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: \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + (\beta_1 + 2) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + (\beta_1 + 2) q^{5} + q^{6} + q^{8} + q^{9} + (\beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{11} + q^{12} + ( - \beta_{3} - \beta_1 + 3) q^{13} + (\beta_1 + 2) q^{15} + q^{16} + ( - \beta_{3} + \beta_1 - 1) q^{17} + q^{18} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{19} + (\beta_1 + 2) q^{20} + (\beta_{2} + 1) q^{22} + q^{23} + q^{24} + (4 \beta_1 + 1) q^{25} + ( - \beta_{3} - \beta_1 + 3) q^{26} + q^{27} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_1 + 2) q^{30} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + q^{32} + (\beta_{2} + 1) q^{33} + ( - \beta_{3} + \beta_1 - 1) q^{34} + q^{36} + (\beta_{3} - \beta_1 + 1) q^{37} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_{3} - \beta_1 + 3) q^{39} + (\beta_1 + 2) q^{40} + ( - 2 \beta_{2} - \beta_1 - 2) q^{41} + (3 \beta_{3} + \beta_1 + 1) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_1 + 2) q^{45} + q^{46} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{47} + q^{48} + (4 \beta_1 + 1) q^{50} + ( - \beta_{3} + \beta_1 - 1) q^{51} + ( - \beta_{3} - \beta_1 + 3) q^{52} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{53} + q^{54} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{55} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + (\beta_1 + 2) q^{60} + ( - 2 \beta_{2} - \beta_1 + 4) q^{61} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{62} + q^{64} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{65} + (\beta_{2} + 1) q^{66} + ( - \beta_{3} + 5 \beta_1 + 5) q^{67} + ( - \beta_{3} + \beta_1 - 1) q^{68} + q^{69} + ( - \beta_{3} + \beta_{2} - \beta_1 - 6) q^{71} + q^{72} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{73} + (\beta_{3} - \beta_1 + 1) q^{74} + (4 \beta_1 + 1) q^{75} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{76} + ( - \beta_{3} - \beta_1 + 3) q^{78} + (\beta_{3} - \beta_{2} - 3 \beta_1 - 8) q^{79} + (\beta_1 + 2) q^{80} + q^{81} + ( - 2 \beta_{2} - \beta_1 - 2) q^{82} + (\beta_{2} + \beta_1 - 1) q^{83} + ( - 3 \beta_{3} - \beta_{2} + \beta_1) q^{85} + (3 \beta_{3} + \beta_1 + 1) q^{86} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{87} + (\beta_{2} + 1) q^{88} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{89} + (\beta_1 + 2) q^{90} + q^{92} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{93} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{94} + (7 \beta_{3} + 3 \beta_{2} - \beta_1) q^{95} + q^{96} + (\beta_{3} - 7 \beta_1 + 1) q^{97} + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 8 q^{10} + 4 q^{11} + 4 q^{12} + 12 q^{13} + 8 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{18} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 4 q^{23} + 4 q^{24} + 4 q^{25} + 12 q^{26} + 4 q^{27} + 8 q^{29} + 8 q^{30} - 4 q^{31} + 4 q^{32} + 4 q^{33} - 4 q^{34} + 4 q^{36} + 4 q^{37} + 4 q^{38} + 12 q^{39} + 8 q^{40} - 8 q^{41} + 4 q^{43} + 4 q^{44} + 8 q^{45} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} + 4 q^{54} + 8 q^{55} + 4 q^{57} + 8 q^{58} + 16 q^{59} + 8 q^{60} + 16 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} + 4 q^{66} + 20 q^{67} - 4 q^{68} + 4 q^{69} - 24 q^{71} + 4 q^{72} + 8 q^{73} + 4 q^{74} + 4 q^{75} + 4 q^{76} + 12 q^{78} - 32 q^{79} + 8 q^{80} + 4 q^{81} - 8 q^{82} - 4 q^{83} + 4 q^{86} + 8 q^{87} + 4 q^{88} + 20 q^{89} + 8 q^{90} + 4 q^{92} - 4 q^{93} + 12 q^{94} + 4 q^{96} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_{2} + 4\beta _1 + 5 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

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

Hecke characteristic polynomials

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