Properties

Label 1440.2.f.h
Level $1440$
Weight $2$
Character orbit 1440.f
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + \beta_{2} q^{7} -\beta_{3} q^{11} + 2 \beta_{1} q^{13} -2 \beta_{1} q^{17} + 2 \beta_{2} q^{23} -5 q^{25} + 4 q^{29} -2 \beta_{3} q^{31} -\beta_{3} q^{35} + 2 \beta_{1} q^{37} -10 q^{41} + 2 \beta_{2} q^{43} -4 \beta_{2} q^{47} + 3 q^{49} + 2 \beta_{1} q^{53} -5 \beta_{2} q^{55} -3 \beta_{3} q^{59} + 10 q^{61} + 10 q^{65} -4 \beta_{2} q^{67} -2 \beta_{3} q^{71} -4 \beta_{1} q^{73} + 4 \beta_{1} q^{77} -2 \beta_{3} q^{79} -2 \beta_{2} q^{83} -10 q^{85} + 6 q^{89} + 2 \beta_{3} q^{91} -8 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 20q^{25} + 16q^{29} - 40q^{41} + 12q^{49} + 40q^{61} + 40q^{65} - 40q^{85} + 24q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} - 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\(\beta_{2} + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.618034i
1.61803i
1.61803i
0.618034i
0 0 0 2.23607i 0 2.00000i 0 0 0
289.2 0 0 0 2.23607i 0 2.00000i 0 0 0
289.3 0 0 0 2.23607i 0 2.00000i 0 0 0
289.4 0 0 0 2.23607i 0 2.00000i 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 inner
5.b even 2 1 inner
20.d 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.f.h 4
3.b odd 2 1 480.2.f.e 4
4.b odd 2 1 inner 1440.2.f.h 4
5.b even 2 1 inner 1440.2.f.h 4
5.c odd 4 1 7200.2.a.cc 2
5.c odd 4 1 7200.2.a.cq 2
8.b even 2 1 2880.2.f.v 4
8.d odd 2 1 2880.2.f.v 4
12.b even 2 1 480.2.f.e 4
15.d odd 2 1 480.2.f.e 4
15.e even 4 1 2400.2.a.bi 2
15.e even 4 1 2400.2.a.bj 2
20.d odd 2 1 inner 1440.2.f.h 4
20.e even 4 1 7200.2.a.cc 2
20.e even 4 1 7200.2.a.cq 2
24.f even 2 1 960.2.f.k 4
24.h odd 2 1 960.2.f.k 4
40.e odd 2 1 2880.2.f.v 4
40.f even 2 1 2880.2.f.v 4
48.i odd 4 1 3840.2.d.bg 4
48.i odd 4 1 3840.2.d.bh 4
48.k even 4 1 3840.2.d.bg 4
48.k even 4 1 3840.2.d.bh 4
60.h even 2 1 480.2.f.e 4
60.l odd 4 1 2400.2.a.bi 2
60.l odd 4 1 2400.2.a.bj 2
120.i odd 2 1 960.2.f.k 4
120.m even 2 1 960.2.f.k 4
120.q odd 4 1 4800.2.a.cu 2
120.q odd 4 1 4800.2.a.cv 2
120.w even 4 1 4800.2.a.cu 2
120.w even 4 1 4800.2.a.cv 2
240.t even 4 1 3840.2.d.bg 4
240.t even 4 1 3840.2.d.bh 4
240.bm odd 4 1 3840.2.d.bg 4
240.bm odd 4 1 3840.2.d.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 3.b odd 2 1
480.2.f.e 4 12.b even 2 1
480.2.f.e 4 15.d odd 2 1
480.2.f.e 4 60.h even 2 1
960.2.f.k 4 24.f even 2 1
960.2.f.k 4 24.h odd 2 1
960.2.f.k 4 120.i odd 2 1
960.2.f.k 4 120.m even 2 1
1440.2.f.h 4 1.a even 1 1 trivial
1440.2.f.h 4 4.b odd 2 1 inner
1440.2.f.h 4 5.b even 2 1 inner
1440.2.f.h 4 20.d odd 2 1 inner
2400.2.a.bi 2 15.e even 4 1
2400.2.a.bi 2 60.l odd 4 1
2400.2.a.bj 2 15.e even 4 1
2400.2.a.bj 2 60.l odd 4 1
2880.2.f.v 4 8.b even 2 1
2880.2.f.v 4 8.d odd 2 1
2880.2.f.v 4 40.e odd 2 1
2880.2.f.v 4 40.f even 2 1
3840.2.d.bg 4 48.i odd 4 1
3840.2.d.bg 4 48.k even 4 1
3840.2.d.bg 4 240.t even 4 1
3840.2.d.bg 4 240.bm odd 4 1
3840.2.d.bh 4 48.i odd 4 1
3840.2.d.bh 4 48.k even 4 1
3840.2.d.bh 4 240.t even 4 1
3840.2.d.bh 4 240.bm odd 4 1
4800.2.a.cu 2 120.q odd 4 1
4800.2.a.cu 2 120.w even 4 1
4800.2.a.cv 2 120.q odd 4 1
4800.2.a.cv 2 120.w even 4 1
7200.2.a.cc 2 5.c odd 4 1
7200.2.a.cc 2 20.e even 4 1
7200.2.a.cq 2 5.c odd 4 1
7200.2.a.cq 2 20.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} - 20 \)
\( T_{17}^{2} + 20 \)
\( T_{19} \)
\( T_{29} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( -20 + T^{2} )^{2} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( ( -80 + T^{2} )^{2} \)
$37$ \( ( 20 + T^{2} )^{2} \)
$41$ \( ( 10 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( ( 64 + T^{2} )^{2} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( ( -180 + T^{2} )^{2} \)
$61$ \( ( -10 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( 80 + T^{2} )^{2} \)
$79$ \( ( -80 + T^{2} )^{2} \)
$83$ \( ( 16 + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( 320 + T^{2} )^{2} \)
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