Properties

Label 1440.1.bh.a
Level 1440
Weight 1
Character orbit 1440.bh
Analytic conductor 0.719
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -4
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.bh (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.18000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - q^{5} +O(q^{10})\) \( q - q^{5} + ( 1 + i ) q^{13} + ( 1 - i ) q^{17} + q^{25} + 2 i q^{29} + ( 1 - i ) q^{37} + 2 q^{41} + i q^{49} + ( -1 - i ) q^{53} + ( -1 - i ) q^{65} + ( -1 - i ) q^{73} + ( -1 + i ) q^{85} + 2 i q^{89} + ( -1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{13} + 2q^{17} + 2q^{25} + 2q^{37} + 4q^{41} - 2q^{53} - 2q^{65} - 2q^{73} - 2q^{85} - 2q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 0 0 −1.00000 0 0 0 0 0
1153.1 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.bh.a 2
3.b odd 2 1 1440.1.bh.c yes 2
4.b odd 2 1 CM 1440.1.bh.a 2
5.c odd 4 1 inner 1440.1.bh.a 2
8.b even 2 1 2880.1.bh.c 2
8.d odd 2 1 2880.1.bh.c 2
12.b even 2 1 1440.1.bh.c yes 2
15.e even 4 1 1440.1.bh.c yes 2
20.e even 4 1 inner 1440.1.bh.a 2
24.f even 2 1 2880.1.bh.a 2
24.h odd 2 1 2880.1.bh.a 2
40.i odd 4 1 2880.1.bh.c 2
40.k even 4 1 2880.1.bh.c 2
60.l odd 4 1 1440.1.bh.c yes 2
120.q odd 4 1 2880.1.bh.a 2
120.w even 4 1 2880.1.bh.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.bh.a 2 1.a even 1 1 trivial
1440.1.bh.a 2 4.b odd 2 1 CM
1440.1.bh.a 2 5.c odd 4 1 inner
1440.1.bh.a 2 20.e even 4 1 inner
1440.1.bh.c yes 2 3.b odd 2 1
1440.1.bh.c yes 2 12.b even 2 1
1440.1.bh.c yes 2 15.e even 4 1
1440.1.bh.c yes 2 60.l odd 4 1
2880.1.bh.a 2 24.f even 2 1
2880.1.bh.a 2 24.h odd 2 1
2880.1.bh.a 2 120.q odd 4 1
2880.1.bh.a 2 120.w even 4 1
2880.1.bh.c 2 8.b even 2 1
2880.1.bh.c 2 8.d odd 2 1
2880.1.bh.c 2 40.i odd 4 1
2880.1.bh.c 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1440, [\chi])\):

\( T_{13}^{2} - 2 T_{13} + 2 \)
\( T_{17}^{2} - 2 T_{17} + 2 \)

Hecke Characteristic Polynomials

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