Properties

Label 6762.2.a.cg
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: 4.4.138768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} - q^{8} + q^{9} -\beta_{1} q^{10} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} - q^{12} + ( -1 + \beta_{1} - \beta_{3} ) q^{13} -\beta_{1} q^{15} + q^{16} + ( -\beta_{1} + \beta_{2} ) q^{17} - q^{18} + ( -1 - \beta_{1} - \beta_{3} ) q^{19} + \beta_{1} q^{20} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} + q^{23} + q^{24} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{25} + ( 1 - \beta_{1} + \beta_{3} ) q^{26} - q^{27} + ( -2 - \beta_{1} - \beta_{3} ) q^{29} + \beta_{1} q^{30} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{31} - q^{32} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( \beta_{1} - \beta_{2} ) q^{34} + q^{36} + ( 1 + \beta_{1} - \beta_{3} ) q^{37} + ( 1 + \beta_{1} + \beta_{3} ) q^{38} + ( 1 - \beta_{1} + \beta_{3} ) q^{39} -\beta_{1} q^{40} + ( 3 - \beta_{1} - \beta_{3} ) q^{41} + 2 q^{43} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + \beta_{1} q^{45} - q^{46} + ( -3 + \beta_{1} - 2 \beta_{3} ) q^{47} - q^{48} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( \beta_{1} - \beta_{2} ) q^{51} + ( -1 + \beta_{1} - \beta_{3} ) q^{52} + ( 2 - \beta_{1} ) q^{53} + q^{54} + ( 4 + 3 \beta_{1} ) q^{55} + ( 1 + \beta_{1} + \beta_{3} ) q^{57} + ( 2 + \beta_{1} + \beta_{3} ) q^{58} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{59} -\beta_{1} q^{60} + ( -8 + 2 \beta_{1} ) q^{61} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{62} + q^{64} + ( 5 - \beta_{1} - \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{66} + ( 6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -\beta_{1} + \beta_{2} ) q^{68} - q^{69} + ( 4 - \beta_{3} ) q^{71} - q^{72} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -1 - \beta_{1} + \beta_{3} ) q^{74} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{75} + ( -1 - \beta_{1} - \beta_{3} ) q^{76} + ( -1 + \beta_{1} - \beta_{3} ) q^{78} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( -3 + \beta_{1} + \beta_{3} ) q^{82} + ( 4 - \beta_{1} + 2 \beta_{3} ) q^{83} + ( -7 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{85} -2 q^{86} + ( 2 + \beta_{1} + \beta_{3} ) q^{87} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( -5 + 3 \beta_{1} + \beta_{3} ) q^{89} -\beta_{1} q^{90} + q^{92} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{93} + ( 3 - \beta_{1} + 2 \beta_{3} ) q^{94} + ( -7 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{95} + q^{96} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 2q^{5} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{10} + 6q^{11} - 4q^{12} - 2q^{13} - 2q^{15} + 4q^{16} - 2q^{17} - 4q^{18} - 6q^{19} + 2q^{20} - 6q^{22} + 4q^{23} + 4q^{24} + 6q^{25} + 2q^{26} - 4q^{27} - 10q^{29} + 2q^{30} - 14q^{31} - 4q^{32} - 6q^{33} + 2q^{34} + 4q^{36} + 6q^{37} + 6q^{38} + 2q^{39} - 2q^{40} + 10q^{41} + 8q^{43} + 6q^{44} + 2q^{45} - 4q^{46} - 10q^{47} - 4q^{48} - 6q^{50} + 2q^{51} - 2q^{52} + 6q^{53} + 4q^{54} + 22q^{55} + 6q^{57} + 10q^{58} + 24q^{59} - 2q^{60} - 28q^{61} + 14q^{62} + 4q^{64} + 18q^{65} + 6q^{66} + 28q^{67} - 2q^{68} - 4q^{69} + 16q^{71} - 4q^{72} + 6q^{73} - 6q^{74} - 6q^{75} - 6q^{76} - 2q^{78} - 4q^{79} + 2q^{80} + 4q^{81} - 10q^{82} + 14q^{83} - 26q^{85} - 8q^{86} + 10q^{87} - 6q^{88} - 14q^{89} - 2q^{90} + 4q^{92} + 14q^{93} + 10q^{94} - 34q^{95} + 4q^{96} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8\):

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

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
−2.89728
−0.479029
1.47903
3.89728
−1.00000 −1.00000 1.00000 −2.89728 1.00000 0 −1.00000 1.00000 2.89728
1.2 −1.00000 −1.00000 1.00000 −0.479029 1.00000 0 −1.00000 1.00000 0.479029
1.3 −1.00000 −1.00000 1.00000 1.47903 1.00000 0 −1.00000 1.00000 −1.47903
1.4 −1.00000 −1.00000 1.00000 3.89728 1.00000 0 −1.00000 1.00000 −3.89728
\(n\): 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.cg 4
7.b odd 2 1 6762.2.a.ch 4
7.c even 3 2 966.2.i.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.n 8 7.c even 3 2
6762.2.a.cg 4 1.a even 1 1 trivial
6762.2.a.ch 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 11 T_{5}^{2} + 12 T_{5} + 8 \)
\( T_{11}^{4} - 6 T_{11}^{3} - 22 T_{11}^{2} + 170 T_{11} - 199 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 34 T_{13}^{2} - 112 T_{13} - 56 \)
\( T_{17}^{4} + 2 T_{17}^{3} - 25 T_{17}^{2} - 82 T_{17} - 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 8 + 12 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -199 + 170 T - 22 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( -56 - 112 T - 34 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( -62 - 82 T - 25 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 296 - 80 T - 34 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 189 - 126 T - 10 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( -636 - 420 T + 5 T^{2} + 14 T^{3} + T^{4} \)
$37$ \( 32 + 16 T - 22 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( -56 + 224 T - 10 T^{2} - 10 T^{3} + T^{4} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( -2588 - 1052 T - 79 T^{2} + 10 T^{3} + T^{4} \)
$53$ \( -12 + 24 T + T^{2} - 6 T^{3} + T^{4} \)
$59$ \( -7104 + 1152 T + 97 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( 128 + 672 T + 244 T^{2} + 28 T^{3} + T^{4} \)
$67$ \( -9008 + 1232 T + 152 T^{2} - 28 T^{3} + T^{4} \)
$71$ \( 18 - 66 T + 67 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( -1624 + 868 T - 103 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 1213 - 1244 T - 226 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( -3708 + 1176 T - 43 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( -3072 - 1344 T - 86 T^{2} + 14 T^{3} + T^{4} \)
$97$ \( 23888 + 112 T - 319 T^{2} + T^{4} \)
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