Properties

Label 6.16.a.a.1.1
Level $6$
Weight $16$
Character 6.1
Self dual yes
Analytic conductor $8.562$
Analytic rank $0$
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] = [6,16,Mod(1,6)] 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("6.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-128,-2187] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.56161030600\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6.1

$q$-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)
 
\(f(q)\) \(=\) \(q-128.000 q^{2} -2187.00 q^{3} +16384.0 q^{4} -314490. q^{5} +279936. q^{6} +2.02506e6 q^{7} -2.09715e6 q^{8} +4.78297e6 q^{9} +4.02547e7 q^{10} +1.10255e8 q^{11} -3.58318e7 q^{12} +5.60479e7 q^{13} -2.59207e8 q^{14} +6.87790e8 q^{15} +2.68435e8 q^{16} -1.93010e9 q^{17} -6.12220e8 q^{18} +2.16319e9 q^{19} -5.15260e9 q^{20} -4.42880e9 q^{21} -1.41126e10 q^{22} +6.22897e9 q^{23} +4.58647e9 q^{24} +6.83864e10 q^{25} -7.17413e9 q^{26} -1.04604e10 q^{27} +3.31785e10 q^{28} +6.47437e10 q^{29} -8.80371e10 q^{30} -2.02376e10 q^{31} -3.43597e10 q^{32} -2.41128e11 q^{33} +2.47053e11 q^{34} -6.36860e11 q^{35} +7.83642e10 q^{36} +4.88968e11 q^{37} -2.76888e11 q^{38} -1.22577e11 q^{39} +6.59533e11 q^{40} -7.72359e11 q^{41} +5.66886e11 q^{42} +1.30677e12 q^{43} +1.80642e12 q^{44} -1.50420e12 q^{45} -7.97309e11 q^{46} +3.35182e12 q^{47} -5.87068e11 q^{48} -6.46710e11 q^{49} -8.75346e12 q^{50} +4.22114e12 q^{51} +9.18288e11 q^{52} +9.38781e12 q^{53} +1.33893e12 q^{54} -3.46741e13 q^{55} -4.24685e12 q^{56} -4.73089e12 q^{57} -8.28720e12 q^{58} +2.89304e13 q^{59} +1.12687e13 q^{60} +4.23931e13 q^{61} +2.59041e12 q^{62} +9.68578e12 q^{63} +4.39805e12 q^{64} -1.76265e13 q^{65} +3.08644e13 q^{66} -5.22472e13 q^{67} -3.16228e13 q^{68} -1.36228e13 q^{69} +8.15181e13 q^{70} -2.71945e13 q^{71} -1.00306e13 q^{72} -9.16042e13 q^{73} -6.25879e13 q^{74} -1.49561e14 q^{75} +3.54417e13 q^{76} +2.23273e14 q^{77} +1.56898e13 q^{78} +6.28821e13 q^{79} -8.44203e13 q^{80} +2.28768e13 q^{81} +9.88620e13 q^{82} -2.23567e14 q^{83} -7.25614e13 q^{84} +6.06999e14 q^{85} -1.67266e14 q^{86} -1.41595e14 q^{87} -2.31222e14 q^{88} +5.54199e14 q^{89} +1.92537e14 q^{90} +1.13500e14 q^{91} +1.02056e14 q^{92} +4.42597e13 q^{93} -4.29033e14 q^{94} -6.80301e14 q^{95} +7.51447e13 q^{96} -1.38887e15 q^{97} +8.27788e13 q^{98} +5.27346e14 q^{99} +O(q^{100})\)

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
\(3\) −2187.00 −0.577350
\(4\) 16384.0 0.500000
\(5\) −314490. −1.80025 −0.900123 − 0.435636i \(-0.856523\pi\)
−0.900123 + 0.435636i \(0.856523\pi\)
\(6\) 279936. 0.408248
\(7\) 2.02506e6 0.929398 0.464699 − 0.885469i \(-0.346163\pi\)
0.464699 + 0.885469i \(0.346163\pi\)
\(8\) −2.09715e6 −0.353553
\(9\) 4.78297e6 0.333333
\(10\) 4.02547e7 1.27297
\(11\) 1.10255e8 1.70590 0.852950 − 0.521993i \(-0.174811\pi\)
0.852950 + 0.521993i \(0.174811\pi\)
\(12\) −3.58318e7 −0.288675
\(13\) 5.60479e7 0.247733 0.123867 − 0.992299i \(-0.460471\pi\)
0.123867 + 0.992299i \(0.460471\pi\)
\(14\) −2.59207e8 −0.657184
\(15\) 6.87790e8 1.03937
\(16\) 2.68435e8 0.250000
\(17\) −1.93010e9 −1.14081 −0.570406 − 0.821363i \(-0.693214\pi\)
−0.570406 + 0.821363i \(0.693214\pi\)
\(18\) −6.12220e8 −0.235702
\(19\) 2.16319e9 0.555191 0.277595 − 0.960698i \(-0.410463\pi\)
0.277595 + 0.960698i \(0.410463\pi\)
\(20\) −5.15260e9 −0.900123
\(21\) −4.42880e9 −0.536588
\(22\) −1.41126e10 −1.20625
\(23\) 6.22897e9 0.381468 0.190734 − 0.981642i \(-0.438913\pi\)
0.190734 + 0.981642i \(0.438913\pi\)
\(24\) 4.58647e9 0.204124
\(25\) 6.83864e10 2.24088
\(26\) −7.17413e9 −0.175174
\(27\) −1.04604e10 −0.192450
\(28\) 3.31785e10 0.464699
\(29\) 6.47437e10 0.696968 0.348484 − 0.937315i \(-0.386697\pi\)
0.348484 + 0.937315i \(0.386697\pi\)
\(30\) −8.80371e10 −0.734947
\(31\) −2.02376e10 −0.132113 −0.0660567 − 0.997816i \(-0.521042\pi\)
−0.0660567 + 0.997816i \(0.521042\pi\)
\(32\) −3.43597e10 −0.176777
\(33\) −2.41128e11 −0.984901
\(34\) 2.47053e11 0.806676
\(35\) −6.36860e11 −1.67314
\(36\) 7.83642e10 0.166667
\(37\) 4.88968e11 0.846773 0.423387 − 0.905949i \(-0.360841\pi\)
0.423387 + 0.905949i \(0.360841\pi\)
\(38\) −2.76888e11 −0.392579
\(39\) −1.22577e11 −0.143029
\(40\) 6.59533e11 0.636483
\(41\) −7.72359e11 −0.619356 −0.309678 − 0.950841i \(-0.600221\pi\)
−0.309678 + 0.950841i \(0.600221\pi\)
\(42\) 5.66886e11 0.379425
\(43\) 1.30677e12 0.733136 0.366568 − 0.930391i \(-0.380533\pi\)
0.366568 + 0.930391i \(0.380533\pi\)
\(44\) 1.80642e12 0.852950
\(45\) −1.50420e12 −0.600082
\(46\) −7.97309e11 −0.269738
\(47\) 3.35182e12 0.965044 0.482522 − 0.875884i \(-0.339721\pi\)
0.482522 + 0.875884i \(0.339721\pi\)
\(48\) −5.87068e11 −0.144338
\(49\) −6.46710e11 −0.136219
\(50\) −8.75346e12 −1.58454
\(51\) 4.22114e12 0.658648
\(52\) 9.18288e11 0.123867
\(53\) 9.38781e12 1.09773 0.548865 − 0.835911i \(-0.315060\pi\)
0.548865 + 0.835911i \(0.315060\pi\)
\(54\) 1.33893e12 0.136083
\(55\) −3.46741e13 −3.07104
\(56\) −4.24685e12 −0.328592
\(57\) −4.73089e12 −0.320540
\(58\) −8.28720e12 −0.492831
\(59\) 2.89304e13 1.51343 0.756717 − 0.653742i \(-0.226802\pi\)
0.756717 + 0.653742i \(0.226802\pi\)
\(60\) 1.12687e13 0.519686
\(61\) 4.23931e13 1.72711 0.863557 − 0.504251i \(-0.168231\pi\)
0.863557 + 0.504251i \(0.168231\pi\)
\(62\) 2.59041e12 0.0934182
\(63\) 9.68578e12 0.309799
\(64\) 4.39805e12 0.125000
\(65\) −1.76265e13 −0.445980
\(66\) 3.08644e13 0.696430
\(67\) −5.22472e13 −1.05318 −0.526590 − 0.850120i \(-0.676530\pi\)
−0.526590 + 0.850120i \(0.676530\pi\)
\(68\) −3.16228e13 −0.570406
\(69\) −1.36228e13 −0.220240
\(70\) 8.15181e13 1.18309
\(71\) −2.71945e13 −0.354849 −0.177425 − 0.984134i \(-0.556777\pi\)
−0.177425 + 0.984134i \(0.556777\pi\)
\(72\) −1.00306e13 −0.117851
\(73\) −9.16042e13 −0.970496 −0.485248 − 0.874376i \(-0.661271\pi\)
−0.485248 + 0.874376i \(0.661271\pi\)
\(74\) −6.25879e13 −0.598759
\(75\) −1.49561e14 −1.29378
\(76\) 3.54417e13 0.277595
\(77\) 2.23273e14 1.58546
\(78\) 1.56898e13 0.101137
\(79\) 6.28821e13 0.368404 0.184202 − 0.982888i \(-0.441030\pi\)
0.184202 + 0.982888i \(0.441030\pi\)
\(80\) −8.44203e13 −0.450061
\(81\) 2.28768e13 0.111111
\(82\) 9.88620e13 0.437951
\(83\) −2.23567e14 −0.904321 −0.452161 − 0.891937i \(-0.649347\pi\)
−0.452161 + 0.891937i \(0.649347\pi\)
\(84\) −7.25614e13 −0.268294
\(85\) 6.06999e14 2.05374
\(86\) −1.67266e14 −0.518405
\(87\) −1.41595e14 −0.402394
\(88\) −2.31222e14 −0.603126
\(89\) 5.54199e14 1.32813 0.664065 − 0.747675i \(-0.268830\pi\)
0.664065 + 0.747675i \(0.268830\pi\)
\(90\) 1.92537e14 0.424322
\(91\) 1.13500e14 0.230243
\(92\) 1.02056e14 0.190734
\(93\) 4.42597e13 0.0762756
\(94\) −4.29033e14 −0.682389
\(95\) −6.80301e14 −0.999480
\(96\) 7.51447e13 0.102062
\(97\) −1.38887e15 −1.74531 −0.872657 − 0.488333i \(-0.837605\pi\)
−0.872657 + 0.488333i \(0.837605\pi\)
\(98\) 8.27788e13 0.0963216
\(99\) 5.27346e14 0.568633
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 6.16.a.a.1.1 ✓ 1
3.2 odd 2 18.16.a.f.1.1 1
4.3 odd 2 48.16.a.c.1.1 1
5.2 odd 4 150.16.c.i.49.1 2
5.3 odd 4 150.16.c.i.49.2 2
5.4 even 2 150.16.a.h.1.1 1
12.11 even 2 144.16.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.a.1.1 ✓ 1 1.1 even 1 trivial
18.16.a.f.1.1 1 3.2 odd 2
48.16.a.c.1.1 1 4.3 odd 2
144.16.a.o.1.1 1 12.11 even 2
150.16.a.h.1.1 1 5.4 even 2
150.16.c.i.49.1 2 5.2 odd 4
150.16.c.i.49.2 2 5.3 odd 4