LMFDB - Embedded newform 1690.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.00000 q^{10} -3.00000 q^{11} +0.732051 q^{12} -3.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} -8.19615 q^{17} +2.46410 q^{18} +0.464102 q^{19} +1.00000 q^{20} +2.19615 q^{21} +3.00000 q^{22} -9.46410 q^{23} -0.732051 q^{24} +1.00000 q^{25} -4.00000 q^{27} +3.00000 q^{28} -2.53590 q^{29} -0.732051 q^{30} -4.73205 q^{31} -1.00000 q^{32} -2.19615 q^{33} +8.19615 q^{34} +3.00000 q^{35} -2.46410 q^{36} +0.803848 q^{37} -0.464102 q^{38} -1.00000 q^{40} -10.3923 q^{41} -2.19615 q^{42} -2.00000 q^{43} -3.00000 q^{44} -2.46410 q^{45} +9.46410 q^{46} +3.00000 q^{47} +0.732051 q^{48} +2.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +0.464102 q^{53} +4.00000 q^{54} -3.00000 q^{55} -3.00000 q^{56} +0.339746 q^{57} +2.53590 q^{58} +10.3923 q^{59} +0.732051 q^{60} +6.19615 q^{61} +4.73205 q^{62} -7.39230 q^{63} +1.00000 q^{64} +2.19615 q^{66} -8.19615 q^{68} -6.92820 q^{69} -3.00000 q^{70} +6.00000 q^{71} +2.46410 q^{72} +11.6603 q^{73} -0.803848 q^{74} +0.732051 q^{75} +0.464102 q^{76} -9.00000 q^{77} -4.19615 q^{79} +1.00000 q^{80} +4.46410 q^{81} +10.3923 q^{82} +8.19615 q^{83} +2.19615 q^{84} -8.19615 q^{85} +2.00000 q^{86} -1.85641 q^{87} +3.00000 q^{88} -6.80385 q^{89} +2.46410 q^{90} -9.46410 q^{92} -3.46410 q^{93} -3.00000 q^{94} +0.464102 q^{95} -0.732051 q^{96} -9.12436 q^{97} -2.00000 q^{98} +7.39230 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{15} + 2 q^{16} - 6 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 6 q^{21} + 6 q^{22} - 12 q^{23} + 2 q^{24} + 2 q^{25} - 8 q^{27} + 6 q^{28} - 12 q^{29} + 2 q^{30} - 6 q^{31} - 2 q^{32} + 6 q^{33} + 6 q^{34} + 6 q^{35} + 2 q^{36} + 12 q^{37} + 6 q^{38} - 2 q^{40} + 6 q^{42} - 4 q^{43} - 6 q^{44} + 2 q^{45} + 12 q^{46} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 2 q^{50} - 12 q^{51} - 6 q^{53} + 8 q^{54} - 6 q^{55} - 6 q^{56} + 18 q^{57} + 12 q^{58} - 2 q^{60} + 2 q^{61} + 6 q^{62} + 6 q^{63} + 2 q^{64} - 6 q^{66} - 6 q^{68} - 6 q^{70} + 12 q^{71} - 2 q^{72} + 6 q^{73} - 12 q^{74} - 2 q^{75} - 6 q^{76} - 18 q^{77} + 2 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 4 q^{86} + 24 q^{87} + 6 q^{88} - 24 q^{89} - 2 q^{90} - 12 q^{92} - 6 q^{94} - 6 q^{95} + 2 q^{96} + 6 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^15 + 2 * q^16 - 6 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 6 * q^21 + 6 * q^22 - 12 * q^23 + 2 * q^24 + 2 * q^25 - 8 * q^27 + 6 * q^28 - 12 * q^29 + 2 * q^30 - 6 * q^31 - 2 * q^32 + 6 * q^33 + 6 * q^34 + 6 * q^35 + 2 * q^36 + 12 * q^37 + 6 * q^38 - 2 * q^40 + 6 * q^42 - 4 * q^43 - 6 * q^44 + 2 * q^45 + 12 * q^46 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 2 * q^50 - 12 * q^51 - 6 * q^53 + 8 * q^54 - 6 * q^55 - 6 * q^56 + 18 * q^57 + 12 * q^58 - 2 * q^60 + 2 * q^61 + 6 * q^62 + 6 * q^63 + 2 * q^64 - 6 * q^66 - 6 * q^68 - 6 * q^70 + 12 * q^71 - 2 * q^72 + 6 * q^73 - 12 * q^74 - 2 * q^75 - 6 * q^76 - 18 * q^77 + 2 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^83 - 6 * q^84 - 6 * q^85 + 4 * q^86 + 24 * q^87 + 6 * q^88 - 24 * q^89 - 2 * q^90 - 12 * q^92 - 6 * q^94 - 6 * q^95 + 2 * q^96 + 6 * q^97 - 4 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.732051	−0.298858
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	−2.46410	−0.821367
$10$	−1.00000	−0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0.732051	0.211325
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	0.732051	0.189015
$16$	1.00000	0.250000
$17$	−8.19615	−1.98786	−0.993929	−	0.110019i	$-0.964909\pi$
$17$	−8.19615	−1.98786	−0.993929	+	0.110019i	$0.964909\pi$
$18$	2.46410	0.580794
$19$	0.464102	0.106472	0.0532361	−	0.998582i	$-0.483046\pi$
$19$	0.464102	0.106472	0.0532361	+	0.998582i	$0.483046\pi$
$20$	1.00000	0.223607
$21$	2.19615	0.479240
$22$	3.00000	0.639602
$23$	−9.46410	−1.97340	−0.986701	−	0.162547i	$-0.948029\pi$
$23$	−9.46410	−1.97340	−0.986701	+	0.162547i	$0.948029\pi$
$24$	−0.732051	−0.149429
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	3.00000	0.566947
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	−0.732051	−0.133654
$31$	−4.73205	−0.849901	−0.424951	−	0.905216i	$-0.639709\pi$
$31$	−4.73205	−0.849901	−0.424951	+	0.905216i	$0.639709\pi$
$32$	−1.00000	−0.176777
$33$	−2.19615	−0.382301
$34$	8.19615	1.40563
$35$	3.00000	0.507093
$36$	−2.46410	−0.410684
$37$	0.803848	0.132152	0.0660759	−	0.997815i	$-0.478952\pi$
$37$	0.803848	0.132152	0.0660759	+	0.997815i	$0.478952\pi$
$38$	−0.464102	−0.0752872
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.3923	−1.62301	−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923	−1.62301	−0.811503	+	0.584349i	$0.801350\pi$
$42$	−2.19615	−0.338874
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−3.00000	−0.452267
$45$	−2.46410	−0.367327
$46$	9.46410	1.39541
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0.732051	0.105662
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	0	0
$53$	0.464102	0.0637493	0.0318746	−	0.999492i	$-0.489852\pi$
$53$	0.464102	0.0637493	0.0318746	+	0.999492i	$0.489852\pi$
$54$	4.00000	0.544331
$55$	−3.00000	−0.404520
$56$	−3.00000	−0.400892
$57$	0.339746	0.0450005
$58$	2.53590	0.332980
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0.732051	0.0945074
$61$	6.19615	0.793336	0.396668	−	0.917962i	$-0.370166\pi$
$61$	6.19615	0.793336	0.396668	+	0.917962i	$0.370166\pi$
$62$	4.73205	0.600971
$63$	−7.39230	−0.931343
$64$	1.00000	0.125000
$65$	0	0
$66$	2.19615	0.270328
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−8.19615	−0.993929
$69$	−6.92820	−0.834058
$70$	−3.00000	−0.358569
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.46410	0.290397
$73$	11.6603	1.36473	0.682365	−	0.731012i	$-0.260952\pi$
$73$	11.6603	1.36473	0.682365	+	0.731012i	$0.260952\pi$
$74$	−0.803848	−0.0934454
$75$	0.732051	0.0845299
$76$	0.464102	0.0532361
$77$	−9.00000	−1.02565
$78$	0	0
$79$	−4.19615	−0.472104	−0.236052	−	0.971740i	$-0.575854\pi$
$79$	−4.19615	−0.472104	−0.236052	+	0.971740i	$0.575854\pi$
$80$	1.00000	0.111803
$81$	4.46410	0.496011
$82$	10.3923	1.14764
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	2.19615	0.239620
$85$	−8.19615	−0.888998
$86$	2.00000	0.215666
$87$	−1.85641	−0.199028
$88$	3.00000	0.319801
$89$	−6.80385	−0.721206	−0.360603	−	0.932719i	$-0.617429\pi$
$89$	−6.80385	−0.721206	−0.360603	+	0.932719i	$0.617429\pi$
$90$	2.46410	0.259739
$91$	0	0
$92$	−9.46410	−0.986701
$93$	−3.46410	−0.359211
$94$	−3.00000	−0.309426
$95$	0.464102	0.0476158
$96$	−0.732051	−0.0747146
$97$	−9.12436	−0.926438	−0.463219	−	0.886244i	$-0.653306\pi$
$97$	−9.12436	−0.926438	−0.463219	+	0.886244i	$0.653306\pi$
$98$	−2.00000	−0.202031
$99$	7.39230	0.742955
$100$	1.00000	0.100000
$101$	−10.7321	−1.06788	−0.533939	−	0.845523i	$-0.679289\pi$
$101$	−10.7321	−1.06788	−0.533939	+	0.845523i	$0.679289\pi$
$102$	6.00000	0.594089
$103$	9.19615	0.906124	0.453062	−	0.891479i	$-0.350332\pi$
$103$	9.19615	0.906124	0.453062	+	0.891479i	$0.350332\pi$
$104$	0	0
$105$	2.19615	0.214323
$106$	−0.464102	−0.0450775
$107$	17.6603	1.70728	0.853641	−	0.520862i	$-0.174390\pi$
$107$	17.6603	1.70728	0.853641	+	0.520862i	$0.174390\pi$
$108$	−4.00000	−0.384900
$109$	8.53590	0.817591	0.408795	−	0.912626i	$-0.365949\pi$
$109$	8.53590	0.817591	0.408795	+	0.912626i	$0.365949\pi$
$110$	3.00000	0.286039
$111$	0.588457	0.0558539
$112$	3.00000	0.283473
$113$	6.92820	0.651751	0.325875	−	0.945413i	$-0.394341\pi$
$113$	6.92820	0.651751	0.325875	+	0.945413i	$0.394341\pi$
$114$	−0.339746	−0.0318201
$115$	−9.46410	−0.882532
$116$	−2.53590	−0.235452
$117$	0	0
$118$	−10.3923	−0.956689
$119$	−24.5885	−2.25402
$120$	−0.732051	−0.0668268
$121$	−2.00000	−0.181818
$122$	−6.19615	−0.560973
$123$	−7.60770	−0.685963
$124$	−4.73205	−0.424951
$125$	1.00000	0.0894427
$126$	7.39230	0.658559
$127$	−10.8038	−0.958686	−0.479343	−	0.877628i	$-0.659125\pi$
$127$	−10.8038	−0.958686	−0.479343	+	0.877628i	$0.659125\pi$
$128$	−1.00000	−0.0883883
$129$	−1.46410	−0.128907
$130$	0	0
$131$	6.12436	0.535087	0.267544	−	0.963546i	$-0.413788\pi$
$131$	6.12436	0.535087	0.267544	+	0.963546i	$0.413788\pi$
$132$	−2.19615	−0.191151
$133$	1.39230	0.120728
$134$	0	0
$135$	−4.00000	−0.344265
$136$	8.19615	0.702814
$137$	2.19615	0.187630	0.0938150	−	0.995590i	$-0.470094\pi$
$137$	2.19615	0.187630	0.0938150	+	0.995590i	$0.470094\pi$
$138$	6.92820	0.589768
$139$	1.19615	0.101456	0.0507282	−	0.998712i	$-0.483846\pi$
$139$	1.19615	0.101456	0.0507282	+	0.998712i	$0.483846\pi$
$140$	3.00000	0.253546
$141$	2.19615	0.184949
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−2.46410	−0.205342
$145$	−2.53590	−0.210595
$146$	−11.6603	−0.965009
$147$	1.46410	0.120757
$148$	0.803848	0.0660759
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−0.732051	−0.0597717
$151$	−23.6603	−1.92544	−0.962722	−	0.270492i	$-0.912813\pi$
$151$	−23.6603	−1.92544	−0.962722	+	0.270492i	$0.912813\pi$
$152$	−0.464102	−0.0376436
$153$	20.1962	1.63276
$154$	9.00000	0.725241
$155$	−4.73205	−0.380087
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	4.19615	0.333828
$159$	0.339746	0.0269436
$160$	−1.00000	−0.0790569
$161$	−28.3923	−2.23763
$162$	−4.46410	−0.350733
$163$	−10.7321	−0.840599	−0.420300	−	0.907385i	$-0.638075\pi$
$163$	−10.7321	−0.840599	−0.420300	+	0.907385i	$0.638075\pi$
$164$	−10.3923	−0.811503
$165$	−2.19615	−0.170970
$166$	−8.19615	−0.636145
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	−2.19615	−0.169437
$169$	0	0
$170$	8.19615	0.628616
$171$	−1.14359	−0.0874528
$172$	−2.00000	−0.152499
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	1.85641	0.140734
$175$	3.00000	0.226779
$176$	−3.00000	−0.226134
$177$	7.60770	0.571829
$178$	6.80385	0.509970
$179$	9.46410	0.707380	0.353690	−	0.935363i	$-0.384927\pi$
$179$	9.46410	0.707380	0.353690	+	0.935363i	$0.384927\pi$
$180$	−2.46410	−0.183663
$181$	−14.5885	−1.08435	−0.542176	−	0.840265i	$-0.682399\pi$
$181$	−14.5885	−1.08435	−0.542176	+	0.840265i	$0.682399\pi$
$182$	0	0
$183$	4.53590	0.335303
$184$	9.46410	0.697703
$185$	0.803848	0.0591000
$186$	3.46410	0.254000
$187$	24.5885	1.79809
$188$	3.00000	0.218797
$189$	−12.0000	−0.872872
$190$	−0.464102	−0.0336695
$191$	−22.7321	−1.64483	−0.822417	−	0.568886i	$-0.807375\pi$
$191$	−22.7321	−1.64483	−0.822417	+	0.568886i	$0.807375\pi$
$192$	0.732051	0.0528312
$193$	16.3923	1.17994	0.589972	−	0.807424i	$-0.299139\pi$
$193$	16.3923	1.17994	0.589972	+	0.807424i	$0.299139\pi$
$194$	9.12436	0.655091
$195$	0	0
$196$	2.00000	0.142857
$197$	21.5885	1.53811	0.769057	−	0.639180i	$-0.220726\pi$
$197$	21.5885	1.53811	0.769057	+	0.639180i	$0.220726\pi$
$198$	−7.39230	−0.525348
$199$	6.39230	0.453138	0.226569	−	0.973995i	$-0.427249\pi$
$199$	6.39230	0.453138	0.226569	+	0.973995i	$0.427249\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.7321	0.755104
$203$	−7.60770	−0.533956
$204$	−6.00000	−0.420084
$205$	−10.3923	−0.725830
$206$	−9.19615	−0.640726
$207$	23.3205	1.62089
$208$	0	0
$209$	−1.39230	−0.0963077
$210$	−2.19615	−0.151549
$211$	−17.5885	−1.21084	−0.605420	−	0.795906i	$-0.706995\pi$
$211$	−17.5885	−1.21084	−0.605420	+	0.795906i	$0.706995\pi$
$212$	0.464102	0.0318746
$213$	4.39230	0.300956
$214$	−17.6603	−1.20723
$215$	−2.00000	−0.136399
$216$	4.00000	0.272166
$217$	−14.1962	−0.963698
$218$	−8.53590	−0.578124
$219$	8.53590	0.576803
$220$	−3.00000	−0.202260
$221$	0	0
$222$	−0.588457	−0.0394947
$223$	−6.46410	−0.432868	−0.216434	−	0.976297i	$-0.569443\pi$
$223$	−6.46410	−0.432868	−0.216434	+	0.976297i	$0.569443\pi$
$224$	−3.00000	−0.200446
$225$	−2.46410	−0.164273
$226$	−6.92820	−0.460857
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0.339746	0.0225002
$229$	−1.26795	−0.0837884	−0.0418942	−	0.999122i	$-0.513339\pi$
$229$	−1.26795	−0.0837884	−0.0418942	+	0.999122i	$0.513339\pi$
$230$	9.46410	0.624044
$231$	−6.58846	−0.433489
$232$	2.53590	0.166490
$233$	−4.73205	−0.310007	−0.155003	−	0.987914i	$-0.549539\pi$
$233$	−4.73205	−0.310007	−0.155003	+	0.987914i	$0.549539\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	10.3923	0.676481
$237$	−3.07180	−0.199535
$238$	24.5885	1.59383
$239$	−2.19615	−0.142057	−0.0710286	−	0.997474i	$-0.522628\pi$
$239$	−2.19615	−0.142057	−0.0710286	+	0.997474i	$0.522628\pi$
$240$	0.732051	0.0472537
$241$	−8.66025	−0.557856	−0.278928	−	0.960312i	$-0.589979\pi$
$241$	−8.66025	−0.557856	−0.278928	+	0.960312i	$0.589979\pi$
$242$	2.00000	0.128565
$243$	15.2679	0.979439
$244$	6.19615	0.396668
$245$	2.00000	0.127775
$246$	7.60770	0.485049
$247$	0	0
$248$	4.73205	0.300486
$249$	6.00000	0.380235
$250$	−1.00000	−0.0632456
$251$	−17.1962	−1.08541	−0.542706	−	0.839923i	$-0.682600\pi$
$251$	−17.1962	−1.08541	−0.542706	+	0.839923i	$0.682600\pi$
$252$	−7.39230	−0.465671
$253$	28.3923	1.78501
$254$	10.8038	0.677894
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	13.8564	0.864339	0.432169	−	0.901792i	$-0.357748\pi$
$257$	13.8564	0.864339	0.432169	+	0.901792i	$0.357748\pi$
$258$	1.46410	0.0911510
$259$	2.41154	0.149846
$260$	0	0
$261$	6.24871	0.386786
$262$	−6.12436	−0.378364
$263$	3.58846	0.221274	0.110637	−	0.993861i	$-0.464711\pi$
$263$	3.58846	0.221274	0.110637	+	0.993861i	$0.464711\pi$
$264$	2.19615	0.135164
$265$	0.464102	0.0285095
$266$	−1.39230	−0.0853677
$267$	−4.98076	−0.304818
$268$	0	0
$269$	−27.1244	−1.65380	−0.826901	−	0.562348i	$-0.809898\pi$
$269$	−27.1244	−1.65380	−0.826901	+	0.562348i	$0.809898\pi$
$270$	4.00000	0.243432
$271$	−9.46410	−0.574903	−0.287452	−	0.957795i	$-0.592808\pi$
$271$	−9.46410	−0.574903	−0.287452	+	0.957795i	$0.592808\pi$
$272$	−8.19615	−0.496965
$273$	0	0
$274$	−2.19615	−0.132674
$275$	−3.00000	−0.180907
$276$	−6.92820	−0.417029
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−1.19615	−0.0717405
$279$	11.6603	0.698081
$280$	−3.00000	−0.179284
$281$	−10.3923	−0.619953	−0.309976	−	0.950744i	$-0.600321\pi$
$281$	−10.3923	−0.619953	−0.309976	+	0.950744i	$0.600321\pi$
$282$	−2.19615	−0.130779
$283$	9.60770	0.571118	0.285559	−	0.958361i	$-0.407821\pi$
$283$	9.60770	0.571118	0.285559	+	0.958361i	$0.407821\pi$
$284$	6.00000	0.356034
$285$	0.339746	0.0201248
$286$	0	0
$287$	−31.1769	−1.84032
$288$	2.46410	0.145199
$289$	50.1769	2.95158
$290$	2.53590	0.148913
$291$	−6.67949	−0.391559
$292$	11.6603	0.682365
$293$	11.1962	0.654086	0.327043	−	0.945009i	$-0.393948\pi$
$293$	11.1962	0.654086	0.327043	+	0.945009i	$0.393948\pi$
$294$	−1.46410	−0.0853881
$295$	10.3923	0.605063
$296$	−0.803848	−0.0467227
$297$	12.0000	0.696311
$298$	6.00000	0.347571
$299$	0	0
$300$	0.732051	0.0422650
$301$	−6.00000	−0.345834
$302$	23.6603	1.36149
$303$	−7.85641	−0.451339
$304$	0.464102	0.0266181
$305$	6.19615	0.354791
$306$	−20.1962	−1.15454
$307$	−27.4641	−1.56746	−0.783730	−	0.621102i	$-0.786685\pi$
$307$	−27.4641	−1.56746	−0.783730	+	0.621102i	$0.786685\pi$
$308$	−9.00000	−0.512823
$309$	6.73205	0.382973
$310$	4.73205	0.268762
$311$	9.12436	0.517395	0.258697	−	0.965958i	$-0.416707\pi$
$311$	9.12436	0.517395	0.258697	+	0.965958i	$0.416707\pi$
$312$	0	0
$313$	26.3923	1.49178	0.745891	−	0.666068i	$-0.232024\pi$
$313$	26.3923	1.49178	0.745891	+	0.666068i	$0.232024\pi$
$314$	13.0000	0.733632
$315$	−7.39230	−0.416509
$316$	−4.19615	−0.236052
$317$	23.1962	1.30283	0.651413	−	0.758723i	$-0.274177\pi$
$317$	23.1962	1.30283	0.651413	+	0.758723i	$0.274177\pi$
$318$	−0.339746	−0.0190520
$319$	7.60770	0.425949
$320$	1.00000	0.0559017
$321$	12.9282	0.721582
$322$	28.3923	1.58224
$323$	−3.80385	−0.211652
$324$	4.46410	0.248006
$325$	0	0
$326$	10.7321	0.594393
$327$	6.24871	0.345555
$328$	10.3923	0.573819
$329$	9.00000	0.496186
$330$	2.19615	0.120894
$331$	−12.9282	−0.710598	−0.355299	−	0.934753i	$-0.615621\pi$
$331$	−12.9282	−0.710598	−0.355299	+	0.934753i	$0.615621\pi$
$332$	8.19615	0.449822
$333$	−1.98076	−0.108545
$334$	3.00000	0.164153
$335$	0	0
$336$	2.19615	0.119810
$337$	−6.19615	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$337$	−6.19615	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$338$	0	0
$339$	5.07180	0.275462
$340$	−8.19615	−0.444499
$341$	14.1962	0.768765
$342$	1.14359	0.0618385
$343$	−15.0000	−0.809924
$344$	2.00000	0.107833
$345$	−6.92820	−0.373002
$346$	−15.0000	−0.806405
$347$	−28.7321	−1.54242	−0.771209	−	0.636582i	$-0.780347\pi$
$347$	−28.7321	−1.54242	−0.771209	+	0.636582i	$0.780347\pi$
$348$	−1.85641	−0.0995138
$349$	4.05256	0.216929	0.108464	−	0.994100i	$-0.465407\pi$
$349$	4.05256	0.216929	0.108464	+	0.994100i	$0.465407\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	3.00000	0.159901
$353$	9.80385	0.521806	0.260903	−	0.965365i	$-0.415980\pi$
$353$	9.80385	0.521806	0.260903	+	0.965365i	$0.415980\pi$
$354$	−7.60770	−0.404344
$355$	6.00000	0.318447
$356$	−6.80385	−0.360603
$357$	−18.0000	−0.952661
$358$	−9.46410	−0.500193
$359$	1.60770	0.0848509	0.0424255	−	0.999100i	$-0.486492\pi$
$359$	1.60770	0.0848509	0.0424255	+	0.999100i	$0.486492\pi$
$360$	2.46410	0.129870
$361$	−18.7846	−0.988664
$362$	14.5885	0.766752
$363$	−1.46410	−0.0768454
$364$	0	0
$365$	11.6603	0.610326
$366$	−4.53590	−0.237095
$367$	−5.60770	−0.292719	−0.146360	−	0.989231i	$-0.546756\pi$
$367$	−5.60770	−0.292719	−0.146360	+	0.989231i	$0.546756\pi$
$368$	−9.46410	−0.493350
$369$	25.6077	1.33308
$370$	−0.803848	−0.0417900
$371$	1.39230	0.0722849
$372$	−3.46410	−0.179605
$373$	−0.392305	−0.0203128	−0.0101564	−	0.999948i	$-0.503233\pi$
$373$	−0.392305	−0.0203128	−0.0101564	+	0.999948i	$0.503233\pi$
$374$	−24.5885	−1.27144
$375$	0.732051	0.0378029
$376$	−3.00000	−0.154713
$377$	0	0
$378$	12.0000	0.617213
$379$	29.7846	1.52993	0.764966	−	0.644070i	$-0.222756\pi$
$379$	29.7846	1.52993	0.764966	+	0.644070i	$0.222756\pi$
$380$	0.464102	0.0238079
$381$	−7.90897	−0.405189
$382$	22.7321	1.16307
$383$	−22.3923	−1.14419	−0.572097	−	0.820186i	$-0.693870\pi$
$383$	−22.3923	−1.14419	−0.572097	+	0.820186i	$0.693870\pi$
$384$	−0.732051	−0.0373573
$385$	−9.00000	−0.458682
$386$	−16.3923	−0.834346
$387$	4.92820	0.250515
$388$	−9.12436	−0.463219
$389$	−22.7321	−1.15256	−0.576280	−	0.817252i	$-0.695496\pi$
$389$	−22.7321	−1.15256	−0.576280	+	0.817252i	$0.695496\pi$
$390$	0	0
$391$	77.5692	3.92284
$392$	−2.00000	−0.101015
$393$	4.48334	0.226155
$394$	−21.5885	−1.08761
$395$	−4.19615	−0.211131
$396$	7.39230	0.371477
$397$	−11.1962	−0.561919	−0.280959	−	0.959720i	$-0.590653\pi$
$397$	−11.1962	−0.561919	−0.280959	+	0.959720i	$0.590653\pi$
$398$	−6.39230	−0.320417
$399$	1.01924	0.0510257
$400$	1.00000	0.0500000
$401$	5.19615	0.259483	0.129742	−	0.991548i	$-0.458585\pi$
$401$	5.19615	0.259483	0.129742	+	0.991548i	$0.458585\pi$
$402$	0	0
$403$	0	0
$404$	−10.7321	−0.533939
$405$	4.46410	0.221823
$406$	7.60770	0.377564
$407$	−2.41154	−0.119536
$408$	6.00000	0.297044
$409$	16.2679	0.804398	0.402199	−	0.915552i	$-0.368246\pi$
$409$	16.2679	0.804398	0.402199	+	0.915552i	$0.368246\pi$
$410$	10.3923	0.513239
$411$	1.60770	0.0793018
$412$	9.19615	0.453062
$413$	31.1769	1.53412
$414$	−23.3205	−1.14614
$415$	8.19615	0.402333
$416$	0	0
$417$	0.875644	0.0428805
$418$	1.39230	0.0680999
$419$	−17.3205	−0.846162	−0.423081	−	0.906092i	$-0.639051\pi$
$419$	−17.3205	−0.846162	−0.423081	+	0.906092i	$0.639051\pi$
$420$	2.19615	0.107161
$421$	−2.87564	−0.140150	−0.0700752	−	0.997542i	$-0.522324\pi$
$421$	−2.87564	−0.140150	−0.0700752	+	0.997542i	$0.522324\pi$
$422$	17.5885	0.856193
$423$	−7.39230	−0.359426
$424$	−0.464102	−0.0225388
$425$	−8.19615	−0.397572
$426$	−4.39230	−0.212808
$427$	18.5885	0.899558
$428$	17.6603	0.853641
$429$	0	0
$430$	2.00000	0.0964486
$431$	−8.19615	−0.394795	−0.197397	−	0.980324i	$-0.563249\pi$
$431$	−8.19615	−0.394795	−0.197397	+	0.980324i	$0.563249\pi$
$432$	−4.00000	−0.192450
$433$	−8.39230	−0.403308	−0.201654	−	0.979457i	$-0.564632\pi$
$433$	−8.39230	−0.403308	−0.201654	+	0.979457i	$0.564632\pi$
$434$	14.1962	0.681437
$435$	−1.85641	−0.0890079
$436$	8.53590	0.408795
$437$	−4.39230	−0.210112
$438$	−8.53590	−0.407861
$439$	3.41154	0.162824	0.0814120	−	0.996681i	$-0.474057\pi$
$439$	3.41154	0.162824	0.0814120	+	0.996681i	$0.474057\pi$
$440$	3.00000	0.143019
$441$	−4.92820	−0.234676
$442$	0	0
$443$	−22.3923	−1.06389	−0.531945	−	0.846779i	$-0.678539\pi$
$443$	−22.3923	−1.06389	−0.531945	+	0.846779i	$0.678539\pi$
$444$	0.588457	0.0279269
$445$	−6.80385	−0.322533
$446$	6.46410	0.306084
$447$	−4.39230	−0.207749
$448$	3.00000	0.141737
$449$	15.5885	0.735665	0.367832	−	0.929892i	$-0.380100\pi$
$449$	15.5885	0.735665	0.367832	+	0.929892i	$0.380100\pi$
$450$	2.46410	0.116159
$451$	31.1769	1.46806
$452$	6.92820	0.325875
$453$	−17.3205	−0.813788
$454$	16.3923	0.769329
$455$	0	0
$456$	−0.339746	−0.0159101
$457$	18.3397	0.857897	0.428949	−	0.903329i	$-0.358884\pi$
$457$	18.3397	0.857897	0.428949	+	0.903329i	$0.358884\pi$
$458$	1.26795	0.0592474
$459$	32.7846	1.53025
$460$	−9.46410	−0.441266
$461$	30.5885	1.42465	0.712323	−	0.701852i	$-0.247643\pi$
$461$	30.5885	1.42465	0.712323	+	0.701852i	$0.247643\pi$
$462$	6.58846	0.306523
$463$	12.9282	0.600825	0.300412	−	0.953809i	$-0.402876\pi$
$463$	12.9282	0.600825	0.300412	+	0.953809i	$0.402876\pi$
$464$	−2.53590	−0.117726
$465$	−3.46410	−0.160644
$466$	4.73205	0.219208
$467$	37.8564	1.75179	0.875893	−	0.482506i	$-0.160273\pi$
$467$	37.8564	1.75179	0.875893	+	0.482506i	$0.160273\pi$
$468$	0	0
$469$	0	0
$470$	−3.00000	−0.138380
$471$	−9.51666	−0.438505
$472$	−10.3923	−0.478345
$473$	6.00000	0.275880
$474$	3.07180	0.141092
$475$	0.464102	0.0212944
$476$	−24.5885	−1.12701
$477$	−1.14359	−0.0523616
$478$	2.19615	0.100450
$479$	40.9808	1.87246	0.936229	−	0.351389i	$-0.114291\pi$
$479$	40.9808	1.87246	0.936229	+	0.351389i	$0.114291\pi$
$480$	−0.732051	−0.0334134
$481$	0	0
$482$	8.66025	0.394464
$483$	−20.7846	−0.945732
$484$	−2.00000	−0.0909091
$485$	−9.12436	−0.414316
$486$	−15.2679	−0.692568
$487$	15.2487	0.690985	0.345493	−	0.938421i	$-0.387712\pi$
$487$	15.2487	0.690985	0.345493	+	0.938421i	$0.387712\pi$
$488$	−6.19615	−0.280487
$489$	−7.85641	−0.355279
$490$	−2.00000	−0.0903508
$491$	−20.6603	−0.932384	−0.466192	−	0.884684i	$-0.654374\pi$
$491$	−20.6603	−0.932384	−0.466192	+	0.884684i	$0.654374\pi$
$492$	−7.60770	−0.342981
$493$	20.7846	0.936092
$494$	0	0
$495$	7.39230	0.332259
$496$	−4.73205	−0.212475
$497$	18.0000	0.807410
$498$	−6.00000	−0.268866
$499$	−25.8564	−1.15749	−0.578746	−	0.815508i	$-0.696458\pi$
$499$	−25.8564	−1.15749	−0.578746	+	0.815508i	$0.696458\pi$
$500$	1.00000	0.0447214
$501$	−2.19615	−0.0981169
$502$	17.1962	0.767502
$503$	−26.6603	−1.18872	−0.594361	−	0.804198i	$-0.702595\pi$
$503$	−26.6603	−1.18872	−0.594361	+	0.804198i	$0.702595\pi$
$504$	7.39230	0.329279
$505$	−10.7321	−0.477570
$506$	−28.3923	−1.26219
$507$	0	0
$508$	−10.8038	−0.479343
$509$	−22.3923	−0.992521	−0.496261	−	0.868174i	$-0.665294\pi$
$509$	−22.3923	−0.992521	−0.496261	+	0.868174i	$0.665294\pi$
$510$	6.00000	0.265684
$511$	34.9808	1.54746
$512$	−1.00000	−0.0441942
$513$	−1.85641	−0.0819623
$514$	−13.8564	−0.611180
$515$	9.19615	0.405231
$516$	−1.46410	−0.0644535
$517$	−9.00000	−0.395820
$518$	−2.41154	−0.105957
$519$	10.9808	0.482002
$520$	0	0
$521$	6.46410	0.283197	0.141599	−	0.989924i	$-0.454776\pi$
$521$	6.46410	0.283197	0.141599	+	0.989924i	$0.454776\pi$
$522$	−6.24871	−0.273499
$523$	−2.39230	−0.104608	−0.0523041	−	0.998631i	$-0.516657\pi$
$523$	−2.39230	−0.104608	−0.0523041	+	0.998631i	$0.516657\pi$
$524$	6.12436	0.267544
$525$	2.19615	0.0958479
$526$	−3.58846	−0.156464
$527$	38.7846	1.68948
$528$	−2.19615	−0.0955753
$529$	66.5692	2.89431
$530$	−0.464102	−0.0201593
$531$	−25.6077	−1.11128
$532$	1.39230	0.0603641
$533$	0	0
$534$	4.98076	0.215539
$535$	17.6603	0.763519
$536$	0	0
$537$	6.92820	0.298974
$538$	27.1244	1.16941
$539$	−6.00000	−0.258438
$540$	−4.00000	−0.172133
$541$	22.0526	0.948114	0.474057	−	0.880494i	$-0.342789\pi$
$541$	22.0526	0.948114	0.474057	+	0.880494i	$0.342789\pi$
$542$	9.46410	0.406518
$543$	−10.6795	−0.458301
$544$	8.19615	0.351407
$545$	8.53590	0.365638
$546$	0	0
$547$	−6.78461	−0.290089	−0.145044	−	0.989425i	$-0.546333\pi$
$547$	−6.78461	−0.290089	−0.145044	+	0.989425i	$0.546333\pi$
$548$	2.19615	0.0938150
$549$	−15.2679	−0.651620
$550$	3.00000	0.127920
$551$	−1.17691	−0.0501382
$552$	6.92820	0.294884
$553$	−12.5885	−0.535316
$554$	−1.00000	−0.0424859
$555$	0.588457	0.0249786
$556$	1.19615	0.0507282
$557$	6.80385	0.288288	0.144144	−	0.989557i	$-0.453957\pi$
$557$	6.80385	0.288288	0.144144	+	0.989557i	$0.453957\pi$
$558$	−11.6603	−0.493618
$559$	0	0
$560$	3.00000	0.126773
$561$	18.0000	0.759961
$562$	10.3923	0.438373
$563$	−14.5359	−0.612615	−0.306308	−	0.951933i	$-0.599094\pi$
$563$	−14.5359	−0.612615	−0.306308	+	0.951933i	$0.599094\pi$
$564$	2.19615	0.0924747
$565$	6.92820	0.291472
$566$	−9.60770	−0.403842
$567$	13.3923	0.562424
$568$	−6.00000	−0.251754
$569$	27.2487	1.14233	0.571163	−	0.820837i	$-0.306493\pi$
$569$	27.2487	1.14233	0.571163	+	0.820837i	$0.306493\pi$
$570$	−0.339746	−0.0142304
$571$	38.3731	1.60586	0.802931	−	0.596071i	$-0.203272\pi$
$571$	38.3731	1.60586	0.802931	+	0.596071i	$0.203272\pi$
$572$	0	0
$573$	−16.6410	−0.695188
$574$	31.1769	1.30130
$575$	−9.46410	−0.394680
$576$	−2.46410	−0.102671
$577$	26.4449	1.10091	0.550457	−	0.834863i	$-0.314453\pi$
$577$	26.4449	1.10091	0.550457	+	0.834863i	$0.314453\pi$
$578$	−50.1769	−2.08708
$579$	12.0000	0.498703
$580$	−2.53590	−0.105297
$581$	24.5885	1.02010
$582$	6.67949	0.276874
$583$	−1.39230	−0.0576634
$584$	−11.6603	−0.482505
$585$	0	0
$586$	−11.1962	−0.462509
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	1.46410	0.0603785
$589$	−2.19615	−0.0904909
$590$	−10.3923	−0.427844
$591$	15.8038	0.650083
$592$	0.803848	0.0330379
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	−12.0000	−0.492366
$595$	−24.5885	−1.00803
$596$	−6.00000	−0.245770
$597$	4.67949	0.191519
$598$	0	0
$599$	−19.8564	−0.811311	−0.405655	−	0.914026i	$-0.632957\pi$
$599$	−19.8564	−0.811311	−0.405655	+	0.914026i	$0.632957\pi$
$600$	−0.732051	−0.0298858
$601$	−37.7846	−1.54127	−0.770633	−	0.637279i	$-0.780060\pi$
$601$	−37.7846	−1.54127	−0.770633	+	0.637279i	$0.780060\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	−23.6603	−0.962722
$605$	−2.00000	−0.0813116
$606$	7.85641	0.319145
$607$	−37.5885	−1.52567	−0.762834	−	0.646594i	$-0.776193\pi$
$607$	−37.5885	−1.52567	−0.762834	+	0.646594i	$0.776193\pi$
$608$	−0.464102	−0.0188218
$609$	−5.56922	−0.225676
$610$	−6.19615	−0.250875
$611$	0	0
$612$	20.1962	0.816381
$613$	38.9090	1.57152	0.785759	−	0.618533i	$-0.212273\pi$
$613$	38.9090	1.57152	0.785759	+	0.618533i	$0.212273\pi$
$614$	27.4641	1.10836
$615$	−7.60770	−0.306772
$616$	9.00000	0.362620
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−6.73205	−0.270803
$619$	−31.3923	−1.26176	−0.630882	−	0.775879i	$-0.717307\pi$
$619$	−31.3923	−1.26176	−0.630882	+	0.775879i	$0.717307\pi$
$620$	−4.73205	−0.190044
$621$	37.8564	1.51913
$622$	−9.12436	−0.365853
$623$	−20.4115	−0.817771
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.3923	−1.05485
$627$	−1.01924	−0.0407044
$628$	−13.0000	−0.518756
$629$	−6.58846	−0.262699
$630$	7.39230	0.294516
$631$	−20.7846	−0.827422	−0.413711	−	0.910408i	$-0.635768\pi$
$631$	−20.7846	−0.827422	−0.413711	+	0.910408i	$0.635768\pi$
$632$	4.19615	0.166914
$633$	−12.8756	−0.511761
$634$	−23.1962	−0.921237
$635$	−10.8038	−0.428738
$636$	0.339746	0.0134718
$637$	0	0
$638$	−7.60770	−0.301192
$639$	−14.7846	−0.584870
$640$	−1.00000	−0.0395285
$641$	−32.0718	−1.26676	−0.633380	−	0.773841i	$-0.718333\pi$
$641$	−32.0718	−1.26676	−0.633380	+	0.773841i	$0.718333\pi$
$642$	−12.9282	−0.510235
$643$	10.7321	0.423231	0.211615	−	0.977353i	$-0.432128\pi$
$643$	10.7321	0.423231	0.211615	+	0.977353i	$0.432128\pi$
$644$	−28.3923	−1.11881
$645$	−1.46410	−0.0576489
$646$	3.80385	0.149660
$647$	0.803848	0.0316025	0.0158013	−	0.999875i	$-0.494970\pi$
$647$	0.803848	0.0316025	0.0158013	+	0.999875i	$0.494970\pi$
$648$	−4.46410	−0.175366
$649$	−31.1769	−1.22380
$650$	0	0
$651$	−10.3923	−0.407307
$652$	−10.7321	−0.420300
$653$	1.39230	0.0544851	0.0272425	−	0.999629i	$-0.491327\pi$
$653$	1.39230	0.0544851	0.0272425	+	0.999629i	$0.491327\pi$
$654$	−6.24871	−0.244344
$655$	6.12436	0.239298
$656$	−10.3923	−0.405751
$657$	−28.7321	−1.12094
$658$	−9.00000	−0.350857
$659$	−5.32051	−0.207258	−0.103629	−	0.994616i	$-0.533045\pi$
$659$	−5.32051	−0.207258	−0.103629	+	0.994616i	$0.533045\pi$
$660$	−2.19615	−0.0854851
$661$	−35.9090	−1.39670	−0.698348	−	0.715758i	$-0.746081\pi$
$661$	−35.9090	−1.39670	−0.698348	+	0.715758i	$0.746081\pi$
$662$	12.9282	0.502469
$663$	0	0
$664$	−8.19615	−0.318072
$665$	1.39230	0.0539913
$666$	1.98076	0.0767530
$667$	24.0000	0.929284
$668$	−3.00000	−0.116073
$669$	−4.73205	−0.182952
$670$	0	0
$671$	−18.5885	−0.717599
$672$	−2.19615	−0.0847184
$673$	−24.3923	−0.940254	−0.470127	−	0.882599i	$-0.655792\pi$
$673$	−24.3923	−0.940254	−0.470127	+	0.882599i	$0.655792\pi$
$674$	6.19615	0.238667
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−25.8564	−0.993742	−0.496871	−	0.867824i	$-0.665518\pi$
$677$	−25.8564	−0.993742	−0.496871	+	0.867824i	$0.665518\pi$
$678$	−5.07180	−0.194781
$679$	−27.3731	−1.05048
$680$	8.19615	0.314308
$681$	−12.0000	−0.459841
$682$	−14.1962	−0.543599
$683$	31.1769	1.19295	0.596476	−	0.802631i	$-0.296567\pi$
$683$	31.1769	1.19295	0.596476	+	0.802631i	$0.296567\pi$
$684$	−1.14359	−0.0437264
$685$	2.19615	0.0839107
$686$	15.0000	0.572703
$687$	−0.928203	−0.0354132
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	6.92820	0.263752
$691$	−36.7128	−1.39662	−0.698311	−	0.715795i	$-0.746065\pi$
$691$	−36.7128	−1.39662	−0.698311	+	0.715795i	$0.746065\pi$
$692$	15.0000	0.570214
$693$	22.1769	0.842431
$694$	28.7321	1.09065
$695$	1.19615	0.0453726
$696$	1.85641	0.0703669
$697$	85.1769	3.22631
$698$	−4.05256	−0.153392
$699$	−3.46410	−0.131024
$700$	3.00000	0.113389
$701$	−35.9090	−1.35626	−0.678131	−	0.734941i	$-0.737210\pi$
$701$	−35.9090	−1.35626	−0.678131	+	0.734941i	$0.737210\pi$
$702$	0	0
$703$	0.373067	0.0140705
$704$	−3.00000	−0.113067
$705$	2.19615	0.0827119
$706$	−9.80385	−0.368973
$707$	−32.1962	−1.21086
$708$	7.60770	0.285915
$709$	10.9808	0.412391	0.206196	−	0.978511i	$-0.433892\pi$
$709$	10.9808	0.412391	0.206196	+	0.978511i	$0.433892\pi$
$710$	−6.00000	−0.225176
$711$	10.3397	0.387771
$712$	6.80385	0.254985
$713$	44.7846	1.67720
$714$	18.0000	0.673633
$715$	0	0
$716$	9.46410	0.353690
$717$	−1.60770	−0.0600405
$718$	−1.60770	−0.0599987
$719$	−25.8564	−0.964281	−0.482141	−	0.876094i	$-0.660141\pi$
$719$	−25.8564	−0.964281	−0.482141	+	0.876094i	$0.660141\pi$
$720$	−2.46410	−0.0918316
$721$	27.5885	1.02745
$722$	18.7846	0.699091
$723$	−6.33975	−0.235778
$724$	−14.5885	−0.542176
$725$	−2.53590	−0.0941809
$726$	1.46410	0.0543379
$727$	−34.3731	−1.27483	−0.637413	−	0.770522i	$-0.719996\pi$
$727$	−34.3731	−1.27483	−0.637413	+	0.770522i	$0.719996\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	−11.6603	−0.431565
$731$	16.3923	0.606291
$732$	4.53590	0.167652
$733$	−14.9090	−0.550675	−0.275338	−	0.961348i	$-0.588790\pi$
$733$	−14.9090	−0.550675	−0.275338	+	0.961348i	$0.588790\pi$
$734$	5.60770	0.206984
$735$	1.46410	0.0540042
$736$	9.46410	0.348851
$737$	0	0
$738$	−25.6077	−0.942632
$739$	−32.5692	−1.19808	−0.599039	−	0.800720i	$-0.704451\pi$
$739$	−32.5692	−1.19808	−0.599039	+	0.800720i	$0.704451\pi$
$740$	0.803848	0.0295500
$741$	0	0
$742$	−1.39230	−0.0511131
$743$	−13.6077	−0.499218	−0.249609	−	0.968347i	$-0.580302\pi$
$743$	−13.6077	−0.499218	−0.249609	+	0.968347i	$0.580302\pi$
$744$	3.46410	0.127000
$745$	−6.00000	−0.219823
$746$	0.392305	0.0143633
$747$	−20.1962	−0.738939
$748$	24.5885	0.899043
$749$	52.9808	1.93587
$750$	−0.732051	−0.0267307
$751$	20.3923	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$751$	20.3923	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$752$	3.00000	0.109399
$753$	−12.5885	−0.458749
$754$	0	0
$755$	−23.6603	−0.861085
$756$	−12.0000	−0.436436
$757$	−37.7846	−1.37330	−0.686652	−	0.726986i	$-0.740921\pi$
$757$	−37.7846	−1.37330	−0.686652	+	0.726986i	$0.740921\pi$
$758$	−29.7846	−1.08183
$759$	20.7846	0.754434
$760$	−0.464102	−0.0168347
$761$	18.8038	0.681639	0.340819	−	0.940129i	$-0.389296\pi$
$761$	18.8038	0.681639	0.340819	+	0.940129i	$0.389296\pi$
$762$	7.90897	0.286512
$763$	25.6077	0.927061
$764$	−22.7321	−0.822417
$765$	20.1962	0.730193
$766$	22.3923	0.809067
$767$	0	0
$768$	0.732051	0.0264156
$769$	−38.1051	−1.37411	−0.687053	−	0.726607i	$-0.741096\pi$
$769$	−38.1051	−1.37411	−0.687053	+	0.726607i	$0.741096\pi$
$770$	9.00000	0.324337
$771$	10.1436	0.365313
$772$	16.3923	0.589972
$773$	−2.41154	−0.0867372	−0.0433686	−	0.999059i	$-0.513809\pi$
$773$	−2.41154	−0.0867372	−0.0433686	+	0.999059i	$0.513809\pi$
$774$	−4.92820	−0.177141
$775$	−4.73205	−0.169980
$776$	9.12436	0.327545
$777$	1.76537	0.0633324
$778$	22.7321	0.814984
$779$	−4.82309	−0.172805
$780$	0	0
$781$	−18.0000	−0.644091
$782$	−77.5692	−2.77387
$783$	10.1436	0.362502
$784$	2.00000	0.0714286
$785$	−13.0000	−0.463990
$786$	−4.48334	−0.159915
$787$	−42.8372	−1.52698	−0.763490	−	0.645820i	$-0.776516\pi$
$787$	−42.8372	−1.52698	−0.763490	+	0.645820i	$0.776516\pi$
$788$	21.5885	0.769057
$789$	2.62693	0.0935213
$790$	4.19615	0.149292
$791$	20.7846	0.739016
$792$	−7.39230	−0.262674
$793$	0	0
$794$	11.1962	0.397337
$795$	0.339746	0.0120495
$796$	6.39230	0.226569
$797$	12.9282	0.457940	0.228970	−	0.973433i	$-0.426464\pi$
$797$	12.9282	0.457940	0.228970	+	0.973433i	$0.426464\pi$
$798$	−1.01924	−0.0360806
$799$	−24.5885	−0.869877
$800$	−1.00000	−0.0353553
$801$	16.7654	0.592375
$802$	−5.19615	−0.183483
$803$	−34.9808	−1.23444
$804$	0	0
$805$	−28.3923	−1.00070
$806$	0	0
$807$	−19.8564	−0.698979
$808$	10.7321	0.377552
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	−4.46410	−0.156853
$811$	−43.3923	−1.52371	−0.761855	−	0.647748i	$-0.775711\pi$
$811$	−43.3923	−1.52371	−0.761855	+	0.647748i	$0.775711\pi$
$812$	−7.60770	−0.266978
$813$	−6.92820	−0.242983
$814$	2.41154	0.0845245
$815$	−10.7321	−0.375927
$816$	−6.00000	−0.210042
$817$	−0.928203	−0.0324737
$818$	−16.2679	−0.568796
$819$	0	0
$820$	−10.3923	−0.362915
$821$	20.1962	0.704851	0.352425	−	0.935840i	$-0.385357\pi$
$821$	20.1962	0.704851	0.352425	+	0.935840i	$0.385357\pi$
$822$	−1.60770	−0.0560748
$823$	27.1962	0.947998	0.473999	−	0.880525i	$-0.342810\pi$
$823$	27.1962	0.947998	0.473999	+	0.880525i	$0.342810\pi$
$824$	−9.19615	−0.320363
$825$	−2.19615	−0.0764602
$826$	−31.1769	−1.08478
$827$	−42.5885	−1.48095	−0.740473	−	0.672086i	$-0.765398\pi$
$827$	−42.5885	−1.48095	−0.740473	+	0.672086i	$0.765398\pi$
$828$	23.3205	0.810444
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−8.19615	−0.284493
$831$	0.732051	0.0253946
$832$	0	0
$833$	−16.3923	−0.567960
$834$	−0.875644	−0.0303211
$835$	−3.00000	−0.103819
$836$	−1.39230	−0.0481539
$837$	18.9282	0.654254
$838$	17.3205	0.598327
$839$	21.8038	0.752752	0.376376	−	0.926467i	$-0.377170\pi$
$839$	21.8038	0.752752	0.376376	+	0.926467i	$0.377170\pi$
$840$	−2.19615	−0.0757745
$841$	−22.5692	−0.778249
$842$	2.87564	0.0991012
$843$	−7.60770	−0.262023
$844$	−17.5885	−0.605420
$845$	0	0
$846$	7.39230	0.254153
$847$	−6.00000	−0.206162
$848$	0.464102	0.0159373
$849$	7.03332	0.241383
$850$	8.19615	0.281126
$851$	−7.60770	−0.260788
$852$	4.39230	0.150478
$853$	13.8564	0.474434	0.237217	−	0.971457i	$-0.423765\pi$
$853$	13.8564	0.474434	0.237217	+	0.971457i	$0.423765\pi$
$854$	−18.5885	−0.636084
$855$	−1.14359	−0.0391101
$856$	−17.6603	−0.603615
$857$	16.7321	0.571556	0.285778	−	0.958296i	$-0.407748\pi$
$857$	16.7321	0.571556	0.285778	+	0.958296i	$0.407748\pi$
$858$	0	0
$859$	−20.3731	−0.695120	−0.347560	−	0.937658i	$-0.612990\pi$
$859$	−20.3731	−0.695120	−0.347560	+	0.937658i	$0.612990\pi$
$860$	−2.00000	−0.0681994
$861$	−22.8231	−0.777809
$862$	8.19615	0.279162
$863$	−43.1769	−1.46976	−0.734880	−	0.678198i	$-0.762761\pi$
$863$	−43.1769	−1.46976	−0.734880	+	0.678198i	$0.762761\pi$
$864$	4.00000	0.136083
$865$	15.0000	0.510015
$866$	8.39230	0.285182
$867$	36.7321	1.24749
$868$	−14.1962	−0.481849
$869$	12.5885	0.427034
$870$	1.85641	0.0629381
$871$	0	0
$872$	−8.53590	−0.289062
$873$	22.4833	0.760946
$874$	4.39230	0.148572
$875$	3.00000	0.101419
$876$	8.53590	0.288401
$877$	27.4641	0.927397	0.463698	−	0.885993i	$-0.346522\pi$
$877$	27.4641	0.927397	0.463698	+	0.885993i	$0.346522\pi$
$878$	−3.41154	−0.115134
$879$	8.19615	0.276449
$880$	−3.00000	−0.101130
$881$	−26.3205	−0.886760	−0.443380	−	0.896334i	$-0.646221\pi$
$881$	−26.3205	−0.886760	−0.443380	+	0.896334i	$0.646221\pi$
$882$	4.92820	0.165941
$883$	−4.58846	−0.154414	−0.0772069	−	0.997015i	$-0.524600\pi$
$883$	−4.58846	−0.154414	−0.0772069	+	0.997015i	$0.524600\pi$
$884$	0	0
$885$	7.60770	0.255730
$886$	22.3923	0.752284
$887$	−40.7654	−1.36877	−0.684384	−	0.729122i	$-0.739929\pi$
$887$	−40.7654	−1.36877	−0.684384	+	0.729122i	$0.739929\pi$
$888$	−0.588457	−0.0197473
$889$	−32.4115	−1.08705
$890$	6.80385	0.228065
$891$	−13.3923	−0.448659
$892$	−6.46410	−0.216434
$893$	1.39230	0.0465917
$894$	4.39230	0.146901
$895$	9.46410	0.316350
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−15.5885	−0.520194
$899$	12.0000	0.400222
$900$	−2.46410	−0.0821367
$901$	−3.80385	−0.126725
$902$	−31.1769	−1.03808
$903$	−4.39230	−0.146167
$904$	−6.92820	−0.230429
$905$	−14.5885	−0.484937
$906$	17.3205	0.575435
$907$	−4.58846	−0.152357	−0.0761786	−	0.997094i	$-0.524272\pi$
$907$	−4.58846	−0.152357	−0.0761786	+	0.997094i	$0.524272\pi$
$908$	−16.3923	−0.543998
$909$	26.4449	0.877121
$910$	0	0
$911$	−21.4641	−0.711137	−0.355569	−	0.934650i	$-0.615713\pi$
$911$	−21.4641	−0.711137	−0.355569	+	0.934650i	$0.615713\pi$
$912$	0.339746	0.0112501
$913$	−24.5885	−0.813759
$914$	−18.3397	−0.606625
$915$	4.53590	0.149952
$916$	−1.26795	−0.0418942
$917$	18.3731	0.606732
$918$	−32.7846	−1.08205
$919$	30.7846	1.01549	0.507745	−	0.861507i	$-0.330479\pi$
$919$	30.7846	1.01549	0.507745	+	0.861507i	$0.330479\pi$
$920$	9.46410	0.312022
$921$	−20.1051	−0.662486
$922$	−30.5885	−1.00738
$923$	0	0
$924$	−6.58846	−0.216744
$925$	0.803848	0.0264303
$926$	−12.9282	−0.424847
$927$	−22.6603	−0.744260
$928$	2.53590	0.0832449
$929$	3.21539	0.105494	0.0527468	−	0.998608i	$-0.483202\pi$
$929$	3.21539	0.105494	0.0527468	+	0.998608i	$0.483202\pi$
$930$	3.46410	0.113592
$931$	0.928203	0.0304206
$932$	−4.73205	−0.155003
$933$	6.67949	0.218677
$934$	−37.8564	−1.23870
$935$	24.5885	0.804129
$936$	0	0
$937$	−9.60770	−0.313870	−0.156935	−	0.987609i	$-0.550161\pi$
$937$	−9.60770	−0.313870	−0.156935	+	0.987609i	$0.550161\pi$
$938$	0	0
$939$	19.3205	0.630501
$940$	3.00000	0.0978492
$941$	−3.21539	−0.104819	−0.0524094	−	0.998626i	$-0.516690\pi$
$941$	−3.21539	−0.104819	−0.0524094	+	0.998626i	$0.516690\pi$
$942$	9.51666	0.310070
$943$	98.3538	3.20284
$944$	10.3923	0.338241
$945$	−12.0000	−0.390360
$946$	−6.00000	−0.195077
$947$	−15.3731	−0.499558	−0.249779	−	0.968303i	$-0.580358\pi$
$947$	−15.3731	−0.499558	−0.249779	+	0.968303i	$0.580358\pi$
$948$	−3.07180	−0.0997673
$949$	0	0
$950$	−0.464102	−0.0150574
$951$	16.9808	0.550639
$952$	24.5885	0.796916
$953$	6.58846	0.213421	0.106711	−	0.994290i	$-0.465968\pi$
$953$	6.58846	0.213421	0.106711	+	0.994290i	$0.465968\pi$
$954$	1.14359	0.0370252
$955$	−22.7321	−0.735592
$956$	−2.19615	−0.0710286
$957$	5.56922	0.180027
$958$	−40.9808	−1.32403
$959$	6.58846	0.212752
$960$	0.732051	0.0236268
$961$	−8.60770	−0.277668
$962$	0	0
$963$	−43.5167	−1.40230
$964$	−8.66025	−0.278928
$965$	16.3923	0.527687
$966$	20.7846	0.668734
$967$	44.5692	1.43325	0.716625	−	0.697459i	$-0.245686\pi$
$967$	44.5692	1.43325	0.716625	+	0.697459i	$0.245686\pi$
$968$	2.00000	0.0642824
$969$	−2.78461	−0.0894546
$970$	9.12436	0.292965
$971$	46.7654	1.50077	0.750386	−	0.661000i	$-0.229868\pi$
$971$	46.7654	1.50077	0.750386	+	0.661000i	$0.229868\pi$
$972$	15.2679	0.489720
$973$	3.58846	0.115041
$974$	−15.2487	−0.488600
$975$	0	0
$976$	6.19615	0.198334
$977$	16.3923	0.524436	0.262218	−	0.965009i	$-0.415546\pi$
$977$	16.3923	0.524436	0.262218	+	0.965009i	$0.415546\pi$
$978$	7.85641	0.251220
$979$	20.4115	0.652356
$980$	2.00000	0.0638877
$981$	−21.0333	−0.671542
$982$	20.6603	0.659295
$983$	44.5692	1.42154	0.710769	−	0.703426i	$-0.248347\pi$
$983$	44.5692	1.42154	0.710769	+	0.703426i	$0.248347\pi$
$984$	7.60770	0.242524
$985$	21.5885	0.687865
$986$	−20.7846	−0.661917
$987$	6.58846	0.209713
$988$	0	0
$989$	18.9282	0.601882
$990$	−7.39230	−0.234943
$991$	−11.1769	−0.355046	−0.177523	−	0.984117i	$-0.556808\pi$
$991$	−11.1769	−0.355046	−0.177523	+	0.984117i	$0.556808\pi$
$992$	4.73205	0.150243
$993$	−9.46410	−0.300334
$994$	−18.0000	−0.570925
$995$	6.39230	0.202650
$996$	6.00000	0.190117
$997$	40.5692	1.28484	0.642420	−	0.766353i	$-0.277930\pi$
$997$	40.5692	1.28484	0.642420	+	0.766353i	$0.277930\pi$
$998$	25.8564	0.818470
$999$	−3.21539	−0.101730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.j.1.2		2
5.4	even	2		8450.2.a.bm.1.1		2
13.2	odd	12		130.2.l.a.121.1	yes	4
13.3	even	3		1690.2.e.n.191.1		4
13.4	even	6		1690.2.e.l.991.1		4
13.5	odd	4		1690.2.d.f.1351.4		4
13.6	odd	12		1690.2.l.g.361.2		4
13.7	odd	12		130.2.l.a.101.1	✓	4
13.8	odd	4		1690.2.d.f.1351.2		4
13.9	even	3		1690.2.e.n.991.1		4
13.10	even	6		1690.2.e.l.191.1		4
13.11	odd	12		1690.2.l.g.1161.2		4
13.12	even	2		1690.2.a.m.1.2		2
39.2	even	12		1170.2.bs.c.901.2		4
39.20	even	12		1170.2.bs.c.361.2		4
52.7	even	12		1040.2.da.a.881.2		4
52.15	even	12		1040.2.da.a.641.2		4
65.2	even	12		650.2.n.b.199.1		4
65.7	even	12		650.2.n.a.49.2		4
65.28	even	12		650.2.n.a.199.2		4
65.33	even	12		650.2.n.b.49.1		4
65.54	odd	12		650.2.m.a.251.2		4
65.59	odd	12		650.2.m.a.101.2		4
65.64	even	2		8450.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.l.a.101.1	✓	4	13.7	odd	12
130.2.l.a.121.1	yes	4	13.2	odd	12
650.2.m.a.101.2		4	65.59	odd	12
650.2.m.a.251.2		4	65.54	odd	12
650.2.n.a.49.2		4	65.7	even	12
650.2.n.a.199.2		4	65.28	even	12
650.2.n.b.49.1		4	65.33	even	12
650.2.n.b.199.1		4	65.2	even	12
1040.2.da.a.641.2		4	52.15	even	12
1040.2.da.a.881.2		4	52.7	even	12
1170.2.bs.c.361.2		4	39.20	even	12
1170.2.bs.c.901.2		4	39.2	even	12
1690.2.a.j.1.2		2	1.1	even	1	trivial
1690.2.a.m.1.2		2	13.12	even	2
1690.2.d.f.1351.2		4	13.8	odd	4
1690.2.d.f.1351.4		4	13.5	odd	4
1690.2.e.l.191.1		4	13.10	even	6
1690.2.e.l.991.1		4	13.4	even	6
1690.2.e.n.191.1		4	13.3	even	3
1690.2.e.n.991.1		4	13.9	even	3
1690.2.l.g.361.2		4	13.6	odd	12
1690.2.l.g.1161.2		4	13.11	odd	12
8450.2.a.bf.1.1		2	65.64	even	2
8450.2.a.bm.1.1		2	5.4	even	2

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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} +3.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.00000 q^{10} -3.00000 q^{11} +0.732051 q^{12} -3.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} -8.19615 q^{17} +2.46410 q^{18} +0.464102 q^{19} +1.00000 q^{20} +2.19615 q^{21} +3.00000 q^{22} -9.46410 q^{23} -0.732051 q^{24} +1.00000 q^{25} -4.00000 q^{27} +3.00000 q^{28} -2.53590 q^{29} -0.732051 q^{30} -4.73205 q^{31} -1.00000 q^{32} -2.19615 q^{33} +8.19615 q^{34} +3.00000 q^{35} -2.46410 q^{36} +0.803848 q^{37} -0.464102 q^{38} -1.00000 q^{40} -10.3923 q^{41} -2.19615 q^{42} -2.00000 q^{43} -3.00000 q^{44} -2.46410 q^{45} +9.46410 q^{46} +3.00000 q^{47} +0.732051 q^{48} +2.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +0.464102 q^{53} +4.00000 q^{54} -3.00000 q^{55} -3.00000 q^{56} +0.339746 q^{57} +2.53590 q^{58} +10.3923 q^{59} +0.732051 q^{60} +6.19615 q^{61} +4.73205 q^{62} -7.39230 q^{63} +1.00000 q^{64} +2.19615 q^{66} -8.19615 q^{68} -6.92820 q^{69} -3.00000 q^{70} +6.00000 q^{71} +2.46410 q^{72} +11.6603 q^{73} -0.803848 q^{74} +0.732051 q^{75} +0.464102 q^{76} -9.00000 q^{77} -4.19615 q^{79} +1.00000 q^{80} +4.46410 q^{81} +10.3923 q^{82} +8.19615 q^{83} +2.19615 q^{84} -8.19615 q^{85} +2.00000 q^{86} -1.85641 q^{87} +3.00000 q^{88} -6.80385 q^{89} +2.46410 q^{90} -9.46410 q^{92} -3.46410 q^{93} -3.00000 q^{94} +0.464102 q^{95} -0.732051 q^{96} -9.12436 q^{97} -2.00000 q^{98} +7.39230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{15} + 2 q^{16} - 6 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 6 q^{21} + 6 q^{22} - 12 q^{23} + 2 q^{24} + 2 q^{25} - 8 q^{27} + 6 q^{28} - 12 q^{29} + 2 q^{30} - 6 q^{31} - 2 q^{32} + 6 q^{33} + 6 q^{34} + 6 q^{35} + 2 q^{36} + 12 q^{37} + 6 q^{38} - 2 q^{40} + 6 q^{42} - 4 q^{43} - 6 q^{44} + 2 q^{45} + 12 q^{46} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 2 q^{50} - 12 q^{51} - 6 q^{53} + 8 q^{54} - 6 q^{55} - 6 q^{56} + 18 q^{57} + 12 q^{58} - 2 q^{60} + 2 q^{61} + 6 q^{62} + 6 q^{63} + 2 q^{64} - 6 q^{66} - 6 q^{68} - 6 q^{70} + 12 q^{71} - 2 q^{72} + 6 q^{73} - 12 q^{74} - 2 q^{75} - 6 q^{76} - 18 q^{77} + 2 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 4 q^{86} + 24 q^{87} + 6 q^{88} - 24 q^{89} - 2 q^{90} - 12 q^{92} - 6 q^{94} - 6 q^{95} + 2 q^{96} + 6 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^15 + 2 * q^16 - 6 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 6 * q^21 + 6 * q^22 - 12 * q^23 + 2 * q^24 + 2 * q^25 - 8 * q^27 + 6 * q^28 - 12 * q^29 + 2 * q^30 - 6 * q^31 - 2 * q^32 + 6 * q^33 + 6 * q^34 + 6 * q^35 + 2 * q^36 + 12 * q^37 + 6 * q^38 - 2 * q^40 + 6 * q^42 - 4 * q^43 - 6 * q^44 + 2 * q^45 + 12 * q^46 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 2 * q^50 - 12 * q^51 - 6 * q^53 + 8 * q^54 - 6 * q^55 - 6 * q^56 + 18 * q^57 + 12 * q^58 - 2 * q^60 + 2 * q^61 + 6 * q^62 + 6 * q^63 + 2 * q^64 - 6 * q^66 - 6 * q^68 - 6 * q^70 + 12 * q^71 - 2 * q^72 + 6 * q^73 - 12 * q^74 - 2 * q^75 - 6 * q^76 - 18 * q^77 + 2 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^83 - 6 * q^84 - 6 * q^85 + 4 * q^86 + 24 * q^87 + 6 * q^88 - 24 * q^89 - 2 * q^90 - 12 * q^92 - 6 * q^94 - 6 * q^95 + 2 * q^96 + 6 * q^97 - 4 * q^98 - 6 * q^99

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