Properties

Label 1792.2.e.h
Level $1792$
Weight $2$
Character orbit 1792.e
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 896)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{5} + ( \beta_{4} + \beta_{7} ) q^{7} -\beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{5} + ( \beta_{4} + \beta_{7} ) q^{7} -\beta_{2} q^{9} + ( -1 + \beta_{2} ) q^{11} + \beta_{6} q^{13} -\beta_{4} q^{15} + ( -\beta_{1} + \beta_{3} ) q^{17} + ( \beta_{1} + 2 \beta_{3} ) q^{19} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{21} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{23} + ( 2 - 3 \beta_{2} ) q^{25} + ( -\beta_{1} + \beta_{3} ) q^{27} + ( \beta_{5} - \beta_{7} ) q^{29} + ( 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( -4 \beta_{4} + \beta_{5} - \beta_{7} ) q^{37} + ( \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{39} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 5 + 3 \beta_{2} ) q^{43} + ( 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{45} + ( -\beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{47} + ( -3 + \beta_{1} - 3 \beta_{3} ) q^{49} -4 q^{51} + ( -2 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{53} + ( -3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{55} + ( -5 - 3 \beta_{2} ) q^{57} + 3 \beta_{1} q^{59} + ( -\beta_{5} - 7 \beta_{6} - \beta_{7} ) q^{61} + ( 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{63} + ( 1 - \beta_{2} ) q^{65} + ( 3 - 3 \beta_{2} ) q^{67} + ( \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{69} + ( 5 \beta_{5} - 5 \beta_{7} ) q^{71} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{73} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{75} + ( -\beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{77} + ( -2 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{79} + ( -4 - 3 \beta_{2} ) q^{81} + ( -4 \beta_{1} - 3 \beta_{3} ) q^{83} + ( -4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{85} + 2 \beta_{6} q^{87} + ( \beta_{1} - 7 \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{91} + ( -4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{93} + ( \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{95} + ( \beta_{1} - \beta_{3} ) q^{97} + ( -5 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 16q^{25} + 24q^{35} + 40q^{43} - 24q^{49} - 32q^{51} - 40q^{57} + 8q^{65} + 24q^{67} - 32q^{81} - 8q^{91} - 40q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 5 \nu^{3} + 2 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{7} + \nu^{5} + 21 \nu^{3} + 8 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} - 10 \nu^{2} \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} + \nu^{6} + \nu^{5} - 13 \nu^{3} + 8 \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - \nu^{5} + 21 \nu^{3} - 8 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{6} + \nu^{5} - 13 \nu^{3} - 8 \nu^{2} + 5 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \beta_{3} - 3 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(4 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} - \beta_{3} + 4 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{7} - 5 \beta_{5} - 8 \beta_{4}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-21 \beta_{7} - 26 \beta_{6} - 21 \beta_{5} - 5 \beta_{3} + 21 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
−0.437016 0.437016i
0.437016 0.437016i
1.14412 1.14412i
−1.14412 1.14412i
1.14412 + 1.14412i
−1.14412 + 1.14412i
−0.437016 + 0.437016i
0.437016 + 0.437016i
0 2.28825i 0 −0.540182 0 −1.41421 2.23607i 0 −2.23607 0
895.2 0 2.28825i 0 0.540182 0 1.41421 + 2.23607i 0 −2.23607 0
895.3 0 0.874032i 0 −3.70246 0 −1.41421 2.23607i 0 2.23607 0
895.4 0 0.874032i 0 3.70246 0 1.41421 + 2.23607i 0 2.23607 0
895.5 0 0.874032i 0 −3.70246 0 −1.41421 + 2.23607i 0 2.23607 0
895.6 0 0.874032i 0 3.70246 0 1.41421 2.23607i 0 2.23607 0
895.7 0 2.28825i 0 −0.540182 0 −1.41421 + 2.23607i 0 −2.23607 0
895.8 0 2.28825i 0 0.540182 0 1.41421 2.23607i 0 −2.23607 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.e.h 8
4.b odd 2 1 1792.2.e.i 8
7.b odd 2 1 inner 1792.2.e.h 8
8.b even 2 1 1792.2.e.i 8
8.d odd 2 1 inner 1792.2.e.h 8
16.e even 4 1 896.2.f.a 8
16.e even 4 1 896.2.f.b yes 8
16.f odd 4 1 896.2.f.a 8
16.f odd 4 1 896.2.f.b yes 8
28.d even 2 1 1792.2.e.i 8
56.e even 2 1 inner 1792.2.e.h 8
56.h odd 2 1 1792.2.e.i 8
112.j even 4 1 896.2.f.a 8
112.j even 4 1 896.2.f.b yes 8
112.l odd 4 1 896.2.f.a 8
112.l odd 4 1 896.2.f.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.f.a 8 16.e even 4 1
896.2.f.a 8 16.f odd 4 1
896.2.f.a 8 112.j even 4 1
896.2.f.a 8 112.l odd 4 1
896.2.f.b yes 8 16.e even 4 1
896.2.f.b yes 8 16.f odd 4 1
896.2.f.b yes 8 112.j even 4 1
896.2.f.b yes 8 112.l odd 4 1
1792.2.e.h 8 1.a even 1 1 trivial
1792.2.e.h 8 7.b odd 2 1 inner
1792.2.e.h 8 8.d odd 2 1 inner
1792.2.e.h 8 56.e even 2 1 inner
1792.2.e.i 8 4.b odd 2 1
1792.2.e.i 8 8.b even 2 1
1792.2.e.i 8 28.d even 2 1
1792.2.e.i 8 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{4} + 6 T_{3}^{2} + 4 \)
\( T_{11}^{2} + 2 T_{11} - 4 \)
\( T_{31}^{4} - 120 T_{31}^{2} + 1600 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 4 + 6 T^{2} + T^{4} )^{2} \)
$5$ \( ( 4 - 14 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 6 T^{2} + T^{4} )^{2} \)
$11$ \( ( -4 + 2 T + T^{2} )^{4} \)
$13$ \( ( 4 - 6 T^{2} + T^{4} )^{2} \)
$17$ \( ( 64 + 24 T^{2} + T^{4} )^{2} \)
$19$ \( ( 100 + 30 T^{2} + T^{4} )^{2} \)
$23$ \( ( 16 + 12 T^{2} + T^{4} )^{2} \)
$29$ \( ( 4 + T^{2} )^{4} \)
$31$ \( ( 1600 - 120 T^{2} + T^{4} )^{2} \)
$37$ \( ( 5776 + 168 T^{2} + T^{4} )^{2} \)
$41$ \( ( 1600 + 120 T^{2} + T^{4} )^{2} \)
$43$ \( ( -20 - 10 T + T^{2} )^{4} \)
$47$ \( ( 7744 - 184 T^{2} + T^{4} )^{2} \)
$53$ \( ( 16 + 72 T^{2} + T^{4} )^{2} \)
$59$ \( ( 324 + 126 T^{2} + T^{4} )^{2} \)
$61$ \( ( 13924 - 254 T^{2} + T^{4} )^{2} \)
$67$ \( ( -36 - 6 T + T^{2} )^{4} \)
$71$ \( ( 100 + T^{2} )^{4} \)
$73$ \( ( 200 + T^{2} )^{4} \)
$79$ \( ( 1936 + 168 T^{2} + T^{4} )^{2} \)
$83$ \( ( 12100 + 230 T^{2} + T^{4} )^{2} \)
$89$ \( ( 23104 + 336 T^{2} + T^{4} )^{2} \)
$97$ \( ( 64 + 24 T^{2} + T^{4} )^{2} \)
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