Properties

Label 18.16.a.f.1.1
Level $18$
Weight $16$
Character 18.1
Self dual yes
Analytic conductor $25.685$
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] = [18,16,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,128,0,16384,314490] 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: \(25.6848309180\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q+128.000 q^{2} +16384.0 q^{4} +314490. q^{5} +2.02506e6 q^{7} +2.09715e6 q^{8} +4.02547e7 q^{10} -1.10255e8 q^{11} +5.60479e7 q^{13} +2.59207e8 q^{14} +2.68435e8 q^{16} +1.93010e9 q^{17} +2.16319e9 q^{19} +5.15260e9 q^{20} -1.41126e10 q^{22} -6.22897e9 q^{23} +6.83864e10 q^{25} +7.17413e9 q^{26} +3.31785e10 q^{28} -6.47437e10 q^{29} -2.02376e10 q^{31} +3.43597e10 q^{32} +2.47053e11 q^{34} +6.36860e11 q^{35} +4.88968e11 q^{37} +2.76888e11 q^{38} +6.59533e11 q^{40} +7.72359e11 q^{41} +1.30677e12 q^{43} -1.80642e12 q^{44} -7.97309e11 q^{46} -3.35182e12 q^{47} -6.46710e11 q^{49} +8.75346e12 q^{50} +9.18288e11 q^{52} -9.38781e12 q^{53} -3.46741e13 q^{55} +4.24685e12 q^{56} -8.28720e12 q^{58} -2.89304e13 q^{59} +4.23931e13 q^{61} -2.59041e12 q^{62} +4.39805e12 q^{64} +1.76265e13 q^{65} -5.22472e13 q^{67} +3.16228e13 q^{68} +8.15181e13 q^{70} +2.71945e13 q^{71} -9.16042e13 q^{73} +6.25879e13 q^{74} +3.54417e13 q^{76} -2.23273e14 q^{77} +6.28821e13 q^{79} +8.44203e13 q^{80} +9.88620e13 q^{82} +2.23567e14 q^{83} +6.06999e14 q^{85} +1.67266e14 q^{86} -2.31222e14 q^{88} -5.54199e14 q^{89} +1.13500e14 q^{91} -1.02056e14 q^{92} -4.29033e14 q^{94} +6.80301e14 q^{95} -1.38887e15 q^{97} -8.27788e13 q^{98} +O(q^{100})\)

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
\(5\) 314490. 1.80025 0.900123 − 0.435636i \(-0.143477\pi\)
0.900123 + 0.435636i \(0.143477\pi\)
\(6\) 0 0
\(7\) 2.02506e6 0.929398 0.464699 − 0.885469i \(-0.346163\pi\)
0.464699 + 0.885469i \(0.346163\pi\)
\(8\) 2.09715e6 0.353553
\(9\) 0 0
\(10\) 4.02547e7 1.27297
\(11\) −1.10255e8 −1.70590 −0.852950 − 0.521993i \(-0.825189\pi\)
−0.852950 + 0.521993i \(0.825189\pi\)
\(12\) 0 0
\(13\) 5.60479e7 0.247733 0.123867 − 0.992299i \(-0.460471\pi\)
0.123867 + 0.992299i \(0.460471\pi\)
\(14\) 2.59207e8 0.657184
\(15\) 0 0
\(16\) 2.68435e8 0.250000
\(17\) 1.93010e9 1.14081 0.570406 − 0.821363i \(-0.306786\pi\)
0.570406 + 0.821363i \(0.306786\pi\)
\(18\) 0 0
\(19\) 2.16319e9 0.555191 0.277595 − 0.960698i \(-0.410463\pi\)
0.277595 + 0.960698i \(0.410463\pi\)
\(20\) 5.15260e9 0.900123
\(21\) 0 0
\(22\) −1.41126e10 −1.20625
\(23\) −6.22897e9 −0.381468 −0.190734 − 0.981642i \(-0.561087\pi\)
−0.190734 + 0.981642i \(0.561087\pi\)
\(24\) 0 0
\(25\) 6.83864e10 2.24088
\(26\) 7.17413e9 0.175174
\(27\) 0 0
\(28\) 3.31785e10 0.464699
\(29\) −6.47437e10 −0.696968 −0.348484 − 0.937315i \(-0.613303\pi\)
−0.348484 + 0.937315i \(0.613303\pi\)
\(30\) 0 0
\(31\) −2.02376e10 −0.132113 −0.0660567 − 0.997816i \(-0.521042\pi\)
−0.0660567 + 0.997816i \(0.521042\pi\)
\(32\) 3.43597e10 0.176777
\(33\) 0 0
\(34\) 2.47053e11 0.806676
\(35\) 6.36860e11 1.67314
\(36\) 0 0
\(37\) 4.88968e11 0.846773 0.423387 − 0.905949i \(-0.360841\pi\)
0.423387 + 0.905949i \(0.360841\pi\)
\(38\) 2.76888e11 0.392579
\(39\) 0 0
\(40\) 6.59533e11 0.636483
\(41\) 7.72359e11 0.619356 0.309678 − 0.950841i \(-0.399779\pi\)
0.309678 + 0.950841i \(0.399779\pi\)
\(42\) 0 0
\(43\) 1.30677e12 0.733136 0.366568 − 0.930391i \(-0.380533\pi\)
0.366568 + 0.930391i \(0.380533\pi\)
\(44\) −1.80642e12 −0.852950
\(45\) 0 0
\(46\) −7.97309e11 −0.269738
\(47\) −3.35182e12 −0.965044 −0.482522 − 0.875884i \(-0.660279\pi\)
−0.482522 + 0.875884i \(0.660279\pi\)
\(48\) 0 0
\(49\) −6.46710e11 −0.136219
\(50\) 8.75346e12 1.58454
\(51\) 0 0
\(52\) 9.18288e11 0.123867
\(53\) −9.38781e12 −1.09773 −0.548865 − 0.835911i \(-0.684940\pi\)
−0.548865 + 0.835911i \(0.684940\pi\)
\(54\) 0 0
\(55\) −3.46741e13 −3.07104
\(56\) 4.24685e12 0.328592
\(57\) 0 0
\(58\) −8.28720e12 −0.492831
\(59\) −2.89304e13 −1.51343 −0.756717 − 0.653742i \(-0.773198\pi\)
−0.756717 + 0.653742i \(0.773198\pi\)
\(60\) 0 0
\(61\) 4.23931e13 1.72711 0.863557 − 0.504251i \(-0.168231\pi\)
0.863557 + 0.504251i \(0.168231\pi\)
\(62\) −2.59041e12 −0.0934182
\(63\) 0 0
\(64\) 4.39805e12 0.125000
\(65\) 1.76265e13 0.445980
\(66\) 0 0
\(67\) −5.22472e13 −1.05318 −0.526590 − 0.850120i \(-0.676530\pi\)
−0.526590 + 0.850120i \(0.676530\pi\)
\(68\) 3.16228e13 0.570406
\(69\) 0 0
\(70\) 8.15181e13 1.18309
\(71\) 2.71945e13 0.354849 0.177425 − 0.984134i \(-0.443223\pi\)
0.177425 + 0.984134i \(0.443223\pi\)
\(72\) 0 0
\(73\) −9.16042e13 −0.970496 −0.485248 − 0.874376i \(-0.661271\pi\)
−0.485248 + 0.874376i \(0.661271\pi\)
\(74\) 6.25879e13 0.598759
\(75\) 0 0
\(76\) 3.54417e13 0.277595
\(77\) −2.23273e14 −1.58546
\(78\) 0 0
\(79\) 6.28821e13 0.368404 0.184202 − 0.982888i \(-0.441030\pi\)
0.184202 + 0.982888i \(0.441030\pi\)
\(80\) 8.44203e13 0.450061
\(81\) 0 0
\(82\) 9.88620e13 0.437951
\(83\) 2.23567e14 0.904321 0.452161 − 0.891937i \(-0.350653\pi\)
0.452161 + 0.891937i \(0.350653\pi\)
\(84\) 0 0
\(85\) 6.06999e14 2.05374
\(86\) 1.67266e14 0.518405
\(87\) 0 0
\(88\) −2.31222e14 −0.603126
\(89\) −5.54199e14 −1.32813 −0.664065 − 0.747675i \(-0.731170\pi\)
−0.664065 + 0.747675i \(0.731170\pi\)
\(90\) 0 0
\(91\) 1.13500e14 0.230243
\(92\) −1.02056e14 −0.190734
\(93\) 0 0
\(94\) −4.29033e14 −0.682389
\(95\) 6.80301e14 0.999480
\(96\) 0 0
\(97\) −1.38887e15 −1.74531 −0.872657 − 0.488333i \(-0.837605\pi\)
−0.872657 + 0.488333i \(0.837605\pi\)
\(98\) −8.27788e13 −0.0963216
\(99\) 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 18.16.a.f.1.1 1
3.2 odd 2 6.16.a.a.1.1 ✓ 1
4.3 odd 2 144.16.a.o.1.1 1
12.11 even 2 48.16.a.c.1.1 1
15.2 even 4 150.16.c.i.49.1 2
15.8 even 4 150.16.c.i.49.2 2
15.14 odd 2 150.16.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.a.1.1 ✓ 1 3.2 odd 2
18.16.a.f.1.1 1 1.1 even 1 trivial
48.16.a.c.1.1 1 12.11 even 2
144.16.a.o.1.1 1 4.3 odd 2
150.16.a.h.1.1 1 15.14 odd 2
150.16.c.i.49.1 2 15.2 even 4
150.16.c.i.49.2 2 15.8 even 4