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Properties

Label 384.5.h.h
Level $384$
Weight $5$
Character orbit 384.h
Analytic conductor $39.694$
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] = [384,5,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: \(39.6940658242\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 224 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 224 q^{9} + 5472 q^{25} - 3712 q^{33} + 13664 q^{49} - 17344 q^{57} - 17472 q^{73} - 10976 q^{81} - 39488 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −8.73496 − 2.16806i 0 −14.5048 0 −70.6373 0 71.5990 + 37.8759i 0
65.2 0 −8.73496 − 2.16806i 0 14.5048 0 70.6373 0 71.5990 + 37.8759i 0
65.3 0 −8.73496 + 2.16806i 0 −14.5048 0 −70.6373 0 71.5990 − 37.8759i 0
65.4 0 −8.73496 + 2.16806i 0 14.5048 0 70.6373 0 71.5990 − 37.8759i 0
65.5 0 −8.10449 − 3.91372i 0 −29.9378 0 −5.14305 0 50.3656 + 63.4374i 0
65.6 0 −8.10449 − 3.91372i 0 29.9378 0 5.14305 0 50.3656 + 63.4374i 0
65.7 0 −8.10449 + 3.91372i 0 −29.9378 0 −5.14305 0 50.3656 − 63.4374i 0
65.8 0 −8.10449 + 3.91372i 0 29.9378 0 5.14305 0 50.3656 − 63.4374i 0
65.9 0 −4.86830 − 7.56965i 0 −45.0402 0 68.3576 0 −33.5992 + 73.7027i 0
65.10 0 −4.86830 − 7.56965i 0 45.0402 0 −68.3576 0 −33.5992 + 73.7027i 0
65.11 0 −4.86830 + 7.56965i 0 −45.0402 0 68.3576 0 −33.5992 − 73.7027i 0
65.12 0 −4.86830 + 7.56965i 0 45.0402 0 −68.3576 0 −33.5992 − 73.7027i 0
65.13 0 −3.21206 − 8.40730i 0 −6.97974 0 −40.2885 0 −60.3654 + 54.0095i 0
65.14 0 −3.21206 − 8.40730i 0 6.97974 0 40.2885 0 −60.3654 + 54.0095i 0
65.15 0 −3.21206 + 8.40730i 0 −6.97974 0 −40.2885 0 −60.3654 − 54.0095i 0
65.16 0 −3.21206 + 8.40730i 0 6.97974 0 40.2885 0 −60.3654 − 54.0095i 0
65.17 0 3.21206 − 8.40730i 0 −6.97974 0 40.2885 0 −60.3654 − 54.0095i 0
65.18 0 3.21206 − 8.40730i 0 6.97974 0 −40.2885 0 −60.3654 − 54.0095i 0
65.19 0 3.21206 + 8.40730i 0 −6.97974 0 40.2885 0 −60.3654 + 54.0095i 0
65.20 0 3.21206 + 8.40730i 0 6.97974 0 −40.2885 0 −60.3654 + 54.0095i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.32
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 384.5.h.h ✓ 32
3.b odd 2 1 inner 384.5.h.h ✓ 32
4.b odd 2 1 inner 384.5.h.h ✓ 32
8.b even 2 1 inner 384.5.h.h ✓ 32
8.d odd 2 1 inner 384.5.h.h ✓ 32
12.b even 2 1 inner 384.5.h.h ✓ 32
24.f even 2 1 inner 384.5.h.h ✓ 32
24.h odd 2 1 inner 384.5.h.h ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.h.h ✓ 32 1.a even 1 1 trivial
384.5.h.h ✓ 32 3.b odd 2 1 inner
384.5.h.h ✓ 32 4.b odd 2 1 inner
384.5.h.h ✓ 32 8.b even 2 1 inner
384.5.h.h ✓ 32 8.d odd 2 1 inner
384.5.h.h ✓ 32 12.b even 2 1 inner
384.5.h.h ✓ 32 24.f even 2 1 inner
384.5.h.h ✓ 32 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{8} - 3184T_{5}^{6} + 2586304T_{5}^{4} - 501083136T_{5}^{2} + 18635526144 \) Copy content Toggle raw display
\( T_{11}^{8} - 64768T_{11}^{6} + 1284243808T_{11}^{4} - 7206386503680T_{11}^{2} + 1389563396815104 \) Copy content Toggle raw display