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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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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.59351616.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 12x^{4} + 21x^{2} - 9 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{4} + \beta_{2} + 1) q^{4} + \beta_1 q^{5} + (\beta_{4} + \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{4} + \beta_{2} + 1) q^{4} + \beta_1 q^{5} + (\beta_{4} + \beta_{2} + 2) q^{8} + (\beta_{5} + 2 \beta_1) q^{10} + ( - \beta_{4} + 1) q^{11} + \beta_{3} q^{13} + ( - \beta_{4} + 2 \beta_{2}) q^{16} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{17} + ( - \beta_{5} + \beta_1) q^{19} + (2 \beta_{5} + \beta_{3} + 3 \beta_1) q^{20} + q^{22} + (2 \beta_{4} + \beta_{2} + 1) q^{23} + (3 \beta_{2} + 1) q^{25} + (\beta_{5} - \beta_{3}) q^{26} + (2 \beta_{4} + \beta_{2} + 4) q^{29} + ( - 2 \beta_{5} - \beta_1) q^{31} + ( - \beta_{2} + 3) q^{32} + ( - 2 \beta_{3} - 3 \beta_1) q^{34} + ( - \beta_{4} + 2 \beta_{2} - 2) q^{37} + (\beta_{5} - \beta_{3} + \beta_1) q^{38} + (2 \beta_{5} + \beta_{3} + 4 \beta_1) q^{40} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{41} + (\beta_{4} - 2 \beta_{2} + 2) q^{43} + (2 \beta_{4} + \beta_{2} - 2) q^{44} + (\beta_{4} + 4 \beta_{2} + 1) q^{46} + (\beta_{5} + 2 \beta_{3} + \beta_1) q^{47} + (3 \beta_{4} + 4 \beta_{2} + 9) q^{50} + ( - \beta_{5} + \beta_1) q^{52} + ( - 2 \beta_{4} - 3 \beta_{2} + 5) q^{53} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{55} + (\beta_{4} + 7 \beta_{2} + 1) q^{58} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{59} + (\beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{61} + ( - \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{62} + (\beta_{4} - 2 \beta_{2} - 3) q^{64} + 3 \beta_{4} q^{65} + ( - 3 \beta_{4} + 3 \beta_{2} - 2) q^{67} + ( - 3 \beta_{5} - 4 \beta_1) q^{68} + (\beta_{2} + 6) q^{71} + (\beta_{5} - \beta_1) q^{73} + (2 \beta_{4} - \beta_{2} + 7) q^{74} + (2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{76} + (3 \beta_{4} + 3 \beta_{2} - 1) q^{79} + (\beta_{5} - \beta_{3} + 4 \beta_1) q^{80} + ( - 2 \beta_{5} - \beta_{3} - 4 \beta_1) q^{82} + ( - \beta_{5} + \beta_{3} - 3 \beta_1) q^{83} + ( - 6 \beta_{2} - 3) q^{85} + ( - 2 \beta_{4} + \beta_{2} - 7) q^{86} + (\beta_{4} + \beta_{2} - 1) q^{88} + ( - 2 \beta_{5} - \beta_{3} - 3 \beta_1) q^{89} + (4 \beta_{2} + 9) q^{92} + (3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{94} + ( - 3 \beta_{4} + 9) q^{95} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 6 q^{4} + 12 q^{8} + 8 q^{11} + 6 q^{16} + 6 q^{22} + 4 q^{23} + 12 q^{25} + 22 q^{29} + 16 q^{32} - 6 q^{37} + 6 q^{43} - 14 q^{44} + 12 q^{46} + 56 q^{50} + 28 q^{53} + 18 q^{58} - 24 q^{64} - 6 q^{65} + 38 q^{71} + 36 q^{74} - 6 q^{79} - 30 q^{85} - 36 q^{86} - 6 q^{88} + 62 q^{92} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 12\nu^{3} + 21\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 12\nu^{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 21\nu^{3} + 15\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} - 21\nu^{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 11\nu^{3} + 9\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 4\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{4} - 21\beta_{2} + 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{5} + 41\beta_{3} + 8\beta_1 \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

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

Hecke characteristic polynomials

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