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Properties

Label 544.4.h.a
Level $544$
Weight $4$
Character orbit 544.h
Analytic conductor $32.097$
Analytic rank $0$
Dimension $52$
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] = [544,4,Mod(305,544)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(544, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("544.305");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 544.h (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: \(32.0970390431\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: no (minimal twist has level 136)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 428 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 428 q^{9} + 232 q^{15} - 28 q^{17} + 1052 q^{25} - 344 q^{33} - 936 q^{47} - 1964 q^{49} + 1384 q^{55} + 1884 q^{81} - 1976 q^{87} - 224 q^{89}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1 0 −9.79402 0 −7.09681 0 25.9378i 0 68.9229 0
305.2 0 −9.79402 0 −7.09681 0 − 25.9378i 0 68.9229 0
305.3 0 −9.08790 0 10.4971 0 − 8.08718i 0 55.5899 0
305.4 0 −9.08790 0 10.4971 0 8.08718i 0 55.5899 0
305.5 0 −7.65507 0 −9.38706 0 6.18241i 0 31.6001 0
305.6 0 −7.65507 0 −9.38706 0 − 6.18241i 0 31.6001 0
305.7 0 −7.41202 0 −21.3965 0 − 20.9219i 0 27.9380 0
305.8 0 −7.41202 0 −21.3965 0 20.9219i 0 27.9380 0
305.9 0 −6.83091 0 11.3915 0 − 3.09707i 0 19.6614 0
305.10 0 −6.83091 0 11.3915 0 3.09707i 0 19.6614 0
305.11 0 −6.38011 0 16.9716 0 31.4494i 0 13.7058 0
305.12 0 −6.38011 0 16.9716 0 − 31.4494i 0 13.7058 0
305.13 0 −5.95403 0 0.689457 0 − 29.5114i 0 8.45048 0
305.14 0 −5.95403 0 0.689457 0 29.5114i 0 8.45048 0
305.15 0 −3.96792 0 −5.58507 0 − 3.60918i 0 −11.2556 0
305.16 0 −3.96792 0 −5.58507 0 3.60918i 0 −11.2556 0
305.17 0 −3.72680 0 −7.30821 0 5.81849i 0 −13.1110 0
305.18 0 −3.72680 0 −7.30821 0 − 5.81849i 0 −13.1110 0
305.19 0 −2.90587 0 −7.76266 0 − 28.9758i 0 −18.5559 0
305.20 0 −2.90587 0 −7.76266 0 28.9758i 0 −18.5559 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.b even 2 1 inner
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.4.h.a 52
4.b odd 2 1 136.4.h.a ✓ 52
8.b even 2 1 inner 544.4.h.a 52
8.d odd 2 1 136.4.h.a ✓ 52
17.b even 2 1 inner 544.4.h.a 52
68.d odd 2 1 136.4.h.a ✓ 52
136.e odd 2 1 136.4.h.a ✓ 52
136.h even 2 1 inner 544.4.h.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.h.a ✓ 52 4.b odd 2 1
136.4.h.a ✓ 52 8.d odd 2 1
136.4.h.a ✓ 52 68.d odd 2 1
136.4.h.a ✓ 52 136.e odd 2 1
544.4.h.a 52 1.a even 1 1 trivial
544.4.h.a 52 8.b even 2 1 inner
544.4.h.a 52 17.b even 2 1 inner
544.4.h.a 52 136.h even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(544, [\chi])\).