Properties

Label 6004.2.a.e
Level $6004$
Weight $2$
Character orbit 6004.a
Self dual yes
Analytic conductor $47.942$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6004,2,Mod(1,6004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.a (trivial)

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: yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.35141 0 −3.29178 0 −3.21633 0 8.23192 0
1.2 0 −3.19734 0 3.86202 0 0.0940904 0 7.22301 0
1.3 0 −3.17665 0 0.661665 0 4.43999 0 7.09108 0
1.4 0 −2.94455 0 3.51674 0 −3.98269 0 5.67040 0
1.5 0 −2.58747 0 −3.96262 0 2.03599 0 3.69501 0
1.6 0 −2.48790 0 −1.86011 0 −0.813337 0 3.18963 0
1.7 0 −2.24876 0 −0.730072 0 −4.54869 0 2.05693 0
1.8 0 −1.95303 0 2.03258 0 1.63097 0 0.814318 0
1.9 0 −1.93761 0 0.795901 0 1.65858 0 0.754317 0
1.10 0 −1.53751 0 2.83612 0 −2.00746 0 −0.636048 0
1.11 0 −1.34737 0 −0.232989 0 3.14804 0 −1.18461 0
1.12 0 −1.33751 0 −1.75116 0 −1.54762 0 −1.21107 0
1.13 0 1.34688 0 3.43943 0 0.297446 0 −1.18593 0
1.14 0 1.70311 0 −2.13957 0 −3.96784 0 −0.0994128 0
1.15 0 1.84886 0 0.871764 0 −3.24231 0 0.418286 0
1.16 0 1.98454 0 1.21429 0 4.02832 0 0.938380 0
1.17 0 2.05400 0 −2.46181 0 3.02591 0 1.21890 0
1.18 0 2.22506 0 −3.09603 0 −0.873879 0 1.95088 0
1.19 0 2.59829 0 0.131830 0 4.07797 0 3.75113 0
1.20 0 2.68661 0 2.87584 0 2.53518 0 4.21786 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6004.2.a.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.a.e 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3}^{24} - T_{3}^{23} - 73 T_{3}^{22} + 70 T_{3}^{21} + 2383 T_{3}^{20} - 2172 T_{3}^{19} - 45944 T_{3}^{18} + 39380 T_{3}^{17} + 582024 T_{3}^{16} - 462924 T_{3}^{15} - 5098571 T_{3}^{14} + 3699519 T_{3}^{13} + \cdots + 409318924 \) Copy content Toggle raw display
\( T_{5}^{24} - 9 T_{5}^{23} - 32 T_{5}^{22} + 495 T_{5}^{21} - 80 T_{5}^{20} - 11125 T_{5}^{19} + 16668 T_{5}^{18} + 132059 T_{5}^{17} - 308391 T_{5}^{16} - 883223 T_{5}^{15} + 2801155 T_{5}^{14} + 3156093 T_{5}^{13} + \cdots + 205761 \) Copy content Toggle raw display