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Properties

Label 192.9.l.a
Level $192$
Weight $9$
Character orbit 192.l
Analytic conductor $78.217$
Analytic rank $0$
Dimension $64$
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] = [192,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.79");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 39552 q^{11} + 167552 q^{19} - 1691136 q^{23} - 2132352 q^{29} + 2415744 q^{35} - 4720512 q^{37} + 7244672 q^{43} + 52706752 q^{49} - 13862016 q^{51} - 5358720 q^{53} + 46326784 q^{55} - 44938752 q^{59} + 24476032 q^{61} + 29941632 q^{65} + 44244736 q^{67} - 8636544 q^{69} - 159664128 q^{71} - 12918528 q^{75} - 94964352 q^{77} - 306110016 q^{81} - 209328000 q^{83} + 106960000 q^{85} + 45401472 q^{91} - 86500224 q^{99}+O(q^{100}) \) 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 −33.0681 − 33.0681i 0 −825.217 − 825.217i 0 −1803.45 0 2187.00i 0
79.2 0 −33.0681 − 33.0681i 0 −633.321 − 633.321i 0 416.943 0 2187.00i 0
79.3 0 −33.0681 − 33.0681i 0 −642.261 − 642.261i 0 3321.99 0 2187.00i 0
79.4 0 −33.0681 − 33.0681i 0 −488.559 − 488.559i 0 90.0632 0 2187.00i 0
79.5 0 −33.0681 − 33.0681i 0 −384.834 − 384.834i 0 −1641.15 0 2187.00i 0
79.6 0 −33.0681 − 33.0681i 0 306.323 + 306.323i 0 −4330.12 0 2187.00i 0
79.7 0 −33.0681 − 33.0681i 0 −171.868 − 171.868i 0 1880.59 0 2187.00i 0
79.8 0 −33.0681 − 33.0681i 0 −173.172 − 173.172i 0 −3867.21 0 2187.00i 0
79.9 0 −33.0681 − 33.0681i 0 135.282 + 135.282i 0 2730.70 0 2187.00i 0
79.10 0 −33.0681 − 33.0681i 0 −24.8301 − 24.8301i 0 −2222.06 0 2187.00i 0
79.11 0 −33.0681 − 33.0681i 0 47.4496 + 47.4496i 0 2246.74 0 2187.00i 0
79.12 0 −33.0681 − 33.0681i 0 508.668 + 508.668i 0 3253.34 0 2187.00i 0
79.13 0 −33.0681 − 33.0681i 0 531.627 + 531.627i 0 3207.17 0 2187.00i 0
79.14 0 −33.0681 − 33.0681i 0 541.894 + 541.894i 0 −1146.54 0 2187.00i 0
79.15 0 −33.0681 − 33.0681i 0 625.993 + 625.993i 0 −873.478 0 2187.00i 0
79.16 0 −33.0681 − 33.0681i 0 646.826 + 646.826i 0 −1263.53 0 2187.00i 0
79.17 0 33.0681 + 33.0681i 0 −851.062 − 851.062i 0 2002.62 0 2187.00i 0
79.18 0 33.0681 + 33.0681i 0 −679.500 − 679.500i 0 −4528.78 0 2187.00i 0
79.19 0 33.0681 + 33.0681i 0 −500.017 − 500.017i 0 988.562 0 2187.00i 0
79.20 0 33.0681 + 33.0681i 0 −341.571 − 341.571i 0 −1351.24 0 2187.00i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.32
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 192.9.l.a 64
4.b odd 2 1 48.9.l.a ✓ 64
16.e even 4 1 48.9.l.a ✓ 64
16.f odd 4 1 inner 192.9.l.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.l.a ✓ 64 4.b odd 2 1
48.9.l.a ✓ 64 16.e even 4 1
192.9.l.a 64 1.a even 1 1 trivial
192.9.l.a 64 16.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(192, [\chi])\).