LMFDB - Embedded newform 169.10.a.h.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,10,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [27,65] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$87.0410563117$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.82532 q^{2} -93.8913 q^{3} -488.716 q^{4} +1259.17 q^{5} +453.056 q^{6} -5921.56 q^{7} +4828.78 q^{8} -10867.4 q^{9} -6075.91 q^{10} +20085.5 q^{11} +45886.2 q^{12} +28573.5 q^{14} -118225. q^{15} +226922. q^{16} -89577.2 q^{17} +52438.9 q^{18} -817804. q^{19} -615378. q^{20} +555983. q^{21} -96919.0 q^{22} -1.17564e6 q^{23} -453380. q^{24} -367612. q^{25} +2.86842e6 q^{27} +2.89396e6 q^{28} +4.02987e6 q^{29} +570475. q^{30} -6.85443e6 q^{31} -3.56731e6 q^{32} -1.88585e6 q^{33} +432239. q^{34} -7.45626e6 q^{35} +5.31109e6 q^{36} -1.56577e7 q^{37} +3.94617e6 q^{38} +6.08026e6 q^{40} +6.67447e6 q^{41} -2.68280e6 q^{42} -2.83685e6 q^{43} -9.81611e6 q^{44} -1.36840e7 q^{45} +5.67284e6 q^{46} -4.23495e7 q^{47} -2.13060e7 q^{48} -5.28871e6 q^{49} +1.77385e6 q^{50} +8.41052e6 q^{51} -9.54736e7 q^{53} -1.38411e7 q^{54} +2.52911e7 q^{55} -2.85939e7 q^{56} +7.67846e7 q^{57} -1.94454e7 q^{58} +9.03136e7 q^{59} +5.77786e7 q^{60} -7.48746e7 q^{61} +3.30748e7 q^{62} +6.43521e7 q^{63} -9.89708e7 q^{64} +9.09985e6 q^{66} -6.65968e6 q^{67} +4.37778e7 q^{68} +1.10382e8 q^{69} +3.59789e7 q^{70} +2.26902e8 q^{71} -5.24764e7 q^{72} -1.95234e8 q^{73} +7.55534e7 q^{74} +3.45156e7 q^{75} +3.99674e8 q^{76} -1.18937e8 q^{77} +5.42837e8 q^{79} +2.85734e8 q^{80} -5.54160e7 q^{81} -3.22065e7 q^{82} +4.31319e8 q^{83} -2.71718e8 q^{84} -1.12793e8 q^{85} +1.36887e7 q^{86} -3.78370e8 q^{87} +9.69884e7 q^{88} -2.11702e8 q^{89} +6.60295e7 q^{90} +5.74554e8 q^{92} +6.43571e8 q^{93} +2.04350e8 q^{94} -1.02975e9 q^{95} +3.34939e8 q^{96} +7.77181e8 q^{97} +2.55197e7 q^{98} -2.18278e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 65 q^{2} + q^{3} + 7169 q^{4} + 3238 q^{5} + 8490 q^{6} + 17378 q^{7} + 54204 q^{8} + 191118 q^{9} + 11697 q^{10} + 164171 q^{11} - 181941 q^{12} - 77651 q^{14} + 614110 q^{15} + 3012565 q^{16}+ \cdots + 5866875443 q^{99}+O(q^{100}) $ 27 * q + 65 * q^2 + q^3 + 7169 * q^4 + 3238 * q^5 + 8490 * q^6 + 17378 * q^7 + 54204 * q^8 + 191118 * q^9 + 11697 * q^10 + 164171 * q^11 - 181941 * q^12 - 77651 * q^14 + 614110 * q^15 + 3012565 * q^16 + 105667 * q^17 + 584849 * q^18 + 2140031 * q^19 + 3722382 * q^20 + 2953252 * q^21 + 3789362 * q^22 + 3586420 * q^23 + 6334338 * q^24 + 10278737 * q^25 - 4332980 * q^27 + 5293616 * q^28 - 4203616 * q^29 - 34788079 * q^30 + 18634294 * q^31 + 38778128 * q^32 + 15170356 * q^33 + 10546914 * q^34 + 16260796 * q^35 + 112071987 * q^36 + 7214770 * q^37 - 60128169 * q^38 - 46270051 * q^40 + 67544533 * q^41 - 31249012 * q^42 - 55585357 * q^43 + 246583161 * q^44 + 172236942 * q^45 + 168715272 * q^46 + 175569972 * q^47 - 48573948 * q^48 + 144680057 * q^49 + 3455750 * q^50 + 122191382 * q^51 + 105504478 * q^53 - 359394409 * q^54 - 2471426 * q^55 + 260765426 * q^56 - 307372146 * q^57 - 777873375 * q^58 + 267562255 * q^59 - 745460915 * q^60 - 371334848 * q^61 + 52175323 * q^62 + 431551952 * q^63 + 642079732 * q^64 + 973549503 * q^66 + 213848603 * q^67 - 278629116 * q^68 - 201174310 * q^69 - 122562066 * q^70 + 970709202 * q^71 + 773619646 * q^72 + 271733895 * q^73 - 1068711482 * q^74 + 1236853937 * q^75 + 2585634844 * q^76 - 244285066 * q^77 + 1052001020 * q^79 + 3293699123 * q^80 - 456846001 * q^81 - 3832425 * q^82 + 1491498691 * q^83 + 5153327787 * q^84 + 2713007958 * q^85 + 3421126005 * q^86 - 1622618562 * q^87 + 6574070508 * q^88 + 4616501291 * q^89 - 2822436128 * q^90 + 3684890562 * q^92 + 3324407374 * q^93 - 5582614303 * q^94 - 2892286998 * q^95 + 12902380548 * q^96 + 3944038157 * q^97 + 7822213576 * q^98 + 5866875443 * q^99

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 $

$2$	−4.82532	−0.213251	−0.106626	−	0.994299i	$-0.534005\pi$
$2$	−4.82532	−0.213251	−0.106626	+	0.994299i	$0.534005\pi$
$3$	−93.8913	−0.669237	−0.334618	−	0.942354i	$-0.608607\pi$
$3$	−93.8913	−0.669237	−0.334618	+	0.942354i	$0.608607\pi$
$4$	−488.716	−0.954524
$5$	1259.17	0.900990	0.450495	−	0.892779i	$-0.351248\pi$
$5$	1259.17	0.900990	0.450495	+	0.892779i	$0.351248\pi$
$6$	453.056	0.142716
$7$	−5921.56	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$7$	−5921.56	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$8$	4828.78	0.416805
$9$	−10867.4	−0.552122
$10$	−6075.91	−0.192137
$11$	20085.5	0.413633	0.206817	−	0.978380i	$-0.433690\pi$
$11$	20085.5	0.413633	0.206817	+	0.978380i	$0.433690\pi$
$12$	45886.2	0.638802
$13$	0	0
$13$	0	0
$14$	28573.5	0.198786
$15$	−118225.	−0.602975
$16$	226922.	0.865640
$17$	−89577.2	−0.260122	−0.130061	−	0.991506i	$-0.541517\pi$
$17$	−89577.2	−0.260122	−0.130061	+	0.991506i	$0.541517\pi$
$18$	52438.9	0.117741
$19$	−817804.	−1.43965	−0.719826	−	0.694154i	$-0.755779\pi$
$19$	−817804.	−1.43965	−0.719826	+	0.694154i	$0.755779\pi$
$20$	−615378.	−0.860016
$21$	555983.	0.623842
$22$	−96919.0	−0.0882078
$23$	−1.17564e6	−0.875989	−0.437994	−	0.898978i	$-0.644311\pi$
$23$	−1.17564e6	−0.875989	−0.437994	+	0.898978i	$0.644311\pi$
$24$	−453380.	−0.278941
$25$	−367612.	−0.188217
$26$	0	0
$27$	2.86842e6	1.03874
$28$	2.89396e6	0.889779
$29$	4.02987e6	1.05804	0.529018	−	0.848611i	$-0.322561\pi$
$29$	4.02987e6	1.05804	0.529018	+	0.848611i	$0.322561\pi$
$30$	570475.	0.128585
$31$	−6.85443e6	−1.33304	−0.666521	−	0.745487i	$-0.732217\pi$
$31$	−6.85443e6	−1.33304	−0.666521	+	0.745487i	$0.732217\pi$
$32$	−3.56731e6	−0.601403
$33$	−1.88585e6	−0.276818
$34$	432239.	0.0554714
$35$	−7.45626e6	−0.839876
$36$	5.31109e6	0.527014
$37$	−1.56577e7	−1.37347	−0.686736	−	0.726907i	$-0.740957\pi$
$37$	−1.56577e7	−1.37347	−0.686736	+	0.726907i	$0.740957\pi$
$38$	3.94617e6	0.307008
$39$	0	0
$40$	6.08026e6	0.375537
$41$	6.67447e6	0.368883	0.184442	−	0.982843i	$-0.440952\pi$
$41$	6.67447e6	0.368883	0.184442	+	0.982843i	$0.440952\pi$
$42$	−2.68280e6	−0.133035
$43$	−2.83685e6	−0.126540	−0.0632700	−	0.997996i	$-0.520153\pi$
$43$	−2.83685e6	−0.126540	−0.0632700	+	0.997996i	$0.520153\pi$
$44$	−9.81611e6	−0.394823
$45$	−1.36840e7	−0.497457
$46$	5.67284e6	0.186806
$47$	−4.23495e7	−1.26593	−0.632963	−	0.774182i	$-0.718161\pi$
$47$	−4.23495e7	−1.26593	−0.632963	+	0.774182i	$0.718161\pi$
$48$	−2.13060e7	−0.579318
$49$	−5.28871e6	−0.131059
$50$	1.77385e6	0.0401376
$51$	8.41052e6	0.174083
$52$	0	0
$53$	−9.54736e7	−1.66204	−0.831020	−	0.556242i	$-0.812243\pi$
$53$	−9.54736e7	−1.66204	−0.831020	+	0.556242i	$0.812243\pi$
$54$	−1.38411e7	−0.221512
$55$	2.52911e7	0.372679
$56$	−2.85939e7	−0.388533
$57$	7.67846e7	0.963468
$58$	−1.94454e7	−0.225627
$59$	9.03136e7	0.970329	0.485164	−	0.874423i	$-0.338760\pi$
$59$	9.03136e7	0.970329	0.485164	+	0.874423i	$0.338760\pi$
$60$	5.77786e7	0.575554
$61$	−7.48746e7	−0.692389	−0.346195	−	0.938163i	$-0.612526\pi$
$61$	−7.48746e7	−0.692389	−0.346195	+	0.938163i	$0.612526\pi$
$62$	3.30748e7	0.284273
$63$	6.43521e7	0.514672
$64$	−9.89708e7	−0.737390
$65$	0	0
$66$	9.09985e6	0.0590319
$67$	−6.65968e6	−0.0403754	−0.0201877	−	0.999796i	$-0.506426\pi$
$67$	−6.65968e6	−0.0403754	−0.0201877	+	0.999796i	$0.506426\pi$
$68$	4.37778e7	0.248293
$69$	1.10382e8	0.586244
$70$	3.59789e7	0.179105
$71$	2.26902e8	1.05968	0.529841	−	0.848097i	$-0.322252\pi$
$71$	2.26902e8	1.05968	0.529841	+	0.848097i	$0.322252\pi$
$72$	−5.24764e7	−0.230127
$73$	−1.95234e8	−0.804643	−0.402321	−	0.915499i	$-0.631797\pi$
$73$	−1.95234e8	−0.804643	−0.402321	+	0.915499i	$0.631797\pi$
$74$	7.55534e7	0.292894
$75$	3.45156e7	0.125962
$76$	3.99674e8	1.37418
$77$	−1.18937e8	−0.385576
$78$	0	0
$79$	5.42837e8	1.56801	0.784003	−	0.620757i	$-0.213175\pi$
$79$	5.42837e8	1.56801	0.784003	+	0.620757i	$0.213175\pi$
$80$	2.85734e8	0.779933
$81$	−5.54160e7	−0.143038
$82$	−3.22065e7	−0.0786648
$83$	4.31319e8	0.997579	0.498789	−	0.866723i	$-0.333778\pi$
$83$	4.31319e8	0.997579	0.498789	+	0.866723i	$0.333778\pi$
$84$	−2.71718e8	−0.595472
$85$	−1.12793e8	−0.234368
$86$	1.36887e7	0.0269848
$87$	−3.78370e8	−0.708076
$88$	9.69884e7	0.172404
$89$	−2.11702e8	−0.357659	−0.178829	−	0.983880i	$-0.557231\pi$
$89$	−2.11702e8	−0.357659	−0.178829	+	0.983880i	$0.557231\pi$
$90$	6.60295e7	0.106083
$91$	0	0
$92$	5.74554e8	0.836152
$93$	6.43571e8	0.892120
$94$	2.04350e8	0.269960
$95$	−1.02975e9	−1.29711
$96$	3.34939e8	0.402481
$97$	7.77181e8	0.891352	0.445676	−	0.895194i	$-0.352963\pi$
$97$	7.77181e8	0.891352	0.445676	+	0.895194i	$0.352963\pi$
$98$	2.55197e7	0.0279485
$99$	−2.18278e8	−0.228376
$100$	1.79658e8	0.179658
$101$	−1.17457e9	−1.12313	−0.561566	−	0.827432i	$-0.689801\pi$
$101$	−1.17457e9	−1.12313	−0.561566	+	0.827432i	$0.689801\pi$
$102$	−4.05835e7	−0.0371235
$103$	1.02324e9	0.895796	0.447898	−	0.894085i	$-0.352173\pi$
$103$	1.02324e9	0.895796	0.447898	+	0.894085i	$0.352173\pi$
$104$	0	0
$105$	7.00078e8	0.562075
$106$	4.60691e8	0.354432
$107$	1.58126e9	1.16621	0.583104	−	0.812397i	$-0.301838\pi$
$107$	1.58126e9	1.16621	0.583104	+	0.812397i	$0.301838\pi$
$108$	−1.40184e9	−0.991499
$109$	2.55758e9	1.73544	0.867721	−	0.497052i	$-0.165584\pi$
$109$	2.55758e9	1.73544	0.867721	+	0.497052i	$0.165584\pi$
$110$	−1.22038e8	−0.0794743
$111$	1.47012e9	0.919177
$112$	−1.34373e9	−0.806923
$113$	−2.65325e9	−1.53083	−0.765413	−	0.643539i	$-0.777465\pi$
$113$	−2.65325e9	−1.53083	−0.765413	+	0.643539i	$0.777465\pi$
$114$	−3.70511e8	−0.205461
$115$	−1.48033e9	−0.789257
$116$	−1.96946e9	−1.00992
$117$	0	0
$118$	−4.35792e8	−0.206924
$119$	5.30437e8	0.242478
$120$	−5.70884e8	−0.251323
$121$	−1.95452e9	−0.828908
$122$	3.61294e8	0.147653
$123$	−6.26674e8	−0.246870
$124$	3.34987e9	1.27242
$125$	−2.92221e9	−1.07057
$126$	−3.10520e8	−0.109754
$127$	5.15530e9	1.75848	0.879239	−	0.476380i	$-0.158052\pi$
$127$	5.15530e9	1.75848	0.879239	+	0.476380i	$0.158052\pi$
$128$	2.30403e9	0.758653
$129$	2.66355e8	0.0846852
$130$	0	0
$131$	6.25888e9	1.85685	0.928423	−	0.371525i	$-0.121165\pi$
$131$	6.25888e9	1.85685	0.928423	+	0.371525i	$0.121165\pi$
$132$	9.21647e8	0.264230
$133$	4.84267e9	1.34200
$134$	3.21351e7	0.00861011
$135$	3.61183e9	0.935892
$136$	−4.32549e8	−0.108420
$137$	4.50387e9	1.09230	0.546152	−	0.837686i	$-0.316092\pi$
$137$	4.50387e9	1.09230	0.546152	+	0.837686i	$0.316092\pi$
$138$	−5.32630e8	−0.125017
$139$	−2.91929e9	−0.663302	−0.331651	−	0.943402i	$-0.607606\pi$
$139$	−2.91929e9	−0.663302	−0.331651	+	0.943402i	$0.607606\pi$
$140$	3.64400e9	0.801681
$141$	3.97625e9	0.847204
$142$	−1.09487e9	−0.225978
$143$	0	0
$144$	−2.46606e9	−0.477939
$145$	5.07430e9	0.953279
$146$	9.42068e8	0.171591
$147$	4.96564e8	0.0877096
$148$	7.65216e9	1.31101
$149$	−2.24179e9	−0.372612	−0.186306	−	0.982492i	$-0.559652\pi$
$149$	−2.24179e9	−0.372612	−0.186306	+	0.982492i	$0.559652\pi$
$150$	−1.66549e8	−0.0268615
$151$	1.56790e9	0.245427	0.122713	−	0.992442i	$-0.460840\pi$
$151$	1.56790e9	0.245427	0.122713	+	0.992442i	$0.460840\pi$
$152$	−3.94899e9	−0.600054
$153$	9.73474e8	0.143619
$154$	5.73912e8	0.0822247
$155$	−8.63090e9	−1.20106
$156$	0	0
$157$	9.90334e9	1.30087	0.650434	−	0.759563i	$-0.274587\pi$
$157$	9.90334e9	1.30087	0.650434	+	0.759563i	$0.274587\pi$
$158$	−2.61937e9	−0.334379
$159$	8.96414e9	1.11230
$160$	−4.49185e9	−0.541858
$161$	6.96162e9	0.816570
$162$	2.67400e8	0.0305031
$163$	2.05141e9	0.227619	0.113809	−	0.993503i	$-0.463695\pi$
$163$	2.05141e9	0.227619	0.113809	+	0.993503i	$0.463695\pi$
$164$	−3.26192e9	−0.352108
$165$	−2.37461e9	−0.249411
$166$	−2.08125e9	−0.212735
$167$	−4.09006e9	−0.406917	−0.203458	−	0.979084i	$-0.565218\pi$
$167$	−4.09006e9	−0.406917	−0.203458	+	0.979084i	$0.565218\pi$
$168$	2.68472e9	0.260020
$169$	0	0
$170$	5.44263e8	0.0499792
$171$	8.88742e9	0.794865
$172$	1.38641e9	0.120785
$173$	−1.45760e10	−1.23718	−0.618588	−	0.785715i	$-0.712295\pi$
$173$	−1.45760e10	−1.23718	−0.618588	+	0.785715i	$0.712295\pi$
$174$	1.82576e9	0.150998
$175$	2.17684e9	0.175451
$176$	4.55785e9	0.358057
$177$	−8.47966e9	−0.649379
$178$	1.02153e9	0.0762712
$179$	−2.41619e10	−1.75911	−0.879555	−	0.475798i	$-0.842159\pi$
$179$	−2.41619e10	−1.75911	−0.879555	+	0.475798i	$0.842159\pi$
$180$	6.68757e9	0.474834
$181$	1.63442e10	1.13190	0.565952	−	0.824438i	$-0.308509\pi$
$181$	1.63442e10	1.13190	0.565952	+	0.824438i	$0.308509\pi$
$182$	0	0
$183$	7.03007e9	0.463372
$184$	−5.67690e9	−0.365116
$185$	−1.97157e10	−1.23748
$186$	−3.10544e9	−0.190246
$187$	−1.79920e9	−0.107595
$188$	2.06969e10	1.20836
$189$	−1.69855e10	−0.968280
$190$	4.96890e9	0.276611
$191$	−4.60576e9	−0.250410	−0.125205	−	0.992131i	$-0.539959\pi$
$191$	−4.60576e9	−0.250410	−0.125205	+	0.992131i	$0.539959\pi$
$192$	9.29249e9	0.493488
$193$	−1.12002e10	−0.581055	−0.290528	−	0.956867i	$-0.593831\pi$
$193$	−1.12002e10	−0.581055	−0.290528	+	0.956867i	$0.593831\pi$
$194$	−3.75015e9	−0.190082
$195$	0	0
$196$	2.58468e9	0.125099
$197$	2.25309e10	1.06581	0.532907	−	0.846174i	$-0.321100\pi$
$197$	2.25309e10	1.06581	0.532907	+	0.846174i	$0.321100\pi$
$198$	1.05326e9	0.0487015
$199$	−3.01489e9	−0.136280	−0.0681402	−	0.997676i	$-0.521707\pi$
$199$	−3.01489e9	−0.136280	−0.0681402	+	0.997676i	$0.521707\pi$
$200$	−1.77512e9	−0.0784498
$201$	6.25286e8	0.0270207
$202$	5.66766e9	0.239509
$203$	−2.38631e10	−0.986269
$204$	−4.11036e9	−0.166167
$205$	8.40430e9	0.332360
$206$	−4.93745e9	−0.191030
$207$	1.27762e10	0.483653
$208$	0	0
$209$	−1.64260e10	−0.595488
$210$	−3.37810e9	−0.119863
$211$	−2.38129e9	−0.0827067	−0.0413533	−	0.999145i	$-0.513167\pi$
$211$	−2.38129e9	−0.0827067	−0.0413533	+	0.999145i	$0.513167\pi$
$212$	4.66595e10	1.58646
$213$	−2.13041e10	−0.709177
$214$	−7.63009e9	−0.248695
$215$	−3.57208e9	−0.114011
$216$	1.38510e10	0.432950
$217$	4.05889e10	1.24262
$218$	−1.23412e10	−0.370085
$219$	1.83308e10	0.538496
$220$	−1.23602e10	−0.355731
$221$	0	0
$222$	−7.09380e9	−0.196016
$223$	−2.79569e10	−0.757037	−0.378518	−	0.925594i	$-0.623566\pi$
$223$	−2.79569e10	−0.757037	−0.378518	+	0.925594i	$0.623566\pi$
$224$	2.11240e10	0.560610
$225$	3.99500e9	0.103919
$226$	1.28028e10	0.326451
$227$	−2.12282e10	−0.530635	−0.265317	−	0.964161i	$-0.585477\pi$
$227$	−2.12282e10	−0.530635	−0.265317	+	0.964161i	$0.585477\pi$
$228$	−3.75259e10	−0.919654
$229$	6.99418e10	1.68065	0.840324	−	0.542084i	$-0.182365\pi$
$229$	6.99418e10	1.68065	0.840324	+	0.542084i	$0.182365\pi$
$230$	7.14307e9	0.168310
$231$	1.11672e10	0.258042
$232$	1.94594e10	0.440994
$233$	−5.67672e10	−1.26182	−0.630908	−	0.775857i	$-0.717318\pi$
$233$	−5.67672e10	−1.26182	−0.630908	+	0.775857i	$0.717318\pi$
$234$	0	0
$235$	−5.33253e10	−1.14059
$236$	−4.41377e10	−0.926202
$237$	−5.09677e10	−1.04937
$238$	−2.55953e9	−0.0517088
$239$	−6.89132e10	−1.36619	−0.683096	−	0.730328i	$-0.739367\pi$
$239$	−6.89132e10	−1.36619	−0.683096	+	0.730328i	$0.739367\pi$
$240$	−2.68279e10	−0.521959
$241$	1.83805e10	0.350979	0.175490	−	0.984481i	$-0.443849\pi$
$241$	1.83805e10	0.350979	0.175490	+	0.984481i	$0.443849\pi$
$242$	9.43120e9	0.176766
$243$	−5.12560e10	−0.943011
$244$	3.65924e10	0.660902
$245$	−6.65939e9	−0.118083
$246$	3.02391e9	0.0526454
$247$	0	0
$248$	−3.30985e10	−0.555618
$249$	−4.04971e10	−0.667616
$250$	1.41006e10	0.228301
$251$	1.07167e11	1.70424	0.852119	−	0.523349i	$-0.175318\pi$
$251$	1.07167e11	1.70424	0.852119	+	0.523349i	$0.175318\pi$
$252$	−3.14499e10	−0.491267
$253$	−2.36133e10	−0.362338
$254$	−2.48760e10	−0.374998
$255$	1.05903e10	0.156847
$256$	3.95554e10	0.575606
$257$	−1.96553e10	−0.281048	−0.140524	−	0.990077i	$-0.544879\pi$
$257$	−1.96553e10	−0.281048	−0.140524	+	0.990077i	$0.544879\pi$
$258$	−1.28525e9	−0.0180592
$259$	9.27179e10	1.28031
$260$	0	0
$261$	−4.37943e10	−0.584165
$262$	−3.02011e10	−0.395975
$263$	−2.04545e10	−0.263626	−0.131813	−	0.991275i	$-0.542080\pi$
$263$	−2.04545e10	−0.263626	−0.131813	+	0.991275i	$0.542080\pi$
$264$	−9.10637e9	−0.115379
$265$	−1.20218e11	−1.49748
$266$	−2.33675e10	−0.286183
$267$	1.98769e10	0.239358
$268$	3.25470e9	0.0385393
$269$	1.32223e11	1.53965	0.769827	−	0.638253i	$-0.220343\pi$
$269$	1.32223e11	1.53965	0.769827	+	0.638253i	$0.220343\pi$
$270$	−1.74283e10	−0.199580
$271$	−1.28886e11	−1.45159	−0.725796	−	0.687910i	$-0.758528\pi$
$271$	−1.28886e11	−1.45159	−0.725796	+	0.687910i	$0.758528\pi$
$272$	−2.03271e10	−0.225172
$273$	0	0
$274$	−2.17326e10	−0.232935
$275$	−7.38367e9	−0.0778529
$276$	−5.39456e10	−0.559584
$277$	−1.06500e10	−0.108691	−0.0543453	−	0.998522i	$-0.517307\pi$
$277$	−1.06500e10	−0.108691	−0.0543453	+	0.998522i	$0.517307\pi$
$278$	1.40865e10	0.141450
$279$	7.44900e10	0.736002
$280$	−3.60047e10	−0.350064
$281$	−2.51296e10	−0.240440	−0.120220	−	0.992747i	$-0.538360\pi$
$281$	−2.51296e10	−0.240440	−0.120220	+	0.992747i	$0.538360\pi$
$282$	−1.91867e10	−0.180667
$283$	1.19785e10	0.111010	0.0555052	−	0.998458i	$-0.482323\pi$
$283$	1.19785e10	0.111010	0.0555052	+	0.998458i	$0.482323\pi$
$284$	−1.10891e11	−1.01149
$285$	9.66850e10	0.868075
$286$	0	0
$287$	−3.95233e10	−0.343862
$288$	3.87675e10	0.332048
$289$	−1.10564e11	−0.932336
$290$	−2.44851e10	−0.203288
$291$	−7.29705e10	−0.596525
$292$	9.54141e10	0.768051
$293$	−1.44831e11	−1.14804	−0.574019	−	0.818842i	$-0.694616\pi$
$293$	−1.44831e11	−1.14804	−0.574019	+	0.818842i	$0.694616\pi$
$294$	−2.39608e9	−0.0187042
$295$	1.13720e11	0.874256
$296$	−7.56075e10	−0.572469
$297$	5.76136e10	0.429656
$298$	1.08174e10	0.0794599
$299$	0	0
$300$	−1.68683e10	−0.120234
$301$	1.67986e10	0.117957
$302$	−7.56563e9	−0.0523376
$303$	1.10281e11	0.751641
$304$	−1.85578e11	−1.24622
$305$	−9.42800e10	−0.623835
$306$	−4.69733e9	−0.0306270
$307$	2.22808e11	1.43155	0.715776	−	0.698330i	$-0.246073\pi$
$307$	2.22808e11	1.43155	0.715776	+	0.698330i	$0.246073\pi$
$308$	5.81267e10	0.368042
$309$	−9.60731e10	−0.599499
$310$	4.16469e10	0.256127
$311$	1.75046e10	0.106104	0.0530519	−	0.998592i	$-0.483105\pi$
$311$	1.75046e10	0.106104	0.0530519	+	0.998592i	$0.483105\pi$
$312$	0	0
$313$	9.85218e10	0.580207	0.290104	−	0.956995i	$-0.406310\pi$
$313$	9.85218e10	0.580207	0.290104	+	0.956995i	$0.406310\pi$
$314$	−4.77868e10	−0.277412
$315$	8.10304e10	0.463714
$316$	−2.65293e11	−1.49670
$317$	2.42788e11	1.35039	0.675197	−	0.737637i	$-0.264059\pi$
$317$	2.42788e11	1.35039	0.675197	+	0.737637i	$0.264059\pi$
$318$	−4.32549e10	−0.237199
$319$	8.09419e10	0.437639
$320$	−1.24621e11	−0.664381
$321$	−1.48467e11	−0.780470
$322$	−3.35921e10	−0.174135
$323$	7.32566e10	0.374486
$324$	2.70827e10	0.136534
$325$	0	0
$326$	−9.89870e9	−0.0485399
$327$	−2.40134e11	−1.16142
$328$	3.22295e10	0.153752
$329$	2.50775e11	1.18006
$330$	1.14583e10	0.0531871
$331$	1.05646e11	0.483755	0.241878	−	0.970307i	$-0.422237\pi$
$331$	1.05646e11	0.483755	0.241878	+	0.970307i	$0.422237\pi$
$332$	−2.10793e11	−0.952213
$333$	1.70159e11	0.758324
$334$	1.97359e10	0.0867755
$335$	−8.38568e9	−0.0363778
$336$	1.26165e11	0.540023
$337$	−3.70603e11	−1.56522	−0.782608	−	0.622515i	$-0.786111\pi$
$337$	−3.70603e11	−1.56522	−0.782608	+	0.622515i	$0.786111\pi$
$338$	0	0
$339$	2.49118e11	1.02449
$340$	5.51238e10	0.223709
$341$	−1.37675e11	−0.551390
$342$	−4.28847e10	−0.169506
$343$	2.70274e11	1.05434
$344$	−1.36985e10	−0.0527424
$345$	1.38990e11	0.528200
$346$	7.03341e10	0.263829
$347$	7.47684e10	0.276844	0.138422	−	0.990373i	$-0.455797\pi$
$347$	7.47684e10	0.276844	0.138422	+	0.990373i	$0.455797\pi$
$348$	1.84915e11	0.675875
$349$	2.03456e11	0.734102	0.367051	−	0.930201i	$-0.380367\pi$
$349$	2.03456e11	0.734102	0.367051	+	0.930201i	$0.380367\pi$
$350$	−1.05039e10	−0.0374150
$351$	0	0
$352$	−7.16512e10	−0.248760
$353$	3.87575e11	1.32853	0.664263	−	0.747499i	$-0.268745\pi$
$353$	3.87575e11	1.32853	0.664263	+	0.747499i	$0.268745\pi$
$354$	4.09171e10	0.138481
$355$	2.85708e11	0.954762
$356$	1.03462e11	0.341394
$357$	−4.98034e10	−0.162275
$358$	1.16589e11	0.375132
$359$	−4.61401e11	−1.46607	−0.733033	−	0.680193i	$-0.761896\pi$
$359$	−4.61401e11	−1.46607	−0.733033	+	0.680193i	$0.761896\pi$
$360$	−6.60768e10	−0.207342
$361$	3.46115e11	1.07260
$362$	−7.88660e10	−0.241380
$363$	1.83512e11	0.554735
$364$	0	0
$365$	−2.45833e11	−0.724975
$366$	−3.39224e10	−0.0988147
$367$	3.67136e11	1.05640	0.528202	−	0.849119i	$-0.322866\pi$
$367$	3.67136e11	1.05640	0.528202	+	0.849119i	$0.322866\pi$
$368$	−2.66779e11	−0.758291
$369$	−7.25343e10	−0.203669
$370$	9.51347e10	0.263895
$371$	5.65353e11	1.54930
$372$	−3.14524e11	−0.851550
$373$	−6.37812e11	−1.70609	−0.853047	−	0.521833i	$-0.825248\pi$
$373$	−6.37812e11	−1.70609	−0.853047	+	0.521833i	$0.825248\pi$
$374$	8.68174e9	0.0229448
$375$	2.74370e11	0.716466
$376$	−2.04497e11	−0.527644
$377$	0	0
$378$	8.19606e10	0.206487
$379$	4.57826e11	1.13979	0.569894	−	0.821718i	$-0.306984\pi$
$379$	4.57826e11	1.13979	0.569894	+	0.821718i	$0.306984\pi$
$380$	5.03258e11	1.23812
$381$	−4.84038e11	−1.17684
$382$	2.22243e10	0.0534001
$383$	−4.94333e10	−0.117388	−0.0586942	−	0.998276i	$-0.518694\pi$
$383$	−4.94333e10	−0.117388	−0.0586942	+	0.998276i	$0.518694\pi$
$384$	−2.16328e11	−0.507718
$385$	−1.49763e11	−0.347400
$386$	5.40445e10	0.123911
$387$	3.08292e10	0.0698656
$388$	−3.79821e11	−0.850817
$389$	9.71174e10	0.215042	0.107521	−	0.994203i	$-0.465709\pi$
$389$	9.71174e10	0.215042	0.107521	+	0.994203i	$0.465709\pi$
$390$	0	0
$391$	1.05310e11	0.227864
$392$	−2.55380e10	−0.0546260
$393$	−5.87654e11	−1.24267
$394$	−1.08719e11	−0.227286
$395$	6.83525e11	1.41276
$396$	1.06676e11	0.217991
$397$	5.03362e11	1.01701	0.508503	−	0.861060i	$-0.330199\pi$
$397$	5.03362e11	1.01701	0.508503	+	0.861060i	$0.330199\pi$
$398$	1.45478e10	0.0290619
$399$	−4.54685e11	−0.898116
$400$	−8.34193e10	−0.162928
$401$	−4.42201e11	−0.854025	−0.427012	−	0.904246i	$-0.640434\pi$
$401$	−4.42201e11	−0.854025	−0.427012	+	0.904246i	$0.640434\pi$
$402$	−3.01721e9	−0.00576220
$403$	0	0
$404$	5.74029e11	1.07206
$405$	−6.97782e10	−0.128876
$406$	1.15147e11	0.210323
$407$	−3.14492e11	−0.568113
$408$	4.06126e10	0.0725587
$409$	−3.35222e11	−0.592349	−0.296174	−	0.955134i	$-0.595711\pi$
$409$	−3.35222e11	−0.592349	−0.296174	+	0.955134i	$0.595711\pi$
$410$	−4.05535e10	−0.0708762
$411$	−4.22874e11	−0.731009
$412$	−5.00073e11	−0.855059
$413$	−5.34797e11	−0.904511
$414$	−6.16491e10	−0.103140
$415$	5.43105e11	0.898809
$416$	0	0
$417$	2.74096e11	0.443906
$418$	7.92607e10	0.126989
$419$	7.66942e11	1.21562	0.607812	−	0.794081i	$-0.292047\pi$
$419$	7.66942e11	1.21562	0.607812	+	0.794081i	$0.292047\pi$
$420$	−3.42140e11	−0.536515
$421$	5.09433e11	0.790347	0.395173	−	0.918607i	$-0.370684\pi$
$421$	5.09433e11	0.790347	0.395173	+	0.918607i	$0.370684\pi$
$422$	1.14905e10	0.0176373
$423$	4.60230e11	0.698946
$424$	−4.61021e11	−0.692746
$425$	3.29297e10	0.0489595
$426$	1.02799e11	0.151233
$427$	4.43375e11	0.645424
$428$	−7.72788e11	−1.11317
$429$	0	0
$430$	1.72364e10	0.0243130
$431$	7.64884e11	1.06770	0.533849	−	0.845580i	$-0.320745\pi$
$431$	7.64884e11	1.06770	0.533849	+	0.845580i	$0.320745\pi$
$432$	6.50908e11	0.899172
$433$	−2.93420e11	−0.401138	−0.200569	−	0.979680i	$-0.564279\pi$
$433$	−2.93420e11	−0.401138	−0.200569	+	0.979680i	$0.564279\pi$
$434$	−1.95855e11	−0.264990
$435$	−4.76432e11	−0.637969
$436$	−1.24993e12	−1.65652
$437$	9.61441e11	1.26112
$438$	−8.84520e10	−0.114835
$439$	−8.40043e11	−1.07947	−0.539736	−	0.841835i	$-0.681476\pi$
$439$	−8.40043e11	−1.07947	−0.539736	+	0.841835i	$0.681476\pi$
$440$	1.22125e11	0.155334
$441$	5.74746e10	0.0723607
$442$	0	0
$443$	9.03059e11	1.11404	0.557018	−	0.830500i	$-0.311945\pi$
$443$	9.03059e11	1.11404	0.557018	+	0.830500i	$0.311945\pi$
$444$	−7.18471e11	−0.877377
$445$	−2.66569e11	−0.322247
$446$	1.34901e11	0.161439
$447$	2.10484e11	0.249365
$448$	5.86062e11	0.687373
$449$	7.34784e11	0.853201	0.426600	−	0.904440i	$-0.359711\pi$
$449$	7.34784e11	0.853201	0.426600	+	0.904440i	$0.359711\pi$
$450$	−1.92771e10	−0.0221609
$451$	1.34060e11	0.152582
$452$	1.29669e12	1.46121
$453$	−1.47212e11	−0.164249
$454$	1.02433e11	0.113159
$455$	0	0
$456$	3.70776e11	0.401578
$457$	1.14353e12	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$457$	1.14353e12	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$458$	−3.37492e11	−0.358400
$459$	−2.56945e11	−0.270199
$460$	7.23461e11	0.753365
$461$	4.78349e11	0.493276	0.246638	−	0.969108i	$-0.420674\pi$
$461$	4.78349e11	0.493276	0.246638	+	0.969108i	$0.420674\pi$
$462$	−5.38853e10	−0.0550277
$463$	3.62233e11	0.366331	0.183165	−	0.983082i	$-0.441366\pi$
$463$	3.62233e11	0.366331	0.183165	+	0.983082i	$0.441366\pi$
$464$	9.14467e11	0.915877
$465$	8.10366e11	0.803791
$466$	2.73920e11	0.269084
$467$	−4.24688e11	−0.413185	−0.206592	−	0.978427i	$-0.566237\pi$
$467$	−4.24688e11	−0.413185	−0.206592	+	0.978427i	$0.566237\pi$
$468$	0	0
$469$	3.94357e10	0.0376367
$470$	2.57312e11	0.243231
$471$	−9.29837e11	−0.870588
$472$	4.36104e11	0.404437
$473$	−5.69795e10	−0.0523411
$474$	2.45936e11	0.223779
$475$	3.00634e11	0.270968
$476$	−2.59233e11	−0.231451
$477$	1.03755e12	0.917650
$478$	3.32529e11	0.291342
$479$	1.50822e12	1.30904	0.654522	−	0.756043i	$-0.272870\pi$
$479$	1.50822e12	1.30904	0.654522	+	0.756043i	$0.272870\pi$
$480$	4.21746e11	0.362631
$481$	0	0
$482$	−8.86920e10	−0.0748467
$483$	−6.53635e11	−0.546479
$484$	9.55206e11	0.791212
$485$	9.78604e11	0.803099
$486$	2.47327e11	0.201098
$487$	1.80244e11	0.145205	0.0726023	−	0.997361i	$-0.476870\pi$
$487$	1.80244e11	0.145205	0.0726023	+	0.997361i	$0.476870\pi$
$488$	−3.61553e11	−0.288591
$489$	−1.92609e11	−0.152331
$490$	3.21337e10	0.0251813
$491$	−1.34172e12	−1.04183	−0.520913	−	0.853610i	$-0.674409\pi$
$491$	−1.34172e12	−1.04183	−0.520913	+	0.853610i	$0.674409\pi$
$492$	3.06266e11	0.235644
$493$	−3.60985e11	−0.275218
$494$	0	0
$495$	−2.74849e11	−0.205765
$496$	−1.55542e12	−1.15393
$497$	−1.34361e12	−0.987803
$498$	1.95412e11	0.142370
$499$	−5.70585e11	−0.411972	−0.205986	−	0.978555i	$-0.566040\pi$
$499$	−5.70585e11	−0.411972	−0.205986	+	0.978555i	$0.566040\pi$
$500$	1.42813e12	1.02189
$501$	3.84021e11	0.272324
$502$	−5.17116e11	−0.363431
$503$	−2.77701e11	−0.193429	−0.0967146	−	0.995312i	$-0.530833\pi$
$503$	−2.77701e11	−0.193429	−0.0967146	+	0.995312i	$0.530833\pi$
$504$	3.10742e11	0.214518
$505$	−1.47898e12	−1.01193
$506$	1.13942e11	0.0772690
$507$	0	0
$508$	−2.51948e12	−1.67851
$509$	−2.01748e12	−1.33223	−0.666114	−	0.745850i	$-0.732043\pi$
$509$	−2.01748e12	−1.33223	−0.666114	+	0.745850i	$0.732043\pi$
$510$	−5.11016e10	−0.0334479
$511$	1.15609e12	0.750064
$512$	−1.37053e12	−0.881401
$513$	−2.34580e12	−1.49542
$514$	9.48430e10	0.0599337
$515$	1.28843e12	0.807103
$516$	−1.30172e11	−0.0808340
$517$	−8.50611e11	−0.523629
$518$	−4.47394e11	−0.273027
$519$	1.36856e12	0.827964
$520$	0	0
$521$	−1.35040e12	−0.802956	−0.401478	−	0.915869i	$-0.631503\pi$
$521$	−1.35040e12	−0.802956	−0.401478	+	0.915869i	$0.631503\pi$
$522$	2.11322e11	0.124574
$523$	−1.32309e12	−0.773271	−0.386635	−	0.922233i	$-0.626363\pi$
$523$	−1.32309e12	−0.773271	−0.386635	+	0.922233i	$0.626363\pi$
$524$	−3.05882e12	−1.77240
$525$	−2.04386e11	−0.117418
$526$	9.86997e10	0.0562186
$527$	6.14001e11	0.346754
$528$	−4.27942e11	−0.239625
$529$	−4.19027e11	−0.232644
$530$	5.80089e11	0.319340
$531$	−9.81476e11	−0.535740
$532$	−2.36669e12	−1.28097
$533$	0	0
$534$	−9.59127e10	−0.0510435
$535$	1.99108e12	1.05074
$536$	−3.21581e10	−0.0168287
$537$	2.26859e12	1.17726
$538$	−6.38021e11	−0.328333
$539$	−1.06226e11	−0.0542104
$540$	−1.76516e12	−0.893331
$541$	2.36940e12	1.18919	0.594593	−	0.804027i	$-0.297313\pi$
$541$	2.36940e12	1.18919	0.594593	+	0.804027i	$0.297313\pi$
$542$	6.21918e11	0.309554
$543$	−1.53458e12	−0.757511
$544$	3.19550e11	0.156438
$545$	3.22043e12	1.56361
$546$	0	0
$547$	−7.57609e11	−0.361828	−0.180914	−	0.983499i	$-0.557906\pi$
$547$	−7.57609e11	−0.361828	−0.180914	+	0.983499i	$0.557906\pi$
$548$	−2.20111e12	−1.04263
$549$	8.13694e11	0.382284
$550$	3.56286e10	0.0166022
$551$	−3.29564e12	−1.52320
$552$	5.33011e11	0.244349
$553$	−3.21444e12	−1.46165
$554$	5.13899e10	0.0231784
$555$	1.85113e12	0.828169
$556$	1.42671e12	0.633138
$557$	−2.09058e12	−0.920276	−0.460138	−	0.887847i	$-0.652200\pi$
$557$	−2.09058e12	−0.920276	−0.460138	+	0.887847i	$0.652200\pi$
$558$	−3.59438e11	−0.156953
$559$	0	0
$560$	−1.69199e12	−0.727030
$561$	1.68929e11	0.0720066
$562$	1.21259e11	0.0512742
$563$	1.07631e12	0.451493	0.225747	−	0.974186i	$-0.427518\pi$
$563$	1.07631e12	0.451493	0.225747	+	0.974186i	$0.427518\pi$
$564$	−1.94326e12	−0.808676
$565$	−3.34090e12	−1.37926
$566$	−5.78001e10	−0.0236731
$567$	3.28149e11	0.133336
$568$	1.09566e12	0.441680
$569$	−1.75994e12	−0.703868	−0.351934	−	0.936025i	$-0.614476\pi$
$569$	−1.75994e12	−0.703868	−0.351934	+	0.936025i	$0.614476\pi$
$570$	−4.66537e11	−0.185118
$571$	2.74377e11	0.108015	0.0540077	−	0.998541i	$-0.482800\pi$
$571$	2.74377e11	0.108015	0.0540077	+	0.998541i	$0.482800\pi$
$572$	0	0
$573$	4.32441e11	0.167583
$574$	1.90713e11	0.0733290
$575$	4.32179e11	0.164876
$576$	1.07556e12	0.407129
$577$	−3.08560e12	−1.15891	−0.579454	−	0.815005i	$-0.696734\pi$
$577$	−3.08560e12	−1.15891	−0.579454	+	0.815005i	$0.696734\pi$
$578$	5.33506e11	0.198822
$579$	1.05160e12	0.388863
$580$	−2.47989e12	−0.909928
$581$	−2.55408e12	−0.929913
$582$	3.52106e11	0.127210
$583$	−1.91763e12	−0.687475
$584$	−9.42743e11	−0.335379
$585$	0	0
$586$	6.98855e11	0.244821
$587$	1.84717e12	0.642149	0.321074	−	0.947054i	$-0.395956\pi$
$587$	1.84717e12	0.642149	0.321074	+	0.947054i	$0.395956\pi$
$588$	−2.42679e11	−0.0837209
$589$	5.60558e12	1.91912
$590$	−5.48737e11	−0.186436
$591$	−2.11546e12	−0.713282
$592$	−3.55308e12	−1.18893
$593$	4.09008e12	1.35827	0.679134	−	0.734015i	$-0.262356\pi$
$593$	4.09008e12	1.35827	0.679134	+	0.734015i	$0.262356\pi$
$594$	−2.78004e11	−0.0916247
$595$	6.67911e11	0.218470
$596$	1.09560e12	0.355667
$597$	2.83072e11	0.0912038
$598$	0	0
$599$	1.58861e12	0.504191	0.252096	−	0.967702i	$-0.418880\pi$
$599$	1.58861e12	0.504191	0.252096	+	0.967702i	$0.418880\pi$
$600$	1.66668e11	0.0525015
$601$	−4.24952e12	−1.32863	−0.664316	−	0.747452i	$-0.731277\pi$
$601$	−4.24952e12	−1.32863	−0.664316	+	0.747452i	$0.731277\pi$
$602$	−8.10585e10	−0.0251544
$603$	7.23736e10	0.0222922
$604$	−7.66258e11	−0.234266
$605$	−2.46108e12	−0.746837
$606$	−5.32144e11	−0.160288
$607$	9.97768e11	0.298319	0.149159	−	0.988813i	$-0.452343\pi$
$607$	9.97768e11	0.298319	0.149159	+	0.988813i	$0.452343\pi$
$608$	2.91736e12	0.865812
$609$	2.24054e12	0.660047
$610$	4.54931e11	0.133034
$611$	0	0
$612$	−4.75753e11	−0.137088
$613$	−4.36254e12	−1.24786	−0.623932	−	0.781479i	$-0.714466\pi$
$613$	−4.36254e12	−1.24786	−0.623932	+	0.781479i	$0.714466\pi$
$614$	−1.07512e12	−0.305280
$615$	−7.89090e11	−0.222428
$616$	−5.74323e11	−0.160710
$617$	6.71209e12	1.86455	0.932276	−	0.361747i	$-0.117819\pi$
$617$	6.71209e12	1.86455	0.932276	+	0.361747i	$0.117819\pi$
$618$	4.63584e11	0.127844
$619$	−1.04194e12	−0.285255	−0.142628	−	0.989776i	$-0.545555\pi$
$619$	−1.04194e12	−0.285255	−0.142628	+	0.989776i	$0.545555\pi$
$620$	4.21806e12	1.14644
$621$	−3.37222e12	−0.909922
$622$	−8.44655e10	−0.0226268
$623$	1.25360e12	0.333399
$624$	0	0
$625$	−2.96157e12	−0.776357
$626$	−4.75400e11	−0.123730
$627$	1.54226e12	0.398523
$628$	−4.83992e12	−1.24171
$629$	1.40257e12	0.357270
$630$	−3.90998e11	−0.0988876
$631$	7.49233e12	1.88142	0.940708	−	0.339216i	$-0.110162\pi$
$631$	7.49233e12	1.88142	0.940708	+	0.339216i	$0.110162\pi$
$632$	2.62124e12	0.653552
$633$	2.23582e11	0.0553503
$634$	−1.17153e12	−0.287973
$635$	6.49141e12	1.58437
$636$	−4.38092e12	−1.06172
$637$	0	0
$638$	−3.90571e11	−0.0933269
$639$	−2.46584e12	−0.585074
$640$	2.90117e12	0.683538
$641$	1.47128e12	0.344218	0.172109	−	0.985078i	$-0.444942\pi$
$641$	1.47128e12	0.344218	0.172109	+	0.985078i	$0.444942\pi$
$642$	7.16399e11	0.166436
$643$	2.87216e10	0.00662611	0.00331306	−	0.999995i	$-0.498945\pi$
$643$	2.87216e10	0.00662611	0.00331306	+	0.999995i	$0.498945\pi$
$644$	−3.40225e12	−0.779436
$645$	3.35387e11	0.0763005
$646$	−3.53487e11	−0.0798595
$647$	−2.79660e12	−0.627424	−0.313712	−	0.949518i	$-0.601573\pi$
$647$	−2.79660e12	−0.627424	−0.313712	+	0.949518i	$0.601573\pi$
$648$	−2.67592e11	−0.0596190
$649$	1.81399e12	0.401360
$650$	0	0
$651$	−3.81095e12	−0.831608
$652$	−1.00256e12	−0.217267
$653$	−3.09232e12	−0.665542	−0.332771	−	0.943008i	$-0.607984\pi$
$653$	−3.09232e12	−0.665542	−0.332771	+	0.943008i	$0.607984\pi$
$654$	1.15873e12	0.247674
$655$	7.88101e12	1.67300
$656$	1.51459e12	0.319320
$657$	2.12169e12	0.444261
$658$	−1.21007e12	−0.251649
$659$	−2.13820e12	−0.441636	−0.220818	−	0.975315i	$-0.570873\pi$
$659$	−2.13820e12	−0.441636	−0.220818	+	0.975315i	$0.570873\pi$
$660$	1.16051e12	0.238068
$661$	−2.41601e12	−0.492258	−0.246129	−	0.969237i	$-0.579159\pi$
$661$	−2.41601e12	−0.492258	−0.246129	+	0.969237i	$0.579159\pi$
$662$	−5.09774e11	−0.103161
$663$	0	0
$664$	2.08275e12	0.415796
$665$	6.09776e12	1.20913
$666$	−8.21071e11	−0.161714
$667$	−4.73767e12	−0.926827
$668$	1.99888e12	0.388412
$669$	2.62491e12	0.506637
$670$	4.04636e10	0.00775762
$671$	−1.50389e12	−0.286395
$672$	−1.98336e12	−0.375181
$673$	3.77066e12	0.708516	0.354258	−	0.935148i	$-0.384733\pi$
$673$	3.77066e12	0.708516	0.354258	+	0.935148i	$0.384733\pi$
$674$	1.78828e12	0.333784
$675$	−1.05447e12	−0.195508
$676$	0	0
$677$	−8.47642e11	−0.155083	−0.0775414	−	0.996989i	$-0.524707\pi$
$677$	−8.47642e11	−0.155083	−0.0775414	+	0.996989i	$0.524707\pi$
$678$	−1.20207e12	−0.218473
$679$	−4.60212e12	−0.830892
$680$	−5.44653e11	−0.0976855
$681$	1.99314e12	0.355120
$682$	6.64325e11	0.117585
$683$	1.68006e12	0.295415	0.147707	−	0.989031i	$-0.452811\pi$
$683$	1.68006e12	0.295415	0.147707	+	0.989031i	$0.452811\pi$
$684$	−4.34343e12	−0.758717
$685$	5.67114e12	0.984154
$686$	−1.30416e12	−0.224839
$687$	−6.56692e12	−1.12475
$688$	−6.43744e11	−0.109538
$689$	0	0
$690$	−6.70672e11	−0.112639
$691$	3.78914e12	0.632250	0.316125	−	0.948718i	$-0.397618\pi$
$691$	3.78914e12	0.632250	0.316125	+	0.948718i	$0.397618\pi$
$692$	7.12354e12	1.18091
$693$	1.29254e12	0.212885
$694$	−3.60782e11	−0.0590374
$695$	−3.67589e12	−0.597628
$696$	−1.82706e12	−0.295129
$697$	−5.97880e11	−0.0959548
$698$	−9.81742e11	−0.156548
$699$	5.32995e12	0.844454
$700$	−1.06386e12	−0.167472
$701$	6.91499e12	1.08159	0.540793	−	0.841156i	$-0.318124\pi$
$701$	6.91499e12	1.08159	0.540793	+	0.841156i	$0.318124\pi$
$702$	0	0
$703$	1.28049e13	1.97732
$704$	−1.98788e12	−0.305009
$705$	5.00678e12	0.763322
$706$	−1.87018e12	−0.283310
$707$	6.95526e12	1.04695
$708$	4.14415e12	0.619848
$709$	9.24886e12	1.37461	0.687306	−	0.726368i	$-0.258793\pi$
$709$	9.24886e12	1.37461	0.687306	+	0.726368i	$0.258793\pi$
$710$	−1.37863e12	−0.203604
$711$	−5.89924e12	−0.865731
$712$	−1.02226e12	−0.149074
$713$	8.05833e12	1.16773
$714$	2.40318e11	0.0346054
$715$	0	0
$716$	1.18083e13	1.67911
$717$	6.47035e12	0.914306
$718$	2.22641e12	0.312640
$719$	−6.22992e12	−0.869365	−0.434683	−	0.900584i	$-0.643139\pi$
$719$	−6.22992e12	−0.869365	−0.434683	+	0.900584i	$0.643139\pi$
$720$	−3.10519e12	−0.430618
$721$	−6.05917e12	−0.835034
$722$	−1.67012e12	−0.228733
$723$	−1.72577e12	−0.234888
$724$	−7.98767e12	−1.08043
$725$	−1.48143e12	−0.199141
$726$	−8.85507e11	−0.118298
$727$	−9.64911e12	−1.28110	−0.640549	−	0.767917i	$-0.721293\pi$
$727$	−9.64911e12	−1.28110	−0.640549	+	0.767917i	$0.721293\pi$
$728$	0	0
$729$	5.90325e12	0.774135
$730$	1.18623e12	0.154602
$731$	2.54117e11	0.0329159
$732$	−3.43571e12	−0.442300
$733$	−5.05649e12	−0.646965	−0.323483	−	0.946234i	$-0.604854\pi$
$733$	−5.05649e12	−0.646965	−0.323483	+	0.946234i	$0.604854\pi$
$734$	−1.77155e12	−0.225279
$735$	6.25259e11	0.0790254
$736$	4.19387e12	0.526822
$737$	−1.33763e11	−0.0167006
$738$	3.50001e11	0.0434326
$739$	−2.42102e12	−0.298605	−0.149303	−	0.988792i	$-0.547703\pi$
$739$	−2.42102e12	−0.298605	−0.149303	+	0.988792i	$0.547703\pi$
$740$	9.63538e12	1.18121
$741$	0	0
$742$	−2.72801e12	−0.330391
$743$	7.51012e12	0.904060	0.452030	−	0.892003i	$-0.350700\pi$
$743$	7.51012e12	0.904060	0.452030	+	0.892003i	$0.350700\pi$
$744$	3.10766e12	0.371840
$745$	−2.82280e12	−0.335719
$746$	3.07765e12	0.363827
$747$	−4.68733e12	−0.550786
$748$	8.79300e11	0.102702
$749$	−9.36353e12	−1.08710
$750$	−1.32392e12	−0.152787
$751$	6.27886e12	0.720279	0.360139	−	0.932898i	$-0.382729\pi$
$751$	6.27886e12	0.720279	0.360139	+	0.932898i	$0.382729\pi$
$752$	−9.61005e12	−1.09584
$753$	−1.00621e13	−1.14054
$754$	0	0
$755$	1.97426e12	0.221127
$756$	8.30110e12	0.924246
$757$	1.11936e13	1.23891	0.619455	−	0.785032i	$-0.287354\pi$
$757$	1.11936e13	1.23891	0.619455	+	0.785032i	$0.287354\pi$
$758$	−2.20916e12	−0.243061
$759$	2.21708e12	0.242490
$760$	−4.97246e12	−0.540642
$761$	1.51704e12	0.163970	0.0819852	−	0.996634i	$-0.473874\pi$
$761$	1.51704e12	0.163970	0.0819852	+	0.996634i	$0.473874\pi$
$762$	2.33564e12	0.250962
$763$	−1.51449e13	−1.61773
$764$	2.25091e12	0.239022
$765$	1.22577e12	0.129400
$766$	2.38532e11	0.0250332
$767$	0	0
$768$	−3.71390e12	−0.385217
$769$	−1.39392e13	−1.43737	−0.718686	−	0.695335i	$-0.755256\pi$
$769$	−1.39392e13	−1.43737	−0.718686	+	0.695335i	$0.755256\pi$
$770$	7.22654e11	0.0740836
$771$	1.84546e12	0.188087
$772$	5.47371e12	0.554631
$773$	−7.63028e11	−0.0768657	−0.0384329	−	0.999261i	$-0.512237\pi$
$773$	−7.63028e11	−0.0768657	−0.0384329	+	0.999261i	$0.512237\pi$
$774$	−1.48761e11	−0.0148989
$775$	2.51977e12	0.250901
$776$	3.75284e12	0.371520
$777$	−8.70540e12	−0.856829
$778$	−4.68623e11	−0.0458580
$779$	−5.45840e12	−0.531064
$780$	0	0
$781$	4.55743e12	0.438319
$782$	−5.08157e11	−0.0485923
$783$	1.15594e13	1.09902
$784$	−1.20013e12	−0.113450
$785$	1.24700e13	1.17207
$786$	2.83562e12	0.265001
$787$	6.74913e12	0.627136	0.313568	−	0.949566i	$-0.398476\pi$
$787$	6.74913e12	0.627136	0.313568	+	0.949566i	$0.398476\pi$
$788$	−1.10112e13	−1.01734
$789$	1.92050e12	0.176428
$790$	−3.29823e12	−0.301272
$791$	1.57114e13	1.42699
$792$	−1.05401e12	−0.0951882
$793$	0	0
$794$	−2.42889e12	−0.216878
$795$	1.12874e13	1.00217
$796$	1.47343e12	0.130083
$797$	−4.49809e12	−0.394880	−0.197440	−	0.980315i	$-0.563263\pi$
$797$	−4.49809e12	−0.394880	−0.197440	+	0.980315i	$0.563263\pi$
$798$	2.19400e12	0.191524
$799$	3.79355e12	0.329295
$800$	1.31139e12	0.113195
$801$	2.30065e12	0.197472
$802$	2.13377e12	0.182122
$803$	−3.92137e12	−0.332827
$804$	−3.05588e11	−0.0257919
$805$	8.76587e12	0.735722
$806$	0	0
$807$	−1.24146e13	−1.03039
$808$	−5.67172e12	−0.468127
$809$	−2.23154e13	−1.83162	−0.915811	−	0.401610i	$-0.868451\pi$
$809$	−2.23154e13	−1.83162	−0.915811	+	0.401610i	$0.868451\pi$
$810$	3.36703e11	0.0274830
$811$	9.05043e12	0.734641	0.367321	−	0.930094i	$-0.380275\pi$
$811$	9.05043e12	0.734641	0.367321	+	0.930094i	$0.380275\pi$
$812$	1.16623e13	0.941417
$813$	1.21013e13	0.971459
$814$	1.51753e12	0.121151
$815$	2.58307e12	0.205082
$816$	1.90853e12	0.150693
$817$	2.31998e12	0.182174
$818$	1.61755e12	0.126319
$819$	0	0
$820$	−4.10732e12	−0.317246
$821$	−7.09709e12	−0.545175	−0.272588	−	0.962131i	$-0.587879\pi$
$821$	−7.09709e12	−0.545175	−0.272588	+	0.962131i	$0.587879\pi$
$822$	2.04050e12	0.155889
$823$	2.20210e12	0.167316	0.0836582	−	0.996495i	$-0.473340\pi$
$823$	2.20210e12	0.167316	0.0836582	+	0.996495i	$0.473340\pi$
$824$	4.94099e12	0.373372
$825$	6.93262e11	0.0521020
$826$	2.58057e12	0.192888
$827$	−4.53701e12	−0.337283	−0.168642	−	0.985677i	$-0.553938\pi$
$827$	−4.53701e12	−0.337283	−0.168642	+	0.985677i	$0.553938\pi$
$828$	−6.24392e12	−0.461658
$829$	2.38397e13	1.75309	0.876546	−	0.481319i	$-0.159842\pi$
$829$	2.38397e13	1.75309	0.876546	+	0.481319i	$0.159842\pi$
$830$	−2.62066e12	−0.191672
$831$	9.99945e11	0.0727397
$832$	0	0
$833$	4.73748e11	0.0340914
$834$	−1.32260e12	−0.0946635
$835$	−5.15009e12	−0.366628
$836$	8.02765e12	0.568408
$837$	−1.96614e13	−1.38468
$838$	−3.70074e12	−0.259233
$839$	−2.47153e13	−1.72202	−0.861009	−	0.508590i	$-0.830167\pi$
$839$	−2.47153e13	−1.72202	−0.861009	+	0.508590i	$0.830167\pi$
$840$	3.38052e12	0.234276
$841$	1.73271e12	0.119438
$842$	−2.45818e12	−0.168542
$843$	2.35945e12	0.160912
$844$	1.16377e12	0.0789455
$845$	0	0
$846$	−2.22076e12	−0.149051
$847$	1.15738e13	0.772683
$848$	−2.16651e13	−1.43873
$849$	−1.12468e12	−0.0742922
$850$	−1.58896e11	−0.0104407
$851$	1.84078e13	1.20315
$852$	1.04117e13	0.676927
$853$	−1.14224e13	−0.738732	−0.369366	−	0.929284i	$-0.620425\pi$
$853$	−1.14224e13	−0.738732	−0.369366	+	0.929284i	$0.620425\pi$
$854$	−2.13943e12	−0.137638
$855$	1.11908e13	0.716165
$856$	7.63556e12	0.486081
$857$	−1.57250e13	−0.995811	−0.497906	−	0.867231i	$-0.665897\pi$
$857$	−1.57250e13	−0.995811	−0.497906	+	0.867231i	$0.665897\pi$
$858$	0	0
$859$	−3.30451e11	−0.0207080	−0.0103540	−	0.999946i	$-0.503296\pi$
$859$	−3.30451e11	−0.0207080	−0.0103540	+	0.999946i	$0.503296\pi$
$860$	1.74573e12	0.108826
$861$	3.71089e12	0.230125
$862$	−3.69082e12	−0.227688
$863$	2.92581e12	0.179555	0.0897774	−	0.995962i	$-0.471384\pi$
$863$	2.92581e12	0.179555	0.0897774	+	0.995962i	$0.471384\pi$
$864$	−1.02325e13	−0.624700
$865$	−1.83537e13	−1.11468
$866$	1.41585e12	0.0855432
$867$	1.03810e13	0.623954
$868$	−1.98365e13	−1.18611
$869$	1.09032e13	0.648579
$870$	2.29894e12	0.136048
$871$	0	0
$872$	1.23500e13	0.723340
$873$	−8.44595e12	−0.492135
$874$	−4.63927e12	−0.268935
$875$	1.73040e13	0.997955
$876$	−8.95855e12	−0.514007
$877$	1.03579e13	0.591251	0.295626	−	0.955304i	$-0.404472\pi$
$877$	1.03579e13	0.591251	0.295626	+	0.955304i	$0.404472\pi$
$878$	4.05348e12	0.230199
$879$	1.35983e13	0.768309
$880$	5.73911e12	0.322606
$881$	−3.38326e13	−1.89210	−0.946048	−	0.324026i	$-0.894963\pi$
$881$	−3.38326e13	−1.89210	−0.946048	+	0.324026i	$0.894963\pi$
$882$	−2.77334e11	−0.0154310
$883$	1.58243e13	0.875992	0.437996	−	0.898977i	$-0.355688\pi$
$883$	1.58243e13	0.875992	0.437996	+	0.898977i	$0.355688\pi$
$884$	0	0
$885$	−1.06773e13	−0.585084
$886$	−4.35755e12	−0.237570
$887$	9.51456e12	0.516098	0.258049	−	0.966132i	$-0.416920\pi$
$887$	9.51456e12	0.516098	0.258049	+	0.966132i	$0.416920\pi$
$888$	7.09888e12	0.383117
$889$	−3.05274e13	−1.63920
$890$	1.28628e12	0.0687196
$891$	−1.11306e12	−0.0591654
$892$	1.36630e13	0.722610
$893$	3.46336e13	1.82249
$894$	−1.01566e12	−0.0531775
$895$	−3.04240e13	−1.58494
$896$	−1.36434e13	−0.707193
$897$	0	0
$898$	−3.54557e12	−0.181946
$899$	−2.76225e13	−1.41040
$900$	−1.95242e12	−0.0991932
$901$	8.55226e12	0.432334
$902$	−6.46883e11	−0.0325384
$903$	−1.57724e12	−0.0789410
$904$	−1.28120e13	−0.638056
$905$	2.05801e13	1.01983
$906$	7.10347e11	0.0350262
$907$	1.55462e13	0.762764	0.381382	−	0.924417i	$-0.375448\pi$
$907$	1.55462e13	0.762764	0.381382	+	0.924417i	$0.375448\pi$
$908$	1.03745e13	0.506504
$909$	1.27645e13	0.620107
$910$	0	0
$911$	2.23975e13	1.07737	0.538687	−	0.842506i	$-0.318921\pi$
$911$	2.23975e13	1.07737	0.538687	+	0.842506i	$0.318921\pi$
$912$	1.74241e13	0.834017
$913$	8.66326e12	0.412632
$914$	−5.51789e12	−0.261526
$915$	8.85207e12	0.417494
$916$	−3.41817e13	−1.60422
$917$	−3.70624e13	−1.73090
$918$	1.23984e12	0.0576202
$919$	−2.37718e13	−1.09937	−0.549683	−	0.835373i	$-0.685252\pi$
$919$	−2.37718e13	−1.09937	−0.549683	+	0.835373i	$0.685252\pi$
$920$	−7.14819e12	−0.328966
$921$	−2.09197e13	−0.958047
$922$	−2.30819e12	−0.105192
$923$	0	0
$924$	−5.45759e12	−0.246307
$925$	5.75595e12	0.258511
$926$	−1.74789e12	−0.0781205
$927$	−1.11200e13	−0.494589
$928$	−1.43758e13	−0.636306
$929$	−5.22801e12	−0.230285	−0.115143	−	0.993349i	$-0.536732\pi$
$929$	−5.22801e12	−0.230285	−0.115143	+	0.993349i	$0.536732\pi$
$930$	−3.91028e12	−0.171409
$931$	4.32512e12	0.188680
$932$	2.77431e13	1.20443
$933$	−1.64353e12	−0.0710086
$934$	2.04926e12	0.0881121
$935$	−2.26551e12	−0.0969422
$936$	0	0
$937$	−4.08690e13	−1.73207	−0.866035	−	0.499983i	$-0.833340\pi$
$937$	−4.08690e13	−1.73207	−0.866035	+	0.499983i	$0.833340\pi$
$938$	−1.90290e11	−0.00802608
$939$	−9.25034e12	−0.388296
$940$	2.60610e13	1.08872
$941$	−3.10598e13	−1.29136	−0.645678	−	0.763610i	$-0.723425\pi$
$941$	−3.10598e13	−1.29136	−0.645678	+	0.763610i	$0.723425\pi$
$942$	4.48677e12	0.185654
$943$	−7.84676e12	−0.323138
$944$	2.04942e13	0.839955
$945$	−2.13877e13	−0.872410
$946$	2.74944e11	0.0111618
$947$	−3.25313e13	−1.31440	−0.657199	−	0.753717i	$-0.728259\pi$
$947$	−3.25313e13	−1.31440	−0.657199	+	0.753717i	$0.728259\pi$
$948$	2.49087e13	1.00165
$949$	0	0
$950$	−1.45066e12	−0.0577842
$951$	−2.27957e13	−0.903733
$952$	2.56136e12	0.101066
$953$	−5.19426e12	−0.203989	−0.101994	−	0.994785i	$-0.532522\pi$
$953$	−5.19426e12	−0.203989	−0.101994	+	0.994785i	$0.532522\pi$
$954$	−5.00652e12	−0.195690
$955$	−5.79944e12	−0.225616
$956$	3.36790e13	1.30406
$957$	−7.59974e12	−0.292884
$958$	−7.27764e12	−0.279155
$959$	−2.66699e13	−1.01821
$960$	1.17008e13	0.444628
$961$	2.05436e13	0.776999
$962$	0	0
$963$	−1.71842e13	−0.643890
$964$	−8.98286e12	−0.335018
$965$	−1.41030e13	−0.523525
$966$	3.15400e12	0.116537
$967$	−3.73620e13	−1.37408	−0.687039	−	0.726620i	$-0.741090\pi$
$967$	−3.73620e13	−1.37408	−0.687039	+	0.726620i	$0.741090\pi$
$968$	−9.43795e12	−0.345492
$969$	−6.87815e12	−0.250620
$970$	−4.72208e12	−0.171262
$971$	3.68002e13	1.32851	0.664254	−	0.747507i	$-0.268749\pi$
$971$	3.68002e13	1.32851	0.664254	+	0.747507i	$0.268749\pi$
$972$	2.50496e13	0.900126
$973$	1.72868e13	0.618310
$974$	−8.69735e11	−0.0309650
$975$	0	0
$976$	−1.69907e13	−0.599360
$977$	−2.41910e13	−0.849430	−0.424715	−	0.905327i	$-0.639626\pi$
$977$	−2.41910e13	−0.849430	−0.424715	+	0.905327i	$0.639626\pi$
$978$	9.29402e11	0.0324847
$979$	−4.25213e12	−0.147940
$980$	3.25455e12	0.112713
$981$	−2.77943e13	−0.958176
$982$	6.47424e12	0.222171
$983$	−1.73443e13	−0.592468	−0.296234	−	0.955115i	$-0.595731\pi$
$983$	−1.73443e13	−0.592468	−0.296234	+	0.955115i	$0.595731\pi$
$984$	−3.02607e12	−0.102897
$985$	2.83703e13	0.960287
$986$	1.74187e12	0.0586907
$987$	−2.35456e13	−0.789738
$988$	0	0
$989$	3.33511e12	0.110848
$990$	1.32624e12	0.0438796
$991$	−1.10802e13	−0.364936	−0.182468	−	0.983212i	$-0.558409\pi$
$991$	−1.10802e13	−0.364936	−0.182468	+	0.983212i	$0.558409\pi$
$992$	2.44519e13	0.801696
$993$	−9.91920e12	−0.323747
$994$	6.48337e12	0.210650
$995$	−3.79627e12	−0.122787
$996$	1.97916e13	0.637256
$997$	9.37866e12	0.300616	0.150308	−	0.988639i	$-0.451973\pi$
$997$	9.37866e12	0.300616	0.150308	+	0.988639i	$0.451973\pi$
$998$	2.75326e12	0.0878536
$999$	−4.49128e13	−1.42668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.h.1.12	yes	27
13.12	even	2		169.10.a.g.1.16	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.10.a.g.1.16	✓	27	13.12	even	2
169.10.a.h.1.12	yes	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-4.82532 q^{2} -93.8913 q^{3} -488.716 q^{4} +1259.17 q^{5} +453.056 q^{6} -5921.56 q^{7} +4828.78 q^{8} -10867.4 q^{9} -6075.91 q^{10} +20085.5 q^{11} +45886.2 q^{12} +28573.5 q^{14} -118225. q^{15} +226922. q^{16} -89577.2 q^{17} +52438.9 q^{18} -817804. q^{19} -615378. q^{20} +555983. q^{21} -96919.0 q^{22} -1.17564e6 q^{23} -453380. q^{24} -367612. q^{25} +2.86842e6 q^{27} +2.89396e6 q^{28} +4.02987e6 q^{29} +570475. q^{30} -6.85443e6 q^{31} -3.56731e6 q^{32} -1.88585e6 q^{33} +432239. q^{34} -7.45626e6 q^{35} +5.31109e6 q^{36} -1.56577e7 q^{37} +3.94617e6 q^{38} +6.08026e6 q^{40} +6.67447e6 q^{41} -2.68280e6 q^{42} -2.83685e6 q^{43} -9.81611e6 q^{44} -1.36840e7 q^{45} +5.67284e6 q^{46} -4.23495e7 q^{47} -2.13060e7 q^{48} -5.28871e6 q^{49} +1.77385e6 q^{50} +8.41052e6 q^{51} -9.54736e7 q^{53} -1.38411e7 q^{54} +2.52911e7 q^{55} -2.85939e7 q^{56} +7.67846e7 q^{57} -1.94454e7 q^{58} +9.03136e7 q^{59} +5.77786e7 q^{60} -7.48746e7 q^{61} +3.30748e7 q^{62} +6.43521e7 q^{63} -9.89708e7 q^{64} +9.09985e6 q^{66} -6.65968e6 q^{67} +4.37778e7 q^{68} +1.10382e8 q^{69} +3.59789e7 q^{70} +2.26902e8 q^{71} -5.24764e7 q^{72} -1.95234e8 q^{73} +7.55534e7 q^{74} +3.45156e7 q^{75} +3.99674e8 q^{76} -1.18937e8 q^{77} +5.42837e8 q^{79} +2.85734e8 q^{80} -5.54160e7 q^{81} -3.22065e7 q^{82} +4.31319e8 q^{83} -2.71718e8 q^{84} -1.12793e8 q^{85} +1.36887e7 q^{86} -3.78370e8 q^{87} +9.69884e7 q^{88} -2.11702e8 q^{89} +6.60295e7 q^{90} +5.74554e8 q^{92} +6.43571e8 q^{93} +2.04350e8 q^{94} -1.02975e9 q^{95} +3.34939e8 q^{96} +7.77181e8 q^{97} +2.55197e7 q^{98} -2.18278e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 65 q^{2} + q^{3} + 7169 q^{4} + 3238 q^{5} + 8490 q^{6} + 17378 q^{7} + 54204 q^{8} + 191118 q^{9} + 11697 q^{10} + 164171 q^{11} - 181941 q^{12} - 77651 q^{14} + 614110 q^{15} + 3012565 q^{16}+ \cdots + 5866875443 q^{99}+O(q^{100}) \) 27 * q + 65 * q^2 + q^3 + 7169 * q^4 + 3238 * q^5 + 8490 * q^6 + 17378 * q^7 + 54204 * q^8 + 191118 * q^9 + 11697 * q^10 + 164171 * q^11 - 181941 * q^12 - 77651 * q^14 + 614110 * q^15 + 3012565 * q^16 + 105667 * q^17 + 584849 * q^18 + 2140031 * q^19 + 3722382 * q^20 + 2953252 * q^21 + 3789362 * q^22 + 3586420 * q^23 + 6334338 * q^24 + 10278737 * q^25 - 4332980 * q^27 + 5293616 * q^28 - 4203616 * q^29 - 34788079 * q^30 + 18634294 * q^31 + 38778128 * q^32 + 15170356 * q^33 + 10546914 * q^34 + 16260796 * q^35 + 112071987 * q^36 + 7214770 * q^37 - 60128169 * q^38 - 46270051 * q^40 + 67544533 * q^41 - 31249012 * q^42 - 55585357 * q^43 + 246583161 * q^44 + 172236942 * q^45 + 168715272 * q^46 + 175569972 * q^47 - 48573948 * q^48 + 144680057 * q^49 + 3455750 * q^50 + 122191382 * q^51 + 105504478 * q^53 - 359394409 * q^54 - 2471426 * q^55 + 260765426 * q^56 - 307372146 * q^57 - 777873375 * q^58 + 267562255 * q^59 - 745460915 * q^60 - 371334848 * q^61 + 52175323 * q^62 + 431551952 * q^63 + 642079732 * q^64 + 973549503 * q^66 + 213848603 * q^67 - 278629116 * q^68 - 201174310 * q^69 - 122562066 * q^70 + 970709202 * q^71 + 773619646 * q^72 + 271733895 * q^73 - 1068711482 * q^74 + 1236853937 * q^75 + 2585634844 * q^76 - 244285066 * q^77 + 1052001020 * q^79 + 3293699123 * q^80 - 456846001 * q^81 - 3832425 * q^82 + 1491498691 * q^83 + 5153327787 * q^84 + 2713007958 * q^85 + 3421126005 * q^86 - 1622618562 * q^87 + 6574070508 * q^88 + 4616501291 * q^89 - 2822436128 * q^90 + 3684890562 * q^92 + 3324407374 * q^93 - 5582614303 * q^94 - 2892286998 * q^95 + 12902380548 * q^96 + 3944038157 * q^97 + 7822213576 * q^98 + 5866875443 * q^99

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