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Properties

Label 550.2.bh.b
Level $550$
Weight $2$
Character orbit 550.bh
Analytic conductor $4.392$
Analytic rank $0$
Dimension $48$
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] = [550,2,Mod(7,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.bh (of order \(20\), degree \(8\), 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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 20 q^{7} + 12 q^{11} + 16 q^{12} + 12 q^{16} + 20 q^{17} + 4 q^{22} + 8 q^{23} + 8 q^{26} - 8 q^{27} + 20 q^{28} + 16 q^{31} + 104 q^{33} - 4 q^{36} - 20 q^{37} + 36 q^{38} - 20 q^{41} + 20 q^{42} + 40 q^{46} - 40 q^{47} - 4 q^{48} + 40 q^{51} - 40 q^{52} - 8 q^{56} - 48 q^{58} + 80 q^{61} - 40 q^{62} - 100 q^{63} + 24 q^{66} - 20 q^{68} - 56 q^{71} + 20 q^{73} + 96 q^{77} - 16 q^{78} - 68 q^{81} - 56 q^{86} + 4 q^{88} - 68 q^{91} + 12 q^{92} - 76 q^{93} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.987688 + 0.156434i −1.18907 − 2.33368i 0.951057 − 0.309017i 0 1.53950 + 2.11894i 2.88379 + 1.46937i −0.891007 + 0.453990i −2.26883 + 3.12278i 0
7.2 −0.987688 + 0.156434i 0.757609 + 1.48689i 0.951057 − 0.309017i 0 −0.980883 − 1.35007i −2.92930 − 1.49255i −0.891007 + 0.453990i 0.126480 − 0.174085i 0
7.3 −0.987688 + 0.156434i 1.07350 + 2.10686i 0.951057 − 0.309017i 0 −1.38987 − 1.91299i 3.37294 + 1.71860i −0.891007 + 0.453990i −1.52312 + 2.09639i 0
7.4 0.987688 − 0.156434i −0.992087 − 1.94708i 0.951057 − 0.309017i 0 −1.28446 − 1.76791i 3.05681 + 1.55752i 0.891007 − 0.453990i −1.04353 + 1.43630i 0
7.5 0.987688 − 0.156434i 0.176422 + 0.346247i 0.951057 − 0.309017i 0 0.228415 + 0.314386i 0.0136870 + 0.00697389i 0.891007 − 0.453990i 1.67459 − 2.30488i 0
7.6 0.987688 − 0.156434i 1.45771 + 2.86091i 0.951057 − 0.309017i 0 1.88730 + 2.59765i 1.35828 + 0.692077i 0.891007 − 0.453990i −4.29653 + 5.91367i 0
57.1 −0.453990 − 0.891007i −0.256351 + 1.61854i −0.587785 + 0.809017i 0 1.55851 − 0.506390i 0.753160 − 0.119289i 0.987688 + 0.156434i 0.299227 + 0.0972249i 0
57.2 −0.453990 − 0.891007i 0.0342750 − 0.216404i −0.587785 + 0.809017i 0 −0.208378 + 0.0677061i −0.104772 + 0.0165943i 0.987688 + 0.156434i 2.80751 + 0.912216i 0
57.3 −0.453990 − 0.891007i 0.443308 − 2.79893i −0.587785 + 0.809017i 0 −2.69513 + 0.875699i −3.26134 + 0.516545i 0.987688 + 0.156434i −4.78434 − 1.55453i 0
57.4 0.453990 + 0.891007i −0.322293 + 2.03488i −0.587785 + 0.809017i 0 −1.95941 + 0.636650i −0.386047 + 0.0611439i −0.987688 − 0.156434i −1.18368 − 0.384601i 0
57.5 0.453990 + 0.891007i 0.194853 − 1.23025i −0.587785 + 0.809017i 0 1.18463 − 0.384908i −2.94829 + 0.466963i −0.987688 − 0.156434i 1.37761 + 0.447613i 0
57.6 0.453990 + 0.891007i 0.348671 − 2.20142i −0.587785 + 0.809017i 0 2.11978 − 0.688757i 3.91761 − 0.620489i −0.987688 − 0.156434i −1.87153 − 0.608097i 0
107.1 −0.156434 + 0.987688i −2.86091 − 1.45771i −0.951057 − 0.309017i 0 1.88730 − 2.59765i 0.692077 + 1.35828i 0.453990 − 0.891007i 4.29653 + 5.91367i 0
107.2 −0.156434 + 0.987688i −0.346247 − 0.176422i −0.951057 − 0.309017i 0 0.228415 − 0.314386i 0.00697389 + 0.0136870i 0.453990 − 0.891007i −1.67459 − 2.30488i 0
107.3 −0.156434 + 0.987688i 1.94708 + 0.992087i −0.951057 − 0.309017i 0 −1.28446 + 1.76791i 1.55752 + 3.05681i 0.453990 − 0.891007i 1.04353 + 1.43630i 0
107.4 0.156434 − 0.987688i −2.10686 − 1.07350i −0.951057 − 0.309017i 0 −1.38987 + 1.91299i 1.71860 + 3.37294i −0.453990 + 0.891007i 1.52312 + 2.09639i 0
107.5 0.156434 − 0.987688i −1.48689 − 0.757609i −0.951057 − 0.309017i 0 −0.980883 + 1.35007i −1.49255 − 2.92930i −0.453990 + 0.891007i −0.126480 − 0.174085i 0
107.6 0.156434 − 0.987688i 2.33368 + 1.18907i −0.951057 − 0.309017i 0 1.53950 − 2.11894i 1.46937 + 2.88379i −0.453990 + 0.891007i 2.26883 + 3.12278i 0
193.1 −0.453990 + 0.891007i −0.256351 − 1.61854i −0.587785 − 0.809017i 0 1.55851 + 0.506390i 0.753160 + 0.119289i 0.987688 − 0.156434i 0.299227 − 0.0972249i 0
193.2 −0.453990 + 0.891007i 0.0342750 + 0.216404i −0.587785 − 0.809017i 0 −0.208378 − 0.0677061i −0.104772 − 0.0165943i 0.987688 − 0.156434i 2.80751 − 0.912216i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.bh.b 48
5.b even 2 1 110.2.k.a ✓ 48
5.c odd 4 1 110.2.k.a ✓ 48
5.c odd 4 1 inner 550.2.bh.b 48
11.d odd 10 1 inner 550.2.bh.b 48
15.d odd 2 1 990.2.bh.c 48
15.e even 4 1 990.2.bh.c 48
20.d odd 2 1 880.2.cm.c 48
20.e even 4 1 880.2.cm.c 48
55.h odd 10 1 110.2.k.a ✓ 48
55.l even 20 1 110.2.k.a ✓ 48
55.l even 20 1 inner 550.2.bh.b 48
165.r even 10 1 990.2.bh.c 48
165.u odd 20 1 990.2.bh.c 48
220.o even 10 1 880.2.cm.c 48
220.w odd 20 1 880.2.cm.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.k.a ✓ 48 5.b even 2 1
110.2.k.a ✓ 48 5.c odd 4 1
110.2.k.a ✓ 48 55.h odd 10 1
110.2.k.a ✓ 48 55.l even 20 1
550.2.bh.b 48 1.a even 1 1 trivial
550.2.bh.b 48 5.c odd 4 1 inner
550.2.bh.b 48 11.d odd 10 1 inner
550.2.bh.b 48 55.l even 20 1 inner
880.2.cm.c 48 20.d odd 2 1
880.2.cm.c 48 20.e even 4 1
880.2.cm.c 48 220.o even 10 1
880.2.cm.c 48 220.w odd 20 1
990.2.bh.c 48 15.d odd 2 1
990.2.bh.c 48 15.e even 4 1
990.2.bh.c 48 165.r even 10 1
990.2.bh.c 48 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 4 T_{3}^{47} + 8 T_{3}^{46} - 60 T_{3}^{44} - 64 T_{3}^{43} + 736 T_{3}^{42} + \cdots + 723394816 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display