Properties

Label 1152.1.b.b
Level $1152$
Weight $1$
Character orbit 1152.b
Analytic conductor $0.575$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -24, 24
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{6})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.191102976.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -2 i q^{5} +O(q^{10})\) \( q -2 i q^{5} -3 q^{25} -2 i q^{29} + q^{49} + 2 i q^{53} + 2 q^{73} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 6q^{25} + 2q^{49} + 4q^{73} + 4q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 2.00000i 0 0 0 0 0
703.2 0 0 0 2.00000i 0 0 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 CM by \(\Q(\sqrt{-1}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
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 1152.1.b.b 2
3.b odd 2 1 inner 1152.1.b.b 2
4.b odd 2 1 CM 1152.1.b.b 2
8.b even 2 1 inner 1152.1.b.b 2
8.d odd 2 1 inner 1152.1.b.b 2
12.b even 2 1 inner 1152.1.b.b 2
16.e even 4 1 2304.1.g.a 1
16.e even 4 1 2304.1.g.c 1
16.f odd 4 1 2304.1.g.a 1
16.f odd 4 1 2304.1.g.c 1
24.f even 2 1 RM 1152.1.b.b 2
24.h odd 2 1 CM 1152.1.b.b 2
48.i odd 4 1 2304.1.g.a 1
48.i odd 4 1 2304.1.g.c 1
48.k even 4 1 2304.1.g.a 1
48.k even 4 1 2304.1.g.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.1.b.b 2 1.a even 1 1 trivial
1152.1.b.b 2 3.b odd 2 1 inner
1152.1.b.b 2 4.b odd 2 1 CM
1152.1.b.b 2 8.b even 2 1 inner
1152.1.b.b 2 8.d odd 2 1 inner
1152.1.b.b 2 12.b even 2 1 inner
1152.1.b.b 2 24.f even 2 1 RM
1152.1.b.b 2 24.h odd 2 1 CM
2304.1.g.a 1 16.e even 4 1
2304.1.g.a 1 16.f odd 4 1
2304.1.g.a 1 48.i odd 4 1
2304.1.g.a 1 48.k even 4 1
2304.1.g.c 1 16.e even 4 1
2304.1.g.c 1 16.f odd 4 1
2304.1.g.c 1 48.i odd 4 1
2304.1.g.c 1 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1152, [\chi])\).

Hecke characteristic polynomials

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