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Properties

Label 42.8.a.a
Level 4242
Weight 88
Character orbit 42.a
Self dual yes
Analytic conductor 13.12013.120
Analytic rank 00
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] = [42,8,Mod(1,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 42=2⋅3⋅7 42 = 2 \cdot 3 \cdot 7
Weight: k k == 8 8
Character orbit: [χ][\chi] == 42.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: 13.120171070313.1201710703
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: yes
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−8q2−27q3+64q4−410q5+216q6−343q7−512q8+729q9+3280q10−5548q11−1728q12−1698q13+2744q14+11070q15+4096q16+9506q17+⋯−4044492q99+O(q100) q - 8 q^{2} - 27 q^{3} + 64 q^{4} - 410 q^{5} + 216 q^{6} - 343 q^{7} - 512 q^{8} + 729 q^{9} + 3280 q^{10} - 5548 q^{11} - 1728 q^{12} - 1698 q^{13} + 2744 q^{14} + 11070 q^{15} + 4096 q^{16} + 9506 q^{17}+ \cdots - 4044492 q^{99}+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
−8.00000 −27.0000 64.0000 −410.000 216.000 −343.000 −512.000 729.000 3280.00
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
33 +1 +1
77 +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 42.8.a.a ✓ 1
3.b odd 2 1 126.8.a.h 1
4.b odd 2 1 336.8.a.g 1
7.b odd 2 1 294.8.a.j 1
7.c even 3 2 294.8.e.r 2
7.d odd 6 2 294.8.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.8.a.a ✓ 1 1.a even 1 1 trivial
126.8.a.h 1 3.b odd 2 1
294.8.a.j 1 7.b odd 2 1
294.8.e.g 2 7.d odd 6 2
294.8.e.r 2 7.c even 3 2
336.8.a.g 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5+410 T_{5} + 410 acting on S8new(Γ0(42))S_{8}^{\mathrm{new}}(\Gamma_0(42)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T+8 T + 8 Copy content Toggle raw display
33 T+27 T + 27 Copy content Toggle raw display
55 T+410 T + 410 Copy content Toggle raw display
77 T+343 T + 343 Copy content Toggle raw display
1111 T+5548 T + 5548 Copy content Toggle raw display
1313 T+1698 T + 1698 Copy content Toggle raw display
1717 T−9506 T - 9506 Copy content Toggle raw display
1919 T−57620 T - 57620 Copy content Toggle raw display
2323 T−59192 T - 59192 Copy content Toggle raw display
2929 T−27190 T - 27190 Copy content Toggle raw display
3131 T+252288 T + 252288 Copy content Toggle raw display
3737 T−524126 T - 524126 Copy content Toggle raw display
4141 T+463638 T + 463638 Copy content Toggle raw display
4343 T−96212 T - 96212 Copy content Toggle raw display
4747 T+1068864 T + 1068864 Copy content Toggle raw display
5353 T−125262 T - 125262 Copy content Toggle raw display
5959 T−932620 T - 932620 Copy content Toggle raw display
6161 T−546942 T - 546942 Copy content Toggle raw display
6767 T−3472396 T - 3472396 Copy content Toggle raw display
7171 T+33928 T + 33928 Copy content Toggle raw display
7373 T+3640838 T + 3640838 Copy content Toggle raw display
7979 T−4272960 T - 4272960 Copy content Toggle raw display
8383 T−9841172 T - 9841172 Copy content Toggle raw display
8989 T+1681590 T + 1681590 Copy content Toggle raw display
9797 T+10813454 T + 10813454 Copy content Toggle raw display
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