Properties

Label 6912.2.a.bv
Level 6912
Weight 2
Character orbit 6912.a
Self dual yes
Analytic conductor 55.193
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6912 = 2^{8} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.1925978771\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.64575
−2.64575
0 0 0 1.00000 0 1.00000 0 0 0
1.2 0 0 0 1.00000 0 1.00000 0 0 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
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6912.2.a.bv 2
3.b odd 2 1 6912.2.a.bd 2
4.b odd 2 1 6912.2.a.bu 2
8.b even 2 1 6912.2.a.bd 2
8.d odd 2 1 6912.2.a.bc 2
12.b even 2 1 6912.2.a.bc 2
16.e even 4 2 864.2.d.a 4
16.f odd 4 2 216.2.d.b 4
24.f even 2 1 6912.2.a.bu 2
24.h odd 2 1 inner 6912.2.a.bv 2
48.i odd 4 2 864.2.d.a 4
48.k even 4 2 216.2.d.b 4
144.u even 12 4 648.2.n.n 8
144.v odd 12 4 648.2.n.n 8
144.w odd 12 4 2592.2.r.p 8
144.x even 12 4 2592.2.r.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 16.f odd 4 2
216.2.d.b 4 48.k even 4 2
648.2.n.n 8 144.u even 12 4
648.2.n.n 8 144.v odd 12 4
864.2.d.a 4 16.e even 4 2
864.2.d.a 4 48.i odd 4 2
2592.2.r.p 8 144.w odd 12 4
2592.2.r.p 8 144.x even 12 4
6912.2.a.bc 2 8.d odd 2 1
6912.2.a.bc 2 12.b even 2 1
6912.2.a.bd 2 3.b odd 2 1
6912.2.a.bd 2 8.b even 2 1
6912.2.a.bu 2 4.b odd 2 1
6912.2.a.bu 2 24.f even 2 1
6912.2.a.bv 2 1.a even 1 1 trivial
6912.2.a.bv 2 24.h odd 2 1 inner

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6912))\):

\( T_{5} - 1 \)
\( T_{7} - 1 \)
\( T_{11} + 3 \)
\( T_{13}^{2} - 28 \)
\( T_{17}^{2} - 28 \)
\( T_{19}^{2} - 28 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 2 T^{2} + 169 T^{4} \)
$17$ \( 1 + 6 T^{2} + 289 T^{4} \)
$19$ \( 1 + 10 T^{2} + 361 T^{4} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 46 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 54 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 26 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 9 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{2} \)
$71$ \( 1 - 110 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 + 3 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 7 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 66 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{2} \)
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