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Properties

Label 6040.2.a.r
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $0$
Dimension $23$
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: \(23\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9} + 5 q^{11} - 2 q^{15} + 16 q^{17} - 14 q^{19} + 3 q^{21} + 11 q^{23} + 23 q^{25} + 14 q^{27} + 15 q^{29} - 14 q^{31} + 23 q^{33} - 3 q^{35} + 10 q^{37} - 5 q^{39} + 28 q^{41} + q^{43} - 31 q^{45} + 24 q^{47} + 40 q^{49} - 6 q^{51} - 11 q^{53} - 5 q^{55} + 42 q^{57} - 9 q^{59} - 14 q^{61} + 33 q^{63} + 9 q^{67} - 7 q^{69} + 22 q^{71} + 29 q^{73} + 2 q^{75} - 10 q^{79} + 67 q^{81} + 46 q^{83} - 16 q^{85} + 22 q^{87} + 33 q^{89} - 32 q^{91} + q^{93} + 14 q^{95} + 57 q^{97} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.35416 0 −1.00000 0 2.39898 0 8.25038 0
1.2 0 −3.04170 0 −1.00000 0 −1.27793 0 6.25193 0
1.3 0 −2.47861 0 −1.00000 0 −4.84176 0 3.14349 0
1.4 0 −2.42679 0 −1.00000 0 −1.20395 0 2.88931 0
1.5 0 −2.28476 0 −1.00000 0 4.31809 0 2.22015 0
1.6 0 −1.86185 0 −1.00000 0 −1.14229 0 0.466481 0
1.7 0 −1.63718 0 −1.00000 0 2.53297 0 −0.319632 0
1.8 0 −1.02367 0 −1.00000 0 3.11653 0 −1.95210 0
1.9 0 −1.00618 0 −1.00000 0 0.737056 0 −1.98759 0
1.10 0 −0.579257 0 −1.00000 0 −2.62931 0 −2.66446 0
1.11 0 −0.0712098 0 −1.00000 0 0.143994 0 −2.99493 0
1.12 0 −0.0188980 0 −1.00000 0 4.41010 0 −2.99964 0
1.13 0 0.0845518 0 −1.00000 0 −0.761778 0 −2.99285 0
1.14 0 1.08343 0 −1.00000 0 −5.06106 0 −1.82619 0
1.15 0 1.30289 0 −1.00000 0 −2.66956 0 −1.30249 0
1.16 0 1.31203 0 −1.00000 0 −4.08175 0 −1.27858 0
1.17 0 1.69908 0 −1.00000 0 2.43676 0 −0.113117 0
1.18 0 1.83763 0 −1.00000 0 1.58536 0 0.376872 0
1.19 0 2.26595 0 −1.00000 0 0.518518 0 2.13452 0
1.20 0 2.58192 0 −1.00000 0 4.44988 0 3.66631 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
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.r ✓ 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.r ✓ 23 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{23} - 2 T_{3}^{22} - 48 T_{3}^{21} + 90 T_{3}^{20} + 978 T_{3}^{19} - 1705 T_{3}^{18} - 11083 T_{3}^{17} + \cdots - 32 \) Copy content Toggle raw display
\( T_{7}^{23} - 3 T_{7}^{22} - 96 T_{7}^{21} + 285 T_{7}^{20} + 3815 T_{7}^{19} - 10958 T_{7}^{18} + \cdots + 5934848 \) Copy content Toggle raw display
\( T_{11}^{23} - 5 T_{11}^{22} - 142 T_{11}^{21} + 722 T_{11}^{20} + 8376 T_{11}^{19} - 43585 T_{11}^{18} + \cdots + 6585892864 \) Copy content Toggle raw display