Properties

Label 1152.2.k.e
Level $1152$
Weight $2$
Character orbit 1152.k
Analytic conductor $9.199$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{7} ) q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + ( -\beta_{6} + \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{5} ) q^{17} + ( 1 - \beta_{3} + \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{23} + ( \beta_{3} - 2 \beta_{7} ) q^{25} + ( -\beta_{4} - 2 \beta_{5} ) q^{29} + ( -3 - \beta_{6} ) q^{31} + ( 3 \beta_{4} - \beta_{5} ) q^{35} + ( -2 - 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{41} + ( 3 + 3 \beta_{3} - \beta_{6} - \beta_{7} ) q^{43} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{6} ) q^{49} + ( -2 \beta_{1} + \beta_{2} ) q^{53} + 4 \beta_{3} q^{55} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{65} + ( -4 + 4 \beta_{3} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -2 \beta_{3} + 2 \beta_{7} ) q^{73} -2 \beta_{5} q^{77} + ( 7 + \beta_{6} ) q^{79} + ( -\beta_{4} - \beta_{5} ) q^{83} + ( 2 + 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{85} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{89} + ( -7 - 7 \beta_{3} + \beta_{6} + \beta_{7} ) q^{91} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{95} -4 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{19} - 24q^{31} - 16q^{37} + 24q^{43} - 8q^{49} - 16q^{61} - 32q^{67} + 56q^{79} + 16q^{85} - 56q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 2 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 4 \nu \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4 \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} - 2 \nu \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{3} + 12 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} - \nu^{4} + 4 \nu^{2} - 7 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{3} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{4} + 2 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 3 \beta_{3}\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} - \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(\beta_{7} - \beta_{6} + 5 \beta_{3} + 5\)
\(\nu^{7}\)\(=\)\(-\beta_{5} - 4 \beta_{4} + \beta_{2} + 2 \beta_{1}\)

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.767178 + 1.18804i
−1.38255 + 0.297594i
1.38255 0.297594i
−0.767178 1.18804i
0.767178 1.18804i
−1.38255 0.297594i
1.38255 + 0.297594i
−0.767178 + 1.18804i
0 0 0 −2.37608 + 2.37608i 0 3.64575i 0 0 0
289.2 0 0 0 −0.595188 + 0.595188i 0 1.64575i 0 0 0
289.3 0 0 0 0.595188 0.595188i 0 1.64575i 0 0 0
289.4 0 0 0 2.37608 2.37608i 0 3.64575i 0 0 0
865.1 0 0 0 −2.37608 2.37608i 0 3.64575i 0 0 0
865.2 0 0 0 −0.595188 0.595188i 0 1.64575i 0 0 0
865.3 0 0 0 0.595188 + 0.595188i 0 1.64575i 0 0 0
865.4 0 0 0 2.37608 + 2.37608i 0 3.64575i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.k.e 8
3.b odd 2 1 inner 1152.2.k.e 8
4.b odd 2 1 1152.2.k.d 8
8.b even 2 1 144.2.k.c 8
8.d odd 2 1 576.2.k.c 8
12.b even 2 1 1152.2.k.d 8
16.e even 4 1 144.2.k.c 8
16.e even 4 1 inner 1152.2.k.e 8
16.f odd 4 1 576.2.k.c 8
16.f odd 4 1 1152.2.k.d 8
24.f even 2 1 576.2.k.c 8
24.h odd 2 1 144.2.k.c 8
32.g even 8 2 9216.2.a.bq 8
32.h odd 8 2 9216.2.a.bt 8
48.i odd 4 1 144.2.k.c 8
48.i odd 4 1 inner 1152.2.k.e 8
48.k even 4 1 576.2.k.c 8
48.k even 4 1 1152.2.k.d 8
96.o even 8 2 9216.2.a.bt 8
96.p odd 8 2 9216.2.a.bq 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 8.b even 2 1
144.2.k.c 8 16.e even 4 1
144.2.k.c 8 24.h odd 2 1
144.2.k.c 8 48.i odd 4 1
576.2.k.c 8 8.d odd 2 1
576.2.k.c 8 16.f odd 4 1
576.2.k.c 8 24.f even 2 1
576.2.k.c 8 48.k even 4 1
1152.2.k.d 8 4.b odd 2 1
1152.2.k.d 8 12.b even 2 1
1152.2.k.d 8 16.f odd 4 1
1152.2.k.d 8 48.k even 4 1
1152.2.k.e 8 1.a even 1 1 trivial
1152.2.k.e 8 3.b odd 2 1 inner
1152.2.k.e 8 16.e even 4 1 inner
1152.2.k.e 8 48.i odd 4 1 inner
9216.2.a.bq 8 32.g even 8 2
9216.2.a.bq 8 96.p odd 8 2
9216.2.a.bt 8 32.h odd 8 2
9216.2.a.bt 8 96.o even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{8} + 128 T_{5}^{4} + 64 \)
\( T_{11}^{8} + 512 T_{11}^{4} + 1024 \)
\( T_{19}^{4} - 4 T_{19}^{3} + 8 T_{19}^{2} + 48 T_{19} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 64 + 128 T^{4} + T^{8} \)
$7$ \( ( 36 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( 1024 + 512 T^{4} + T^{8} \)
$13$ \( ( 196 + T^{4} )^{2} \)
$17$ \( ( 288 - 40 T^{2} + T^{4} )^{2} \)
$19$ \( ( 144 + 48 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( ( 1152 + 80 T^{2} + T^{4} )^{2} \)
$29$ \( 5184 + 5632 T^{4} + T^{8} \)
$31$ \( ( 2 + 6 T + T^{2} )^{4} \)
$37$ \( ( 36 - 48 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$41$ \( ( 2592 + 104 T^{2} + T^{4} )^{2} \)
$43$ \( ( 16 - 48 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$47$ \( ( 10368 - 208 T^{2} + T^{4} )^{2} \)
$53$ \( 8340544 + 5888 T^{4} + T^{8} \)
$59$ \( 21233664 + 16384 T^{4} + T^{8} \)
$61$ \( ( 36 - 48 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( ( 32 + 8 T + T^{2} )^{4} \)
$71$ \( ( 2048 + 192 T^{2} + T^{4} )^{2} \)
$73$ \( ( 576 + 64 T^{2} + T^{4} )^{2} \)
$79$ \( ( 42 - 14 T + T^{2} )^{4} \)
$83$ \( 1024 + 512 T^{4} + T^{8} \)
$89$ \( ( 512 + 96 T^{2} + T^{4} )^{2} \)
$97$ \( ( -112 + T^{2} )^{4} \)
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