Properties

Label 6040.2.a.s
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
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] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.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: \(48.2296428209\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 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 −3.28894 0 1.00000 0 4.04677 0 7.81711 0
1.2 0 −3.20286 0 1.00000 0 −3.67370 0 7.25831 0
1.3 0 −3.03909 0 1.00000 0 −1.69120 0 6.23604 0
1.4 0 −2.50341 0 1.00000 0 −2.70762 0 3.26705 0
1.5 0 −2.45370 0 1.00000 0 −1.79083 0 3.02066 0
1.6 0 −2.29572 0 1.00000 0 4.99611 0 2.27034 0
1.7 0 −2.03886 0 1.00000 0 1.67649 0 1.15693 0
1.8 0 −1.08470 0 1.00000 0 4.94762 0 −1.82343 0
1.9 0 −0.875996 0 1.00000 0 −1.42220 0 −2.23263 0
1.10 0 −0.559764 0 1.00000 0 −0.837547 0 −2.68666 0
1.11 0 0.0496662 0 1.00000 0 1.60144 0 −2.99753 0
1.12 0 0.161432 0 1.00000 0 −4.19061 0 −2.97394 0
1.13 0 0.225898 0 1.00000 0 2.54851 0 −2.94897 0
1.14 0 0.549309 0 1.00000 0 −0.285162 0 −2.69826 0
1.15 0 0.568728 0 1.00000 0 −5.03281 0 −2.67655 0
1.16 0 1.21987 0 1.00000 0 0.325030 0 −1.51192 0
1.17 0 1.49843 0 1.00000 0 1.73584 0 −0.754697 0
1.18 0 1.94109 0 1.00000 0 3.18961 0 0.767827 0
1.19 0 2.40154 0 1.00000 0 4.37326 0 2.76742 0
1.20 0 2.57003 0 1.00000 0 −3.15345 0 3.60506 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
\(2\) \(-1\)
\(5\) \(-1\)
\(151\) \(-1\)

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 6040.2.a.s 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{24} - 2 T_{3}^{23} - 54 T_{3}^{22} + 108 T_{3}^{21} + 1242 T_{3}^{20} - 2487 T_{3}^{19} - 15853 T_{3}^{18} + 31874 T_{3}^{17} + 122485 T_{3}^{16} - 248751 T_{3}^{15} - 584720 T_{3}^{14} + 1214112 T_{3}^{13} + \cdots + 512 \) Copy content Toggle raw display
\( T_{7}^{24} - 3 T_{7}^{23} - 110 T_{7}^{22} + 329 T_{7}^{21} + 5053 T_{7}^{20} - 15020 T_{7}^{19} - 126624 T_{7}^{18} + 372827 T_{7}^{17} + 1902930 T_{7}^{16} - 5527710 T_{7}^{15} - 17830623 T_{7}^{14} + \cdots + 58789888 \) Copy content Toggle raw display
\( T_{11}^{24} - 17 T_{11}^{23} - 45 T_{11}^{22} + 2247 T_{11}^{21} - 6181 T_{11}^{20} - 109944 T_{11}^{19} + 604245 T_{11}^{18} + 2170225 T_{11}^{17} - 21327696 T_{11}^{16} + 533851 T_{11}^{15} + 356758849 T_{11}^{14} + \cdots + 628228096 \) Copy content Toggle raw display