LMFDB - Embedded newform 4002.2.a.bj.1.7

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} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 $ x^8 - x^7 - 26x^6 + 4x^5 + 209x^4 + 113x^3 - 436x^2 - 360x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.49512$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.49512 q^{5} -1.00000 q^{6} -1.05589 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.49512 q^{5} -1.00000 q^{6} -1.05589 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.49512 q^{10} +4.23999 q^{11} -1.00000 q^{12} +5.43785 q^{13} -1.05589 q^{14} -3.49512 q^{15} +1.00000 q^{16} -3.65785 q^{17} +1.00000 q^{18} -1.05589 q^{19} +3.49512 q^{20} +1.05589 q^{21} +4.23999 q^{22} +1.00000 q^{23} -1.00000 q^{24} +7.21586 q^{25} +5.43785 q^{26} -1.00000 q^{27} -1.05589 q^{28} +1.00000 q^{29} -3.49512 q^{30} -4.03347 q^{31} +1.00000 q^{32} -4.23999 q^{33} -3.65785 q^{34} -3.69045 q^{35} +1.00000 q^{36} +7.11494 q^{37} -1.05589 q^{38} -5.43785 q^{39} +3.49512 q^{40} +4.28297 q^{41} +1.05589 q^{42} +8.48558 q^{43} +4.23999 q^{44} +3.49512 q^{45} +1.00000 q^{46} -4.96374 q^{47} -1.00000 q^{48} -5.88511 q^{49} +7.21586 q^{50} +3.65785 q^{51} +5.43785 q^{52} +9.06888 q^{53} -1.00000 q^{54} +14.8193 q^{55} -1.05589 q^{56} +1.05589 q^{57} +1.00000 q^{58} -13.2516 q^{59} -3.49512 q^{60} -6.93145 q^{61} -4.03347 q^{62} -1.05589 q^{63} +1.00000 q^{64} +19.0060 q^{65} -4.23999 q^{66} -12.9486 q^{67} -3.65785 q^{68} -1.00000 q^{69} -3.69045 q^{70} +9.18374 q^{71} +1.00000 q^{72} -12.3054 q^{73} +7.11494 q^{74} -7.21586 q^{75} -1.05589 q^{76} -4.47695 q^{77} -5.43785 q^{78} +4.66957 q^{79} +3.49512 q^{80} +1.00000 q^{81} +4.28297 q^{82} -4.60197 q^{83} +1.05589 q^{84} -12.7846 q^{85} +8.48558 q^{86} -1.00000 q^{87} +4.23999 q^{88} -13.8513 q^{89} +3.49512 q^{90} -5.74175 q^{91} +1.00000 q^{92} +4.03347 q^{93} -4.96374 q^{94} -3.69045 q^{95} -1.00000 q^{96} +4.83622 q^{97} -5.88511 q^{98} +4.23999 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * 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
$5$	3.49512	1.56307	0.781533	−	0.623864i	$-0.214438\pi$
$5$	3.49512	1.56307	0.781533	+	0.623864i	$0.214438\pi$
$6$	−1.00000	−0.408248
$7$	−1.05589	−0.399087	−0.199544	−	0.979889i	$-0.563946\pi$
$7$	−1.05589	−0.399087	−0.199544	+	0.979889i	$0.563946\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.49512	1.10525
$11$	4.23999	1.27841	0.639203	−	0.769038i	$-0.279264\pi$
$11$	4.23999	1.27841	0.639203	+	0.769038i	$0.279264\pi$
$12$	−1.00000	−0.288675
$13$	5.43785	1.50819	0.754095	−	0.656766i	$-0.228076\pi$
$13$	5.43785	1.50819	0.754095	+	0.656766i	$0.228076\pi$
$14$	−1.05589	−0.282197
$15$	−3.49512	−0.902436
$16$	1.00000	0.250000
$17$	−3.65785	−0.887159	−0.443580	−	0.896235i	$-0.646292\pi$
$17$	−3.65785	−0.887159	−0.443580	+	0.896235i	$0.646292\pi$
$18$	1.00000	0.235702
$19$	−1.05589	−0.242237	−0.121118	−	0.992638i	$-0.538648\pi$
$19$	−1.05589	−0.242237	−0.121118	+	0.992638i	$0.538648\pi$
$20$	3.49512	0.781533
$21$	1.05589	0.230413
$22$	4.23999	0.903970
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	7.21586	1.44317
$26$	5.43785	1.06645
$27$	−1.00000	−0.192450
$28$	−1.05589	−0.199544
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.49512	−0.638119
$31$	−4.03347	−0.724432	−0.362216	−	0.932094i	$-0.617980\pi$
$31$	−4.03347	−0.724432	−0.362216	+	0.932094i	$0.617980\pi$
$32$	1.00000	0.176777
$33$	−4.23999	−0.738088
$34$	−3.65785	−0.627316
$35$	−3.69045	−0.623799
$36$	1.00000	0.166667
$37$	7.11494	1.16969	0.584844	−	0.811145i	$-0.301156\pi$
$37$	7.11494	1.16969	0.584844	+	0.811145i	$0.301156\pi$
$38$	−1.05589	−0.171287
$39$	−5.43785	−0.870754
$40$	3.49512	0.552627
$41$	4.28297	0.668887	0.334443	−	0.942416i	$-0.391452\pi$
$41$	4.28297	0.668887	0.334443	+	0.942416i	$0.391452\pi$
$42$	1.05589	0.162927
$43$	8.48558	1.29404	0.647019	−	0.762474i	$-0.276015\pi$
$43$	8.48558	1.29404	0.647019	+	0.762474i	$0.276015\pi$
$44$	4.23999	0.639203
$45$	3.49512	0.521022
$46$	1.00000	0.147442
$47$	−4.96374	−0.724036	−0.362018	−	0.932171i	$-0.617912\pi$
$47$	−4.96374	−0.724036	−0.362018	+	0.932171i	$0.617912\pi$
$48$	−1.00000	−0.144338
$49$	−5.88511	−0.840729
$50$	7.21586	1.02048
$51$	3.65785	0.512202
$52$	5.43785	0.754095
$53$	9.06888	1.24571	0.622853	−	0.782339i	$-0.285974\pi$
$53$	9.06888	1.24571	0.622853	+	0.782339i	$0.285974\pi$
$54$	−1.00000	−0.136083
$55$	14.8193	1.99823
$56$	−1.05589	−0.141099
$57$	1.05589	0.139855
$58$	1.00000	0.131306
$59$	−13.2516	−1.72521	−0.862606	−	0.505877i	$-0.831169\pi$
$59$	−13.2516	−1.72521	−0.862606	+	0.505877i	$0.831169\pi$
$60$	−3.49512	−0.451218
$61$	−6.93145	−0.887481	−0.443741	−	0.896155i	$-0.646349\pi$
$61$	−6.93145	−0.887481	−0.443741	+	0.896155i	$0.646349\pi$
$62$	−4.03347	−0.512251
$63$	−1.05589	−0.133029
$64$	1.00000	0.125000
$65$	19.0060	2.35740
$66$	−4.23999	−0.521907
$67$	−12.9486	−1.58192	−0.790959	−	0.611870i	$-0.790418\pi$
$67$	−12.9486	−1.58192	−0.790959	+	0.611870i	$0.790418\pi$
$68$	−3.65785	−0.443580
$69$	−1.00000	−0.120386
$70$	−3.69045	−0.441093
$71$	9.18374	1.08991	0.544955	−	0.838466i	$-0.316547\pi$
$71$	9.18374	1.08991	0.544955	+	0.838466i	$0.316547\pi$
$72$	1.00000	0.117851
$73$	−12.3054	−1.44024	−0.720119	−	0.693850i	$-0.755913\pi$
$73$	−12.3054	−1.44024	−0.720119	+	0.693850i	$0.755913\pi$
$74$	7.11494	0.827095
$75$	−7.21586	−0.833216
$76$	−1.05589	−0.121118
$77$	−4.47695	−0.510195
$78$	−5.43785	−0.615716
$79$	4.66957	0.525368	0.262684	−	0.964882i	$-0.415392\pi$
$79$	4.66957	0.525368	0.262684	+	0.964882i	$0.415392\pi$
$80$	3.49512	0.390766
$81$	1.00000	0.111111
$82$	4.28297	0.472974
$83$	−4.60197	−0.505131	−0.252566	−	0.967580i	$-0.581274\pi$
$83$	−4.60197	−0.505131	−0.252566	+	0.967580i	$0.581274\pi$
$84$	1.05589	0.115207
$85$	−12.7846	−1.38669
$86$	8.48558	0.915023
$87$	−1.00000	−0.107211
$88$	4.23999	0.451985
$89$	−13.8513	−1.46823	−0.734115	−	0.679025i	$-0.762403\pi$
$89$	−13.8513	−1.46823	−0.734115	+	0.679025i	$0.762403\pi$
$90$	3.49512	0.368418
$91$	−5.74175	−0.601899
$92$	1.00000	0.104257
$93$	4.03347	0.418251
$94$	−4.96374	−0.511971
$95$	−3.69045	−0.378632
$96$	−1.00000	−0.102062
$97$	4.83622	0.491044	0.245522	−	0.969391i	$-0.421041\pi$
$97$	4.83622	0.491044	0.245522	+	0.969391i	$0.421041\pi$
$98$	−5.88511	−0.594485
$99$	4.23999	0.426135
$100$	7.21586	0.721586
$101$	19.1308	1.90359	0.951793	−	0.306741i	$-0.0992385\pi$
$101$	19.1308	1.90359	0.951793	+	0.306741i	$0.0992385\pi$
$102$	3.65785	0.362181
$103$	0.827424	0.0815285	0.0407643	−	0.999169i	$-0.487021\pi$
$103$	0.827424	0.0815285	0.0407643	+	0.999169i	$0.487021\pi$
$104$	5.43785	0.533225
$105$	3.69045	0.360151
$106$	9.06888	0.880847
$107$	−14.2650	−1.37905	−0.689524	−	0.724262i	$-0.742180\pi$
$107$	−14.2650	−1.37905	−0.689524	+	0.724262i	$0.742180\pi$
$108$	−1.00000	−0.0962250
$109$	9.21892	0.883012	0.441506	−	0.897258i	$-0.354444\pi$
$109$	9.21892	0.883012	0.441506	+	0.897258i	$0.354444\pi$
$110$	14.8193	1.41296
$111$	−7.11494	−0.675320
$112$	−1.05589	−0.0997718
$113$	15.0393	1.41478	0.707388	−	0.706825i	$-0.249873\pi$
$113$	15.0393	1.41478	0.707388	+	0.706825i	$0.249873\pi$
$114$	1.05589	0.0988927
$115$	3.49512	0.325922
$116$	1.00000	0.0928477
$117$	5.43785	0.502730
$118$	−13.2516	−1.21991
$119$	3.86227	0.354054
$120$	−3.49512	−0.319059
$121$	6.97755	0.634322
$122$	−6.93145	−0.627544
$123$	−4.28297	−0.386182
$124$	−4.03347	−0.362216
$125$	7.74471	0.692708
$126$	−1.05589	−0.0940657
$127$	2.73857	0.243009	0.121504	−	0.992591i	$-0.461228\pi$
$127$	2.73857	0.243009	0.121504	+	0.992591i	$0.461228\pi$
$128$	1.00000	0.0883883
$129$	−8.48558	−0.747114
$130$	19.0060	1.66693
$131$	21.1991	1.85217	0.926085	−	0.377315i	$-0.123152\pi$
$131$	21.1991	1.85217	0.926085	+	0.377315i	$0.123152\pi$
$132$	−4.23999	−0.369044
$133$	1.11489	0.0966736
$134$	−12.9486	−1.11858
$135$	−3.49512	−0.300812
$136$	−3.65785	−0.313658
$137$	−5.58255	−0.476950	−0.238475	−	0.971149i	$-0.576647\pi$
$137$	−5.58255	−0.476950	−0.238475	+	0.971149i	$0.576647\pi$
$138$	−1.00000	−0.0851257
$139$	−20.7073	−1.75637	−0.878185	−	0.478321i	$-0.841245\pi$
$139$	−20.7073	−1.75637	−0.878185	+	0.478321i	$0.841245\pi$
$140$	−3.69045	−0.311900
$141$	4.96374	0.418022
$142$	9.18374	0.770682
$143$	23.0565	1.92808
$144$	1.00000	0.0833333
$145$	3.49512	0.290254
$146$	−12.3054	−1.01840
$147$	5.88511	0.485395
$148$	7.11494	0.584844
$149$	1.18480	0.0970629	0.0485315	−	0.998822i	$-0.484546\pi$
$149$	1.18480	0.0970629	0.0485315	+	0.998822i	$0.484546\pi$
$150$	−7.21586	−0.589173
$151$	−4.28192	−0.348458	−0.174229	−	0.984705i	$-0.555743\pi$
$151$	−4.28192	−0.348458	−0.174229	+	0.984705i	$0.555743\pi$
$152$	−1.05589	−0.0856436
$153$	−3.65785	−0.295720
$154$	−4.47695	−0.360763
$155$	−14.0974	−1.13233
$156$	−5.43785	−0.435377
$157$	12.8331	1.02420	0.512098	−	0.858927i	$-0.328868\pi$
$157$	12.8331	1.02420	0.512098	+	0.858927i	$0.328868\pi$
$158$	4.66957	0.371491
$159$	−9.06888	−0.719209
$160$	3.49512	0.276313
$161$	−1.05589	−0.0832154
$162$	1.00000	0.0785674
$163$	7.51658	0.588744	0.294372	−	0.955691i	$-0.404890\pi$
$163$	7.51658	0.588744	0.294372	+	0.955691i	$0.404890\pi$
$164$	4.28297	0.334443
$165$	−14.8193	−1.15368
$166$	−4.60197	−0.357182
$167$	−9.25414	−0.716107	−0.358054	−	0.933701i	$-0.616559\pi$
$167$	−9.25414	−0.716107	−0.358054	+	0.933701i	$0.616559\pi$
$168$	1.05589	0.0814633
$169$	16.5703	1.27464
$170$	−12.7846	−0.980536
$171$	−1.05589	−0.0807456
$172$	8.48558	0.647019
$173$	−9.51083	−0.723095	−0.361548	−	0.932354i	$-0.617751\pi$
$173$	−9.51083	−0.723095	−0.361548	+	0.932354i	$0.617751\pi$
$174$	−1.00000	−0.0758098
$175$	−7.61912	−0.575952
$176$	4.23999	0.319602
$177$	13.2516	0.996051
$178$	−13.8513	−1.03820
$179$	13.3185	0.995469	0.497734	−	0.867330i	$-0.334165\pi$
$179$	13.3185	0.995469	0.497734	+	0.867330i	$0.334165\pi$
$180$	3.49512	0.260511
$181$	−21.7224	−1.61461	−0.807306	−	0.590133i	$-0.799075\pi$
$181$	−21.7224	−1.61461	−0.807306	+	0.590133i	$0.799075\pi$
$182$	−5.74175	−0.425607
$183$	6.93145	0.512388
$184$	1.00000	0.0737210
$185$	24.8676	1.82830
$186$	4.03347	0.295748
$187$	−15.5093	−1.13415
$188$	−4.96374	−0.362018
$189$	1.05589	0.0768044
$190$	−3.69045	−0.267733
$191$	10.9414	0.791692	0.395846	−	0.918317i	$-0.370451\pi$
$191$	10.9414	0.791692	0.395846	+	0.918317i	$0.370451\pi$
$192$	−1.00000	−0.0721688
$193$	−3.93212	−0.283040	−0.141520	−	0.989935i	$-0.545199\pi$
$193$	−3.93212	−0.283040	−0.141520	+	0.989935i	$0.545199\pi$
$194$	4.83622	0.347220
$195$	−19.0060	−1.36104
$196$	−5.88511	−0.420365
$197$	−5.87495	−0.418573	−0.209286	−	0.977854i	$-0.567114\pi$
$197$	−5.87495	−0.418573	−0.209286	+	0.977854i	$0.567114\pi$
$198$	4.23999	0.301323
$199$	−5.40515	−0.383161	−0.191580	−	0.981477i	$-0.561361\pi$
$199$	−5.40515	−0.383161	−0.191580	+	0.981477i	$0.561361\pi$
$200$	7.21586	0.510239
$201$	12.9486	0.913320
$202$	19.1308	1.34604
$203$	−1.05589	−0.0741086
$204$	3.65785	0.256101
$205$	14.9695	1.04551
$206$	0.827424	0.0576494
$207$	1.00000	0.0695048
$208$	5.43785	0.377047
$209$	−4.47695	−0.309677
$210$	3.69045	0.254665
$211$	−2.39162	−0.164646	−0.0823230	−	0.996606i	$-0.526234\pi$
$211$	−2.39162	−0.164646	−0.0823230	+	0.996606i	$0.526234\pi$
$212$	9.06888	0.622853
$213$	−9.18374	−0.629259
$214$	−14.2650	−0.975135
$215$	29.6581	2.02267
$216$	−1.00000	−0.0680414
$217$	4.25888	0.289112
$218$	9.21892	0.624384
$219$	12.3054	0.831522
$220$	14.8193	0.999116
$221$	−19.8909	−1.33800
$222$	−7.11494	−0.477524
$223$	1.17667	0.0787957	0.0393979	−	0.999224i	$-0.487456\pi$
$223$	1.17667	0.0787957	0.0393979	+	0.999224i	$0.487456\pi$
$224$	−1.05589	−0.0705493
$225$	7.21586	0.481058
$226$	15.0393	1.00040
$227$	−5.93573	−0.393968	−0.196984	−	0.980407i	$-0.563115\pi$
$227$	−5.93573	−0.393968	−0.196984	+	0.980407i	$0.563115\pi$
$228$	1.05589	0.0699277
$229$	1.70563	0.112711	0.0563555	−	0.998411i	$-0.482052\pi$
$229$	1.70563	0.112711	0.0563555	+	0.998411i	$0.482052\pi$
$230$	3.49512	0.230461
$231$	4.47695	0.294562
$232$	1.00000	0.0656532
$233$	6.92781	0.453856	0.226928	−	0.973912i	$-0.427132\pi$
$233$	6.92781	0.453856	0.226928	+	0.973912i	$0.427132\pi$
$234$	5.43785	0.355484
$235$	−17.3489	−1.13172
$236$	−13.2516	−0.862606
$237$	−4.66957	−0.303321
$238$	3.86227	0.250354
$239$	10.6965	0.691899	0.345950	−	0.938253i	$-0.387557\pi$
$239$	10.6965	0.691899	0.345950	+	0.938253i	$0.387557\pi$
$240$	−3.49512	−0.225609
$241$	12.1453	0.782346	0.391173	−	0.920317i	$-0.372069\pi$
$241$	12.1453	0.782346	0.391173	+	0.920317i	$0.372069\pi$
$242$	6.97755	0.448534
$243$	−1.00000	−0.0641500
$244$	−6.93145	−0.443741
$245$	−20.5692	−1.31411
$246$	−4.28297	−0.273072
$247$	−5.74175	−0.365339
$248$	−4.03347	−0.256125
$249$	4.60197	0.291638
$250$	7.74471	0.489818
$251$	14.0329	0.885750	0.442875	−	0.896583i	$-0.353959\pi$
$251$	14.0329	0.885750	0.442875	+	0.896583i	$0.353959\pi$
$252$	−1.05589	−0.0665145
$253$	4.23999	0.266566
$254$	2.73857	0.171833
$255$	12.7846	0.800604
$256$	1.00000	0.0625000
$257$	−0.413438	−0.0257895	−0.0128948	−	0.999917i	$-0.504105\pi$
$257$	−0.413438	−0.0257895	−0.0128948	+	0.999917i	$0.504105\pi$
$258$	−8.48558	−0.528289
$259$	−7.51256	−0.466808
$260$	19.0060	1.17870
$261$	1.00000	0.0618984
$262$	21.1991	1.30968
$263$	12.9642	0.799406	0.399703	−	0.916645i	$-0.369113\pi$
$263$	12.9642	0.799406	0.399703	+	0.916645i	$0.369113\pi$
$264$	−4.23999	−0.260954
$265$	31.6968	1.94712
$266$	1.11489	0.0683585
$267$	13.8513	0.847683
$268$	−12.9486	−0.790959
$269$	16.6963	1.01799	0.508995	−	0.860770i	$-0.330017\pi$
$269$	16.6963	1.01799	0.508995	+	0.860770i	$0.330017\pi$
$270$	−3.49512	−0.212706
$271$	27.1117	1.64692	0.823460	−	0.567375i	$-0.192041\pi$
$271$	27.1117	1.64692	0.823460	+	0.567375i	$0.192041\pi$
$272$	−3.65785	−0.221790
$273$	5.74175	0.347507
$274$	−5.58255	−0.337254
$275$	30.5952	1.84496
$276$	−1.00000	−0.0601929
$277$	18.4920	1.11108	0.555540	−	0.831490i	$-0.312512\pi$
$277$	18.4920	1.11108	0.555540	+	0.831490i	$0.312512\pi$
$278$	−20.7073	−1.24194
$279$	−4.03347	−0.241477
$280$	−3.69045	−0.220546
$281$	−11.5054	−0.686355	−0.343178	−	0.939271i	$-0.611503\pi$
$281$	−11.5054	−0.686355	−0.343178	+	0.939271i	$0.611503\pi$
$282$	4.96374	0.295587
$283$	−21.2173	−1.26124	−0.630619	−	0.776092i	$-0.717199\pi$
$283$	−21.2173	−1.26124	−0.630619	+	0.776092i	$0.717199\pi$
$284$	9.18374	0.544955
$285$	3.69045	0.218603
$286$	23.0565	1.36336
$287$	−4.52232	−0.266944
$288$	1.00000	0.0589256
$289$	−3.62013	−0.212949
$290$	3.49512	0.205241
$291$	−4.83622	−0.283504
$292$	−12.3054	−0.720119
$293$	−8.00326	−0.467555	−0.233778	−	0.972290i	$-0.575109\pi$
$293$	−8.00326	−0.467555	−0.233778	+	0.972290i	$0.575109\pi$
$294$	5.88511	0.343226
$295$	−46.3159	−2.69662
$296$	7.11494	0.413547
$297$	−4.23999	−0.246029
$298$	1.18480	0.0686339
$299$	5.43785	0.314479
$300$	−7.21586	−0.416608
$301$	−8.95980	−0.516434
$302$	−4.28192	−0.246397
$303$	−19.1308	−1.09904
$304$	−1.05589	−0.0605592
$305$	−24.2262	−1.38719
$306$	−3.65785	−0.209105
$307$	−18.8924	−1.07824	−0.539122	−	0.842228i	$-0.681244\pi$
$307$	−18.8924	−1.07824	−0.539122	+	0.842228i	$0.681244\pi$
$308$	−4.47695	−0.255098
$309$	−0.827424	−0.0470705
$310$	−14.0974	−0.800681
$311$	−0.789316	−0.0447580	−0.0223790	−	0.999750i	$-0.507124\pi$
$311$	−0.789316	−0.0447580	−0.0223790	+	0.999750i	$0.507124\pi$
$312$	−5.43785	−0.307858
$313$	6.14077	0.347097	0.173548	−	0.984825i	$-0.444477\pi$
$313$	6.14077	0.347097	0.173548	+	0.984825i	$0.444477\pi$
$314$	12.8331	0.724216
$315$	−3.69045	−0.207933
$316$	4.66957	0.262684
$317$	−1.88403	−0.105818	−0.0529089	−	0.998599i	$-0.516849\pi$
$317$	−1.88403	−0.105818	−0.0529089	+	0.998599i	$0.516849\pi$
$318$	−9.06888	−0.508557
$319$	4.23999	0.237394
$320$	3.49512	0.195383
$321$	14.2650	0.796194
$322$	−1.05589	−0.0588422
$323$	3.86227	0.214903
$324$	1.00000	0.0555556
$325$	39.2388	2.17658
$326$	7.51658	0.416305
$327$	−9.21892	−0.509807
$328$	4.28297	0.236487
$329$	5.24114	0.288954
$330$	−14.8193	−0.815775
$331$	−1.53412	−0.0843227	−0.0421613	−	0.999111i	$-0.513424\pi$
$331$	−1.53412	−0.0843227	−0.0421613	+	0.999111i	$0.513424\pi$
$332$	−4.60197	−0.252566
$333$	7.11494	0.389896
$334$	−9.25414	−0.506364
$335$	−45.2567	−2.47264
$336$	1.05589	0.0576033
$337$	0.685212	0.0373259	0.0186629	−	0.999826i	$-0.494059\pi$
$337$	0.685212	0.0373259	0.0186629	+	0.999826i	$0.494059\pi$
$338$	16.5703	0.901303
$339$	−15.0393	−0.816822
$340$	−12.7846	−0.693344
$341$	−17.1019	−0.926118
$342$	−1.05589	−0.0570957
$343$	13.6052	0.734611
$344$	8.48558	0.457512
$345$	−3.49512	−0.188171
$346$	−9.51083	−0.511305
$347$	−33.4817	−1.79739	−0.898695	−	0.438575i	$-0.855483\pi$
$347$	−33.4817	−1.79739	−0.898695	+	0.438575i	$0.855483\pi$
$348$	−1.00000	−0.0536056
$349$	−1.04546	−0.0559622	−0.0279811	−	0.999608i	$-0.508908\pi$
$349$	−1.04546	−0.0559622	−0.0279811	+	0.999608i	$0.508908\pi$
$350$	−7.61912	−0.407259
$351$	−5.43785	−0.290251
$352$	4.23999	0.225992
$353$	−21.5087	−1.14479	−0.572395	−	0.819978i	$-0.693986\pi$
$353$	−21.5087	−1.14479	−0.572395	+	0.819978i	$0.693986\pi$
$354$	13.2516	0.704315
$355$	32.0983	1.70360
$356$	−13.8513	−0.734115
$357$	−3.86227	−0.204413
$358$	13.3185	0.703903
$359$	−35.5193	−1.87464	−0.937318	−	0.348476i	$-0.886699\pi$
$359$	−35.5193	−1.87464	−0.937318	+	0.348476i	$0.886699\pi$
$360$	3.49512	0.184209
$361$	−17.8851	−0.941321
$362$	−21.7224	−1.14170
$363$	−6.97755	−0.366226
$364$	−5.74175	−0.300950
$365$	−43.0089	−2.25119
$366$	6.93145	0.362313
$367$	14.9554	0.780667	0.390333	−	0.920674i	$-0.372360\pi$
$367$	14.9554	0.780667	0.390333	+	0.920674i	$0.372360\pi$
$368$	1.00000	0.0521286
$369$	4.28297	0.222962
$370$	24.8676	1.29280
$371$	−9.57569	−0.497145
$372$	4.03347	0.209125
$373$	22.6229	1.17137	0.585685	−	0.810539i	$-0.300826\pi$
$373$	22.6229	1.17137	0.585685	+	0.810539i	$0.300826\pi$
$374$	−15.5093	−0.801965
$375$	−7.74471	−0.399935
$376$	−4.96374	−0.255985
$377$	5.43785	0.280064
$378$	1.05589	0.0543089
$379$	2.96130	0.152112	0.0760559	−	0.997104i	$-0.475767\pi$
$379$	2.96130	0.152112	0.0760559	+	0.997104i	$0.475767\pi$
$380$	−3.69045	−0.189316
$381$	−2.73857	−0.140301
$382$	10.9414	0.559811
$383$	8.51598	0.435146	0.217573	−	0.976044i	$-0.430186\pi$
$383$	8.51598	0.435146	0.217573	+	0.976044i	$0.430186\pi$
$384$	−1.00000	−0.0510310
$385$	−15.6475	−0.797469
$386$	−3.93212	−0.200139
$387$	8.48558	0.431346
$388$	4.83622	0.245522
$389$	−36.8685	−1.86931	−0.934654	−	0.355558i	$-0.884291\pi$
$389$	−36.8685	−1.86931	−0.934654	+	0.355558i	$0.884291\pi$
$390$	−19.0060	−0.962404
$391$	−3.65785	−0.184985
$392$	−5.88511	−0.297243
$393$	−21.1991	−1.06935
$394$	−5.87495	−0.295976
$395$	16.3207	0.821185
$396$	4.23999	0.213068
$397$	19.9790	1.00272	0.501358	−	0.865240i	$-0.332834\pi$
$397$	19.9790	1.00272	0.501358	+	0.865240i	$0.332834\pi$
$398$	−5.40515	−0.270936
$399$	−1.11489	−0.0558145
$400$	7.21586	0.360793
$401$	29.8040	1.48834	0.744170	−	0.667990i	$-0.232845\pi$
$401$	29.8040	1.48834	0.744170	+	0.667990i	$0.232845\pi$
$402$	12.9486	0.645815
$403$	−21.9334	−1.09258
$404$	19.1308	0.951793
$405$	3.49512	0.173674
$406$	−1.05589	−0.0524027
$407$	30.1673	1.49534
$408$	3.65785	0.181091
$409$	24.0039	1.18691	0.593457	−	0.804866i	$-0.297763\pi$
$409$	24.0039	1.18691	0.593457	+	0.804866i	$0.297763\pi$
$410$	14.9695	0.739290
$411$	5.58255	0.275367
$412$	0.827424	0.0407643
$413$	13.9922	0.688510
$414$	1.00000	0.0491473
$415$	−16.0844	−0.789553
$416$	5.43785	0.266613
$417$	20.7073	1.01404
$418$	−4.47695	−0.218975
$419$	−38.8474	−1.89782	−0.948911	−	0.315544i	$-0.897813\pi$
$419$	−38.8474	−1.89782	−0.948911	+	0.315544i	$0.897813\pi$
$420$	3.69045	0.180075
$421$	14.0747	0.685960	0.342980	−	0.939343i	$-0.388564\pi$
$421$	14.0747	0.685960	0.342980	+	0.939343i	$0.388564\pi$
$422$	−2.39162	−0.116422
$423$	−4.96374	−0.241345
$424$	9.06888	0.440424
$425$	−26.3945	−1.28032
$426$	−9.18374	−0.444954
$427$	7.31882	0.354182
$428$	−14.2650	−0.689524
$429$	−23.0565	−1.11318
$430$	29.6581	1.43024
$431$	−4.04478	−0.194830	−0.0974152	−	0.995244i	$-0.531057\pi$
$431$	−4.04478	−0.194830	−0.0974152	+	0.995244i	$0.531057\pi$
$432$	−1.00000	−0.0481125
$433$	16.8889	0.811628	0.405814	−	0.913956i	$-0.366988\pi$
$433$	16.8889	0.811628	0.405814	+	0.913956i	$0.366988\pi$
$434$	4.25888	0.204433
$435$	−3.49512	−0.167578
$436$	9.21892	0.441506
$437$	−1.05589	−0.0505098
$438$	12.3054	0.587975
$439$	−18.9116	−0.902601	−0.451300	−	0.892372i	$-0.649040\pi$
$439$	−18.9116	−0.902601	−0.451300	+	0.892372i	$0.649040\pi$
$440$	14.8193	0.706482
$441$	−5.88511	−0.280243
$442$	−19.8909	−0.946112
$443$	−38.5977	−1.83383	−0.916916	−	0.399080i	$-0.869329\pi$
$443$	−38.5977	−1.83383	−0.916916	+	0.399080i	$0.869329\pi$
$444$	−7.11494	−0.337660
$445$	−48.4118	−2.29494
$446$	1.17667	0.0557170
$447$	−1.18480	−0.0560393
$448$	−1.05589	−0.0498859
$449$	27.5904	1.30207	0.651036	−	0.759047i	$-0.274335\pi$
$449$	27.5904	1.30207	0.651036	+	0.759047i	$0.274335\pi$
$450$	7.21586	0.340159
$451$	18.1597	0.855109
$452$	15.0393	0.707388
$453$	4.28192	0.201182
$454$	−5.93573	−0.278577
$455$	−20.0681	−0.940807
$456$	1.05589	0.0494464
$457$	22.5461	1.05466	0.527332	−	0.849660i	$-0.323193\pi$
$457$	22.5461	1.05466	0.527332	+	0.849660i	$0.323193\pi$
$458$	1.70563	0.0796988
$459$	3.65785	0.170734
$460$	3.49512	0.162961
$461$	−27.2823	−1.27066	−0.635331	−	0.772240i	$-0.719137\pi$
$461$	−27.2823	−1.27066	−0.635331	+	0.772240i	$0.719137\pi$
$462$	4.47695	0.208286
$463$	−4.47035	−0.207755	−0.103877	−	0.994590i	$-0.533125\pi$
$463$	−4.47035	−0.207755	−0.103877	+	0.994590i	$0.533125\pi$
$464$	1.00000	0.0464238
$465$	14.0974	0.653754
$466$	6.92781	0.320925
$467$	−0.457011	−0.0211479	−0.0105740	−	0.999944i	$-0.503366\pi$
$467$	−0.457011	−0.0211479	−0.0105740	+	0.999944i	$0.503366\pi$
$468$	5.43785	0.251365
$469$	13.6722	0.631323
$470$	−17.3489	−0.800244
$471$	−12.8331	−0.591320
$472$	−13.2516	−0.609954
$473$	35.9788	1.65431
$474$	−4.66957	−0.214481
$475$	−7.61912	−0.349589
$476$	3.86227	0.177027
$477$	9.06888	0.415235
$478$	10.6965	0.489247
$479$	−0.135314	−0.00618266	−0.00309133	−	0.999995i	$-0.500984\pi$
$479$	−0.135314	−0.00618266	−0.00309133	+	0.999995i	$0.500984\pi$
$480$	−3.49512	−0.159530
$481$	38.6900	1.76411
$482$	12.1453	0.553202
$483$	1.05589	0.0480444
$484$	6.97755	0.317161
$485$	16.9032	0.767533
$486$	−1.00000	−0.0453609
$487$	13.4945	0.611494	0.305747	−	0.952113i	$-0.401094\pi$
$487$	13.4945	0.611494	0.305747	+	0.952113i	$0.401094\pi$
$488$	−6.93145	−0.313772
$489$	−7.51658	−0.339911
$490$	−20.5692	−0.929220
$491$	38.8989	1.75548	0.877741	−	0.479135i	$-0.159050\pi$
$491$	38.8989	1.75548	0.877741	+	0.479135i	$0.159050\pi$
$492$	−4.28297	−0.193091
$493$	−3.65785	−0.164741
$494$	−5.74175	−0.258334
$495$	14.8193	0.666077
$496$	−4.03347	−0.181108
$497$	−9.69697	−0.434969
$498$	4.60197	0.206219
$499$	−40.7876	−1.82590	−0.912952	−	0.408068i	$-0.866203\pi$
$499$	−40.7876	−1.82590	−0.912952	+	0.408068i	$0.866203\pi$
$500$	7.74471	0.346354
$501$	9.25414	0.413445
$502$	14.0329	0.626320
$503$	22.4789	1.00228	0.501142	−	0.865365i	$-0.332913\pi$
$503$	22.4789	1.00228	0.501142	+	0.865365i	$0.332913\pi$
$504$	−1.05589	−0.0470329
$505$	66.8645	2.97543
$506$	4.23999	0.188491
$507$	−16.5703	−0.735911
$508$	2.73857	0.121504
$509$	6.85472	0.303830	0.151915	−	0.988394i	$-0.451456\pi$
$509$	6.85472	0.303830	0.151915	+	0.988394i	$0.451456\pi$
$510$	12.7846	0.566113
$511$	12.9931	0.574781
$512$	1.00000	0.0441942
$513$	1.05589	0.0466185
$514$	−0.413438	−0.0182360
$515$	2.89195	0.127434
$516$	−8.48558	−0.373557
$517$	−21.0462	−0.925612
$518$	−7.51256	−0.330083
$519$	9.51083	0.417479
$520$	19.0060	0.833466
$521$	−36.3360	−1.59191	−0.795955	−	0.605355i	$-0.793031\pi$
$521$	−36.3360	−1.59191	−0.795955	+	0.605355i	$0.793031\pi$
$522$	1.00000	0.0437688
$523$	5.64973	0.247045	0.123523	−	0.992342i	$-0.460581\pi$
$523$	5.64973	0.247045	0.123523	+	0.992342i	$0.460581\pi$
$524$	21.1991	0.926085
$525$	7.61912	0.332526
$526$	12.9642	0.565266
$527$	14.7538	0.642686
$528$	−4.23999	−0.184522
$529$	1.00000	0.0434783
$530$	31.6968	1.37682
$531$	−13.2516	−0.575071
$532$	1.11489	0.0483368
$533$	23.2901	1.00881
$534$	13.8513	0.599403
$535$	−49.8579	−2.15554
$536$	−12.9486	−0.559292
$537$	−13.3185	−0.574734
$538$	16.6963	0.719827
$539$	−24.9528	−1.07479
$540$	−3.49512	−0.150406
$541$	−32.9877	−1.41825	−0.709125	−	0.705082i	$-0.750910\pi$
$541$	−32.9877	−1.41825	−0.709125	+	0.705082i	$0.750910\pi$
$542$	27.1117	1.16455
$543$	21.7224	0.932197
$544$	−3.65785	−0.156829
$545$	32.2212	1.38021
$546$	5.74175	0.245724
$547$	−1.57869	−0.0675000	−0.0337500	−	0.999430i	$-0.510745\pi$
$547$	−1.57869	−0.0675000	−0.0337500	+	0.999430i	$0.510745\pi$
$548$	−5.58255	−0.238475
$549$	−6.93145	−0.295827
$550$	30.5952	1.30458
$551$	−1.05589	−0.0449822
$552$	−1.00000	−0.0425628
$553$	−4.93054	−0.209668
$554$	18.4920	0.785652
$555$	−24.8676	−1.05557
$556$	−20.7073	−0.878185
$557$	−21.8487	−0.925757	−0.462879	−	0.886422i	$-0.653183\pi$
$557$	−21.8487	−0.925757	−0.462879	+	0.886422i	$0.653183\pi$
$558$	−4.03347	−0.170750
$559$	46.1433	1.95166
$560$	−3.69045	−0.155950
$561$	15.5093	0.654802
$562$	−11.5054	−0.485326
$563$	−40.1340	−1.69145	−0.845723	−	0.533622i	$-0.820830\pi$
$563$	−40.1340	−1.69145	−0.845723	+	0.533622i	$0.820830\pi$
$564$	4.96374	0.209011
$565$	52.5641	2.21139
$566$	−21.2173	−0.891830
$567$	−1.05589	−0.0443430
$568$	9.18374	0.385341
$569$	−18.0124	−0.755119	−0.377560	−	0.925985i	$-0.623237\pi$
$569$	−18.0124	−0.755119	−0.377560	+	0.925985i	$0.623237\pi$
$570$	3.69045	0.154576
$571$	−37.3882	−1.56465	−0.782323	−	0.622873i	$-0.785965\pi$
$571$	−37.3882	−1.56465	−0.782323	+	0.622873i	$0.785965\pi$
$572$	23.0565	0.964039
$573$	−10.9414	−0.457084
$574$	−4.52232	−0.188758
$575$	7.21586	0.300922
$576$	1.00000	0.0416667
$577$	42.9285	1.78714	0.893569	−	0.448927i	$-0.148194\pi$
$577$	42.9285	1.78714	0.893569	+	0.448927i	$0.148194\pi$
$578$	−3.62013	−0.150577
$579$	3.93212	0.163413
$580$	3.49512	0.145127
$581$	4.85915	0.201591
$582$	−4.83622	−0.200468
$583$	38.4520	1.59252
$584$	−12.3054	−0.509201
$585$	19.0060	0.785799
$586$	−8.00326	−0.330612
$587$	26.7238	1.10301	0.551505	−	0.834171i	$-0.314054\pi$
$587$	26.7238	1.10301	0.551505	+	0.834171i	$0.314054\pi$
$588$	5.88511	0.242698
$589$	4.25888	0.175484
$590$	−46.3159	−1.90680
$591$	5.87495	0.241663
$592$	7.11494	0.292422
$593$	−46.4606	−1.90791	−0.953955	−	0.299951i	$-0.903030\pi$
$593$	−46.4606	−1.90791	−0.953955	+	0.299951i	$0.903030\pi$
$594$	−4.23999	−0.173969
$595$	13.4991	0.553409
$596$	1.18480	0.0485315
$597$	5.40515	0.221218
$598$	5.43785	0.222370
$599$	−27.8443	−1.13769	−0.568844	−	0.822446i	$-0.692609\pi$
$599$	−27.8443	−1.13769	−0.568844	+	0.822446i	$0.692609\pi$
$600$	−7.21586	−0.294586
$601$	−39.5453	−1.61309	−0.806544	−	0.591174i	$-0.798665\pi$
$601$	−39.5453	−1.61309	−0.806544	+	0.591174i	$0.798665\pi$
$602$	−8.95980	−0.365174
$603$	−12.9486	−0.527306
$604$	−4.28192	−0.174229
$605$	24.3874	0.991487
$606$	−19.1308	−0.777136
$607$	4.48213	0.181924	0.0909621	−	0.995854i	$-0.471006\pi$
$607$	4.48213	0.181924	0.0909621	+	0.995854i	$0.471006\pi$
$608$	−1.05589	−0.0428218
$609$	1.05589	0.0427866
$610$	−24.2262	−0.980892
$611$	−26.9921	−1.09198
$612$	−3.65785	−0.147860
$613$	14.0743	0.568455	0.284227	−	0.958757i	$-0.408263\pi$
$613$	14.0743	0.568455	0.284227	+	0.958757i	$0.408263\pi$
$614$	−18.8924	−0.762433
$615$	−14.9695	−0.603628
$616$	−4.47695	−0.180381
$617$	−13.1038	−0.527539	−0.263770	−	0.964586i	$-0.584966\pi$
$617$	−13.1038	−0.527539	−0.263770	+	0.964586i	$0.584966\pi$
$618$	−0.827424	−0.0332839
$619$	−36.2462	−1.45686	−0.728429	−	0.685121i	$-0.759749\pi$
$619$	−36.2462	−1.45686	−0.728429	+	0.685121i	$0.759749\pi$
$620$	−14.0974	−0.566167
$621$	−1.00000	−0.0401286
$622$	−0.789316	−0.0316487
$623$	14.6253	0.585952
$624$	−5.43785	−0.217688
$625$	−9.01064	−0.360425
$626$	6.14077	0.245434
$627$	4.47695	0.178792
$628$	12.8331	0.512098
$629$	−26.0254	−1.03770
$630$	−3.69045	−0.147031
$631$	−37.2024	−1.48100	−0.740502	−	0.672054i	$-0.765412\pi$
$631$	−37.2024	−1.48100	−0.740502	+	0.672054i	$0.765412\pi$
$632$	4.66957	0.185746
$633$	2.39162	0.0950585
$634$	−1.88403	−0.0748245
$635$	9.57162	0.379838
$636$	−9.06888	−0.359604
$637$	−32.0023	−1.26798
$638$	4.23999	0.167863
$639$	9.18374	0.363303
$640$	3.49512	0.138157
$641$	−17.9456	−0.708808	−0.354404	−	0.935092i	$-0.615316\pi$
$641$	−17.9456	−0.708808	−0.354404	+	0.935092i	$0.615316\pi$
$642$	14.2650	0.562994
$643$	28.8684	1.13846	0.569228	−	0.822179i	$-0.307242\pi$
$643$	28.8684	1.13846	0.569228	+	0.822179i	$0.307242\pi$
$644$	−1.05589	−0.0416077
$645$	−29.6581	−1.16779
$646$	3.86227	0.151959
$647$	38.8044	1.52556	0.762779	−	0.646659i	$-0.223834\pi$
$647$	38.8044	1.52556	0.762779	+	0.646659i	$0.223834\pi$
$648$	1.00000	0.0392837
$649$	−56.1867	−2.20552
$650$	39.2388	1.53907
$651$	−4.25888	−0.166919
$652$	7.51658	0.294372
$653$	−7.59637	−0.297269	−0.148634	−	0.988892i	$-0.547488\pi$
$653$	−7.59637	−0.297269	−0.148634	+	0.988892i	$0.547488\pi$
$654$	−9.21892	−0.360488
$655$	74.0932	2.89506
$656$	4.28297	0.167222
$657$	−12.3054	−0.480079
$658$	5.24114	0.204321
$659$	−39.7713	−1.54927	−0.774636	−	0.632408i	$-0.782067\pi$
$659$	−39.7713	−1.54927	−0.774636	+	0.632408i	$0.782067\pi$
$660$	−14.8193	−0.576840
$661$	−26.1141	−1.01572	−0.507861	−	0.861439i	$-0.669564\pi$
$661$	−26.1141	−1.01572	−0.507861	+	0.861439i	$0.669564\pi$
$662$	−1.53412	−0.0596251
$663$	19.8909	0.772497
$664$	−4.60197	−0.178591
$665$	3.89669	0.151107
$666$	7.11494	0.275698
$667$	1.00000	0.0387202
$668$	−9.25414	−0.358054
$669$	−1.17667	−0.0454927
$670$	−45.2567	−1.74842
$671$	−29.3893	−1.13456
$672$	1.05589	0.0407317
$673$	10.3950	0.400696	0.200348	−	0.979725i	$-0.435793\pi$
$673$	10.3950	0.400696	0.200348	+	0.979725i	$0.435793\pi$
$674$	0.685212	0.0263934
$675$	−7.21586	−0.277739
$676$	16.5703	0.637318
$677$	3.20100	0.123024	0.0615122	−	0.998106i	$-0.480408\pi$
$677$	3.20100	0.123024	0.0615122	+	0.998106i	$0.480408\pi$
$678$	−15.0393	−0.577580
$679$	−5.10649	−0.195969
$680$	−12.7846	−0.490268
$681$	5.93573	0.227458
$682$	−17.1019	−0.654865
$683$	−19.0993	−0.730813	−0.365407	−	0.930848i	$-0.619070\pi$
$683$	−19.0993	−0.730813	−0.365407	+	0.930848i	$0.619070\pi$
$684$	−1.05589	−0.0403728
$685$	−19.5117	−0.745503
$686$	13.6052	0.519449
$687$	−1.70563	−0.0650738
$688$	8.48558	0.323510
$689$	49.3152	1.87876
$690$	−3.49512	−0.133057
$691$	−2.67577	−0.101791	−0.0508956	−	0.998704i	$-0.516208\pi$
$691$	−2.67577	−0.101791	−0.0508956	+	0.998704i	$0.516208\pi$
$692$	−9.51083	−0.361548
$693$	−4.47695	−0.170065
$694$	−33.4817	−1.27095
$695$	−72.3745	−2.74532
$696$	−1.00000	−0.0379049
$697$	−15.6665	−0.593409
$698$	−1.04546	−0.0395712
$699$	−6.92781	−0.262034
$700$	−7.61912	−0.287976
$701$	8.19951	0.309691	0.154846	−	0.987939i	$-0.450512\pi$
$701$	8.19951	0.309691	0.154846	+	0.987939i	$0.450512\pi$
$702$	−5.43785	−0.205239
$703$	−7.51256	−0.283342
$704$	4.23999	0.159801
$705$	17.3489	0.653396
$706$	−21.5087	−0.809489
$707$	−20.1999	−0.759697
$708$	13.2516	0.498026
$709$	10.6630	0.400458	0.200229	−	0.979749i	$-0.435831\pi$
$709$	10.6630	0.400458	0.200229	+	0.979749i	$0.435831\pi$
$710$	32.0983	1.20463
$711$	4.66957	0.175123
$712$	−13.8513	−0.519098
$713$	−4.03347	−0.151055
$714$	−3.86227	−0.144542
$715$	80.5851	3.01371
$716$	13.3185	0.497734
$717$	−10.6965	−0.399468
$718$	−35.5193	−1.32557
$719$	0.0184633	0.000688564 0	0.000344282	−	1.00000i	$-0.499890\pi$
$719$	0.0184633	0.000688564 0	0.000344282		1.00000i	$0.499890\pi$
$720$	3.49512	0.130255
$721$	−0.873665	−0.0325370
$722$	−17.8851	−0.665615
$723$	−12.1453	−0.451688
$724$	−21.7224	−0.807306
$725$	7.21586	0.267990
$726$	−6.97755	−0.258961
$727$	−16.9955	−0.630330	−0.315165	−	0.949037i	$-0.602060\pi$
$727$	−16.9955	−0.630330	−0.315165	+	0.949037i	$0.602060\pi$
$728$	−5.74175	−0.212803
$729$	1.00000	0.0370370
$730$	−43.0089	−1.59183
$731$	−31.0390	−1.14802
$732$	6.93145	0.256194
$733$	44.0222	1.62600	0.812998	−	0.582266i	$-0.197834\pi$
$733$	44.0222	1.62600	0.812998	+	0.582266i	$0.197834\pi$
$734$	14.9554	0.552015
$735$	20.5692	0.758705
$736$	1.00000	0.0368605
$737$	−54.9018	−2.02233
$738$	4.28297	0.157658
$739$	35.2198	1.29558	0.647791	−	0.761818i	$-0.275693\pi$
$739$	35.2198	1.29558	0.647791	+	0.761818i	$0.275693\pi$
$740$	24.8676	0.914150
$741$	5.74175	0.210928
$742$	−9.57569	−0.351535
$743$	1.30868	0.0480109	0.0240054	−	0.999712i	$-0.492358\pi$
$743$	1.30868	0.0480109	0.0240054	+	0.999712i	$0.492358\pi$
$744$	4.03347	0.147874
$745$	4.14103	0.151716
$746$	22.6229	0.828284
$747$	−4.60197	−0.168377
$748$	−15.5093	−0.567075
$749$	15.0622	0.550361
$750$	−7.74471	−0.282797
$751$	6.90025	0.251794	0.125897	−	0.992043i	$-0.459819\pi$
$751$	6.90025	0.251794	0.125897	+	0.992043i	$0.459819\pi$
$752$	−4.96374	−0.181009
$753$	−14.0329	−0.511388
$754$	5.43785	0.198035
$755$	−14.9658	−0.544663
$756$	1.05589	0.0384022
$757$	43.5338	1.58226	0.791132	−	0.611646i	$-0.209492\pi$
$757$	43.5338	1.58226	0.791132	+	0.611646i	$0.209492\pi$
$758$	2.96130	0.107559
$759$	−4.23999	−0.153902
$760$	−3.69045	−0.133867
$761$	10.3834	0.376398	0.188199	−	0.982131i	$-0.439735\pi$
$761$	10.3834	0.376398	0.188199	+	0.982131i	$0.439735\pi$
$762$	−2.73857	−0.0992079
$763$	−9.73412	−0.352399
$764$	10.9414	0.395846
$765$	−12.7846	−0.462229
$766$	8.51598	0.307695
$767$	−72.0603	−2.60195
$768$	−1.00000	−0.0360844
$769$	−26.2358	−0.946089	−0.473044	−	0.881039i	$-0.656845\pi$
$769$	−26.2358	−0.946089	−0.473044	+	0.881039i	$0.656845\pi$
$770$	−15.6475	−0.563896
$771$	0.413438	0.0148896
$772$	−3.93212	−0.141520
$773$	−11.8771	−0.427190	−0.213595	−	0.976922i	$-0.568517\pi$
$773$	−11.8771	−0.427190	−0.213595	+	0.976922i	$0.568517\pi$
$774$	8.48558	0.305008
$775$	−29.1049	−1.04548
$776$	4.83622	0.173610
$777$	7.51256	0.269512
$778$	−36.8685	−1.32180
$779$	−4.52232	−0.162029
$780$	−19.0060	−0.680522
$781$	38.9390	1.39335
$782$	−3.65785	−0.130804
$783$	−1.00000	−0.0357371
$784$	−5.88511	−0.210182
$785$	44.8534	1.60089
$786$	−21.1991	−0.756145
$787$	−2.81021	−0.100173	−0.0500866	−	0.998745i	$-0.515950\pi$
$787$	−2.81021	−0.100173	−0.0500866	+	0.998745i	$0.515950\pi$
$788$	−5.87495	−0.209286
$789$	−12.9642	−0.461537
$790$	16.3207	0.580665
$791$	−15.8798	−0.564619
$792$	4.23999	0.150662
$793$	−37.6922	−1.33849
$794$	19.9790	0.709027
$795$	−31.6968	−1.12417
$796$	−5.40515	−0.191580
$797$	43.2628	1.53245	0.766223	−	0.642575i	$-0.222134\pi$
$797$	43.2628	1.53245	0.766223	+	0.642575i	$0.222134\pi$
$798$	−1.11489	−0.0394668
$799$	18.1566	0.642335
$800$	7.21586	0.255119
$801$	−13.8513	−0.489410
$802$	29.8040	1.05242
$803$	−52.1748	−1.84121
$804$	12.9486	0.456660
$805$	−3.69045	−0.130071
$806$	−21.9334	−0.772571
$807$	−16.6963	−0.587736
$808$	19.1308	0.673019
$809$	−40.4069	−1.42063	−0.710315	−	0.703883i	$-0.751448\pi$
$809$	−40.4069	−1.42063	−0.710315	+	0.703883i	$0.751448\pi$
$810$	3.49512	0.122806
$811$	5.44429	0.191175	0.0955874	−	0.995421i	$-0.469527\pi$
$811$	5.44429	0.191175	0.0955874	+	0.995421i	$0.469527\pi$
$812$	−1.05589	−0.0370543
$813$	−27.1117	−0.950850
$814$	30.1673	1.05736
$815$	26.2713	0.920245
$816$	3.65785	0.128050
$817$	−8.95980	−0.313464
$818$	24.0039	0.839275
$819$	−5.74175	−0.200633
$820$	14.9695	0.522757
$821$	−38.8105	−1.35449	−0.677247	−	0.735755i	$-0.736827\pi$
$821$	−38.8105	−1.35449	−0.677247	+	0.735755i	$0.736827\pi$
$822$	5.58255	0.194714
$823$	−29.0821	−1.01374	−0.506869	−	0.862023i	$-0.669197\pi$
$823$	−29.0821	−1.01374	−0.506869	+	0.862023i	$0.669197\pi$
$824$	0.827424	0.0288247
$825$	−30.5952	−1.06519
$826$	13.9922	0.486850
$827$	31.7654	1.10459	0.552296	−	0.833648i	$-0.313752\pi$
$827$	31.7654	1.10459	0.552296	+	0.833648i	$0.313752\pi$
$828$	1.00000	0.0347524
$829$	−31.5168	−1.09462	−0.547311	−	0.836929i	$-0.684349\pi$
$829$	−31.5168	−1.09462	−0.547311	+	0.836929i	$0.684349\pi$
$830$	−16.0844	−0.558298
$831$	−18.4920	−0.641482
$832$	5.43785	0.188524
$833$	21.5268	0.745861
$834$	20.7073	0.717035
$835$	−32.3443	−1.11932
$836$	−4.47695	−0.154838
$837$	4.03347	0.139417
$838$	−38.8474	−1.34196
$839$	8.92396	0.308089	0.154045	−	0.988064i	$-0.450770\pi$
$839$	8.92396	0.308089	0.154045	+	0.988064i	$0.450770\pi$
$840$	3.69045	0.127332
$841$	1.00000	0.0344828
$842$	14.0747	0.485047
$843$	11.5054	0.396267
$844$	−2.39162	−0.0823230
$845$	57.9150	1.99234
$846$	−4.96374	−0.170657
$847$	−7.36749	−0.253150
$848$	9.06888	0.311426
$849$	21.2173	0.728176
$850$	−26.3945	−0.905326
$851$	7.11494	0.243897
$852$	−9.18374	−0.314630
$853$	−20.7711	−0.711188	−0.355594	−	0.934641i	$-0.615721\pi$
$853$	−20.7711	−0.711188	−0.355594	+	0.934641i	$0.615721\pi$
$854$	7.31882	0.250445
$855$	−3.69045	−0.126211
$856$	−14.2650	−0.487567
$857$	8.91820	0.304640	0.152320	−	0.988331i	$-0.451326\pi$
$857$	8.91820	0.304640	0.152320	+	0.988331i	$0.451326\pi$
$858$	−23.0565	−0.787135
$859$	48.2760	1.64716	0.823579	−	0.567202i	$-0.191974\pi$
$859$	48.2760	1.64716	0.823579	+	0.567202i	$0.191974\pi$
$860$	29.6581	1.01133
$861$	4.52232	0.154120
$862$	−4.04478	−0.137766
$863$	−53.5057	−1.82136	−0.910678	−	0.413117i	$-0.864440\pi$
$863$	−53.5057	−1.82136	−0.910678	+	0.413117i	$0.864440\pi$
$864$	−1.00000	−0.0340207
$865$	−33.2415	−1.13024
$866$	16.8889	0.573908
$867$	3.62013	0.122946
$868$	4.25888	0.144556
$869$	19.7990	0.671634
$870$	−3.49512	−0.118496
$871$	−70.4123	−2.38583
$872$	9.21892	0.312192
$873$	4.83622	0.163681
$874$	−1.05589	−0.0357159
$875$	−8.17752	−0.276451
$876$	12.3054	0.415761
$877$	−24.0632	−0.812556	−0.406278	−	0.913750i	$-0.633174\pi$
$877$	−24.0632	−0.812556	−0.406278	+	0.913750i	$0.633174\pi$
$878$	−18.9116	−0.638235
$879$	8.00326	0.269943
$880$	14.8193	0.499558
$881$	−6.20301	−0.208985	−0.104492	−	0.994526i	$-0.533322\pi$
$881$	−6.20301	−0.208985	−0.104492	+	0.994526i	$0.533322\pi$
$882$	−5.88511	−0.198162
$883$	−12.8830	−0.433546	−0.216773	−	0.976222i	$-0.569553\pi$
$883$	−12.8830	−0.433546	−0.216773	+	0.976222i	$0.569553\pi$
$884$	−19.8909	−0.669002
$885$	46.3159	1.55689
$886$	−38.5977	−1.29672
$887$	−32.0058	−1.07465	−0.537325	−	0.843376i	$-0.680565\pi$
$887$	−32.0058	−1.07465	−0.537325	+	0.843376i	$0.680565\pi$
$888$	−7.11494	−0.238762
$889$	−2.89161	−0.0969816
$890$	−48.4118	−1.62277
$891$	4.23999	0.142045
$892$	1.17667	0.0393979
$893$	5.24114	0.175388
$894$	−1.18480	−0.0396258
$895$	46.5496	1.55598
$896$	−1.05589	−0.0352747
$897$	−5.43785	−0.181565
$898$	27.5904	0.920704
$899$	−4.03347	−0.134524
$900$	7.21586	0.240529
$901$	−33.1726	−1.10514
$902$	18.1597	0.604654
$903$	8.95980	0.298163
$904$	15.0393	0.500199
$905$	−75.9223	−2.52374
$906$	4.28192	0.142257
$907$	−9.95932	−0.330694	−0.165347	−	0.986235i	$-0.552874\pi$
$907$	−9.95932	−0.330694	−0.165347	+	0.986235i	$0.552874\pi$
$908$	−5.93573	−0.196984
$909$	19.1308	0.634529
$910$	−20.0681	−0.665251
$911$	−42.5141	−1.40856	−0.704278	−	0.709924i	$-0.748729\pi$
$911$	−42.5141	−1.40856	−0.704278	+	0.709924i	$0.748729\pi$
$912$	1.05589	0.0349639
$913$	−19.5123	−0.645763
$914$	22.5461	0.745759
$915$	24.2262	0.800895
$916$	1.70563	0.0563555
$917$	−22.3838	−0.739177
$918$	3.65785	0.120727
$919$	−12.5938	−0.415430	−0.207715	−	0.978189i	$-0.566603\pi$
$919$	−12.5938	−0.415430	−0.207715	+	0.978189i	$0.566603\pi$
$920$	3.49512	0.115231
$921$	18.8924	0.622524
$922$	−27.2823	−0.898494
$923$	49.9398	1.64379
$924$	4.47695	0.147281
$925$	51.3404	1.68806
$926$	−4.47035	−0.146905
$927$	0.827424	0.0271762
$928$	1.00000	0.0328266
$929$	−33.7555	−1.10748	−0.553742	−	0.832688i	$-0.686801\pi$
$929$	−33.7555	−1.10748	−0.553742	+	0.832688i	$0.686801\pi$
$930$	14.0974	0.462274
$931$	6.21400	0.203656
$932$	6.92781	0.226928
$933$	0.789316	0.0258411
$934$	−0.457011	−0.0149538
$935$	−54.2067	−1.77275
$936$	5.43785	0.177742
$937$	−15.4473	−0.504640	−0.252320	−	0.967644i	$-0.581194\pi$
$937$	−15.4473	−0.504640	−0.252320	+	0.967644i	$0.581194\pi$
$938$	13.6722	0.446413
$939$	−6.14077	−0.200396
$940$	−17.3489	−0.565858
$941$	−8.95014	−0.291766	−0.145883	−	0.989302i	$-0.546602\pi$
$941$	−8.95014	−0.291766	−0.145883	+	0.989302i	$0.546602\pi$
$942$	−12.8331	−0.418127
$943$	4.28297	0.139473
$944$	−13.2516	−0.431303
$945$	3.69045	0.120050
$946$	35.9788	1.16977
$947$	3.32810	0.108149	0.0540744	−	0.998537i	$-0.482779\pi$
$947$	3.32810	0.108149	0.0540744	+	0.998537i	$0.482779\pi$
$948$	−4.66957	−0.151661
$949$	−66.9150	−2.17215
$950$	−7.61912	−0.247197
$951$	1.88403	0.0610939
$952$	3.86227	0.125177
$953$	−4.89734	−0.158640	−0.0793202	−	0.996849i	$-0.525275\pi$
$953$	−4.89734	−0.158640	−0.0793202	+	0.996849i	$0.525275\pi$
$954$	9.06888	0.293616
$955$	38.2415	1.23747
$956$	10.6965	0.345950
$957$	−4.23999	−0.137060
$958$	−0.135314	−0.00437180
$959$	5.89453	0.190344
$960$	−3.49512	−0.112805
$961$	−14.7311	−0.475198
$962$	38.6900	1.24742
$963$	−14.2650	−0.459683
$964$	12.1453	0.391173
$965$	−13.7432	−0.442410
$966$	1.05589	0.0339726
$967$	−22.4265	−0.721187	−0.360593	−	0.932723i	$-0.617426\pi$
$967$	−22.4265	−0.721187	−0.360593	+	0.932723i	$0.617426\pi$
$968$	6.97755	0.224267
$969$	−3.86227	−0.124074
$970$	16.9032	0.542728
$971$	−30.9397	−0.992901	−0.496451	−	0.868065i	$-0.665364\pi$
$971$	−30.9397	−0.992901	−0.496451	+	0.868065i	$0.665364\pi$
$972$	−1.00000	−0.0320750
$973$	21.8645	0.700945
$974$	13.4945	0.432391
$975$	−39.2388	−1.25665
$976$	−6.93145	−0.221870
$977$	7.18204	0.229774	0.114887	−	0.993379i	$-0.463349\pi$
$977$	7.18204	0.229774	0.114887	+	0.993379i	$0.463349\pi$
$978$	−7.51658	−0.240354
$979$	−58.7293	−1.87700
$980$	−20.5692	−0.657057
$981$	9.21892	0.294337
$982$	38.8989	1.24131
$983$	14.2025	0.452991	0.226495	−	0.974012i	$-0.427273\pi$
$983$	14.2025	0.452991	0.226495	+	0.974012i	$0.427273\pi$
$984$	−4.28297	−0.136536
$985$	−20.5337	−0.654257
$986$	−3.65785	−0.116490
$987$	−5.24114	−0.166827
$988$	−5.74175	−0.182669
$989$	8.48558	0.269826
$990$	14.8193	0.470988
$991$	3.46798	0.110164	0.0550821	−	0.998482i	$-0.482458\pi$
$991$	3.46798	0.110164	0.0550821	+	0.998482i	$0.482458\pi$
$992$	−4.03347	−0.128063
$993$	1.53412	0.0486837
$994$	−9.69697	−0.307569
$995$	−18.8916	−0.598905
$996$	4.60197	0.145819
$997$	0.983869	0.0311595	0.0155797	−	0.999879i	$-0.495041\pi$
$997$	0.983869	0.0311595	0.0155797	+	0.999879i	$0.495041\pi$
$998$	−40.7876	−1.29111
$999$	−7.11494	−0.225107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bj.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.49512 q^{5} -1.00000 q^{6} -1.05589 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.49512 q^{5} -1.00000 q^{6} -1.05589 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.49512 q^{10} +4.23999 q^{11} -1.00000 q^{12} +5.43785 q^{13} -1.05589 q^{14} -3.49512 q^{15} +1.00000 q^{16} -3.65785 q^{17} +1.00000 q^{18} -1.05589 q^{19} +3.49512 q^{20} +1.05589 q^{21} +4.23999 q^{22} +1.00000 q^{23} -1.00000 q^{24} +7.21586 q^{25} +5.43785 q^{26} -1.00000 q^{27} -1.05589 q^{28} +1.00000 q^{29} -3.49512 q^{30} -4.03347 q^{31} +1.00000 q^{32} -4.23999 q^{33} -3.65785 q^{34} -3.69045 q^{35} +1.00000 q^{36} +7.11494 q^{37} -1.05589 q^{38} -5.43785 q^{39} +3.49512 q^{40} +4.28297 q^{41} +1.05589 q^{42} +8.48558 q^{43} +4.23999 q^{44} +3.49512 q^{45} +1.00000 q^{46} -4.96374 q^{47} -1.00000 q^{48} -5.88511 q^{49} +7.21586 q^{50} +3.65785 q^{51} +5.43785 q^{52} +9.06888 q^{53} -1.00000 q^{54} +14.8193 q^{55} -1.05589 q^{56} +1.05589 q^{57} +1.00000 q^{58} -13.2516 q^{59} -3.49512 q^{60} -6.93145 q^{61} -4.03347 q^{62} -1.05589 q^{63} +1.00000 q^{64} +19.0060 q^{65} -4.23999 q^{66} -12.9486 q^{67} -3.65785 q^{68} -1.00000 q^{69} -3.69045 q^{70} +9.18374 q^{71} +1.00000 q^{72} -12.3054 q^{73} +7.11494 q^{74} -7.21586 q^{75} -1.05589 q^{76} -4.47695 q^{77} -5.43785 q^{78} +4.66957 q^{79} +3.49512 q^{80} +1.00000 q^{81} +4.28297 q^{82} -4.60197 q^{83} +1.05589 q^{84} -12.7846 q^{85} +8.48558 q^{86} -1.00000 q^{87} +4.23999 q^{88} -13.8513 q^{89} +3.49512 q^{90} -5.74175 q^{91} +1.00000 q^{92} +4.03347 q^{93} -4.96374 q^{94} -3.69045 q^{95} -1.00000 q^{96} +4.83622 q^{97} -5.88511 q^{98} +4.23999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

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