Properties

Label 2646.2.a.bp
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 3 + \beta ) q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 3 + \beta ) q^{5} + q^{8} + ( 3 + \beta ) q^{10} + q^{11} -3 \beta q^{13} + q^{16} -2 \beta q^{17} + ( 3 + 2 \beta ) q^{19} + ( 3 + \beta ) q^{20} + q^{22} + ( -1 - 3 \beta ) q^{23} + ( 6 + 6 \beta ) q^{25} -3 \beta q^{26} + ( -4 + 3 \beta ) q^{29} + ( 3 + 3 \beta ) q^{31} + q^{32} -2 \beta q^{34} + ( 1 + 3 \beta ) q^{37} + ( 3 + 2 \beta ) q^{38} + ( 3 + \beta ) q^{40} + ( 9 - 2 \beta ) q^{41} + ( 2 - 6 \beta ) q^{43} + q^{44} + ( -1 - 3 \beta ) q^{46} + ( 6 + 4 \beta ) q^{47} + ( 6 + 6 \beta ) q^{50} -3 \beta q^{52} -8 q^{53} + ( 3 + \beta ) q^{55} + ( -4 + 3 \beta ) q^{58} + ( 6 + 4 \beta ) q^{59} + ( -12 - 2 \beta ) q^{61} + ( 3 + 3 \beta ) q^{62} + q^{64} + ( -6 - 9 \beta ) q^{65} + ( -6 + 3 \beta ) q^{67} -2 \beta q^{68} + ( -3 + 3 \beta ) q^{71} + ( 6 - 2 \beta ) q^{73} + ( 1 + 3 \beta ) q^{74} + ( 3 + 2 \beta ) q^{76} + ( 2 - 6 \beta ) q^{79} + ( 3 + \beta ) q^{80} + ( 9 - 2 \beta ) q^{82} + ( 6 - 7 \beta ) q^{83} + ( -4 - 6 \beta ) q^{85} + ( 2 - 6 \beta ) q^{86} + q^{88} + ( 9 - 4 \beta ) q^{89} + ( -1 - 3 \beta ) q^{92} + ( 6 + 4 \beta ) q^{94} + ( 13 + 9 \beta ) q^{95} + 8 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 6q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 6q^{5} + 2q^{8} + 6q^{10} + 2q^{11} + 2q^{16} + 6q^{19} + 6q^{20} + 2q^{22} - 2q^{23} + 12q^{25} - 8q^{29} + 6q^{31} + 2q^{32} + 2q^{37} + 6q^{38} + 6q^{40} + 18q^{41} + 4q^{43} + 2q^{44} - 2q^{46} + 12q^{47} + 12q^{50} - 16q^{53} + 6q^{55} - 8q^{58} + 12q^{59} - 24q^{61} + 6q^{62} + 2q^{64} - 12q^{65} - 12q^{67} - 6q^{71} + 12q^{73} + 2q^{74} + 6q^{76} + 4q^{79} + 6q^{80} + 18q^{82} + 12q^{83} - 8q^{85} + 4q^{86} + 2q^{88} + 18q^{89} - 2q^{92} + 12q^{94} + 26q^{95} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 0 1.00000 1.58579 0 0 1.00000 0 1.58579
1.2 1.00000 0 1.00000 4.41421 0 0 1.00000 0 4.41421
\(n\): 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.bp yes 2
3.b odd 2 1 2646.2.a.be 2
7.b odd 2 1 2646.2.a.bk yes 2
21.c even 2 1 2646.2.a.bj yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2646.2.a.be 2 3.b odd 2 1
2646.2.a.bj yes 2 21.c even 2 1
2646.2.a.bk yes 2 7.b odd 2 1
2646.2.a.bp yes 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{2} - 6 T_{5} + 7 \)
\( T_{11} - 1 \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 7 - 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -18 + T^{2} \)
$17$ \( -8 + T^{2} \)
$19$ \( 1 - 6 T + T^{2} \)
$23$ \( -17 + 2 T + T^{2} \)
$29$ \( -2 + 8 T + T^{2} \)
$31$ \( -9 - 6 T + T^{2} \)
$37$ \( -17 - 2 T + T^{2} \)
$41$ \( 73 - 18 T + T^{2} \)
$43$ \( -68 - 4 T + T^{2} \)
$47$ \( 4 - 12 T + T^{2} \)
$53$ \( ( 8 + T )^{2} \)
$59$ \( 4 - 12 T + T^{2} \)
$61$ \( 136 + 24 T + T^{2} \)
$67$ \( 18 + 12 T + T^{2} \)
$71$ \( -9 + 6 T + T^{2} \)
$73$ \( 28 - 12 T + T^{2} \)
$79$ \( -68 - 4 T + T^{2} \)
$83$ \( -62 - 12 T + T^{2} \)
$89$ \( 49 - 18 T + T^{2} \)
$97$ \( -128 + T^{2} \)
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