↑

Properties

Label 192.4.a.c
Level 192192
Weight 44
Character orbit 192.a
Self dual yes
Analytic conductor 11.32811.328
Analytic rank 11
Dimension 11
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,4,Mod(1,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 192=26⋅3 192 = 2^{6} \cdot 3
Weight: k k == 4 4
Character orbit: [χ][\chi] == 192.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: 11.328366721111.3283667211
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 6)
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−3q3−6q5+16q7+9q9+12q11−38q13+18q15−126q17+20q19−48q21−168q23−89q25−27q27−30q29+88q31−36q33−96q35+⋯+108q99+O(q100) q - 3 q^{3} - 6 q^{5} + 16 q^{7} + 9 q^{9} + 12 q^{11} - 38 q^{13} + 18 q^{15} - 126 q^{17} + 20 q^{19} - 48 q^{21} - 168 q^{23} - 89 q^{25} - 27 q^{27} - 30 q^{29} + 88 q^{31} - 36 q^{33} - 96 q^{35}+ \cdots + 108 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
0 −3.00000 0 −6.00000 0 16.0000 0 9.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 −1 -1
33 +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 192.4.a.c 1
3.b odd 2 1 576.4.a.r 1
4.b odd 2 1 192.4.a.i 1
8.b even 2 1 48.4.a.c 1
8.d odd 2 1 6.4.a.a ✓ 1
12.b even 2 1 576.4.a.q 1
16.e even 4 2 768.4.d.c 2
16.f odd 4 2 768.4.d.n 2
24.f even 2 1 18.4.a.a 1
24.h odd 2 1 144.4.a.c 1
40.e odd 2 1 150.4.a.i 1
40.f even 2 1 1200.4.a.b 1
40.i odd 4 2 1200.4.f.j 2
40.k even 4 2 150.4.c.d 2
56.e even 2 1 294.4.a.e 1
56.h odd 2 1 2352.4.a.e 1
56.k odd 6 2 294.4.e.h 2
56.m even 6 2 294.4.e.g 2
72.l even 6 2 162.4.c.c 2
72.p odd 6 2 162.4.c.f 2
88.g even 2 1 726.4.a.f 1
104.h odd 2 1 1014.4.a.g 1
104.m even 4 2 1014.4.b.d 2
120.m even 2 1 450.4.a.h 1
120.q odd 4 2 450.4.c.e 2
136.e odd 2 1 1734.4.a.d 1
152.b even 2 1 2166.4.a.i 1
168.e odd 2 1 882.4.a.n 1
168.v even 6 2 882.4.g.i 2
168.be odd 6 2 882.4.g.f 2
264.p odd 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a ✓ 1 8.d odd 2 1
18.4.a.a 1 24.f even 2 1
48.4.a.c 1 8.b even 2 1
144.4.a.c 1 24.h odd 2 1
150.4.a.i 1 40.e odd 2 1
150.4.c.d 2 40.k even 4 2
162.4.c.c 2 72.l even 6 2
162.4.c.f 2 72.p odd 6 2
192.4.a.c 1 1.a even 1 1 trivial
192.4.a.i 1 4.b odd 2 1
294.4.a.e 1 56.e even 2 1
294.4.e.g 2 56.m even 6 2
294.4.e.h 2 56.k odd 6 2
450.4.a.h 1 120.m even 2 1
450.4.c.e 2 120.q odd 4 2
576.4.a.q 1 12.b even 2 1
576.4.a.r 1 3.b odd 2 1
726.4.a.f 1 88.g even 2 1
768.4.d.c 2 16.e even 4 2
768.4.d.n 2 16.f odd 4 2
882.4.a.n 1 168.e odd 2 1
882.4.g.f 2 168.be odd 6 2
882.4.g.i 2 168.v even 6 2
1014.4.a.g 1 104.h odd 2 1
1014.4.b.d 2 104.m even 4 2
1200.4.a.b 1 40.f even 2 1
1200.4.f.j 2 40.i odd 4 2
1734.4.a.d 1 136.e odd 2 1
2166.4.a.i 1 152.b even 2 1
2178.4.a.e 1 264.p odd 2 1
2352.4.a.e 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(192))S_{4}^{\mathrm{new}}(\Gamma_0(192)):

T5+6 T_{5} + 6 Copy content Toggle raw display
T7−16 T_{7} - 16 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T T Copy content Toggle raw display
33 T+3 T + 3 Copy content Toggle raw display
55 T+6 T + 6 Copy content Toggle raw display
77 T−16 T - 16 Copy content Toggle raw display
1111 T−12 T - 12 Copy content Toggle raw display
1313 T+38 T + 38 Copy content Toggle raw display
1717 T+126 T + 126 Copy content Toggle raw display
1919 T−20 T - 20 Copy content Toggle raw display
2323 T+168 T + 168 Copy content Toggle raw display
2929 T+30 T + 30 Copy content Toggle raw display
3131 T−88 T - 88 Copy content Toggle raw display
3737 T+254 T + 254 Copy content Toggle raw display
4141 T−42 T - 42 Copy content Toggle raw display
4343 T+52 T + 52 Copy content Toggle raw display
4747 T−96 T - 96 Copy content Toggle raw display
5353 T+198 T + 198 Copy content Toggle raw display
5959 T+660 T + 660 Copy content Toggle raw display
6161 T−538 T - 538 Copy content Toggle raw display
6767 T−884 T - 884 Copy content Toggle raw display
7171 T+792 T + 792 Copy content Toggle raw display
7373 T−218 T - 218 Copy content Toggle raw display
7979 T−520 T - 520 Copy content Toggle raw display
8383 T+492 T + 492 Copy content Toggle raw display
8989 T−810 T - 810 Copy content Toggle raw display
9797 T−1154 T - 1154 Copy content Toggle raw display
show more
show less