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Properties

Label 3024.2.cz.h
Level $3024$
Weight $2$
Character orbit 3024.cz
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(1279,3024)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1279");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cz (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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 4 q^{7} + 9 q^{11} - 3 q^{13} + 3 q^{17} - 4 q^{19} + 6 q^{23} + 15 q^{25} - 18 q^{29} + 34 q^{31} + 42 q^{35} - 3 q^{37} - 36 q^{41} + 24 q^{43} + 42 q^{47} + 30 q^{49} + 12 q^{53} - 30 q^{55} + 12 q^{59} + 48 q^{73} + 48 q^{77} + 48 q^{83} - 21 q^{85} - 39 q^{89} + 9 q^{91} + 3 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1 0 0 0 −2.87107 + 1.65762i 0 −2.01336 − 1.71651i 0 0 0
1279.2 0 0 0 −2.47441 + 1.42860i 0 −2.59845 + 0.498066i 0 0 0
1279.3 0 0 0 −2.43562 + 1.40621i 0 −0.717569 − 2.54659i 0 0 0
1279.4 0 0 0 −1.73177 + 0.999840i 0 2.38101 − 1.15360i 0 0 0
1279.5 0 0 0 −1.53539 + 0.886460i 0 2.64391 + 0.0987102i 0 0 0
1279.6 0 0 0 0.243063 − 0.140332i 0 2.20451 + 1.46292i 0 0 0
1279.7 0 0 0 0.679706 − 0.392428i 0 −2.49091 + 0.891831i 0 0 0
1279.8 0 0 0 1.27943 − 0.738680i 0 1.34028 − 2.28115i 0 0 0
1279.9 0 0 0 2.08545 − 1.20403i 0 −2.53309 − 0.763830i 0 0 0
1279.10 0 0 0 2.10605 − 1.21593i 0 0.347128 + 2.62288i 0 0 0
1279.11 0 0 0 2.47996 − 1.43180i 0 1.38867 − 2.25202i 0 0 0
1279.12 0 0 0 3.67461 − 2.12154i 0 2.04787 + 1.67518i 0 0 0
2719.1 0 0 0 −2.87107 − 1.65762i 0 −2.01336 + 1.71651i 0 0 0
2719.2 0 0 0 −2.47441 − 1.42860i 0 −2.59845 − 0.498066i 0 0 0
2719.3 0 0 0 −2.43562 − 1.40621i 0 −0.717569 + 2.54659i 0 0 0
2719.4 0 0 0 −1.73177 − 0.999840i 0 2.38101 + 1.15360i 0 0 0
2719.5 0 0 0 −1.53539 − 0.886460i 0 2.64391 − 0.0987102i 0 0 0
2719.6 0 0 0 0.243063 + 0.140332i 0 2.20451 − 1.46292i 0 0 0
2719.7 0 0 0 0.679706 + 0.392428i 0 −2.49091 − 0.891831i 0 0 0
2719.8 0 0 0 1.27943 + 0.738680i 0 1.34028 + 2.28115i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.bj 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.cz.h 24
3.b odd 2 1 1008.2.cz.g yes 24
4.b odd 2 1 3024.2.cz.g 24
7.d odd 6 1 3024.2.bf.h 24
9.c even 3 1 3024.2.bf.g 24
9.d odd 6 1 1008.2.bf.h yes 24
12.b even 2 1 1008.2.cz.h yes 24
21.g even 6 1 1008.2.bf.g ✓ 24
28.f even 6 1 3024.2.bf.g 24
36.f odd 6 1 3024.2.bf.h 24
36.h even 6 1 1008.2.bf.g ✓ 24
63.i even 6 1 1008.2.cz.h yes 24
63.t odd 6 1 3024.2.cz.g 24
84.j odd 6 1 1008.2.bf.h yes 24
252.r odd 6 1 1008.2.cz.g yes 24
252.bj even 6 1 inner 3024.2.cz.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.g ✓ 24 21.g even 6 1
1008.2.bf.g ✓ 24 36.h even 6 1
1008.2.bf.h yes 24 9.d odd 6 1
1008.2.bf.h yes 24 84.j odd 6 1
1008.2.cz.g yes 24 3.b odd 2 1
1008.2.cz.g yes 24 252.r odd 6 1
1008.2.cz.h yes 24 12.b even 2 1
1008.2.cz.h yes 24 63.i even 6 1
3024.2.bf.g 24 9.c even 3 1
3024.2.bf.g 24 28.f even 6 1
3024.2.bf.h 24 7.d odd 6 1
3024.2.bf.h 24 36.f odd 6 1
3024.2.cz.g 24 4.b odd 2 1
3024.2.cz.g 24 63.t odd 6 1
3024.2.cz.h 24 1.a even 1 1 trivial
3024.2.cz.h 24 252.bj 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 \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{24} - 3 T_{5}^{23} - 33 T_{5}^{22} + 108 T_{5}^{21} + 723 T_{5}^{20} - 2178 T_{5}^{19} + \cdots + 4782969 \) Copy content Toggle raw display
\( T_{11}^{24} - 9 T_{11}^{23} - 36 T_{11}^{22} + 567 T_{11}^{21} + 984 T_{11}^{20} - 20682 T_{11}^{19} + \cdots + 109592116209 \) Copy content Toggle raw display