Properties

Label 3969.2.a.bc
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
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: \(0\)
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
−2.05365
−0.670333
0.495868
1.84124
2.38687
−2.05365 0 2.21746 0.146246 0 0 −0.446582 0 −0.300337
1.2 −0.670333 0 −1.55065 −1.42494 0 0 2.38012 0 0.955182
1.3 0.495868 0 −1.75411 3.69258 0 0 −1.86155 0 1.83103
1.4 1.84124 0 1.39017 −1.33475 0 0 −1.12285 0 −2.45760
1.5 2.38687 0 3.69714 2.92087 0 0 4.05086 0 6.97172
\(n\): 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.bc 5
3.b odd 2 1 3969.2.a.z 5
7.b odd 2 1 3969.2.a.bb 5
7.c even 3 2 567.2.e.e 10
9.c even 3 2 1323.2.f.e 10
9.d odd 6 2 441.2.f.e 10
21.c even 2 1 3969.2.a.ba 5
21.h odd 6 2 567.2.e.f 10
63.g even 3 2 189.2.g.b 10
63.h even 3 2 189.2.h.b 10
63.i even 6 2 441.2.h.f 10
63.j odd 6 2 63.2.h.b yes 10
63.k odd 6 2 1323.2.g.f 10
63.l odd 6 2 1323.2.f.f 10
63.n odd 6 2 63.2.g.b 10
63.o even 6 2 441.2.f.f 10
63.s even 6 2 441.2.g.f 10
63.t odd 6 2 1323.2.h.f 10
252.o even 6 2 1008.2.t.i 10
252.u odd 6 2 3024.2.q.i 10
252.bb even 6 2 1008.2.q.i 10
252.bl odd 6 2 3024.2.t.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 63.n odd 6 2
63.2.h.b yes 10 63.j odd 6 2
189.2.g.b 10 63.g even 3 2
189.2.h.b 10 63.h even 3 2
441.2.f.e 10 9.d odd 6 2
441.2.f.f 10 63.o even 6 2
441.2.g.f 10 63.s even 6 2
441.2.h.f 10 63.i even 6 2
567.2.e.e 10 7.c even 3 2
567.2.e.f 10 21.h odd 6 2
1008.2.q.i 10 252.bb even 6 2
1008.2.t.i 10 252.o even 6 2
1323.2.f.e 10 9.c even 3 2
1323.2.f.f 10 63.l odd 6 2
1323.2.g.f 10 63.k odd 6 2
1323.2.h.f 10 63.t odd 6 2
3024.2.q.i 10 252.u odd 6 2
3024.2.t.i 10 252.bl odd 6 2
3969.2.a.z 5 3.b odd 2 1
3969.2.a.ba 5 21.c even 2 1
3969.2.a.bb 5 7.b odd 2 1
3969.2.a.bc 5 1.a even 1 1 trivial

Hecke kernels

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

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