LMFDB - Embedded newform 8022.2.a.q.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8022,2,Mod(1,8022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8022, 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("8022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8022 = 2 \cdot 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8022.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:	$64.0559925015$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 11x^{7} + 36x^{6} + 50x^{5} - 70x^{4} - 73x^{3} + 14x^{2} + 22x + 4 $ x^9 - 4x^8 - 11x^7 + 36x^6 + 50x^5 - 70x^4 - 73x^3 + 14x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.15530$ of defining polynomial
Character	$\chi$	$=$	8022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.491351 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.491351 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.491351 q^{10} -3.78362 q^{11} -1.00000 q^{12} -4.72535 q^{13} -1.00000 q^{14} -0.491351 q^{15} +1.00000 q^{16} +3.78096 q^{17} +1.00000 q^{18} +1.59589 q^{19} +0.491351 q^{20} +1.00000 q^{21} -3.78362 q^{22} +2.70694 q^{23} -1.00000 q^{24} -4.75857 q^{25} -4.72535 q^{26} -1.00000 q^{27} -1.00000 q^{28} +5.17983 q^{29} -0.491351 q^{30} +2.97193 q^{31} +1.00000 q^{32} +3.78362 q^{33} +3.78096 q^{34} -0.491351 q^{35} +1.00000 q^{36} -0.0475190 q^{37} +1.59589 q^{38} +4.72535 q^{39} +0.491351 q^{40} +0.784995 q^{41} +1.00000 q^{42} +6.93665 q^{43} -3.78362 q^{44} +0.491351 q^{45} +2.70694 q^{46} +6.75113 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.75857 q^{50} -3.78096 q^{51} -4.72535 q^{52} -3.59500 q^{53} -1.00000 q^{54} -1.85908 q^{55} -1.00000 q^{56} -1.59589 q^{57} +5.17983 q^{58} +5.45743 q^{59} -0.491351 q^{60} -10.8934 q^{61} +2.97193 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.32180 q^{65} +3.78362 q^{66} -7.47271 q^{67} +3.78096 q^{68} -2.70694 q^{69} -0.491351 q^{70} -10.7881 q^{71} +1.00000 q^{72} +1.85043 q^{73} -0.0475190 q^{74} +4.75857 q^{75} +1.59589 q^{76} +3.78362 q^{77} +4.72535 q^{78} -11.7890 q^{79} +0.491351 q^{80} +1.00000 q^{81} +0.784995 q^{82} -4.32320 q^{83} +1.00000 q^{84} +1.85778 q^{85} +6.93665 q^{86} -5.17983 q^{87} -3.78362 q^{88} -11.0249 q^{89} +0.491351 q^{90} +4.72535 q^{91} +2.70694 q^{92} -2.97193 q^{93} +6.75113 q^{94} +0.784140 q^{95} -1.00000 q^{96} -18.1300 q^{97} +1.00000 q^{98} -3.78362 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 + 6 * q^5 - 9 * q^6 - 9 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9} + 6 q^{10} - 7 q^{11} - 9 q^{12} - q^{13} - 9 q^{14} - 6 q^{15} + 9 q^{16} - 10 q^{17} + 9 q^{18} + 6 q^{20} + 9 q^{21} - 7 q^{22} - 19 q^{23} - 9 q^{24} + 5 q^{25} - q^{26} - 9 q^{27} - 9 q^{28} - 21 q^{29} - 6 q^{30} + 6 q^{31} + 9 q^{32} + 7 q^{33} - 10 q^{34} - 6 q^{35} + 9 q^{36} - 14 q^{37} + q^{39} + 6 q^{40} + 12 q^{41} + 9 q^{42} - 17 q^{43} - 7 q^{44} + 6 q^{45} - 19 q^{46} - 11 q^{47} - 9 q^{48} + 9 q^{49} + 5 q^{50} + 10 q^{51} - q^{52} - 12 q^{53} - 9 q^{54} - 21 q^{55} - 9 q^{56} - 21 q^{58} - 8 q^{59} - 6 q^{60} + q^{61} + 6 q^{62} - 9 q^{63} + 9 q^{64} - 8 q^{65} + 7 q^{66} - 31 q^{67} - 10 q^{68} + 19 q^{69} - 6 q^{70} - 41 q^{71} + 9 q^{72} - q^{73} - 14 q^{74} - 5 q^{75} + 7 q^{77} + q^{78} - 17 q^{79} + 6 q^{80} + 9 q^{81} + 12 q^{82} - 36 q^{83} + 9 q^{84} - 30 q^{85} - 17 q^{86} + 21 q^{87} - 7 q^{88} - 17 q^{89} + 6 q^{90} + q^{91} - 19 q^{92} - 6 q^{93} - 11 q^{94} - 50 q^{95} - 9 q^{96} - 14 q^{97} + 9 q^{98} - 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 + 6 * q^5 - 9 * q^6 - 9 * q^7 + 9 * q^8 + 9 * q^9 + 6 * q^10 - 7 * q^11 - 9 * q^12 - q^13 - 9 * q^14 - 6 * q^15 + 9 * q^16 - 10 * q^17 + 9 * q^18 + 6 * q^20 + 9 * q^21 - 7 * q^22 - 19 * q^23 - 9 * q^24 + 5 * q^25 - q^26 - 9 * q^27 - 9 * q^28 - 21 * q^29 - 6 * q^30 + 6 * q^31 + 9 * q^32 + 7 * q^33 - 10 * q^34 - 6 * q^35 + 9 * q^36 - 14 * q^37 + q^39 + 6 * q^40 + 12 * q^41 + 9 * q^42 - 17 * q^43 - 7 * q^44 + 6 * q^45 - 19 * q^46 - 11 * q^47 - 9 * q^48 + 9 * q^49 + 5 * q^50 + 10 * q^51 - q^52 - 12 * q^53 - 9 * q^54 - 21 * q^55 - 9 * q^56 - 21 * q^58 - 8 * q^59 - 6 * q^60 + q^61 + 6 * q^62 - 9 * q^63 + 9 * q^64 - 8 * q^65 + 7 * q^66 - 31 * q^67 - 10 * q^68 + 19 * q^69 - 6 * q^70 - 41 * q^71 + 9 * q^72 - q^73 - 14 * q^74 - 5 * q^75 + 7 * q^77 + q^78 - 17 * q^79 + 6 * q^80 + 9 * q^81 + 12 * q^82 - 36 * q^83 + 9 * q^84 - 30 * q^85 - 17 * q^86 + 21 * q^87 - 7 * q^88 - 17 * q^89 + 6 * q^90 + q^91 - 19 * q^92 - 6 * q^93 - 11 * q^94 - 50 * q^95 - 9 * q^96 - 14 * q^97 + 9 * q^98 - 7 * 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
$5$	0.491351	0.219739	0.109869	−	0.993946i	$-0.464957\pi$
$5$	0.491351	0.219739	0.109869	+	0.993946i	$0.464957\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.491351	0.155379
$11$	−3.78362	−1.14080	−0.570402	−	0.821366i	$-0.693213\pi$
$11$	−3.78362	−1.14080	−0.570402	+	0.821366i	$0.693213\pi$
$12$	−1.00000	−0.288675
$13$	−4.72535	−1.31058	−0.655288	−	0.755379i	$-0.727453\pi$
$13$	−4.72535	−1.31058	−0.655288	+	0.755379i	$0.727453\pi$
$14$	−1.00000	−0.267261
$15$	−0.491351	−0.126866
$16$	1.00000	0.250000
$17$	3.78096	0.917018	0.458509	−	0.888690i	$-0.348384\pi$
$17$	3.78096	0.917018	0.458509	+	0.888690i	$0.348384\pi$
$18$	1.00000	0.235702
$19$	1.59589	0.366121	0.183061	−	0.983102i	$-0.441400\pi$
$19$	1.59589	0.366121	0.183061	+	0.983102i	$0.441400\pi$
$20$	0.491351	0.109869
$21$	1.00000	0.218218
$22$	−3.78362	−0.806670
$23$	2.70694	0.564436	0.282218	−	0.959350i	$-0.408930\pi$
$23$	2.70694	0.564436	0.282218	+	0.959350i	$0.408930\pi$
$24$	−1.00000	−0.204124
$25$	−4.75857	−0.951715
$26$	−4.72535	−0.926717
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	5.17983	0.961870	0.480935	−	0.876756i	$-0.340297\pi$
$29$	5.17983	0.961870	0.480935	+	0.876756i	$0.340297\pi$
$30$	−0.491351	−0.0897080
$31$	2.97193	0.533775	0.266887	−	0.963728i	$-0.414005\pi$
$31$	2.97193	0.533775	0.266887	+	0.963728i	$0.414005\pi$
$32$	1.00000	0.176777
$33$	3.78362	0.658643
$34$	3.78096	0.648430
$35$	−0.491351	−0.0830535
$36$	1.00000	0.166667
$37$	−0.0475190	−0.00781208	−0.00390604	−	0.999992i	$-0.501243\pi$
$37$	−0.0475190	−0.00781208	−0.00390604	+	0.999992i	$0.501243\pi$
$38$	1.59589	0.258887
$39$	4.72535	0.756661
$40$	0.491351	0.0776894
$41$	0.784995	0.122596	0.0612978	−	0.998120i	$-0.480476\pi$
$41$	0.784995	0.122596	0.0612978	+	0.998120i	$0.480476\pi$
$42$	1.00000	0.154303
$43$	6.93665	1.05783	0.528914	−	0.848675i	$-0.322599\pi$
$43$	6.93665	1.05783	0.528914	+	0.848675i	$0.322599\pi$
$44$	−3.78362	−0.570402
$45$	0.491351	0.0732463
$46$	2.70694	0.399116
$47$	6.75113	0.984753	0.492377	−	0.870382i	$-0.336128\pi$
$47$	6.75113	0.984753	0.492377	+	0.870382i	$0.336128\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.75857	−0.672964
$51$	−3.78096	−0.529441
$52$	−4.72535	−0.655288
$53$	−3.59500	−0.493812	−0.246906	−	0.969039i	$-0.579414\pi$
$53$	−3.59500	−0.493812	−0.246906	+	0.969039i	$0.579414\pi$
$54$	−1.00000	−0.136083
$55$	−1.85908	−0.250679
$56$	−1.00000	−0.133631
$57$	−1.59589	−0.211380
$58$	5.17983	0.680145
$59$	5.45743	0.710497	0.355249	−	0.934772i	$-0.384396\pi$
$59$	5.45743	0.710497	0.355249	+	0.934772i	$0.384396\pi$
$60$	−0.491351	−0.0634331
$61$	−10.8934	−1.39476	−0.697378	−	0.716704i	$-0.745650\pi$
$61$	−10.8934	−1.39476	−0.697378	+	0.716704i	$0.745650\pi$
$62$	2.97193	0.377436
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.32180	−0.287984
$66$	3.78362	0.465731
$67$	−7.47271	−0.912936	−0.456468	−	0.889740i	$-0.650886\pi$
$67$	−7.47271	−0.912936	−0.456468	+	0.889740i	$0.650886\pi$
$68$	3.78096	0.458509
$69$	−2.70694	−0.325877
$70$	−0.491351	−0.0587277
$71$	−10.7881	−1.28031	−0.640157	−	0.768244i	$-0.721131\pi$
$71$	−10.7881	−1.28031	−0.640157	+	0.768244i	$0.721131\pi$
$72$	1.00000	0.117851
$73$	1.85043	0.216576	0.108288	−	0.994120i	$-0.465463\pi$
$73$	1.85043	0.216576	0.108288	+	0.994120i	$0.465463\pi$
$74$	−0.0475190	−0.00552397
$75$	4.75857	0.549473
$76$	1.59589	0.183061
$77$	3.78362	0.431183
$78$	4.72535	0.535040
$79$	−11.7890	−1.32636	−0.663182	−	0.748458i	$-0.730795\pi$
$79$	−11.7890	−1.32636	−0.663182	+	0.748458i	$0.730795\pi$
$80$	0.491351	0.0549347
$81$	1.00000	0.111111
$82$	0.784995	0.0866882
$83$	−4.32320	−0.474533	−0.237266	−	0.971445i	$-0.576251\pi$
$83$	−4.32320	−0.474533	−0.237266	+	0.971445i	$0.576251\pi$
$84$	1.00000	0.109109
$85$	1.85778	0.201505
$86$	6.93665	0.747998
$87$	−5.17983	−0.555336
$88$	−3.78362	−0.403335
$89$	−11.0249	−1.16864	−0.584318	−	0.811525i	$-0.698638\pi$
$89$	−11.0249	−1.16864	−0.584318	+	0.811525i	$0.698638\pi$
$90$	0.491351	0.0517929
$91$	4.72535	0.495351
$92$	2.70694	0.282218
$93$	−2.97193	−0.308175
$94$	6.75113	0.696326
$95$	0.784140	0.0804510
$96$	−1.00000	−0.102062
$97$	−18.1300	−1.84082	−0.920410	−	0.390954i	$-0.872145\pi$
$97$	−18.1300	−1.84082	−0.920410	+	0.390954i	$0.872145\pi$
$98$	1.00000	0.101015
$99$	−3.78362	−0.380268
$100$	−4.75857	−0.475857
$101$	−9.20150	−0.915583	−0.457792	−	0.889059i	$-0.651359\pi$
$101$	−9.20150	−0.915583	−0.457792	+	0.889059i	$0.651359\pi$
$102$	−3.78096	−0.374371
$103$	5.83586	0.575024	0.287512	−	0.957777i	$-0.407172\pi$
$103$	5.83586	0.575024	0.287512	+	0.957777i	$0.407172\pi$
$104$	−4.72535	−0.463358
$105$	0.491351	0.0479509
$106$	−3.59500	−0.349178
$107$	2.78692	0.269421	0.134711	−	0.990885i	$-0.456990\pi$
$107$	2.78692	0.269421	0.134711	+	0.990885i	$0.456990\pi$
$108$	−1.00000	−0.0962250
$109$	1.11465	0.106764	0.0533822	−	0.998574i	$-0.483000\pi$
$109$	1.11465	0.106764	0.0533822	+	0.998574i	$0.483000\pi$
$110$	−1.85908	−0.177257
$111$	0.0475190	0.00451030
$112$	−1.00000	−0.0944911
$113$	9.14874	0.860641	0.430321	−	0.902676i	$-0.358400\pi$
$113$	9.14874	0.860641	0.430321	+	0.902676i	$0.358400\pi$
$114$	−1.59589	−0.149468
$115$	1.33006	0.124028
$116$	5.17983	0.480935
$117$	−4.72535	−0.436858
$118$	5.45743	0.502398
$119$	−3.78096	−0.346600
$120$	−0.491351	−0.0448540
$121$	3.31577	0.301433
$122$	−10.8934	−0.986241
$123$	−0.784995	−0.0707806
$124$	2.97193	0.266887
$125$	−4.79489	−0.428868
$126$	−1.00000	−0.0890871
$127$	−16.4295	−1.45788	−0.728941	−	0.684577i	$-0.759987\pi$
$127$	−16.4295	−1.45788	−0.728941	+	0.684577i	$0.759987\pi$
$128$	1.00000	0.0883883
$129$	−6.93665	−0.610738
$130$	−2.32180	−0.203636
$131$	−17.3837	−1.51882	−0.759411	−	0.650611i	$-0.774513\pi$
$131$	−17.3837	−1.51882	−0.759411	+	0.650611i	$0.774513\pi$
$132$	3.78362	0.329322
$133$	−1.59589	−0.138381
$134$	−7.47271	−0.645544
$135$	−0.491351	−0.0422888
$136$	3.78096	0.324215
$137$	−8.95090	−0.764727	−0.382364	−	0.924012i	$-0.624890\pi$
$137$	−8.95090	−0.764727	−0.382364	+	0.924012i	$0.624890\pi$
$138$	−2.70694	−0.230430
$139$	−19.0559	−1.61630	−0.808150	−	0.588977i	$-0.799531\pi$
$139$	−19.0559	−1.61630	−0.808150	+	0.588977i	$0.799531\pi$
$140$	−0.491351	−0.0415267
$141$	−6.75113	−0.568548
$142$	−10.7881	−0.905319
$143$	17.8789	1.49511
$144$	1.00000	0.0833333
$145$	2.54511	0.211360
$146$	1.85043	0.153143
$147$	−1.00000	−0.0824786
$148$	−0.0475190	−0.00390604
$149$	−10.9154	−0.894226	−0.447113	−	0.894478i	$-0.647548\pi$
$149$	−10.9154	−0.894226	−0.447113	+	0.894478i	$0.647548\pi$
$150$	4.75857	0.388536
$151$	−19.9340	−1.62220	−0.811101	−	0.584906i	$-0.801132\pi$
$151$	−19.9340	−1.62220	−0.811101	+	0.584906i	$0.801132\pi$
$152$	1.59589	0.129443
$153$	3.78096	0.305673
$154$	3.78362	0.304893
$155$	1.46026	0.117291
$156$	4.72535	0.378330
$157$	16.0874	1.28391	0.641957	−	0.766741i	$-0.278123\pi$
$157$	16.0874	1.28391	0.641957	+	0.766741i	$0.278123\pi$
$158$	−11.7890	−0.937882
$159$	3.59500	0.285102
$160$	0.491351	0.0388447
$161$	−2.70694	−0.213337
$162$	1.00000	0.0785674
$163$	−8.80865	−0.689947	−0.344973	−	0.938613i	$-0.612112\pi$
$163$	−8.80865	−0.689947	−0.344973	+	0.938613i	$0.612112\pi$
$164$	0.784995	0.0612978
$165$	1.85908	0.144730
$166$	−4.32320	−0.335545
$167$	−14.0118	−1.08426	−0.542132	−	0.840293i	$-0.682383\pi$
$167$	−14.0118	−1.08426	−0.542132	+	0.840293i	$0.682383\pi$
$168$	1.00000	0.0771517
$169$	9.32890	0.717607
$170$	1.85778	0.142485
$171$	1.59589	0.122040
$172$	6.93665	0.528914
$173$	18.7414	1.42488	0.712441	−	0.701732i	$-0.247590\pi$
$173$	18.7414	1.42488	0.712441	+	0.701732i	$0.247590\pi$
$174$	−5.17983	−0.392682
$175$	4.75857	0.359714
$176$	−3.78362	−0.285201
$177$	−5.45743	−0.410206
$178$	−11.0249	−0.826351
$179$	13.3221	0.995742	0.497871	−	0.867251i	$-0.334115\pi$
$179$	13.3221	0.995742	0.497871	+	0.867251i	$0.334115\pi$
$180$	0.491351	0.0366231
$181$	25.6112	1.90367	0.951834	−	0.306613i	$-0.0991957\pi$
$181$	25.6112	1.90367	0.951834	+	0.306613i	$0.0991957\pi$
$182$	4.72535	0.350266
$183$	10.8934	0.805262
$184$	2.70694	0.199558
$185$	−0.0233485	−0.00171662
$186$	−2.97193	−0.217913
$187$	−14.3057	−1.04614
$188$	6.75113	0.492377
$189$	1.00000	0.0727393
$190$	0.784140	0.0568875
$191$	1.00000	0.0723575
$191$	1.00000	0.0723575
$192$	−1.00000	−0.0721688
$193$	−7.10649	−0.511536	−0.255768	−	0.966738i	$-0.582328\pi$
$193$	−7.10649	−0.511536	−0.255768	+	0.966738i	$0.582328\pi$
$194$	−18.1300	−1.30166
$195$	2.32180	0.166268
$196$	1.00000	0.0714286
$197$	7.97056	0.567879	0.283939	−	0.958842i	$-0.408359\pi$
$197$	7.97056	0.567879	0.283939	+	0.958842i	$0.408359\pi$
$198$	−3.78362	−0.268890
$199$	−7.43367	−0.526959	−0.263479	−	0.964665i	$-0.584870\pi$
$199$	−7.43367	−0.526959	−0.263479	+	0.964665i	$0.584870\pi$
$200$	−4.75857	−0.336482
$201$	7.47271	0.527084
$202$	−9.20150	−0.647415
$203$	−5.17983	−0.363553
$204$	−3.78096	−0.264720
$205$	0.385708	0.0269390
$206$	5.83586	0.406603
$207$	2.70694	0.188145
$208$	−4.72535	−0.327644
$209$	−6.03822	−0.417672
$210$	0.491351	0.0339064
$211$	1.71470	0.118045	0.0590223	−	0.998257i	$-0.481202\pi$
$211$	1.71470	0.118045	0.0590223	+	0.998257i	$0.481202\pi$
$212$	−3.59500	−0.246906
$213$	10.7881	0.739190
$214$	2.78692	0.190510
$215$	3.40833	0.232446
$216$	−1.00000	−0.0680414
$217$	−2.97193	−0.201748
$218$	1.11465	0.0754939
$219$	−1.85043	−0.125040
$220$	−1.85908	−0.125339
$221$	−17.8664	−1.20182
$222$	0.0475190	0.00318927
$223$	−19.2003	−1.28575	−0.642874	−	0.765972i	$-0.722258\pi$
$223$	−19.2003	−1.28575	−0.642874	+	0.765972i	$0.722258\pi$
$224$	−1.00000	−0.0668153
$225$	−4.75857	−0.317238
$226$	9.14874	0.608565
$227$	−1.36716	−0.0907416	−0.0453708	−	0.998970i	$-0.514447\pi$
$227$	−1.36716	−0.0907416	−0.0453708	+	0.998970i	$0.514447\pi$
$228$	−1.59589	−0.105690
$229$	15.5407	1.02696	0.513480	−	0.858101i	$-0.328356\pi$
$229$	15.5407	1.02696	0.513480	+	0.858101i	$0.328356\pi$
$230$	1.33006	0.0877014
$231$	−3.78362	−0.248944
$232$	5.17983	0.340072
$233$	−20.0353	−1.31256	−0.656278	−	0.754519i	$-0.727870\pi$
$233$	−20.0353	−1.31256	−0.656278	+	0.754519i	$0.727870\pi$
$234$	−4.72535	−0.308906
$235$	3.31717	0.216389
$236$	5.45743	0.355249
$237$	11.7890	0.765777
$238$	−3.78096	−0.245083
$239$	−22.8767	−1.47977	−0.739884	−	0.672734i	$-0.765120\pi$
$239$	−22.8767	−1.47977	−0.739884	+	0.672734i	$0.765120\pi$
$240$	−0.491351	−0.0317166
$241$	28.2471	1.81956	0.909778	−	0.415095i	$-0.136252\pi$
$241$	28.2471	1.81956	0.909778	+	0.415095i	$0.136252\pi$
$242$	3.31577	0.213146
$243$	−1.00000	−0.0641500
$244$	−10.8934	−0.697378
$245$	0.491351	0.0313913
$246$	−0.784995	−0.0500495
$247$	−7.54111	−0.479829
$248$	2.97193	0.188718
$249$	4.32320	0.273972
$250$	−4.79489	−0.303255
$251$	−12.2938	−0.775981	−0.387990	−	0.921663i	$-0.626831\pi$
$251$	−12.2938	−0.775981	−0.387990	+	0.921663i	$0.626831\pi$
$252$	−1.00000	−0.0629941
$253$	−10.2420	−0.643910
$254$	−16.4295	−1.03088
$255$	−1.85778	−0.116339
$256$	1.00000	0.0625000
$257$	22.4212	1.39860	0.699299	−	0.714829i	$-0.253496\pi$
$257$	22.4212	1.39860	0.699299	+	0.714829i	$0.253496\pi$
$258$	−6.93665	−0.431857
$259$	0.0475190	0.00295269
$260$	−2.32180	−0.143992
$261$	5.17983	0.320623
$262$	−17.3837	−1.07397
$263$	−0.731020	−0.0450766	−0.0225383	−	0.999746i	$-0.507175\pi$
$263$	−0.731020	−0.0450766	−0.0225383	+	0.999746i	$0.507175\pi$
$264$	3.78362	0.232866
$265$	−1.76641	−0.108510
$266$	−1.59589	−0.0978500
$267$	11.0249	0.674712
$268$	−7.47271	−0.456468
$269$	9.63429	0.587413	0.293707	−	0.955896i	$-0.405111\pi$
$269$	9.63429	0.587413	0.293707	+	0.955896i	$0.405111\pi$
$270$	−0.491351	−0.0299027
$271$	5.67323	0.344624	0.172312	−	0.985042i	$-0.444876\pi$
$271$	5.67323	0.344624	0.172312	+	0.985042i	$0.444876\pi$
$272$	3.78096	0.229255
$273$	−4.72535	−0.285991
$274$	−8.95090	−0.540744
$275$	18.0046	1.08572
$276$	−2.70694	−0.162939
$277$	10.8082	0.649404	0.324702	−	0.945816i	$-0.394736\pi$
$277$	10.8082	0.649404	0.324702	+	0.945816i	$0.394736\pi$
$278$	−19.0559	−1.14290
$279$	2.97193	0.177925
$280$	−0.491351	−0.0293638
$281$	−7.41685	−0.442452	−0.221226	−	0.975223i	$-0.571006\pi$
$281$	−7.41685	−0.442452	−0.221226	+	0.975223i	$0.571006\pi$
$282$	−6.75113	−0.402024
$283$	28.1875	1.67557	0.837787	−	0.545998i	$-0.183849\pi$
$283$	28.1875	1.67557	0.837787	+	0.545998i	$0.183849\pi$
$284$	−10.7881	−0.640157
$285$	−0.784140	−0.0464484
$286$	17.8789	1.05720
$287$	−0.784995	−0.0463368
$288$	1.00000	0.0589256
$289$	−2.70431	−0.159077
$290$	2.54511	0.149454
$291$	18.1300	1.06280
$292$	1.85043	0.108288
$293$	−21.7190	−1.26884	−0.634418	−	0.772990i	$-0.718760\pi$
$293$	−21.7190	−1.26884	−0.634418	+	0.772990i	$0.718760\pi$
$294$	−1.00000	−0.0583212
$295$	2.68152	0.156124
$296$	−0.0475190	−0.00276199
$297$	3.78362	0.219548
$298$	−10.9154	−0.632313
$299$	−12.7912	−0.739735
$300$	4.75857	0.274736
$301$	−6.93665	−0.399822
$302$	−19.9340	−1.14707
$303$	9.20150	0.528612
$304$	1.59589	0.0915303
$305$	−5.35248	−0.306482
$306$	3.78096	0.216143
$307$	−22.0441	−1.25812	−0.629062	−	0.777355i	$-0.716561\pi$
$307$	−22.0441	−1.25812	−0.629062	+	0.777355i	$0.716561\pi$
$308$	3.78362	0.215592
$309$	−5.83586	−0.331990
$310$	1.46026	0.0829373
$311$	−18.2434	−1.03449	−0.517244	−	0.855838i	$-0.673042\pi$
$311$	−18.2434	−1.03449	−0.517244	+	0.855838i	$0.673042\pi$
$312$	4.72535	0.267520
$313$	4.88185	0.275938	0.137969	−	0.990437i	$-0.455943\pi$
$313$	4.88185	0.275938	0.137969	+	0.990437i	$0.455943\pi$
$314$	16.0874	0.907864
$315$	−0.491351	−0.0276845
$316$	−11.7890	−0.663182
$317$	−26.1740	−1.47008	−0.735038	−	0.678026i	$-0.762836\pi$
$317$	−26.1740	−1.47008	−0.735038	+	0.678026i	$0.762836\pi$
$318$	3.59500	0.201598
$319$	−19.5985	−1.09731
$320$	0.491351	0.0274674
$321$	−2.78692	−0.155550
$322$	−2.70694	−0.150852
$323$	6.03398	0.335740
$324$	1.00000	0.0555556
$325$	22.4859	1.24729
$326$	−8.80865	−0.487866
$327$	−1.11465	−0.0616405
$328$	0.784995	0.0433441
$329$	−6.75113	−0.372202
$330$	1.85908	0.102339
$331$	−2.79751	−0.153765	−0.0768827	−	0.997040i	$-0.524497\pi$
$331$	−2.79751	−0.153765	−0.0768827	+	0.997040i	$0.524497\pi$
$332$	−4.32320	−0.237266
$333$	−0.0475190	−0.00260403
$334$	−14.0118	−0.766691
$335$	−3.67172	−0.200608
$336$	1.00000	0.0545545
$337$	−25.7887	−1.40480	−0.702401	−	0.711782i	$-0.747888\pi$
$337$	−25.7887	−1.40480	−0.702401	+	0.711782i	$0.747888\pi$
$338$	9.32890	0.507425
$339$	−9.14874	−0.496892
$340$	1.85778	0.100752
$341$	−11.2447	−0.608932
$342$	1.59589	0.0862956
$343$	−1.00000	−0.0539949
$344$	6.93665	0.373999
$345$	−1.33006	−0.0716079
$346$	18.7414	1.00754
$347$	−4.38898	−0.235613	−0.117806	−	0.993037i	$-0.537586\pi$
$347$	−4.38898	−0.235613	−0.117806	+	0.993037i	$0.537586\pi$
$348$	−5.17983	−0.277668
$349$	27.1516	1.45339	0.726695	−	0.686960i	$-0.241055\pi$
$349$	27.1516	1.45339	0.726695	+	0.686960i	$0.241055\pi$
$350$	4.75857	0.254356
$351$	4.72535	0.252220
$352$	−3.78362	−0.201668
$353$	−0.708243	−0.0376960	−0.0188480	−	0.999822i	$-0.506000\pi$
$353$	−0.708243	−0.0376960	−0.0188480	+	0.999822i	$0.506000\pi$
$354$	−5.45743	−0.290059
$355$	−5.30075	−0.281335
$356$	−11.0249	−0.584318
$357$	3.78096	0.200110
$358$	13.3221	0.704096
$359$	−12.5222	−0.660894	−0.330447	−	0.943825i	$-0.607199\pi$
$359$	−12.5222	−0.660894	−0.330447	+	0.943825i	$0.607199\pi$
$360$	0.491351	0.0258965
$361$	−16.4532	−0.865955
$362$	25.6112	1.34610
$363$	−3.31577	−0.174033
$364$	4.72535	0.247675
$365$	0.909210	0.0475902
$366$	10.8934	0.569406
$367$	0.654025	0.0341398	0.0170699	−	0.999854i	$-0.494566\pi$
$367$	0.654025	0.0341398	0.0170699	+	0.999854i	$0.494566\pi$
$368$	2.70694	0.141109
$369$	0.784995	0.0408652
$370$	−0.0233485	−0.00121383
$371$	3.59500	0.186643
$372$	−2.97193	−0.154087
$373$	−12.7907	−0.662277	−0.331138	−	0.943582i	$-0.607433\pi$
$373$	−12.7907	−0.662277	−0.331138	+	0.943582i	$0.607433\pi$
$374$	−14.3057	−0.739731
$375$	4.79489	0.247607
$376$	6.75113	0.348163
$377$	−24.4765	−1.26060
$378$	1.00000	0.0514344
$379$	−27.8720	−1.43169	−0.715844	−	0.698260i	$-0.753958\pi$
$379$	−27.8720	−1.43169	−0.715844	+	0.698260i	$0.753958\pi$
$380$	0.784140	0.0402255
$381$	16.4295	0.841708
$382$	1.00000	0.0511645
$383$	−8.31758	−0.425008	−0.212504	−	0.977160i	$-0.568162\pi$
$383$	−8.31758	−0.425008	−0.212504	+	0.977160i	$0.568162\pi$
$384$	−1.00000	−0.0510310
$385$	1.85908	0.0947477
$386$	−7.10649	−0.361711
$387$	6.93665	0.352610
$388$	−18.1300	−0.920410
$389$	9.91260	0.502589	0.251294	−	0.967911i	$-0.419144\pi$
$389$	9.91260	0.502589	0.251294	+	0.967911i	$0.419144\pi$
$390$	2.32180	0.117569
$391$	10.2348	0.517598
$392$	1.00000	0.0505076
$393$	17.3837	0.876893
$394$	7.97056	0.401551
$395$	−5.79253	−0.291454
$396$	−3.78362	−0.190134
$397$	24.1592	1.21251	0.606257	−	0.795269i	$-0.292670\pi$
$397$	24.1592	1.21251	0.606257	+	0.795269i	$0.292670\pi$
$398$	−7.43367	−0.372616
$399$	1.59589	0.0798942
$400$	−4.75857	−0.237929
$401$	32.3763	1.61680	0.808398	−	0.588636i	$-0.200335\pi$
$401$	32.3763	1.61680	0.808398	+	0.588636i	$0.200335\pi$
$402$	7.47271	0.372705
$403$	−14.0434	−0.699552
$404$	−9.20150	−0.457792
$405$	0.491351	0.0244154
$406$	−5.17983	−0.257071
$407$	0.179794	0.00891205
$408$	−3.78096	−0.187186
$409$	25.5867	1.26518	0.632591	−	0.774486i	$-0.281991\pi$
$409$	25.5867	1.26518	0.632591	+	0.774486i	$0.281991\pi$
$410$	0.385708	0.0190488
$411$	8.95090	0.441515
$412$	5.83586	0.287512
$413$	−5.45743	−0.268543
$414$	2.70694	0.133039
$415$	−2.12421	−0.104273
$416$	−4.72535	−0.231679
$417$	19.0559	0.933171
$418$	−6.03822	−0.295339
$419$	33.9919	1.66061	0.830306	−	0.557308i	$-0.188166\pi$
$419$	33.9919	1.66061	0.830306	+	0.557308i	$0.188166\pi$
$420$	0.491351	0.0239755
$421$	20.9468	1.02088	0.510442	−	0.859912i	$-0.329482\pi$
$421$	20.9468	1.02088	0.510442	+	0.859912i	$0.329482\pi$
$422$	1.71470	0.0834701
$423$	6.75113	0.328251
$424$	−3.59500	−0.174589
$425$	−17.9920	−0.872740
$426$	10.7881	0.522686
$427$	10.8934	0.527168
$428$	2.78692	0.134711
$429$	−17.8789	−0.863202
$430$	3.40833	0.164364
$431$	34.6569	1.66937	0.834683	−	0.550731i	$-0.185651\pi$
$431$	34.6569	1.66937	0.834683	+	0.550731i	$0.185651\pi$
$432$	−1.00000	−0.0481125
$433$	11.2101	0.538723	0.269362	−	0.963039i	$-0.413187\pi$
$433$	11.2101	0.538723	0.269362	+	0.963039i	$0.413187\pi$
$434$	−2.97193	−0.142657
$435$	−2.54511	−0.122029
$436$	1.11465	0.0533822
$437$	4.31996	0.206652
$438$	−1.85043	−0.0884169
$439$	−20.7615	−0.990894	−0.495447	−	0.868638i	$-0.664996\pi$
$439$	−20.7615	−0.990894	−0.495447	+	0.868638i	$0.664996\pi$
$440$	−1.85908	−0.0886284
$441$	1.00000	0.0476190
$442$	−17.8664	−0.849816
$443$	−9.22301	−0.438198	−0.219099	−	0.975703i	$-0.570312\pi$
$443$	−9.22301	−0.438198	−0.219099	+	0.975703i	$0.570312\pi$
$444$	0.0475190	0.00225515
$445$	−5.41709	−0.256795
$446$	−19.2003	−0.909162
$447$	10.9154	0.516282
$448$	−1.00000	−0.0472456
$449$	27.9076	1.31704	0.658521	−	0.752562i	$-0.271182\pi$
$449$	27.9076	1.31704	0.658521	+	0.752562i	$0.271182\pi$
$450$	−4.75857	−0.224321
$451$	−2.97012	−0.139858
$452$	9.14874	0.430321
$453$	19.9340	0.936579
$454$	−1.36716	−0.0641640
$455$	2.32180	0.108848
$456$	−1.59589	−0.0747342
$457$	1.13930	0.0532941	0.0266471	−	0.999645i	$-0.491517\pi$
$457$	1.13930	0.0532941	0.0266471	+	0.999645i	$0.491517\pi$
$458$	15.5407	0.726171
$459$	−3.78096	−0.176480
$460$	1.33006	0.0620142
$461$	23.5770	1.09809	0.549046	−	0.835792i	$-0.314991\pi$
$461$	23.5770	1.09809	0.549046	+	0.835792i	$0.314991\pi$
$462$	−3.78362	−0.176030
$463$	10.9563	0.509181	0.254590	−	0.967049i	$-0.418059\pi$
$463$	10.9563	0.509181	0.254590	+	0.967049i	$0.418059\pi$
$464$	5.17983	0.240468
$465$	−1.46026	−0.0677180
$466$	−20.0353	−0.928118
$467$	5.22685	0.241870	0.120935	−	0.992660i	$-0.461411\pi$
$467$	5.22685	0.241870	0.120935	+	0.992660i	$0.461411\pi$
$468$	−4.72535	−0.218429
$469$	7.47271	0.345058
$470$	3.31717	0.153010
$471$	−16.0874	−0.741268
$472$	5.45743	0.251199
$473$	−26.2456	−1.20677
$474$	11.7890	0.541486
$475$	−7.59414	−0.348443
$476$	−3.78096	−0.173300
$477$	−3.59500	−0.164604
$478$	−22.8767	−1.04635
$479$	−9.94457	−0.454379	−0.227190	−	0.973851i	$-0.572954\pi$
$479$	−9.94457	−0.454379	−0.227190	+	0.973851i	$0.572954\pi$
$480$	−0.491351	−0.0224270
$481$	0.224544	0.0102383
$482$	28.2471	1.28662
$483$	2.70694	0.123170
$484$	3.31577	0.150717
$485$	−8.90818	−0.404500
$486$	−1.00000	−0.0453609
$487$	22.5868	1.02351	0.511754	−	0.859132i	$-0.328996\pi$
$487$	22.5868	1.02351	0.511754	+	0.859132i	$0.328996\pi$
$488$	−10.8934	−0.493120
$489$	8.80865	0.398341
$490$	0.491351	0.0221970
$491$	−8.26919	−0.373184	−0.186592	−	0.982438i	$-0.559744\pi$
$491$	−8.26919	−0.373184	−0.186592	+	0.982438i	$0.559744\pi$
$492$	−0.784995	−0.0353903
$493$	19.5847	0.882053
$494$	−7.54111	−0.339291
$495$	−1.85908	−0.0835596
$496$	2.97193	0.133444
$497$	10.7881	0.483913
$498$	4.32320	0.193727
$499$	4.37240	0.195736	0.0978678	−	0.995199i	$-0.468798\pi$
$499$	4.37240	0.195736	0.0978678	+	0.995199i	$0.468798\pi$
$500$	−4.79489	−0.214434
$501$	14.0118	0.626000
$502$	−12.2938	−0.548701
$503$	−35.7232	−1.59282	−0.796408	−	0.604759i	$-0.793269\pi$
$503$	−35.7232	−1.59282	−0.796408	+	0.604759i	$0.793269\pi$
$504$	−1.00000	−0.0445435
$505$	−4.52117	−0.201189
$506$	−10.2420	−0.455313
$507$	−9.32890	−0.414311
$508$	−16.4295	−0.728941
$509$	24.4260	1.08266	0.541331	−	0.840809i	$-0.317921\pi$
$509$	24.4260	1.08266	0.541331	+	0.840809i	$0.317921\pi$
$510$	−1.85778	−0.0822639
$511$	−1.85043	−0.0818581
$512$	1.00000	0.0441942
$513$	−1.59589	−0.0704600
$514$	22.4212	0.988958
$515$	2.86745	0.126355
$516$	−6.93665	−0.305369
$517$	−25.5437	−1.12341
$518$	0.0475190	0.00208787
$519$	−18.7414	−0.822656
$520$	−2.32180	−0.101818
$521$	−22.0181	−0.964629	−0.482315	−	0.875998i	$-0.660204\pi$
$521$	−22.0181	−0.964629	−0.482315	+	0.875998i	$0.660204\pi$
$522$	5.17983	0.226715
$523$	−33.9030	−1.48247	−0.741237	−	0.671244i	$-0.765760\pi$
$523$	−33.9030	−1.48247	−0.741237	+	0.671244i	$0.765760\pi$
$524$	−17.3837	−0.759411
$525$	−4.75857	−0.207681
$526$	−0.731020	−0.0318740
$527$	11.2368	0.489481
$528$	3.78362	0.164661
$529$	−15.6725	−0.681412
$530$	−1.76641	−0.0767279
$531$	5.45743	0.236832
$532$	−1.59589	−0.0691904
$533$	−3.70937	−0.160671
$534$	11.0249	0.477094
$535$	1.36935	0.0592023
$536$	−7.47271	−0.322772
$537$	−13.3221	−0.574892
$538$	9.63429	0.415364
$539$	−3.78362	−0.162972
$540$	−0.491351	−0.0211444
$541$	13.7620	0.591673	0.295837	−	0.955239i	$-0.404402\pi$
$541$	13.7620	0.591673	0.295837	+	0.955239i	$0.404402\pi$
$542$	5.67323	0.243686
$543$	−25.6112	−1.09908
$544$	3.78096	0.162107
$545$	0.547686	0.0234603
$546$	−4.72535	−0.202226
$547$	−13.7811	−0.589237	−0.294619	−	0.955615i	$-0.595193\pi$
$547$	−13.7811	−0.589237	−0.294619	+	0.955615i	$0.595193\pi$
$548$	−8.95090	−0.382364
$549$	−10.8934	−0.464918
$550$	18.0046	0.767720
$551$	8.26641	0.352161
$552$	−2.70694	−0.115215
$553$	11.7890	0.501319
$554$	10.8082	0.459198
$555$	0.0233485	0.000991089 0
$556$	−19.0559	−0.808150
$557$	−19.7617	−0.837330	−0.418665	−	0.908141i	$-0.637502\pi$
$557$	−19.7617	−0.837330	−0.418665	+	0.908141i	$0.637502\pi$
$558$	2.97193	0.125812
$559$	−32.7781	−1.38636
$560$	−0.491351	−0.0207634
$561$	14.3057	0.603988
$562$	−7.41685	−0.312861
$563$	1.49446	0.0629839	0.0314920	−	0.999504i	$-0.489974\pi$
$563$	1.49446	0.0629839	0.0314920	+	0.999504i	$0.489974\pi$
$564$	−6.75113	−0.284274
$565$	4.49524	0.189116
$566$	28.1875	1.18481
$567$	−1.00000	−0.0419961
$568$	−10.7881	−0.452660
$569$	14.8244	0.621471	0.310736	−	0.950496i	$-0.399425\pi$
$569$	14.8244	0.621471	0.310736	+	0.950496i	$0.399425\pi$
$570$	−0.784140	−0.0328440
$571$	40.4849	1.69424	0.847121	−	0.531401i	$-0.178334\pi$
$571$	40.4849	1.69424	0.847121	+	0.531401i	$0.178334\pi$
$572$	17.8789	0.747555
$573$	−1.00000	−0.0417756
$574$	−0.784995	−0.0327651
$575$	−12.8812	−0.537182
$576$	1.00000	0.0416667
$577$	−19.4545	−0.809902	−0.404951	−	0.914338i	$-0.632711\pi$
$577$	−19.4545	−0.809902	−0.404951	+	0.914338i	$0.632711\pi$
$578$	−2.70431	−0.112484
$579$	7.10649	0.295336
$580$	2.54511	0.105680
$581$	4.32320	0.179357
$582$	18.1300	0.751512
$583$	13.6021	0.563342
$584$	1.85043	0.0765713
$585$	−2.32180	−0.0959948
$586$	−21.7190	−0.897203
$587$	22.3170	0.921120	0.460560	−	0.887629i	$-0.347649\pi$
$587$	22.3170	0.921120	0.460560	+	0.887629i	$0.347649\pi$
$588$	−1.00000	−0.0412393
$589$	4.74286	0.195426
$590$	2.68152	0.110396
$591$	−7.97056	−0.327865
$592$	−0.0475190	−0.00195302
$593$	21.5811	0.886229	0.443115	−	0.896465i	$-0.353873\pi$
$593$	21.5811	0.886229	0.443115	+	0.896465i	$0.353873\pi$
$594$	3.78362	0.155244
$595$	−1.85778	−0.0761616
$596$	−10.9154	−0.447113
$597$	7.43367	0.304240
$598$	−12.7912	−0.523072
$599$	−7.47232	−0.305311	−0.152655	−	0.988279i	$-0.548782\pi$
$599$	−7.47232	−0.305311	−0.152655	+	0.988279i	$0.548782\pi$
$600$	4.75857	0.194268
$601$	−15.2144	−0.620610	−0.310305	−	0.950637i	$-0.600431\pi$
$601$	−15.2144	−0.620610	−0.310305	+	0.950637i	$0.600431\pi$
$602$	−6.93665	−0.282717
$603$	−7.47271	−0.304312
$604$	−19.9340	−0.811101
$605$	1.62921	0.0662366
$606$	9.20150	0.373785
$607$	−44.1611	−1.79245	−0.896223	−	0.443605i	$-0.853699\pi$
$607$	−44.1611	−1.79245	−0.896223	+	0.443605i	$0.853699\pi$
$608$	1.59589	0.0647217
$609$	5.17983	0.209897
$610$	−5.35248	−0.216715
$611$	−31.9014	−1.29059
$612$	3.78096	0.152836
$613$	1.41714	0.0572379	0.0286190	−	0.999590i	$-0.490889\pi$
$613$	1.41714	0.0572379	0.0286190	+	0.999590i	$0.490889\pi$
$614$	−22.0441	−0.889628
$615$	−0.385708	−0.0155533
$616$	3.78362	0.152446
$617$	−6.75094	−0.271783	−0.135891	−	0.990724i	$-0.543390\pi$
$617$	−6.75094	−0.271783	−0.135891	+	0.990724i	$0.543390\pi$
$618$	−5.83586	−0.234753
$619$	−17.5407	−0.705019	−0.352510	−	0.935808i	$-0.614672\pi$
$619$	−17.5407	−0.705019	−0.352510	+	0.935808i	$0.614672\pi$
$620$	1.46026	0.0586455
$621$	−2.70694	−0.108626
$622$	−18.2434	−0.731494
$623$	11.0249	0.441703
$624$	4.72535	0.189165
$625$	21.4369	0.857476
$626$	4.88185	0.195118
$627$	6.03822	0.241143
$628$	16.0874	0.641957
$629$	−0.179668	−0.00716382
$630$	−0.491351	−0.0195759
$631$	40.7939	1.62398	0.811990	−	0.583672i	$-0.198385\pi$
$631$	40.7939	1.62398	0.811990	+	0.583672i	$0.198385\pi$
$632$	−11.7890	−0.468941
$633$	−1.71470	−0.0681531
$634$	−26.1740	−1.03950
$635$	−8.07265	−0.320353
$636$	3.59500	0.142551
$637$	−4.72535	−0.187225
$638$	−19.5985	−0.775912
$639$	−10.7881	−0.426771
$640$	0.491351	0.0194224
$641$	−25.1854	−0.994764	−0.497382	−	0.867532i	$-0.665705\pi$
$641$	−25.1854	−0.994764	−0.497382	+	0.867532i	$0.665705\pi$
$642$	−2.78692	−0.109991
$643$	11.0560	0.436004	0.218002	−	0.975948i	$-0.430046\pi$
$643$	11.0560	0.436004	0.218002	+	0.975948i	$0.430046\pi$
$644$	−2.70694	−0.106668
$645$	−3.40833	−0.134203
$646$	6.03398	0.237404
$647$	−45.7621	−1.79910	−0.899548	−	0.436823i	$-0.856104\pi$
$647$	−45.7621	−1.79910	−0.899548	+	0.436823i	$0.856104\pi$
$648$	1.00000	0.0392837
$649$	−20.6488	−0.810538
$650$	22.4859	0.881970
$651$	2.97193	0.116479
$652$	−8.80865	−0.344973
$653$	−1.93681	−0.0757934	−0.0378967	−	0.999282i	$-0.512066\pi$
$653$	−1.93681	−0.0757934	−0.0378967	+	0.999282i	$0.512066\pi$
$654$	−1.11465	−0.0435864
$655$	−8.54151	−0.333744
$656$	0.784995	0.0306489
$657$	1.85043	0.0721921
$658$	−6.75113	−0.263186
$659$	−38.6537	−1.50574	−0.752868	−	0.658172i	$-0.771330\pi$
$659$	−38.6537	−1.50574	−0.752868	+	0.658172i	$0.771330\pi$
$660$	1.85908	0.0723648
$661$	20.3207	0.790384	0.395192	−	0.918598i	$-0.370678\pi$
$661$	20.3207	0.790384	0.395192	+	0.918598i	$0.370678\pi$
$662$	−2.79751	−0.108729
$663$	17.8664	0.693872
$664$	−4.32320	−0.167773
$665$	−0.784140	−0.0304076
$666$	−0.0475190	−0.00184132
$667$	14.0215	0.542914
$668$	−14.0118	−0.542132
$669$	19.2003	0.742328
$670$	−3.67172	−0.141851
$671$	41.2164	1.59114
$672$	1.00000	0.0385758
$673$	−31.7973	−1.22570	−0.612849	−	0.790200i	$-0.709977\pi$
$673$	−31.7973	−1.22570	−0.612849	+	0.790200i	$0.709977\pi$
$674$	−25.7887	−0.993345
$675$	4.75857	0.183158
$676$	9.32890	0.358804
$677$	−32.2512	−1.23952	−0.619758	−	0.784793i	$-0.712769\pi$
$677$	−32.2512	−1.23952	−0.619758	+	0.784793i	$0.712769\pi$
$678$	−9.14874	−0.351355
$679$	18.1300	0.695765
$680$	1.85778	0.0712426
$681$	1.36716	0.0523897
$682$	−11.2447	−0.430580
$683$	13.8593	0.530310	0.265155	−	0.964206i	$-0.414577\pi$
$683$	13.8593	0.530310	0.265155	+	0.964206i	$0.414577\pi$
$684$	1.59589	0.0610202
$685$	−4.39803	−0.168040
$686$	−1.00000	−0.0381802
$687$	−15.5407	−0.592916
$688$	6.93665	0.264457
$689$	16.9876	0.647177
$690$	−1.33006	−0.0506344
$691$	7.54282	0.286943	0.143471	−	0.989654i	$-0.454174\pi$
$691$	7.54282	0.286943	0.143471	+	0.989654i	$0.454174\pi$
$692$	18.7414	0.712441
$693$	3.78362	0.143728
$694$	−4.38898	−0.166603
$695$	−9.36313	−0.355164
$696$	−5.17983	−0.196341
$697$	2.96804	0.112423
$698$	27.1516	1.02770
$699$	20.0353	0.757805
$700$	4.75857	0.179857
$701$	−29.1388	−1.10056	−0.550279	−	0.834981i	$-0.685479\pi$
$701$	−29.1388	−1.10056	−0.550279	+	0.834981i	$0.685479\pi$
$702$	4.72535	0.178347
$703$	−0.0758349	−0.00286017
$704$	−3.78362	−0.142600
$705$	−3.31717	−0.124932
$706$	−0.708243	−0.0266551
$707$	9.20150	0.346058
$708$	−5.45743	−0.205103
$709$	−43.4950	−1.63349	−0.816744	−	0.577000i	$-0.804223\pi$
$709$	−43.4950	−1.63349	−0.816744	+	0.577000i	$0.804223\pi$
$710$	−5.30075	−0.198934
$711$	−11.7890	−0.442122
$712$	−11.0249	−0.413175
$713$	8.04483	0.301281
$714$	3.78096	0.141499
$715$	8.78482	0.328534
$716$	13.3221	0.497871
$717$	22.8767	0.854345
$718$	−12.5222	−0.467323
$719$	−24.1431	−0.900385	−0.450193	−	0.892932i	$-0.648645\pi$
$719$	−24.1431	−0.900385	−0.450193	+	0.892932i	$0.648645\pi$
$720$	0.491351	0.0183116
$721$	−5.83586	−0.217339
$722$	−16.4532	−0.612323
$723$	−28.2471	−1.05052
$724$	25.6112	0.951834
$725$	−24.6486	−0.915426
$726$	−3.31577	−0.123060
$727$	2.86576	0.106285	0.0531426	−	0.998587i	$-0.483076\pi$
$727$	2.86576	0.106285	0.0531426	+	0.998587i	$0.483076\pi$
$728$	4.72535	0.175133
$729$	1.00000	0.0370370
$730$	0.909210	0.0336514
$731$	26.2272	0.970048
$732$	10.8934	0.402631
$733$	28.3601	1.04750	0.523752	−	0.851871i	$-0.324532\pi$
$733$	28.3601	1.04750	0.523752	+	0.851871i	$0.324532\pi$
$734$	0.654025	0.0241405
$735$	−0.491351	−0.0181238
$736$	2.70694	0.0997791
$737$	28.2739	1.04148
$738$	0.784995	0.0288961
$739$	−35.0380	−1.28889	−0.644446	−	0.764650i	$-0.722912\pi$
$739$	−35.0380	−1.28889	−0.644446	+	0.764650i	$0.722912\pi$
$740$	−0.0233485	−0.000858308 0
$741$	7.54111	0.277030
$742$	3.59500	0.131977
$743$	−12.8484	−0.471361	−0.235680	−	0.971831i	$-0.575732\pi$
$743$	−12.8484	−0.471361	−0.235680	+	0.971831i	$0.575732\pi$
$744$	−2.97193	−0.108956
$745$	−5.36330	−0.196496
$746$	−12.7907	−0.468300
$747$	−4.32320	−0.158178
$748$	−14.3057	−0.523069
$749$	−2.78692	−0.101832
$750$	4.79489	0.175084
$751$	−23.9140	−0.872635	−0.436317	−	0.899793i	$-0.643717\pi$
$751$	−23.9140	−0.872635	−0.436317	+	0.899793i	$0.643717\pi$
$752$	6.75113	0.246188
$753$	12.2938	0.448013
$754$	−24.4765	−0.891381
$755$	−9.79457	−0.356461
$756$	1.00000	0.0363696
$757$	−50.3052	−1.82838	−0.914188	−	0.405291i	$-0.867170\pi$
$757$	−50.3052	−1.82838	−0.914188	+	0.405291i	$0.867170\pi$
$758$	−27.8720	−1.01236
$759$	10.2420	0.371762
$760$	0.784140	0.0284437
$761$	−11.5733	−0.419532	−0.209766	−	0.977752i	$-0.567270\pi$
$761$	−11.5733	−0.419532	−0.209766	+	0.977752i	$0.567270\pi$
$762$	16.4295	0.595178
$763$	−1.11465	−0.0403532
$764$	1.00000	0.0361787
$765$	1.85778	0.0671682
$766$	−8.31758	−0.300526
$767$	−25.7883	−0.931160
$768$	−1.00000	−0.0360844
$769$	18.8372	0.679286	0.339643	−	0.940554i	$-0.389694\pi$
$769$	18.8372	0.679286	0.339643	+	0.940554i	$0.389694\pi$
$770$	1.85908	0.0669968
$771$	−22.4212	−0.807481
$772$	−7.10649	−0.255768
$773$	−25.6562	−0.922789	−0.461394	−	0.887195i	$-0.652651\pi$
$773$	−25.6562	−0.922789	−0.461394	+	0.887195i	$0.652651\pi$
$774$	6.93665	0.249333
$775$	−14.1422	−0.508001
$776$	−18.1300	−0.650828
$777$	−0.0475190	−0.00170473
$778$	9.91260	0.355384
$779$	1.25276	0.0448849
$780$	2.32180	0.0831339
$781$	40.8181	1.46059
$782$	10.2348	0.365997
$783$	−5.17983	−0.185112
$784$	1.00000	0.0357143
$785$	7.90455	0.282126
$786$	17.3837	0.620057
$787$	28.6566	1.02150	0.510748	−	0.859730i	$-0.329369\pi$
$787$	28.6566	1.02150	0.510748	+	0.859730i	$0.329369\pi$
$788$	7.97056	0.283939
$789$	0.731020	0.0260250
$790$	−5.79253	−0.206089
$791$	−9.14874	−0.325292
$792$	−3.78362	−0.134445
$793$	51.4750	1.82793
$794$	24.1592	0.857377
$795$	1.76641	0.0626481
$796$	−7.43367	−0.263479
$797$	39.8070	1.41003	0.705017	−	0.709190i	$-0.250939\pi$
$797$	39.8070	1.41003	0.705017	+	0.709190i	$0.250939\pi$
$798$	1.59589	0.0564937
$799$	25.5258	0.903037
$800$	−4.75857	−0.168241
$801$	−11.0249	−0.389545
$802$	32.3763	1.14325
$803$	−7.00131	−0.247071
$804$	7.47271	0.263542
$805$	−1.33006	−0.0468783
$806$	−14.0434	−0.494658
$807$	−9.63429	−0.339143
$808$	−9.20150	−0.323708
$809$	−29.9481	−1.05292	−0.526460	−	0.850200i	$-0.676481\pi$
$809$	−29.9481	−1.05292	−0.526460	+	0.850200i	$0.676481\pi$
$810$	0.491351	0.0172643
$811$	40.7863	1.43220	0.716101	−	0.697997i	$-0.245925\pi$
$811$	40.7863	1.43220	0.716101	+	0.697997i	$0.245925\pi$
$812$	−5.17983	−0.181776
$813$	−5.67323	−0.198969
$814$	0.179794	0.00630177
$815$	−4.32814	−0.151608
$816$	−3.78096	−0.132360
$817$	11.0701	0.387293
$818$	25.5867	0.894619
$819$	4.72535	0.165117
$820$	0.385708	0.0134695
$821$	−32.5990	−1.13771	−0.568856	−	0.822437i	$-0.692614\pi$
$821$	−32.5990	−1.13771	−0.568856	+	0.822437i	$0.692614\pi$
$822$	8.95090	0.312199
$823$	5.59012	0.194859	0.0974297	−	0.995242i	$-0.468938\pi$
$823$	5.59012	0.194859	0.0974297	+	0.995242i	$0.468938\pi$
$824$	5.83586	0.203302
$825$	−18.0046	−0.626841
$826$	−5.45743	−0.189888
$827$	7.70843	0.268048	0.134024	−	0.990978i	$-0.457210\pi$
$827$	7.70843	0.268048	0.134024	+	0.990978i	$0.457210\pi$
$828$	2.70694	0.0940726
$829$	51.6498	1.79387	0.896935	−	0.442162i	$-0.145788\pi$
$829$	51.6498	1.79387	0.896935	+	0.442162i	$0.145788\pi$
$830$	−2.12421	−0.0737323
$831$	−10.8082	−0.374934
$832$	−4.72535	−0.163822
$833$	3.78096	0.131003
$834$	19.0559	0.659851
$835$	−6.88470	−0.238255
$836$	−6.03822	−0.208836
$837$	−2.97193	−0.102725
$838$	33.9919	1.17423
$839$	46.4278	1.60287	0.801433	−	0.598085i	$-0.204071\pi$
$839$	46.4278	1.60287	0.801433	+	0.598085i	$0.204071\pi$
$840$	0.491351	0.0169532
$841$	−2.16937	−0.0748060
$842$	20.9468	0.721874
$843$	7.41685	0.255450
$844$	1.71470	0.0590223
$845$	4.58376	0.157686
$846$	6.75113	0.232109
$847$	−3.31577	−0.113931
$848$	−3.59500	−0.123453
$849$	−28.1875	−0.967393
$850$	−17.9920	−0.617120
$851$	−0.128631	−0.00440941
$852$	10.7881	0.369595
$853$	14.3483	0.491276	0.245638	−	0.969362i	$-0.421003\pi$
$853$	14.3483	0.491276	0.245638	+	0.969362i	$0.421003\pi$
$854$	10.8934	0.372764
$855$	0.784140	0.0268170
$856$	2.78692	0.0952548
$857$	−25.3326	−0.865347	−0.432673	−	0.901551i	$-0.642430\pi$
$857$	−25.3326	−0.865347	−0.432673	+	0.901551i	$0.642430\pi$
$858$	−17.8789	−0.610376
$859$	35.3397	1.20578	0.602888	−	0.797826i	$-0.294017\pi$
$859$	35.3397	1.20578	0.602888	+	0.797826i	$0.294017\pi$
$860$	3.40833	0.116223
$861$	0.784995	0.0267526
$862$	34.6569	1.18042
$863$	−21.0581	−0.716827	−0.358413	−	0.933563i	$-0.616682\pi$
$863$	−21.0581	−0.716827	−0.358413	+	0.933563i	$0.616682\pi$
$864$	−1.00000	−0.0340207
$865$	9.20860	0.313102
$866$	11.2101	0.380935
$867$	2.70431	0.0918432
$868$	−2.97193	−0.100874
$869$	44.6050	1.51312
$870$	−2.54511	−0.0862874
$871$	35.3111	1.19647
$872$	1.11465	0.0377469
$873$	−18.1300	−0.613607
$874$	4.31996	0.146125
$875$	4.79489	0.162097
$876$	−1.85043	−0.0625202
$877$	8.63014	0.291419	0.145709	−	0.989327i	$-0.453454\pi$
$877$	8.63014	0.291419	0.145709	+	0.989327i	$0.453454\pi$
$878$	−20.7615	−0.700668
$879$	21.7190	0.732563
$880$	−1.85908	−0.0626697
$881$	56.8049	1.91381	0.956903	−	0.290409i	$-0.0937913\pi$
$881$	56.8049	1.91381	0.956903	+	0.290409i	$0.0937913\pi$
$882$	1.00000	0.0336718
$883$	5.12443	0.172451	0.0862254	−	0.996276i	$-0.472519\pi$
$883$	5.12443	0.172451	0.0862254	+	0.996276i	$0.472519\pi$
$884$	−17.8664	−0.600911
$885$	−2.68152	−0.0901382
$886$	−9.22301	−0.309853
$887$	−24.6712	−0.828378	−0.414189	−	0.910191i	$-0.635935\pi$
$887$	−24.6712	−0.828378	−0.414189	+	0.910191i	$0.635935\pi$
$888$	0.0475190	0.00159463
$889$	16.4295	0.551027
$890$	−5.41709	−0.181581
$891$	−3.78362	−0.126756
$892$	−19.2003	−0.642874
$893$	10.7740	0.360539
$894$	10.9154	0.365066
$895$	6.54584	0.218803
$896$	−1.00000	−0.0334077
$897$	12.7912	0.427086
$898$	27.9076	0.931290
$899$	15.3941	0.513422
$900$	−4.75857	−0.158619
$901$	−13.5926	−0.452834
$902$	−2.97012	−0.0988943
$903$	6.93665	0.230837
$904$	9.14874	0.304283
$905$	12.5841	0.418310
$906$	19.9340	0.662261
$907$	−43.6222	−1.44845	−0.724226	−	0.689562i	$-0.757803\pi$
$907$	−43.6222	−1.44845	−0.724226	+	0.689562i	$0.757803\pi$
$908$	−1.36716	−0.0453708
$909$	−9.20150	−0.305194
$910$	2.32180	0.0769670
$911$	11.0617	0.366490	0.183245	−	0.983067i	$-0.441340\pi$
$911$	11.0617	0.366490	0.183245	+	0.983067i	$0.441340\pi$
$912$	−1.59589	−0.0528450
$913$	16.3573	0.541349
$914$	1.13930	0.0376846
$915$	5.35248	0.176947
$916$	15.5407	0.513480
$917$	17.3837	0.574061
$918$	−3.78096	−0.124790
$919$	−40.2249	−1.32690	−0.663449	−	0.748222i	$-0.730908\pi$
$919$	−40.2249	−1.32690	−0.663449	+	0.748222i	$0.730908\pi$
$920$	1.33006	0.0438507
$921$	22.0441	0.726378
$922$	23.5770	0.776469
$923$	50.9776	1.67795
$924$	−3.78362	−0.124472
$925$	0.226123	0.00743487
$926$	10.9563	0.360045
$927$	5.83586	0.191675
$928$	5.17983	0.170036
$929$	−49.6113	−1.62769	−0.813847	−	0.581079i	$-0.802631\pi$
$929$	−49.6113	−1.62769	−0.813847	+	0.581079i	$0.802631\pi$
$930$	−1.46026	−0.0478839
$931$	1.59589	0.0523030
$932$	−20.0353	−0.656278
$933$	18.2434	0.597262
$934$	5.22685	0.171028
$935$	−7.02913	−0.229877
$936$	−4.72535	−0.154453
$937$	14.1063	0.460831	0.230416	−	0.973092i	$-0.425991\pi$
$937$	14.1063	0.460831	0.230416	+	0.973092i	$0.425991\pi$
$938$	7.47271	0.243993
$939$	−4.88185	−0.159313
$940$	3.31717	0.108194
$941$	−29.9600	−0.976667	−0.488333	−	0.872657i	$-0.662395\pi$
$941$	−29.9600	−0.976667	−0.488333	+	0.872657i	$0.662395\pi$
$942$	−16.0874	−0.524155
$943$	2.12493	0.0691974
$944$	5.45743	0.177624
$945$	0.491351	0.0159836
$946$	−26.2456	−0.853319
$947$	10.2254	0.332280	0.166140	−	0.986102i	$-0.446870\pi$
$947$	10.2254	0.332280	0.166140	+	0.986102i	$0.446870\pi$
$948$	11.7890	0.382889
$949$	−8.74391	−0.283839
$950$	−7.59414	−0.246386
$951$	26.1740	0.848749
$952$	−3.78096	−0.122542
$953$	42.1640	1.36583	0.682913	−	0.730500i	$-0.260713\pi$
$953$	42.1640	1.36583	0.682913	+	0.730500i	$0.260713\pi$
$954$	−3.59500	−0.116393
$955$	0.491351	0.0158997
$956$	−22.8767	−0.739884
$957$	19.5985	0.633529
$958$	−9.94457	−0.321295
$959$	8.95090	0.289040
$960$	−0.491351	−0.0158583
$961$	−22.1676	−0.715085
$962$	0.224544	0.00723958
$963$	2.78692	0.0898071
$964$	28.2471	0.909778
$965$	−3.49178	−0.112404
$966$	2.70694	0.0870943
$967$	−45.2889	−1.45639	−0.728196	−	0.685369i	$-0.759641\pi$
$967$	−45.2889	−1.45639	−0.728196	+	0.685369i	$0.759641\pi$
$968$	3.31577	0.106573
$969$	−6.03398	−0.193839
$970$	−8.90818	−0.286024
$971$	−22.7241	−0.729252	−0.364626	−	0.931154i	$-0.618803\pi$
$971$	−22.7241	−0.729252	−0.364626	+	0.931154i	$0.618803\pi$
$972$	−1.00000	−0.0320750
$973$	19.0559	0.610904
$974$	22.5868	0.723729
$975$	−22.4859	−0.720125
$976$	−10.8934	−0.348689
$977$	−29.8001	−0.953391	−0.476696	−	0.879068i	$-0.658166\pi$
$977$	−29.8001	−0.953391	−0.476696	+	0.879068i	$0.658166\pi$
$978$	8.80865	0.281670
$979$	41.7140	1.33318
$980$	0.491351	0.0156956
$981$	1.11465	0.0355882
$982$	−8.26919	−0.263881
$983$	−18.0395	−0.575370	−0.287685	−	0.957725i	$-0.592886\pi$
$983$	−18.0395	−0.575370	−0.287685	+	0.957725i	$0.592886\pi$
$984$	−0.784995	−0.0250247
$985$	3.91634	0.124785
$986$	19.5847	0.623705
$987$	6.75113	0.214891
$988$	−7.54111	−0.239915
$989$	18.7771	0.597076
$990$	−1.85908	−0.0590856
$991$	18.4111	0.584847	0.292424	−	0.956289i	$-0.405538\pi$
$991$	18.4111	0.584847	0.292424	+	0.956289i	$0.405538\pi$
$992$	2.97193	0.0943589
$993$	2.79751	0.0887764
$994$	10.7881	0.342178
$995$	−3.65254	−0.115793
$996$	4.32320	0.136986
$997$	18.2214	0.577079	0.288540	−	0.957468i	$-0.406830\pi$
$997$	18.2214	0.577079	0.288540	+	0.957468i	$0.406830\pi$
$998$	4.37240	0.138406
$999$	0.0475190	0.00150343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8022.2.a.q.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8022.2.a.q.1.5	✓	9	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:	\( N \)	\(=\)	\( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.491351 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.491351 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.491351 q^{10} -3.78362 q^{11} -1.00000 q^{12} -4.72535 q^{13} -1.00000 q^{14} -0.491351 q^{15} +1.00000 q^{16} +3.78096 q^{17} +1.00000 q^{18} +1.59589 q^{19} +0.491351 q^{20} +1.00000 q^{21} -3.78362 q^{22} +2.70694 q^{23} -1.00000 q^{24} -4.75857 q^{25} -4.72535 q^{26} -1.00000 q^{27} -1.00000 q^{28} +5.17983 q^{29} -0.491351 q^{30} +2.97193 q^{31} +1.00000 q^{32} +3.78362 q^{33} +3.78096 q^{34} -0.491351 q^{35} +1.00000 q^{36} -0.0475190 q^{37} +1.59589 q^{38} +4.72535 q^{39} +0.491351 q^{40} +0.784995 q^{41} +1.00000 q^{42} +6.93665 q^{43} -3.78362 q^{44} +0.491351 q^{45} +2.70694 q^{46} +6.75113 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.75857 q^{50} -3.78096 q^{51} -4.72535 q^{52} -3.59500 q^{53} -1.00000 q^{54} -1.85908 q^{55} -1.00000 q^{56} -1.59589 q^{57} +5.17983 q^{58} +5.45743 q^{59} -0.491351 q^{60} -10.8934 q^{61} +2.97193 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.32180 q^{65} +3.78362 q^{66} -7.47271 q^{67} +3.78096 q^{68} -2.70694 q^{69} -0.491351 q^{70} -10.7881 q^{71} +1.00000 q^{72} +1.85043 q^{73} -0.0475190 q^{74} +4.75857 q^{75} +1.59589 q^{76} +3.78362 q^{77} +4.72535 q^{78} -11.7890 q^{79} +0.491351 q^{80} +1.00000 q^{81} +0.784995 q^{82} -4.32320 q^{83} +1.00000 q^{84} +1.85778 q^{85} +6.93665 q^{86} -5.17983 q^{87} -3.78362 q^{88} -11.0249 q^{89} +0.491351 q^{90} +4.72535 q^{91} +2.70694 q^{92} -2.97193 q^{93} +6.75113 q^{94} +0.784140 q^{95} -1.00000 q^{96} -18.1300 q^{97} +1.00000 q^{98} -3.78362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 + 6 * q^5 - 9 * q^6 - 9 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9} + 6 q^{10} - 7 q^{11} - 9 q^{12} - q^{13} - 9 q^{14} - 6 q^{15} + 9 q^{16} - 10 q^{17} + 9 q^{18} + 6 q^{20} + 9 q^{21} - 7 q^{22} - 19 q^{23} - 9 q^{24} + 5 q^{25} - q^{26} - 9 q^{27} - 9 q^{28} - 21 q^{29} - 6 q^{30} + 6 q^{31} + 9 q^{32} + 7 q^{33} - 10 q^{34} - 6 q^{35} + 9 q^{36} - 14 q^{37} + q^{39} + 6 q^{40} + 12 q^{41} + 9 q^{42} - 17 q^{43} - 7 q^{44} + 6 q^{45} - 19 q^{46} - 11 q^{47} - 9 q^{48} + 9 q^{49} + 5 q^{50} + 10 q^{51} - q^{52} - 12 q^{53} - 9 q^{54} - 21 q^{55} - 9 q^{56} - 21 q^{58} - 8 q^{59} - 6 q^{60} + q^{61} + 6 q^{62} - 9 q^{63} + 9 q^{64} - 8 q^{65} + 7 q^{66} - 31 q^{67} - 10 q^{68} + 19 q^{69} - 6 q^{70} - 41 q^{71} + 9 q^{72} - q^{73} - 14 q^{74} - 5 q^{75} + 7 q^{77} + q^{78} - 17 q^{79} + 6 q^{80} + 9 q^{81} + 12 q^{82} - 36 q^{83} + 9 q^{84} - 30 q^{85} - 17 q^{86} + 21 q^{87} - 7 q^{88} - 17 q^{89} + 6 q^{90} + q^{91} - 19 q^{92} - 6 q^{93} - 11 q^{94} - 50 q^{95} - 9 q^{96} - 14 q^{97} + 9 q^{98} - 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 + 6 * q^5 - 9 * q^6 - 9 * q^7 + 9 * q^8 + 9 * q^9 + 6 * q^10 - 7 * q^11 - 9 * q^12 - q^13 - 9 * q^14 - 6 * q^15 + 9 * q^16 - 10 * q^17 + 9 * q^18 + 6 * q^20 + 9 * q^21 - 7 * q^22 - 19 * q^23 - 9 * q^24 + 5 * q^25 - q^26 - 9 * q^27 - 9 * q^28 - 21 * q^29 - 6 * q^30 + 6 * q^31 + 9 * q^32 + 7 * q^33 - 10 * q^34 - 6 * q^35 + 9 * q^36 - 14 * q^37 + q^39 + 6 * q^40 + 12 * q^41 + 9 * q^42 - 17 * q^43 - 7 * q^44 + 6 * q^45 - 19 * q^46 - 11 * q^47 - 9 * q^48 + 9 * q^49 + 5 * q^50 + 10 * q^51 - q^52 - 12 * q^53 - 9 * q^54 - 21 * q^55 - 9 * q^56 - 21 * q^58 - 8 * q^59 - 6 * q^60 + q^61 + 6 * q^62 - 9 * q^63 + 9 * q^64 - 8 * q^65 + 7 * q^66 - 31 * q^67 - 10 * q^68 + 19 * q^69 - 6 * q^70 - 41 * q^71 + 9 * q^72 - q^73 - 14 * q^74 - 5 * q^75 + 7 * q^77 + q^78 - 17 * q^79 + 6 * q^80 + 9 * q^81 + 12 * q^82 - 36 * q^83 + 9 * q^84 - 30 * q^85 - 17 * q^86 + 21 * q^87 - 7 * q^88 - 17 * q^89 + 6 * q^90 + q^91 - 19 * q^92 - 6 * q^93 - 11 * q^94 - 50 * q^95 - 9 * q^96 - 14 * q^97 + 9 * q^98 - 7 * q^99

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