Properties

Label 2646.2.a.bl
Level $2646$
Weight $2$
Character orbit 2646.a
Self dual yes
Analytic conductor $21.128$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 + \beta ) q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 + \beta ) q^{5} + q^{8} + ( -1 + \beta ) q^{10} + ( 1 - \beta ) q^{11} + ( -2 + \beta ) q^{13} + q^{16} + ( 1 - \beta ) q^{17} + 2 q^{19} + ( -1 + \beta ) q^{20} + ( 1 - \beta ) q^{22} + ( 4 + 2 \beta ) q^{23} + ( 3 - 2 \beta ) q^{25} + ( -2 + \beta ) q^{26} + ( 5 + \beta ) q^{29} + ( -2 + \beta ) q^{31} + q^{32} + ( 1 - \beta ) q^{34} + ( -4 + 3 \beta ) q^{37} + 2 q^{38} + ( -1 + \beta ) q^{40} + ( 3 - 3 \beta ) q^{41} + 5 q^{43} + ( 1 - \beta ) q^{44} + ( 4 + 2 \beta ) q^{46} + ( -3 - 3 \beta ) q^{47} + ( 3 - 2 \beta ) q^{50} + ( -2 + \beta ) q^{52} + 6 q^{53} + ( -8 + 2 \beta ) q^{55} + ( 5 + \beta ) q^{58} + ( 11 + \beta ) q^{59} + ( 10 + \beta ) q^{61} + ( -2 + \beta ) q^{62} + q^{64} + ( 9 - 3 \beta ) q^{65} + ( 3 + 2 \beta ) q^{67} + ( 1 - \beta ) q^{68} + ( 13 - \beta ) q^{71} -4 \beta q^{73} + ( -4 + 3 \beta ) q^{74} + 2 q^{76} + ( -2 - 5 \beta ) q^{79} + ( -1 + \beta ) q^{80} + ( 3 - 3 \beta ) q^{82} + ( -8 + 2 \beta ) q^{83} + ( -8 + 2 \beta ) q^{85} + 5 q^{86} + ( 1 - \beta ) q^{88} + ( -3 - 3 \beta ) q^{89} + ( 4 + 2 \beta ) q^{92} + ( -3 - 3 \beta ) q^{94} + ( -2 + 2 \beta ) q^{95} + ( 3 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} - 2q^{10} + 2q^{11} - 4q^{13} + 2q^{16} + 2q^{17} + 4q^{19} - 2q^{20} + 2q^{22} + 8q^{23} + 6q^{25} - 4q^{26} + 10q^{29} - 4q^{31} + 2q^{32} + 2q^{34} - 8q^{37} + 4q^{38} - 2q^{40} + 6q^{41} + 10q^{43} + 2q^{44} + 8q^{46} - 6q^{47} + 6q^{50} - 4q^{52} + 12q^{53} - 16q^{55} + 10q^{58} + 22q^{59} + 20q^{61} - 4q^{62} + 2q^{64} + 18q^{65} + 6q^{67} + 2q^{68} + 26q^{71} - 8q^{74} + 4q^{76} - 4q^{79} - 2q^{80} + 6q^{82} - 16q^{83} - 16q^{85} + 10q^{86} + 2q^{88} - 6q^{89} + 8q^{92} - 6q^{94} - 4q^{95} + 6q^{97} + O(q^{100}) \)

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
1.00000 0 1.00000 −3.64575 0 0 1.00000 0 −3.64575
1.2 1.00000 0 1.00000 1.64575 0 0 1.00000 0 1.64575
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 2646.2.a.bl 2
3.b odd 2 1 2646.2.a.bi 2
7.b odd 2 1 2646.2.a.bo 2
7.c even 3 2 378.2.g.g 4
21.c even 2 1 2646.2.a.bf 2
21.h odd 6 2 378.2.g.h yes 4
63.g even 3 2 1134.2.h.q 4
63.h even 3 2 1134.2.e.t 4
63.j odd 6 2 1134.2.e.q 4
63.n odd 6 2 1134.2.h.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 7.c even 3 2
378.2.g.h yes 4 21.h odd 6 2
1134.2.e.q 4 63.j odd 6 2
1134.2.e.t 4 63.h even 3 2
1134.2.h.q 4 63.g even 3 2
1134.2.h.t 4 63.n odd 6 2
2646.2.a.bf 2 21.c even 2 1
2646.2.a.bi 2 3.b odd 2 1
2646.2.a.bl 2 1.a even 1 1 trivial
2646.2.a.bo 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2646))\):

\( T_{5}^{2} + 2 T_{5} - 6 \)
\( T_{11}^{2} - 2 T_{11} - 6 \)
\( T_{13}^{2} + 4 T_{13} - 3 \)
\( T_{17}^{2} - 2 T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -6 - 2 T + T^{2} \)
$13$ \( -3 + 4 T + T^{2} \)
$17$ \( -6 - 2 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -12 - 8 T + T^{2} \)
$29$ \( 18 - 10 T + T^{2} \)
$31$ \( -3 + 4 T + T^{2} \)
$37$ \( -47 + 8 T + T^{2} \)
$41$ \( -54 - 6 T + T^{2} \)
$43$ \( ( -5 + T )^{2} \)
$47$ \( -54 + 6 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 114 - 22 T + T^{2} \)
$61$ \( 93 - 20 T + T^{2} \)
$67$ \( -19 - 6 T + T^{2} \)
$71$ \( 162 - 26 T + T^{2} \)
$73$ \( -112 + T^{2} \)
$79$ \( -171 + 4 T + T^{2} \)
$83$ \( 36 + 16 T + T^{2} \)
$89$ \( -54 + 6 T + T^{2} \)
$97$ \( -103 - 6 T + T^{2} \)
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