Properties

Label 9016.2.a.z
Level $9016$
Weight $2$
Character orbit 9016.a
Self dual yes
Analytic conductor $71.993$
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] = [9016,2,Mod(1,9016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9016, 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("9016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9016.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: \(71.9931224624\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 1288)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{2} + 2 \beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{2} + 2 \beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1) q^{9} + (\beta_{2} - \beta_1 - 1) q^{11} + 2 \beta_1 q^{13} + ( - 3 \beta_{2} + \beta_1 + 1) q^{15} + (\beta_{2} - 2 \beta_1 - 4) q^{17} + q^{23} + ( - \beta_{2} - \beta_1 + 6) q^{25} + ( - 2 \beta_{2} - 2) q^{27} + (\beta_{2} + \beta_1 + 3) q^{29} + (\beta_{2} - 4 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{33} + ( - 2 \beta_1 + 8) q^{37} + (2 \beta_1 - 2) q^{39} + (4 \beta_{2} + 4 \beta_1 - 2) q^{41} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{43} + (\beta_{2} - 2 \beta_1 - 4) q^{45} + (\beta_{2} - 2 \beta_1) q^{47} + ( - 5 \beta_{2} - 3 \beta_1 + 5) q^{51} + (4 \beta_{2} + 4 \beta_1 - 6) q^{53} + ( - 6 \beta_{2} - 2 \beta_1) q^{55} + ( - 3 \beta_{2} + 2 \beta_1 + 6) q^{59} + ( - 7 \beta_{2} - 4 \beta_1) q^{61} + (4 \beta_{2} + 2 \beta_1 + 6) q^{65} + (\beta_{2} - 5 \beta_1 - 1) q^{67} + \beta_{2} q^{69} + 4 \beta_{2} q^{71} + ( - 6 \beta_{2} - 2) q^{73} + (7 \beta_{2} - 2) q^{75} + ( - \beta_{2} - 3 \beta_1 + 5) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + (2 \beta_1 + 2) q^{83} + ( - 11 \beta_{2} - 9 \beta_1 + 3) q^{85} + (2 \beta_{2} + 2) q^{87} + ( - 5 \beta_{2} - 6) q^{89} + ( - 3 \beta_{2} - 5 \beta_1 + 7) q^{93} + ( - 9 \beta_{2} - 2 \beta_1 - 2) q^{97} + (3 \beta_{2} + 3 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{5} - q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{15} - 14 q^{17} + 3 q^{23} + 17 q^{25} - 6 q^{27} + 10 q^{29} - 10 q^{31} + 10 q^{33} + 22 q^{37} - 4 q^{39} - 2 q^{41} - 8 q^{43} - 14 q^{45} - 2 q^{47} + 12 q^{51} - 14 q^{53} - 2 q^{55} + 20 q^{59} - 4 q^{61} + 20 q^{65} - 8 q^{67} - 6 q^{73} - 6 q^{75} + 12 q^{79} - 13 q^{81} + 8 q^{83} + 6 q^{87} - 18 q^{89} + 16 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
0 −2.21432 0 −3.59210 0 0 0 1.90321 0
1.2 0 0.539189 0 2.87936 0 0 0 −2.70928 0
1.3 0 1.67513 0 −3.28726 0 0 0 −0.193937 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 9016.2.a.z 3
7.b odd 2 1 1288.2.a.l 3
28.d even 2 1 2576.2.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.a.l 3 7.b odd 2 1
2576.2.a.y 3 28.d even 2 1
9016.2.a.z 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9016))\):

\( T_{3}^{3} - 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - 8T_{5} - 34 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 4T_{11} - 20 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 12T_{13} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T + 2 \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 8 T - 34 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 4 T - 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 14 T^{2} + 44 T - 34 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 28 T - 20 \) Copy content Toggle raw display
$31$ \( T^{3} + 10 T^{2} - 32 T - 310 \) Copy content Toggle raw display
$37$ \( T^{3} - 22 T^{2} + 148 T - 296 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 84 T - 104 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 20 T - 50 \) Copy content Toggle raw display
$53$ \( T^{3} + 14 T^{2} - 20 T - 344 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + 72 T + 230 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 188 T - 1030 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 76 T - 436 \) Copy content Toggle raw display
$71$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 132 T - 712 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + 20 T + 100 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} + 8 T - 634 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 280 T - 2734 \) Copy content Toggle raw display
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