Properties

Label 6762.2.a.cl
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: 4.4.473376.1
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 30\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 30\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 7 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 8 \nu + 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 3 \beta_{2} + 14 \beta_{1} + 11\)

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
4.22985
1.83178
−1.52852
−2.53310
1.00000 −1.00000 1.00000 −3.22985 −1.00000 0 1.00000 1.00000 −3.22985
1.2 1.00000 −1.00000 1.00000 −0.831776 −1.00000 0 1.00000 1.00000 −0.831776
1.3 1.00000 −1.00000 1.00000 2.52852 −1.00000 0 1.00000 1.00000 2.52852
1.4 1.00000 −1.00000 1.00000 3.53310 −1.00000 0 1.00000 1.00000 3.53310
\(n\): 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.cl 4
7.b odd 2 1 6762.2.a.cr 4
7.c even 3 2 966.2.i.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.m 8 7.c even 3 2
6762.2.a.cl 4 1.a even 1 1 trivial
6762.2.a.cr 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 13 T_{5}^{2} + 20 T_{5} + 24 \)
\( T_{11}^{4} - 31 T_{11}^{2} + 12 T_{11} + 180 \)
\( T_{13}^{4} - 6 T_{13}^{3} - 27 T_{13}^{2} + 164 T_{13} - 60 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 28 T_{17}^{2} - 34 T_{17} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 24 + 20 T - 13 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 180 + 12 T - 31 T^{2} + T^{4} \)
$13$ \( -60 + 164 T - 27 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 3 - 34 T - 28 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -40 + 56 T + 33 T^{2} - 14 T^{3} + T^{4} \)
$31$ \( ( 2 + T )^{4} \)
$37$ \( -576 + 128 T + 92 T^{2} - 20 T^{3} + T^{4} \)
$41$ \( 480 + 64 T - 52 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( 2661 - 340 T - 118 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( -324 + 1264 T - 33 T^{2} - 18 T^{3} + T^{4} \)
$59$ \( 6144 + 1280 T - 208 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( 2048 - 128 T - 156 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 3308 - 392 T - 153 T^{2} + 2 T^{3} + T^{4} \)
$71$ \( 8181 + 612 T - 246 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( 2025 + 168 T - 138 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( 2388 - 1564 T + 353 T^{2} - 32 T^{3} + T^{4} \)
$83$ \( 48 + 272 T - 112 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( 2304 - 960 T - 152 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 8 + T )^{4} \)
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