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Properties

Label 2016.2.p.a
Level $2016$
Weight $2$
Character orbit 2016.p
Analytic conductor $16.098$
Analytic rank $1$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2016,2,Mod(559,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.559"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,-10,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0978410475\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{7} - 4 q^{11} - 2 \beta q^{23} - 5 q^{25} + 4 \beta q^{29} + 4 \beta q^{37} - 12 q^{43} - 7 q^{49} - 4 \beta q^{53} - 4 q^{67} + 2 \beta q^{71} + 4 \beta q^{77} + 6 \beta q^{79} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{11} - 10 q^{25} - 24 q^{43} - 14 q^{49} - 8 q^{67}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.500000 + 1.32288i
0.500000 − 1.32288i
0 0 0 0 0 − 2.64575i 0 0 0
559.2 0 0 0 0 0 2.64575i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.d odd 2 1 inner
56.e 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.p.a 2
3.b odd 2 1 224.2.e.a 2
4.b odd 2 1 504.2.p.a 2
7.b odd 2 1 CM 2016.2.p.a 2
8.b even 2 1 504.2.p.a 2
8.d odd 2 1 inner 2016.2.p.a 2
12.b even 2 1 56.2.e.a ✓ 2
21.c even 2 1 224.2.e.a 2
21.g even 6 2 1568.2.q.a 4
21.h odd 6 2 1568.2.q.a 4
24.f even 2 1 224.2.e.a 2
24.h odd 2 1 56.2.e.a ✓ 2
28.d even 2 1 504.2.p.a 2
48.i odd 4 2 1792.2.f.d 4
48.k even 4 2 1792.2.f.d 4
56.e even 2 1 inner 2016.2.p.a 2
56.h odd 2 1 504.2.p.a 2
84.h odd 2 1 56.2.e.a ✓ 2
84.j odd 6 2 392.2.m.a 4
84.n even 6 2 392.2.m.a 4
168.e odd 2 1 224.2.e.a 2
168.i even 2 1 56.2.e.a ✓ 2
168.s odd 6 2 392.2.m.a 4
168.v even 6 2 1568.2.q.a 4
168.ba even 6 2 392.2.m.a 4
168.be odd 6 2 1568.2.q.a 4
336.v odd 4 2 1792.2.f.d 4
336.y even 4 2 1792.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.a ✓ 2 12.b even 2 1
56.2.e.a ✓ 2 24.h odd 2 1
56.2.e.a ✓ 2 84.h odd 2 1
56.2.e.a ✓ 2 168.i even 2 1
224.2.e.a 2 3.b odd 2 1
224.2.e.a 2 21.c even 2 1
224.2.e.a 2 24.f even 2 1
224.2.e.a 2 168.e odd 2 1
392.2.m.a 4 84.j odd 6 2
392.2.m.a 4 84.n even 6 2
392.2.m.a 4 168.s odd 6 2
392.2.m.a 4 168.ba even 6 2
504.2.p.a 2 4.b odd 2 1
504.2.p.a 2 8.b even 2 1
504.2.p.a 2 28.d even 2 1
504.2.p.a 2 56.h odd 2 1
1568.2.q.a 4 21.g even 6 2
1568.2.q.a 4 21.h odd 6 2
1568.2.q.a 4 168.v even 6 2
1568.2.q.a 4 168.be odd 6 2
1792.2.f.d 4 48.i odd 4 2
1792.2.f.d 4 48.k even 4 2
1792.2.f.d 4 336.v odd 4 2
1792.2.f.d 4 336.y even 4 2
2016.2.p.a 2 1.a even 1 1 trivial
2016.2.p.a 2 7.b odd 2 1 CM
2016.2.p.a 2 8.d odd 2 1 inner
2016.2.p.a 2 56.e even 2 1 inner

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28 \) Copy content Toggle raw display
$29$ \( T^{2} + 112 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 112 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 112 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 28 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 252 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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