Properties

Label 8016.2.a.n
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{2} + 2) q^{5} + ( - 2 \beta_1 + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{2} + 2) q^{5} + ( - 2 \beta_1 + 2) q^{7} + q^{9} + ( - \beta_{2} + \beta_1 + 1) q^{11} + 2 q^{13} + ( - \beta_{2} + 2) q^{15} + 3 \beta_{2} q^{17} + (2 \beta_{2} + 2 \beta_1 - 2) q^{19} + ( - 2 \beta_1 + 2) q^{21} + 8 q^{23} + ( - 5 \beta_{2} - \beta_1 + 2) q^{25} + q^{27} + (2 \beta_{2} - 2 \beta_1 - 4) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{33} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{35} + (2 \beta_1 - 4) q^{37} + 2 q^{39} + ( - 3 \beta_{2} - 6 \beta_1 + 6) q^{41} + (3 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - \beta_{2} + 2) q^{45} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{47} + (4 \beta_{2} - 4 \beta_1 + 5) q^{49} + 3 \beta_{2} q^{51} + (3 \beta_{2} + 4 \beta_1 - 4) q^{53} + ( - 4 \beta_{2} + 6) q^{55} + (2 \beta_{2} + 2 \beta_1 - 2) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (5 \beta_{2} + 5 \beta_1 - 1) q^{61} + ( - 2 \beta_1 + 2) q^{63} + ( - 2 \beta_{2} + 4) q^{65} + (\beta_{2} + 2 \beta_1 + 8) q^{67} + 8 q^{69} + (6 \beta_{2} + 4 \beta_1 + 4) q^{71} + ( - 4 \beta_{2} - 2 \beta_1 + 8) q^{73} + ( - 5 \beta_{2} - \beta_1 + 2) q^{75} + ( - 4 \beta_{2} - 4) q^{77} + ( - 5 \beta_{2} + 2 \beta_1 - 4) q^{79} + q^{81} + ( - 6 \beta_{2} - 8 \beta_1 + 4) q^{83} + (9 \beta_{2} + 3 \beta_1 - 9) q^{85} + (2 \beta_{2} - 2 \beta_1 - 4) q^{87} + (2 \beta_{2} - 6 \beta_1) q^{89} + ( - 4 \beta_1 + 4) q^{91} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + (8 \beta_{2} + 4 \beta_1 - 8) q^{95} + ( - 6 \beta_{2} + 2) q^{97} + ( - \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{5} + 4 q^{7} + 3 q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{15} - 4 q^{19} + 4 q^{21} + 24 q^{23} + 5 q^{25} + 3 q^{27} - 14 q^{29} + 4 q^{31} + 4 q^{33} + 4 q^{35} - 10 q^{37} + 6 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} + 18 q^{55} - 4 q^{57} - 4 q^{59} + 2 q^{61} + 4 q^{63} + 12 q^{65} + 26 q^{67} + 24 q^{69} + 16 q^{71} + 22 q^{73} + 5 q^{75} - 12 q^{77} - 10 q^{79} + 3 q^{81} + 4 q^{83} - 24 q^{85} - 14 q^{87} - 6 q^{89} + 8 q^{91} + 4 q^{93} - 20 q^{95} + 6 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : 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
−1.48119
2.17009
0.311108
0 1.00000 0 0.324869 0 4.96239 0 1.00000 0
1.2 0 1.00000 0 1.46081 0 −2.34017 0 1.00000 0
1.3 0 1.00000 0 4.21432 0 1.37778 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(167\) \(1\)

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 8016.2.a.n 3
4.b odd 2 1 4008.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.e 3 4.b odd 2 1
8016.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{3} - 6T_{5}^{2} + 8T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 8T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 4T_{11} + 20 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 6 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 4 T + 20 \) Copy content Toggle raw display
$13$ \( (T - 2)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 36T + 54 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 16 T - 32 \) Copy content Toggle raw display
$23$ \( (T - 8)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 14 T^{2} + 28 T - 152 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} + 10 T^{2} + 20 T - 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 12 T^{2} - 72 T + 918 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 16 T + 130 \) Copy content Toggle raw display
$47$ \( T^{3} - 84T + 268 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 44 T - 290 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 132 T - 4 \) Copy content Toggle raw display
$67$ \( T^{3} - 26 T^{2} + 212 T - 554 \) Copy content Toggle raw display
$71$ \( T^{3} - 16 T^{2} - 64 T + 1040 \) Copy content Toggle raw display
$73$ \( T^{3} - 22 T^{2} + 100 T - 104 \) Copy content Toggle raw display
$79$ \( T^{3} + 10 T^{2} - 100 T - 278 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 256 T + 1424 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 148 T - 920 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 132 T - 152 \) Copy content Toggle raw display
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