Properties

Label 3969.2.a.y
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: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{2} - 1) q^{8} + \beta_{3} q^{10} + q^{11} - 2 \beta_1 q^{13} + (3 \beta_{2} - 3) q^{16} + ( - \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{3} + 3 \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{20} + ( - \beta_{2} + 1) q^{22} + ( - 2 \beta_{2} - 4) q^{23} + (2 \beta_{2} - 3) q^{25} + 2 \beta_{3} q^{26} + (2 \beta_{2} - 2) q^{29} + (\beta_{3} + 2 \beta_1) q^{31} + ( - \beta_{2} - 4) q^{32} + ( - 3 \beta_{3} + \beta_1) q^{34} + ( - 6 \beta_{2} - 1) q^{37} + ( - 5 \beta_{3} + 2 \beta_1) q^{38} + ( - 2 \beta_{3} - \beta_1) q^{40} + ( - 4 \beta_{3} + 5 \beta_1) q^{41} + (8 \beta_{2} - 7) q^{43} - \beta_{2} q^{44} + (4 \beta_{2} - 2) q^{46} + (7 \beta_{3} - 3 \beta_1) q^{47} + (3 \beta_{2} - 5) q^{50} + (2 \beta_{3} + 2 \beta_1) q^{52} + (6 \beta_{2} + 1) q^{53} - \beta_1 q^{55} + (2 \beta_{2} - 4) q^{58} + (7 \beta_{3} + 2 \beta_1) q^{59} + (5 \beta_{3} - 7 \beta_1) q^{61} + ( - \beta_{3} - \beta_1) q^{62} + ( - 2 \beta_{2} + 3) q^{64} + (4 \beta_{2} + 4) q^{65} + (4 \beta_{2} - 5) q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + (4 \beta_{2} - 1) q^{71} + (8 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{2} + 5) q^{74} + ( - 3 \beta_{3} - \beta_1) q^{76} + (2 \beta_{2} - 11) q^{79} - 3 \beta_{3} q^{80} + ( - 9 \beta_{3} + 4 \beta_1) q^{82} + ( - 2 \beta_{3} + 5 \beta_1) q^{83} + ( - 2 \beta_{2} - 4) q^{85} + (7 \beta_{2} - 15) q^{86} + (2 \beta_{2} - 1) q^{88} + (2 \beta_{3} - 4 \beta_1) q^{89} + (6 \beta_{2} + 2) q^{92} + (10 \beta_{3} - 7 \beta_1) q^{94} + ( - 2 \beta_{2} - 6) q^{95} + ( - 6 \beta_{3} + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_1 \) Copy content Toggle raw display

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
2.28825
−2.28825
0.874032
−0.874032
−0.618034 0 −1.61803 −2.28825 0 0 2.23607 0 1.41421
1.2 −0.618034 0 −1.61803 2.28825 0 0 2.23607 0 −1.41421
1.3 1.61803 0 0.618034 −0.874032 0 0 −2.23607 0 −1.41421
1.4 1.61803 0 0.618034 0.874032 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.y yes 4
3.b odd 2 1 3969.2.a.r 4
7.b odd 2 1 inner 3969.2.a.y yes 4
21.c even 2 1 3969.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.2.a.r 4 3.b odd 2 1
3969.2.a.r 4 21.c even 2 1
3969.2.a.y yes 4 1.a even 1 1 trivial
3969.2.a.y yes 4 7.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display
\( T_{13}^{4} - 24T_{13}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

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