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Properties

Label 1160.1.o.c
Level 11601160
Weight 11
Character orbit 1160.o
Self dual yes
Analytic conductor 0.5790.579
Analytic rank 00
Dimension 22
Projective image D5D_{5}
CM discriminant -1160
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1160,1,Mod(579,1160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1160.579");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1160=23⋅5⋅29 1160 = 2^{3} \cdot 5 \cdot 29
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1160.o (of order 22, degree 11, 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: yes
Analytic conductor: 0.5789154146540.578915414654
Analytic rank: 00
Dimension: 22
Coefficient field: Q(5)\Q(\sqrt{5})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x−1 x^{2} - x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D5D_{5}
Projective field: Galois closure of 5.1.1345600.2

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of β=12(1+5)\beta = \frac{1}{2}(1 + \sqrt{5}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+q2+(β−1)q3+q4−q5+(β−1)q6+βq7+q8+(−β+1)q9−q10+(β−1)q12+(−β+1)q13+βq14+(−β+1)q15+⋯+βq98+O(q100) q + q^{2} + (\beta - 1) q^{3} + q^{4} - q^{5} + (\beta - 1) q^{6} + \beta q^{7} + q^{8} + ( - \beta + 1) q^{9} - q^{10} + (\beta - 1) q^{12} + ( - \beta + 1) q^{13} + \beta q^{14} + ( - \beta + 1) q^{15}+ \cdots + \beta q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q+2q2−q3+2q4−2q5−q6+q7+2q8+q9−2q10−q12+q13+q14+q15+2q16−q17+q18−2q20+2q21+q23−q24+⋯+q98+O(q100) 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} + q^{9} - 2 q^{10} - q^{12} + q^{13} + q^{14} + q^{15} + 2 q^{16} - q^{17} + q^{18} - 2 q^{20} + 2 q^{21} + q^{23} - q^{24}+ \cdots + q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1160Z)×\left(\mathbb{Z}/1160\mathbb{Z}\right)^\times.

nn 321321 581581 697697 871871
χ(n)\chi(n) −1-1 −1-1 −1-1 −1-1

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}
579.1
−0.618034
1.61803
1.00000 −1.61803 1.00000 −1.00000 −1.61803 −0.618034 1.00000 1.61803 −1.00000
579.2 1.00000 0.618034 1.00000 −1.00000 0.618034 1.61803 1.00000 −0.618034 −1.00000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1160.o odd 2 1 CM by Q(−290)\Q(\sqrt{-290})

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1160.1.o.c yes 2
5.b even 2 1 1160.1.o.b yes 2
8.d odd 2 1 1160.1.o.d yes 2
29.b even 2 1 1160.1.o.a ✓ 2
40.e odd 2 1 1160.1.o.a ✓ 2
145.d even 2 1 1160.1.o.d yes 2
232.b odd 2 1 1160.1.o.b yes 2
1160.o odd 2 1 CM 1160.1.o.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.1.o.a ✓ 2 29.b even 2 1
1160.1.o.a ✓ 2 40.e odd 2 1
1160.1.o.b yes 2 5.b even 2 1
1160.1.o.b yes 2 232.b odd 2 1
1160.1.o.c yes 2 1.a even 1 1 trivial
1160.1.o.c yes 2 1160.o odd 2 1 CM
1160.1.o.d yes 2 8.d odd 2 1
1160.1.o.d yes 2 145.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(1160,[χ])S_{1}^{\mathrm{new}}(1160, [\chi]):

T32+T3−1 T_{3}^{2} + T_{3} - 1 Copy content Toggle raw display
T72−T7−1 T_{7}^{2} - T_{7} - 1 Copy content Toggle raw display

Hecke characteristic polynomials

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