LMFDB - Embedded newform 4027.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, 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("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$2$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} -1.00000 q^{5} -2.00000 q^{6} -4.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} -1.00000 q^{5} -2.00000 q^{6} -4.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +1.61803 q^{10} -1.00000 q^{11} +0.763932 q^{12} -2.61803 q^{13} +7.47214 q^{14} -1.23607 q^{15} -4.85410 q^{16} -5.09017 q^{17} +2.38197 q^{18} -5.00000 q^{19} -0.618034 q^{20} -5.70820 q^{21} +1.61803 q^{22} -6.61803 q^{23} +2.76393 q^{24} -4.00000 q^{25} +4.23607 q^{26} -5.52786 q^{27} -2.85410 q^{28} -3.38197 q^{29} +2.00000 q^{30} -3.76393 q^{31} +3.38197 q^{32} -1.23607 q^{33} +8.23607 q^{34} +4.61803 q^{35} -0.909830 q^{36} +6.94427 q^{37} +8.09017 q^{38} -3.23607 q^{39} -2.23607 q^{40} -2.76393 q^{41} +9.23607 q^{42} +8.70820 q^{43} -0.618034 q^{44} +1.47214 q^{45} +10.7082 q^{46} +0.472136 q^{47} -6.00000 q^{48} +14.3262 q^{49} +6.47214 q^{50} -6.29180 q^{51} -1.61803 q^{52} +1.76393 q^{53} +8.94427 q^{54} +1.00000 q^{55} -10.3262 q^{56} -6.18034 q^{57} +5.47214 q^{58} +4.70820 q^{59} -0.763932 q^{60} -11.1803 q^{61} +6.09017 q^{62} +6.79837 q^{63} +4.23607 q^{64} +2.61803 q^{65} +2.00000 q^{66} -5.70820 q^{67} -3.14590 q^{68} -8.18034 q^{69} -7.47214 q^{70} -4.14590 q^{71} -3.29180 q^{72} +0.0901699 q^{73} -11.2361 q^{74} -4.94427 q^{75} -3.09017 q^{76} +4.61803 q^{77} +5.23607 q^{78} +13.0000 q^{79} +4.85410 q^{80} -2.41641 q^{81} +4.47214 q^{82} -16.5623 q^{83} -3.52786 q^{84} +5.09017 q^{85} -14.0902 q^{86} -4.18034 q^{87} -2.23607 q^{88} -4.14590 q^{89} -2.38197 q^{90} +12.0902 q^{91} -4.09017 q^{92} -4.65248 q^{93} -0.763932 q^{94} +5.00000 q^{95} +4.18034 q^{96} +4.52786 q^{97} -23.1803 q^{98} +1.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9} + q^{10} - 2 q^{11} + 6 q^{12} - 3 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + 7 q^{18} - 10 q^{19} + q^{20} + 2 q^{21} + q^{22} - 11 q^{23} + 10 q^{24} - 8 q^{25} + 4 q^{26} - 20 q^{27} + q^{28} - 9 q^{29} + 4 q^{30} - 12 q^{31} + 9 q^{32} + 2 q^{33} + 12 q^{34} + 7 q^{35} - 13 q^{36} - 4 q^{37} + 5 q^{38} - 2 q^{39} - 10 q^{41} + 14 q^{42} + 4 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} - 8 q^{47} - 12 q^{48} + 13 q^{49} + 4 q^{50} - 26 q^{51} - q^{52} + 8 q^{53} + 2 q^{55} - 5 q^{56} + 10 q^{57} + 2 q^{58} - 4 q^{59} - 6 q^{60} + q^{62} - 11 q^{63} + 4 q^{64} + 3 q^{65} + 4 q^{66} + 2 q^{67} - 13 q^{68} + 6 q^{69} - 6 q^{70} - 15 q^{71} - 20 q^{72} - 11 q^{73} - 18 q^{74} + 8 q^{75} + 5 q^{76} + 7 q^{77} + 6 q^{78} + 26 q^{79} + 3 q^{80} + 22 q^{81} - 13 q^{83} - 16 q^{84} - q^{85} - 17 q^{86} + 14 q^{87} - 15 q^{89} - 7 q^{90} + 13 q^{91} + 3 q^{92} + 22 q^{93} - 6 q^{94} + 10 q^{95} - 14 q^{96} + 18 q^{97} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9 + q^10 - 2 * q^11 + 6 * q^12 - 3 * q^13 + 6 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + 7 * q^18 - 10 * q^19 + q^20 + 2 * q^21 + q^22 - 11 * q^23 + 10 * q^24 - 8 * q^25 + 4 * q^26 - 20 * q^27 + q^28 - 9 * q^29 + 4 * q^30 - 12 * q^31 + 9 * q^32 + 2 * q^33 + 12 * q^34 + 7 * q^35 - 13 * q^36 - 4 * q^37 + 5 * q^38 - 2 * q^39 - 10 * q^41 + 14 * q^42 + 4 * q^43 + q^44 - 6 * q^45 + 8 * q^46 - 8 * q^47 - 12 * q^48 + 13 * q^49 + 4 * q^50 - 26 * q^51 - q^52 + 8 * q^53 + 2 * q^55 - 5 * q^56 + 10 * q^57 + 2 * q^58 - 4 * q^59 - 6 * q^60 + q^62 - 11 * q^63 + 4 * q^64 + 3 * q^65 + 4 * q^66 + 2 * q^67 - 13 * q^68 + 6 * q^69 - 6 * q^70 - 15 * q^71 - 20 * q^72 - 11 * q^73 - 18 * q^74 + 8 * q^75 + 5 * q^76 + 7 * q^77 + 6 * q^78 + 26 * q^79 + 3 * q^80 + 22 * q^81 - 13 * q^83 - 16 * q^84 - q^85 - 17 * q^86 + 14 * q^87 - 15 * q^89 - 7 * q^90 + 13 * q^91 + 3 * q^92 + 22 * q^93 - 6 * q^94 + 10 * q^95 - 14 * q^96 + 18 * q^97 - 24 * q^98 - 6 * 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0.618034	0.309017
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−2.00000	−0.816497
$7$	−4.61803	−1.74545	−0.872726	−	0.488210i	$-0.837650\pi$
$7$	−4.61803	−1.74545	−0.872726	+	0.488210i	$0.837650\pi$
$8$	2.23607	0.790569
$9$	−1.47214	−0.490712
$10$	1.61803	0.511667
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0.763932	0.220528
$13$	−2.61803	−0.726112	−0.363056	−	0.931767i	$-0.618267\pi$
$13$	−2.61803	−0.726112	−0.363056	+	0.931767i	$0.618267\pi$
$14$	7.47214	1.99701
$15$	−1.23607	−0.319151
$16$	−4.85410	−1.21353
$17$	−5.09017	−1.23455	−0.617274	−	0.786748i	$-0.711763\pi$
$17$	−5.09017	−1.23455	−0.617274	+	0.786748i	$0.711763\pi$
$18$	2.38197	0.561435
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−0.618034	−0.138197
$21$	−5.70820	−1.24563
$22$	1.61803	0.344966
$23$	−6.61803	−1.37996	−0.689978	−	0.723831i	$-0.742380\pi$
$23$	−6.61803	−1.37996	−0.689978	+	0.723831i	$0.742380\pi$
$24$	2.76393	0.564185
$25$	−4.00000	−0.800000
$26$	4.23607	0.830761
$27$	−5.52786	−1.06384
$28$	−2.85410	−0.539375
$29$	−3.38197	−0.628015	−0.314008	−	0.949420i	$-0.601672\pi$
$29$	−3.38197	−0.628015	−0.314008	+	0.949420i	$0.601672\pi$
$30$	2.00000	0.365148
$31$	−3.76393	−0.676022	−0.338011	−	0.941142i	$-0.609754\pi$
$31$	−3.76393	−0.676022	−0.338011	+	0.941142i	$0.609754\pi$
$32$	3.38197	0.597853
$33$	−1.23607	−0.215172
$34$	8.23607	1.41247
$35$	4.61803	0.780590
$36$	−0.909830	−0.151638
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	8.09017	1.31240
$39$	−3.23607	−0.518186
$40$	−2.23607	−0.353553
$41$	−2.76393	−0.431654	−0.215827	−	0.976432i	$-0.569245\pi$
$41$	−2.76393	−0.431654	−0.215827	+	0.976432i	$0.569245\pi$
$42$	9.23607	1.42516
$43$	8.70820	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$43$	8.70820	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$44$	−0.618034	−0.0931721
$45$	1.47214	0.219453
$46$	10.7082	1.57884
$47$	0.472136	0.0688681	0.0344341	−	0.999407i	$-0.489037\pi$
$47$	0.472136	0.0688681	0.0344341	+	0.999407i	$0.489037\pi$
$48$	−6.00000	−0.866025
$49$	14.3262	2.04661
$50$	6.47214	0.915298
$51$	−6.29180	−0.881028
$52$	−1.61803	−0.224381
$53$	1.76393	0.242295	0.121147	−	0.992635i	$-0.461343\pi$
$53$	1.76393	0.242295	0.121147	+	0.992635i	$0.461343\pi$
$54$	8.94427	1.21716
$55$	1.00000	0.134840
$56$	−10.3262	−1.37990
$57$	−6.18034	−0.818606
$58$	5.47214	0.718527
$59$	4.70820	0.612956	0.306478	−	0.951878i	$-0.400849\pi$
$59$	4.70820	0.612956	0.306478	+	0.951878i	$0.400849\pi$
$60$	−0.763932	−0.0986232
$61$	−11.1803	−1.43150	−0.715748	−	0.698359i	$-0.753914\pi$
$61$	−11.1803	−1.43150	−0.715748	+	0.698359i	$0.753914\pi$
$62$	6.09017	0.773452
$63$	6.79837	0.856515
$64$	4.23607	0.529508
$65$	2.61803	0.324727
$66$	2.00000	0.246183
$67$	−5.70820	−0.697368	−0.348684	−	0.937240i	$-0.613371\pi$
$67$	−5.70820	−0.697368	−0.348684	+	0.937240i	$0.613371\pi$
$68$	−3.14590	−0.381496
$69$	−8.18034	−0.984797
$70$	−7.47214	−0.893091
$71$	−4.14590	−0.492028	−0.246014	−	0.969266i	$-0.579121\pi$
$71$	−4.14590	−0.492028	−0.246014	+	0.969266i	$0.579121\pi$
$72$	−3.29180	−0.387942
$73$	0.0901699	0.0105536	0.00527680	−	0.999986i	$-0.498320\pi$
$73$	0.0901699	0.0105536	0.00527680	+	0.999986i	$0.498320\pi$
$74$	−11.2361	−1.30617
$75$	−4.94427	−0.570915
$76$	−3.09017	−0.354467
$77$	4.61803	0.526274
$78$	5.23607	0.592868
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	4.85410	0.542705
$81$	−2.41641	−0.268490
$82$	4.47214	0.493865
$83$	−16.5623	−1.81795	−0.908975	−	0.416851i	$-0.863134\pi$
$83$	−16.5623	−1.81795	−0.908975	+	0.416851i	$0.863134\pi$
$84$	−3.52786	−0.384922
$85$	5.09017	0.552106
$86$	−14.0902	−1.51938
$87$	−4.18034	−0.448179
$88$	−2.23607	−0.238366
$89$	−4.14590	−0.439464	−0.219732	−	0.975560i	$-0.570518\pi$
$89$	−4.14590	−0.439464	−0.219732	+	0.975560i	$0.570518\pi$
$90$	−2.38197	−0.251081
$91$	12.0902	1.26739
$92$	−4.09017	−0.426430
$93$	−4.65248	−0.482439
$94$	−0.763932	−0.0787936
$95$	5.00000	0.512989
$96$	4.18034	0.426654
$97$	4.52786	0.459735	0.229867	−	0.973222i	$-0.426171\pi$
$97$	4.52786	0.459735	0.229867	+	0.973222i	$0.426171\pi$
$98$	−23.1803	−2.34157
$99$	1.47214	0.147955
$100$	−2.47214	−0.247214
$101$	0.854102	0.0849863	0.0424932	−	0.999097i	$-0.486470\pi$
$101$	0.854102	0.0849863	0.0424932	+	0.999097i	$0.486470\pi$
$102$	10.1803	1.00800
$103$	−15.3262	−1.51014	−0.755070	−	0.655645i	$-0.772397\pi$
$103$	−15.3262	−1.51014	−0.755070	+	0.655645i	$0.772397\pi$
$104$	−5.85410	−0.574042
$105$	5.70820	0.557064
$106$	−2.85410	−0.277215
$107$	6.32624	0.611581	0.305790	−	0.952099i	$-0.401079\pi$
$107$	6.32624	0.611581	0.305790	+	0.952099i	$0.401079\pi$
$108$	−3.41641	−0.328744
$109$	−3.47214	−0.332570	−0.166285	−	0.986078i	$-0.553177\pi$
$109$	−3.47214	−0.332570	−0.166285	+	0.986078i	$0.553177\pi$
$110$	−1.61803	−0.154273
$111$	8.58359	0.814719
$112$	22.4164	2.11815
$113$	−1.76393	−0.165937	−0.0829684	−	0.996552i	$-0.526440\pi$
$113$	−1.76393	−0.165937	−0.0829684	+	0.996552i	$0.526440\pi$
$114$	10.0000	0.936586
$115$	6.61803	0.617135
$116$	−2.09017	−0.194067
$117$	3.85410	0.356312
$118$	−7.61803	−0.701297
$119$	23.5066	2.15484
$120$	−2.76393	−0.252311
$121$	−10.0000	−0.909091
$122$	18.0902	1.63781
$123$	−3.41641	−0.308047
$124$	−2.32624	−0.208902
$125$	9.00000	0.804984
$126$	−11.0000	−0.979958
$127$	−10.7639	−0.955145	−0.477572	−	0.878592i	$-0.658483\pi$
$127$	−10.7639	−0.955145	−0.477572	+	0.878592i	$0.658483\pi$
$128$	−13.6180	−1.20368
$129$	10.7639	0.947711
$130$	−4.23607	−0.371528
$131$	−19.7984	−1.72979	−0.864896	−	0.501951i	$-0.832616\pi$
$131$	−19.7984	−1.72979	−0.864896	+	0.501951i	$0.832616\pi$
$132$	−0.763932	−0.0664917
$133$	23.0902	2.00217
$134$	9.23607	0.797875
$135$	5.52786	0.475763
$136$	−11.3820	−0.975996
$137$	−9.38197	−0.801555	−0.400778	−	0.916175i	$-0.631260\pi$
$137$	−9.38197	−0.801555	−0.400778	+	0.916175i	$0.631260\pi$
$138$	13.2361	1.12673
$139$	20.2361	1.71640	0.858200	−	0.513315i	$-0.171583\pi$
$139$	20.2361	1.71640	0.858200	+	0.513315i	$0.171583\pi$
$140$	2.85410	0.241216
$141$	0.583592	0.0491473
$142$	6.70820	0.562940
$143$	2.61803	0.218931
$144$	7.14590	0.595492
$145$	3.38197	0.280857
$146$	−0.145898	−0.0120746
$147$	17.7082	1.46055
$148$	4.29180	0.352783
$149$	8.09017	0.662773	0.331386	−	0.943495i	$-0.392484\pi$
$149$	8.09017	0.662773	0.331386	+	0.943495i	$0.392484\pi$
$150$	8.00000	0.653197
$151$	6.70820	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$151$	6.70820	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$152$	−11.1803	−0.906845
$153$	7.49342	0.605807
$154$	−7.47214	−0.602122
$155$	3.76393	0.302326
$156$	−2.00000	−0.160128
$157$	−5.85410	−0.467208	−0.233604	−	0.972332i	$-0.575052\pi$
$157$	−5.85410	−0.467208	−0.233604	+	0.972332i	$0.575052\pi$
$158$	−21.0344	−1.67341
$159$	2.18034	0.172912
$160$	−3.38197	−0.267368
$161$	30.5623	2.40865
$162$	3.90983	0.307185
$163$	−5.90983	−0.462894	−0.231447	−	0.972848i	$-0.574346\pi$
$163$	−5.90983	−0.462894	−0.231447	+	0.972848i	$0.574346\pi$
$164$	−1.70820	−0.133388
$165$	1.23607	0.0962278
$166$	26.7984	2.07996
$167$	−2.29180	−0.177345	−0.0886723	−	0.996061i	$-0.528262\pi$
$167$	−2.29180	−0.177345	−0.0886723	+	0.996061i	$0.528262\pi$
$168$	−12.7639	−0.984759
$169$	−6.14590	−0.472761
$170$	−8.23607	−0.631678
$171$	7.36068	0.562885
$172$	5.38197	0.410371
$173$	16.4721	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$173$	16.4721	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$174$	6.76393	0.512772
$175$	18.4721	1.39636
$176$	4.85410	0.365892
$177$	5.81966	0.437432
$178$	6.70820	0.502801
$179$	−15.7984	−1.18083	−0.590413	−	0.807101i	$-0.701035\pi$
$179$	−15.7984	−1.18083	−0.590413	+	0.807101i	$0.701035\pi$
$180$	0.909830	0.0678147
$181$	−4.47214	−0.332411	−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214	−0.332411	−0.166206	+	0.986091i	$0.553152\pi$
$182$	−19.5623	−1.45005
$183$	−13.8197	−1.02158
$184$	−14.7984	−1.09095
$185$	−6.94427	−0.510553
$186$	7.52786	0.551970
$187$	5.09017	0.372230
$188$	0.291796	0.0212814
$189$	25.5279	1.85688
$190$	−8.09017	−0.586923
$191$	8.41641	0.608990	0.304495	−	0.952514i	$-0.401512\pi$
$191$	8.41641	0.608990	0.304495	+	0.952514i	$0.401512\pi$
$192$	5.23607	0.377881
$193$	−5.18034	−0.372889	−0.186445	−	0.982465i	$-0.559696\pi$
$193$	−5.18034	−0.372889	−0.186445	+	0.982465i	$0.559696\pi$
$194$	−7.32624	−0.525993
$195$	3.23607	0.231740
$196$	8.85410	0.632436
$197$	14.7082	1.04792	0.523958	−	0.851744i	$-0.324455\pi$
$197$	14.7082	1.04792	0.523958	+	0.851744i	$0.324455\pi$
$198$	−2.38197	−0.169279
$199$	−7.05573	−0.500167	−0.250084	−	0.968224i	$-0.580458\pi$
$199$	−7.05573	−0.500167	−0.250084	+	0.968224i	$0.580458\pi$
$200$	−8.94427	−0.632456
$201$	−7.05573	−0.497673
$202$	−1.38197	−0.0972348
$203$	15.6180	1.09617
$204$	−3.88854	−0.272253
$205$	2.76393	0.193041
$206$	24.7984	1.72778
$207$	9.74265	0.677161
$208$	12.7082	0.881155
$209$	5.00000	0.345857
$210$	−9.23607	−0.637349
$211$	−25.9443	−1.78608	−0.893039	−	0.449980i	$-0.851431\pi$
$211$	−25.9443	−1.78608	−0.893039	+	0.449980i	$0.851431\pi$
$212$	1.09017	0.0748732
$213$	−5.12461	−0.351133
$214$	−10.2361	−0.699723
$215$	−8.70820	−0.593895
$216$	−12.3607	−0.841038
$217$	17.3820	1.17996
$218$	5.61803	0.380501
$219$	0.111456	0.00753151
$220$	0.618034	0.0416678
$221$	13.3262	0.896420
$222$	−13.8885	−0.932138
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−15.6180	−1.04352
$225$	5.88854	0.392570
$226$	2.85410	0.189852
$227$	7.41641	0.492244	0.246122	−	0.969239i	$-0.420844\pi$
$227$	7.41641	0.492244	0.246122	+	0.969239i	$0.420844\pi$
$228$	−3.81966	−0.252963
$229$	−28.9787	−1.91497	−0.957484	−	0.288488i	$-0.906848\pi$
$229$	−28.9787	−1.91497	−0.957484	+	0.288488i	$0.906848\pi$
$230$	−10.7082	−0.706078
$231$	5.70820	0.375572
$232$	−7.56231	−0.496490
$233$	9.56231	0.626447	0.313224	−	0.949679i	$-0.398591\pi$
$233$	9.56231	0.626447	0.313224	+	0.949679i	$0.398591\pi$
$234$	−6.23607	−0.407665
$235$	−0.472136	−0.0307988
$236$	2.90983	0.189414
$237$	16.0689	1.04379
$238$	−38.0344	−2.46541
$239$	−15.1246	−0.978330	−0.489165	−	0.872191i	$-0.662698\pi$
$239$	−15.1246	−0.978330	−0.489165	+	0.872191i	$0.662698\pi$
$240$	6.00000	0.387298
$241$	27.2705	1.75665	0.878324	−	0.478066i	$-0.158662\pi$
$241$	27.2705	1.75665	0.878324	+	0.478066i	$0.158662\pi$
$242$	16.1803	1.04011
$243$	13.5967	0.872232
$244$	−6.90983	−0.442357
$245$	−14.3262	−0.915270
$246$	5.52786	0.352444
$247$	13.0902	0.832908
$248$	−8.41641	−0.534442
$249$	−20.4721	−1.29737
$250$	−14.5623	−0.921001
$251$	11.2361	0.709214	0.354607	−	0.935015i	$-0.384615\pi$
$251$	11.2361	0.709214	0.354607	+	0.935015i	$0.384615\pi$
$252$	4.20163	0.264678
$253$	6.61803	0.416072
$254$	17.4164	1.09280
$255$	6.29180	0.394008
$256$	13.5623	0.847644
$257$	−22.0344	−1.37447	−0.687235	−	0.726435i	$-0.741176\pi$
$257$	−22.0344	−1.37447	−0.687235	+	0.726435i	$0.741176\pi$
$258$	−17.4164	−1.08430
$259$	−32.0689	−1.99266
$260$	1.61803	0.100346
$261$	4.97871	0.308175
$262$	32.0344	1.97909
$263$	17.3820	1.07182	0.535909	−	0.844276i	$-0.319969\pi$
$263$	17.3820	1.07182	0.535909	+	0.844276i	$0.319969\pi$
$264$	−2.76393	−0.170108
$265$	−1.76393	−0.108357
$266$	−37.3607	−2.29073
$267$	−5.12461	−0.313621
$268$	−3.52786	−0.215499
$269$	6.23607	0.380220	0.190110	−	0.981763i	$-0.439116\pi$
$269$	6.23607	0.380220	0.190110	+	0.981763i	$0.439116\pi$
$270$	−8.94427	−0.544331
$271$	−2.67376	−0.162419	−0.0812097	−	0.996697i	$-0.525878\pi$
$271$	−2.67376	−0.162419	−0.0812097	+	0.996697i	$0.525878\pi$
$272$	24.7082	1.49815
$273$	14.9443	0.904468
$274$	15.1803	0.917078
$275$	4.00000	0.241209
$276$	−5.05573	−0.304319
$277$	−7.52786	−0.452306	−0.226153	−	0.974092i	$-0.572615\pi$
$277$	−7.52786	−0.452306	−0.226153	+	0.974092i	$0.572615\pi$
$278$	−32.7426	−1.96377
$279$	5.54102	0.331732
$280$	10.3262	0.617111
$281$	−15.7639	−0.940397	−0.470199	−	0.882561i	$-0.655818\pi$
$281$	−15.7639	−0.940397	−0.470199	+	0.882561i	$0.655818\pi$
$282$	−0.944272	−0.0562306
$283$	13.4721	0.800835	0.400418	−	0.916333i	$-0.368865\pi$
$283$	13.4721	0.800835	0.400418	+	0.916333i	$0.368865\pi$
$284$	−2.56231	−0.152045
$285$	6.18034	0.366092
$286$	−4.23607	−0.250484
$287$	12.7639	0.753431
$288$	−4.97871	−0.293374
$289$	8.90983	0.524108
$290$	−5.47214	−0.321335
$291$	5.59675	0.328087
$292$	0.0557281	0.00326124
$293$	−6.32624	−0.369583	−0.184791	−	0.982778i	$-0.559161\pi$
$293$	−6.32624	−0.369583	−0.184791	+	0.982778i	$0.559161\pi$
$294$	−28.6525	−1.67105
$295$	−4.70820	−0.274122
$296$	15.5279	0.902539
$297$	5.52786	0.320759
$298$	−13.0902	−0.758293
$299$	17.3262	1.00200
$300$	−3.05573	−0.176423
$301$	−40.2148	−2.31794
$302$	−10.8541	−0.624583
$303$	1.05573	0.0606500
$304$	24.2705	1.39201
$305$	11.1803	0.640184
$306$	−12.1246	−0.693118
$307$	13.5279	0.772076	0.386038	−	0.922483i	$-0.373843\pi$
$307$	13.5279	0.772076	0.386038	+	0.922483i	$0.373843\pi$
$308$	2.85410	0.162628
$309$	−18.9443	−1.07770
$310$	−6.09017	−0.345898
$311$	−3.09017	−0.175227	−0.0876137	−	0.996155i	$-0.527924\pi$
$311$	−3.09017	−0.175227	−0.0876137	+	0.996155i	$0.527924\pi$
$312$	−7.23607	−0.409662
$313$	1.65248	0.0934035	0.0467017	−	0.998909i	$-0.485129\pi$
$313$	1.65248	0.0934035	0.0467017	+	0.998909i	$0.485129\pi$
$314$	9.47214	0.534544
$315$	−6.79837	−0.383045
$316$	8.03444	0.451973
$317$	−23.1246	−1.29881	−0.649404	−	0.760444i	$-0.724981\pi$
$317$	−23.1246	−1.29881	−0.649404	+	0.760444i	$0.724981\pi$
$318$	−3.52786	−0.197833
$319$	3.38197	0.189354
$320$	−4.23607	−0.236803
$321$	7.81966	0.436451
$322$	−49.4508	−2.75579
$323$	25.4508	1.41612
$324$	−1.49342	−0.0829679
$325$	10.4721	0.580890
$326$	9.56231	0.529607
$327$	−4.29180	−0.237337
$328$	−6.18034	−0.341252
$329$	−2.18034	−0.120206
$330$	−2.00000	−0.110096
$331$	−29.2361	−1.60696	−0.803480	−	0.595332i	$-0.797021\pi$
$331$	−29.2361	−1.60696	−0.803480	+	0.595332i	$0.797021\pi$
$332$	−10.2361	−0.561777
$333$	−10.2229	−0.560212
$334$	3.70820	0.202904
$335$	5.70820	0.311872
$336$	27.7082	1.51161
$337$	−4.03444	−0.219770	−0.109885	−	0.993944i	$-0.535048\pi$
$337$	−4.03444	−0.219770	−0.109885	+	0.993944i	$0.535048\pi$
$338$	9.94427	0.540897
$339$	−2.18034	−0.118420
$340$	3.14590	0.170610
$341$	3.76393	0.203828
$342$	−11.9098	−0.644010
$343$	−33.8328	−1.82680
$344$	19.4721	1.04987
$345$	8.18034	0.440415
$346$	−26.6525	−1.43285
$347$	4.90983	0.263573	0.131787	−	0.991278i	$-0.457929\pi$
$347$	4.90983	0.263573	0.131787	+	0.991278i	$0.457929\pi$
$348$	−2.58359	−0.138495
$349$	29.9443	1.60288	0.801440	−	0.598075i	$-0.204068\pi$
$349$	29.9443	1.60288	0.801440	+	0.598075i	$0.204068\pi$
$350$	−29.8885	−1.59761
$351$	14.4721	0.772465
$352$	−3.38197	−0.180259
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−9.41641	−0.500476
$355$	4.14590	0.220041
$356$	−2.56231	−0.135802
$357$	29.0557	1.53779
$358$	25.5623	1.35101
$359$	−7.27051	−0.383723	−0.191861	−	0.981422i	$-0.561452\pi$
$359$	−7.27051	−0.383723	−0.191861	+	0.981422i	$0.561452\pi$
$360$	3.29180	0.173493
$361$	6.00000	0.315789
$362$	7.23607	0.380319
$363$	−12.3607	−0.648767
$364$	7.47214	0.391646
$365$	−0.0901699	−0.00471971
$366$	22.3607	1.16881
$367$	−8.03444	−0.419394	−0.209697	−	0.977766i	$-0.567248\pi$
$367$	−8.03444	−0.419394	−0.209697	+	0.977766i	$0.567248\pi$
$368$	32.1246	1.67461
$369$	4.06888	0.211818
$370$	11.2361	0.584135
$371$	−8.14590	−0.422914
$372$	−2.87539	−0.149082
$373$	−17.5066	−0.906456	−0.453228	−	0.891395i	$-0.649728\pi$
$373$	−17.5066	−0.906456	−0.453228	+	0.891395i	$0.649728\pi$
$374$	−8.23607	−0.425877
$375$	11.1246	0.574472
$376$	1.05573	0.0544450
$377$	8.85410	0.456009
$378$	−41.3050	−2.12450
$379$	−18.5967	−0.955251	−0.477625	−	0.878564i	$-0.658502\pi$
$379$	−18.5967	−0.955251	−0.477625	+	0.878564i	$0.658502\pi$
$380$	3.09017	0.158522
$381$	−13.3050	−0.681633
$382$	−13.6180	−0.696759
$383$	30.7984	1.57372	0.786862	−	0.617129i	$-0.211704\pi$
$383$	30.7984	1.57372	0.786862	+	0.617129i	$0.211704\pi$
$384$	−16.8328	−0.858996
$385$	−4.61803	−0.235357
$386$	8.38197	0.426631
$387$	−12.8197	−0.651660
$388$	2.79837	0.142066
$389$	29.0689	1.47385	0.736925	−	0.675974i	$-0.236277\pi$
$389$	29.0689	1.47385	0.736925	+	0.675974i	$0.236277\pi$
$390$	−5.23607	−0.265139
$391$	33.6869	1.70362
$392$	32.0344	1.61798
$393$	−24.4721	−1.23446
$394$	−23.7984	−1.19894
$395$	−13.0000	−0.654101
$396$	0.909830	0.0457207
$397$	24.5967	1.23448	0.617238	−	0.786777i	$-0.288252\pi$
$397$	24.5967	1.23448	0.617238	+	0.786777i	$0.288252\pi$
$398$	11.4164	0.572253
$399$	28.5410	1.42884
$400$	19.4164	0.970820
$401$	−16.6738	−0.832648	−0.416324	−	0.909216i	$-0.636682\pi$
$401$	−16.6738	−0.832648	−0.416324	+	0.909216i	$0.636682\pi$
$402$	11.4164	0.569399
$403$	9.85410	0.490868
$404$	0.527864	0.0262622
$405$	2.41641	0.120072
$406$	−25.2705	−1.25415
$407$	−6.94427	−0.344215
$408$	−14.0689	−0.696514
$409$	−38.6525	−1.91124	−0.955621	−	0.294599i	$-0.904814\pi$
$409$	−38.6525	−1.91124	−0.955621	+	0.294599i	$0.904814\pi$
$410$	−4.47214	−0.220863
$411$	−11.5967	−0.572025
$412$	−9.47214	−0.466659
$413$	−21.7426	−1.06989
$414$	−15.7639	−0.774755
$415$	16.5623	0.813012
$416$	−8.85410	−0.434108
$417$	25.0132	1.22490
$418$	−8.09017	−0.395703
$419$	−11.7639	−0.574706	−0.287353	−	0.957825i	$-0.592775\pi$
$419$	−11.7639	−0.574706	−0.287353	+	0.957825i	$0.592775\pi$
$420$	3.52786	0.172142
$421$	−36.8328	−1.79512	−0.897561	−	0.440891i	$-0.854663\pi$
$421$	−36.8328	−1.79512	−0.897561	+	0.440891i	$0.854663\pi$
$422$	41.9787	2.04349
$423$	−0.695048	−0.0337944
$424$	3.94427	0.191551
$425$	20.3607	0.987638
$426$	8.29180	0.401739
$427$	51.6312	2.49861
$428$	3.90983	0.188989
$429$	3.23607	0.156239
$430$	14.0902	0.679488
$431$	−32.1459	−1.54841	−0.774207	−	0.632933i	$-0.781851\pi$
$431$	−32.1459	−1.54841	−0.774207	+	0.632933i	$0.781851\pi$
$432$	26.8328	1.29099
$433$	−17.3262	−0.832646	−0.416323	−	0.909217i	$-0.636681\pi$
$433$	−17.3262	−0.832646	−0.416323	+	0.909217i	$0.636681\pi$
$434$	−28.1246	−1.35002
$435$	4.18034	0.200432
$436$	−2.14590	−0.102770
$437$	33.0902	1.58292
$438$	−0.180340	−0.00861697
$439$	−26.8885	−1.28332	−0.641660	−	0.766989i	$-0.721754\pi$
$439$	−26.8885	−1.28332	−0.641660	+	0.766989i	$0.721754\pi$
$440$	2.23607	0.106600
$441$	−21.0902	−1.00429
$442$	−21.5623	−1.02561
$443$	−34.1591	−1.62295	−0.811473	−	0.584390i	$-0.801334\pi$
$443$	−34.1591	−1.62295	−0.811473	+	0.584390i	$0.801334\pi$
$444$	5.30495	0.251762
$445$	4.14590	0.196534
$446$	6.47214	0.306465
$447$	10.0000	0.472984
$448$	−19.5623	−0.924232
$449$	2.65248	0.125178	0.0625890	−	0.998039i	$-0.480064\pi$
$449$	2.65248	0.125178	0.0625890	+	0.998039i	$0.480064\pi$
$450$	−9.52786	−0.449148
$451$	2.76393	0.130148
$452$	−1.09017	−0.0512773
$453$	8.29180	0.389583
$454$	−12.0000	−0.563188
$455$	−12.0902	−0.566796
$456$	−13.8197	−0.647165
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	46.8885	2.19096
$459$	28.1378	1.31336
$460$	4.09017	0.190705
$461$	33.3607	1.55376	0.776881	−	0.629648i	$-0.216801\pi$
$461$	33.3607	1.55376	0.776881	+	0.629648i	$0.216801\pi$
$462$	−9.23607	−0.429701
$463$	26.8885	1.24962	0.624808	−	0.780778i	$-0.285177\pi$
$463$	26.8885	1.24962	0.624808	+	0.780778i	$0.285177\pi$
$464$	16.4164	0.762113
$465$	4.65248	0.215753
$466$	−15.4721	−0.716733
$467$	0.875388	0.0405081	0.0202541	−	0.999795i	$-0.493552\pi$
$467$	0.875388	0.0405081	0.0202541	+	0.999795i	$0.493552\pi$
$468$	2.38197	0.110106
$469$	26.3607	1.21722
$470$	0.763932	0.0352376
$471$	−7.23607	−0.333420
$472$	10.5279	0.484584
$473$	−8.70820	−0.400404
$474$	−26.0000	−1.19422
$475$	20.0000	0.917663
$476$	14.5279	0.665884
$477$	−2.59675	−0.118897
$478$	24.4721	1.11933
$479$	32.9787	1.50684	0.753418	−	0.657542i	$-0.228404\pi$
$479$	32.9787	1.50684	0.753418	+	0.657542i	$0.228404\pi$
$480$	−4.18034	−0.190806
$481$	−18.1803	−0.828952
$482$	−44.1246	−2.00982
$483$	37.7771	1.71892
$484$	−6.18034	−0.280925
$485$	−4.52786	−0.205600
$486$	−22.0000	−0.997940
$487$	−0.111456	−0.00505056	−0.00252528	−	0.999997i	$-0.500804\pi$
$487$	−0.111456	−0.00505056	−0.00252528	+	0.999997i	$0.500804\pi$
$488$	−25.0000	−1.13170
$489$	−7.30495	−0.330341
$490$	23.1803	1.04718
$491$	−17.0000	−0.767199	−0.383600	−	0.923499i	$-0.625316\pi$
$491$	−17.0000	−0.767199	−0.383600	+	0.923499i	$0.625316\pi$
$492$	−2.11146	−0.0951918
$493$	17.2148	0.775315
$494$	−21.1803	−0.952949
$495$	−1.47214	−0.0661676
$496$	18.2705	0.820370
$497$	19.1459	0.858811
$498$	33.1246	1.48435
$499$	−31.6869	−1.41850	−0.709251	−	0.704956i	$-0.750967\pi$
$499$	−31.6869	−1.41850	−0.709251	+	0.704956i	$0.750967\pi$
$500$	5.56231	0.248754
$501$	−2.83282	−0.126561
$502$	−18.1803	−0.811428
$503$	28.2705	1.26052	0.630260	−	0.776384i	$-0.282948\pi$
$503$	28.2705	1.26052	0.630260	+	0.776384i	$0.282948\pi$
$504$	15.2016	0.677134
$505$	−0.854102	−0.0380070
$506$	−10.7082	−0.476038
$507$	−7.59675	−0.337383
$508$	−6.65248	−0.295156
$509$	38.3820	1.70125	0.850625	−	0.525772i	$-0.176224\pi$
$509$	38.3820	1.70125	0.850625	+	0.525772i	$0.176224\pi$
$510$	−10.1803	−0.450793
$511$	−0.416408	−0.0184208
$512$	5.29180	0.233867
$513$	27.6393	1.22031
$514$	35.6525	1.57256
$515$	15.3262	0.675355
$516$	6.65248	0.292859
$517$	−0.472136	−0.0207645
$518$	51.8885	2.27985
$519$	20.3607	0.893735
$520$	5.85410	0.256719
$521$	12.1459	0.532121	0.266061	−	0.963956i	$-0.414278\pi$
$521$	12.1459	0.532121	0.266061	+	0.963956i	$0.414278\pi$
$522$	−8.05573	−0.352590
$523$	−25.9443	−1.13446	−0.567232	−	0.823558i	$-0.691986\pi$
$523$	−25.9443	−1.13446	−0.567232	+	0.823558i	$0.691986\pi$
$524$	−12.2361	−0.534535
$525$	22.8328	0.996506
$526$	−28.1246	−1.22629
$527$	19.1591	0.834581
$528$	6.00000	0.261116
$529$	20.7984	0.904277
$530$	2.85410	0.123974
$531$	−6.93112	−0.300785
$532$	14.2705	0.618705
$533$	7.23607	0.313429
$534$	8.29180	0.358821
$535$	−6.32624	−0.273507
$536$	−12.7639	−0.551318
$537$	−19.5279	−0.842690
$538$	−10.0902	−0.435018
$539$	−14.3262	−0.617075
$540$	3.41641	0.147019
$541$	36.7984	1.58209	0.791043	−	0.611761i	$-0.209538\pi$
$541$	36.7984	1.58209	0.791043	+	0.611761i	$0.209538\pi$
$542$	4.32624	0.185828
$543$	−5.52786	−0.237223
$544$	−17.2148	−0.738078
$545$	3.47214	0.148730
$546$	−24.1803	−1.03482
$547$	28.2148	1.20638	0.603188	−	0.797599i	$-0.293897\pi$
$547$	28.2148	1.20638	0.603188	+	0.797599i	$0.293897\pi$
$548$	−5.79837	−0.247694
$549$	16.4590	0.702452
$550$	−6.47214	−0.275973
$551$	16.9098	0.720383
$552$	−18.2918	−0.778551
$553$	−60.0344	−2.55292
$554$	12.1803	0.517493
$555$	−8.58359	−0.364353
$556$	12.5066	0.530397
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	−8.96556	−0.379542
$559$	−22.7984	−0.964268
$560$	−22.4164	−0.947266
$561$	6.29180	0.265640
$562$	25.5066	1.07593
$563$	−18.3262	−0.772359	−0.386179	−	0.922424i	$-0.626205\pi$
$563$	−18.3262	−0.772359	−0.386179	+	0.922424i	$0.626205\pi$
$564$	0.360680	0.0151874
$565$	1.76393	0.0742092
$566$	−21.7984	−0.916254
$567$	11.1591	0.468636
$568$	−9.27051	−0.388982
$569$	8.74265	0.366511	0.183255	−	0.983065i	$-0.441336\pi$
$569$	8.74265	0.366511	0.183255	+	0.983065i	$0.441336\pi$
$570$	−10.0000	−0.418854
$571$	−14.6738	−0.614078	−0.307039	−	0.951697i	$-0.599338\pi$
$571$	−14.6738	−0.614078	−0.307039	+	0.951697i	$0.599338\pi$
$572$	1.61803	0.0676534
$573$	10.4033	0.434602
$574$	−20.6525	−0.862018
$575$	26.4721	1.10396
$576$	−6.23607	−0.259836
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	−14.4164	−0.599644
$579$	−6.40325	−0.266110
$580$	2.09017	0.0867896
$581$	76.4853	3.17314
$582$	−9.05573	−0.375372
$583$	−1.76393	−0.0730546
$584$	0.201626	0.00834335
$585$	−3.85410	−0.159348
$586$	10.2361	0.422848
$587$	−2.27051	−0.0937140	−0.0468570	−	0.998902i	$-0.514921\pi$
$587$	−2.27051	−0.0937140	−0.0468570	+	0.998902i	$0.514921\pi$
$588$	10.9443	0.451334
$589$	18.8197	0.775451
$590$	7.61803	0.313629
$591$	18.1803	0.747839
$592$	−33.7082	−1.38540
$593$	−3.18034	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$593$	−3.18034	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$594$	−8.94427	−0.366988
$595$	−23.5066	−0.963676
$596$	5.00000	0.204808
$597$	−8.72136	−0.356941
$598$	−28.0344	−1.14641
$599$	8.81966	0.360362	0.180181	−	0.983634i	$-0.442332\pi$
$599$	8.81966	0.360362	0.180181	+	0.983634i	$0.442332\pi$
$600$	−11.0557	−0.451348
$601$	−4.36068	−0.177876	−0.0889379	−	0.996037i	$-0.528347\pi$
$601$	−4.36068	−0.177876	−0.0889379	+	0.996037i	$0.528347\pi$
$602$	65.0689	2.65201
$603$	8.40325	0.342207
$604$	4.14590	0.168694
$605$	10.0000	0.406558
$606$	−1.70820	−0.0693910
$607$	−41.0689	−1.66693	−0.833467	−	0.552569i	$-0.813648\pi$
$607$	−41.0689	−1.66693	−0.833467	+	0.552569i	$0.813648\pi$
$608$	−16.9098	−0.685784
$609$	19.3050	0.782276
$610$	−18.0902	−0.732450
$611$	−1.23607	−0.0500060
$612$	4.63119	0.187205
$613$	−32.2705	−1.30339	−0.651697	−	0.758480i	$-0.725943\pi$
$613$	−32.2705	−1.30339	−0.651697	+	0.758480i	$0.725943\pi$
$614$	−21.8885	−0.883350
$615$	3.41641	0.137763
$616$	10.3262	0.416056
$617$	−4.27051	−0.171924	−0.0859621	−	0.996298i	$-0.527396\pi$
$617$	−4.27051	−0.171924	−0.0859621	+	0.996298i	$0.527396\pi$
$618$	30.6525	1.23302
$619$	−39.6180	−1.59238	−0.796192	−	0.605045i	$-0.793155\pi$
$619$	−39.6180	−1.59238	−0.796192	+	0.605045i	$0.793155\pi$
$620$	2.32624	0.0934240
$621$	36.5836	1.46805
$622$	5.00000	0.200482
$623$	19.1459	0.767064
$624$	15.7082	0.628831
$625$	11.0000	0.440000
$626$	−2.67376	−0.106865
$627$	6.18034	0.246819
$628$	−3.61803	−0.144375
$629$	−35.3475	−1.40940
$630$	11.0000	0.438250
$631$	30.9098	1.23050	0.615250	−	0.788332i	$-0.289055\pi$
$631$	30.9098	1.23050	0.615250	+	0.788332i	$0.289055\pi$
$632$	29.0689	1.15630
$633$	−32.0689	−1.27462
$634$	37.4164	1.48600
$635$	10.7639	0.427154
$636$	1.34752	0.0534328
$637$	−37.5066	−1.48606
$638$	−5.47214	−0.216644
$639$	6.10333	0.241444
$640$	13.6180	0.538300
$641$	46.5410	1.83826	0.919130	−	0.393955i	$-0.128893\pi$
$641$	46.5410	1.83826	0.919130	+	0.393955i	$0.128893\pi$
$642$	−12.6525	−0.499353
$643$	22.8197	0.899920	0.449960	−	0.893049i	$-0.351438\pi$
$643$	22.8197	0.899920	0.449960	+	0.893049i	$0.351438\pi$
$644$	18.8885	0.744313
$645$	−10.7639	−0.423829
$646$	−41.1803	−1.62022
$647$	28.6525	1.12645	0.563223	−	0.826305i	$-0.309561\pi$
$647$	28.6525	1.12645	0.563223	+	0.826305i	$0.309561\pi$
$648$	−5.40325	−0.212260
$649$	−4.70820	−0.184813
$650$	−16.9443	−0.664609
$651$	21.4853	0.842075
$652$	−3.65248	−0.143042
$653$	−7.38197	−0.288879	−0.144439	−	0.989514i	$-0.546138\pi$
$653$	−7.38197	−0.288879	−0.144439	+	0.989514i	$0.546138\pi$
$654$	6.94427	0.271543
$655$	19.7984	0.773586
$656$	13.4164	0.523823
$657$	−0.132742	−0.00517877
$658$	3.52786	0.137530
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0.763932	0.0297360
$661$	−13.4377	−0.522666	−0.261333	−	0.965249i	$-0.584162\pi$
$661$	−13.4377	−0.522666	−0.261333	+	0.965249i	$0.584162\pi$
$662$	47.3050	1.83856
$663$	16.4721	0.639725
$664$	−37.0344	−1.43722
$665$	−23.0902	−0.895398
$666$	16.5410	0.640951
$667$	22.3820	0.866633
$668$	−1.41641	−0.0548025
$669$	−4.94427	−0.191157
$670$	−9.23607	−0.356820
$671$	11.1803	0.431612
$672$	−19.3050	−0.744705
$673$	11.7984	0.454794	0.227397	−	0.973802i	$-0.426979\pi$
$673$	11.7984	0.454794	0.227397	+	0.973802i	$0.426979\pi$
$674$	6.52786	0.251444
$675$	22.1115	0.851070
$676$	−3.79837	−0.146091
$677$	−40.3607	−1.55119	−0.775593	−	0.631233i	$-0.782549\pi$
$677$	−40.3607	−1.55119	−0.775593	+	0.631233i	$0.782549\pi$
$678$	3.52786	0.135487
$679$	−20.9098	−0.802446
$680$	11.3820	0.436478
$681$	9.16718	0.351287
$682$	−6.09017	−0.233205
$683$	−6.76393	−0.258815	−0.129407	−	0.991592i	$-0.541307\pi$
$683$	−6.76393	−0.258815	−0.129407	+	0.991592i	$0.541307\pi$
$684$	4.54915	0.173941
$685$	9.38197	0.358466
$686$	54.7426	2.09008
$687$	−35.8197	−1.36661
$688$	−42.2705	−1.61155
$689$	−4.61803	−0.175933
$690$	−13.2361	−0.503888
$691$	−45.9443	−1.74780	−0.873901	−	0.486104i	$-0.838418\pi$
$691$	−45.9443	−1.74780	−0.873901	+	0.486104i	$0.838418\pi$
$692$	10.1803	0.386998
$693$	−6.79837	−0.258249
$694$	−7.94427	−0.301560
$695$	−20.2361	−0.767598
$696$	−9.34752	−0.354317
$697$	14.0689	0.532897
$698$	−48.4508	−1.83389
$699$	11.8197	0.447061
$700$	11.4164	0.431500
$701$	−34.5623	−1.30540	−0.652700	−	0.757616i	$-0.726364\pi$
$701$	−34.5623	−1.30540	−0.652700	+	0.757616i	$0.726364\pi$
$702$	−23.4164	−0.883795
$703$	−34.7214	−1.30954
$704$	−4.23607	−0.159653
$705$	−0.583592	−0.0219794
$706$	−9.70820	−0.365373
$707$	−3.94427	−0.148340
$708$	3.59675	0.135174
$709$	−40.1803	−1.50900	−0.754502	−	0.656298i	$-0.772122\pi$
$709$	−40.1803	−1.50900	−0.754502	+	0.656298i	$0.772122\pi$
$710$	−6.70820	−0.251754
$711$	−19.1378	−0.717722
$712$	−9.27051	−0.347427
$713$	24.9098	0.932880
$714$	−47.0132	−1.75942
$715$	−2.61803	−0.0979089
$716$	−9.76393	−0.364895
$717$	−18.6950	−0.698179
$718$	11.7639	0.439026
$719$	35.3262	1.31745	0.658723	−	0.752385i	$-0.271097\pi$
$719$	35.3262	1.31745	0.658723	+	0.752385i	$0.271097\pi$
$720$	−7.14590	−0.266312
$721$	70.7771	2.63588
$722$	−9.70820	−0.361302
$723$	33.7082	1.25362
$724$	−2.76393	−0.102721
$725$	13.5279	0.502412
$726$	20.0000	0.742270
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	27.0344	1.00196
$729$	24.0557	0.890953
$730$	0.145898	0.00539993
$731$	−44.3262	−1.63947
$732$	−8.54102	−0.315685
$733$	−49.7082	−1.83601	−0.918007	−	0.396564i	$-0.870203\pi$
$733$	−49.7082	−1.83601	−0.918007	+	0.396564i	$0.870203\pi$
$734$	13.0000	0.479839
$735$	−17.7082	−0.653177
$736$	−22.3820	−0.825010
$737$	5.70820	0.210264
$738$	−6.58359	−0.242345
$739$	−16.0557	−0.590620	−0.295310	−	0.955402i	$-0.595423\pi$
$739$	−16.0557	−0.590620	−0.295310	+	0.955402i	$0.595423\pi$
$740$	−4.29180	−0.157770
$741$	16.1803	0.594400
$742$	13.1803	0.483865
$743$	−40.2492	−1.47660	−0.738300	−	0.674472i	$-0.764371\pi$
$743$	−40.2492	−1.47660	−0.738300	+	0.674472i	$0.764371\pi$
$744$	−10.4033	−0.381402
$745$	−8.09017	−0.296401
$746$	28.3262	1.03710
$747$	24.3820	0.892089
$748$	3.14590	0.115025
$749$	−29.2148	−1.06748
$750$	−18.0000	−0.657267
$751$	30.6312	1.11775	0.558874	−	0.829253i	$-0.311234\pi$
$751$	30.6312	1.11775	0.558874	+	0.829253i	$0.311234\pi$
$752$	−2.29180	−0.0835732
$753$	13.8885	0.506127
$754$	−14.3262	−0.521731
$755$	−6.70820	−0.244137
$756$	15.7771	0.573807
$757$	−2.88854	−0.104986	−0.0524930	−	0.998621i	$-0.516717\pi$
$757$	−2.88854	−0.104986	−0.0524930	+	0.998621i	$0.516717\pi$
$758$	30.0902	1.09292
$759$	8.18034	0.296928
$760$	11.1803	0.405554
$761$	−5.72949	−0.207694	−0.103847	−	0.994593i	$-0.533115\pi$
$761$	−5.72949	−0.207694	−0.103847	+	0.994593i	$0.533115\pi$
$762$	21.5279	0.779872
$763$	16.0344	0.580486
$764$	5.20163	0.188188
$765$	−7.49342	−0.270925
$766$	−49.8328	−1.80053
$767$	−12.3262	−0.445075
$768$	16.7639	0.604916
$769$	15.7082	0.566452	0.283226	−	0.959053i	$-0.408595\pi$
$769$	15.7082	0.566452	0.283226	+	0.959053i	$0.408595\pi$
$770$	7.47214	0.269277
$771$	−27.2361	−0.980883
$772$	−3.20163	−0.115229
$773$	20.0344	0.720589	0.360294	−	0.932839i	$-0.382676\pi$
$773$	20.0344	0.720589	0.360294	+	0.932839i	$0.382676\pi$
$774$	20.7426	0.745579
$775$	15.0557	0.540818
$776$	10.1246	0.363452
$777$	−39.6393	−1.42205
$778$	−47.0344	−1.68627
$779$	13.8197	0.495141
$780$	2.00000	0.0716115
$781$	4.14590	0.148352
$782$	−54.5066	−1.94915
$783$	18.6950	0.668107
$784$	−69.5410	−2.48361
$785$	5.85410	0.208942
$786$	39.5967	1.41237
$787$	47.8673	1.70628	0.853142	−	0.521679i	$-0.174694\pi$
$787$	47.8673	1.70628	0.853142	+	0.521679i	$0.174694\pi$
$788$	9.09017	0.323824
$789$	21.4853	0.764897
$790$	21.0344	0.748372
$791$	8.14590	0.289635
$792$	3.29180	0.116969
$793$	29.2705	1.03943
$794$	−39.7984	−1.41239
$795$	−2.18034	−0.0773287
$796$	−4.36068	−0.154560
$797$	−28.1246	−0.996225	−0.498112	−	0.867112i	$-0.665973\pi$
$797$	−28.1246	−0.996225	−0.498112	+	0.867112i	$0.665973\pi$
$798$	−46.1803	−1.63477
$799$	−2.40325	−0.0850210
$800$	−13.5279	−0.478282
$801$	6.10333	0.215650
$802$	26.9787	0.952652
$803$	−0.0901699	−0.00318203
$804$	−4.36068	−0.153789
$805$	−30.5623	−1.07718
$806$	−15.9443	−0.561613
$807$	7.70820	0.271342
$808$	1.90983	0.0671876
$809$	26.9787	0.948521	0.474261	−	0.880385i	$-0.342715\pi$
$809$	26.9787	0.948521	0.474261	+	0.880385i	$0.342715\pi$
$810$	−3.90983	−0.137377
$811$	−5.72949	−0.201190	−0.100595	−	0.994927i	$-0.532075\pi$
$811$	−5.72949	−0.201190	−0.100595	+	0.994927i	$0.532075\pi$
$812$	9.65248	0.338735
$813$	−3.30495	−0.115910
$814$	11.2361	0.393824
$815$	5.90983	0.207012
$816$	30.5410	1.06915
$817$	−43.5410	−1.52331
$818$	62.5410	2.18670
$819$	−17.7984	−0.621926
$820$	1.70820	0.0596531
$821$	−14.5279	−0.507026	−0.253513	−	0.967332i	$-0.581586\pi$
$821$	−14.5279	−0.507026	−0.253513	+	0.967332i	$0.581586\pi$
$822$	18.7639	0.654467
$823$	−20.3475	−0.709270	−0.354635	−	0.935005i	$-0.615395\pi$
$823$	−20.3475	−0.709270	−0.354635	+	0.935005i	$0.615395\pi$
$824$	−34.2705	−1.19387
$825$	4.94427	0.172137
$826$	35.1803	1.22408
$827$	−32.7082	−1.13738	−0.568688	−	0.822553i	$-0.692549\pi$
$827$	−32.7082	−1.13738	−0.568688	+	0.822553i	$0.692549\pi$
$828$	6.02129	0.209254
$829$	−10.1459	−0.352382	−0.176191	−	0.984356i	$-0.556378\pi$
$829$	−10.1459	−0.352382	−0.176191	+	0.984356i	$0.556378\pi$
$830$	−26.7984	−0.930185
$831$	−9.30495	−0.322785
$832$	−11.0902	−0.384482
$833$	−72.9230	−2.52663
$834$	−40.4721	−1.40144
$835$	2.29180	0.0793109
$836$	3.09017	0.106876
$837$	20.8065	0.719178
$838$	19.0344	0.657534
$839$	−49.6525	−1.71419	−0.857097	−	0.515155i	$-0.827734\pi$
$839$	−49.6525	−1.71419	−0.857097	+	0.515155i	$0.827734\pi$
$840$	12.7639	0.440397
$841$	−17.5623	−0.605597
$842$	59.5967	2.05384
$843$	−19.4853	−0.671109
$844$	−16.0344	−0.551928
$845$	6.14590	0.211425
$846$	1.12461	0.0386650
$847$	46.1803	1.58678
$848$	−8.56231	−0.294031
$849$	16.6525	0.571511
$850$	−32.9443	−1.12998
$851$	−45.9574	−1.57540
$852$	−3.16718	−0.108506
$853$	−15.2918	−0.523581	−0.261791	−	0.965125i	$-0.584313\pi$
$853$	−15.2918	−0.523581	−0.261791	+	0.965125i	$0.584313\pi$
$854$	−83.5410	−2.85871
$855$	−7.36068	−0.251730
$856$	14.1459	0.483497
$857$	47.0689	1.60784	0.803921	−	0.594736i	$-0.202743\pi$
$857$	47.0689	1.60784	0.803921	+	0.594736i	$0.202743\pi$
$858$	−5.23607	−0.178756
$859$	29.8885	1.01978	0.509892	−	0.860238i	$-0.329685\pi$
$859$	29.8885	1.01978	0.509892	+	0.860238i	$0.329685\pi$
$860$	−5.38197	−0.183524
$861$	15.7771	0.537682
$862$	52.0132	1.77158
$863$	−48.1935	−1.64053	−0.820263	−	0.571987i	$-0.806173\pi$
$863$	−48.1935	−1.64053	−0.820263	+	0.571987i	$0.806173\pi$
$864$	−18.6950	−0.636018
$865$	−16.4721	−0.560069
$866$	28.0344	0.952649
$867$	11.0132	0.374026
$868$	10.7426	0.364629
$869$	−13.0000	−0.440995
$870$	−6.76393	−0.229319
$871$	14.9443	0.506367
$872$	−7.76393	−0.262920
$873$	−6.66563	−0.225597
$874$	−53.5410	−1.81105
$875$	−41.5623	−1.40506
$876$	0.0688837	0.00232736
$877$	37.0689	1.25173	0.625864	−	0.779933i	$-0.284747\pi$
$877$	37.0689	1.25173	0.625864	+	0.779933i	$0.284747\pi$
$878$	43.5066	1.46828
$879$	−7.81966	−0.263751
$880$	−4.85410	−0.163632
$881$	−27.3262	−0.920644	−0.460322	−	0.887752i	$-0.652266\pi$
$881$	−27.3262	−0.920644	−0.460322	+	0.887752i	$0.652266\pi$
$882$	34.1246	1.14904
$883$	−46.4853	−1.56435	−0.782177	−	0.623056i	$-0.785891\pi$
$883$	−46.4853	−1.56435	−0.782177	+	0.623056i	$0.785891\pi$
$884$	8.23607	0.277009
$885$	−5.81966	−0.195626
$886$	55.2705	1.85685
$887$	11.4508	0.384482	0.192241	−	0.981348i	$-0.438424\pi$
$887$	11.4508	0.384482	0.192241	+	0.981348i	$0.438424\pi$
$888$	19.1935	0.644092
$889$	49.7082	1.66716
$890$	−6.70820	−0.224860
$891$	2.41641	0.0809527
$892$	−2.47214	−0.0827732
$893$	−2.36068	−0.0789971
$894$	−16.1803	−0.541152
$895$	15.7984	0.528081
$896$	62.8885	2.10096
$897$	21.4164	0.715073
$898$	−4.29180	−0.143219
$899$	12.7295	0.424552
$900$	3.63932	0.121311
$901$	−8.97871	−0.299124
$902$	−4.47214	−0.148906
$903$	−49.7082	−1.65419
$904$	−3.94427	−0.131185
$905$	4.47214	0.148659
$906$	−13.4164	−0.445730
$907$	−24.5066	−0.813728	−0.406864	−	0.913489i	$-0.633378\pi$
$907$	−24.5066	−0.813728	−0.406864	+	0.913489i	$0.633378\pi$
$908$	4.58359	0.152112
$909$	−1.25735	−0.0417038
$910$	19.5623	0.648484
$911$	2.83282	0.0938554	0.0469277	−	0.998898i	$-0.485057\pi$
$911$	2.83282	0.0938554	0.0469277	+	0.998898i	$0.485057\pi$
$912$	30.0000	0.993399
$913$	16.5623	0.548132
$914$	−21.0344	−0.695757
$915$	13.8197	0.456864
$916$	−17.9098	−0.591757
$917$	91.4296	3.01927
$918$	−45.5279	−1.50264
$919$	−11.6180	−0.383244	−0.191622	−	0.981469i	$-0.561375\pi$
$919$	−11.6180	−0.383244	−0.191622	+	0.981469i	$0.561375\pi$
$920$	14.7984	0.487888
$921$	16.7214	0.550988
$922$	−53.9787	−1.77769
$923$	10.8541	0.357267
$924$	3.52786	0.116058
$925$	−27.7771	−0.913305
$926$	−43.5066	−1.42971
$927$	22.5623	0.741043
$928$	−11.4377	−0.375461
$929$	25.6525	0.841630	0.420815	−	0.907146i	$-0.361744\pi$
$929$	25.6525	0.841630	0.420815	+	0.907146i	$0.361744\pi$
$930$	−7.52786	−0.246848
$931$	−71.6312	−2.34762
$932$	5.90983	0.193583
$933$	−3.81966	−0.125050
$934$	−1.41641	−0.0463463
$935$	−5.09017	−0.166466
$936$	8.61803	0.281689
$937$	42.1033	1.37546	0.687728	−	0.725969i	$-0.258608\pi$
$937$	42.1033	1.37546	0.687728	+	0.725969i	$0.258608\pi$
$938$	−42.6525	−1.39265
$939$	2.04257	0.0666568
$940$	−0.291796	−0.00951734
$941$	41.2705	1.34538	0.672690	−	0.739924i	$-0.265139\pi$
$941$	41.2705	1.34538	0.672690	+	0.739924i	$0.265139\pi$
$942$	11.7082	0.381474
$943$	18.2918	0.595663
$944$	−22.8541	−0.743838
$945$	−25.5279	−0.830421
$946$	14.0902	0.458111
$947$	10.7082	0.347970	0.173985	−	0.984748i	$-0.444336\pi$
$947$	10.7082	0.347970	0.173985	+	0.984748i	$0.444336\pi$
$948$	9.93112	0.322548
$949$	−0.236068	−0.00766309
$950$	−32.3607	−1.04992
$951$	−28.5836	−0.926886
$952$	52.5623	1.70355
$953$	−9.45085	−0.306143	−0.153072	−	0.988215i	$-0.548916\pi$
$953$	−9.45085	−0.306143	−0.153072	+	0.988215i	$0.548916\pi$
$954$	4.20163	0.136033
$955$	−8.41641	−0.272349
$956$	−9.34752	−0.302321
$957$	4.18034	0.135131
$958$	−53.3607	−1.72401
$959$	43.3262	1.39908
$960$	−5.23607	−0.168993
$961$	−16.8328	−0.542994
$962$	29.4164	0.948423
$963$	−9.31308	−0.300110
$964$	16.8541	0.542834
$965$	5.18034	0.166761
$966$	−61.1246	−1.96665
$967$	−0.652476	−0.0209822	−0.0104911	−	0.999945i	$-0.503339\pi$
$967$	−0.652476	−0.0209822	−0.0104911	+	0.999945i	$0.503339\pi$
$968$	−22.3607	−0.718699
$969$	31.4590	1.01061
$970$	7.32624	0.235231
$971$	−30.7639	−0.987262	−0.493631	−	0.869672i	$-0.664331\pi$
$971$	−30.7639	−0.987262	−0.493631	+	0.869672i	$0.664331\pi$
$972$	8.40325	0.269534
$973$	−93.4508	−2.99590
$974$	0.180340	0.00577846
$975$	12.9443	0.414548
$976$	54.2705	1.73716
$977$	−6.09017	−0.194842	−0.0974209	−	0.995243i	$-0.531059\pi$
$977$	−6.09017	−0.194842	−0.0974209	+	0.995243i	$0.531059\pi$
$978$	11.8197	0.377951
$979$	4.14590	0.132503
$980$	−8.85410	−0.282834
$981$	5.11146	0.163196
$982$	27.5066	0.877770
$983$	−1.18034	−0.0376470	−0.0188235	−	0.999823i	$-0.505992\pi$
$983$	−1.18034	−0.0376470	−0.0188235	+	0.999823i	$0.505992\pi$
$984$	−7.63932	−0.243533
$985$	−14.7082	−0.468642
$986$	−27.8541	−0.887055
$987$	−2.69505	−0.0857843
$988$	8.09017	0.257383
$989$	−57.6312	−1.83257
$990$	2.38197	0.0757038
$991$	−20.4164	−0.648549	−0.324274	−	0.945963i	$-0.605120\pi$
$991$	−20.4164	−0.648549	−0.324274	+	0.945963i	$0.605120\pi$
$992$	−12.7295	−0.404162
$993$	−36.1378	−1.14680
$994$	−30.9787	−0.982585
$995$	7.05573	0.223682
$996$	−12.6525	−0.400909
$997$	−33.9787	−1.07612	−0.538058	−	0.842908i	$-0.680842\pi$
$997$	−33.9787	−1.07612	−0.538058	+	0.842908i	$0.680842\pi$
$998$	51.2705	1.62294
$999$	−38.3870	−1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.a.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.a.1.1	✓	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} -1.00000 q^{5} -2.00000 q^{6} -4.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} +1.23607 q^{3} +0.618034 q^{4} -1.00000 q^{5} -2.00000 q^{6} -4.61803 q^{7} +2.23607 q^{8} -1.47214 q^{9} +1.61803 q^{10} -1.00000 q^{11} +0.763932 q^{12} -2.61803 q^{13} +7.47214 q^{14} -1.23607 q^{15} -4.85410 q^{16} -5.09017 q^{17} +2.38197 q^{18} -5.00000 q^{19} -0.618034 q^{20} -5.70820 q^{21} +1.61803 q^{22} -6.61803 q^{23} +2.76393 q^{24} -4.00000 q^{25} +4.23607 q^{26} -5.52786 q^{27} -2.85410 q^{28} -3.38197 q^{29} +2.00000 q^{30} -3.76393 q^{31} +3.38197 q^{32} -1.23607 q^{33} +8.23607 q^{34} +4.61803 q^{35} -0.909830 q^{36} +6.94427 q^{37} +8.09017 q^{38} -3.23607 q^{39} -2.23607 q^{40} -2.76393 q^{41} +9.23607 q^{42} +8.70820 q^{43} -0.618034 q^{44} +1.47214 q^{45} +10.7082 q^{46} +0.472136 q^{47} -6.00000 q^{48} +14.3262 q^{49} +6.47214 q^{50} -6.29180 q^{51} -1.61803 q^{52} +1.76393 q^{53} +8.94427 q^{54} +1.00000 q^{55} -10.3262 q^{56} -6.18034 q^{57} +5.47214 q^{58} +4.70820 q^{59} -0.763932 q^{60} -11.1803 q^{61} +6.09017 q^{62} +6.79837 q^{63} +4.23607 q^{64} +2.61803 q^{65} +2.00000 q^{66} -5.70820 q^{67} -3.14590 q^{68} -8.18034 q^{69} -7.47214 q^{70} -4.14590 q^{71} -3.29180 q^{72} +0.0901699 q^{73} -11.2361 q^{74} -4.94427 q^{75} -3.09017 q^{76} +4.61803 q^{77} +5.23607 q^{78} +13.0000 q^{79} +4.85410 q^{80} -2.41641 q^{81} +4.47214 q^{82} -16.5623 q^{83} -3.52786 q^{84} +5.09017 q^{85} -14.0902 q^{86} -4.18034 q^{87} -2.23607 q^{88} -4.14590 q^{89} -2.38197 q^{90} +12.0902 q^{91} -4.09017 q^{92} -4.65248 q^{93} -0.763932 q^{94} +5.00000 q^{95} +4.18034 q^{96} +4.52786 q^{97} -23.1803 q^{98} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 4 q^{6} - 7 q^{7} + 6 q^{9} + q^{10} - 2 q^{11} + 6 q^{12} - 3 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + 7 q^{18} - 10 q^{19} + q^{20} + 2 q^{21} + q^{22} - 11 q^{23} + 10 q^{24} - 8 q^{25} + 4 q^{26} - 20 q^{27} + q^{28} - 9 q^{29} + 4 q^{30} - 12 q^{31} + 9 q^{32} + 2 q^{33} + 12 q^{34} + 7 q^{35} - 13 q^{36} - 4 q^{37} + 5 q^{38} - 2 q^{39} - 10 q^{41} + 14 q^{42} + 4 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} - 8 q^{47} - 12 q^{48} + 13 q^{49} + 4 q^{50} - 26 q^{51} - q^{52} + 8 q^{53} + 2 q^{55} - 5 q^{56} + 10 q^{57} + 2 q^{58} - 4 q^{59} - 6 q^{60} + q^{62} - 11 q^{63} + 4 q^{64} + 3 q^{65} + 4 q^{66} + 2 q^{67} - 13 q^{68} + 6 q^{69} - 6 q^{70} - 15 q^{71} - 20 q^{72} - 11 q^{73} - 18 q^{74} + 8 q^{75} + 5 q^{76} + 7 q^{77} + 6 q^{78} + 26 q^{79} + 3 q^{80} + 22 q^{81} - 13 q^{83} - 16 q^{84} - q^{85} - 17 q^{86} + 14 q^{87} - 15 q^{89} - 7 q^{90} + 13 q^{91} + 3 q^{92} + 22 q^{93} - 6 q^{94} + 10 q^{95} - 14 q^{96} + 18 q^{97} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 - 4 * q^6 - 7 * q^7 + 6 * q^9 + q^10 - 2 * q^11 + 6 * q^12 - 3 * q^13 + 6 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + 7 * q^18 - 10 * q^19 + q^20 + 2 * q^21 + q^22 - 11 * q^23 + 10 * q^24 - 8 * q^25 + 4 * q^26 - 20 * q^27 + q^28 - 9 * q^29 + 4 * q^30 - 12 * q^31 + 9 * q^32 + 2 * q^33 + 12 * q^34 + 7 * q^35 - 13 * q^36 - 4 * q^37 + 5 * q^38 - 2 * q^39 - 10 * q^41 + 14 * q^42 + 4 * q^43 + q^44 - 6 * q^45 + 8 * q^46 - 8 * q^47 - 12 * q^48 + 13 * q^49 + 4 * q^50 - 26 * q^51 - q^52 + 8 * q^53 + 2 * q^55 - 5 * q^56 + 10 * q^57 + 2 * q^58 - 4 * q^59 - 6 * q^60 + q^62 - 11 * q^63 + 4 * q^64 + 3 * q^65 + 4 * q^66 + 2 * q^67 - 13 * q^68 + 6 * q^69 - 6 * q^70 - 15 * q^71 - 20 * q^72 - 11 * q^73 - 18 * q^74 + 8 * q^75 + 5 * q^76 + 7 * q^77 + 6 * q^78 + 26 * q^79 + 3 * q^80 + 22 * q^81 - 13 * q^83 - 16 * q^84 - q^85 - 17 * q^86 + 14 * q^87 - 15 * q^89 - 7 * q^90 + 13 * q^91 + 3 * q^92 + 22 * q^93 - 6 * q^94 + 10 * q^95 - 14 * q^96 + 18 * q^97 - 24 * q^98 - 6 * q^99

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