Properties

Label 2240.2.a.w
Level $2240$
Weight $2$
Character orbit 2240.a
Self dual yes
Analytic conductor $17.886$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.8864900528\)
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

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} + q^{7} - 2q^{9} - q^{11} - 3q^{13} + q^{15} - 7q^{17} - 4q^{19} + q^{21} + q^{25} - 5q^{27} + 5q^{29} - 10q^{31} - q^{33} + q^{35} + 4q^{37} - 3q^{39} - 10q^{41} - 8q^{43} - 2q^{45} - q^{47} + q^{49} - 7q^{51} + 4q^{53} - q^{55} - 4q^{57} + 10q^{61} - 2q^{63} - 3q^{65} + 12q^{67} - 12q^{71} - 2q^{73} + q^{75} - q^{77} + 11q^{79} + q^{81} + 8q^{83} - 7q^{85} + 5q^{87} + 6q^{89} - 3q^{91} - 10q^{93} - 4q^{95} + q^{97} + 2q^{99} + O(q^{100}) \)

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.

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 1.00000 0 1.00000 0 1.00000 0 −2.00000 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 2240.2.a.w 1
4.b odd 2 1 2240.2.a.h 1
8.b even 2 1 1120.2.a.d 1
8.d odd 2 1 1120.2.a.l yes 1
40.e odd 2 1 5600.2.a.f 1
40.f even 2 1 5600.2.a.p 1
56.e even 2 1 7840.2.a.i 1
56.h odd 2 1 7840.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.a.d 1 8.b even 2 1
1120.2.a.l yes 1 8.d odd 2 1
2240.2.a.h 1 4.b odd 2 1
2240.2.a.w 1 1.a even 1 1 trivial
5600.2.a.f 1 40.e odd 2 1
5600.2.a.p 1 40.f even 2 1
7840.2.a.i 1 56.e even 2 1
7840.2.a.t 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3} - 1 \)
\( T_{11} + 1 \)
\( T_{13} + 3 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( 1 + T \)
$13$ \( 3 + T \)
$17$ \( 7 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( -5 + T \)
$31$ \( 10 + T \)
$37$ \( -4 + T \)
$41$ \( 10 + T \)
$43$ \( 8 + T \)
$47$ \( 1 + T \)
$53$ \( -4 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( -12 + T \)
$71$ \( 12 + T \)
$73$ \( 2 + T \)
$79$ \( -11 + T \)
$83$ \( -8 + T \)
$89$ \( -6 + T \)
$97$ \( -1 + T \)
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