LMFDB - Embedded newform 8034.2.a.x.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} + \cdots + 8832 $ x^13 - x^12 - 40x^11 + 34x^10 + 604x^9 - 381x^8 - 4352x^7 + 1474x^6 + 15809x^5 - 368x^4 - 27369x^3 - 7621x^2 + 17300*x + 8832
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.22001$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.22001 q^{5} -1.00000 q^{6} +3.96620 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.22001 q^{5} -1.00000 q^{6} +3.96620 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.22001 q^{10} +1.38625 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.96620 q^{14} -1.22001 q^{15} +1.00000 q^{16} +5.91083 q^{17} -1.00000 q^{18} -3.46427 q^{19} -1.22001 q^{20} +3.96620 q^{21} -1.38625 q^{22} -1.88200 q^{23} -1.00000 q^{24} -3.51157 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.96620 q^{28} +6.86205 q^{29} +1.22001 q^{30} -10.2736 q^{31} -1.00000 q^{32} +1.38625 q^{33} -5.91083 q^{34} -4.83881 q^{35} +1.00000 q^{36} -5.10201 q^{37} +3.46427 q^{38} +1.00000 q^{39} +1.22001 q^{40} -0.267033 q^{41} -3.96620 q^{42} +10.6311 q^{43} +1.38625 q^{44} -1.22001 q^{45} +1.88200 q^{46} -1.56018 q^{47} +1.00000 q^{48} +8.73071 q^{49} +3.51157 q^{50} +5.91083 q^{51} +1.00000 q^{52} +11.8248 q^{53} -1.00000 q^{54} -1.69124 q^{55} -3.96620 q^{56} -3.46427 q^{57} -6.86205 q^{58} +0.429893 q^{59} -1.22001 q^{60} +0.390728 q^{61} +10.2736 q^{62} +3.96620 q^{63} +1.00000 q^{64} -1.22001 q^{65} -1.38625 q^{66} +5.38174 q^{67} +5.91083 q^{68} -1.88200 q^{69} +4.83881 q^{70} +11.7903 q^{71} -1.00000 q^{72} +8.64854 q^{73} +5.10201 q^{74} -3.51157 q^{75} -3.46427 q^{76} +5.49813 q^{77} -1.00000 q^{78} +10.3409 q^{79} -1.22001 q^{80} +1.00000 q^{81} +0.267033 q^{82} +13.6379 q^{83} +3.96620 q^{84} -7.21129 q^{85} -10.6311 q^{86} +6.86205 q^{87} -1.38625 q^{88} -13.7322 q^{89} +1.22001 q^{90} +3.96620 q^{91} -1.88200 q^{92} -10.2736 q^{93} +1.56018 q^{94} +4.22646 q^{95} -1.00000 q^{96} -10.4310 q^{97} -8.73071 q^{98} +1.38625 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9} - q^{10} + 6 q^{11} + 13 q^{12} + 13 q^{13} + q^{14} + q^{15} + 13 q^{16} + 18 q^{17} - 13 q^{18} - q^{19} + q^{20} - q^{21} - 6 q^{22} + 20 q^{23} - 13 q^{24} + 16 q^{25} - 13 q^{26} + 13 q^{27} - q^{28} + 25 q^{29} - q^{30} - 5 q^{31} - 13 q^{32} + 6 q^{33} - 18 q^{34} + 26 q^{35} + 13 q^{36} - 6 q^{37} + q^{38} + 13 q^{39} - q^{40} + 13 q^{41} + q^{42} - 2 q^{43} + 6 q^{44} + q^{45} - 20 q^{46} + 5 q^{47} + 13 q^{48} - 16 q^{50} + 18 q^{51} + 13 q^{52} + 21 q^{53} - 13 q^{54} + 2 q^{55} + q^{56} - q^{57} - 25 q^{58} + 22 q^{59} + q^{60} + 2 q^{61} + 5 q^{62} - q^{63} + 13 q^{64} + q^{65} - 6 q^{66} - q^{67} + 18 q^{68} + 20 q^{69} - 26 q^{70} + 32 q^{71} - 13 q^{72} - 13 q^{73} + 6 q^{74} + 16 q^{75} - q^{76} + 33 q^{77} - 13 q^{78} - 7 q^{79} + q^{80} + 13 q^{81} - 13 q^{82} + 17 q^{83} - q^{84} - 25 q^{85} + 2 q^{86} + 25 q^{87} - 6 q^{88} - 12 q^{89} - q^{90} - q^{91} + 20 q^{92} - 5 q^{93} - 5 q^{94} + 36 q^{95} - 13 q^{96} - 42 q^{97} + 6 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9 - q^10 + 6 * q^11 + 13 * q^12 + 13 * q^13 + q^14 + q^15 + 13 * q^16 + 18 * q^17 - 13 * q^18 - q^19 + q^20 - q^21 - 6 * q^22 + 20 * q^23 - 13 * q^24 + 16 * q^25 - 13 * q^26 + 13 * q^27 - q^28 + 25 * q^29 - q^30 - 5 * q^31 - 13 * q^32 + 6 * q^33 - 18 * q^34 + 26 * q^35 + 13 * q^36 - 6 * q^37 + q^38 + 13 * q^39 - q^40 + 13 * q^41 + q^42 - 2 * q^43 + 6 * q^44 + q^45 - 20 * q^46 + 5 * q^47 + 13 * q^48 - 16 * q^50 + 18 * q^51 + 13 * q^52 + 21 * q^53 - 13 * q^54 + 2 * q^55 + q^56 - q^57 - 25 * q^58 + 22 * q^59 + q^60 + 2 * q^61 + 5 * q^62 - q^63 + 13 * q^64 + q^65 - 6 * q^66 - q^67 + 18 * q^68 + 20 * q^69 - 26 * q^70 + 32 * q^71 - 13 * q^72 - 13 * q^73 + 6 * q^74 + 16 * q^75 - q^76 + 33 * q^77 - 13 * q^78 - 7 * q^79 + q^80 + 13 * q^81 - 13 * q^82 + 17 * q^83 - q^84 - 25 * q^85 + 2 * q^86 + 25 * q^87 - 6 * q^88 - 12 * q^89 - q^90 - q^91 + 20 * q^92 - 5 * q^93 - 5 * q^94 + 36 * q^95 - 13 * q^96 - 42 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.22001	−0.545607	−0.272803	−	0.962070i	$-0.587951\pi$
$5$	−1.22001	−0.545607	−0.272803	+	0.962070i	$0.587951\pi$
$6$	−1.00000	−0.408248
$7$	3.96620	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$7$	3.96620	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.22001	0.385802
$11$	1.38625	0.417969	0.208985	−	0.977919i	$-0.432984\pi$
$11$	1.38625	0.417969	0.208985	+	0.977919i	$0.432984\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.96620	−1.06001
$15$	−1.22001	−0.315006
$16$	1.00000	0.250000
$17$	5.91083	1.43359	0.716793	−	0.697286i	$-0.245609\pi$
$17$	5.91083	1.43359	0.716793	+	0.697286i	$0.245609\pi$
$18$	−1.00000	−0.235702
$19$	−3.46427	−0.794758	−0.397379	−	0.917655i	$-0.630080\pi$
$19$	−3.46427	−0.794758	−0.397379	+	0.917655i	$0.630080\pi$
$20$	−1.22001	−0.272803
$21$	3.96620	0.865495
$22$	−1.38625	−0.295549
$23$	−1.88200	−0.392425	−0.196212	−	0.980561i	$-0.562864\pi$
$23$	−1.88200	−0.392425	−0.196212	+	0.980561i	$0.562864\pi$
$24$	−1.00000	−0.204124
$25$	−3.51157	−0.702314
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	3.96620	0.749541
$29$	6.86205	1.27425	0.637125	−	0.770760i	$-0.280123\pi$
$29$	6.86205	1.27425	0.637125	+	0.770760i	$0.280123\pi$
$30$	1.22001	0.222743
$31$	−10.2736	−1.84520	−0.922600	−	0.385758i	$-0.873940\pi$
$31$	−10.2736	−1.84520	−0.922600	+	0.385758i	$0.873940\pi$
$32$	−1.00000	−0.176777
$33$	1.38625	0.241315
$34$	−5.91083	−1.01370
$35$	−4.83881	−0.817908
$36$	1.00000	0.166667
$37$	−5.10201	−0.838765	−0.419382	−	0.907810i	$-0.637753\pi$
$37$	−5.10201	−0.838765	−0.419382	+	0.907810i	$0.637753\pi$
$38$	3.46427	0.561979
$39$	1.00000	0.160128
$40$	1.22001	0.192901
$41$	−0.267033	−0.0417035	−0.0208518	−	0.999783i	$-0.506638\pi$
$41$	−0.267033	−0.0417035	−0.0208518	+	0.999783i	$0.506638\pi$
$42$	−3.96620	−0.611997
$43$	10.6311	1.62123	0.810617	−	0.585576i	$-0.199132\pi$
$43$	10.6311	1.62123	0.810617	+	0.585576i	$0.199132\pi$
$44$	1.38625	0.208985
$45$	−1.22001	−0.181869
$46$	1.88200	0.277486
$47$	−1.56018	−0.227576	−0.113788	−	0.993505i	$-0.536299\pi$
$47$	−1.56018	−0.227576	−0.113788	+	0.993505i	$0.536299\pi$
$48$	1.00000	0.144338
$49$	8.73071	1.24724
$50$	3.51157	0.496611
$51$	5.91083	0.827682
$52$	1.00000	0.138675
$53$	11.8248	1.62426	0.812132	−	0.583473i	$-0.198307\pi$
$53$	11.8248	1.62426	0.812132	+	0.583473i	$0.198307\pi$
$54$	−1.00000	−0.136083
$55$	−1.69124	−0.228047
$56$	−3.96620	−0.530005
$57$	−3.46427	−0.458854
$58$	−6.86205	−0.901031
$59$	0.429893	0.0559673	0.0279837	−	0.999608i	$-0.491091\pi$
$59$	0.429893	0.0559673	0.0279837	+	0.999608i	$0.491091\pi$
$60$	−1.22001	−0.157503
$61$	0.390728	0.0500276	0.0250138	−	0.999687i	$-0.492037\pi$
$61$	0.390728	0.0500276	0.0250138	+	0.999687i	$0.492037\pi$
$62$	10.2736	1.30475
$63$	3.96620	0.499694
$64$	1.00000	0.125000
$65$	−1.22001	−0.151324
$66$	−1.38625	−0.170635
$67$	5.38174	0.657484	0.328742	−	0.944420i	$-0.393375\pi$
$67$	5.38174	0.657484	0.328742	+	0.944420i	$0.393375\pi$
$68$	5.91083	0.716793
$69$	−1.88200	−0.226567
$70$	4.83881	0.578349
$71$	11.7903	1.39925	0.699627	−	0.714508i	$-0.253349\pi$
$71$	11.7903	1.39925	0.699627	+	0.714508i	$0.253349\pi$
$72$	−1.00000	−0.117851
$73$	8.64854	1.01223	0.506117	−	0.862465i	$-0.331080\pi$
$73$	8.64854	1.01223	0.506117	+	0.862465i	$0.331080\pi$
$74$	5.10201	0.593096
$75$	−3.51157	−0.405481
$76$	−3.46427	−0.397379
$77$	5.49813	0.626570
$78$	−1.00000	−0.113228
$79$	10.3409	1.16344	0.581722	−	0.813388i	$-0.302379\pi$
$79$	10.3409	1.16344	0.581722	+	0.813388i	$0.302379\pi$
$80$	−1.22001	−0.136402
$81$	1.00000	0.111111
$82$	0.267033	0.0294888
$83$	13.6379	1.49695	0.748476	−	0.663162i	$-0.230786\pi$
$83$	13.6379	1.49695	0.748476	+	0.663162i	$0.230786\pi$
$84$	3.96620	0.432747
$85$	−7.21129	−0.782174
$86$	−10.6311	−1.14639
$87$	6.86205	0.735689
$88$	−1.38625	−0.147774
$89$	−13.7322	−1.45561	−0.727806	−	0.685783i	$-0.759460\pi$
$89$	−13.7322	−1.45561	−0.727806	+	0.685783i	$0.759460\pi$
$90$	1.22001	0.128601
$91$	3.96620	0.415770
$92$	−1.88200	−0.196212
$93$	−10.2736	−1.06533
$94$	1.56018	0.160921
$95$	4.22646	0.433625
$96$	−1.00000	−0.102062
$97$	−10.4310	−1.05911	−0.529556	−	0.848275i	$-0.677641\pi$
$97$	−10.4310	−1.05911	−0.529556	+	0.848275i	$0.677641\pi$
$98$	−8.73071	−0.881935
$99$	1.38625	0.139323
$100$	−3.51157	−0.351157
$101$	2.28727	0.227592	0.113796	−	0.993504i	$-0.463699\pi$
$101$	2.28727	0.227592	0.113796	+	0.993504i	$0.463699\pi$
$102$	−5.91083	−0.585259
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	−4.83881	−0.472220
$106$	−11.8248	−1.14853
$107$	−0.569973	−0.0551014	−0.0275507	−	0.999620i	$-0.508771\pi$
$107$	−0.569973	−0.0551014	−0.0275507	+	0.999620i	$0.508771\pi$
$108$	1.00000	0.0962250
$109$	−6.79818	−0.651148	−0.325574	−	0.945517i	$-0.605557\pi$
$109$	−6.79818	−0.651148	−0.325574	+	0.945517i	$0.605557\pi$
$110$	1.69124	0.161253
$111$	−5.10201	−0.484261
$112$	3.96620	0.374770
$113$	6.70233	0.630503	0.315251	−	0.949008i	$-0.397911\pi$
$113$	6.70233	0.630503	0.315251	+	0.949008i	$0.397911\pi$
$114$	3.46427	0.324459
$115$	2.29607	0.214110
$116$	6.86205	0.637125
$117$	1.00000	0.0924500
$118$	−0.429893	−0.0395749
$119$	23.4435	2.14906
$120$	1.22001	0.111371
$121$	−9.07832	−0.825302
$122$	−0.390728	−0.0353749
$123$	−0.267033	−0.0240775
$124$	−10.2736	−0.922600
$125$	10.3842	0.928793
$126$	−3.96620	−0.353337
$127$	0.307298	0.0272683	0.0136341	−	0.999907i	$-0.495660\pi$
$127$	0.307298	0.0272683	0.0136341	+	0.999907i	$0.495660\pi$
$128$	−1.00000	−0.0883883
$129$	10.6311	0.936020
$130$	1.22001	0.107002
$131$	−13.5792	−1.18642	−0.593210	−	0.805048i	$-0.702140\pi$
$131$	−13.5792	−1.18642	−0.593210	+	0.805048i	$0.702140\pi$
$132$	1.38625	0.120657
$133$	−13.7400	−1.19141
$134$	−5.38174	−0.464911
$135$	−1.22001	−0.105002
$136$	−5.91083	−0.506849
$137$	11.2890	0.964481	0.482240	−	0.876039i	$-0.339823\pi$
$137$	11.2890	0.964481	0.482240	+	0.876039i	$0.339823\pi$
$138$	1.88200	0.160207
$139$	−10.7537	−0.912118	−0.456059	−	0.889950i	$-0.650739\pi$
$139$	−10.7537	−0.912118	−0.456059	+	0.889950i	$0.650739\pi$
$140$	−4.83881	−0.408954
$141$	−1.56018	−0.131391
$142$	−11.7903	−0.989423
$143$	1.38625	0.115924
$144$	1.00000	0.0833333
$145$	−8.37179	−0.695239
$146$	−8.64854	−0.715758
$147$	8.73071	0.720097
$148$	−5.10201	−0.419382
$149$	−4.46743	−0.365986	−0.182993	−	0.983114i	$-0.558579\pi$
$149$	−4.46743	−0.365986	−0.182993	+	0.983114i	$0.558579\pi$
$150$	3.51157	0.286718
$151$	13.0784	1.06431	0.532153	−	0.846648i	$-0.321383\pi$
$151$	13.0784	1.06431	0.532153	+	0.846648i	$0.321383\pi$
$152$	3.46427	0.280989
$153$	5.91083	0.477862
$154$	−5.49813	−0.443052
$155$	12.5340	1.00675
$156$	1.00000	0.0800641
$157$	−3.95084	−0.315311	−0.157655	−	0.987494i	$-0.550394\pi$
$157$	−3.95084	−0.315311	−0.157655	+	0.987494i	$0.550394\pi$
$158$	−10.3409	−0.822679
$159$	11.8248	0.937769
$160$	1.22001	0.0964505
$161$	−7.46440	−0.588277
$162$	−1.00000	−0.0785674
$163$	−2.40839	−0.188640	−0.0943200	−	0.995542i	$-0.530068\pi$
$163$	−2.40839	−0.188640	−0.0943200	+	0.995542i	$0.530068\pi$
$164$	−0.267033	−0.0208518
$165$	−1.69124	−0.131663
$166$	−13.6379	−1.05850
$167$	4.36766	0.337980	0.168990	−	0.985618i	$-0.445949\pi$
$167$	4.36766	0.337980	0.168990	+	0.985618i	$0.445949\pi$
$168$	−3.96620	−0.305999
$169$	1.00000	0.0769231
$170$	7.21129	0.553081
$171$	−3.46427	−0.264919
$172$	10.6311	0.810617
$173$	1.17943	0.0896706	0.0448353	−	0.998994i	$-0.485724\pi$
$173$	1.17943	0.0896706	0.0448353	+	0.998994i	$0.485724\pi$
$174$	−6.86205	−0.520210
$175$	−13.9276	−1.05282
$176$	1.38625	0.104492
$177$	0.429893	0.0323128
$178$	13.7322	1.02927
$179$	19.9083	1.48801	0.744007	−	0.668171i	$-0.232923\pi$
$179$	19.9083	1.48801	0.744007	+	0.668171i	$0.232923\pi$
$180$	−1.22001	−0.0909344
$181$	−5.75480	−0.427751	−0.213875	−	0.976861i	$-0.568609\pi$
$181$	−5.75480	−0.427751	−0.213875	+	0.976861i	$0.568609\pi$
$182$	−3.96620	−0.293994
$183$	0.390728	0.0288835
$184$	1.88200	0.138743
$185$	6.22452	0.457635
$186$	10.2736	0.753300
$187$	8.19387	0.599195
$188$	−1.56018	−0.113788
$189$	3.96620	0.288498
$190$	−4.22646	−0.306619
$191$	−14.1566	−1.02433	−0.512166	−	0.858886i	$-0.671157\pi$
$191$	−14.1566	−1.02433	−0.512166	+	0.858886i	$0.671157\pi$
$192$	1.00000	0.0721688
$193$	−20.9342	−1.50688	−0.753438	−	0.657519i	$-0.771606\pi$
$193$	−20.9342	−1.50688	−0.753438	+	0.657519i	$0.771606\pi$
$194$	10.4310	0.748905
$195$	−1.22001	−0.0873670
$196$	8.73071	0.623622
$197$	15.0664	1.07344	0.536719	−	0.843761i	$-0.319664\pi$
$197$	15.0664	1.07344	0.536719	+	0.843761i	$0.319664\pi$
$198$	−1.38625	−0.0985163
$199$	5.22284	0.370238	0.185119	−	0.982716i	$-0.440733\pi$
$199$	5.22284	0.370238	0.185119	+	0.982716i	$0.440733\pi$
$200$	3.51157	0.248305
$201$	5.38174	0.379599
$202$	−2.28727	−0.160932
$203$	27.2162	1.91020
$204$	5.91083	0.413841
$205$	0.325783	0.0227537
$206$	−1.00000	−0.0696733
$207$	−1.88200	−0.130808
$208$	1.00000	0.0693375
$209$	−4.80233	−0.332184
$210$	4.83881	0.333910
$211$	−14.8912	−1.02515	−0.512575	−	0.858643i	$-0.671308\pi$
$211$	−14.8912	−1.02515	−0.512575	+	0.858643i	$0.671308\pi$
$212$	11.8248	0.812132
$213$	11.7903	0.807860
$214$	0.569973	0.0389625
$215$	−12.9701	−0.884556
$216$	−1.00000	−0.0680414
$217$	−40.7473	−2.76610
$218$	6.79818	0.460431
$219$	8.64854	0.584414
$220$	−1.69124	−0.114023
$221$	5.91083	0.397605
$222$	5.10201	0.342424
$223$	14.7195	0.985692	0.492846	−	0.870117i	$-0.335957\pi$
$223$	14.7195	0.985692	0.492846	+	0.870117i	$0.335957\pi$
$224$	−3.96620	−0.265003
$225$	−3.51157	−0.234105
$226$	−6.70233	−0.445833
$227$	−23.0313	−1.52864	−0.764321	−	0.644836i	$-0.776926\pi$
$227$	−23.0313	−1.52864	−0.764321	+	0.644836i	$0.776926\pi$
$228$	−3.46427	−0.229427
$229$	−0.774678	−0.0511922	−0.0255961	−	0.999672i	$-0.508148\pi$
$229$	−0.774678	−0.0511922	−0.0255961	+	0.999672i	$0.508148\pi$
$230$	−2.29607	−0.151398
$231$	5.49813	0.361750
$232$	−6.86205	−0.450515
$233$	20.4913	1.34243	0.671215	−	0.741262i	$-0.265773\pi$
$233$	20.4913	1.34243	0.671215	+	0.741262i	$0.265773\pi$
$234$	−1.00000	−0.0653720
$235$	1.90345	0.124167
$236$	0.429893	0.0279837
$237$	10.3409	0.671715
$238$	−23.4435	−1.51962
$239$	−8.63229	−0.558376	−0.279188	−	0.960236i	$-0.590065\pi$
$239$	−8.63229	−0.558376	−0.279188	+	0.960236i	$0.590065\pi$
$240$	−1.22001	−0.0787515
$241$	6.66735	0.429482	0.214741	−	0.976671i	$-0.431109\pi$
$241$	6.66735	0.429482	0.214741	+	0.976671i	$0.431109\pi$
$242$	9.07832	0.583576
$243$	1.00000	0.0641500
$244$	0.390728	0.0250138
$245$	−10.6516	−0.680504
$246$	0.267033	0.0170254
$247$	−3.46427	−0.220426
$248$	10.2736	0.652377
$249$	13.6379	0.864265
$250$	−10.3842	−0.656756
$251$	11.9835	0.756390	0.378195	−	0.925726i	$-0.376545\pi$
$251$	11.9835	0.756390	0.378195	+	0.925726i	$0.376545\pi$
$252$	3.96620	0.249847
$253$	−2.60892	−0.164022
$254$	−0.307298	−0.0192816
$255$	−7.21129	−0.451589
$256$	1.00000	0.0625000
$257$	20.6472	1.28794	0.643970	−	0.765051i	$-0.277286\pi$
$257$	20.6472	1.28794	0.643970	+	0.765051i	$0.277286\pi$
$258$	−10.6311	−0.661866
$259$	−20.2356	−1.25738
$260$	−1.22001	−0.0756620
$261$	6.86205	0.424750
$262$	13.5792	0.838926
$263$	1.38466	0.0853817	0.0426909	−	0.999088i	$-0.486407\pi$
$263$	1.38466	0.0853817	0.0426909	+	0.999088i	$0.486407\pi$
$264$	−1.38625	−0.0853176
$265$	−14.4264	−0.886209
$266$	13.7400	0.842452
$267$	−13.7322	−0.840398
$268$	5.38174	0.328742
$269$	−29.0128	−1.76894	−0.884470	−	0.466597i	$-0.845480\pi$
$269$	−29.0128	−1.76894	−0.884470	+	0.466597i	$0.845480\pi$
$270$	1.22001	0.0742476
$271$	−14.1431	−0.859133	−0.429566	−	0.903035i	$-0.641334\pi$
$271$	−14.1431	−0.859133	−0.429566	+	0.903035i	$0.641334\pi$
$272$	5.91083	0.358397
$273$	3.96620	0.240045
$274$	−11.2890	−0.681991
$275$	−4.86790	−0.293545
$276$	−1.88200	−0.113283
$277$	16.8638	1.01325	0.506623	−	0.862168i	$-0.330894\pi$
$277$	16.8638	1.01325	0.506623	+	0.862168i	$0.330894\pi$
$278$	10.7537	0.644965
$279$	−10.2736	−0.615067
$280$	4.83881	0.289174
$281$	0.278548	0.0166168	0.00830839	−	0.999965i	$-0.497355\pi$
$281$	0.278548	0.0166168	0.00830839	+	0.999965i	$0.497355\pi$
$282$	1.56018	0.0929076
$283$	−11.4750	−0.682116	−0.341058	−	0.940042i	$-0.610785\pi$
$283$	−11.4750	−0.682116	−0.341058	+	0.940042i	$0.610785\pi$
$284$	11.7903	0.699627
$285$	4.22646	0.250354
$286$	−1.38625	−0.0819705
$287$	−1.05910	−0.0625169
$288$	−1.00000	−0.0589256
$289$	17.9379	1.05517
$290$	8.37179	0.491608
$291$	−10.4310	−0.611479
$292$	8.64854	0.506117
$293$	−21.2819	−1.24330	−0.621650	−	0.783295i	$-0.713538\pi$
$293$	−21.2819	−1.24330	−0.621650	+	0.783295i	$0.713538\pi$
$294$	−8.73071	−0.509185
$295$	−0.524475	−0.0305361
$296$	5.10201	0.296548
$297$	1.38625	0.0804382
$298$	4.46743	0.258791
$299$	−1.88200	−0.108839
$300$	−3.51157	−0.202740
$301$	42.1652	2.43036
$302$	−13.0784	−0.752578
$303$	2.28727	0.131400
$304$	−3.46427	−0.198690
$305$	−0.476694	−0.0272954
$306$	−5.91083	−0.337900
$307$	25.5688	1.45929	0.729643	−	0.683828i	$-0.239686\pi$
$307$	25.5688	1.45929	0.729643	+	0.683828i	$0.239686\pi$
$308$	5.49813	0.313285
$309$	1.00000	0.0568880
$310$	−12.5340	−0.711882
$311$	32.1386	1.82242	0.911208	−	0.411948i	$-0.135151\pi$
$311$	32.1386	1.82242	0.911208	+	0.411948i	$0.135151\pi$
$312$	−1.00000	−0.0566139
$313$	6.56333	0.370982	0.185491	−	0.982646i	$-0.440613\pi$
$313$	6.56333	0.370982	0.185491	+	0.982646i	$0.440613\pi$
$314$	3.95084	0.222959
$315$	−4.83881	−0.272636
$316$	10.3409	0.581722
$317$	17.8362	1.00178	0.500889	−	0.865511i	$-0.333006\pi$
$317$	17.8362	1.00178	0.500889	+	0.865511i	$0.333006\pi$
$318$	−11.8248	−0.663103
$319$	9.51249	0.532597
$320$	−1.22001	−0.0682008
$321$	−0.569973	−0.0318128
$322$	7.46440	0.415975
$323$	−20.4767	−1.13935
$324$	1.00000	0.0555556
$325$	−3.51157	−0.194787
$326$	2.40839	0.133389
$327$	−6.79818	−0.375940
$328$	0.267033	0.0147444
$329$	−6.18799	−0.341155
$330$	1.69124	0.0930997
$331$	7.88185	0.433226	0.216613	−	0.976258i	$-0.430499\pi$
$331$	7.88185	0.433226	0.216613	+	0.976258i	$0.430499\pi$
$332$	13.6379	0.748476
$333$	−5.10201	−0.279588
$334$	−4.36766	−0.238988
$335$	−6.56579	−0.358728
$336$	3.96620	0.216374
$337$	21.6220	1.17783	0.588913	−	0.808196i	$-0.299556\pi$
$337$	21.6220	1.17783	0.588913	+	0.808196i	$0.299556\pi$
$338$	−1.00000	−0.0543928
$339$	6.70233	0.364021
$340$	−7.21129	−0.391087
$341$	−14.2418	−0.771237
$342$	3.46427	0.187326
$343$	6.86433	0.370639
$344$	−10.6311	−0.573193
$345$	2.29607	0.123616
$346$	−1.17943	−0.0634067
$347$	21.0040	1.12756	0.563778	−	0.825927i	$-0.309347\pi$
$347$	21.0040	1.12756	0.563778	+	0.825927i	$0.309347\pi$
$348$	6.86205	0.367844
$349$	3.29346	0.176295	0.0881476	−	0.996107i	$-0.471905\pi$
$349$	3.29346	0.176295	0.0881476	+	0.996107i	$0.471905\pi$
$350$	13.9276	0.744460
$351$	1.00000	0.0533761
$352$	−1.38625	−0.0738872
$353$	12.9671	0.690168	0.345084	−	0.938572i	$-0.387850\pi$
$353$	12.9671	0.690168	0.345084	+	0.938572i	$0.387850\pi$
$354$	−0.429893	−0.0228486
$355$	−14.3844	−0.763443
$356$	−13.7322	−0.727806
$357$	23.4435	1.24076
$358$	−19.9083	−1.05219
$359$	−25.1310	−1.32636	−0.663180	−	0.748460i	$-0.730794\pi$
$359$	−25.1310	−1.32636	−0.663180	+	0.748460i	$0.730794\pi$
$360$	1.22001	0.0643003
$361$	−6.99883	−0.368359
$362$	5.75480	0.302465
$363$	−9.07832	−0.476488
$364$	3.96620	0.207885
$365$	−10.5513	−0.552282
$366$	−0.390728	−0.0204237
$367$	−24.1479	−1.26051	−0.630255	−	0.776388i	$-0.717050\pi$
$367$	−24.1479	−1.26051	−0.630255	+	0.776388i	$0.717050\pi$
$368$	−1.88200	−0.0981062
$369$	−0.267033	−0.0139012
$370$	−6.22452	−0.323597
$371$	46.8996	2.43490
$372$	−10.2736	−0.532663
$373$	−16.3082	−0.844405	−0.422203	−	0.906501i	$-0.638743\pi$
$373$	−16.3082	−0.844405	−0.422203	+	0.906501i	$0.638743\pi$
$374$	−8.19387	−0.423695
$375$	10.3842	0.536239
$376$	1.56018	0.0804603
$377$	6.86205	0.353413
$378$	−3.96620	−0.203999
$379$	−9.74651	−0.500645	−0.250322	−	0.968163i	$-0.580537\pi$
$379$	−9.74651	−0.500645	−0.250322	+	0.968163i	$0.580537\pi$
$380$	4.22646	0.216813
$381$	0.307298	0.0157433
$382$	14.1566	0.724312
$383$	23.5660	1.20417	0.602084	−	0.798433i	$-0.294337\pi$
$383$	23.5660	1.20417	0.602084	+	0.798433i	$0.294337\pi$
$384$	−1.00000	−0.0510310
$385$	−6.70779	−0.341861
$386$	20.9342	1.06552
$387$	10.6311	0.540412
$388$	−10.4310	−0.529556
$389$	35.3692	1.79329	0.896645	−	0.442751i	$-0.145997\pi$
$389$	35.3692	1.79329	0.896645	+	0.442751i	$0.145997\pi$
$390$	1.22001	0.0617778
$391$	−11.1242	−0.562575
$392$	−8.73071	−0.440967
$393$	−13.5792	−0.684980
$394$	−15.0664	−0.759035
$395$	−12.6161	−0.634783
$396$	1.38625	0.0696615
$397$	14.4456	0.725005	0.362502	−	0.931983i	$-0.381922\pi$
$397$	14.4456	0.725005	0.362502	+	0.931983i	$0.381922\pi$
$398$	−5.22284	−0.261797
$399$	−13.7400	−0.687859
$400$	−3.51157	−0.175578
$401$	−10.6754	−0.533104	−0.266552	−	0.963821i	$-0.585884\pi$
$401$	−10.6754	−0.533104	−0.266552	+	0.963821i	$0.585884\pi$
$402$	−5.38174	−0.268417
$403$	−10.2736	−0.511766
$404$	2.28727	0.113796
$405$	−1.22001	−0.0606229
$406$	−27.2162	−1.35072
$407$	−7.07264	−0.350578
$408$	−5.91083	−0.292630
$409$	−14.2025	−0.702269	−0.351134	−	0.936325i	$-0.614204\pi$
$409$	−14.2025	−0.702269	−0.351134	+	0.936325i	$0.614204\pi$
$410$	−0.325783	−0.0160893
$411$	11.2890	0.556843
$412$	1.00000	0.0492665
$413$	1.70504	0.0838996
$414$	1.88200	0.0924954
$415$	−16.6384	−0.816746
$416$	−1.00000	−0.0490290
$417$	−10.7537	−0.526611
$418$	4.80233	0.234890
$419$	−35.1352	−1.71646	−0.858232	−	0.513262i	$-0.828437\pi$
$419$	−35.1352	−1.71646	−0.858232	+	0.513262i	$0.828437\pi$
$420$	−4.83881	−0.236110
$421$	32.3829	1.57825	0.789123	−	0.614235i	$-0.210535\pi$
$421$	32.3829	1.57825	0.789123	+	0.614235i	$0.210535\pi$
$422$	14.8912	0.724890
$423$	−1.56018	−0.0758587
$424$	−11.8248	−0.574264
$425$	−20.7563	−1.00683
$426$	−11.7903	−0.571243
$427$	1.54970	0.0749955
$428$	−0.569973	−0.0275507
$429$	1.38625	0.0669286
$430$	12.9701	0.625476
$431$	6.59612	0.317724	0.158862	−	0.987301i	$-0.449217\pi$
$431$	6.59612	0.317724	0.158862	+	0.987301i	$0.449217\pi$
$432$	1.00000	0.0481125
$433$	15.4865	0.744236	0.372118	−	0.928185i	$-0.378632\pi$
$433$	15.4865	0.744236	0.372118	+	0.928185i	$0.378632\pi$
$434$	40.7473	1.95593
$435$	−8.37179	−0.401396
$436$	−6.79818	−0.325574
$437$	6.51977	0.311883
$438$	−8.64854	−0.413243
$439$	9.26529	0.442208	0.221104	−	0.975250i	$-0.429034\pi$
$439$	9.26529	0.442208	0.221104	+	0.975250i	$0.429034\pi$
$440$	1.69124	0.0806267
$441$	8.73071	0.415748
$442$	−5.91083	−0.281150
$443$	−37.1180	−1.76353	−0.881765	−	0.471689i	$-0.843645\pi$
$443$	−37.1180	−1.76353	−0.881765	+	0.471689i	$0.843645\pi$
$444$	−5.10201	−0.242131
$445$	16.7535	0.794192
$446$	−14.7195	−0.696989
$447$	−4.46743	−0.211302
$448$	3.96620	0.187385
$449$	23.0717	1.08882	0.544410	−	0.838819i	$-0.316754\pi$
$449$	23.0717	1.08882	0.544410	+	0.838819i	$0.316754\pi$
$450$	3.51157	0.165537
$451$	−0.370173	−0.0174308
$452$	6.70233	0.315251
$453$	13.0784	0.614478
$454$	23.0313	1.08091
$455$	−4.83881	−0.226847
$456$	3.46427	0.162229
$457$	−3.77668	−0.176666	−0.0883329	−	0.996091i	$-0.528154\pi$
$457$	−3.77668	−0.176666	−0.0883329	+	0.996091i	$0.528154\pi$
$458$	0.774678	0.0361983
$459$	5.91083	0.275894
$460$	2.29607	0.107055
$461$	−11.1347	−0.518594	−0.259297	−	0.965798i	$-0.583491\pi$
$461$	−11.1347	−0.518594	−0.259297	+	0.965798i	$0.583491\pi$
$462$	−5.49813	−0.255796
$463$	22.2118	1.03227	0.516134	−	0.856508i	$-0.327371\pi$
$463$	22.2118	1.03227	0.516134	+	0.856508i	$0.327371\pi$
$464$	6.86205	0.318562
$465$	12.5340	0.581249
$466$	−20.4913	−0.949242
$467$	3.17808	0.147064	0.0735321	−	0.997293i	$-0.476573\pi$
$467$	3.17808	0.147064	0.0735321	+	0.997293i	$0.476573\pi$
$468$	1.00000	0.0462250
$469$	21.3450	0.985622
$470$	−1.90345	−0.0877994
$471$	−3.95084	−0.182045
$472$	−0.429893	−0.0197874
$473$	14.7374	0.677626
$474$	−10.3409	−0.474974
$475$	12.1650	0.558169
$476$	23.4435	1.07453
$477$	11.8248	0.541421
$478$	8.63229	0.394832
$479$	−10.9280	−0.499313	−0.249657	−	0.968334i	$-0.580318\pi$
$479$	−10.9280	−0.499313	−0.249657	+	0.968334i	$0.580318\pi$
$480$	1.22001	0.0556857
$481$	−5.10201	−0.232631
$482$	−6.66735	−0.303690
$483$	−7.46440	−0.339642
$484$	−9.07832	−0.412651
$485$	12.7260	0.577858
$486$	−1.00000	−0.0453609
$487$	9.97466	0.451995	0.225997	−	0.974128i	$-0.427436\pi$
$487$	9.97466	0.451995	0.225997	+	0.974128i	$0.427436\pi$
$488$	−0.390728	−0.0176874
$489$	−2.40839	−0.108911
$490$	10.6516	0.481189
$491$	−36.7153	−1.65694	−0.828470	−	0.560034i	$-0.810788\pi$
$491$	−36.7153	−1.65694	−0.828470	+	0.560034i	$0.810788\pi$
$492$	−0.267033	−0.0120388
$493$	40.5604	1.82675
$494$	3.46427	0.155865
$495$	−1.69124	−0.0760156
$496$	−10.2736	−0.461300
$497$	46.7628	2.09760
$498$	−13.6379	−0.611128
$499$	33.5174	1.50044	0.750222	−	0.661186i	$-0.229947\pi$
$499$	33.5174	1.50044	0.750222	+	0.661186i	$0.229947\pi$
$500$	10.3842	0.464397
$501$	4.36766	0.195133
$502$	−11.9835	−0.534849
$503$	21.6781	0.966581	0.483290	−	0.875460i	$-0.339442\pi$
$503$	21.6781	0.966581	0.483290	+	0.875460i	$0.339442\pi$
$504$	−3.96620	−0.176668
$505$	−2.79050	−0.124176
$506$	2.60892	0.115981
$507$	1.00000	0.0444116
$508$	0.307298	0.0136341
$509$	13.3389	0.591235	0.295618	−	0.955306i	$-0.404475\pi$
$509$	13.3389	0.591235	0.295618	+	0.955306i	$0.404475\pi$
$510$	7.21129	0.319321
$511$	34.3018	1.51742
$512$	−1.00000	−0.0441942
$513$	−3.46427	−0.152951
$514$	−20.6472	−0.910711
$515$	−1.22001	−0.0537602
$516$	10.6311	0.468010
$517$	−2.16280	−0.0951199
$518$	20.2356	0.889099
$519$	1.17943	0.0517713
$520$	1.22001	0.0535011
$521$	5.21254	0.228366	0.114183	−	0.993460i	$-0.463575\pi$
$521$	5.21254	0.228366	0.114183	+	0.993460i	$0.463575\pi$
$522$	−6.86205	−0.300344
$523$	34.1514	1.49334	0.746669	−	0.665195i	$-0.231652\pi$
$523$	34.1514	1.49334	0.746669	+	0.665195i	$0.231652\pi$
$524$	−13.5792	−0.593210
$525$	−13.9276	−0.607849
$526$	−1.38466	−0.0603740
$527$	−60.7257	−2.64525
$528$	1.38625	0.0603287
$529$	−19.4581	−0.846003
$530$	14.4264	0.626645
$531$	0.429893	0.0186558
$532$	−13.7400	−0.595703
$533$	−0.267033	−0.0115665
$534$	13.7322	0.594251
$535$	0.695375	0.0300637
$536$	−5.38174	−0.232456
$537$	19.9083	0.859106
$538$	29.0128	1.25083
$539$	12.1029	0.521310
$540$	−1.22001	−0.0525010
$541$	7.40860	0.318521	0.159260	−	0.987237i	$-0.449089\pi$
$541$	7.40860	0.318521	0.159260	+	0.987237i	$0.449089\pi$
$542$	14.1431	0.607499
$543$	−5.75480	−0.246962
$544$	−5.91083	−0.253425
$545$	8.29387	0.355271
$546$	−3.96620	−0.169738
$547$	45.2631	1.93531	0.967654	−	0.252279i	$-0.0811801\pi$
$547$	45.2631	1.93531	0.967654	+	0.252279i	$0.0811801\pi$
$548$	11.2890	0.482240
$549$	0.390728	0.0166759
$550$	4.86790	0.207568
$551$	−23.7720	−1.01272
$552$	1.88200	0.0801034
$553$	41.0141	1.74410
$554$	−16.8638	−0.716473
$555$	6.22452	0.264216
$556$	−10.7537	−0.456059
$557$	10.8797	0.460990	0.230495	−	0.973074i	$-0.425966\pi$
$557$	10.8797	0.460990	0.230495	+	0.973074i	$0.425966\pi$
$558$	10.2736	0.434918
$559$	10.6311	0.449650
$560$	−4.83881	−0.204477
$561$	8.19387	0.345945
$562$	−0.278548	−0.0117498
$563$	−32.6754	−1.37711	−0.688553	−	0.725186i	$-0.741754\pi$
$563$	−32.6754	−1.37711	−0.688553	+	0.725186i	$0.741754\pi$
$564$	−1.56018	−0.0656956
$565$	−8.17694	−0.344006
$566$	11.4750	0.482329
$567$	3.96620	0.166565
$568$	−11.7903	−0.494711
$569$	27.6377	1.15863	0.579316	−	0.815103i	$-0.303320\pi$
$569$	27.6377	1.15863	0.579316	+	0.815103i	$0.303320\pi$
$570$	−4.22646	−0.177027
$571$	−22.9438	−0.960170	−0.480085	−	0.877222i	$-0.659394\pi$
$571$	−22.9438	−0.960170	−0.480085	+	0.877222i	$0.659394\pi$
$572$	1.38625	0.0579619
$573$	−14.1566	−0.591399
$574$	1.05910	0.0442061
$575$	6.60878	0.275605
$576$	1.00000	0.0416667
$577$	−46.2258	−1.92441	−0.962203	−	0.272333i	$-0.912205\pi$
$577$	−46.2258	−1.92441	−0.962203	+	0.272333i	$0.912205\pi$
$578$	−17.9379	−0.746119
$579$	−20.9342	−0.869995
$580$	−8.37179	−0.347620
$581$	54.0905	2.24405
$582$	10.4310	0.432381
$583$	16.3921	0.678893
$584$	−8.64854	−0.357879
$585$	−1.22001	−0.0504413
$586$	21.2819	0.879146
$587$	8.82359	0.364188	0.182094	−	0.983281i	$-0.441712\pi$
$587$	8.82359	0.364188	0.182094	+	0.983281i	$0.441712\pi$
$588$	8.73071	0.360048
$589$	35.5907	1.46649
$590$	0.524475	0.0215923
$591$	15.0664	0.619749
$592$	−5.10201	−0.209691
$593$	−19.0058	−0.780475	−0.390237	−	0.920714i	$-0.627607\pi$
$593$	−19.0058	−0.780475	−0.390237	+	0.920714i	$0.627607\pi$
$594$	−1.38625	−0.0568784
$595$	−28.6014	−1.17254
$596$	−4.46743	−0.182993
$597$	5.22284	0.213757
$598$	1.88200	0.0769609
$599$	25.9910	1.06196	0.530981	−	0.847383i	$-0.321823\pi$
$599$	25.9910	1.06196	0.530981	+	0.847383i	$0.321823\pi$
$600$	3.51157	0.143359
$601$	−13.7317	−0.560127	−0.280063	−	0.959982i	$-0.590355\pi$
$601$	−13.7317	−0.560127	−0.280063	+	0.959982i	$0.590355\pi$
$602$	−42.1652	−1.71853
$603$	5.38174	0.219161
$604$	13.0784	0.532153
$605$	11.0757	0.450290
$606$	−2.28727	−0.0929141
$607$	−13.5019	−0.548027	−0.274014	−	0.961726i	$-0.588351\pi$
$607$	−13.5019	−0.548027	−0.274014	+	0.961726i	$0.588351\pi$
$608$	3.46427	0.140495
$609$	27.2162	1.10286
$610$	0.476694	0.0193008
$611$	−1.56018	−0.0631183
$612$	5.91083	0.238931
$613$	−31.2414	−1.26183	−0.630914	−	0.775853i	$-0.717320\pi$
$613$	−31.2414	−1.26183	−0.630914	+	0.775853i	$0.717320\pi$
$614$	−25.5688	−1.03187
$615$	0.325783	0.0131369
$616$	−5.49813	−0.221526
$617$	−20.9825	−0.844723	−0.422362	−	0.906427i	$-0.638799\pi$
$617$	−20.9825	−0.844723	−0.422362	+	0.906427i	$0.638799\pi$
$618$	−1.00000	−0.0402259
$619$	−47.3376	−1.90266	−0.951329	−	0.308177i	$-0.900281\pi$
$619$	−47.3376	−1.90266	−0.951329	+	0.308177i	$0.900281\pi$
$620$	12.5340	0.503377
$621$	−1.88200	−0.0755222
$622$	−32.1386	−1.28864
$623$	−54.4647	−2.18208
$624$	1.00000	0.0400320
$625$	4.88895	0.195558
$626$	−6.56333	−0.262324
$627$	−4.80233	−0.191787
$628$	−3.95084	−0.157655
$629$	−30.1571	−1.20244
$630$	4.83881	0.192783
$631$	14.5320	0.578510	0.289255	−	0.957252i	$-0.406592\pi$
$631$	14.5320	0.578510	0.289255	+	0.957252i	$0.406592\pi$
$632$	−10.3409	−0.411340
$633$	−14.8912	−0.591870
$634$	−17.8362	−0.708364
$635$	−0.374907	−0.0148777
$636$	11.8248	0.468885
$637$	8.73071	0.345923
$638$	−9.51249	−0.376603
$639$	11.7903	0.466418
$640$	1.22001	0.0482253
$641$	−31.6213	−1.24897	−0.624483	−	0.781038i	$-0.714690\pi$
$641$	−31.6213	−1.24897	−0.624483	+	0.781038i	$0.714690\pi$
$642$	0.569973	0.0224950
$643$	23.9207	0.943340	0.471670	−	0.881775i	$-0.343651\pi$
$643$	23.9207	0.943340	0.471670	+	0.881775i	$0.343651\pi$
$644$	−7.46440	−0.294138
$645$	−12.9701	−0.510699
$646$	20.4767	0.805645
$647$	11.6719	0.458870	0.229435	−	0.973324i	$-0.426312\pi$
$647$	11.6719	0.458870	0.229435	+	0.973324i	$0.426312\pi$
$648$	−1.00000	−0.0392837
$649$	0.595938	0.0233926
$650$	3.51157	0.137735
$651$	−40.7473	−1.59701
$652$	−2.40839	−0.0943200
$653$	−31.6544	−1.23873	−0.619367	−	0.785102i	$-0.712611\pi$
$653$	−31.6544	−1.23873	−0.619367	+	0.785102i	$0.712611\pi$
$654$	6.79818	0.265830
$655$	16.5668	0.647319
$656$	−0.267033	−0.0104259
$657$	8.64854	0.337411
$658$	6.18799	0.241233
$659$	6.04475	0.235470	0.117735	−	0.993045i	$-0.462437\pi$
$659$	6.04475	0.235470	0.117735	+	0.993045i	$0.462437\pi$
$660$	−1.69124	−0.0658314
$661$	−12.1685	−0.473298	−0.236649	−	0.971595i	$-0.576049\pi$
$661$	−12.1685	−0.473298	−0.236649	+	0.971595i	$0.576049\pi$
$662$	−7.88185	−0.306337
$663$	5.91083	0.229558
$664$	−13.6379	−0.529252
$665$	16.7630	0.650039
$666$	5.10201	0.197699
$667$	−12.9144	−0.500048
$668$	4.36766	0.168990
$669$	14.7195	0.569090
$670$	6.56579	0.253659
$671$	0.541646	0.0209100
$672$	−3.96620	−0.152999
$673$	36.9946	1.42604	0.713019	−	0.701145i	$-0.247328\pi$
$673$	36.9946	1.42604	0.713019	+	0.701145i	$0.247328\pi$
$674$	−21.6220	−0.832849
$675$	−3.51157	−0.135160
$676$	1.00000	0.0384615
$677$	−19.5453	−0.751188	−0.375594	−	0.926784i	$-0.622561\pi$
$677$	−19.5453	−0.751188	−0.375594	+	0.926784i	$0.622561\pi$
$678$	−6.70233	−0.257402
$679$	−41.3716	−1.58769
$680$	7.21129	0.276540
$681$	−23.0313	−0.882562
$682$	14.2418	0.545347
$683$	14.8558	0.568441	0.284220	−	0.958759i	$-0.408265\pi$
$683$	14.8558	0.568441	0.284220	+	0.958759i	$0.408265\pi$
$684$	−3.46427	−0.132460
$685$	−13.7727	−0.526227
$686$	−6.86433	−0.262081
$687$	−0.774678	−0.0295558
$688$	10.6311	0.405309
$689$	11.8248	0.450490
$690$	−2.29607	−0.0874099
$691$	−25.9305	−0.986442	−0.493221	−	0.869904i	$-0.664181\pi$
$691$	−25.9305	−0.986442	−0.493221	+	0.869904i	$0.664181\pi$
$692$	1.17943	0.0448353
$693$	5.49813	0.208857
$694$	−21.0040	−0.797302
$695$	13.1197	0.497657
$696$	−6.86205	−0.260105
$697$	−1.57838	−0.0597856
$698$	−3.29346	−0.124659
$699$	20.4913	0.775053
$700$	−13.9276	−0.526412
$701$	22.1803	0.837740	0.418870	−	0.908046i	$-0.362426\pi$
$701$	22.1803	0.837740	0.418870	+	0.908046i	$0.362426\pi$
$702$	−1.00000	−0.0377426
$703$	17.6747	0.666615
$704$	1.38625	0.0522462
$705$	1.90345	0.0716879
$706$	−12.9671	−0.488022
$707$	9.07177	0.341179
$708$	0.429893	0.0161564
$709$	9.81618	0.368654	0.184327	−	0.982865i	$-0.440989\pi$
$709$	9.81618	0.368654	0.184327	+	0.982865i	$0.440989\pi$
$710$	14.3844	0.539835
$711$	10.3409	0.387815
$712$	13.7322	0.514637
$713$	19.3350	0.724103
$714$	−23.4435	−0.877351
$715$	−1.69124	−0.0632488
$716$	19.9083	0.744007
$717$	−8.63229	−0.322379
$718$	25.1310	0.937879
$719$	−14.1908	−0.529228	−0.264614	−	0.964354i	$-0.585245\pi$
$719$	−14.1908	−0.529228	−0.264614	+	0.964354i	$0.585245\pi$
$720$	−1.22001	−0.0454672
$721$	3.96620	0.147709
$722$	6.99883	0.260469
$723$	6.66735	0.247961
$724$	−5.75480	−0.213875
$725$	−24.0965	−0.894923
$726$	9.07832	0.336928
$727$	47.4448	1.75963	0.879816	−	0.475315i	$-0.157666\pi$
$727$	47.4448	1.75963	0.879816	+	0.475315i	$0.157666\pi$
$728$	−3.96620	−0.146997
$729$	1.00000	0.0370370
$730$	10.5513	0.390522
$731$	62.8389	2.32418
$732$	0.390728	0.0144417
$733$	−24.4861	−0.904416	−0.452208	−	0.891913i	$-0.649364\pi$
$733$	−24.4861	−0.904416	−0.452208	+	0.891913i	$0.649364\pi$
$734$	24.1479	0.891315
$735$	−10.6516	−0.392889
$736$	1.88200	0.0693716
$737$	7.46042	0.274808
$738$	0.267033	0.00982961
$739$	7.53683	0.277247	0.138623	−	0.990345i	$-0.455732\pi$
$739$	7.53683	0.277247	0.138623	+	0.990345i	$0.455732\pi$
$740$	6.22452	0.228818
$741$	−3.46427	−0.127263
$742$	−46.8996	−1.72174
$743$	9.86342	0.361854	0.180927	−	0.983497i	$-0.442090\pi$
$743$	9.86342	0.361854	0.180927	+	0.983497i	$0.442090\pi$
$744$	10.2736	0.376650
$745$	5.45032	0.199684
$746$	16.3082	0.597085
$747$	13.6379	0.498984
$748$	8.19387	0.299598
$749$	−2.26062	−0.0826014
$750$	−10.3842	−0.379178
$751$	−15.7052	−0.573090	−0.286545	−	0.958067i	$-0.592507\pi$
$751$	−15.7052	−0.573090	−0.286545	+	0.958067i	$0.592507\pi$
$752$	−1.56018	−0.0568941
$753$	11.9835	0.436702
$754$	−6.86205	−0.249901
$755$	−15.9558	−0.580692
$756$	3.96620	0.144249
$757$	23.2266	0.844185	0.422092	−	0.906553i	$-0.361296\pi$
$757$	23.2266	0.844185	0.422092	+	0.906553i	$0.361296\pi$
$758$	9.74651	0.354009
$759$	−2.60892	−0.0946979
$760$	−4.22646	−0.153310
$761$	−19.0599	−0.690920	−0.345460	−	0.938433i	$-0.612277\pi$
$761$	−19.0599	−0.690920	−0.345460	+	0.938433i	$0.612277\pi$
$762$	−0.307298	−0.0111322
$763$	−26.9629	−0.976123
$764$	−14.1566	−0.512166
$765$	−7.21129	−0.260725
$766$	−23.5660	−0.851475
$767$	0.429893	0.0155225
$768$	1.00000	0.0360844
$769$	−49.7513	−1.79408	−0.897038	−	0.441953i	$-0.854286\pi$
$769$	−49.7513	−1.79408	−0.897038	+	0.441953i	$0.854286\pi$
$770$	6.70779	0.241732
$771$	20.6472	0.743592
$772$	−20.9342	−0.753438
$773$	1.40416	0.0505040	0.0252520	−	0.999681i	$-0.491961\pi$
$773$	1.40416	0.0505040	0.0252520	+	0.999681i	$0.491961\pi$
$774$	−10.6311	−0.382129
$775$	36.0766	1.29591
$776$	10.4310	0.374453
$777$	−20.2356	−0.725947
$778$	−35.3692	−1.26805
$779$	0.925074	0.0331442
$780$	−1.22001	−0.0436835
$781$	16.3443	0.584846
$782$	11.1242	0.397801
$783$	6.86205	0.245230
$784$	8.73071	0.311811
$785$	4.82007	0.172036
$786$	13.5792	0.484354
$787$	−27.3037	−0.973272	−0.486636	−	0.873605i	$-0.661776\pi$
$787$	−27.3037	−0.973272	−0.486636	+	0.873605i	$0.661776\pi$
$788$	15.0664	0.536719
$789$	1.38466	0.0492952
$790$	12.6161	0.448859
$791$	26.5828	0.945174
$792$	−1.38625	−0.0492581
$793$	0.390728	0.0138752
$794$	−14.4456	−0.512656
$795$	−14.4264	−0.511653
$796$	5.22284	0.185119
$797$	−6.17931	−0.218882	−0.109441	−	0.993993i	$-0.534906\pi$
$797$	−6.17931	−0.218882	−0.109441	+	0.993993i	$0.534906\pi$
$798$	13.7400	0.486390
$799$	−9.22198	−0.326250
$800$	3.51157	0.124153
$801$	−13.7322	−0.485204
$802$	10.6754	0.376961
$803$	11.9890	0.423083
$804$	5.38174	0.189799
$805$	9.10666	0.320968
$806$	10.2736	0.361873
$807$	−29.0128	−1.02130
$808$	−2.28727	−0.0804660
$809$	−5.72977	−0.201448	−0.100724	−	0.994914i	$-0.532116\pi$
$809$	−5.72977	−0.201448	−0.100724	+	0.994914i	$0.532116\pi$
$810$	1.22001	0.0428669
$811$	18.6632	0.655354	0.327677	−	0.944790i	$-0.393734\pi$
$811$	18.6632	0.655354	0.327677	+	0.944790i	$0.393734\pi$
$812$	27.2162	0.955102
$813$	−14.1431	−0.496021
$814$	7.07264	0.247896
$815$	2.93827	0.102923
$816$	5.91083	0.206920
$817$	−36.8292	−1.28849
$818$	14.2025	0.496579
$819$	3.96620	0.138590
$820$	0.325783	0.0113769
$821$	−30.2459	−1.05559	−0.527795	−	0.849372i	$-0.676981\pi$
$821$	−30.2459	−1.05559	−0.527795	+	0.849372i	$0.676981\pi$
$822$	−11.2890	−0.393748
$823$	−50.6368	−1.76509	−0.882543	−	0.470231i	$-0.844170\pi$
$823$	−50.6368	−1.76509	−0.882543	+	0.470231i	$0.844170\pi$
$824$	−1.00000	−0.0348367
$825$	−4.86790	−0.169479
$826$	−1.70504	−0.0593259
$827$	43.1640	1.50096	0.750479	−	0.660895i	$-0.229823\pi$
$827$	43.1640	1.50096	0.750479	+	0.660895i	$0.229823\pi$
$828$	−1.88200	−0.0654042
$829$	3.31601	0.115170	0.0575849	−	0.998341i	$-0.481660\pi$
$829$	3.31601	0.115170	0.0575849	+	0.998341i	$0.481660\pi$
$830$	16.6384	0.577527
$831$	16.8638	0.584998
$832$	1.00000	0.0346688
$833$	51.6057	1.78803
$834$	10.7537	0.372370
$835$	−5.32861	−0.184404
$836$	−4.80233	−0.166092
$837$	−10.2736	−0.355109
$838$	35.1352	1.21372
$839$	−51.0538	−1.76257	−0.881287	−	0.472581i	$-0.843322\pi$
$839$	−51.0538	−1.76257	−0.881287	+	0.472581i	$0.843322\pi$
$840$	4.83881	0.166955
$841$	18.0877	0.623713
$842$	−32.3829	−1.11599
$843$	0.278548	0.00959370
$844$	−14.8912	−0.512575
$845$	−1.22001	−0.0419697
$846$	1.56018	0.0536402
$847$	−36.0064	−1.23719
$848$	11.8248	0.406066
$849$	−11.4750	−0.393820
$850$	20.7563	0.711934
$851$	9.60200	0.329152
$852$	11.7903	0.403930
$853$	31.1817	1.06764	0.533821	−	0.845597i	$-0.320756\pi$
$853$	31.1817	1.06764	0.533821	+	0.845597i	$0.320756\pi$
$854$	−1.54970	−0.0530298
$855$	4.22646	0.144542
$856$	0.569973	0.0194813
$857$	19.3410	0.660676	0.330338	−	0.943863i	$-0.392837\pi$
$857$	19.3410	0.660676	0.330338	+	0.943863i	$0.392837\pi$
$858$	−1.38625	−0.0473257
$859$	−49.9646	−1.70477	−0.852385	−	0.522915i	$-0.824845\pi$
$859$	−49.9646	−1.70477	−0.852385	+	0.522915i	$0.824845\pi$
$860$	−12.9701	−0.442278
$861$	−1.05910	−0.0360942
$862$	−6.59612	−0.224665
$863$	46.1391	1.57059	0.785297	−	0.619119i	$-0.212510\pi$
$863$	46.1391	1.57059	0.785297	+	0.619119i	$0.212510\pi$
$864$	−1.00000	−0.0340207
$865$	−1.43892	−0.0489249
$866$	−15.4865	−0.526254
$867$	17.9379	0.609203
$868$	−40.7473	−1.38305
$869$	14.3351	0.486284
$870$	8.37179	0.283830
$871$	5.38174	0.182353
$872$	6.79818	0.230216
$873$	−10.4310	−0.353037
$874$	−6.51977	−0.220535
$875$	41.1859	1.39234
$876$	8.64854	0.292207
$877$	35.2700	1.19098	0.595492	−	0.803361i	$-0.296957\pi$
$877$	35.2700	1.19098	0.595492	+	0.803361i	$0.296957\pi$
$878$	−9.26529	−0.312688
$879$	−21.2819	−0.717820
$880$	−1.69124	−0.0570117
$881$	31.2362	1.05237	0.526187	−	0.850369i	$-0.323621\pi$
$881$	31.2362	1.05237	0.526187	+	0.850369i	$0.323621\pi$
$882$	−8.73071	−0.293978
$883$	−30.1506	−1.01465	−0.507324	−	0.861756i	$-0.669365\pi$
$883$	−30.1506	−1.01465	−0.507324	+	0.861756i	$0.669365\pi$
$884$	5.91083	0.198803
$885$	−0.524475	−0.0176300
$886$	37.1180	1.24700
$887$	−30.5899	−1.02711	−0.513555	−	0.858057i	$-0.671672\pi$
$887$	−30.5899	−1.02711	−0.513555	+	0.858057i	$0.671672\pi$
$888$	5.10201	0.171212
$889$	1.21880	0.0408773
$890$	−16.7535	−0.561578
$891$	1.38625	0.0464410
$892$	14.7195	0.492846
$893$	5.40490	0.180868
$894$	4.46743	0.149413
$895$	−24.2884	−0.811870
$896$	−3.96620	−0.132501
$897$	−1.88200	−0.0628383
$898$	−23.0717	−0.769912
$899$	−70.4982	−2.35125
$900$	−3.51157	−0.117052
$901$	69.8945	2.32852
$902$	0.370173	0.0123254
$903$	42.1652	1.40317
$904$	−6.70233	−0.222916
$905$	7.02093	0.233384
$906$	−13.0784	−0.434501
$907$	50.7840	1.68625	0.843127	−	0.537714i	$-0.180712\pi$
$907$	50.7840	1.68625	0.843127	+	0.537714i	$0.180712\pi$
$908$	−23.0313	−0.764321
$909$	2.28727	0.0758640
$910$	4.83881	0.160405
$911$	6.45528	0.213873	0.106937	−	0.994266i	$-0.465896\pi$
$911$	6.45528	0.213873	0.106937	+	0.994266i	$0.465896\pi$
$912$	−3.46427	−0.114713
$913$	18.9055	0.625680
$914$	3.77668	0.124922
$915$	−0.476694	−0.0157590
$916$	−0.774678	−0.0255961
$917$	−53.8578	−1.77854
$918$	−5.91083	−0.195086
$919$	31.9161	1.05281	0.526407	−	0.850232i	$-0.323539\pi$
$919$	31.9161	1.05281	0.526407	+	0.850232i	$0.323539\pi$
$920$	−2.29607	−0.0756992
$921$	25.5688	0.842519
$922$	11.1347	0.366702
$923$	11.7903	0.388083
$924$	5.49813	0.180875
$925$	17.9160	0.589076
$926$	−22.2118	−0.729924
$927$	1.00000	0.0328443
$928$	−6.86205	−0.225258
$929$	−5.20287	−0.170701	−0.0853503	−	0.996351i	$-0.527201\pi$
$929$	−5.20287	−0.170701	−0.0853503	+	0.996351i	$0.527201\pi$
$930$	−12.5340	−0.411005
$931$	−30.2455	−0.991257
$932$	20.4913	0.671215
$933$	32.1386	1.05217
$934$	−3.17808	−0.103990
$935$	−9.99663	−0.326925
$936$	−1.00000	−0.0326860
$937$	3.27990	0.107150	0.0535748	−	0.998564i	$-0.482938\pi$
$937$	3.27990	0.107150	0.0535748	+	0.998564i	$0.482938\pi$
$938$	−21.3450	−0.696940
$939$	6.56333	0.214186
$940$	1.90345	0.0620835
$941$	42.0934	1.37220	0.686102	−	0.727505i	$-0.259320\pi$
$941$	42.0934	1.37220	0.686102	+	0.727505i	$0.259320\pi$
$942$	3.95084	0.128725
$943$	0.502557	0.0163655
$944$	0.429893	0.0139918
$945$	−4.83881	−0.157407
$946$	−14.7374	−0.479154
$947$	−10.6665	−0.346616	−0.173308	−	0.984868i	$-0.555446\pi$
$947$	−10.6665	−0.346616	−0.173308	+	0.984868i	$0.555446\pi$
$948$	10.3409	0.335857
$949$	8.64854	0.280743
$950$	−12.1650	−0.394685
$951$	17.8362	0.578377
$952$	−23.4435	−0.759808
$953$	−50.0777	−1.62218	−0.811088	−	0.584924i	$-0.801124\pi$
$953$	−50.0777	−1.62218	−0.811088	+	0.584924i	$0.801124\pi$
$954$	−11.8248	−0.382843
$955$	17.2712	0.558882
$956$	−8.63229	−0.279188
$957$	9.51249	0.307495
$958$	10.9280	0.353068
$959$	44.7742	1.44583
$960$	−1.22001	−0.0393758
$961$	74.5477	2.40476
$962$	5.10201	0.164495
$963$	−0.569973	−0.0183671
$964$	6.66735	0.214741
$965$	25.5400	0.822161
$966$	7.46440	0.240163
$967$	−36.8995	−1.18661	−0.593303	−	0.804979i	$-0.702177\pi$
$967$	−36.8995	−1.18661	−0.593303	+	0.804979i	$0.702177\pi$
$968$	9.07832	0.291788
$969$	−20.4767	−0.657807
$970$	−12.7260	−0.408608
$971$	35.4743	1.13843	0.569213	−	0.822190i	$-0.307248\pi$
$971$	35.4743	1.13843	0.569213	+	0.822190i	$0.307248\pi$
$972$	1.00000	0.0320750
$973$	−42.6513	−1.36734
$974$	−9.97466	−0.319609
$975$	−3.51157	−0.112460
$976$	0.390728	0.0125069
$977$	40.6664	1.30103	0.650517	−	0.759491i	$-0.274552\pi$
$977$	40.6664	1.30103	0.650517	+	0.759491i	$0.274552\pi$
$978$	2.40839	0.0770119
$979$	−19.0362	−0.608401
$980$	−10.6516	−0.340252
$981$	−6.79818	−0.217049
$982$	36.7153	1.17163
$983$	−5.12676	−0.163518	−0.0817592	−	0.996652i	$-0.526054\pi$
$983$	−5.12676	−0.163518	−0.0817592	+	0.996652i	$0.526054\pi$
$984$	0.267033	0.00851269
$985$	−18.3812	−0.585674
$986$	−40.5604	−1.29171
$987$	−6.18799	−0.196966
$988$	−3.46427	−0.110213
$989$	−20.0079	−0.636213
$990$	1.69124	0.0537511
$991$	34.7057	1.10246	0.551232	−	0.834352i	$-0.314158\pi$
$991$	34.7057	1.10246	0.551232	+	0.834352i	$0.314158\pi$
$992$	10.2736	0.326188
$993$	7.88185	0.250123
$994$	−46.7628	−1.48322
$995$	−6.37194	−0.202004
$996$	13.6379	0.432133
$997$	−56.8697	−1.80108	−0.900541	−	0.434772i	$-0.856829\pi$
$997$	−56.8697	−1.80108	−0.900541	+	0.434772i	$0.856829\pi$
$998$	−33.5174	−1.06097
$999$	−5.10201	−0.161420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.x.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.x.1.5	✓	13	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.22001 q^{5} -1.00000 q^{6} +3.96620 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.22001 q^{5} -1.00000 q^{6} +3.96620 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.22001 q^{10} +1.38625 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.96620 q^{14} -1.22001 q^{15} +1.00000 q^{16} +5.91083 q^{17} -1.00000 q^{18} -3.46427 q^{19} -1.22001 q^{20} +3.96620 q^{21} -1.38625 q^{22} -1.88200 q^{23} -1.00000 q^{24} -3.51157 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.96620 q^{28} +6.86205 q^{29} +1.22001 q^{30} -10.2736 q^{31} -1.00000 q^{32} +1.38625 q^{33} -5.91083 q^{34} -4.83881 q^{35} +1.00000 q^{36} -5.10201 q^{37} +3.46427 q^{38} +1.00000 q^{39} +1.22001 q^{40} -0.267033 q^{41} -3.96620 q^{42} +10.6311 q^{43} +1.38625 q^{44} -1.22001 q^{45} +1.88200 q^{46} -1.56018 q^{47} +1.00000 q^{48} +8.73071 q^{49} +3.51157 q^{50} +5.91083 q^{51} +1.00000 q^{52} +11.8248 q^{53} -1.00000 q^{54} -1.69124 q^{55} -3.96620 q^{56} -3.46427 q^{57} -6.86205 q^{58} +0.429893 q^{59} -1.22001 q^{60} +0.390728 q^{61} +10.2736 q^{62} +3.96620 q^{63} +1.00000 q^{64} -1.22001 q^{65} -1.38625 q^{66} +5.38174 q^{67} +5.91083 q^{68} -1.88200 q^{69} +4.83881 q^{70} +11.7903 q^{71} -1.00000 q^{72} +8.64854 q^{73} +5.10201 q^{74} -3.51157 q^{75} -3.46427 q^{76} +5.49813 q^{77} -1.00000 q^{78} +10.3409 q^{79} -1.22001 q^{80} +1.00000 q^{81} +0.267033 q^{82} +13.6379 q^{83} +3.96620 q^{84} -7.21129 q^{85} -10.6311 q^{86} +6.86205 q^{87} -1.38625 q^{88} -13.7322 q^{89} +1.22001 q^{90} +3.96620 q^{91} -1.88200 q^{92} -10.2736 q^{93} +1.56018 q^{94} +4.22646 q^{95} -1.00000 q^{96} -10.4310 q^{97} -8.73071 q^{98} +1.38625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9	\( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9} - q^{10} + 6 q^{11} + 13 q^{12} + 13 q^{13} + q^{14} + q^{15} + 13 q^{16} + 18 q^{17} - 13 q^{18} - q^{19} + q^{20} - q^{21} - 6 q^{22} + 20 q^{23} - 13 q^{24} + 16 q^{25} - 13 q^{26} + 13 q^{27} - q^{28} + 25 q^{29} - q^{30} - 5 q^{31} - 13 q^{32} + 6 q^{33} - 18 q^{34} + 26 q^{35} + 13 q^{36} - 6 q^{37} + q^{38} + 13 q^{39} - q^{40} + 13 q^{41} + q^{42} - 2 q^{43} + 6 q^{44} + q^{45} - 20 q^{46} + 5 q^{47} + 13 q^{48} - 16 q^{50} + 18 q^{51} + 13 q^{52} + 21 q^{53} - 13 q^{54} + 2 q^{55} + q^{56} - q^{57} - 25 q^{58} + 22 q^{59} + q^{60} + 2 q^{61} + 5 q^{62} - q^{63} + 13 q^{64} + q^{65} - 6 q^{66} - q^{67} + 18 q^{68} + 20 q^{69} - 26 q^{70} + 32 q^{71} - 13 q^{72} - 13 q^{73} + 6 q^{74} + 16 q^{75} - q^{76} + 33 q^{77} - 13 q^{78} - 7 q^{79} + q^{80} + 13 q^{81} - 13 q^{82} + 17 q^{83} - q^{84} - 25 q^{85} + 2 q^{86} + 25 q^{87} - 6 q^{88} - 12 q^{89} - q^{90} - q^{91} + 20 q^{92} - 5 q^{93} - 5 q^{94} + 36 q^{95} - 13 q^{96} - 42 q^{97} + 6 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 + 13 * q^3 + 13 * q^4 + q^5 - 13 * q^6 - q^7 - 13 * q^8 + 13 * q^9 - q^10 + 6 * q^11 + 13 * q^12 + 13 * q^13 + q^14 + q^15 + 13 * q^16 + 18 * q^17 - 13 * q^18 - q^19 + q^20 - q^21 - 6 * q^22 + 20 * q^23 - 13 * q^24 + 16 * q^25 - 13 * q^26 + 13 * q^27 - q^28 + 25 * q^29 - q^30 - 5 * q^31 - 13 * q^32 + 6 * q^33 - 18 * q^34 + 26 * q^35 + 13 * q^36 - 6 * q^37 + q^38 + 13 * q^39 - q^40 + 13 * q^41 + q^42 - 2 * q^43 + 6 * q^44 + q^45 - 20 * q^46 + 5 * q^47 + 13 * q^48 - 16 * q^50 + 18 * q^51 + 13 * q^52 + 21 * q^53 - 13 * q^54 + 2 * q^55 + q^56 - q^57 - 25 * q^58 + 22 * q^59 + q^60 + 2 * q^61 + 5 * q^62 - q^63 + 13 * q^64 + q^65 - 6 * q^66 - q^67 + 18 * q^68 + 20 * q^69 - 26 * q^70 + 32 * q^71 - 13 * q^72 - 13 * q^73 + 6 * q^74 + 16 * q^75 - q^76 + 33 * q^77 - 13 * q^78 - 7 * q^79 + q^80 + 13 * q^81 - 13 * q^82 + 17 * q^83 - q^84 - 25 * q^85 + 2 * q^86 + 25 * q^87 - 6 * q^88 - 12 * q^89 - q^90 - q^91 + 20 * q^92 - 5 * q^93 - 5 * q^94 + 36 * q^95 - 13 * q^96 - 42 * q^97 + 6 * q^99

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