Properties

Label 1440.2.a.o
Level $1440$
Weight $2$
Character orbit 1440.a
Self dual yes
Analytic conductor $11.498$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4984578911\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7} +O(q^{10})\) \( q - q^{5} + \beta q^{7} -2 \beta q^{11} -2 q^{13} -2 q^{17} + \beta q^{23} + q^{25} -6 q^{29} -2 \beta q^{31} -\beta q^{35} -10 q^{37} -2 q^{41} + 3 \beta q^{43} -\beta q^{47} + q^{49} -6 q^{53} + 2 \beta q^{55} + 4 \beta q^{59} -2 q^{61} + 2 q^{65} + \beta q^{67} + 2 \beta q^{71} -6 q^{73} -16 q^{77} -4 \beta q^{79} + \beta q^{83} + 2 q^{85} -10 q^{89} -2 \beta q^{91} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} - 4 q^{13} - 4 q^{17} + 2 q^{25} - 12 q^{29} - 20 q^{37} - 4 q^{41} + 2 q^{49} - 12 q^{53} - 4 q^{61} + 4 q^{65} - 12 q^{73} - 32 q^{77} + 4 q^{85} - 20 q^{89} + 4 q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −1.00000 0 −2.82843 0 0 0
1.2 0 0 0 −1.00000 0 2.82843 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.a.o 2
3.b odd 2 1 160.2.a.c 2
4.b odd 2 1 inner 1440.2.a.o 2
5.b even 2 1 7200.2.a.cm 2
5.c odd 4 2 7200.2.f.bh 4
8.b even 2 1 2880.2.a.bk 2
8.d odd 2 1 2880.2.a.bk 2
12.b even 2 1 160.2.a.c 2
15.d odd 2 1 800.2.a.m 2
15.e even 4 2 800.2.c.f 4
20.d odd 2 1 7200.2.a.cm 2
20.e even 4 2 7200.2.f.bh 4
21.c even 2 1 7840.2.a.bf 2
24.f even 2 1 320.2.a.g 2
24.h odd 2 1 320.2.a.g 2
48.i odd 4 2 1280.2.d.l 4
48.k even 4 2 1280.2.d.l 4
60.h even 2 1 800.2.a.m 2
60.l odd 4 2 800.2.c.f 4
84.h odd 2 1 7840.2.a.bf 2
120.i odd 2 1 1600.2.a.bc 2
120.m even 2 1 1600.2.a.bc 2
120.q odd 4 2 1600.2.c.n 4
120.w even 4 2 1600.2.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 3.b odd 2 1
160.2.a.c 2 12.b even 2 1
320.2.a.g 2 24.f even 2 1
320.2.a.g 2 24.h odd 2 1
800.2.a.m 2 15.d odd 2 1
800.2.a.m 2 60.h even 2 1
800.2.c.f 4 15.e even 4 2
800.2.c.f 4 60.l odd 4 2
1280.2.d.l 4 48.i odd 4 2
1280.2.d.l 4 48.k even 4 2
1440.2.a.o 2 1.a even 1 1 trivial
1440.2.a.o 2 4.b odd 2 1 inner
1600.2.a.bc 2 120.i odd 2 1
1600.2.a.bc 2 120.m even 2 1
1600.2.c.n 4 120.q odd 4 2
1600.2.c.n 4 120.w even 4 2
2880.2.a.bk 2 8.b even 2 1
2880.2.a.bk 2 8.d odd 2 1
7200.2.a.cm 2 5.b even 2 1
7200.2.a.cm 2 20.d odd 2 1
7200.2.f.bh 4 5.c odd 4 2
7200.2.f.bh 4 20.e even 4 2
7840.2.a.bf 2 21.c even 2 1
7840.2.a.bf 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7}^{2} - 8 \)
\( T_{11}^{2} - 32 \)
\( T_{17} + 2 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( -32 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -72 + T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -8 + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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