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: \(x^{4} - x^{3} - 6 x^{2} + 6 x + 3\)
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.

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

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

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

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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 6 T - 6 T^{2} + T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( -21 + 33 T - 12 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -249 - 174 T - 24 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( -7 + 20 T - 5 T^{3} + T^{4} \)
$17$ \( -567 + 354 T - 45 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( -49 + 47 T - 8 T^{3} + T^{4} \)
$23$ \( 9 - 33 T + 36 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 63 + 90 T - 10 T^{3} + T^{4} \)
$31$ \( -21 - 136 T + 93 T^{2} - 18 T^{3} + T^{4} \)
$37$ \( 951 + 37 T - 78 T^{2} + T^{4} \)
$41$ \( 441 - 126 T - 72 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( 49 - 176 T - 30 T^{2} + 7 T^{3} + T^{4} \)
$47$ \( 441 + 447 T + 153 T^{2} + 21 T^{3} + T^{4} \)
$53$ \( -81 + 105 T - 12 T^{3} + T^{4} \)
$59$ \( 189 + 201 T - 108 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( 1043 + 650 T + 27 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( 353 - 74 T - 72 T^{2} + 5 T^{3} + T^{4} \)
$71$ \( 243 + 135 T - 27 T^{2} - 9 T^{3} + T^{4} \)
$73$ \( 2289 - 73 T - 150 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 7 - 5 T - 84 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( -5103 + 1710 T - 126 T^{2} - 9 T^{3} + T^{4} \)
$89$ \( -21 + 51 T + 114 T^{2} + 22 T^{3} + T^{4} \)
$97$ \( 2877 + 2456 T - 258 T^{2} - 9 T^{3} + T^{4} \)
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