Properties

Label 1440.4.a.o
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $1$
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: \(1\)
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

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

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

Hecke kernels

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

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

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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