Properties

Label 45.6.a.c
Level 4545
Weight 66
Character orbit 45.a
Self dual yes
Analytic conductor 7.2177.217
Analytic rank 11
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,6,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 45=325 45 = 3^{2} \cdot 5
Weight: k k == 6 6
Character orbit: [χ][\chi] == 45.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: 7.217271891587.21727189158
Analytic rank: 11
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 15)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+2q228q4+25q5132q7120q8+50q10472q11686q13264q14+656q16+1562q172180q19700q20944q22264q23+625q25++1234q98+O(q100) q + 2 q^{2} - 28 q^{4} + 25 q^{5} - 132 q^{7} - 120 q^{8} + 50 q^{10} - 472 q^{11} - 686 q^{13} - 264 q^{14} + 656 q^{16} + 1562 q^{17} - 2180 q^{19} - 700 q^{20} - 944 q^{22} - 264 q^{23} + 625 q^{25}+ \cdots + 1234 q^{98}+O(q^{100}) Copy content Toggle raw display

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}
1.1
0
2.00000 0 −28.0000 25.0000 0 −132.000 −120.000 0 50.0000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 1 -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 45.6.a.c 1
3.b odd 2 1 15.6.a.a 1
4.b odd 2 1 720.6.a.w 1
5.b even 2 1 225.6.a.c 1
5.c odd 4 2 225.6.b.d 2
12.b even 2 1 240.6.a.k 1
15.d odd 2 1 75.6.a.c 1
15.e even 4 2 75.6.b.d 2
21.c even 2 1 735.6.a.a 1
24.f even 2 1 960.6.a.m 1
24.h odd 2 1 960.6.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.a 1 3.b odd 2 1
45.6.a.c 1 1.a even 1 1 trivial
75.6.a.c 1 15.d odd 2 1
75.6.b.d 2 15.e even 4 2
225.6.a.c 1 5.b even 2 1
225.6.b.d 2 5.c odd 4 2
240.6.a.k 1 12.b even 2 1
720.6.a.w 1 4.b odd 2 1
735.6.a.a 1 21.c even 2 1
960.6.a.m 1 24.f even 2 1
960.6.a.v 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22 T_{2} - 2 acting on S6new(Γ0(45))S_{6}^{\mathrm{new}}(\Gamma_0(45)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T - 2 Copy content Toggle raw display
33 T T Copy content Toggle raw display
55 T25 T - 25 Copy content Toggle raw display
77 T+132 T + 132 Copy content Toggle raw display
1111 T+472 T + 472 Copy content Toggle raw display
1313 T+686 T + 686 Copy content Toggle raw display
1717 T1562 T - 1562 Copy content Toggle raw display
1919 T+2180 T + 2180 Copy content Toggle raw display
2323 T+264 T + 264 Copy content Toggle raw display
2929 T+170 T + 170 Copy content Toggle raw display
3131 T7272 T - 7272 Copy content Toggle raw display
3737 T+142 T + 142 Copy content Toggle raw display
4141 T16198 T - 16198 Copy content Toggle raw display
4343 T+10316 T + 10316 Copy content Toggle raw display
4747 T+18568 T + 18568 Copy content Toggle raw display
5353 T+21514 T + 21514 Copy content Toggle raw display
5959 T+34600 T + 34600 Copy content Toggle raw display
6161 T+35738 T + 35738 Copy content Toggle raw display
6767 T+5772 T + 5772 Copy content Toggle raw display
7171 T69088 T - 69088 Copy content Toggle raw display
7373 T+70526 T + 70526 Copy content Toggle raw display
7979 T47640 T - 47640 Copy content Toggle raw display
8383 T+74004 T + 74004 Copy content Toggle raw display
8989 T90030 T - 90030 Copy content Toggle raw display
9797 T+33502 T + 33502 Copy content Toggle raw display
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