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Properties

Label 3024.2.t.k
Level $3024$
Weight $2$
Character orbit 3024.t
Analytic conductor $24.147$
Analytic rank $0$
Dimension $22$
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(289,3024)]
 
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([0, 0, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.t (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{5} + q^{7} - 6 q^{11} + 7 q^{13} + q^{17} - 13 q^{19} + 44 q^{25} + 7 q^{29} - 6 q^{31} + 2 q^{35} + 6 q^{37} - 4 q^{41} - 2 q^{43} + 17 q^{47} + 29 q^{49} - q^{53} - 2 q^{55} - 21 q^{59} + 31 q^{61} + 3 q^{65} + 26 q^{67} - 32 q^{71} + 17 q^{73} + 4 q^{77} + 16 q^{79} - 36 q^{83} + 28 q^{85} + 2 q^{89} - 15 q^{91} - 24 q^{95} + 19 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 0 0 −3.40736 0 2.05842 − 1.66220i 0 0 0
289.2 0 0 0 −3.19500 0 −2.61289 − 0.415693i 0 0 0
289.3 0 0 0 −2.77180 0 −0.855737 + 2.50354i 0 0 0
289.4 0 0 0 −2.10440 0 1.78475 + 1.95312i 0 0 0
289.5 0 0 0 −0.526004 0 −2.43963 + 1.02383i 0 0 0
289.6 0 0 0 0.0619693 0 1.63689 − 2.07860i 0 0 0
289.7 0 0 0 0.468169 0 2.39007 − 1.13471i 0 0 0
289.8 0 0 0 1.78355 0 −1.90167 − 1.83948i 0 0 0
289.9 0 0 0 2.66851 0 0.654882 + 2.56342i 0 0 0
289.10 0 0 0 3.79940 0 −2.59312 + 0.525101i 0 0 0
289.11 0 0 0 4.22296 0 2.37802 + 1.15974i 0 0 0
1873.1 0 0 0 −3.40736 0 2.05842 + 1.66220i 0 0 0
1873.2 0 0 0 −3.19500 0 −2.61289 + 0.415693i 0 0 0
1873.3 0 0 0 −2.77180 0 −0.855737 − 2.50354i 0 0 0
1873.4 0 0 0 −2.10440 0 1.78475 − 1.95312i 0 0 0
1873.5 0 0 0 −0.526004 0 −2.43963 − 1.02383i 0 0 0
1873.6 0 0 0 0.0619693 0 1.63689 + 2.07860i 0 0 0
1873.7 0 0 0 0.468169 0 2.39007 + 1.13471i 0 0 0
1873.8 0 0 0 1.78355 0 −1.90167 + 1.83948i 0 0 0
1873.9 0 0 0 2.66851 0 0.654882 − 2.56342i 0 0 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.t.k 22
3.b odd 2 1 1008.2.t.l 22
4.b odd 2 1 1512.2.t.c 22
7.c even 3 1 3024.2.q.l 22
9.c even 3 1 3024.2.q.l 22
9.d odd 6 1 1008.2.q.l 22
12.b even 2 1 504.2.t.c yes 22
21.h odd 6 1 1008.2.q.l 22
28.g odd 6 1 1512.2.q.d 22
36.f odd 6 1 1512.2.q.d 22
36.h even 6 1 504.2.q.c ✓ 22
63.g even 3 1 inner 3024.2.t.k 22
63.n odd 6 1 1008.2.t.l 22
84.n even 6 1 504.2.q.c ✓ 22
252.o even 6 1 504.2.t.c yes 22
252.bl odd 6 1 1512.2.t.c 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.c ✓ 22 36.h even 6 1
504.2.q.c ✓ 22 84.n even 6 1
504.2.t.c yes 22 12.b even 2 1
504.2.t.c yes 22 252.o even 6 1
1008.2.q.l 22 9.d odd 6 1
1008.2.q.l 22 21.h odd 6 1
1008.2.t.l 22 3.b odd 2 1
1008.2.t.l 22 63.n odd 6 1
1512.2.q.d 22 28.g odd 6 1
1512.2.q.d 22 36.f odd 6 1
1512.2.t.c 22 4.b odd 2 1
1512.2.t.c 22 252.bl odd 6 1
3024.2.q.l 22 7.c even 3 1
3024.2.q.l 22 9.c even 3 1
3024.2.t.k 22 1.a even 1 1 trivial
3024.2.t.k 22 63.g even 3 1 inner

Hecke kernels

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

\( T_{5}^{11} - T_{5}^{10} - 38 T_{5}^{9} + 21 T_{5}^{8} + 513 T_{5}^{7} - 108 T_{5}^{6} - 2841 T_{5}^{5} + \cdots + 74 \) Copy content Toggle raw display
\( T_{11}^{11} + 3 T_{11}^{10} - 71 T_{11}^{9} - 231 T_{11}^{8} + 1675 T_{11}^{7} + 6195 T_{11}^{6} + \cdots - 65124 \) Copy content Toggle raw display