Properties

Label 8016.2.a.u
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.38569.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 5x^{3} + 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 4\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 5\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + \nu^{3} - 5\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} + 5\beta_{2} + 7 \) 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
−1.95408
0.790734
−1.15098
0.275834
2.03850
0 1.00000 0 −3.11102 0 1.66149 0 1.00000 0
1.2 0 1.00000 0 −1.61314 0 −2.77861 0 1.00000 0
1.3 0 1.00000 0 0.0593478 0 1.53510 0 1.00000 0
1.4 0 1.00000 0 1.81885 0 0.619101 0 1.00000 0
1.5 0 1.00000 0 1.84596 0 2.96293 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.u 5
4.b odd 2 1 501.2.a.c 5
12.b even 2 1 1503.2.a.c 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.c 5 4.b odd 2 1
1503.2.a.c 5 12.b even 2 1
8016.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} + T_{5}^{4} - 9T_{5}^{3} - 2T_{5}^{2} + 17T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{5} - 4T_{7}^{4} - 3T_{7}^{3} + 29T_{7}^{2} - 37T_{7} + 13 \) Copy content Toggle raw display
\( T_{11}^{5} - 7T_{11}^{4} - 3T_{11}^{3} + 92T_{11}^{2} - 101T_{11} - 121 \) Copy content Toggle raw display
\( T_{13}^{5} + 8T_{13}^{4} - T_{13}^{3} - 66T_{13}^{2} - 48T_{13} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + T^{4} - 9 T^{3} - 2 T^{2} + 17 T - 1 \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} - 3 T^{3} + 29 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$11$ \( T^{5} - 7 T^{4} - 3 T^{3} + 92 T^{2} + \cdots - 121 \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} - T^{3} - 66 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$17$ \( T^{5} - 5 T^{4} - 34 T^{3} + 234 T^{2} + \cdots - 293 \) Copy content Toggle raw display
$19$ \( T^{5} - 20 T^{4} + 153 T^{3} + \cdots - 599 \) Copy content Toggle raw display
$23$ \( T^{5} + T^{4} - 77 T^{3} - 26 T^{2} + \cdots - 1793 \) Copy content Toggle raw display
$29$ \( T^{5} + 5 T^{4} - 71 T^{3} + \cdots + 4001 \) Copy content Toggle raw display
$31$ \( T^{5} - 18 T^{4} + 118 T^{3} + \cdots - 181 \) Copy content Toggle raw display
$37$ \( T^{5} + 17 T^{4} + 23 T^{3} + \cdots - 13913 \) Copy content Toggle raw display
$41$ \( T^{5} - 6 T^{4} - 40 T^{3} - 55 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{5} - 10 T^{4} - 52 T^{3} + \cdots - 4159 \) Copy content Toggle raw display
$47$ \( T^{5} - 3 T^{4} - 63 T^{3} + 223 T^{2} + \cdots - 547 \) Copy content Toggle raw display
$53$ \( T^{5} - 15 T^{4} - 65 T^{3} + \cdots + 707 \) Copy content Toggle raw display
$59$ \( T^{5} - 17 T^{4} - 73 T^{3} + \cdots + 5737 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} - 147 T^{3} + \cdots - 8609 \) Copy content Toggle raw display
$67$ \( T^{5} - 24 T^{4} - 3 T^{3} + \cdots - 23449 \) Copy content Toggle raw display
$71$ \( T^{5} + T^{4} - 292 T^{3} + \cdots + 28039 \) Copy content Toggle raw display
$73$ \( T^{5} - 2 T^{4} - 109 T^{3} + \cdots + 8743 \) Copy content Toggle raw display
$79$ \( T^{5} - 6 T^{4} - 186 T^{3} + \cdots - 31489 \) Copy content Toggle raw display
$83$ \( T^{5} - 7 T^{4} - 129 T^{3} + \cdots + 7703 \) Copy content Toggle raw display
$89$ \( T^{5} - 155 T^{3} - 322 T^{2} + \cdots + 17071 \) Copy content Toggle raw display
$97$ \( T^{5} + 13 T^{4} - 112 T^{3} + \cdots - 869 \) Copy content Toggle raw display
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