Properties

Label 1440.2.d.b
Level $1440$
Weight $2$
Character orbit 1440.d
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + ( -1 - \beta_{3} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{29} -10 q^{31} + ( -\beta_{1} + 5 \beta_{2} ) q^{35} -17 q^{49} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{53} + ( -6 - \beta_{3} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{59} -2 \beta_{3} q^{73} + ( -6 \beta_{1} + 6 \beta_{2} ) q^{77} + 10 q^{79} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{83} -4 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{25} - 40q^{31} - 68q^{49} - 24q^{55} + 40q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 2 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i
0 0 0 −1.41421 1.73205i 0 4.89898i 0 0 0
1009.2 0 0 0 −1.41421 + 1.73205i 0 4.89898i 0 0 0
1009.3 0 0 0 1.41421 1.73205i 0 4.89898i 0 0 0
1009.4 0 0 0 1.41421 + 1.73205i 0 4.89898i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.d.b 4
3.b odd 2 1 inner 1440.2.d.b 4
4.b odd 2 1 360.2.d.d 4
5.b even 2 1 inner 1440.2.d.b 4
5.c odd 4 2 7200.2.k.k 4
8.b even 2 1 inner 1440.2.d.b 4
8.d odd 2 1 360.2.d.d 4
12.b even 2 1 360.2.d.d 4
15.d odd 2 1 inner 1440.2.d.b 4
15.e even 4 2 7200.2.k.k 4
20.d odd 2 1 360.2.d.d 4
20.e even 4 2 1800.2.k.k 4
24.f even 2 1 360.2.d.d 4
24.h odd 2 1 CM 1440.2.d.b 4
40.e odd 2 1 360.2.d.d 4
40.f even 2 1 inner 1440.2.d.b 4
40.i odd 4 2 7200.2.k.k 4
40.k even 4 2 1800.2.k.k 4
60.h even 2 1 360.2.d.d 4
60.l odd 4 2 1800.2.k.k 4
120.i odd 2 1 inner 1440.2.d.b 4
120.m even 2 1 360.2.d.d 4
120.q odd 4 2 1800.2.k.k 4
120.w even 4 2 7200.2.k.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.d 4 4.b odd 2 1
360.2.d.d 4 8.d odd 2 1
360.2.d.d 4 12.b even 2 1
360.2.d.d 4 20.d odd 2 1
360.2.d.d 4 24.f even 2 1
360.2.d.d 4 40.e odd 2 1
360.2.d.d 4 60.h even 2 1
360.2.d.d 4 120.m even 2 1
1440.2.d.b 4 1.a even 1 1 trivial
1440.2.d.b 4 3.b odd 2 1 inner
1440.2.d.b 4 5.b even 2 1 inner
1440.2.d.b 4 8.b even 2 1 inner
1440.2.d.b 4 15.d odd 2 1 inner
1440.2.d.b 4 24.h odd 2 1 CM
1440.2.d.b 4 40.f even 2 1 inner
1440.2.d.b 4 120.i odd 2 1 inner
1800.2.k.k 4 20.e even 4 2
1800.2.k.k 4 40.k even 4 2
1800.2.k.k 4 60.l odd 4 2
1800.2.k.k 4 120.q odd 4 2
7200.2.k.k 4 5.c odd 4 2
7200.2.k.k 4 15.e even 4 2
7200.2.k.k 4 40.i odd 4 2
7200.2.k.k 4 120.w even 4 2

Hecke kernels

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

\( T_{7}^{2} + 24 \)
\( T_{11}^{2} + 12 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 2 T^{2} + T^{4} \)
$7$ \( ( 24 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 108 + T^{2} )^{2} \)
$31$ \( ( 10 + T )^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -200 + T^{2} )^{2} \)
$59$ \( ( 108 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 96 + T^{2} )^{2} \)
$79$ \( ( -10 + T )^{4} \)
$83$ \( ( -32 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 384 + T^{2} )^{2} \)
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