LMFDB - Embedded newform 4002.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12580$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.12580 q^{5} -1.00000 q^{6} -0.350474 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.12580 q^{5} -1.00000 q^{6} -0.350474 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.12580 q^{10} -2.98960 q^{11} -1.00000 q^{12} +2.39608 q^{13} -0.350474 q^{14} +1.12580 q^{15} +1.00000 q^{16} -4.11540 q^{17} +1.00000 q^{18} +8.36517 q^{19} -1.12580 q^{20} +0.350474 q^{21} -2.98960 q^{22} -1.00000 q^{23} -1.00000 q^{24} -3.73257 q^{25} +2.39608 q^{26} -1.00000 q^{27} -0.350474 q^{28} -1.00000 q^{29} +1.12580 q^{30} +7.08620 q^{31} +1.00000 q^{32} +2.98960 q^{33} -4.11540 q^{34} +0.394564 q^{35} +1.00000 q^{36} -8.11825 q^{37} +8.36517 q^{38} -2.39608 q^{39} -1.12580 q^{40} +8.02165 q^{41} +0.350474 q^{42} +5.37579 q^{43} -2.98960 q^{44} -1.12580 q^{45} -1.00000 q^{46} -1.42528 q^{47} -1.00000 q^{48} -6.87717 q^{49} -3.73257 q^{50} +4.11540 q^{51} +2.39608 q^{52} +1.66480 q^{53} -1.00000 q^{54} +3.36569 q^{55} -0.350474 q^{56} -8.36517 q^{57} -1.00000 q^{58} -12.0108 q^{59} +1.12580 q^{60} +4.98960 q^{61} +7.08620 q^{62} -0.350474 q^{63} +1.00000 q^{64} -2.69751 q^{65} +2.98960 q^{66} +13.7129 q^{67} -4.11540 q^{68} +1.00000 q^{69} +0.394564 q^{70} +4.56540 q^{71} +1.00000 q^{72} +16.5331 q^{73} -8.11825 q^{74} +3.73257 q^{75} +8.36517 q^{76} +1.04778 q^{77} -2.39608 q^{78} +12.4519 q^{79} -1.12580 q^{80} +1.00000 q^{81} +8.02165 q^{82} +1.75023 q^{83} +0.350474 q^{84} +4.63312 q^{85} +5.37579 q^{86} +1.00000 q^{87} -2.98960 q^{88} -7.49934 q^{89} -1.12580 q^{90} -0.839764 q^{91} -1.00000 q^{92} -7.08620 q^{93} -1.42528 q^{94} -9.41752 q^{95} -1.00000 q^{96} +14.1692 q^{97} -6.87717 q^{98} -2.98960 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * 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
$5$	−1.12580	−0.503474	−0.251737	−	0.967796i	$-0.581002\pi$
$5$	−1.12580	−0.503474	−0.251737	+	0.967796i	$0.581002\pi$
$6$	−1.00000	−0.408248
$7$	−0.350474	−0.132467	−0.0662334	−	0.997804i	$-0.521098\pi$
$7$	−0.350474	−0.132467	−0.0662334	+	0.997804i	$0.521098\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.12580	−0.356010
$11$	−2.98960	−0.901397	−0.450698	−	0.892676i	$-0.648825\pi$
$11$	−2.98960	−0.901397	−0.450698	+	0.892676i	$0.648825\pi$
$12$	−1.00000	−0.288675
$13$	2.39608	0.664553	0.332277	−	0.943182i	$-0.392183\pi$
$13$	2.39608	0.664553	0.332277	+	0.943182i	$0.392183\pi$
$14$	−0.350474	−0.0936681
$15$	1.12580	0.290681
$16$	1.00000	0.250000
$17$	−4.11540	−0.998130	−0.499065	−	0.866564i	$-0.666323\pi$
$17$	−4.11540	−0.998130	−0.499065	+	0.866564i	$0.666323\pi$
$18$	1.00000	0.235702
$19$	8.36517	1.91910	0.959551	−	0.281536i	$-0.0908438\pi$
$19$	8.36517	1.91910	0.959551	+	0.281536i	$0.0908438\pi$
$20$	−1.12580	−0.251737
$21$	0.350474	0.0764797
$22$	−2.98960	−0.637384
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−3.73257	−0.746514
$26$	2.39608	0.469910
$27$	−1.00000	−0.192450
$28$	−0.350474	−0.0662334
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.12580	0.205542
$31$	7.08620	1.27272	0.636359	−	0.771393i	$-0.280440\pi$
$31$	7.08620	1.27272	0.636359	+	0.771393i	$0.280440\pi$
$32$	1.00000	0.176777
$33$	2.98960	0.520422
$34$	−4.11540	−0.705785
$35$	0.394564	0.0666935
$36$	1.00000	0.166667
$37$	−8.11825	−1.33463	−0.667316	−	0.744775i	$-0.732557\pi$
$37$	−8.11825	−1.33463	−0.667316	+	0.744775i	$0.732557\pi$
$38$	8.36517	1.35701
$39$	−2.39608	−0.383680
$40$	−1.12580	−0.178005
$41$	8.02165	1.25277	0.626385	−	0.779514i	$-0.284534\pi$
$41$	8.02165	1.25277	0.626385	+	0.779514i	$0.284534\pi$
$42$	0.350474	0.0540793
$43$	5.37579	0.819801	0.409900	−	0.912130i	$-0.365563\pi$
$43$	5.37579	0.819801	0.409900	+	0.912130i	$0.365563\pi$
$44$	−2.98960	−0.450698
$45$	−1.12580	−0.167825
$46$	−1.00000	−0.147442
$47$	−1.42528	−0.207898	−0.103949	−	0.994583i	$-0.533148\pi$
$47$	−1.42528	−0.207898	−0.103949	+	0.994583i	$0.533148\pi$
$48$	−1.00000	−0.144338
$49$	−6.87717	−0.982453
$50$	−3.73257	−0.527865
$51$	4.11540	0.576271
$52$	2.39608	0.332277
$53$	1.66480	0.228678	0.114339	−	0.993442i	$-0.463525\pi$
$53$	1.66480	0.228678	0.114339	+	0.993442i	$0.463525\pi$
$54$	−1.00000	−0.136083
$55$	3.36569	0.453830
$56$	−0.350474	−0.0468341
$57$	−8.36517	−1.10799
$58$	−1.00000	−0.131306
$59$	−12.0108	−1.56367	−0.781837	−	0.623482i	$-0.785717\pi$
$59$	−12.0108	−1.56367	−0.781837	+	0.623482i	$0.785717\pi$
$60$	1.12580	0.145340
$61$	4.98960	0.638852	0.319426	−	0.947611i	$-0.396510\pi$
$61$	4.98960	0.638852	0.319426	+	0.947611i	$0.396510\pi$
$62$	7.08620	0.899948
$63$	−0.350474	−0.0441556
$64$	1.00000	0.125000
$65$	−2.69751	−0.334585
$66$	2.98960	0.367994
$67$	13.7129	1.67529	0.837646	−	0.546214i	$-0.183931\pi$
$67$	13.7129	1.67529	0.837646	+	0.546214i	$0.183931\pi$
$68$	−4.11540	−0.499065
$69$	1.00000	0.120386
$70$	0.394564	0.0471594
$71$	4.56540	0.541814	0.270907	−	0.962606i	$-0.412676\pi$
$71$	4.56540	0.541814	0.270907	+	0.962606i	$0.412676\pi$
$72$	1.00000	0.117851
$73$	16.5331	1.93506	0.967528	−	0.252764i	$-0.0813398\pi$
$73$	16.5331	1.93506	0.967528	+	0.252764i	$0.0813398\pi$
$74$	−8.11825	−0.943727
$75$	3.73257	0.431000
$76$	8.36517	0.959551
$77$	1.04778	0.119405
$78$	−2.39608	−0.271303
$79$	12.4519	1.40095	0.700473	−	0.713678i	$-0.252972\pi$
$79$	12.4519	1.40095	0.700473	+	0.713678i	$0.252972\pi$
$80$	−1.12580	−0.125868
$81$	1.00000	0.111111
$82$	8.02165	0.885843
$83$	1.75023	0.192112	0.0960562	−	0.995376i	$-0.469377\pi$
$83$	1.75023	0.192112	0.0960562	+	0.995376i	$0.469377\pi$
$84$	0.350474	0.0382399
$85$	4.63312	0.502532
$86$	5.37579	0.579687
$87$	1.00000	0.107211
$88$	−2.98960	−0.318692
$89$	−7.49934	−0.794928	−0.397464	−	0.917618i	$-0.630110\pi$
$89$	−7.49934	−0.794928	−0.397464	+	0.917618i	$0.630110\pi$
$90$	−1.12580	−0.118670
$91$	−0.839764	−0.0880312
$92$	−1.00000	−0.104257
$93$	−7.08620	−0.734804
$94$	−1.42528	−0.147006
$95$	−9.41752	−0.966217
$96$	−1.00000	−0.102062
$97$	14.1692	1.43867	0.719334	−	0.694664i	$-0.244447\pi$
$97$	14.1692	1.43867	0.719334	+	0.694664i	$0.244447\pi$
$98$	−6.87717	−0.694699
$99$	−2.98960	−0.300466
$100$	−3.73257	−0.373257
$101$	1.79712	0.178820	0.0894102	−	0.995995i	$-0.471502\pi$
$101$	1.79712	0.178820	0.0894102	+	0.995995i	$0.471502\pi$
$102$	4.11540	0.407485
$103$	0.657566	0.0647919	0.0323959	−	0.999475i	$-0.489686\pi$
$103$	0.657566	0.0647919	0.0323959	+	0.999475i	$0.489686\pi$
$104$	2.39608	0.234955
$105$	−0.394564	−0.0385055
$106$	1.66480	0.161699
$107$	1.62485	0.157080	0.0785402	−	0.996911i	$-0.474974\pi$
$107$	1.62485	0.157080	0.0785402	+	0.996911i	$0.474974\pi$
$108$	−1.00000	−0.0962250
$109$	−0.828964	−0.0794003	−0.0397002	−	0.999212i	$-0.512640\pi$
$109$	−0.828964	−0.0794003	−0.0397002	+	0.999212i	$0.512640\pi$
$110$	3.36569	0.320906
$111$	8.11825	0.770550
$112$	−0.350474	−0.0331167
$113$	15.9989	1.50505	0.752526	−	0.658562i	$-0.228835\pi$
$113$	15.9989	1.50505	0.752526	+	0.658562i	$0.228835\pi$
$114$	−8.36517	−0.783470
$115$	1.12580	0.104982
$116$	−1.00000	−0.0928477
$117$	2.39608	0.221518
$118$	−12.0108	−1.10569
$119$	1.44234	0.132219
$120$	1.12580	0.102771
$121$	−2.06232	−0.187483
$122$	4.98960	0.451737
$123$	−8.02165	−0.723287
$124$	7.08620	0.636359
$125$	9.83114	0.879324
$126$	−0.350474	−0.0312227
$127$	−14.0607	−1.24769	−0.623844	−	0.781549i	$-0.714430\pi$
$127$	−14.0607	−1.24769	−0.623844	+	0.781549i	$0.714430\pi$
$128$	1.00000	0.0883883
$129$	−5.37579	−0.473312
$130$	−2.69751	−0.236587
$131$	8.53005	0.745274	0.372637	−	0.927977i	$-0.378454\pi$
$131$	8.53005	0.745274	0.372637	+	0.927977i	$0.378454\pi$
$132$	2.98960	0.260211
$133$	−2.93177	−0.254217
$134$	13.7129	1.18461
$135$	1.12580	0.0968936
$136$	−4.11540	−0.352892
$137$	14.4519	1.23471	0.617354	−	0.786685i	$-0.288205\pi$
$137$	14.4519	1.23471	0.617354	+	0.786685i	$0.288205\pi$
$138$	1.00000	0.0851257
$139$	−21.4164	−1.81652	−0.908259	−	0.418407i	$-0.862588\pi$
$139$	−21.4164	−1.81652	−0.908259	+	0.418407i	$0.862588\pi$
$140$	0.394564	0.0333468
$141$	1.42528	0.120030
$142$	4.56540	0.383120
$143$	−7.16331	−0.599026
$144$	1.00000	0.0833333
$145$	1.12580	0.0934927
$146$	16.5331	1.36829
$147$	6.87717	0.567219
$148$	−8.11825	−0.667316
$149$	12.6490	1.03625	0.518124	−	0.855306i	$-0.326631\pi$
$149$	12.6490	1.03625	0.518124	+	0.855306i	$0.326631\pi$
$150$	3.73257	0.304763
$151$	3.81570	0.310517	0.155259	−	0.987874i	$-0.450379\pi$
$151$	3.81570	0.310517	0.155259	+	0.987874i	$0.450379\pi$
$152$	8.36517	0.678505
$153$	−4.11540	−0.332710
$154$	1.04778	0.0844322
$155$	−7.97765	−0.640780
$156$	−2.39608	−0.191840
$157$	23.5540	1.87981	0.939906	−	0.341434i	$-0.110913\pi$
$157$	23.5540	1.87981	0.939906	+	0.341434i	$0.110913\pi$
$158$	12.4519	0.990619
$159$	−1.66480	−0.132027
$160$	−1.12580	−0.0890024
$161$	0.350474	0.0276212
$162$	1.00000	0.0785674
$163$	−0.0185758	−0.00145497	−0.000727486	−	1.00000i	$-0.500232\pi$
$163$	−0.0185758	−0.00145497	−0.000727486		1.00000i	$0.500232\pi$
$164$	8.02165	0.626385
$165$	−3.36569	−0.262019
$166$	1.75023	0.135844
$167$	−0.936111	−0.0724385	−0.0362192	−	0.999344i	$-0.511531\pi$
$167$	−0.936111	−0.0724385	−0.0362192	+	0.999344i	$0.511531\pi$
$168$	0.350474	0.0270397
$169$	−7.25880	−0.558369
$170$	4.63312	0.355344
$171$	8.36517	0.639700
$172$	5.37579	0.409900
$173$	18.2665	1.38878	0.694390	−	0.719599i	$-0.255674\pi$
$173$	18.2665	1.38878	0.694390	+	0.719599i	$0.255674\pi$
$174$	1.00000	0.0758098
$175$	1.30817	0.0988883
$176$	−2.98960	−0.225349
$177$	12.0108	0.902788
$178$	−7.49934	−0.562099
$179$	−1.65718	−0.123864	−0.0619319	−	0.998080i	$-0.519726\pi$
$179$	−1.65718	−0.123864	−0.0619319	+	0.998080i	$0.519726\pi$
$180$	−1.12580	−0.0839123
$181$	5.72942	0.425864	0.212932	−	0.977067i	$-0.431699\pi$
$181$	5.72942	0.425864	0.212932	+	0.977067i	$0.431699\pi$
$182$	−0.839764	−0.0622474
$183$	−4.98960	−0.368842
$184$	−1.00000	−0.0737210
$185$	9.13953	0.671952
$186$	−7.08620	−0.519585
$187$	12.3034	0.899712
$188$	−1.42528	−0.103949
$189$	0.350474	0.0254932
$190$	−9.41752	−0.683219
$191$	−17.0860	−1.23630	−0.618151	−	0.786060i	$-0.712118\pi$
$191$	−17.0860	−1.23630	−0.618151	+	0.786060i	$0.712118\pi$
$192$	−1.00000	−0.0721688
$193$	1.53464	0.110466	0.0552328	−	0.998474i	$-0.482410\pi$
$193$	1.53464	0.110466	0.0552328	+	0.998474i	$0.482410\pi$
$194$	14.1692	1.01729
$195$	2.69751	0.193173
$196$	−6.87717	−0.491226
$197$	−6.17610	−0.440029	−0.220014	−	0.975497i	$-0.570610\pi$
$197$	−6.17610	−0.440029	−0.220014	+	0.975497i	$0.570610\pi$
$198$	−2.98960	−0.212461
$199$	17.1526	1.21591	0.607956	−	0.793971i	$-0.291990\pi$
$199$	17.1526	1.21591	0.607956	+	0.793971i	$0.291990\pi$
$200$	−3.73257	−0.263933
$201$	−13.7129	−0.967230
$202$	1.79712	0.126445
$203$	0.350474	0.0245985
$204$	4.11540	0.288135
$205$	−9.03078	−0.630737
$206$	0.657566	0.0458148
$207$	−1.00000	−0.0695048
$208$	2.39608	0.166138
$209$	−25.0085	−1.72987
$210$	−0.394564	−0.0272275
$211$	13.9524	0.960524	0.480262	−	0.877125i	$-0.340542\pi$
$211$	13.9524	0.960524	0.480262	+	0.877125i	$0.340542\pi$
$212$	1.66480	0.114339
$213$	−4.56540	−0.312816
$214$	1.62485	0.111073
$215$	−6.05208	−0.412748
$216$	−1.00000	−0.0680414
$217$	−2.48353	−0.168593
$218$	−0.828964	−0.0561445
$219$	−16.5331	−1.11720
$220$	3.36569	0.226915
$221$	−9.86082	−0.663311
$222$	8.11825	0.544861
$223$	−20.6482	−1.38270	−0.691352	−	0.722519i	$-0.742984\pi$
$223$	−20.6482	−1.38270	−0.691352	+	0.722519i	$0.742984\pi$
$224$	−0.350474	−0.0234170
$225$	−3.73257	−0.248838
$226$	15.9989	1.06423
$227$	−4.64032	−0.307989	−0.153995	−	0.988072i	$-0.549214\pi$
$227$	−4.64032	−0.307989	−0.153995	+	0.988072i	$0.549214\pi$
$228$	−8.36517	−0.553997
$229$	−7.32879	−0.484300	−0.242150	−	0.970239i	$-0.577853\pi$
$229$	−7.32879	−0.484300	−0.242150	+	0.970239i	$0.577853\pi$
$230$	1.12580	0.0742331
$231$	−1.04778	−0.0689386
$232$	−1.00000	−0.0656532
$233$	21.9859	1.44035	0.720173	−	0.693794i	$-0.244062\pi$
$233$	21.9859	1.44035	0.720173	+	0.693794i	$0.244062\pi$
$234$	2.39608	0.156637
$235$	1.60458	0.104671
$236$	−12.0108	−0.781837
$237$	−12.4519	−0.808837
$238$	1.44234	0.0934930
$239$	7.54702	0.488176	0.244088	−	0.969753i	$-0.421511\pi$
$239$	7.54702	0.488176	0.244088	+	0.969753i	$0.421511\pi$
$240$	1.12580	0.0726702
$241$	−2.82443	−0.181937	−0.0909686	−	0.995854i	$-0.528996\pi$
$241$	−2.82443	−0.181937	−0.0909686	+	0.995854i	$0.528996\pi$
$242$	−2.06232	−0.132571
$243$	−1.00000	−0.0641500
$244$	4.98960	0.319426
$245$	7.74232	0.494639
$246$	−8.02165	−0.511441
$247$	20.0436	1.27534
$248$	7.08620	0.449974
$249$	−1.75023	−0.110916
$250$	9.83114	0.621776
$251$	13.0232	0.822015	0.411008	−	0.911632i	$-0.365177\pi$
$251$	13.0232	0.822015	0.411008	+	0.911632i	$0.365177\pi$
$252$	−0.350474	−0.0220778
$253$	2.98960	0.187954
$254$	−14.0607	−0.882249
$255$	−4.63312	−0.290137
$256$	1.00000	0.0625000
$257$	16.4400	1.02550	0.512750	−	0.858538i	$-0.328627\pi$
$257$	16.4400	1.02550	0.512750	+	0.858538i	$0.328627\pi$
$258$	−5.37579	−0.334682
$259$	2.84523	0.176794
$260$	−2.69751	−0.167293
$261$	−1.00000	−0.0618984
$262$	8.53005	0.526988
$263$	−15.0908	−0.930538	−0.465269	−	0.885169i	$-0.654042\pi$
$263$	−15.0908	−0.930538	−0.465269	+	0.885169i	$0.654042\pi$
$264$	2.98960	0.183997
$265$	−1.87423	−0.115133
$266$	−2.93177	−0.179759
$267$	7.49934	0.458952
$268$	13.7129	0.837646
$269$	−6.06664	−0.369889	−0.184945	−	0.982749i	$-0.559211\pi$
$269$	−6.06664	−0.369889	−0.184945	+	0.982749i	$0.559211\pi$
$270$	1.12580	0.0685141
$271$	−14.0182	−0.851544	−0.425772	−	0.904831i	$-0.639997\pi$
$271$	−14.0182	−0.851544	−0.425772	+	0.904831i	$0.639997\pi$
$272$	−4.11540	−0.249533
$273$	0.839764	0.0508248
$274$	14.4519	0.873070
$275$	11.1589	0.672906
$276$	1.00000	0.0601929
$277$	20.7988	1.24968	0.624840	−	0.780753i	$-0.285164\pi$
$277$	20.7988	1.24968	0.624840	+	0.780753i	$0.285164\pi$
$278$	−21.4164	−1.28447
$279$	7.08620	0.424240
$280$	0.394564	0.0235797
$281$	9.21258	0.549576	0.274788	−	0.961505i	$-0.411392\pi$
$281$	9.21258	0.549576	0.274788	+	0.961505i	$0.411392\pi$
$282$	1.42528	0.0848742
$283$	−21.7252	−1.29143	−0.645714	−	0.763579i	$-0.723440\pi$
$283$	−21.7252	−1.29143	−0.645714	+	0.763579i	$0.723440\pi$
$284$	4.56540	0.270907
$285$	9.41752	0.557846
$286$	−7.16331	−0.423575
$287$	−2.81138	−0.165950
$288$	1.00000	0.0589256
$289$	−0.0635084	−0.00373579
$290$	1.12580	0.0661093
$291$	−14.1692	−0.830616
$292$	16.5331	0.967528
$293$	13.1872	0.770406	0.385203	−	0.922832i	$-0.374131\pi$
$293$	13.1872	0.770406	0.385203	+	0.922832i	$0.374131\pi$
$294$	6.87717	0.401085
$295$	13.5218	0.787269
$296$	−8.11825	−0.471864
$297$	2.98960	0.173474
$298$	12.6490	0.732738
$299$	−2.39608	−0.138569
$300$	3.73257	0.215500
$301$	−1.88408	−0.108596
$302$	3.81570	0.219569
$303$	−1.79712	−0.103242
$304$	8.36517	0.479775
$305$	−5.61729	−0.321645
$306$	−4.11540	−0.235262
$307$	−23.6286	−1.34855	−0.674277	−	0.738478i	$-0.735545\pi$
$307$	−23.6286	−1.34855	−0.674277	+	0.738478i	$0.735545\pi$
$308$	1.04778	0.0597026
$309$	−0.657566	−0.0374076
$310$	−7.97765	−0.453100
$311$	−21.8984	−1.24175	−0.620873	−	0.783911i	$-0.713222\pi$
$311$	−21.8984	−1.24175	−0.620873	+	0.783911i	$0.713222\pi$
$312$	−2.39608	−0.135651
$313$	8.88472	0.502194	0.251097	−	0.967962i	$-0.419209\pi$
$313$	8.88472	0.502194	0.251097	+	0.967962i	$0.419209\pi$
$314$	23.5540	1.32923
$315$	0.394564	0.0222312
$316$	12.4519	0.700473
$317$	13.6485	0.766578	0.383289	−	0.923628i	$-0.374791\pi$
$317$	13.6485	0.766578	0.383289	+	0.923628i	$0.374791\pi$
$318$	−1.66480	−0.0933572
$319$	2.98960	0.167385
$320$	−1.12580	−0.0629342
$321$	−1.62485	−0.0906905
$322$	0.350474	0.0195312
$323$	−34.4260	−1.91551
$324$	1.00000	0.0555556
$325$	−8.94354	−0.496098
$326$	−0.0185758	−0.00102882
$327$	0.828964	0.0458418
$328$	8.02165	0.442921
$329$	0.499524	0.0275396
$330$	−3.36569	−0.185275
$331$	−25.9849	−1.42826	−0.714129	−	0.700014i	$-0.753177\pi$
$331$	−25.9849	−1.42826	−0.714129	+	0.700014i	$0.753177\pi$
$332$	1.75023	0.0960562
$333$	−8.11825	−0.444877
$334$	−0.936111	−0.0512217
$335$	−15.4379	−0.843465
$336$	0.350474	0.0191199
$337$	12.2127	0.665270	0.332635	−	0.943056i	$-0.392062\pi$
$337$	12.2127	0.665270	0.332635	+	0.943056i	$0.392062\pi$
$338$	−7.25880	−0.394827
$339$	−15.9989	−0.868942
$340$	4.63312	0.251266
$341$	−21.1849	−1.14722
$342$	8.36517	0.452337
$343$	4.86359	0.262609
$344$	5.37579	0.289843
$345$	−1.12580	−0.0606111
$346$	18.2665	0.982015
$347$	5.61559	0.301461	0.150730	−	0.988575i	$-0.451838\pi$
$347$	5.61559	0.301461	0.150730	+	0.988575i	$0.451838\pi$
$348$	1.00000	0.0536056
$349$	7.96379	0.426292	0.213146	−	0.977020i	$-0.431629\pi$
$349$	7.96379	0.426292	0.213146	+	0.977020i	$0.431629\pi$
$350$	1.30817	0.0699246
$351$	−2.39608	−0.127893
$352$	−2.98960	−0.159346
$353$	−0.748592	−0.0398435	−0.0199218	−	0.999802i	$-0.506342\pi$
$353$	−0.748592	−0.0398435	−0.0199218	+	0.999802i	$0.506342\pi$
$354$	12.0108	0.638368
$355$	−5.13974	−0.272789
$356$	−7.49934	−0.397464
$357$	−1.44234	−0.0763367
$358$	−1.65718	−0.0875849
$359$	31.6199	1.66883	0.834417	−	0.551133i	$-0.185804\pi$
$359$	31.6199	1.66883	0.834417	+	0.551133i	$0.185804\pi$
$360$	−1.12580	−0.0593349
$361$	50.9761	2.68295
$362$	5.72942	0.301132
$363$	2.06232	0.108244
$364$	−0.839764	−0.0440156
$365$	−18.6130	−0.974250
$366$	−4.98960	−0.260810
$367$	−30.4244	−1.58814	−0.794072	−	0.607824i	$-0.792043\pi$
$367$	−30.4244	−1.58814	−0.794072	+	0.607824i	$0.792043\pi$
$368$	−1.00000	−0.0521286
$369$	8.02165	0.417590
$370$	9.13953	0.475142
$371$	−0.583468	−0.0302922
$372$	−7.08620	−0.367402
$373$	26.7664	1.38591	0.692955	−	0.720981i	$-0.256309\pi$
$373$	26.7664	1.38591	0.692955	+	0.720981i	$0.256309\pi$
$374$	12.3034	0.636192
$375$	−9.83114	−0.507678
$376$	−1.42528	−0.0735032
$377$	−2.39608	−0.123404
$378$	0.350474	0.0180264
$379$	12.5308	0.643666	0.321833	−	0.946796i	$-0.395701\pi$
$379$	12.5308	0.643666	0.321833	+	0.946796i	$0.395701\pi$
$380$	−9.41752	−0.483109
$381$	14.0607	0.720353
$382$	−17.0860	−0.874197
$383$	26.9873	1.37899	0.689494	−	0.724292i	$-0.257833\pi$
$383$	26.9873	1.37899	0.689494	+	0.724292i	$0.257833\pi$
$384$	−1.00000	−0.0510310
$385$	−1.17959	−0.0601173
$386$	1.53464	0.0781110
$387$	5.37579	0.273267
$388$	14.1692	0.719334
$389$	−35.3771	−1.79369	−0.896844	−	0.442346i	$-0.854146\pi$
$389$	−35.3771	−1.79369	−0.896844	+	0.442346i	$0.854146\pi$
$390$	2.69751	0.136594
$391$	4.11540	0.208125
$392$	−6.87717	−0.347349
$393$	−8.53005	−0.430284
$394$	−6.17610	−0.311147
$395$	−14.0184	−0.705340
$396$	−2.98960	−0.150233
$397$	−7.82589	−0.392770	−0.196385	−	0.980527i	$-0.562920\pi$
$397$	−7.82589	−0.392770	−0.196385	+	0.980527i	$0.562920\pi$
$398$	17.1526	0.859780
$399$	2.93177	0.146772
$400$	−3.73257	−0.186629
$401$	−21.9564	−1.09645	−0.548226	−	0.836330i	$-0.684697\pi$
$401$	−21.9564	−1.09645	−0.548226	+	0.836330i	$0.684697\pi$
$402$	−13.7129	−0.683935
$403$	16.9791	0.845789
$404$	1.79712	0.0894102
$405$	−1.12580	−0.0559415
$406$	0.350474	0.0173937
$407$	24.2703	1.20303
$408$	4.11540	0.203743
$409$	−31.0578	−1.53571	−0.767855	−	0.640624i	$-0.778676\pi$
$409$	−31.0578	−1.53571	−0.767855	+	0.640624i	$0.778676\pi$
$410$	−9.03078	−0.445998
$411$	−14.4519	−0.712859
$412$	0.657566	0.0323959
$413$	4.20948	0.207135
$414$	−1.00000	−0.0491473
$415$	−1.97041	−0.0967236
$416$	2.39608	0.117478
$417$	21.4164	1.04877
$418$	−25.0085	−1.22320
$419$	17.5352	0.856650	0.428325	−	0.903625i	$-0.359104\pi$
$419$	17.5352	0.856650	0.428325	+	0.903625i	$0.359104\pi$
$420$	−0.394564	−0.0192528
$421$	16.1908	0.789092	0.394546	−	0.918876i	$-0.370902\pi$
$421$	16.1908	0.789092	0.394546	+	0.918876i	$0.370902\pi$
$422$	13.9524	0.679193
$423$	−1.42528	−0.0692995
$424$	1.66480	0.0808497
$425$	15.3610	0.745119
$426$	−4.56540	−0.221195
$427$	−1.74872	−0.0846267
$428$	1.62485	0.0785402
$429$	7.16331	0.345848
$430$	−6.05208	−0.291857
$431$	−32.9913	−1.58914	−0.794569	−	0.607174i	$-0.792303\pi$
$431$	−32.9913	−1.58914	−0.794569	+	0.607174i	$0.792303\pi$
$432$	−1.00000	−0.0481125
$433$	−11.1102	−0.533920	−0.266960	−	0.963708i	$-0.586019\pi$
$433$	−11.1102	−0.533920	−0.266960	+	0.963708i	$0.586019\pi$
$434$	−2.48353	−0.119213
$435$	−1.12580	−0.0539780
$436$	−0.828964	−0.0397002
$437$	−8.36517	−0.400160
$438$	−16.5331	−0.789983
$439$	19.6712	0.938854	0.469427	−	0.882971i	$-0.344460\pi$
$439$	19.6712	0.938854	0.469427	+	0.882971i	$0.344460\pi$
$440$	3.36569	0.160453
$441$	−6.87717	−0.327484
$442$	−9.86082	−0.469031
$443$	−5.05453	−0.240148	−0.120074	−	0.992765i	$-0.538313\pi$
$443$	−5.05453	−0.240148	−0.120074	+	0.992765i	$0.538313\pi$
$444$	8.11825	0.385275
$445$	8.44276	0.400225
$446$	−20.6482	−0.977719
$447$	−12.6490	−0.598278
$448$	−0.350474	−0.0165583
$449$	−21.3222	−1.00626	−0.503129	−	0.864211i	$-0.667818\pi$
$449$	−21.3222	−1.00626	−0.503129	+	0.864211i	$0.667818\pi$
$450$	−3.73257	−0.175955
$451$	−23.9815	−1.12924
$452$	15.9989	0.752526
$453$	−3.81570	−0.179277
$454$	−4.64032	−0.217781
$455$	0.945407	0.0443214
$456$	−8.36517	−0.391735
$457$	28.3969	1.32835	0.664176	−	0.747577i	$-0.268783\pi$
$457$	28.3969	1.32835	0.664176	+	0.747577i	$0.268783\pi$
$458$	−7.32879	−0.342452
$459$	4.11540	0.192090
$460$	1.12580	0.0524908
$461$	−26.2260	−1.22147	−0.610733	−	0.791837i	$-0.709125\pi$
$461$	−26.2260	−1.22147	−0.610733	+	0.791837i	$0.709125\pi$
$462$	−1.04778	−0.0487469
$463$	13.1757	0.612325	0.306163	−	0.951979i	$-0.400955\pi$
$463$	13.1757	0.612325	0.306163	+	0.951979i	$0.400955\pi$
$464$	−1.00000	−0.0464238
$465$	7.97765	0.369955
$466$	21.9859	1.01848
$467$	39.2223	1.81499	0.907495	−	0.420064i	$-0.137992\pi$
$467$	39.2223	1.81499	0.907495	+	0.420064i	$0.137992\pi$
$468$	2.39608	0.110759
$469$	−4.80600	−0.221920
$470$	1.60458	0.0740139
$471$	−23.5540	−1.08531
$472$	−12.0108	−0.552843
$473$	−16.0714	−0.738966
$474$	−12.4519	−0.571934
$475$	−31.2236	−1.43264
$476$	1.44234	0.0661095
$477$	1.66480	0.0762259
$478$	7.54702	0.345193
$479$	−32.1503	−1.46898	−0.734492	−	0.678617i	$-0.762580\pi$
$479$	−32.1503	−1.46898	−0.734492	+	0.678617i	$0.762580\pi$
$480$	1.12580	0.0513856
$481$	−19.4520	−0.886934
$482$	−2.82443	−0.128649
$483$	−0.350474	−0.0159471
$484$	−2.06232	−0.0937417
$485$	−15.9518	−0.724332
$486$	−1.00000	−0.0453609
$487$	−22.2810	−1.00965	−0.504825	−	0.863222i	$-0.668443\pi$
$487$	−22.2810	−1.00965	−0.504825	+	0.863222i	$0.668443\pi$
$488$	4.98960	0.225868
$489$	0.0185758	0.000840028 0
$490$	7.74232	0.349763
$491$	−14.0078	−0.632163	−0.316082	−	0.948732i	$-0.602367\pi$
$491$	−14.0078	−0.632163	−0.316082	+	0.948732i	$0.602367\pi$
$492$	−8.02165	−0.361644
$493$	4.11540	0.185348
$494$	20.0436	0.901805
$495$	3.36569	0.151277
$496$	7.08620	0.318180
$497$	−1.60006	−0.0717723
$498$	−1.75023	−0.0784296
$499$	−12.9964	−0.581800	−0.290900	−	0.956753i	$-0.593955\pi$
$499$	−12.9964	−0.581800	−0.290900	+	0.956753i	$0.593955\pi$
$500$	9.83114	0.439662
$501$	0.936111	0.0418224
$502$	13.0232	0.581253
$503$	20.9364	0.933510	0.466755	−	0.884387i	$-0.345423\pi$
$503$	20.9364	0.933510	0.466755	+	0.884387i	$0.345423\pi$
$504$	−0.350474	−0.0156114
$505$	−2.02320	−0.0900314
$506$	2.98960	0.132904
$507$	7.25880	0.322375
$508$	−14.0607	−0.623844
$509$	28.7917	1.27617	0.638086	−	0.769965i	$-0.279727\pi$
$509$	28.7917	1.27617	0.638086	+	0.769965i	$0.279727\pi$
$510$	−4.63312	−0.205158
$511$	−5.79443	−0.256330
$512$	1.00000	0.0441942
$513$	−8.36517	−0.369331
$514$	16.4400	0.725138
$515$	−0.740289	−0.0326210
$516$	−5.37579	−0.236656
$517$	4.26101	0.187399
$518$	2.84523	0.125012
$519$	−18.2665	−0.801812
$520$	−2.69751	−0.118294
$521$	−27.3793	−1.19951	−0.599754	−	0.800184i	$-0.704735\pi$
$521$	−27.3793	−1.19951	−0.599754	+	0.800184i	$0.704735\pi$
$522$	−1.00000	−0.0437688
$523$	−2.59764	−0.113587	−0.0567935	−	0.998386i	$-0.518088\pi$
$523$	−2.59764	−0.113587	−0.0567935	+	0.998386i	$0.518088\pi$
$524$	8.53005	0.372637
$525$	−1.30817	−0.0570932
$526$	−15.0908	−0.657990
$527$	−29.1625	−1.27034
$528$	2.98960	0.130105
$529$	1.00000	0.0434783
$530$	−1.87423	−0.0814114
$531$	−12.0108	−0.521225
$532$	−2.93177	−0.127109
$533$	19.2205	0.832533
$534$	7.49934	0.324528
$535$	−1.82926	−0.0790859
$536$	13.7129	0.592305
$537$	1.65718	0.0715128
$538$	−6.06664	−0.261551
$539$	20.5600	0.885580
$540$	1.12580	0.0484468
$541$	−5.79191	−0.249014	−0.124507	−	0.992219i	$-0.539735\pi$
$541$	−5.79191	−0.249014	−0.124507	+	0.992219i	$0.539735\pi$
$542$	−14.0182	−0.602132
$543$	−5.72942	−0.245873
$544$	−4.11540	−0.176446
$545$	0.933249	0.0399760
$546$	0.839764	0.0359386
$547$	−1.93162	−0.0825902	−0.0412951	−	0.999147i	$-0.513148\pi$
$547$	−1.93162	−0.0825902	−0.0412951	+	0.999147i	$0.513148\pi$
$548$	14.4519	0.617354
$549$	4.98960	0.212951
$550$	11.1589	0.475816
$551$	−8.36517	−0.356368
$552$	1.00000	0.0425628
$553$	−4.36406	−0.185579
$554$	20.7988	0.883658
$555$	−9.13953	−0.387952
$556$	−21.4164	−0.908259
$557$	−30.0279	−1.27232	−0.636162	−	0.771555i	$-0.719479\pi$
$557$	−30.0279	−1.27232	−0.636162	+	0.771555i	$0.719479\pi$
$558$	7.08620	0.299983
$559$	12.8808	0.544801
$560$	0.394564	0.0166734
$561$	−12.3034	−0.519449
$562$	9.21258	0.388609
$563$	−19.7433	−0.832081	−0.416040	−	0.909346i	$-0.636583\pi$
$563$	−19.7433	−0.832081	−0.416040	+	0.909346i	$0.636583\pi$
$564$	1.42528	0.0600151
$565$	−18.0116	−0.757754
$566$	−21.7252	−0.913178
$567$	−0.350474	−0.0147185
$568$	4.56540	0.191560
$569$	−42.8764	−1.79747	−0.898736	−	0.438489i	$-0.855514\pi$
$569$	−42.8764	−1.79747	−0.898736	+	0.438489i	$0.855514\pi$
$570$	9.41752	0.394456
$571$	−11.4734	−0.480147	−0.240074	−	0.970755i	$-0.577172\pi$
$571$	−11.4734	−0.480147	−0.240074	+	0.970755i	$0.577172\pi$
$572$	−7.16331	−0.299513
$573$	17.0860	0.713779
$574$	−2.81138	−0.117345
$575$	3.73257	0.155659
$576$	1.00000	0.0416667
$577$	−0.0751962	−0.00313046	−0.00156523	−	0.999999i	$-0.500498\pi$
$577$	−0.0751962	−0.00313046	−0.00156523	+	0.999999i	$0.500498\pi$
$578$	−0.0635084	−0.00264160
$579$	−1.53464	−0.0637774
$580$	1.12580	0.0467464
$581$	−0.613409	−0.0254485
$582$	−14.1692	−0.587334
$583$	−4.97707	−0.206129
$584$	16.5331	0.684145
$585$	−2.69751	−0.111528
$586$	13.1872	0.544759
$587$	4.43290	0.182965	0.0914827	−	0.995807i	$-0.470839\pi$
$587$	4.43290	0.182965	0.0914827	+	0.995807i	$0.470839\pi$
$588$	6.87717	0.283610
$589$	59.2772	2.44248
$590$	13.5218	0.556683
$591$	6.17610	0.254051
$592$	−8.11825	−0.333658
$593$	−44.3929	−1.82300	−0.911500	−	0.411301i	$-0.865074\pi$
$593$	−44.3929	−1.82300	−0.911500	+	0.411301i	$0.865074\pi$
$594$	2.98960	0.122665
$595$	−1.62379	−0.0665688
$596$	12.6490	0.518124
$597$	−17.1526	−0.702007
$598$	−2.39608	−0.0979830
$599$	−4.00033	−0.163449	−0.0817245	−	0.996655i	$-0.526043\pi$
$599$	−4.00033	−0.163449	−0.0817245	+	0.996655i	$0.526043\pi$
$600$	3.73257	0.152382
$601$	14.2952	0.583112	0.291556	−	0.956554i	$-0.405827\pi$
$601$	14.2952	0.583112	0.291556	+	0.956554i	$0.405827\pi$
$602$	−1.88408	−0.0767892
$603$	13.7129	0.558430
$604$	3.81570	0.155259
$605$	2.32176	0.0943930
$606$	−1.79712	−0.0730031
$607$	−11.6776	−0.473981	−0.236990	−	0.971512i	$-0.576161\pi$
$607$	−11.6776	−0.473981	−0.236990	+	0.971512i	$0.576161\pi$
$608$	8.36517	0.339252
$609$	−0.350474	−0.0142019
$610$	−5.61729	−0.227438
$611$	−3.41509	−0.138160
$612$	−4.11540	−0.166355
$613$	−49.2290	−1.98834	−0.994169	−	0.107829i	$-0.965610\pi$
$613$	−49.2290	−1.98834	−0.994169	+	0.107829i	$0.965610\pi$
$614$	−23.6286	−0.953572
$615$	9.03078	0.364156
$616$	1.04778	0.0422161
$617$	−31.6489	−1.27414	−0.637069	−	0.770807i	$-0.719853\pi$
$617$	−31.6489	−1.27414	−0.637069	+	0.770807i	$0.719853\pi$
$618$	−0.657566	−0.0264512
$619$	−24.7636	−0.995332	−0.497666	−	0.867369i	$-0.665810\pi$
$619$	−24.7636	−0.995332	−0.497666	+	0.867369i	$0.665810\pi$
$620$	−7.97765	−0.320390
$621$	1.00000	0.0401286
$622$	−21.8984	−0.878047
$623$	2.62832	0.105302
$624$	−2.39608	−0.0959200
$625$	7.59495	0.303798
$626$	8.88472	0.355105
$627$	25.0085	0.998742
$628$	23.5540	0.939906
$629$	33.4098	1.33214
$630$	0.394564	0.0157198
$631$	7.48180	0.297846	0.148923	−	0.988849i	$-0.452419\pi$
$631$	7.48180	0.297846	0.148923	+	0.988849i	$0.452419\pi$
$632$	12.4519	0.495310
$633$	−13.9524	−0.554559
$634$	13.6485	0.542053
$635$	15.8296	0.628178
$636$	−1.66480	−0.0660135
$637$	−16.4782	−0.652892
$638$	2.98960	0.118359
$639$	4.56540	0.180605
$640$	−1.12580	−0.0445012
$641$	44.1319	1.74311	0.871553	−	0.490302i	$-0.163113\pi$
$641$	44.1319	1.74311	0.871553	+	0.490302i	$0.163113\pi$
$642$	−1.62485	−0.0641278
$643$	20.7923	0.819966	0.409983	−	0.912093i	$-0.365535\pi$
$643$	20.7923	0.819966	0.409983	+	0.912093i	$0.365535\pi$
$644$	0.350474	0.0138106
$645$	6.05208	0.238300
$646$	−34.4260	−1.35447
$647$	−14.8099	−0.582239	−0.291119	−	0.956687i	$-0.594028\pi$
$647$	−14.8099	−0.582239	−0.291119	+	0.956687i	$0.594028\pi$
$648$	1.00000	0.0392837
$649$	35.9075	1.40949
$650$	−8.94354	−0.350795
$651$	2.48353	0.0973371
$652$	−0.0185758	−0.000727486 0
$653$	5.40775	0.211622	0.105811	−	0.994386i	$-0.466256\pi$
$653$	5.40775	0.211622	0.105811	+	0.994386i	$0.466256\pi$
$654$	0.828964	0.0324151
$655$	−9.60315	−0.375226
$656$	8.02165	0.313193
$657$	16.5331	0.645019
$658$	0.499524	0.0194735
$659$	−3.51771	−0.137030	−0.0685152	−	0.997650i	$-0.521826\pi$
$659$	−3.51771	−0.137030	−0.0685152	+	0.997650i	$0.521826\pi$
$660$	−3.36569	−0.131009
$661$	27.1072	1.05435	0.527173	−	0.849758i	$-0.323252\pi$
$661$	27.1072	1.05435	0.527173	+	0.849758i	$0.323252\pi$
$662$	−25.9849	−1.00993
$663$	9.86082	0.382963
$664$	1.75023	0.0679220
$665$	3.30060	0.127992
$666$	−8.11825	−0.314576
$667$	1.00000	0.0387202
$668$	−0.936111	−0.0362192
$669$	20.6482	0.798304
$670$	−15.4379	−0.596420
$671$	−14.9169	−0.575859
$672$	0.350474	0.0135198
$673$	12.6152	0.486280	0.243140	−	0.969991i	$-0.421822\pi$
$673$	12.6152	0.486280	0.243140	+	0.969991i	$0.421822\pi$
$674$	12.2127	0.470417
$675$	3.73257	0.143667
$676$	−7.25880	−0.279185
$677$	−17.4004	−0.668751	−0.334376	−	0.942440i	$-0.608525\pi$
$677$	−17.4004	−0.668751	−0.334376	+	0.942440i	$0.608525\pi$
$678$	−15.9989	−0.614435
$679$	−4.96595	−0.190576
$680$	4.63312	0.177672
$681$	4.64032	0.177818
$682$	−21.1849	−0.811210
$683$	30.2111	1.15600	0.577998	−	0.816038i	$-0.303834\pi$
$683$	30.2111	1.15600	0.577998	+	0.816038i	$0.303834\pi$
$684$	8.36517	0.319850
$685$	−16.2700	−0.621643
$686$	4.86359	0.185693
$687$	7.32879	0.279611
$688$	5.37579	0.204950
$689$	3.98899	0.151968
$690$	−1.12580	−0.0428585
$691$	14.7690	0.561838	0.280919	−	0.959731i	$-0.409361\pi$
$691$	14.7690	0.561838	0.280919	+	0.959731i	$0.409361\pi$
$692$	18.2665	0.694390
$693$	1.04778	0.0398017
$694$	5.61559	0.213165
$695$	24.1107	0.914569
$696$	1.00000	0.0379049
$697$	−33.0123	−1.25043
$698$	7.96379	0.301434
$699$	−21.9859	−0.831584
$700$	1.30817	0.0494442
$701$	−4.58767	−0.173274	−0.0866369	−	0.996240i	$-0.527612\pi$
$701$	−4.58767	−0.173274	−0.0866369	+	0.996240i	$0.527612\pi$
$702$	−2.39608	−0.0904342
$703$	−67.9105	−2.56129
$704$	−2.98960	−0.112675
$705$	−1.60458	−0.0604321
$706$	−0.748592	−0.0281736
$707$	−0.629845	−0.0236878
$708$	12.0108	0.451394
$709$	13.0893	0.491579	0.245789	−	0.969323i	$-0.420953\pi$
$709$	13.0893	0.491579	0.245789	+	0.969323i	$0.420953\pi$
$710$	−5.13974	−0.192891
$711$	12.4519	0.466982
$712$	−7.49934	−0.281050
$713$	−7.08620	−0.265380
$714$	−1.44234	−0.0539782
$715$	8.06447	0.301594
$716$	−1.65718	−0.0619319
$717$	−7.54702	−0.281849
$718$	31.6199	1.18004
$719$	−16.0950	−0.600241	−0.300120	−	0.953901i	$-0.597027\pi$
$719$	−16.0950	−0.600241	−0.300120	+	0.953901i	$0.597027\pi$
$720$	−1.12580	−0.0419561
$721$	−0.230460	−0.00858277
$722$	50.9761	1.89713
$723$	2.82443	0.105042
$724$	5.72942	0.212932
$725$	3.73257	0.138624
$726$	2.06232	0.0765398
$727$	51.5776	1.91291	0.956454	−	0.291882i	$-0.0942816\pi$
$727$	51.5776	1.91291	0.956454	+	0.291882i	$0.0942816\pi$
$728$	−0.839764	−0.0311237
$729$	1.00000	0.0370370
$730$	−18.6130	−0.688898
$731$	−22.1235	−0.818268
$732$	−4.98960	−0.184421
$733$	15.2078	0.561711	0.280856	−	0.959750i	$-0.409382\pi$
$733$	15.2078	0.561711	0.280856	+	0.959750i	$0.409382\pi$
$734$	−30.4244	−1.12299
$735$	−7.74232	−0.285580
$736$	−1.00000	−0.0368605
$737$	−40.9959	−1.51010
$738$	8.02165	0.295281
$739$	−23.5676	−0.866948	−0.433474	−	0.901166i	$-0.642712\pi$
$739$	−23.5676	−0.866948	−0.433474	+	0.901166i	$0.642712\pi$
$740$	9.13953	0.335976
$741$	−20.0436	−0.736321
$742$	−0.583468	−0.0214198
$743$	7.55971	0.277339	0.138669	−	0.990339i	$-0.455717\pi$
$743$	7.55971	0.277339	0.138669	+	0.990339i	$0.455717\pi$
$744$	−7.08620	−0.259793
$745$	−14.2403	−0.521724
$746$	26.7664	0.979986
$747$	1.75023	0.0640375
$748$	12.3034	0.449856
$749$	−0.569469	−0.0208079
$750$	−9.83114	−0.358982
$751$	8.40461	0.306689	0.153344	−	0.988173i	$-0.450996\pi$
$751$	8.40461	0.306689	0.153344	+	0.988173i	$0.450996\pi$
$752$	−1.42528	−0.0519746
$753$	−13.0232	−0.474591
$754$	−2.39608	−0.0872601
$755$	−4.29572	−0.156337
$756$	0.350474	0.0127466
$757$	1.17043	0.0425398	0.0212699	−	0.999774i	$-0.493229\pi$
$757$	1.17043	0.0425398	0.0212699	+	0.999774i	$0.493229\pi$
$758$	12.5308	0.455141
$759$	−2.98960	−0.108515
$760$	−9.41752	−0.341609
$761$	17.7561	0.643658	0.321829	−	0.946798i	$-0.395702\pi$
$761$	17.7561	0.643658	0.321829	+	0.946798i	$0.395702\pi$
$762$	14.0607	0.509367
$763$	0.290530	0.0105179
$764$	−17.0860	−0.618151
$765$	4.63312	0.167511
$766$	26.9873	0.975091
$767$	−28.7789	−1.03915
$768$	−1.00000	−0.0360844
$769$	−15.9240	−0.574234	−0.287117	−	0.957896i	$-0.592697\pi$
$769$	−15.9240	−0.574234	−0.287117	+	0.957896i	$0.592697\pi$
$770$	−1.17959	−0.0425094
$771$	−16.4400	−0.592073
$772$	1.53464	0.0552328
$773$	−22.6674	−0.815289	−0.407644	−	0.913141i	$-0.633650\pi$
$773$	−22.6674	−0.815289	−0.407644	+	0.913141i	$0.633650\pi$
$774$	5.37579	0.193229
$775$	−26.4497	−0.950103
$776$	14.1692	0.508646
$777$	−2.84523	−0.102072
$778$	−35.3771	−1.26833
$779$	67.1024	2.40419
$780$	2.69751	0.0965864
$781$	−13.6487	−0.488389
$782$	4.11540	0.147166
$783$	1.00000	0.0357371
$784$	−6.87717	−0.245613
$785$	−26.5171	−0.946436
$786$	−8.53005	−0.304257
$787$	13.5643	0.483514	0.241757	−	0.970337i	$-0.422276\pi$
$787$	13.5643	0.483514	0.241757	+	0.970337i	$0.422276\pi$
$788$	−6.17610	−0.220014
$789$	15.0908	0.537246
$790$	−14.0184	−0.498751
$791$	−5.60721	−0.199369
$792$	−2.98960	−0.106231
$793$	11.9555	0.424551
$794$	−7.82589	−0.277731
$795$	1.87423	0.0664721
$796$	17.1526	0.607956
$797$	−12.6901	−0.449508	−0.224754	−	0.974416i	$-0.572158\pi$
$797$	−12.6901	−0.449508	−0.224754	+	0.974416i	$0.572158\pi$
$798$	2.93177	0.103784
$799$	5.86559	0.207510
$800$	−3.73257	−0.131966
$801$	−7.49934	−0.264976
$802$	−21.9564	−0.775309
$803$	−49.4274	−1.74425
$804$	−13.7129	−0.483615
$805$	−0.394564	−0.0139066
$806$	16.9791	0.598063
$807$	6.06664	0.213556
$808$	1.79712	0.0632226
$809$	−55.1732	−1.93979	−0.969893	−	0.243532i	$-0.921694\pi$
$809$	−55.1732	−1.93979	−0.969893	+	0.243532i	$0.921694\pi$
$810$	−1.12580	−0.0395566
$811$	−20.0491	−0.704018	−0.352009	−	0.935997i	$-0.614501\pi$
$811$	−20.0491	−0.704018	−0.352009	+	0.935997i	$0.614501\pi$
$812$	0.350474	0.0122992
$813$	14.0182	0.491639
$814$	24.2703	0.850673
$815$	0.0209127	0.000732540 0
$816$	4.11540	0.144068
$817$	44.9694	1.57328
$818$	−31.0578	−1.08591
$819$	−0.839764	−0.0293437
$820$	−9.03078	−0.315368
$821$	4.98902	0.174118	0.0870590	−	0.996203i	$-0.472253\pi$
$821$	4.98902	0.174118	0.0870590	+	0.996203i	$0.472253\pi$
$822$	−14.4519	−0.504067
$823$	−1.30163	−0.0453720	−0.0226860	−	0.999743i	$-0.507222\pi$
$823$	−1.30163	−0.0453720	−0.0226860	+	0.999743i	$0.507222\pi$
$824$	0.657566	0.0229074
$825$	−11.1589	−0.388502
$826$	4.20948	0.146466
$827$	27.2286	0.946832	0.473416	−	0.880839i	$-0.343021\pi$
$827$	27.2286	0.946832	0.473416	+	0.880839i	$0.343021\pi$
$828$	−1.00000	−0.0347524
$829$	7.29576	0.253392	0.126696	−	0.991942i	$-0.459563\pi$
$829$	7.29576	0.253392	0.126696	+	0.991942i	$0.459563\pi$
$830$	−1.97041	−0.0683939
$831$	−20.7988	−0.721503
$832$	2.39608	0.0830691
$833$	28.3023	0.980616
$834$	21.4164	0.741591
$835$	1.05388	0.0364709
$836$	−25.0085	−0.864936
$837$	−7.08620	−0.244935
$838$	17.5352	0.605743
$839$	−33.0457	−1.14087	−0.570433	−	0.821344i	$-0.693224\pi$
$839$	−33.0457	−1.14087	−0.570433	+	0.821344i	$0.693224\pi$
$840$	−0.394564	−0.0136138
$841$	1.00000	0.0344828
$842$	16.1908	0.557972
$843$	−9.21258	−0.317298
$844$	13.9524	0.480262
$845$	8.17197	0.281124
$846$	−1.42528	−0.0490021
$847$	0.722789	0.0248353
$848$	1.66480	0.0571694
$849$	21.7252	0.745606
$850$	15.3610	0.526878
$851$	8.11825	0.278290
$852$	−4.56540	−0.156408
$853$	−40.2058	−1.37662	−0.688310	−	0.725417i	$-0.741647\pi$
$853$	−40.2058	−1.37662	−0.688310	+	0.725417i	$0.741647\pi$
$854$	−1.74872	−0.0598401
$855$	−9.41752	−0.322072
$856$	1.62485	0.0555363
$857$	6.91411	0.236181	0.118091	−	0.993003i	$-0.462323\pi$
$857$	6.91411	0.236181	0.118091	+	0.993003i	$0.462323\pi$
$858$	7.16331	0.244551
$859$	−0.0440757	−0.00150384	−0.000751922	−	1.00000i	$-0.500239\pi$
$859$	−0.0440757	−0.00150384	−0.000751922		1.00000i	$0.500239\pi$
$860$	−6.05208	−0.206374
$861$	2.81138	0.0958115
$862$	−32.9913	−1.12369
$863$	40.7084	1.38573	0.692866	−	0.721067i	$-0.256348\pi$
$863$	40.7084	1.38573	0.692866	+	0.721067i	$0.256348\pi$
$864$	−1.00000	−0.0340207
$865$	−20.5645	−0.699214
$866$	−11.1102	−0.377539
$867$	0.0635084	0.00215686
$868$	−2.48353	−0.0842964
$869$	−37.2261	−1.26281
$870$	−1.12580	−0.0381682
$871$	32.8571	1.11332
$872$	−0.828964	−0.0280723
$873$	14.1692	0.479556
$874$	−8.36517	−0.282956
$875$	−3.44556	−0.116481
$876$	−16.5331	−0.558602
$877$	8.41308	0.284090	0.142045	−	0.989860i	$-0.454632\pi$
$877$	8.41308	0.284090	0.142045	+	0.989860i	$0.454632\pi$
$878$	19.6712	0.663870
$879$	−13.1872	−0.444794
$880$	3.36569	0.113457
$881$	12.4581	0.419723	0.209862	−	0.977731i	$-0.432699\pi$
$881$	12.4581	0.419723	0.209862	+	0.977731i	$0.432699\pi$
$882$	−6.87717	−0.231566
$883$	13.7322	0.462127	0.231063	−	0.972939i	$-0.425779\pi$
$883$	13.7322	0.462127	0.231063	+	0.972939i	$0.425779\pi$
$884$	−9.86082	−0.331655
$885$	−13.5218	−0.454530
$886$	−5.05453	−0.169810
$887$	5.91013	0.198443	0.0992214	−	0.995065i	$-0.468365\pi$
$887$	5.91013	0.198443	0.0992214	+	0.995065i	$0.468365\pi$
$888$	8.11825	0.272431
$889$	4.92792	0.165277
$890$	8.44276	0.283002
$891$	−2.98960	−0.100155
$892$	−20.6482	−0.691352
$893$	−11.9227	−0.398978
$894$	−12.6490	−0.423046
$895$	1.86566	0.0623622
$896$	−0.350474	−0.0117085
$897$	2.39608	0.0800028
$898$	−21.3222	−0.711532
$899$	−7.08620	−0.236338
$900$	−3.73257	−0.124419
$901$	−6.85130	−0.228250
$902$	−23.9815	−0.798496
$903$	1.88408	0.0626981
$904$	15.9989	0.532116
$905$	−6.45019	−0.214411
$906$	−3.81570	−0.126768
$907$	10.7702	0.357619	0.178809	−	0.983884i	$-0.442775\pi$
$907$	10.7702	0.357619	0.178809	+	0.983884i	$0.442775\pi$
$908$	−4.64032	−0.153995
$909$	1.79712	0.0596068
$910$	0.945407	0.0313399
$911$	0.159611	0.00528814	0.00264407	−	0.999997i	$-0.499158\pi$
$911$	0.159611	0.00528814	0.00264407	+	0.999997i	$0.499158\pi$
$912$	−8.36517	−0.276998
$913$	−5.23247	−0.173170
$914$	28.3969	0.939286
$915$	5.61729	0.185702
$916$	−7.32879	−0.242150
$917$	−2.98956	−0.0987240
$918$	4.11540	0.135828
$919$	−12.0752	−0.398325	−0.199162	−	0.979967i	$-0.563822\pi$
$919$	−12.0752	−0.398325	−0.199162	+	0.979967i	$0.563822\pi$
$920$	1.12580	0.0371166
$921$	23.6286	0.778588
$922$	−26.2260	−0.863707
$923$	10.9391	0.360064
$924$	−1.04778	−0.0344693
$925$	30.3019	0.996322
$926$	13.1757	0.432979
$927$	0.657566	0.0215973
$928$	−1.00000	−0.0328266
$929$	4.33661	0.142279	0.0711397	−	0.997466i	$-0.477336\pi$
$929$	4.33661	0.142279	0.0711397	+	0.997466i	$0.477336\pi$
$930$	7.97765	0.261597
$931$	−57.5287	−1.88543
$932$	21.9859	0.720173
$933$	21.8984	0.716923
$934$	39.2223	1.28339
$935$	−13.8512	−0.452981
$936$	2.39608	0.0783183
$937$	−57.9298	−1.89248	−0.946242	−	0.323460i	$-0.895154\pi$
$937$	−57.9298	−1.89248	−0.946242	+	0.323460i	$0.895154\pi$
$938$	−4.80600	−0.156921
$939$	−8.88472	−0.289942
$940$	1.60458	0.0523357
$941$	10.1336	0.330345	0.165173	−	0.986265i	$-0.447182\pi$
$941$	10.1336	0.330345	0.165173	+	0.986265i	$0.447182\pi$
$942$	−23.5540	−0.767430
$943$	−8.02165	−0.261221
$944$	−12.0108	−0.390919
$945$	−0.394564	−0.0128352
$946$	−16.0714	−0.522528
$947$	2.99423	0.0972995	0.0486497	−	0.998816i	$-0.484508\pi$
$947$	2.99423	0.0972995	0.0486497	+	0.998816i	$0.484508\pi$
$948$	−12.4519	−0.404419
$949$	39.6147	1.28595
$950$	−31.2236	−1.01303
$951$	−13.6485	−0.442584
$952$	1.44234	0.0467465
$953$	2.85277	0.0924102	0.0462051	−	0.998932i	$-0.485287\pi$
$953$	2.85277	0.0924102	0.0462051	+	0.998932i	$0.485287\pi$
$954$	1.66480	0.0538998
$955$	19.2355	0.622445
$956$	7.54702	0.244088
$957$	−2.98960	−0.0966399
$958$	−32.1503	−1.03873
$959$	−5.06501	−0.163558
$960$	1.12580	0.0363351
$961$	19.2142	0.619813
$962$	−19.4520	−0.627157
$963$	1.62485	0.0523602
$964$	−2.82443	−0.0909686
$965$	−1.72770	−0.0556166
$966$	−0.350474	−0.0112763
$967$	−33.4656	−1.07618	−0.538090	−	0.842887i	$-0.680854\pi$
$967$	−33.4656	−1.07618	−0.538090	+	0.842887i	$0.680854\pi$
$968$	−2.06232	−0.0662854
$969$	34.4260	1.10592
$970$	−15.9518	−0.512180
$971$	−16.7581	−0.537793	−0.268896	−	0.963169i	$-0.586659\pi$
$971$	−16.7581	−0.537793	−0.268896	+	0.963169i	$0.586659\pi$
$972$	−1.00000	−0.0320750
$973$	7.50591	0.240628
$974$	−22.2810	−0.713930
$975$	8.94354	0.286423
$976$	4.98960	0.159713
$977$	34.0443	1.08917	0.544586	−	0.838705i	$-0.316687\pi$
$977$	34.0443	1.08917	0.544586	+	0.838705i	$0.316687\pi$
$978$	0.0185758	0.000593990 0
$979$	22.4200	0.716546
$980$	7.74232	0.247319
$981$	−0.828964	−0.0264668
$982$	−14.0078	−0.447007
$983$	−30.6621	−0.977968	−0.488984	−	0.872293i	$-0.662632\pi$
$983$	−30.6621	−0.977968	−0.488984	+	0.872293i	$0.662632\pi$
$984$	−8.02165	−0.255721
$985$	6.95306	0.221543
$986$	4.11540	0.131061
$987$	−0.499524	−0.0159000
$988$	20.0436	0.637672
$989$	−5.37579	−0.170940
$990$	3.36569	0.106969
$991$	54.7946	1.74061	0.870304	−	0.492516i	$-0.163923\pi$
$991$	54.7946	1.74061	0.870304	+	0.492516i	$0.163923\pi$
$992$	7.08620	0.224987
$993$	25.9849	0.824605
$994$	−1.60006	−0.0507507
$995$	−19.3104	−0.612180
$996$	−1.75023	−0.0554581
$997$	−38.9758	−1.23438	−0.617189	−	0.786815i	$-0.711729\pi$
$997$	−38.9758	−1.23438	−0.617189	+	0.786815i	$0.711729\pi$
$998$	−12.9964	−0.411395
$999$	8.11825	0.256850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.3	✓	8	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.12580 q^{5} -1.00000 q^{6} -0.350474 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.12580 q^{5} -1.00000 q^{6} -0.350474 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.12580 q^{10} -2.98960 q^{11} -1.00000 q^{12} +2.39608 q^{13} -0.350474 q^{14} +1.12580 q^{15} +1.00000 q^{16} -4.11540 q^{17} +1.00000 q^{18} +8.36517 q^{19} -1.12580 q^{20} +0.350474 q^{21} -2.98960 q^{22} -1.00000 q^{23} -1.00000 q^{24} -3.73257 q^{25} +2.39608 q^{26} -1.00000 q^{27} -0.350474 q^{28} -1.00000 q^{29} +1.12580 q^{30} +7.08620 q^{31} +1.00000 q^{32} +2.98960 q^{33} -4.11540 q^{34} +0.394564 q^{35} +1.00000 q^{36} -8.11825 q^{37} +8.36517 q^{38} -2.39608 q^{39} -1.12580 q^{40} +8.02165 q^{41} +0.350474 q^{42} +5.37579 q^{43} -2.98960 q^{44} -1.12580 q^{45} -1.00000 q^{46} -1.42528 q^{47} -1.00000 q^{48} -6.87717 q^{49} -3.73257 q^{50} +4.11540 q^{51} +2.39608 q^{52} +1.66480 q^{53} -1.00000 q^{54} +3.36569 q^{55} -0.350474 q^{56} -8.36517 q^{57} -1.00000 q^{58} -12.0108 q^{59} +1.12580 q^{60} +4.98960 q^{61} +7.08620 q^{62} -0.350474 q^{63} +1.00000 q^{64} -2.69751 q^{65} +2.98960 q^{66} +13.7129 q^{67} -4.11540 q^{68} +1.00000 q^{69} +0.394564 q^{70} +4.56540 q^{71} +1.00000 q^{72} +16.5331 q^{73} -8.11825 q^{74} +3.73257 q^{75} +8.36517 q^{76} +1.04778 q^{77} -2.39608 q^{78} +12.4519 q^{79} -1.12580 q^{80} +1.00000 q^{81} +8.02165 q^{82} +1.75023 q^{83} +0.350474 q^{84} +4.63312 q^{85} +5.37579 q^{86} +1.00000 q^{87} -2.98960 q^{88} -7.49934 q^{89} -1.12580 q^{90} -0.839764 q^{91} -1.00000 q^{92} -7.08620 q^{93} -1.42528 q^{94} -9.41752 q^{95} -1.00000 q^{96} +14.1692 q^{97} -6.87717 q^{98} -2.98960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

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