Properties

Label 3969.2.a.z
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.574857.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} + ( 2 + \beta_{3} + \beta_{4} ) q^{10} + ( -1 + \beta_{2} - \beta_{3} ) q^{11} + ( 1 + \beta_{1} - \beta_{4} ) q^{13} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -3 + \beta_{1} - \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{22} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{26} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{31} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{32} + ( -2 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{38} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{40} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{43} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{44} + ( -4 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{46} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -6 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{52} + ( 5 - \beta_{1} + 2 \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{55} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{58} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{62} + ( -6 + \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{64} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{67} + ( -7 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -4 + \beta_{2} - 3 \beta_{3} ) q^{73} + ( 10 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} + \beta_{4} ) q^{76} + ( 3 - 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{79} + ( -3 - 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{80} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{82} + ( -1 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{86} + ( 4 - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{88} + ( -7 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{89} + ( -5 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{92} + ( -3 + 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{94} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{95} + ( 2 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} + 4q^{4} - 4q^{5} - 3q^{8} + O(q^{10}) \) \( 5q - 2q^{2} + 4q^{4} - 4q^{5} - 3q^{8} + 7q^{10} - 4q^{11} + 8q^{13} - 2q^{16} - 12q^{17} - q^{19} - 5q^{20} + q^{22} - 3q^{23} + q^{25} - 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 20q^{38} + 3q^{40} - 5q^{41} + 7q^{43} + 10q^{44} - 3q^{46} - 27q^{47} - 19q^{50} + 10q^{52} + 21q^{53} + 2q^{55} + 10q^{58} - 30q^{59} + 14q^{61} - 6q^{62} - 25q^{64} + 11q^{65} + 2q^{67} - 27q^{68} - 3q^{71} - 15q^{73} + 36q^{74} - 5q^{76} + 4q^{79} - 20q^{80} + 5q^{82} - 9q^{83} + 6q^{85} + 8q^{86} + 18q^{88} - 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 4 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 14\)

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
2.38687
1.84124
0.495868
−0.670333
−2.05365
−2.38687 0 3.69714 −2.92087 0 0 −4.05086 0 6.97172
1.2 −1.84124 0 1.39017 1.33475 0 0 1.12285 0 −2.45760
1.3 −0.495868 0 −1.75411 −3.69258 0 0 1.86155 0 1.83103
1.4 0.670333 0 −1.55065 1.42494 0 0 −2.38012 0 0.955182
1.5 2.05365 0 2.21746 −0.146246 0 0 0.446582 0 −0.300337
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.z 5
3.b odd 2 1 3969.2.a.bc 5
7.b odd 2 1 3969.2.a.ba 5
7.c even 3 2 567.2.e.f 10
9.c even 3 2 441.2.f.e 10
9.d odd 6 2 1323.2.f.e 10
21.c even 2 1 3969.2.a.bb 5
21.h odd 6 2 567.2.e.e 10
63.g even 3 2 63.2.g.b 10
63.h even 3 2 63.2.h.b yes 10
63.i even 6 2 1323.2.h.f 10
63.j odd 6 2 189.2.h.b 10
63.k odd 6 2 441.2.g.f 10
63.l odd 6 2 441.2.f.f 10
63.n odd 6 2 189.2.g.b 10
63.o even 6 2 1323.2.f.f 10
63.s even 6 2 1323.2.g.f 10
63.t odd 6 2 441.2.h.f 10
252.o even 6 2 3024.2.t.i 10
252.u odd 6 2 1008.2.q.i 10
252.bb even 6 2 3024.2.q.i 10
252.bl odd 6 2 1008.2.t.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 63.g even 3 2
63.2.h.b yes 10 63.h even 3 2
189.2.g.b 10 63.n odd 6 2
189.2.h.b 10 63.j odd 6 2
441.2.f.e 10 9.c even 3 2
441.2.f.f 10 63.l odd 6 2
441.2.g.f 10 63.k odd 6 2
441.2.h.f 10 63.t odd 6 2
567.2.e.e 10 21.h odd 6 2
567.2.e.f 10 7.c even 3 2
1008.2.q.i 10 252.u odd 6 2
1008.2.t.i 10 252.bl odd 6 2
1323.2.f.e 10 9.d odd 6 2
1323.2.f.f 10 63.o even 6 2
1323.2.g.f 10 63.s even 6 2
1323.2.h.f 10 63.i even 6 2
3024.2.q.i 10 252.bb even 6 2
3024.2.t.i 10 252.o even 6 2
3969.2.a.z 5 1.a even 1 1 trivial
3969.2.a.ba 5 7.b odd 2 1
3969.2.a.bb 5 21.c even 2 1
3969.2.a.bc 5 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{5} + 2 T_{2}^{4} - 5 T_{2}^{3} - 9 T_{2}^{2} + 3 T_{2} + 3 \)
\( T_{5}^{5} + 4 T_{5}^{4} - 5 T_{5}^{3} - 18 T_{5}^{2} + 18 T_{5} + 3 \)
\( T_{11}^{5} + 4 T_{11}^{4} - 8 T_{11}^{3} - 15 T_{11}^{2} + 12 T_{11} + 15 \)
\( T_{13}^{5} - 8 T_{13}^{4} + 13 T_{13}^{3} + 13 T_{13}^{2} - 23 T_{13} - 5 \)