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Properties

Label 3024.2.cj.d
Level $3024$
Weight $2$
Character orbit 3024.cj
Analytic conductor $24.147$
Analytic rank $0$
Dimension $30$
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(1439,3024)]
 
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, 1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cj (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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 12 q^{17} + 9 q^{19} + 12 q^{23} + 18 q^{25} - 27 q^{29} - 6 q^{35} + 6 q^{37} - 9 q^{41} + 21 q^{43} - 3 q^{49} - 3 q^{53} - 6 q^{59} + 6 q^{61} + 18 q^{71} + 21 q^{73} - 72 q^{77} - 15 q^{83} - 3 q^{85} + 6 q^{89} - 26 q^{91} + 54 q^{95} - 6 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1439.1 0 0 0 −3.50422 − 2.02316i 0 0.621968 + 2.57161i 0 0 0
1439.2 0 0 0 −2.76761 − 1.59788i 0 2.07525 − 1.64114i 0 0 0
1439.3 0 0 0 −2.64251 − 1.52565i 0 1.91400 − 1.82664i 0 0 0
1439.4 0 0 0 −2.50753 − 1.44772i 0 −0.0917386 + 2.64416i 0 0 0
1439.5 0 0 0 −1.63811 − 0.945766i 0 −2.17603 − 1.50496i 0 0 0
1439.6 0 0 0 −0.747767 − 0.431724i 0 −0.316457 − 2.62676i 0 0 0
1439.7 0 0 0 −0.702858 − 0.405795i 0 −2.57092 + 0.624796i 0 0 0
1439.8 0 0 0 0.110170 + 0.0636065i 0 2.62201 + 0.353650i 0 0 0
1439.9 0 0 0 0.126591 + 0.0730873i 0 0.562042 + 2.58536i 0 0 0
1439.10 0 0 0 0.789399 + 0.455760i 0 2.37421 + 1.16752i 0 0 0
1439.11 0 0 0 1.10138 + 0.635882i 0 −1.35874 − 2.27020i 0 0 0
1439.12 0 0 0 2.05161 + 1.18450i 0 −1.79697 + 1.94188i 0 0 0
1439.13 0 0 0 2.34742 + 1.35528i 0 2.57527 + 0.606609i 0 0 0
1439.14 0 0 0 2.92699 + 1.68990i 0 1.39970 − 2.24518i 0 0 0
1439.15 0 0 0 3.55705 + 2.05366i 0 −2.33360 − 1.24673i 0 0 0
2879.1 0 0 0 −3.50422 + 2.02316i 0 0.621968 − 2.57161i 0 0 0
2879.2 0 0 0 −2.76761 + 1.59788i 0 2.07525 + 1.64114i 0 0 0
2879.3 0 0 0 −2.64251 + 1.52565i 0 1.91400 + 1.82664i 0 0 0
2879.4 0 0 0 −2.50753 + 1.44772i 0 −0.0917386 − 2.64416i 0 0 0
2879.5 0 0 0 −1.63811 + 0.945766i 0 −2.17603 + 1.50496i 0 0 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1439.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.bb 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.cj.d 30
3.b odd 2 1 1008.2.cj.d yes 30
4.b odd 2 1 3024.2.cj.c 30
7.c even 3 1 3024.2.bh.d 30
9.c even 3 1 1008.2.bh.c ✓ 30
9.d odd 6 1 3024.2.bh.c 30
12.b even 2 1 1008.2.cj.c yes 30
21.h odd 6 1 1008.2.bh.d yes 30
28.g odd 6 1 3024.2.bh.c 30
36.f odd 6 1 1008.2.bh.d yes 30
36.h even 6 1 3024.2.bh.d 30
63.h even 3 1 1008.2.cj.c yes 30
63.j odd 6 1 3024.2.cj.c 30
84.n even 6 1 1008.2.bh.c ✓ 30
252.u odd 6 1 1008.2.cj.d yes 30
252.bb even 6 1 inner 3024.2.cj.d 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bh.c ✓ 30 9.c even 3 1
1008.2.bh.c ✓ 30 84.n even 6 1
1008.2.bh.d yes 30 21.h odd 6 1
1008.2.bh.d yes 30 36.f odd 6 1
1008.2.cj.c yes 30 12.b even 2 1
1008.2.cj.c yes 30 63.h even 3 1
1008.2.cj.d yes 30 3.b odd 2 1
1008.2.cj.d yes 30 252.u odd 6 1
3024.2.bh.c 30 9.d odd 6 1
3024.2.bh.c 30 28.g odd 6 1
3024.2.bh.d 30 7.c even 3 1
3024.2.bh.d 30 36.h even 6 1
3024.2.cj.c 30 4.b odd 2 1
3024.2.cj.c 30 63.j odd 6 1
3024.2.cj.d 30 1.a even 1 1 trivial
3024.2.cj.d 30 252.bb even 6 1 inner

Hecke kernels

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

\( T_{5}^{30} + 3 T_{5}^{29} - 42 T_{5}^{28} - 135 T_{5}^{27} + 1137 T_{5}^{26} + 3795 T_{5}^{25} + \cdots + 84672 \) Copy content Toggle raw display
\( T_{11}^{30} + 3 T_{11}^{29} + 93 T_{11}^{28} + 78 T_{11}^{27} + 4725 T_{11}^{26} - 192 T_{11}^{25} + \cdots + 71911930896 \) Copy content Toggle raw display