Properties

Label 110.2.f.b
Level 110110
Weight 22
Character orbit 110.f
Analytic conductor 0.8780.878
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,2,Mod(43,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 110=2511 110 = 2 \cdot 5 \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 110.f (of order 44, 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: 0.8783544222340.878354422234
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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 a primitive root of unity ζ8\zeta_{8}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+ζ8q2+(ζ82ζ8+1)q3+ζ82q4+(ζ83+2ζ8)q5+(ζ83ζ82+ζ8)q63ζ8q7+ζ83q8++(6ζ83++6ζ8)q99+O(q100) q + \zeta_{8} q^{2} + (\zeta_{8}^{2} - \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{6} - 3 \zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + \cdots + ( - 6 \zeta_{8}^{3} + \cdots + 6 \zeta_{8}) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q34q104q12+12q13+4q154q1612q17+8q1812q19+4q2212q23+4q2416q25+4q27+12q2912q3012q3116q33++8q97+O(q100) 4 q + 4 q^{3} - 4 q^{10} - 4 q^{12} + 12 q^{13} + 4 q^{15} - 4 q^{16} - 12 q^{17} + 8 q^{18} - 12 q^{19} + 4 q^{22} - 12 q^{23} + 4 q^{24} - 16 q^{25} + 4 q^{27} + 12 q^{29} - 12 q^{30} - 12 q^{31} - 16 q^{33}+ \cdots + 8 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/110Z)×\left(\mathbb{Z}/110\mathbb{Z}\right)^\times.

nn 6767 101101
χ(n)\chi(n) ζ82-\zeta_{8}^{2} 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}
43.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 1.70711 + 1.70711i 1.00000i −0.707107 2.12132i 2.41421i 2.12132 + 2.12132i 0.707107 0.707107i 2.82843i −1.00000 + 2.00000i
43.2 0.707107 + 0.707107i 0.292893 + 0.292893i 1.00000i 0.707107 + 2.12132i 0.414214i −2.12132 2.12132i −0.707107 + 0.707107i 2.82843i −1.00000 + 2.00000i
87.1 −0.707107 + 0.707107i 1.70711 1.70711i 1.00000i −0.707107 + 2.12132i 2.41421i 2.12132 2.12132i 0.707107 + 0.707107i 2.82843i −1.00000 2.00000i
87.2 0.707107 0.707107i 0.292893 0.292893i 1.00000i 0.707107 2.12132i 0.414214i −2.12132 + 2.12132i −0.707107 0.707107i 2.82843i −1.00000 2.00000i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.2.f.b 4
3.b odd 2 1 990.2.m.c 4
4.b odd 2 1 880.2.bd.c 4
5.b even 2 1 550.2.f.a 4
5.c odd 4 1 110.2.f.c yes 4
5.c odd 4 1 550.2.f.b 4
11.b odd 2 1 110.2.f.c yes 4
15.e even 4 1 990.2.m.d 4
20.e even 4 1 880.2.bd.b 4
33.d even 2 1 990.2.m.d 4
44.c even 2 1 880.2.bd.b 4
55.d odd 2 1 550.2.f.b 4
55.e even 4 1 inner 110.2.f.b 4
55.e even 4 1 550.2.f.a 4
165.l odd 4 1 990.2.m.c 4
220.i odd 4 1 880.2.bd.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.f.b 4 1.a even 1 1 trivial
110.2.f.b 4 55.e even 4 1 inner
110.2.f.c yes 4 5.c odd 4 1
110.2.f.c yes 4 11.b odd 2 1
550.2.f.a 4 5.b even 2 1
550.2.f.a 4 55.e even 4 1
550.2.f.b 4 5.c odd 4 1
550.2.f.b 4 55.d odd 2 1
880.2.bd.b 4 20.e even 4 1
880.2.bd.b 4 44.c even 2 1
880.2.bd.c 4 4.b odd 2 1
880.2.bd.c 4 220.i odd 4 1
990.2.m.c 4 3.b odd 2 1
990.2.m.c 4 165.l odd 4 1
990.2.m.d 4 15.e even 4 1
990.2.m.d 4 33.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 S2new(110,[χ])S_{2}^{\mathrm{new}}(110, [\chi]):

T344T33+8T324T3+1 T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} - 4T_{3} + 1 Copy content Toggle raw display
T1326T13+18 T_{13}^{2} - 6T_{13} + 18 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+1 T^{4} + 1 Copy content Toggle raw display
33 T44T3++1 T^{4} - 4 T^{3} + \cdots + 1 Copy content Toggle raw display
55 T4+8T2+25 T^{4} + 8T^{2} + 25 Copy content Toggle raw display
77 T4+81 T^{4} + 81 Copy content Toggle raw display
1111 T4+14T2+121 T^{4} + 14T^{2} + 121 Copy content Toggle raw display
1313 (T26T+18)2 (T^{2} - 6 T + 18)^{2} Copy content Toggle raw display
1717 T4+12T3++81 T^{4} + 12 T^{3} + \cdots + 81 Copy content Toggle raw display
1919 (T+3)4 (T + 3)^{4} Copy content Toggle raw display
2323 T4+12T3++4 T^{4} + 12 T^{3} + \cdots + 4 Copy content Toggle raw display
2929 (T26T9)2 (T^{2} - 6 T - 9)^{2} Copy content Toggle raw display
3131 (T2+6T9)2 (T^{2} + 6 T - 9)^{2} Copy content Toggle raw display
3737 T4+8T3++1 T^{4} + 8 T^{3} + \cdots + 1 Copy content Toggle raw display
4141 T4+108T2+324 T^{4} + 108T^{2} + 324 Copy content Toggle raw display
4343 T4+12T3++324 T^{4} + 12 T^{3} + \cdots + 324 Copy content Toggle raw display
4747 T412T3++196 T^{4} - 12 T^{3} + \cdots + 196 Copy content Toggle raw display
5353 T424T3++2209 T^{4} - 24 T^{3} + \cdots + 2209 Copy content Toggle raw display
5959 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
6161 T4+54T2+81 T^{4} + 54T^{2} + 81 Copy content Toggle raw display
6767 (T2+8T+32)2 (T^{2} + 8 T + 32)^{2} Copy content Toggle raw display
7171 (T26T9)2 (T^{2} - 6 T - 9)^{2} Copy content Toggle raw display
7373 (T212T+72)2 (T^{2} - 12 T + 72)^{2} Copy content Toggle raw display
7979 (T212T+18)2 (T^{2} - 12 T + 18)^{2} Copy content Toggle raw display
8383 T4+12T3++324 T^{4} + 12 T^{3} + \cdots + 324 Copy content Toggle raw display
8989 T4+162T2+3969 T^{4} + 162T^{2} + 3969 Copy content Toggle raw display
9797 T48T3++784 T^{4} - 8 T^{3} + \cdots + 784 Copy content Toggle raw display
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