LMFDB - Embedded newform 8022.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8022,2,Mod(1,8022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8022, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8022 = 2 \cdot 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8022.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0559925015$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + \cdots + 4704 $ x^15 - 53x^13 - x^12 + 1068x^11 + 45x^10 - 10139x^9 - 615x^8 + 45390x^7 + 2130x^6 - 84842x^5 + 7822x^4 + 62828x^3 - 16144x^2 - 13616x + 4704
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.90136$ of defining polynomial
Character	$\chi$	$=$	8022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.90136 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.90136 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.90136 q^{10} +4.70071 q^{11} +1.00000 q^{12} -0.944322 q^{13} -1.00000 q^{14} -2.90136 q^{15} +1.00000 q^{16} +7.13056 q^{17} -1.00000 q^{18} -2.37991 q^{19} -2.90136 q^{20} +1.00000 q^{21} -4.70071 q^{22} +1.52132 q^{23} -1.00000 q^{24} +3.41788 q^{25} +0.944322 q^{26} +1.00000 q^{27} +1.00000 q^{28} +3.78318 q^{29} +2.90136 q^{30} +5.22599 q^{31} -1.00000 q^{32} +4.70071 q^{33} -7.13056 q^{34} -2.90136 q^{35} +1.00000 q^{36} +6.79141 q^{37} +2.37991 q^{38} -0.944322 q^{39} +2.90136 q^{40} -8.73569 q^{41} -1.00000 q^{42} +1.23018 q^{43} +4.70071 q^{44} -2.90136 q^{45} -1.52132 q^{46} -3.08198 q^{47} +1.00000 q^{48} +1.00000 q^{49} -3.41788 q^{50} +7.13056 q^{51} -0.944322 q^{52} +1.58993 q^{53} -1.00000 q^{54} -13.6384 q^{55} -1.00000 q^{56} -2.37991 q^{57} -3.78318 q^{58} +1.73938 q^{59} -2.90136 q^{60} +12.4300 q^{61} -5.22599 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.73982 q^{65} -4.70071 q^{66} -3.46074 q^{67} +7.13056 q^{68} +1.52132 q^{69} +2.90136 q^{70} -10.1674 q^{71} -1.00000 q^{72} +10.0199 q^{73} -6.79141 q^{74} +3.41788 q^{75} -2.37991 q^{76} +4.70071 q^{77} +0.944322 q^{78} -4.47680 q^{79} -2.90136 q^{80} +1.00000 q^{81} +8.73569 q^{82} -15.0816 q^{83} +1.00000 q^{84} -20.6883 q^{85} -1.23018 q^{86} +3.78318 q^{87} -4.70071 q^{88} -2.70912 q^{89} +2.90136 q^{90} -0.944322 q^{91} +1.52132 q^{92} +5.22599 q^{93} +3.08198 q^{94} +6.90497 q^{95} -1.00000 q^{96} -2.93516 q^{97} -1.00000 q^{98} +4.70071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.90136	−1.29753	−0.648764	−	0.760990i	$-0.724714\pi$
$5$	−2.90136	−1.29753	−0.648764	+	0.760990i	$0.724714\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.90136	0.917490
$11$	4.70071	1.41732	0.708658	−	0.705552i	$-0.249301\pi$
$11$	4.70071	1.41732	0.708658	+	0.705552i	$0.249301\pi$
$12$	1.00000	0.288675
$13$	−0.944322	−0.261908	−0.130954	−	0.991388i	$-0.541804\pi$
$13$	−0.944322	−0.261908	−0.130954	+	0.991388i	$0.541804\pi$
$14$	−1.00000	−0.267261
$15$	−2.90136	−0.749128
$16$	1.00000	0.250000
$17$	7.13056	1.72941	0.864707	−	0.502277i	$-0.167504\pi$
$17$	7.13056	1.72941	0.864707	+	0.502277i	$0.167504\pi$
$18$	−1.00000	−0.235702
$19$	−2.37991	−0.545989	−0.272994	−	0.962016i	$-0.588014\pi$
$19$	−2.37991	−0.545989	−0.272994	+	0.962016i	$0.588014\pi$
$20$	−2.90136	−0.648764
$21$	1.00000	0.218218
$22$	−4.70071	−1.00219
$23$	1.52132	0.317217	0.158608	−	0.987342i	$-0.449299\pi$
$23$	1.52132	0.317217	0.158608	+	0.987342i	$0.449299\pi$
$24$	−1.00000	−0.204124
$25$	3.41788	0.683577
$26$	0.944322	0.185197
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	3.78318	0.702519	0.351260	−	0.936278i	$-0.385753\pi$
$29$	3.78318	0.702519	0.351260	+	0.936278i	$0.385753\pi$
$30$	2.90136	0.529713
$31$	5.22599	0.938616	0.469308	−	0.883035i	$-0.344503\pi$
$31$	5.22599	0.938616	0.469308	+	0.883035i	$0.344503\pi$
$32$	−1.00000	−0.176777
$33$	4.70071	0.818288
$34$	−7.13056	−1.22288
$35$	−2.90136	−0.490419
$36$	1.00000	0.166667
$37$	6.79141	1.11650	0.558251	−	0.829672i	$-0.311473\pi$
$37$	6.79141	1.11650	0.558251	+	0.829672i	$0.311473\pi$
$38$	2.37991	0.386072
$39$	−0.944322	−0.151213
$40$	2.90136	0.458745
$41$	−8.73569	−1.36429	−0.682143	−	0.731219i	$-0.738952\pi$
$41$	−8.73569	−1.36429	−0.682143	+	0.731219i	$0.738952\pi$
$42$	−1.00000	−0.154303
$43$	1.23018	0.187601	0.0938004	−	0.995591i	$-0.470098\pi$
$43$	1.23018	0.187601	0.0938004	+	0.995591i	$0.470098\pi$
$44$	4.70071	0.708658
$45$	−2.90136	−0.432509
$46$	−1.52132	−0.224306
$47$	−3.08198	−0.449553	−0.224776	−	0.974410i	$-0.572165\pi$
$47$	−3.08198	−0.449553	−0.224776	+	0.974410i	$0.572165\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−3.41788	−0.483362
$51$	7.13056	0.998478
$52$	−0.944322	−0.130954
$53$	1.58993	0.218394	0.109197	−	0.994020i	$-0.465172\pi$
$53$	1.58993	0.218394	0.109197	+	0.994020i	$0.465172\pi$
$54$	−1.00000	−0.136083
$55$	−13.6384	−1.83901
$56$	−1.00000	−0.133631
$57$	−2.37991	−0.315227
$58$	−3.78318	−0.496756
$59$	1.73938	0.226448	0.113224	−	0.993570i	$-0.463882\pi$
$59$	1.73938	0.226448	0.113224	+	0.993570i	$0.463882\pi$
$60$	−2.90136	−0.374564
$61$	12.4300	1.59150	0.795750	−	0.605625i	$-0.207077\pi$
$61$	12.4300	1.59150	0.795750	+	0.605625i	$0.207077\pi$
$62$	−5.22599	−0.663702
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	2.73982	0.339833
$66$	−4.70071	−0.578617
$67$	−3.46074	−0.422797	−0.211398	−	0.977400i	$-0.567802\pi$
$67$	−3.46074	−0.422797	−0.211398	+	0.977400i	$0.567802\pi$
$68$	7.13056	0.864707
$69$	1.52132	0.183145
$70$	2.90136	0.346779
$71$	−10.1674	−1.20665	−0.603323	−	0.797497i	$-0.706157\pi$
$71$	−10.1674	−1.20665	−0.603323	+	0.797497i	$0.706157\pi$
$72$	−1.00000	−0.117851
$73$	10.0199	1.17274	0.586370	−	0.810043i	$-0.300556\pi$
$73$	10.0199	1.17274	0.586370	+	0.810043i	$0.300556\pi$
$74$	−6.79141	−0.789486
$75$	3.41788	0.394663
$76$	−2.37991	−0.272994
$77$	4.70071	0.535695
$78$	0.944322	0.106923
$79$	−4.47680	−0.503679	−0.251839	−	0.967769i	$-0.581035\pi$
$79$	−4.47680	−0.503679	−0.251839	+	0.967769i	$0.581035\pi$
$80$	−2.90136	−0.324382
$81$	1.00000	0.111111
$82$	8.73569	0.964695
$83$	−15.0816	−1.65542	−0.827708	−	0.561159i	$-0.810355\pi$
$83$	−15.0816	−1.65542	−0.827708	+	0.561159i	$0.810355\pi$
$84$	1.00000	0.109109
$85$	−20.6883	−2.24396
$86$	−1.23018	−0.132654
$87$	3.78318	0.405600
$88$	−4.70071	−0.501097
$89$	−2.70912	−0.287166	−0.143583	−	0.989638i	$-0.545862\pi$
$89$	−2.70912	−0.287166	−0.143583	+	0.989638i	$0.545862\pi$
$90$	2.90136	0.305830
$91$	−0.944322	−0.0989919
$92$	1.52132	0.158608
$93$	5.22599	0.541910
$94$	3.08198	0.317882
$95$	6.90497	0.708435
$96$	−1.00000	−0.102062
$97$	−2.93516	−0.298021	−0.149010	−	0.988836i	$-0.547609\pi$
$97$	−2.93516	−0.298021	−0.149010	+	0.988836i	$0.547609\pi$
$98$	−1.00000	−0.101015
$99$	4.70071	0.472439
$100$	3.41788	0.341788
$101$	10.9623	1.09079	0.545397	−	0.838178i	$-0.316379\pi$
$101$	10.9623	1.09079	0.545397	+	0.838178i	$0.316379\pi$
$102$	−7.13056	−0.706030
$103$	16.1529	1.59159	0.795794	−	0.605567i	$-0.207054\pi$
$103$	16.1529	1.59159	0.795794	+	0.605567i	$0.207054\pi$
$104$	0.944322	0.0925984
$105$	−2.90136	−0.283144
$106$	−1.58993	−0.154428
$107$	−18.2243	−1.76181	−0.880904	−	0.473295i	$-0.843064\pi$
$107$	−18.2243	−1.76181	−0.880904	+	0.473295i	$0.843064\pi$
$108$	1.00000	0.0962250
$109$	−3.93789	−0.377181	−0.188591	−	0.982056i	$-0.560392\pi$
$109$	−3.93789	−0.377181	−0.188591	+	0.982056i	$0.560392\pi$
$110$	13.6384	1.30037
$111$	6.79141	0.644613
$112$	1.00000	0.0944911
$113$	−0.167450	−0.0157524	−0.00787620	−	0.999969i	$-0.502507\pi$
$113$	−0.167450	−0.0157524	−0.00787620	+	0.999969i	$0.502507\pi$
$114$	2.37991	0.222899
$115$	−4.41389	−0.411597
$116$	3.78318	0.351260
$117$	−0.944322	−0.0873026
$118$	−1.73938	−0.160123
$119$	7.13056	0.653657
$120$	2.90136	0.264857
$121$	11.0966	1.00878
$122$	−12.4300	−1.12536
$123$	−8.73569	−0.787670
$124$	5.22599	0.469308
$125$	4.59029	0.410568
$126$	−1.00000	−0.0890871
$127$	12.6904	1.12609	0.563044	−	0.826427i	$-0.309630\pi$
$127$	12.6904	1.12609	0.563044	+	0.826427i	$0.309630\pi$
$128$	−1.00000	−0.0883883
$129$	1.23018	0.108311
$130$	−2.73982	−0.240298
$131$	15.1245	1.32144	0.660718	−	0.750634i	$-0.270252\pi$
$131$	15.1245	1.32144	0.660718	+	0.750634i	$0.270252\pi$
$132$	4.70071	0.409144
$133$	−2.37991	−0.206364
$134$	3.46074	0.298962
$135$	−2.90136	−0.249709
$136$	−7.13056	−0.611440
$137$	−5.67567	−0.484906	−0.242453	−	0.970163i	$-0.577952\pi$
$137$	−5.67567	−0.484906	−0.242453	+	0.970163i	$0.577952\pi$
$138$	−1.52132	−0.129503
$139$	11.7399	0.995768	0.497884	−	0.867244i	$-0.334110\pi$
$139$	11.7399	0.995768	0.497884	+	0.867244i	$0.334110\pi$
$140$	−2.90136	−0.245210
$141$	−3.08198	−0.259550
$142$	10.1674	0.853228
$143$	−4.43898	−0.371206
$144$	1.00000	0.0833333
$145$	−10.9764	−0.911538
$146$	−10.0199	−0.829252
$147$	1.00000	0.0824786
$148$	6.79141	0.558251
$149$	−13.2097	−1.08218	−0.541089	−	0.840965i	$-0.681988\pi$
$149$	−13.2097	−1.08218	−0.541089	+	0.840965i	$0.681988\pi$
$150$	−3.41788	−0.279069
$151$	−15.7639	−1.28285	−0.641424	−	0.767187i	$-0.721656\pi$
$151$	−15.7639	−1.28285	−0.641424	+	0.767187i	$0.721656\pi$
$152$	2.37991	0.193036
$153$	7.13056	0.576471
$154$	−4.70071	−0.378794
$155$	−15.1625	−1.21788
$156$	−0.944322	−0.0756063
$157$	9.05763	0.722878	0.361439	−	0.932396i	$-0.382286\pi$
$157$	9.05763	0.722878	0.361439	+	0.932396i	$0.382286\pi$
$158$	4.47680	0.356155
$159$	1.58993	0.126090
$160$	2.90136	0.229373
$161$	1.52132	0.119897
$162$	−1.00000	−0.0785674
$163$	−20.9680	−1.64234	−0.821172	−	0.570681i	$-0.806679\pi$
$163$	−20.9680	−1.64234	−0.821172	+	0.570681i	$0.806679\pi$
$164$	−8.73569	−0.682143
$165$	−13.6384	−1.06175
$166$	15.0816	1.17056
$167$	−1.05712	−0.0818024	−0.0409012	−	0.999163i	$-0.513023\pi$
$167$	−1.05712	−0.0818024	−0.0409012	+	0.999163i	$0.513023\pi$
$168$	−1.00000	−0.0771517
$169$	−12.1083	−0.931404
$170$	20.6883	1.58672
$171$	−2.37991	−0.181996
$172$	1.23018	0.0938004
$173$	−14.4696	−1.10010	−0.550051	−	0.835131i	$-0.685392\pi$
$173$	−14.4696	−1.10010	−0.550051	+	0.835131i	$0.685392\pi$
$174$	−3.78318	−0.286802
$175$	3.41788	0.258368
$176$	4.70071	0.354329
$177$	1.73938	0.130740
$178$	2.70912	0.203057
$179$	−9.34868	−0.698753	−0.349376	−	0.936982i	$-0.613607\pi$
$179$	−9.34868	−0.698753	−0.349376	+	0.936982i	$0.613607\pi$
$180$	−2.90136	−0.216255
$181$	19.1516	1.42353	0.711764	−	0.702419i	$-0.247896\pi$
$181$	19.1516	1.42353	0.711764	+	0.702419i	$0.247896\pi$
$182$	0.944322	0.0699978
$183$	12.4300	0.918853
$184$	−1.52132	−0.112153
$185$	−19.7043	−1.44869
$186$	−5.22599	−0.383188
$187$	33.5186	2.45113
$188$	−3.08198	−0.224776
$189$	1.00000	0.0727393
$190$	−6.90497	−0.500939
$191$	1.00000	0.0723575
$191$	1.00000	0.0723575
$192$	1.00000	0.0721688
$193$	17.1086	1.23150	0.615751	−	0.787941i	$-0.288853\pi$
$193$	17.1086	1.23150	0.615751	+	0.787941i	$0.288853\pi$
$194$	2.93516	0.210732
$195$	2.73982	0.196202
$196$	1.00000	0.0714286
$197$	12.0808	0.860719	0.430359	−	0.902658i	$-0.358387\pi$
$197$	12.0808	0.860719	0.430359	+	0.902658i	$0.358387\pi$
$198$	−4.70071	−0.334065
$199$	6.91546	0.490224	0.245112	−	0.969495i	$-0.421175\pi$
$199$	6.91546	0.490224	0.245112	+	0.969495i	$0.421175\pi$
$200$	−3.41788	−0.241681
$201$	−3.46074	−0.244102
$202$	−10.9623	−0.771307
$203$	3.78318	0.265527
$204$	7.13056	0.499239
$205$	25.3454	1.77020
$206$	−16.1529	−1.12542
$207$	1.52132	0.105739
$208$	−0.944322	−0.0654770
$209$	−11.1873	−0.773839
$210$	2.90136	0.200213
$211$	15.5349	1.06947	0.534733	−	0.845021i	$-0.320412\pi$
$211$	15.5349	1.06947	0.534733	+	0.845021i	$0.320412\pi$
$212$	1.58993	0.109197
$213$	−10.1674	−0.696658
$214$	18.2243	1.24579
$215$	−3.56920	−0.243417
$216$	−1.00000	−0.0680414
$217$	5.22599	0.354764
$218$	3.93789	0.266708
$219$	10.0199	0.677082
$220$	−13.6384	−0.919503
$221$	−6.73354	−0.452947
$222$	−6.79141	−0.455810
$223$	−5.47520	−0.366646	−0.183323	−	0.983053i	$-0.558685\pi$
$223$	−5.47520	−0.366646	−0.183323	+	0.983053i	$0.558685\pi$
$224$	−1.00000	−0.0668153
$225$	3.41788	0.227859
$226$	0.167450	0.0111386
$227$	12.7369	0.845380	0.422690	−	0.906274i	$-0.361086\pi$
$227$	12.7369	0.845380	0.422690	+	0.906274i	$0.361086\pi$
$228$	−2.37991	−0.157613
$229$	−8.87822	−0.586689	−0.293345	−	0.956007i	$-0.594768\pi$
$229$	−8.87822	−0.586689	−0.293345	+	0.956007i	$0.594768\pi$
$230$	4.41389	0.291043
$231$	4.70071	0.309284
$232$	−3.78318	−0.248378
$233$	−14.6354	−0.958796	−0.479398	−	0.877598i	$-0.659145\pi$
$233$	−14.6354	−0.958796	−0.479398	+	0.877598i	$0.659145\pi$
$234$	0.944322	0.0617323
$235$	8.94193	0.583307
$236$	1.73938	0.113224
$237$	−4.47680	−0.290799
$238$	−7.13056	−0.462205
$239$	−20.8714	−1.35006	−0.675029	−	0.737791i	$-0.735869\pi$
$239$	−20.8714	−1.35006	−0.675029	+	0.737791i	$0.735869\pi$
$240$	−2.90136	−0.187282
$241$	−10.8268	−0.697413	−0.348706	−	0.937232i	$-0.613379\pi$
$241$	−10.8268	−0.697413	−0.348706	+	0.937232i	$0.613379\pi$
$242$	−11.0966	−0.713319
$243$	1.00000	0.0641500
$244$	12.4300	0.795750
$245$	−2.90136	−0.185361
$246$	8.73569	0.556967
$247$	2.24740	0.142999
$248$	−5.22599	−0.331851
$249$	−15.0816	−0.955755
$250$	−4.59029	−0.290315
$251$	2.19278	0.138407	0.0692036	−	0.997603i	$-0.477954\pi$
$251$	2.19278	0.138407	0.0692036	+	0.997603i	$0.477954\pi$
$252$	1.00000	0.0629941
$253$	7.15126	0.449596
$254$	−12.6904	−0.796265
$255$	−20.6883	−1.29555
$256$	1.00000	0.0625000
$257$	13.3233	0.831083	0.415542	−	0.909574i	$-0.363592\pi$
$257$	13.3233	0.831083	0.415542	+	0.909574i	$0.363592\pi$
$258$	−1.23018	−0.0765877
$259$	6.79141	0.421998
$260$	2.73982	0.169916
$261$	3.78318	0.234173
$262$	−15.1245	−0.934397
$263$	−0.151771	−0.00935858	−0.00467929	−	0.999989i	$-0.501489\pi$
$263$	−0.151771	−0.00935858	−0.00467929	+	0.999989i	$0.501489\pi$
$264$	−4.70071	−0.289308
$265$	−4.61297	−0.283372
$266$	2.37991	0.145922
$267$	−2.70912	−0.165795
$268$	−3.46074	−0.211398
$269$	−13.7133	−0.836116	−0.418058	−	0.908420i	$-0.637289\pi$
$269$	−13.7133	−0.836116	−0.418058	+	0.908420i	$0.637289\pi$
$270$	2.90136	0.176571
$271$	13.9161	0.845343	0.422671	−	0.906283i	$-0.361092\pi$
$271$	13.9161	0.845343	0.422671	+	0.906283i	$0.361092\pi$
$272$	7.13056	0.432354
$273$	−0.944322	−0.0571530
$274$	5.67567	0.342880
$275$	16.0665	0.968845
$276$	1.52132	0.0915725
$277$	3.18992	0.191664	0.0958318	−	0.995398i	$-0.469449\pi$
$277$	3.18992	0.191664	0.0958318	+	0.995398i	$0.469449\pi$
$278$	−11.7399	−0.704115
$279$	5.22599	0.312872
$280$	2.90136	0.173389
$281$	2.80853	0.167543	0.0837715	−	0.996485i	$-0.473303\pi$
$281$	2.80853	0.167543	0.0837715	+	0.996485i	$0.473303\pi$
$282$	3.08198	0.183529
$283$	0.517631	0.0307700	0.0153850	−	0.999882i	$-0.495103\pi$
$283$	0.517631	0.0307700	0.0153850	+	0.999882i	$0.495103\pi$
$284$	−10.1674	−0.603323
$285$	6.90497	0.409015
$286$	4.43898	0.262482
$287$	−8.73569	−0.515651
$288$	−1.00000	−0.0589256
$289$	33.8448	1.99087
$290$	10.9764	0.644555
$291$	−2.93516	−0.172062
$292$	10.0199	0.586370
$293$	4.27164	0.249552	0.124776	−	0.992185i	$-0.460179\pi$
$293$	4.27164	0.249552	0.124776	+	0.992185i	$0.460179\pi$
$294$	−1.00000	−0.0583212
$295$	−5.04656	−0.293822
$296$	−6.79141	−0.394743
$297$	4.70071	0.272763
$298$	13.2097	0.765215
$299$	−1.43661	−0.0830815
$300$	3.41788	0.197332
$301$	1.23018	0.0709065
$302$	15.7639	0.907110
$303$	10.9623	0.629770
$304$	−2.37991	−0.136497
$305$	−36.0639	−2.06501
$306$	−7.13056	−0.407627
$307$	−21.5583	−1.23040	−0.615200	−	0.788371i	$-0.710925\pi$
$307$	−21.5583	−1.23040	−0.615200	+	0.788371i	$0.710925\pi$
$308$	4.70071	0.267848
$309$	16.1529	0.918904
$310$	15.1625	0.861171
$311$	30.0780	1.70557	0.852785	−	0.522263i	$-0.174912\pi$
$311$	30.0780	1.70557	0.852785	+	0.522263i	$0.174912\pi$
$312$	0.944322	0.0534617
$313$	22.8729	1.29285	0.646427	−	0.762976i	$-0.276262\pi$
$313$	22.8729	1.29285	0.646427	+	0.762976i	$0.276262\pi$
$314$	−9.05763	−0.511152
$315$	−2.90136	−0.163473
$316$	−4.47680	−0.251839
$317$	−9.24977	−0.519519	−0.259759	−	0.965673i	$-0.583643\pi$
$317$	−9.24977	−0.519519	−0.259759	+	0.965673i	$0.583643\pi$
$318$	−1.58993	−0.0891590
$319$	17.7836	0.995692
$320$	−2.90136	−0.162191
$321$	−18.2243	−1.01718
$322$	−1.52132	−0.0847797
$323$	−16.9701	−0.944241
$324$	1.00000	0.0555556
$325$	−3.22758	−0.179034
$326$	20.9680	1.16131
$327$	−3.93789	−0.217766
$328$	8.73569	0.482348
$329$	−3.08198	−0.169915
$330$	13.6384	0.750771
$331$	29.9235	1.64475	0.822373	−	0.568949i	$-0.192650\pi$
$331$	29.9235	1.64475	0.822373	+	0.568949i	$0.192650\pi$
$332$	−15.0816	−0.827708
$333$	6.79141	0.372167
$334$	1.05712	0.0578430
$335$	10.0409	0.548590
$336$	1.00000	0.0545545
$337$	12.5558	0.683958	0.341979	−	0.939708i	$-0.388903\pi$
$337$	12.5558	0.683958	0.341979	+	0.939708i	$0.388903\pi$
$338$	12.1083	0.658602
$339$	−0.167450	−0.00909465
$340$	−20.6883	−1.12198
$341$	24.5659	1.33032
$342$	2.37991	0.128691
$343$	1.00000	0.0539949
$344$	−1.23018	−0.0663269
$345$	−4.41389	−0.237636
$346$	14.4696	0.777889
$347$	14.2284	0.763819	0.381910	−	0.924200i	$-0.375267\pi$
$347$	14.2284	0.763819	0.381910	+	0.924200i	$0.375267\pi$
$348$	3.78318	0.202800
$349$	27.0754	1.44931	0.724657	−	0.689109i	$-0.241998\pi$
$349$	27.0754	1.44931	0.724657	+	0.689109i	$0.241998\pi$
$350$	−3.41788	−0.182694
$351$	−0.944322	−0.0504042
$352$	−4.70071	−0.250548
$353$	6.39811	0.340537	0.170268	−	0.985398i	$-0.445537\pi$
$353$	6.39811	0.340537	0.170268	+	0.985398i	$0.445537\pi$
$354$	−1.73938	−0.0924469
$355$	29.4992	1.56566
$356$	−2.70912	−0.143583
$357$	7.13056	0.377389
$358$	9.34868	0.494093
$359$	23.1559	1.22212	0.611061	−	0.791584i	$-0.290743\pi$
$359$	23.1559	1.22212	0.611061	+	0.791584i	$0.290743\pi$
$360$	2.90136	0.152915
$361$	−13.3360	−0.701896
$362$	−19.1516	−1.00659
$363$	11.0966	0.582422
$364$	−0.944322	−0.0494959
$365$	−29.0713	−1.52166
$366$	−12.4300	−0.649727
$367$	−14.4992	−0.756852	−0.378426	−	0.925632i	$-0.623535\pi$
$367$	−14.4992	−0.756852	−0.378426	+	0.925632i	$0.623535\pi$
$368$	1.52132	0.0793041
$369$	−8.73569	−0.454762
$370$	19.7043	1.02438
$371$	1.58993	0.0825452
$372$	5.22599	0.270955
$373$	12.7635	0.660870	0.330435	−	0.943829i	$-0.392804\pi$
$373$	12.7635	0.660870	0.330435	+	0.943829i	$0.392804\pi$
$374$	−33.5186	−1.73321
$375$	4.59029	0.237041
$376$	3.08198	0.158941
$377$	−3.57254	−0.183995
$378$	−1.00000	−0.0514344
$379$	6.09821	0.313244	0.156622	−	0.987659i	$-0.449940\pi$
$379$	6.09821	0.313244	0.156622	+	0.987659i	$0.449940\pi$
$380$	6.90497	0.354218
$381$	12.6904	0.650147
$382$	−1.00000	−0.0511645
$383$	16.2849	0.832119	0.416060	−	0.909337i	$-0.363411\pi$
$383$	16.2849	0.832119	0.416060	+	0.909337i	$0.363411\pi$
$384$	−1.00000	−0.0510310
$385$	−13.6384	−0.695079
$386$	−17.1086	−0.870804
$387$	1.23018	0.0625336
$388$	−2.93516	−0.149010
$389$	−0.606424	−0.0307469	−0.0153735	−	0.999882i	$-0.504894\pi$
$389$	−0.606424	−0.0307469	−0.0153735	+	0.999882i	$0.504894\pi$
$390$	−2.73982	−0.138736
$391$	10.8478	0.548599
$392$	−1.00000	−0.0505076
$393$	15.1245	0.762932
$394$	−12.0808	−0.608620
$395$	12.9888	0.653537
$396$	4.70071	0.236219
$397$	15.0713	0.756407	0.378204	−	0.925722i	$-0.376542\pi$
$397$	15.0713	0.756407	0.378204	+	0.925722i	$0.376542\pi$
$398$	−6.91546	−0.346641
$399$	−2.37991	−0.119145
$400$	3.41788	0.170894
$401$	19.6022	0.978887	0.489444	−	0.872035i	$-0.337200\pi$
$401$	19.6022	0.978887	0.489444	+	0.872035i	$0.337200\pi$
$402$	3.46074	0.172606
$403$	−4.93502	−0.245831
$404$	10.9623	0.545397
$405$	−2.90136	−0.144170
$406$	−3.78318	−0.187756
$407$	31.9244	1.58244
$408$	−7.13056	−0.353015
$409$	26.6165	1.31610	0.658052	−	0.752973i	$-0.271381\pi$
$409$	26.6165	1.31610	0.658052	+	0.752973i	$0.271381\pi$
$410$	−25.3454	−1.25172
$411$	−5.67567	−0.279960
$412$	16.1529	0.795794
$413$	1.73938	0.0855892
$414$	−1.52132	−0.0747687
$415$	43.7570	2.14795
$416$	0.944322	0.0462992
$417$	11.7399	0.574907
$418$	11.1873	0.547187
$419$	−5.73349	−0.280099	−0.140050	−	0.990144i	$-0.544726\pi$
$419$	−5.73349	−0.280099	−0.140050	+	0.990144i	$0.544726\pi$
$420$	−2.90136	−0.141572
$421$	−4.30363	−0.209746	−0.104873	−	0.994486i	$-0.533444\pi$
$421$	−4.30363	−0.209746	−0.104873	+	0.994486i	$0.533444\pi$
$422$	−15.5349	−0.756227
$423$	−3.08198	−0.149851
$424$	−1.58993	−0.0772140
$425$	24.3714	1.18219
$426$	10.1674	0.492612
$427$	12.4300	0.601530
$428$	−18.2243	−0.880904
$429$	−4.43898	−0.214316
$430$	3.56920	0.172122
$431$	15.6670	0.754653	0.377327	−	0.926080i	$-0.376843\pi$
$431$	15.6670	0.754653	0.377327	+	0.926080i	$0.376843\pi$
$432$	1.00000	0.0481125
$433$	31.0182	1.49064	0.745320	−	0.666707i	$-0.232297\pi$
$433$	31.0182	1.49064	0.745320	+	0.666707i	$0.232297\pi$
$434$	−5.22599	−0.250856
$435$	−10.9764	−0.526277
$436$	−3.93789	−0.188591
$437$	−3.62060	−0.173197
$438$	−10.0199	−0.478769
$439$	2.44466	0.116677	0.0583385	−	0.998297i	$-0.481420\pi$
$439$	2.44466	0.116677	0.0583385	+	0.998297i	$0.481420\pi$
$440$	13.6384	0.650187
$441$	1.00000	0.0476190
$442$	6.73354	0.320282
$443$	−2.91170	−0.138339	−0.0691696	−	0.997605i	$-0.522035\pi$
$443$	−2.91170	−0.138339	−0.0691696	+	0.997605i	$0.522035\pi$
$444$	6.79141	0.322306
$445$	7.86013	0.372606
$446$	5.47520	0.259258
$447$	−13.2097	−0.624796
$448$	1.00000	0.0472456
$449$	−13.4539	−0.634929	−0.317465	−	0.948270i	$-0.602831\pi$
$449$	−13.4539	−0.634929	−0.317465	+	0.948270i	$0.602831\pi$
$450$	−3.41788	−0.161121
$451$	−41.0639	−1.93362
$452$	−0.167450	−0.00787620
$453$	−15.7639	−0.740653
$454$	−12.7369	−0.597774
$455$	2.73982	0.128445
$456$	2.37991	0.111450
$457$	−6.66194	−0.311632	−0.155816	−	0.987786i	$-0.549801\pi$
$457$	−6.66194	−0.311632	−0.155816	+	0.987786i	$0.549801\pi$
$458$	8.87822	0.414852
$459$	7.13056	0.332826
$460$	−4.41389	−0.205799
$461$	−13.1021	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$461$	−13.1021	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$462$	−4.70071	−0.218697
$463$	9.20717	0.427894	0.213947	−	0.976845i	$-0.431368\pi$
$463$	9.20717	0.427894	0.213947	+	0.976845i	$0.431368\pi$
$464$	3.78318	0.175630
$465$	−15.1625	−0.703143
$466$	14.6354	0.677971
$467$	−27.4038	−1.26810	−0.634048	−	0.773293i	$-0.718608\pi$
$467$	−27.4038	−1.26810	−0.634048	+	0.773293i	$0.718608\pi$
$468$	−0.944322	−0.0436513
$469$	−3.46074	−0.159802
$470$	−8.94193	−0.412460
$471$	9.05763	0.417354
$472$	−1.73938	−0.0800613
$473$	5.78272	0.265890
$474$	4.47680	0.205626
$475$	−8.13426	−0.373225
$476$	7.13056	0.326829
$477$	1.58993	0.0727980
$478$	20.8714	0.954635
$479$	27.0405	1.23551	0.617757	−	0.786369i	$-0.288042\pi$
$479$	27.0405	1.23551	0.617757	+	0.786369i	$0.288042\pi$
$480$	2.90136	0.132428
$481$	−6.41328	−0.292421
$482$	10.8268	0.493145
$483$	1.52132	0.0692223
$484$	11.0966	0.504392
$485$	8.51596	0.386690
$486$	−1.00000	−0.0453609
$487$	−18.1540	−0.822638	−0.411319	−	0.911492i	$-0.634932\pi$
$487$	−18.1540	−0.822638	−0.411319	+	0.911492i	$0.634932\pi$
$488$	−12.4300	−0.562680
$489$	−20.9680	−0.948207
$490$	2.90136	0.131070
$491$	17.4511	0.787557	0.393778	−	0.919205i	$-0.371168\pi$
$491$	17.4511	0.787557	0.393778	+	0.919205i	$0.371168\pi$
$492$	−8.73569	−0.393835
$493$	26.9762	1.21495
$494$	−2.24740	−0.101115
$495$	−13.6384	−0.613002
$496$	5.22599	0.234654
$497$	−10.1674	−0.456070
$498$	15.0816	0.675821
$499$	−27.4262	−1.22776	−0.613882	−	0.789398i	$-0.710393\pi$
$499$	−27.4262	−1.22776	−0.613882	+	0.789398i	$0.710393\pi$
$500$	4.59029	0.205284
$501$	−1.05712	−0.0472286
$502$	−2.19278	−0.0978686
$503$	−17.1471	−0.764552	−0.382276	−	0.924048i	$-0.624860\pi$
$503$	−17.1471	−0.764552	−0.382276	+	0.924048i	$0.624860\pi$
$504$	−1.00000	−0.0445435
$505$	−31.8057	−1.41533
$506$	−7.15126	−0.317912
$507$	−12.1083	−0.537747
$508$	12.6904	0.563044
$509$	40.8197	1.80930	0.904650	−	0.426156i	$-0.140133\pi$
$509$	40.8197	1.80930	0.904650	+	0.426156i	$0.140133\pi$
$510$	20.6883	0.916094
$511$	10.0199	0.443254
$512$	−1.00000	−0.0441942
$513$	−2.37991	−0.105076
$514$	−13.3233	−0.587665
$515$	−46.8652	−2.06513
$516$	1.23018	0.0541557
$517$	−14.4875	−0.637159
$518$	−6.79141	−0.298398
$519$	−14.4696	−0.635144
$520$	−2.73982	−0.120149
$521$	−18.8831	−0.827282	−0.413641	−	0.910440i	$-0.635743\pi$
$521$	−18.8831	−0.827282	−0.413641	+	0.910440i	$0.635743\pi$
$522$	−3.78318	−0.165585
$523$	10.3024	0.450491	0.225245	−	0.974302i	$-0.427682\pi$
$523$	10.3024	0.450491	0.225245	+	0.974302i	$0.427682\pi$
$524$	15.1245	0.660718
$525$	3.41788	0.149169
$526$	0.151771	0.00661752
$527$	37.2642	1.62326
$528$	4.70071	0.204572
$529$	−20.6856	−0.899374
$530$	4.61297	0.200374
$531$	1.73938	0.0754826
$532$	−2.37991	−0.103182
$533$	8.24930	0.357317
$534$	2.70912	0.117235
$535$	52.8752	2.28599
$536$	3.46074	0.149481
$537$	−9.34868	−0.403425
$538$	13.7133	0.591224
$539$	4.70071	0.202474
$540$	−2.90136	−0.124855
$541$	−19.9748	−0.858783	−0.429392	−	0.903118i	$-0.641272\pi$
$541$	−19.9748	−0.858783	−0.429392	+	0.903118i	$0.641272\pi$
$542$	−13.9161	−0.597748
$543$	19.1516	0.821874
$544$	−7.13056	−0.305720
$545$	11.4252	0.489403
$546$	0.944322	0.0404133
$547$	4.36422	0.186601	0.0933004	−	0.995638i	$-0.470258\pi$
$547$	4.36422	0.186601	0.0933004	+	0.995638i	$0.470258\pi$
$548$	−5.67567	−0.242453
$549$	12.4300	0.530500
$550$	−16.0665	−0.685077
$551$	−9.00364	−0.383568
$552$	−1.52132	−0.0647516
$553$	−4.47680	−0.190373
$554$	−3.18992	−0.135527
$555$	−19.7043	−0.836402
$556$	11.7399	0.497884
$557$	−28.2424	−1.19667	−0.598334	−	0.801247i	$-0.704170\pi$
$557$	−28.2424	−1.19667	−0.598334	+	0.801247i	$0.704170\pi$
$558$	−5.22599	−0.221234
$559$	−1.16169	−0.0491341
$560$	−2.90136	−0.122605
$561$	33.5186	1.41516
$562$	−2.80853	−0.118471
$563$	14.2977	0.602577	0.301288	−	0.953533i	$-0.402583\pi$
$563$	14.2977	0.602577	0.301288	+	0.953533i	$0.402583\pi$
$564$	−3.08198	−0.129775
$565$	0.485833	0.0204392
$566$	−0.517631	−0.0217577
$567$	1.00000	0.0419961
$568$	10.1674	0.426614
$569$	−31.0345	−1.30103	−0.650517	−	0.759492i	$-0.725448\pi$
$569$	−31.0345	−1.30103	−0.650517	+	0.759492i	$0.725448\pi$
$570$	−6.90497	−0.289218
$571$	−27.8577	−1.16581	−0.582904	−	0.812541i	$-0.698083\pi$
$571$	−27.8577	−1.16581	−0.582904	+	0.812541i	$0.698083\pi$
$572$	−4.43898	−0.185603
$573$	1.00000	0.0417756
$574$	8.73569	0.364621
$575$	5.19969	0.216842
$576$	1.00000	0.0416667
$577$	−11.1829	−0.465550	−0.232775	−	0.972531i	$-0.574781\pi$
$577$	−11.1829	−0.465550	−0.232775	+	0.972531i	$0.574781\pi$
$578$	−33.8448	−1.40776
$579$	17.1086	0.711008
$580$	−10.9764	−0.455769
$581$	−15.0816	−0.625688
$582$	2.93516	0.121666
$583$	7.47381	0.309533
$584$	−10.0199	−0.414626
$585$	2.73982	0.113278
$586$	−4.27164	−0.176460
$587$	10.3187	0.425899	0.212950	−	0.977063i	$-0.431693\pi$
$587$	10.3187	0.425899	0.212950	+	0.977063i	$0.431693\pi$
$588$	1.00000	0.0412393
$589$	−12.4374	−0.512474
$590$	5.04656	0.207764
$591$	12.0808	0.496936
$592$	6.79141	0.279125
$593$	−19.1464	−0.786249	−0.393125	−	0.919485i	$-0.628606\pi$
$593$	−19.1464	−0.786249	−0.393125	+	0.919485i	$0.628606\pi$
$594$	−4.70071	−0.192872
$595$	−20.6883	−0.848138
$596$	−13.2097	−0.541089
$597$	6.91546	0.283031
$598$	1.43661	0.0587475
$599$	38.5006	1.57309	0.786546	−	0.617532i	$-0.211867\pi$
$599$	38.5006	1.57309	0.786546	+	0.617532i	$0.211867\pi$
$600$	−3.41788	−0.139535
$601$	−9.15082	−0.373270	−0.186635	−	0.982429i	$-0.559758\pi$
$601$	−9.15082	−0.373270	−0.186635	+	0.982429i	$0.559758\pi$
$602$	−1.23018	−0.0501384
$603$	−3.46074	−0.140932
$604$	−15.7639	−0.641424
$605$	−32.1953	−1.30893
$606$	−10.9623	−0.445314
$607$	−13.2677	−0.538519	−0.269259	−	0.963068i	$-0.586779\pi$
$607$	−13.2677	−0.538519	−0.269259	+	0.963068i	$0.586779\pi$
$608$	2.37991	0.0965181
$609$	3.78318	0.153302
$610$	36.0639	1.46019
$611$	2.91038	0.117741
$612$	7.13056	0.288236
$613$	45.4400	1.83530	0.917652	−	0.397385i	$-0.130082\pi$
$613$	45.4400	1.83530	0.917652	+	0.397385i	$0.130082\pi$
$614$	21.5583	0.870024
$615$	25.3454	1.02202
$616$	−4.70071	−0.189397
$617$	23.6574	0.952412	0.476206	−	0.879334i	$-0.342012\pi$
$617$	23.6574	0.952412	0.476206	+	0.879334i	$0.342012\pi$
$618$	−16.1529	−0.649763
$619$	32.1163	1.29086	0.645432	−	0.763817i	$-0.276677\pi$
$619$	32.1163	1.29086	0.645432	+	0.763817i	$0.276677\pi$
$620$	−15.1625	−0.608940
$621$	1.52132	0.0610484
$622$	−30.0780	−1.20602
$623$	−2.70912	−0.108539
$624$	−0.944322	−0.0378031
$625$	−30.4075	−1.21630
$626$	−22.8729	−0.914186
$627$	−11.1873	−0.446776
$628$	9.05763	0.361439
$629$	48.4266	1.93089
$630$	2.90136	0.115593
$631$	−9.57249	−0.381075	−0.190537	−	0.981680i	$-0.561023\pi$
$631$	−9.57249	−0.381075	−0.190537	+	0.981680i	$0.561023\pi$
$632$	4.47680	0.178077
$633$	15.5349	0.617457
$634$	9.24977	0.367355
$635$	−36.8193	−1.46113
$636$	1.58993	0.0630449
$637$	−0.944322	−0.0374154
$638$	−17.7836	−0.704061
$639$	−10.1674	−0.402216
$640$	2.90136	0.114686
$641$	19.0229	0.751359	0.375679	−	0.926750i	$-0.377409\pi$
$641$	19.0229	0.751359	0.375679	+	0.926750i	$0.377409\pi$
$642$	18.2243	0.719255
$643$	5.48873	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$643$	5.48873	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$644$	1.52132	0.0599483
$645$	−3.56920	−0.140537
$646$	16.9701	0.667679
$647$	6.81282	0.267840	0.133920	−	0.990992i	$-0.457244\pi$
$647$	6.81282	0.267840	0.133920	+	0.990992i	$0.457244\pi$
$648$	−1.00000	−0.0392837
$649$	8.17630	0.320948
$650$	3.22758	0.126596
$651$	5.22599	0.204823
$652$	−20.9680	−0.821172
$653$	−8.33131	−0.326029	−0.163015	−	0.986624i	$-0.552122\pi$
$653$	−8.33131	−0.326029	−0.163015	+	0.986624i	$0.552122\pi$
$654$	3.93789	0.153984
$655$	−43.8817	−1.71460
$656$	−8.73569	−0.341071
$657$	10.0199	0.390913
$658$	3.08198	0.120148
$659$	7.87831	0.306895	0.153448	−	0.988157i	$-0.450962\pi$
$659$	7.87831	0.306895	0.153448	+	0.988157i	$0.450962\pi$
$660$	−13.6384	−0.530875
$661$	−6.71482	−0.261176	−0.130588	−	0.991437i	$-0.541687\pi$
$661$	−6.71482	−0.261176	−0.130588	+	0.991437i	$0.541687\pi$
$662$	−29.9235	−1.16301
$663$	−6.73354	−0.261509
$664$	15.0816	0.585278
$665$	6.90497	0.267763
$666$	−6.79141	−0.263162
$667$	5.75542	0.222851
$668$	−1.05712	−0.0409012
$669$	−5.47520	−0.211683
$670$	−10.0409	−0.387912
$671$	58.4298	2.25566
$672$	−1.00000	−0.0385758
$673$	−35.2728	−1.35966	−0.679832	−	0.733368i	$-0.737947\pi$
$673$	−35.2728	−1.35966	−0.679832	+	0.733368i	$0.737947\pi$
$674$	−12.5558	−0.483631
$675$	3.41788	0.131554
$676$	−12.1083	−0.465702
$677$	27.0851	1.04096	0.520482	−	0.853872i	$-0.325752\pi$
$677$	27.0851	1.04096	0.520482	+	0.853872i	$0.325752\pi$
$678$	0.167450	0.00643089
$679$	−2.93516	−0.112641
$680$	20.6883	0.793360
$681$	12.7369	0.488080
$682$	−24.5659	−0.940675
$683$	20.1347	0.770432	0.385216	−	0.922826i	$-0.374127\pi$
$683$	20.1347	0.770432	0.385216	+	0.922826i	$0.374127\pi$
$684$	−2.37991	−0.0909981
$685$	16.4672	0.629178
$686$	−1.00000	−0.0381802
$687$	−8.87822	−0.338725
$688$	1.23018	0.0469002
$689$	−1.50141	−0.0571991
$690$	4.41389	0.168034
$691$	−37.2105	−1.41555	−0.707777	−	0.706436i	$-0.750302\pi$
$691$	−37.2105	−1.41555	−0.707777	+	0.706436i	$0.750302\pi$
$692$	−14.4696	−0.550051
$693$	4.70071	0.178565
$694$	−14.2284	−0.540102
$695$	−34.0618	−1.29204
$696$	−3.78318	−0.143401
$697$	−62.2903	−2.35941
$698$	−27.0754	−1.02482
$699$	−14.6354	−0.553561
$700$	3.41788	0.129184
$701$	−0.991213	−0.0374376	−0.0187188	−	0.999825i	$-0.505959\pi$
$701$	−0.991213	−0.0374376	−0.0187188	+	0.999825i	$0.505959\pi$
$702$	0.944322	0.0356411
$703$	−16.1630	−0.609597
$704$	4.70071	0.177165
$705$	8.94193	0.336773
$706$	−6.39811	−0.240796
$707$	10.9623	0.412281
$708$	1.73938	0.0653698
$709$	−22.6768	−0.851646	−0.425823	−	0.904806i	$-0.640015\pi$
$709$	−22.6768	−0.851646	−0.425823	+	0.904806i	$0.640015\pi$
$710$	−29.4992	−1.10709
$711$	−4.47680	−0.167893
$712$	2.70912	0.101529
$713$	7.95039	0.297745
$714$	−7.13056	−0.266854
$715$	12.8791	0.481650
$716$	−9.34868	−0.349376
$717$	−20.8714	−0.779456
$718$	−23.1559	−0.864170
$719$	−50.0463	−1.86641	−0.933207	−	0.359340i	$-0.883002\pi$
$719$	−50.0463	−1.86641	−0.933207	+	0.359340i	$0.883002\pi$
$720$	−2.90136	−0.108127
$721$	16.1529	0.601564
$722$	13.3360	0.496316
$723$	−10.8268	−0.402651
$724$	19.1516	0.711764
$725$	12.9305	0.480226
$726$	−11.0966	−0.411835
$727$	−22.8690	−0.848166	−0.424083	−	0.905623i	$-0.639403\pi$
$727$	−22.8690	−0.848166	−0.424083	+	0.905623i	$0.639403\pi$
$728$	0.944322	0.0349989
$729$	1.00000	0.0370370
$730$	29.0713	1.07598
$731$	8.77188	0.324440
$732$	12.4300	0.459426
$733$	−1.04846	−0.0387258	−0.0193629	−	0.999813i	$-0.506164\pi$
$733$	−1.04846	−0.0387258	−0.0193629	+	0.999813i	$0.506164\pi$
$734$	14.4992	0.535175
$735$	−2.90136	−0.107018
$736$	−1.52132	−0.0560765
$737$	−16.2679	−0.599237
$738$	8.73569	0.321565
$739$	−36.5253	−1.34360	−0.671802	−	0.740731i	$-0.734479\pi$
$739$	−36.5253	−1.34360	−0.671802	+	0.740731i	$0.734479\pi$
$740$	−19.7043	−0.724346
$741$	2.24740	0.0825604
$742$	−1.58993	−0.0583683
$743$	34.8744	1.27942	0.639710	−	0.768617i	$-0.279055\pi$
$743$	34.8744	1.27942	0.639710	+	0.768617i	$0.279055\pi$
$744$	−5.22599	−0.191594
$745$	38.3260	1.40416
$746$	−12.7635	−0.467306
$747$	−15.0816	−0.551805
$748$	33.5186	1.22556
$749$	−18.2243	−0.665901
$750$	−4.59029	−0.167614
$751$	36.6668	1.33799	0.668996	−	0.743266i	$-0.266724\pi$
$751$	36.6668	1.33799	0.668996	+	0.743266i	$0.266724\pi$
$752$	−3.08198	−0.112388
$753$	2.19278	0.0799094
$754$	3.57254	0.130104
$755$	45.7367	1.66453
$756$	1.00000	0.0363696
$757$	2.67133	0.0970910	0.0485455	−	0.998821i	$-0.484541\pi$
$757$	2.67133	0.0970910	0.0485455	+	0.998821i	$0.484541\pi$
$758$	−6.09821	−0.221497
$759$	7.15126	0.259574
$760$	−6.90497	−0.250470
$761$	17.9909	0.652171	0.326086	−	0.945340i	$-0.394270\pi$
$761$	17.9909	0.652171	0.326086	+	0.945340i	$0.394270\pi$
$762$	−12.6904	−0.459724
$763$	−3.93789	−0.142561
$764$	1.00000	0.0361787
$765$	−20.6883	−0.747987
$766$	−16.2849	−0.588397
$767$	−1.64253	−0.0593084
$768$	1.00000	0.0360844
$769$	−39.6480	−1.42974	−0.714872	−	0.699256i	$-0.753515\pi$
$769$	−39.6480	−1.42974	−0.714872	+	0.699256i	$0.753515\pi$
$770$	13.6384	0.491495
$771$	13.3233	0.479826
$772$	17.1086	0.615751
$773$	34.9948	1.25868	0.629338	−	0.777132i	$-0.283326\pi$
$773$	34.9948	1.25868	0.629338	+	0.777132i	$0.283326\pi$
$774$	−1.23018	−0.0442179
$775$	17.8618	0.641616
$776$	2.93516	0.105366
$777$	6.79141	0.243641
$778$	0.606424	0.0217414
$779$	20.7902	0.744885
$780$	2.73982	0.0981012
$781$	−47.7939	−1.71020
$782$	−10.8478	−0.387918
$783$	3.78318	0.135200
$784$	1.00000	0.0357143
$785$	−26.2794	−0.937954
$786$	−15.1245	−0.539474
$787$	53.9587	1.92342	0.961710	−	0.274070i	$-0.0883701\pi$
$787$	53.9587	1.92342	0.961710	+	0.274070i	$0.0883701\pi$
$788$	12.0808	0.430359
$789$	−0.151771	−0.00540318
$790$	−12.9888	−0.462121
$791$	−0.167450	−0.00595384
$792$	−4.70071	−0.167032
$793$	−11.7379	−0.416826
$794$	−15.0713	−0.534861
$795$	−4.61297	−0.163605
$796$	6.91546	0.245112
$797$	0.507485	0.0179760	0.00898802	−	0.999960i	$-0.497139\pi$
$797$	0.507485	0.0179760	0.00898802	+	0.999960i	$0.497139\pi$
$798$	2.37991	0.0842479
$799$	−21.9762	−0.777463
$800$	−3.41788	−0.120840
$801$	−2.70912	−0.0957220
$802$	−19.6022	−0.692178
$803$	47.1006	1.66214
$804$	−3.46074	−0.122051
$805$	−4.41389	−0.155569
$806$	4.93502	0.173829
$807$	−13.7133	−0.482732
$808$	−10.9623	−0.385654
$809$	−33.0725	−1.16277	−0.581384	−	0.813629i	$-0.697488\pi$
$809$	−33.0725	−1.16277	−0.581384	+	0.813629i	$0.697488\pi$
$810$	2.90136	0.101943
$811$	33.2631	1.16803	0.584013	−	0.811744i	$-0.301482\pi$
$811$	33.2631	1.16803	0.584013	+	0.811744i	$0.301482\pi$
$812$	3.78318	0.132764
$813$	13.9161	0.488059
$814$	−31.9244	−1.11895
$815$	60.8358	2.13098
$816$	7.13056	0.249619
$817$	−2.92772	−0.102428
$818$	−26.6165	−0.930626
$819$	−0.944322	−0.0329973
$820$	25.3454	0.885099
$821$	17.9103	0.625075	0.312538	−	0.949905i	$-0.398821\pi$
$821$	17.9103	0.625075	0.312538	+	0.949905i	$0.398821\pi$
$822$	5.67567	0.197962
$823$	13.3533	0.465468	0.232734	−	0.972540i	$-0.425233\pi$
$823$	13.3533	0.465468	0.232734	+	0.972540i	$0.425233\pi$
$824$	−16.1529	−0.562712
$825$	16.0665	0.559363
$826$	−1.73938	−0.0605207
$827$	24.8359	0.863628	0.431814	−	0.901963i	$-0.357874\pi$
$827$	24.8359	0.863628	0.431814	+	0.901963i	$0.357874\pi$
$828$	1.52132	0.0528694
$829$	10.9243	0.379418	0.189709	−	0.981840i	$-0.439246\pi$
$829$	10.9243	0.379418	0.189709	+	0.981840i	$0.439246\pi$
$830$	−43.7570	−1.51883
$831$	3.18992	0.110657
$832$	−0.944322	−0.0327385
$833$	7.13056	0.247059
$834$	−11.7399	−0.406521
$835$	3.06708	0.106141
$836$	−11.1873	−0.386919
$837$	5.22599	0.180637
$838$	5.73349	0.198060
$839$	−18.6522	−0.643947	−0.321973	−	0.946749i	$-0.604346\pi$
$839$	−18.6522	−0.643947	−0.321973	+	0.946749i	$0.604346\pi$
$840$	2.90136	0.100106
$841$	−14.6875	−0.506466
$842$	4.30363	0.148313
$843$	2.80853	0.0967310
$844$	15.5349	0.534733
$845$	35.1304	1.20852
$846$	3.08198	0.105961
$847$	11.0966	0.381285
$848$	1.58993	0.0545985
$849$	0.517631	0.0177651
$850$	−24.3714	−0.835933
$851$	10.3319	0.354173
$852$	−10.1674	−0.348329
$853$	−19.3369	−0.662084	−0.331042	−	0.943616i	$-0.607400\pi$
$853$	−19.3369	−0.662084	−0.331042	+	0.943616i	$0.607400\pi$
$854$	−12.4300	−0.425346
$855$	6.90497	0.236145
$856$	18.2243	0.622893
$857$	−17.1714	−0.586564	−0.293282	−	0.956026i	$-0.594747\pi$
$857$	−17.1714	−0.586564	−0.293282	+	0.956026i	$0.594747\pi$
$858$	4.43898	0.151544
$859$	2.09765	0.0715708	0.0357854	−	0.999359i	$-0.488607\pi$
$859$	2.09765	0.0715708	0.0357854	+	0.999359i	$0.488607\pi$
$860$	−3.56920	−0.121709
$861$	−8.73569	−0.297711
$862$	−15.6670	−0.533620
$863$	−10.5952	−0.360663	−0.180332	−	0.983606i	$-0.557717\pi$
$863$	−10.5952	−0.360663	−0.180332	+	0.983606i	$0.557717\pi$
$864$	−1.00000	−0.0340207
$865$	41.9814	1.42741
$866$	−31.0182	−1.05404
$867$	33.8448	1.14943
$868$	5.22599	0.177382
$869$	−21.0441	−0.713872
$870$	10.9764	0.372134
$871$	3.26805	0.110734
$872$	3.93789	0.133354
$873$	−2.93516	−0.0993402
$874$	3.62060	0.122469
$875$	4.59029	0.155180
$876$	10.0199	0.338541
$877$	44.6044	1.50618	0.753092	−	0.657915i	$-0.228561\pi$
$877$	44.6044	1.50618	0.753092	+	0.657915i	$0.228561\pi$
$878$	−2.44466	−0.0825031
$879$	4.27164	0.144079
$880$	−13.6384	−0.459752
$881$	5.33519	0.179747	0.0898735	−	0.995953i	$-0.471354\pi$
$881$	5.33519	0.179747	0.0898735	+	0.995953i	$0.471354\pi$
$882$	−1.00000	−0.0336718
$883$	44.1261	1.48496	0.742481	−	0.669867i	$-0.233649\pi$
$883$	44.1261	1.48496	0.742481	+	0.669867i	$0.233649\pi$
$884$	−6.73354	−0.226474
$885$	−5.04656	−0.169638
$886$	2.91170	0.0978206
$887$	15.8883	0.533477	0.266739	−	0.963769i	$-0.414054\pi$
$887$	15.8883	0.533477	0.266739	+	0.963769i	$0.414054\pi$
$888$	−6.79141	−0.227905
$889$	12.6904	0.425621
$890$	−7.86013	−0.263472
$891$	4.70071	0.157480
$892$	−5.47520	−0.183323
$893$	7.33483	0.245451
$894$	13.2097	0.441797
$895$	27.1239	0.906651
$896$	−1.00000	−0.0334077
$897$	−1.43661	−0.0479671
$898$	13.4539	0.448963
$899$	19.7709	0.659396
$900$	3.41788	0.113929
$901$	11.3371	0.377694
$902$	41.0639	1.36728
$903$	1.23018	0.0409379
$904$	0.167450	0.00556931
$905$	−55.5657	−1.84707
$906$	15.7639	0.523720
$907$	1.14487	0.0380146	0.0190073	−	0.999819i	$-0.493949\pi$
$907$	1.14487	0.0380146	0.0190073	+	0.999819i	$0.493949\pi$
$908$	12.7369	0.422690
$909$	10.9623	0.363598
$910$	−2.73982	−0.0908241
$911$	−5.45399	−0.180699	−0.0903493	−	0.995910i	$-0.528798\pi$
$911$	−5.45399	−0.180699	−0.0903493	+	0.995910i	$0.528798\pi$
$912$	−2.37991	−0.0788067
$913$	−70.8939	−2.34625
$914$	6.66194	0.220357
$915$	−36.0639	−1.19224
$916$	−8.87822	−0.293345
$917$	15.1245	0.499456
$918$	−7.13056	−0.235343
$919$	24.1592	0.796940	0.398470	−	0.917181i	$-0.369541\pi$
$919$	24.1592	0.796940	0.398470	+	0.917181i	$0.369541\pi$
$920$	4.41389	0.145522
$921$	−21.5583	−0.710371
$922$	13.1021	0.431495
$923$	9.60129	0.316030
$924$	4.70071	0.154642
$925$	23.2123	0.763215
$926$	−9.20717	−0.302566
$927$	16.1529	0.530530
$928$	−3.78318	−0.124189
$929$	21.8164	0.715774	0.357887	−	0.933765i	$-0.383497\pi$
$929$	21.8164	0.715774	0.357887	+	0.933765i	$0.383497\pi$
$930$	15.1625	0.497197
$931$	−2.37991	−0.0779984
$932$	−14.6354	−0.479398
$933$	30.0780	0.984711
$934$	27.4038	0.896680
$935$	−97.2496	−3.18040
$936$	0.944322	0.0308661
$937$	−31.1862	−1.01881	−0.509405	−	0.860527i	$-0.670134\pi$
$937$	−31.1862	−1.01881	−0.509405	+	0.860527i	$0.670134\pi$
$938$	3.46074	0.112997
$939$	22.8729	0.746430
$940$	8.94193	0.291654
$941$	−4.20370	−0.137037	−0.0685183	−	0.997650i	$-0.521827\pi$
$941$	−4.20370	−0.137037	−0.0685183	+	0.997650i	$0.521827\pi$
$942$	−9.05763	−0.295114
$943$	−13.2898	−0.432774
$944$	1.73938	0.0566119
$945$	−2.90136	−0.0943812
$946$	−5.78272	−0.188012
$947$	35.6219	1.15756	0.578779	−	0.815485i	$-0.303529\pi$
$947$	35.6219	1.15756	0.578779	+	0.815485i	$0.303529\pi$
$948$	−4.47680	−0.145400
$949$	−9.46201	−0.307150
$950$	8.13426	0.263910
$951$	−9.24977	−0.299944
$952$	−7.13056	−0.231103
$953$	−47.8316	−1.54942	−0.774709	−	0.632318i	$-0.782103\pi$
$953$	−47.8316	−1.54942	−0.774709	+	0.632318i	$0.782103\pi$
$954$	−1.58993	−0.0514760
$955$	−2.90136	−0.0938858
$956$	−20.8714	−0.675029
$957$	17.7836	0.574863
$958$	−27.0405	−0.873640
$959$	−5.67567	−0.183277
$960$	−2.90136	−0.0936410
$961$	−3.68899	−0.119000
$962$	6.41328	0.206773
$963$	−18.2243	−0.587269
$964$	−10.8268	−0.348706
$965$	−49.6381	−1.59791
$966$	−1.52132	−0.0489476
$967$	−16.7404	−0.538336	−0.269168	−	0.963093i	$-0.586749\pi$
$967$	−16.7404	−0.538336	−0.269168	+	0.963093i	$0.586749\pi$
$968$	−11.0966	−0.356659
$969$	−16.9701	−0.545158
$970$	−8.51596	−0.273431
$971$	−48.7899	−1.56574	−0.782872	−	0.622183i	$-0.786246\pi$
$971$	−48.7899	−1.56574	−0.782872	+	0.622183i	$0.786246\pi$
$972$	1.00000	0.0320750
$973$	11.7399	0.376365
$974$	18.1540	0.581693
$975$	−3.22758	−0.103365
$976$	12.4300	0.397875
$977$	29.9765	0.959034	0.479517	−	0.877533i	$-0.340812\pi$
$977$	29.9765	0.959034	0.479517	+	0.877533i	$0.340812\pi$
$978$	20.9680	0.670484
$979$	−12.7348	−0.407005
$980$	−2.90136	−0.0926805
$981$	−3.93789	−0.125727
$982$	−17.4511	−0.556887
$983$	39.3578	1.25532	0.627659	−	0.778489i	$-0.284013\pi$
$983$	39.3578	1.25532	0.627659	+	0.778489i	$0.284013\pi$
$984$	8.73569	0.278484
$985$	−35.0506	−1.11681
$986$	−26.9762	−0.859097
$987$	−3.08198	−0.0981005
$988$	2.24740	0.0714994
$989$	1.87150	0.0595101
$990$	13.6384	0.433458
$991$	29.7409	0.944750	0.472375	−	0.881398i	$-0.343397\pi$
$991$	29.7409	0.944750	0.472375	+	0.881398i	$0.343397\pi$
$992$	−5.22599	−0.165925
$993$	29.9235	0.949594
$994$	10.1674	0.322490
$995$	−20.0642	−0.636079
$996$	−15.0816	−0.477877
$997$	11.3081	0.358130	0.179065	−	0.983837i	$-0.442693\pi$
$997$	11.3081	0.358130	0.179065	+	0.983837i	$0.442693\pi$
$998$	27.4262	0.868160
$999$	6.79141	0.214871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8022.2.a.z.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8022.2.a.z.1.4	✓	15	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:	\( N \)	\(=\)	\( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.90136 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.90136 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.90136 q^{10} +4.70071 q^{11} +1.00000 q^{12} -0.944322 q^{13} -1.00000 q^{14} -2.90136 q^{15} +1.00000 q^{16} +7.13056 q^{17} -1.00000 q^{18} -2.37991 q^{19} -2.90136 q^{20} +1.00000 q^{21} -4.70071 q^{22} +1.52132 q^{23} -1.00000 q^{24} +3.41788 q^{25} +0.944322 q^{26} +1.00000 q^{27} +1.00000 q^{28} +3.78318 q^{29} +2.90136 q^{30} +5.22599 q^{31} -1.00000 q^{32} +4.70071 q^{33} -7.13056 q^{34} -2.90136 q^{35} +1.00000 q^{36} +6.79141 q^{37} +2.37991 q^{38} -0.944322 q^{39} +2.90136 q^{40} -8.73569 q^{41} -1.00000 q^{42} +1.23018 q^{43} +4.70071 q^{44} -2.90136 q^{45} -1.52132 q^{46} -3.08198 q^{47} +1.00000 q^{48} +1.00000 q^{49} -3.41788 q^{50} +7.13056 q^{51} -0.944322 q^{52} +1.58993 q^{53} -1.00000 q^{54} -13.6384 q^{55} -1.00000 q^{56} -2.37991 q^{57} -3.78318 q^{58} +1.73938 q^{59} -2.90136 q^{60} +12.4300 q^{61} -5.22599 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.73982 q^{65} -4.70071 q^{66} -3.46074 q^{67} +7.13056 q^{68} +1.52132 q^{69} +2.90136 q^{70} -10.1674 q^{71} -1.00000 q^{72} +10.0199 q^{73} -6.79141 q^{74} +3.41788 q^{75} -2.37991 q^{76} +4.70071 q^{77} +0.944322 q^{78} -4.47680 q^{79} -2.90136 q^{80} +1.00000 q^{81} +8.73569 q^{82} -15.0816 q^{83} +1.00000 q^{84} -20.6883 q^{85} -1.23018 q^{86} +3.78318 q^{87} -4.70071 q^{88} -2.70912 q^{89} +2.90136 q^{90} -0.944322 q^{91} +1.52132 q^{92} +5.22599 q^{93} +3.08198 q^{94} +6.90497 q^{95} -1.00000 q^{96} -2.93516 q^{97} -1.00000 q^{98} +4.70071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * q^99

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