LMFDB - Embedded newform 2005.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.609831 q^{2} -1.81759 q^{3} -1.62811 q^{4} -1.00000 q^{5} -1.10842 q^{6} -0.599157 q^{7} -2.21253 q^{8} +0.303632 q^{9} +O(q^{10})$	\(q+0.609831 q^{2} -1.81759 q^{3} -1.62811 q^{4} -1.00000 q^{5} -1.10842 q^{6} -0.599157 q^{7} -2.21253 q^{8} +0.303632 q^{9} -0.609831 q^{10} +4.34337 q^{11} +2.95923 q^{12} +3.98107 q^{13} -0.365385 q^{14} +1.81759 q^{15} +1.90694 q^{16} -2.99169 q^{17} +0.185164 q^{18} +0.628674 q^{19} +1.62811 q^{20} +1.08902 q^{21} +2.64872 q^{22} -0.714627 q^{23} +4.02148 q^{24} +1.00000 q^{25} +2.42778 q^{26} +4.90089 q^{27} +0.975491 q^{28} -5.57591 q^{29} +1.10842 q^{30} +2.81364 q^{31} +5.58798 q^{32} -7.89446 q^{33} -1.82443 q^{34} +0.599157 q^{35} -0.494345 q^{36} +2.41696 q^{37} +0.383385 q^{38} -7.23595 q^{39} +2.21253 q^{40} -6.21248 q^{41} +0.664119 q^{42} +9.39058 q^{43} -7.07147 q^{44} -0.303632 q^{45} -0.435802 q^{46} -6.03028 q^{47} -3.46603 q^{48} -6.64101 q^{49} +0.609831 q^{50} +5.43767 q^{51} -6.48160 q^{52} +4.23138 q^{53} +2.98872 q^{54} -4.34337 q^{55} +1.32565 q^{56} -1.14267 q^{57} -3.40036 q^{58} -5.81298 q^{59} -2.95923 q^{60} +1.24473 q^{61} +1.71584 q^{62} -0.181923 q^{63} -0.406155 q^{64} -3.98107 q^{65} -4.81429 q^{66} -7.79466 q^{67} +4.87079 q^{68} +1.29890 q^{69} +0.365385 q^{70} -15.3825 q^{71} -0.671796 q^{72} +0.118745 q^{73} +1.47394 q^{74} -1.81759 q^{75} -1.02355 q^{76} -2.60236 q^{77} -4.41271 q^{78} -5.01829 q^{79} -1.90694 q^{80} -9.81870 q^{81} -3.78856 q^{82} -2.68854 q^{83} -1.77304 q^{84} +2.99169 q^{85} +5.72667 q^{86} +10.1347 q^{87} -9.60985 q^{88} +6.39100 q^{89} -0.185164 q^{90} -2.38529 q^{91} +1.16349 q^{92} -5.11404 q^{93} -3.67745 q^{94} -0.628674 q^{95} -10.1566 q^{96} -1.66687 q^{97} -4.04990 q^{98} +1.31879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9}+O(q^{10}) $ 29 * q - 5 * q^2 - 3 * q^3 + 19 * q^4 - 29 * q^5 - 6 * q^6 + 12 * q^7 - 15 * q^8 + 14 * q^9	\( 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9} + 5 q^{10} - 38 q^{11} - 6 q^{12} + 5 q^{13} - 18 q^{14} + 3 q^{15} + 7 q^{16} - 16 q^{17} - 2 q^{18} - 18 q^{19} - 19 q^{20} - 20 q^{21} - 2 q^{22} - 19 q^{23} - 19 q^{24} + 29 q^{25} - 21 q^{26} - 21 q^{27} + 26 q^{28} - 31 q^{29} + 6 q^{30} - 13 q^{31} - 30 q^{32} + 2 q^{33} - 14 q^{34} - 12 q^{35} - 29 q^{36} - q^{37} - 23 q^{38} - 39 q^{39} + 15 q^{40} - 24 q^{41} - 20 q^{42} - 27 q^{43} - 76 q^{44} - 14 q^{45} - 11 q^{46} - 5 q^{47} - 2 q^{48} - 11 q^{49} - 5 q^{50} - 58 q^{51} + 11 q^{52} - 37 q^{53} - 18 q^{54} + 38 q^{55} - 50 q^{56} - 6 q^{57} + 31 q^{58} - 67 q^{59} + 6 q^{60} - 31 q^{61} - 19 q^{62} - 2 q^{63} - 13 q^{64} - 5 q^{65} + 6 q^{66} - 17 q^{67} - 16 q^{68} - 48 q^{69} + 18 q^{70} - 53 q^{71} + 9 q^{72} + 29 q^{73} - 59 q^{74} - 3 q^{75} - 21 q^{76} - 62 q^{77} - 12 q^{78} - 13 q^{79} - 7 q^{80} - 11 q^{81} + 32 q^{82} - 72 q^{83} - 58 q^{84} + 16 q^{85} - 43 q^{86} + 4 q^{87} + 12 q^{88} - 38 q^{89} + 2 q^{90} - 45 q^{91} - 37 q^{92} - 27 q^{93} - 44 q^{94} + 18 q^{95} - 21 q^{96} + 32 q^{97} - 32 q^{98} - 107 q^{99}+O(q^{100}) \) 29 * q - 5 * q^2 - 3 * q^3 + 19 * q^4 - 29 * q^5 - 6 * q^6 + 12 * q^7 - 15 * q^8 + 14 * q^9 + 5 * q^10 - 38 * q^11 - 6 * q^12 + 5 * q^13 - 18 * q^14 + 3 * q^15 + 7 * q^16 - 16 * q^17 - 2 * q^18 - 18 * q^19 - 19 * q^20 - 20 * q^21 - 2 * q^22 - 19 * q^23 - 19 * q^24 + 29 * q^25 - 21 * q^26 - 21 * q^27 + 26 * q^28 - 31 * q^29 + 6 * q^30 - 13 * q^31 - 30 * q^32 + 2 * q^33 - 14 * q^34 - 12 * q^35 - 29 * q^36 - q^37 - 23 * q^38 - 39 * q^39 + 15 * q^40 - 24 * q^41 - 20 * q^42 - 27 * q^43 - 76 * q^44 - 14 * q^45 - 11 * q^46 - 5 * q^47 - 2 * q^48 - 11 * q^49 - 5 * q^50 - 58 * q^51 + 11 * q^52 - 37 * q^53 - 18 * q^54 + 38 * q^55 - 50 * q^56 - 6 * q^57 + 31 * q^58 - 67 * q^59 + 6 * q^60 - 31 * q^61 - 19 * q^62 - 2 * q^63 - 13 * q^64 - 5 * q^65 + 6 * q^66 - 17 * q^67 - 16 * q^68 - 48 * q^69 + 18 * q^70 - 53 * q^71 + 9 * q^72 + 29 * q^73 - 59 * q^74 - 3 * q^75 - 21 * q^76 - 62 * q^77 - 12 * q^78 - 13 * q^79 - 7 * q^80 - 11 * q^81 + 32 * q^82 - 72 * q^83 - 58 * q^84 + 16 * q^85 - 43 * q^86 + 4 * q^87 + 12 * q^88 - 38 * q^89 + 2 * q^90 - 45 * q^91 - 37 * q^92 - 27 * q^93 - 44 * q^94 + 18 * q^95 - 21 * q^96 + 32 * q^97 - 32 * q^98 - 107 * 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 $

$2$	0.609831	0.431216	0.215608	−	0.976480i	$-0.430827\pi$
$2$	0.609831	0.431216	0.215608	+	0.976480i	$0.430827\pi$
$3$	−1.81759	−1.04939	−0.524693	−	0.851292i	$-0.675820\pi$
$3$	−1.81759	−1.04939	−0.524693	+	0.851292i	$0.675820\pi$
$4$	−1.62811	−0.814053
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.10842	−0.452512
$7$	−0.599157	−0.226460	−0.113230	−	0.993569i	$-0.536120\pi$
$7$	−0.599157	−0.226460	−0.113230	+	0.993569i	$0.536120\pi$
$8$	−2.21253	−0.782248
$9$	0.303632	0.101211
$10$	−0.609831	−0.192846
$11$	4.34337	1.30958	0.654788	−	0.755813i	$-0.272758\pi$
$11$	4.34337	1.30958	0.654788	+	0.755813i	$0.272758\pi$
$12$	2.95923	0.854256
$13$	3.98107	1.10415	0.552075	−	0.833794i	$-0.313836\pi$
$13$	3.98107	1.10415	0.552075	+	0.833794i	$0.313836\pi$
$14$	−0.365385	−0.0976532
$15$	1.81759	0.469300
$16$	1.90694	0.476735
$17$	−2.99169	−0.725592	−0.362796	−	0.931869i	$-0.618178\pi$
$17$	−2.99169	−0.725592	−0.362796	+	0.931869i	$0.618178\pi$
$18$	0.185164	0.0436437
$19$	0.628674	0.144228	0.0721138	−	0.997396i	$-0.477026\pi$
$19$	0.628674	0.144228	0.0721138	+	0.997396i	$0.477026\pi$
$20$	1.62811	0.364056
$21$	1.08902	0.237644
$22$	2.64872	0.564710
$23$	−0.714627	−0.149010	−0.0745051	−	0.997221i	$-0.523738\pi$
$23$	−0.714627	−0.149010	−0.0745051	+	0.997221i	$0.523738\pi$
$24$	4.02148	0.820880
$25$	1.00000	0.200000
$26$	2.42778	0.476127
$27$	4.90089	0.943177
$28$	0.975491	0.184350
$29$	−5.57591	−1.03542	−0.517710	−	0.855556i	$-0.673215\pi$
$29$	−5.57591	−1.03542	−0.517710	+	0.855556i	$0.673215\pi$
$30$	1.10842	0.202369
$31$	2.81364	0.505344	0.252672	−	0.967552i	$-0.418691\pi$
$31$	2.81364	0.505344	0.252672	+	0.967552i	$0.418691\pi$
$32$	5.58798	0.987824
$33$	−7.89446	−1.37425
$34$	−1.82443	−0.312887
$35$	0.599157	0.101276
$36$	−0.494345	−0.0823909
$37$	2.41696	0.397346	0.198673	−	0.980066i	$-0.436337\pi$
$37$	2.41696	0.397346	0.198673	+	0.980066i	$0.436337\pi$
$38$	0.383385	0.0621932
$39$	−7.23595	−1.15868
$40$	2.21253	0.349832
$41$	−6.21248	−0.970226	−0.485113	−	0.874451i	$-0.661222\pi$
$41$	−6.21248	−0.970226	−0.485113	+	0.874451i	$0.661222\pi$
$42$	0.664119	0.102476
$43$	9.39058	1.43205	0.716025	−	0.698075i	$-0.245960\pi$
$43$	9.39058	1.43205	0.716025	+	0.698075i	$0.245960\pi$
$44$	−7.07147	−1.06606
$45$	−0.303632	−0.0452628
$46$	−0.435802	−0.0642555
$47$	−6.03028	−0.879607	−0.439803	−	0.898094i	$-0.644952\pi$
$47$	−6.03028	−0.879607	−0.439803	+	0.898094i	$0.644952\pi$
$48$	−3.46603	−0.500279
$49$	−6.64101	−0.948716
$50$	0.609831	0.0862432
$51$	5.43767	0.761426
$52$	−6.48160	−0.898837
$53$	4.23138	0.581225	0.290612	−	0.956841i	$-0.406141\pi$
$53$	4.23138	0.581225	0.290612	+	0.956841i	$0.406141\pi$
$54$	2.98872	0.406713
$55$	−4.34337	−0.585660
$56$	1.32565	0.177148
$57$	−1.14267	−0.151350
$58$	−3.40036	−0.446489
$59$	−5.81298	−0.756786	−0.378393	−	0.925645i	$-0.623523\pi$
$59$	−5.81298	−0.756786	−0.378393	+	0.925645i	$0.623523\pi$
$60$	−2.95923	−0.382035
$61$	1.24473	0.159371	0.0796856	−	0.996820i	$-0.474608\pi$
$61$	1.24473	0.159371	0.0796856	+	0.996820i	$0.474608\pi$
$62$	1.71584	0.217912
$63$	−0.181923	−0.0229202
$64$	−0.406155	−0.0507694
$65$	−3.98107	−0.493791
$66$	−4.81429	−0.592598
$67$	−7.79466	−0.952270	−0.476135	−	0.879372i	$-0.657963\pi$
$67$	−7.79466	−0.952270	−0.476135	+	0.879372i	$0.657963\pi$
$68$	4.87079	0.590671
$69$	1.29890	0.156369
$70$	0.365385	0.0436718
$71$	−15.3825	−1.82556	−0.912782	−	0.408447i	$-0.866070\pi$
$71$	−15.3825	−1.82556	−0.912782	+	0.408447i	$0.866070\pi$
$72$	−0.671796	−0.0791719
$73$	0.118745	0.0138980	0.00694901	−	0.999976i	$-0.497788\pi$
$73$	0.118745	0.0138980	0.00694901	+	0.999976i	$0.497788\pi$
$74$	1.47394	0.171342
$75$	−1.81759	−0.209877
$76$	−1.02355	−0.117409
$77$	−2.60236	−0.296566
$78$	−4.41271	−0.499641
$79$	−5.01829	−0.564602	−0.282301	−	0.959326i	$-0.591098\pi$
$79$	−5.01829	−0.564602	−0.282301	+	0.959326i	$0.591098\pi$
$80$	−1.90694	−0.213202
$81$	−9.81870	−1.09097
$82$	−3.78856	−0.418377
$83$	−2.68854	−0.295105	−0.147553	−	0.989054i	$-0.547140\pi$
$83$	−2.68854	−0.295105	−0.147553	+	0.989054i	$0.547140\pi$
$84$	−1.77304	−0.193455
$85$	2.99169	0.324495
$86$	5.72667	0.617523
$87$	10.1347	1.08655
$88$	−9.60985	−1.02441
$89$	6.39100	0.677445	0.338722	−	0.940886i	$-0.390005\pi$
$89$	6.39100	0.677445	0.338722	+	0.940886i	$0.390005\pi$
$90$	−0.185164	−0.0195180
$91$	−2.38529	−0.250046
$92$	1.16349	0.121302
$93$	−5.11404	−0.530301
$94$	−3.67745	−0.379300
$95$	−0.628674	−0.0645005
$96$	−10.1566	−1.03661
$97$	−1.66687	−0.169245	−0.0846225	−	0.996413i	$-0.526968\pi$
$97$	−1.66687	−0.169245	−0.0846225	+	0.996413i	$0.526968\pi$
$98$	−4.04990	−0.409101
$99$	1.31879	0.132543
$100$	−1.62811	−0.162811
$101$	1.77940	0.177057	0.0885286	−	0.996074i	$-0.471784\pi$
$101$	1.77940	0.177057	0.0885286	+	0.996074i	$0.471784\pi$
$102$	3.31606	0.328339
$103$	−4.62488	−0.455703	−0.227851	−	0.973696i	$-0.573170\pi$
$103$	−4.62488	−0.455703	−0.227851	+	0.973696i	$0.573170\pi$
$104$	−8.80825	−0.863720
$105$	−1.08902	−0.106278
$106$	2.58043	0.250633
$107$	−10.7846	−1.04258	−0.521291	−	0.853379i	$-0.674549\pi$
$107$	−10.7846	−1.04258	−0.521291	+	0.853379i	$0.674549\pi$
$108$	−7.97917	−0.767796
$109$	−16.7011	−1.59967	−0.799836	−	0.600218i	$-0.795080\pi$
$109$	−16.7011	−1.59967	−0.799836	+	0.600218i	$0.795080\pi$
$110$	−2.64872	−0.252546
$111$	−4.39305	−0.416970
$112$	−1.14256	−0.107961
$113$	−7.57950	−0.713020	−0.356510	−	0.934292i	$-0.616033\pi$
$113$	−7.57950	−0.713020	−0.356510	+	0.934292i	$0.616033\pi$
$114$	−0.696836	−0.0652647
$115$	0.714627	0.0666393
$116$	9.07816	0.842886
$117$	1.20878	0.111752
$118$	−3.54494	−0.326338
$119$	1.79249	0.164318
$120$	−4.02148	−0.367109
$121$	7.86486	0.714988
$122$	0.759075	0.0687234
$123$	11.2917	1.01814
$124$	−4.58090	−0.411377
$125$	−1.00000	−0.0894427
$126$	−0.110943	−0.00988355
$127$	7.42279	0.658666	0.329333	−	0.944214i	$-0.393176\pi$
$127$	7.42279	0.658666	0.329333	+	0.944214i	$0.393176\pi$
$128$	−11.4236	−1.00972
$129$	−17.0682	−1.50277
$130$	−2.42778	−0.212931
$131$	15.5523	1.35881	0.679404	−	0.733764i	$-0.262238\pi$
$131$	15.5523	1.35881	0.679404	+	0.733764i	$0.262238\pi$
$132$	12.8530	1.11871
$133$	−0.376674	−0.0326618
$134$	−4.75343	−0.410634
$135$	−4.90089	−0.421801
$136$	6.61922	0.567594
$137$	−4.59323	−0.392426	−0.196213	−	0.980561i	$-0.562864\pi$
$137$	−4.59323	−0.392426	−0.196213	+	0.980561i	$0.562864\pi$
$138$	0.792110	0.0674288
$139$	1.05777	0.0897186	0.0448593	−	0.998993i	$-0.485716\pi$
$139$	1.05777	0.0897186	0.0448593	+	0.998993i	$0.485716\pi$
$140$	−0.975491	−0.0824440
$141$	10.9606	0.923047
$142$	−9.38072	−0.787212
$143$	17.2913	1.44597
$144$	0.579008	0.0482507
$145$	5.57591	0.463054
$146$	0.0724143	0.00599305
$147$	12.0706	0.995569
$148$	−3.93507	−0.323461
$149$	2.01682	0.165224	0.0826121	−	0.996582i	$-0.473674\pi$
$149$	2.01682	0.165224	0.0826121	+	0.996582i	$0.473674\pi$
$150$	−1.10842	−0.0905024
$151$	6.76147	0.550241	0.275121	−	0.961410i	$-0.411282\pi$
$151$	6.76147	0.550241	0.275121	+	0.961410i	$0.411282\pi$
$152$	−1.39096	−0.112822
$153$	−0.908375	−0.0734377
$154$	−1.58700	−0.127884
$155$	−2.81364	−0.225997
$156$	11.7809	0.943227
$157$	−7.20307	−0.574867	−0.287434	−	0.957801i	$-0.592802\pi$
$157$	−7.20307	−0.574867	−0.287434	+	0.957801i	$0.592802\pi$
$158$	−3.06031	−0.243465
$159$	−7.69091	−0.609929
$160$	−5.58798	−0.441768
$161$	0.428174	0.0337448
$162$	−5.98775	−0.470442
$163$	1.66689	0.130561	0.0652803	−	0.997867i	$-0.479206\pi$
$163$	1.66689	0.130561	0.0652803	+	0.997867i	$0.479206\pi$
$164$	10.1146	0.789815
$165$	7.89446	0.614583
$166$	−1.63956	−0.127254
$167$	2.04549	0.158285	0.0791423	−	0.996863i	$-0.474782\pi$
$167$	2.04549	0.158285	0.0791423	+	0.996863i	$0.474782\pi$
$168$	−2.40950	−0.185897
$169$	2.84892	0.219148
$170$	1.82443	0.139927
$171$	0.190886	0.0145974
$172$	−15.2889	−1.16576
$173$	22.2539	1.69193	0.845967	−	0.533236i	$-0.179024\pi$
$173$	22.2539	1.69193	0.845967	+	0.533236i	$0.179024\pi$
$174$	6.18046	0.468540
$175$	−0.599157	−0.0452920
$176$	8.28254	0.624320
$177$	10.5656	0.794161
$178$	3.89743	0.292125
$179$	−16.3522	−1.22222	−0.611109	−	0.791547i	$-0.709276\pi$
$179$	−16.3522	−1.22222	−0.611109	+	0.791547i	$0.709276\pi$
$180$	0.494345	0.0368463
$181$	8.85686	0.658325	0.329163	−	0.944273i	$-0.393234\pi$
$181$	8.85686	0.658325	0.329163	+	0.944273i	$0.393234\pi$
$182$	−1.45462	−0.107824
$183$	−2.26241	−0.167242
$184$	1.58114	0.116563
$185$	−2.41696	−0.177699
$186$	−3.11870	−0.228674
$187$	−12.9940	−0.950218
$188$	9.81793	0.716046
$189$	−2.93640	−0.213592
$190$	−0.383385	−0.0278137
$191$	−17.8707	−1.29308	−0.646540	−	0.762880i	$-0.723784\pi$
$191$	−17.8707	−1.29308	−0.646540	+	0.762880i	$0.723784\pi$
$192$	0.738224	0.0532767
$193$	9.68228	0.696945	0.348473	−	0.937319i	$-0.386700\pi$
$193$	9.68228	0.696945	0.348473	+	0.937319i	$0.386700\pi$
$194$	−1.01651	−0.0729811
$195$	7.23595	0.518177
$196$	10.8123	0.772305
$197$	20.6012	1.46777	0.733887	−	0.679272i	$-0.237704\pi$
$197$	20.6012	1.46777	0.733887	+	0.679272i	$0.237704\pi$
$198$	0.804238	0.0571547
$199$	−5.58335	−0.395793	−0.197897	−	0.980223i	$-0.563411\pi$
$199$	−5.58335	−0.395793	−0.197897	+	0.980223i	$0.563411\pi$
$200$	−2.21253	−0.156450
$201$	14.1675	0.999298
$202$	1.08514	0.0763498
$203$	3.34084	0.234481
$204$	−8.85311	−0.619841
$205$	6.21248	0.433898
$206$	−2.82040	−0.196506
$207$	−0.216984	−0.0150814
$208$	7.59166	0.526387
$209$	2.73056	0.188877
$210$	−0.664119	−0.0458286
$211$	6.44464	0.443667	0.221834	−	0.975085i	$-0.428796\pi$
$211$	6.44464	0.443667	0.221834	+	0.975085i	$0.428796\pi$
$212$	−6.88913	−0.473147
$213$	27.9590	1.91572
$214$	−6.57676	−0.449578
$215$	−9.39058	−0.640432
$216$	−10.8434	−0.737799
$217$	−1.68581	−0.114440
$218$	−10.1848	−0.689804
$219$	−0.215829	−0.0145844
$220$	7.07147	0.476758
$221$	−11.9101	−0.801163
$222$	−2.67902	−0.179804
$223$	−26.6940	−1.78757	−0.893783	−	0.448500i	$-0.851958\pi$
$223$	−26.6940	−1.78757	−0.893783	+	0.448500i	$0.851958\pi$
$224$	−3.34807	−0.223703
$225$	0.303632	0.0202421
$226$	−4.62222	−0.307465
$227$	−11.8067	−0.783640	−0.391820	−	0.920042i	$-0.628154\pi$
$227$	−11.8067	−0.783640	−0.391820	+	0.920042i	$0.628154\pi$
$228$	1.86039	0.123207
$229$	−8.43915	−0.557675	−0.278837	−	0.960338i	$-0.589949\pi$
$229$	−8.43915	−0.557675	−0.278837	+	0.960338i	$0.589949\pi$
$230$	0.435802	0.0287359
$231$	4.73002	0.311213
$232$	12.3369	0.809955
$233$	−24.5778	−1.61014	−0.805072	−	0.593177i	$-0.797873\pi$
$233$	−24.5778	−1.61014	−0.805072	+	0.593177i	$0.797873\pi$
$234$	0.737153	0.0481892
$235$	6.03028	0.393372
$236$	9.46415	0.616064
$237$	9.12120	0.592485
$238$	1.09312	0.0708564
$239$	−9.05326	−0.585607	−0.292803	−	0.956173i	$-0.594588\pi$
$239$	−9.05326	−0.585607	−0.292803	+	0.956173i	$0.594588\pi$
$240$	3.46603	0.223732
$241$	−1.78559	−0.115020	−0.0575099	−	0.998345i	$-0.518316\pi$
$241$	−1.78559	−0.115020	−0.0575099	+	0.998345i	$0.518316\pi$
$242$	4.79624	0.308314
$243$	3.14370	0.201669
$244$	−2.02655	−0.129737
$245$	6.64101	0.424279
$246$	6.88605	0.439039
$247$	2.50279	0.159249
$248$	−6.22526	−0.395305
$249$	4.88666	0.309679
$250$	−0.609831	−0.0385691
$251$	−20.1842	−1.27402	−0.637008	−	0.770857i	$-0.719828\pi$
$251$	−20.1842	−1.27402	−0.637008	+	0.770857i	$0.719828\pi$
$252$	0.296190	0.0186582
$253$	−3.10389	−0.195140
$254$	4.52665	0.284027
$255$	−5.43767	−0.340520
$256$	−6.15418	−0.384636
$257$	23.0905	1.44035	0.720174	−	0.693793i	$-0.244062\pi$
$257$	23.0905	1.44035	0.720174	+	0.693793i	$0.244062\pi$
$258$	−10.4087	−0.648020
$259$	−1.44814	−0.0899831
$260$	6.48160	0.401972
$261$	−1.69302	−0.104796
$262$	9.48427	0.585940
$263$	−8.11044	−0.500111	−0.250055	−	0.968232i	$-0.580449\pi$
$263$	−8.11044	−0.500111	−0.250055	+	0.968232i	$0.580449\pi$
$264$	17.4668	1.07500
$265$	−4.23138	−0.259932
$266$	−0.229708	−0.0140843
$267$	−11.6162	−0.710901
$268$	12.6905	0.775198
$269$	−18.9360	−1.15455	−0.577275	−	0.816549i	$-0.695884\pi$
$269$	−18.9360	−1.15455	−0.577275	+	0.816549i	$0.695884\pi$
$270$	−2.98872	−0.181888
$271$	−13.3038	−0.808148	−0.404074	−	0.914726i	$-0.632406\pi$
$271$	−13.3038	−0.808148	−0.404074	+	0.914726i	$0.632406\pi$
$272$	−5.70498	−0.345915
$273$	4.33547	0.262395
$274$	−2.80109	−0.169220
$275$	4.34337	0.261915
$276$	−2.11475	−0.127293
$277$	20.6578	1.24120	0.620602	−	0.784126i	$-0.286888\pi$
$277$	20.6578	1.24120	0.620602	+	0.784126i	$0.286888\pi$
$278$	0.645060	0.0386881
$279$	0.854311	0.0511463
$280$	−1.32565	−0.0792230
$281$	−8.21424	−0.490020	−0.245010	−	0.969520i	$-0.578791\pi$
$281$	−8.21424	−0.490020	−0.245010	+	0.969520i	$0.578791\pi$
$282$	6.68410	0.398032
$283$	−1.24195	−0.0738264	−0.0369132	−	0.999318i	$-0.511753\pi$
$283$	−1.24195	−0.0738264	−0.0369132	+	0.999318i	$0.511753\pi$
$284$	25.0443	1.48611
$285$	1.14267	0.0676860
$286$	10.5448	0.623524
$287$	3.72225	0.219717
$288$	1.69669	0.0999784
$289$	−8.04977	−0.473516
$290$	3.40036	0.199676
$291$	3.02968	0.177603
$292$	−0.193329	−0.0113137
$293$	26.1954	1.53035	0.765177	−	0.643820i	$-0.222651\pi$
$293$	26.1954	1.53035	0.765177	+	0.643820i	$0.222651\pi$
$294$	7.36105	0.429305
$295$	5.81298	0.338445
$296$	−5.34761	−0.310824
$297$	21.2864	1.23516
$298$	1.22992	0.0712473
$299$	−2.84498	−0.164530
$300$	2.95923	0.170851
$301$	−5.62643	−0.324302
$302$	4.12336	0.237273
$303$	−3.23422	−0.185801
$304$	1.19884	0.0687583
$305$	−1.24473	−0.0712730
$306$	−0.553955	−0.0316675
$307$	−18.5224	−1.05713	−0.528565	−	0.848893i	$-0.677270\pi$
$307$	−18.5224	−1.05713	−0.528565	+	0.848893i	$0.677270\pi$
$308$	4.23692	0.241421
$309$	8.40613	0.478208
$310$	−1.71584	−0.0974534
$311$	−11.7864	−0.668347	−0.334174	−	0.942511i	$-0.608457\pi$
$311$	−11.7864	−0.668347	−0.334174	+	0.942511i	$0.608457\pi$
$312$	16.0098	0.906375
$313$	−19.1459	−1.08219	−0.541095	−	0.840961i	$-0.681990\pi$
$313$	−19.1459	−1.08219	−0.541095	+	0.840961i	$0.681990\pi$
$314$	−4.39266	−0.247892
$315$	0.181923	0.0102502
$316$	8.17031	0.459616
$317$	−0.700543	−0.0393464	−0.0196732	−	0.999806i	$-0.506263\pi$
$317$	−0.700543	−0.0393464	−0.0196732	+	0.999806i	$0.506263\pi$
$318$	−4.69016	−0.263011
$319$	−24.2182	−1.35596
$320$	0.406155	0.0227048
$321$	19.6019	1.09407
$322$	0.261114	0.0145513
$323$	−1.88080	−0.104650
$324$	15.9859	0.888105
$325$	3.98107	0.220830
$326$	1.01652	0.0562998
$327$	30.3557	1.67867
$328$	13.7453	0.758958
$329$	3.61308	0.199196
$330$	4.81429	0.265018
$331$	0.302119	0.0166060	0.00830298	−	0.999966i	$-0.497357\pi$
$331$	0.302119	0.0166060	0.00830298	+	0.999966i	$0.497357\pi$
$332$	4.37723	0.240231
$333$	0.733868	0.0402157
$334$	1.24740	0.0682548
$335$	7.79466	0.425868
$336$	2.07670	0.113293
$337$	21.2098	1.15537	0.577684	−	0.816260i	$-0.303956\pi$
$337$	21.2098	1.15537	0.577684	+	0.816260i	$0.303956\pi$
$338$	1.73736	0.0945000
$339$	13.7764	0.748233
$340$	−4.87079	−0.264156
$341$	12.2207	0.661786
$342$	0.116408	0.00629462
$343$	8.17311	0.441306
$344$	−20.7770	−1.12022
$345$	−1.29890	−0.0699304
$346$	13.5711	0.729589
$347$	12.6695	0.680137	0.340069	−	0.940401i	$-0.389550\pi$
$347$	12.6695	0.680137	0.340069	+	0.940401i	$0.389550\pi$
$348$	−16.5004	−0.884513
$349$	9.33452	0.499665	0.249833	−	0.968289i	$-0.419624\pi$
$349$	9.33452	0.499665	0.249833	+	0.968289i	$0.419624\pi$
$350$	−0.365385	−0.0195306
$351$	19.5108	1.04141
$352$	24.2707	1.29363
$353$	26.9486	1.43433	0.717165	−	0.696904i	$-0.245440\pi$
$353$	26.9486	1.43433	0.717165	+	0.696904i	$0.245440\pi$
$354$	6.44325	0.342455
$355$	15.3825	0.816417
$356$	−10.4052	−0.551476
$357$	−3.25802	−0.172433
$358$	−9.97206	−0.527040
$359$	22.2423	1.17390	0.586951	−	0.809622i	$-0.300328\pi$
$359$	22.2423	1.17390	0.586951	+	0.809622i	$0.300328\pi$
$360$	0.671796	0.0354068
$361$	−18.6048	−0.979198
$362$	5.40119	0.283880
$363$	−14.2951	−0.750298
$364$	3.88350	0.203551
$365$	−0.118745	−0.00621538
$366$	−1.37969	−0.0721174
$367$	−34.8526	−1.81929	−0.909646	−	0.415384i	$-0.863647\pi$
$367$	−34.8526	−1.81929	−0.909646	+	0.415384i	$0.863647\pi$
$368$	−1.36275	−0.0710383
$369$	−1.88631	−0.0981973
$370$	−1.47394	−0.0766265
$371$	−2.53526	−0.131624
$372$	8.32619	0.431693
$373$	−8.97522	−0.464719	−0.232360	−	0.972630i	$-0.574645\pi$
$373$	−8.97522	−0.464719	−0.232360	+	0.972630i	$0.574645\pi$
$374$	−7.92417	−0.409749
$375$	1.81759	0.0938599
$376$	13.3422	0.688071
$377$	−22.1981	−1.14326
$378$	−1.79071	−0.0921042
$379$	34.1467	1.75400	0.877000	−	0.480491i	$-0.159542\pi$
$379$	34.1467	1.75400	0.877000	+	0.480491i	$0.159542\pi$
$380$	1.02355	0.0525068
$381$	−13.4916	−0.691195
$382$	−10.8981	−0.557596
$383$	0.563200	0.0287782	0.0143891	−	0.999896i	$-0.495420\pi$
$383$	0.563200	0.0287782	0.0143891	+	0.999896i	$0.495420\pi$
$384$	20.7635	1.05958
$385$	2.60236	0.132629
$386$	5.90456	0.300534
$387$	2.85128	0.144939
$388$	2.71384	0.137774
$389$	−29.2715	−1.48413	−0.742063	−	0.670330i	$-0.766153\pi$
$389$	−29.2715	−1.48413	−0.742063	+	0.670330i	$0.766153\pi$
$390$	4.41271	0.223446
$391$	2.13795	0.108121
$392$	14.6935	0.742131
$393$	−28.2677	−1.42591
$394$	12.5633	0.632927
$395$	5.01829	0.252498
$396$	−2.14712	−0.107897
$397$	−6.57461	−0.329970	−0.164985	−	0.986296i	$-0.552758\pi$
$397$	−6.57461	−0.329970	−0.164985	+	0.986296i	$0.552758\pi$
$398$	−3.40490	−0.170672
$399$	0.684639	0.0342748
$400$	1.90694	0.0953470
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	8.63979	0.430913
$403$	11.2013	0.557976
$404$	−2.89705	−0.144134
$405$	9.81870	0.487895
$406$	2.03735	0.101112
$407$	10.4978	0.520355
$408$	−12.0310	−0.595625
$409$	−9.36855	−0.463245	−0.231623	−	0.972806i	$-0.574403\pi$
$409$	−9.36855	−0.463245	−0.231623	+	0.972806i	$0.574403\pi$
$410$	3.78856	0.187104
$411$	8.34860	0.411806
$412$	7.52979	0.370966
$413$	3.48289	0.171382
$414$	−0.132324	−0.00650335
$415$	2.68854	0.131975
$416$	22.2461	1.09071
$417$	−1.92259	−0.0941495
$418$	1.66518	0.0814467
$419$	−11.4427	−0.559011	−0.279505	−	0.960144i	$-0.590171\pi$
$419$	−11.4427	−0.559011	−0.279505	+	0.960144i	$0.590171\pi$
$420$	1.77304	0.0865156
$421$	23.1259	1.12709	0.563543	−	0.826087i	$-0.309438\pi$
$421$	23.1259	1.12709	0.563543	+	0.826087i	$0.309438\pi$
$422$	3.93014	0.191316
$423$	−1.83099	−0.0890256
$424$	−9.36206	−0.454662
$425$	−2.99169	−0.145118
$426$	17.0503	0.826089
$427$	−0.745788	−0.0360912
$428$	17.5584	0.848717
$429$	−31.4284	−1.51738
$430$	−5.72667	−0.276165
$431$	−24.7704	−1.19315	−0.596574	−	0.802558i	$-0.703472\pi$
$431$	−24.7704	−1.19315	−0.596574	+	0.802558i	$0.703472\pi$
$432$	9.34570	0.449645
$433$	−0.171931	−0.00826246	−0.00413123	−	0.999991i	$-0.501315\pi$
$433$	−0.171931	−0.00826246	−0.00413123	+	0.999991i	$0.501315\pi$
$434$	−1.02806	−0.0493485
$435$	−10.1347	−0.485922
$436$	27.1911	1.30222
$437$	−0.449267	−0.0214914
$438$	−0.131619	−0.00628902
$439$	14.8253	0.707571	0.353786	−	0.935327i	$-0.384894\pi$
$439$	14.8253	0.707571	0.353786	+	0.935327i	$0.384894\pi$
$440$	9.60985	0.458132
$441$	−2.01642	−0.0960202
$442$	−7.26318	−0.345474
$443$	−21.5373	−1.02327	−0.511635	−	0.859203i	$-0.670960\pi$
$443$	−21.5373	−1.02327	−0.511635	+	0.859203i	$0.670960\pi$
$444$	7.15235	0.339435
$445$	−6.39100	−0.302963
$446$	−16.2789	−0.770827
$447$	−3.66575	−0.173384
$448$	0.243351	0.0114972
$449$	3.78806	0.178769	0.0893847	−	0.995997i	$-0.471510\pi$
$449$	3.78806	0.178769	0.0893847	+	0.995997i	$0.471510\pi$
$450$	0.185164	0.00872874
$451$	−26.9831	−1.27058
$452$	12.3402	0.580436
$453$	−12.2896	−0.577415
$454$	−7.20012	−0.337918
$455$	2.38529	0.111824
$456$	2.52820	0.118394
$457$	−3.07643	−0.143909	−0.0719547	−	0.997408i	$-0.522924\pi$
$457$	−3.07643	−0.143909	−0.0719547	+	0.997408i	$0.522924\pi$
$458$	−5.14646	−0.240478
$459$	−14.6620	−0.684362
$460$	−1.16349	−0.0542480
$461$	−15.0298	−0.700006	−0.350003	−	0.936749i	$-0.613819\pi$
$461$	−15.0298	−0.700006	−0.350003	+	0.936749i	$0.613819\pi$
$462$	2.88452	0.134200
$463$	2.83308	0.131664	0.0658322	−	0.997831i	$-0.479030\pi$
$463$	2.83308	0.131664	0.0658322	+	0.997831i	$0.479030\pi$
$464$	−10.6329	−0.493621
$465$	5.11404	0.237158
$466$	−14.9883	−0.694320
$467$	−3.14801	−0.145672	−0.0728362	−	0.997344i	$-0.523205\pi$
$467$	−3.14801	−0.145672	−0.0728362	+	0.997344i	$0.523205\pi$
$468$	−1.96802	−0.0909719
$469$	4.67023	0.215651
$470$	3.67745	0.169628
$471$	13.0922	0.603258
$472$	12.8614	0.591995
$473$	40.7868	1.87538
$474$	5.56239	0.255489
$475$	0.628674	0.0288455
$476$	−2.91837	−0.133763
$477$	1.28478	0.0588262
$478$	−5.52096	−0.252523
$479$	−14.8301	−0.677606	−0.338803	−	0.940857i	$-0.610022\pi$
$479$	−14.8301	−0.677606	−0.338803	+	0.940857i	$0.610022\pi$
$480$	10.1566	0.463585
$481$	9.62210	0.438730
$482$	−1.08891	−0.0495983
$483$	−0.778245	−0.0354114
$484$	−12.8048	−0.582038
$485$	1.66687	0.0756886
$486$	1.91713	0.0869628
$487$	11.4534	0.519003	0.259502	−	0.965743i	$-0.416442\pi$
$487$	11.4534	0.519003	0.259502	+	0.965743i	$0.416442\pi$
$488$	−2.75400	−0.124668
$489$	−3.02972	−0.137009
$490$	4.04990	0.182956
$491$	−39.0189	−1.76090	−0.880449	−	0.474141i	$-0.842759\pi$
$491$	−39.0189	−1.76090	−0.880449	+	0.474141i	$0.842759\pi$
$492$	−18.3841	−0.828821
$493$	16.6814	0.751293
$494$	1.52628	0.0686707
$495$	−1.31879	−0.0592751
$496$	5.36544	0.240915
$497$	9.21652	0.413417
$498$	2.98004	0.133539
$499$	−13.4611	−0.602600	−0.301300	−	0.953529i	$-0.597421\pi$
$499$	−13.4611	−0.602600	−0.301300	+	0.953529i	$0.597421\pi$
$500$	1.62811	0.0728111
$501$	−3.71786	−0.166102
$502$	−12.3090	−0.549376
$503$	−13.0492	−0.581835	−0.290918	−	0.956748i	$-0.593961\pi$
$503$	−13.0492	−0.581835	−0.290918	+	0.956748i	$0.593961\pi$
$504$	0.402511	0.0179293
$505$	−1.77940	−0.0791824
$506$	−1.89285	−0.0841475
$507$	−5.17817	−0.229971
$508$	−12.0851	−0.536189
$509$	9.61162	0.426028	0.213014	−	0.977049i	$-0.431672\pi$
$509$	9.61162	0.426028	0.213014	+	0.977049i	$0.431672\pi$
$510$	−3.31606	−0.146838
$511$	−0.0711467	−0.00314735
$512$	19.0943	0.843855
$513$	3.08106	0.136032
$514$	14.0813	0.621101
$515$	4.62488	0.203796
$516$	27.7889	1.22334
$517$	−26.1917	−1.15191
$518$	−0.883121	−0.0388021
$519$	−40.4485	−1.77549
$520$	8.80825	0.386267
$521$	5.21836	0.228621	0.114310	−	0.993445i	$-0.463534\pi$
$521$	5.21836	0.228621	0.114310	+	0.993445i	$0.463534\pi$
$522$	−1.03246	−0.0451895
$523$	−5.59176	−0.244510	−0.122255	−	0.992499i	$-0.539013\pi$
$523$	−5.59176	−0.244510	−0.122255	+	0.992499i	$0.539013\pi$
$524$	−25.3208	−1.10614
$525$	1.08902	0.0475288
$526$	−4.94600	−0.215656
$527$	−8.41754	−0.366674
$528$	−15.0543	−0.655153
$529$	−22.4893	−0.977796
$530$	−2.58043	−0.112087
$531$	−1.76501	−0.0765949
$532$	0.613265	0.0265884
$533$	−24.7323	−1.07128
$534$	−7.08394	−0.306552
$535$	10.7846	0.466257
$536$	17.2459	0.744911
$537$	29.7215	1.28258
$538$	−11.5478	−0.497861
$539$	−28.8444	−1.24241
$540$	7.97917	0.343369
$541$	−25.4373	−1.09363	−0.546817	−	0.837252i	$-0.684161\pi$
$541$	−25.4373	−1.09363	−0.546817	+	0.837252i	$0.684161\pi$
$542$	−8.11307	−0.348486
$543$	−16.0981	−0.690837
$544$	−16.7175	−0.716758
$545$	16.7011	0.715395
$546$	2.64391	0.113149
$547$	23.9478	1.02393	0.511967	−	0.859005i	$-0.328917\pi$
$547$	23.9478	1.02393	0.511967	+	0.859005i	$0.328917\pi$
$548$	7.47826	0.319456
$549$	0.377940	0.0161301
$550$	2.64872	0.112942
$551$	−3.50542	−0.149336
$552$	−2.87386	−0.122319
$553$	3.00675	0.127860
$554$	12.5978	0.535227
$555$	4.39305	0.186475
$556$	−1.72216	−0.0730357
$557$	27.6949	1.17347	0.586735	−	0.809779i	$-0.300413\pi$
$557$	27.6949	1.17347	0.586735	+	0.809779i	$0.300413\pi$
$558$	0.520986	0.0220551
$559$	37.3846	1.58120
$560$	1.14256	0.0482818
$561$	23.6178	0.997145
$562$	−5.00930	−0.211305
$563$	−15.4394	−0.650694	−0.325347	−	0.945595i	$-0.605481\pi$
$563$	−15.4394	−0.650694	−0.325347	+	0.945595i	$0.605481\pi$
$564$	−17.8450	−0.751409
$565$	7.57950	0.318872
$566$	−0.757381	−0.0318351
$567$	5.88294	0.247060
$568$	34.0342	1.42804
$569$	−25.5404	−1.07071	−0.535355	−	0.844627i	$-0.679822\pi$
$569$	−25.5404	−1.07071	−0.535355	+	0.844627i	$0.679822\pi$
$570$	0.696836	0.0291873
$571$	−16.6291	−0.695906	−0.347953	−	0.937512i	$-0.613123\pi$
$571$	−16.6291	−0.695906	−0.347953	+	0.937512i	$0.613123\pi$
$572$	−28.1520	−1.17709
$573$	32.4816	1.35694
$574$	2.26994	0.0947456
$575$	−0.714627	−0.0298020
$576$	−0.123322	−0.00513841
$577$	−19.5996	−0.815943	−0.407971	−	0.912995i	$-0.633764\pi$
$577$	−19.5996	−0.815943	−0.407971	+	0.912995i	$0.633764\pi$
$578$	−4.90900	−0.204187
$579$	−17.5984	−0.731365
$580$	−9.07816	−0.376950
$581$	1.61086	0.0668296
$582$	1.84760	0.0765854
$583$	18.3784	0.761157
$584$	−0.262727	−0.0108717
$585$	−1.20878	−0.0499770
$586$	15.9748	0.659913
$587$	19.3470	0.798534	0.399267	−	0.916835i	$-0.369265\pi$
$587$	19.3470	0.798534	0.399267	+	0.916835i	$0.369265\pi$
$588$	−19.6523	−0.810446
$589$	1.76886	0.0728846
$590$	3.54494	0.145943
$591$	−37.4445	−1.54026
$592$	4.60900	0.189429
$593$	22.6233	0.929026	0.464513	−	0.885566i	$-0.346229\pi$
$593$	22.6233	0.929026	0.464513	+	0.885566i	$0.346229\pi$
$594$	12.9811	0.532621
$595$	−1.79249	−0.0734851
$596$	−3.28359	−0.134501
$597$	10.1482	0.415340
$598$	−1.73496	−0.0709478
$599$	27.1862	1.11080	0.555398	−	0.831585i	$-0.312566\pi$
$599$	27.1862	1.11080	0.555398	+	0.831585i	$0.312566\pi$
$600$	4.02148	0.164176
$601$	14.2904	0.582918	0.291459	−	0.956583i	$-0.405859\pi$
$601$	14.2904	0.582918	0.291459	+	0.956583i	$0.405859\pi$
$602$	−3.43117	−0.139844
$603$	−2.36671	−0.0963799
$604$	−11.0084	−0.447925
$605$	−7.86486	−0.319752
$606$	−1.97233	−0.0801205
$607$	9.54044	0.387235	0.193617	−	0.981077i	$-0.437978\pi$
$607$	9.54044	0.387235	0.193617	+	0.981077i	$0.437978\pi$
$608$	3.51301	0.142471
$609$	−6.07228	−0.246061
$610$	−0.759075	−0.0307341
$611$	−24.0070	−0.971218
$612$	1.47893	0.0597822
$613$	−4.00536	−0.161775	−0.0808875	−	0.996723i	$-0.525775\pi$
$613$	−4.00536	−0.161775	−0.0808875	+	0.996723i	$0.525775\pi$
$614$	−11.2955	−0.455851
$615$	−11.2917	−0.455327
$616$	5.75781	0.231989
$617$	28.2740	1.13827	0.569134	−	0.822245i	$-0.307279\pi$
$617$	28.2740	1.13827	0.569134	+	0.822245i	$0.307279\pi$
$618$	5.12632	0.206211
$619$	42.6719	1.71513	0.857564	−	0.514377i	$-0.171977\pi$
$619$	42.6719	1.71513	0.857564	+	0.514377i	$0.171977\pi$
$620$	4.58090	0.183973
$621$	−3.50231	−0.140543
$622$	−7.18774	−0.288202
$623$	−3.82921	−0.153414
$624$	−13.7985	−0.552383
$625$	1.00000	0.0400000
$626$	−11.6758	−0.466658
$627$	−4.96304	−0.198205
$628$	11.7274	0.467973
$629$	−7.23082	−0.288311
$630$	0.110943	0.00442006
$631$	−7.50062	−0.298595	−0.149298	−	0.988792i	$-0.547701\pi$
$631$	−7.50062	−0.298595	−0.149298	+	0.988792i	$0.547701\pi$
$632$	11.1031	0.441659
$633$	−11.7137	−0.465578
$634$	−0.427213	−0.0169668
$635$	−7.42279	−0.294564
$636$	12.5216	0.496514
$637$	−26.4383	−1.04752
$638$	−14.7690	−0.584712
$639$	−4.67062	−0.184767
$640$	11.4236	0.451559
$641$	−37.9920	−1.50059	−0.750296	−	0.661102i	$-0.770089\pi$
$641$	−37.9920	−1.50059	−0.750296	+	0.661102i	$0.770089\pi$
$642$	11.9539	0.471781
$643$	−6.20303	−0.244624	−0.122312	−	0.992492i	$-0.539031\pi$
$643$	−6.20303	−0.244624	−0.122312	+	0.992492i	$0.539031\pi$
$644$	−0.697112	−0.0274701
$645$	17.0682	0.672061
$646$	−1.14697	−0.0451269
$647$	4.58111	0.180102	0.0900509	−	0.995937i	$-0.471297\pi$
$647$	4.58111	0.180102	0.0900509	+	0.995937i	$0.471297\pi$
$648$	21.7242	0.853407
$649$	−25.2479	−0.991068
$650$	2.42778	0.0952254
$651$	3.06411	0.120092
$652$	−2.71387	−0.106283
$653$	−39.3734	−1.54080	−0.770401	−	0.637560i	$-0.779944\pi$
$653$	−39.3734	−1.54080	−0.770401	+	0.637560i	$0.779944\pi$
$654$	18.5119	0.723871
$655$	−15.5523	−0.607678
$656$	−11.8468	−0.462541
$657$	0.0360547	0.00140663
$658$	2.20337	0.0858964
$659$	0.674449	0.0262728	0.0131364	−	0.999914i	$-0.495818\pi$
$659$	0.674449	0.0262728	0.0131364	+	0.999914i	$0.495818\pi$
$660$	−12.8530	−0.500303
$661$	−28.6226	−1.11329	−0.556645	−	0.830750i	$-0.687912\pi$
$661$	−28.6226	−1.11329	−0.556645	+	0.830750i	$0.687912\pi$
$662$	0.184242	0.00716075
$663$	21.6478	0.840729
$664$	5.94848	0.230846
$665$	0.376674	0.0146068
$666$	0.447536	0.0173417
$667$	3.98469	0.154288
$668$	−3.33027	−0.128852
$669$	48.5188	1.87585
$670$	4.75343	0.183641
$671$	5.40632	0.208709
$672$	6.08543	0.234750
$673$	34.0012	1.31065	0.655324	−	0.755347i	$-0.272532\pi$
$673$	34.0012	1.31065	0.655324	+	0.755347i	$0.272532\pi$
$674$	12.9344	0.498213
$675$	4.90089	0.188635
$676$	−4.63835	−0.178398
$677$	−36.8098	−1.41471	−0.707357	−	0.706856i	$-0.750113\pi$
$677$	−36.8098	−1.41471	−0.707357	+	0.706856i	$0.750113\pi$
$678$	8.40130	0.322650
$679$	0.998716	0.0383272
$680$	−6.61922	−0.253836
$681$	21.4598	0.822341
$682$	7.45255	0.285373
$683$	−45.6755	−1.74772	−0.873862	−	0.486174i	$-0.838392\pi$
$683$	−45.6755	−1.74772	−0.873862	+	0.486174i	$0.838392\pi$
$684$	−0.310782	−0.0118830
$685$	4.59323	0.175498
$686$	4.98422	0.190298
$687$	15.3389	0.585216
$688$	17.9073	0.682708
$689$	16.8454	0.641759
$690$	−0.792110	−0.0301551
$691$	−19.4179	−0.738691	−0.369346	−	0.929292i	$-0.620418\pi$
$691$	−19.4179	−0.738691	−0.369346	+	0.929292i	$0.620418\pi$
$692$	−36.2317	−1.37732
$693$	−0.790160	−0.0300157
$694$	7.72629	0.293286
$695$	−1.05777	−0.0401234
$696$	−22.4234	−0.849956
$697$	18.5858	0.703989
$698$	5.69248	0.215464
$699$	44.6723	1.68966
$700$	0.975491	0.0368701
$701$	−14.1061	−0.532779	−0.266390	−	0.963865i	$-0.585831\pi$
$701$	−14.1061	−0.532779	−0.266390	+	0.963865i	$0.585831\pi$
$702$	11.8983	0.449072
$703$	1.51948	0.0573083
$704$	−1.76408	−0.0664864
$705$	−10.9606	−0.412799
$706$	16.4341	0.618505
$707$	−1.06614	−0.0400964
$708$	−17.2019	−0.646489
$709$	−0.330646	−0.0124177	−0.00620884	−	0.999981i	$-0.501976\pi$
$709$	−0.330646	−0.0124177	−0.00620884	+	0.999981i	$0.501976\pi$
$710$	9.38072	0.352052
$711$	−1.52372	−0.0571438
$712$	−14.1403	−0.529930
$713$	−2.01070	−0.0753014
$714$	−1.98684	−0.0743557
$715$	−17.2913	−0.646657
$716$	26.6230	0.994950
$717$	16.4551	0.614527
$718$	13.5640	0.506205
$719$	33.1389	1.23587	0.617937	−	0.786228i	$-0.287969\pi$
$719$	33.1389	1.23587	0.617937	+	0.786228i	$0.287969\pi$
$720$	−0.579008	−0.0215784
$721$	2.77103	0.103198
$722$	−11.3458	−0.422246
$723$	3.24546	0.120700
$724$	−14.4199	−0.535912
$725$	−5.57591	−0.207084
$726$	−8.71760	−0.323540
$727$	14.6370	0.542855	0.271427	−	0.962459i	$-0.412504\pi$
$727$	14.6370	0.542855	0.271427	+	0.962459i	$0.412504\pi$
$728$	5.27752	0.195598
$729$	23.7421	0.879339
$730$	−0.0724143	−0.00268017
$731$	−28.0937	−1.03908
$732$	3.68344	0.136144
$733$	13.0046	0.480337	0.240168	−	0.970731i	$-0.422797\pi$
$733$	13.0046	0.480337	0.240168	+	0.970731i	$0.422797\pi$
$734$	−21.2542	−0.784508
$735$	−12.0706	−0.445232
$736$	−3.99332	−0.147196
$737$	−33.8551	−1.24707
$738$	−1.15033	−0.0423442
$739$	−50.3789	−1.85322	−0.926609	−	0.376026i	$-0.877290\pi$
$739$	−50.3789	−1.85322	−0.926609	+	0.376026i	$0.877290\pi$
$740$	3.93507	0.144656
$741$	−4.54905	−0.167114
$742$	−1.54608	−0.0567584
$743$	−28.1094	−1.03123	−0.515617	−	0.856819i	$-0.672437\pi$
$743$	−28.1094	−1.03123	−0.515617	+	0.856819i	$0.672437\pi$
$744$	11.3150	0.414827
$745$	−2.01682	−0.0738905
$746$	−5.47337	−0.200394
$747$	−0.816327	−0.0298678
$748$	21.1557	0.773528
$749$	6.46164	0.236103
$750$	1.10842	0.0404739
$751$	16.7795	0.612293	0.306146	−	0.951984i	$-0.400960\pi$
$751$	16.7795	0.612293	0.306146	+	0.951984i	$0.400960\pi$
$752$	−11.4994	−0.419339
$753$	36.6866	1.33693
$754$	−13.5371	−0.492991
$755$	−6.76147	−0.246075
$756$	4.78077	0.173875
$757$	38.4283	1.39670	0.698351	−	0.715756i	$-0.253918\pi$
$757$	38.4283	1.39670	0.698351	+	0.715756i	$0.253918\pi$
$758$	20.8237	0.756352
$759$	5.64160	0.204777
$760$	1.39096	0.0504554
$761$	29.3007	1.06215	0.531074	−	0.847325i	$-0.321789\pi$
$761$	29.3007	1.06215	0.531074	+	0.847325i	$0.321789\pi$
$762$	−8.22759	−0.298054
$763$	10.0066	0.362262
$764$	29.0954	1.05263
$765$	0.908375	0.0328424
$766$	0.343457	0.0124096
$767$	−23.1419	−0.835606
$768$	11.1858	0.403632
$769$	36.7688	1.32592	0.662958	−	0.748657i	$-0.269301\pi$
$769$	36.7688	1.32592	0.662958	+	0.748657i	$0.269301\pi$
$770$	1.58700	0.0571915
$771$	−41.9691	−1.51148
$772$	−15.7638	−0.567350
$773$	−32.7677	−1.17857	−0.589287	−	0.807924i	$-0.700591\pi$
$773$	−32.7677	−1.17857	−0.589287	+	0.807924i	$0.700591\pi$
$774$	1.73880	0.0624999
$775$	2.81364	0.101069
$776$	3.68800	0.132392
$777$	2.63213	0.0944270
$778$	−17.8507	−0.639979
$779$	−3.90562	−0.139933
$780$	−11.7809	−0.421824
$781$	−66.8118	−2.39071
$782$	1.30379	0.0466233
$783$	−27.3269	−0.976584
$784$	−12.6640	−0.452286
$785$	7.20307	0.257089
$786$	−17.2385	−0.614877
$787$	−27.0284	−0.963457	−0.481728	−	0.876321i	$-0.659991\pi$
$787$	−27.0284	−0.963457	−0.481728	+	0.876321i	$0.659991\pi$
$788$	−33.5409	−1.19485
$789$	14.7414	0.524809
$790$	3.06031	0.108881
$791$	4.54131	0.161470
$792$	−2.91786	−0.103682
$793$	4.95536	0.175970
$794$	−4.00940	−0.142288
$795$	7.69091	0.272768
$796$	9.09028	0.322197
$797$	28.3409	1.00388	0.501942	−	0.864901i	$-0.332619\pi$
$797$	28.3409	1.00388	0.501942	+	0.864901i	$0.332619\pi$
$798$	0.417514	0.0147798
$799$	18.0408	0.638236
$800$	5.58798	0.197565
$801$	1.94051	0.0685647
$802$	−0.609831	−0.0215339
$803$	0.515752	0.0182005
$804$	−23.0662	−0.813482
$805$	−0.428174	−0.0150911
$806$	6.83090	0.240608
$807$	34.4180	1.21157
$808$	−3.93699	−0.138503
$809$	−4.66346	−0.163958	−0.0819792	−	0.996634i	$-0.526124\pi$
$809$	−4.66346	−0.163958	−0.0819792	+	0.996634i	$0.526124\pi$
$810$	5.98775	0.210388
$811$	45.2668	1.58953	0.794766	−	0.606916i	$-0.207594\pi$
$811$	45.2668	1.58953	0.794766	+	0.606916i	$0.207594\pi$
$812$	−5.43924	−0.190880
$813$	24.1808	0.848059
$814$	6.40187	0.224385
$815$	−1.66689	−0.0583885
$816$	10.3693	0.362999
$817$	5.90361	0.206541
$818$	−5.71324	−0.199759
$819$	−0.724250	−0.0253073
$820$	−10.1146	−0.353216
$821$	33.1415	1.15665	0.578323	−	0.815808i	$-0.303707\pi$
$821$	33.1415	1.15665	0.578323	+	0.815808i	$0.303707\pi$
$822$	5.09124	0.177577
$823$	2.74047	0.0955268	0.0477634	−	0.998859i	$-0.484791\pi$
$823$	2.74047	0.0955268	0.0477634	+	0.998859i	$0.484791\pi$
$824$	10.2327	0.356473
$825$	−7.89446	−0.274850
$826$	2.12398	0.0739026
$827$	−28.8745	−1.00406	−0.502032	−	0.864849i	$-0.667414\pi$
$827$	−28.8745	−1.00406	−0.502032	+	0.864849i	$0.667414\pi$
$828$	0.353273	0.0122771
$829$	−24.2871	−0.843524	−0.421762	−	0.906707i	$-0.638588\pi$
$829$	−24.2871	−0.843524	−0.421762	+	0.906707i	$0.638588\pi$
$830$	1.63956	0.0569098
$831$	−37.5473	−1.30250
$832$	−1.61693	−0.0560571
$833$	19.8679	0.688381
$834$	−1.17245	−0.0405987
$835$	−2.04549	−0.0707870
$836$	−4.44564	−0.153756
$837$	13.7893	0.476629
$838$	−6.97810	−0.241054
$839$	−24.9261	−0.860544	−0.430272	−	0.902699i	$-0.641582\pi$
$839$	−24.9261	−0.860544	−0.430272	+	0.902699i	$0.641582\pi$
$840$	2.40950	0.0831355
$841$	2.09072	0.0720938
$842$	14.1029	0.486018
$843$	14.9301	0.514221
$844$	−10.4926	−0.361169
$845$	−2.84892	−0.0980059
$846$	−1.11659	−0.0383893
$847$	−4.71229	−0.161916
$848$	8.06898	0.277090
$849$	2.25736	0.0774724
$850$	−1.82443	−0.0625774
$851$	−1.72723	−0.0592086
$852$	−45.5203	−1.55950
$853$	42.5650	1.45740	0.728699	−	0.684834i	$-0.240125\pi$
$853$	42.5650	1.45740	0.728699	+	0.684834i	$0.240125\pi$
$854$	−0.454805	−0.0155631
$855$	−0.190886	−0.00652815
$856$	23.8612	0.815558
$857$	−22.8883	−0.781850	−0.390925	−	0.920422i	$-0.627845\pi$
$857$	−22.8883	−0.781850	−0.390925	+	0.920422i	$0.627845\pi$
$858$	−19.1660	−0.654318
$859$	−5.28923	−0.180466	−0.0902331	−	0.995921i	$-0.528761\pi$
$859$	−5.28923	−0.180466	−0.0902331	+	0.995921i	$0.528761\pi$
$860$	15.2889	0.521346
$861$	−6.76552	−0.230568
$862$	−15.1058	−0.514504
$863$	−30.6945	−1.04485	−0.522427	−	0.852684i	$-0.674973\pi$
$863$	−30.6945	−1.04485	−0.522427	+	0.852684i	$0.674973\pi$
$864$	27.3861	0.931693
$865$	−22.2539	−0.756656
$866$	−0.104849	−0.00356291
$867$	14.6312	0.496901
$868$	2.74468	0.0931604
$869$	−21.7963	−0.739389
$870$	−6.18046	−0.209537
$871$	−31.0311	−1.05145
$872$	36.9517	1.25134
$873$	−0.506115	−0.0171294
$874$	−0.273977	−0.00926742
$875$	0.599157	0.0202552
$876$	0.351393	0.0118725
$877$	−34.3571	−1.16016	−0.580079	−	0.814560i	$-0.696978\pi$
$877$	−34.3571	−1.16016	−0.580079	+	0.814560i	$0.696978\pi$
$878$	9.04091	0.305116
$879$	−47.6126	−1.60593
$880$	−8.28254	−0.279205
$881$	−25.3637	−0.854526	−0.427263	−	0.904127i	$-0.640522\pi$
$881$	−25.3637	−0.854526	−0.427263	+	0.904127i	$0.640522\pi$
$882$	−1.22968	−0.0414055
$883$	−10.5742	−0.355850	−0.177925	−	0.984044i	$-0.556938\pi$
$883$	−10.5742	−0.355850	−0.177925	+	0.984044i	$0.556938\pi$
$884$	19.3910	0.652189
$885$	−10.5656	−0.355159
$886$	−13.1341	−0.441250
$887$	53.3712	1.79203	0.896014	−	0.444025i	$-0.146450\pi$
$887$	53.3712	1.79203	0.896014	+	0.444025i	$0.146450\pi$
$888$	9.71976	0.326174
$889$	−4.44741	−0.149161
$890$	−3.89743	−0.130642
$891$	−42.6463	−1.42870
$892$	43.4607	1.45517
$893$	−3.79108	−0.126864
$894$	−2.23549	−0.0747659
$895$	16.3522	0.546592
$896$	6.84455	0.228660
$897$	5.17101	0.172655
$898$	2.31008	0.0770882
$899$	−15.6886	−0.523243
$900$	−0.494345	−0.0164782
$901$	−12.6590	−0.421732
$902$	−16.4551	−0.547896
$903$	10.2265	0.340318
$904$	16.7699	0.557759
$905$	−8.85686	−0.294412
$906$	−7.49458	−0.248991
$907$	−20.0045	−0.664237	−0.332119	−	0.943238i	$-0.607763\pi$
$907$	−20.0045	−0.664237	−0.332119	+	0.943238i	$0.607763\pi$
$908$	19.2226	0.637925
$909$	0.540284	0.0179201
$910$	1.45462	0.0482203
$911$	−3.59709	−0.119177	−0.0595885	−	0.998223i	$-0.518979\pi$
$911$	−3.59709	−0.119177	−0.0595885	+	0.998223i	$0.518979\pi$
$912$	−2.17900	−0.0721540
$913$	−11.6773	−0.386463
$914$	−1.87610	−0.0620560
$915$	2.26241	0.0747929
$916$	13.7398	0.453977
$917$	−9.31826	−0.307716
$918$	−8.94133	−0.295108
$919$	7.25886	0.239448	0.119724	−	0.992807i	$-0.461799\pi$
$919$	7.25886	0.239448	0.119724	+	0.992807i	$0.461799\pi$
$920$	−1.58114	−0.0521285
$921$	33.6661	1.10934
$922$	−9.16562	−0.301854
$923$	−61.2387	−2.01570
$924$	−7.70098	−0.253344
$925$	2.41696	0.0794693
$926$	1.72770	0.0567758
$927$	−1.40426	−0.0461220
$928$	−31.1580	−1.02281
$929$	14.1907	0.465583	0.232791	−	0.972527i	$-0.425214\pi$
$929$	14.1907	0.465583	0.232791	+	0.972527i	$0.425214\pi$
$930$	3.11870	0.102266
$931$	−4.17503	−0.136831
$932$	40.0152	1.31074
$933$	21.4229	0.701354
$934$	−1.91975	−0.0628162
$935$	12.9940	0.424950
$936$	−2.67447	−0.0874177
$937$	59.1978	1.93391	0.966953	−	0.254953i	$-0.0820600\pi$
$937$	59.1978	1.93391	0.966953	+	0.254953i	$0.0820600\pi$
$938$	2.84805	0.0929921
$939$	34.7994	1.13564
$940$	−9.81793	−0.320226
$941$	42.8428	1.39664	0.698318	−	0.715788i	$-0.253932\pi$
$941$	42.8428	1.39664	0.698318	+	0.715788i	$0.253932\pi$
$942$	7.98405	0.260134
$943$	4.43961	0.144573
$944$	−11.0850	−0.360786
$945$	2.93640	0.0955212
$946$	24.8730	0.808693
$947$	−16.6887	−0.542309	−0.271155	−	0.962536i	$-0.587405\pi$
$947$	−16.6887	−0.542309	−0.271155	+	0.962536i	$0.587405\pi$
$948$	−14.8503	−0.482315
$949$	0.472731	0.0153455
$950$	0.383385	0.0124386
$951$	1.27330	0.0412896
$952$	−3.96595	−0.128537
$953$	44.9265	1.45531	0.727657	−	0.685942i	$-0.240609\pi$
$953$	44.9265	1.45531	0.727657	+	0.685942i	$0.240609\pi$
$954$	0.783501	0.0253668
$955$	17.8707	0.578283
$956$	14.7397	0.476715
$957$	44.0188	1.42293
$958$	−9.04388	−0.292195
$959$	2.75206	0.0888688
$960$	−0.738224	−0.0238261
$961$	−23.0834	−0.744627
$962$	5.86786	0.189187
$963$	−3.27454	−0.105521
$964$	2.90712	0.0936321
$965$	−9.68228	−0.311683
$966$	−0.474598	−0.0152699
$967$	−22.5062	−0.723752	−0.361876	−	0.932226i	$-0.617864\pi$
$967$	−22.5062	−0.723752	−0.361876	+	0.932226i	$0.617864\pi$
$968$	−17.4013	−0.559298
$969$	3.41852	0.109819
$970$	1.01651	0.0326381
$971$	−0.309122	−0.00992019	−0.00496009	−	0.999988i	$-0.501579\pi$
$971$	−0.309122	−0.00992019	−0.00496009	+	0.999988i	$0.501579\pi$
$972$	−5.11828	−0.164169
$973$	−0.633768	−0.0203177
$974$	6.98465	0.223803
$975$	−7.23595	−0.231736
$976$	2.37362	0.0759779
$977$	18.4297	0.589619	0.294809	−	0.955556i	$-0.404744\pi$
$977$	18.4297	0.589619	0.294809	+	0.955556i	$0.404744\pi$
$978$	−1.84762	−0.0590803
$979$	27.7585	0.887165
$980$	−10.8123	−0.345385
$981$	−5.07098	−0.161904
$982$	−23.7949	−0.759327
$983$	−11.9008	−0.379577	−0.189788	−	0.981825i	$-0.560780\pi$
$983$	−11.9008	−0.379577	−0.189788	+	0.981825i	$0.560780\pi$
$984$	−24.9833	−0.796440
$985$	−20.6012	−0.656408
$986$	10.1728	0.323969
$987$	−6.56710	−0.209033
$988$	−4.07481	−0.129637
$989$	−6.71077	−0.213390
$990$	−0.804238	−0.0255604
$991$	−9.79988	−0.311303	−0.155652	−	0.987812i	$-0.549748\pi$
$991$	−9.79988	−0.311303	−0.155652	+	0.987812i	$0.549748\pi$
$992$	15.7225	0.499191
$993$	−0.549128	−0.0174261
$994$	5.62052	0.178272
$995$	5.58335	0.177004
$996$	−7.95600	−0.252095
$997$	−19.7062	−0.624100	−0.312050	−	0.950066i	$-0.601016\pi$
$997$	−19.7062	−0.624100	−0.312050	+	0.950066i	$0.601016\pi$
$998$	−8.20898	−0.259851
$999$	11.8453	0.374768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.e.1.19	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.e.1.19	✓	29	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.609831 q^{2} -1.81759 q^{3} -1.62811 q^{4} -1.00000 q^{5} -1.10842 q^{6} -0.599157 q^{7} -2.21253 q^{8} +0.303632 q^{9} +O(q^{10})\)	\(q+0.609831 q^{2} -1.81759 q^{3} -1.62811 q^{4} -1.00000 q^{5} -1.10842 q^{6} -0.599157 q^{7} -2.21253 q^{8} +0.303632 q^{9} -0.609831 q^{10} +4.34337 q^{11} +2.95923 q^{12} +3.98107 q^{13} -0.365385 q^{14} +1.81759 q^{15} +1.90694 q^{16} -2.99169 q^{17} +0.185164 q^{18} +0.628674 q^{19} +1.62811 q^{20} +1.08902 q^{21} +2.64872 q^{22} -0.714627 q^{23} +4.02148 q^{24} +1.00000 q^{25} +2.42778 q^{26} +4.90089 q^{27} +0.975491 q^{28} -5.57591 q^{29} +1.10842 q^{30} +2.81364 q^{31} +5.58798 q^{32} -7.89446 q^{33} -1.82443 q^{34} +0.599157 q^{35} -0.494345 q^{36} +2.41696 q^{37} +0.383385 q^{38} -7.23595 q^{39} +2.21253 q^{40} -6.21248 q^{41} +0.664119 q^{42} +9.39058 q^{43} -7.07147 q^{44} -0.303632 q^{45} -0.435802 q^{46} -6.03028 q^{47} -3.46603 q^{48} -6.64101 q^{49} +0.609831 q^{50} +5.43767 q^{51} -6.48160 q^{52} +4.23138 q^{53} +2.98872 q^{54} -4.34337 q^{55} +1.32565 q^{56} -1.14267 q^{57} -3.40036 q^{58} -5.81298 q^{59} -2.95923 q^{60} +1.24473 q^{61} +1.71584 q^{62} -0.181923 q^{63} -0.406155 q^{64} -3.98107 q^{65} -4.81429 q^{66} -7.79466 q^{67} +4.87079 q^{68} +1.29890 q^{69} +0.365385 q^{70} -15.3825 q^{71} -0.671796 q^{72} +0.118745 q^{73} +1.47394 q^{74} -1.81759 q^{75} -1.02355 q^{76} -2.60236 q^{77} -4.41271 q^{78} -5.01829 q^{79} -1.90694 q^{80} -9.81870 q^{81} -3.78856 q^{82} -2.68854 q^{83} -1.77304 q^{84} +2.99169 q^{85} +5.72667 q^{86} +10.1347 q^{87} -9.60985 q^{88} +6.39100 q^{89} -0.185164 q^{90} -2.38529 q^{91} +1.16349 q^{92} -5.11404 q^{93} -3.67745 q^{94} -0.628674 q^{95} -10.1566 q^{96} -1.66687 q^{97} -4.04990 q^{98} +1.31879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9}+O(q^{10}) \) 29 * q - 5 * q^2 - 3 * q^3 + 19 * q^4 - 29 * q^5 - 6 * q^6 + 12 * q^7 - 15 * q^8 + 14 * q^9	\( 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9} + 5 q^{10} - 38 q^{11} - 6 q^{12} + 5 q^{13} - 18 q^{14} + 3 q^{15} + 7 q^{16} - 16 q^{17} - 2 q^{18} - 18 q^{19} - 19 q^{20} - 20 q^{21} - 2 q^{22} - 19 q^{23} - 19 q^{24} + 29 q^{25} - 21 q^{26} - 21 q^{27} + 26 q^{28} - 31 q^{29} + 6 q^{30} - 13 q^{31} - 30 q^{32} + 2 q^{33} - 14 q^{34} - 12 q^{35} - 29 q^{36} - q^{37} - 23 q^{38} - 39 q^{39} + 15 q^{40} - 24 q^{41} - 20 q^{42} - 27 q^{43} - 76 q^{44} - 14 q^{45} - 11 q^{46} - 5 q^{47} - 2 q^{48} - 11 q^{49} - 5 q^{50} - 58 q^{51} + 11 q^{52} - 37 q^{53} - 18 q^{54} + 38 q^{55} - 50 q^{56} - 6 q^{57} + 31 q^{58} - 67 q^{59} + 6 q^{60} - 31 q^{61} - 19 q^{62} - 2 q^{63} - 13 q^{64} - 5 q^{65} + 6 q^{66} - 17 q^{67} - 16 q^{68} - 48 q^{69} + 18 q^{70} - 53 q^{71} + 9 q^{72} + 29 q^{73} - 59 q^{74} - 3 q^{75} - 21 q^{76} - 62 q^{77} - 12 q^{78} - 13 q^{79} - 7 q^{80} - 11 q^{81} + 32 q^{82} - 72 q^{83} - 58 q^{84} + 16 q^{85} - 43 q^{86} + 4 q^{87} + 12 q^{88} - 38 q^{89} + 2 q^{90} - 45 q^{91} - 37 q^{92} - 27 q^{93} - 44 q^{94} + 18 q^{95} - 21 q^{96} + 32 q^{97} - 32 q^{98} - 107 q^{99}+O(q^{100}) \) 29 * q - 5 * q^2 - 3 * q^3 + 19 * q^4 - 29 * q^5 - 6 * q^6 + 12 * q^7 - 15 * q^8 + 14 * q^9 + 5 * q^10 - 38 * q^11 - 6 * q^12 + 5 * q^13 - 18 * q^14 + 3 * q^15 + 7 * q^16 - 16 * q^17 - 2 * q^18 - 18 * q^19 - 19 * q^20 - 20 * q^21 - 2 * q^22 - 19 * q^23 - 19 * q^24 + 29 * q^25 - 21 * q^26 - 21 * q^27 + 26 * q^28 - 31 * q^29 + 6 * q^30 - 13 * q^31 - 30 * q^32 + 2 * q^33 - 14 * q^34 - 12 * q^35 - 29 * q^36 - q^37 - 23 * q^38 - 39 * q^39 + 15 * q^40 - 24 * q^41 - 20 * q^42 - 27 * q^43 - 76 * q^44 - 14 * q^45 - 11 * q^46 - 5 * q^47 - 2 * q^48 - 11 * q^49 - 5 * q^50 - 58 * q^51 + 11 * q^52 - 37 * q^53 - 18 * q^54 + 38 * q^55 - 50 * q^56 - 6 * q^57 + 31 * q^58 - 67 * q^59 + 6 * q^60 - 31 * q^61 - 19 * q^62 - 2 * q^63 - 13 * q^64 - 5 * q^65 + 6 * q^66 - 17 * q^67 - 16 * q^68 - 48 * q^69 + 18 * q^70 - 53 * q^71 + 9 * q^72 + 29 * q^73 - 59 * q^74 - 3 * q^75 - 21 * q^76 - 62 * q^77 - 12 * q^78 - 13 * q^79 - 7 * q^80 - 11 * q^81 + 32 * q^82 - 72 * q^83 - 58 * q^84 + 16 * q^85 - 43 * q^86 + 4 * q^87 + 12 * q^88 - 38 * q^89 + 2 * q^90 - 45 * q^91 - 37 * q^92 - 27 * q^93 - 44 * q^94 + 18 * q^95 - 21 * q^96 + 32 * q^97 - 32 * q^98 - 107 * q^99

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