Properties

Label 6762.2.a.ct
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: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 2 \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 4 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + \beta_{2} + 4 \beta_{1} + 5\)

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
0.814115
−1.22833
−0.360409
2.77462
1.00000 1.00000 1.00000 0.585786 1.00000 0 1.00000 1.00000 0.585786
1.2 1.00000 1.00000 1.00000 0.585786 1.00000 0 1.00000 1.00000 0.585786
1.3 1.00000 1.00000 1.00000 3.41421 1.00000 0 1.00000 1.00000 3.41421
1.4 1.00000 1.00000 1.00000 3.41421 1.00000 0 1.00000 1.00000 3.41421
\(n\): 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.ct yes 4
7.b odd 2 1 6762.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6762.2.a.ci 4 7.b odd 2 1
6762.2.a.ct yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{2} - 4 T_{5} + 2 \)
\( T_{11}^{4} - 4 T_{11}^{3} - 20 T_{11}^{2} + 48 T_{11} + 16 \)
\( T_{13}^{4} - 12 T_{13}^{3} + 36 T_{13}^{2} + 16 T_{13} - 112 \)
\( T_{17}^{4} + 4 T_{17}^{3} - 12 T_{17}^{2} - 48 T_{17} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 2 - 4 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 + 48 T - 20 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( -112 + 16 T + 36 T^{2} - 12 T^{3} + T^{4} \)
$17$ \( -16 - 48 T - 12 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( -292 + 280 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 784 + 736 T - 104 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( 28 + 40 T - 56 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( -16 + 48 T - 12 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( -476 - 656 T - 84 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( 3088 + 112 T - 124 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 412 + 712 T - 56 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 1648 + 272 T - 92 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( -1024 + 1024 T - 32 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 4 + 352 T - 12 T^{2} - 16 T^{3} + T^{4} \)
$67$ \( 16 + 560 T + 36 T^{2} - 20 T^{3} + T^{4} \)
$71$ \( -1424 + 64 T + 160 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( -316 + 464 T - 68 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( -5056 + 256 T + 296 T^{2} + 32 T^{3} + T^{4} \)
$83$ \( 28 + 8 T - 24 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( -9968 + 4656 T - 188 T^{2} - 20 T^{3} + T^{4} \)
$97$ \( 7952 + 528 T - 204 T^{2} - 4 T^{3} + T^{4} \)
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