Properties

Label 8016.2.a.o
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2777.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{7} + q^{9} + (\beta_{3} + 2) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{2} + \beta_1 + 3) q^{17} + 2 \beta_{2} q^{19} + (\beta_{3} - \beta_{2}) q^{21} - 2 \beta_{2} q^{23} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{25} + q^{27} + (2 \beta_1 + 4) q^{29} + ( - \beta_{3} - \beta_{2}) q^{31} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{35} + ( - 3 \beta_{2} + 2) q^{37} + (\beta_{3} + 2) q^{39} + (3 \beta_{2} + \beta_1 + 3) q^{41} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{43} + ( - \beta_1 + 1) q^{45} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{47} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + \beta_1 + 3) q^{51} + ( - 2 \beta_{2} - \beta_1 + 1) q^{53} + 2 \beta_{2} q^{57} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{59} - 2 \beta_1 q^{61} + (\beta_{3} - \beta_{2}) q^{63} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{67} - 2 \beta_{2} q^{69} + (4 \beta_{2} - 2 \beta_1 - 2) q^{71} + ( - 2 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{75} + (\beta_{2} + 3 \beta_1 + 3) q^{79} + q^{81} + ( - \beta_{2} - 4 \beta_1 - 4) q^{83} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{85} + (2 \beta_1 + 4) q^{87} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{89} + (2 \beta_{3} - 2 \beta_{2} + 8) q^{91} + ( - \beta_{3} - \beta_{2}) q^{93} + (2 \beta_{3} + 2 \beta_{2}) q^{95} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 5\beta _1 + 7 ) / 2 \) 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
2.36234
−0.679643
0.825785
−1.50848
0 1.00000 0 −1.87806 0 3.28324 0 1.00000 0
1.2 0 1.00000 0 0.416566 0 −4.65960 0 1.00000 0
1.3 0 1.00000 0 2.77037 0 −1.86579 0 1.00000 0
1.4 0 1.00000 0 3.69113 0 2.24216 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.o 4
4.b odd 2 1 1002.2.a.i 4
12.b even 2 1 3006.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.i 4 4.b odd 2 1
3006.2.a.s 4 12.b even 2 1
8016.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{4} - 5T_{5}^{3} + 20T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 20T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} + 4T_{13}^{2} + 56T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + 20 T - 8 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 20 T^{2} + 64 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 4 T^{2} + 56 T - 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + 20 T^{2} + 16 T - 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 32 T^{2} + 32 T + 128 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 32 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$29$ \( T^{4} - 14 T^{3} + 36 T^{2} + 56 T - 32 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 36 T^{2} - 96 T - 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} - 66 T^{2} + 184 T + 568 \) Copy content Toggle raw display
$41$ \( T^{4} - 14 T^{3} - 12 T^{2} + \cdots - 1072 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} - 32 T^{2} + 40 T + 32 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} - 32 T^{2} - 208 T - 64 \) Copy content Toggle raw display
$53$ \( T^{4} - 3 T^{3} - 40 T^{2} + 36 T + 8 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} - 116 T^{2} + \cdots + 808 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} - 36 T^{2} + 72 T + 128 \) Copy content Toggle raw display
$67$ \( T^{4} - 7 T^{3} - 186 T^{2} + \cdots - 3208 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} - 168 T^{2} + \cdots + 6784 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} - 132 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} - 56 T^{2} + \cdots + 1472 \) Copy content Toggle raw display
$83$ \( T^{4} + 13 T^{3} - 96 T^{2} + \cdots + 4072 \) Copy content Toggle raw display
$89$ \( T^{4} - 13 T^{3} - 2 T^{2} + \cdots - 1448 \) Copy content Toggle raw display
$97$ \( T^{4} - 5 T^{3} - 146 T^{2} + \cdots + 3256 \) Copy content Toggle raw display
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