LMFDB - Embedded newform 4029.2.a.k.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.959467 q^{2} +1.00000 q^{3} -1.07942 q^{4} -3.93035 q^{5} +0.959467 q^{6} -0.0159719 q^{7} -2.95460 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.959467 q^{2} +1.00000 q^{3} -1.07942 q^{4} -3.93035 q^{5} +0.959467 q^{6} -0.0159719 q^{7} -2.95460 q^{8} +1.00000 q^{9} -3.77104 q^{10} -0.0651658 q^{11} -1.07942 q^{12} -1.13960 q^{13} -0.0153245 q^{14} -3.93035 q^{15} -0.675997 q^{16} +1.00000 q^{17} +0.959467 q^{18} -6.81607 q^{19} +4.24251 q^{20} -0.0159719 q^{21} -0.0625244 q^{22} -3.11319 q^{23} -2.95460 q^{24} +10.4477 q^{25} -1.09341 q^{26} +1.00000 q^{27} +0.0172404 q^{28} +0.147476 q^{29} -3.77104 q^{30} -3.00436 q^{31} +5.26061 q^{32} -0.0651658 q^{33} +0.959467 q^{34} +0.0627751 q^{35} -1.07942 q^{36} +0.619732 q^{37} -6.53980 q^{38} -1.13960 q^{39} +11.6126 q^{40} +6.70124 q^{41} -0.0153245 q^{42} +0.482967 q^{43} +0.0703415 q^{44} -3.93035 q^{45} -2.98700 q^{46} -1.80113 q^{47} -0.675997 q^{48} -6.99974 q^{49} +10.0242 q^{50} +1.00000 q^{51} +1.23011 q^{52} +2.47166 q^{53} +0.959467 q^{54} +0.256125 q^{55} +0.0471906 q^{56} -6.81607 q^{57} +0.141499 q^{58} -0.178477 q^{59} +4.24251 q^{60} +2.91148 q^{61} -2.88259 q^{62} -0.0159719 q^{63} +6.39938 q^{64} +4.47904 q^{65} -0.0625244 q^{66} +14.4005 q^{67} -1.07942 q^{68} -3.11319 q^{69} +0.0602306 q^{70} +7.32560 q^{71} -2.95460 q^{72} +4.12876 q^{73} +0.594612 q^{74} +10.4477 q^{75} +7.35743 q^{76} +0.00104082 q^{77} -1.09341 q^{78} +1.00000 q^{79} +2.65691 q^{80} +1.00000 q^{81} +6.42962 q^{82} +10.0814 q^{83} +0.0172404 q^{84} -3.93035 q^{85} +0.463390 q^{86} +0.147476 q^{87} +0.192539 q^{88} +3.56683 q^{89} -3.77104 q^{90} +0.0182016 q^{91} +3.36045 q^{92} -3.00436 q^{93} -1.72812 q^{94} +26.7896 q^{95} +5.26061 q^{96} -0.920142 q^{97} -6.71602 q^{98} -0.0651658 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * 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 $

$2$	0.959467	0.678445	0.339223	−	0.940706i	$-0.389836\pi$
$2$	0.959467	0.678445	0.339223	+	0.940706i	$0.389836\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.07942	−0.539712
$5$	−3.93035	−1.75771	−0.878853	−	0.477092i	$-0.841691\pi$
$5$	−3.93035	−1.75771	−0.878853	+	0.477092i	$0.841691\pi$
$6$	0.959467	0.391701
$7$	−0.0159719	−0.00603680	−0.00301840	−	0.999995i	$-0.500961\pi$
$7$	−0.0159719	−0.00603680	−0.00301840	+	0.999995i	$0.500961\pi$
$8$	−2.95460	−1.04461
$9$	1.00000	0.333333
$10$	−3.77104	−1.19251
$11$	−0.0651658	−0.0196482	−0.00982412	−	0.999952i	$-0.503127\pi$
$11$	−0.0651658	−0.0196482	−0.00982412	+	0.999952i	$0.503127\pi$
$12$	−1.07942	−0.311603
$13$	−1.13960	−0.316069	−0.158034	−	0.987434i	$-0.550516\pi$
$13$	−1.13960	−0.316069	−0.158034	+	0.987434i	$0.550516\pi$
$14$	−0.0153245	−0.00409564
$15$	−3.93035	−1.01481
$16$	−0.675997	−0.168999
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.959467	0.226148
$19$	−6.81607	−1.56371	−0.781857	−	0.623457i	$-0.785728\pi$
$19$	−6.81607	−1.56371	−0.781857	+	0.623457i	$0.785728\pi$
$20$	4.24251	0.948655
$21$	−0.0159719	−0.00348535
$22$	−0.0625244	−0.0133303
$23$	−3.11319	−0.649144	−0.324572	−	0.945861i	$-0.605220\pi$
$23$	−3.11319	−0.649144	−0.324572	+	0.945861i	$0.605220\pi$
$24$	−2.95460	−0.603106
$25$	10.4477	2.08953
$26$	−1.09341	−0.214435
$27$	1.00000	0.192450
$28$	0.0172404	0.00325813
$29$	0.147476	0.0273857	0.0136928	−	0.999906i	$-0.495641\pi$
$29$	0.147476	0.0273857	0.0136928	+	0.999906i	$0.495641\pi$
$30$	−3.77104	−0.688495
$31$	−3.00436	−0.539599	−0.269800	−	0.962916i	$-0.586957\pi$
$31$	−3.00436	−0.539599	−0.269800	+	0.962916i	$0.586957\pi$
$32$	5.26061	0.929954
$33$	−0.0651658	−0.0113439
$34$	0.959467	0.164547
$35$	0.0627751	0.0106109
$36$	−1.07942	−0.179904
$37$	0.619732	0.101883	0.0509416	−	0.998702i	$-0.483778\pi$
$37$	0.619732	0.101883	0.0509416	+	0.998702i	$0.483778\pi$
$38$	−6.53980	−1.06090
$39$	−1.13960	−0.182482
$40$	11.6126	1.83612
$41$	6.70124	1.04656	0.523279	−	0.852161i	$-0.324709\pi$
$41$	6.70124	1.04656	0.523279	+	0.852161i	$0.324709\pi$
$42$	−0.0153245	−0.00236462
$43$	0.482967	0.0736517	0.0368259	−	0.999322i	$-0.488275\pi$
$43$	0.482967	0.0736517	0.0368259	+	0.999322i	$0.488275\pi$
$44$	0.0703415	0.0106044
$45$	−3.93035	−0.585902
$46$	−2.98700	−0.440409
$47$	−1.80113	−0.262721	−0.131361	−	0.991335i	$-0.541935\pi$
$47$	−1.80113	−0.262721	−0.131361	+	0.991335i	$0.541935\pi$
$48$	−0.675997	−0.0975718
$49$	−6.99974	−0.999964
$50$	10.0242	1.41763
$51$	1.00000	0.140028
$52$	1.23011	0.170586
$53$	2.47166	0.339509	0.169754	−	0.985486i	$-0.445703\pi$
$53$	2.47166	0.339509	0.169754	+	0.985486i	$0.445703\pi$
$54$	0.959467	0.130567
$55$	0.256125	0.0345358
$56$	0.0471906	0.00630611
$57$	−6.81607	−0.902811
$58$	0.141499	0.0185797
$59$	−0.178477	−0.0232358	−0.0116179	−	0.999933i	$-0.503698\pi$
$59$	−0.178477	−0.0232358	−0.0116179	+	0.999933i	$0.503698\pi$
$60$	4.24251	0.547706
$61$	2.91148	0.372777	0.186388	−	0.982476i	$-0.440322\pi$
$61$	2.91148	0.372777	0.186388	+	0.982476i	$0.440322\pi$
$62$	−2.88259	−0.366089
$63$	−0.0159719	−0.00201227
$64$	6.39938	0.799922
$65$	4.47904	0.555556
$66$	−0.0625244	−0.00769622
$67$	14.4005	1.75930	0.879650	−	0.475622i	$-0.157777\pi$
$67$	14.4005	1.75930	0.879650	+	0.475622i	$0.157777\pi$
$68$	−1.07942	−0.130899
$69$	−3.11319	−0.374784
$70$	0.0602306	0.00719893
$71$	7.32560	0.869389	0.434695	−	0.900578i	$-0.356856\pi$
$71$	7.32560	0.869389	0.434695	+	0.900578i	$0.356856\pi$
$72$	−2.95460	−0.348203
$73$	4.12876	0.483234	0.241617	−	0.970372i	$-0.422322\pi$
$73$	4.12876	0.483234	0.241617	+	0.970372i	$0.422322\pi$
$74$	0.594612	0.0691222
$75$	10.4477	1.20639
$76$	7.35743	0.843955
$77$	0.00104082	0.000118612 0
$78$	−1.09341	−0.123804
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	2.65691	0.297051
$81$	1.00000	0.111111
$82$	6.42962	0.710033
$83$	10.0814	1.10657	0.553287	−	0.832991i	$-0.313373\pi$
$83$	10.0814	1.10657	0.553287	+	0.832991i	$0.313373\pi$
$84$	0.0172404	0.00188108
$85$	−3.93035	−0.426306
$86$	0.463390	0.0499687
$87$	0.147476	0.0158111
$88$	0.192539	0.0205247
$89$	3.56683	0.378084	0.189042	−	0.981969i	$-0.439462\pi$
$89$	3.56683	0.378084	0.189042	+	0.981969i	$0.439462\pi$
$90$	−3.77104	−0.397503
$91$	0.0182016	0.00190804
$92$	3.36045	0.350351
$93$	−3.00436	−0.311538
$94$	−1.72812	−0.178242
$95$	26.7896	2.74855
$96$	5.26061	0.536909
$97$	−0.920142	−0.0934263	−0.0467131	−	0.998908i	$-0.514875\pi$
$97$	−0.920142	−0.0934263	−0.0467131	+	0.998908i	$0.514875\pi$
$98$	−6.71602	−0.678421
$99$	−0.0651658	−0.00654941
$100$	−11.2775	−1.12775
$101$	8.15477	0.811430	0.405715	−	0.914000i	$-0.367023\pi$
$101$	8.15477	0.811430	0.405715	+	0.914000i	$0.367023\pi$
$102$	0.959467	0.0950014
$103$	3.69861	0.364435	0.182218	−	0.983258i	$-0.441672\pi$
$103$	3.69861	0.364435	0.182218	+	0.983258i	$0.441672\pi$
$104$	3.36707	0.330169
$105$	0.0627751	0.00612622
$106$	2.37148	0.230338
$107$	−10.5464	−1.01955	−0.509777	−	0.860307i	$-0.670272\pi$
$107$	−10.5464	−1.01955	−0.509777	+	0.860307i	$0.670272\pi$
$108$	−1.07942	−0.103868
$109$	−12.0152	−1.15084	−0.575421	−	0.817857i	$-0.695162\pi$
$109$	−12.0152	−1.15084	−0.575421	+	0.817857i	$0.695162\pi$
$110$	0.245743	0.0234307
$111$	0.619732	0.0588223
$112$	0.0107969	0.00102021
$113$	12.3319	1.16009	0.580045	−	0.814585i	$-0.303035\pi$
$113$	12.3319	1.16009	0.580045	+	0.814585i	$0.303035\pi$
$114$	−6.53980	−0.612508
$115$	12.2359	1.14101
$116$	−0.159189	−0.0147804
$117$	−1.13960	−0.105356
$118$	−0.171243	−0.0157642
$119$	−0.0159719	−0.00146414
$120$	11.6126	1.06008
$121$	−10.9958	−0.999614
$122$	2.79347	0.252909
$123$	6.70124	0.604231
$124$	3.24298	0.291228
$125$	−21.4112	−1.91508
$126$	−0.0153245	−0.00136521
$127$	12.0038	1.06517	0.532583	−	0.846378i	$-0.321221\pi$
$127$	12.0038	1.06517	0.532583	+	0.846378i	$0.321221\pi$
$128$	−4.38124	−0.387250
$129$	0.482967	0.0425228
$130$	4.29749	0.376915
$131$	16.6459	1.45436	0.727181	−	0.686446i	$-0.240830\pi$
$131$	16.6459	1.45436	0.727181	+	0.686446i	$0.240830\pi$
$132$	0.0703415	0.00612244
$133$	0.108865	0.00943984
$134$	13.8168	1.19359
$135$	−3.93035	−0.338271
$136$	−2.95460	−0.253355
$137$	−10.2100	−0.872298	−0.436149	−	0.899874i	$-0.643658\pi$
$137$	−10.2100	−0.872298	−0.436149	+	0.899874i	$0.643658\pi$
$138$	−2.98700	−0.254270
$139$	6.65275	0.564279	0.282140	−	0.959373i	$-0.408956\pi$
$139$	6.65275	0.564279	0.282140	+	0.959373i	$0.408956\pi$
$140$	−0.0677609	−0.00572684
$141$	−1.80113	−0.151682
$142$	7.02867	0.589833
$143$	0.0742631	0.00621019
$144$	−0.675997	−0.0563331
$145$	−0.579634	−0.0481360
$146$	3.96140	0.327848
$147$	−6.99974	−0.577329
$148$	−0.668953	−0.0549876
$149$	3.47475	0.284663	0.142332	−	0.989819i	$-0.454540\pi$
$149$	3.47475	0.284663	0.142332	+	0.989819i	$0.454540\pi$
$150$	10.0242	0.818471
$151$	−4.73716	−0.385505	−0.192752	−	0.981247i	$-0.561741\pi$
$151$	−4.73716	−0.385505	−0.192752	+	0.981247i	$0.561741\pi$
$152$	20.1388	1.63347
$153$	1.00000	0.0808452
$154$	0.000998632 0	8.04721e−5 0
$155$	11.8082	0.948457
$156$	1.23011	0.0984879
$157$	0.886583	0.0707570	0.0353785	−	0.999374i	$-0.488736\pi$
$157$	0.886583	0.0707570	0.0353785	+	0.999374i	$0.488736\pi$
$158$	0.959467	0.0763311
$159$	2.47166	0.196015
$160$	−20.6761	−1.63459
$161$	0.0497234	0.00391876
$162$	0.959467	0.0753828
$163$	−3.22247	−0.252403	−0.126202	−	0.992005i	$-0.540279\pi$
$163$	−3.22247	−0.252403	−0.126202	+	0.992005i	$0.540279\pi$
$164$	−7.23348	−0.564840
$165$	0.256125	0.0199393
$166$	9.67274	0.750750
$167$	13.3747	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$167$	13.3747	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$168$	0.0471906	0.00364083
$169$	−11.7013	−0.900101
$170$	−3.77104	−0.289226
$171$	−6.81607	−0.521238
$172$	−0.521326	−0.0397507
$173$	15.8197	1.20275	0.601374	−	0.798967i	$-0.294620\pi$
$173$	15.8197	1.20275	0.601374	+	0.798967i	$0.294620\pi$
$174$	0.141499	0.0107270
$175$	−0.166869	−0.0126141
$176$	0.0440519	0.00332054
$177$	−0.178477	−0.0134152
$178$	3.42226	0.256509
$179$	11.7407	0.877539	0.438769	−	0.898600i	$-0.355415\pi$
$179$	11.7407	0.877539	0.438769	+	0.898600i	$0.355415\pi$
$180$	4.24251	0.316218
$181$	12.5486	0.932728	0.466364	−	0.884593i	$-0.345564\pi$
$181$	12.5486	0.932728	0.466364	+	0.884593i	$0.345564\pi$
$182$	0.0174638	0.00129450
$183$	2.91148	0.215223
$184$	9.19824	0.678103
$185$	−2.43576	−0.179081
$186$	−2.88259	−0.211361
$187$	−0.0651658	−0.00476540
$188$	1.94418	0.141794
$189$	−0.0159719	−0.00116178
$190$	25.7037	1.86474
$191$	−7.42412	−0.537190	−0.268595	−	0.963253i	$-0.586559\pi$
$191$	−7.42412	−0.537190	−0.268595	+	0.963253i	$0.586559\pi$
$192$	6.39938	0.461835
$193$	−16.8962	−1.21621	−0.608107	−	0.793855i	$-0.708071\pi$
$193$	−16.8962	−1.21621	−0.608107	+	0.793855i	$0.708071\pi$
$194$	−0.882845	−0.0633846
$195$	4.47904	0.320751
$196$	7.55569	0.539692
$197$	6.58428	0.469110	0.234555	−	0.972103i	$-0.424637\pi$
$197$	6.58428	0.469110	0.234555	+	0.972103i	$0.424637\pi$
$198$	−0.0625244	−0.00444342
$199$	−24.6876	−1.75006	−0.875029	−	0.484071i	$-0.839158\pi$
$199$	−24.6876	−1.75006	−0.875029	+	0.484071i	$0.839158\pi$
$200$	−30.8687	−2.18275
$201$	14.4005	1.01573
$202$	7.82423	0.550511
$203$	−0.00235547	−0.000165322 0
$204$	−1.07942	−0.0755748
$205$	−26.3382	−1.83954
$206$	3.54870	0.247249
$207$	−3.11319	−0.216381
$208$	0.770368	0.0534154
$209$	0.444175	0.0307242
$210$	0.0602306	0.00415631
$211$	0.866183	0.0596305	0.0298153	−	0.999555i	$-0.490508\pi$
$211$	0.866183	0.0596305	0.0298153	+	0.999555i	$0.490508\pi$
$212$	−2.66797	−0.183237
$213$	7.32560	0.501942
$214$	−10.1189	−0.691712
$215$	−1.89823	−0.129458
$216$	−2.95460	−0.201035
$217$	0.0479853	0.00325745
$218$	−11.5281	−0.780784
$219$	4.12876	0.278995
$220$	−0.276467	−0.0186394
$221$	−1.13960	−0.0766579
$222$	0.594612	0.0399077
$223$	8.48759	0.568371	0.284185	−	0.958769i	$-0.408277\pi$
$223$	8.48759	0.568371	0.284185	+	0.958769i	$0.408277\pi$
$224$	−0.0840218	−0.00561395
$225$	10.4477	0.696511
$226$	11.8321	0.787057
$227$	7.23551	0.480238	0.240119	−	0.970743i	$-0.422814\pi$
$227$	7.23551	0.480238	0.240119	+	0.970743i	$0.422814\pi$
$228$	7.35743	0.487258
$229$	8.20090	0.541931	0.270965	−	0.962589i	$-0.412657\pi$
$229$	8.20090	0.541931	0.270965	+	0.962589i	$0.412657\pi$
$230$	11.7400	0.774110
$231$	0.00104082	6.84809e−5 0
$232$	−0.435734	−0.0286074
$233$	−3.88340	−0.254410	−0.127205	−	0.991876i	$-0.540601\pi$
$233$	−3.88340	−0.254410	−0.127205	+	0.991876i	$0.540601\pi$
$234$	−1.09341	−0.0714785
$235$	7.07906	0.461787
$236$	0.192653	0.0125406
$237$	1.00000	0.0649570
$238$	−0.0153245	−0.000993339 0
$239$	−6.99870	−0.452708	−0.226354	−	0.974045i	$-0.572681\pi$
$239$	−6.99870	−0.452708	−0.226354	+	0.974045i	$0.572681\pi$
$240$	2.65691	0.171503
$241$	−8.91752	−0.574428	−0.287214	−	0.957866i	$-0.592729\pi$
$241$	−8.91752	−0.574428	−0.287214	+	0.957866i	$0.592729\pi$
$242$	−10.5501	−0.678183
$243$	1.00000	0.0641500
$244$	−3.14272	−0.201192
$245$	27.5115	1.75764
$246$	6.42962	0.409937
$247$	7.76761	0.494241
$248$	8.87670	0.563671
$249$	10.0814	0.638881
$250$	−20.5434	−1.29928
$251$	25.1611	1.58816	0.794078	−	0.607815i	$-0.207954\pi$
$251$	25.1611	1.58816	0.794078	+	0.607815i	$0.207954\pi$
$252$	0.0172404	0.00108604
$253$	0.202873	0.0127545
$254$	11.5173	0.722657
$255$	−3.93035	−0.246128
$256$	−17.0024	−1.06265
$257$	−6.43201	−0.401218	−0.200609	−	0.979671i	$-0.564292\pi$
$257$	−6.43201	−0.401218	−0.200609	+	0.979671i	$0.564292\pi$
$258$	0.463390	0.0288494
$259$	−0.00989828	−0.000615049 0
$260$	−4.83478	−0.299840
$261$	0.147476	0.00912856
$262$	15.9712	0.986705
$263$	−19.2936	−1.18969	−0.594846	−	0.803840i	$-0.702787\pi$
$263$	−19.2936	−1.18969	−0.594846	+	0.803840i	$0.702787\pi$
$264$	0.192539	0.0118500
$265$	−9.71449	−0.596757
$266$	0.104453	0.00640441
$267$	3.56683	0.218287
$268$	−15.5442	−0.949515
$269$	−24.7396	−1.50840	−0.754201	−	0.656643i	$-0.771976\pi$
$269$	−24.7396	−1.50840	−0.754201	+	0.656643i	$0.771976\pi$
$270$	−3.77104	−0.229498
$271$	−22.1634	−1.34633	−0.673166	−	0.739491i	$-0.735066\pi$
$271$	−22.1634	−1.34633	−0.673166	+	0.739491i	$0.735066\pi$
$272$	−0.675997	−0.0409883
$273$	0.0182016	0.00110161
$274$	−9.79614	−0.591807
$275$	−0.680831	−0.0410556
$276$	3.36045	0.202275
$277$	−19.4586	−1.16916	−0.584578	−	0.811337i	$-0.698740\pi$
$277$	−19.4586	−1.16916	−0.584578	+	0.811337i	$0.698740\pi$
$278$	6.38309	0.382832
$279$	−3.00436	−0.179866
$280$	−0.185476	−0.0110843
$281$	25.1970	1.50313	0.751565	−	0.659659i	$-0.229299\pi$
$281$	25.1970	1.50313	0.751565	+	0.659659i	$0.229299\pi$
$282$	−1.72812	−0.102908
$283$	23.7291	1.41055	0.705274	−	0.708935i	$-0.250824\pi$
$283$	23.7291	1.41055	0.705274	+	0.708935i	$0.250824\pi$
$284$	−7.90743	−0.469220
$285$	26.7896	1.58688
$286$	0.0712530	0.00421328
$287$	−0.107031	−0.00631786
$288$	5.26061	0.309985
$289$	1.00000	0.0588235
$290$	−0.556139	−0.0326576
$291$	−0.920142	−0.0539397
$292$	−4.45668	−0.260807
$293$	22.4175	1.30964	0.654822	−	0.755783i	$-0.272744\pi$
$293$	22.4175	1.30964	0.654822	+	0.755783i	$0.272744\pi$
$294$	−6.71602	−0.391686
$295$	0.701479	0.0408417
$296$	−1.83106	−0.106428
$297$	−0.0651658	−0.00378130
$298$	3.33391	0.193128
$299$	3.54779	0.205174
$300$	−11.2775	−0.651104
$301$	−0.00771388	−0.000444621 0
$302$	−4.54515	−0.261544
$303$	8.15477	0.468479
$304$	4.60765	0.264267
$305$	−11.4431	−0.655233
$306$	0.959467	0.0548491
$307$	7.21037	0.411518	0.205759	−	0.978603i	$-0.434034\pi$
$307$	7.21037	0.411518	0.205759	+	0.978603i	$0.434034\pi$
$308$	−0.00112349	−6.40166e−5 0
$309$	3.69861	0.210407
$310$	11.3296	0.643477
$311$	29.7883	1.68914	0.844570	−	0.535445i	$-0.179856\pi$
$311$	29.7883	1.68914	0.844570	+	0.535445i	$0.179856\pi$
$312$	3.36707	0.190623
$313$	20.1310	1.13787	0.568935	−	0.822383i	$-0.307356\pi$
$313$	20.1310	1.13787	0.568935	+	0.822383i	$0.307356\pi$
$314$	0.850646	0.0480048
$315$	0.0627751	0.00353698
$316$	−1.07942	−0.0607223
$317$	13.3012	0.747067	0.373534	−	0.927617i	$-0.378146\pi$
$317$	13.3012	0.747067	0.373534	+	0.927617i	$0.378146\pi$
$318$	2.37148	0.132986
$319$	−0.00961042	−0.000538080 0
$320$	−25.1518	−1.40603
$321$	−10.5464	−0.588640
$322$	0.0477080	0.00265866
$323$	−6.81607	−0.379257
$324$	−1.07942	−0.0599680
$325$	−11.9062	−0.660436
$326$	−3.09185	−0.171242
$327$	−12.0152	−0.664439
$328$	−19.7995	−1.09325
$329$	0.0287674	0.00158600
$330$	0.245743	0.0135277
$331$	−14.5188	−0.798028	−0.399014	−	0.916945i	$-0.630648\pi$
$331$	−14.5188	−0.798028	−0.399014	+	0.916945i	$0.630648\pi$
$332$	−10.8821	−0.597231
$333$	0.619732	0.0339611
$334$	12.8325	0.702165
$335$	−56.5990	−3.09233
$336$	0.0107969	0.000589021 0
$337$	1.09532	0.0596660	0.0298330	−	0.999555i	$-0.490502\pi$
$337$	1.09532	0.0596660	0.0298330	+	0.999555i	$0.490502\pi$
$338$	−11.2270	−0.610669
$339$	12.3319	0.669778
$340$	4.24251	0.230083
$341$	0.195782	0.0106022
$342$	−6.53980	−0.353632
$343$	0.223602	0.0120734
$344$	−1.42698	−0.0769374
$345$	12.2359	0.658760
$346$	15.1785	0.815999
$347$	−14.3275	−0.769140	−0.384570	−	0.923096i	$-0.625650\pi$
$347$	−14.3275	−0.769140	−0.384570	+	0.923096i	$0.625650\pi$
$348$	−0.159189	−0.00853345
$349$	−18.4260	−0.986321	−0.493160	−	0.869938i	$-0.664158\pi$
$349$	−18.4260	−0.986321	−0.493160	+	0.869938i	$0.664158\pi$
$350$	−0.160105	−0.00855797
$351$	−1.13960	−0.0608275
$352$	−0.342812	−0.0182719
$353$	−5.62999	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$353$	−5.62999	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$354$	−0.171243	−0.00910147
$355$	−28.7922	−1.52813
$356$	−3.85012	−0.204056
$357$	−0.0159719	−0.000845321 0
$358$	11.2648	0.595362
$359$	−29.6945	−1.56722	−0.783609	−	0.621254i	$-0.786623\pi$
$359$	−29.6945	−1.56722	−0.783609	+	0.621254i	$0.786623\pi$
$360$	11.6126	0.612040
$361$	27.4589	1.44520
$362$	12.0399	0.632805
$363$	−10.9958	−0.577127
$364$	−0.0196472	−0.00102979
$365$	−16.2275	−0.849384
$366$	2.79347	0.146017
$367$	14.1238	0.737257	0.368628	−	0.929577i	$-0.379828\pi$
$367$	14.1238	0.737257	0.368628	+	0.929577i	$0.379828\pi$
$368$	2.10451	0.109705
$369$	6.70124	0.348853
$370$	−2.33703	−0.121497
$371$	−0.0394770	−0.00204955
$372$	3.24298	0.168141
$373$	26.1016	1.35149	0.675744	−	0.737137i	$-0.263823\pi$
$373$	26.1016	1.35149	0.675744	+	0.737137i	$0.263823\pi$
$374$	−0.0625244	−0.00323306
$375$	−21.4112	−1.10567
$376$	5.32162	0.274441
$377$	−0.168064	−0.00865575
$378$	−0.0153245	−0.000788206 0
$379$	21.4221	1.10038	0.550191	−	0.835039i	$-0.314555\pi$
$379$	21.4221	1.10038	0.550191	+	0.835039i	$0.314555\pi$
$380$	−28.9173	−1.48343
$381$	12.0038	0.614974
$382$	−7.12319	−0.364454
$383$	−28.5364	−1.45814	−0.729071	−	0.684438i	$-0.760048\pi$
$383$	−28.5364	−1.45814	−0.729071	+	0.684438i	$0.760048\pi$
$384$	−4.38124	−0.223579
$385$	−0.00409079	−0.000208486 0
$386$	−16.2113	−0.825135
$387$	0.482967	0.0245506
$388$	0.993223	0.0504233
$389$	−14.3090	−0.725495	−0.362747	−	0.931887i	$-0.618161\pi$
$389$	−14.3090	−0.725495	−0.362747	+	0.931887i	$0.618161\pi$
$390$	4.29749	0.217612
$391$	−3.11319	−0.157441
$392$	20.6815	1.04457
$393$	16.6459	0.839676
$394$	6.31740	0.318266
$395$	−3.93035	−0.197757
$396$	0.0703415	0.00353479
$397$	−13.6764	−0.686399	−0.343200	−	0.939263i	$-0.611511\pi$
$397$	−13.6764	−0.686399	−0.343200	+	0.939263i	$0.611511\pi$
$398$	−23.6869	−1.18732
$399$	0.108865	0.00545009
$400$	−7.06259	−0.353130
$401$	−2.74041	−0.136849	−0.0684246	−	0.997656i	$-0.521797\pi$
$401$	−2.74041	−0.136849	−0.0684246	+	0.997656i	$0.521797\pi$
$402$	13.8168	0.689119
$403$	3.42378	0.170551
$404$	−8.80245	−0.437938
$405$	−3.93035	−0.195301
$406$	−0.00226000	−0.000112162 0
$407$	−0.0403853	−0.00200183
$408$	−2.95460	−0.146275
$409$	13.0519	0.645377	0.322688	−	0.946505i	$-0.395413\pi$
$409$	13.0519	0.645377	0.322688	+	0.946505i	$0.395413\pi$
$410$	−25.2707	−1.24803
$411$	−10.2100	−0.503622
$412$	−3.99237	−0.196690
$413$	0.00285062	0.000140270 0
$414$	−2.98700	−0.146803
$415$	−39.6233	−1.94503
$416$	−5.99500	−0.293929
$417$	6.65275	0.325787
$418$	0.426171	0.0208447
$419$	23.1403	1.13048	0.565239	−	0.824927i	$-0.308784\pi$
$419$	23.1403	1.13048	0.565239	+	0.824927i	$0.308784\pi$
$420$	−0.0677609	−0.00330639
$421$	−7.23885	−0.352800	−0.176400	−	0.984319i	$-0.556445\pi$
$421$	−7.23885	−0.352800	−0.176400	+	0.984319i	$0.556445\pi$
$422$	0.831074	0.0404560
$423$	−1.80113	−0.0875738
$424$	−7.30278	−0.354654
$425$	10.4477	0.506786
$426$	7.02867	0.340540
$427$	−0.0465018	−0.00225038
$428$	11.3840	0.550265
$429$	0.0742631	0.00358546
$430$	−1.82129	−0.0878303
$431$	−29.6728	−1.42929	−0.714644	−	0.699489i	$-0.753411\pi$
$431$	−29.6728	−1.42929	−0.714644	+	0.699489i	$0.753411\pi$
$432$	−0.675997	−0.0325239
$433$	22.2900	1.07119	0.535596	−	0.844475i	$-0.320087\pi$
$433$	22.2900	1.07119	0.535596	+	0.844475i	$0.320087\pi$
$434$	0.0460403	0.00221000
$435$	−0.579634	−0.0277913
$436$	12.9694	0.621123
$437$	21.2197	1.01508
$438$	3.96140	0.189283
$439$	−10.0081	−0.477661	−0.238831	−	0.971061i	$-0.576764\pi$
$439$	−10.0081	−0.477661	−0.238831	+	0.971061i	$0.576764\pi$
$440$	−0.756747	−0.0360765
$441$	−6.99974	−0.333321
$442$	−1.09341	−0.0520082
$443$	−29.7876	−1.41525	−0.707625	−	0.706588i	$-0.750233\pi$
$443$	−29.7876	−1.41525	−0.707625	+	0.706588i	$0.750233\pi$
$444$	−0.668953	−0.0317471
$445$	−14.0189	−0.664560
$446$	8.14356	0.385609
$447$	3.47475	0.164350
$448$	−0.102210	−0.00482897
$449$	0.598430	0.0282416	0.0141208	−	0.999900i	$-0.495505\pi$
$449$	0.598430	0.0282416	0.0141208	+	0.999900i	$0.495505\pi$
$450$	10.0242	0.472545
$451$	−0.436692	−0.0205630
$452$	−13.3114	−0.626114
$453$	−4.73716	−0.222571
$454$	6.94223	0.325815
$455$	−0.0715386	−0.00335378
$456$	20.1388	0.943086
$457$	−31.3201	−1.46509	−0.732547	−	0.680717i	$-0.761668\pi$
$457$	−31.3201	−1.46509	−0.732547	+	0.680717i	$0.761668\pi$
$458$	7.86849	0.367671
$459$	1.00000	0.0466760
$460$	−13.2077	−0.615814
$461$	−21.3776	−0.995654	−0.497827	−	0.867276i	$-0.665868\pi$
$461$	−21.3776	−0.995654	−0.497827	+	0.867276i	$0.665868\pi$
$462$	0.000998632 0	4.64606e−5 0
$463$	22.3273	1.03764	0.518818	−	0.854885i	$-0.326372\pi$
$463$	22.3273	1.03764	0.518818	+	0.854885i	$0.326372\pi$
$464$	−0.0996936	−0.00462816
$465$	11.8082	0.547592
$466$	−3.72600	−0.172603
$467$	25.7610	1.19208	0.596039	−	0.802956i	$-0.296740\pi$
$467$	25.7610	1.19208	0.596039	+	0.802956i	$0.296740\pi$
$468$	1.23011	0.0568620
$469$	−0.230003	−0.0106205
$470$	6.79212	0.313297
$471$	0.886583	0.0408516
$472$	0.527330	0.0242723
$473$	−0.0314729	−0.00144713
$474$	0.959467	0.0440698
$475$	−71.2121	−3.26743
$476$	0.0172404	0.000790213 0
$477$	2.47166	0.113170
$478$	−6.71502	−0.307138
$479$	17.4209	0.795979	0.397990	−	0.917390i	$-0.369708\pi$
$479$	17.4209	0.795979	0.397990	+	0.917390i	$0.369708\pi$
$480$	−20.6761	−0.943729
$481$	−0.706247	−0.0322021
$482$	−8.55606	−0.389718
$483$	0.0497234	0.00226249
$484$	11.8691	0.539503
$485$	3.61648	0.164216
$486$	0.959467	0.0435223
$487$	1.10880	0.0502444	0.0251222	−	0.999684i	$-0.492003\pi$
$487$	1.10880	0.0502444	0.0251222	+	0.999684i	$0.492003\pi$
$488$	−8.60228	−0.389407
$489$	−3.22247	−0.145725
$490$	26.3963	1.19246
$491$	36.3792	1.64177	0.820884	−	0.571094i	$-0.193481\pi$
$491$	36.3792	1.64177	0.820884	+	0.571094i	$0.193481\pi$
$492$	−7.23348	−0.326110
$493$	0.147476	0.00664200
$494$	7.45277	0.335316
$495$	0.256125	0.0115119
$496$	2.03094	0.0911919
$497$	−0.117004	−0.00524833
$498$	9.67274	0.433446
$499$	5.10789	0.228661	0.114330	−	0.993443i	$-0.463528\pi$
$499$	5.10789	0.228661	0.114330	+	0.993443i	$0.463528\pi$
$500$	23.1118	1.03359
$501$	13.3747	0.597536
$502$	24.1413	1.07748
$503$	−17.2101	−0.767359	−0.383680	−	0.923466i	$-0.625343\pi$
$503$	−17.2101	−0.767359	−0.383680	+	0.923466i	$0.625343\pi$
$504$	0.0471906	0.00210204
$505$	−32.0511	−1.42626
$506$	0.194650	0.00865326
$507$	−11.7013	−0.519673
$508$	−12.9572	−0.574883
$509$	−7.65504	−0.339304	−0.169652	−	0.985504i	$-0.554264\pi$
$509$	−7.65504	−0.339304	−0.169652	+	0.985504i	$0.554264\pi$
$510$	−3.77104	−0.166985
$511$	−0.0659440	−0.00291719
$512$	−7.55077	−0.333700
$513$	−6.81607	−0.300937
$514$	−6.17130	−0.272204
$515$	−14.5368	−0.640570
$516$	−0.521326	−0.0229501
$517$	0.117372	0.00516201
$518$	−0.00949707	−0.000417277 0
$519$	15.8197	0.694407
$520$	−13.2338	−0.580340
$521$	13.8703	0.607668	0.303834	−	0.952725i	$-0.401733\pi$
$521$	13.8703	0.607668	0.303834	+	0.952725i	$0.401733\pi$
$522$	0.141499	0.00619323
$523$	−14.7109	−0.643264	−0.321632	−	0.946865i	$-0.604231\pi$
$523$	−14.7109	−0.643264	−0.321632	+	0.946865i	$0.604231\pi$
$524$	−17.9680	−0.784936
$525$	−0.166869	−0.00728275
$526$	−18.5115	−0.807141
$527$	−3.00436	−0.130872
$528$	0.0440519	0.00191711
$529$	−13.3081	−0.578612
$530$	−9.32073	−0.404867
$531$	−0.178477	−0.00774526
$532$	−0.117512	−0.00509479
$533$	−7.63675	−0.330784
$534$	3.42226	0.148096
$535$	41.4509	1.79208
$536$	−42.5477	−1.83778
$537$	11.7407	0.506647
$538$	−23.7369	−1.02337
$539$	0.456144	0.0196475
$540$	4.24251	0.182569
$541$	−30.8850	−1.32785	−0.663926	−	0.747799i	$-0.731111\pi$
$541$	−30.8850	−1.32785	−0.663926	+	0.747799i	$0.731111\pi$
$542$	−21.2651	−0.913413
$543$	12.5486	0.538511
$544$	5.26061	0.225547
$545$	47.2238	2.02284
$546$	0.0174638	0.000747382 0
$547$	−8.19687	−0.350473	−0.175236	−	0.984526i	$-0.556069\pi$
$547$	−8.19687	−0.350473	−0.175236	+	0.984526i	$0.556069\pi$
$548$	11.0209	0.470790
$549$	2.91148	0.124259
$550$	−0.653234	−0.0278540
$551$	−1.00521	−0.0428234
$552$	9.19824	0.391503
$553$	−0.0159719	−0.000679193 0
$554$	−18.6699	−0.793209
$555$	−2.43576	−0.103392
$556$	−7.18114	−0.304548
$557$	30.7603	1.30335	0.651677	−	0.758497i	$-0.274066\pi$
$557$	30.7603	1.30335	0.651677	+	0.758497i	$0.274066\pi$
$558$	−2.88259	−0.122030
$559$	−0.550390	−0.0232790
$560$	−0.0424358	−0.00179324
$561$	−0.0651658	−0.00275130
$562$	24.1757	1.01979
$563$	−36.1210	−1.52232	−0.761159	−	0.648566i	$-0.775369\pi$
$563$	−36.1210	−1.52232	−0.761159	+	0.648566i	$0.775369\pi$
$564$	1.94418	0.0818647
$565$	−48.4688	−2.03910
$566$	22.7673	0.956980
$567$	−0.0159719	−0.000670756 0
$568$	−21.6443	−0.908173
$569$	8.53839	0.357948	0.178974	−	0.983854i	$-0.442722\pi$
$569$	8.53839	0.357948	0.178974	+	0.983854i	$0.442722\pi$
$570$	25.7037	1.07661
$571$	23.0989	0.966659	0.483329	−	0.875439i	$-0.339427\pi$
$571$	23.0989	0.966659	0.483329	+	0.875439i	$0.339427\pi$
$572$	−0.0801613	−0.00335171
$573$	−7.42412	−0.310147
$574$	−0.102693	−0.00428633
$575$	−32.5255	−1.35641
$576$	6.39938	0.266641
$577$	−9.68378	−0.403141	−0.201571	−	0.979474i	$-0.564605\pi$
$577$	−9.68378	−0.403141	−0.201571	+	0.979474i	$0.564605\pi$
$578$	0.959467	0.0399086
$579$	−16.8962	−0.702181
$580$	0.625671	0.0259796
$581$	−0.161018	−0.00668017
$582$	−0.882845	−0.0365951
$583$	−0.161068	−0.00667074
$584$	−12.1988	−0.504792
$585$	4.47904	0.185185
$586$	21.5088	0.888522
$587$	10.9296	0.451113	0.225557	−	0.974230i	$-0.427580\pi$
$587$	10.9296	0.451113	0.225557	+	0.974230i	$0.427580\pi$
$588$	7.55569	0.311591
$589$	20.4780	0.843780
$590$	0.673045	0.0277088
$591$	6.58428	0.270841
$592$	−0.418937	−0.0172182
$593$	−9.83762	−0.403983	−0.201991	−	0.979387i	$-0.564741\pi$
$593$	−9.83762	−0.403983	−0.201991	+	0.979387i	$0.564741\pi$
$594$	−0.0625244	−0.00256541
$595$	0.0627751	0.00257353
$596$	−3.75073	−0.153636
$597$	−24.6876	−1.01040
$598$	3.40399	0.139200
$599$	3.57030	0.145878	0.0729392	−	0.997336i	$-0.476762\pi$
$599$	3.57030	0.145878	0.0729392	+	0.997336i	$0.476762\pi$
$600$	−30.8687	−1.26021
$601$	10.4946	0.428083	0.214041	−	0.976825i	$-0.431337\pi$
$601$	10.4946	0.428083	0.214041	+	0.976825i	$0.431337\pi$
$602$	−0.00740121	−0.000301651 0
$603$	14.4005	0.586433
$604$	5.11340	0.208061
$605$	43.2172	1.75703
$606$	7.82423	0.317838
$607$	−24.7558	−1.00481	−0.502403	−	0.864634i	$-0.667551\pi$
$607$	−24.7558	−1.00481	−0.502403	+	0.864634i	$0.667551\pi$
$608$	−35.8567	−1.45418
$609$	−0.00235547	−9.54486e−5 0
$610$	−10.9793	−0.444540
$611$	2.05257	0.0830380
$612$	−1.07942	−0.0436331
$613$	−0.389380	−0.0157269	−0.00786346	−	0.999969i	$-0.502503\pi$
$613$	−0.389380	−0.0157269	−0.00786346	+	0.999969i	$0.502503\pi$
$614$	6.91811	0.279192
$615$	−26.3382	−1.06206
$616$	−0.00307521	−0.000123904 0
$617$	31.5848	1.27156	0.635779	−	0.771871i	$-0.280679\pi$
$617$	31.5848	1.27156	0.635779	+	0.771871i	$0.280679\pi$
$618$	3.54870	0.142749
$619$	−5.87315	−0.236062	−0.118031	−	0.993010i	$-0.537658\pi$
$619$	−5.87315	−0.236062	−0.118031	+	0.993010i	$0.537658\pi$
$620$	−12.7461	−0.511894
$621$	−3.11319	−0.124928
$622$	28.5809	1.14599
$623$	−0.0569690	−0.00228242
$624$	0.770368	0.0308394
$625$	31.9154	1.27661
$626$	19.3150	0.771982
$627$	0.444175	0.0177386
$628$	−0.956998	−0.0381884
$629$	0.619732	0.0247103
$630$	0.0602306	0.00239964
$631$	−36.9771	−1.47203	−0.736017	−	0.676963i	$-0.763295\pi$
$631$	−36.9771	−1.47203	−0.736017	+	0.676963i	$0.763295\pi$
$632$	−2.95460	−0.117528
$633$	0.866183	0.0344277
$634$	12.7620	0.506844
$635$	−47.1792	−1.87225
$636$	−2.66797	−0.105792
$637$	7.97692	0.316057
$638$	−0.00922087	−0.000365058 0
$639$	7.32560	0.289796
$640$	17.2198	0.680672
$641$	11.8230	0.466980	0.233490	−	0.972359i	$-0.424985\pi$
$641$	11.8230	0.466980	0.233490	+	0.972359i	$0.424985\pi$
$642$	−10.1189	−0.399360
$643$	7.95713	0.313798	0.156899	−	0.987615i	$-0.449850\pi$
$643$	7.95713	0.313798	0.156899	+	0.987615i	$0.449850\pi$
$644$	−0.0536727	−0.00211500
$645$	−1.89823	−0.0747427
$646$	−6.53980	−0.257305
$647$	−44.7463	−1.75916	−0.879580	−	0.475751i	$-0.842176\pi$
$647$	−44.7463	−1.75916	−0.879580	+	0.475751i	$0.842176\pi$
$648$	−2.95460	−0.116068
$649$	0.0116306	0.000456542 0
$650$	−11.4236	−0.448070
$651$	0.0479853	0.00188069
$652$	3.47841	0.136225
$653$	3.73814	0.146285	0.0731423	−	0.997322i	$-0.476697\pi$
$653$	3.73814	0.146285	0.0731423	+	0.997322i	$0.476697\pi$
$654$	−11.5281	−0.450786
$655$	−65.4244	−2.55634
$656$	−4.53002	−0.176868
$657$	4.12876	0.161078
$658$	0.0276013	0.00107601
$659$	12.7381	0.496207	0.248104	−	0.968733i	$-0.420193\pi$
$659$	12.7381	0.496207	0.248104	+	0.968733i	$0.420193\pi$
$660$	−0.276467	−0.0107615
$661$	−29.0859	−1.13131	−0.565655	−	0.824642i	$-0.691377\pi$
$661$	−29.0859	−1.13131	−0.565655	+	0.824642i	$0.691377\pi$
$662$	−13.9303	−0.541418
$663$	−1.13960	−0.0442585
$664$	−29.7865	−1.15594
$665$	−0.427880	−0.0165925
$666$	0.594612	0.0230407
$667$	−0.459121	−0.0177773
$668$	−14.4369	−0.558581
$669$	8.48759	0.328149
$670$	−54.3048	−2.09798
$671$	−0.189729	−0.00732441
$672$	−0.0840218	−0.00324121
$673$	6.07420	0.234143	0.117072	−	0.993123i	$-0.462649\pi$
$673$	6.07420	0.234143	0.117072	+	0.993123i	$0.462649\pi$
$674$	1.05093	0.0404801
$675$	10.4477	0.402131
$676$	12.6307	0.485795
$677$	42.3898	1.62917	0.814586	−	0.580043i	$-0.196964\pi$
$677$	42.3898	1.62917	0.814586	+	0.580043i	$0.196964\pi$
$678$	11.8321	0.454408
$679$	0.0146964	0.000563996 0
$680$	11.6126	0.445324
$681$	7.23551	0.277265
$682$	0.187846	0.00719300
$683$	37.5410	1.43647	0.718234	−	0.695801i	$-0.244951\pi$
$683$	37.5410	1.43647	0.718234	+	0.695801i	$0.244951\pi$
$684$	7.35743	0.281318
$685$	40.1288	1.53324
$686$	0.214539	0.00819113
$687$	8.20090	0.312884
$688$	−0.326484	−0.0124471
$689$	−2.81671	−0.107308
$690$	11.7400	0.446933
$691$	22.6172	0.860400	0.430200	−	0.902734i	$-0.358443\pi$
$691$	22.6172	0.860400	0.430200	+	0.902734i	$0.358443\pi$
$692$	−17.0761	−0.649138
$693$	0.00104082	3.95375e−5 0
$694$	−13.7468	−0.521820
$695$	−26.1477	−0.991837
$696$	−0.435734	−0.0165165
$697$	6.70124	0.253828
$698$	−17.6791	−0.669165
$699$	−3.88340	−0.146884
$700$	0.180122	0.00680798
$701$	25.4839	0.962512	0.481256	−	0.876580i	$-0.340181\pi$
$701$	25.4839	0.962512	0.481256	+	0.876580i	$0.340181\pi$
$702$	−1.09341	−0.0412681
$703$	−4.22414	−0.159316
$704$	−0.417021	−0.0157171
$705$	7.07906	0.266613
$706$	−5.40179	−0.203299
$707$	−0.130247	−0.00489844
$708$	0.192653	0.00724033
$709$	8.91381	0.334765	0.167383	−	0.985892i	$-0.446468\pi$
$709$	8.91381	0.334765	0.167383	+	0.985892i	$0.446468\pi$
$710$	−27.6252	−1.03675
$711$	1.00000	0.0375029
$712$	−10.5386	−0.394950
$713$	9.35314	0.350278
$714$	−0.0153245	−0.000573504 0
$715$	−0.291880	−0.0109157
$716$	−12.6732	−0.473618
$717$	−6.99870	−0.261371
$718$	−28.4909	−1.06327
$719$	48.7025	1.81630	0.908148	−	0.418648i	$-0.137496\pi$
$719$	48.7025	1.81630	0.908148	+	0.418648i	$0.137496\pi$
$720$	2.65691	0.0990170
$721$	−0.0590738	−0.00220002
$722$	26.3459	0.980492
$723$	−8.91752	−0.331646
$724$	−13.5452	−0.503404
$725$	1.54078	0.0572233
$726$	−10.5501	−0.391549
$727$	38.5277	1.42891	0.714456	−	0.699680i	$-0.246674\pi$
$727$	38.5277	1.42891	0.714456	+	0.699680i	$0.246674\pi$
$728$	−0.0537785	−0.00199316
$729$	1.00000	0.0370370
$730$	−15.5697	−0.576261
$731$	0.482967	0.0178632
$732$	−3.14272	−0.116158
$733$	27.5212	1.01652	0.508260	−	0.861204i	$-0.330289\pi$
$733$	27.5212	1.01652	0.508260	+	0.861204i	$0.330289\pi$
$734$	13.5513	0.500188
$735$	27.5115	1.01478
$736$	−16.3773	−0.603674
$737$	−0.938419	−0.0345671
$738$	6.42962	0.236678
$739$	34.8732	1.28283	0.641416	−	0.767194i	$-0.278347\pi$
$739$	34.8732	1.28283	0.641416	+	0.767194i	$0.278347\pi$
$740$	2.62922	0.0966521
$741$	7.76761	0.285350
$742$	−0.0378769	−0.00139051
$743$	1.40984	0.0517219	0.0258610	−	0.999666i	$-0.491767\pi$
$743$	1.40984	0.0517219	0.0258610	+	0.999666i	$0.491767\pi$
$744$	8.87670	0.325436
$745$	−13.6570	−0.500354
$746$	25.0436	0.916910
$747$	10.0814	0.368858
$748$	0.0703415	0.00257194
$749$	0.168445	0.00615485
$750$	−20.5434	−0.750138
$751$	40.3381	1.47196	0.735979	−	0.677005i	$-0.236722\pi$
$751$	40.3381	1.47196	0.735979	+	0.677005i	$0.236722\pi$
$752$	1.21756	0.0443997
$753$	25.1611	0.916923
$754$	−0.161252	−0.00587246
$755$	18.6187	0.677604
$756$	0.0172404	0.000627028 0
$757$	−2.61955	−0.0952090	−0.0476045	−	0.998866i	$-0.515159\pi$
$757$	−2.61955	−0.0952090	−0.0476045	+	0.998866i	$0.515159\pi$
$758$	20.5538	0.746549
$759$	0.202873	0.00736384
$760$	−79.1526	−2.87117
$761$	15.4011	0.558289	0.279144	−	0.960249i	$-0.409949\pi$
$761$	15.4011	0.558289	0.279144	+	0.960249i	$0.409949\pi$
$762$	11.5173	0.417226
$763$	0.191904	0.00694741
$764$	8.01377	0.289928
$765$	−3.93035	−0.142102
$766$	−27.3797	−0.989270
$767$	0.203393	0.00734410
$768$	−17.0024	−0.613521
$769$	15.2773	0.550913	0.275457	−	0.961314i	$-0.411171\pi$
$769$	15.2773	0.550913	0.275457	+	0.961314i	$0.411171\pi$
$770$	−0.00392498	−0.000141446 0
$771$	−6.43201	−0.231643
$772$	18.2381	0.656405
$773$	23.9573	0.861684	0.430842	−	0.902427i	$-0.358217\pi$
$773$	23.9573	0.861684	0.430842	+	0.902427i	$0.358217\pi$
$774$	0.463390	0.0166562
$775$	−31.3886	−1.12751
$776$	2.71866	0.0975940
$777$	−0.00989828	−0.000355099 0
$778$	−13.7290	−0.492209
$779$	−45.6762	−1.63652
$780$	−4.83478	−0.173113
$781$	−0.477379	−0.0170820
$782$	−2.98700	−0.106815
$783$	0.147476	0.00527037
$784$	4.73181	0.168993
$785$	−3.48458	−0.124370
$786$	15.9712	0.569674
$787$	27.8730	0.993567	0.496783	−	0.867875i	$-0.334514\pi$
$787$	27.8730	0.993567	0.496783	+	0.867875i	$0.334514\pi$
$788$	−7.10723	−0.253184
$789$	−19.2936	−0.686869
$790$	−3.77104	−0.134168
$791$	−0.196964	−0.00700323
$792$	0.192539	0.00684158
$793$	−3.31793	−0.117823
$794$	−13.1221	−0.465684
$795$	−9.71449	−0.344538
$796$	26.6484	0.944527
$797$	19.4277	0.688163	0.344082	−	0.938940i	$-0.388190\pi$
$797$	19.4277	0.688163	0.344082	+	0.938940i	$0.388190\pi$
$798$	0.104453	0.00369759
$799$	−1.80113	−0.0637193
$800$	54.9611	1.94317
$801$	3.56683	0.126028
$802$	−2.62933	−0.0928448
$803$	−0.269054	−0.00949470
$804$	−15.5442	−0.548202
$805$	−0.195431	−0.00688802
$806$	3.28500	0.115709
$807$	−24.7396	−0.870877
$808$	−24.0941	−0.847628
$809$	−14.9533	−0.525731	−0.262866	−	0.964832i	$-0.584668\pi$
$809$	−14.9533	−0.525731	−0.262866	+	0.964832i	$0.584668\pi$
$810$	−3.77104	−0.132501
$811$	44.0460	1.54666	0.773332	−	0.634001i	$-0.218589\pi$
$811$	44.0460	1.54666	0.773332	+	0.634001i	$0.218589\pi$
$812$	0.00254255	8.92262e−5 0
$813$	−22.1634	−0.777305
$814$	−0.0387484	−0.00135813
$815$	12.6654	0.443651
$816$	−0.675997	−0.0236646
$817$	−3.29194	−0.115170
$818$	12.5229	0.437853
$819$	0.0182016	0.000636015 0
$820$	28.4301	0.992823
$821$	34.6371	1.20884	0.604421	−	0.796665i	$-0.293405\pi$
$821$	34.6371	1.20884	0.604421	+	0.796665i	$0.293405\pi$
$822$	−9.79614	−0.341680
$823$	23.1865	0.808231	0.404116	−	0.914708i	$-0.367579\pi$
$823$	23.1865	0.808231	0.404116	+	0.914708i	$0.367579\pi$
$824$	−10.9279	−0.380693
$825$	−0.680831	−0.0237035
$826$	0.00273507	9.51654e−5 0
$827$	9.59765	0.333743	0.166871	−	0.985979i	$-0.446634\pi$
$827$	9.59765	0.333743	0.166871	+	0.985979i	$0.446634\pi$
$828$	3.36045	0.116784
$829$	34.6244	1.20255	0.601277	−	0.799040i	$-0.294659\pi$
$829$	34.6244	1.20255	0.601277	+	0.799040i	$0.294659\pi$
$830$	−38.0173	−1.31960
$831$	−19.4586	−0.675013
$832$	−7.29274	−0.252830
$833$	−6.99974	−0.242527
$834$	6.38309	0.221028
$835$	−52.5671	−1.81916
$836$	−0.479453	−0.0165822
$837$	−3.00436	−0.103846
$838$	22.2024	0.766968
$839$	−7.76267	−0.267997	−0.133999	−	0.990982i	$-0.542782\pi$
$839$	−7.76267	−0.267997	−0.133999	+	0.990982i	$0.542782\pi$
$840$	−0.185476	−0.00639951
$841$	−28.9783	−0.999250
$842$	−6.94543	−0.239355
$843$	25.1970	0.867832
$844$	−0.934979	−0.0321833
$845$	45.9903	1.58211
$846$	−1.72812	−0.0594140
$847$	0.175623	0.00603447
$848$	−1.67084	−0.0573767
$849$	23.7291	0.814380
$850$	10.0242	0.343827
$851$	−1.92934	−0.0661369
$852$	−7.90743	−0.270904
$853$	−33.4815	−1.14638	−0.573192	−	0.819421i	$-0.694295\pi$
$853$	−33.4815	−1.14638	−0.573192	+	0.819421i	$0.694295\pi$
$854$	−0.0446169	−0.00152676
$855$	26.7896	0.916184
$856$	31.1603	1.06504
$857$	20.5662	0.702527	0.351263	−	0.936277i	$-0.385752\pi$
$857$	20.5662	0.702527	0.351263	+	0.936277i	$0.385752\pi$
$858$	0.0712530	0.00243254
$859$	33.3077	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$859$	33.3077	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$860$	2.04899	0.0698701
$861$	−0.107031	−0.00364762
$862$	−28.4700	−0.969693
$863$	−21.8597	−0.744111	−0.372056	−	0.928210i	$-0.621347\pi$
$863$	−21.8597	−0.744111	−0.372056	+	0.928210i	$0.621347\pi$
$864$	5.26061	0.178970
$865$	−62.1769	−2.11408
$866$	21.3866	0.726745
$867$	1.00000	0.0339618
$868$	−0.0517965	−0.00175809
$869$	−0.0651658	−0.00221060
$870$	−0.556139	−0.0188549
$871$	−16.4108	−0.556059
$872$	35.5000	1.20218
$873$	−0.920142	−0.0311421
$874$	20.3596	0.688674
$875$	0.341978	0.0115610
$876$	−4.45668	−0.150577
$877$	9.72757	0.328477	0.164238	−	0.986421i	$-0.447483\pi$
$877$	9.72757	0.328477	0.164238	+	0.986421i	$0.447483\pi$
$878$	−9.60245	−0.324067
$879$	22.4175	0.756123
$880$	−0.173139	−0.00583653
$881$	42.8279	1.44291	0.721455	−	0.692461i	$-0.243474\pi$
$881$	42.8279	1.44291	0.721455	+	0.692461i	$0.243474\pi$
$882$	−6.71602	−0.226140
$883$	−2.78237	−0.0936341	−0.0468171	−	0.998903i	$-0.514908\pi$
$883$	−2.78237	−0.0936341	−0.0468171	+	0.998903i	$0.514908\pi$
$884$	1.23011	0.0413732
$885$	0.701479	0.0235799
$886$	−28.5802	−0.960170
$887$	31.6063	1.06124	0.530618	−	0.847611i	$-0.321960\pi$
$887$	31.6063	1.06124	0.530618	+	0.847611i	$0.321960\pi$
$888$	−1.83106	−0.0614464
$889$	−0.191723	−0.00643020
$890$	−13.4507	−0.450868
$891$	−0.0651658	−0.00218314
$892$	−9.16170	−0.306757
$893$	12.2766	0.410821
$894$	3.33391	0.111503
$895$	−46.1449	−1.54246
$896$	0.0699766	0.00233775
$897$	3.54779	0.118457
$898$	0.574173	0.0191604
$899$	−0.443072	−0.0147773
$900$	−11.2775	−0.375915
$901$	2.47166	0.0823429
$902$	−0.418991	−0.0139509
$903$	−0.00771388	−0.000256702 0
$904$	−36.4359	−1.21184
$905$	−49.3203	−1.63946
$906$	−4.54515	−0.151002
$907$	−42.8161	−1.42168	−0.710842	−	0.703352i	$-0.751686\pi$
$907$	−42.8161	−1.42168	−0.710842	+	0.703352i	$0.751686\pi$
$908$	−7.81018	−0.259190
$909$	8.15477	0.270477
$910$	−0.0686389	−0.00227536
$911$	33.8733	1.12227	0.561136	−	0.827723i	$-0.310364\pi$
$911$	33.8733	1.12227	0.561136	+	0.827723i	$0.310364\pi$
$912$	4.60765	0.152574
$913$	−0.656961	−0.0217422
$914$	−30.0506	−0.993986
$915$	−11.4431	−0.378299
$916$	−8.85225	−0.292487
$917$	−0.265867	−0.00877969
$918$	0.959467	0.0316671
$919$	−29.2074	−0.963464	−0.481732	−	0.876319i	$-0.659992\pi$
$919$	−29.2074	−0.963464	−0.481732	+	0.876319i	$0.659992\pi$
$920$	−36.1523	−1.19191
$921$	7.21037	0.237590
$922$	−20.5111	−0.675497
$923$	−8.34827	−0.274787
$924$	−0.00112349	−3.69600e−5 0
$925$	6.47475	0.212888
$926$	21.4223	0.703979
$927$	3.69861	0.121478
$928$	0.775816	0.0254674
$929$	33.9710	1.11455	0.557277	−	0.830327i	$-0.311846\pi$
$929$	33.9710	1.11455	0.557277	+	0.830327i	$0.311846\pi$
$930$	11.3296	0.371511
$931$	47.7108	1.56366
$932$	4.19184	0.137308
$933$	29.7883	0.975226
$934$	24.7168	0.808759
$935$	0.256125	0.00837617
$936$	3.36707	0.110056
$937$	−6.02058	−0.196684	−0.0983418	−	0.995153i	$-0.531354\pi$
$937$	−6.02058	−0.196684	−0.0983418	+	0.995153i	$0.531354\pi$
$938$	−0.220680	−0.00720546
$939$	20.1310	0.656949
$940$	−7.64130	−0.249232
$941$	−21.1006	−0.687860	−0.343930	−	0.938995i	$-0.611758\pi$
$941$	−21.1006	−0.687860	−0.343930	+	0.938995i	$0.611758\pi$
$942$	0.850646	0.0277156
$943$	−20.8622	−0.679367
$944$	0.120650	0.00392683
$945$	0.0627751	0.00204207
$946$	−0.0301972	−0.000981796 0
$947$	23.6547	0.768675	0.384338	−	0.923193i	$-0.374430\pi$
$947$	23.6547	0.768675	0.384338	+	0.923193i	$0.374430\pi$
$948$	−1.07942	−0.0350581
$949$	−4.70514	−0.152735
$950$	−68.3256	−2.21678
$951$	13.3012	0.431319
$952$	0.0471906	0.00152946
$953$	24.4904	0.793323	0.396661	−	0.917965i	$-0.370169\pi$
$953$	24.4904	0.793323	0.396661	+	0.917965i	$0.370169\pi$
$954$	2.37148	0.0767794
$955$	29.1794	0.944223
$956$	7.55456	0.244332
$957$	−0.00961042	−0.000310661 0
$958$	16.7147	0.540028
$959$	0.163073	0.00526589
$960$	−25.1518	−0.811771
$961$	−21.9738	−0.708832
$962$	−0.677621	−0.0218474
$963$	−10.5464	−0.339851
$964$	9.62578	0.310025
$965$	66.4079	2.13775
$966$	0.0477080	0.00153498
$967$	−9.73216	−0.312965	−0.156483	−	0.987681i	$-0.550015\pi$
$967$	−9.73216	−0.312965	−0.156483	+	0.987681i	$0.550015\pi$
$968$	32.4881	1.04421
$969$	−6.81607	−0.218964
$970$	3.46989	0.111412
$971$	5.94716	0.190854	0.0954268	−	0.995436i	$-0.469578\pi$
$971$	5.94716	0.190854	0.0954268	+	0.995436i	$0.469578\pi$
$972$	−1.07942	−0.0346225
$973$	−0.106257	−0.00340644
$974$	1.06385	0.0340881
$975$	−11.9062	−0.381303
$976$	−1.96815	−0.0629990
$977$	3.55776	0.113823	0.0569114	−	0.998379i	$-0.481875\pi$
$977$	3.55776	0.113823	0.0569114	+	0.998379i	$0.481875\pi$
$978$	−3.09185	−0.0988666
$979$	−0.232436	−0.00742867
$980$	−29.6965	−0.948621
$981$	−12.0152	−0.383614
$982$	34.9046	1.11385
$983$	−0.0183782	−0.000586173 0	−0.000293087	−	1.00000i	$-0.500093\pi$
$983$	−0.0183782	−0.000586173 0	−0.000293087		1.00000i	$0.500093\pi$
$984$	−19.7995	−0.631186
$985$	−25.8785	−0.824559
$986$	0.141499	0.00450623
$987$	0.0287674	0.000915675 0
$988$	−8.38455	−0.266748
$989$	−1.50357	−0.0478106
$990$	0.245743	0.00781022
$991$	−49.7820	−1.58138	−0.790689	−	0.612218i	$-0.790278\pi$
$991$	−49.7820	−1.58138	−0.790689	+	0.612218i	$0.790278\pi$
$992$	−15.8048	−0.501802
$993$	−14.5188	−0.460742
$994$	−0.112261	−0.00356070
$995$	97.0309	3.07609
$996$	−10.8821	−0.344812
$997$	13.7478	0.435398	0.217699	−	0.976016i	$-0.430145\pi$
$997$	13.7478	0.435398	0.217699	+	0.976016i	$0.430145\pi$
$998$	4.90085	0.155134
$999$	0.619732	0.0196074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.20	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.20	✓	31	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+0.959467 q^{2} +1.00000 q^{3} -1.07942 q^{4} -3.93035 q^{5} +0.959467 q^{6} -0.0159719 q^{7} -2.95460 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.959467 q^{2} +1.00000 q^{3} -1.07942 q^{4} -3.93035 q^{5} +0.959467 q^{6} -0.0159719 q^{7} -2.95460 q^{8} +1.00000 q^{9} -3.77104 q^{10} -0.0651658 q^{11} -1.07942 q^{12} -1.13960 q^{13} -0.0153245 q^{14} -3.93035 q^{15} -0.675997 q^{16} +1.00000 q^{17} +0.959467 q^{18} -6.81607 q^{19} +4.24251 q^{20} -0.0159719 q^{21} -0.0625244 q^{22} -3.11319 q^{23} -2.95460 q^{24} +10.4477 q^{25} -1.09341 q^{26} +1.00000 q^{27} +0.0172404 q^{28} +0.147476 q^{29} -3.77104 q^{30} -3.00436 q^{31} +5.26061 q^{32} -0.0651658 q^{33} +0.959467 q^{34} +0.0627751 q^{35} -1.07942 q^{36} +0.619732 q^{37} -6.53980 q^{38} -1.13960 q^{39} +11.6126 q^{40} +6.70124 q^{41} -0.0153245 q^{42} +0.482967 q^{43} +0.0703415 q^{44} -3.93035 q^{45} -2.98700 q^{46} -1.80113 q^{47} -0.675997 q^{48} -6.99974 q^{49} +10.0242 q^{50} +1.00000 q^{51} +1.23011 q^{52} +2.47166 q^{53} +0.959467 q^{54} +0.256125 q^{55} +0.0471906 q^{56} -6.81607 q^{57} +0.141499 q^{58} -0.178477 q^{59} +4.24251 q^{60} +2.91148 q^{61} -2.88259 q^{62} -0.0159719 q^{63} +6.39938 q^{64} +4.47904 q^{65} -0.0625244 q^{66} +14.4005 q^{67} -1.07942 q^{68} -3.11319 q^{69} +0.0602306 q^{70} +7.32560 q^{71} -2.95460 q^{72} +4.12876 q^{73} +0.594612 q^{74} +10.4477 q^{75} +7.35743 q^{76} +0.00104082 q^{77} -1.09341 q^{78} +1.00000 q^{79} +2.65691 q^{80} +1.00000 q^{81} +6.42962 q^{82} +10.0814 q^{83} +0.0172404 q^{84} -3.93035 q^{85} +0.463390 q^{86} +0.147476 q^{87} +0.192539 q^{88} +3.56683 q^{89} -3.77104 q^{90} +0.0182016 q^{91} +3.36045 q^{92} -3.00436 q^{93} -1.72812 q^{94} +26.7896 q^{95} +5.26061 q^{96} -0.920142 q^{97} -6.71602 q^{98} -0.0651658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

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