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Properties

Label 3969.2.a.bf
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.114612039936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 34x^{4} - 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 567)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{8} + (\beta_{6} - \beta_{2} - 2) q^{10} - \beta_{4} q^{11} + ( - 2 \beta_{6} - \beta_{2} - 1) q^{13} + (\beta_{7} + \beta_{6} + 2 \beta_{2} + 1) q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{2} - 3) q^{19} + ( - \beta_{4} - 3 \beta_1) q^{20} + ( - \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{22} + ( - \beta_{4} - 2 \beta_{3} - \beta_1) q^{23} - \beta_{6} q^{25} + ( - 3 \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{26}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{4} - 14 q^{10} - 6 q^{13} + 6 q^{16} - 24 q^{19} + 2 q^{22} - 20 q^{31} - 4 q^{37} - 36 q^{40} + 10 q^{43} - 36 q^{46} - 34 q^{52} - 4 q^{55} - 22 q^{58} - 36 q^{61} + 38 q^{64} - 18 q^{67} - 32 q^{73} - 58 q^{76} - 32 q^{79} + 2 q^{82} + 30 q^{85} - 72 q^{88} - 54 q^{94} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 34x^{4} - 36x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 7\nu^{3} + 12\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - 31\nu^{3} + 18\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 19\nu^{5} + 41\nu^{3} - 21\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 9\nu^{4} + 16\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{6} + 10\nu^{4} - 24\nu^{2} + 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 9\beta_{4} - 7\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} + 10\beta_{6} + 56\beta_{2} + 91 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 57\beta_{5} + 65\beta_{4} - 46\beta_{3} + 193\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.58737
−1.52818
−1.27020
−0.597336
0.597336
1.27020
1.52818
2.58737
−2.58737 0 4.69447 2.28320 0 0 −6.97158 0 −5.90748
1.2 −1.52818 0 0.335323 2.82528 0 0 2.54392 0 −4.31752
1.3 −1.27020 0 −0.386601 −1.55350 0 0 3.03145 0 1.97325
1.4 −0.597336 0 −1.64319 −2.09557 0 0 2.17621 0 1.25176
1.5 0.597336 0 −1.64319 2.09557 0 0 −2.17621 0 1.25176
1.6 1.27020 0 −0.386601 1.55350 0 0 −3.03145 0 1.97325
1.7 1.52818 0 0.335323 −2.82528 0 0 −2.54392 0 −4.31752
1.8 2.58737 0 4.69447 −2.28320 0 0 6.97158 0 −5.90748
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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
3.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.bf 8
3.b odd 2 1 inner 3969.2.a.bf 8
7.b odd 2 1 3969.2.a.bg 8
7.d odd 6 2 567.2.e.g ✓ 16
21.c even 2 1 3969.2.a.bg 8
21.g even 6 2 567.2.e.g ✓ 16
63.i even 6 2 567.2.h.l 16
63.k odd 6 2 567.2.g.l 16
63.s even 6 2 567.2.g.l 16
63.t odd 6 2 567.2.h.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.g ✓ 16 7.d odd 6 2
567.2.e.g ✓ 16 21.g even 6 2
567.2.g.l 16 63.k odd 6 2
567.2.g.l 16 63.s even 6 2
567.2.h.l 16 63.i even 6 2
567.2.h.l 16 63.t odd 6 2
3969.2.a.bf 8 1.a even 1 1 trivial
3969.2.a.bf 8 3.b odd 2 1 inner
3969.2.a.bg 8 7.b odd 2 1
3969.2.a.bg 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{8} - 11T_{2}^{6} + 34T_{2}^{4} - 36T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{8} - 20T_{5}^{6} + 142T_{5}^{4} - 423T_{5}^{2} + 441 \) Copy content Toggle raw display
\( T_{11}^{8} - 35T_{11}^{6} + 430T_{11}^{4} - 2124T_{11}^{2} + 3249 \) Copy content Toggle raw display
\( T_{13}^{4} + 3T_{13}^{3} - 32T_{13}^{2} - 102T_{13} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 35 T^{6} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} - 32 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 54 T^{6} + \cdots + 3969 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots - 791)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 132 T^{6} + \cdots + 670761 \) Copy content Toggle raw display
$29$ \( T^{8} - 92 T^{6} + \cdots + 74529 \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + \cdots - 651)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 44 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 191 T^{6} + \cdots + 159201 \) Copy content Toggle raw display
$43$ \( (T^{4} - 5 T^{3} + \cdots - 317)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 189 T^{6} + \cdots + 321489 \) Copy content Toggle raw display
$53$ \( T^{8} - 288 T^{6} + \cdots + 927369 \) Copy content Toggle raw display
$59$ \( T^{8} - 60 T^{6} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( (T^{4} + 18 T^{3} + \cdots - 3269)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9 T^{3} + \cdots - 2699)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 357 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{3} + \cdots - 21)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{3} + \cdots - 71)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 111 T^{6} + \cdots + 3969 \) Copy content Toggle raw display
$89$ \( T^{8} - 716 T^{6} + \cdots + 122478489 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7 T^{3} + \cdots + 987)^{2} \) Copy content Toggle raw display
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