi$
$373$	21.0344	1.08912	0.544561	+	0.838721i	$0.316696\pi$
$374$	−22.0000	−1.13759
$375$	7.34752	0.379425
$376$	0.763932	0.0393968
$377$	−6.47214	−0.333332
$378$	21.3050	1.09581
$379$	5.38197	0.276453	0.138227	−	0.990401i	$-0.455860\pi$
$379$	5.38197	0.276453	0.138227	+	0.990401i	$0.455860\pi$
$380$	−3.00000	−0.153897
$381$	8.36068	0.428331
$382$	5.43769	0.278217
$383$	29.0902	1.48644	0.743219	−	0.669048i	$-0.233298\pi$
$383$	29.0902	1.48644	0.743219	+	0.669048i	$0.233298\pi$
$384$	1.23607	0.0630778
$385$	−9.18034	−0.467873
$386$	−2.38197	−0.121239
$387$	−6.58359	−0.334663
$388$	−17.2361	−0.875029
$389$	−12.3607	−0.626711	−0.313356	−	0.949636i	$-0.601453\pi$
$389$	−12.3607	−0.626711	−0.313356	+	0.949636i	$0.601453\pi$
$390$	2.47214	0.125181
$391$	−29.0557	−1.46941
$392$	7.85410	0.396692
$393$	−22.9443	−1.15739
$394$	−10.0000	−0.503793
$395$	−2.47214	−0.124387
$396$	5.67376	0.285117
$397$	20.4721	1.02747	0.513734	−	0.857950i	$-0.328262\pi$
$397$	20.4721	1.02747	0.513734	+	0.857950i	$0.328262\pi$
$398$	10.3820	0.520401
$399$	−23.1246	−1.15768
$400$	−4.61803	−0.230902
$401$	−25.7082	−1.28381	−0.641903	−	0.766786i	$-0.721855\pi$
$401$	−25.7082	−1.28381	−0.641903	+	0.766786i	$0.721855\pi$
$402$	2.00000	0.0997509
$403$	4.00000	0.199254
$404$	5.23607	0.260504
$405$	1.49342	0.0742087
$406$	−7.70820	−0.382552
$407$	−35.5967	−1.76447
$408$	7.05573	0.349311
$409$	−11.6738	−0.577230	−0.288615	−	0.957445i	$-0.593195\pi$
$409$	−11.6738	−0.577230	−0.288615	+	0.957445i	$0.593195\pi$
$410$	−4.18034	−0.206452
$411$	−25.3050	−1.24820
$412$	19.2361	0.947693
$413$	−52.8328	−2.59973
$414$	7.49342	0.368282
$415$	0.0901699	0.00442627
$416$	−3.23607	−0.158661
$417$	8.00000	0.391762
$418$	−18.7082	−0.915048
$419$	−26.8328	−1.31087	−0.655434	−	0.755252i	$-0.727514\pi$
$419$	−26.8328	−1.31087	−0.655434	+	0.755252i	$0.727514\pi$
$420$	2.94427	0.143666
$421$	0.0901699	0.00439461	0.00219731	−	0.999998i	$-0.499301\pi$
$421$	0.0901699	0.00439461	0.00219731	+	0.999998i	$0.499301\pi$
$422$	−11.3262	−0.551353
$423$	−1.12461	−0.0546805
$424$	0.909830	0.0441853
$425$	−26.3607	−1.27868
$426$	15.8197	0.766465
$427$	−23.1246	−1.11908
$428$	0	0
$429$	15.4164	0.744311
$430$	−2.76393	−0.133289
$431$	20.3607	0.980739	0.490370	−	0.871515i	$-0.336862\pi$
$431$	20.3607	0.980739	0.490370	+	0.871515i	$0.336862\pi$
$432$	−5.52786	−0.265959
$433$	29.1246	1.39964	0.699820	−	0.714319i	$-0.253264\pi$
$433$	29.1246	1.39964	0.699820	+	0.714319i	$0.253264\pi$
$434$	4.76393	0.228676
$435$	−1.52786	−0.0732555
$436$	1.52786	0.0731714
$437$	−24.7082	−1.18195
$438$	6.00000	0.286691
$439$	8.18034	0.390426	0.195213	−	0.980761i	$-0.437460\pi$
$439$	8.18034	0.390426	0.195213	+	0.980761i	$0.437460\pi$
$440$	2.38197	0.113556
$441$	−11.5623	−0.550586
$442$	−18.4721	−0.878630
$443$	−35.8885	−1.70512	−0.852558	−	0.522632i	$-0.824950\pi$
$443$	−35.8885	−1.70512	−0.852558	+	0.522632i	$0.824950\pi$
$444$	11.4164	0.541799
$445$	2.09017	0.0990836
$446$	−5.61803	−0.266022
$447$	−3.41641	−0.161591
$448$	−3.85410	−0.182089
$449$	−12.9443	−0.610878	−0.305439	−	0.952212i	$-0.598803\pi$
$449$	−12.9443	−0.610878	−0.305439	+	0.952212i	$0.598803\pi$
$450$	6.79837	0.320478
$451$	−26.0689	−1.22754
$452$	−4.61803	−0.217214
$453$	−7.05573	−0.331507
$454$	−0.763932	−0.0358531
$455$	−7.70820	−0.361366
$456$	6.00000	0.280976
$457$	−34.1803	−1.59889	−0.799444	−	0.600740i	$-0.794873\pi$
$457$	−34.1803	−1.59889	−0.799444	+	0.600740i	$0.794873\pi$
$458$	−2.38197	−0.111302
$459$	−31.5542	−1.47282
$460$	3.14590	0.146678
$461$	35.4164	1.64951	0.824753	−	0.565493i	$-0.191314\pi$
$461$	35.4164	1.64951	0.824753	+	0.565493i	$0.191314\pi$
$462$	18.3607	0.854216
$463$	−27.4164	−1.27415	−0.637074	−	0.770802i	$-0.719856\pi$
$463$	−27.4164	−1.27415	−0.637074	+	0.770802i	$0.719856\pi$
$464$	2.00000	0.0928477
$465$	0.944272	0.0437896
$466$	3.09017	0.143149
$467$	−2.47214	−0.114397	−0.0571984	−	0.998363i	$-0.518217\pi$
$467$	−2.47214	−0.114397	−0.0571984	+	0.998363i	$0.518217\pi$
$468$	4.76393	0.220213
$469$	−6.23607	−0.287955
$470$	−0.472136	−0.0217780
$471$	−13.1672	−0.606712
$472$	13.7082	0.630971
$473$	−17.2361	−0.792515
$474$	4.94427	0.227098
$475$	−22.4164	−1.02854
$476$	−22.0000	−1.00837
$477$	−1.33939	−0.0613266
$478$	29.7082	1.35882
$479$	13.1246	0.599679	0.299839	−	0.953990i	$-0.403067\pi$
$479$	13.1246	0.599679	0.299839	+	0.953990i	$0.403067\pi$
$480$	−0.763932	−0.0348686
$481$	−29.8885	−1.36280
$482$	22.9443	1.04508
$483$	24.2492	1.10338
$484$	3.85410	0.175186
$485$	10.6525	0.483704
$486$	13.5967	0.616761
$487$	−23.4164	−1.06110	−0.530549	−	0.847654i	$-0.678014\pi$
$487$	−23.4164	−1.06110	−0.530549	+	0.847654i	$0.678014\pi$
$488$	6.00000	0.271607
$489$	−7.81966	−0.353617
$490$	−4.85410	−0.219286
$491$	38.5623	1.74029	0.870146	−	0.492794i	$-0.164024\pi$
$491$	38.5623	1.74029	0.870146	+	0.492794i	$0.164024\pi$
$492$	8.36068	0.376929
$493$	11.4164	0.514169
$494$	−15.7082	−0.706746
$495$	−3.50658	−0.157609
$496$	−1.23607	−0.0555011
$497$	−49.3262	−2.21258
$498$	−0.180340	−0.00808122
$499$	−5.52786	−0.247461	−0.123731	−	0.992316i	$-0.539486\pi$
$499$	−5.52786	−0.247461	−0.123731	+	0.992316i	$0.539486\pi$
$500$	5.94427	0.265836
$501$	9.05573	0.404580
$502$	−5.70820	−0.254770
$503$	15.0557	0.671302	0.335651	−	0.941986i	$-0.391044\pi$
$503$	15.0557	0.671302	0.335651	+	0.941986i	$0.391044\pi$
$504$	5.67376	0.252729
$505$	−3.23607	−0.144003
$506$	19.6180	0.872128
$507$	−3.12461	−0.138769
$508$	6.76393	0.300101
$509$	−0.472136	−0.0209271	−0.0104635	−	0.999945i	$-0.503331\pi$
$509$	−0.472136	−0.0209271	−0.0104635	+	0.999945i	$0.503331\pi$
$510$	−4.36068	−0.193094
$511$	−18.7082	−0.827602
$512$	1.00000	0.0441942
$513$	−26.8328	−1.18470
$514$	9.52786	0.420256
$515$	−11.8885	−0.523872
$516$	5.52786	0.243351
$517$	−2.94427	−0.129489
$518$	−35.5967	−1.56403
$519$	−6.24922	−0.274310
$520$	2.00000	0.0877058
$521$	−9.20163	−0.403131	−0.201565	−	0.979475i	$-0.564603\pi$
$521$	−9.20163	−0.403131	−0.201565	+	0.979475i	$0.564603\pi$
$522$	−2.94427	−0.128867
$523$	12.1803	0.532609	0.266305	−	0.963889i	$-0.414197\pi$
$523$	12.1803	0.532609	0.266305	+	0.963889i	$0.414197\pi$
$524$	−18.5623	−0.810898
$525$	22.0000	0.960159
$526$	−5.41641	−0.236167
$527$	−7.05573	−0.307352
$528$	−4.76393	−0.207324
$529$	2.90983	0.126514
$530$	−0.562306	−0.0244250
$531$	−20.1803	−0.875752
$532$	−18.7082	−0.811104
$533$	−21.8885	−0.948098
$534$	−4.18034	−0.180901
$535$	0	0
$536$	1.61803	0.0698884
$537$	12.3607	0.533403
$538$	−4.09017	−0.176340
$539$	−30.2705	−1.30384
$540$	3.41641	0.147019
$541$	32.0902	1.37966	0.689832	−	0.723969i	$-0.257684\pi$
$541$	32.0902	1.37966	0.689832	+	0.723969i	$0.257684\pi$
$542$	9.23607	0.396723
$543$	8.94427	0.383835
$544$	5.70820	0.244737
$545$	−0.944272	−0.0404482
$546$	15.4164	0.659761
$547$	31.3262	1.33941	0.669707	−	0.742626i	$-0.266420\pi$
$547$	31.3262	1.33941	0.669707	+	0.742626i	$0.266420\pi$
$548$	−20.4721	−0.874526
$549$	−8.83282	−0.376975
$550$	17.7984	0.758925
$551$	9.70820	0.413583
$552$	−6.29180	−0.267797
$553$	−15.4164	−0.655572
$554$	−2.58359	−0.109766
$555$	−7.05573	−0.299499
$556$	6.47214	0.274480
$557$	−12.6738	−0.537005	−0.268502	−	0.963279i	$-0.586529\pi$
$557$	−12.6738	−0.537005	−0.268502	+	0.963279i	$0.586529\pi$
$558$	1.81966	0.0770324
$559$	−14.4721	−0.612106
$560$	2.38197	0.100656
$561$	−27.1935	−1.14811
$562$	−8.29180	−0.349768
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0.944272	0.0397610
$565$	2.85410	0.120073
$566$	24.9443	1.04849
$567$	9.31308	0.391113
$568$	12.7984	0.537008
$569$	29.7082	1.24543	0.622716	−	0.782448i	$-0.286029\pi$
$569$	29.7082	1.24543	0.622716	+	0.782448i	$0.286029\pi$
$570$	−3.70820	−0.155320
$571$	31.5279	1.31940	0.659700	−	0.751529i	$-0.270683\pi$
$571$	31.5279	1.31940	0.659700	+	0.751529i	$0.270683\pi$
$572$	12.4721	0.521486
$573$	6.72136	0.280789
$574$	−26.0689	−1.08809
$575$	23.5066	0.980292
$576$	−1.47214	−0.0613390
$577$	16.0344	0.667523	0.333761	−	0.942658i	$-0.391682\pi$
$577$	16.0344	0.667523	0.333761	+	0.942658i	$0.391682\pi$
$578$	15.5836	0.648192
$579$	−2.94427	−0.122360
$580$	−1.23607	−0.0513249
$581$	0.562306	0.0233284
$582$	−21.3050	−0.883119
$583$	−3.50658	−0.145228
$584$	4.85410	0.200864
$585$	−2.94427	−0.121731
$586$	−2.29180	−0.0946732
$587$	33.7426	1.39271	0.696354	−	0.717698i	$-0.254804\pi$
$587$	33.7426	1.39271	0.696354	+	0.717698i	$0.254804\pi$
$588$	9.70820	0.400360
$589$	−6.00000	−0.247226
$590$	−8.47214	−0.348792
$591$	−12.3607	−0.508450
$592$	9.23607	0.379600
$593$	−15.0902	−0.619679	−0.309840	−	0.950789i	$-0.600275\pi$
$593$	−15.0902	−0.619679	−0.309840	+	0.950789i	$0.600275\pi$
$594$	21.3050	0.874153
$595$	13.5967	0.557412
$596$	−2.76393	−0.113215
$597$	12.8328	0.525212
$598$	16.4721	0.673596
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	−5.70820	−0.233036
$601$	−14.4377	−0.588926	−0.294463	−	0.955663i	$-0.595141\pi$
$601$	−14.4377	−0.588926	−0.294463	+	0.955663i	$0.595141\pi$
$602$	−17.2361	−0.702490
$603$	−2.38197	−0.0970012
$604$	−5.70820	−0.232264
$605$	−2.38197	−0.0968407
$606$	6.47214	0.262913
$607$	−3.20163	−0.129950	−0.0649750	−	0.997887i	$-0.520697\pi$
$607$	−3.20163	−0.129950	−0.0649750	+	0.997887i	$0.520697\pi$
$608$	4.85410	0.196860
$609$	−9.52786	−0.386089
$610$	−3.70820	−0.150141
$611$	−2.47214	−0.100012
$612$	−8.40325	−0.339681
$613$	−1.70820	−0.0689937	−0.0344969	−	0.999405i	$-0.510983\pi$
$613$	−1.70820	−0.0689937	−0.0344969	+	0.999405i	$0.510983\pi$
$614$	−18.3820	−0.741836
$615$	−5.16718	−0.208361
$616$	14.8541	0.598489
$617$	12.0902	0.486732	0.243366	−	0.969935i	$-0.421748\pi$
$617$	12.0902	0.486732	0.243366	+	0.969935i	$0.421748\pi$
$618$	23.7771	0.956455
$619$	13.6738	0.549595	0.274797	−	0.961502i	$-0.411389\pi$
$619$	13.6738	0.549595	0.274797	+	0.961502i	$0.411389\pi$
$620$	0.763932	0.0306802
$621$	28.1378	1.12913
$622$	27.5066	1.10291
$623$	13.0344	0.522214
$624$	−4.00000	−0.160128
$625$	19.4164	0.776656
$626$	11.5279	0.460746
$627$	−23.1246	−0.923508
$628$	−10.6525	−0.425080
$629$	52.7214	2.10214
$630$	−3.50658	−0.139705
$631$	−8.09017	−0.322065	−0.161032	−	0.986949i	$-0.551482\pi$
$631$	−8.09017	−0.322065	−0.161032	+	0.986949i	$0.551482\pi$
$632$	4.00000	0.159111
$633$	−14.0000	−0.556450
$634$	−7.50658	−0.298124
$635$	−4.18034	−0.165892
$636$	1.12461	0.0445938
$637$	−25.4164	−1.00703
$638$	−7.70820	−0.305171
$639$	−18.8409	−0.745336
$640$	−0.618034	−0.0244299
$641$	45.4164	1.79384	0.896920	−	0.442193i	$-0.145799\pi$
$641$	45.4164	1.79384	0.896920	+	0.442193i	$0.145799\pi$
$642$	0	0
$643$	32.6738	1.28853	0.644264	−	0.764803i	$-0.277164\pi$
$643$	32.6738	1.28853	0.644264	+	0.764803i	$0.277164\pi$
$644$	19.6180	0.773059
$645$	−3.41641	−0.134521
$646$	27.7082	1.09016
$647$	4.85410	0.190834	0.0954172	−	0.995437i	$-0.469581\pi$
$647$	4.85410	0.190834	0.0954172	+	0.995437i	$0.469581\pi$
$648$	−2.41641	−0.0949255
$649$	−52.8328	−2.07387
$650$	14.9443	0.586163
$651$	5.88854	0.230790
$652$	−6.32624	−0.247755
$653$	−11.5066	−0.450287	−0.225144	−	0.974326i	$-0.572285\pi$
$653$	−11.5066	−0.450287	−0.225144	+	0.974326i	$0.572285\pi$
$654$	1.88854	0.0738479
$655$	11.4721	0.448253
$656$	6.76393	0.264087
$657$	−7.14590	−0.278788
$658$	−2.94427	−0.114780
$659$	−37.3050	−1.45319	−0.726597	−	0.687064i	$-0.758899\pi$
$659$	−37.3050	−1.45319	−0.726597	+	0.687064i	$0.758899\pi$
$660$	2.94427	0.114606
$661$	13.2361	0.514823	0.257412	−	0.966302i	$-0.417130\pi$
$661$	13.2361	0.514823	0.257412	+	0.966302i	$0.417130\pi$
$662$	−26.8328	−1.04289
$663$	−22.8328	−0.886753
$664$	−0.145898	−0.00566194
$665$	11.5623	0.448367
$666$	−13.5967	−0.526863
$667$	−10.1803	−0.394184
$668$	7.32624	0.283461
$669$	−6.94427	−0.268481
$670$	−1.00000	−0.0386334
$671$	−23.1246	−0.892716
$672$	−4.76393	−0.183773
$673$	30.2492	1.16602	0.583011	−	0.812464i	$-0.301874\pi$
$673$	30.2492	1.16602	0.583011	+	0.812464i	$0.301874\pi$
$674$	−3.50658	−0.135068
$675$	25.5279	0.982568
$676$	−2.52786	−0.0972255
$677$	−27.8885	−1.07184	−0.535922	−	0.844268i	$-0.680036\pi$
$677$	−27.8885	−1.07184	−0.535922	+	0.844268i	$0.680036\pi$
$678$	−5.70820	−0.219222
$679$	66.4296	2.54933
$680$	−3.52786	−0.135287
$681$	−0.944272	−0.0361846
$682$	4.76393	0.182420
$683$	−39.4164	−1.50823	−0.754113	−	0.656744i	$-0.771933\pi$
$683$	−39.4164	−1.50823	−0.754113	+	0.656744i	$0.771933\pi$
$684$	−7.14590	−0.273230
$685$	12.6525	0.483426
$686$	−3.29180	−0.125681
$687$	−2.94427	−0.112331
$688$	4.47214	0.170499
$689$	−2.94427	−0.112168
$690$	3.88854	0.148034
$691$	−28.0689	−1.06779	−0.533895	−	0.845551i	$-0.679272\pi$
$691$	−28.0689	−1.06779	−0.533895	+	0.845551i	$0.679272\pi$
$692$	−5.05573	−0.192190
$693$	−21.8673	−0.830668
$694$	2.94427	0.111763
$695$	−4.00000	−0.151729
$696$	2.47214	0.0937061
$697$	38.6099	1.46245
$698$	18.3607	0.694962
$699$	3.81966	0.144473
$700$	17.7984	0.672715
$701$	17.8197	0.673039	0.336520	−	0.941676i	$-0.390750\pi$
$701$	17.8197	0.673039	0.336520	+	0.941676i	$0.390750\pi$
$702$	17.8885	0.675160
$703$	44.8328	1.69090
$704$	−3.85410	−0.145257
$705$	−0.583592	−0.0219794
$706$	−8.00000	−0.301084
$707$	−20.1803	−0.758960
$708$	16.9443	0.636805
$709$	−33.7771	−1.26853	−0.634263	−	0.773118i	$-0.718696\pi$
$709$	−33.7771	−1.26853	−0.634263	+	0.773118i	$0.718696\pi$
$710$	−7.90983	−0.296851
$711$	−5.88854	−0.220838
$712$	−3.38197	−0.126745
$713$	6.29180	0.235630
$714$	−27.1935	−1.01769
$715$	−7.70820	−0.288270
$716$	10.0000	0.373718
$717$	36.7214	1.37138
$718$	23.1459	0.863797
$719$	51.7082	1.92839	0.964195	−	0.265193i	$-0.0854357\pi$
$719$	51.7082	1.92839	0.964195	+	0.265193i	$0.0854357\pi$
$720$	0.909830	0.0339074
$721$	−74.1378	−2.76103
$722$	4.56231	0.169791
$723$	28.3607	1.05475
$724$	7.23607	0.268926
$725$	−9.23607	−0.343019
$726$	4.76393	0.176806
$727$	28.3607	1.05184	0.525920	−	0.850534i	$-0.323721\pi$
$727$	28.3607	1.05184	0.525920	+	0.850534i	$0.323721\pi$
$728$	12.4721	0.462248
$729$	24.0557	0.890953
$730$	−3.00000	−0.111035
$731$	25.5279	0.944182
$732$	7.41641	0.274118
$733$	−15.3820	−0.568146	−0.284073	−	0.958803i	$-0.591686\pi$
$733$	−15.3820	−0.568146	−0.284073	+	0.958803i	$0.591686\pi$
$734$	13.7426	0.507250
$735$	−6.00000	−0.221313
$736$	−5.09017	−0.187626
$737$	−6.23607	−0.229709
$738$	−9.95743	−0.366538
$739$	11.5279	0.424059	0.212030	−	0.977263i	$-0.431993\pi$
$739$	11.5279	0.424059	0.212030	+	0.977263i	$0.431993\pi$
$740$	−5.70820	−0.209838
$741$	−19.4164	−0.713280
$742$	−3.50658	−0.128731
$743$	41.7984	1.53343	0.766717	−	0.641985i	$-0.221889\pi$
$743$	41.7984	1.53343	0.766717	+	0.641985i	$0.221889\pi$
$744$	−1.52786	−0.0560142
$745$	1.70820	0.0625837
$746$	21.0344	0.770126
$747$	0.214782	0.00785845
$748$	−22.0000	−0.804400
$749$	0	0
$750$	7.34752	0.268294
$751$	35.6312	1.30020	0.650100	−	0.759848i	$-0.274727\pi$
$751$	35.6312	1.30020	0.650100	+	0.759848i	$0.274727\pi$
$752$	0.763932	0.0278577
$753$	−7.05573	−0.257125
$754$	−6.47214	−0.235701
$755$	3.52786	0.128392
$756$	21.3050	0.774854
$757$	24.1803	0.878849	0.439425	−	0.898279i	$-0.355182\pi$
$757$	24.1803	0.878849	0.439425	+	0.898279i	$0.355182\pi$
$758$	5.38197	0.195482
$759$	24.2492	0.880191
$760$	−3.00000	−0.108821
$761$	−11.7295	−0.425194	−0.212597	−	0.977140i	$-0.568192\pi$
$761$	−11.7295	−0.425194	−0.212597	+	0.977140i	$0.568192\pi$
$762$	8.36068	0.302875
$763$	−5.88854	−0.213180
$764$	5.43769	0.196729
$765$	5.19350	0.187771
$766$	29.0902	1.05107
$767$	−44.3607	−1.60177
$768$	1.23607	0.0446028
$769$	5.43769	0.196088	0.0980441	−	0.995182i	$-0.468741\pi$
$769$	5.43769	0.196088	0.0980441	+	0.995182i	$0.468741\pi$
$770$	−9.18034	−0.330836
$771$	11.7771	0.424141
$772$	−2.38197	−0.0857288
$773$	12.8328	0.461564	0.230782	−	0.973005i	$-0.425872\pi$
$773$	12.8328	0.461564	0.230782	+	0.973005i	$0.425872\pi$
$774$	−6.58359	−0.236642
$775$	5.70820	0.205045
$776$	−17.2361	−0.618739
$777$	−44.0000	−1.57849
$778$	−12.3607	−0.443152
$779$	32.8328	1.17636
$780$	2.47214	0.0885167
$781$	−49.3262	−1.76503
$782$	−29.0557	−1.03903
$783$	−11.0557	−0.395099
$784$	7.85410	0.280504
$785$	6.58359	0.234978
$786$	−22.9443	−0.818395
$787$	14.1803	0.505475	0.252737	−	0.967535i	$-0.418669\pi$
$787$	14.1803	0.505475	0.252737	+	0.967535i	$0.418669\pi$
$788$	−10.0000	−0.356235
$789$	−6.69505	−0.238350
$790$	−2.47214	−0.0879547
$791$	17.7984	0.632837
$792$	5.67376	0.201608
$793$	−19.4164	−0.689497
$794$	20.4721	0.726529
$795$	−0.695048	−0.0246508
$796$	10.3820	0.367979
$797$	−37.7984	−1.33889	−0.669444	−	0.742863i	$-0.733467\pi$
$797$	−37.7984	−1.33889	−0.669444	+	0.742863i	$0.733467\pi$
$798$	−23.1246	−0.818602
$799$	4.36068	0.154270
$800$	−4.61803	−0.163272
$801$	4.97871	0.175914
$802$	−25.7082	−0.907788
$803$	−18.7082	−0.660198
$804$	2.00000	0.0705346
$805$	−12.1246	−0.427336
$806$	4.00000	0.140894
$807$	−5.05573	−0.177970
$808$	5.23607	0.184204
$809$	10.3607	0.364262	0.182131	−	0.983274i	$-0.441700\pi$
$809$	10.3607	0.364262	0.182131	+	0.983274i	$0.441700\pi$
$810$	1.49342	0.0524735
$811$	32.1459	1.12880	0.564398	−	0.825503i	$-0.309109\pi$
$811$	32.1459	1.12880	0.564398	+	0.825503i	$0.309109\pi$
$812$	−7.70820	−0.270505
$813$	11.4164	0.400391
$814$	−35.5967	−1.24767
$815$	3.90983	0.136955
$816$	7.05573	0.247000
$817$	21.7082	0.759474
$818$	−11.6738	−0.408164
$819$	−18.3607	−0.641574
$820$	−4.18034	−0.145984
$821$	43.0902	1.50386	0.751929	−	0.659244i	$-0.229124\pi$
$821$	43.0902	1.50386	0.751929	+	0.659244i	$0.229124\pi$
$822$	−25.3050	−0.882612
$823$	−30.1803	−1.05202	−0.526010	−	0.850478i	$-0.676313\pi$
$823$	−30.1803	−1.05202	−0.526010	+	0.850478i	$0.676313\pi$
$824$	19.2361	0.670120
$825$	22.0000	0.765942
$826$	−52.8328	−1.83829
$827$	−1.74265	−0.0605977	−0.0302989	−	0.999541i	$-0.509646\pi$
$827$	−1.74265	−0.0605977	−0.0302989	+	0.999541i	$0.509646\pi$
$828$	7.49342	0.260414
$829$	−37.6312	−1.30699	−0.653493	−	0.756933i	$-0.726697\pi$
$829$	−37.6312	−1.30699	−0.653493	+	0.756933i	$0.726697\pi$
$830$	0.0901699	0.00312984
$831$	−3.19350	−0.110781
$832$	−3.23607	−0.112190
$833$	44.8328	1.55336
$834$	8.00000	0.277017
$835$	−4.52786	−0.156693
$836$	−18.7082	−0.647037
$837$	6.83282	0.236177
$838$	−26.8328	−0.926924
$839$	5.52786	0.190843	0.0954215	−	0.995437i	$-0.469580\pi$
$839$	5.52786	0.190843	0.0954215	+	0.995437i	$0.469580\pi$
$840$	2.94427	0.101587
$841$	−25.0000	−0.862069
$842$	0.0901699	0.00310746
$843$	−10.2492	−0.353002
$844$	−11.3262	−0.389865
$845$	1.56231	0.0537450
$846$	−1.12461	−0.0386650
$847$	−14.8541	−0.510393
$848$	0.909830	0.0312437
$849$	30.8328	1.05818
$850$	−26.3607	−0.904164
$851$	−47.0132	−1.61159
$852$	15.8197	0.541973
$853$	25.9230	0.887586	0.443793	−	0.896129i	$-0.353633\pi$
$853$	25.9230	0.887586	0.443793	+	0.896129i	$0.353633\pi$
$854$	−23.1246	−0.791308
$855$	4.41641	0.151038
$856$	0	0
$857$	31.1246	1.06320	0.531598	−	0.846997i	$-0.321592\pi$
$857$	31.1246	1.06320	0.531598	+	0.846997i	$0.321592\pi$
$858$	15.4164	0.526307
$859$	18.3262	0.625283	0.312642	−	0.949871i	$-0.398786\pi$
$859$	18.3262	0.625283	0.312642	+	0.949871i	$0.398786\pi$
$860$	−2.76393	−0.0942493
$861$	−32.2229	−1.09815
$862$	20.3607	0.693488
$863$	−33.7426	−1.14861	−0.574307	−	0.818640i	$-0.694728\pi$
$863$	−33.7426	−1.14861	−0.574307	+	0.818640i	$0.694728\pi$
$864$	−5.52786	−0.188062
$865$	3.12461	0.106240
$866$	29.1246	0.989695
$867$	19.2624	0.654185
$868$	4.76393	0.161698
$869$	−15.4164	−0.522966
$870$	−1.52786	−0.0517994
$871$	−5.23607	−0.177417
$872$	1.52786	0.0517400
$873$	25.3738	0.858774
$874$	−24.7082	−0.835767
$875$	−22.9098	−0.774494
$876$	6.00000	0.202721
$877$	35.8673	1.21115	0.605576	−	0.795788i	$-0.292943\pi$
$877$	35.8673	1.21115	0.605576	+	0.795788i	$0.292943\pi$
$878$	8.18034	0.276073
$879$	−2.83282	−0.0955485
$880$	2.38197	0.0802961
$881$	−20.6525	−0.695800	−0.347900	−	0.937532i	$-0.613105\pi$
$881$	−20.6525	−0.695800	−0.347900	+	0.937532i	$0.613105\pi$
$882$	−11.5623	−0.389323
$883$	−53.4853	−1.79992	−0.899962	−	0.435969i	$-0.856406\pi$
$883$	−53.4853	−1.79992	−0.899962	+	0.435969i	$0.856406\pi$
$884$	−18.4721	−0.621285
$885$	−10.4721	−0.352017
$886$	−35.8885	−1.20570
$887$	−24.6738	−0.828464	−0.414232	−	0.910171i	$-0.635950\pi$
$887$	−24.6738	−0.828464	−0.414232	+	0.910171i	$0.635950\pi$
$888$	11.4164	0.383110
$889$	−26.0689	−0.874322
$890$	2.09017	0.0700627
$891$	9.31308	0.312000
$892$	−5.61803	−0.188106
$893$	3.70820	0.124090
$894$	−3.41641	−0.114262
$895$	−6.18034	−0.206586
$896$	−3.85410	−0.128757
$897$	20.3607	0.679823
$898$	−12.9443	−0.431956
$899$	−2.47214	−0.0824504
$900$	6.79837	0.226612
$901$	5.19350	0.173020
$902$	−26.0689	−0.867999
$903$	−21.3050	−0.708984
$904$	−4.61803	−0.153594
$905$	−4.47214	−0.148659
$906$	−7.05573	−0.234411
$907$	−31.8197	−1.05655	−0.528277	−	0.849072i	$-0.677162\pi$
$907$	−31.8197	−1.05655	−0.528277	+	0.849072i	$0.677162\pi$
$908$	−0.763932	−0.0253520
$909$	−7.70820	−0.255665
$910$	−7.70820	−0.255524
$911$	−9.63932	−0.319365	−0.159682	−	0.987168i	$-0.551047\pi$
$911$	−9.63932	−0.319365	−0.159682	+	0.987168i	$0.551047\pi$
$912$	6.00000	0.198680
$913$	0.562306	0.0186096
$914$	−34.1803	−1.13059
$915$	−4.58359	−0.151529
$916$	−2.38197	−0.0787024
$917$	71.5410	2.36249
$918$	−31.5542	−1.04144
$919$	6.29180	0.207547	0.103774	−	0.994601i	$-0.466908\pi$
$919$	6.29180	0.207547	0.103774	+	0.994601i	$0.466908\pi$
$920$	3.14590	0.103717
$921$	−22.7214	−0.748694
$922$	35.4164	1.16638
$923$	−41.4164	−1.36324
$924$	18.3607	0.604022
$925$	−42.6525	−1.40240
$926$	−27.4164	−0.900959
$927$	−28.3181	−0.930089
$928$	2.00000	0.0656532
$929$	31.4508	1.03187	0.515934	−	0.856628i	$-0.327445\pi$
$929$	31.4508	1.03187	0.515934	+	0.856628i	$0.327445\pi$
$930$	0.944272	0.0309639
$931$	38.1246	1.24948
$932$	3.09017	0.101222
$933$	34.0000	1.11311
$934$	−2.47214	−0.0808908
$935$	13.5967	0.444661
$936$	4.76393	0.155714
$937$	10.0689	0.328936	0.164468	−	0.986382i	$-0.447409\pi$
$937$	10.0689	0.328936	0.164468	+	0.986382i	$0.447409\pi$
$938$	−6.23607	−0.203615
$939$	14.2492	0.465006
$940$	−0.472136	−0.0153994
$941$	−31.8885	−1.03954	−0.519768	−	0.854307i	$-0.673982\pi$
$941$	−31.8885	−1.03954	−0.519768	+	0.854307i	$0.673982\pi$
$942$	−13.1672	−0.429010
$943$	−34.4296	−1.12118
$944$	13.7082	0.446164
$945$	−13.1672	−0.428329
$946$	−17.2361	−0.560393
$947$	35.6869	1.15967	0.579835	−	0.814734i	$-0.303117\pi$
$947$	35.6869	1.15967	0.579835	+	0.814734i	$0.303117\pi$
$948$	4.94427	0.160582
$949$	−15.7082	−0.509910
$950$	−22.4164	−0.727284
$951$	−9.27864	−0.300881
$952$	−22.0000	−0.713024
$953$	−45.3951	−1.47049	−0.735246	−	0.677800i	$-0.762933\pi$
$953$	−45.3951	−1.47049	−0.735246	+	0.677800i	$0.762933\pi$
$954$	−1.33939	−0.0433645
$955$	−3.36068	−0.108749
$956$	29.7082	0.960832
$957$	−9.52786	−0.307992
$958$	13.1246	0.424037
$959$	78.9017	2.54787
$960$	−0.763932	−0.0246558
$961$	−29.4721	−0.950714
$962$	−29.8885	−0.963645
$963$	0	0
$964$	22.9443	0.738985
$965$	1.47214	0.0473897
$966$	24.2492	0.780206
$967$	−32.5410	−1.04645	−0.523224	−	0.852195i	$-0.675271\pi$
$967$	−32.5410	−1.04645	−0.523224	+	0.852195i	$0.675271\pi$
$968$	3.85410	0.123876
$969$	34.2492	1.10024
$970$	10.6525	0.342030
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	13.5967	0.436116
$973$	−24.9443	−0.799677
$974$	−23.4164	−0.750310
$975$	18.4721	0.591582
$976$	6.00000	0.192055
$977$	23.5279	0.752723	0.376362	−	0.926473i	$-0.377175\pi$
$977$	23.5279	0.752723	0.376362	+	0.926473i	$0.377175\pi$
$978$	−7.81966	−0.250045
$979$	13.0344	0.416583
$980$	−4.85410	−0.155059
$981$	−2.24922	−0.0718122
$982$	38.5623	1.23057
$983$	1.70820	0.0544832	0.0272416	−	0.999629i	$-0.491328\pi$
$983$	1.70820	0.0544832	0.0272416	+	0.999629i	$0.491328\pi$
$984$	8.36068	0.266529
$985$	6.18034	0.196922
$986$	11.4164	0.363572
$987$	−3.63932	−0.115841
$988$	−15.7082	−0.499745
$989$	−22.7639	−0.723851
$990$	−3.50658	−0.111446
$991$	22.4721	0.713851	0.356925	−	0.934133i	$-0.383825\pi$
$991$	22.4721	0.713851	0.356925	+	0.934133i	$0.383825\pi$
$992$	−1.23607	−0.0392452
$993$	−33.1672	−1.05253
$994$	−49.3262	−1.56453
$995$	−6.41641	−0.203414
$996$	−0.180340	−0.00571429
$997$	−0.944272	−0.0299054	−0.0149527	−	0.999888i	$-0.504760\pi$
$997$	−0.944272	−0.0299054	−0.0149527	+	0.999888i	$0.504760\pi$
$998$	−5.52786	−0.174981
$999$	−51.0557	−1.61533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8014.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8014.2.a.a.1.2	✓	2	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:	\( N \)	\(=\)	\( 8014 = 2 \cdot 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.23607 q^{6} -3.85410 q^{7} +1.00000 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.23607 q^{6} -3.85410 q^{7} +1.00000 q^{8} -1.47214 q^{9} -0.618034 q^{10} -3.85410 q^{11} +1.23607 q^{12} -3.23607 q^{13} -3.85410 q^{14} -0.763932 q^{15} +1.00000 q^{16} +5.70820 q^{17} -1.47214 q^{18} +4.85410 q^{19} -0.618034 q^{20} -4.76393 q^{21} -3.85410 q^{22} -5.09017 q^{23} +1.23607 q^{24} -4.61803 q^{25} -3.23607 q^{26} -5.52786 q^{27} -3.85410 q^{28} +2.00000 q^{29} -0.763932 q^{30} -1.23607 q^{31} +1.00000 q^{32} -4.76393 q^{33} +5.70820 q^{34} +2.38197 q^{35} -1.47214 q^{36} +9.23607 q^{37} +4.85410 q^{38} -4.00000 q^{39} -0.618034 q^{40} +6.76393 q^{41} -4.76393 q^{42} +4.47214 q^{43} -3.85410 q^{44} +0.909830 q^{45} -5.09017 q^{46} +0.763932 q^{47} +1.23607 q^{48} +7.85410 q^{49} -4.61803 q^{50} +7.05573 q^{51} -3.23607 q^{52} +0.909830 q^{53} -5.52786 q^{54} +2.38197 q^{55} -3.85410 q^{56} +6.00000 q^{57} +2.00000 q^{58} +13.7082 q^{59} -0.763932 q^{60} +6.00000 q^{61} -1.23607 q^{62} +5.67376 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.76393 q^{66} +1.61803 q^{67} +5.70820 q^{68} -6.29180 q^{69} +2.38197 q^{70} +12.7984 q^{71} -1.47214 q^{72} +4.85410 q^{73} +9.23607 q^{74} -5.70820 q^{75} +4.85410 q^{76} +14.8541 q^{77} -4.00000 q^{78} +4.00000 q^{79} -0.618034 q^{80} -2.41641 q^{81} +6.76393 q^{82} -0.145898 q^{83} -4.76393 q^{84} -3.52786 q^{85} +4.47214 q^{86} +2.47214 q^{87} -3.85410 q^{88} -3.38197 q^{89} +0.909830 q^{90} +12.4721 q^{91} -5.09017 q^{92} -1.52786 q^{93} +0.763932 q^{94} -3.00000 q^{95} +1.23607 q^{96} -17.2361 q^{97} +7.85410 q^{98} +5.67376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9} + q^{10} - q^{11} - 2 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{17} + 6 q^{18} + 3 q^{19} + q^{20} - 14 q^{21} - q^{22} + q^{23} - 2 q^{24} - 7 q^{25} - 2 q^{26} - 20 q^{27} - q^{28} + 4 q^{29} - 6 q^{30} + 2 q^{31} + 2 q^{32} - 14 q^{33} - 2 q^{34} + 7 q^{35} + 6 q^{36} + 14 q^{37} + 3 q^{38} - 8 q^{39} + q^{40} + 18 q^{41} - 14 q^{42} - q^{44} + 13 q^{45} + q^{46} + 6 q^{47} - 2 q^{48} + 9 q^{49} - 7 q^{50} + 32 q^{51} - 2 q^{52} + 13 q^{53} - 20 q^{54} + 7 q^{55} - q^{56} + 12 q^{57} + 4 q^{58} + 14 q^{59} - 6 q^{60} + 12 q^{61} + 2 q^{62} + 27 q^{63} + 2 q^{64} + 4 q^{65} - 14 q^{66} + q^{67} - 2 q^{68} - 26 q^{69} + 7 q^{70} + q^{71} + 6 q^{72} + 3 q^{73} + 14 q^{74} + 2 q^{75} + 3 q^{76} + 23 q^{77} - 8 q^{78} + 8 q^{79} + q^{80} + 22 q^{81} + 18 q^{82} - 7 q^{83} - 14 q^{84} - 16 q^{85} - 4 q^{87} - q^{88} - 9 q^{89} + 13 q^{90} + 16 q^{91} + q^{92} - 12 q^{93} + 6 q^{94} - 6 q^{95} - 2 q^{96} - 30 q^{97} + 9 q^{98} + 27 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - q^7 + 2 * q^8 + 6 * q^9 + q^10 - q^11 - 2 * q^12 - 2 * q^13 - q^14 - 6 * q^15 + 2 * q^16 - 2 * q^17 + 6 * q^18 + 3 * q^19 + q^20 - 14 * q^21 - q^22 + q^23 - 2 * q^24 - 7 * q^25 - 2 * q^26 - 20 * q^27 - q^28 + 4 * q^29 - 6 * q^30 + 2 * q^31 + 2 * q^32 - 14 * q^33 - 2 * q^34 + 7 * q^35 + 6 * q^36 + 14 * q^37 + 3 * q^38 - 8 * q^39 + q^40 + 18 * q^41 - 14 * q^42 - q^44 + 13 * q^45 + q^46 + 6 * q^47 - 2 * q^48 + 9 * q^49 - 7 * q^50 + 32 * q^51 - 2 * q^52 + 13 * q^53 - 20 * q^54 + 7 * q^55 - q^56 + 12 * q^57 + 4 * q^58 + 14 * q^59 - 6 * q^60 + 12 * q^61 + 2 * q^62 + 27 * q^63 + 2 * q^64 + 4 * q^65 - 14 * q^66 + q^67 - 2 * q^68 - 26 * q^69 + 7 * q^70 + q^71 + 6 * q^72 + 3 * q^73 + 14 * q^74 + 2 * q^75 + 3 * q^76 + 23 * q^77 - 8 * q^78 + 8 * q^79 + q^80 + 22 * q^81 + 18 * q^82 - 7 * q^83 - 14 * q^84 - 16 * q^85 - 4 * q^87 - q^88 - 9 * q^89 + 13 * q^90 + 16 * q^91 + q^92 - 12 * q^93 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 30 * q^97 + 9 * q^98 + 27 * q^99

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