Properties

Label 1152.3.g.e
Level $1152$
Weight $3$
Character orbit 1152.g
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.g (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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} -\beta_{5} q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} -\beta_{5} q^{7} + \beta_{1} q^{11} + ( 6 - \beta_{7} ) q^{13} + ( \beta_{3} + \beta_{4} ) q^{17} + ( \beta_{2} + 2 \beta_{5} ) q^{19} + ( 2 \beta_{1} + \beta_{6} ) q^{23} + ( 11 - 2 \beta_{7} ) q^{25} + ( -\beta_{3} + 2 \beta_{4} ) q^{29} + ( -2 \beta_{2} - \beta_{5} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{6} ) q^{35} + ( 2 + \beta_{7} ) q^{37} + ( -5 \beta_{3} + 3 \beta_{4} ) q^{41} + ( -3 \beta_{2} + 2 \beta_{5} ) q^{43} + ( -6 \beta_{1} + \beta_{6} ) q^{47} + ( -11 + 2 \beta_{7} ) q^{49} + ( -\beta_{3} + 6 \beta_{4} ) q^{53} + ( 6 \beta_{2} + 4 \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{59} + ( -14 + \beta_{7} ) q^{61} + ( -19 \beta_{3} + 5 \beta_{4} ) q^{65} + ( 6 \beta_{2} - 12 \beta_{5} ) q^{67} + 4 \beta_{6} q^{71} + ( -38 + 6 \beta_{7} ) q^{73} + ( -10 \beta_{3} + 6 \beta_{4} ) q^{77} + ( -8 \beta_{2} + 3 \beta_{5} ) q^{79} + ( -\beta_{1} - 4 \beta_{6} ) q^{83} -40 q^{85} + ( 2 \beta_{3} + 10 \beta_{4} ) q^{89} + ( -13 \beta_{2} - 10 \beta_{5} ) q^{91} + ( 10 \beta_{1} + 5 \beta_{6} ) q^{95} + ( 14 - 4 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 48q^{13} + 88q^{25} + 16q^{37} - 88q^{49} - 112q^{61} - 304q^{73} - 320q^{85} + 112q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 8 \nu^{4} + 4 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{6} - 10 \nu^{2} \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{6} + 4 \nu^{5} - 7 \nu^{3} + 18 \nu^{2} - 4 \nu \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 4 \nu^{5} + 7 \nu^{3} + 6 \nu^{2} + 4 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} - 2 \nu^{6} + 4 \nu^{5} + 13 \nu^{3} - 10 \nu^{2} + 44 \nu \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - 8 \nu^{5} - 14 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{7} + \beta_{6} + 8 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 3 \beta_{3} - 3 \beta_{2}\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} - 2 \beta_{2}\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 4\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(4 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 22 \beta_{4} + 22 \beta_{3} - 2 \beta_{2}\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{4} - 15 \beta_{3} - 9 \beta_{2}\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(-28 \beta_{7} + 13 \beta_{6} + 56 \beta_{5} - 26 \beta_{4} + 26 \beta_{3} - 14 \beta_{2}\)\()/64\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.28897 + 0.581861i
−1.28897 0.581861i
1.28897 + 0.581861i
1.28897 0.581861i
0.581861 + 1.28897i
0.581861 1.28897i
−0.581861 + 1.28897i
−0.581861 1.28897i
0 0 0 −8.11993 0 9.48331i 0 0 0
127.2 0 0 0 −8.11993 0 9.48331i 0 0 0
127.3 0 0 0 −2.46308 0 5.48331i 0 0 0
127.4 0 0 0 −2.46308 0 5.48331i 0 0 0
127.5 0 0 0 2.46308 0 5.48331i 0 0 0
127.6 0 0 0 2.46308 0 5.48331i 0 0 0
127.7 0 0 0 8.11993 0 9.48331i 0 0 0
127.8 0 0 0 8.11993 0 9.48331i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
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
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 1152.3.g.e yes 8
3.b odd 2 1 inner 1152.3.g.e yes 8
4.b odd 2 1 inner 1152.3.g.e yes 8
8.b even 2 1 1152.3.g.d 8
8.d odd 2 1 1152.3.g.d 8
12.b even 2 1 inner 1152.3.g.e yes 8
16.e even 4 1 2304.3.b.r 8
16.e even 4 1 2304.3.b.s 8
16.f odd 4 1 2304.3.b.r 8
16.f odd 4 1 2304.3.b.s 8
24.f even 2 1 1152.3.g.d 8
24.h odd 2 1 1152.3.g.d 8
48.i odd 4 1 2304.3.b.r 8
48.i odd 4 1 2304.3.b.s 8
48.k even 4 1 2304.3.b.r 8
48.k even 4 1 2304.3.b.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.g.d 8 8.b even 2 1
1152.3.g.d 8 8.d odd 2 1
1152.3.g.d 8 24.f even 2 1
1152.3.g.d 8 24.h odd 2 1
1152.3.g.e yes 8 1.a even 1 1 trivial
1152.3.g.e yes 8 3.b odd 2 1 inner
1152.3.g.e yes 8 4.b odd 2 1 inner
1152.3.g.e yes 8 12.b even 2 1 inner
2304.3.b.r 8 16.e even 4 1
2304.3.b.r 8 16.f odd 4 1
2304.3.b.r 8 48.i odd 4 1
2304.3.b.r 8 48.k even 4 1
2304.3.b.s 8 16.e even 4 1
2304.3.b.s 8 16.f odd 4 1
2304.3.b.s 8 48.i odd 4 1
2304.3.b.s 8 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} - 72 T_{5}^{2} + 400 \)
\( T_{13}^{2} - 12 T_{13} - 188 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 400 - 72 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2704 + 120 T^{2} + T^{4} )^{2} \)
$11$ \( ( 112 + T^{2} )^{4} \)
$13$ \( ( -188 - 12 T + T^{2} )^{4} \)
$17$ \( ( 6400 - 288 T^{2} + T^{4} )^{2} \)
$19$ \( ( 6400 + 736 T^{2} + T^{4} )^{2} \)
$23$ \( ( 4096 + 1920 T^{2} + T^{4} )^{2} \)
$29$ \( ( 132496 - 840 T^{2} + T^{4} )^{2} \)
$31$ \( ( 71824 + 760 T^{2} + T^{4} )^{2} \)
$37$ \( ( -220 - 4 T + T^{2} )^{4} \)
$41$ \( ( 2119936 - 3360 T^{2} + T^{4} )^{2} \)
$43$ \( ( 30976 + 1248 T^{2} + T^{4} )^{2} \)
$47$ \( ( 12390400 + 9088 T^{2} + T^{4} )^{2} \)
$53$ \( ( 4787344 - 7176 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2560000 + 4992 T^{2} + T^{4} )^{2} \)
$61$ \( ( -28 + 28 T + T^{2} )^{4} \)
$67$ \( ( 56070144 + 17280 T^{2} + T^{4} )^{2} \)
$71$ \( ( 8192 + T^{2} )^{4} \)
$73$ \( ( -6620 + 76 T + T^{2} )^{4} \)
$79$ \( ( 8179600 + 7736 T^{2} + T^{4} )^{2} \)
$83$ \( ( 65286400 + 16608 T^{2} + T^{4} )^{2} \)
$89$ \( ( 5017600 - 20608 T^{2} + T^{4} )^{2} \)
$97$ \( ( -3388 - 28 T + T^{2} )^{4} \)
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