Properties

Label 1792.2.f.g
Level $1792$
Weight $2$
Character orbit 1792.f
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{7} - q^{9} + 2 \beta_{3} q^{11} -\beta_{3} q^{13} -2 \beta_{2} q^{15} -4 \beta_{2} q^{17} -3 \beta_{1} q^{19} + ( -2 \beta_{1} + \beta_{3} ) q^{21} -4 \beta_{2} q^{23} - q^{25} + 4 \beta_{1} q^{27} -4 \beta_{1} q^{29} + 4 q^{31} + 4 \beta_{2} q^{33} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{35} -2 \beta_{2} q^{39} + 2 \beta_{3} q^{43} + \beta_{3} q^{45} + 12 q^{47} + ( 1 + 4 \beta_{2} ) q^{49} -4 \beta_{3} q^{51} -4 \beta_{1} q^{53} + 12 q^{55} + 6 q^{57} + 7 \beta_{1} q^{59} -3 \beta_{3} q^{61} + ( -2 - \beta_{2} ) q^{63} -6 q^{65} + 2 \beta_{3} q^{67} -4 \beta_{3} q^{69} -2 \beta_{2} q^{71} + 4 \beta_{2} q^{73} + \beta_{1} q^{75} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{77} -6 \beta_{2} q^{79} -5 q^{81} -\beta_{1} q^{83} + 12 \beta_{1} q^{85} + 8 q^{87} -4 \beta_{2} q^{89} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{91} -4 \beta_{1} q^{93} -6 \beta_{2} q^{95} -4 \beta_{2} q^{97} -2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{7} - 4q^{9} - 4q^{25} + 16q^{31} + 48q^{47} + 4q^{49} + 48q^{55} + 24q^{57} - 8q^{63} - 24q^{65} - 20q^{81} + 32q^{87} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

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
−0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0.707107 1.22474i
0 −1.41421 0 2.44949i 0 2.00000 1.73205i 0 −1.00000 0
1791.2 0 −1.41421 0 2.44949i 0 2.00000 + 1.73205i 0 −1.00000 0
1791.3 0 1.41421 0 2.44949i 0 2.00000 + 1.73205i 0 −1.00000 0
1791.4 0 1.41421 0 2.44949i 0 2.00000 1.73205i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 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.g 4
4.b odd 2 1 1792.2.f.c 4
7.b odd 2 1 1792.2.f.c 4
8.b even 2 1 inner 1792.2.f.g 4
8.d odd 2 1 1792.2.f.c 4
16.e even 4 2 896.2.e.c 4
16.f odd 4 2 896.2.e.d yes 4
28.d even 2 1 inner 1792.2.f.g 4
56.e even 2 1 inner 1792.2.f.g 4
56.h odd 2 1 1792.2.f.c 4
112.j even 4 2 896.2.e.c 4
112.l odd 4 2 896.2.e.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.c 4 16.e even 4 2
896.2.e.c 4 112.j even 4 2
896.2.e.d yes 4 16.f odd 4 2
896.2.e.d yes 4 112.l odd 4 2
1792.2.f.c 4 4.b odd 2 1
1792.2.f.c 4 7.b odd 2 1
1792.2.f.c 4 8.d odd 2 1
1792.2.f.c 4 56.h odd 2 1
1792.2.f.g 4 1.a even 1 1 trivial
1792.2.f.g 4 8.b even 2 1 inner
1792.2.f.g 4 28.d even 2 1 inner
1792.2.f.g 4 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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} + 6 \)
\( T_{29}^{2} - 32 \)
\( T_{31} - 4 \)
\( T_{37} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( ( 6 + T^{2} )^{2} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( ( 24 + T^{2} )^{2} \)
$13$ \( ( 6 + T^{2} )^{2} \)
$17$ \( ( 48 + T^{2} )^{2} \)
$19$ \( ( -18 + T^{2} )^{2} \)
$23$ \( ( 48 + T^{2} )^{2} \)
$29$ \( ( -32 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 24 + T^{2} )^{2} \)
$47$ \( ( -12 + T )^{4} \)
$53$ \( ( -32 + T^{2} )^{2} \)
$59$ \( ( -98 + T^{2} )^{2} \)
$61$ \( ( 54 + T^{2} )^{2} \)
$67$ \( ( 24 + T^{2} )^{2} \)
$71$ \( ( 12 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 108 + T^{2} )^{2} \)
$83$ \( ( -2 + T^{2} )^{2} \)
$89$ \( ( 48 + T^{2} )^{2} \)
$97$ \( ( 48 + T^{2} )^{2} \)
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