Properties

Label 1152.2.f.b
Level $1152$
Weight $2$
Character orbit 1152.f
Analytic conductor $9.199$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 6 \zeta_{8}^{2} q^{13} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} -3 q^{25} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{29} + 12 \zeta_{8}^{2} q^{37} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{41} + 7 q^{49} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{53} + 12 \zeta_{8}^{2} q^{61} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{65} -16 q^{73} + 6 \zeta_{8}^{2} q^{85} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{89} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{25} + 28q^{49} - 64q^{73} + 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −1.41421 0 0 0 0 0
575.2 0 0 0 −1.41421 0 0 0 0 0
575.3 0 0 0 1.41421 0 0 0 0 0
575.4 0 0 0 1.41421 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 \(\Q(\sqrt{-1}) \)
3.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
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.f.b 4
3.b odd 2 1 inner 1152.2.f.b 4
4.b odd 2 1 CM 1152.2.f.b 4
8.b even 2 1 inner 1152.2.f.b 4
8.d odd 2 1 inner 1152.2.f.b 4
12.b even 2 1 inner 1152.2.f.b 4
16.e even 4 1 2304.2.c.d 2
16.e even 4 1 2304.2.c.f 2
16.f odd 4 1 2304.2.c.d 2
16.f odd 4 1 2304.2.c.f 2
24.f even 2 1 inner 1152.2.f.b 4
24.h odd 2 1 inner 1152.2.f.b 4
48.i odd 4 1 2304.2.c.d 2
48.i odd 4 1 2304.2.c.f 2
48.k even 4 1 2304.2.c.d 2
48.k even 4 1 2304.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.b 4 1.a even 1 1 trivial
1152.2.f.b 4 3.b odd 2 1 inner
1152.2.f.b 4 4.b odd 2 1 CM
1152.2.f.b 4 8.b even 2 1 inner
1152.2.f.b 4 8.d odd 2 1 inner
1152.2.f.b 4 12.b even 2 1 inner
1152.2.f.b 4 24.f even 2 1 inner
1152.2.f.b 4 24.h odd 2 1 inner
2304.2.c.d 2 16.e even 4 1
2304.2.c.d 2 16.f odd 4 1
2304.2.c.d 2 48.i odd 4 1
2304.2.c.d 2 48.k even 4 1
2304.2.c.f 2 16.e even 4 1
2304.2.c.f 2 16.f odd 4 1
2304.2.c.f 2 48.i odd 4 1
2304.2.c.f 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 \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{7} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -2 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 36 + T^{2} )^{2} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -98 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 144 + T^{2} )^{2} \)
$41$ \( ( 162 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -50 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 144 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 16 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 18 + T^{2} )^{2} \)
$97$ \( ( -8 + T )^{4} \)
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