Properties

Label 1152.3.b.f
Level $1152$
Weight $3$
Character orbit 1152.b
Analytic conductor $31.390$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} + \beta_{3} q^{11} + 21 q^{25} + 25 \beta_{1} q^{29} -5 \beta_{2} q^{31} -\beta_{3} q^{35} -47 q^{49} + 47 \beta_{1} q^{53} + 4 \beta_{2} q^{55} -6 \beta_{3} q^{59} + 50 q^{73} + 96 \beta_{1} q^{77} + 15 \beta_{2} q^{79} + 5 \beta_{3} q^{83} -190 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 84q^{25} - 188q^{49} + 200q^{73} - 760q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} + 12 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 24 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2}\)\()/16\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 6 \beta_{2}\)\()/16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 0 0 2.00000i 0 9.79796i 0 0 0
703.2 0 0 0 2.00000i 0 9.79796i 0 0 0
703.3 0 0 0 2.00000i 0 9.79796i 0 0 0
703.4 0 0 0 2.00000i 0 9.79796i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.b.f 4
3.b odd 2 1 inner 1152.3.b.f 4
4.b odd 2 1 inner 1152.3.b.f 4
8.b even 2 1 inner 1152.3.b.f 4
8.d odd 2 1 inner 1152.3.b.f 4
12.b even 2 1 inner 1152.3.b.f 4
16.e even 4 1 2304.3.g.i 2
16.e even 4 1 2304.3.g.l 2
16.f odd 4 1 2304.3.g.i 2
16.f odd 4 1 2304.3.g.l 2
24.f even 2 1 inner 1152.3.b.f 4
24.h odd 2 1 CM 1152.3.b.f 4
48.i odd 4 1 2304.3.g.i 2
48.i odd 4 1 2304.3.g.l 2
48.k even 4 1 2304.3.g.i 2
48.k even 4 1 2304.3.g.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.f 4 1.a even 1 1 trivial
1152.3.b.f 4 3.b odd 2 1 inner
1152.3.b.f 4 4.b odd 2 1 inner
1152.3.b.f 4 8.b even 2 1 inner
1152.3.b.f 4 8.d odd 2 1 inner
1152.3.b.f 4 12.b even 2 1 inner
1152.3.b.f 4 24.f even 2 1 inner
1152.3.b.f 4 24.h odd 2 1 CM
2304.3.g.i 2 16.e even 4 1
2304.3.g.i 2 16.f odd 4 1
2304.3.g.i 2 48.i odd 4 1
2304.3.g.i 2 48.k even 4 1
2304.3.g.l 2 16.e even 4 1
2304.3.g.l 2 16.f odd 4 1
2304.3.g.l 2 48.i odd 4 1
2304.3.g.l 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{7}^{2} + 96 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 4 + T^{2} )^{2} \)
$7$ \( ( 96 + T^{2} )^{2} \)
$11$ \( ( -384 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 2500 + T^{2} )^{2} \)
$31$ \( ( 2400 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 8836 + T^{2} )^{2} \)
$59$ \( ( -13824 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -50 + T )^{4} \)
$79$ \( ( 21600 + T^{2} )^{2} \)
$83$ \( ( -9600 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 190 + T )^{4} \)
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