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Properties

Label 192.7.l.a
Level $192$
Weight $7$
Character orbit 192.l
Analytic conductor $44.170$
Analytic rank $0$
Dimension $48$
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] = [192,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.79");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(44.1703840550\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2720 q^{11} + 3936 q^{19} - 26240 q^{23} + 66400 q^{29} - 162336 q^{35} - 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} + 443680 q^{53} - 232704 q^{55} + 886144 q^{59} - 326496 q^{61} - 372832 q^{65} + 962112 q^{67} + 541728 q^{69} - 534016 q^{71} + 1073088 q^{75} - 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} - 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) \) 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 −11.0227 − 11.0227i 0 −159.596 − 159.596i 0 527.602 0 243.000i 0
79.2 0 −11.0227 − 11.0227i 0 −124.587 − 124.587i 0 −200.172 0 243.000i 0
79.3 0 −11.0227 − 11.0227i 0 −73.3870 − 73.3870i 0 −291.401 0 243.000i 0
79.4 0 −11.0227 − 11.0227i 0 −26.8357 − 26.8357i 0 81.4485 0 243.000i 0
79.5 0 −11.0227 − 11.0227i 0 −26.6395 − 26.6395i 0 403.840 0 243.000i 0
79.6 0 −11.0227 − 11.0227i 0 38.1198 + 38.1198i 0 −34.3159 0 243.000i 0
79.7 0 −11.0227 − 11.0227i 0 −9.26689 − 9.26689i 0 320.373 0 243.000i 0
79.8 0 −11.0227 − 11.0227i 0 47.0845 + 47.0845i 0 −400.703 0 243.000i 0
79.9 0 −11.0227 − 11.0227i 0 −99.4317 − 99.4317i 0 −407.926 0 243.000i 0
79.10 0 −11.0227 − 11.0227i 0 132.791 + 132.791i 0 560.492 0 243.000i 0
79.11 0 −11.0227 − 11.0227i 0 145.472 + 145.472i 0 −535.877 0 243.000i 0
79.12 0 −11.0227 − 11.0227i 0 156.277 + 156.277i 0 −23.3623 0 243.000i 0
79.13 0 11.0227 + 11.0227i 0 −160.484 − 160.484i 0 53.5181 0 243.000i 0
79.14 0 11.0227 + 11.0227i 0 128.299 + 128.299i 0 −76.4178 0 243.000i 0
79.15 0 11.0227 + 11.0227i 0 −98.7574 − 98.7574i 0 218.381 0 243.000i 0
79.16 0 11.0227 + 11.0227i 0 −95.7535 − 95.7535i 0 −338.697 0 243.000i 0
79.17 0 11.0227 + 11.0227i 0 −59.6079 − 59.6079i 0 661.000 0 243.000i 0
79.18 0 11.0227 + 11.0227i 0 45.7781 + 45.7781i 0 565.544 0 243.000i 0
79.19 0 11.0227 + 11.0227i 0 −55.7840 − 55.7840i 0 −496.753 0 243.000i 0
79.20 0 11.0227 + 11.0227i 0 29.4894 + 29.4894i 0 −461.206 0 243.000i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.24
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 192.7.l.a 48
4.b odd 2 1 48.7.l.a ✓ 48
8.b even 2 1 384.7.l.a 48
8.d odd 2 1 384.7.l.b 48
16.e even 4 1 48.7.l.a ✓ 48
16.e even 4 1 384.7.l.b 48
16.f odd 4 1 inner 192.7.l.a 48
16.f odd 4 1 384.7.l.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.l.a ✓ 48 4.b odd 2 1
48.7.l.a ✓ 48 16.e even 4 1
192.7.l.a 48 1.a even 1 1 trivial
192.7.l.a 48 16.f odd 4 1 inner
384.7.l.a 48 8.b even 2 1
384.7.l.a 48 16.f odd 4 1
384.7.l.b 48 8.d odd 2 1
384.7.l.b 48 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(192, [\chi])\).