Properties

Label 2240.4.a.k
Level $2240$
Weight $4$
Character orbit 2240.a
Self dual yes
Analytic conductor $132.164$
Analytic rank $1$
Dimension $1$
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] = [2240,4,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(132.164278413\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{3} - 5 q^{5} - 7 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} - 5 q^{5} - 7 q^{7} - 11 q^{9} - 36 q^{11} + 82 q^{13} + 20 q^{15} + 6 q^{17} - 56 q^{19} + 28 q^{21} - 16 q^{23} + 25 q^{25} + 152 q^{27} - 118 q^{29} - 104 q^{31} + 144 q^{33} + 35 q^{35} + 230 q^{37} - 328 q^{39} - 30 q^{41} + 228 q^{43} + 55 q^{45} - 72 q^{47} + 49 q^{49} - 24 q^{51} + 606 q^{53} + 180 q^{55} + 224 q^{57} + 40 q^{59} - 102 q^{61} + 77 q^{63} - 410 q^{65} + 836 q^{67} + 64 q^{69} + 1020 q^{71} - 578 q^{73} - 100 q^{75} + 252 q^{77} - 1324 q^{79} - 311 q^{81} + 244 q^{83} - 30 q^{85} + 472 q^{87} + 722 q^{89} - 574 q^{91} + 416 q^{93} + 280 q^{95} + 166 q^{97} + 396 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −4.00000 0 −5.00000 0 −7.00000 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 2240.4.a.k 1
4.b odd 2 1 2240.4.a.ba 1
8.b even 2 1 1120.4.a.d yes 1
8.d odd 2 1 1120.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.4.a.b 1 8.d odd 2 1
1120.4.a.d yes 1 8.b even 2 1
2240.4.a.k 1 1.a even 1 1 trivial
2240.4.a.ba 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3} + 4 \) Copy content Toggle raw display
\( T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T - 82 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 56 \) Copy content Toggle raw display
$23$ \( T + 16 \) Copy content Toggle raw display
$29$ \( T + 118 \) Copy content Toggle raw display
$31$ \( T + 104 \) Copy content Toggle raw display
$37$ \( T - 230 \) Copy content Toggle raw display
$41$ \( T + 30 \) Copy content Toggle raw display
$43$ \( T - 228 \) Copy content Toggle raw display
$47$ \( T + 72 \) Copy content Toggle raw display
$53$ \( T - 606 \) Copy content Toggle raw display
$59$ \( T - 40 \) Copy content Toggle raw display
$61$ \( T + 102 \) Copy content Toggle raw display
$67$ \( T - 836 \) Copy content Toggle raw display
$71$ \( T - 1020 \) Copy content Toggle raw display
$73$ \( T + 578 \) Copy content Toggle raw display
$79$ \( T + 1324 \) Copy content Toggle raw display
$83$ \( T - 244 \) Copy content Toggle raw display
$89$ \( T - 722 \) Copy content Toggle raw display
$97$ \( T - 166 \) Copy content Toggle raw display
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