Properties

Label 3969.2.a.v
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $4$
CM discriminant -7
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{16} + ( -5 + 3 \beta_{2} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{23} -5 q^{25} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{29} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -1 + 4 \beta_{2} ) q^{37} + ( -5 + 2 \beta_{2} ) q^{43} + ( -\beta_{1} - 8 \beta_{3} ) q^{44} + ( -6 + 2 \beta_{2} ) q^{46} -5 \beta_{3} q^{50} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 12 - 4 \beta_{2} ) q^{58} + ( -7 + 2 \beta_{2} ) q^{64} + ( -5 - 6 \beta_{2} ) q^{67} + ( 9 \beta_{1} + 7 \beta_{3} ) q^{71} + ( -4 \beta_{1} - 13 \beta_{3} ) q^{74} + ( -1 + 6 \beta_{2} ) q^{79} + ( -2 \beta_{1} - 11 \beta_{3} ) q^{86} + ( -5 + 2 \beta_{2} ) q^{88} + ( -6 \beta_{1} - 8 \beta_{3} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + O(q^{10}) \) \( 4 q + 2 q^{4} + 2 q^{16} - 26 q^{22} - 20 q^{25} - 12 q^{37} - 24 q^{43} - 28 q^{46} + 56 q^{58} - 32 q^{64} - 8 q^{67} - 16 q^{79} - 24 q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 0 2.79129 0 0 0 −1.73205 0 0
1.2 −0.456850 0 −1.79129 0 0 0 1.73205 0 0
1.3 0.456850 0 −1.79129 0 0 0 −1.73205 0 0
1.4 2.18890 0 2.79129 0 0 0 1.73205 0 0
\(n\): 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 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 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.v 4
3.b odd 2 1 inner 3969.2.a.v 4
7.b odd 2 1 CM 3969.2.a.v 4
21.c even 2 1 inner 3969.2.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.2.a.v 4 1.a even 1 1 trivial
3969.2.a.v 4 3.b odd 2 1 inner
3969.2.a.v 4 7.b odd 2 1 CM
3969.2.a.v 4 21.c even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} + 1 \)
\( T_{5} \)
\( T_{11}^{4} - 38 T_{11}^{2} + 25 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 25 - 38 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -28 + T^{2} )^{2} \)
$29$ \( ( -112 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -75 + 6 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 15 + 12 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 2209 - 206 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -185 + 4 T + T^{2} )^{2} \)
$71$ \( 34225 - 398 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -173 + 8 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
show more
show less