Properties

Label 1440.4.a.n
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5q^{5} - 6q^{7} + O(q^{10}) \) \( q + 5q^{5} - 6q^{7} + 60q^{11} + 50q^{13} + 30q^{17} - 40q^{19} + 178q^{23} + 25q^{25} - 166q^{29} - 20q^{31} - 30q^{35} + 10q^{37} + 250q^{41} - 142q^{43} + 214q^{47} - 307q^{49} - 490q^{53} + 300q^{55} - 800q^{59} + 250q^{61} + 250q^{65} + 774q^{67} + 100q^{71} - 230q^{73} - 360q^{77} + 1320q^{79} + 982q^{83} + 150q^{85} - 874q^{89} - 300q^{91} - 200q^{95} - 310q^{97} + O(q^{100}) \)

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 5.00000 0 −6.00000 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 1440.4.a.n 1
3.b odd 2 1 160.4.a.b yes 1
4.b odd 2 1 1440.4.a.o 1
12.b even 2 1 160.4.a.a 1
15.d odd 2 1 800.4.a.d 1
15.e even 4 2 800.4.c.e 2
24.f even 2 1 320.4.a.i 1
24.h odd 2 1 320.4.a.f 1
48.i odd 4 2 1280.4.d.k 2
48.k even 4 2 1280.4.d.f 2
60.h even 2 1 800.4.a.h 1
60.l odd 4 2 800.4.c.f 2
120.i odd 2 1 1600.4.a.bj 1
120.m even 2 1 1600.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 12.b even 2 1
160.4.a.b yes 1 3.b odd 2 1
320.4.a.f 1 24.h odd 2 1
320.4.a.i 1 24.f even 2 1
800.4.a.d 1 15.d odd 2 1
800.4.a.h 1 60.h even 2 1
800.4.c.e 2 15.e even 4 2
800.4.c.f 2 60.l odd 4 2
1280.4.d.f 2 48.k even 4 2
1280.4.d.k 2 48.i odd 4 2
1440.4.a.n 1 1.a even 1 1 trivial
1440.4.a.o 1 4.b odd 2 1
1600.4.a.r 1 120.m even 2 1
1600.4.a.bj 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7} + 6 \)
\( T_{11} - 60 \)
\( T_{17} - 30 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -5 + T \)
$7$ \( 6 + T \)
$11$ \( -60 + T \)
$13$ \( -50 + T \)
$17$ \( -30 + T \)
$19$ \( 40 + T \)
$23$ \( -178 + T \)
$29$ \( 166 + T \)
$31$ \( 20 + T \)
$37$ \( -10 + T \)
$41$ \( -250 + T \)
$43$ \( 142 + T \)
$47$ \( -214 + T \)
$53$ \( 490 + T \)
$59$ \( 800 + T \)
$61$ \( -250 + T \)
$67$ \( -774 + T \)
$71$ \( -100 + T \)
$73$ \( 230 + T \)
$79$ \( -1320 + T \)
$83$ \( -982 + T \)
$89$ \( 874 + T \)
$97$ \( 310 + T \)
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