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Properties

Label 126.16.a.e.1.1
Level $126$
Weight $16$
Character 126.1
Self dual yes
Analytic conductor $179.794$
Analytic rank $1$
Dimension $1$
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] = [126,16,Mod(1,126)] 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("126.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,128,0,16384,81060] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(179.793816426\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 126.1

$q$-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)
 
\(f(q)\) \(=\) \(q+128.000 q^{2} +16384.0 q^{4} +81060.0 q^{5} +823543. q^{7} +2.09715e6 q^{8} +1.03757e7 q^{10} -7.01212e7 q^{11} +1.51470e8 q^{13} +1.05414e8 q^{14} +2.68435e8 q^{16} +2.49757e8 q^{17} -6.47686e9 q^{19} +1.32809e9 q^{20} -8.97551e9 q^{22} +2.11292e10 q^{23} -2.39469e10 q^{25} +1.93881e10 q^{26} +1.34929e10 q^{28} -7.79483e9 q^{29} -9.50321e10 q^{31} +3.43597e10 q^{32} +3.19688e10 q^{34} +6.67564e10 q^{35} -8.70082e11 q^{37} -8.29038e11 q^{38} +1.69995e11 q^{40} -1.00767e12 q^{41} +1.55008e11 q^{43} -1.14887e12 q^{44} +2.70454e12 q^{46} +2.55197e12 q^{47} +6.78223e11 q^{49} -3.06520e12 q^{50} +2.48168e12 q^{52} -4.04765e12 q^{53} -5.68402e12 q^{55} +1.72709e12 q^{56} -9.97738e11 q^{58} +1.25992e13 q^{59} -3.99250e13 q^{61} -1.21641e13 q^{62} +4.39805e12 q^{64} +1.22781e13 q^{65} -4.84238e13 q^{67} +4.09201e12 q^{68} +8.54482e12 q^{70} -3.76931e13 q^{71} +1.41416e14 q^{73} -1.11371e14 q^{74} -1.06117e14 q^{76} -5.77478e13 q^{77} +2.47021e14 q^{79} +2.17594e13 q^{80} -1.28981e14 q^{82} -2.78879e12 q^{83} +2.02453e13 q^{85} +1.98410e13 q^{86} -1.47055e14 q^{88} +5.83963e12 q^{89} +1.24742e14 q^{91} +3.46181e14 q^{92} +3.26652e14 q^{94} -5.25014e14 q^{95} +2.78027e14 q^{97} +8.68126e13 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 128.000 0.707107
\(3\) 0 0
\(4\) 16384.0 0.500000
\(5\) 81060.0 0.464015 0.232007 − 0.972714i \(-0.425471\pi\)
0.232007 + 0.972714i \(0.425471\pi\)
\(6\) 0 0
\(7\) 823543. 0.377964
\(8\) 2.09715e6 0.353553
\(9\) 0 0
\(10\) 1.03757e7 0.328108
\(11\) −7.01212e7 −1.08494 −0.542468 − 0.840076i \(-0.682510\pi\)
−0.542468 + 0.840076i \(0.682510\pi\)
\(12\) 0 0
\(13\) 1.51470e8 0.669499 0.334750 − 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(14\) 1.05414e8 0.267261
\(15\) 0 0
\(16\) 2.68435e8 0.250000
\(17\) 2.49757e8 0.147622 0.0738108 − 0.997272i \(-0.476484\pi\)
0.0738108 + 0.997272i \(0.476484\pi\)
\(18\) 0 0
\(19\) −6.47686e9 −1.66231 −0.831155 − 0.556040i \(-0.812320\pi\)
−0.831155 + 0.556040i \(0.812320\pi\)
\(20\) 1.32809e9 0.232007
\(21\) 0 0
\(22\) −8.97551e9 −0.767166
\(23\) 2.11292e10 1.29397 0.646985 − 0.762503i \(-0.276030\pi\)
0.646985 + 0.762503i \(0.276030\pi\)
\(24\) 0 0
\(25\) −2.39469e10 −0.784691
\(26\) 1.93881e10 0.473408
\(27\) 0 0
\(28\) 1.34929e10 0.188982
\(29\) −7.79483e9 −0.0839115 −0.0419557 − 0.999119i \(-0.513359\pi\)
−0.0419557 + 0.999119i \(0.513359\pi\)
\(30\) 0 0
\(31\) −9.50321e10 −0.620379 −0.310190 − 0.950675i \(-0.600393\pi\)
−0.310190 + 0.950675i \(0.600393\pi\)
\(32\) 3.43597e10 0.176777
\(33\) 0 0
\(34\) 3.19688e10 0.104384
\(35\) 6.67564e10 0.175381
\(36\) 0 0
\(37\) −8.70082e11 −1.50677 −0.753386 − 0.657579i \(-0.771581\pi\)
−0.753386 + 0.657579i \(0.771581\pi\)
\(38\) −8.29038e11 −1.17543
\(39\) 0 0
\(40\) 1.69995e11 0.164054
\(41\) −1.00767e12 −0.808049 −0.404025 − 0.914748i \(-0.632389\pi\)
−0.404025 + 0.914748i \(0.632389\pi\)
\(42\) 0 0
\(43\) 1.55008e11 0.0869640 0.0434820 − 0.999054i \(-0.486155\pi\)
0.0434820 + 0.999054i \(0.486155\pi\)
\(44\) −1.14887e12 −0.542468
\(45\) 0 0
\(46\) 2.70454e12 0.914975
\(47\) 2.55197e12 0.734753 0.367377 − 0.930072i \(-0.380256\pi\)
0.367377 + 0.930072i \(0.380256\pi\)
\(48\) 0 0
\(49\) 6.78223e11 0.142857
\(50\) −3.06520e12 −0.554860
\(51\) 0 0
\(52\) 2.48168e12 0.334750
\(53\) −4.04765e12 −0.473297 −0.236648 − 0.971595i \(-0.576049\pi\)
−0.236648 + 0.971595i \(0.576049\pi\)
\(54\) 0 0
\(55\) −5.68402e12 −0.503426
\(56\) 1.72709e12 0.133631
\(57\) 0 0
\(58\) −9.97738e11 −0.0593344
\(59\) 1.25992e13 0.659105 0.329552 − 0.944137i \(-0.393102\pi\)
0.329552 + 0.944137i \(0.393102\pi\)
\(60\) 0 0
\(61\) −3.99250e13 −1.62657 −0.813283 − 0.581869i \(-0.802322\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(62\) −1.21641e13 −0.438675
\(63\) 0 0
\(64\) 4.39805e12 0.125000
\(65\) 1.22781e13 0.310657
\(66\) 0 0
\(67\) −4.84238e13 −0.976108 −0.488054 − 0.872814i \(-0.662293\pi\)
−0.488054 + 0.872814i \(0.662293\pi\)
\(68\) 4.09201e12 0.0738108
\(69\) 0 0
\(70\) 8.54482e12 0.124013
\(71\) −3.76931e13 −0.491840 −0.245920 − 0.969290i \(-0.579090\pi\)
−0.245920 + 0.969290i \(0.579090\pi\)
\(72\) 0 0
\(73\) 1.41416e14 1.49823 0.749114 − 0.662442i \(-0.230480\pi\)
0.749114 + 0.662442i \(0.230480\pi\)
\(74\) −1.11371e14 −1.06545
\(75\) 0 0
\(76\) −1.06117e14 −0.831155
\(77\) −5.77478e13 −0.410067
\(78\) 0 0
\(79\) 2.47021e14 1.44720 0.723602 − 0.690217i \(-0.242485\pi\)
0.723602 + 0.690217i \(0.242485\pi\)
\(80\) 2.17594e13 0.116004
\(81\) 0 0
\(82\) −1.28981e14 −0.571377
\(83\) −2.78879e12 −0.0112805 −0.00564027 − 0.999984i \(-0.501795\pi\)
−0.00564027 + 0.999984i \(0.501795\pi\)
\(84\) 0 0
\(85\) 2.02453e13 0.0684986
\(86\) 1.98410e13 0.0614928
\(87\) 0 0
\(88\) −1.47055e14 −0.383583
\(89\) 5.83963e12 0.0139946 0.00699730 − 0.999976i \(-0.497773\pi\)
0.00699730 + 0.999976i \(0.497773\pi\)
\(90\) 0 0
\(91\) 1.24742e14 0.253047
\(92\) 3.46181e14 0.646985
\(93\) 0 0
\(94\) 3.26652e14 0.519549
\(95\) −5.25014e14 −0.771336
\(96\) 0 0
\(97\) 2.78027e14 0.349381 0.174690 − 0.984623i \(-0.444108\pi\)
0.174690 + 0.984623i \(0.444108\pi\)
\(98\) 8.68126e13 0.101015
\(99\) 0 0
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 126.16.a.e.1.1 1
3.2 odd 2 14.16.a.a.1.1 ✓ 1
12.11 even 2 112.16.a.a.1.1 1
21.2 odd 6 98.16.c.b.67.1 2
21.5 even 6 98.16.c.c.67.1 2
21.11 odd 6 98.16.c.b.79.1 2
21.17 even 6 98.16.c.c.79.1 2
21.20 even 2 98.16.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.16.a.a.1.1 ✓ 1 3.2 odd 2
98.16.a.b.1.1 1 21.20 even 2
98.16.c.b.67.1 2 21.2 odd 6
98.16.c.b.79.1 2 21.11 odd 6
98.16.c.c.67.1 2 21.5 even 6
98.16.c.c.79.1 2 21.17 even 6
112.16.a.a.1.1 1 12.11 even 2
126.16.a.e.1.1 1 1.1 even 1 trivial