Properties

Label 2016.3.e.a
Level $2016$
Weight $3$
Character orbit 2016.e
Analytic conductor $54.932$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,3,Mod(1007,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1007");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + (\beta_{5} + 3 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} + (\beta_{5} + 3 \beta_1) q^{7} + 7 \beta_{2} q^{11} + ( - 4 \beta_{5} - 2 \beta_1) q^{13} - \beta_{3} q^{17} + \beta_{7} q^{19} + 29 \beta_{4} q^{23} - 23 q^{25} - 13 \beta_{4} q^{29} + ( - 8 \beta_{5} - 4 \beta_1) q^{31} + (5 \beta_{3} + 24 \beta_{2}) q^{35} - 13 \beta_1 q^{37} - 9 \beta_{3} q^{41} - 48 q^{43} + 4 \beta_{6} q^{47} + (5 \beta_{7} - 1) q^{49} + 37 \beta_{4} q^{53} + ( - 14 \beta_{5} - 7 \beta_1) q^{55} - 10 \beta_{3} q^{59} + (6 \beta_{5} + 3 \beta_1) q^{61} - 96 \beta_{2} q^{65} + 14 q^{67} - 29 \beta_{4} q^{71} - 3 \beta_{7} q^{73} + ( - 7 \beta_{6} + 35 \beta_{4}) q^{77} + 24 \beta_1 q^{79} + 16 \beta_{3} q^{83} + 24 \beta_1 q^{85} - 21 \beta_{3} q^{89} + ( - 10 \beta_{7} - 96) q^{91} + 48 \beta_{4} q^{95} - 17 \beta_{7} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 184 q^{25} - 384 q^{43} - 8 q^{49} + 112 q^{67} - 768 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{24}^{6} + 8\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} - \zeta_{24}^{6} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 8\zeta_{24}^{4} - 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\zeta_{24}^{7} + 4\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{5} + 4\beta_{4} + 4\beta_{2} + \beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{6} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - 4\beta_{4} - 4\beta_{2} + \beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{5} + 4\beta_{4} - 4\beta_{2} - \beta_1 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1007.1
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
0 0 0 6.92820i 0 −4.89898 5.00000i 0 0 0
1007.2 0 0 0 6.92820i 0 −4.89898 + 5.00000i 0 0 0
1007.3 0 0 0 6.92820i 0 4.89898 5.00000i 0 0 0
1007.4 0 0 0 6.92820i 0 4.89898 + 5.00000i 0 0 0
1007.5 0 0 0 6.92820i 0 −4.89898 5.00000i 0 0 0
1007.6 0 0 0 6.92820i 0 −4.89898 + 5.00000i 0 0 0
1007.7 0 0 0 6.92820i 0 4.89898 5.00000i 0 0 0
1007.8 0 0 0 6.92820i 0 4.89898 + 5.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1007.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
56.e even 2 1 inner
168.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.3.e.a 8
3.b odd 2 1 inner 2016.3.e.a 8
4.b odd 2 1 504.3.e.a 8
7.b odd 2 1 inner 2016.3.e.a 8
8.b even 2 1 504.3.e.a 8
8.d odd 2 1 inner 2016.3.e.a 8
12.b even 2 1 504.3.e.a 8
21.c even 2 1 inner 2016.3.e.a 8
24.f even 2 1 inner 2016.3.e.a 8
24.h odd 2 1 504.3.e.a 8
28.d even 2 1 504.3.e.a 8
56.e even 2 1 inner 2016.3.e.a 8
56.h odd 2 1 504.3.e.a 8
84.h odd 2 1 504.3.e.a 8
168.e odd 2 1 inner 2016.3.e.a 8
168.i even 2 1 504.3.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.e.a 8 4.b odd 2 1
504.3.e.a 8 8.b even 2 1
504.3.e.a 8 12.b even 2 1
504.3.e.a 8 24.h odd 2 1
504.3.e.a 8 28.d even 2 1
504.3.e.a 8 56.h odd 2 1
504.3.e.a 8 84.h odd 2 1
504.3.e.a 8 168.i even 2 1
2016.3.e.a 8 1.a even 1 1 trivial
2016.3.e.a 8 3.b odd 2 1 inner
2016.3.e.a 8 7.b odd 2 1 inner
2016.3.e.a 8 8.d odd 2 1 inner
2016.3.e.a 8 21.c even 2 1 inner
2016.3.e.a 8 24.f even 2 1 inner
2016.3.e.a 8 56.e even 2 1 inner
2016.3.e.a 8 168.e odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 48 \) acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 384)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 1682)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 338)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 1536)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 676)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3888)^{4} \) Copy content Toggle raw display
$43$ \( (T + 48)^{8} \) Copy content Toggle raw display
$47$ \( (T^{2} + 768)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2738)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4800)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 864)^{4} \) Copy content Toggle raw display
$67$ \( (T - 14)^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1682)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 864)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2304)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12288)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 21168)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27744)^{4} \) Copy content Toggle raw display
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