Properties

Label 1152.2.k.f
Level $1152$
Weight $2$
Character orbit 1152.k
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: 8.0.18939904.2
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{4}) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{4}) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{7} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{7} + \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{17} + (2 \beta_{7} + \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{7} + \beta_{6}) q^{23} + (\beta_{5} + 2 \beta_1) q^{25} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - 2) q^{29} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{31} + (\beta_{7} + 3 \beta_{5} + \beta_{2} + \beta_1 - 3) q^{35} + (2 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 + 2) q^{37} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{41} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{43} + ( - \beta_{7} - \beta_{6}) q^{47} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{49} + (3 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2) q^{53} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5}) q^{55} + ( - 4 \beta_{5} - 4) q^{59} + ( - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{61} + (4 \beta_{7} + 4 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{65} + (2 \beta_{5} + 2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{71} + (4 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{73} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{77} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 3 \beta_{2} + 3) q^{79} + (3 \beta_{7} - 5 \beta_{5} + 4 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{83} + ( - 6 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 2) q^{85} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{89} + (4 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{91} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 6) q^{95} + ( - 4 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 8 q^{19} - 16 q^{29} - 24 q^{31} - 24 q^{35} + 16 q^{37} - 8 q^{43} - 8 q^{49} + 16 q^{53} - 32 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} + 16 q^{77} + 24 q^{79} + 40 q^{83} + 16 q^{85} - 8 q^{91} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 2\nu^{3} - 5\nu^{2} + 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{7} - 13\nu^{6} + 45\nu^{5} - 75\nu^{4} + 105\nu^{3} - 84\nu^{2} + 42\nu - 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\nu^{7} + 15\nu^{6} - 51\nu^{5} + 95\nu^{4} - 135\nu^{3} + 120\nu^{2} - 64\nu + 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -10\nu^{7} + 34\nu^{6} - 120\nu^{5} + 210\nu^{4} - 310\nu^{3} + 266\nu^{2} - 154\nu + 38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 36\nu^{6} + 126\nu^{5} - 230\nu^{4} + 340\nu^{3} - 304\nu^{2} + 178\nu - 46 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 5\beta_{6} - 6\beta_{5} - 4\beta_{4} - 2\beta_{3} - 4\beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 6\beta_{5} + \beta_{4} + 3\beta_{3} - 2\beta_{2} - 5\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} + 21\beta_{6} + 20\beta_{5} + 20\beta_{4} + 10\beta_{3} - 10\beta_{2} + 6\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{7} + 12\beta_{6} + 45\beta_{5} + 9\beta_{4} - 11\beta_{3} + 5\beta_{2} + 22\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -71\beta_{7} - 69\beta_{6} - 66\beta_{4} - 60\beta_{3} + 70\beta_{2} + 14\beta _1 - 236 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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
0.500000 2.10607i
0.500000 + 1.44392i
0.500000 + 0.691860i
0.500000 0.0297061i
0.500000 + 2.10607i
0.500000 1.44392i
0.500000 0.691860i
0.500000 + 0.0297061i
0 0 0 −2.68554 + 2.68554i 0 2.15894i 0 0 0
289.2 0 0 0 −0.334904 + 0.334904i 0 4.55765i 0 0 0
289.3 0 0 0 1.27133 1.27133i 0 0.158942i 0 0 0
289.4 0 0 0 1.74912 1.74912i 0 2.55765i 0 0 0
865.1 0 0 0 −2.68554 2.68554i 0 2.15894i 0 0 0
865.2 0 0 0 −0.334904 0.334904i 0 4.55765i 0 0 0
865.3 0 0 0 1.27133 + 1.27133i 0 0.158942i 0 0 0
865.4 0 0 0 1.74912 + 1.74912i 0 2.55765i 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
16.e even 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.f 8
3.b odd 2 1 384.2.j.a 8
4.b odd 2 1 1152.2.k.c 8
8.b even 2 1 576.2.k.b 8
8.d odd 2 1 144.2.k.b 8
12.b even 2 1 384.2.j.b 8
16.e even 4 1 576.2.k.b 8
16.e even 4 1 inner 1152.2.k.f 8
16.f odd 4 1 144.2.k.b 8
16.f odd 4 1 1152.2.k.c 8
24.f even 2 1 48.2.j.a 8
24.h odd 2 1 192.2.j.a 8
32.g even 8 1 9216.2.a.x 4
32.g even 8 1 9216.2.a.bn 4
32.h odd 8 1 9216.2.a.y 4
32.h odd 8 1 9216.2.a.bo 4
48.i odd 4 1 192.2.j.a 8
48.i odd 4 1 384.2.j.a 8
48.k even 4 1 48.2.j.a 8
48.k even 4 1 384.2.j.b 8
96.o even 8 1 3072.2.a.i 4
96.o even 8 1 3072.2.a.t 4
96.o even 8 2 3072.2.d.f 8
96.p odd 8 1 3072.2.a.n 4
96.p odd 8 1 3072.2.a.o 4
96.p odd 8 2 3072.2.d.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 24.f even 2 1
48.2.j.a 8 48.k even 4 1
144.2.k.b 8 8.d odd 2 1
144.2.k.b 8 16.f odd 4 1
192.2.j.a 8 24.h odd 2 1
192.2.j.a 8 48.i odd 4 1
384.2.j.a 8 3.b odd 2 1
384.2.j.a 8 48.i odd 4 1
384.2.j.b 8 12.b even 2 1
384.2.j.b 8 48.k even 4 1
576.2.k.b 8 8.b even 2 1
576.2.k.b 8 16.e even 4 1
1152.2.k.c 8 4.b odd 2 1
1152.2.k.c 8 16.f odd 4 1
1152.2.k.f 8 1.a even 1 1 trivial
1152.2.k.f 8 16.e even 4 1 inner
3072.2.a.i 4 96.o even 8 1
3072.2.a.n 4 96.p odd 8 1
3072.2.a.o 4 96.p odd 8 1
3072.2.a.t 4 96.o even 8 1
3072.2.d.f 8 96.o even 8 2
3072.2.d.i 8 96.p odd 8 2
9216.2.a.x 4 32.g even 8 1
9216.2.a.y 4 32.h odd 8 1
9216.2.a.bn 4 32.g even 8 1
9216.2.a.bo 4 32.h odd 8 1

Hecke kernels

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

\( T_{5}^{8} - 16T_{5}^{5} + 128T_{5}^{4} - 192T_{5}^{3} + 128T_{5}^{2} + 128T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} - 8T_{11}^{7} + 32T_{11}^{6} - 256T_{11}^{3} + 2048T_{11}^{2} + 2048T_{11} + 1024 \) Copy content Toggle raw display
\( T_{19}^{8} + 8T_{19}^{7} + 32T_{19}^{6} - 32T_{19}^{5} + 224T_{19}^{4} + 1408T_{19}^{3} + 4608T_{19}^{2} - 1536T_{19} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{5} + 128 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} + 64 T^{5} + 776 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} - 64 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 32 T^{6} - 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$41$ \( T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216 \) Copy content Toggle raw display
$97$ \( (T^{4} - 224 T^{2} + 768 T + 512)^{2} \) Copy content Toggle raw display
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