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Properties

Label 338.8.a.d
Level 338338
Weight 88
Character orbit 338.a
Self dual yes
Analytic conductor 105.586105.586
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] = [338,8,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 338=2⋅132 338 = 2 \cdot 13^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 338.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: 105.586138614105.586138614
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 2)
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+12q3+64q4+210q5+96q6−1016q7+512q8−2043q9+1680q10−1092q11+768q12−8128q14+2520q15+4096q16+14706q17−16344q18+⋯+2230956q99+O(q100) q + 8 q^{2} + 12 q^{3} + 64 q^{4} + 210 q^{5} + 96 q^{6} - 1016 q^{7} + 512 q^{8} - 2043 q^{9} + 1680 q^{10} - 1092 q^{11} + 768 q^{12} - 8128 q^{14} + 2520 q^{15} + 4096 q^{16} + 14706 q^{17} - 16344 q^{18}+ \cdots + 2230956 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 12.0000 64.0000 210.000 96.0000 −1016.00 512.000 −2043.00 1680.00
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 −1 -1
1313 +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 338.8.a.d 1
13.b even 2 1 2.8.a.a ✓ 1
13.d odd 4 2 338.8.b.d 2
39.d odd 2 1 18.8.a.b 1
52.b odd 2 1 16.8.a.b 1
65.d even 2 1 50.8.a.g 1
65.h odd 4 2 50.8.b.c 2
91.b odd 2 1 98.8.a.a 1
91.r even 6 2 98.8.c.d 2
91.s odd 6 2 98.8.c.e 2
104.e even 2 1 64.8.a.c 1
104.h odd 2 1 64.8.a.e 1
117.n odd 6 2 162.8.c.a 2
117.t even 6 2 162.8.c.l 2
143.d odd 2 1 242.8.a.e 1
156.h even 2 1 144.8.a.i 1
195.e odd 2 1 450.8.a.c 1
195.s even 4 2 450.8.c.g 2
208.o odd 4 2 256.8.b.f 2
208.p even 4 2 256.8.b.b 2
221.b even 2 1 578.8.a.b 1
260.g odd 2 1 400.8.a.l 1
260.p even 4 2 400.8.c.j 2
312.b odd 2 1 576.8.a.g 1
312.h even 2 1 576.8.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a ✓ 1 13.b even 2 1
16.8.a.b 1 52.b odd 2 1
18.8.a.b 1 39.d odd 2 1
50.8.a.g 1 65.d even 2 1
50.8.b.c 2 65.h odd 4 2
64.8.a.c 1 104.e even 2 1
64.8.a.e 1 104.h odd 2 1
98.8.a.a 1 91.b odd 2 1
98.8.c.d 2 91.r even 6 2
98.8.c.e 2 91.s odd 6 2
144.8.a.i 1 156.h even 2 1
162.8.c.a 2 117.n odd 6 2
162.8.c.l 2 117.t even 6 2
242.8.a.e 1 143.d odd 2 1
256.8.b.b 2 208.p even 4 2
256.8.b.f 2 208.o odd 4 2
338.8.a.d 1 1.a even 1 1 trivial
338.8.b.d 2 13.d odd 4 2
400.8.a.l 1 260.g odd 2 1
400.8.c.j 2 260.p even 4 2
450.8.a.c 1 195.e odd 2 1
450.8.c.g 2 195.s even 4 2
576.8.a.f 1 312.h even 2 1
576.8.a.g 1 312.b odd 2 1
578.8.a.b 1 221.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S8new(Γ0(338))S_{8}^{\mathrm{new}}(\Gamma_0(338)):

T3−12 T_{3} - 12 Copy content Toggle raw display
T5−210 T_{5} - 210 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−12 T - 12 Copy content Toggle raw display
55 T−210 T - 210 Copy content Toggle raw display
77 T+1016 T + 1016 Copy content Toggle raw display
1111 T+1092 T + 1092 Copy content Toggle raw display
1313 T T Copy content Toggle raw display
1717 T−14706 T - 14706 Copy content Toggle raw display
1919 T−39940 T - 39940 Copy content Toggle raw display
2323 T−68712 T - 68712 Copy content Toggle raw display
2929 T+102570 T + 102570 Copy content Toggle raw display
3131 T+227552 T + 227552 Copy content Toggle raw display
3737 T+160526 T + 160526 Copy content Toggle raw display
4141 T+10842 T + 10842 Copy content Toggle raw display
4343 T+630748 T + 630748 Copy content Toggle raw display
4747 T+472656 T + 472656 Copy content Toggle raw display
5353 T+1494018 T + 1494018 Copy content Toggle raw display
5959 T+2640660 T + 2640660 Copy content Toggle raw display
6161 T−827702 T - 827702 Copy content Toggle raw display
6767 T−126004 T - 126004 Copy content Toggle raw display
7171 T−1414728 T - 1414728 Copy content Toggle raw display
7373 T+980282 T + 980282 Copy content Toggle raw display
7979 T+3566800 T + 3566800 Copy content Toggle raw display
8383 T+5672892 T + 5672892 Copy content Toggle raw display
8989 T−11951190 T - 11951190 Copy content Toggle raw display
9797 T+8682146 T + 8682146 Copy content Toggle raw display
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