Properties

Label 2016.2.a.r
Level $2016$
Weight $2$
Character orbit 2016.a
Self dual yes
Analytic conductor $16.098$
Analytic rank $0$
Dimension $2$
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] = [2016,2,Mod(1,2016)]
 
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([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("2016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 224)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{5} + q^{7} + (2 \beta + 2) q^{11} + ( - \beta + 3) q^{13} - 2 \beta q^{17} + ( - \beta + 1) q^{19} - 4 q^{23} + (2 \beta + 1) q^{25} + 2 \beta q^{29} + ( - 2 \beta + 2) q^{31} + ( - \beta - 1) q^{35} - 2 \beta q^{37} + (2 \beta + 4) q^{41} + (2 \beta + 2) q^{43} + ( - 2 \beta - 6) q^{47} + q^{49} + 10 q^{53} + ( - 4 \beta - 12) q^{55} + (\beta + 7) q^{59} + (\beta + 9) q^{61} + ( - 2 \beta + 2) q^{65} + 4 q^{67} + (4 \beta - 4) q^{71} + ( - 4 \beta + 6) q^{73} + (2 \beta + 2) q^{77} + (4 \beta + 4) q^{79} + (\beta + 7) q^{83} + (2 \beta + 10) q^{85} + 6 q^{89} + ( - \beta + 3) q^{91} + 4 q^{95} + (2 \beta + 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} + 6 q^{13} + 2 q^{19} - 8 q^{23} + 2 q^{25} + 4 q^{31} - 2 q^{35} + 8 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 20 q^{53} - 24 q^{55} + 14 q^{59} + 18 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{71} + 12 q^{73} + 4 q^{77} + 8 q^{79} + 14 q^{83} + 20 q^{85} + 12 q^{89} + 6 q^{91} + 8 q^{95} + 16 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 0 0 −3.23607 0 1.00000 0 0 0
1.2 0 0 0 1.23607 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

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 2016.2.a.r 2
3.b odd 2 1 224.2.a.c 2
4.b odd 2 1 2016.2.a.o 2
8.b even 2 1 4032.2.a.bw 2
8.d odd 2 1 4032.2.a.bv 2
12.b even 2 1 224.2.a.d yes 2
15.d odd 2 1 5600.2.a.bk 2
21.c even 2 1 1568.2.a.v 2
21.g even 6 2 1568.2.i.n 4
21.h odd 6 2 1568.2.i.v 4
24.f even 2 1 448.2.a.i 2
24.h odd 2 1 448.2.a.j 2
48.i odd 4 2 1792.2.b.k 4
48.k even 4 2 1792.2.b.m 4
60.h even 2 1 5600.2.a.z 2
84.h odd 2 1 1568.2.a.k 2
84.j odd 6 2 1568.2.i.w 4
84.n even 6 2 1568.2.i.m 4
168.e odd 2 1 3136.2.a.by 2
168.i even 2 1 3136.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 3.b odd 2 1
224.2.a.d yes 2 12.b even 2 1
448.2.a.i 2 24.f even 2 1
448.2.a.j 2 24.h odd 2 1
1568.2.a.k 2 84.h odd 2 1
1568.2.a.v 2 21.c even 2 1
1568.2.i.m 4 84.n even 6 2
1568.2.i.n 4 21.g even 6 2
1568.2.i.v 4 21.h odd 6 2
1568.2.i.w 4 84.j odd 6 2
1792.2.b.k 4 48.i odd 4 2
1792.2.b.m 4 48.k even 4 2
2016.2.a.o 2 4.b odd 2 1
2016.2.a.r 2 1.a even 1 1 trivial
3136.2.a.bf 2 168.i even 2 1
3136.2.a.by 2 168.e odd 2 1
4032.2.a.bv 2 8.d odd 2 1
4032.2.a.bw 2 8.b even 2 1
5600.2.a.z 2 60.h even 2 1
5600.2.a.bk 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2016))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 4 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 20 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
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