Properties

Label 1792.2.a.r
Level $1792$
Weight $2$
Character orbit 1792.a
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 3 + 2 \beta ) q^{9} + 4 q^{11} + ( 1 + \beta ) q^{13} + ( -6 - 2 \beta ) q^{15} -2 q^{17} + ( -1 - \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( 2 - 2 \beta ) q^{23} + ( 1 + 2 \beta ) q^{25} + ( 10 + 2 \beta ) q^{27} + ( 6 + 2 \beta ) q^{29} + ( 4 + 4 \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( 2 - 2 \beta ) q^{37} + ( 6 + 2 \beta ) q^{39} -2 q^{41} -4 \beta q^{43} + ( -13 - 5 \beta ) q^{45} + ( -4 + 4 \beta ) q^{47} + q^{49} + ( -2 - 2 \beta ) q^{51} + 4 \beta q^{53} + ( -4 - 4 \beta ) q^{55} + ( -6 - 2 \beta ) q^{57} + ( 7 - \beta ) q^{59} + ( -1 - \beta ) q^{61} + ( 3 + 2 \beta ) q^{63} + ( -6 - 2 \beta ) q^{65} + ( 6 + 2 \beta ) q^{67} -8 q^{69} + ( 4 + 4 \beta ) q^{71} + ( 6 - 4 \beta ) q^{73} + ( 11 + 3 \beta ) q^{75} + 4 q^{77} + ( -4 - 4 \beta ) q^{79} + ( 11 + 6 \beta ) q^{81} + ( 7 - \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( 16 + 8 \beta ) q^{87} + 2 q^{89} + ( 1 + \beta ) q^{91} + ( 6 + 2 \beta ) q^{95} + ( 2 - 4 \beta ) q^{97} + ( 12 + 8 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 6q^{9} + 8q^{11} + 2q^{13} - 12q^{15} - 4q^{17} - 2q^{19} + 2q^{21} + 4q^{23} + 2q^{25} + 20q^{27} + 12q^{29} + 8q^{33} - 2q^{35} + 4q^{37} + 12q^{39} - 4q^{41} - 26q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 8q^{55} - 12q^{57} + 14q^{59} - 2q^{61} + 6q^{63} - 12q^{65} + 12q^{67} - 16q^{69} + 8q^{71} + 12q^{73} + 22q^{75} + 8q^{77} - 8q^{79} + 22q^{81} + 14q^{83} + 4q^{85} + 32q^{87} + 4q^{89} + 2q^{91} + 12q^{95} + 4q^{97} + 24q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 −1.23607 0 1.23607 0 1.00000 0 −1.47214 0
1.2 0 3.23607 0 −3.23607 0 1.00000 0 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.a.r 2
4.b odd 2 1 1792.2.a.j 2
8.b even 2 1 1792.2.a.l 2
8.d odd 2 1 1792.2.a.t 2
16.e even 4 2 896.2.b.e 4
16.f odd 4 2 896.2.b.g yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.e 4 16.e even 4 2
896.2.b.g yes 4 16.f odd 4 2
1792.2.a.j 2 4.b odd 2 1
1792.2.a.l 2 8.b even 2 1
1792.2.a.r 2 1.a even 1 1 trivial
1792.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1792))\):

\( T_{3}^{2} - 2 T_{3} - 4 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{23}^{2} - 4 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( -4 + 2 T + T^{2} \)
$23$ \( -16 - 4 T + T^{2} \)
$29$ \( 16 - 12 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( -16 - 4 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -80 + T^{2} \)
$47$ \( -64 + 8 T + T^{2} \)
$53$ \( -80 + T^{2} \)
$59$ \( 44 - 14 T + T^{2} \)
$61$ \( -4 + 2 T + T^{2} \)
$67$ \( 16 - 12 T + T^{2} \)
$71$ \( -64 - 8 T + T^{2} \)
$73$ \( -44 - 12 T + T^{2} \)
$79$ \( -64 + 8 T + T^{2} \)
$83$ \( 44 - 14 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( -76 - 4 T + T^{2} \)
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