Properties

Label 1440.2.a.f
Level $1440$
Weight $2$
Character orbit 1440.a
Self dual yes
Analytic conductor $11.498$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + 4 q^{7} + O(q^{10}) \) \( q - q^{5} + 4 q^{7} - 2 q^{13} + 6 q^{17} - 4 q^{23} + q^{25} + 2 q^{29} + 8 q^{31} - 4 q^{35} + 6 q^{37} + 6 q^{41} - 12 q^{43} - 12 q^{47} + 9 q^{49} + 10 q^{53} + 8 q^{59} - 10 q^{61} + 2 q^{65} + 12 q^{67} + 8 q^{71} + 10 q^{73} - 16 q^{79} + 12 q^{83} - 6 q^{85} + 6 q^{89} - 8 q^{91} + 18 q^{97} + O(q^{100}) \)

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 0 0 −1.00000 0 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

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 1440.2.a.f 1
3.b odd 2 1 480.2.a.h yes 1
4.b odd 2 1 1440.2.a.a 1
5.b even 2 1 7200.2.a.b 1
5.c odd 4 2 7200.2.f.p 2
8.b even 2 1 2880.2.a.bh 1
8.d odd 2 1 2880.2.a.s 1
12.b even 2 1 480.2.a.c 1
15.d odd 2 1 2400.2.a.b 1
15.e even 4 2 2400.2.f.j 2
20.d odd 2 1 7200.2.a.bz 1
20.e even 4 2 7200.2.f.o 2
24.f even 2 1 960.2.a.i 1
24.h odd 2 1 960.2.a.d 1
48.i odd 4 2 3840.2.k.d 2
48.k even 4 2 3840.2.k.w 2
60.h even 2 1 2400.2.a.bg 1
60.l odd 4 2 2400.2.f.i 2
120.i odd 2 1 4800.2.a.bm 1
120.m even 2 1 4800.2.a.bi 1
120.q odd 4 2 4800.2.f.r 2
120.w even 4 2 4800.2.f.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.c 1 12.b even 2 1
480.2.a.h yes 1 3.b odd 2 1
960.2.a.d 1 24.h odd 2 1
960.2.a.i 1 24.f even 2 1
1440.2.a.a 1 4.b odd 2 1
1440.2.a.f 1 1.a even 1 1 trivial
2400.2.a.b 1 15.d odd 2 1
2400.2.a.bg 1 60.h even 2 1
2400.2.f.i 2 60.l odd 4 2
2400.2.f.j 2 15.e even 4 2
2880.2.a.s 1 8.d odd 2 1
2880.2.a.bh 1 8.b even 2 1
3840.2.k.d 2 48.i odd 4 2
3840.2.k.w 2 48.k even 4 2
4800.2.a.bi 1 120.m even 2 1
4800.2.a.bm 1 120.i odd 2 1
4800.2.f.r 2 120.q odd 4 2
4800.2.f.s 2 120.w even 4 2
7200.2.a.b 1 5.b even 2 1
7200.2.a.bz 1 20.d odd 2 1
7200.2.f.o 2 20.e even 4 2
7200.2.f.p 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7} - 4 \)
\( T_{11} \)
\( T_{17} - 6 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -4 + T \)
$11$ \( T \)
$13$ \( 2 + T \)
$17$ \( -6 + T \)
$19$ \( T \)
$23$ \( 4 + T \)
$29$ \( -2 + T \)
$31$ \( -8 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( 12 + T \)
$47$ \( 12 + T \)
$53$ \( -10 + T \)
$59$ \( -8 + T \)
$61$ \( 10 + T \)
$67$ \( -12 + T \)
$71$ \( -8 + T \)
$73$ \( -10 + T \)
$79$ \( 16 + T \)
$83$ \( -12 + T \)
$89$ \( -6 + T \)
$97$ \( -18 + T \)
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