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Properties

Label 550.2.bh.c
Level $550$
Weight $2$
Character orbit 550.bh
Analytic conductor $4.392$
Analytic rank $0$
Dimension $64$
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] = [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: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 20 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 20 q^{6} - 60 q^{11} + 16 q^{16} + 40 q^{26} + 8 q^{31} + 60 q^{36} + 160 q^{41} - 80 q^{46} - 100 q^{51} + 32 q^{56} - 240 q^{61} - 148 q^{66} - 8 q^{71} + 124 q^{81} + 16 q^{86} + 120 q^{91}+O(q^{100}) \) 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 −0.603093 − 1.18364i 0.951057 − 0.309017i 0 0.780830 + 1.07472i −3.89622 − 1.98522i −0.891007 + 0.453990i 0.726080 − 0.999363i 0
7.2 −0.987688 + 0.156434i −0.259233 − 0.508774i 0.951057 − 0.309017i 0 0.335632 + 0.461957i −0.904276 − 0.460751i −0.891007 + 0.453990i 1.57171 − 2.16327i 0
7.3 −0.987688 + 0.156434i 0.233784 + 0.458828i 0.951057 − 0.309017i 0 −0.302683 − 0.416607i 3.00521 + 1.53123i −0.891007 + 0.453990i 1.60749 − 2.21252i 0
7.4 −0.987688 + 0.156434i 1.16224 + 2.28103i 0.951057 − 0.309017i 0 −1.50476 − 2.07113i 0.0132690 + 0.00676089i −0.891007 + 0.453990i −2.08892 + 2.87515i 0
7.5 0.987688 − 0.156434i −1.16224 − 2.28103i 0.951057 − 0.309017i 0 −1.50476 − 2.07113i −0.0132690 − 0.00676089i 0.891007 − 0.453990i −2.08892 + 2.87515i 0
7.6 0.987688 − 0.156434i −0.233784 − 0.458828i 0.951057 − 0.309017i 0 −0.302683 − 0.416607i −3.00521 − 1.53123i 0.891007 − 0.453990i 1.60749 − 2.21252i 0
7.7 0.987688 − 0.156434i 0.259233 + 0.508774i 0.951057 − 0.309017i 0 0.335632 + 0.461957i 0.904276 + 0.460751i 0.891007 − 0.453990i 1.57171 − 2.16327i 0
7.8 0.987688 − 0.156434i 0.603093 + 1.18364i 0.951057 − 0.309017i 0 0.780830 + 1.07472i 3.89622 + 1.98522i 0.891007 − 0.453990i 0.726080 − 0.999363i 0
57.1 −0.453990 − 0.891007i −0.431084 + 2.72176i −0.587785 + 0.809017i 0 2.62081 − 0.851554i −1.78322 + 0.282434i 0.987688 + 0.156434i −4.36897 − 1.41956i 0
57.2 −0.453990 − 0.891007i −0.0114765 + 0.0724597i −0.587785 + 0.809017i 0 0.0697723 − 0.0226704i −3.25690 + 0.515842i 0.987688 + 0.156434i 2.84805 + 0.925388i 0
57.3 −0.453990 − 0.891007i 0.243983 − 1.54045i −0.587785 + 0.809017i 0 −1.48332 + 0.481959i 4.24459 − 0.672277i 0.987688 + 0.156434i 0.539709 + 0.175362i 0
57.4 −0.453990 − 0.891007i 0.496133 − 3.13246i −0.587785 + 0.809017i 0 −3.01628 + 0.980050i 2.77090 − 0.438868i 0.987688 + 0.156434i −6.71300 − 2.18119i 0
57.5 0.453990 + 0.891007i −0.496133 + 3.13246i −0.587785 + 0.809017i 0 −3.01628 + 0.980050i −2.77090 + 0.438868i −0.987688 − 0.156434i −6.71300 − 2.18119i 0
57.6 0.453990 + 0.891007i −0.243983 + 1.54045i −0.587785 + 0.809017i 0 −1.48332 + 0.481959i −4.24459 + 0.672277i −0.987688 − 0.156434i 0.539709 + 0.175362i 0
57.7 0.453990 + 0.891007i 0.0114765 − 0.0724597i −0.587785 + 0.809017i 0 0.0697723 − 0.0226704i 3.25690 − 0.515842i −0.987688 − 0.156434i 2.84805 + 0.925388i 0
57.8 0.453990 + 0.891007i 0.431084 − 2.72176i −0.587785 + 0.809017i 0 2.62081 − 0.851554i 1.78322 − 0.282434i −0.987688 − 0.156434i −4.36897 − 1.41956i 0
107.1 −0.156434 + 0.987688i −1.18364 − 0.603093i −0.951057 − 0.309017i 0 0.780830 − 1.07472i 1.98522 + 3.89622i 0.453990 − 0.891007i −0.726080 − 0.999363i 0
107.2 −0.156434 + 0.987688i −0.508774 − 0.259233i −0.951057 − 0.309017i 0 0.335632 − 0.461957i 0.460751 + 0.904276i 0.453990 − 0.891007i −1.57171 − 2.16327i 0
107.3 −0.156434 + 0.987688i 0.458828 + 0.233784i −0.951057 − 0.309017i 0 −0.302683 + 0.416607i −1.53123 − 3.00521i 0.453990 − 0.891007i −1.60749 − 2.21252i 0
107.4 −0.156434 + 0.987688i 2.28103 + 1.16224i −0.951057 − 0.309017i 0 −1.50476 + 2.07113i −0.00676089 − 0.0132690i 0.453990 − 0.891007i 2.08892 + 2.87515i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
11.d odd 10 1 inner
55.h odd 10 1 inner
55.l even 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.bh.c ✓ 64
5.b even 2 1 inner 550.2.bh.c ✓ 64
5.c odd 4 2 inner 550.2.bh.c ✓ 64
11.d odd 10 1 inner 550.2.bh.c ✓ 64
55.h odd 10 1 inner 550.2.bh.c ✓ 64
55.l even 20 2 inner 550.2.bh.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.bh.c ✓ 64 1.a even 1 1 trivial
550.2.bh.c ✓ 64 5.b even 2 1 inner
550.2.bh.c ✓ 64 5.c odd 4 2 inner
550.2.bh.c ✓ 64 11.d odd 10 1 inner
550.2.bh.c ✓ 64 55.h odd 10 1 inner
550.2.bh.c ✓ 64 55.l even 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 238 T_{3}^{60} + 25621 T_{3}^{56} - 1393907 T_{3}^{52} + 57157136 T_{3}^{48} - 2251894309 T_{3}^{44} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display