Properties

Label 1792.2.e.e
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: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{3} ) q^{9} + ( 1 - \beta_{3} ) q^{11} + ( -\beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{13} + ( -3 \beta_{4} + \beta_{7} ) q^{15} + ( -\beta_{1} - \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{6} ) q^{19} + ( -\beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{21} + \beta_{7} q^{23} -\beta_{3} q^{25} + ( -3 \beta_{1} + \beta_{6} ) q^{27} -3 \beta_{4} q^{29} + ( \beta_{4} + 2 \beta_{5} ) q^{31} + ( 5 \beta_{1} - \beta_{6} ) q^{33} + ( 3 + \beta_{1} - \beta_{3} ) q^{35} + \beta_{4} q^{37} + ( \beta_{4} - \beta_{7} ) q^{39} + 2 \beta_{1} q^{41} + ( -1 + \beta_{3} ) q^{43} + ( -7 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{45} + ( \beta_{4} + 2 \beta_{5} ) q^{47} + ( 5 + \beta_{1} - \beta_{6} ) q^{49} + 4 q^{51} + ( \beta_{4} - 2 \beta_{7} ) q^{53} + ( 6 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{55} + ( 9 - \beta_{3} ) q^{57} + \beta_{6} q^{59} + ( -3 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{61} + ( -5 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{63} + ( -1 + \beta_{3} ) q^{65} + ( 9 - \beta_{3} ) q^{67} + ( -4 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{69} + ( -\beta_{4} + 2 \beta_{7} ) q^{71} + ( -3 \beta_{1} - \beta_{6} ) q^{73} + ( 4 \beta_{1} - \beta_{6} ) q^{75} + ( 4 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{77} + \beta_{4} q^{79} + ( 10 - \beta_{3} ) q^{81} + \beta_{1} q^{83} + 2 \beta_{4} q^{85} -6 \beta_{2} q^{87} + ( -3 \beta_{1} - \beta_{6} ) q^{89} + ( -9 - \beta_{3} + \beta_{6} ) q^{91} -2 \beta_{4} q^{93} + ( 5 \beta_{4} - \beta_{7} ) q^{95} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{97} + ( -23 + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{9} + O(q^{10}) \) \( 8q - 16q^{9} + 8q^{11} + 24q^{35} - 8q^{43} + 40q^{49} + 32q^{51} + 72q^{57} - 8q^{65} + 72q^{67} + 80q^{81} - 72q^{91} - 184q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 24 \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 24 \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{4} + 23 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} - 48 \nu^{2} \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + \nu^{6} + \nu^{5} - 115 \nu^{3} + 24 \nu^{2} + 24 \nu \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} + 2 \nu^{5} + 206 \nu^{3} + 43 \nu \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -6 \nu^{6} - 134 \nu^{2} \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 3 \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{6} + 2 \beta_{5} + \beta_{4} - 10 \beta_{2} + 9 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{3} - 23\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} - 48 \beta_{2} - 43 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-24 \beta_{7} + 67 \beta_{4}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(12 \beta_{6} - 24 \beta_{5} - 12 \beta_{4} + 115 \beta_{2} - 103 \beta_{1}\)\()/2\)

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
−1.54779 1.54779i
1.54779 1.54779i
−0.323042 0.323042i
0.323042 0.323042i
−0.323042 + 0.323042i
0.323042 + 0.323042i
−1.54779 + 1.54779i
1.54779 + 1.54779i
0 3.09557i 0 −3.09557 0 −2.44949 + 1.00000i 0 −6.58258 0
895.2 0 3.09557i 0 3.09557 0 2.44949 1.00000i 0 −6.58258 0
895.3 0 0.646084i 0 −0.646084 0 2.44949 + 1.00000i 0 2.58258 0
895.4 0 0.646084i 0 0.646084 0 −2.44949 1.00000i 0 2.58258 0
895.5 0 0.646084i 0 −0.646084 0 2.44949 1.00000i 0 2.58258 0
895.6 0 0.646084i 0 0.646084 0 −2.44949 + 1.00000i 0 2.58258 0
895.7 0 3.09557i 0 −3.09557 0 −2.44949 1.00000i 0 −6.58258 0
895.8 0 3.09557i 0 3.09557 0 2.44949 + 1.00000i 0 −6.58258 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.e 8
4.b odd 2 1 1792.2.e.d 8
7.b odd 2 1 inner 1792.2.e.e 8
8.b even 2 1 1792.2.e.d 8
8.d odd 2 1 inner 1792.2.e.e 8
16.e even 4 1 896.2.f.c 8
16.e even 4 1 896.2.f.d yes 8
16.f odd 4 1 896.2.f.c 8
16.f odd 4 1 896.2.f.d yes 8
28.d even 2 1 1792.2.e.d 8
56.e even 2 1 inner 1792.2.e.e 8
56.h odd 2 1 1792.2.e.d 8
112.j even 4 1 896.2.f.c 8
112.j even 4 1 896.2.f.d yes 8
112.l odd 4 1 896.2.f.c 8
112.l odd 4 1 896.2.f.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.f.c 8 16.e even 4 1
896.2.f.c 8 16.f odd 4 1
896.2.f.c 8 112.j even 4 1
896.2.f.c 8 112.l odd 4 1
896.2.f.d yes 8 16.e even 4 1
896.2.f.d yes 8 16.f odd 4 1
896.2.f.d yes 8 112.j even 4 1
896.2.f.d yes 8 112.l odd 4 1
1792.2.e.d 8 4.b odd 2 1
1792.2.e.d 8 8.b even 2 1
1792.2.e.d 8 28.d even 2 1
1792.2.e.d 8 56.h odd 2 1
1792.2.e.e 8 1.a even 1 1 trivial
1792.2.e.e 8 7.b odd 2 1 inner
1792.2.e.e 8 8.d odd 2 1 inner
1792.2.e.e 8 56.e even 2 1 inner

Hecke kernels

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

\( T_{3}^{4} + 10 T_{3}^{2} + 4 \)
\( T_{11}^{2} - 2 T_{11} - 20 \)
\( T_{31}^{4} - 40 T_{31}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 4 + 10 T^{2} + T^{4} )^{2} \)
$5$ \( ( 4 - 10 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 - 10 T^{2} + T^{4} )^{2} \)
$11$ \( ( -20 - 2 T + T^{2} )^{4} \)
$13$ \( ( 100 - 34 T^{2} + T^{4} )^{2} \)
$17$ \( ( 64 + 40 T^{2} + T^{4} )^{2} \)
$19$ \( ( 900 + 66 T^{2} + T^{4} )^{2} \)
$23$ \( ( 400 + 44 T^{2} + T^{4} )^{2} \)
$29$ \( ( 36 + T^{2} )^{4} \)
$31$ \( ( 64 - 40 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4 + T^{2} )^{4} \)
$41$ \( ( 64 + 40 T^{2} + T^{4} )^{2} \)
$43$ \( ( -20 + 2 T + T^{2} )^{4} \)
$47$ \( ( 64 - 40 T^{2} + T^{4} )^{2} \)
$53$ \( ( 84 + T^{2} )^{4} \)
$59$ \( ( 100 + 34 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4900 - 154 T^{2} + T^{4} )^{2} \)
$67$ \( ( 60 - 18 T + T^{2} )^{4} \)
$71$ \( ( 84 + T^{2} )^{4} \)
$73$ \( ( 56 + T^{2} )^{4} \)
$79$ \( ( 4 + T^{2} )^{4} \)
$83$ \( ( 4 + 10 T^{2} + T^{4} )^{2} \)
$89$ \( ( 56 + T^{2} )^{4} \)
$97$ \( ( 5184 + 360 T^{2} + T^{4} )^{2} \)
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