Properties

Label 3969.2.a.be
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.59351616.1
Defining polynomial: \(x^{6} - 12 x^{4} + 21 x^{2} - 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{1} q^{5} + ( 2 + \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{1} q^{5} + ( 2 + \beta_{2} + \beta_{4} ) q^{8} + ( 2 \beta_{1} + \beta_{5} ) q^{10} + ( 1 - \beta_{4} ) q^{11} + \beta_{3} q^{13} + ( 2 \beta_{2} - \beta_{4} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{1} - \beta_{5} ) q^{19} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{20} + q^{22} + ( 1 + \beta_{2} + 2 \beta_{4} ) q^{23} + ( 1 + 3 \beta_{2} ) q^{25} + ( -\beta_{3} + \beta_{5} ) q^{26} + ( 4 + \beta_{2} + 2 \beta_{4} ) q^{29} + ( -\beta_{1} - 2 \beta_{5} ) q^{31} + ( 3 - \beta_{2} ) q^{32} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -2 + 2 \beta_{2} - \beta_{4} ) q^{37} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{38} + ( 4 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{40} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{41} + ( 2 - 2 \beta_{2} + \beta_{4} ) q^{43} + ( -2 + \beta_{2} + 2 \beta_{4} ) q^{44} + ( 1 + 4 \beta_{2} + \beta_{4} ) q^{46} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{47} + ( 9 + 4 \beta_{2} + 3 \beta_{4} ) q^{50} + ( \beta_{1} - \beta_{5} ) q^{52} + ( 5 - 3 \beta_{2} - 2 \beta_{4} ) q^{53} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{55} + ( 1 + 7 \beta_{2} + \beta_{4} ) q^{58} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{62} + ( -3 - 2 \beta_{2} + \beta_{4} ) q^{64} + 3 \beta_{4} q^{65} + ( -2 + 3 \beta_{2} - 3 \beta_{4} ) q^{67} + ( -4 \beta_{1} - 3 \beta_{5} ) q^{68} + ( 6 + \beta_{2} ) q^{71} + ( -\beta_{1} + \beta_{5} ) q^{73} + ( 7 - \beta_{2} + 2 \beta_{4} ) q^{74} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{76} + ( -1 + 3 \beta_{2} + 3 \beta_{4} ) q^{79} + ( 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{80} + ( -4 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{82} + ( -3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{83} + ( -3 - 6 \beta_{2} ) q^{85} + ( -7 + \beta_{2} - 2 \beta_{4} ) q^{86} + ( -1 + \beta_{2} + \beta_{4} ) q^{88} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{89} + ( 9 + 4 \beta_{2} ) q^{92} + ( 3 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{94} + ( 9 - 3 \beta_{4} ) q^{95} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 12 x^{4} + 21 x^{2} - 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 12 \nu^{3} + 21 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 12 \nu^{2} + 15 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 21 \nu^{3} + 15 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} - 21 \nu^{2} + 15 \)\()/3\)
\(\beta_{5}\)\(=\)\( \nu^{5} - 11 \nu^{3} + 9 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{3} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 2 \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + 4 \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{4} - 21 \beta_{2} + 45\)
\(\nu^{5}\)\(=\)\(-29 \beta_{5} + 41 \beta_{3} + 8 \beta_{1}\)

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
−3.16032
3.16032
−1.15750
1.15750
−0.820103
0.820103
−1.69963 0 0.888736 −0.949271 0 0 1.88874 0 1.61341
1.2 −1.69963 0 0.888736 0.949271 0 0 1.88874 0 −1.61341
1.3 0.239123 0 −1.94282 −2.59179 0 0 −0.942820 0 −0.619757
1.4 0.239123 0 −1.94282 2.59179 0 0 −0.942820 0 0.619757
1.5 2.46050 0 4.05408 −3.65808 0 0 5.05408 0 −9.00071
1.6 2.46050 0 4.05408 3.65808 0 0 5.05408 0 9.00071
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.be 6
3.b odd 2 1 3969.2.a.bd 6
7.b odd 2 1 inner 3969.2.a.be 6
9.c even 3 2 441.2.f.g 12
9.d odd 6 2 1323.2.f.g 12
21.c even 2 1 3969.2.a.bd 6
63.g even 3 2 441.2.g.g 12
63.h even 3 2 441.2.h.g 12
63.i even 6 2 1323.2.h.g 12
63.j odd 6 2 1323.2.h.g 12
63.k odd 6 2 441.2.g.g 12
63.l odd 6 2 441.2.f.g 12
63.n odd 6 2 1323.2.g.g 12
63.o even 6 2 1323.2.f.g 12
63.s even 6 2 1323.2.g.g 12
63.t odd 6 2 441.2.h.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 9.c even 3 2
441.2.f.g 12 63.l odd 6 2
441.2.g.g 12 63.g even 3 2
441.2.g.g 12 63.k odd 6 2
441.2.h.g 12 63.h even 3 2
441.2.h.g 12 63.t odd 6 2
1323.2.f.g 12 9.d odd 6 2
1323.2.f.g 12 63.o even 6 2
1323.2.g.g 12 63.n odd 6 2
1323.2.g.g 12 63.s even 6 2
1323.2.h.g 12 63.i even 6 2
1323.2.h.g 12 63.j odd 6 2
3969.2.a.bd 6 3.b odd 2 1
3969.2.a.bd 6 21.c even 2 1
3969.2.a.be 6 1.a even 1 1 trivial
3969.2.a.be 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{5}^{6} - 21 T_{5}^{4} + 108 T_{5}^{2} - 81 \)
\( T_{11}^{3} - 4 T_{11}^{2} - T_{11} + 1 \)
\( T_{13}^{6} - 39 T_{13}^{4} + 351 T_{13}^{2} - 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T - T^{2} + T^{3} )^{2} \)
$3$ \( T^{6} \)
$5$ \( -81 + 108 T^{2} - 21 T^{4} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 1 - T - 4 T^{2} + T^{3} )^{2} \)
$13$ \( -81 + 351 T^{2} - 39 T^{4} + T^{6} \)
$17$ \( -3969 + 1593 T^{2} - 84 T^{4} + T^{6} \)
$19$ \( -3969 + 1296 T^{2} - 75 T^{4} + T^{6} \)
$23$ \( ( 59 - 25 T - 2 T^{2} + T^{3} )^{2} \)
$29$ \( ( 89 + 14 T - 11 T^{2} + T^{3} )^{2} \)
$31$ \( -77841 + 5508 T^{2} - 129 T^{4} + T^{6} \)
$37$ \( ( 27 - 24 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( -6561 + 6075 T^{2} - 162 T^{4} + T^{6} \)
$43$ \( ( -27 - 24 T - 3 T^{2} + T^{3} )^{2} \)
$47$ \( -194481 + 10584 T^{2} - 183 T^{4} + T^{6} \)
$53$ \( ( 263 + 11 T - 14 T^{2} + T^{3} )^{2} \)
$59$ \( -149769 + 9450 T^{2} - 183 T^{4} + T^{6} \)
$61$ \( -558009 + 21519 T^{2} - 264 T^{4} + T^{6} \)
$67$ \( ( 353 - 111 T + T^{3} )^{2} \)
$71$ \( ( -227 + 116 T - 19 T^{2} + T^{3} )^{2} \)
$73$ \( -3969 + 1296 T^{2} - 75 T^{4} + T^{6} \)
$79$ \( ( -107 - 78 T + 3 T^{2} + T^{3} )^{2} \)
$83$ \( -227529 + 13365 T^{2} - 228 T^{4} + T^{6} \)
$89$ \( -3969 + 5751 T^{2} - 246 T^{4} + T^{6} \)
$97$ \( -9801 + 1998 T^{2} - 111 T^{4} + T^{6} \)
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