Properties

Label 3969.2.a.u
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: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} -4 q^{13} + ( 1 + \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + 2 \beta_{2} q^{19} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 3 - \beta_{2} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 7 + 4 \beta_{2} ) q^{25} -4 \beta_{3} q^{26} -4 \beta_{3} q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -2 + 2 \beta_{2} ) q^{34} + 3 q^{37} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{38} + ( 4 + 2 \beta_{2} ) q^{40} -2 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{44} + ( -2 + 2 \beta_{2} ) q^{46} + 6 \beta_{3} q^{47} + ( -4 \beta_{1} - 5 \beta_{3} ) q^{50} + 4 \beta_{2} q^{52} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{53} + ( -8 - 2 \beta_{2} ) q^{55} + ( -8 + 4 \beta_{2} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -6 + 2 \beta_{2} ) q^{61} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{62} + ( -7 + 2 \beta_{2} ) q^{64} -8 \beta_{1} q^{65} + ( 3 - 2 \beta_{2} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 5 \beta_{1} - \beta_{3} ) q^{71} + ( -4 + 4 \beta_{2} ) q^{73} + 3 \beta_{3} q^{74} + ( -10 + 2 \beta_{2} ) q^{76} + ( -5 - 2 \beta_{2} ) q^{79} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{80} + 2 q^{82} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{83} + ( -8 - 4 \beta_{2} ) q^{85} + ( 2 \beta_{1} + \beta_{3} ) q^{86} + ( -1 - 2 \beta_{2} ) q^{88} -4 \beta_{1} q^{89} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 12 - 6 \beta_{2} ) q^{94} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{95} + ( -4 - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{10} - 16q^{13} + 2q^{16} - 4q^{19} + 14q^{22} + 20q^{25} - 12q^{34} + 12q^{37} + 12q^{40} - 16q^{43} - 12q^{46} - 8q^{52} - 28q^{55} - 40q^{58} - 28q^{61} - 32q^{64} + 16q^{67} - 24q^{73} - 44q^{76} - 16q^{79} + 8q^{82} - 24q^{85} + 60q^{94} - 12q^{97} + O(q^{100}) \)

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 0.913701 0 0 −1.73205 0 −2.00000
1.2 −0.456850 0 −1.79129 4.37780 0 0 1.73205 0 −2.00000
1.3 0.456850 0 −1.79129 −4.37780 0 0 −1.73205 0 −2.00000
1.4 2.18890 0 2.79129 −0.913701 0 0 1.73205 0 −2.00000
\(n\): 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
3.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.u 4
3.b odd 2 1 inner 3969.2.a.u 4
7.b odd 2 1 567.2.a.i 4
21.c even 2 1 567.2.a.i 4
28.d even 2 1 9072.2.a.ci 4
63.l odd 6 2 567.2.f.n 8
63.o even 6 2 567.2.f.n 8
84.h odd 2 1 9072.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.i 4 7.b odd 2 1
567.2.a.i 4 21.c even 2 1
567.2.f.n 8 63.l odd 6 2
567.2.f.n 8 63.o even 6 2
3969.2.a.u 4 1.a even 1 1 trivial
3969.2.a.u 4 3.b odd 2 1 inner
9072.2.a.ci 4 28.d even 2 1
9072.2.a.ci 4 84.h odd 2 1

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} + 1 \)
\( T_{5}^{4} - 20 T_{5}^{2} + 16 \)
\( T_{11}^{2} - 7 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 16 - 20 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -7 + T^{2} )^{2} \)
$13$ \( ( 4 + T )^{4} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( -20 + 2 T + T^{2} )^{2} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( 256 - 80 T^{2} + T^{4} \)
$31$ \( ( -84 + T^{2} )^{2} \)
$37$ \( ( -3 + T )^{4} \)
$41$ \( 16 - 20 T^{2} + T^{4} \)
$43$ \( ( -5 + 8 T + T^{2} )^{2} \)
$47$ \( 1296 - 180 T^{2} + T^{4} \)
$53$ \( ( -75 + T^{2} )^{2} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( ( 28 + 14 T + T^{2} )^{2} \)
$67$ \( ( -5 - 8 T + T^{2} )^{2} \)
$71$ \( 2601 - 150 T^{2} + T^{4} \)
$73$ \( ( -48 + 12 T + T^{2} )^{2} \)
$79$ \( ( -5 + 8 T + T^{2} )^{2} \)
$83$ \( 3600 - 132 T^{2} + T^{4} \)
$89$ \( 256 - 80 T^{2} + T^{4} \)
$97$ \( ( -12 + 6 T + T^{2} )^{2} \)
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