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Properties

Label 1288.2.f.b
Level $1288$
Weight $2$
Character orbit 1288.f
Analytic conductor $10.285$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1288,2,Mod(321,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.f (of order \(2\), degree \(1\), 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: \(10.2847317803\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{9} + 4 q^{23} + 16 q^{25} + 28 q^{35} - 24 q^{39} - 12 q^{49} - 16 q^{71} + 20 q^{77} + 32 q^{81} + 16 q^{85} - 8 q^{93} - 24 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 0 − 1.72145i 0 −4.01868 0 −2.27145 + 1.35665i 0 0.0366007 0
321.2 0 1.72145i 0 −4.01868 0 −2.27145 − 1.35665i 0 0.0366007 0
321.3 0 − 1.55077i 0 −2.86607 0 2.61257 + 0.417713i 0 0.595102 0
321.4 0 1.55077i 0 −2.86607 0 2.61257 − 0.417713i 0 0.595102 0
321.5 0 − 2.81002i 0 2.47124 0 0.859611 + 2.50221i 0 −4.89623 0
321.6 0 2.81002i 0 2.47124 0 0.859611 − 2.50221i 0 −4.89623 0
321.7 0 − 3.09717i 0 −0.561074 0 −1.40237 + 2.24352i 0 −6.59244 0
321.8 0 3.09717i 0 −0.561074 0 −1.40237 − 2.24352i 0 −6.59244 0
321.9 0 − 1.01374i 0 −0.921258 0 0.770210 + 2.53116i 0 1.97234 0
321.10 0 1.01374i 0 −0.921258 0 0.770210 − 2.53116i 0 1.97234 0
321.11 0 − 0.339662i 0 1.53798 0 2.05334 − 1.66848i 0 2.88463 0
321.12 0 0.339662i 0 1.53798 0 2.05334 + 1.66848i 0 2.88463 0
321.13 0 − 0.339662i 0 −1.53798 0 −2.05334 + 1.66848i 0 2.88463 0
321.14 0 0.339662i 0 −1.53798 0 −2.05334 − 1.66848i 0 2.88463 0
321.15 0 − 1.01374i 0 0.921258 0 −0.770210 − 2.53116i 0 1.97234 0
321.16 0 1.01374i 0 0.921258 0 −0.770210 + 2.53116i 0 1.97234 0
321.17 0 − 3.09717i 0 0.561074 0 1.40237 − 2.24352i 0 −6.59244 0
321.18 0 3.09717i 0 0.561074 0 1.40237 + 2.24352i 0 −6.59244 0
321.19 0 − 2.81002i 0 −2.47124 0 −0.859611 − 2.50221i 0 −4.89623 0
321.20 0 2.81002i 0 −2.47124 0 −0.859611 + 2.50221i 0 −4.89623 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.2.f.b ✓ 24
4.b odd 2 1 2576.2.f.i 24
7.b odd 2 1 inner 1288.2.f.b ✓ 24
23.b odd 2 1 inner 1288.2.f.b ✓ 24
28.d even 2 1 2576.2.f.i 24
92.b even 2 1 2576.2.f.i 24
161.c even 2 1 inner 1288.2.f.b ✓ 24
644.h odd 2 1 2576.2.f.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.f.b ✓ 24 1.a even 1 1 trivial
1288.2.f.b ✓ 24 7.b odd 2 1 inner
1288.2.f.b ✓ 24 23.b odd 2 1 inner
1288.2.f.b ✓ 24 161.c even 2 1 inner
2576.2.f.i 24 4.b odd 2 1
2576.2.f.i 24 28.d even 2 1
2576.2.f.i 24 92.b even 2 1
2576.2.f.i 24 644.h odd 2 1