Properties

Label 1440.2.d.a
Level $1440$
Weight $2$
Character orbit 1440.d
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM discriminant -120
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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} -2 \beta_{2} q^{11} -\beta_{3} q^{13} + \beta_{1} q^{17} + 2 \beta_{1} q^{23} -5 q^{25} + 2 \beta_{2} q^{29} -2 q^{31} -\beta_{3} q^{37} -2 \beta_{3} q^{43} -4 \beta_{1} q^{47} + 7 q^{49} + 10 q^{55} -2 \beta_{2} q^{59} -5 \beta_{1} q^{65} -2 \beta_{3} q^{67} -14 q^{79} -\beta_{3} q^{85} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 20q^{25} - 8q^{31} + 28q^{49} + 40q^{55} - 56q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} - 2 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 14 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 7 \beta_{1}\)\()/4\)

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
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
−1.58114 0.707107i
0 0 0 2.23607i 0 0 0 0 0
1009.2 0 0 0 2.23607i 0 0 0 0 0
1009.3 0 0 0 2.23607i 0 0 0 0 0
1009.4 0 0 0 2.23607i 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
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 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.a 4
3.b odd 2 1 inner 1440.2.d.a 4
4.b odd 2 1 360.2.d.a 4
5.b even 2 1 inner 1440.2.d.a 4
5.c odd 4 2 7200.2.k.m 4
8.b even 2 1 inner 1440.2.d.a 4
8.d odd 2 1 360.2.d.a 4
12.b even 2 1 360.2.d.a 4
15.d odd 2 1 inner 1440.2.d.a 4
15.e even 4 2 7200.2.k.m 4
20.d odd 2 1 360.2.d.a 4
20.e even 4 2 1800.2.k.o 4
24.f even 2 1 360.2.d.a 4
24.h odd 2 1 inner 1440.2.d.a 4
40.e odd 2 1 360.2.d.a 4
40.f even 2 1 inner 1440.2.d.a 4
40.i odd 4 2 7200.2.k.m 4
40.k even 4 2 1800.2.k.o 4
60.h even 2 1 360.2.d.a 4
60.l odd 4 2 1800.2.k.o 4
120.i odd 2 1 CM 1440.2.d.a 4
120.m even 2 1 360.2.d.a 4
120.q odd 4 2 1800.2.k.o 4
120.w even 4 2 7200.2.k.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.a 4 4.b odd 2 1
360.2.d.a 4 8.d odd 2 1
360.2.d.a 4 12.b even 2 1
360.2.d.a 4 20.d odd 2 1
360.2.d.a 4 24.f even 2 1
360.2.d.a 4 40.e odd 2 1
360.2.d.a 4 60.h even 2 1
360.2.d.a 4 120.m even 2 1
1440.2.d.a 4 1.a even 1 1 trivial
1440.2.d.a 4 3.b odd 2 1 inner
1440.2.d.a 4 5.b even 2 1 inner
1440.2.d.a 4 8.b even 2 1 inner
1440.2.d.a 4 15.d odd 2 1 inner
1440.2.d.a 4 24.h odd 2 1 inner
1440.2.d.a 4 40.f even 2 1 inner
1440.2.d.a 4 120.i odd 2 1 CM
1800.2.k.o 4 20.e even 4 2
1800.2.k.o 4 40.k even 4 2
1800.2.k.o 4 60.l odd 4 2
1800.2.k.o 4 120.q odd 4 2
7200.2.k.m 4 5.c odd 4 2
7200.2.k.m 4 15.e even 4 2
7200.2.k.m 4 40.i odd 4 2
7200.2.k.m 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 \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7} \)
\( T_{11}^{2} + 20 \)
\( T_{13}^{2} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( -40 + T^{2} )^{2} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 32 + T^{2} )^{2} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( 2 + T )^{4} \)
$37$ \( ( -40 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -160 + T^{2} )^{2} \)
$47$ \( ( 128 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 20 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -160 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 14 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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