LMFDB - Embedded newform 1690.2.a.v.1.3

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:	$6$
Coefficient field:	6.6.20439713.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 15x^{4} + 22x^{3} + 70x^{2} - 56x - 91 $ x^6 - 2x^5 - 15x^4 + 22x^3 + 70x^2 - 56*x - 91
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77061$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.77061 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.77061 q^{6} -3.83691 q^{7} -1.00000 q^{8} +0.135043 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.77061 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.77061 q^{6} -3.83691 q^{7} -1.00000 q^{8} +0.135043 q^{9} +1.00000 q^{10} +4.58239 q^{11} -1.77061 q^{12} +3.83691 q^{14} +1.77061 q^{15} +1.00000 q^{16} +3.86860 q^{17} -0.135043 q^{18} -5.15145 q^{19} -1.00000 q^{20} +6.79365 q^{21} -4.58239 q^{22} +4.98463 q^{23} +1.77061 q^{24} +1.00000 q^{25} +5.07271 q^{27} -3.83691 q^{28} +0.976494 q^{29} -1.77061 q^{30} +8.74896 q^{31} -1.00000 q^{32} -8.11360 q^{33} -3.86860 q^{34} +3.83691 q^{35} +0.135043 q^{36} -6.45918 q^{37} +5.15145 q^{38} +1.00000 q^{40} +2.57254 q^{41} -6.79365 q^{42} -12.0727 q^{43} +4.58239 q^{44} -0.135043 q^{45} -4.98463 q^{46} -5.52245 q^{47} -1.77061 q^{48} +7.72188 q^{49} -1.00000 q^{50} -6.84976 q^{51} -0.238957 q^{53} -5.07271 q^{54} -4.58239 q^{55} +3.83691 q^{56} +9.12118 q^{57} -0.976494 q^{58} -1.77799 q^{59} +1.77061 q^{60} +13.0381 q^{61} -8.74896 q^{62} -0.518148 q^{63} +1.00000 q^{64} +8.11360 q^{66} +4.50850 q^{67} +3.86860 q^{68} -8.82582 q^{69} -3.83691 q^{70} -15.7017 q^{71} -0.135043 q^{72} -2.09597 q^{73} +6.45918 q^{74} -1.77061 q^{75} -5.15145 q^{76} -17.5822 q^{77} +7.75196 q^{79} -1.00000 q^{80} -9.38689 q^{81} -2.57254 q^{82} -13.7597 q^{83} +6.79365 q^{84} -3.86860 q^{85} +12.0727 q^{86} -1.72898 q^{87} -4.58239 q^{88} -11.1506 q^{89} +0.135043 q^{90} +4.98463 q^{92} -15.4910 q^{93} +5.52245 q^{94} +5.15145 q^{95} +1.77061 q^{96} +7.79046 q^{97} -7.72188 q^{98} +0.618819 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.77061	−1.02226	−0.511130	−	0.859504i	$-0.670773\pi$
$3$	−1.77061	−1.02226	−0.511130	+	0.859504i	$0.670773\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.77061	0.722847
$7$	−3.83691	−1.45022	−0.725108	−	0.688635i	$-0.758210\pi$
$7$	−3.83691	−1.45022	−0.725108	+	0.688635i	$0.758210\pi$
$8$	−1.00000	−0.353553
$9$	0.135043	0.0450143
$10$	1.00000	0.316228
$11$	4.58239	1.38164	0.690821	−	0.723026i	$-0.257249\pi$
$11$	4.58239	1.38164	0.690821	+	0.723026i	$0.257249\pi$
$12$	−1.77061	−0.511130
$13$	0	0
$13$	0	0
$14$	3.83691	1.02546
$15$	1.77061	0.457168
$16$	1.00000	0.250000
$17$	3.86860	0.938273	0.469136	−	0.883126i	$-0.344565\pi$
$17$	3.86860	0.938273	0.469136	+	0.883126i	$0.344565\pi$
$18$	−0.135043	−0.0318299
$19$	−5.15145	−1.18182	−0.590912	−	0.806736i	$-0.701232\pi$
$19$	−5.15145	−1.18182	−0.590912	+	0.806736i	$0.701232\pi$
$20$	−1.00000	−0.223607
$21$	6.79365	1.48250
$22$	−4.58239	−0.976969
$23$	4.98463	1.03937	0.519684	−	0.854359i	$-0.326050\pi$
$23$	4.98463	1.03937	0.519684	+	0.854359i	$0.326050\pi$
$24$	1.77061	0.361423
$25$	1.00000	0.200000
$26$	0	0
$27$	5.07271	0.976243
$28$	−3.83691	−0.725108
$29$	0.976494	0.181330	0.0906651	−	0.995881i	$-0.471101\pi$
$29$	0.976494	0.181330	0.0906651	+	0.995881i	$0.471101\pi$
$30$	−1.77061	−0.323267
$31$	8.74896	1.57136	0.785680	−	0.618633i	$-0.212313\pi$
$31$	8.74896	1.57136	0.785680	+	0.618633i	$0.212313\pi$
$32$	−1.00000	−0.176777
$33$	−8.11360	−1.41240
$34$	−3.86860	−0.663459
$35$	3.83691	0.648556
$36$	0.135043	0.0225072
$37$	−6.45918	−1.06188	−0.530942	−	0.847408i	$-0.678162\pi$
$37$	−6.45918	−1.06188	−0.530942	+	0.847408i	$0.678162\pi$
$38$	5.15145	0.835675
$39$	0	0
$40$	1.00000	0.158114
$41$	2.57254	0.401764	0.200882	−	0.979615i	$-0.435619\pi$
$41$	2.57254	0.401764	0.200882	+	0.979615i	$0.435619\pi$
$42$	−6.79365	−1.04828
$43$	−12.0727	−1.84107	−0.920535	−	0.390659i	$-0.872247\pi$
$43$	−12.0727	−1.84107	−0.920535	+	0.390659i	$0.872247\pi$
$44$	4.58239	0.690821
$45$	−0.135043	−0.0201310
$46$	−4.98463	−0.734944
$47$	−5.52245	−0.805532	−0.402766	−	0.915303i	$-0.631951\pi$
$47$	−5.52245	−0.805532	−0.402766	+	0.915303i	$0.631951\pi$
$48$	−1.77061	−0.255565
$49$	7.72188	1.10313
$50$	−1.00000	−0.141421
$51$	−6.84976	−0.959158
$52$	0	0
$53$	−0.238957	−0.0328233	−0.0164116	−	0.999865i	$-0.505224\pi$
$53$	−0.238957	−0.0328233	−0.0164116	+	0.999865i	$0.505224\pi$
$54$	−5.07271	−0.690308
$55$	−4.58239	−0.617889
$56$	3.83691	0.512729
$57$	9.12118	1.20813
$58$	−0.976494	−0.128220
$59$	−1.77799	−0.231475	−0.115737	−	0.993280i	$-0.536923\pi$
$59$	−1.77799	−0.231475	−0.115737	+	0.993280i	$0.536923\pi$
$60$	1.77061	0.228584
$61$	13.0381	1.66936	0.834678	−	0.550738i	$-0.185654\pi$
$61$	13.0381	1.66936	0.834678	+	0.550738i	$0.185654\pi$
$62$	−8.74896	−1.11112
$63$	−0.518148	−0.0652805
$64$	1.00000	0.125000
$65$	0	0
$66$	8.11360	0.998715
$67$	4.50850	0.550801	0.275401	−	0.961330i	$-0.411189\pi$
$67$	4.50850	0.550801	0.275401	+	0.961330i	$0.411189\pi$
$68$	3.86860	0.469136
$69$	−8.82582	−1.06250
$70$	−3.83691	−0.458599
$71$	−15.7017	−1.86345	−0.931725	−	0.363164i	$-0.881696\pi$
$71$	−15.7017	−1.86345	−0.931725	+	0.363164i	$0.881696\pi$
$72$	−0.135043	−0.0159150
$73$	−2.09597	−0.245315	−0.122657	−	0.992449i	$-0.539142\pi$
$73$	−2.09597	−0.245315	−0.122657	+	0.992449i	$0.539142\pi$
$74$	6.45918	0.750865
$75$	−1.77061	−0.204452
$76$	−5.15145	−0.590912
$77$	−17.5822	−2.00368
$78$	0	0
$79$	7.75196	0.872164	0.436082	−	0.899907i	$-0.356366\pi$
$79$	7.75196	0.872164	0.436082	+	0.899907i	$0.356366\pi$
$80$	−1.00000	−0.111803
$81$	−9.38689	−1.04299
$82$	−2.57254	−0.284090
$83$	−13.7597	−1.51032	−0.755162	−	0.655538i	$-0.772442\pi$
$83$	−13.7597	−1.51032	−0.755162	+	0.655538i	$0.772442\pi$
$84$	6.79365	0.741248
$85$	−3.86860	−0.419608
$86$	12.0727	1.30183
$87$	−1.72898	−0.185367
$88$	−4.58239	−0.488484
$89$	−11.1506	−1.18196	−0.590978	−	0.806688i	$-0.701258\pi$
$89$	−11.1506	−1.18196	−0.590978	+	0.806688i	$0.701258\pi$
$90$	0.135043	0.0142348
$91$	0	0
$92$	4.98463	0.519684
$93$	−15.4910	−1.60634
$94$	5.52245	0.569597
$95$	5.15145	0.528528
$96$	1.77061	0.180712
$97$	7.79046	0.791001	0.395500	−	0.918466i	$-0.370571\pi$
$97$	7.79046	0.791001	0.395500	+	0.918466i	$0.370571\pi$
$98$	−7.72188	−0.780028
$99$	0.618819	0.0621937
$100$	1.00000	0.100000
$101$	7.52902	0.749165	0.374583	−	0.927193i	$-0.377786\pi$
$101$	7.52902	0.749165	0.374583	+	0.927193i	$0.377786\pi$
$102$	6.84976	0.678227
$103$	−10.6767	−1.05201	−0.526005	−	0.850481i	$-0.676311\pi$
$103$	−10.6767	−1.05201	−0.526005	+	0.850481i	$0.676311\pi$
$104$	0	0
$105$	−6.79365	−0.662993
$106$	0.238957	0.0232096
$107$	−2.70774	−0.261767	−0.130884	−	0.991398i	$-0.541781\pi$
$107$	−2.70774	−0.261767	−0.130884	+	0.991398i	$0.541781\pi$
$108$	5.07271	0.488122
$109$	15.5231	1.48685	0.743424	−	0.668821i	$-0.233201\pi$
$109$	15.5231	1.48685	0.743424	+	0.668821i	$0.233201\pi$
$110$	4.58239	0.436914
$111$	11.4367	1.08552
$112$	−3.83691	−0.362554
$113$	11.0209	1.03676	0.518380	−	0.855150i	$-0.326535\pi$
$113$	11.0209	1.03676	0.518380	+	0.855150i	$0.326535\pi$
$114$	−9.12118	−0.854277
$115$	−4.98463	−0.464819
$116$	0.976494	0.0906651
$117$	0	0
$118$	1.77799	0.163677
$119$	−14.8435	−1.36070
$120$	−1.77061	−0.161633
$121$	9.99829	0.908936
$122$	−13.0381	−1.18041
$123$	−4.55496	−0.410707
$124$	8.74896	0.785680
$125$	−1.00000	−0.0894427
$126$	0.518148	0.0461603
$127$	−20.2183	−1.79409	−0.897044	−	0.441941i	$-0.854290\pi$
$127$	−20.2183	−1.79409	−0.897044	+	0.441941i	$0.854290\pi$
$128$	−1.00000	−0.0883883
$129$	21.3760	1.88205
$130$	0	0
$131$	−8.24571	−0.720431	−0.360216	−	0.932869i	$-0.617297\pi$
$131$	−8.24571	−0.720431	−0.360216	+	0.932869i	$0.617297\pi$
$132$	−8.11360	−0.706198
$133$	19.7657	1.71390
$134$	−4.50850	−0.389475
$135$	−5.07271	−0.436589
$136$	−3.86860	−0.331730
$137$	−19.1981	−1.64020	−0.820102	−	0.572218i	$-0.806083\pi$
$137$	−19.1981	−1.64020	−0.820102	+	0.572218i	$0.806083\pi$
$138$	8.82582	0.751303
$139$	−3.23295	−0.274215	−0.137108	−	0.990556i	$-0.543781\pi$
$139$	−3.23295	−0.274215	−0.137108	+	0.990556i	$0.543781\pi$
$140$	3.83691	0.324278
$141$	9.77807	0.823462
$142$	15.7017	1.31766
$143$	0	0
$144$	0.135043	0.0112536
$145$	−0.976494	−0.0810934
$146$	2.09597	0.173464
$147$	−13.6724	−1.12768
$148$	−6.45918	−0.530942
$149$	−13.5100	−1.10678	−0.553390	−	0.832922i	$-0.686666\pi$
$149$	−13.5100	−1.10678	−0.553390	+	0.832922i	$0.686666\pi$
$150$	1.77061	0.144569
$151$	−11.6044	−0.944356	−0.472178	−	0.881503i	$-0.656532\pi$
$151$	−11.6044	−0.944356	−0.472178	+	0.881503i	$0.656532\pi$
$152$	5.15145	0.417838
$153$	0.522427	0.0422357
$154$	17.5822	1.41682
$155$	−8.74896	−0.702733
$156$	0	0
$157$	2.15393	0.171902	0.0859510	−	0.996299i	$-0.472607\pi$
$157$	2.15393	0.171902	0.0859510	+	0.996299i	$0.472607\pi$
$158$	−7.75196	−0.616713
$159$	0.423098	0.0335539
$160$	1.00000	0.0790569
$161$	−19.1256	−1.50731
$162$	9.38689	0.737504
$163$	9.80188	0.767743	0.383871	−	0.923387i	$-0.374591\pi$
$163$	9.80188	0.767743	0.383871	+	0.923387i	$0.374591\pi$
$164$	2.57254	0.200882
$165$	8.11360	0.631643
$166$	13.7597	1.06796
$167$	−4.79005	−0.370665	−0.185333	−	0.982676i	$-0.559336\pi$
$167$	−4.79005	−0.370665	−0.185333	+	0.982676i	$0.559336\pi$
$168$	−6.79365	−0.524142
$169$	0	0
$170$	3.86860	0.296708
$171$	−0.695667	−0.0531990
$172$	−12.0727	−0.920535
$173$	−9.24324	−0.702750	−0.351375	−	0.936235i	$-0.614286\pi$
$173$	−9.24324	−0.702750	−0.351375	+	0.936235i	$0.614286\pi$
$174$	1.72898	0.131074
$175$	−3.83691	−0.290043
$176$	4.58239	0.345411
$177$	3.14812	0.236627
$178$	11.1506	0.835769
$179$	−17.9476	−1.34147	−0.670734	−	0.741698i	$-0.734021\pi$
$179$	−17.9476	−1.34147	−0.670734	+	0.741698i	$0.734021\pi$
$180$	−0.135043	−0.0100655
$181$	−13.6207	−1.01242	−0.506208	−	0.862411i	$-0.668953\pi$
$181$	−13.6207	−1.01242	−0.506208	+	0.862411i	$0.668953\pi$
$182$	0	0
$183$	−23.0853	−1.70652
$184$	−4.98463	−0.367472
$185$	6.45918	0.474889
$186$	15.4910	1.13585
$187$	17.7274	1.29636
$188$	−5.52245	−0.402766
$189$	−19.4635	−1.41576
$190$	−5.15145	−0.373725
$191$	2.19510	0.158832	0.0794160	−	0.996842i	$-0.474694\pi$
$191$	2.19510	0.158832	0.0794160	+	0.996842i	$0.474694\pi$
$192$	−1.77061	−0.127782
$193$	−1.22231	−0.0879838	−0.0439919	−	0.999032i	$-0.514008\pi$
$193$	−1.22231	−0.0879838	−0.0439919	+	0.999032i	$0.514008\pi$
$194$	−7.79046	−0.559322
$195$	0	0
$196$	7.72188	0.551563
$197$	15.3850	1.09614	0.548068	−	0.836434i	$-0.315363\pi$
$197$	15.3850	1.09614	0.548068	+	0.836434i	$0.315363\pi$
$198$	−0.618819	−0.0439776
$199$	−11.0785	−0.785332	−0.392666	−	0.919681i	$-0.628447\pi$
$199$	−11.0785	−0.785332	−0.392666	+	0.919681i	$0.628447\pi$
$200$	−1.00000	−0.0707107
$201$	−7.98278	−0.563062
$202$	−7.52902	−0.529740
$203$	−3.74672	−0.262968
$204$	−6.84976	−0.479579
$205$	−2.57254	−0.179674
$206$	10.6767	0.743884
$207$	0.673140	0.0467864
$208$	0	0
$209$	−23.6059	−1.63286
$210$	6.79365	0.468807
$211$	−19.7495	−1.35961	−0.679807	−	0.733391i	$-0.737936\pi$
$211$	−19.7495	−1.35961	−0.679807	+	0.733391i	$0.737936\pi$
$212$	−0.238957	−0.0164116
$213$	27.8015	1.90493
$214$	2.70774	0.185098
$215$	12.0727	0.823352
$216$	−5.07271	−0.345154
$217$	−33.5690	−2.27881
$218$	−15.5231	−1.05136
$219$	3.71114	0.250776
$220$	−4.58239	−0.308945
$221$	0	0
$222$	−11.4367	−0.767579
$223$	12.9244	0.865485	0.432742	−	0.901518i	$-0.357546\pi$
$223$	12.9244	0.865485	0.432742	+	0.901518i	$0.357546\pi$
$224$	3.83691	0.256364
$225$	0.135043	0.00900286
$226$	−11.0209	−0.733100
$227$	22.2184	1.47468	0.737342	−	0.675520i	$-0.236081\pi$
$227$	22.2184	1.47468	0.737342	+	0.675520i	$0.236081\pi$
$228$	9.12118	0.604065
$229$	−24.5775	−1.62413	−0.812064	−	0.583568i	$-0.801656\pi$
$229$	−24.5775	−1.62413	−0.812064	+	0.583568i	$0.801656\pi$
$230$	4.98463	0.328677
$231$	31.1312	2.04828
$232$	−0.976494	−0.0641099
$233$	13.0303	0.853644	0.426822	−	0.904336i	$-0.359633\pi$
$233$	13.0303	0.853644	0.426822	+	0.904336i	$0.359633\pi$
$234$	0	0
$235$	5.52245	0.360245
$236$	−1.77799	−0.115737
$237$	−13.7257	−0.891578
$238$	14.8435	0.962159
$239$	−4.81106	−0.311201	−0.155601	−	0.987820i	$-0.549731\pi$
$239$	−4.81106	−0.311201	−0.155601	+	0.987820i	$0.549731\pi$
$240$	1.77061	0.114292
$241$	1.43938	0.0927187	0.0463593	−	0.998925i	$-0.485238\pi$
$241$	1.43938	0.0927187	0.0463593	+	0.998925i	$0.485238\pi$
$242$	−9.99829	−0.642715
$243$	1.40236	0.0899612
$244$	13.0381	0.834678
$245$	−7.72188	−0.493333
$246$	4.55496	0.290414
$247$	0	0
$248$	−8.74896	−0.555560
$249$	24.3630	1.54394
$250$	1.00000	0.0632456
$251$	−11.8658	−0.748960	−0.374480	−	0.927235i	$-0.622179\pi$
$251$	−11.8658	−0.748960	−0.374480	+	0.927235i	$0.622179\pi$
$252$	−0.518148	−0.0326402
$253$	22.8415	1.43603
$254$	20.2183	1.26861
$255$	6.84976	0.428949
$256$	1.00000	0.0625000
$257$	−11.1225	−0.693805	−0.346902	−	0.937901i	$-0.612766\pi$
$257$	−11.1225	−0.693805	−0.346902	+	0.937901i	$0.612766\pi$
$258$	−21.3760	−1.33081
$259$	24.7833	1.53996
$260$	0	0
$261$	0.131869	0.00816246
$262$	8.24571	0.509422
$263$	1.28305	0.0791163	0.0395581	−	0.999217i	$-0.487405\pi$
$263$	1.28305	0.0791163	0.0395581	+	0.999217i	$0.487405\pi$
$264$	8.11360	0.499358
$265$	0.238957	0.0146790
$266$	−19.7657	−1.21191
$267$	19.7432	1.20827
$268$	4.50850	0.275401
$269$	−17.9493	−1.09439	−0.547195	−	0.837005i	$-0.684304\pi$
$269$	−17.9493	−1.09439	−0.547195	+	0.837005i	$0.684304\pi$
$270$	5.07271	0.308715
$271$	25.3959	1.54269	0.771347	−	0.636415i	$-0.219584\pi$
$271$	25.3959	1.54269	0.771347	+	0.636415i	$0.219584\pi$
$272$	3.86860	0.234568
$273$	0	0
$274$	19.1981	1.15980
$275$	4.58239	0.276328
$276$	−8.82582	−0.531252
$277$	5.42610	0.326023	0.163011	−	0.986624i	$-0.447879\pi$
$277$	5.42610	0.326023	0.163011	+	0.986624i	$0.447879\pi$
$278$	3.23295	0.193899
$279$	1.18149	0.0707337
$280$	−3.83691	−0.229299
$281$	−2.71086	−0.161716	−0.0808581	−	0.996726i	$-0.525766\pi$
$281$	−2.71086	−0.161716	−0.0808581	+	0.996726i	$0.525766\pi$
$282$	−9.77807	−0.582276
$283$	−13.4333	−0.798525	−0.399263	−	0.916837i	$-0.630734\pi$
$283$	−13.4333	−0.798525	−0.399263	+	0.916837i	$0.630734\pi$
$284$	−15.7017	−0.931725
$285$	−9.12118	−0.540292
$286$	0	0
$287$	−9.87062	−0.582644
$288$	−0.135043	−0.00795748
$289$	−2.03395	−0.119644
$290$	0.976494	0.0573417
$291$	−13.7938	−0.808608
$292$	−2.09597	−0.122657
$293$	14.0044	0.818148	0.409074	−	0.912501i	$-0.365852\pi$
$293$	14.0044	0.818148	0.409074	+	0.912501i	$0.365852\pi$
$294$	13.6724	0.797391
$295$	1.77799	0.103519
$296$	6.45918	0.375432
$297$	23.2451	1.34882
$298$	13.5100	0.782612
$299$	0	0
$300$	−1.77061	−0.102226
$301$	46.3219	2.66995
$302$	11.6044	0.667761
$303$	−13.3309	−0.765841
$304$	−5.15145	−0.295456
$305$	−13.0381	−0.746559
$306$	−0.522427	−0.0298652
$307$	4.42199	0.252376	0.126188	−	0.992006i	$-0.459726\pi$
$307$	4.42199	0.252376	0.126188	+	0.992006i	$0.459726\pi$
$308$	−17.5822	−1.00184
$309$	18.9043	1.07543
$310$	8.74896	0.496908
$311$	15.0706	0.854575	0.427287	−	0.904116i	$-0.359469\pi$
$311$	15.0706	0.854575	0.427287	+	0.904116i	$0.359469\pi$
$312$	0	0
$313$	27.8961	1.57678	0.788392	−	0.615174i	$-0.210914\pi$
$313$	27.8961	1.57678	0.788392	+	0.615174i	$0.210914\pi$
$314$	−2.15393	−0.121553
$315$	0.518148	0.0291943
$316$	7.75196	0.436082
$317$	21.1887	1.19007	0.595037	−	0.803699i	$-0.297138\pi$
$317$	21.1887	1.19007	0.595037	+	0.803699i	$0.297138\pi$
$318$	−0.423098	−0.0237262
$319$	4.47467	0.250534
$320$	−1.00000	−0.0559017
$321$	4.79435	0.267594
$322$	19.1256	1.06583
$323$	−19.9289	−1.10887
$324$	−9.38689	−0.521494
$325$	0	0
$326$	−9.80188	−0.542876
$327$	−27.4854	−1.51994
$328$	−2.57254	−0.142045
$329$	21.1891	1.16819
$330$	−8.11360	−0.446639
$331$	−24.7539	−1.36060	−0.680298	−	0.732935i	$-0.738150\pi$
$331$	−24.7539	−1.36060	−0.680298	+	0.732935i	$0.738150\pi$
$332$	−13.7597	−0.755162
$333$	−0.872267	−0.0477999
$334$	4.79005	0.262100
$335$	−4.50850	−0.246326
$336$	6.79365	0.370624
$337$	21.2095	1.15535	0.577677	−	0.816266i	$-0.303959\pi$
$337$	21.2095	1.15535	0.577677	+	0.816266i	$0.303959\pi$
$338$	0	0
$339$	−19.5137	−1.05984
$340$	−3.86860	−0.209804
$341$	40.0911	2.17106
$342$	0.695667	0.0376174
$343$	−2.76981	−0.149556
$344$	12.0727	0.650917
$345$	8.82582	0.475166
$346$	9.24324	0.496919
$347$	−16.8348	−0.903740	−0.451870	−	0.892084i	$-0.649243\pi$
$347$	−16.8348	−0.903740	−0.451870	+	0.892084i	$0.649243\pi$
$348$	−1.72898	−0.0926833
$349$	12.0036	0.642541	0.321270	−	0.946988i	$-0.395890\pi$
$349$	12.0036	0.642541	0.321270	+	0.946988i	$0.395890\pi$
$350$	3.83691	0.205092
$351$	0	0
$352$	−4.58239	−0.244242
$353$	−19.2651	−1.02538	−0.512688	−	0.858575i	$-0.671350\pi$
$353$	−19.2651	−1.02538	−0.512688	+	0.858575i	$0.671350\pi$
$354$	−3.14812	−0.167321
$355$	15.7017	0.833360
$356$	−11.1506	−0.590978
$357$	26.2819	1.39099
$358$	17.9476	0.948561
$359$	25.7690	1.36003	0.680017	−	0.733196i	$-0.261972\pi$
$359$	25.7690	1.36003	0.680017	+	0.733196i	$0.261972\pi$
$360$	0.135043	0.00711739
$361$	7.53743	0.396707
$362$	13.6207	0.715887
$363$	−17.7030	−0.929168
$364$	0	0
$365$	2.09597	0.109708
$366$	23.0853	1.20669
$367$	−8.62414	−0.450177	−0.225088	−	0.974338i	$-0.572267\pi$
$367$	−8.62414	−0.450177	−0.225088	+	0.974338i	$0.572267\pi$
$368$	4.98463	0.259842
$369$	0.347404	0.0180851
$370$	−6.45918	−0.335797
$371$	0.916857	0.0476008
$372$	−15.4910	−0.803169
$373$	−13.7724	−0.713106	−0.356553	−	0.934275i	$-0.616048\pi$
$373$	−13.7724	−0.713106	−0.356553	+	0.934275i	$0.616048\pi$
$374$	−17.7274	−0.916663
$375$	1.77061	0.0914337
$376$	5.52245	0.284798
$377$	0	0
$378$	19.4635	1.00110
$379$	−17.4269	−0.895160	−0.447580	−	0.894244i	$-0.647714\pi$
$379$	−17.4269	−0.895160	−0.447580	+	0.894244i	$0.647714\pi$
$380$	5.15145	0.264264
$381$	35.7987	1.83402
$382$	−2.19510	−0.112311
$383$	28.9137	1.47742	0.738711	−	0.674022i	$-0.235435\pi$
$383$	28.9137	1.47742	0.738711	+	0.674022i	$0.235435\pi$
$384$	1.77061	0.0903558
$385$	17.5822	0.896073
$386$	1.22231	0.0622140
$387$	−1.63033	−0.0828745
$388$	7.79046	0.395500
$389$	−21.0537	−1.06747	−0.533734	−	0.845653i	$-0.679211\pi$
$389$	−21.0537	−1.06747	−0.533734	+	0.845653i	$0.679211\pi$
$390$	0	0
$391$	19.2835	0.975210
$392$	−7.72188	−0.390014
$393$	14.5999	0.736468
$394$	−15.3850	−0.775085
$395$	−7.75196	−0.390044
$396$	0.618819	0.0310968
$397$	29.8791	1.49959	0.749795	−	0.661671i	$-0.230152\pi$
$397$	29.8791	1.49959	0.749795	+	0.661671i	$0.230152\pi$
$398$	11.0785	0.555314
$399$	−34.9972	−1.75205
$400$	1.00000	0.0500000
$401$	−22.5805	−1.12762	−0.563808	−	0.825906i	$-0.690664\pi$
$401$	−22.5805	−1.12762	−0.563808	+	0.825906i	$0.690664\pi$
$402$	7.98278	0.398145
$403$	0	0
$404$	7.52902	0.374583
$405$	9.38689	0.466438
$406$	3.74672	0.185947
$407$	−29.5985	−1.46714
$408$	6.84976	0.339114
$409$	27.3927	1.35448	0.677241	−	0.735761i	$-0.263175\pi$
$409$	27.3927	1.35448	0.677241	+	0.735761i	$0.263175\pi$
$410$	2.57254	0.127049
$411$	33.9922	1.67671
$412$	−10.6767	−0.526005
$413$	6.82199	0.335688
$414$	−0.673140	−0.0330830
$415$	13.7597	0.675437
$416$	0	0
$417$	5.72427	0.280319
$418$	23.6059	1.15460
$419$	−6.37956	−0.311662	−0.155831	−	0.987784i	$-0.549805\pi$
$419$	−6.37956	−0.311662	−0.155831	+	0.987784i	$0.549805\pi$
$420$	−6.79365	−0.331496
$421$	−13.6636	−0.665923	−0.332961	−	0.942940i	$-0.608048\pi$
$421$	−13.6636	−0.665923	−0.332961	+	0.942940i	$0.608048\pi$
$422$	19.7495	0.961392
$423$	−0.745768	−0.0362605
$424$	0.238957	0.0116048
$425$	3.86860	0.187655
$426$	−27.8015	−1.34699
$427$	−50.0260	−2.42093
$428$	−2.70774	−0.130884
$429$	0	0
$430$	−12.0727	−0.582198
$431$	30.3618	1.46248	0.731238	−	0.682122i	$-0.238943\pi$
$431$	30.3618	1.46248	0.731238	+	0.682122i	$0.238943\pi$
$432$	5.07271	0.244061
$433$	21.9597	1.05532	0.527658	−	0.849457i	$-0.323070\pi$
$433$	21.9597	1.05532	0.527658	+	0.849457i	$0.323070\pi$
$434$	33.5690	1.61136
$435$	1.72898	0.0828985
$436$	15.5231	0.743424
$437$	−25.6781	−1.22835
$438$	−3.71114	−0.177325
$439$	5.52827	0.263850	0.131925	−	0.991260i	$-0.457884\pi$
$439$	5.52827	0.263850	0.131925	+	0.991260i	$0.457884\pi$
$440$	4.58239	0.218457
$441$	1.04279	0.0496565
$442$	0	0
$443$	−23.9693	−1.13882	−0.569409	−	0.822054i	$-0.692828\pi$
$443$	−23.9693	−1.13882	−0.569409	+	0.822054i	$0.692828\pi$
$444$	11.4367	0.542760
$445$	11.1506	0.528587
$446$	−12.9244	−0.611990
$447$	23.9208	1.13142
$448$	−3.83691	−0.181277
$449$	−34.3810	−1.62254	−0.811270	−	0.584672i	$-0.801223\pi$
$449$	−34.3810	−1.62254	−0.811270	+	0.584672i	$0.801223\pi$
$450$	−0.135043	−0.00636599
$451$	11.7884	0.555094
$452$	11.0209	0.518380
$453$	20.5469	0.965377
$454$	−22.2184	−1.04276
$455$	0	0
$456$	−9.12118	−0.427139
$457$	−18.3436	−0.858075	−0.429038	−	0.903287i	$-0.641147\pi$
$457$	−18.3436	−0.858075	−0.429038	+	0.903287i	$0.641147\pi$
$458$	24.5775	1.14843
$459$	19.6243	0.915982
$460$	−4.98463	−0.232410
$461$	23.1178	1.07670	0.538352	−	0.842720i	$-0.319047\pi$
$461$	23.1178	1.07670	0.538352	+	0.842720i	$0.319047\pi$
$462$	−31.1312	−1.44835
$463$	−12.6863	−0.589583	−0.294791	−	0.955562i	$-0.595250\pi$
$463$	−12.6863	−0.589583	−0.294791	+	0.955562i	$0.595250\pi$
$464$	0.976494	0.0453326
$465$	15.4910	0.718376
$466$	−13.0303	−0.603617
$467$	−4.88813	−0.226196	−0.113098	−	0.993584i	$-0.536077\pi$
$467$	−4.88813	−0.226196	−0.113098	+	0.993584i	$0.536077\pi$
$468$	0	0
$469$	−17.2987	−0.798781
$470$	−5.52245	−0.254731
$471$	−3.81375	−0.175729
$472$	1.77799	0.0818386
$473$	−55.3218	−2.54370
$474$	13.7257	0.630441
$475$	−5.15145	−0.236365
$476$	−14.8435	−0.680349
$477$	−0.0322695	−0.00147752
$478$	4.81106	0.220053
$479$	−13.2019	−0.603210	−0.301605	−	0.953433i	$-0.597522\pi$
$479$	−13.2019	−0.603210	−0.301605	+	0.953433i	$0.597522\pi$
$480$	−1.77061	−0.0808167
$481$	0	0
$482$	−1.43938	−0.0655620
$483$	33.8639	1.54086
$484$	9.99829	0.454468
$485$	−7.79046	−0.353746
$486$	−1.40236	−0.0636122
$487$	−29.9840	−1.35871	−0.679353	−	0.733812i	$-0.737740\pi$
$487$	−29.9840	−1.35871	−0.679353	+	0.733812i	$0.737740\pi$
$488$	−13.0381	−0.590207
$489$	−17.3553	−0.784832
$490$	7.72188	0.348839
$491$	29.3475	1.32444	0.662218	−	0.749311i	$-0.269615\pi$
$491$	29.3475	1.32444	0.662218	+	0.749311i	$0.269615\pi$
$492$	−4.55496	−0.205353
$493$	3.77766	0.170137
$494$	0	0
$495$	−0.618819	−0.0278139
$496$	8.74896	0.392840
$497$	60.2461	2.70241
$498$	−24.3630	−1.09173
$499$	−38.5398	−1.72528	−0.862639	−	0.505819i	$-0.831190\pi$
$499$	−38.5398	−1.72528	−0.862639	+	0.505819i	$0.831190\pi$
$500$	−1.00000	−0.0447214
$501$	8.48129	0.378916
$502$	11.8658	0.529594
$503$	−2.39660	−0.106859	−0.0534296	−	0.998572i	$-0.517015\pi$
$503$	−2.39660	−0.106859	−0.0534296	+	0.998572i	$0.517015\pi$
$504$	0.518148	0.0230801
$505$	−7.52902	−0.335037
$506$	−22.8415	−1.01543
$507$	0	0
$508$	−20.2183	−0.897044
$509$	−0.818442	−0.0362768	−0.0181384	−	0.999835i	$-0.505774\pi$
$509$	−0.818442	−0.0362768	−0.0181384	+	0.999835i	$0.505774\pi$
$510$	−6.84976	−0.303312
$511$	8.04206	0.355760
$512$	−1.00000	−0.0441942
$513$	−26.1318	−1.15375
$514$	11.1225	0.490594
$515$	10.6767	0.470473
$516$	21.3760	0.941026
$517$	−25.3060	−1.11296
$518$	−24.7833	−1.08892
$519$	16.3661	0.718393
$520$	0	0
$521$	−3.03837	−0.133113	−0.0665566	−	0.997783i	$-0.521201\pi$
$521$	−3.03837	−0.133113	−0.0665566	+	0.997783i	$0.521201\pi$
$522$	−0.131869	−0.00577173
$523$	−42.8553	−1.87393	−0.936966	−	0.349419i	$-0.886379\pi$
$523$	−42.8553	−1.87393	−0.936966	+	0.349419i	$0.886379\pi$
$524$	−8.24571	−0.360216
$525$	6.79365	0.296499
$526$	−1.28305	−0.0559437
$527$	33.8462	1.47436
$528$	−8.11360	−0.353099
$529$	1.84656	0.0802851
$530$	−0.238957	−0.0103796
$531$	−0.240105	−0.0104197
$532$	19.7657	0.856950
$533$	0	0
$534$	−19.7432	−0.854373
$535$	2.70774	0.117066
$536$	−4.50850	−0.194738
$537$	31.7782	1.37133
$538$	17.9493	0.773850
$539$	35.3847	1.52413
$540$	−5.07271	−0.218295
$541$	−10.8975	−0.468522	−0.234261	−	0.972174i	$-0.575267\pi$
$541$	−10.8975	−0.468522	−0.234261	+	0.972174i	$0.575267\pi$
$542$	−25.3959	−1.09085
$543$	24.1168	1.03495
$544$	−3.86860	−0.165865
$545$	−15.5231	−0.664938
$546$	0	0
$547$	−9.47621	−0.405173	−0.202587	−	0.979264i	$-0.564935\pi$
$547$	−9.47621	−0.405173	−0.202587	+	0.979264i	$0.564935\pi$
$548$	−19.1981	−0.820102
$549$	1.76070	0.0751449
$550$	−4.58239	−0.195394
$551$	−5.03036	−0.214300
$552$	8.82582	0.375652
$553$	−29.7436	−1.26483
$554$	−5.42610	−0.230533
$555$	−11.4367	−0.485459
$556$	−3.23295	−0.137108
$557$	−9.21099	−0.390282	−0.195141	−	0.980775i	$-0.562516\pi$
$557$	−9.21099	−0.390282	−0.195141	+	0.980775i	$0.562516\pi$
$558$	−1.18149	−0.0500163
$559$	0	0
$560$	3.83691	0.162139
$561$	−31.3883	−1.32521
$562$	2.71086	0.114351
$563$	−5.79613	−0.244278	−0.122139	−	0.992513i	$-0.538975\pi$
$563$	−5.79613	−0.244278	−0.122139	+	0.992513i	$0.538975\pi$
$564$	9.77807	0.411731
$565$	−11.0209	−0.463653
$566$	13.4333	0.564643
$567$	36.0167	1.51256
$568$	15.7017	0.658829
$569$	−42.4763	−1.78070	−0.890349	−	0.455279i	$-0.849540\pi$
$569$	−42.4763	−1.78070	−0.890349	+	0.455279i	$0.849540\pi$
$570$	9.12118	0.382044
$571$	−26.1436	−1.09408	−0.547038	−	0.837107i	$-0.684245\pi$
$571$	−26.1436	−1.09408	−0.547038	+	0.837107i	$0.684245\pi$
$572$	0	0
$573$	−3.88666	−0.162367
$574$	9.87062	0.411992
$575$	4.98463	0.207874
$576$	0.135043	0.00562679
$577$	36.3946	1.51513	0.757564	−	0.652761i	$-0.226389\pi$
$577$	36.3946	1.51513	0.757564	+	0.652761i	$0.226389\pi$
$578$	2.03395	0.0846013
$579$	2.16423	0.0899423
$580$	−0.976494	−0.0405467
$581$	52.7948	2.19030
$582$	13.7938	0.571772
$583$	−1.09499	−0.0453500
$584$	2.09597	0.0867319
$585$	0	0
$586$	−14.0044	−0.578518
$587$	−28.1702	−1.16271	−0.581355	−	0.813650i	$-0.697477\pi$
$587$	−28.1702	−1.16271	−0.581355	+	0.813650i	$0.697477\pi$
$588$	−13.6724	−0.563841
$589$	−45.0698	−1.85707
$590$	−1.77799	−0.0731987
$591$	−27.2408	−1.12054
$592$	−6.45918	−0.265471
$593$	−21.0238	−0.863342	−0.431671	−	0.902031i	$-0.642076\pi$
$593$	−21.0238	−0.863342	−0.431671	+	0.902031i	$0.642076\pi$
$594$	−23.2451	−0.953759
$595$	14.8435	0.608523
$596$	−13.5100	−0.553390
$597$	19.6156	0.802813
$598$	0	0
$599$	21.1905	0.865822	0.432911	−	0.901437i	$-0.357487\pi$
$599$	21.1905	0.865822	0.432911	+	0.901437i	$0.357487\pi$
$600$	1.77061	0.0722847
$601$	13.5018	0.550750	0.275375	−	0.961337i	$-0.411198\pi$
$601$	13.5018	0.550750	0.275375	+	0.961337i	$0.411198\pi$
$602$	−46.3219	−1.88794
$603$	0.608842	0.0247939
$604$	−11.6044	−0.472178
$605$	−9.99829	−0.406488
$606$	13.3309	0.541532
$607$	9.26458	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$607$	9.26458	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$608$	5.15145	0.208919
$609$	6.63396	0.268822
$610$	13.0381	0.527897
$611$	0	0
$612$	0.522427	0.0211179
$613$	7.16088	0.289225	0.144613	−	0.989488i	$-0.453806\pi$
$613$	7.16088	0.289225	0.144613	+	0.989488i	$0.453806\pi$
$614$	−4.42199	−0.178457
$615$	4.55496	0.183674
$616$	17.5822	0.708408
$617$	−14.5619	−0.586239	−0.293120	−	0.956076i	$-0.594693\pi$
$617$	−14.5619	−0.586239	−0.293120	+	0.956076i	$0.594693\pi$
$618$	−18.9043	−0.760442
$619$	−24.5298	−0.985937	−0.492969	−	0.870047i	$-0.664088\pi$
$619$	−24.5298	−0.985937	−0.492969	+	0.870047i	$0.664088\pi$
$620$	−8.74896	−0.351367
$621$	25.2856	1.01468
$622$	−15.0706	−0.604276
$623$	42.7837	1.71409
$624$	0	0
$625$	1.00000	0.0400000
$626$	−27.8961	−1.11495
$627$	41.7968	1.66920
$628$	2.15393	0.0859510
$629$	−24.9880	−0.996336
$630$	−0.518148	−0.0206435
$631$	19.7298	0.785430	0.392715	−	0.919660i	$-0.371536\pi$
$631$	19.7298	0.785430	0.392715	+	0.919660i	$0.371536\pi$
$632$	−7.75196	−0.308357
$633$	34.9686	1.38988
$634$	−21.1887	−0.841509
$635$	20.2183	0.802341
$636$	0.423098	0.0167769
$637$	0	0
$638$	−4.47467	−0.177154
$639$	−2.12041	−0.0838819
$640$	1.00000	0.0395285
$641$	5.92290	0.233940	0.116970	−	0.993135i	$-0.462682\pi$
$641$	5.92290	0.233940	0.116970	+	0.993135i	$0.462682\pi$
$642$	−4.79435	−0.189218
$643$	26.5771	1.04810	0.524049	−	0.851688i	$-0.324421\pi$
$643$	26.5771	1.04810	0.524049	+	0.851688i	$0.324421\pi$
$644$	−19.1256	−0.753654
$645$	−21.3760	−0.841679
$646$	19.9289	0.784091
$647$	−38.5174	−1.51427	−0.757137	−	0.653256i	$-0.773403\pi$
$647$	−38.5174	−1.51427	−0.757137	+	0.653256i	$0.773403\pi$
$648$	9.38689	0.368752
$649$	−8.14744	−0.319815
$650$	0	0
$651$	59.4374	2.32954
$652$	9.80188	0.383871
$653$	−27.8131	−1.08841	−0.544205	−	0.838953i	$-0.683168\pi$
$653$	−27.8131	−1.08841	−0.544205	+	0.838953i	$0.683168\pi$
$654$	27.4854	1.07476
$655$	8.24571	0.322187
$656$	2.57254	0.100441
$657$	−0.283046	−0.0110427
$658$	−21.1891	−0.826039
$659$	20.0095	0.779461	0.389730	−	0.920929i	$-0.372568\pi$
$659$	20.0095	0.779461	0.389730	+	0.920929i	$0.372568\pi$
$660$	8.11360	0.315822
$661$	−2.76586	−0.107579	−0.0537897	−	0.998552i	$-0.517130\pi$
$661$	−2.76586	−0.107579	−0.0537897	+	0.998552i	$0.517130\pi$
$662$	24.7539	0.962087
$663$	0	0
$664$	13.7597	0.533980
$665$	−19.7657	−0.766479
$666$	0.872267	0.0337997
$667$	4.86746	0.188469
$668$	−4.79005	−0.185333
$669$	−22.8841	−0.884750
$670$	4.50850	0.174179
$671$	59.7456	2.30645
$672$	−6.79365	−0.262071
$673$	7.95736	0.306734	0.153367	−	0.988169i	$-0.450988\pi$
$673$	7.95736	0.306734	0.153367	+	0.988169i	$0.450988\pi$
$674$	−21.2095	−0.816958
$675$	5.07271	0.195249
$676$	0	0
$677$	21.4308	0.823653	0.411827	−	0.911262i	$-0.364891\pi$
$677$	21.4308	0.823653	0.411827	+	0.911262i	$0.364891\pi$
$678$	19.5137	0.749418
$679$	−29.8913	−1.14712
$680$	3.86860	0.148354
$681$	−39.3399	−1.50751
$682$	−40.0911	−1.53517
$683$	4.55605	0.174333	0.0871663	−	0.996194i	$-0.472219\pi$
$683$	4.55605	0.174333	0.0871663	+	0.996194i	$0.472219\pi$
$684$	−0.695667	−0.0265995
$685$	19.1981	0.733521
$686$	2.76981	0.105752
$687$	43.5171	1.66028
$688$	−12.0727	−0.460268
$689$	0	0
$690$	−8.82582	−0.335993
$691$	−0.851349	−0.0323868	−0.0161934	−	0.999869i	$-0.505155\pi$
$691$	−0.851349	−0.0323868	−0.0161934	+	0.999869i	$0.505155\pi$
$692$	−9.24324	−0.351375
$693$	−2.37436	−0.0901943
$694$	16.8348	0.639040
$695$	3.23295	0.122633
$696$	1.72898	0.0655370
$697$	9.95213	0.376964
$698$	−12.0036	−0.454345
$699$	−23.0715	−0.872645
$700$	−3.83691	−0.145022
$701$	−12.8804	−0.486486	−0.243243	−	0.969965i	$-0.578211\pi$
$701$	−12.8804	−0.486486	−0.243243	+	0.969965i	$0.578211\pi$
$702$	0	0
$703$	33.2742	1.25496
$704$	4.58239	0.172705
$705$	−9.77807	−0.368264
$706$	19.2651	0.725050
$707$	−28.8882	−1.08645
$708$	3.14812	0.118313
$709$	−6.62860	−0.248942	−0.124471	−	0.992223i	$-0.539723\pi$
$709$	−6.62860	−0.248942	−0.124471	+	0.992223i	$0.539723\pi$
$710$	−15.7017	−0.589275
$711$	1.04685	0.0392599
$712$	11.1506	0.417885
$713$	43.6104	1.63322
$714$	−26.2819	−0.983576
$715$	0	0
$716$	−17.9476	−0.670734
$717$	8.51848	0.318129
$718$	−25.7690	−0.961690
$719$	−15.2830	−0.569958	−0.284979	−	0.958534i	$-0.591987\pi$
$719$	−15.2830	−0.569958	−0.284979	+	0.958534i	$0.591987\pi$
$720$	−0.135043	−0.00503275
$721$	40.9657	1.52564
$722$	−7.53743	−0.280514
$723$	−2.54858	−0.0947825
$724$	−13.6207	−0.506208
$725$	0.976494	0.0362661
$726$	17.7030	0.657021
$727$	−11.1149	−0.412230	−0.206115	−	0.978528i	$-0.566082\pi$
$727$	−11.1149	−0.412230	−0.206115	+	0.978528i	$0.566082\pi$
$728$	0	0
$729$	25.6777	0.951024
$730$	−2.09597	−0.0775754
$731$	−46.7044	−1.72743
$732$	−23.0853	−0.853258
$733$	−12.2571	−0.452727	−0.226363	−	0.974043i	$-0.572684\pi$
$733$	−12.2571	−0.452727	−0.226363	+	0.974043i	$0.572684\pi$
$734$	8.62414	0.318323
$735$	13.6724	0.504314
$736$	−4.98463	−0.183736
$737$	20.6597	0.761010
$738$	−0.347404	−0.0127881
$739$	12.6213	0.464283	0.232142	−	0.972682i	$-0.425427\pi$
$739$	12.6213	0.464283	0.232142	+	0.972682i	$0.425427\pi$
$740$	6.45918	0.237444
$741$	0	0
$742$	−0.916857	−0.0336589
$743$	−19.1676	−0.703192	−0.351596	−	0.936152i	$-0.614361\pi$
$743$	−19.1676	−0.703192	−0.351596	+	0.936152i	$0.614361\pi$
$744$	15.4910	0.567926
$745$	13.5100	0.494967
$746$	13.7724	0.504242
$747$	−1.85815	−0.0679862
$748$	17.7274	0.648179
$749$	10.3894	0.379619
$750$	−1.77061	−0.0646534
$751$	−12.2558	−0.447221	−0.223611	−	0.974679i	$-0.571784\pi$
$751$	−12.2558	−0.447221	−0.223611	+	0.974679i	$0.571784\pi$
$752$	−5.52245	−0.201383
$753$	21.0096	0.765631
$754$	0	0
$755$	11.6044	0.422329
$756$	−19.4635	−0.707882
$757$	−22.6792	−0.824291	−0.412145	−	0.911118i	$-0.635220\pi$
$757$	−22.6792	−0.824291	−0.412145	+	0.911118i	$0.635220\pi$
$758$	17.4269	0.632974
$759$	−40.4433	−1.46800
$760$	−5.15145	−0.186863
$761$	8.77264	0.318008	0.159004	−	0.987278i	$-0.449172\pi$
$761$	8.77264	0.318008	0.159004	+	0.987278i	$0.449172\pi$
$762$	−35.7987	−1.29685
$763$	−59.5609	−2.15625
$764$	2.19510	0.0794160
$765$	−0.522427	−0.0188884
$766$	−28.9137	−1.04469
$767$	0	0
$768$	−1.77061	−0.0638912
$769$	−0.193591	−0.00698109	−0.00349054	−	0.999994i	$-0.501111\pi$
$769$	−0.193591	−0.00698109	−0.00349054	+	0.999994i	$0.501111\pi$
$770$	−17.5822	−0.633619
$771$	19.6936	0.709248
$772$	−1.22231	−0.0439919
$773$	−6.26878	−0.225472	−0.112736	−	0.993625i	$-0.535961\pi$
$773$	−6.26878	−0.225472	−0.112736	+	0.993625i	$0.535961\pi$
$774$	1.63033	0.0586012
$775$	8.74896	0.314272
$776$	−7.79046	−0.279661
$777$	−43.8815	−1.57424
$778$	21.0537	0.754813
$779$	−13.2523	−0.474814
$780$	0	0
$781$	−71.9513	−2.57462
$782$	−19.2835	−0.689578
$783$	4.95347	0.177022
$784$	7.72188	0.275782
$785$	−2.15393	−0.0768770
$786$	−14.5999	−0.520761
$787$	−35.8546	−1.27808	−0.639040	−	0.769174i	$-0.720668\pi$
$787$	−35.8546	−1.27808	−0.639040	+	0.769174i	$0.720668\pi$
$788$	15.3850	0.548068
$789$	−2.27178	−0.0808774
$790$	7.75196	0.275802
$791$	−42.2862	−1.50353
$792$	−0.618819	−0.0219888
$793$	0	0
$794$	−29.8791	−1.06037
$795$	−0.423098	−0.0150058
$796$	−11.0785	−0.392666
$797$	−12.1067	−0.428843	−0.214421	−	0.976741i	$-0.568787\pi$
$797$	−12.1067	−0.428843	−0.214421	+	0.976741i	$0.568787\pi$
$798$	34.9972	1.23889
$799$	−21.3641	−0.755808
$800$	−1.00000	−0.0353553
$801$	−1.50580	−0.0532049
$802$	22.5805	0.797345
$803$	−9.60456	−0.338938
$804$	−7.98278	−0.281531
$805$	19.1256	0.674088
$806$	0	0
$807$	31.7812	1.11875
$808$	−7.52902	−0.264870
$809$	−34.1027	−1.19899	−0.599494	−	0.800379i	$-0.704631\pi$
$809$	−34.1027	−1.19899	−0.599494	+	0.800379i	$0.704631\pi$
$810$	−9.38689	−0.329822
$811$	31.4988	1.10607	0.553036	−	0.833157i	$-0.313469\pi$
$811$	31.4988	1.10607	0.553036	+	0.833157i	$0.313469\pi$
$812$	−3.74672	−0.131484
$813$	−44.9662	−1.57703
$814$	29.5985	1.03743
$815$	−9.80188	−0.343345
$816$	−6.84976	−0.239790
$817$	62.1919	2.17582
$818$	−27.3927	−0.957764
$819$	0	0
$820$	−2.57254	−0.0898371
$821$	−37.3400	−1.30318	−0.651588	−	0.758573i	$-0.725897\pi$
$821$	−37.3400	−1.30318	−0.651588	+	0.758573i	$0.725897\pi$
$822$	−33.9922	−1.18562
$823$	−4.90354	−0.170927	−0.0854634	−	0.996341i	$-0.527237\pi$
$823$	−4.90354	−0.170927	−0.0854634	+	0.996341i	$0.527237\pi$
$824$	10.6767	0.371942
$825$	−8.11360	−0.282479
$826$	−6.82199	−0.237367
$827$	−4.16585	−0.144861	−0.0724303	−	0.997373i	$-0.523075\pi$
$827$	−4.16585	−0.144861	−0.0724303	+	0.997373i	$0.523075\pi$
$828$	0.673140	0.0233932
$829$	15.3834	0.534288	0.267144	−	0.963657i	$-0.413920\pi$
$829$	15.3834	0.534288	0.267144	+	0.963657i	$0.413920\pi$
$830$	−13.7597	−0.477606
$831$	−9.60748	−0.333280
$832$	0	0
$833$	29.8729	1.03503
$834$	−5.72427	−0.198215
$835$	4.79005	0.165767
$836$	−23.6059	−0.816429
$837$	44.3809	1.53403
$838$	6.37956	0.220378
$839$	−29.4604	−1.01709	−0.508543	−	0.861037i	$-0.669816\pi$
$839$	−29.4604	−1.01709	−0.508543	+	0.861037i	$0.669816\pi$
$840$	6.79365	0.234403
$841$	−28.0465	−0.967119
$842$	13.6636	0.470878
$843$	4.79986	0.165316
$844$	−19.7495	−0.679807
$845$	0	0
$846$	0.745768	0.0256400
$847$	−38.3626	−1.31815
$848$	−0.238957	−0.00820582
$849$	23.7850	0.816300
$850$	−3.86860	−0.132692
$851$	−32.1967	−1.10369
$852$	27.8015	0.952465
$853$	9.55849	0.327276	0.163638	−	0.986520i	$-0.447677\pi$
$853$	9.55849	0.327276	0.163638	+	0.986520i	$0.447677\pi$
$854$	50.0260	1.71185
$855$	0.695667	0.0237913
$856$	2.70774	0.0925488
$857$	−26.9134	−0.919345	−0.459672	−	0.888089i	$-0.652033\pi$
$857$	−26.9134	−0.919345	−0.459672	+	0.888089i	$0.652033\pi$
$858$	0	0
$859$	46.5363	1.58780	0.793899	−	0.608049i	$-0.208048\pi$
$859$	46.5363	1.58780	0.793899	+	0.608049i	$0.208048\pi$
$860$	12.0727	0.411676
$861$	17.4770	0.595613
$862$	−30.3618	−1.03413
$863$	32.5981	1.10965	0.554826	−	0.831966i	$-0.312785\pi$
$863$	32.5981	1.10965	0.554826	+	0.831966i	$0.312785\pi$
$864$	−5.07271	−0.172577
$865$	9.24324	0.314279
$866$	−21.9597	−0.746221
$867$	3.60133	0.122307
$868$	−33.5690	−1.13941
$869$	35.5225	1.20502
$870$	−1.72898	−0.0586181
$871$	0	0
$872$	−15.5231	−0.525680
$873$	1.05205	0.0356064
$874$	25.6781	0.868574
$875$	3.83691	0.129711
$876$	3.71114	0.125388
$877$	−3.03252	−0.102401	−0.0512004	−	0.998688i	$-0.516305\pi$
$877$	−3.03252	−0.102401	−0.0512004	+	0.998688i	$0.516305\pi$
$878$	−5.52827	−0.186570
$879$	−24.7963	−0.836359
$880$	−4.58239	−0.154472
$881$	−43.1633	−1.45421	−0.727104	−	0.686527i	$-0.759134\pi$
$881$	−43.1633	−1.45421	−0.727104	+	0.686527i	$0.759134\pi$
$882$	−1.04279	−0.0351124
$883$	7.62736	0.256681	0.128341	−	0.991730i	$-0.459035\pi$
$883$	7.62736	0.256681	0.128341	+	0.991730i	$0.459035\pi$
$884$	0	0
$885$	−3.14812	−0.105823
$886$	23.9693	0.805266
$887$	37.1547	1.24753	0.623767	−	0.781610i	$-0.285601\pi$
$887$	37.1547	1.24753	0.623767	+	0.781610i	$0.285601\pi$
$888$	−11.4367	−0.383789
$889$	77.5760	2.60182
$890$	−11.1506	−0.373767
$891$	−43.0144	−1.44104
$892$	12.9244	0.432742
$893$	28.4486	0.951996
$894$	−23.9208	−0.800033
$895$	17.9476	0.599923
$896$	3.83691	0.128182
$897$	0	0
$898$	34.3810	1.14731
$899$	8.54330	0.284935
$900$	0.135043	0.00450143
$901$	−0.924428	−0.0307972
$902$	−11.7884	−0.392511
$903$	−82.0178	−2.72938
$904$	−11.0209	−0.366550
$905$	13.6207	0.452767
$906$	−20.5469	−0.682625
$907$	−2.37206	−0.0787629	−0.0393814	−	0.999224i	$-0.512539\pi$
$907$	−2.37206	−0.0787629	−0.0393814	+	0.999224i	$0.512539\pi$
$908$	22.2184	0.737342
$909$	1.01674	0.0337232
$910$	0	0
$911$	−12.3071	−0.407751	−0.203875	−	0.978997i	$-0.565354\pi$
$911$	−12.3071	−0.407751	−0.203875	+	0.978997i	$0.565354\pi$
$912$	9.12118	0.302033
$913$	−63.0523	−2.08673
$914$	18.3436	0.606751
$915$	23.0853	0.763177
$916$	−24.5775	−0.812064
$917$	31.6381	1.04478
$918$	−19.6243	−0.647697
$919$	13.2770	0.437967	0.218983	−	0.975729i	$-0.429726\pi$
$919$	13.2770	0.437967	0.218983	+	0.975729i	$0.429726\pi$
$920$	4.98463	0.164338
$921$	−7.82960	−0.257994
$922$	−23.1178	−0.761345
$923$	0	0
$924$	31.1312	1.02414
$925$	−6.45918	−0.212377
$926$	12.6863	0.416898
$927$	−1.44182	−0.0473555
$928$	−0.976494	−0.0320550
$929$	18.5536	0.608724	0.304362	−	0.952556i	$-0.401557\pi$
$929$	18.5536	0.608724	0.304362	+	0.952556i	$0.401557\pi$
$930$	−15.4910	−0.507968
$931$	−39.7789	−1.30370
$932$	13.0303	0.426822
$933$	−26.6841	−0.873597
$934$	4.88813	0.159944
$935$	−17.7274	−0.579749
$936$	0	0
$937$	−12.3292	−0.402778	−0.201389	−	0.979511i	$-0.564545\pi$
$937$	−12.3292	−0.402778	−0.201389	+	0.979511i	$0.564545\pi$
$938$	17.2987	0.564823
$939$	−49.3931	−1.61188
$940$	5.52245	0.180122
$941$	−14.2254	−0.463736	−0.231868	−	0.972747i	$-0.574484\pi$
$941$	−14.2254	−0.463736	−0.231868	+	0.972747i	$0.574484\pi$
$942$	3.81375	0.124259
$943$	12.8232	0.417580
$944$	−1.77799	−0.0578686
$945$	19.4635	0.633149
$946$	55.3218	1.79867
$947$	−55.5452	−1.80498	−0.902488	−	0.430715i	$-0.858261\pi$
$947$	−55.5452	−1.80498	−0.902488	+	0.430715i	$0.858261\pi$
$948$	−13.7257	−0.445789
$949$	0	0
$950$	5.15145	0.167135
$951$	−37.5167	−1.21656
$952$	14.8435	0.481079
$953$	41.9688	1.35950	0.679751	−	0.733443i	$-0.262088\pi$
$953$	41.9688	1.35950	0.679751	+	0.733443i	$0.262088\pi$
$954$	0.0322695	0.00104476
$955$	−2.19510	−0.0710318
$956$	−4.81106	−0.155601
$957$	−7.92288	−0.256110
$958$	13.2019	0.426534
$959$	73.6613	2.37865
$960$	1.77061	0.0571460
$961$	45.5443	1.46917
$962$	0	0
$963$	−0.365662	−0.0117833
$964$	1.43938	0.0463593
$965$	1.22231	0.0393476
$966$	−33.8639	−1.08955
$967$	6.51932	0.209647	0.104824	−	0.994491i	$-0.466572\pi$
$967$	6.51932	0.209647	0.104824	+	0.994491i	$0.466572\pi$
$968$	−9.99829	−0.321357
$969$	35.2862	1.13356
$970$	7.79046	0.250136
$971$	42.3972	1.36059	0.680295	−	0.732938i	$-0.261851\pi$
$971$	42.3972	1.36059	0.680295	+	0.732938i	$0.261851\pi$
$972$	1.40236	0.0449806
$973$	12.4045	0.397671
$974$	29.9840	0.960750
$975$	0	0
$976$	13.0381	0.417339
$977$	−5.09733	−0.163078	−0.0815390	−	0.996670i	$-0.525984\pi$
$977$	−5.09733	−0.163078	−0.0815390	+	0.996670i	$0.525984\pi$
$978$	17.3553	0.554960
$979$	−51.0962	−1.63304
$980$	−7.72188	−0.246667
$981$	2.09629	0.0669294
$982$	−29.3475	−0.936518
$983$	26.2289	0.836572	0.418286	−	0.908315i	$-0.362631\pi$
$983$	26.2289	0.836572	0.418286	+	0.908315i	$0.362631\pi$
$984$	4.55496	0.145207
$985$	−15.3850	−0.490207
$986$	−3.77766	−0.120305
$987$	−37.5176	−1.19420
$988$	0	0
$989$	−60.1780	−1.91355
$990$	0.618819	0.0196674
$991$	0.539832	0.0171483	0.00857416	−	0.999963i	$-0.497271\pi$
$991$	0.539832	0.0171483	0.00857416	+	0.999963i	$0.497271\pi$
$992$	−8.74896	−0.277780
$993$	43.8294	1.39088
$994$	−60.2461	−1.91089
$995$	11.0785	0.351211
$996$	24.3630	0.771971
$997$	−11.7059	−0.370728	−0.185364	−	0.982670i	$-0.559346\pi$
$997$	−11.7059	−0.370728	−0.185364	+	0.982670i	$0.559346\pi$
$998$	38.5398	1.21996
$999$	−32.7655	−1.03666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.v.1.3	✓	6
5.4	even	2		8450.2.a.cq.1.4		6
13.2	odd	12		1690.2.l.n.1161.4		24
13.3	even	3		1690.2.e.v.191.4		12
13.4	even	6		1690.2.e.u.991.4		12
13.5	odd	4		1690.2.d.l.1351.9		12
13.6	odd	12		1690.2.l.n.361.12		24
13.7	odd	12		1690.2.l.n.361.4		24
13.8	odd	4		1690.2.d.l.1351.3		12
13.9	even	3		1690.2.e.v.991.4		12
13.10	even	6		1690.2.e.u.191.4		12
13.11	odd	12		1690.2.l.n.1161.12		24
13.12	even	2		1690.2.a.w.1.3	yes	6
65.64	even	2		8450.2.a.cp.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.v.1.3	✓	6	1.1	even	1	trivial
1690.2.a.w.1.3	yes	6	13.12	even	2
1690.2.d.l.1351.3		12	13.8	odd	4
1690.2.d.l.1351.9		12	13.5	odd	4
1690.2.e.u.191.4		12	13.10	even	6
1690.2.e.u.991.4		12	13.4	even	6
1690.2.e.v.191.4		12	13.3	even	3
1690.2.e.v.991.4		12	13.9	even	3
1690.2.l.n.361.4		24	13.7	odd	12
1690.2.l.n.361.12		24	13.6	odd	12
1690.2.l.n.1161.4		24	13.2	odd	12
1690.2.l.n.1161.12		24	13.11	odd	12
8450.2.a.cp.1.4		6	65.64	even	2
8450.2.a.cq.1.4		6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.77061 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.77061 q^{6} -3.83691 q^{7} -1.00000 q^{8} +0.135043 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.77061 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.77061 q^{6} -3.83691 q^{7} -1.00000 q^{8} +0.135043 q^{9} +1.00000 q^{10} +4.58239 q^{11} -1.77061 q^{12} +3.83691 q^{14} +1.77061 q^{15} +1.00000 q^{16} +3.86860 q^{17} -0.135043 q^{18} -5.15145 q^{19} -1.00000 q^{20} +6.79365 q^{21} -4.58239 q^{22} +4.98463 q^{23} +1.77061 q^{24} +1.00000 q^{25} +5.07271 q^{27} -3.83691 q^{28} +0.976494 q^{29} -1.77061 q^{30} +8.74896 q^{31} -1.00000 q^{32} -8.11360 q^{33} -3.86860 q^{34} +3.83691 q^{35} +0.135043 q^{36} -6.45918 q^{37} +5.15145 q^{38} +1.00000 q^{40} +2.57254 q^{41} -6.79365 q^{42} -12.0727 q^{43} +4.58239 q^{44} -0.135043 q^{45} -4.98463 q^{46} -5.52245 q^{47} -1.77061 q^{48} +7.72188 q^{49} -1.00000 q^{50} -6.84976 q^{51} -0.238957 q^{53} -5.07271 q^{54} -4.58239 q^{55} +3.83691 q^{56} +9.12118 q^{57} -0.976494 q^{58} -1.77799 q^{59} +1.77061 q^{60} +13.0381 q^{61} -8.74896 q^{62} -0.518148 q^{63} +1.00000 q^{64} +8.11360 q^{66} +4.50850 q^{67} +3.86860 q^{68} -8.82582 q^{69} -3.83691 q^{70} -15.7017 q^{71} -0.135043 q^{72} -2.09597 q^{73} +6.45918 q^{74} -1.77061 q^{75} -5.15145 q^{76} -17.5822 q^{77} +7.75196 q^{79} -1.00000 q^{80} -9.38689 q^{81} -2.57254 q^{82} -13.7597 q^{83} +6.79365 q^{84} -3.86860 q^{85} +12.0727 q^{86} -1.72898 q^{87} -4.58239 q^{88} -11.1506 q^{89} +0.135043 q^{90} +4.98463 q^{92} -15.4910 q^{93} +5.52245 q^{94} +5.15145 q^{95} +1.77061 q^{96} +7.79046 q^{97} -7.72188 q^{98} +0.618819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * q^99

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