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Properties

Label 192.8.f.d
Level $192$
Weight $8$
Character orbit 192.f
Analytic conductor $59.978$
Analytic rank $0$
Dimension $32$
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] = [192,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.95");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(59.9779248930\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 15360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 15360 q^{9} + 923168 q^{25} - 424800 q^{33} - 6986592 q^{49} - 1707360 q^{57} - 12976960 q^{73} - 57198816 q^{81} - 93958400 q^{97}+O(q^{100}) \) 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 −41.1706 − 22.1806i 0 −190.624 0 895.264i 0 1203.04 + 1826.38i 0
95.2 0 −41.1706 − 22.1806i 0 190.624 0 − 895.264i 0 1203.04 + 1826.38i 0
95.3 0 −41.1706 + 22.1806i 0 −190.624 0 − 895.264i 0 1203.04 − 1826.38i 0
95.4 0 −41.1706 + 22.1806i 0 190.624 0 895.264i 0 1203.04 − 1826.38i 0
95.5 0 −30.6573 − 35.3147i 0 −283.364 0 1485.59i 0 −307.262 + 2165.31i 0
95.6 0 −30.6573 − 35.3147i 0 283.364 0 − 1485.59i 0 −307.262 + 2165.31i 0
95.7 0 −30.6573 + 35.3147i 0 −283.364 0 − 1485.59i 0 −307.262 − 2165.31i 0
95.8 0 −30.6573 + 35.3147i 0 283.364 0 1485.59i 0 −307.262 − 2165.31i 0
95.9 0 −25.2145 − 39.3856i 0 −467.619 0 − 1073.86i 0 −915.456 + 1986.18i 0
95.10 0 −25.2145 − 39.3856i 0 467.619 0 1073.86i 0 −915.456 + 1986.18i 0
95.11 0 −25.2145 + 39.3856i 0 −467.619 0 1073.86i 0 −915.456 − 1986.18i 0
95.12 0 −25.2145 + 39.3856i 0 467.619 0 − 1073.86i 0 −915.456 − 1986.18i 0
95.13 0 −11.9724 − 45.2069i 0 −304.295 0 76.4703i 0 −1900.32 + 1082.47i 0
95.14 0 −11.9724 − 45.2069i 0 304.295 0 − 76.4703i 0 −1900.32 + 1082.47i 0
95.15 0 −11.9724 + 45.2069i 0 −304.295 0 − 76.4703i 0 −1900.32 − 1082.47i 0
95.16 0 −11.9724 + 45.2069i 0 304.295 0 76.4703i 0 −1900.32 − 1082.47i 0
95.17 0 11.9724 − 45.2069i 0 −304.295 0 76.4703i 0 −1900.32 − 1082.47i 0
95.18 0 11.9724 − 45.2069i 0 304.295 0 − 76.4703i 0 −1900.32 − 1082.47i 0
95.19 0 11.9724 + 45.2069i 0 −304.295 0 − 76.4703i 0 −1900.32 + 1082.47i 0
95.20 0 11.9724 + 45.2069i 0 304.295 0 76.4703i 0 −1900.32 + 1082.47i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.32
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 192.8.f.d ✓ 32
3.b odd 2 1 inner 192.8.f.d ✓ 32
4.b odd 2 1 inner 192.8.f.d ✓ 32
8.b even 2 1 inner 192.8.f.d ✓ 32
8.d odd 2 1 inner 192.8.f.d ✓ 32
12.b even 2 1 inner 192.8.f.d ✓ 32
24.f even 2 1 inner 192.8.f.d ✓ 32
24.h odd 2 1 inner 192.8.f.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.8.f.d ✓ 32 1.a even 1 1 trivial
192.8.f.d ✓ 32 3.b odd 2 1 inner
192.8.f.d ✓ 32 4.b odd 2 1 inner
192.8.f.d ✓ 32 8.b even 2 1 inner
192.8.f.d ✓ 32 8.d odd 2 1 inner
192.8.f.d ✓ 32 12.b even 2 1 inner
192.8.f.d ✓ 32 24.f even 2 1 inner
192.8.f.d ✓ 32 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 427896T_{5}^{6} + 59468913936T_{5}^{4} - 3269728171929600T_{5}^{2} + 59077437362012160000 \) acting on \(S_{8}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display