Properties

Label 1440.2.h.d
Level $1440$
Weight $2$
Character orbit 1440.h
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} - q^{25} + 6 \zeta_{8}^{2} q^{29} + ( -6 \zeta_{8} - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{31} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{35} + ( -2 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{37} + ( -5 \zeta_{8} + 4 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} + ( 1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{49} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{55} + ( 8 + \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( 10 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{65} + 8 \zeta_{8}^{2} q^{67} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{73} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{79} + ( 12 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{91} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{11} - 8q^{13} + 8q^{23} - 4q^{25} + 8q^{35} - 8q^{37} + 4q^{49} + 32q^{59} + 40q^{61} - 8q^{73} + 48q^{83} + 16q^{85} - 24q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 1.00000i 0 0.585786i 0 0 0
1151.2 0 0 0 1.00000i 0 3.41421i 0 0 0
1151.3 0 0 0 1.00000i 0 3.41421i 0 0 0
1151.4 0 0 0 1.00000i 0 0.585786i 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
12.b 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.h.d yes 4
3.b odd 2 1 1440.2.h.a 4
4.b odd 2 1 1440.2.h.a 4
5.b even 2 1 7200.2.h.i 4
5.c odd 4 1 7200.2.o.e 4
5.c odd 4 1 7200.2.o.m 4
8.b even 2 1 2880.2.h.a 4
8.d odd 2 1 2880.2.h.d 4
12.b even 2 1 inner 1440.2.h.d yes 4
15.d odd 2 1 7200.2.h.c 4
15.e even 4 1 7200.2.o.b 4
15.e even 4 1 7200.2.o.j 4
20.d odd 2 1 7200.2.h.c 4
20.e even 4 1 7200.2.o.b 4
20.e even 4 1 7200.2.o.j 4
24.f even 2 1 2880.2.h.a 4
24.h odd 2 1 2880.2.h.d 4
60.h even 2 1 7200.2.h.i 4
60.l odd 4 1 7200.2.o.e 4
60.l odd 4 1 7200.2.o.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.h.a 4 3.b odd 2 1
1440.2.h.a 4 4.b odd 2 1
1440.2.h.d yes 4 1.a even 1 1 trivial
1440.2.h.d yes 4 12.b even 2 1 inner
2880.2.h.a 4 8.b even 2 1
2880.2.h.a 4 24.f even 2 1
2880.2.h.d 4 8.d odd 2 1
2880.2.h.d 4 24.h odd 2 1
7200.2.h.c 4 15.d odd 2 1
7200.2.h.c 4 20.d odd 2 1
7200.2.h.i 4 5.b even 2 1
7200.2.h.i 4 60.h even 2 1
7200.2.o.b 4 15.e even 4 1
7200.2.o.b 4 20.e even 4 1
7200.2.o.e 4 5.c odd 4 1
7200.2.o.e 4 60.l odd 4 1
7200.2.o.j 4 15.e even 4 1
7200.2.o.j 4 20.e even 4 1
7200.2.o.m 4 5.c odd 4 1
7200.2.o.m 4 60.l odd 4 1

Hecke kernels

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

\( T_{11}^{2} - 8 T_{11} + 14 \)
\( T_{23}^{2} - 4 T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 4 + 12 T^{2} + T^{4} \)
$11$ \( ( 14 - 8 T + T^{2} )^{2} \)
$13$ \( ( 2 + 4 T + T^{2} )^{2} \)
$17$ \( 64 + 48 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -4 - 4 T + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( 4624 + 152 T^{2} + T^{4} \)
$37$ \( ( -46 + 4 T + T^{2} )^{2} \)
$41$ \( 1156 + 132 T^{2} + T^{4} \)
$43$ \( 64 + 48 T^{2} + T^{4} \)
$47$ \( ( -32 + T^{2} )^{2} \)
$53$ \( 64 + 48 T^{2} + T^{4} \)
$59$ \( ( 62 - 16 T + T^{2} )^{2} \)
$61$ \( ( 92 - 20 T + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( ( -68 + 4 T + T^{2} )^{2} \)
$79$ \( 1296 + 216 T^{2} + T^{4} \)
$83$ \( ( 136 - 24 T + T^{2} )^{2} \)
$89$ \( ( 18 + T^{2} )^{2} \)
$97$ \( ( -36 + 12 T + T^{2} )^{2} \)
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