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Properties

Label 39.4.a.b.1.1
Level $39$
Weight $4$
Character 39.1
Self dual yes
Analytic conductor $2.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [39,4,Mod(1,39)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("39.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(39, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.74166\) of defining polynomial
Character \(\chi\) \(=\) 39.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.74166 q^{2} -3.00000 q^{3} -0.483315 q^{4} +19.4833 q^{5} +8.22497 q^{6} +7.48331 q^{7} +23.2583 q^{8} +9.00000 q^{9} -53.4166 q^{10} +22.8999 q^{11} +1.44994 q^{12} -13.0000 q^{13} -20.5167 q^{14} -58.4499 q^{15} -59.8999 q^{16} +67.0334 q^{17} -24.6749 q^{18} +16.5167 q^{19} -9.41657 q^{20} -22.4499 q^{21} -62.7836 q^{22} -175.600 q^{23} -69.7750 q^{24} +254.600 q^{25} +35.6415 q^{26} -27.0000 q^{27} -3.61680 q^{28} +291.800 q^{29} +160.250 q^{30} +117.283 q^{31} -21.8418 q^{32} -68.6997 q^{33} -183.783 q^{34} +145.800 q^{35} -4.34983 q^{36} -154.766 q^{37} -45.2831 q^{38} +39.0000 q^{39} +453.150 q^{40} -251.716 q^{41} +61.5501 q^{42} -502.566 q^{43} -11.0679 q^{44} +175.350 q^{45} +481.434 q^{46} -281.733 q^{47} +179.700 q^{48} -287.000 q^{49} -698.025 q^{50} -201.100 q^{51} +6.28309 q^{52} +366.999 q^{53} +74.0247 q^{54} +446.166 q^{55} +174.049 q^{56} -49.5501 q^{57} -800.015 q^{58} -79.6663 q^{59} +28.2497 q^{60} -194.865 q^{61} -321.550 q^{62} +67.3498 q^{63} +539.082 q^{64} -253.283 q^{65} +188.351 q^{66} +400.082 q^{67} -32.3982 q^{68} +526.799 q^{69} -399.733 q^{70} +528.299 q^{71} +209.325 q^{72} -734.366 q^{73} +424.316 q^{74} -763.799 q^{75} -7.98276 q^{76} +171.367 q^{77} -106.925 q^{78} +113.266 q^{79} -1167.05 q^{80} +81.0000 q^{81} +690.118 q^{82} -933.466 q^{83} +10.8504 q^{84} +1306.03 q^{85} +1377.86 q^{86} -875.399 q^{87} +532.613 q^{88} +1190.91 q^{89} -480.749 q^{90} -97.2831 q^{91} +84.8699 q^{92} -351.849 q^{93} +772.415 q^{94} +321.800 q^{95} +65.5253 q^{96} +557.165 q^{97} +786.856 q^{98} +206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20}+ \cdots - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74166 −0.969322 −0.484661 − 0.874702i \(-0.661057\pi\)
−0.484661 + 0.874702i \(0.661057\pi\)
\(3\) −3.00000 −0.577350
\(4\) −0.483315 −0.0604143
\(5\) 19.4833 1.74264 0.871320 − 0.490715i \(-0.163264\pi\)
0.871320 + 0.490715i \(0.163264\pi\)
\(6\) 8.22497 0.559638
\(7\) 7.48331 0.404061 0.202031 − 0.979379i \(-0.435246\pi\)
0.202031 + 0.979379i \(0.435246\pi\)
\(8\) 23.2583 1.02788
\(9\) 9.00000 0.333333
\(10\) −53.4166 −1.68918
\(11\) 22.8999 0.627689 0.313844 − 0.949474i \(-0.398383\pi\)
0.313844 + 0.949474i \(0.398383\pi\)
\(12\) 1.44994 0.0348802
\(13\) −13.0000 −0.277350
\(14\) −20.5167 −0.391665
\(15\) −58.4499 −1.00611
\(16\) −59.8999 −0.935936
\(17\) 67.0334 0.956352 0.478176 − 0.878264i \(-0.341298\pi\)
0.478176 + 0.878264i \(0.341298\pi\)
\(18\) −24.6749 −0.323107
\(19\) 16.5167 0.199431 0.0997155 − 0.995016i \(-0.468207\pi\)
0.0997155 + 0.995016i \(0.468207\pi\)
\(20\) −9.41657 −0.105280
\(21\) −22.4499 −0.233285
\(22\) −62.7836 −0.608433
\(23\) −175.600 −1.59196 −0.795979 − 0.605324i \(-0.793044\pi\)
−0.795979 + 0.605324i \(0.793044\pi\)
\(24\) −69.7750 −0.593449
\(25\) 254.600 2.03680
\(26\) 35.6415 0.268842
\(27\) −27.0000 −0.192450
\(28\) −3.61680 −0.0244111
\(29\) 291.800 1.86848 0.934239 − 0.356648i \(-0.116080\pi\)
0.934239 + 0.356648i \(0.116080\pi\)
\(30\) 160.250 0.975249
\(31\) 117.283 0.679505 0.339753 − 0.940515i \(-0.389657\pi\)
0.339753 + 0.940515i \(0.389657\pi\)
\(32\) −21.8418 −0.120660
\(33\) −68.6997 −0.362396
\(34\) −183.783 −0.927013
\(35\) 145.800 0.704133
\(36\) −4.34983 −0.0201381
\(37\) −154.766 −0.687661 −0.343830 − 0.939032i \(-0.611724\pi\)
−0.343830 + 0.939032i \(0.611724\pi\)
\(38\) −45.2831 −0.193313
\(39\) 39.0000 0.160128
\(40\) 453.150 1.79123
\(41\) −251.716 −0.958815 −0.479407 − 0.877592i \(-0.659148\pi\)
−0.479407 + 0.877592i \(0.659148\pi\)
\(42\) 61.5501 0.226128
\(43\) −502.566 −1.78234 −0.891170 − 0.453669i \(-0.850115\pi\)
−0.891170 + 0.453669i \(0.850115\pi\)
\(44\) −11.0679 −0.0379214
\(45\) 175.350 0.580880
\(46\) 481.434 1.54312
\(47\) −281.733 −0.874361 −0.437181 − 0.899374i \(-0.644023\pi\)
−0.437181 + 0.899374i \(0.644023\pi\)
\(48\) 179.700 0.540363
\(49\) −287.000 −0.836735
\(50\) −698.025 −1.97431
\(51\) −201.100 −0.552150
\(52\) 6.28309 0.0167559
\(53\) 366.999 0.951154 0.475577 − 0.879674i \(-0.342239\pi\)
0.475577 + 0.879674i \(0.342239\pi\)
\(54\) 74.0247 0.186546
\(55\) 446.166 1.09384
\(56\) 174.049 0.415328
\(57\) −49.5501 −0.115141
\(58\) −800.015 −1.81116
\(59\) −79.6663 −0.175791 −0.0878955 − 0.996130i \(-0.528014\pi\)
−0.0878955 + 0.996130i \(0.528014\pi\)
\(60\) 28.2497 0.0607837
\(61\) −194.865 −0.409016 −0.204508 − 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) −321.550 −0.658660
\(63\) 67.3498 0.134687
\(64\) 539.082 1.05289
\(65\) −253.283 −0.483322
\(66\) 188.351 0.351279
\(67\) 400.082 0.729519 0.364759 − 0.931102i \(-0.381151\pi\)
0.364759 + 0.931102i \(0.381151\pi\)
\(68\) −32.3982 −0.0577774
\(69\) 526.799 0.919117
\(70\) −399.733 −0.682532
\(71\) 528.299 0.883065 0.441532 − 0.897245i \(-0.354435\pi\)
0.441532 + 0.897245i \(0.354435\pi\)
\(72\) 209.325 0.342628
\(73\) −734.366 −1.17741 −0.588706 − 0.808347i \(-0.700362\pi\)
−0.588706 + 0.808347i \(0.700362\pi\)
\(74\) 424.316 0.666565
\(75\) −763.799 −1.17594
\(76\) −7.98276 −0.0120485
\(77\) 171.367 0.253625
\(78\) −106.925 −0.155216
\(79\) 113.266 0.161309 0.0806545 − 0.996742i \(-0.474299\pi\)
0.0806545 + 0.996742i \(0.474299\pi\)
\(80\) −1167.05 −1.63100
\(81\) 81.0000 0.111111
\(82\) 690.118 0.929400
\(83\) −933.466 −1.23447 −0.617236 − 0.786778i \(-0.711748\pi\)
−0.617236 + 0.786778i \(0.711748\pi\)
\(84\) 10.8504 0.0140937
\(85\) 1306.03 1.66658
\(86\) 1377.86 1.72766
\(87\) −875.399 −1.07877
\(88\) 532.613 0.645191
\(89\) 1190.91 1.41839 0.709195 − 0.705012i \(-0.249059\pi\)
0.709195 + 0.705012i \(0.249059\pi\)
\(90\) −480.749 −0.563060
\(91\) −97.2831 −0.112066
\(92\) 84.8699 0.0961771
\(93\) −351.849 −0.392313
\(94\) 772.415 0.847538
\(95\) 321.800 0.347536
\(96\) 65.5253 0.0696630
\(97\) 557.165 0.583211 0.291606 − 0.956539i \(-0.405811\pi\)
0.291606 + 0.956539i \(0.405811\pi\)
\(98\) 786.856 0.811066
\(99\) 206.099 0.209230
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 39.4.a.b.1.1 ✓ 2
3.2 odd 2 117.4.a.c.1.2 2
4.3 odd 2 624.4.a.r.1.2 2
5.4 even 2 975.4.a.j.1.2 2
7.6 odd 2 1911.4.a.h.1.1 2
8.3 odd 2 2496.4.a.s.1.1 2
8.5 even 2 2496.4.a.bc.1.1 2
12.11 even 2 1872.4.a.t.1.1 2
13.5 odd 4 507.4.b.f.337.3 4
13.8 odd 4 507.4.b.f.337.2 4
13.12 even 2 507.4.a.f.1.2 2
39.38 odd 2 1521.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 ✓ 2 1.1 even 1 trivial
117.4.a.c.1.2 2 3.2 odd 2
507.4.a.f.1.2 2 13.12 even 2
507.4.b.f.337.2 4 13.8 odd 4
507.4.b.f.337.3 4 13.5 odd 4
624.4.a.r.1.2 2 4.3 odd 2
975.4.a.j.1.2 2 5.4 even 2
1521.4.a.s.1.1 2 39.38 odd 2
1872.4.a.t.1.1 2 12.11 even 2
1911.4.a.h.1.1 2 7.6 odd 2
2496.4.a.s.1.1 2 8.3 odd 2
2496.4.a.bc.1.1 2 8.5 even 2