Properties

Label 384.1.h.a.65.1
Level $384$
Weight $1$
Character 384.65
Self dual yes
Analytic conductor $0.192$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -8, -24, 12
Inner twists $4$

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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] = [384,1,Mod(65,384)] 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("384.65"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1] 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: \(0.191640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt{1 + \sqrt{-2}})\)
Stark unit: Root of $x^{4} - 12x^{3} - 10x^{2} - 12x + 1$

Embedding invariants

Embedding label 65.1
Character \(\chi\) \(=\) 384.65

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{33} -1.00000 q^{49} -2.00000 q^{59} -2.00000 q^{73} +1.00000 q^{75} +1.00000 q^{81} +2.00000 q^{83} +2.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −1.00000 −1.00000
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −2.00000 −2.00000
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 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
\(59\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(98\) 0 0
\(99\) 2.00000 2.00000
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 384.1.h.a.65.1 ✓ 1
3.2 odd 2 384.1.h.b.65.1 yes 1
4.3 odd 2 384.1.h.b.65.1 yes 1
8.3 odd 2 CM 384.1.h.a.65.1 ✓ 1
8.5 even 2 384.1.h.b.65.1 yes 1
12.11 even 2 RM 384.1.h.a.65.1 ✓ 1
16.3 odd 4 768.1.e.c.257.2 2
16.5 even 4 768.1.e.c.257.2 2
16.11 odd 4 768.1.e.c.257.1 2
16.13 even 4 768.1.e.c.257.1 2
24.5 odd 2 CM 384.1.h.a.65.1 ✓ 1
24.11 even 2 384.1.h.b.65.1 yes 1
32.3 odd 8 3072.1.i.f.1793.2 4
32.5 even 8 3072.1.i.f.257.2 4
32.11 odd 8 3072.1.i.f.257.2 4
32.13 even 8 3072.1.i.f.1793.2 4
32.19 odd 8 3072.1.i.f.1793.1 4
32.21 even 8 3072.1.i.f.257.1 4
32.27 odd 8 3072.1.i.f.257.1 4
32.29 even 8 3072.1.i.f.1793.1 4
48.5 odd 4 768.1.e.c.257.1 2
48.11 even 4 768.1.e.c.257.2 2
48.29 odd 4 768.1.e.c.257.2 2
48.35 even 4 768.1.e.c.257.1 2
96.5 odd 8 3072.1.i.f.257.1 4
96.11 even 8 3072.1.i.f.257.1 4
96.29 odd 8 3072.1.i.f.1793.2 4
96.35 even 8 3072.1.i.f.1793.1 4
96.53 odd 8 3072.1.i.f.257.2 4
96.59 even 8 3072.1.i.f.257.2 4
96.77 odd 8 3072.1.i.f.1793.1 4
96.83 even 8 3072.1.i.f.1793.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.1.h.a.65.1 ✓ 1 1.1 even 1 trivial
384.1.h.a.65.1 ✓ 1 8.3 odd 2 CM
384.1.h.a.65.1 ✓ 1 12.11 even 2 RM
384.1.h.a.65.1 ✓ 1 24.5 odd 2 CM
384.1.h.b.65.1 yes 1 3.2 odd 2
384.1.h.b.65.1 yes 1 4.3 odd 2
384.1.h.b.65.1 yes 1 8.5 even 2
384.1.h.b.65.1 yes 1 24.11 even 2
768.1.e.c.257.1 2 16.11 odd 4
768.1.e.c.257.1 2 16.13 even 4
768.1.e.c.257.1 2 48.5 odd 4
768.1.e.c.257.1 2 48.35 even 4
768.1.e.c.257.2 2 16.3 odd 4
768.1.e.c.257.2 2 16.5 even 4
768.1.e.c.257.2 2 48.11 even 4
768.1.e.c.257.2 2 48.29 odd 4
3072.1.i.f.257.1 4 32.21 even 8
3072.1.i.f.257.1 4 32.27 odd 8
3072.1.i.f.257.1 4 96.5 odd 8
3072.1.i.f.257.1 4 96.11 even 8
3072.1.i.f.257.2 4 32.5 even 8
3072.1.i.f.257.2 4 32.11 odd 8
3072.1.i.f.257.2 4 96.53 odd 8
3072.1.i.f.257.2 4 96.59 even 8
3072.1.i.f.1793.1 4 32.19 odd 8
3072.1.i.f.1793.1 4 32.29 even 8
3072.1.i.f.1793.1 4 96.35 even 8
3072.1.i.f.1793.1 4 96.77 odd 8
3072.1.i.f.1793.2 4 32.3 odd 8
3072.1.i.f.1793.2 4 32.13 even 8
3072.1.i.f.1793.2 4 96.29 odd 8
3072.1.i.f.1793.2 4 96.83 even 8