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Properties

Label 14.16.a.a.1.1
Level $14$
Weight $16$
Character 14.1
Self dual yes
Analytic conductor $19.977$
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] = [14,16,Mod(1,14)] 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("14.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.9770907140\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 14.1

$q$-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)
 
\(f(q)\) \(=\) \(q-128.000 q^{2} +1350.00 q^{3} +16384.0 q^{4} -81060.0 q^{5} -172800. q^{6} +823543. q^{7} -2.09715e6 q^{8} -1.25264e7 q^{9} +1.03757e7 q^{10} +7.01212e7 q^{11} +2.21184e7 q^{12} +1.51470e8 q^{13} -1.05414e8 q^{14} -1.09431e8 q^{15} +2.68435e8 q^{16} -2.49757e8 q^{17} +1.60338e9 q^{18} -6.47686e9 q^{19} -1.32809e9 q^{20} +1.11178e9 q^{21} -8.97551e9 q^{22} -2.11292e10 q^{23} -2.83116e9 q^{24} -2.39469e10 q^{25} -1.93881e10 q^{26} -3.62817e10 q^{27} +1.34929e10 q^{28} +7.79483e9 q^{29} +1.40072e10 q^{30} -9.50321e10 q^{31} -3.43597e10 q^{32} +9.46636e10 q^{33} +3.19688e10 q^{34} -6.67564e10 q^{35} -2.05233e11 q^{36} -8.70082e11 q^{37} +8.29038e11 q^{38} +2.04484e11 q^{39} +1.69995e11 q^{40} +1.00767e12 q^{41} -1.42308e11 q^{42} +1.55008e11 q^{43} +1.14887e12 q^{44} +1.01539e12 q^{45} +2.70454e12 q^{46} -2.55197e12 q^{47} +3.62388e11 q^{48} +6.78223e11 q^{49} +3.06520e12 q^{50} -3.37171e11 q^{51} +2.48168e12 q^{52} +4.04765e12 q^{53} +4.64405e12 q^{54} -5.68402e12 q^{55} -1.72709e12 q^{56} -8.74376e12 q^{57} -9.97738e11 q^{58} -1.25992e13 q^{59} -1.79292e12 q^{60} -3.99250e13 q^{61} +1.21641e13 q^{62} -1.03160e13 q^{63} +4.39805e12 q^{64} -1.22781e13 q^{65} -1.21169e13 q^{66} -4.84238e13 q^{67} -4.09201e12 q^{68} -2.85244e13 q^{69} +8.54482e12 q^{70} +3.76931e13 q^{71} +2.62698e13 q^{72} +1.41416e14 q^{73} +1.11371e14 q^{74} -3.23283e13 q^{75} -1.06117e14 q^{76} +5.77478e13 q^{77} -2.61739e13 q^{78} +2.47021e14 q^{79} -2.17594e13 q^{80} +1.30760e14 q^{81} -1.28981e14 q^{82} +2.78879e12 q^{83} +1.82155e13 q^{84} +2.02453e13 q^{85} -1.98410e13 q^{86} +1.05230e13 q^{87} -1.47055e14 q^{88} -5.83963e12 q^{89} -1.29970e14 q^{90} +1.24742e14 q^{91} -3.46181e14 q^{92} -1.28293e14 q^{93} +3.26652e14 q^{94} +5.25014e14 q^{95} -4.63856e13 q^{96} +2.78027e14 q^{97} -8.68126e13 q^{98} -8.78366e14 q^{99} +O(q^{100})\)

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
\(3\) 1350.00 0.356389 0.178195 − 0.983995i \(-0.442974\pi\)
0.178195 + 0.983995i \(0.442974\pi\)
\(4\) 16384.0 0.500000
\(5\) −81060.0 −0.464015 −0.232007 − 0.972714i \(-0.574529\pi\)
−0.232007 + 0.972714i \(0.574529\pi\)
\(6\) −172800. −0.252005
\(7\) 823543. 0.377964
\(8\) −2.09715e6 −0.353553
\(9\) −1.25264e7 −0.872987
\(10\) 1.03757e7 0.328108
\(11\) 7.01212e7 1.08494 0.542468 − 0.840076i \(-0.317490\pi\)
0.542468 + 0.840076i \(0.317490\pi\)
\(12\) 2.21184e7 0.178195
\(13\) 1.51470e8 0.669499 0.334750 − 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(14\) −1.05414e8 −0.267261
\(15\) −1.09431e8 −0.165370
\(16\) 2.68435e8 0.250000
\(17\) −2.49757e8 −0.147622 −0.0738108 − 0.997272i \(-0.523516\pi\)
−0.0738108 + 0.997272i \(0.523516\pi\)
\(18\) 1.60338e9 0.617295
\(19\) −6.47686e9 −1.66231 −0.831155 − 0.556040i \(-0.812320\pi\)
−0.831155 + 0.556040i \(0.812320\pi\)
\(20\) −1.32809e9 −0.232007
\(21\) 1.11178e9 0.134702
\(22\) −8.97551e9 −0.767166
\(23\) −2.11292e10 −1.29397 −0.646985 − 0.762503i \(-0.723970\pi\)
−0.646985 + 0.762503i \(0.723970\pi\)
\(24\) −2.83116e9 −0.126003
\(25\) −2.39469e10 −0.784691
\(26\) −1.93881e10 −0.473408
\(27\) −3.62817e10 −0.667512
\(28\) 1.34929e10 0.188982
\(29\) 7.79483e9 0.0839115 0.0419557 − 0.999119i \(-0.486641\pi\)
0.0419557 + 0.999119i \(0.486641\pi\)
\(30\) 1.40072e10 0.116934
\(31\) −9.50321e10 −0.620379 −0.310190 − 0.950675i \(-0.600393\pi\)
−0.310190 + 0.950675i \(0.600393\pi\)
\(32\) −3.43597e10 −0.176777
\(33\) 9.46636e10 0.386659
\(34\) 3.19688e10 0.104384
\(35\) −6.67564e10 −0.175381
\(36\) −2.05233e11 −0.436493
\(37\) −8.70082e11 −1.50677 −0.753386 − 0.657579i \(-0.771581\pi\)
−0.753386 + 0.657579i \(0.771581\pi\)
\(38\) 8.29038e11 1.17543
\(39\) 2.04484e11 0.238602
\(40\) 1.69995e11 0.164054
\(41\) 1.00767e12 0.808049 0.404025 − 0.914748i \(-0.367611\pi\)
0.404025 + 0.914748i \(0.367611\pi\)
\(42\) −1.42308e11 −0.0952490
\(43\) 1.55008e11 0.0869640 0.0434820 − 0.999054i \(-0.486155\pi\)
0.0434820 + 0.999054i \(0.486155\pi\)
\(44\) 1.14887e12 0.542468
\(45\) 1.01539e12 0.405079
\(46\) 2.70454e12 0.914975
\(47\) −2.55197e12 −0.734753 −0.367377 − 0.930072i \(-0.619744\pi\)
−0.367377 + 0.930072i \(0.619744\pi\)
\(48\) 3.62388e11 0.0890973
\(49\) 6.78223e11 0.142857
\(50\) 3.06520e12 0.554860
\(51\) −3.37171e11 −0.0526107
\(52\) 2.48168e12 0.334750
\(53\) 4.04765e12 0.473297 0.236648 − 0.971595i \(-0.423951\pi\)
0.236648 + 0.971595i \(0.423951\pi\)
\(54\) 4.64405e12 0.472002
\(55\) −5.68402e12 −0.503426
\(56\) −1.72709e12 −0.133631
\(57\) −8.74376e12 −0.592429
\(58\) −9.97738e11 −0.0593344
\(59\) −1.25992e13 −0.659105 −0.329552 − 0.944137i \(-0.606898\pi\)
−0.329552 + 0.944137i \(0.606898\pi\)
\(60\) −1.79292e12 −0.0826848
\(61\) −3.99250e13 −1.62657 −0.813283 − 0.581869i \(-0.802322\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(62\) 1.21641e13 0.438675
\(63\) −1.03160e13 −0.329958
\(64\) 4.39805e12 0.125000
\(65\) −1.22781e13 −0.310657
\(66\) −1.21169e13 −0.273409
\(67\) −4.84238e13 −0.976108 −0.488054 − 0.872814i \(-0.662293\pi\)
−0.488054 + 0.872814i \(0.662293\pi\)
\(68\) −4.09201e12 −0.0738108
\(69\) −2.85244e13 −0.461157
\(70\) 8.54482e12 0.124013
\(71\) 3.76931e13 0.491840 0.245920 − 0.969290i \(-0.420910\pi\)
0.245920 + 0.969290i \(0.420910\pi\)
\(72\) 2.62698e13 0.308647
\(73\) 1.41416e14 1.49823 0.749114 − 0.662442i \(-0.230480\pi\)
0.749114 + 0.662442i \(0.230480\pi\)
\(74\) 1.11371e14 1.06545
\(75\) −3.23283e13 −0.279655
\(76\) −1.06117e14 −0.831155
\(77\) 5.77478e13 0.410067
\(78\) −2.61739e13 −0.168717
\(79\) 2.47021e14 1.44720 0.723602 − 0.690217i \(-0.242485\pi\)
0.723602 + 0.690217i \(0.242485\pi\)
\(80\) −2.17594e13 −0.116004
\(81\) 1.30760e14 0.635093
\(82\) −1.28981e14 −0.571377
\(83\) 2.78879e12 0.0112805 0.00564027 − 0.999984i \(-0.498205\pi\)
0.00564027 + 0.999984i \(0.498205\pi\)
\(84\) 1.82155e13 0.0673512
\(85\) 2.02453e13 0.0684986
\(86\) −1.98410e13 −0.0614928
\(87\) 1.05230e13 0.0299051
\(88\) −1.47055e14 −0.383583
\(89\) −5.83963e12 −0.0139946 −0.00699730 − 0.999976i \(-0.502227\pi\)
−0.00699730 + 0.999976i \(0.502227\pi\)
\(90\) −1.29970e14 −0.286434
\(91\) 1.24742e14 0.253047
\(92\) −3.46181e14 −0.646985
\(93\) −1.28293e14 −0.221096
\(94\) 3.26652e14 0.519549
\(95\) 5.25014e14 0.771336
\(96\) −4.63856e13 −0.0630013
\(97\) 2.78027e14 0.349381 0.174690 − 0.984623i \(-0.444108\pi\)
0.174690 + 0.984623i \(0.444108\pi\)
\(98\) −8.68126e13 −0.101015
\(99\) −8.78366e14 −0.947135
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 14.16.a.a.1.1 ✓ 1
3.2 odd 2 126.16.a.e.1.1 1
4.3 odd 2 112.16.a.a.1.1 1
7.2 even 3 98.16.c.b.67.1 2
7.3 odd 6 98.16.c.c.79.1 2
7.4 even 3 98.16.c.b.79.1 2
7.5 odd 6 98.16.c.c.67.1 2
7.6 odd 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 1.1 even 1 trivial
98.16.a.b.1.1 1 7.6 odd 2
98.16.c.b.67.1 2 7.2 even 3
98.16.c.b.79.1 2 7.4 even 3
98.16.c.c.67.1 2 7.5 odd 6
98.16.c.c.79.1 2 7.3 odd 6
112.16.a.a.1.1 1 4.3 odd 2
126.16.a.e.1.1 1 3.2 odd 2