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Properties

Label 288.2.a.e.1.1
Level $288$
Weight $2$
Character 288.1
Self dual yes
Analytic conductor $2.300$
Analytic rank $0$
Dimension $1$
CM discriminant -4
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] = [288,2,Mod(1,288)] 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("288.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.29969157821\)
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: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 288.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{5} -6.00000 q^{13} +8.00000 q^{17} +11.0000 q^{25} -4.00000 q^{29} -2.00000 q^{37} -8.00000 q^{41} -7.00000 q^{49} -4.00000 q^{53} -10.0000 q^{61} -24.0000 q^{65} +6.00000 q^{73} +32.0000 q^{85} +16.0000 q^{89} -18.0000 q^{97} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 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
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −24.0000 −2.97683
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 32.0000 3.47089
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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 288.2.a.e.1.1 yes 1
3.2 odd 2 288.2.a.a.1.1 ✓ 1
4.3 odd 2 CM 288.2.a.e.1.1 yes 1
5.2 odd 4 7200.2.f.q.6049.2 2
5.3 odd 4 7200.2.f.q.6049.1 2
5.4 even 2 7200.2.a.be.1.1 1
8.3 odd 2 576.2.a.a.1.1 1
8.5 even 2 576.2.a.a.1.1 1
9.2 odd 6 2592.2.i.x.1729.1 2
9.4 even 3 2592.2.i.a.865.1 2
9.5 odd 6 2592.2.i.x.865.1 2
9.7 even 3 2592.2.i.a.1729.1 2
12.11 even 2 288.2.a.a.1.1 ✓ 1
15.2 even 4 7200.2.f.n.6049.2 2
15.8 even 4 7200.2.f.n.6049.1 2
15.14 odd 2 7200.2.a.bf.1.1 1
16.3 odd 4 2304.2.d.l.1153.1 2
16.5 even 4 2304.2.d.l.1153.2 2
16.11 odd 4 2304.2.d.l.1153.2 2
16.13 even 4 2304.2.d.l.1153.1 2
20.3 even 4 7200.2.f.q.6049.1 2
20.7 even 4 7200.2.f.q.6049.2 2
20.19 odd 2 7200.2.a.be.1.1 1
24.5 odd 2 576.2.a.i.1.1 1
24.11 even 2 576.2.a.i.1.1 1
36.7 odd 6 2592.2.i.a.1729.1 2
36.11 even 6 2592.2.i.x.1729.1 2
36.23 even 6 2592.2.i.x.865.1 2
36.31 odd 6 2592.2.i.a.865.1 2
48.5 odd 4 2304.2.d.h.1153.1 2
48.11 even 4 2304.2.d.h.1153.1 2
48.29 odd 4 2304.2.d.h.1153.2 2
48.35 even 4 2304.2.d.h.1153.2 2
60.23 odd 4 7200.2.f.n.6049.1 2
60.47 odd 4 7200.2.f.n.6049.2 2
60.59 even 2 7200.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.a.a.1.1 ✓ 1 3.2 odd 2
288.2.a.a.1.1 ✓ 1 12.11 even 2
288.2.a.e.1.1 yes 1 1.1 even 1 trivial
288.2.a.e.1.1 yes 1 4.3 odd 2 CM
576.2.a.a.1.1 1 8.3 odd 2
576.2.a.a.1.1 1 8.5 even 2
576.2.a.i.1.1 1 24.5 odd 2
576.2.a.i.1.1 1 24.11 even 2
2304.2.d.h.1153.1 2 48.5 odd 4
2304.2.d.h.1153.1 2 48.11 even 4
2304.2.d.h.1153.2 2 48.29 odd 4
2304.2.d.h.1153.2 2 48.35 even 4
2304.2.d.l.1153.1 2 16.3 odd 4
2304.2.d.l.1153.1 2 16.13 even 4
2304.2.d.l.1153.2 2 16.5 even 4
2304.2.d.l.1153.2 2 16.11 odd 4
2592.2.i.a.865.1 2 9.4 even 3
2592.2.i.a.865.1 2 36.31 odd 6
2592.2.i.a.1729.1 2 9.7 even 3
2592.2.i.a.1729.1 2 36.7 odd 6
2592.2.i.x.865.1 2 9.5 odd 6
2592.2.i.x.865.1 2 36.23 even 6
2592.2.i.x.1729.1 2 9.2 odd 6
2592.2.i.x.1729.1 2 36.11 even 6
7200.2.a.be.1.1 1 5.4 even 2
7200.2.a.be.1.1 1 20.19 odd 2
7200.2.a.bf.1.1 1 15.14 odd 2
7200.2.a.bf.1.1 1 60.59 even 2
7200.2.f.n.6049.1 2 15.8 even 4
7200.2.f.n.6049.1 2 60.23 odd 4
7200.2.f.n.6049.2 2 15.2 even 4
7200.2.f.n.6049.2 2 60.47 odd 4
7200.2.f.q.6049.1 2 5.3 odd 4
7200.2.f.q.6049.1 2 20.3 even 4
7200.2.f.q.6049.2 2 5.2 odd 4
7200.2.f.q.6049.2 2 20.7 even 4