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Properties

Label 2016.1.dd.a
Level 20162016
Weight 11
Character orbit 2016.dd
Analytic conductor 1.0061.006
Analytic rank 00
Dimension 44
Projective image A4A_{4}
CM/RM no
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,1,Mod(319,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.319");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 2016=25⋅32⋅7 2016 = 2^{5} \cdot 3^{2} \cdot 7
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2016.dd (of order 66, degree 22, 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: 1.006115065471.00611506547
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4−x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: A4A_{4}
Projective field: Galois closure of 4.0.254016.1

qq-expansion

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

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ123q3−ζ123q7−q9−ζ122q13+ζ122q17−ζ12q19−q21−2ζ123q23−q25+ζ123q27+ζ124q29+⋯−ζ124q97+O(q100) q - \zeta_{12}^{3} q^{3} - \zeta_{12}^{3} q^{7} - q^{9} - \zeta_{12}^{2} q^{13} + \zeta_{12}^{2} q^{17} - \zeta_{12} q^{19} - q^{21} - 2 \zeta_{12}^{3} q^{23} - q^{25} + \zeta_{12}^{3} q^{27} + \zeta_{12}^{4} q^{29} + \cdots - \zeta_{12}^{4} q^{97} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q−4q9−2q13+2q17−4q21−4q25−2q29+2q37+2q41−4q49−2q53−2q57−2q61−8q69+2q73+4q81+2q89+2q93+2q97+O(q100) 4 q - 4 q^{9} - 2 q^{13} + 2 q^{17} - 4 q^{21} - 4 q^{25} - 2 q^{29} + 2 q^{37} + 2 q^{41} - 4 q^{49} - 2 q^{53} - 2 q^{57} - 2 q^{61} - 8 q^{69} + 2 q^{73} + 4 q^{81} + 2 q^{89} + 2 q^{93} + 2 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 127127 577577 17651765 17931793
χ(n)\chi(n) −1-1 ζ124\zeta_{12}^{4} 11 −ζ122-\zeta_{12}^{2}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
319.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 − 1.00000i 0 0 0 − 1.00000i 0 −1.00000 0
319.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
1087.1 0 − 1.00000i 0 0 0 − 1.00000i 0 −1.00000 0
1087.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.g even 3 1 inner
252.bl odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.1.dd.a yes 4
4.b odd 2 1 inner 2016.1.dd.a yes 4
7.c even 3 1 2016.1.bw.a ✓ 4
9.c even 3 1 2016.1.bw.a ✓ 4
28.g odd 6 1 2016.1.bw.a ✓ 4
36.f odd 6 1 2016.1.bw.a ✓ 4
63.g even 3 1 inner 2016.1.dd.a yes 4
252.bl odd 6 1 inner 2016.1.dd.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.1.bw.a ✓ 4 7.c even 3 1
2016.1.bw.a ✓ 4 9.c even 3 1
2016.1.bw.a ✓ 4 28.g odd 6 1
2016.1.bw.a ✓ 4 36.f odd 6 1
2016.1.dd.a yes 4 1.a even 1 1 trivial
2016.1.dd.a yes 4 4.b odd 2 1 inner
2016.1.dd.a yes 4 63.g even 3 1 inner
2016.1.dd.a yes 4 252.bl odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace S1new(2016,[χ])S_{1}^{\mathrm{new}}(2016, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
1717 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
1919 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
2323 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
2929 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
3131 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
3737 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
4141 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
4343 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
4747 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
5353 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
5959 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
6161 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
6767 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
7979 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
8383 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
8989 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
9797 (T2−T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
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