Properties

Label 1152.3.b.i
Level $1152$
Weight $3$
Character orbit 1152.b
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} -\beta_{5} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{5} q^{7} -\beta_{3} q^{11} -\beta_{2} q^{13} -\beta_{4} q^{17} -\beta_{7} q^{19} -\beta_{6} q^{23} -3 q^{25} -\beta_{1} q^{29} -3 \beta_{5} q^{31} -7 \beta_{3} q^{35} + 7 \beta_{2} q^{37} + 3 \beta_{4} q^{41} + \beta_{7} q^{43} -3 \beta_{6} q^{47} -7 q^{49} + 9 \beta_{1} q^{53} + 4 \beta_{5} q^{55} -18 \beta_{3} q^{59} -19 \beta_{2} q^{61} -\beta_{4} q^{65} + 2 \beta_{7} q^{67} -4 \beta_{6} q^{71} + 26 q^{73} + 8 \beta_{1} q^{77} + 17 \beta_{5} q^{79} -21 \beta_{3} q^{83} + 28 \beta_{2} q^{85} -2 \beta_{4} q^{89} -\beta_{7} q^{91} -7 \beta_{6} q^{95} + 18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} - 56q^{49} + 208q^{73} + 144q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{4} + 2 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\( -2 \nu^{6} + 6 \nu^{2} \)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - 8 \nu^{5} - 14 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 8 \beta_{5} + 4 \beta_{3}\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 6 \beta_{2}\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} + 20 \beta_{3}\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 44 \beta_{3}\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{4} + 18 \beta_{2}\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(-14 \beta_{7} + 13 \beta_{6} + 56 \beta_{5} - 52 \beta_{3}\)\()/64\)

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
1.28897 + 0.581861i
−0.581861 + 1.28897i
−1.28897 0.581861i
0.581861 1.28897i
0.581861 + 1.28897i
−1.28897 + 0.581861i
−0.581861 1.28897i
1.28897 0.581861i
0 0 0 5.29150i 0 7.48331i 0 0 0
703.2 0 0 0 5.29150i 0 7.48331i 0 0 0
703.3 0 0 0 5.29150i 0 7.48331i 0 0 0
703.4 0 0 0 5.29150i 0 7.48331i 0 0 0
703.5 0 0 0 5.29150i 0 7.48331i 0 0 0
703.6 0 0 0 5.29150i 0 7.48331i 0 0 0
703.7 0 0 0 5.29150i 0 7.48331i 0 0 0
703.8 0 0 0 5.29150i 0 7.48331i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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
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 1152.3.b.i 8
3.b odd 2 1 inner 1152.3.b.i 8
4.b odd 2 1 inner 1152.3.b.i 8
8.b even 2 1 inner 1152.3.b.i 8
8.d odd 2 1 inner 1152.3.b.i 8
12.b even 2 1 inner 1152.3.b.i 8
16.e even 4 1 2304.3.g.p 4
16.e even 4 1 2304.3.g.w 4
16.f odd 4 1 2304.3.g.p 4
16.f odd 4 1 2304.3.g.w 4
24.f even 2 1 inner 1152.3.b.i 8
24.h odd 2 1 inner 1152.3.b.i 8
48.i odd 4 1 2304.3.g.p 4
48.i odd 4 1 2304.3.g.w 4
48.k even 4 1 2304.3.g.p 4
48.k even 4 1 2304.3.g.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.i 8 1.a even 1 1 trivial
1152.3.b.i 8 3.b odd 2 1 inner
1152.3.b.i 8 4.b odd 2 1 inner
1152.3.b.i 8 8.b even 2 1 inner
1152.3.b.i 8 8.d odd 2 1 inner
1152.3.b.i 8 12.b even 2 1 inner
1152.3.b.i 8 24.f even 2 1 inner
1152.3.b.i 8 24.h odd 2 1 inner
2304.3.g.p 4 16.e even 4 1
2304.3.g.p 4 16.f odd 4 1
2304.3.g.p 4 48.i odd 4 1
2304.3.g.p 4 48.k even 4 1
2304.3.g.w 4 16.e even 4 1
2304.3.g.w 4 16.f odd 4 1
2304.3.g.w 4 48.i odd 4 1
2304.3.g.w 4 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 28 \)
\( T_{7}^{2} + 56 \)
\( T_{17}^{2} - 448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 28 + T^{2} )^{4} \)
$7$ \( ( 56 + T^{2} )^{4} \)
$11$ \( ( -32 + T^{2} )^{4} \)
$13$ \( ( 16 + T^{2} )^{4} \)
$17$ \( ( -448 + T^{2} )^{4} \)
$19$ \( ( -896 + T^{2} )^{4} \)
$23$ \( ( 512 + T^{2} )^{4} \)
$29$ \( ( 28 + T^{2} )^{4} \)
$31$ \( ( 504 + T^{2} )^{4} \)
$37$ \( ( 784 + T^{2} )^{4} \)
$41$ \( ( -4032 + T^{2} )^{4} \)
$43$ \( ( -896 + T^{2} )^{4} \)
$47$ \( ( 4608 + T^{2} )^{4} \)
$53$ \( ( 2268 + T^{2} )^{4} \)
$59$ \( ( -10368 + T^{2} )^{4} \)
$61$ \( ( 5776 + T^{2} )^{4} \)
$67$ \( ( -3584 + T^{2} )^{4} \)
$71$ \( ( 8192 + T^{2} )^{4} \)
$73$ \( ( -26 + T )^{8} \)
$79$ \( ( 16184 + T^{2} )^{4} \)
$83$ \( ( -14112 + T^{2} )^{4} \)
$89$ \( ( -1792 + T^{2} )^{4} \)
$97$ \( ( -18 + T )^{8} \)
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