Properties

Label 3969.2.a.r
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} -\beta_{1} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} -\beta_{1} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} -\beta_{3} q^{10} - q^{11} + 2 \beta_{1} q^{13} + ( -3 + 3 \beta_{2} ) q^{16} + ( 2 \beta_{1} - \beta_{3} ) q^{17} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( 1 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} + ( -3 + 2 \beta_{2} ) q^{25} + 2 \beta_{3} q^{26} + ( 2 - 2 \beta_{2} ) q^{29} + ( -2 \beta_{1} - \beta_{3} ) q^{31} + ( 4 + \beta_{2} ) q^{32} + ( -\beta_{1} + 3 \beta_{3} ) q^{34} + ( -1 - 6 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 5 \beta_{3} ) q^{38} + ( \beta_{1} + 2 \beta_{3} ) q^{40} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -7 + 8 \beta_{2} ) q^{43} + \beta_{2} q^{44} + ( -2 + 4 \beta_{2} ) q^{46} + ( -3 \beta_{1} + 7 \beta_{3} ) q^{47} + ( 5 - 3 \beta_{2} ) q^{50} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{52} + ( -1 - 6 \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( -4 + 2 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 7 \beta_{3} ) q^{59} + ( 7 \beta_{1} - 5 \beta_{3} ) q^{61} + ( -\beta_{1} - \beta_{3} ) q^{62} + ( 3 - 2 \beta_{2} ) q^{64} + ( -4 - 4 \beta_{2} ) q^{65} + ( -5 + 4 \beta_{2} ) q^{67} + ( -\beta_{1} - 2 \beta_{3} ) q^{68} + ( 1 - 4 \beta_{2} ) q^{71} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{73} + ( -5 - \beta_{2} ) q^{74} + ( \beta_{1} + 3 \beta_{3} ) q^{76} + ( -11 + 2 \beta_{2} ) q^{79} -3 \beta_{3} q^{80} + ( -4 \beta_{1} + 9 \beta_{3} ) q^{82} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -4 - 2 \beta_{2} ) q^{85} + ( 15 - 7 \beta_{2} ) q^{86} + ( -1 + 2 \beta_{2} ) q^{88} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 - 6 \beta_{2} ) q^{92} + ( 7 \beta_{1} - 10 \beta_{3} ) q^{94} + ( 6 + 2 \beta_{2} ) q^{95} + ( -\beta_{1} + 6 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + O(q^{10}) \) \( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{11} - 6 q^{16} + 2 q^{22} + 20 q^{23} - 8 q^{25} + 4 q^{29} + 18 q^{32} - 16 q^{37} - 12 q^{43} + 2 q^{44} + 14 q^{50} - 16 q^{53} - 12 q^{58} + 8 q^{64} - 24 q^{65} - 12 q^{67} - 4 q^{71} - 22 q^{74} - 40 q^{79} - 20 q^{85} + 46 q^{86} - 20 q^{92} + 28 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

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

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
0.874032
−0.874032
2.28825
−2.28825
−1.61803 0 0.618034 −0.874032 0 0 2.23607 0 1.41421
1.2 −1.61803 0 0.618034 0.874032 0 0 2.23607 0 −1.41421
1.3 0.618034 0 −1.61803 −2.28825 0 0 −2.23607 0 −1.41421
1.4 0.618034 0 −1.61803 2.28825 0 0 −2.23607 0 1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.r 4
3.b odd 2 1 3969.2.a.y yes 4
7.b odd 2 1 inner 3969.2.a.r 4
21.c even 2 1 3969.2.a.y yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.2.a.r 4 1.a even 1 1 trivial
3969.2.a.r 4 7.b odd 2 1 inner
3969.2.a.y yes 4 3.b odd 2 1
3969.2.a.y yes 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5}^{4} - 6 T_{5}^{2} + 4 \)
\( T_{11} + 1 \)
\( T_{13}^{4} - 24 T_{13}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 4 - 6 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 64 - 24 T^{2} + T^{4} \)
$17$ \( ( -10 + T^{2} )^{2} \)
$19$ \( 484 - 46 T^{2} + T^{4} \)
$23$ \( ( 20 - 10 T + T^{2} )^{2} \)
$29$ \( ( -4 - 2 T + T^{2} )^{2} \)
$31$ \( 4 - 36 T^{2} + T^{4} \)
$37$ \( ( -29 + 8 T + T^{2} )^{2} \)
$41$ \( 3364 - 134 T^{2} + T^{4} \)
$43$ \( ( -71 + 6 T + T^{2} )^{2} \)
$47$ \( 1444 - 166 T^{2} + T^{4} \)
$53$ \( ( -29 + 8 T + T^{2} )^{2} \)
$59$ \( 13924 - 276 T^{2} + T^{4} \)
$61$ \( 13924 - 254 T^{2} + T^{4} \)
$67$ \( ( -11 + 6 T + T^{2} )^{2} \)
$71$ \( ( -19 + 2 T + T^{2} )^{2} \)
$73$ \( 7744 - 216 T^{2} + T^{4} \)
$79$ \( ( 95 + 20 T + T^{2} )^{2} \)
$83$ \( 3844 - 126 T^{2} + T^{4} \)
$89$ \( ( -40 + T^{2} )^{2} \)
$97$ \( 3364 - 126 T^{2} + T^{4} \)
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