Properties

Label 1792.2.a.m
Level $1792$
Weight $2$
Character orbit 1792.a
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 896)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta q^{5} - q^{7} -3 q^{9} + \beta q^{11} -\beta q^{13} -2 q^{17} + 2 \beta q^{19} + 8 q^{23} + 3 q^{25} -2 \beta q^{29} + 8 q^{31} -\beta q^{35} + 2 \beta q^{37} + 6 q^{41} + \beta q^{43} -3 \beta q^{45} + 8 q^{47} + q^{49} + 4 \beta q^{53} + 8 q^{55} -4 \beta q^{59} + \beta q^{61} + 3 q^{63} -8 q^{65} -3 \beta q^{67} + 8 q^{71} -6 q^{73} -\beta q^{77} -8 q^{79} + 9 q^{81} -2 \beta q^{83} -2 \beta q^{85} -6 q^{89} + \beta q^{91} + 16 q^{95} + 14 q^{97} -3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 2q^{7} - 6q^{9} - 4q^{17} + 16q^{23} + 6q^{25} + 16q^{31} + 12q^{41} + 16q^{47} + 2q^{49} + 16q^{55} + 6q^{63} - 16q^{65} + 16q^{71} - 12q^{73} - 16q^{79} + 18q^{81} - 12q^{89} + 32q^{95} + 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0 0 −2.82843 0 −1.00000 0 −3.00000 0
1.2 0 0 0 2.82843 0 −1.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.a.m 2
4.b odd 2 1 1792.2.a.o 2
8.b even 2 1 inner 1792.2.a.m 2
8.d odd 2 1 1792.2.a.o 2
16.e even 4 2 896.2.b.d yes 2
16.f odd 4 2 896.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.b 2 16.f odd 4 2
896.2.b.d yes 2 16.e even 4 2
1792.2.a.m 2 1.a even 1 1 trivial
1792.2.a.m 2 8.b even 2 1 inner
1792.2.a.o 2 4.b odd 2 1
1792.2.a.o 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1792))\):

\( T_{3} \)
\( T_{5}^{2} - 8 \)
\( T_{23} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( -32 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( -32 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( -32 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -8 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -128 + T^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( -8 + T^{2} \)
$67$ \( -72 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -32 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( ( -14 + T )^{2} \)
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