Properties

Label 6762.2.a.cf
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3132.1
Defining polynomial: \(x^{3} - 15 x - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + \beta_{1} q^{11} + q^{12} + ( 1 + \beta_{1} + \beta_{2} ) q^{13} - q^{15} + q^{16} + ( -1 - \beta_{1} + \beta_{2} ) q^{17} - q^{18} + ( -2 - \beta_{2} ) q^{19} - q^{20} -\beta_{1} q^{22} - q^{23} - q^{24} -4 q^{25} + ( -1 - \beta_{1} - \beta_{2} ) q^{26} + q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + q^{30} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} - q^{32} + \beta_{1} q^{33} + ( 1 + \beta_{1} - \beta_{2} ) q^{34} + q^{36} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( 2 + \beta_{2} ) q^{38} + ( 1 + \beta_{1} + \beta_{2} ) q^{39} + q^{40} + ( 6 - \beta_{2} ) q^{41} -2 \beta_{1} q^{43} + \beta_{1} q^{44} - q^{45} + q^{46} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{47} + q^{48} + 4 q^{50} + ( -1 - \beta_{1} + \beta_{2} ) q^{51} + ( 1 + \beta_{1} + \beta_{2} ) q^{52} + ( -1 - 2 \beta_{1} ) q^{53} - q^{54} -\beta_{1} q^{55} + ( -2 - \beta_{2} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} ) q^{58} + ( 4 + \beta_{1} - \beta_{2} ) q^{59} - q^{60} -10 q^{61} + ( \beta_{1} + 2 \beta_{2} ) q^{62} + q^{64} + ( -1 - \beta_{1} - \beta_{2} ) q^{65} -\beta_{1} q^{66} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( -1 - \beta_{1} + \beta_{2} ) q^{68} - q^{69} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{71} - q^{72} + ( -5 - \beta_{1} ) q^{73} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{74} -4 q^{75} + ( -2 - \beta_{2} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} ) q^{78} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{79} - q^{80} + q^{81} + ( -6 + \beta_{2} ) q^{82} + ( 4 - \beta_{1} ) q^{83} + ( 1 + \beta_{1} - \beta_{2} ) q^{85} + 2 \beta_{1} q^{86} + ( 4 - \beta_{1} - \beta_{2} ) q^{87} -\beta_{1} q^{88} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{89} + q^{90} - q^{92} + ( -\beta_{1} - 2 \beta_{2} ) q^{93} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{94} + ( 2 + \beta_{2} ) q^{95} - q^{96} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{97} + \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} - 6q^{19} - 3q^{20} - 3q^{23} - 3q^{24} - 12q^{25} - 3q^{26} + 3q^{27} + 12q^{29} + 3q^{30} - 3q^{32} + 3q^{34} + 3q^{36} - 12q^{37} + 6q^{38} + 3q^{39} + 3q^{40} + 18q^{41} - 3q^{45} + 3q^{46} + 3q^{47} + 3q^{48} + 12q^{50} - 3q^{51} + 3q^{52} - 3q^{53} - 3q^{54} - 6q^{57} - 12q^{58} + 12q^{59} - 3q^{60} - 30q^{61} + 3q^{64} - 3q^{65} - 21q^{67} - 3q^{68} - 3q^{69} - 3q^{71} - 3q^{72} - 15q^{73} + 12q^{74} - 12q^{75} - 6q^{76} - 3q^{78} - 12q^{79} - 3q^{80} + 3q^{81} - 18q^{82} + 12q^{83} + 3q^{85} + 12q^{87} - 12q^{89} + 3q^{90} - 3q^{92} - 3q^{94} + 6q^{95} - 3q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 15 x - 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 10\)

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
−3.65491
−0.404409
4.05932
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.3 −1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 6762.2.a.cf 3
7.b odd 2 1 6762.2.a.ce 3
7.d odd 6 2 966.2.i.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.k 6 7.d odd 6 2
6762.2.a.ce 3 7.b odd 2 1
6762.2.a.cf 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} + 1 \)
\( T_{11}^{3} - 15 T_{11} - 6 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 22 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 36 T_{17} - 126 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( T^{3} \)
$11$ \( -6 - 15 T + T^{3} \)
$13$ \( 22 - 24 T - 3 T^{2} + T^{3} \)
$17$ \( -126 - 36 T + 3 T^{2} + T^{3} \)
$19$ \( -48 - 6 T + 6 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 48 + 21 T - 12 T^{2} + T^{3} \)
$31$ \( -214 - 75 T + T^{3} \)
$37$ \( -588 - 42 T + 12 T^{2} + T^{3} \)
$41$ \( -128 + 90 T - 18 T^{2} + T^{3} \)
$43$ \( 48 - 60 T + T^{3} \)
$47$ \( -140 - 72 T - 3 T^{2} + T^{3} \)
$53$ \( -11 - 57 T + 3 T^{2} + T^{3} \)
$59$ \( 180 + 9 T - 12 T^{2} + T^{3} \)
$61$ \( ( 10 + T )^{3} \)
$67$ \( -396 + 72 T + 21 T^{2} + T^{3} \)
$71$ \( -726 - 132 T + 3 T^{2} + T^{3} \)
$73$ \( 56 + 60 T + 15 T^{2} + T^{3} \)
$79$ \( -1068 - 87 T + 12 T^{2} + T^{3} \)
$83$ \( 2 + 33 T - 12 T^{2} + T^{3} \)
$89$ \( -588 - 42 T + 12 T^{2} + T^{3} \)
$97$ \( -108 - 243 T + T^{3} \)
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