Properties

Label 192.8.c.f
Level $192$
Weight $8$
Character orbit 192.c
Analytic conductor $59.978$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,8,Mod(191,192)]
 
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, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.191");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.c (of order \(2\), degree \(1\), 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: \(59.9779248930\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 96)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2236 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2236 q^{9} + 7064 q^{13} - 2344 q^{21} - 417012 q^{25} + 278672 q^{33} + 965112 q^{37} + 2114464 q^{45} - 4821612 q^{49} - 3683352 q^{57} - 402664 q^{61} + 1987424 q^{69} + 223128 q^{73} - 6688900 q^{81} - 7477248 q^{85} - 34969576 q^{93} + 12115480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −45.5869 10.4324i 0 183.679i 0 1238.80i 0 1969.33 + 951.162i 0
191.2 0 −45.5869 + 10.4324i 0 183.679i 0 1238.80i 0 1969.33 951.162i 0
191.3 0 −43.8532 16.2450i 0 235.987i 0 481.583i 0 1659.20 + 1424.79i 0
191.4 0 −43.8532 + 16.2450i 0 235.987i 0 481.583i 0 1659.20 1424.79i 0
191.5 0 −39.5690 24.9257i 0 508.011i 0 1121.87i 0 944.419 + 1972.57i 0
191.6 0 −39.5690 + 24.9257i 0 508.011i 0 1121.87i 0 944.419 1972.57i 0
191.7 0 −36.3965 29.3649i 0 163.559i 0 284.735i 0 462.403 + 2137.56i 0
191.8 0 −36.3965 + 29.3649i 0 163.559i 0 284.735i 0 462.403 2137.56i 0
191.9 0 −28.9719 36.7101i 0 331.036i 0 1721.03i 0 −508.262 + 2127.12i 0
191.10 0 −28.9719 + 36.7101i 0 331.036i 0 1721.03i 0 −508.262 2127.12i 0
191.11 0 −12.9879 44.9256i 0 361.105i 0 928.878i 0 −1849.63 + 1166.98i 0
191.12 0 −12.9879 + 44.9256i 0 361.105i 0 928.878i 0 −1849.63 1166.98i 0
191.13 0 −5.85395 46.3975i 0 192.070i 0 198.065i 0 −2118.46 + 543.217i 0
191.14 0 −5.85395 + 46.3975i 0 192.070i 0 198.065i 0 −2118.46 543.217i 0
191.15 0 5.85395 46.3975i 0 192.070i 0 198.065i 0 −2118.46 543.217i 0
191.16 0 5.85395 + 46.3975i 0 192.070i 0 198.065i 0 −2118.46 + 543.217i 0
191.17 0 12.9879 44.9256i 0 361.105i 0 928.878i 0 −1849.63 1166.98i 0
191.18 0 12.9879 + 44.9256i 0 361.105i 0 928.878i 0 −1849.63 + 1166.98i 0
191.19 0 28.9719 36.7101i 0 331.036i 0 1721.03i 0 −508.262 2127.12i 0
191.20 0 28.9719 + 36.7101i 0 331.036i 0 1721.03i 0 −508.262 + 2127.12i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.28
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.8.c.f 28
3.b odd 2 1 inner 192.8.c.f 28
4.b odd 2 1 inner 192.8.c.f 28
8.b even 2 1 96.8.c.a 28
8.d odd 2 1 96.8.c.a 28
12.b even 2 1 inner 192.8.c.f 28
24.f even 2 1 96.8.c.a 28
24.h odd 2 1 96.8.c.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.8.c.a 28 8.b even 2 1
96.8.c.a 28 8.d odd 2 1
96.8.c.a 28 24.f even 2 1
96.8.c.a 28 24.h odd 2 1
192.8.c.f 28 1.a even 1 1 trivial
192.8.c.f 28 3.b odd 2 1 inner
192.8.c.f 28 4.b odd 2 1 inner
192.8.c.f 28 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 651128 T_{5}^{12} + 161018125632 T_{5}^{10} + \cdots + 68\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display