LMFDB - Embedded newform 8049.2.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$104$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10025 q^{2} -1.00000 q^{3} +2.41107 q^{4} +2.84518 q^{5} +2.10025 q^{6} +2.53466 q^{7} -0.863342 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.10025 q^{2} -1.00000 q^{3} +2.41107 q^{4} +2.84518 q^{5} +2.10025 q^{6} +2.53466 q^{7} -0.863342 q^{8} +1.00000 q^{9} -5.97560 q^{10} -5.48488 q^{11} -2.41107 q^{12} +5.03357 q^{13} -5.32343 q^{14} -2.84518 q^{15} -3.00889 q^{16} -3.89360 q^{17} -2.10025 q^{18} -6.18791 q^{19} +6.85992 q^{20} -2.53466 q^{21} +11.5196 q^{22} -7.09350 q^{23} +0.863342 q^{24} +3.09506 q^{25} -10.5718 q^{26} -1.00000 q^{27} +6.11123 q^{28} -4.76892 q^{29} +5.97560 q^{30} +7.68396 q^{31} +8.04612 q^{32} +5.48488 q^{33} +8.17755 q^{34} +7.21157 q^{35} +2.41107 q^{36} +0.825683 q^{37} +12.9962 q^{38} -5.03357 q^{39} -2.45636 q^{40} +3.84243 q^{41} +5.32343 q^{42} +4.63842 q^{43} -13.2244 q^{44} +2.84518 q^{45} +14.8982 q^{46} -3.27197 q^{47} +3.00889 q^{48} -0.575491 q^{49} -6.50041 q^{50} +3.89360 q^{51} +12.1363 q^{52} +1.24022 q^{53} +2.10025 q^{54} -15.6055 q^{55} -2.18828 q^{56} +6.18791 q^{57} +10.0159 q^{58} +4.07736 q^{59} -6.85992 q^{60} +13.4034 q^{61} -16.1383 q^{62} +2.53466 q^{63} -10.8811 q^{64} +14.3214 q^{65} -11.5196 q^{66} -8.59718 q^{67} -9.38773 q^{68} +7.09350 q^{69} -15.1461 q^{70} +14.6122 q^{71} -0.863342 q^{72} +12.7949 q^{73} -1.73414 q^{74} -3.09506 q^{75} -14.9195 q^{76} -13.9023 q^{77} +10.5718 q^{78} -7.32304 q^{79} -8.56085 q^{80} +1.00000 q^{81} -8.07008 q^{82} -4.40997 q^{83} -6.11123 q^{84} -11.0780 q^{85} -9.74186 q^{86} +4.76892 q^{87} +4.73532 q^{88} -9.48428 q^{89} -5.97560 q^{90} +12.7584 q^{91} -17.1029 q^{92} -7.68396 q^{93} +6.87197 q^{94} -17.6057 q^{95} -8.04612 q^{96} -8.48940 q^{97} +1.20868 q^{98} -5.48488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) $ 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.10025	−1.48510	−0.742552	−	0.669789i	$-0.766385\pi$
$2$	−2.10025	−1.48510	−0.742552	+	0.669789i	$0.766385\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.41107	1.20553
$5$	2.84518	1.27240	0.636202	−	0.771523i	$-0.280505\pi$
$5$	2.84518	1.27240	0.636202	+	0.771523i	$0.280505\pi$
$6$	2.10025	0.857425
$7$	2.53466	0.958012	0.479006	−	0.877812i	$-0.340997\pi$
$7$	2.53466	0.958012	0.479006	+	0.877812i	$0.340997\pi$
$8$	−0.863342	−0.305237
$9$	1.00000	0.333333
$10$	−5.97560	−1.88965
$11$	−5.48488	−1.65375	−0.826876	−	0.562384i	$-0.809884\pi$
$11$	−5.48488	−1.65375	−0.826876	+	0.562384i	$0.809884\pi$
$12$	−2.41107	−0.696015
$13$	5.03357	1.39606	0.698030	−	0.716068i	$-0.254060\pi$
$13$	5.03357	1.39606	0.698030	+	0.716068i	$0.254060\pi$
$14$	−5.32343	−1.42275
$15$	−2.84518	−0.734623
$16$	−3.00889	−0.752224
$17$	−3.89360	−0.944337	−0.472169	−	0.881508i	$-0.656529\pi$
$17$	−3.89360	−0.944337	−0.472169	+	0.881508i	$0.656529\pi$
$18$	−2.10025	−0.495035
$19$	−6.18791	−1.41960	−0.709802	−	0.704401i	$-0.751216\pi$
$19$	−6.18791	−1.41960	−0.709802	+	0.704401i	$0.751216\pi$
$20$	6.85992	1.53392
$21$	−2.53466	−0.553108
$22$	11.5196	2.45599
$23$	−7.09350	−1.47910	−0.739549	−	0.673103i	$-0.764961\pi$
$23$	−7.09350	−1.47910	−0.739549	+	0.673103i	$0.764961\pi$
$24$	0.863342	0.176229
$25$	3.09506	0.619012
$26$	−10.5718	−2.07330
$27$	−1.00000	−0.192450
$28$	6.11123	1.15491
$29$	−4.76892	−0.885567	−0.442783	−	0.896629i	$-0.646009\pi$
$29$	−4.76892	−0.885567	−0.442783	+	0.896629i	$0.646009\pi$
$30$	5.97560	1.09099
$31$	7.68396	1.38008	0.690040	−	0.723772i	$-0.257593\pi$
$31$	7.68396	1.38008	0.690040	+	0.723772i	$0.257593\pi$
$32$	8.04612	1.42237
$33$	5.48488	0.954794
$34$	8.17755	1.40244
$35$	7.21157	1.21898
$36$	2.41107	0.401844
$37$	0.825683	0.135741	0.0678707	−	0.997694i	$-0.478379\pi$
$37$	0.825683	0.135741	0.0678707	+	0.997694i	$0.478379\pi$
$38$	12.9962	2.10826
$39$	−5.03357	−0.806016
$40$	−2.45636	−0.388385
$41$	3.84243	0.600087	0.300044	−	0.953925i	$-0.402999\pi$
$41$	3.84243	0.600087	0.300044	+	0.953925i	$0.402999\pi$
$42$	5.32343	0.821423
$43$	4.63842	0.707353	0.353676	−	0.935368i	$-0.384931\pi$
$43$	4.63842	0.707353	0.353676	+	0.935368i	$0.384931\pi$
$44$	−13.2244	−1.99365
$45$	2.84518	0.424135
$46$	14.8982	2.19661
$47$	−3.27197	−0.477266	−0.238633	−	0.971110i	$-0.576699\pi$
$47$	−3.27197	−0.477266	−0.238633	+	0.971110i	$0.576699\pi$
$48$	3.00889	0.434296
$49$	−0.575491	−0.0822130
$50$	−6.50041	−0.919296
$51$	3.89360	0.545213
$52$	12.1363	1.68300
$53$	1.24022	0.170357	0.0851784	−	0.996366i	$-0.472854\pi$
$53$	1.24022	0.170357	0.0851784	+	0.996366i	$0.472854\pi$
$54$	2.10025	0.285808
$55$	−15.6055	−2.10424
$56$	−2.18828	−0.292421
$57$	6.18791	0.819609
$58$	10.0159	1.31516
$59$	4.07736	0.530827	0.265414	−	0.964135i	$-0.414492\pi$
$59$	4.07736	0.530827	0.265414	+	0.964135i	$0.414492\pi$
$60$	−6.85992	−0.885612
$61$	13.4034	1.71613	0.858065	−	0.513541i	$-0.171667\pi$
$61$	13.4034	1.71613	0.858065	+	0.513541i	$0.171667\pi$
$62$	−16.1383	−2.04956
$63$	2.53466	0.319337
$64$	−10.8811	−1.36014
$65$	14.3214	1.77635
$66$	−11.5196	−1.41797
$67$	−8.59718	−1.05031	−0.525156	−	0.851006i	$-0.675993\pi$
$67$	−8.59718	−1.05031	−0.525156	+	0.851006i	$0.675993\pi$
$68$	−9.38773	−1.13843
$69$	7.09350	0.853957
$70$	−15.1461	−1.81031
$71$	14.6122	1.73415	0.867073	−	0.498181i	$-0.165999\pi$
$71$	14.6122	1.73415	0.867073	+	0.498181i	$0.165999\pi$
$72$	−0.863342	−0.101746
$73$	12.7949	1.49753	0.748763	−	0.662837i	$-0.230648\pi$
$73$	12.7949	1.49753	0.748763	+	0.662837i	$0.230648\pi$
$74$	−1.73414	−0.201590
$75$	−3.09506	−0.357387
$76$	−14.9195	−1.71138
$77$	−13.9023	−1.58431
$78$	10.5718	1.19702
$79$	−7.32304	−0.823906	−0.411953	−	0.911205i	$-0.635153\pi$
$79$	−7.32304	−0.823906	−0.411953	+	0.911205i	$0.635153\pi$
$80$	−8.56085	−0.957132
$81$	1.00000	0.111111
$82$	−8.07008	−0.891192
$83$	−4.40997	−0.484057	−0.242028	−	0.970269i	$-0.577813\pi$
$83$	−4.40997	−0.484057	−0.242028	+	0.970269i	$0.577813\pi$
$84$	−6.11123	−0.666790
$85$	−11.0780	−1.20158
$86$	−9.74186	−1.05049
$87$	4.76892	0.511282
$88$	4.73532	0.504787
$89$	−9.48428	−1.00533	−0.502666	−	0.864481i	$-0.667647\pi$
$89$	−9.48428	−1.00533	−0.502666	+	0.864481i	$0.667647\pi$
$90$	−5.97560	−0.629884
$91$	12.7584	1.33744
$92$	−17.1029	−1.78310
$93$	−7.68396	−0.796789
$94$	6.87197	0.708790
$95$	−17.6057	−1.80631
$96$	−8.04612	−0.821204
$97$	−8.48940	−0.861968	−0.430984	−	0.902360i	$-0.641833\pi$
$97$	−8.48940	−0.861968	−0.430984	+	0.902360i	$0.641833\pi$
$98$	1.20868	0.122095
$99$	−5.48488	−0.551251
$100$	7.46239	0.746239
$101$	13.7392	1.36711	0.683553	−	0.729901i	$-0.260434\pi$
$101$	13.7392	1.36711	0.683553	+	0.729901i	$0.260434\pi$
$102$	−8.17755	−0.809698
$103$	8.75982	0.863130	0.431565	−	0.902082i	$-0.357962\pi$
$103$	8.75982	0.863130	0.431565	+	0.902082i	$0.357962\pi$
$104$	−4.34569	−0.426130
$105$	−7.21157	−0.703777
$106$	−2.60477	−0.252997
$107$	−5.77299	−0.558096	−0.279048	−	0.960277i	$-0.590019\pi$
$107$	−5.77299	−0.558096	−0.279048	+	0.960277i	$0.590019\pi$
$108$	−2.41107	−0.232005
$109$	9.31746	0.892451	0.446225	−	0.894921i	$-0.352768\pi$
$109$	9.31746	0.892451	0.446225	+	0.894921i	$0.352768\pi$
$110$	32.7754	3.12502
$111$	−0.825683	−0.0783703
$112$	−7.62653	−0.720639
$113$	−11.7153	−1.10208	−0.551040	−	0.834479i	$-0.685769\pi$
$113$	−11.7153	−1.10208	−0.551040	+	0.834479i	$0.685769\pi$
$114$	−12.9962	−1.21720
$115$	−20.1823	−1.88201
$116$	−11.4982	−1.06758
$117$	5.03357	0.465354
$118$	−8.56349	−0.788333
$119$	−9.86896	−0.904686
$120$	2.45636	0.224234
$121$	19.0839	1.73490
$122$	−28.1506	−2.54863
$123$	−3.84243	−0.346460
$124$	18.5265	1.66373
$125$	−5.41991	−0.484771
$126$	−5.32343	−0.474249
$127$	−17.6959	−1.57026	−0.785128	−	0.619334i	$-0.787403\pi$
$127$	−17.6959	−1.57026	−0.785128	+	0.619334i	$0.787403\pi$
$128$	6.76085	0.597581
$129$	−4.63842	−0.408390
$130$	−30.0786	−2.63807
$131$	16.5267	1.44395	0.721973	−	0.691921i	$-0.243235\pi$
$131$	16.5267	1.44395	0.721973	+	0.691921i	$0.243235\pi$
$132$	13.2244	1.15104
$133$	−15.6843	−1.36000
$134$	18.0563	1.55982
$135$	−2.84518	−0.244874
$136$	3.36151	0.288247
$137$	16.9893	1.45149	0.725745	−	0.687964i	$-0.241495\pi$
$137$	16.9893	1.45149	0.725745	+	0.687964i	$0.241495\pi$
$138$	−14.8982	−1.26821
$139$	−14.8062	−1.25584	−0.627921	−	0.778277i	$-0.716094\pi$
$139$	−14.8062	−1.25584	−0.627921	+	0.778277i	$0.716094\pi$
$140$	17.3876	1.46952
$141$	3.27197	0.275550
$142$	−30.6893	−2.57539
$143$	−27.6085	−2.30874
$144$	−3.00889	−0.250741
$145$	−13.5685	−1.12680
$146$	−26.8725	−2.22398
$147$	0.575491	0.0474657
$148$	1.99078	0.163641
$149$	−13.7147	−1.12355	−0.561777	−	0.827289i	$-0.689882\pi$
$149$	−13.7147	−1.12355	−0.561777	+	0.827289i	$0.689882\pi$
$150$	6.50041	0.530756
$151$	−12.8451	−1.04532	−0.522659	−	0.852542i	$-0.675060\pi$
$151$	−12.8451	−1.04532	−0.522659	+	0.852542i	$0.675060\pi$
$152$	5.34229	0.433317
$153$	−3.89360	−0.314779
$154$	29.1984	2.35287
$155$	21.8623	1.75602
$156$	−12.1363	−0.971679
$157$	4.76577	0.380350	0.190175	−	0.981750i	$-0.439095\pi$
$157$	4.76577	0.380350	0.190175	+	0.981750i	$0.439095\pi$
$158$	15.3802	1.22359
$159$	−1.24022	−0.0983555
$160$	22.8927	1.80983
$161$	−17.9796	−1.41699
$162$	−2.10025	−0.165012
$163$	8.25661	0.646708	0.323354	−	0.946278i	$-0.395190\pi$
$163$	8.25661	0.646708	0.323354	+	0.946278i	$0.395190\pi$
$164$	9.26436	0.723425
$165$	15.6055	1.21488
$166$	9.26205	0.718875
$167$	−7.74892	−0.599630	−0.299815	−	0.953997i	$-0.596925\pi$
$167$	−7.74892	−0.599630	−0.299815	+	0.953997i	$0.596925\pi$
$168$	2.18828	0.168829
$169$	12.3368	0.948986
$170$	23.2666	1.78447
$171$	−6.18791	−0.473202
$172$	11.1835	0.852737
$173$	−11.8453	−0.900583	−0.450291	−	0.892882i	$-0.648680\pi$
$173$	−11.8453	−0.900583	−0.450291	+	0.892882i	$0.648680\pi$
$174$	−10.0159	−0.759307
$175$	7.84492	0.593021
$176$	16.5034	1.24399
$177$	−4.07736	−0.306473
$178$	19.9194	1.49302
$179$	−1.05881	−0.0791393	−0.0395697	−	0.999217i	$-0.512599\pi$
$179$	−1.05881	−0.0791393	−0.0395697	+	0.999217i	$0.512599\pi$
$180$	6.85992	0.511308
$181$	−7.20691	−0.535685	−0.267843	−	0.963463i	$-0.586311\pi$
$181$	−7.20691	−0.535685	−0.267843	+	0.963463i	$0.586311\pi$
$182$	−26.7959	−1.98624
$183$	−13.4034	−0.990808
$184$	6.12412	0.451476
$185$	2.34922	0.172718
$186$	16.1383	1.18331
$187$	21.3559	1.56170
$188$	−7.88894	−0.575360
$189$	−2.53466	−0.184369
$190$	36.9765	2.68256
$191$	21.9628	1.58917	0.794586	−	0.607151i	$-0.207688\pi$
$191$	21.9628	1.58917	0.794586	+	0.607151i	$0.207688\pi$
$192$	10.8811	0.785277
$193$	4.79152	0.344901	0.172451	−	0.985018i	$-0.444832\pi$
$193$	4.79152	0.344901	0.172451	+	0.985018i	$0.444832\pi$
$194$	17.8299	1.28011
$195$	−14.3214	−1.02558
$196$	−1.38755	−0.0991105
$197$	−27.4381	−1.95488	−0.977441	−	0.211207i	$-0.932260\pi$
$197$	−27.4381	−1.95488	−0.977441	+	0.211207i	$0.932260\pi$
$198$	11.5196	0.818665
$199$	−11.4372	−0.810760	−0.405380	−	0.914148i	$-0.632861\pi$
$199$	−11.4372	−0.810760	−0.405380	+	0.914148i	$0.632861\pi$
$200$	−2.67209	−0.188946
$201$	8.59718	0.606398
$202$	−28.8559	−2.03029
$203$	−12.0876	−0.848384
$204$	9.38773	0.657273
$205$	10.9324	0.763553
$206$	−18.3978	−1.28184
$207$	−7.09350	−0.493032
$208$	−15.1455	−1.05015
$209$	33.9399	2.34768
$210$	15.1461	1.04518
$211$	−8.43010	−0.580352	−0.290176	−	0.956973i	$-0.593714\pi$
$211$	−8.43010	−0.580352	−0.290176	+	0.956973i	$0.593714\pi$
$212$	2.99024	0.205371
$213$	−14.6122	−1.00121
$214$	12.1247	0.828830
$215$	13.1971	0.900038
$216$	0.863342	0.0587430
$217$	19.4762	1.32213
$218$	−19.5690	−1.32538
$219$	−12.7949	−0.864597
$220$	−37.6258	−2.53673
$221$	−19.5987	−1.31835
$222$	1.73414	0.116388
$223$	−6.30026	−0.421896	−0.210948	−	0.977497i	$-0.567655\pi$
$223$	−6.30026	−0.421896	−0.210948	+	0.977497i	$0.567655\pi$
$224$	20.3942	1.36264
$225$	3.09506	0.206337
$226$	24.6050	1.63670
$227$	−3.15628	−0.209490	−0.104745	−	0.994499i	$-0.533403\pi$
$227$	−3.15628	−0.209490	−0.104745	+	0.994499i	$0.533403\pi$
$228$	14.9195	0.988066
$229$	−22.5495	−1.49011	−0.745057	−	0.667000i	$-0.767578\pi$
$229$	−22.5495	−1.49011	−0.745057	+	0.667000i	$0.767578\pi$
$230$	42.3879	2.79498
$231$	13.9023	0.914704
$232$	4.11721	0.270308
$233$	3.05586	0.200196	0.100098	−	0.994978i	$-0.468084\pi$
$233$	3.05586	0.200196	0.100098	+	0.994978i	$0.468084\pi$
$234$	−10.5718	−0.691098
$235$	−9.30935	−0.607275
$236$	9.83078	0.639930
$237$	7.32304	0.475683
$238$	20.7273	1.34355
$239$	−21.6335	−1.39936	−0.699679	−	0.714458i	$-0.746674\pi$
$239$	−21.6335	−1.39936	−0.699679	+	0.714458i	$0.746674\pi$
$240$	8.56085	0.552600
$241$	7.09572	0.457075	0.228538	−	0.973535i	$-0.426606\pi$
$241$	7.09572	0.457075	0.228538	+	0.973535i	$0.426606\pi$
$242$	−40.0810	−2.57650
$243$	−1.00000	−0.0641500
$244$	32.3165	2.06885
$245$	−1.63738	−0.104608
$246$	8.07008	0.514530
$247$	−31.1473	−1.98185
$248$	−6.63388	−0.421252
$249$	4.40997	0.279470
$250$	11.3832	0.719935
$251$	−10.0470	−0.634164	−0.317082	−	0.948398i	$-0.602703\pi$
$251$	−10.0470	−0.634164	−0.317082	+	0.948398i	$0.602703\pi$
$252$	6.11123	0.384972
$253$	38.9070	2.44606
$254$	37.1658	2.33199
$255$	11.0780	0.693732
$256$	7.56273	0.472670
$257$	−17.8931	−1.11614	−0.558069	−	0.829795i	$-0.688458\pi$
$257$	−17.8931	−1.11614	−0.558069	+	0.829795i	$0.688458\pi$
$258$	9.74186	0.606502
$259$	2.09283	0.130042
$260$	34.5299	2.14145
$261$	−4.76892	−0.295189
$262$	−34.7103	−2.14441
$263$	1.29865	0.0800785	0.0400392	−	0.999198i	$-0.487252\pi$
$263$	1.29865	0.0800785	0.0400392	+	0.999198i	$0.487252\pi$
$264$	−4.73532	−0.291439
$265$	3.52864	0.216763
$266$	32.9409	2.01974
$267$	9.48428	0.580428
$268$	−20.7284	−1.26619
$269$	7.71366	0.470310	0.235155	−	0.971958i	$-0.424440\pi$
$269$	7.71366	0.470310	0.235155	+	0.971958i	$0.424440\pi$
$270$	5.97560	0.363664
$271$	−13.0743	−0.794205	−0.397103	−	0.917774i	$-0.629984\pi$
$271$	−13.0743	−0.794205	−0.397103	+	0.917774i	$0.629984\pi$
$272$	11.7154	0.710353
$273$	−12.7584	−0.772173
$274$	−35.6817	−2.15561
$275$	−16.9760	−1.02369
$276$	17.1029	1.02947
$277$	−18.4336	−1.10757	−0.553783	−	0.832661i	$-0.686816\pi$
$277$	−18.4336	−1.10757	−0.553783	+	0.832661i	$0.686816\pi$
$278$	31.0967	1.86505
$279$	7.68396	0.460026
$280$	−6.22605	−0.372078
$281$	−30.9816	−1.84820	−0.924102	−	0.382145i	$-0.875185\pi$
$281$	−30.9816	−1.84820	−0.924102	+	0.382145i	$0.875185\pi$
$282$	−6.87197	−0.409220
$283$	−26.2539	−1.56063	−0.780315	−	0.625386i	$-0.784941\pi$
$283$	−26.2539	−1.56063	−0.780315	+	0.625386i	$0.784941\pi$
$284$	35.2309	2.09057
$285$	17.6057	1.04287
$286$	57.9849	3.42872
$287$	9.73927	0.574891
$288$	8.04612	0.474122
$289$	−1.83986	−0.108227
$290$	28.4972	1.67341
$291$	8.48940	0.497657
$292$	30.8493	1.80532
$293$	26.7395	1.56214	0.781070	−	0.624443i	$-0.214674\pi$
$293$	26.7395	1.56214	0.781070	+	0.624443i	$0.214674\pi$
$294$	−1.20868	−0.0704915
$295$	11.6008	0.675427
$296$	−0.712847	−0.0414334
$297$	5.48488	0.318265
$298$	28.8044	1.66859
$299$	−35.7056	−2.06491
$300$	−7.46239	−0.430841
$301$	11.7568	0.677652
$302$	26.9779	1.55240
$303$	−13.7392	−0.789299
$304$	18.6188	1.06786
$305$	38.1351	2.18361
$306$	8.17755	0.467480
$307$	18.3577	1.04773	0.523864	−	0.851802i	$-0.324490\pi$
$307$	18.3577	1.04773	0.523864	+	0.851802i	$0.324490\pi$
$308$	−33.5194	−1.90994
$309$	−8.75982	−0.498329
$310$	−45.9163	−2.60787
$311$	5.12756	0.290757	0.145379	−	0.989376i	$-0.453560\pi$
$311$	5.12756	0.290757	0.145379	+	0.989376i	$0.453560\pi$
$312$	4.34569	0.246026
$313$	−6.03424	−0.341076	−0.170538	−	0.985351i	$-0.554550\pi$
$313$	−6.03424	−0.341076	−0.170538	+	0.985351i	$0.554550\pi$
$314$	−10.0093	−0.564859
$315$	7.21157	0.406326
$316$	−17.6563	−0.993246
$317$	−0.827870	−0.0464978	−0.0232489	−	0.999730i	$-0.507401\pi$
$317$	−0.827870	−0.0464978	−0.0232489	+	0.999730i	$0.507401\pi$
$318$	2.60477	0.146068
$319$	26.1570	1.46451
$320$	−30.9587	−1.73065
$321$	5.77299	0.322217
$322$	37.7618	2.10438
$323$	24.0933	1.34059
$324$	2.41107	0.133948
$325$	15.5792	0.864178
$326$	−17.3410	−0.960428
$327$	−9.31746	−0.515257
$328$	−3.31733	−0.183169
$329$	−8.29334	−0.457227
$330$	−32.7754	−1.80423
$331$	−18.2306	−1.00204	−0.501021	−	0.865435i	$-0.667042\pi$
$331$	−18.2306	−1.00204	−0.501021	+	0.865435i	$0.667042\pi$
$332$	−10.6327	−0.583547
$333$	0.825683	0.0452471
$334$	16.2747	0.890512
$335$	−24.4605	−1.33642
$336$	7.62653	0.416061
$337$	−12.8523	−0.700110	−0.350055	−	0.936729i	$-0.613837\pi$
$337$	−12.8523	−0.700110	−0.350055	+	0.936729i	$0.613837\pi$
$338$	−25.9104	−1.40934
$339$	11.7153	0.636286
$340$	−26.7098	−1.44854
$341$	−42.1455	−2.28231
$342$	12.9962	0.702753
$343$	−19.2013	−1.03677
$344$	−4.00454	−0.215910
$345$	20.1823	1.08658
$346$	24.8782	1.33746
$347$	−19.2367	−1.03268	−0.516341	−	0.856383i	$-0.672706\pi$
$347$	−19.2367	−1.03268	−0.516341	+	0.856383i	$0.672706\pi$
$348$	11.4982	0.616368
$349$	34.6180	1.85306	0.926530	−	0.376222i	$-0.122777\pi$
$349$	34.6180	1.85306	0.926530	+	0.376222i	$0.122777\pi$
$350$	−16.4763	−0.880697
$351$	−5.03357	−0.268672
$352$	−44.1320	−2.35224
$353$	22.5674	1.20114	0.600571	−	0.799572i	$-0.294940\pi$
$353$	22.5674	1.20114	0.600571	+	0.799572i	$0.294940\pi$
$354$	8.56349	0.455144
$355$	41.5743	2.20653
$356$	−22.8672	−1.21196
$357$	9.86896	0.522321
$358$	2.22377	0.117530
$359$	−17.8177	−0.940382	−0.470191	−	0.882565i	$-0.655815\pi$
$359$	−17.8177	−0.940382	−0.470191	+	0.882565i	$0.655815\pi$
$360$	−2.45636	−0.129462
$361$	19.2903	1.01528
$362$	15.1363	0.795548
$363$	−19.0839	−1.00164
$364$	30.7613	1.61233
$365$	36.4037	1.90546
$366$	28.1506	1.47145
$367$	7.04661	0.367830	0.183915	−	0.982942i	$-0.441123\pi$
$367$	7.04661	0.367830	0.183915	+	0.982942i	$0.441123\pi$
$368$	21.3436	1.11261
$369$	3.84243	0.200029
$370$	−4.93395	−0.256504
$371$	3.14353	0.163204
$372$	−18.5265	−0.960555
$373$	−26.4927	−1.37174	−0.685870	−	0.727724i	$-0.740578\pi$
$373$	−26.4927	−1.37174	−0.685870	+	0.727724i	$0.740578\pi$
$374$	−44.8529	−2.31929
$375$	5.41991	0.279883
$376$	2.82483	0.145679
$377$	−24.0047	−1.23631
$378$	5.32343	0.273808
$379$	37.3938	1.92079	0.960396	−	0.278638i	$-0.0898828\pi$
$379$	37.3938	1.92079	0.960396	+	0.278638i	$0.0898828\pi$
$380$	−42.4486	−2.17757
$381$	17.6959	0.906587
$382$	−46.1274	−2.36009
$383$	−30.3387	−1.55024	−0.775118	−	0.631816i	$-0.782310\pi$
$383$	−30.3387	−1.55024	−0.775118	+	0.631816i	$0.782310\pi$
$384$	−6.76085	−0.345013
$385$	−39.5546	−2.01589
$386$	−10.0634	−0.512214
$387$	4.63842	0.235784
$388$	−20.4685	−1.03913
$389$	−20.8709	−1.05820	−0.529099	−	0.848560i	$-0.677470\pi$
$389$	−20.8709	−1.05820	−0.529099	+	0.848560i	$0.677470\pi$
$390$	30.0786	1.52309
$391$	27.6193	1.39677
$392$	0.496846	0.0250945
$393$	−16.5267	−0.833663
$394$	57.6269	2.90320
$395$	−20.8354	−1.04834
$396$	−13.2244	−0.664551
$397$	−2.04801	−0.102787	−0.0513933	−	0.998678i	$-0.516366\pi$
$397$	−2.04801	−0.102787	−0.0513933	+	0.998678i	$0.516366\pi$
$398$	24.0210	1.20406
$399$	15.6843	0.785196
$400$	−9.31270	−0.465635
$401$	−28.6818	−1.43230	−0.716150	−	0.697946i	$-0.754097\pi$
$401$	−28.6818	−1.43230	−0.716150	+	0.697946i	$0.754097\pi$
$402$	−18.0563	−0.900564
$403$	38.6777	1.92667
$404$	33.1262	1.64809
$405$	2.84518	0.141378
$406$	25.3870	1.25994
$407$	−4.52877	−0.224483
$408$	−3.36151	−0.166420
$409$	−6.71730	−0.332149	−0.166075	−	0.986113i	$-0.553109\pi$
$409$	−6.71730	−0.332149	−0.166075	+	0.986113i	$0.553109\pi$
$410$	−22.9609	−1.13396
$411$	−16.9893	−0.838018
$412$	21.1205	1.04053
$413$	10.3347	0.508539
$414$	14.8982	0.732204
$415$	−12.5472	−0.615916
$416$	40.5007	1.98571
$417$	14.8062	0.725061
$418$	−71.2825	−3.48654
$419$	−25.1006	−1.22624	−0.613121	−	0.789989i	$-0.710086\pi$
$419$	−25.1006	−1.22624	−0.613121	+	0.789989i	$0.710086\pi$
$420$	−17.3876	−0.848427
$421$	−3.77409	−0.183938	−0.0919690	−	0.995762i	$-0.529316\pi$
$421$	−3.77409	−0.183938	−0.0919690	+	0.995762i	$0.529316\pi$
$422$	17.7053	0.861882
$423$	−3.27197	−0.159089
$424$	−1.07073	−0.0519993
$425$	−12.0509	−0.584556
$426$	30.6893	1.48690
$427$	33.9731	1.64407
$428$	−13.9191	−0.672803
$429$	27.6085	1.33295
$430$	−27.7174	−1.33665
$431$	32.3023	1.55595	0.777974	−	0.628297i	$-0.216248\pi$
$431$	32.3023	1.55595	0.777974	+	0.628297i	$0.216248\pi$
$432$	3.00889	0.144765
$433$	40.4291	1.94290	0.971448	−	0.237251i	$-0.0762464\pi$
$433$	40.4291	1.94290	0.971448	+	0.237251i	$0.0762464\pi$
$434$	−40.9050	−1.96350
$435$	13.5685	0.650558
$436$	22.4650	1.07588
$437$	43.8940	2.09973
$438$	26.8725	1.28402
$439$	−17.6970	−0.844634	−0.422317	−	0.906448i	$-0.638783\pi$
$439$	−17.6970	−0.844634	−0.422317	+	0.906448i	$0.638783\pi$
$440$	13.4729	0.642293
$441$	−0.575491	−0.0274043
$442$	41.1623	1.95789
$443$	−1.85668	−0.0882135	−0.0441068	−	0.999027i	$-0.514044\pi$
$443$	−1.85668	−0.0882135	−0.0441068	+	0.999027i	$0.514044\pi$
$444$	−1.99078	−0.0944780
$445$	−26.9845	−1.27919
$446$	13.2321	0.626560
$447$	13.7147	0.648684
$448$	−27.5799	−1.30303
$449$	−17.4066	−0.821469	−0.410735	−	0.911755i	$-0.634728\pi$
$449$	−17.4066	−0.821469	−0.410735	+	0.911755i	$0.634728\pi$
$450$	−6.50041	−0.306432
$451$	−21.0753	−0.992396
$452$	−28.2463	−1.32859
$453$	12.8451	0.603514
$454$	6.62899	0.311114
$455$	36.2999	1.70177
$456$	−5.34229	−0.250175
$457$	37.7559	1.76615	0.883074	−	0.469234i	$-0.155470\pi$
$457$	37.7559	1.76615	0.883074	+	0.469234i	$0.155470\pi$
$458$	47.3597	2.21297
$459$	3.89360	0.181738
$460$	−48.6608	−2.26882
$461$	−26.4495	−1.23188	−0.615939	−	0.787794i	$-0.711223\pi$
$461$	−26.4495	−1.23188	−0.615939	+	0.787794i	$0.711223\pi$
$462$	−29.1984	−1.35843
$463$	−18.7106	−0.869556	−0.434778	−	0.900538i	$-0.643173\pi$
$463$	−18.7106	−0.869556	−0.434778	+	0.900538i	$0.643173\pi$
$464$	14.3492	0.666144
$465$	−21.8623	−1.01384
$466$	−6.41807	−0.297312
$467$	17.5949	0.814196	0.407098	−	0.913384i	$-0.366541\pi$
$467$	17.5949	0.814196	0.407098	+	0.913384i	$0.366541\pi$
$468$	12.1363	0.560999
$469$	−21.7909	−1.00621
$470$	19.5520	0.901867
$471$	−4.76577	−0.219595
$472$	−3.52016	−0.162028
$473$	−25.4412	−1.16979
$474$	−15.3802	−0.706438
$475$	−19.1520	−0.878752
$476$	−23.7947	−1.09063
$477$	1.24022	0.0567856
$478$	45.4359	2.07819
$479$	−28.1535	−1.28637	−0.643184	−	0.765712i	$-0.722387\pi$
$479$	−28.1535	−1.28637	−0.643184	+	0.765712i	$0.722387\pi$
$480$	−22.8927	−1.04490
$481$	4.15613	0.189503
$482$	−14.9028	−0.678804
$483$	17.9796	0.818101
$484$	46.0125	2.09148
$485$	−24.1539	−1.09677
$486$	2.10025	0.0952694
$487$	−12.3535	−0.559792	−0.279896	−	0.960030i	$-0.590300\pi$
$487$	−12.3535	−0.559792	−0.279896	+	0.960030i	$0.590300\pi$
$488$	−11.5717	−0.523827
$489$	−8.25661	−0.373377
$490$	3.43891	0.155354
$491$	−11.5095	−0.519417	−0.259709	−	0.965687i	$-0.583627\pi$
$491$	−11.5095	−0.519417	−0.259709	+	0.965687i	$0.583627\pi$
$492$	−9.26436	−0.417669
$493$	18.5683	0.836274
$494$	65.4172	2.94326
$495$	−15.6055	−0.701414
$496$	−23.1202	−1.03813
$497$	37.0369	1.66133
$498$	−9.26205	−0.415043
$499$	−13.5765	−0.607767	−0.303884	−	0.952709i	$-0.598283\pi$
$499$	−13.5765	−0.607767	−0.303884	+	0.952709i	$0.598283\pi$
$500$	−13.0677	−0.584407
$501$	7.74892	0.346196
$502$	21.1013	0.941799
$503$	−17.6870	−0.788624	−0.394312	−	0.918977i	$-0.629017\pi$
$503$	−17.6870	−0.788624	−0.394312	+	0.918977i	$0.629017\pi$
$504$	−2.18828	−0.0974737
$505$	39.0906	1.73951
$506$	−81.7145	−3.63265
$507$	−12.3368	−0.547897
$508$	−42.6659	−1.89299
$509$	−27.4306	−1.21584	−0.607919	−	0.793999i	$-0.707996\pi$
$509$	−27.4306	−1.21584	−0.607919	+	0.793999i	$0.707996\pi$
$510$	−23.2666	−1.03026
$511$	32.4307	1.43465
$512$	−29.4053	−1.29954
$513$	6.18791	0.273203
$514$	37.5799	1.65758
$515$	24.9233	1.09825
$516$	−11.1835	−0.492328
$517$	17.9464	0.789280
$518$	−4.39547	−0.193126
$519$	11.8453	0.519952
$520$	−12.3643	−0.542210
$521$	−9.12407	−0.399733	−0.199866	−	0.979823i	$-0.564051\pi$
$521$	−9.12407	−0.399733	−0.199866	+	0.979823i	$0.564051\pi$
$522$	10.0159	0.438386
$523$	−28.8916	−1.26334	−0.631670	−	0.775237i	$-0.717630\pi$
$523$	−28.8916	−1.26334	−0.631670	+	0.775237i	$0.717630\pi$
$524$	39.8470	1.74072
$525$	−7.84492	−0.342381
$526$	−2.72750	−0.118925
$527$	−29.9183	−1.30326
$528$	−16.5034	−0.718219
$529$	27.3178	1.18773
$530$	−7.41104	−0.321915
$531$	4.07736	0.176942
$532$	−37.8158	−1.63952
$533$	19.3412	0.837758
$534$	−19.9194	−0.861996
$535$	−16.4252	−0.710124
$536$	7.42231	0.320595
$537$	1.05881	0.0456911
$538$	−16.2006	−0.698459
$539$	3.15650	0.135960
$540$	−6.85992	−0.295204
$541$	17.9169	0.770309	0.385154	−	0.922852i	$-0.374148\pi$
$541$	17.9169	0.770309	0.385154	+	0.922852i	$0.374148\pi$
$542$	27.4593	1.17948
$543$	7.20691	0.309278
$544$	−31.3284	−1.34319
$545$	26.5099	1.13556
$546$	26.7959	1.14676
$547$	2.65810	0.113652	0.0568261	−	0.998384i	$-0.481902\pi$
$547$	2.65810	0.113652	0.0568261	+	0.998384i	$0.481902\pi$
$548$	40.9622	1.74982
$549$	13.4034	0.572043
$550$	35.6539	1.52029
$551$	29.5097	1.25716
$552$	−6.12412	−0.260660
$553$	−18.5614	−0.789312
$554$	38.7152	1.64485
$555$	−2.34922	−0.0997187
$556$	−35.6986	−1.51396
$557$	32.4585	1.37531	0.687655	−	0.726037i	$-0.258640\pi$
$557$	32.4585	1.37531	0.687655	+	0.726037i	$0.258640\pi$
$558$	−16.1383	−0.683187
$559$	23.3478	0.987507
$560$	−21.6989	−0.916944
$561$	−21.3559	−0.901648
$562$	65.0691	2.74478
$563$	−39.8110	−1.67783	−0.838917	−	0.544259i	$-0.816811\pi$
$563$	−39.8110	−1.67783	−0.838917	+	0.544259i	$0.816811\pi$
$564$	7.88894	0.332184
$565$	−33.3321	−1.40229
$566$	55.1398	2.31770
$567$	2.53466	0.106446
$568$	−12.6153	−0.529326
$569$	−18.2948	−0.766959	−0.383480	−	0.923549i	$-0.625274\pi$
$569$	−18.2948	−0.766959	−0.383480	+	0.923549i	$0.625274\pi$
$570$	−36.9765	−1.54878
$571$	38.5285	1.61237	0.806184	−	0.591665i	$-0.201529\pi$
$571$	38.5285	1.61237	0.806184	+	0.591665i	$0.201529\pi$
$572$	−66.5659	−2.78326
$573$	−21.9628	−0.917509
$574$	−20.4549	−0.853772
$575$	−21.9548	−0.915578
$576$	−10.8811	−0.453380
$577$	17.3640	0.722872	0.361436	−	0.932397i	$-0.382287\pi$
$577$	17.3640	0.722872	0.361436	+	0.932397i	$0.382287\pi$
$578$	3.86418	0.160729
$579$	−4.79152	−0.199129
$580$	−32.7144	−1.35839
$581$	−11.1778	−0.463732
$582$	−17.8299	−0.739073
$583$	−6.80243	−0.281728
$584$	−11.0464	−0.457101
$585$	14.3214	0.592118
$586$	−56.1598	−2.31994
$587$	43.7453	1.80556	0.902780	−	0.430102i	$-0.141522\pi$
$587$	43.7453	1.80556	0.902780	+	0.430102i	$0.141522\pi$
$588$	1.38755	0.0572215
$589$	−47.5477	−1.95917
$590$	−24.3647	−1.00308
$591$	27.4381	1.12865
$592$	−2.48439	−0.102108
$593$	−47.0998	−1.93416	−0.967079	−	0.254478i	$-0.918096\pi$
$593$	−47.0998	−1.93416	−0.967079	+	0.254478i	$0.918096\pi$
$594$	−11.5196	−0.472656
$595$	−28.0790	−1.15113
$596$	−33.0671	−1.35448
$597$	11.4372	0.468092
$598$	74.9909	3.06660
$599$	5.02570	0.205344	0.102672	−	0.994715i	$-0.467261\pi$
$599$	5.02570	0.205344	0.102672	+	0.994715i	$0.467261\pi$
$600$	2.67209	0.109088
$601$	10.0340	0.409295	0.204647	−	0.978836i	$-0.434395\pi$
$601$	10.0340	0.409295	0.204647	+	0.978836i	$0.434395\pi$
$602$	−24.6923	−1.00638
$603$	−8.59718	−0.350104
$604$	−30.9703	−1.26016
$605$	54.2971	2.20749
$606$	28.8559	1.17219
$607$	−9.65412	−0.391849	−0.195924	−	0.980619i	$-0.562771\pi$
$607$	−9.65412	−0.391849	−0.195924	+	0.980619i	$0.562771\pi$
$608$	−49.7887	−2.01920
$609$	12.0876	0.489815
$610$	−80.0934	−3.24289
$611$	−16.4697	−0.666293
$612$	−9.38773	−0.379476
$613$	15.7230	0.635048	0.317524	−	0.948250i	$-0.397149\pi$
$613$	15.7230	0.635048	0.317524	+	0.948250i	$0.397149\pi$
$614$	−38.5558	−1.55598
$615$	−10.9324	−0.440838
$616$	12.0024	0.483592
$617$	34.9342	1.40640	0.703199	−	0.710993i	$-0.251754\pi$
$617$	34.9342	1.40640	0.703199	+	0.710993i	$0.251754\pi$
$618$	18.3978	0.740069
$619$	−5.70389	−0.229259	−0.114629	−	0.993408i	$-0.536568\pi$
$619$	−5.70389	−0.229259	−0.114629	+	0.993408i	$0.536568\pi$
$620$	52.7113	2.11694
$621$	7.09350	0.284652
$622$	−10.7692	−0.431804
$623$	−24.0394	−0.963120
$624$	15.1455	0.606304
$625$	−30.8959	−1.23584
$626$	12.6734	0.506533
$627$	−33.9399	−1.35543
$628$	11.4906	0.458524
$629$	−3.21488	−0.128186
$630$	−15.1461	−0.603436
$631$	8.86299	0.352830	0.176415	−	0.984316i	$-0.443550\pi$
$631$	8.86299	0.352830	0.176415	+	0.984316i	$0.443550\pi$
$632$	6.32229	0.251487
$633$	8.43010	0.335066
$634$	1.73874	0.0690541
$635$	−50.3480	−1.99800
$636$	−2.99024	−0.118571
$637$	−2.89678	−0.114774
$638$	−54.9362	−2.17495
$639$	14.6122	0.578049
$640$	19.2359	0.760364
$641$	−48.6600	−1.92195	−0.960977	−	0.276629i	$-0.910783\pi$
$641$	−48.6600	−1.92195	−0.960977	+	0.276629i	$0.910783\pi$
$642$	−12.1247	−0.478525
$643$	−46.6681	−1.84041	−0.920205	−	0.391437i	$-0.871978\pi$
$643$	−46.6681	−1.84041	−0.920205	+	0.391437i	$0.871978\pi$
$644$	−43.3500	−1.70823
$645$	−13.1971	−0.519637
$646$	−50.6020	−1.99091
$647$	6.84509	0.269108	0.134554	−	0.990906i	$-0.457040\pi$
$647$	6.84509	0.269108	0.134554	+	0.990906i	$0.457040\pi$
$648$	−0.863342	−0.0339153
$649$	−22.3638	−0.877857
$650$	−32.7203	−1.28339
$651$	−19.4762	−0.763334
$652$	19.9072	0.779627
$653$	19.7892	0.774412	0.387206	−	0.921993i	$-0.373440\pi$
$653$	19.7892	0.774412	0.387206	+	0.921993i	$0.373440\pi$
$654$	19.5690	0.765210
$655$	47.0215	1.83728
$656$	−11.5615	−0.451400
$657$	12.7949	0.499176
$658$	17.4181	0.679029
$659$	−29.2900	−1.14098	−0.570488	−	0.821306i	$-0.693246\pi$
$659$	−29.2900	−1.14098	−0.570488	+	0.821306i	$0.693246\pi$
$660$	37.6258	1.46458
$661$	−6.27664	−0.244133	−0.122066	−	0.992522i	$-0.538952\pi$
$661$	−6.27664	−0.244133	−0.122066	+	0.992522i	$0.538952\pi$
$662$	38.2888	1.48814
$663$	19.5987	0.761151
$664$	3.80731	0.147752
$665$	−44.6246	−1.73047
$666$	−1.73414	−0.0671967
$667$	33.8284	1.30984
$668$	−18.6832	−0.722873
$669$	6.30026	0.243582
$670$	51.3733	1.98473
$671$	−73.5160	−2.83805
$672$	−20.3942	−0.786723
$673$	20.9287	0.806740	0.403370	−	0.915037i	$-0.367839\pi$
$673$	20.9287	0.806740	0.403370	+	0.915037i	$0.367839\pi$
$674$	26.9931	1.03974
$675$	−3.09506	−0.119129
$676$	29.7449	1.14403
$677$	−44.1727	−1.69769	−0.848847	−	0.528638i	$-0.822703\pi$
$677$	−44.1727	−1.69769	−0.848847	+	0.528638i	$0.822703\pi$
$678$	−24.6050	−0.944951
$679$	−21.5177	−0.825775
$680$	9.56411	0.366767
$681$	3.15628	0.120949
$682$	88.5163	3.38947
$683$	−6.30497	−0.241253	−0.120626	−	0.992698i	$-0.538490\pi$
$683$	−6.30497	−0.241253	−0.120626	+	0.992698i	$0.538490\pi$
$684$	−14.9195	−0.570460
$685$	48.3375	1.84688
$686$	40.3276	1.53972
$687$	22.5495	0.860318
$688$	−13.9565	−0.532087
$689$	6.24271	0.237828
$690$	−42.3879	−1.61368
$691$	−0.928439	−0.0353195	−0.0176597	−	0.999844i	$-0.505622\pi$
$691$	−0.928439	−0.0353195	−0.0176597	+	0.999844i	$0.505622\pi$
$692$	−28.5598	−1.08568
$693$	−13.9023	−0.528105
$694$	40.4021	1.53364
$695$	−42.1262	−1.59794
$696$	−4.11721	−0.156063
$697$	−14.9609	−0.566685
$698$	−72.7066	−2.75199
$699$	−3.05586	−0.115583
$700$	18.9146	0.714906
$701$	5.91541	0.223422	0.111711	−	0.993741i	$-0.464367\pi$
$701$	5.91541	0.223422	0.111711	+	0.993741i	$0.464367\pi$
$702$	10.5718	0.399006
$703$	−5.10925	−0.192699
$704$	59.6816	2.24933
$705$	9.30935	0.350611
$706$	−47.3973	−1.78382
$707$	34.8243	1.30970
$708$	−9.83078	−0.369463
$709$	−3.97160	−0.149157	−0.0745783	−	0.997215i	$-0.523761\pi$
$709$	−3.97160	−0.149157	−0.0745783	+	0.997215i	$0.523761\pi$
$710$	−87.3165	−3.27693
$711$	−7.32304	−0.274635
$712$	8.18817	0.306865
$713$	−54.5062	−2.04127
$714$	−20.7273	−0.775701
$715$	−78.5512	−2.93765
$716$	−2.55287	−0.0954051
$717$	21.6335	0.807920
$718$	37.4217	1.39656
$719$	−46.4328	−1.73165	−0.865825	−	0.500347i	$-0.833206\pi$
$719$	−46.4328	−1.73165	−0.865825	+	0.500347i	$0.833206\pi$
$720$	−8.56085	−0.319044
$721$	22.2032	0.826889
$722$	−40.5145	−1.50779
$723$	−7.09572	−0.263892
$724$	−17.3763	−0.645786
$725$	−14.7601	−0.548176
$726$	40.0810	1.48754
$727$	−8.34278	−0.309417	−0.154708	−	0.987960i	$-0.549444\pi$
$727$	−8.34278	−0.309417	−0.154708	+	0.987960i	$0.549444\pi$
$728$	−11.0149	−0.408238
$729$	1.00000	0.0370370
$730$	−76.4571	−2.82980
$731$	−18.0602	−0.667979
$732$	−32.3165	−1.19445
$733$	−22.3352	−0.824969	−0.412484	−	0.910965i	$-0.635339\pi$
$733$	−22.3352	−0.824969	−0.412484	+	0.910965i	$0.635339\pi$
$734$	−14.7997	−0.546265
$735$	1.63738	0.0603956
$736$	−57.0752	−2.10382
$737$	47.1545	1.73696
$738$	−8.07008	−0.297064
$739$	−28.0810	−1.03298	−0.516488	−	0.856294i	$-0.672761\pi$
$739$	−28.0810	−1.03298	−0.516488	+	0.856294i	$0.672761\pi$
$740$	5.66412	0.208217
$741$	31.1473	1.14422
$742$	−6.60220	−0.242375
$743$	2.79894	0.102683	0.0513415	−	0.998681i	$-0.483650\pi$
$743$	2.79894	0.102683	0.0513415	+	0.998681i	$0.483650\pi$
$744$	6.63388	0.243210
$745$	−39.0209	−1.42961
$746$	55.6414	2.03718
$747$	−4.40997	−0.161352
$748$	51.4905	1.88268
$749$	−14.6326	−0.534663
$750$	−11.3832	−0.415655
$751$	−30.7729	−1.12292	−0.561460	−	0.827504i	$-0.689760\pi$
$751$	−30.7729	−1.12292	−0.561460	+	0.827504i	$0.689760\pi$
$752$	9.84502	0.359011
$753$	10.0470	0.366135
$754$	50.4160	1.83604
$755$	−36.5466	−1.33007
$756$	−6.11123	−0.222263
$757$	−46.0897	−1.67516	−0.837579	−	0.546316i	$-0.816030\pi$
$757$	−46.0897	−1.67516	−0.837579	+	0.546316i	$0.816030\pi$
$758$	−78.5366	−2.85258
$759$	−38.9070	−1.41223
$760$	15.1998	0.551354
$761$	32.7819	1.18834	0.594172	−	0.804338i	$-0.297480\pi$
$761$	32.7819	1.18834	0.594172	+	0.804338i	$0.297480\pi$
$762$	−37.1658	−1.34638
$763$	23.6166	0.854978
$764$	52.9537	1.91580
$765$	−11.0780	−0.400526
$766$	63.7190	2.30226
$767$	20.5237	0.741067
$768$	−7.56273	−0.272896
$769$	−10.1849	−0.367276	−0.183638	−	0.982994i	$-0.558787\pi$
$769$	−10.1849	−0.367276	−0.183638	+	0.982994i	$0.558787\pi$
$770$	83.0747	2.99380
$771$	17.8931	0.644402
$772$	11.5527	0.415790
$773$	−34.3104	−1.23406	−0.617029	−	0.786940i	$-0.711664\pi$
$773$	−34.3104	−1.23406	−0.617029	+	0.786940i	$0.711664\pi$
$774$	−9.74186	−0.350164
$775$	23.7823	0.854285
$776$	7.32925	0.263105
$777$	−2.09283	−0.0750797
$778$	43.8342	1.57153
$779$	−23.7766	−0.851887
$780$	−34.5299	−1.23637
$781$	−80.1460	−2.86785
$782$	−58.0075	−2.07434
$783$	4.76892	0.170427
$784$	1.73159	0.0618426
$785$	13.5595	0.483958
$786$	34.7103	1.23808
$787$	39.0198	1.39091	0.695454	−	0.718571i	$-0.255203\pi$
$787$	39.0198	1.39091	0.695454	+	0.718571i	$0.255203\pi$
$788$	−66.1550	−2.35667
$789$	−1.29865	−0.0462333
$790$	43.7596	1.55690
$791$	−29.6942	−1.05581
$792$	4.73532	0.168262
$793$	67.4670	2.39582
$794$	4.30134	0.152649
$795$	−3.52864	−0.125148
$796$	−27.5758	−0.977398
$797$	−16.5083	−0.584755	−0.292378	−	0.956303i	$-0.594446\pi$
$797$	−16.5083	−0.584755	−0.292378	+	0.956303i	$0.594446\pi$
$798$	−32.9409	−1.16610
$799$	12.7398	0.450700
$800$	24.9032	0.880462
$801$	−9.48428	−0.335111
$802$	60.2390	2.12711
$803$	−70.1783	−2.47654
$804$	20.7284	0.731033
$805$	−51.1553	−1.80299
$806$	−81.2330	−2.86131
$807$	−7.71366	−0.271534
$808$	−11.8617	−0.417292
$809$	−2.01045	−0.0706836	−0.0353418	−	0.999375i	$-0.511252\pi$
$809$	−2.01045	−0.0706836	−0.0353418	+	0.999375i	$0.511252\pi$
$810$	−5.97560	−0.209961
$811$	−38.5300	−1.35297	−0.676485	−	0.736456i	$-0.736498\pi$
$811$	−38.5300	−1.35297	−0.676485	+	0.736456i	$0.736498\pi$
$812$	−29.1440	−1.02275
$813$	13.0743	0.458535
$814$	9.51156	0.333380
$815$	23.4916	0.822873
$816$	−11.7154	−0.410122
$817$	−28.7022	−1.00416
$818$	14.1080	0.493276
$819$	12.7584	0.445814
$820$	26.3588	0.920488
$821$	26.0139	0.907892	0.453946	−	0.891029i	$-0.350016\pi$
$821$	26.0139	0.907892	0.453946	+	0.891029i	$0.350016\pi$
$822$	35.6817	1.24454
$823$	−45.2023	−1.57565	−0.787826	−	0.615897i	$-0.788794\pi$
$823$	−45.2023	−1.57565	−0.787826	+	0.615897i	$0.788794\pi$
$824$	−7.56272	−0.263460
$825$	16.9760	0.591029
$826$	−21.7056	−0.755233
$827$	26.6084	0.925264	0.462632	−	0.886550i	$-0.346905\pi$
$827$	26.6084	0.925264	0.462632	+	0.886550i	$0.346905\pi$
$828$	−17.1029	−0.594367
$829$	35.5644	1.23520	0.617602	−	0.786491i	$-0.288104\pi$
$829$	35.5644	1.23520	0.617602	+	0.786491i	$0.288104\pi$
$830$	26.3522	0.914699
$831$	18.4336	0.639453
$832$	−54.7708	−1.89884
$833$	2.24073	0.0776368
$834$	−31.0967	−1.07679
$835$	−22.0471	−0.762971
$836$	81.8314	2.83020
$837$	−7.68396	−0.265596
$838$	52.7175	1.82110
$839$	33.5968	1.15989	0.579945	−	0.814655i	$-0.303074\pi$
$839$	33.5968	1.15989	0.579945	+	0.814655i	$0.303074\pi$
$840$	6.22605	0.214819
$841$	−6.25737	−0.215771
$842$	7.92655	0.273167
$843$	30.9816	1.06706
$844$	−20.3255	−0.699633
$845$	35.1005	1.20749
$846$	6.87197	0.236263
$847$	48.3711	1.66205
$848$	−3.73168	−0.128146
$849$	26.2539	0.901031
$850$	25.3100	0.868126
$851$	−5.85698	−0.200775
$852$	−35.2309	−1.20699
$853$	37.1025	1.27037	0.635183	−	0.772362i	$-0.280925\pi$
$853$	37.1025	1.27037	0.635183	+	0.772362i	$0.280925\pi$
$854$	−71.3521	−2.44162
$855$	−17.6057	−0.602104
$856$	4.98406	0.170352
$857$	45.0751	1.53974	0.769868	−	0.638203i	$-0.220322\pi$
$857$	45.0751	1.53974	0.769868	+	0.638203i	$0.220322\pi$
$858$	−57.9849	−1.97957
$859$	−21.6120	−0.737393	−0.368697	−	0.929550i	$-0.620196\pi$
$859$	−21.6120	−0.737393	−0.368697	+	0.929550i	$0.620196\pi$
$860$	31.8192	1.08503
$861$	−9.73927	−0.331913
$862$	−67.8430	−2.31074
$863$	9.05932	0.308383	0.154192	−	0.988041i	$-0.450723\pi$
$863$	9.05932	0.308383	0.154192	+	0.988041i	$0.450723\pi$
$864$	−8.04612	−0.273735
$865$	−33.7021	−1.14590
$866$	−84.9113	−2.88540
$867$	1.83986	0.0624850
$868$	46.9585	1.59387
$869$	40.1660	1.36254
$870$	−28.4972	−0.966145
$871$	−43.2745	−1.46630
$872$	−8.04415	−0.272409
$873$	−8.48940	−0.287323
$874$	−92.1885	−3.11832
$875$	−13.7376	−0.464417
$876$	−30.8493	−1.04230
$877$	−0.123886	−0.00418333	−0.00209166	−	0.999998i	$-0.500666\pi$
$877$	−0.123886	−0.00418333	−0.00209166	+	0.999998i	$0.500666\pi$
$878$	37.1683	1.25437
$879$	−26.7395	−0.901902
$880$	46.9552	1.58286
$881$	−14.1469	−0.476623	−0.238311	−	0.971189i	$-0.576594\pi$
$881$	−14.1469	−0.476623	−0.238311	+	0.971189i	$0.576594\pi$
$882$	1.20868	0.0406983
$883$	−24.2513	−0.816121	−0.408061	−	0.912955i	$-0.633795\pi$
$883$	−24.2513	−0.816121	−0.408061	+	0.912955i	$0.633795\pi$
$884$	−47.2538	−1.58932
$885$	−11.6008	−0.389958
$886$	3.89950	0.131006
$887$	22.1374	0.743301	0.371651	−	0.928373i	$-0.378792\pi$
$887$	22.1374	0.743301	0.371651	+	0.928373i	$0.378792\pi$
$888$	0.712847	0.0239216
$889$	−44.8531	−1.50432
$890$	56.6743	1.89973
$891$	−5.48488	−0.183750
$892$	−15.1903	−0.508610
$893$	20.2467	0.677529
$894$	−28.8044	−0.963363
$895$	−3.01251	−0.100697
$896$	17.1365	0.572489
$897$	35.7056	1.19218
$898$	36.5583	1.21997
$899$	−36.6442	−1.22215
$900$	7.46239	0.248746
$901$	−4.82891	−0.160874
$902$	44.2634	1.47381
$903$	−11.7568	−0.391243
$904$	10.1143	0.336396
$905$	−20.5050	−0.681608
$906$	−26.9779	−0.896281
$907$	−9.25482	−0.307301	−0.153651	−	0.988125i	$-0.549103\pi$
$907$	−9.25482	−0.307301	−0.153651	+	0.988125i	$0.549103\pi$
$908$	−7.60999	−0.252547
$909$	13.7392	0.455702
$910$	−76.2391	−2.52730
$911$	39.6076	1.31226	0.656130	−	0.754648i	$-0.272192\pi$
$911$	39.6076	1.31226	0.656130	+	0.754648i	$0.272192\pi$
$912$	−18.6188	−0.616529
$913$	24.1881	0.800510
$914$	−79.2970	−2.62291
$915$	−38.1351	−1.26071
$916$	−54.3684	−1.79638
$917$	41.8896	1.38332
$918$	−8.17755	−0.269899
$919$	48.5093	1.60018	0.800088	−	0.599883i	$-0.204786\pi$
$919$	48.5093	1.60018	0.800088	+	0.599883i	$0.204786\pi$
$920$	17.4242	0.574460
$921$	−18.3577	−0.604906
$922$	55.5507	1.82947
$923$	73.5514	2.42097
$924$	33.5194	1.10271
$925$	2.55554	0.0840255
$926$	39.2971	1.29138
$927$	8.75982	0.287710
$928$	−38.3714	−1.25960
$929$	1.20749	0.0396165	0.0198083	−	0.999804i	$-0.493694\pi$
$929$	1.20749	0.0396165	0.0198083	+	0.999804i	$0.493694\pi$
$930$	45.9163	1.50565
$931$	3.56109	0.116710
$932$	7.36787	0.241343
$933$	−5.12756	−0.167869
$934$	−36.9538	−1.20917
$935$	60.7615	1.98711
$936$	−4.34569	−0.142043
$937$	29.5751	0.966176	0.483088	−	0.875572i	$-0.339515\pi$
$937$	29.5751	0.966176	0.483088	+	0.875572i	$0.339515\pi$
$938$	45.7665	1.49433
$939$	6.03424	0.196920
$940$	−22.4455	−0.732090
$941$	19.5250	0.636496	0.318248	−	0.948007i	$-0.396906\pi$
$941$	19.5250	0.636496	0.318248	+	0.948007i	$0.396906\pi$
$942$	10.0093	0.326121
$943$	−27.2563	−0.887587
$944$	−12.2683	−0.399301
$945$	−7.21157	−0.234592
$946$	53.4329	1.73725
$947$	35.1864	1.14340	0.571702	−	0.820461i	$-0.306283\pi$
$947$	35.1864	1.14340	0.571702	+	0.820461i	$0.306283\pi$
$948$	17.6563	0.573451
$949$	64.4039	2.09064
$950$	40.2240	1.30504
$951$	0.827870	0.0268455
$952$	8.52029	0.276144
$953$	8.94866	0.289875	0.144938	−	0.989441i	$-0.453702\pi$
$953$	8.94866	0.289875	0.144938	+	0.989441i	$0.453702\pi$
$954$	−2.60477	−0.0843325
$955$	62.4881	2.02207
$956$	−52.1599	−1.68697
$957$	−26.1570	−0.845534
$958$	59.1296	1.91039
$959$	43.0620	1.39054
$960$	30.9587	0.999189
$961$	28.0432	0.904619
$962$	−8.72893	−0.281432
$963$	−5.77299	−0.186032
$964$	17.1082	0.551019
$965$	13.6327	0.438853
$966$	−37.7618	−1.21497
$967$	−26.8745	−0.864227	−0.432114	−	0.901819i	$-0.642232\pi$
$967$	−26.8745	−0.864227	−0.432114	+	0.901819i	$0.642232\pi$
$968$	−16.4759	−0.529556
$969$	−24.0933	−0.773988
$970$	50.7293	1.62882
$971$	3.25044	0.104312	0.0521558	−	0.998639i	$-0.483391\pi$
$971$	3.25044	0.104312	0.0521558	+	0.998639i	$0.483391\pi$
$972$	−2.41107	−0.0773350
$973$	−37.5286	−1.20311
$974$	25.9456	0.831349
$975$	−15.5792	−0.498933
$976$	−40.3294	−1.29091
$977$	12.9139	0.413153	0.206577	−	0.978430i	$-0.433768\pi$
$977$	12.9139	0.413153	0.206577	+	0.978430i	$0.433768\pi$
$978$	17.3410	0.554503
$979$	52.0201	1.66257
$980$	−3.94782	−0.126109
$981$	9.31746	0.297484
$982$	24.1729	0.771389
$983$	50.8038	1.62039	0.810194	−	0.586161i	$-0.199361\pi$
$983$	50.8038	1.62039	0.810194	+	0.586161i	$0.199361\pi$
$984$	3.31733	0.105753
$985$	−78.0663	−2.48740
$986$	−38.9981	−1.24195
$987$	8.29334	0.263980
$988$	−75.0982	−2.38919
$989$	−32.9026	−1.04624
$990$	32.7754	1.04167
$991$	−46.7649	−1.48554	−0.742769	−	0.669548i	$-0.766488\pi$
$991$	−46.7649	−1.48554	−0.742769	+	0.669548i	$0.766488\pi$
$992$	61.8261	1.96298
$993$	18.2306	0.578530
$994$	−77.7869	−2.46725
$995$	−32.5408	−1.03161
$996$	10.6327	0.336911
$997$	10.1826	0.322486	0.161243	−	0.986915i	$-0.448450\pi$
$997$	10.1826	0.322486	0.161243	+	0.986915i	$0.448450\pi$
$998$	28.5141	0.902597
$999$	−0.825683	−0.0261234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.b.1.17	✓	104

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.b.1.17	✓	104	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10025 q^{2} -1.00000 q^{3} +2.41107 q^{4} +2.84518 q^{5} +2.10025 q^{6} +2.53466 q^{7} -0.863342 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.10025 q^{2} -1.00000 q^{3} +2.41107 q^{4} +2.84518 q^{5} +2.10025 q^{6} +2.53466 q^{7} -0.863342 q^{8} +1.00000 q^{9} -5.97560 q^{10} -5.48488 q^{11} -2.41107 q^{12} +5.03357 q^{13} -5.32343 q^{14} -2.84518 q^{15} -3.00889 q^{16} -3.89360 q^{17} -2.10025 q^{18} -6.18791 q^{19} +6.85992 q^{20} -2.53466 q^{21} +11.5196 q^{22} -7.09350 q^{23} +0.863342 q^{24} +3.09506 q^{25} -10.5718 q^{26} -1.00000 q^{27} +6.11123 q^{28} -4.76892 q^{29} +5.97560 q^{30} +7.68396 q^{31} +8.04612 q^{32} +5.48488 q^{33} +8.17755 q^{34} +7.21157 q^{35} +2.41107 q^{36} +0.825683 q^{37} +12.9962 q^{38} -5.03357 q^{39} -2.45636 q^{40} +3.84243 q^{41} +5.32343 q^{42} +4.63842 q^{43} -13.2244 q^{44} +2.84518 q^{45} +14.8982 q^{46} -3.27197 q^{47} +3.00889 q^{48} -0.575491 q^{49} -6.50041 q^{50} +3.89360 q^{51} +12.1363 q^{52} +1.24022 q^{53} +2.10025 q^{54} -15.6055 q^{55} -2.18828 q^{56} +6.18791 q^{57} +10.0159 q^{58} +4.07736 q^{59} -6.85992 q^{60} +13.4034 q^{61} -16.1383 q^{62} +2.53466 q^{63} -10.8811 q^{64} +14.3214 q^{65} -11.5196 q^{66} -8.59718 q^{67} -9.38773 q^{68} +7.09350 q^{69} -15.1461 q^{70} +14.6122 q^{71} -0.863342 q^{72} +12.7949 q^{73} -1.73414 q^{74} -3.09506 q^{75} -14.9195 q^{76} -13.9023 q^{77} +10.5718 q^{78} -7.32304 q^{79} -8.56085 q^{80} +1.00000 q^{81} -8.07008 q^{82} -4.40997 q^{83} -6.11123 q^{84} -11.0780 q^{85} -9.74186 q^{86} +4.76892 q^{87} +4.73532 q^{88} -9.48428 q^{89} -5.97560 q^{90} +12.7584 q^{91} -17.1029 q^{92} -7.68396 q^{93} +6.87197 q^{94} -17.6057 q^{95} -8.04612 q^{96} -8.48940 q^{97} +1.20868 q^{98} -5.48488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

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