Properties

Label 8624.2.a.ci
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $3$
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] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 308)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{2} - \beta_1 + 1) q^{9} + q^{11} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{15} + (\beta_1 - 2) q^{17} + (\beta_1 - 4) q^{19} + (\beta_1 + 1) q^{23} + ( - \beta_{2} - 2 \beta_1 + 4) q^{25} + (2 \beta_{2} + \beta_1 - 7) q^{27} + (\beta_{2} - 3 \beta_1 + 1) q^{29} + (\beta_{2} + 3 \beta_1 - 7) q^{31} + \beta_{2} q^{33} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{37} + (2 \beta_{2} + 3 \beta_1 - 6) q^{39} + ( - \beta_{2} - 2 \beta_1 + 6) q^{41} + (3 \beta_{2} + 2 \beta_1 - 3) q^{43} + (6 \beta_{2} + 3 \beta_1 - 6) q^{45} + ( - \beta_1 - 4) q^{47} + ( - 2 \beta_{2} + \beta_1 - 1) q^{51} + (\beta_{2} + 4 \beta_1 + 2) q^{53} + (\beta_{2} - \beta_1) q^{55} + ( - 4 \beta_{2} + \beta_1 - 1) q^{57} + ( - 3 \beta_{2} - 3) q^{59} + (4 \beta_{2} + 2 \beta_1 + 2) q^{61} + (2 \beta_1 - 13) q^{65} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{67} + (\beta_{2} + \beta_1 - 1) q^{69} + (\beta_1 + 4) q^{71} + ( - \beta_{2} + 7) q^{73} + (6 \beta_{2} - \beta_1 - 2) q^{75} + ( - 3 \beta_{2} - 5 \beta_1 - 1) q^{79} + ( - 5 \beta_{2} + 2 \beta_1 + 4) q^{81} + ( - 4 \beta_{2} - \beta_1 - 2) q^{83} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{85} + ( - \beta_{2} - 4 \beta_1 + 7) q^{87} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{89} + ( - 9 \beta_{2} + 2 \beta_1 + 1) q^{93} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{95} + ( - \beta_{2} - 6 \beta_1 + 10) q^{97} + ( - 2 \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 2 q^{5} + 4 q^{9} + 3 q^{11} + 3 q^{13} + 15 q^{15} - 5 q^{17} - 11 q^{19} + 4 q^{23} + 11 q^{25} - 22 q^{27} - q^{29} - 19 q^{31} - q^{33} - 8 q^{37} - 17 q^{39} + 17 q^{41} - 10 q^{43} - 21 q^{45} - 13 q^{47} + 9 q^{53} - 2 q^{55} + 2 q^{57} - 6 q^{59} + 4 q^{61} - 37 q^{65} - 8 q^{67} - 3 q^{69} + 13 q^{71} + 22 q^{73} - 13 q^{75} - 5 q^{79} + 19 q^{81} - 3 q^{83} - 7 q^{85} + 18 q^{87} - 3 q^{89} + 14 q^{93} - 3 q^{95} + 25 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
0.239123
2.46050
−1.69963
0 −3.18194 0 −3.42107 0 0 0 7.12476 0
1.2 0 0.593579 0 −1.86693 0 0 0 −2.64766 0
1.3 0 1.58836 0 3.28799 0 0 0 −0.477100 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 8624.2.a.ci 3
4.b odd 2 1 2156.2.a.i 3
7.b odd 2 1 8624.2.a.cn 3
7.c even 3 2 1232.2.q.l 6
28.d even 2 1 2156.2.a.h 3
28.f even 6 2 2156.2.i.l 6
28.g odd 6 2 308.2.i.a 6
84.n even 6 2 2772.2.s.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.i.a 6 28.g odd 6 2
1232.2.q.l 6 7.c even 3 2
2156.2.a.h 3 28.d even 2 1
2156.2.a.i 3 4.b odd 2 1
2156.2.i.l 6 28.f even 6 2
2772.2.s.f 6 84.n even 6 2
8624.2.a.ci 3 1.a even 1 1 trivial
8624.2.a.cn 3 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3}^{3} + T_{3}^{2} - 6T_{3} + 3 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 11T_{5} - 21 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 24T_{13} + 79 \) Copy content Toggle raw display
\( T_{17}^{3} + 5T_{17}^{2} + 4T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 6T + 3 \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 11 T - 21 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} - 24 T + 79 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} + 4 T - 3 \) Copy content Toggle raw display
$19$ \( T^{3} + 11 T^{2} + 36 T + 33 \) Copy content Toggle raw display
$23$ \( T^{3} - 4T^{2} + T + 3 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 50 T - 129 \) Copy content Toggle raw display
$31$ \( T^{3} + 19 T^{2} + 80 T - 97 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 17 T^{2} + 76 T - 33 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 31 T - 73 \) Copy content Toggle raw display
$47$ \( T^{3} + 13 T^{2} + 52 T + 63 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 42 T - 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 45 T - 243 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 100 T + 472 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 60 T + 72 \) Copy content Toggle raw display
$71$ \( T^{3} - 13 T^{2} + 52 T - 63 \) Copy content Toggle raw display
$73$ \( T^{3} - 22 T^{2} + 155 T - 353 \) Copy content Toggle raw display
$79$ \( T^{3} + 5 T^{2} - 132 T + 303 \) Copy content Toggle raw display
$83$ \( T^{3} + 3 T^{2} - 96 T - 477 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} - 108 T + 243 \) Copy content Toggle raw display
$97$ \( T^{3} - 25 T^{2} + 56 T + 1171 \) Copy content Toggle raw display
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