Properties

Label 6762.2.a.bo
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.70156
−2.70156
−1.00000 −1.00000 1.00000 −3.70156 1.00000 0 −1.00000 1.00000 3.70156
1.2 −1.00000 −1.00000 1.00000 2.70156 1.00000 0 −1.00000 1.00000 −2.70156
\(n\): 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.bo 2
7.b odd 2 1 966.2.a.n 2
21.c even 2 1 2898.2.a.bb 2
28.d even 2 1 7728.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 7.b odd 2 1
2898.2.a.bb 2 21.c even 2 1
6762.2.a.bo 2 1.a even 1 1 trivial
7728.2.a.bc 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5}^{2} + T_{5} - 10 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 5 T_{13} - 4 \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -4 + 5 T + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( -40 + 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 2 - 7 T + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 46 - 15 T + T^{2} \)
$41$ \( 2 - 7 T + T^{2} \)
$43$ \( -8 - 3 T + T^{2} \)
$47$ \( -90 - 3 T + T^{2} \)
$53$ \( -32 + 6 T + T^{2} \)
$59$ \( -40 - 2 T + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( -160 - 4 T + T^{2} \)
$71$ \( -40 + 2 T + T^{2} \)
$73$ \( -16 - 10 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 8 + 14 T + T^{2} \)
$89$ \( -16 - 10 T + T^{2} \)
$97$ \( 80 + 19 T + T^{2} \)
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