Properties

Label 8712.2.a.cj
Level $8712$
Weight $2$
Character orbit 8712.a
Self dual yes
Analytic conductor $69.566$
Analytic rank $0$
Dimension $6$
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] = [8712,2,Mod(1,8712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8712, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("8712.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8712.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: \(69.5656702409\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.62158000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 14x^{4} + 22x^{3} + 38x^{2} - 60x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 792)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7} + (\beta_{4} + \beta_{2} - 1) q^{13} + (\beta_{4} - \beta_{3} + \beta_1 + 1) q^{17} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 3) q^{23} + (\beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3) q^{25} + (\beta_{5} + 2 \beta_{4} + \beta_1 + 1) q^{29} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{31} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{35} + (2 \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{37} + (\beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + 1) q^{41} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 3) q^{43} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 3) q^{47} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{49} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{59} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{61} + (\beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 10) q^{65} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{67} + (\beta_{5} + 6 \beta_{3} + \beta_{2} + \beta_1) q^{71} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{73} + ( - \beta_{5} - \beta_{4} + 3 \beta_{2} - \beta_1 - 2) q^{79} + (2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{83} + (2 \beta_{5} - \beta_{4} + 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{85} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{89} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{91} + (2 \beta_{5} - \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + \beta_1 + 5) q^{95} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 5 q^{5} - q^{7} - 4 q^{13} + 6 q^{17} + 14 q^{19} + 6 q^{23} + 7 q^{25} + 7 q^{29} - 7 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} + 12 q^{43} + 26 q^{47} + 9 q^{49} + 9 q^{53} + 17 q^{59} - 20 q^{61} - 48 q^{65} + 14 q^{67} + 18 q^{71} - 7 q^{73} - 9 q^{79} + 11 q^{83} - 10 q^{85} + 14 q^{89} + 8 q^{91} + 30 q^{95} + 25 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 14x^{4} + 22x^{3} + 38x^{2} - 60x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + 7\nu^{4} - 13\nu^{3} - 9\nu^{2} + 363\nu - 99 ) / 158 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{5} + 21\nu^{4} + 119\nu^{3} - 185\nu^{2} - 333\nu + 335 ) / 158 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{5} + 15\nu^{4} + 401\nu^{3} + 71\nu^{2} - 915\nu - 167 ) / 158 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\nu^{5} - 29\nu^{4} - 533\nu^{3} + 263\nu^{2} + 1453\nu - 1057 ) / 158 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} - 4\beta_{2} + 10\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{5} + 10\beta_{4} + 23\beta_{3} - 14\beta_{2} + 19\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 23\beta_{5} + 16\beta_{4} + 42\beta_{3} - 65\beta_{2} + 119\beta _1 + 41 \) 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
1.90351
1.17002
−2.95126
0.216081
3.73518
−2.07353
0 0 0 −2.07994 0 −1.39055 0 0 0
1.2 0 0 0 −0.893131 0 2.25714 0 0 0
1.3 0 0 0 −0.823980 0 0.910904 0 0 0
1.4 0 0 0 1.13355 0 −4.44328 0 0 0
1.5 0 0 0 3.30847 0 4.15041 0 0 0
1.6 0 0 0 4.35504 0 −2.48462 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)

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 8712.2.a.cj 6
3.b odd 2 1 8712.2.a.ch 6
11.b odd 2 1 8712.2.a.ck 6
11.c even 5 2 792.2.r.h 12
33.d even 2 1 8712.2.a.ci 6
33.h odd 10 2 792.2.r.i yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.r.h 12 11.c even 5 2
792.2.r.i yes 12 33.h odd 10 2
8712.2.a.ch 6 3.b odd 2 1
8712.2.a.ci 6 33.d even 2 1
8712.2.a.cj 6 1.a even 1 1 trivial
8712.2.a.ck 6 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\):

\( T_{5}^{6} - 5T_{5}^{5} - 6T_{5}^{4} + 35T_{5}^{3} + 24T_{5}^{2} - 35T_{5} - 25 \) Copy content Toggle raw display
\( T_{7}^{6} + T_{7}^{5} - 25T_{7}^{4} - 18T_{7}^{3} + 131T_{7}^{2} + 57T_{7} - 131 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 42T_{13}^{4} - 200T_{13}^{3} + 233T_{13}^{2} + 2084T_{13} + 2416 \) Copy content Toggle raw display
\( T_{17}^{6} - 6T_{17}^{5} - 21T_{17}^{4} + 58T_{17}^{3} + 191T_{17}^{2} + 150T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} - 6 T^{4} + 35 T^{3} + \cdots - 25 \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} - 25 T^{4} - 18 T^{3} + \cdots - 131 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} - 42 T^{4} + \cdots + 2416 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} - 21 T^{4} + 58 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{6} - 14 T^{5} + 11 T^{4} + \cdots + 8244 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} - 52 T^{4} + 476 T^{3} + \cdots - 484 \) Copy content Toggle raw display
$29$ \( T^{6} - 7 T^{5} - 99 T^{4} + \cdots - 36304 \) Copy content Toggle raw display
$31$ \( T^{6} + 7 T^{5} - 70 T^{4} + \cdots + 17281 \) Copy content Toggle raw display
$37$ \( T^{6} - 2 T^{5} - 134 T^{4} + \cdots - 2404 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} - 68 T^{4} - 556 T^{3} + \cdots + 20 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} - 50 T^{4} + \cdots + 11456 \) Copy content Toggle raw display
$47$ \( T^{6} - 26 T^{5} + 191 T^{4} + \cdots - 6124 \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} - 176 T^{4} + \cdots - 21121 \) Copy content Toggle raw display
$59$ \( T^{6} - 17 T^{5} - 40 T^{4} + \cdots + 17141 \) Copy content Toggle raw display
$61$ \( T^{6} + 20 T^{5} - 17 T^{4} + \cdots + 60516 \) Copy content Toggle raw display
$67$ \( T^{6} - 14 T^{5} - 17 T^{4} + \cdots + 10436 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} - 43 T^{4} + \cdots + 6284 \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} - 141 T^{4} + \cdots + 4864 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} - 127 T^{4} + \cdots - 5429 \) Copy content Toggle raw display
$83$ \( T^{6} - 11 T^{5} - 119 T^{4} + \cdots + 27511 \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} - 164 T^{4} + \cdots + 45620 \) Copy content Toggle raw display
$97$ \( T^{6} - 25 T^{5} + 208 T^{4} + \cdots + 3331 \) Copy content Toggle raw display
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