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Properties

Label 2016.2.b.a
Level $2016$
Weight $2$
Character orbit 2016.b
Analytic conductor $16.098$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(1567,2016)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.b (of order \(2\), degree \(1\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 672)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{4} q^{7} + (\beta_{7} + \beta_{4} + \beta_{2}) q^{11} - \beta_{5} q^{13} - \beta_1 q^{17} + (\beta_{6} - \beta_{4} + \beta_{2} - 2) q^{19} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{23}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 8 q^{19} - 16 q^{25} + 16 q^{31} - 8 q^{35} + 8 q^{37} + 16 q^{47} - 16 q^{53} + 8 q^{55} + 32 q^{59} - 16 q^{65} - 32 q^{77} + 16 q^{83} - 56 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 16\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 30\nu^{2} + 4\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 9\nu^{4} - 16\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 11\nu^{4} - 30\nu^{2} + 4\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 11\nu^{3} + 26\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 11\nu^{4} - 26\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 50\nu^{3} + 54\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 5\beta_{4} - 5\beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{6} + 5\beta_{4} + 2\beta_{3} - 5\beta_{2} + 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{5} + 29\beta_{4} + 29\beta_{2} + 22\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47\beta_{6} - 29\beta_{4} - 22\beta_{3} + 29\beta_{2} - 246 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} + 134\beta_{5} - 181\beta_{4} - 181\beta_{2} - 186\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
− 2.06644i
2.63640i
0.222191i
− 1.65222i
1.65222i
− 0.222191i
− 2.63640i
2.06644i
0 0 0 − 4.33660i 0 −1.65222 − 2.06644i 0 0 0
1567.2 0 0 0 − 2.31423i 0 0.222191 + 2.63640i 0 0 0
1567.3 0 0 0 − 1.72844i 0 −2.63640 + 0.222191i 0 0 0
1567.4 0 0 0 − 0.922382i 0 2.06644 − 1.65222i 0 0 0
1567.5 0 0 0 0.922382i 0 2.06644 + 1.65222i 0 0 0
1567.6 0 0 0 1.72844i 0 −2.63640 − 0.222191i 0 0 0
1567.7 0 0 0 2.31423i 0 0.222191 − 2.63640i 0 0 0
1567.8 0 0 0 4.33660i 0 −1.65222 + 2.06644i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.b.a 8
3.b odd 2 1 672.2.b.a ✓ 8
4.b odd 2 1 2016.2.b.c 8
7.b odd 2 1 2016.2.b.c 8
8.b even 2 1 4032.2.b.o 8
8.d odd 2 1 4032.2.b.q 8
12.b even 2 1 672.2.b.b yes 8
21.c even 2 1 672.2.b.b yes 8
24.f even 2 1 1344.2.b.g 8
24.h odd 2 1 1344.2.b.h 8
28.d even 2 1 inner 2016.2.b.a 8
56.e even 2 1 4032.2.b.o 8
56.h odd 2 1 4032.2.b.q 8
84.h odd 2 1 672.2.b.a ✓ 8
168.e odd 2 1 1344.2.b.h 8
168.i even 2 1 1344.2.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.b.a ✓ 8 3.b odd 2 1
672.2.b.a ✓ 8 84.h odd 2 1
672.2.b.b yes 8 12.b even 2 1
672.2.b.b yes 8 21.c even 2 1
1344.2.b.g 8 24.f even 2 1
1344.2.b.g 8 168.i even 2 1
1344.2.b.h 8 24.h odd 2 1
1344.2.b.h 8 168.e odd 2 1
2016.2.b.a 8 1.a even 1 1 trivial
2016.2.b.a 8 28.d even 2 1 inner
2016.2.b.c 8 4.b odd 2 1
2016.2.b.c 8 7.b odd 2 1
4032.2.b.o 8 8.b even 2 1
4032.2.b.o 8 56.e even 2 1
4032.2.b.q 8 8.d odd 2 1
4032.2.b.q 8 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 196T_{5}^{4} + 448T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} - 44T_{19}^{2} - 96T_{19} + 64 \) 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
$5$ \( T^{8} + 28 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 68 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 28 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} - 44 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 84 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{4} - 80 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots - 1088)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 124 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$43$ \( T^{8} + 152 T^{6} + \cdots + 295936 \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + \cdots - 656)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 16)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 304 T^{6} + \cdots + 4734976 \) Copy content Toggle raw display
$67$ \( T^{8} + 296 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
$71$ \( T^{8} + 84 T^{6} + \cdots + 141376 \) Copy content Toggle raw display
$73$ \( T^{8} + 176 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$79$ \( T^{8} + 264 T^{6} + \cdots + 8667136 \) Copy content Toggle raw display
$83$ \( (T^{4} - 8 T^{3} + \cdots + 512)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 764 T^{6} + \cdots + 364351744 \) Copy content Toggle raw display
$97$ \( T^{8} + 432 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
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