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Properties

Label 3024.2.bf.h
Level $3024$
Weight $2$
Character orbit 3024.bf
Analytic conductor $24.147$
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] = [3024,2,Mod(1711,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1711");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.bf (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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1008)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{7} + 3 q^{13} + 3 q^{17} + 4 q^{19} - 30 q^{25} - 18 q^{29} + 17 q^{31} - 42 q^{35} - 3 q^{37} + 36 q^{41} + 24 q^{43} + 21 q^{47} - 24 q^{49} + 12 q^{53} + 30 q^{55} + 6 q^{59} - 48 q^{61} + 42 q^{67} + 48 q^{73} - 36 q^{77} - 30 q^{79} - 48 q^{83} - 21 q^{85} - 39 q^{89} - 9 q^{91} + 33 q^{95} - 3 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1711.1 0 0 0 − 4.24307i 0 −0.426811 − 2.61110i 0 0 0
1711.2 0 0 0 − 2.86361i 0 2.64464 − 0.0766188i 0 0 0
1711.3 0 0 0 − 2.43186i 0 −2.09792 − 1.61206i 0 0 0
1711.4 0 0 0 − 2.40807i 0 −0.605051 + 2.57564i 0 0 0
1711.5 0 0 0 − 1.47736i 0 2.64567 − 0.0201361i 0 0 0
1711.6 0 0 0 − 0.784857i 0 −2.01780 + 1.71128i 0 0 0
1711.7 0 0 0 − 0.280665i 0 −0.164674 − 2.64062i 0 0 0
1711.8 0 0 0 1.77292i 0 1.23647 − 2.33905i 0 0 0
1711.9 0 0 0 1.99968i 0 2.18955 − 1.48522i 0 0 0
1711.10 0 0 0 2.81241i 0 1.84662 + 1.89473i 0 0 0
1711.11 0 0 0 2.85720i 0 −1.73056 + 2.00129i 0 0 0
1711.12 0 0 0 3.31523i 0 0.479859 + 2.60187i 0 0 0
2287.1 0 0 0 − 3.31523i 0 0.479859 − 2.60187i 0 0 0
2287.2 0 0 0 − 2.85720i 0 −1.73056 − 2.00129i 0 0 0
2287.3 0 0 0 − 2.81241i 0 1.84662 − 1.89473i 0 0 0
2287.4 0 0 0 − 1.99968i 0 2.18955 + 1.48522i 0 0 0
2287.5 0 0 0 − 1.77292i 0 1.23647 + 2.33905i 0 0 0
2287.6 0 0 0 0.280665i 0 −0.164674 + 2.64062i 0 0 0
2287.7 0 0 0 0.784857i 0 −2.01780 − 1.71128i 0 0 0
2287.8 0 0 0 1.47736i 0 2.64567 + 0.0201361i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1711.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.bf.h 24
3.b odd 2 1 1008.2.bf.g ✓ 24
4.b odd 2 1 3024.2.bf.g 24
7.d odd 6 1 3024.2.cz.h 24
9.c even 3 1 3024.2.cz.g 24
9.d odd 6 1 1008.2.cz.h yes 24
12.b even 2 1 1008.2.bf.h yes 24
21.g even 6 1 1008.2.cz.g yes 24
28.f even 6 1 3024.2.cz.g 24
36.f odd 6 1 3024.2.cz.h 24
36.h even 6 1 1008.2.cz.g yes 24
63.k odd 6 1 3024.2.bf.g 24
63.s even 6 1 1008.2.bf.h yes 24
84.j odd 6 1 1008.2.cz.h yes 24
252.n even 6 1 inner 3024.2.bf.h 24
252.bn odd 6 1 1008.2.bf.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.g ✓ 24 3.b odd 2 1
1008.2.bf.g ✓ 24 252.bn odd 6 1
1008.2.bf.h yes 24 12.b even 2 1
1008.2.bf.h yes 24 63.s even 6 1
1008.2.cz.g yes 24 21.g even 6 1
1008.2.cz.g yes 24 36.h even 6 1
1008.2.cz.h yes 24 9.d odd 6 1
1008.2.cz.h yes 24 84.j odd 6 1
3024.2.bf.g 24 4.b odd 2 1
3024.2.bf.g 24 63.k odd 6 1
3024.2.bf.h 24 1.a even 1 1 trivial
3024.2.bf.h 24 252.n even 6 1 inner
3024.2.cz.g 24 9.c even 3 1
3024.2.cz.g 24 28.f even 6 1
3024.2.cz.h 24 7.d odd 6 1
3024.2.cz.h 24 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{24} + 75 T_{5}^{22} + 2442 T_{5}^{20} + 45612 T_{5}^{18} + 542673 T_{5}^{16} + 4308822 T_{5}^{14} + \cdots + 4782969 \) Copy content Toggle raw display
\( T_{19}^{24} - 4 T_{19}^{23} + 161 T_{19}^{22} - 738 T_{19}^{21} + 16894 T_{19}^{20} + \cdots + 572426914921 \) Copy content Toggle raw display