Properties

Label 1792.2.f.l
Level $1792$
Weight $2$
Character orbit 1792.f
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
CM discriminant -56
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2517630976.5
Defining polynomial: \(x^{8} - 2 x^{6} + 11 x^{4} + 4 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 896)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} + ( 3 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} + ( 3 + \beta_{1} ) q^{9} + ( -\beta_{4} - \beta_{5} ) q^{13} + ( -2 \beta_{2} + \beta_{6} ) q^{15} -\beta_{7} q^{19} + ( -\beta_{4} + \beta_{5} ) q^{21} -\beta_{6} q^{23} + ( -5 - 3 \beta_{1} ) q^{25} + ( \beta_{3} - \beta_{7} ) q^{27} + ( -2 \beta_{3} + \beta_{7} ) q^{35} + ( -2 \beta_{2} - \beta_{6} ) q^{39} + ( -\beta_{4} + 5 \beta_{5} ) q^{45} -7 q^{49} + ( 2 + 5 \beta_{1} ) q^{57} + ( 4 \beta_{3} + \beta_{7} ) q^{59} + ( 2 \beta_{4} - \beta_{5} ) q^{61} + ( -3 \beta_{2} + \beta_{6} ) q^{63} + ( 6 - \beta_{1} ) q^{65} + ( -\beta_{4} - 6 \beta_{5} ) q^{69} -6 \beta_{2} q^{71} + ( -8 \beta_{3} + 3 \beta_{7} ) q^{75} -2 \beta_{2} q^{79} + ( -1 + 3 \beta_{1} ) q^{81} + ( -\beta_{3} + 2 \beta_{7} ) q^{83} + ( -3 \beta_{3} - 2 \beta_{7} ) q^{91} + ( -6 \beta_{2} + \beta_{6} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 40q^{25} - 56q^{49} + 16q^{57} + 48q^{65} - 8q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 11 x^{4} + 4 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} + \nu^{2} + 26 \)\()/9\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{6} + 10 \nu^{4} - 40 \nu^{2} - 5 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{7} + 10 \nu^{5} - 49 \nu^{3} + 13 \nu \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 8 \nu^{5} - 41 \nu^{3} - 40 \nu \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} - 8 \nu^{5} + 50 \nu^{3} + 49 \nu \)\()/9\)
\(\beta_{6}\)\(=\)\( \nu^{6} - 2 \nu^{4} + 13 \nu^{2} + 2 \)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 22 \nu^{5} + 88 \nu^{3} - 25 \nu \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 2 \beta_{2} - \beta_{1} + 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 6 \beta_{5} + 7 \beta_{4} - 3 \beta_{3}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{2} + 3 \beta_{1} - 9\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-17 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - 31 \beta_{3}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 14 \beta_{2} + 25 \beta_{1} - 70\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-27 \beta_{7} - 62 \beta_{5} - 75 \beta_{4} - 49 \beta_{3}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
−0.435132 0.629640i
−0.435132 + 0.629640i
−1.52009 1.05050i
−1.52009 + 1.05050i
1.52009 1.05050i
1.52009 + 1.05050i
0.435132 0.629640i
0.435132 + 0.629640i
0 −2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.2 0 −2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.3 0 −1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.4 0 −1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.5 0 1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.6 0 1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.7 0 2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.8 0 2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1791.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 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.f.l 8
4.b odd 2 1 inner 1792.2.f.l 8
7.b odd 2 1 inner 1792.2.f.l 8
8.b even 2 1 inner 1792.2.f.l 8
8.d odd 2 1 inner 1792.2.f.l 8
16.e even 4 2 896.2.e.g 8
16.f odd 4 2 896.2.e.g 8
28.d even 2 1 inner 1792.2.f.l 8
56.e even 2 1 inner 1792.2.f.l 8
56.h odd 2 1 CM 1792.2.f.l 8
112.j even 4 2 896.2.e.g 8
112.l odd 4 2 896.2.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.g 8 16.e even 4 2
896.2.e.g 8 16.f odd 4 2
896.2.e.g 8 112.j even 4 2
896.2.e.g 8 112.l odd 4 2
1792.2.f.l 8 1.a even 1 1 trivial
1792.2.f.l 8 4.b odd 2 1 inner
1792.2.f.l 8 7.b odd 2 1 inner
1792.2.f.l 8 8.b even 2 1 inner
1792.2.f.l 8 8.d odd 2 1 inner
1792.2.f.l 8 28.d even 2 1 inner
1792.2.f.l 8 56.e even 2 1 inner
1792.2.f.l 8 56.h odd 2 1 CM

Hecke kernels

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

\( T_{3}^{4} - 12 T_{3}^{2} + 28 \)
\( T_{5}^{4} + 20 T_{5}^{2} + 28 \)
\( T_{29} \)
\( T_{31} \)
\( T_{37} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 28 - 12 T^{2} + T^{4} )^{2} \)
$5$ \( ( 28 + 20 T^{2} + T^{4} )^{2} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( ( 28 + 52 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 1372 - 76 T^{2} + T^{4} )^{2} \)
$23$ \( ( 56 + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 8092 - 236 T^{2} + T^{4} )^{2} \)
$61$ \( ( 14812 + 244 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( ( 252 + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( 28 + T^{2} )^{4} \)
$83$ \( ( 26908 - 332 T^{2} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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