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Properties

Label 3456.2.p.f
Level $3456$
Weight $2$
Character orbit 3456.p
Analytic conductor $27.596$
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] = [3456,2,Mod(575,3456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3456.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.p (of order \(6\), degree \(2\), 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: \(27.5962989386\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{7} - 12 q^{23} - 12 q^{25} + 36 q^{31} + 12 q^{41} + 12 q^{47} + 12 q^{49} + 48 q^{65} + 24 q^{73} - 84 q^{79} + 24 q^{95} - 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1 0 0 0 −2.11539 − 3.66397i 0 −3.25610 − 1.87991i 0 0 0
575.2 0 0 0 −1.43889 − 2.49224i 0 −1.90348 − 1.09898i 0 0 0
575.3 0 0 0 −1.07112 − 1.85524i 0 −1.13037 − 0.652621i 0 0 0
575.4 0 0 0 −0.781546 − 1.35368i 0 0.954503 + 0.551083i 0 0 0
575.5 0 0 0 −0.637560 − 1.10429i 0 −1.73560 − 1.00205i 0 0 0
575.6 0 0 0 −0.538609 − 0.932899i 0 4.07105 + 2.35042i 0 0 0
575.7 0 0 0 0.538609 + 0.932899i 0 4.07105 + 2.35042i 0 0 0
575.8 0 0 0 0.637560 + 1.10429i 0 −1.73560 − 1.00205i 0 0 0
575.9 0 0 0 0.781546 + 1.35368i 0 0.954503 + 0.551083i 0 0 0
575.10 0 0 0 1.07112 + 1.85524i 0 −1.13037 − 0.652621i 0 0 0
575.11 0 0 0 1.43889 + 2.49224i 0 −1.90348 − 1.09898i 0 0 0
575.12 0 0 0 2.11539 + 3.66397i 0 −3.25610 − 1.87991i 0 0 0
2879.1 0 0 0 −2.11539 + 3.66397i 0 −3.25610 + 1.87991i 0 0 0
2879.2 0 0 0 −1.43889 + 2.49224i 0 −1.90348 + 1.09898i 0 0 0
2879.3 0 0 0 −1.07112 + 1.85524i 0 −1.13037 + 0.652621i 0 0 0
2879.4 0 0 0 −0.781546 + 1.35368i 0 0.954503 − 0.551083i 0 0 0
2879.5 0 0 0 −0.637560 + 1.10429i 0 −1.73560 + 1.00205i 0 0 0
2879.6 0 0 0 −0.538609 + 0.932899i 0 4.07105 − 2.35042i 0 0 0
2879.7 0 0 0 0.538609 − 0.932899i 0 4.07105 − 2.35042i 0 0 0
2879.8 0 0 0 0.637560 − 1.10429i 0 −1.73560 + 1.00205i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
36.h even 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.2.p.f 24
3.b odd 2 1 1152.2.p.f ✓ 24
4.b odd 2 1 3456.2.p.g 24
8.b even 2 1 inner 3456.2.p.f 24
8.d odd 2 1 3456.2.p.g 24
9.c even 3 1 1152.2.p.g yes 24
9.d odd 6 1 3456.2.p.g 24
12.b even 2 1 1152.2.p.g yes 24
24.f even 2 1 1152.2.p.g yes 24
24.h odd 2 1 1152.2.p.f ✓ 24
36.f odd 6 1 1152.2.p.f ✓ 24
36.h even 6 1 inner 3456.2.p.f 24
72.j odd 6 1 3456.2.p.g 24
72.l even 6 1 inner 3456.2.p.f 24
72.n even 6 1 1152.2.p.g yes 24
72.p odd 6 1 1152.2.p.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.p.f ✓ 24 3.b odd 2 1
1152.2.p.f ✓ 24 24.h odd 2 1
1152.2.p.f ✓ 24 36.f odd 6 1
1152.2.p.f ✓ 24 72.p odd 6 1
1152.2.p.g yes 24 9.c even 3 1
1152.2.p.g yes 24 12.b even 2 1
1152.2.p.g yes 24 24.f even 2 1
1152.2.p.g yes 24 72.n even 6 1
3456.2.p.f 24 1.a even 1 1 trivial
3456.2.p.f 24 8.b even 2 1 inner
3456.2.p.f 24 36.h even 6 1 inner
3456.2.p.f 24 72.l even 6 1 inner
3456.2.p.g 24 4.b odd 2 1
3456.2.p.g 24 8.d odd 2 1
3456.2.p.g 24 9.d odd 6 1
3456.2.p.g 24 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3456, [\chi])\):

\( T_{5}^{24} + 36 T_{5}^{22} + 858 T_{5}^{20} + 11056 T_{5}^{18} + 100995 T_{5}^{16} + 604704 T_{5}^{14} + \cdots + 9834496 \) Copy content Toggle raw display
\( T_{7}^{12} + 6 T_{7}^{11} - 6 T_{7}^{10} - 108 T_{7}^{9} + 93 T_{7}^{8} + 2532 T_{7}^{7} + 8182 T_{7}^{6} + \cdots + 12544 \) Copy content Toggle raw display
\( T_{11}^{24} - 78 T_{11}^{22} + 4161 T_{11}^{20} - 115682 T_{11}^{18} + 2305878 T_{11}^{16} + \cdots + 13845841 \) Copy content Toggle raw display