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Properties

Label 192.5.l.a
Level $192$
Weight $5$
Character orbit 192.l
Analytic conductor $19.847$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,5,Mod(79,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.79");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 192.l (of order \(4\), degree \(2\), not 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: \(19.8470329121\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 192 q^{11} - 704 q^{19} + 2304 q^{23} - 1728 q^{29} + 5184 q^{35} + 3648 q^{37} - 1088 q^{43} + 10976 q^{49} + 4032 q^{51} + 960 q^{53} - 11776 q^{55} - 13056 q^{59} + 3776 q^{61} + 4032 q^{65} + 896 q^{67} - 9792 q^{69} + 39936 q^{71} + 1152 q^{75} + 9408 q^{77} - 23328 q^{81} - 24000 q^{83} - 11200 q^{85} - 30528 q^{91} + 5184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 0 −3.67423 − 3.67423i 0 17.5812 + 17.5812i 0 −50.9945 0 27.0000i 0
79.2 0 −3.67423 − 3.67423i 0 −17.3646 − 17.3646i 0 −89.8255 0 27.0000i 0
79.3 0 −3.67423 − 3.67423i 0 5.17564 + 5.17564i 0 −6.77541 0 27.0000i 0
79.4 0 −3.67423 − 3.67423i 0 −7.37831 − 7.37831i 0 75.1984 0 27.0000i 0
79.5 0 −3.67423 − 3.67423i 0 −21.2428 − 21.2428i 0 −35.6927 0 27.0000i 0
79.6 0 −3.67423 − 3.67423i 0 −25.9147 − 25.9147i 0 48.6273 0 27.0000i 0
79.7 0 −3.67423 − 3.67423i 0 21.6601 + 21.6601i 0 −15.5177 0 27.0000i 0
79.8 0 −3.67423 − 3.67423i 0 27.4836 + 27.4836i 0 74.9802 0 27.0000i 0
79.9 0 3.67423 + 3.67423i 0 13.5312 + 13.5312i 0 −44.0276 0 27.0000i 0
79.10 0 3.67423 + 3.67423i 0 −0.419719 − 0.419719i 0 40.4181 0 27.0000i 0
79.11 0 3.67423 + 3.67423i 0 −2.10268 − 2.10268i 0 −84.8276 0 27.0000i 0
79.12 0 3.67423 + 3.67423i 0 −9.21870 − 9.21870i 0 −17.0639 0 27.0000i 0
79.13 0 3.67423 + 3.67423i 0 −15.8310 − 15.8310i 0 14.0988 0 27.0000i 0
79.14 0 3.67423 + 3.67423i 0 18.4189 + 18.4189i 0 −4.95460 0 27.0000i 0
79.15 0 3.67423 + 3.67423i 0 30.1272 + 30.1272i 0 80.6454 0 27.0000i 0
79.16 0 3.67423 + 3.67423i 0 −34.5052 − 34.5052i 0 15.7115 0 27.0000i 0
175.1 0 −3.67423 + 3.67423i 0 17.5812 − 17.5812i 0 −50.9945 0 − 27.0000i 0
175.2 0 −3.67423 + 3.67423i 0 −17.3646 + 17.3646i 0 −89.8255 0 − 27.0000i 0
175.3 0 −3.67423 + 3.67423i 0 5.17564 − 5.17564i 0 −6.77541 0 − 27.0000i 0
175.4 0 −3.67423 + 3.67423i 0 −7.37831 + 7.37831i 0 75.1984 0 − 27.0000i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.5.l.a 32
3.b odd 2 1 576.5.m.b 32
4.b odd 2 1 48.5.l.a ✓ 32
8.b even 2 1 384.5.l.a 32
8.d odd 2 1 384.5.l.b 32
12.b even 2 1 144.5.m.c 32
16.e even 4 1 48.5.l.a ✓ 32
16.e even 4 1 384.5.l.b 32
16.f odd 4 1 inner 192.5.l.a 32
16.f odd 4 1 384.5.l.a 32
48.i odd 4 1 144.5.m.c 32
48.k even 4 1 576.5.m.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.5.l.a ✓ 32 4.b odd 2 1
48.5.l.a ✓ 32 16.e even 4 1
144.5.m.c 32 12.b even 2 1
144.5.m.c 32 48.i odd 4 1
192.5.l.a 32 1.a even 1 1 trivial
192.5.l.a 32 16.f odd 4 1 inner
384.5.l.a 32 8.b even 2 1
384.5.l.a 32 16.f odd 4 1
384.5.l.b 32 8.d odd 2 1
384.5.l.b 32 16.e even 4 1
576.5.m.b 32 3.b odd 2 1
576.5.m.b 32 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(192, [\chi])\).