Properties

Label 3969.2.a.t
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14013.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 567)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{8} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{10} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{2} + 1) q^{13} + (\beta_{3} - \beta_1) q^{16} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{19} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{20} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{22} + (\beta_{2} - \beta_1 + 3) q^{23} + (\beta_{2} + 3 \beta_1 + 1) q^{25} + ( - \beta_{3} - 3 \beta_1 + 1) q^{26} + ( - 2 \beta_{2} + 3) q^{29} + ( - \beta_{3} - 2 \beta_1 + 5) q^{31} + (\beta_{3} + 2 \beta_1 + 2) q^{32} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 1) q^{34} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{37} + (2 \beta_{3} - 5) q^{38} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{40} + ( - 2 \beta_{2} - 3 \beta_1) q^{41} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{43} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{44} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{46} + ( - \beta_{3} - \beta_1 - 5) q^{47} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 8) q^{50} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{52} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{53} + ( - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{55} + (2 \beta_{3} + \beta_1 - 2) q^{58} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{59} + (\beta_{3} - 4 \beta_1 + 6) q^{61} + (\beta_{3} + 3 \beta_{2} - 5 \beta_1 + 5) q^{62} + ( - 3 \beta_{3} - 3 \beta_{2} - 5) q^{64} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{65} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{67} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 13) q^{68} + ( - 3 \beta_1 + 3) q^{71} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{73} + (5 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 2) q^{74} + (3 \beta_1 - 2) q^{76} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 1) q^{79} + (4 \beta_{3} + 2 \beta_{2} - 1) q^{80} + (2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 7) q^{82} + (4 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{83} + ( - 5 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{85} + ( - 3 \beta_{3} - 2 \beta_1 + 6) q^{86} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{88} + (3 \beta_{3} + \beta_{2} + \beta_1 - 6) q^{89} + (4 \beta_{2} - 4 \beta_1 + 9) q^{92} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{94} + ( - 5 \beta_{3} - 4 \beta_{2} + 5) q^{95} + (4 \beta_{3} + \beta_{2} - 4 \beta_1 + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 6x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 - 1 \) 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
2.20800
1.53652
−0.372845
−2.37167
−2.20800 0 2.87525 −3.80779 0 0 −1.93254 0 8.40758
1.2 −1.53652 0 0.360904 3.15761 0 0 2.51851 0 −4.85173
1.3 0.372845 0 −1.86099 1.42143 0 0 −1.43955 0 0.529976
1.4 2.37167 0 3.62484 1.22875 0 0 3.85358 0 2.91418
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3969.2.a.t 4
3.b odd 2 1 3969.2.a.w 4
7.b odd 2 1 3969.2.a.s 4
7.d odd 6 2 567.2.e.d yes 8
21.c even 2 1 3969.2.a.x 4
21.g even 6 2 567.2.e.c 8
63.i even 6 2 567.2.h.k 8
63.k odd 6 2 567.2.g.k 8
63.s even 6 2 567.2.g.j 8
63.t odd 6 2 567.2.h.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c 8 21.g even 6 2
567.2.e.d yes 8 7.d odd 6 2
567.2.g.j 8 63.s even 6 2
567.2.g.k 8 63.k odd 6 2
567.2.h.j 8 63.t odd 6 2
567.2.h.k 8 63.i even 6 2
3969.2.a.s 4 7.b odd 2 1
3969.2.a.t 4 1.a even 1 1 trivial
3969.2.a.w 4 3.b odd 2 1
3969.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 6T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} - 12T_{5}^{2} + 33T_{5} - 21 \) Copy content Toggle raw display
\( T_{11}^{4} + 5T_{11}^{3} - 24T_{11}^{2} - 174T_{11} - 249 \) Copy content Toggle raw display
\( T_{13}^{4} - 5T_{13}^{3} + 20T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 6 T^{2} - 6 T + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - 12 T^{2} + 33 T - 21 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} - 24 T^{2} - 174 T - 249 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + 20 T - 7 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} - 45 T^{2} + 354 T - 567 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 47 T - 49 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 36 T^{2} - 33 T + 9 \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 90 T + 63 \) Copy content Toggle raw display
$31$ \( T^{4} - 18 T^{3} + 93 T^{2} - 136 T - 21 \) Copy content Toggle raw display
$37$ \( T^{4} - 78 T^{2} + 37 T + 951 \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} - 72 T^{2} - 126 T + 441 \) Copy content Toggle raw display
$43$ \( T^{4} + 7 T^{3} - 30 T^{2} - 176 T + 49 \) Copy content Toggle raw display
$47$ \( T^{4} + 21 T^{3} + 153 T^{2} + \cdots + 441 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 105 T - 81 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 108 T^{2} + \cdots + 189 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + 27 T^{2} + \cdots + 1043 \) Copy content Toggle raw display
$67$ \( T^{4} + 5 T^{3} - 72 T^{2} - 74 T + 353 \) Copy content Toggle raw display
$71$ \( T^{4} - 9 T^{3} - 27 T^{2} + 135 T + 243 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} - 150 T^{2} + \cdots + 2289 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} - 84 T^{2} - 5 T + 7 \) Copy content Toggle raw display
$83$ \( T^{4} - 9 T^{3} - 126 T^{2} + \cdots - 5103 \) Copy content Toggle raw display
$89$ \( T^{4} + 22 T^{3} + 114 T^{2} + \cdots - 21 \) Copy content Toggle raw display
$97$ \( T^{4} - 9 T^{3} - 258 T^{2} + \cdots + 2877 \) Copy content Toggle raw display
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