Properties

Label 2240.2.a.r
Level $2240$
Weight $2$
Character orbit 2240.a
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} - q^{7} - 2q^{9} + 3q^{11} + q^{13} - q^{15} - 3q^{17} + 2q^{19} - q^{21} + 6q^{23} + q^{25} - 5q^{27} + 9q^{29} - 8q^{31} + 3q^{33} + q^{35} + 10q^{37} + q^{39} + 2q^{43} + 2q^{45} + 3q^{47} + q^{49} - 3q^{51} - 3q^{55} + 2q^{57} + 12q^{59} - 8q^{61} + 2q^{63} - q^{65} + 8q^{67} + 6q^{69} + 14q^{73} + q^{75} - 3q^{77} - 5q^{79} + q^{81} - 12q^{83} + 3q^{85} + 9q^{87} + 12q^{89} - q^{91} - 8q^{93} - 2q^{95} + 17q^{97} - 6q^{99} + O(q^{100}) \)

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.r 1
4.b odd 2 1 2240.2.a.g 1
8.b even 2 1 560.2.a.c 1
8.d odd 2 1 140.2.a.a 1
24.f even 2 1 1260.2.a.c 1
24.h odd 2 1 5040.2.a.h 1
40.e odd 2 1 700.2.a.d 1
40.f even 2 1 2800.2.a.y 1
40.i odd 4 2 2800.2.g.j 2
40.k even 4 2 700.2.e.c 2
56.e even 2 1 980.2.a.c 1
56.h odd 2 1 3920.2.a.u 1
56.k odd 6 2 980.2.i.d 2
56.m even 6 2 980.2.i.h 2
120.m even 2 1 6300.2.a.d 1
120.q odd 4 2 6300.2.k.c 2
168.e odd 2 1 8820.2.a.r 1
280.n even 2 1 4900.2.a.p 1
280.y odd 4 2 4900.2.e.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 8.d odd 2 1
560.2.a.c 1 8.b even 2 1
700.2.a.d 1 40.e odd 2 1
700.2.e.c 2 40.k even 4 2
980.2.a.c 1 56.e even 2 1
980.2.i.d 2 56.k odd 6 2
980.2.i.h 2 56.m even 6 2
1260.2.a.c 1 24.f even 2 1
2240.2.a.g 1 4.b odd 2 1
2240.2.a.r 1 1.a even 1 1 trivial
2800.2.a.y 1 40.f even 2 1
2800.2.g.j 2 40.i odd 4 2
3920.2.a.u 1 56.h odd 2 1
4900.2.a.p 1 280.n even 2 1
4900.2.e.l 2 280.y odd 4 2
5040.2.a.h 1 24.h odd 2 1
6300.2.a.d 1 120.m even 2 1
6300.2.k.c 2 120.q odd 4 2
8820.2.a.r 1 168.e odd 2 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{11} - 3 \)
\( T_{13} - 1 \)
\( T_{19} - 2 \)

Hecke characteristic polynomials

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