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Properties

Label 192.10.f.d
Level $192$
Weight $10$
Character orbit 192.f
Analytic conductor $98.887$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,10,Mod(95,192)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.95");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 192.f (of order \(2\), degree \(1\), 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: \(98.8868805435\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36864 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36864 q^{9} + 16263984 q^{25} - 17348496 q^{33} - 236733840 q^{49} + 22572912 q^{57} + 210002592 q^{73} - 666528336 q^{81} - 5058961536 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −133.007 − 44.6342i 0 −1420.93 0 − 268.525i 0 15698.6 + 11873.3i 0
95.2 0 −133.007 − 44.6342i 0 1420.93 0 268.525i 0 15698.6 + 11873.3i 0
95.3 0 −133.007 + 44.6342i 0 −1420.93 0 268.525i 0 15698.6 − 11873.3i 0
95.4 0 −133.007 + 44.6342i 0 1420.93 0 − 268.525i 0 15698.6 − 11873.3i 0
95.5 0 −127.027 − 59.5579i 0 −2365.68 0 8208.88i 0 12588.7 + 15130.9i 0
95.6 0 −127.027 − 59.5579i 0 2365.68 0 − 8208.88i 0 12588.7 + 15130.9i 0
95.7 0 −127.027 + 59.5579i 0 −2365.68 0 − 8208.88i 0 12588.7 − 15130.9i 0
95.8 0 −127.027 + 59.5579i 0 2365.68 0 8208.88i 0 12588.7 − 15130.9i 0
95.9 0 −120.807 − 71.3357i 0 −275.654 0 3902.31i 0 9505.43 + 17235.6i 0
95.10 0 −120.807 − 71.3357i 0 275.654 0 − 3902.31i 0 9505.43 + 17235.6i 0
95.11 0 −120.807 + 71.3357i 0 −275.654 0 − 3902.31i 0 9505.43 − 17235.6i 0
95.12 0 −120.807 + 71.3357i 0 275.654 0 3902.31i 0 9505.43 − 17235.6i 0
95.13 0 −68.3868 − 122.500i 0 −2066.50 0 − 8585.10i 0 −10329.5 + 16754.8i 0
95.14 0 −68.3868 − 122.500i 0 2066.50 0 8585.10i 0 −10329.5 + 16754.8i 0
95.15 0 −68.3868 + 122.500i 0 −2066.50 0 8585.10i 0 −10329.5 − 16754.8i 0
95.16 0 −68.3868 + 122.500i 0 2066.50 0 − 8585.10i 0 −10329.5 − 16754.8i 0
95.17 0 −50.6063 − 130.851i 0 −1332.54 0 5391.42i 0 −14561.0 + 13243.8i 0
95.18 0 −50.6063 − 130.851i 0 1332.54 0 − 5391.42i 0 −14561.0 + 13243.8i 0
95.19 0 −50.6063 + 130.851i 0 −1332.54 0 − 5391.42i 0 −14561.0 − 13243.8i 0
95.20 0 −50.6063 + 130.851i 0 1332.54 0 5391.42i 0 −14561.0 − 13243.8i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.10.f.d ✓ 48
3.b odd 2 1 inner 192.10.f.d ✓ 48
4.b odd 2 1 inner 192.10.f.d ✓ 48
8.b even 2 1 inner 192.10.f.d ✓ 48
8.d odd 2 1 inner 192.10.f.d ✓ 48
12.b even 2 1 inner 192.10.f.d ✓ 48
24.f even 2 1 inner 192.10.f.d ✓ 48
24.h odd 2 1 inner 192.10.f.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.10.f.d ✓ 48 1.a even 1 1 trivial
192.10.f.d ✓ 48 3.b odd 2 1 inner
192.10.f.d ✓ 48 4.b odd 2 1 inner
192.10.f.d ✓ 48 8.b even 2 1 inner
192.10.f.d ✓ 48 8.d odd 2 1 inner
192.10.f.d ✓ 48 12.b even 2 1 inner
192.10.f.d ✓ 48 24.f even 2 1 inner
192.10.f.d ✓ 48 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 13751748 T_{5}^{10} + 66159185308848 T_{5}^{8} + \cdots + 92\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display