Properties

Label 1792.2.a.x
Level $1792$
Weight $2$
Character orbit 1792.a
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \(x^{4} - 5 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} + q^{7} + ( 2 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} + q^{7} + ( 2 - \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + \beta_{1} q^{13} + ( 1 - \beta_{3} ) q^{15} + 2 q^{17} + \beta_{2} q^{19} + \beta_{2} q^{21} + ( -1 + \beta_{3} ) q^{23} + ( 2 + \beta_{3} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{27} -2 \beta_{2} q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + 4 q^{33} + \beta_{1} q^{35} -2 \beta_{2} q^{37} + ( 1 - \beta_{3} ) q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -\beta_{1} + 4 \beta_{2} ) q^{45} + q^{49} + 2 \beta_{2} q^{51} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + ( 5 - \beta_{3} ) q^{57} + ( 2 \beta_{1} - \beta_{2} ) q^{59} + \beta_{1} q^{61} + ( 2 - \beta_{3} ) q^{63} + ( 7 + \beta_{3} ) q^{65} + ( -5 \beta_{1} - \beta_{2} ) q^{67} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{69} + 8 q^{71} + 6 q^{73} + ( -2 \beta_{1} - \beta_{2} ) q^{75} + ( -\beta_{1} + \beta_{2} ) q^{77} + ( 6 - \beta_{3} ) q^{81} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{83} + 2 \beta_{1} q^{85} + ( -10 + 2 \beta_{3} ) q^{87} + ( 8 - 2 \beta_{3} ) q^{89} + \beta_{1} q^{91} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{93} + ( 1 - \beta_{3} ) q^{95} + ( 4 - 2 \beta_{3} ) q^{97} + ( 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + 8q^{9} + O(q^{10}) \) \( 4q + 4q^{7} + 8q^{9} + 4q^{15} + 8q^{17} - 4q^{23} + 8q^{25} - 8q^{31} + 16q^{33} + 4q^{39} + 16q^{41} + 4q^{49} - 24q^{55} + 20q^{57} + 8q^{63} + 28q^{65} + 32q^{71} + 24q^{73} + 24q^{81} - 40q^{87} + 32q^{89} + 4q^{95} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{2} + 5 \beta_{1}\)\()/2\)

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
0.662153
2.13578
−2.13578
−0.662153
0 −3.02045 0 −1.69614 0 1.00000 0 6.12311 0
1.2 0 −0.936426 0 3.33513 0 1.00000 0 −2.12311 0
1.3 0 0.936426 0 −3.33513 0 1.00000 0 −2.12311 0
1.4 0 3.02045 0 1.69614 0 1.00000 0 6.12311 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.x 4
4.b odd 2 1 1792.2.a.v 4
8.b even 2 1 inner 1792.2.a.x 4
8.d odd 2 1 1792.2.a.v 4
16.e even 4 2 56.2.b.b 4
16.f odd 4 2 224.2.b.b 4
48.i odd 4 2 504.2.c.d 4
48.k even 4 2 2016.2.c.c 4
112.j even 4 2 1568.2.b.d 4
112.l odd 4 2 392.2.b.c 4
112.u odd 12 4 1568.2.t.d 8
112.v even 12 4 1568.2.t.e 8
112.w even 12 4 392.2.p.f 8
112.x odd 12 4 392.2.p.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 16.e even 4 2
224.2.b.b 4 16.f odd 4 2
392.2.b.c 4 112.l odd 4 2
392.2.p.e 8 112.x odd 12 4
392.2.p.f 8 112.w even 12 4
504.2.c.d 4 48.i odd 4 2
1568.2.b.d 4 112.j even 4 2
1568.2.t.d 8 112.u odd 12 4
1568.2.t.e 8 112.v even 12 4
1792.2.a.v 4 4.b odd 2 1
1792.2.a.v 4 8.d odd 2 1
1792.2.a.x 4 1.a even 1 1 trivial
1792.2.a.x 4 8.b even 2 1 inner
2016.2.c.c 4 48.k even 4 2

Hecke kernels

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

\( T_{3}^{4} - 10 T_{3}^{2} + 8 \)
\( T_{5}^{4} - 14 T_{5}^{2} + 32 \)
\( T_{23}^{2} + 2 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 8 - 10 T^{2} + T^{4} \)
$5$ \( 32 - 14 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 32 - 20 T^{2} + T^{4} \)
$13$ \( 32 - 14 T^{2} + T^{4} \)
$17$ \( ( -2 + T )^{4} \)
$19$ \( 8 - 10 T^{2} + T^{4} \)
$23$ \( ( -16 + 2 T + T^{2} )^{2} \)
$29$ \( 128 - 40 T^{2} + T^{4} \)
$31$ \( ( -64 + 4 T + T^{2} )^{2} \)
$37$ \( 128 - 40 T^{2} + T^{4} \)
$41$ \( ( -52 - 8 T + T^{2} )^{2} \)
$43$ \( 32 - 20 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 512 - 80 T^{2} + T^{4} \)
$59$ \( 8 - 58 T^{2} + T^{4} \)
$61$ \( 32 - 14 T^{2} + T^{4} \)
$67$ \( 32768 - 380 T^{2} + T^{4} \)
$71$ \( ( -8 + T )^{4} \)
$73$ \( ( -6 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( 2888 - 122 T^{2} + T^{4} \)
$89$ \( ( -4 - 16 T + T^{2} )^{2} \)
$97$ \( ( -52 - 8 T + T^{2} )^{2} \)
show more
show less