Properties

Label 1440.2.h.a
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: \( x^{4} + 1 \) Copy content Toggle raw display
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{3} - 4) q^{11} + (\beta_{3} - 2) q^{13} + ( - 2 \beta_{2} - 4 \beta_1) q^{17} + (2 \beta_{3} - 2) q^{23} - q^{25} - 6 \beta_1 q^{29} + ( - 6 \beta_{2} - 2 \beta_1) q^{31} + (\beta_{3} - 2) q^{35} + (5 \beta_{3} - 2) q^{37} + (5 \beta_{2} - 4 \beta_1) q^{41} + (2 \beta_{2} - 4 \beta_1) q^{43} - 4 \beta_{3} q^{47} + (4 \beta_{3} + 1) q^{49} + ( - 2 \beta_{2} - 4 \beta_1) q^{53} + ( - \beta_{2} - 4 \beta_1) q^{55} + ( - \beta_{3} - 8) q^{59} + ( - 2 \beta_{3} + 10) q^{61} + (\beta_{2} - 2 \beta_1) q^{65} + 8 \beta_1 q^{67} + 4 \beta_{3} q^{71} + (6 \beta_{3} - 2) q^{73} + (2 \beta_{2} - 6 \beta_1) q^{77} + ( - 6 \beta_{2} + 6 \beta_1) q^{79} + ( - 2 \beta_{3} - 12) q^{83} + (2 \beta_{3} + 4) q^{85} - 3 \beta_{2} q^{89} + (4 \beta_{2} - 6 \beta_1) q^{91} + ( - 6 \beta_{3} - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{11} - 8 q^{13} - 8 q^{23} - 4 q^{25} - 8 q^{35} - 8 q^{37} + 4 q^{49} - 32 q^{59} + 40 q^{61} - 8 q^{73} - 48 q^{83} + 16 q^{85} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

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 3.41421i 0 0 0
1151.2 0 0 0 1.00000i 0 0.585786i 0 0 0
1151.3 0 0 0 1.00000i 0 0.585786i 0 0 0
1151.4 0 0 0 1.00000i 0 3.41421i 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.a 4
3.b odd 2 1 1440.2.h.d yes 4
4.b odd 2 1 1440.2.h.d yes 4
5.b even 2 1 7200.2.h.c 4
5.c odd 4 1 7200.2.o.b 4
5.c odd 4 1 7200.2.o.j 4
8.b even 2 1 2880.2.h.d 4
8.d odd 2 1 2880.2.h.a 4
12.b even 2 1 inner 1440.2.h.a 4
15.d odd 2 1 7200.2.h.i 4
15.e even 4 1 7200.2.o.e 4
15.e even 4 1 7200.2.o.m 4
20.d odd 2 1 7200.2.h.i 4
20.e even 4 1 7200.2.o.e 4
20.e even 4 1 7200.2.o.m 4
24.f even 2 1 2880.2.h.d 4
24.h odd 2 1 2880.2.h.a 4
60.h even 2 1 7200.2.h.c 4
60.l odd 4 1 7200.2.o.b 4
60.l odd 4 1 7200.2.o.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.h.a 4 1.a even 1 1 trivial
1440.2.h.a 4 12.b even 2 1 inner
1440.2.h.d yes 4 3.b odd 2 1
1440.2.h.d yes 4 4.b odd 2 1
2880.2.h.a 4 8.d odd 2 1
2880.2.h.a 4 24.h odd 2 1
2880.2.h.d 4 8.b even 2 1
2880.2.h.d 4 24.f even 2 1
7200.2.h.c 4 5.b even 2 1
7200.2.h.c 4 60.h even 2 1
7200.2.h.i 4 15.d odd 2 1
7200.2.h.i 4 20.d odd 2 1
7200.2.o.b 4 5.c odd 4 1
7200.2.o.b 4 60.l odd 4 1
7200.2.o.e 4 15.e even 4 1
7200.2.o.e 4 20.e even 4 1
7200.2.o.j 4 5.c odd 4 1
7200.2.o.j 4 60.l odd 4 1
7200.2.o.m 4 15.e even 4 1
7200.2.o.m 4 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 \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{11}^{2} + 8T_{11} + 14 \) Copy content Toggle raw display
\( T_{23}^{2} + 4T_{23} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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