Properties

Label 6762.2.a.bz
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
\(n\): 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.bz 2
7.b odd 2 1 966.2.a.p 2
21.c even 2 1 2898.2.a.v 2
28.d even 2 1 7728.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.p 2 7.b odd 2 1
2898.2.a.v 2 21.c even 2 1
6762.2.a.bz 2 1.a even 1 1 trivial
7728.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} + 2 \)
\( T_{11} \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -16 - 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( -76 + 4 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -64 - 8 T + T^{2} \)
$47$ \( -16 + 4 T + T^{2} \)
$53$ \( -76 + 4 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -64 + 8 T + T^{2} \)
$71$ \( -64 + 8 T + T^{2} \)
$73$ \( -44 + 12 T + T^{2} \)
$79$ \( -64 - 8 T + T^{2} \)
$83$ \( -16 - 4 T + T^{2} \)
$89$ \( -164 - 8 T + T^{2} \)
$97$ \( -4 + 8 T + T^{2} \)
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