↑

Properties

Label 384.4.j.a
Level $384$
Weight $4$
Character orbit 384.j
Analytic conductor $22.657$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(97,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.97");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.j (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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} - 16 q^{37} + 1240 q^{43} - 1176 q^{49} + 744 q^{51} - 752 q^{53} - 1376 q^{59} + 912 q^{61} + 504 q^{63} + 976 q^{65} - 2256 q^{67} + 528 q^{69} + 1104 q^{75} - 1904 q^{77} - 5992 q^{79} - 1944 q^{81} + 2680 q^{83} + 240 q^{85} - 3496 q^{91} + 7728 q^{95} - 360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −2.12132 + 2.12132i 0 11.7719 + 11.7719i 0 − 14.7089i 0 − 9.00000i 0
97.2 0 −2.12132 + 2.12132i 0 8.83384 + 8.83384i 0 29.4760i 0 − 9.00000i 0
97.3 0 −2.12132 + 2.12132i 0 −3.22588 − 3.22588i 0 − 24.6080i 0 − 9.00000i 0
97.4 0 −2.12132 + 2.12132i 0 −2.24191 − 2.24191i 0 − 9.00196i 0 − 9.00000i 0
97.5 0 −2.12132 + 2.12132i 0 3.72414 + 3.72414i 0 20.2675i 0 − 9.00000i 0
97.6 0 −2.12132 + 2.12132i 0 −11.7911 − 11.7911i 0 12.5754i 0 − 9.00000i 0
97.7 0 2.12132 − 2.12132i 0 11.8955 + 11.8955i 0 0.485059i 0 − 9.00000i 0
97.8 0 2.12132 − 2.12132i 0 7.29121 + 7.29121i 0 22.1610i 0 − 9.00000i 0
97.9 0 2.12132 − 2.12132i 0 −0.644922 − 0.644922i 0 − 7.13926i 0 − 9.00000i 0
97.10 0 2.12132 − 2.12132i 0 −0.706564 − 0.706564i 0 4.44122i 0 − 9.00000i 0
97.11 0 2.12132 − 2.12132i 0 −10.2951 − 10.2951i 0 − 32.8369i 0 − 9.00000i 0
97.12 0 2.12132 − 2.12132i 0 −14.6111 − 14.6111i 0 26.8889i 0 − 9.00000i 0
289.1 0 −2.12132 − 2.12132i 0 11.7719 − 11.7719i 0 14.7089i 0 9.00000i 0
289.2 0 −2.12132 − 2.12132i 0 8.83384 − 8.83384i 0 − 29.4760i 0 9.00000i 0
289.3 0 −2.12132 − 2.12132i 0 −3.22588 + 3.22588i 0 24.6080i 0 9.00000i 0
289.4 0 −2.12132 − 2.12132i 0 −2.24191 + 2.24191i 0 9.00196i 0 9.00000i 0
289.5 0 −2.12132 − 2.12132i 0 3.72414 − 3.72414i 0 − 20.2675i 0 9.00000i 0
289.6 0 −2.12132 − 2.12132i 0 −11.7911 + 11.7911i 0 − 12.5754i 0 9.00000i 0
289.7 0 2.12132 + 2.12132i 0 11.8955 − 11.8955i 0 − 0.485059i 0 9.00000i 0
289.8 0 2.12132 + 2.12132i 0 7.29121 − 7.29121i 0 − 22.1610i 0 9.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.j.a 24
4.b odd 2 1 384.4.j.b 24
8.b even 2 1 192.4.j.a 24
8.d odd 2 1 48.4.j.a ✓ 24
16.e even 4 1 192.4.j.a 24
16.e even 4 1 inner 384.4.j.a 24
16.f odd 4 1 48.4.j.a ✓ 24
16.f odd 4 1 384.4.j.b 24
24.f even 2 1 144.4.k.b 24
24.h odd 2 1 576.4.k.b 24
48.i odd 4 1 576.4.k.b 24
48.k even 4 1 144.4.k.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.j.a ✓ 24 8.d odd 2 1
48.4.j.a ✓ 24 16.f odd 4 1
144.4.k.b 24 24.f even 2 1
144.4.k.b 24 48.k even 4 1
192.4.j.a 24 8.b even 2 1
192.4.j.a 24 16.e even 4 1
384.4.j.a 24 1.a even 1 1 trivial
384.4.j.a 24 16.e even 4 1 inner
384.4.j.b 24 4.b odd 2 1
384.4.j.b 24 16.f odd 4 1
576.4.k.b 24 24.h odd 2 1
576.4.k.b 24 48.i odd 4 1