Properties

Label 6762.2.a.cm
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.42048.1
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{6} + q^{8} + q^{9} + ( 1 - \beta_{2} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} - q^{12} + ( -1 + \beta_{1} ) q^{13} + ( -1 + \beta_{2} ) q^{15} + q^{16} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + q^{18} + ( 1 + \beta_{2} - \beta_{3} ) q^{19} + ( 1 - \beta_{2} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + q^{23} - q^{24} + ( 4 - 2 \beta_{2} ) q^{25} + ( -1 + \beta_{1} ) q^{26} - q^{27} + ( 2 + \beta_{1} ) q^{29} + ( -1 + \beta_{2} ) q^{30} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + q^{32} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( 3 + \beta_{1} - \beta_{2} ) q^{34} + q^{36} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( 1 + \beta_{2} - \beta_{3} ) q^{38} + ( 1 - \beta_{1} ) q^{39} + ( 1 - \beta_{2} ) q^{40} + ( -1 - \beta_{2} + \beta_{3} ) q^{41} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + ( 1 - \beta_{2} ) q^{45} + q^{46} + ( -4 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} - q^{48} + ( 4 - 2 \beta_{2} ) q^{50} + ( -3 - \beta_{1} + \beta_{2} ) q^{51} + ( -1 + \beta_{1} ) q^{52} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{53} - q^{54} + ( -5 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{55} + ( -1 - \beta_{2} + \beta_{3} ) q^{57} + ( 2 + \beta_{1} ) q^{58} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{59} + ( -1 + \beta_{2} ) q^{60} + ( 2 + 4 \beta_{1} ) q^{61} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{62} + q^{64} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{65} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( 6 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{67} + ( 3 + \beta_{1} - \beta_{2} ) q^{68} - q^{69} + ( 11 + \beta_{1} ) q^{71} + q^{72} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{73} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{74} + ( -4 + 2 \beta_{2} ) q^{75} + ( 1 + \beta_{2} - \beta_{3} ) q^{76} + ( 1 - \beta_{1} ) q^{78} + ( -2 - \beta_{1} ) q^{79} + ( 1 - \beta_{2} ) q^{80} + q^{81} + ( -1 - \beta_{2} + \beta_{3} ) q^{82} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{83} + ( 9 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -2 - \beta_{1} ) q^{87} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{2} ) q^{90} + q^{92} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -4 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{94} + ( -7 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{95} - q^{96} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + 4q^{10} + 6q^{11} - 4q^{12} - 6q^{13} - 4q^{15} + 4q^{16} + 10q^{17} + 4q^{18} + 4q^{19} + 4q^{20} + 6q^{22} + 4q^{23} - 4q^{24} + 16q^{25} - 6q^{26} - 4q^{27} + 6q^{29} - 4q^{30} - 2q^{31} + 4q^{32} - 6q^{33} + 10q^{34} + 4q^{36} + 4q^{38} + 6q^{39} + 4q^{40} - 4q^{41} + 20q^{43} + 6q^{44} + 4q^{45} + 4q^{46} - 10q^{47} - 4q^{48} + 16q^{50} - 10q^{51} - 6q^{52} - 4q^{54} - 10q^{55} - 4q^{57} + 6q^{58} - 6q^{59} - 4q^{60} - 2q^{62} + 4q^{64} - 14q^{65} - 6q^{66} + 18q^{67} + 10q^{68} - 4q^{69} + 42q^{71} + 4q^{72} - 6q^{73} - 16q^{75} + 4q^{76} + 6q^{78} - 6q^{79} + 4q^{80} + 4q^{81} - 4q^{82} + 10q^{83} + 34q^{85} + 20q^{86} - 6q^{87} + 6q^{88} + 4q^{90} + 4q^{92} + 2q^{93} - 10q^{94} - 20q^{95} - 4q^{96} + 10q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 10 \nu + 10 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 6 \nu + 10 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 7 \nu^{2} + 2 \nu - 38 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} - \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + 8 \beta_{2} - 9 \beta_{1} + 11\)

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
4.18558
−1.77137
−2.35854
1.94433
1.00000 −1.00000 1.00000 −1.82843 −1.00000 0 1.00000 1.00000 −1.82843
1.2 1.00000 −1.00000 1.00000 −1.82843 −1.00000 0 1.00000 1.00000 −1.82843
1.3 1.00000 −1.00000 1.00000 3.82843 −1.00000 0 1.00000 1.00000 3.82843
1.4 1.00000 −1.00000 1.00000 3.82843 −1.00000 0 1.00000 1.00000 3.82843
\(n\): 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.cm 4
7.b odd 2 1 6762.2.a.cp 4
7.d odd 6 2 966.2.i.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.l 8 7.d odd 6 2
6762.2.a.cm 4 1.a even 1 1 trivial
6762.2.a.cp 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{2} - 2 T_{5} - 7 \)
\( T_{11}^{4} - 6 T_{11}^{3} - 19 T_{11}^{2} + 156 T_{11} - 188 \)
\( T_{13}^{4} + 6 T_{13}^{3} - T_{13}^{2} - 36 T_{13} - 2 \)
\( T_{17}^{4} - 10 T_{17}^{3} + 15 T_{17}^{2} + 68 T_{17} - 146 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( -7 - 2 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( -188 + 156 T - 19 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( -2 - 36 T - T^{2} + 6 T^{3} + T^{4} \)
$17$ \( -146 + 68 T + 15 T^{2} - 10 T^{3} + T^{4} \)
$19$ \( 64 + 32 T - 30 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 16 + 24 T - T^{2} - 6 T^{3} + T^{4} \)
$31$ \( -188 + 220 T - 67 T^{2} + 2 T^{3} + T^{4} \)
$37$ \( 136 + 72 T - 58 T^{2} + T^{4} \)
$41$ \( 64 - 32 T - 30 T^{2} + 4 T^{3} + T^{4} \)
$43$ \( -5696 + 1376 T + 20 T^{2} - 20 T^{3} + T^{4} \)
$47$ \( 1348 - 500 T - 75 T^{2} + 10 T^{3} + T^{4} \)
$53$ \( 553 + 48 T - 58 T^{2} + T^{4} \)
$59$ \( 112 - 216 T - 49 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( 8848 - 384 T - 232 T^{2} + T^{4} \)
$67$ \( -11196 + 2988 T - 99 T^{2} - 18 T^{3} + T^{4} \)
$71$ \( 10654 - 4332 T + 647 T^{2} - 42 T^{3} + T^{4} \)
$73$ \( -32 + 96 T - 55 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 16 - 24 T - T^{2} + 6 T^{3} + T^{4} \)
$83$ \( 1252 + 220 T - 91 T^{2} - 10 T^{3} + T^{4} \)
$89$ \( -632 - 504 T - 106 T^{2} + T^{4} \)
$97$ \( 64 + 208 T - 25 T^{2} - 10 T^{3} + T^{4} \)
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