Properties

Label 1440.1.p.a
Level $1440$
Weight $1$
Character orbit 1440.p
Self dual yes
Analytic conductor $0.719$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -15, -40, 24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1440.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{6}, \sqrt{-10})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.21600.2

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 2q^{19} + 2q^{23} + q^{25} - 2q^{47} - q^{49} + 2q^{53} - 2q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0
0 0 0 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.p.a 1
3.b odd 2 1 1440.1.p.b 1
4.b odd 2 1 360.1.p.b yes 1
5.b even 2 1 1440.1.p.b 1
8.b even 2 1 360.1.p.a 1
8.d odd 2 1 1440.1.p.b 1
12.b even 2 1 360.1.p.a 1
15.d odd 2 1 CM 1440.1.p.a 1
20.d odd 2 1 360.1.p.a 1
20.e even 4 2 1800.1.g.c 2
24.f even 2 1 RM 1440.1.p.a 1
24.h odd 2 1 360.1.p.b yes 1
36.f odd 6 2 3240.1.z.d 2
36.h even 6 2 3240.1.z.f 2
40.e odd 2 1 CM 1440.1.p.a 1
40.f even 2 1 360.1.p.b yes 1
40.i odd 4 2 1800.1.g.c 2
60.h even 2 1 360.1.p.b yes 1
60.l odd 4 2 1800.1.g.c 2
72.j odd 6 2 3240.1.z.d 2
72.n even 6 2 3240.1.z.f 2
120.i odd 2 1 360.1.p.a 1
120.m even 2 1 1440.1.p.b 1
120.w even 4 2 1800.1.g.c 2
180.n even 6 2 3240.1.z.d 2
180.p odd 6 2 3240.1.z.f 2
360.bh odd 6 2 3240.1.z.f 2
360.bk even 6 2 3240.1.z.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.p.a 1 8.b even 2 1
360.1.p.a 1 12.b even 2 1
360.1.p.a 1 20.d odd 2 1
360.1.p.a 1 120.i odd 2 1
360.1.p.b yes 1 4.b odd 2 1
360.1.p.b yes 1 24.h odd 2 1
360.1.p.b yes 1 40.f even 2 1
360.1.p.b yes 1 60.h even 2 1
1440.1.p.a 1 1.a even 1 1 trivial
1440.1.p.a 1 15.d odd 2 1 CM
1440.1.p.a 1 24.f even 2 1 RM
1440.1.p.a 1 40.e odd 2 1 CM
1440.1.p.b 1 3.b odd 2 1
1440.1.p.b 1 5.b even 2 1
1440.1.p.b 1 8.d odd 2 1
1440.1.p.b 1 120.m even 2 1
1800.1.g.c 2 20.e even 4 2
1800.1.g.c 2 40.i odd 4 2
1800.1.g.c 2 60.l odd 4 2
1800.1.g.c 2 120.w even 4 2
3240.1.z.d 2 36.f odd 6 2
3240.1.z.d 2 72.j odd 6 2
3240.1.z.d 2 180.n even 6 2
3240.1.z.d 2 360.bk even 6 2
3240.1.z.f 2 36.h even 6 2
3240.1.z.f 2 72.n even 6 2
3240.1.z.f 2 180.p odd 6 2
3240.1.z.f 2 360.bh odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1440, [\chi])\).

Hecke characteristic polynomials

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