Properties

Label 6762.2.a.cs
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: 4.4.23252.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 2\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 6 \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 3 \beta_{1} + 3\)

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
−0.634868
0.565882
2.95372
−1.88474
1.00000 1.00000 1.00000 −3.47747 1.00000 0 1.00000 1.00000 −3.47747
1.2 1.00000 1.00000 1.00000 0.722765 1.00000 0 1.00000 1.00000 0.722765
1.3 1.00000 1.00000 1.00000 1.49415 1.00000 0 1.00000 1.00000 1.49415
1.4 1.00000 1.00000 1.00000 4.26056 1.00000 0 1.00000 1.00000 4.26056
\(n\): 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.cs yes 4
7.b odd 2 1 6762.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6762.2.a.cj 4 7.b odd 2 1
6762.2.a.cs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{4} - 3 T_{5}^{3} - 12 T_{5}^{2} + 32 T_{5} - 16 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 24 T_{11}^{2} + 32 \)
\( T_{13}^{4} - 5 T_{13}^{3} - 6 T_{13}^{2} + 20 T_{13} - 8 \)
\( T_{17}^{4} - 10 T_{17}^{3} - 4 T_{17}^{2} + 184 T_{17} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -16 + 32 T - 12 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 32 - 24 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( -8 + 20 T - 6 T^{2} - 5 T^{3} + T^{4} \)
$17$ \( -64 + 184 T - 4 T^{2} - 10 T^{3} + T^{4} \)
$19$ \( ( -2 + T )^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( 2744 + 156 T - 106 T^{2} - 3 T^{3} + T^{4} \)
$31$ \( 2464 - 296 T - 92 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( -328 + 316 T - 74 T^{2} - T^{3} + T^{4} \)
$41$ \( 1024 + 128 T - 80 T^{2} - T^{3} + T^{4} \)
$43$ \( 3472 - 120 T - 128 T^{2} + T^{3} + T^{4} \)
$47$ \( -2456 + 948 T - 2 T^{2} - 17 T^{3} + T^{4} \)
$53$ \( -656 + 544 T - 88 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( 11008 - 128 T - 224 T^{2} + T^{4} \)
$61$ \( 64 - 208 T + 160 T^{2} - 24 T^{3} + T^{4} \)
$67$ \( -2816 - 1168 T - 104 T^{2} + 8 T^{3} + T^{4} \)
$71$ \( 3584 - 640 T - 160 T^{2} + 4 T^{3} + T^{4} \)
$73$ \( -128 + 480 T - 136 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( -128 + 480 T - 136 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( -4384 + 1256 T - 12 T^{2} - 18 T^{3} + T^{4} \)
$89$ \( 32 - 520 T + 212 T^{2} - 26 T^{3} + T^{4} \)
$97$ \( -3512 + 4948 T - 346 T^{2} - 13 T^{3} + T^{4} \)
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