Properties

Label 1728.2.a.p.1.1
Level $1728$
Weight $2$
Character 1728.1
Self dual yes
Analytic conductor $13.798$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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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] = [1728,2,Mod(1,1728)] 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("1728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,5,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{7} +7.00000 q^{13} +1.00000 q^{19} -5.00000 q^{25} -4.00000 q^{31} +1.00000 q^{37} -8.00000 q^{43} +18.0000 q^{49} +13.0000 q^{61} -11.0000 q^{67} +17.0000 q^{73} -13.0000 q^{79} +35.0000 q^{91} +5.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 − 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 7.00000 1.94145 0.970725 − 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 − 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.0000 −1.34386 −0.671932 − 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 17.0000 1.98970 0.994850 − 0.101361i \(-0.0323196\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 35.0000 3.66900
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 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 1728.2.a.p.1.1 1
3.2 odd 2 CM 1728.2.a.p.1.1 1
4.3 odd 2 1728.2.a.m.1.1 1
8.3 odd 2 432.2.a.d.1.1 1
8.5 even 2 108.2.a.a.1.1 ✓ 1
12.11 even 2 1728.2.a.m.1.1 1
24.5 odd 2 108.2.a.a.1.1 ✓ 1
24.11 even 2 432.2.a.d.1.1 1
40.13 odd 4 2700.2.d.g.649.1 2
40.29 even 2 2700.2.a.b.1.1 1
40.37 odd 4 2700.2.d.g.649.2 2
56.13 odd 2 5292.2.a.j.1.1 1
72.5 odd 6 324.2.e.b.217.1 2
72.11 even 6 1296.2.i.j.433.1 2
72.13 even 6 324.2.e.b.217.1 2
72.29 odd 6 324.2.e.b.109.1 2
72.43 odd 6 1296.2.i.j.433.1 2
72.59 even 6 1296.2.i.j.865.1 2
72.61 even 6 324.2.e.b.109.1 2
72.67 odd 6 1296.2.i.j.865.1 2
120.29 odd 2 2700.2.a.b.1.1 1
120.53 even 4 2700.2.d.g.649.1 2
120.77 even 4 2700.2.d.g.649.2 2
168.125 even 2 5292.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.a.a.1.1 ✓ 1 8.5 even 2
108.2.a.a.1.1 ✓ 1 24.5 odd 2
324.2.e.b.109.1 2 72.29 odd 6
324.2.e.b.109.1 2 72.61 even 6
324.2.e.b.217.1 2 72.5 odd 6
324.2.e.b.217.1 2 72.13 even 6
432.2.a.d.1.1 1 8.3 odd 2
432.2.a.d.1.1 1 24.11 even 2
1296.2.i.j.433.1 2 72.11 even 6
1296.2.i.j.433.1 2 72.43 odd 6
1296.2.i.j.865.1 2 72.59 even 6
1296.2.i.j.865.1 2 72.67 odd 6
1728.2.a.m.1.1 1 4.3 odd 2
1728.2.a.m.1.1 1 12.11 even 2
1728.2.a.p.1.1 1 1.1 even 1 trivial
1728.2.a.p.1.1 1 3.2 odd 2 CM
2700.2.a.b.1.1 1 40.29 even 2
2700.2.a.b.1.1 1 120.29 odd 2
2700.2.d.g.649.1 2 40.13 odd 4
2700.2.d.g.649.1 2 120.53 even 4
2700.2.d.g.649.2 2 40.37 odd 4
2700.2.d.g.649.2 2 120.77 even 4
5292.2.a.j.1.1 1 56.13 odd 2
5292.2.a.j.1.1 1 168.125 even 2