Properties

Label 1792.2.a.q
Level $1792$
Weight $2$
Character orbit 1792.a
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 448)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 2 \beta ) q^{9} + ( -2 - 2 \beta ) q^{11} + ( -5 - \beta ) q^{13} + ( -4 - 2 \beta ) q^{15} + 2 q^{17} + ( -3 + \beta ) q^{19} + ( 1 + \beta ) q^{21} -2 \beta q^{23} + ( -1 + 2 \beta ) q^{25} + 4 q^{27} + ( -2 + 2 \beta ) q^{29} -4 q^{31} + ( -8 - 4 \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( -2 + 2 \beta ) q^{37} + ( -8 - 6 \beta ) q^{39} + 2 q^{41} + ( -2 - 2 \beta ) q^{43} + ( -7 - 3 \beta ) q^{45} + ( 4 - 4 \beta ) q^{47} + q^{49} + ( 2 + 2 \beta ) q^{51} -12 q^{53} + ( 8 + 4 \beta ) q^{55} -2 \beta q^{57} + ( 1 + 5 \beta ) q^{59} + ( -1 + 7 \beta ) q^{61} + ( 1 + 2 \beta ) q^{63} + ( 8 + 6 \beta ) q^{65} + 8 q^{67} + ( -6 - 2 \beta ) q^{69} + ( -4 + 4 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} + ( 5 + \beta ) q^{75} + ( -2 - 2 \beta ) q^{77} + ( 4 + 4 \beta ) q^{79} + ( 1 - 2 \beta ) q^{81} + ( 3 - 5 \beta ) q^{83} + ( -2 - 2 \beta ) q^{85} + 4 q^{87} + ( 2 - 8 \beta ) q^{89} + ( -5 - \beta ) q^{91} + ( -4 - 4 \beta ) q^{93} + 2 \beta q^{95} + ( 2 + 4 \beta ) q^{97} + ( -14 - 6 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} - 10q^{13} - 8q^{15} + 4q^{17} - 6q^{19} + 2q^{21} - 2q^{25} + 8q^{27} - 4q^{29} - 8q^{31} - 16q^{33} - 2q^{35} - 4q^{37} - 16q^{39} + 4q^{41} - 4q^{43} - 14q^{45} + 8q^{47} + 2q^{49} + 4q^{51} - 24q^{53} + 16q^{55} + 2q^{59} - 2q^{61} + 2q^{63} + 16q^{65} + 16q^{67} - 12q^{69} - 8q^{71} - 12q^{73} + 10q^{75} - 4q^{77} + 8q^{79} + 2q^{81} + 6q^{83} - 4q^{85} + 8q^{87} + 4q^{89} - 10q^{91} - 8q^{93} + 4q^{97} - 28q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −0.732051 0 0.732051 0 1.00000 0 −2.46410 0
1.2 0 2.73205 0 −2.73205 0 1.00000 0 4.46410 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.q 2
4.b odd 2 1 1792.2.a.i 2
8.b even 2 1 1792.2.a.k 2
8.d odd 2 1 1792.2.a.s 2
16.e even 4 2 448.2.b.c 4
16.f odd 4 2 448.2.b.d yes 4
48.i odd 4 2 4032.2.c.k 4
48.k even 4 2 4032.2.c.n 4
112.j even 4 2 3136.2.b.h 4
112.l odd 4 2 3136.2.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.c 4 16.e even 4 2
448.2.b.d yes 4 16.f odd 4 2
1792.2.a.i 2 4.b odd 2 1
1792.2.a.k 2 8.b even 2 1
1792.2.a.q 2 1.a even 1 1 trivial
1792.2.a.s 2 8.d odd 2 1
3136.2.b.g 4 112.l odd 4 2
3136.2.b.h 4 112.j even 4 2
4032.2.c.k 4 48.i odd 4 2
4032.2.c.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1792))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{5}^{2} + 2 T_{5} - 2 \)
\( T_{23}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( 22 + 10 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 6 + 6 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( -8 + 4 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( -8 + 4 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -8 + 4 T + T^{2} \)
$47$ \( -32 - 8 T + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( -74 - 2 T + T^{2} \)
$61$ \( -146 + 2 T + T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( -32 + 8 T + T^{2} \)
$73$ \( -12 + 12 T + T^{2} \)
$79$ \( -32 - 8 T + T^{2} \)
$83$ \( -66 - 6 T + T^{2} \)
$89$ \( -188 - 4 T + T^{2} \)
$97$ \( -44 - 4 T + T^{2} \)
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