LMFDB - Embedded newform 1690.2.d.i.1351.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(1351,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,-6,0,0,0,0,-8,6,0,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$1.80194i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.i.1351.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.24698i q^{6} +2.49396i q^{7} -1.00000i q^{8} -1.44504 q^{9} +1.00000 q^{10} +5.80194i q^{11} -1.24698 q^{12} -2.49396 q^{14} -1.24698i q^{15} +1.00000 q^{16} -4.29590 q^{17} -1.44504i q^{18} -4.04892i q^{19} +1.00000i q^{20} +3.10992i q^{21} -5.80194 q^{22} +3.10992 q^{23} -1.24698i q^{24} -1.00000 q^{25} -5.54288 q^{27} -2.49396i q^{28} -5.60388 q^{29} +1.24698 q^{30} +7.70171i q^{31} +1.00000i q^{32} +7.23490i q^{33} -4.29590i q^{34} +2.49396 q^{35} +1.44504 q^{36} +2.67025i q^{37} +4.04892 q^{38} -1.00000 q^{40} -12.5429i q^{41} -3.10992 q^{42} -6.98254 q^{43} -5.80194i q^{44} +1.44504i q^{45} +3.10992i q^{46} -3.87800i q^{47} +1.24698 q^{48} +0.780167 q^{49} -1.00000i q^{50} -5.35690 q^{51} -4.93362 q^{53} -5.54288i q^{54} +5.80194 q^{55} +2.49396 q^{56} -5.04892i q^{57} -5.60388i q^{58} +10.0151i q^{59} +1.24698i q^{60} -4.93362 q^{61} -7.70171 q^{62} -3.60388i q^{63} -1.00000 q^{64} -7.23490 q^{66} +8.01507i q^{67} +4.29590 q^{68} +3.87800 q^{69} +2.49396i q^{70} +5.48188i q^{71} +1.44504i q^{72} +8.67456i q^{73} -2.67025 q^{74} -1.24698 q^{75} +4.04892i q^{76} -14.4698 q^{77} -1.82371 q^{79} -1.00000i q^{80} -2.57673 q^{81} +12.5429 q^{82} +14.6407i q^{83} -3.10992i q^{84} +4.29590i q^{85} -6.98254i q^{86} -6.98792 q^{87} +5.80194 q^{88} -0.454731i q^{89} -1.44504 q^{90} -3.10992 q^{92} +9.60388i q^{93} +3.87800 q^{94} -4.04892 q^{95} +1.24698i q^{96} -8.69202i q^{97} +0.780167i q^{98} -8.38404i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{4} - 8 q^{9} + 6 q^{10} + 2 q^{12} + 4 q^{14} + 6 q^{16} + 2 q^{17} - 26 q^{22} + 20 q^{23} - 6 q^{25} + 4 q^{27} - 16 q^{29} - 2 q^{30} - 4 q^{35} + 8 q^{36} + 6 q^{38} - 6 q^{40}+ \cdots - 6 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^4 - 8 * q^9 + 6 * q^10 + 2 * q^12 + 4 * q^14 + 6 * q^16 + 2 * q^17 - 26 * q^22 + 20 * q^23 - 6 * q^25 + 4 * q^27 - 16 * q^29 - 2 * q^30 - 4 * q^35 + 8 * q^36 + 6 * q^38 - 6 * q^40 - 20 * q^42 - 10 * q^43 - 2 * q^48 + 2 * q^49 - 24 * q^51 - 16 * q^53 + 26 * q^55 - 4 * q^56 - 16 * q^61 + 8 * q^62 - 6 * q^64 + 4 * q^66 - 2 * q^68 - 16 * q^69 - 12 * q^74 + 2 * q^75 + 8 * q^77 + 4 * q^79 - 10 * q^81 + 38 * q^82 - 4 * q^87 + 26 * q^88 - 8 * q^90 - 20 * q^92 - 16 * q^94 - 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.24698i	0.509077i
$7$	2.49396i	0.942628i	0.881965	+	0.471314i	$0.156220\pi$
$7$	2.49396i	0.942628i	−0.881965	+	0.471314i	$0.843780\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.44504	−0.481681
$10$	1.00000	0.316228
$11$	5.80194i	1.74935i	0.484710	+	0.874675i	$0.338925\pi$
$11$	5.80194i	1.74935i	−0.484710	+	0.874675i	$0.661075\pi$
$12$	−1.24698	−0.359972
$13$	0	0
$13$	0	0
$14$	−2.49396	−0.666539
$15$	− 1.24698i	− 0.321969i
$16$	1.00000	0.250000
$17$	−4.29590	−1.04191	−0.520954	−	0.853585i	$-0.674424\pi$
$17$	−4.29590	−1.04191	−0.520954	+	0.853585i	$0.674424\pi$
$18$	− 1.44504i	− 0.340600i
$19$	− 4.04892i	− 0.928885i	−0.885603	−	0.464443i	$-0.846255\pi$
$19$	− 4.04892i	− 0.928885i	0.885603	−	0.464443i	$-0.153745\pi$
$20$	1.00000i	0.223607i
$21$	3.10992i	0.678639i
$22$	−5.80194	−1.23698
$23$	3.10992	0.648462	0.324231	−	0.945978i	$-0.394894\pi$
$23$	3.10992	0.648462	0.324231	+	0.945978i	$0.394894\pi$
$24$	− 1.24698i	− 0.254539i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.54288	−1.06673
$28$	− 2.49396i	− 0.471314i
$29$	−5.60388	−1.04061	−0.520307	−	0.853979i	$-0.674182\pi$
$29$	−5.60388	−1.04061	−0.520307	+	0.853979i	$0.674182\pi$
$30$	1.24698	0.227666
$31$	7.70171i	1.38327i	0.722248	+	0.691634i	$0.243109\pi$
$31$	7.70171i	1.38327i	−0.722248	+	0.691634i	$0.756891\pi$
$32$	1.00000i	0.176777i
$33$	7.23490i	1.25943i
$34$	− 4.29590i	− 0.736740i
$35$	2.49396	0.421556
$36$	1.44504	0.240840
$37$	2.67025i	0.438987i	0.975614	+	0.219493i	$0.0704404\pi$
$37$	2.67025i	0.438987i	−0.975614	+	0.219493i	$0.929560\pi$
$38$	4.04892	0.656821
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 12.5429i	− 1.95887i	−0.201764	−	0.979434i	$-0.564667\pi$
$41$	− 12.5429i	− 1.95887i	0.201764	−	0.979434i	$-0.435333\pi$
$42$	−3.10992	−0.479870
$43$	−6.98254	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$43$	−6.98254	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$44$	− 5.80194i	− 0.874675i
$45$	1.44504i	0.215414i
$46$	3.10992i	0.458532i
$47$	− 3.87800i	− 0.565665i	−0.959169	−	0.282832i	$-0.908726\pi$
$47$	− 3.87800i	− 0.565665i	0.959169	−	0.282832i	$-0.0912740\pi$
$48$	1.24698	0.179986
$49$	0.780167	0.111452
$50$	− 1.00000i	− 0.141421i
$51$	−5.35690	−0.750115
$52$	0	0
$53$	−4.93362	−0.677685	−0.338843	−	0.940843i	$-0.610035\pi$
$53$	−4.93362	−0.677685	−0.338843	+	0.940843i	$0.610035\pi$
$54$	− 5.54288i	− 0.754290i
$55$	5.80194	0.782333
$56$	2.49396	0.333269
$57$	− 5.04892i	− 0.668745i
$58$	− 5.60388i	− 0.735825i
$59$	10.0151i	1.30385i	0.758283	+	0.651925i	$0.226038\pi$
$59$	10.0151i	1.30385i	−0.758283	+	0.651925i	$0.773962\pi$
$60$	1.24698i	0.160984i
$61$	−4.93362	−0.631686	−0.315843	−	0.948811i	$-0.602287\pi$
$61$	−4.93362	−0.631686	−0.315843	+	0.948811i	$0.602287\pi$
$62$	−7.70171	−0.978118
$63$	− 3.60388i	− 0.454046i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−7.23490	−0.890554
$67$	8.01507i	0.979196i	0.871948	+	0.489598i	$0.162856\pi$
$67$	8.01507i	0.979196i	−0.871948	+	0.489598i	$0.837144\pi$
$68$	4.29590	0.520954
$69$	3.87800	0.466857
$70$	2.49396i	0.298085i
$71$	5.48188i	0.650579i	0.945614	+	0.325290i	$0.105462\pi$
$71$	5.48188i	0.650579i	−0.945614	+	0.325290i	$0.894538\pi$
$72$	1.44504i	0.170300i
$73$	8.67456i	1.01528i	0.861569	+	0.507640i	$0.169482\pi$
$73$	8.67456i	1.01528i	−0.861569	+	0.507640i	$0.830518\pi$
$74$	−2.67025	−0.310410
$75$	−1.24698	−0.143989
$76$	4.04892i	0.464443i
$77$	−14.4698	−1.64899
$78$	0	0
$79$	−1.82371	−0.205183	−0.102592	−	0.994724i	$-0.532713\pi$
$79$	−1.82371	−0.205183	−0.102592	+	0.994724i	$0.532713\pi$
$80$	− 1.00000i	− 0.111803i
$81$	−2.57673	−0.286303
$82$	12.5429	1.38513
$83$	14.6407i	1.60703i	0.595287	+	0.803513i	$0.297039\pi$
$83$	14.6407i	1.60703i	−0.595287	+	0.803513i	$0.702961\pi$
$84$	− 3.10992i	− 0.339320i
$85$	4.29590i	0.465955i
$86$	− 6.98254i	− 0.752947i
$87$	−6.98792	−0.749183
$88$	5.80194	0.618489
$89$	− 0.454731i	− 0.0482013i	−0.999710	−	0.0241007i	$-0.992328\pi$
$89$	− 0.454731i	− 0.0482013i	0.999710	−	0.0241007i	$-0.00767222\pi$
$90$	−1.44504	−0.152321
$91$	0	0
$92$	−3.10992	−0.324231
$93$	9.60388i	0.995875i
$94$	3.87800	0.399985
$95$	−4.04892	−0.415410
$96$	1.24698i	0.127269i
$97$	− 8.69202i	− 0.882541i	−0.897374	−	0.441271i	$-0.854528\pi$
$97$	− 8.69202i	− 0.882541i	0.897374	−	0.441271i	$-0.145472\pi$
$98$	0.780167i	0.0788088i
$99$	− 8.38404i	− 0.842628i
$100$	1.00000	0.100000
$101$	−15.7453	−1.56671	−0.783355	−	0.621574i	$-0.786494\pi$
$101$	−15.7453	−1.56671	−0.783355	+	0.621574i	$0.786494\pi$
$102$	− 5.35690i	− 0.530412i
$103$	3.70171	0.364740	0.182370	−	0.983230i	$-0.441623\pi$
$103$	3.70171	0.364740	0.182370	+	0.983230i	$0.441623\pi$
$104$	0	0
$105$	3.10992	0.303497
$106$	− 4.93362i	− 0.479196i
$107$	15.0804	1.45788	0.728938	−	0.684580i	$-0.240014\pi$
$107$	15.0804	1.45788	0.728938	+	0.684580i	$0.240014\pi$
$108$	5.54288	0.533364
$109$	17.4819i	1.67446i	0.546851	+	0.837230i	$0.315827\pi$
$109$	17.4819i	1.67446i	−0.546851	+	0.837230i	$0.684173\pi$
$110$	5.80194i	0.553193i
$111$	3.32975i	0.316046i
$112$	2.49396i	0.235657i
$113$	−8.23490	−0.774674	−0.387337	−	0.921938i	$-0.626605\pi$
$113$	−8.23490	−0.774674	−0.387337	+	0.921938i	$0.626605\pi$
$114$	5.04892	0.472874
$115$	− 3.10992i	− 0.290001i
$116$	5.60388	0.520307
$117$	0	0
$118$	−10.0151	−0.921962
$119$	− 10.7138i	− 0.982132i
$120$	−1.24698	−0.113833
$121$	−22.6625	−2.06023
$122$	− 4.93362i	− 0.446669i
$123$	− 15.6407i	− 1.41028i
$124$	− 7.70171i	− 0.691634i
$125$	1.00000i	0.0894427i
$126$	3.60388	0.321059
$127$	13.0315	1.15635	0.578177	−	0.815911i	$-0.303764\pi$
$127$	13.0315	1.15635	0.578177	+	0.815911i	$0.303764\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.70709	−0.766616
$130$	0	0
$131$	9.82908	0.858771	0.429386	−	0.903121i	$-0.358730\pi$
$131$	9.82908	0.858771	0.429386	+	0.903121i	$0.358730\pi$
$132$	− 7.23490i	− 0.629717i
$133$	10.0978	0.875593
$134$	−8.01507	−0.692396
$135$	5.54288i	0.477055i
$136$	4.29590i	0.368370i
$137$	4.75302i	0.406078i	0.979171	+	0.203039i	$0.0650818\pi$
$137$	4.75302i	0.406078i	−0.979171	+	0.203039i	$0.934918\pi$
$138$	3.87800i	0.330117i
$139$	−2.39373	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$139$	−2.39373	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$140$	−2.49396	−0.210778
$141$	− 4.83579i	− 0.407247i
$142$	−5.48188	−0.460029
$143$	0	0
$144$	−1.44504	−0.120420
$145$	5.60388i	0.465377i
$146$	−8.67456	−0.717912
$147$	0.972853	0.0802396
$148$	− 2.67025i	− 0.219493i
$149$	− 20.8116i	− 1.70495i	−0.522764	−	0.852477i	$-0.675099\pi$
$149$	− 20.8116i	− 1.70495i	0.522764	−	0.852477i	$-0.324901\pi$
$150$	− 1.24698i	− 0.101815i
$151$	5.95646i	0.484730i	0.970185	+	0.242365i	$0.0779231\pi$
$151$	5.95646i	0.484730i	−0.970185	+	0.242365i	$0.922077\pi$
$152$	−4.04892	−0.328411
$153$	6.20775	0.501867
$154$	− 14.4698i	− 1.16601i
$155$	7.70171	0.618616
$156$	0	0
$157$	8.19567	0.654086	0.327043	−	0.945010i	$-0.393948\pi$
$157$	8.19567	0.654086	0.327043	+	0.945010i	$0.393948\pi$
$158$	− 1.82371i	− 0.145086i
$159$	−6.15213	−0.487896
$160$	1.00000	0.0790569
$161$	7.75600i	0.611259i
$162$	− 2.57673i	− 0.202447i
$163$	− 16.8877i	− 1.32275i	−0.750057	−	0.661373i	$-0.769974\pi$
$163$	− 16.8877i	− 1.32275i	0.750057	−	0.661373i	$-0.230026\pi$
$164$	12.5429i	0.979434i
$165$	7.23490	0.563236
$166$	−14.6407	−1.13634
$167$	19.6582i	1.52119i	0.649224	+	0.760597i	$0.275094\pi$
$167$	19.6582i	1.52119i	−0.649224	+	0.760597i	$0.724906\pi$
$168$	3.10992	0.239935
$169$	0	0
$170$	−4.29590	−0.329480
$171$	5.85086i	0.447426i
$172$	6.98254	0.532414
$173$	16.8659	1.28229	0.641146	−	0.767419i	$-0.278459\pi$
$173$	16.8659	1.28229	0.641146	+	0.767419i	$0.278459\pi$
$174$	− 6.98792i	− 0.529753i
$175$	− 2.49396i	− 0.188526i
$176$	5.80194i	0.437338i
$177$	12.4886i	0.938699i
$178$	0.454731	0.0340835
$179$	5.99761	0.448282	0.224141	−	0.974557i	$-0.428042\pi$
$179$	5.99761	0.448282	0.224141	+	0.974557i	$0.428042\pi$
$180$	− 1.44504i	− 0.107707i
$181$	16.5676	1.23146	0.615731	−	0.787956i	$-0.288861\pi$
$181$	16.5676	1.23146	0.615731	+	0.787956i	$0.288861\pi$
$182$	0	0
$183$	−6.15213	−0.454778
$184$	− 3.10992i	− 0.229266i
$185$	2.67025	0.196321
$186$	−9.60388	−0.704190
$187$	− 24.9245i	− 1.82266i
$188$	3.87800i	0.282832i
$189$	− 13.8237i	− 1.00553i
$190$	− 4.04892i	− 0.293739i
$191$	12.7922	0.925615	0.462807	−	0.886459i	$-0.346842\pi$
$191$	12.7922	0.925615	0.462807	+	0.886459i	$0.346842\pi$
$192$	−1.24698	−0.0899930
$193$	− 23.5851i	− 1.69769i	−0.528640	−	0.848846i	$-0.677298\pi$
$193$	− 23.5851i	− 1.69769i	0.528640	−	0.848846i	$-0.322702\pi$
$194$	8.69202	0.624051
$195$	0	0
$196$	−0.780167	−0.0557262
$197$	1.97584i	0.140773i	0.997520	+	0.0703863i	$0.0224232\pi$
$197$	1.97584i	0.140773i	−0.997520	+	0.0703863i	$0.977577\pi$
$198$	8.38404	0.595828
$199$	−0.835790	−0.0592476	−0.0296238	−	0.999561i	$-0.509431\pi$
$199$	−0.835790	−0.0592476	−0.0296238	+	0.999561i	$0.509431\pi$
$200$	1.00000i	0.0707107i
$201$	9.99462i	0.704966i
$202$	− 15.7453i	− 1.10783i
$203$	− 13.9758i	− 0.980911i
$204$	5.35690	0.375058
$205$	−12.5429	−0.876032
$206$	3.70171i	0.257910i
$207$	−4.49396	−0.312352
$208$	0	0
$209$	23.4916	1.62495
$210$	3.10992i	0.214605i
$211$	9.36227	0.644525	0.322263	−	0.946650i	$-0.395557\pi$
$211$	9.36227	0.644525	0.322263	+	0.946650i	$0.395557\pi$
$212$	4.93362	0.338843
$213$	6.83579i	0.468381i
$214$	15.0804i	1.03087i
$215$	6.98254i	0.476205i
$216$	5.54288i	0.377145i
$217$	−19.2078	−1.30391
$218$	−17.4819	−1.18402
$219$	10.8170i	0.730945i
$220$	−5.80194	−0.391167
$221$	0	0
$222$	−3.32975	−0.223478
$223$	− 6.31767i	− 0.423062i	−0.977371	−	0.211531i	$-0.932155\pi$
$223$	− 6.31767i	− 0.423062i	0.977371	−	0.211531i	$-0.0678450\pi$
$224$	−2.49396	−0.166635
$225$	1.44504	0.0963361
$226$	− 8.23490i	− 0.547777i
$227$	4.89440i	0.324852i	0.986721	+	0.162426i	$0.0519320\pi$
$227$	4.89440i	0.324852i	−0.986721	+	0.162426i	$0.948068\pi$
$228$	5.04892i	0.334373i
$229$	0.689629i	0.0455719i	0.999740	+	0.0227860i	$0.00725363\pi$
$229$	0.689629i	0.0455719i	−0.999740	+	0.0227860i	$0.992746\pi$
$230$	3.10992	0.205062
$231$	−18.0435	−1.18718
$232$	5.60388i	0.367912i
$233$	−14.1806	−0.929002	−0.464501	−	0.885573i	$-0.653766\pi$
$233$	−14.1806	−0.929002	−0.464501	+	0.885573i	$0.653766\pi$
$234$	0	0
$235$	−3.87800	−0.252973
$236$	− 10.0151i	− 0.651925i
$237$	−2.27413	−0.147720
$238$	10.7138	0.694472
$239$	− 25.7754i	− 1.66727i	−0.552315	−	0.833635i	$-0.686255\pi$
$239$	− 25.7754i	− 1.66727i	0.552315	−	0.833635i	$-0.313745\pi$
$240$	− 1.24698i	− 0.0804922i
$241$	7.86725i	0.506774i	0.967365	+	0.253387i	$0.0815446\pi$
$241$	7.86725i	0.506774i	−0.967365	+	0.253387i	$0.918455\pi$
$242$	− 22.6625i	− 1.45680i
$243$	13.4155	0.860605
$244$	4.93362	0.315843
$245$	− 0.780167i	− 0.0498431i
$246$	15.6407	0.997215
$247$	0	0
$248$	7.70171	0.489059
$249$	18.2567i	1.15697i
$250$	−1.00000	−0.0632456
$251$	−19.9782	−1.26101	−0.630507	−	0.776183i	$-0.717153\pi$
$251$	−19.9782	−1.26101	−0.630507	+	0.776183i	$0.717153\pi$
$252$	3.60388i	0.227023i
$253$	18.0435i	1.13439i
$254$	13.0315i	0.817666i
$255$	5.35690i	0.335462i
$256$	1.00000	0.0625000
$257$	25.4523	1.58767	0.793837	−	0.608131i	$-0.208081\pi$
$257$	25.4523	1.58767	0.793837	+	0.608131i	$0.208081\pi$
$258$	− 8.70709i	− 0.542080i
$259$	−6.65950	−0.413801
$260$	0	0
$261$	8.09783	0.501243
$262$	9.82908i	0.607243i
$263$	−19.9758	−1.23176	−0.615881	−	0.787839i	$-0.711200\pi$
$263$	−19.9758	−1.23176	−0.615881	+	0.787839i	$0.711200\pi$
$264$	7.23490	0.445277
$265$	4.93362i	0.303070i
$266$	10.0978i	0.619138i
$267$	− 0.567040i	− 0.0347023i
$268$	− 8.01507i	− 0.489598i
$269$	−16.1414	−0.984157	−0.492079	−	0.870551i	$-0.663763\pi$
$269$	−16.1414	−0.984157	−0.492079	+	0.870551i	$0.663763\pi$
$270$	−5.54288	−0.337329
$271$	15.1051i	0.917571i	0.888547	+	0.458786i	$0.151715\pi$
$271$	15.1051i	0.917571i	−0.888547	+	0.458786i	$0.848285\pi$
$272$	−4.29590	−0.260477
$273$	0	0
$274$	−4.75302	−0.287140
$275$	− 5.80194i	− 0.349870i
$276$	−3.87800	−0.233428
$277$	24.4940	1.47170	0.735850	−	0.677145i	$-0.236783\pi$
$277$	24.4940	1.47170	0.735850	+	0.677145i	$0.236783\pi$
$278$	− 2.39373i	− 0.143566i
$279$	− 11.1293i	− 0.666293i
$280$	− 2.49396i	− 0.149043i
$281$	23.3381i	1.39223i	0.717928	+	0.696117i	$0.245091\pi$
$281$	23.3381i	1.39223i	−0.717928	+	0.696117i	$0.754909\pi$
$282$	4.83579	0.287967
$283$	11.4222	0.678980	0.339490	−	0.940610i	$-0.389746\pi$
$283$	11.4222	0.678980	0.339490	+	0.940610i	$0.389746\pi$
$284$	− 5.48188i	− 0.325290i
$285$	−5.04892	−0.299072
$286$	0	0
$287$	31.2814	1.84648
$288$	− 1.44504i	− 0.0851499i
$289$	1.45473	0.0855724
$290$	−5.60388	−0.329071
$291$	− 10.8388i	− 0.635380i
$292$	− 8.67456i	− 0.507640i
$293$	− 24.4155i	− 1.42637i	−0.700976	−	0.713184i	$-0.747252\pi$
$293$	− 24.4155i	− 1.42637i	0.700976	−	0.713184i	$-0.252748\pi$
$294$	0.972853i	0.0567379i
$295$	10.0151	0.583100
$296$	2.67025	0.155205
$297$	− 32.1594i	− 1.86608i
$298$	20.8116	1.20559
$299$	0	0
$300$	1.24698	0.0719944
$301$	− 17.4142i	− 1.00374i
$302$	−5.95646	−0.342756
$303$	−19.6340	−1.12794
$304$	− 4.04892i	− 0.232221i
$305$	4.93362i	0.282499i
$306$	6.20775i	0.354874i
$307$	− 7.70410i	− 0.439696i	−0.975534	−	0.219848i	$-0.929444\pi$
$307$	− 7.70410i	− 0.439696i	0.975534	−	0.219848i	$-0.0705562\pi$
$308$	14.4698	0.824493
$309$	4.61596	0.262593
$310$	7.70171i	0.437428i
$311$	−4.71379	−0.267295	−0.133647	−	0.991029i	$-0.542669\pi$
$311$	−4.71379	−0.267295	−0.133647	+	0.991029i	$0.542669\pi$
$312$	0	0
$313$	8.49157	0.479972	0.239986	−	0.970776i	$-0.422857\pi$
$313$	8.49157	0.479972	0.239986	+	0.970776i	$0.422857\pi$
$314$	8.19567i	0.462508i
$315$	−3.60388	−0.203055
$316$	1.82371	0.102592
$317$	− 27.0508i	− 1.51933i	−0.650317	−	0.759663i	$-0.725364\pi$
$317$	− 27.0508i	− 1.51933i	0.650317	−	0.759663i	$-0.274636\pi$
$318$	− 6.15213i	− 0.344994i
$319$	− 32.5133i	− 1.82040i
$320$	1.00000i	0.0559017i
$321$	18.8049	1.04959
$322$	−7.75600	−0.432225
$323$	17.3937i	0.967813i
$324$	2.57673	0.143152
$325$	0	0
$326$	16.8877	0.935323
$327$	21.7995i	1.20552i
$328$	−12.5429	−0.692564
$329$	9.67158	0.533211
$330$	7.23490i	0.398268i
$331$	28.7439i	1.57991i	0.613166	+	0.789954i	$0.289896\pi$
$331$	28.7439i	1.57991i	−0.613166	+	0.789954i	$0.710104\pi$
$332$	− 14.6407i	− 0.803513i
$333$	− 3.85862i	− 0.211451i
$334$	−19.6582	−1.07565
$335$	8.01507	0.437910
$336$	3.10992i	0.169660i
$337$	34.5972	1.88463	0.942314	−	0.334730i	$-0.108645\pi$
$337$	34.5972	1.88463	0.942314	+	0.334730i	$0.108645\pi$
$338$	0	0
$339$	−10.2687	−0.557722
$340$	− 4.29590i	− 0.232978i
$341$	−44.6848	−2.41982
$342$	−5.85086	−0.316378
$343$	19.4034i	1.04769i
$344$	6.98254i	0.376473i
$345$	− 3.87800i	− 0.208785i
$346$	16.8659i	0.906718i
$347$	−28.3043	−1.51945	−0.759726	−	0.650243i	$-0.774667\pi$
$347$	−28.3043	−1.51945	−0.759726	+	0.650243i	$0.774667\pi$
$348$	6.98792	0.374592
$349$	16.8310i	0.900943i	0.892791	+	0.450471i	$0.148744\pi$
$349$	16.8310i	0.900943i	−0.892791	+	0.450471i	$0.851256\pi$
$350$	2.49396	0.133308
$351$	0	0
$352$	−5.80194	−0.309244
$353$	23.7429i	1.26370i	0.775089	+	0.631852i	$0.217705\pi$
$353$	23.7429i	1.26370i	−0.775089	+	0.631852i	$0.782295\pi$
$354$	−12.4886	−0.663761
$355$	5.48188	0.290948
$356$	0.454731i	0.0241007i
$357$	− 13.3599i	− 0.707080i
$358$	5.99761i	0.316983i
$359$	− 26.8310i	− 1.41609i	−0.706169	−	0.708043i	$-0.749578\pi$
$359$	− 26.8310i	− 1.41609i	0.706169	−	0.708043i	$-0.250422\pi$
$360$	1.44504	0.0761604
$361$	2.60627	0.137172
$362$	16.5676i	0.870775i
$363$	−28.2597	−1.48325
$364$	0	0
$365$	8.67456	0.454047
$366$	− 6.15213i	− 0.321577i
$367$	−18.0978	−0.944699	−0.472350	−	0.881411i	$-0.656594\pi$
$367$	−18.0978	−0.944699	−0.472350	+	0.881411i	$0.656594\pi$
$368$	3.10992	0.162116
$369$	18.1250i	0.943549i
$370$	2.67025i	0.138820i
$371$	− 12.3043i	− 0.638805i
$372$	− 9.60388i	− 0.497938i
$373$	14.3720	0.744152	0.372076	−	0.928202i	$-0.378646\pi$
$373$	14.3720	0.744152	0.372076	+	0.928202i	$0.378646\pi$
$374$	24.9245	1.28882
$375$	1.24698i	0.0643937i
$376$	−3.87800	−0.199993
$377$	0	0
$378$	13.8237	0.711015
$379$	26.1806i	1.34481i	0.740185	+	0.672404i	$0.234738\pi$
$379$	26.1806i	1.34481i	−0.740185	+	0.672404i	$0.765262\pi$
$380$	4.04892	0.207705
$381$	16.2500	0.832511
$382$	12.7922i	0.654508i
$383$	− 11.3491i	− 0.579913i	−0.957040	−	0.289957i	$-0.906359\pi$
$383$	− 11.3491i	− 0.579913i	0.957040	−	0.289957i	$-0.0936409\pi$
$384$	− 1.24698i	− 0.0636347i
$385$	14.4698i	0.737449i
$386$	23.5851	1.20045
$387$	10.0901	0.512907
$388$	8.69202i	0.441271i
$389$	12.8465	0.651346	0.325673	−	0.945483i	$-0.394409\pi$
$389$	12.8465	0.651346	0.325673	+	0.945483i	$0.394409\pi$
$390$	0	0
$391$	−13.3599	−0.675638
$392$	− 0.780167i	− 0.0394044i
$393$	12.2567	0.618267
$394$	−1.97584	−0.0995412
$395$	1.82371i	0.0917607i
$396$	8.38404i	0.421314i
$397$	18.7439i	0.940731i	0.882472	+	0.470365i	$0.155878\pi$
$397$	18.7439i	0.940731i	−0.882472	+	0.470365i	$0.844122\pi$
$398$	− 0.835790i	− 0.0418943i
$399$	12.5918	0.630378
$400$	−1.00000	−0.0500000
$401$	19.8213i	0.989829i	0.868942	+	0.494915i	$0.164801\pi$
$401$	19.8213i	0.989829i	−0.868942	+	0.494915i	$0.835199\pi$
$402$	−9.99462	−0.498486
$403$	0	0
$404$	15.7453	0.783355
$405$	2.57673i	0.128039i
$406$	13.9758	0.693609
$407$	−15.4926	−0.767941
$408$	5.35690i	0.265206i
$409$	18.9487i	0.936952i	0.883476	+	0.468476i	$0.155197\pi$
$409$	18.9487i	0.936952i	−0.883476	+	0.468476i	$0.844803\pi$
$410$	− 12.5429i	− 0.619449i
$411$	5.92692i	0.292353i
$412$	−3.70171	−0.182370
$413$	−24.9772	−1.22905
$414$	− 4.49396i	− 0.220866i
$415$	14.6407	0.718684
$416$	0	0
$417$	−2.98493	−0.146173
$418$	23.4916i	1.14901i
$419$	1.86725	0.0912211	0.0456105	−	0.998959i	$-0.485477\pi$
$419$	1.86725	0.0912211	0.0456105	+	0.998959i	$0.485477\pi$
$420$	−3.10992	−0.151748
$421$	27.3250i	1.33174i	0.746069	+	0.665869i	$0.231939\pi$
$421$	27.3250i	1.33174i	−0.746069	+	0.665869i	$0.768061\pi$
$422$	9.36227i	0.455748i
$423$	5.60388i	0.272470i
$424$	4.93362i	0.239598i
$425$	4.29590	0.208382
$426$	−6.83579	−0.331195
$427$	− 12.3043i	− 0.595445i
$428$	−15.0804	−0.728938
$429$	0	0
$430$	−6.98254	−0.336728
$431$	− 1.56033i	− 0.0751587i	−0.999294	−	0.0375793i	$-0.988035\pi$
$431$	− 1.56033i	− 0.0751587i	0.999294	−	0.0375793i	$-0.0119647\pi$
$432$	−5.54288	−0.266682
$433$	24.1564	1.16088	0.580442	−	0.814301i	$-0.302880\pi$
$433$	24.1564	1.16088	0.580442	+	0.814301i	$0.302880\pi$
$434$	− 19.2078i	− 0.922002i
$435$	6.98792i	0.335045i
$436$	− 17.4819i	− 0.837230i
$437$	− 12.5918i	− 0.602347i
$438$	−10.8170	−0.516856
$439$	−21.3599	−1.01945	−0.509726	−	0.860337i	$-0.670253\pi$
$439$	−21.3599	−1.01945	−0.509726	+	0.860337i	$0.670253\pi$
$440$	− 5.80194i	− 0.276597i
$441$	−1.12737	−0.0536845
$442$	0	0
$443$	−16.5351	−0.785607	−0.392803	−	0.919623i	$-0.628495\pi$
$443$	−16.5351	−0.785607	−0.392803	+	0.919623i	$0.628495\pi$
$444$	− 3.32975i	− 0.158023i
$445$	−0.454731	−0.0215563
$446$	6.31767	0.299150
$447$	− 25.9517i	− 1.22747i
$448$	− 2.49396i	− 0.117828i
$449$	6.56704i	0.309918i	0.987921	+	0.154959i	$0.0495245\pi$
$449$	6.56704i	0.309918i	−0.987921	+	0.154959i	$0.950475\pi$
$450$	1.44504i	0.0681199i
$451$	72.7730	3.42675
$452$	8.23490	0.387337
$453$	7.42758i	0.348978i
$454$	−4.89440	−0.229705
$455$	0	0
$456$	−5.04892	−0.236437
$457$	24.1715i	1.13070i	0.824853	+	0.565348i	$0.191258\pi$
$457$	24.1715i	1.13070i	−0.824853	+	0.565348i	$0.808742\pi$
$458$	−0.689629	−0.0322242
$459$	23.8116	1.11143
$460$	3.10992i	0.145001i
$461$	23.5797i	1.09822i	0.835751	+	0.549108i	$0.185033\pi$
$461$	23.5797i	1.09822i	−0.835751	+	0.549108i	$0.814967\pi$
$462$	− 18.0435i	− 0.839461i
$463$	1.68233i	0.0781846i	0.999236	+	0.0390923i	$0.0124466\pi$
$463$	1.68233i	0.0781846i	−0.999236	+	0.0390923i	$0.987553\pi$
$464$	−5.60388	−0.260153
$465$	9.60388	0.445369
$466$	− 14.1806i	− 0.656904i
$467$	−0.376273	−0.0174118	−0.00870592	−	0.999962i	$-0.502771\pi$
$467$	−0.376273	−0.0174118	−0.00870592	+	0.999962i	$0.502771\pi$
$468$	0	0
$469$	−19.9892	−0.923018
$470$	− 3.87800i	− 0.178879i
$471$	10.2198	0.470905
$472$	10.0151	0.460981
$473$	− 40.5123i	− 1.86276i
$474$	− 2.27413i	− 0.104454i
$475$	4.04892i	0.185777i
$476$	10.7138i	0.491066i
$477$	7.12929	0.326428
$478$	25.7754	1.17894
$479$	10.0301i	0.458288i	0.973392	+	0.229144i	$0.0735927\pi$
$479$	10.0301i	0.458288i	−0.973392	+	0.229144i	$0.926407\pi$
$480$	1.24698	0.0569166
$481$	0	0
$482$	−7.86725	−0.358343
$483$	9.67158i	0.440072i
$484$	22.6625	1.03011
$485$	−8.69202	−0.394684
$486$	13.4155i	0.608540i
$487$	− 15.3297i	− 0.694657i	−0.937743	−	0.347329i	$-0.887089\pi$
$487$	− 15.3297i	− 0.694657i	0.937743	−	0.347329i	$-0.112911\pi$
$488$	4.93362i	0.223335i
$489$	− 21.0586i	− 0.952303i
$490$	0.780167	0.0352444
$491$	−9.19269	−0.414860	−0.207430	−	0.978250i	$-0.566510\pi$
$491$	−9.19269	−0.414860	−0.207430	+	0.978250i	$0.566510\pi$
$492$	15.6407i	0.705138i
$493$	24.0737	1.08422
$494$	0	0
$495$	−8.38404	−0.376835
$496$	7.70171i	0.345817i
$497$	−13.6716	−0.613254
$498$	−18.2567	−0.818101
$499$	− 2.74632i	− 0.122942i	−0.998109	−	0.0614710i	$-0.980421\pi$
$499$	− 2.74632i	− 0.122942i	0.998109	−	0.0614710i	$-0.0195792\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	24.5133i	1.09518i
$502$	− 19.9782i	− 0.891672i
$503$	−12.5676	−0.560363	−0.280181	−	0.959947i	$-0.590395\pi$
$503$	−12.5676	−0.560363	−0.280181	+	0.959947i	$0.590395\pi$
$504$	−3.60388	−0.160529
$505$	15.7453i	0.700654i
$506$	−18.0435	−0.802133
$507$	0	0
$508$	−13.0315	−0.578177
$509$	− 0.0677037i	− 0.00300091i	−0.999999	−	0.00150046i	$-0.999522\pi$
$509$	− 0.0677037i	− 0.00300091i	0.999999	−	0.00150046i	$-0.000477610\pi$
$510$	−5.35690	−0.237207
$511$	−21.6340	−0.957032
$512$	1.00000i	0.0441942i
$513$	22.4426i	0.990867i
$514$	25.4523i	1.12265i
$515$	− 3.70171i	− 0.163117i
$516$	8.70709	0.383308
$517$	22.4999	0.989546
$518$	− 6.65950i	− 0.292602i
$519$	21.0315	0.923179
$520$	0	0
$521$	8.18060	0.358399	0.179199	−	0.983813i	$-0.442649\pi$
$521$	8.18060	0.358399	0.179199	+	0.983813i	$0.442649\pi$
$522$	8.09783i	0.354433i
$523$	16.9312	0.740351	0.370176	−	0.928962i	$-0.379298\pi$
$523$	16.9312	0.740351	0.370176	+	0.928962i	$0.379298\pi$
$524$	−9.82908	−0.429386
$525$	− 3.10992i	− 0.135728i
$526$	− 19.9758i	− 0.870988i
$527$	− 33.0858i	− 1.44124i
$528$	7.23490i	0.314859i
$529$	−13.3284	−0.579497
$530$	−4.93362	−0.214303
$531$	− 14.4722i	− 0.628040i
$532$	−10.0978	−0.437797
$533$	0	0
$534$	0.567040	0.0245382
$535$	− 15.0804i	− 0.651982i
$536$	8.01507	0.346198
$537$	7.47889	0.322738
$538$	− 16.1414i	− 0.695904i
$539$	4.52648i	0.194969i
$540$	− 5.54288i	− 0.238527i
$541$	− 16.5676i	− 0.712298i	−0.934429	−	0.356149i	$-0.884090\pi$
$541$	− 16.5676i	− 0.712298i	0.934429	−	0.356149i	$-0.115910\pi$
$542$	−15.1051	−0.648821
$543$	20.6595	0.886584
$544$	− 4.29590i	− 0.184185i
$545$	17.4819	0.748841
$546$	0	0
$547$	−3.57540	−0.152873	−0.0764365	−	0.997074i	$-0.524354\pi$
$547$	−3.57540	−0.152873	−0.0764365	+	0.997074i	$0.524354\pi$
$548$	− 4.75302i	− 0.203039i
$549$	7.12929	0.304271
$550$	5.80194	0.247395
$551$	22.6896i	0.966611i
$552$	− 3.87800i	− 0.165059i
$553$	− 4.54825i	− 0.193411i
$554$	24.4940i	1.04065i
$555$	3.32975	0.141340
$556$	2.39373	0.101517
$557$	− 39.8926i	− 1.69030i	−0.534526	−	0.845152i	$-0.679510\pi$
$557$	− 39.8926i	− 1.69030i	0.534526	−	0.845152i	$-0.320490\pi$
$558$	11.1293	0.471141
$559$	0	0
$560$	2.49396	0.105389
$561$	− 31.0804i	− 1.31221i
$562$	−23.3381	−0.984459
$563$	3.64742	0.153720	0.0768601	−	0.997042i	$-0.475511\pi$
$563$	3.64742	0.153720	0.0768601	+	0.997042i	$0.475511\pi$
$564$	4.83579i	0.203623i
$565$	8.23490i	0.346445i
$566$	11.4222i	0.480111i
$567$	− 6.42626i	− 0.269877i
$568$	5.48188	0.230014
$569$	−18.3521	−0.769360	−0.384680	−	0.923050i	$-0.625688\pi$
$569$	−18.3521	−0.769360	−0.384680	+	0.923050i	$0.625688\pi$
$570$	− 5.04892i	− 0.211476i
$571$	19.1987	0.803439	0.401719	−	0.915763i	$-0.368413\pi$
$571$	19.1987	0.803439	0.401719	+	0.915763i	$0.368413\pi$
$572$	0	0
$573$	15.9517	0.666391
$574$	31.2814i	1.30566i
$575$	−3.10992	−0.129692
$576$	1.44504	0.0602101
$577$	35.2218i	1.46630i	0.680067	+	0.733150i	$0.261951\pi$
$577$	35.2218i	1.46630i	−0.680067	+	0.733150i	$0.738049\pi$
$578$	1.45473i	0.0605088i
$579$	− 29.4101i	− 1.22224i
$580$	− 5.60388i	− 0.232688i
$581$	−36.5133	−1.51483
$582$	10.8388	0.449282
$583$	− 28.6246i	− 1.18551i
$584$	8.67456	0.358956
$585$	0	0
$586$	24.4155	1.00860
$587$	− 32.4596i	− 1.33975i	−0.742473	−	0.669876i	$-0.766347\pi$
$587$	− 32.4596i	− 1.33975i	0.742473	−	0.669876i	$-0.233653\pi$
$588$	−0.972853	−0.0401198
$589$	31.1836	1.28490
$590$	10.0151i	0.412314i
$591$	2.46383i	0.101348i
$592$	2.67025i	0.109747i
$593$	− 9.57135i	− 0.393048i	−0.980499	−	0.196524i	$-0.937035\pi$
$593$	− 9.57135i	− 0.393048i	0.980499	−	0.196524i	$-0.0629654\pi$
$594$	32.1594	1.31952
$595$	−10.7138	−0.439223
$596$	20.8116i	0.852477i
$597$	−1.04221	−0.0426549
$598$	0	0
$599$	−29.0858	−1.18841	−0.594206	−	0.804313i	$-0.702534\pi$
$599$	−29.0858	−1.18841	−0.594206	+	0.804313i	$0.702534\pi$
$600$	1.24698i	0.0509077i
$601$	−6.31229	−0.257484	−0.128742	−	0.991678i	$-0.541094\pi$
$601$	−6.31229	−0.257484	−0.128742	+	0.991678i	$0.541094\pi$
$602$	17.4142	0.709749
$603$	− 11.5821i	− 0.471660i
$604$	− 5.95646i	− 0.242365i
$605$	22.6625i	0.921361i
$606$	− 19.6340i	− 0.797577i
$607$	32.4650	1.31771	0.658857	−	0.752268i	$-0.271040\pi$
$607$	32.4650	1.31771	0.658857	+	0.752268i	$0.271040\pi$
$608$	4.04892	0.164205
$609$	− 17.4276i	− 0.706201i
$610$	−4.93362	−0.199757
$611$	0	0
$612$	−6.20775	−0.250933
$613$	− 29.7017i	− 1.19964i	−0.800135	−	0.599820i	$-0.795239\pi$
$613$	− 29.7017i	− 1.19964i	0.800135	−	0.599820i	$-0.204761\pi$
$614$	7.70410	0.310912
$615$	−15.6407	−0.630694
$616$	14.4698i	0.583005i
$617$	− 10.8556i	− 0.437032i	−0.975833	−	0.218516i	$-0.929878\pi$
$617$	− 10.8556i	− 0.437032i	0.975833	−	0.218516i	$-0.0701215\pi$
$618$	4.61596i	0.185681i
$619$	23.2948i	0.936298i	0.883649	+	0.468149i	$0.155079\pi$
$619$	23.2948i	0.936298i	−0.883649	+	0.468149i	$0.844921\pi$
$620$	−7.70171	−0.309308
$621$	−17.2379	−0.691732
$622$	− 4.71379i	− 0.189006i
$623$	1.13408	0.0454359
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.49157i	0.339391i
$627$	29.2935	1.16987
$628$	−8.19567	−0.327043
$629$	− 11.4711i	− 0.457384i
$630$	− 3.60388i	− 0.143582i
$631$	− 6.72455i	− 0.267700i	−0.991002	−	0.133850i	$-0.957266\pi$
$631$	− 6.72455i	− 0.267700i	0.991002	−	0.133850i	$-0.0427340\pi$
$632$	1.82371i	0.0725432i
$633$	11.6746	0.464022
$634$	27.0508	1.07433
$635$	− 13.0315i	− 0.517138i
$636$	6.15213	0.243948
$637$	0	0
$638$	32.5133	1.28722
$639$	− 7.92154i	− 0.313371i
$640$	−1.00000	−0.0395285
$641$	−9.32437	−0.368291	−0.184145	−	0.982899i	$-0.558952\pi$
$641$	−9.32437	−0.368291	−0.184145	+	0.982899i	$0.558952\pi$
$642$	18.8049i	0.742171i
$643$	− 12.2145i	− 0.481691i	−0.970563	−	0.240846i	$-0.922575\pi$
$643$	− 12.2145i	− 0.481691i	0.970563	−	0.240846i	$-0.0774247\pi$
$644$	− 7.75600i	− 0.305629i
$645$	8.70709i	0.342841i
$646$	−17.3937	−0.684347
$647$	36.8853	1.45011	0.725055	−	0.688691i	$-0.241814\pi$
$647$	36.8853	1.45011	0.725055	+	0.688691i	$0.241814\pi$
$648$	2.57673i	0.101223i
$649$	−58.1068	−2.28089
$650$	0	0
$651$	−23.9517	−0.938740
$652$	16.8877i	0.661373i
$653$	−38.6655	−1.51310	−0.756548	−	0.653938i	$-0.773116\pi$
$653$	−38.6655	−1.51310	−0.756548	+	0.653938i	$0.773116\pi$
$654$	−21.7995	−0.852430
$655$	− 9.82908i	− 0.384054i
$656$	− 12.5429i	− 0.489717i
$657$	− 12.5351i	− 0.489041i
$658$	9.67158i	0.377037i
$659$	−1.19269	−0.0464604	−0.0232302	−	0.999730i	$-0.507395\pi$
$659$	−1.19269	−0.0464604	−0.0232302	+	0.999730i	$0.507395\pi$
$660$	−7.23490	−0.281618
$661$	− 3.93230i	− 0.152949i	−0.997072	−	0.0764743i	$-0.975634\pi$
$661$	− 3.93230i	− 0.152949i	0.997072	−	0.0764743i	$-0.0243663\pi$
$662$	−28.7439	−1.11716
$663$	0	0
$664$	14.6407	0.568170
$665$	− 10.0978i	− 0.391577i
$666$	3.85862	0.149519
$667$	−17.4276	−0.674799
$668$	− 19.6582i	− 0.760597i
$669$	− 7.87800i	− 0.304581i
$670$	8.01507i	0.309649i
$671$	− 28.6246i	− 1.10504i
$672$	−3.10992	−0.119968
$673$	−4.87023	−0.187734	−0.0938668	−	0.995585i	$-0.529923\pi$
$673$	−4.87023	−0.187734	−0.0938668	+	0.995585i	$0.529923\pi$
$674$	34.5972i	1.33263i
$675$	5.54288	0.213345
$676$	0	0
$677$	−14.6461	−0.562895	−0.281447	−	0.959577i	$-0.590815\pi$
$677$	−14.6461	−0.562895	−0.281447	+	0.959577i	$0.590815\pi$
$678$	− 10.2687i	− 0.394369i
$679$	21.6775	0.831908
$680$	4.29590	0.164740
$681$	6.10321i	0.233876i
$682$	− 44.6848i	− 1.71107i
$683$	35.1997i	1.34688i	0.739242	+	0.673440i	$0.235184\pi$
$683$	35.1997i	1.34688i	−0.739242	+	0.673440i	$0.764816\pi$
$684$	− 5.85086i	− 0.223713i
$685$	4.75302	0.181604
$686$	−19.4034	−0.740826
$687$	0.859953i	0.0328092i
$688$	−6.98254	−0.266207
$689$	0	0
$690$	3.87800	0.147633
$691$	41.5851i	1.58197i	0.611835	+	0.790986i	$0.290432\pi$
$691$	41.5851i	1.58197i	−0.611835	+	0.790986i	$0.709568\pi$
$692$	−16.8659	−0.641146
$693$	20.9095	0.794285
$694$	− 28.3043i	− 1.07441i
$695$	2.39373i	0.0907994i
$696$	6.98792i	0.264876i
$697$	53.8829i	2.04096i
$698$	−16.8310	−0.637063
$699$	−17.6829	−0.668830
$700$	2.49396i	0.0942628i
$701$	−5.61463	−0.212062	−0.106031	−	0.994363i	$-0.533814\pi$
$701$	−5.61463	−0.212062	−0.106031	+	0.994363i	$0.533814\pi$
$702$	0	0
$703$	10.8116	0.407768
$704$	− 5.80194i	− 0.218669i
$705$	−4.83579	−0.182126
$706$	−23.7429	−0.893574
$707$	− 39.2680i	− 1.47683i
$708$	− 12.4886i	− 0.469350i
$709$	34.8659i	1.30942i	0.755881	+	0.654709i	$0.227209\pi$
$709$	34.8659i	1.30942i	−0.755881	+	0.654709i	$0.772791\pi$
$710$	5.48188i	0.205731i
$711$	2.63533	0.0988328
$712$	−0.454731	−0.0170417
$713$	23.9517i	0.896997i
$714$	13.3599	0.499981
$715$	0	0
$716$	−5.99761	−0.224141
$717$	− 32.1414i	− 1.20034i
$718$	26.8310	1.00132
$719$	1.90217	0.0709388	0.0354694	−	0.999371i	$-0.488707\pi$
$719$	1.90217	0.0709388	0.0354694	+	0.999371i	$0.488707\pi$
$720$	1.44504i	0.0538535i
$721$	9.23191i	0.343814i
$722$	2.60627i	0.0969953i
$723$	9.81030i	0.364849i
$724$	−16.5676	−0.615731
$725$	5.60388	0.208123
$726$	− 28.2597i	− 1.04881i
$727$	23.7211	0.879766	0.439883	−	0.898055i	$-0.355020\pi$
$727$	23.7211	0.879766	0.439883	+	0.898055i	$0.355020\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	8.67456i	0.321060i
$731$	29.9963	1.10945
$732$	6.15213	0.227389
$733$	13.2862i	0.490737i	0.969430	+	0.245369i	$0.0789090\pi$
$733$	13.2862i	0.490737i	−0.969430	+	0.245369i	$0.921091\pi$
$734$	− 18.0978i	− 0.668003i
$735$	− 0.972853i	− 0.0358842i
$736$	3.10992i	0.114633i
$737$	−46.5029	−1.71296
$738$	−18.1250	−0.667190
$739$	6.34614i	0.233447i	0.993164	+	0.116723i	$0.0372391\pi$
$739$	6.34614i	0.233447i	−0.993164	+	0.116723i	$0.962761\pi$
$740$	−2.67025	−0.0981604
$741$	0	0
$742$	12.3043	0.451704
$743$	− 40.1801i	− 1.47407i	−0.675857	−	0.737033i	$-0.736226\pi$
$743$	− 40.1801i	− 1.47407i	0.675857	−	0.737033i	$-0.263774\pi$
$744$	9.60388	0.352095
$745$	−20.8116	−0.762479
$746$	14.3720i	0.526195i
$747$	− 21.1564i	− 0.774074i
$748$	24.9245i	0.911331i
$749$	37.6098i	1.37423i
$750$	−1.24698	−0.0455333
$751$	−15.5362	−0.566923	−0.283461	−	0.958984i	$-0.591483\pi$
$751$	−15.5362	−0.566923	−0.283461	+	0.958984i	$0.591483\pi$
$752$	− 3.87800i	− 0.141416i
$753$	−24.9124	−0.907860
$754$	0	0
$755$	5.95646	0.216778
$756$	13.8237i	0.502763i
$757$	32.2064	1.17056	0.585281	−	0.810830i	$-0.300984\pi$
$757$	32.2064	1.17056	0.585281	+	0.810830i	$0.300984\pi$
$758$	−26.1806	−0.950922
$759$	22.4999i	0.816696i
$760$	4.04892i	0.146870i
$761$	19.6472i	0.712209i	0.934446	+	0.356104i	$0.115895\pi$
$761$	19.6472i	0.712209i	−0.934446	+	0.356104i	$0.884105\pi$
$762$	16.2500i	0.588674i
$763$	−43.5991	−1.57839
$764$	−12.7922	−0.462807
$765$	− 6.20775i	− 0.224442i
$766$	11.3491	0.410061
$767$	0	0
$768$	1.24698	0.0449965
$769$	− 6.50232i	− 0.234480i	−0.993104	−	0.117240i	$-0.962595\pi$
$769$	− 6.50232i	− 0.234480i	0.993104	−	0.117240i	$-0.0374046\pi$
$770$	−14.4698	−0.521455
$771$	31.7385	1.14304
$772$	23.5851i	0.848846i
$773$	32.2586i	1.16026i	0.814524	+	0.580130i	$0.196998\pi$
$773$	32.2586i	1.16026i	−0.814524	+	0.580130i	$0.803002\pi$
$774$	10.0901i	0.362680i
$775$	− 7.70171i	− 0.276654i
$776$	−8.69202	−0.312025
$777$	−8.30426	−0.297914
$778$	12.8465i	0.460571i
$779$	−50.7851	−1.81956
$780$	0	0
$781$	−31.8055	−1.13809
$782$	− 13.3599i	− 0.477748i
$783$	31.0616	1.11005
$784$	0.780167	0.0278631
$785$	− 8.19567i	− 0.292516i
$786$	12.2567i	0.437181i
$787$	13.6799i	0.487637i	0.969821	+	0.243819i	$0.0784002\pi$
$787$	13.6799i	0.487637i	−0.969821	+	0.243819i	$0.921600\pi$
$788$	− 1.97584i	− 0.0703863i
$789$	−24.9095	−0.886800
$790$	−1.82371	−0.0648846
$791$	− 20.5375i	− 0.730229i
$792$	−8.38404	−0.297914
$793$	0	0
$794$	−18.7439	−0.665197
$795$	6.15213i	0.218194i
$796$	0.835790	0.0296238
$797$	7.77538	0.275418	0.137709	−	0.990473i	$-0.456026\pi$
$797$	7.77538	0.275418	0.137709	+	0.990473i	$0.456026\pi$
$798$	12.5918i	0.445745i
$799$	16.6595i	0.589371i
$800$	− 1.00000i	− 0.0353553i
$801$	0.657105i	0.0232177i
$802$	−19.8213	−0.699915
$803$	−50.3293	−1.77608
$804$	− 9.99462i	− 0.352483i
$805$	7.75600	0.273363
$806$	0	0
$807$	−20.1280	−0.708538
$808$	15.7453i	0.553916i
$809$	−43.1540	−1.51722	−0.758608	−	0.651548i	$-0.774120\pi$
$809$	−43.1540	−1.51722	−0.758608	+	0.651548i	$0.774120\pi$
$810$	−2.57673	−0.0905370
$811$	35.8165i	1.25769i	0.777531	+	0.628844i	$0.216472\pi$
$811$	35.8165i	1.25769i	−0.777531	+	0.628844i	$0.783528\pi$
$812$	13.9758i	0.490456i
$813$	18.8358i	0.660600i
$814$	− 15.4926i	− 0.543016i
$815$	−16.8877	−0.591550
$816$	−5.35690	−0.187529
$817$	28.2717i	0.989103i
$818$	−18.9487	−0.662525
$819$	0	0
$820$	12.5429	0.438016
$821$	− 6.70304i	− 0.233938i	−0.993136	−	0.116969i	$-0.962682\pi$
$821$	− 6.70304i	− 0.233938i	0.993136	−	0.116969i	$-0.0373178\pi$
$822$	−5.92692	−0.206725
$823$	−35.7103	−1.24478	−0.622392	−	0.782706i	$-0.713839\pi$
$823$	−35.7103	−1.24478	−0.622392	+	0.782706i	$0.713839\pi$
$824$	− 3.70171i	− 0.128955i
$825$	− 7.23490i	− 0.251887i
$826$	− 24.9772i	− 0.869067i
$827$	− 1.79523i	− 0.0624264i	−0.999513	−	0.0312132i	$-0.990063\pi$
$827$	− 1.79523i	− 0.0624264i	0.999513	−	0.0312132i	$-0.00993708\pi$
$828$	4.49396	0.156176
$829$	−0.733169	−0.0254640	−0.0127320	−	0.999919i	$-0.504053\pi$
$829$	−0.733169	−0.0254640	−0.0127320	+	0.999919i	$0.504053\pi$
$830$	14.6407i	0.508187i
$831$	30.5435	1.05954
$832$	0	0
$833$	−3.35152	−0.116123
$834$	− 2.98493i	− 0.103360i
$835$	19.6582	0.680299
$836$	−23.4916	−0.812473
$837$	− 42.6896i	− 1.47557i
$838$	1.86725i	0.0645030i
$839$	− 26.6112i	− 0.918720i	−0.888250	−	0.459360i	$-0.848079\pi$
$839$	− 26.6112i	− 0.918720i	0.888250	−	0.459360i	$-0.151921\pi$
$840$	− 3.10992i	− 0.107302i
$841$	2.40342	0.0828766
$842$	−27.3250	−0.941680
$843$	29.1021i	1.00233i
$844$	−9.36227	−0.322263
$845$	0	0
$846$	−5.60388	−0.192665
$847$	− 56.5193i	− 1.94203i
$848$	−4.93362	−0.169421
$849$	14.2433	0.488827
$850$	4.29590i	0.147348i
$851$	8.30426i	0.284666i
$852$	− 6.83579i	− 0.234190i
$853$	− 4.78746i	− 0.163920i	−0.996636	−	0.0819598i	$-0.973882\pi$
$853$	− 4.78746i	− 0.163920i	0.996636	−	0.0819598i	$-0.0261179\pi$
$854$	12.3043	0.421043
$855$	5.85086	0.200095
$856$	− 15.0804i	− 0.515437i
$857$	24.9748	0.853122	0.426561	−	0.904459i	$-0.359725\pi$
$857$	24.9748	0.853122	0.426561	+	0.904459i	$0.359725\pi$
$858$	0	0
$859$	−16.2737	−0.555250	−0.277625	−	0.960690i	$-0.589547\pi$
$859$	−16.2737	−0.555250	−0.277625	+	0.960690i	$0.589547\pi$
$860$	− 6.98254i	− 0.238103i
$861$	39.0073	1.32937
$862$	1.56033	0.0531452
$863$	− 44.3806i	− 1.51073i	−0.655303	−	0.755366i	$-0.727459\pi$
$863$	− 44.3806i	− 1.51073i	0.655303	−	0.755366i	$-0.272541\pi$
$864$	− 5.54288i	− 0.188572i
$865$	− 16.8659i	− 0.573459i
$866$	24.1564i	0.820869i
$867$	1.81402	0.0616073
$868$	19.2078	0.651954
$869$	− 10.5810i	− 0.358937i
$870$	−6.98792	−0.236913
$871$	0	0
$872$	17.4819	0.592011
$873$	12.5603i	0.425103i
$874$	12.5918	0.425924
$875$	−2.49396	−0.0843112
$876$	− 10.8170i	− 0.365473i
$877$	− 37.3900i	− 1.26257i	−0.775551	−	0.631285i	$-0.782528\pi$
$877$	− 37.3900i	− 1.26257i	0.775551	−	0.631285i	$-0.217472\pi$
$878$	− 21.3599i	− 0.720861i
$879$	− 30.4456i	− 1.02691i
$880$	5.80194	0.195583
$881$	−38.1885	−1.28660	−0.643301	−	0.765613i	$-0.722436\pi$
$881$	−38.1885	−1.28660	−0.643301	+	0.765613i	$0.722436\pi$
$882$	− 1.12737i	− 0.0379607i
$883$	−4.45281	−0.149849	−0.0749245	−	0.997189i	$-0.523872\pi$
$883$	−4.45281	−0.149849	−0.0749245	+	0.997189i	$0.523872\pi$
$884$	0	0
$885$	12.4886	0.419799
$886$	− 16.5351i	− 0.555508i
$887$	6.06292	0.203573	0.101786	−	0.994806i	$-0.467544\pi$
$887$	6.06292	0.203573	0.101786	+	0.994806i	$0.467544\pi$
$888$	3.32975	0.111739
$889$	32.4999i	1.09001i
$890$	− 0.454731i	− 0.0152426i
$891$	− 14.9500i	− 0.500844i
$892$	6.31767i	0.211531i
$893$	−15.7017	−0.525438
$894$	25.9517	0.867954
$895$	− 5.99761i	− 0.200478i
$896$	2.49396	0.0833173
$897$	0	0
$898$	−6.56704	−0.219145
$899$	− 43.1594i	− 1.43945i
$900$	−1.44504	−0.0481681
$901$	21.1943	0.706086
$902$	72.7730i	2.42308i
$903$	− 21.7151i	− 0.722634i
$904$	8.23490i	0.273889i
$905$	− 16.5676i	− 0.550727i
$906$	−7.42758	−0.246765
$907$	−25.1116	−0.833816	−0.416908	−	0.908949i	$-0.636886\pi$
$907$	−25.1116	−0.833816	−0.416908	+	0.908949i	$0.636886\pi$
$908$	− 4.89440i	− 0.162426i
$909$	22.7525	0.754654
$910$	0	0
$911$	4.11721	0.136409	0.0682047	−	0.997671i	$-0.478273\pi$
$911$	4.11721	0.136409	0.0682047	+	0.997671i	$0.478273\pi$
$912$	− 5.04892i	− 0.167186i
$913$	−84.9445	−2.81125
$914$	−24.1715	−0.799522
$915$	6.15213i	0.203383i
$916$	− 0.689629i	− 0.0227860i
$917$	24.5133i	0.809502i
$918$	23.8116i	0.785901i
$919$	−34.7633	−1.14674	−0.573368	−	0.819298i	$-0.694363\pi$
$919$	−34.7633	−1.14674	−0.573368	+	0.819298i	$0.694363\pi$
$920$	−3.10992	−0.102531
$921$	− 9.60686i	− 0.316557i
$922$	−23.5797	−0.776556
$923$	0	0
$924$	18.0435	0.593589
$925$	− 2.67025i	− 0.0877973i
$926$	−1.68233	−0.0552849
$927$	−5.34913	−0.175688
$928$	− 5.60388i	− 0.183956i
$929$	− 38.8015i	− 1.27303i	−0.771262	−	0.636517i	$-0.780374\pi$
$929$	− 38.8015i	− 1.27303i	0.771262	−	0.636517i	$-0.219626\pi$
$930$	9.60388i	0.314923i
$931$	− 3.15883i	− 0.103527i
$932$	14.1806	0.464501
$933$	−5.87800	−0.192437
$934$	− 0.376273i	− 0.0123120i
$935$	−24.9245	−0.815119
$936$	0	0
$937$	−31.7808	−1.03823	−0.519116	−	0.854704i	$-0.673739\pi$
$937$	−31.7808	−1.03823	−0.519116	+	0.854704i	$0.673739\pi$
$938$	− 19.9892i	− 0.652672i
$939$	10.5888	0.345553
$940$	3.87800	0.126486
$941$	− 30.2258i	− 0.985333i	−0.870218	−	0.492666i	$-0.836022\pi$
$941$	− 30.2258i	− 0.985333i	0.870218	−	0.492666i	$-0.163978\pi$
$942$	10.2198i	0.332980i
$943$	− 39.0073i	− 1.27025i
$944$	10.0151i	0.325963i
$945$	−13.8237	−0.449685
$946$	40.5123	1.31717
$947$	2.54288i	0.0826324i	0.999146	+	0.0413162i	$0.0131551\pi$
$947$	2.54288i	0.0826324i	−0.999146	+	0.0413162i	$0.986845\pi$
$948$	2.27413	0.0738602
$949$	0	0
$950$	−4.04892	−0.131364
$951$	− 33.7318i	− 1.09383i
$952$	−10.7138	−0.347236
$953$	6.58642	0.213355	0.106677	−	0.994294i	$-0.465979\pi$
$953$	6.58642	0.213355	0.106677	+	0.994294i	$0.465979\pi$
$954$	7.12929i	0.230819i
$955$	− 12.7922i	− 0.413947i
$956$	25.7754i	0.833635i
$957$	− 40.5435i	− 1.31058i
$958$	−10.0301	−0.324059
$959$	−11.8538	−0.382780
$960$	1.24698i	0.0402461i
$961$	−28.3163	−0.913430
$962$	0	0
$963$	−21.7918	−0.702230
$964$	− 7.86725i	− 0.253387i
$965$	−23.5851	−0.759231
$966$	−9.67158	−0.311178
$967$	− 23.1884i	− 0.745688i	−0.927894	−	0.372844i	$-0.878383\pi$
$967$	− 23.1884i	− 0.745688i	0.927894	−	0.372844i	$-0.121617\pi$
$968$	22.6625i	0.728400i
$969$	21.6896i	0.696771i
$970$	− 8.69202i	− 0.279084i
$971$	−4.58642	−0.147185	−0.0735926	−	0.997288i	$-0.523446\pi$
$971$	−4.58642	−0.147185	−0.0735926	+	0.997288i	$0.523446\pi$
$972$	−13.4155	−0.430302
$973$	− 5.96987i	− 0.191385i
$974$	15.3297	0.491197
$975$	0	0
$976$	−4.93362	−0.157921
$977$	30.1575i	0.964824i	0.875944	+	0.482412i	$0.160239\pi$
$977$	30.1575i	0.964824i	−0.875944	+	0.482412i	$0.839761\pi$
$978$	21.0586	0.673380
$979$	2.63832	0.0843210
$980$	0.780167i	0.0249215i
$981$	− 25.2620i	− 0.806555i
$982$	− 9.19269i	− 0.293350i
$983$	− 0.787463i	− 0.0251162i	−0.999921	−	0.0125581i	$-0.996003\pi$
$983$	− 0.787463i	− 0.0251162i	0.999921	−	0.0125581i	$-0.00399747\pi$
$984$	−15.6407	−0.498608
$985$	1.97584	0.0629554
$986$	24.0737i	0.766662i
$987$	12.0603	0.383882
$988$	0	0
$989$	−21.7151	−0.690501
$990$	− 8.38404i	− 0.266462i
$991$	−12.0892	−0.384026	−0.192013	−	0.981392i	$-0.561502\pi$
$991$	−12.0892	−0.384026	−0.192013	+	0.981392i	$0.561502\pi$
$992$	−7.70171	−0.244530
$993$	35.8431i	1.13745i
$994$	− 13.6716i	− 0.433636i
$995$	0.835790i	0.0264963i
$996$	− 18.2567i	− 0.578485i
$997$	46.6607	1.47776	0.738879	−	0.673838i	$-0.235355\pi$
$997$	46.6607	1.47776	0.738879	+	0.673838i	$0.235355\pi$
$998$	2.74632	0.0869331
$999$	− 14.8009i	− 0.468279i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.i.1351.6		6
13.2	odd	12		1690.2.e.p.191.1		6
13.3	even	3		1690.2.l.m.1161.1		12
13.4	even	6		1690.2.l.m.361.1		12
13.5	odd	4		1690.2.a.r.1.3	yes	3
13.6	odd	12		1690.2.e.p.991.1		6
13.7	odd	12		1690.2.e.r.991.1		6
13.8	odd	4		1690.2.a.p.1.3	✓	3
13.9	even	3		1690.2.l.m.361.4		12
13.10	even	6		1690.2.l.m.1161.4		12
13.11	odd	12		1690.2.e.r.191.1		6
13.12	even	2	inner	1690.2.d.i.1351.3		6
65.34	odd	4		8450.2.a.cg.1.1		3
65.44	odd	4		8450.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.p.1.3	✓	3	13.8	odd	4
1690.2.a.r.1.3	yes	3	13.5	odd	4
1690.2.d.i.1351.3		6	13.12	even	2	inner
1690.2.d.i.1351.6		6	1.1	even	1	trivial
1690.2.e.p.191.1		6	13.2	odd	12
1690.2.e.p.991.1		6	13.6	odd	12
1690.2.e.r.191.1		6	13.11	odd	12
1690.2.e.r.991.1		6	13.7	odd	12
1690.2.l.m.361.1		12	13.4	even	6
1690.2.l.m.361.4		12	13.9	even	3
1690.2.l.m.1161.1		12	13.3	even	3
1690.2.l.m.1161.4		12	13.10	even	6
8450.2.a.bv.1.1		3	65.44	odd	4
8450.2.a.cg.1.1		3	65.34	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.24698 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.24698i q^{6} +2.49396i q^{7} -1.00000i q^{8} -1.44504 q^{9} +1.00000 q^{10} +5.80194i q^{11} -1.24698 q^{12} -2.49396 q^{14} -1.24698i q^{15} +1.00000 q^{16} -4.29590 q^{17} -1.44504i q^{18} -4.04892i q^{19} +1.00000i q^{20} +3.10992i q^{21} -5.80194 q^{22} +3.10992 q^{23} -1.24698i q^{24} -1.00000 q^{25} -5.54288 q^{27} -2.49396i q^{28} -5.60388 q^{29} +1.24698 q^{30} +7.70171i q^{31} +1.00000i q^{32} +7.23490i q^{33} -4.29590i q^{34} +2.49396 q^{35} +1.44504 q^{36} +2.67025i q^{37} +4.04892 q^{38} -1.00000 q^{40} -12.5429i q^{41} -3.10992 q^{42} -6.98254 q^{43} -5.80194i q^{44} +1.44504i q^{45} +3.10992i q^{46} -3.87800i q^{47} +1.24698 q^{48} +0.780167 q^{49} -1.00000i q^{50} -5.35690 q^{51} -4.93362 q^{53} -5.54288i q^{54} +5.80194 q^{55} +2.49396 q^{56} -5.04892i q^{57} -5.60388i q^{58} +10.0151i q^{59} +1.24698i q^{60} -4.93362 q^{61} -7.70171 q^{62} -3.60388i q^{63} -1.00000 q^{64} -7.23490 q^{66} +8.01507i q^{67} +4.29590 q^{68} +3.87800 q^{69} +2.49396i q^{70} +5.48188i q^{71} +1.44504i q^{72} +8.67456i q^{73} -2.67025 q^{74} -1.24698 q^{75} +4.04892i q^{76} -14.4698 q^{77} -1.82371 q^{79} -1.00000i q^{80} -2.57673 q^{81} +12.5429 q^{82} +14.6407i q^{83} -3.10992i q^{84} +4.29590i q^{85} -6.98254i q^{86} -6.98792 q^{87} +5.80194 q^{88} -0.454731i q^{89} -1.44504 q^{90} -3.10992 q^{92} +9.60388i q^{93} +3.87800 q^{94} -4.04892 q^{95} +1.24698i q^{96} -8.69202i q^{97} +0.780167i q^{98} -8.38404i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{4} - 8 q^{9} + 6 q^{10} + 2 q^{12} + 4 q^{14} + 6 q^{16} + 2 q^{17} - 26 q^{22} + 20 q^{23} - 6 q^{25} + 4 q^{27} - 16 q^{29} - 2 q^{30} - 4 q^{35} + 8 q^{36} + 6 q^{38} - 6 q^{40}+ \cdots - 6 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^4 - 8 * q^9 + 6 * q^10 + 2 * q^12 + 4 * q^14 + 6 * q^16 + 2 * q^17 - 26 * q^22 + 20 * q^23 - 6 * q^25 + 4 * q^27 - 16 * q^29 - 2 * q^30 - 4 * q^35 + 8 * q^36 + 6 * q^38 - 6 * q^40 - 20 * q^42 - 10 * q^43 - 2 * q^48 + 2 * q^49 - 24 * q^51 - 16 * q^53 + 26 * q^55 - 4 * q^56 - 16 * q^61 + 8 * q^62 - 6 * q^64 + 4 * q^66 - 2 * q^68 - 16 * q^69 - 12 * q^74 + 2 * q^75 + 8 * q^77 + 4 * q^79 - 10 * q^81 + 38 * q^82 - 4 * q^87 + 26 * q^88 - 8 * q^90 - 20 * q^92 - 16 * q^94 - 6 * q^95

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