Properties

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

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{3} q^{5} -\beta_{6} q^{7} +O(q^{10})\) \( q + 3 \beta_{3} q^{5} -\beta_{6} q^{7} + \beta_{5} q^{11} + 5 \beta_{1} q^{13} + 17 \beta_{2} q^{17} -\beta_{7} q^{19} + \beta_{4} q^{23} -7 q^{25} -11 \beta_{3} q^{29} + \beta_{6} q^{31} -3 \beta_{5} q^{35} + 32 \beta_{1} q^{37} + 9 \beta_{2} q^{41} + 2 \beta_{7} q^{43} -\beta_{4} q^{47} + 111 q^{49} -13 \beta_{3} q^{53} + 6 \beta_{6} q^{55} + 30 \beta_{2} q^{65} + 3 \beta_{7} q^{67} -7 \beta_{4} q^{71} + 96 q^{73} -160 \beta_{3} q^{77} -5 \beta_{6} q^{79} + 7 \beta_{5} q^{83} + 51 \beta_{1} q^{85} + 39 \beta_{2} q^{89} -5 \beta_{7} q^{91} + 6 \beta_{4} q^{95} + 64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 56q^{25} + 888q^{49} + 768q^{73} + 512q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} + \nu^{5} - 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\( -8 \nu^{6} - 48 \nu^{2} \)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{4} + 56 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -16 \nu^{7} + 4 \nu^{5} - 116 \nu^{3} + 44 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 32 \nu^{7} + 8 \nu^{5} + 232 \nu^{3} + 88 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} - 8 \beta_{3} - 8 \beta_{2}\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 12 \beta_{1}\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 16 \beta_{3} - 16 \beta_{2}\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{5} - 56\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 10 \beta_{6} + 88 \beta_{3} + 88 \beta_{2}\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{4} - 9 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 26 \beta_{6} - 232 \beta_{3} + 232 \beta_{2}\)\()/32\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.14412 1.14412i
1.14412 + 1.14412i
−0.437016 + 0.437016i
−0.437016 0.437016i
0.437016 0.437016i
0.437016 + 0.437016i
−1.14412 + 1.14412i
−1.14412 1.14412i
0 0 0 −4.24264 0 −12.6491 0 0 0
449.2 0 0 0 −4.24264 0 −12.6491 0 0 0
449.3 0 0 0 −4.24264 0 12.6491 0 0 0
449.4 0 0 0 −4.24264 0 12.6491 0 0 0
449.5 0 0 0 4.24264 0 −12.6491 0 0 0
449.6 0 0 0 4.24264 0 −12.6491 0 0 0
449.7 0 0 0 4.24264 0 12.6491 0 0 0
449.8 0 0 0 4.24264 0 12.6491 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
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.h.e 8
3.b odd 2 1 inner 1152.3.h.e 8
4.b odd 2 1 inner 1152.3.h.e 8
8.b even 2 1 inner 1152.3.h.e 8
8.d odd 2 1 inner 1152.3.h.e 8
12.b even 2 1 inner 1152.3.h.e 8
16.e even 4 1 2304.3.e.f 4
16.e even 4 1 2304.3.e.k 4
16.f odd 4 1 2304.3.e.f 4
16.f odd 4 1 2304.3.e.k 4
24.f even 2 1 inner 1152.3.h.e 8
24.h odd 2 1 inner 1152.3.h.e 8
48.i odd 4 1 2304.3.e.f 4
48.i odd 4 1 2304.3.e.k 4
48.k even 4 1 2304.3.e.f 4
48.k even 4 1 2304.3.e.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.e 8 1.a even 1 1 trivial
1152.3.h.e 8 3.b odd 2 1 inner
1152.3.h.e 8 4.b odd 2 1 inner
1152.3.h.e 8 8.b even 2 1 inner
1152.3.h.e 8 8.d odd 2 1 inner
1152.3.h.e 8 12.b even 2 1 inner
1152.3.h.e 8 24.f even 2 1 inner
1152.3.h.e 8 24.h odd 2 1 inner
2304.3.e.f 4 16.e even 4 1
2304.3.e.f 4 16.f odd 4 1
2304.3.e.f 4 48.i odd 4 1
2304.3.e.f 4 48.k even 4 1
2304.3.e.k 4 16.e even 4 1
2304.3.e.k 4 16.f odd 4 1
2304.3.e.k 4 48.i odd 4 1
2304.3.e.k 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} - 18 \)
\( T_{7}^{2} - 160 \)
\( T_{11}^{2} - 320 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -18 + T^{2} )^{4} \)
$7$ \( ( -160 + T^{2} )^{4} \)
$11$ \( ( -320 + T^{2} )^{4} \)
$13$ \( ( 100 + T^{2} )^{4} \)
$17$ \( ( 578 + T^{2} )^{4} \)
$19$ \( ( 640 + T^{2} )^{4} \)
$23$ \( ( 320 + T^{2} )^{4} \)
$29$ \( ( -242 + T^{2} )^{4} \)
$31$ \( ( -160 + T^{2} )^{4} \)
$37$ \( ( 4096 + T^{2} )^{4} \)
$41$ \( ( 162 + T^{2} )^{4} \)
$43$ \( ( 2560 + T^{2} )^{4} \)
$47$ \( ( 320 + T^{2} )^{4} \)
$53$ \( ( -338 + T^{2} )^{4} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 5760 + T^{2} )^{4} \)
$71$ \( ( 15680 + T^{2} )^{4} \)
$73$ \( ( -96 + T )^{8} \)
$79$ \( ( -4000 + T^{2} )^{4} \)
$83$ \( ( -15680 + T^{2} )^{4} \)
$89$ \( ( 3042 + T^{2} )^{4} \)
$97$ \( ( -64 + T )^{8} \)
show more
show less