Properties

Label 3969.2.a.c
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
−1.00000 0 −1.00000 1.00000 0 0 3.00000 0 −1.00000
\(n\): 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.c 1
3.b odd 2 1 3969.2.a.d 1
7.b odd 2 1 3969.2.a.a 1
7.c even 3 2 567.2.e.b 2
9.c even 3 2 1323.2.f.a 2
9.d odd 6 2 441.2.f.b 2
21.c even 2 1 3969.2.a.f 1
21.h odd 6 2 567.2.e.a 2
63.g even 3 2 189.2.g.a 2
63.h even 3 2 189.2.h.a 2
63.i even 6 2 441.2.h.a 2
63.j odd 6 2 63.2.h.a yes 2
63.k odd 6 2 1323.2.g.a 2
63.l odd 6 2 1323.2.f.b 2
63.n odd 6 2 63.2.g.a 2
63.o even 6 2 441.2.f.a 2
63.s even 6 2 441.2.g.a 2
63.t odd 6 2 1323.2.h.a 2
252.o even 6 2 1008.2.t.d 2
252.u odd 6 2 3024.2.q.b 2
252.bb even 6 2 1008.2.q.c 2
252.bl odd 6 2 3024.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 63.n odd 6 2
63.2.h.a yes 2 63.j odd 6 2
189.2.g.a 2 63.g even 3 2
189.2.h.a 2 63.h even 3 2
441.2.f.a 2 63.o even 6 2
441.2.f.b 2 9.d odd 6 2
441.2.g.a 2 63.s even 6 2
441.2.h.a 2 63.i even 6 2
567.2.e.a 2 21.h odd 6 2
567.2.e.b 2 7.c even 3 2
1008.2.q.c 2 252.bb even 6 2
1008.2.t.d 2 252.o even 6 2
1323.2.f.a 2 9.c even 3 2
1323.2.f.b 2 63.l odd 6 2
1323.2.g.a 2 63.k odd 6 2
1323.2.h.a 2 63.t odd 6 2
3024.2.q.b 2 252.u odd 6 2
3024.2.t.d 2 252.bl odd 6 2
3969.2.a.a 1 7.b odd 2 1
3969.2.a.c 1 1.a even 1 1 trivial
3969.2.a.d 1 3.b odd 2 1
3969.2.a.f 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2} + 1 \)
\( T_{5} - 1 \)
\( T_{11} + 5 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ 1
$5$ \( 1 - T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 + 5 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 - T + 19 T^{2} \)
$23$ \( 1 + 3 T + 23 T^{2} \)
$29$ \( 1 - T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 3 T + 37 T^{2} \)
$41$ \( 1 - 5 T + 41 T^{2} \)
$43$ \( 1 + T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 9 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 14 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 12 T + 71 T^{2} \)
$73$ \( 1 - 3 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 - 9 T + 83 T^{2} \)
$89$ \( 1 - 13 T + 89 T^{2} \)
$97$ \( 1 + 9 T + 97 T^{2} \)
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