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Properties

Label 3456.2.s.c
Level $3456$
Weight $2$
Character orbit 3456.s
Analytic conductor $27.596$
Analytic rank $0$
Dimension $24$
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] = [3456,2,Mod(1151,3456)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3456.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.s (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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{11} + 12 q^{25} - 24 q^{29} - 36 q^{31} - 12 q^{41} - 42 q^{43} + 12 q^{49} - 6 q^{59} + 54 q^{67} - 48 q^{77} + 60 q^{79} - 36 q^{83}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1 0 0 0 −3.37746 − 1.94998i 0 −0.0433891 + 0.0250507i 0 0 0
1151.2 0 0 0 −3.35990 − 1.93984i 0 −3.77468 + 2.17932i 0 0 0
1151.3 0 0 0 −2.20740 − 1.27444i 0 0.695946 − 0.401805i 0 0 0
1151.4 0 0 0 −0.932763 − 0.538531i 0 −1.20495 + 0.695680i 0 0 0
1151.5 0 0 0 −0.860638 − 0.496890i 0 3.59294 − 2.07438i 0 0 0
1151.6 0 0 0 −0.190721 − 0.110113i 0 2.40217 − 1.38689i 0 0 0
1151.7 0 0 0 0.273344 + 0.157815i 0 2.41739 − 1.39568i 0 0 0
1151.8 0 0 0 1.38781 + 0.801253i 0 1.71629 − 0.990903i 0 0 0
1151.9 0 0 0 1.62431 + 0.937794i 0 −0.0301976 + 0.0174346i 0 0 0
1151.10 0 0 0 1.94279 + 1.12167i 0 −3.69568 + 2.13370i 0 0 0
1151.11 0 0 0 2.60329 + 1.50301i 0 −3.54102 + 2.04441i 0 0 0
1151.12 0 0 0 3.09733 + 1.78824i 0 1.46519 − 0.845928i 0 0 0
2303.1 0 0 0 −3.37746 + 1.94998i 0 −0.0433891 − 0.0250507i 0 0 0
2303.2 0 0 0 −3.35990 + 1.93984i 0 −3.77468 − 2.17932i 0 0 0
2303.3 0 0 0 −2.20740 + 1.27444i 0 0.695946 + 0.401805i 0 0 0
2303.4 0 0 0 −0.932763 + 0.538531i 0 −1.20495 − 0.695680i 0 0 0
2303.5 0 0 0 −0.860638 + 0.496890i 0 3.59294 + 2.07438i 0 0 0
2303.6 0 0 0 −0.190721 + 0.110113i 0 2.40217 + 1.38689i 0 0 0
2303.7 0 0 0 0.273344 − 0.157815i 0 2.41739 + 1.39568i 0 0 0
2303.8 0 0 0 1.38781 − 0.801253i 0 1.71629 + 0.990903i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.12
Significant digits:
Format:

Inner twists

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