Properties

Label 2240.2.a.b
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

\(f(q)\) \(=\) \( q - 3q^{3} - q^{5} + q^{7} + 6q^{9} + O(q^{10}) \) \( q - 3q^{3} - q^{5} + q^{7} + 6q^{9} - 3q^{11} - q^{13} + 3q^{15} - q^{17} + 4q^{19} - 3q^{21} - 4q^{23} + q^{25} - 9q^{27} + 9q^{29} - 6q^{31} + 9q^{33} - q^{35} + 8q^{37} + 3q^{39} + 6q^{41} - 8q^{43} - 6q^{45} + 7q^{47} + q^{49} + 3q^{51} + 8q^{53} + 3q^{55} - 12q^{57} + 4q^{59} - 10q^{61} + 6q^{63} + q^{65} - 8q^{67} + 12q^{69} + 12q^{71} - 14q^{73} - 3q^{75} - 3q^{77} + 5q^{79} + 9q^{81} - 16q^{83} + q^{85} - 27q^{87} - 14q^{89} - q^{91} + 18q^{93} - 4q^{95} - 17q^{97} - 18q^{99} + O(q^{100}) \)

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 −3.00000 0 −1.00000 0 1.00000 0 6.00000 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.b 1
4.b odd 2 1 2240.2.a.y 1
8.b even 2 1 1120.2.a.p yes 1
8.d odd 2 1 1120.2.a.b 1
40.e odd 2 1 5600.2.a.u 1
40.f even 2 1 5600.2.a.b 1
56.e even 2 1 7840.2.a.y 1
56.h odd 2 1 7840.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.a.b 1 8.d odd 2 1
1120.2.a.p yes 1 8.b even 2 1
2240.2.a.b 1 1.a even 1 1 trivial
2240.2.a.y 1 4.b odd 2 1
5600.2.a.b 1 40.f even 2 1
5600.2.a.u 1 40.e odd 2 1
7840.2.a.a 1 56.h odd 2 1
7840.2.a.y 1 56.e even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( 3 + T \)
$13$ \( 1 + T \)
$17$ \( 1 + T \)
$19$ \( -4 + T \)
$23$ \( 4 + T \)
$29$ \( -9 + T \)
$31$ \( 6 + T \)
$37$ \( -8 + T \)
$41$ \( -6 + T \)
$43$ \( 8 + T \)
$47$ \( -7 + T \)
$53$ \( -8 + T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( 8 + T \)
$71$ \( -12 + T \)
$73$ \( 14 + T \)
$79$ \( -5 + T \)
$83$ \( 16 + T \)
$89$ \( 14 + T \)
$97$ \( 17 + T \)
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