Properties

Label 3969.2.a.a
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3969,2,Mod(1,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3969.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} - q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} - q^{5} + 3 q^{8} + q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 3 q^{17} - q^{19} + q^{20} + 5 q^{22} - 3 q^{23} - 4 q^{25} - 5 q^{26} + q^{29} - 5 q^{32} - 3 q^{34} + 3 q^{37} + q^{38} - 3 q^{40} - 5 q^{41} - q^{43} + 5 q^{44} + 3 q^{46} + 4 q^{50} - 5 q^{52} + 9 q^{53} + 5 q^{55} - q^{58} + 14 q^{61} + 7 q^{64} - 5 q^{65} + 4 q^{67} - 3 q^{68} + 12 q^{71} - 3 q^{73} - 3 q^{74} + q^{76} + 8 q^{79} + q^{80} + 5 q^{82} - 9 q^{83} - 3 q^{85} + q^{86} - 15 q^{88} - 13 q^{89} + 3 q^{92} + q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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.a 1
3.b odd 2 1 3969.2.a.f 1
7.b odd 2 1 3969.2.a.c 1
7.d odd 6 2 567.2.e.b 2
9.c even 3 2 1323.2.f.b 2
9.d odd 6 2 441.2.f.a 2
21.c even 2 1 3969.2.a.d 1
21.g even 6 2 567.2.e.a 2
63.g even 3 2 1323.2.g.a 2
63.h even 3 2 1323.2.h.a 2
63.i even 6 2 63.2.h.a yes 2
63.j odd 6 2 441.2.h.a 2
63.k odd 6 2 189.2.g.a 2
63.l odd 6 2 1323.2.f.a 2
63.n odd 6 2 441.2.g.a 2
63.o even 6 2 441.2.f.b 2
63.s even 6 2 63.2.g.a 2
63.t odd 6 2 189.2.h.a 2
252.n even 6 2 3024.2.t.d 2
252.r odd 6 2 1008.2.q.c 2
252.bj even 6 2 3024.2.q.b 2
252.bn odd 6 2 1008.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 63.s even 6 2
63.2.h.a yes 2 63.i even 6 2
189.2.g.a 2 63.k odd 6 2
189.2.h.a 2 63.t odd 6 2
441.2.f.a 2 9.d odd 6 2
441.2.f.b 2 63.o even 6 2
441.2.g.a 2 63.n odd 6 2
441.2.h.a 2 63.j odd 6 2
567.2.e.a 2 21.g even 6 2
567.2.e.b 2 7.d odd 6 2
1008.2.q.c 2 252.r odd 6 2
1008.2.t.d 2 252.bn odd 6 2
1323.2.f.a 2 63.l odd 6 2
1323.2.f.b 2 9.c even 3 2
1323.2.g.a 2 63.g even 3 2
1323.2.h.a 2 63.h even 3 2
3024.2.q.b 2 252.bj even 6 2
3024.2.t.d 2 252.n even 6 2
3969.2.a.a 1 1.a even 1 1 trivial
3969.2.a.c 1 7.b odd 2 1
3969.2.a.d 1 21.c even 2 1
3969.2.a.f 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

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