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Properties

Label 384.2.n.a
Level $384$
Weight $2$
Character orbit 384.n
Analytic conductor $3.066$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,2,Mod(49,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.n (of order \(8\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{23} + 48 q^{31} + 48 q^{35} + 16 q^{43} - 16 q^{51} - 32 q^{53} - 32 q^{55} - 64 q^{59} - 32 q^{61} - 16 q^{63} - 16 q^{67} - 32 q^{69} - 64 q^{71} - 32 q^{75} - 32 q^{77} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −0.923880 + 0.382683i 0 −1.35803 + 3.27858i 0 −2.48546 − 2.48546i 0 0.707107 − 0.707107i 0
49.2 0 −0.923880 + 0.382683i 0 −0.155637 + 0.375742i 0 −0.709092 − 0.709092i 0 0.707107 − 0.707107i 0
49.3 0 −0.923880 + 0.382683i 0 −0.00259461 + 0.00626394i 0 2.41880 + 2.41880i 0 0.707107 − 0.707107i 0
49.4 0 −0.923880 + 0.382683i 0 0.750897 − 1.81283i 0 −0.638460 − 0.638460i 0 0.707107 − 0.707107i 0
49.5 0 0.923880 − 0.382683i 0 −1.48656 + 3.58888i 0 −1.03821 − 1.03821i 0 0.707107 − 0.707107i 0
49.6 0 0.923880 − 0.382683i 0 0.184062 − 0.444366i 0 0.134531 + 0.134531i 0 0.707107 − 0.707107i 0
49.7 0 0.923880 − 0.382683i 0 0.705805 − 1.70396i 0 −3.24150 − 3.24150i 0 0.707107 − 0.707107i 0
49.8 0 0.923880 − 0.382683i 0 1.36206 − 3.28830i 0 2.73097 + 2.73097i 0 0.707107 − 0.707107i 0
145.1 0 −0.382683 + 0.923880i 0 −1.46213 + 0.605634i 0 3.54889 − 3.54889i 0 −0.707107 − 0.707107i 0
145.2 0 −0.382683 + 0.923880i 0 −1.20409 + 0.498752i 0 −2.59422 + 2.59422i 0 −0.707107 − 0.707107i 0
145.3 0 −0.382683 + 0.923880i 0 0.825824 − 0.342068i 0 −1.17750 + 1.17750i 0 −0.707107 − 0.707107i 0
145.4 0 −0.382683 + 0.923880i 0 3.68816 − 1.52768i 0 1.63704 − 1.63704i 0 −0.707107 − 0.707107i 0
145.5 0 0.382683 − 0.923880i 0 −3.09318 + 1.28124i 0 1.73503 − 1.73503i 0 −0.707107 − 0.707107i 0
145.6 0 0.382683 − 0.923880i 0 −2.51374 + 1.04122i 0 −2.01027 + 2.01027i 0 −0.707107 − 0.707107i 0
145.7 0 0.382683 − 0.923880i 0 1.60930 − 0.666593i 0 0.589445 − 0.589445i 0 −0.707107 − 0.707107i 0
145.8 0 0.382683 − 0.923880i 0 2.14986 − 0.890503i 0 1.10001 − 1.10001i 0 −0.707107 − 0.707107i 0
241.1 0 −0.382683 − 0.923880i 0 −1.46213 − 0.605634i 0 3.54889 + 3.54889i 0 −0.707107 + 0.707107i 0
241.2 0 −0.382683 − 0.923880i 0 −1.20409 − 0.498752i 0 −2.59422 − 2.59422i 0 −0.707107 + 0.707107i 0
241.3 0 −0.382683 − 0.923880i 0 0.825824 + 0.342068i 0 −1.17750 − 1.17750i 0 −0.707107 + 0.707107i 0
241.4 0 −0.382683 − 0.923880i 0 3.68816 + 1.52768i 0 1.63704 + 1.63704i 0 −0.707107 + 0.707107i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.n.a 32
3.b odd 2 1 1152.2.v.c 32
4.b odd 2 1 96.2.n.a ✓ 32
8.b even 2 1 768.2.n.b 32
8.d odd 2 1 768.2.n.a 32
12.b even 2 1 288.2.v.d 32
32.g even 8 1 inner 384.2.n.a 32
32.g even 8 1 768.2.n.b 32
32.h odd 8 1 96.2.n.a ✓ 32
32.h odd 8 1 768.2.n.a 32
96.o even 8 1 288.2.v.d 32
96.p odd 8 1 1152.2.v.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.n.a ✓ 32 4.b odd 2 1
96.2.n.a ✓ 32 32.h odd 8 1
288.2.v.d 32 12.b even 2 1
288.2.v.d 32 96.o even 8 1
384.2.n.a 32 1.a even 1 1 trivial
384.2.n.a 32 32.g even 8 1 inner
768.2.n.a 32 8.d odd 2 1
768.2.n.a 32 32.h odd 8 1
768.2.n.b 32 8.b even 2 1
768.2.n.b 32 32.g even 8 1
1152.2.v.c 32 3.b odd 2 1
1152.2.v.c 32 96.p odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(384, [\chi])\).