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Properties

Label 660.2.bm.a
Level $660$
Weight $2$
Character orbit 660.bm
Analytic conductor $5.270$
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] = [660,2,Mod(49,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.bm (of order \(10\), degree \(4\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{5} + 12 q^{9} + 8 q^{11} - 2 q^{15} - 8 q^{19} + 32 q^{21} - 26 q^{25} + 20 q^{31} - 4 q^{35} - 8 q^{39} + 12 q^{41} + 4 q^{45} + 52 q^{49} + 4 q^{51} + 48 q^{55} + 28 q^{59} - 8 q^{61} + 68 q^{65} + 20 q^{69} - 80 q^{71} + 28 q^{79} - 12 q^{81} + 42 q^{85} - 24 q^{89} - 64 q^{91} - 54 q^{95} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −0.587785 + 0.809017i 0 −2.18045 − 0.495606i 0 −0.369641 − 0.508767i 0 −0.309017 − 0.951057i 0
49.2 0 −0.587785 + 0.809017i 0 −0.493835 + 2.18085i 0 −0.796941 − 1.09690i 0 −0.309017 − 0.951057i 0
49.3 0 −0.587785 + 0.809017i 0 −0.310113 − 2.21446i 0 −1.16163 − 1.59885i 0 −0.309017 − 0.951057i 0
49.4 0 −0.587785 + 0.809017i 0 0.188984 + 2.22807i 0 2.12366 + 2.92297i 0 −0.309017 − 0.951057i 0
49.5 0 −0.587785 + 0.809017i 0 1.55281 + 1.60897i 0 −2.58300 − 3.55520i 0 −0.309017 − 0.951057i 0
49.6 0 −0.587785 + 0.809017i 0 1.69366 − 1.45997i 0 0.436413 + 0.600671i 0 −0.309017 − 0.951057i 0
49.7 0 0.587785 − 0.809017i 0 −2.20198 + 0.388964i 0 2.58300 + 3.55520i 0 −0.309017 − 0.951057i 0
49.8 0 0.587785 − 0.809017i 0 −1.46252 + 1.69146i 0 −2.12366 − 2.92297i 0 −0.309017 − 0.951057i 0
49.9 0 0.587785 − 0.809017i 0 −0.882354 + 2.05462i 0 0.796941 + 1.09690i 0 −0.309017 − 0.951057i 0
49.10 0 0.587785 − 0.809017i 0 −0.512055 − 2.17665i 0 −0.436413 − 0.600671i 0 −0.309017 − 0.951057i 0
49.11 0 0.587785 − 0.809017i 0 1.55251 − 1.60926i 0 1.16163 + 1.59885i 0 −0.309017 − 0.951057i 0
49.12 0 0.587785 − 0.809017i 0 2.05533 + 0.880685i 0 0.369641 + 0.508767i 0 −0.309017 − 0.951057i 0
169.1 0 −0.951057 − 0.309017i 0 −2.11342 − 0.730384i 0 2.96619 − 0.963772i 0 0.809017 + 0.587785i 0
169.2 0 −0.951057 − 0.309017i 0 −1.74935 − 1.39275i 0 −4.70546 + 1.52890i 0 0.809017 + 0.587785i 0
169.3 0 −0.951057 − 0.309017i 0 −0.391521 + 2.20152i 0 −2.37557 + 0.771870i 0 0.809017 + 0.587785i 0
169.4 0 −0.951057 − 0.309017i 0 0.493301 − 2.18098i 0 1.67290 − 0.543559i 0 0.809017 + 0.587785i 0
169.5 0 −0.951057 − 0.309017i 0 0.499116 + 2.17965i 0 1.68498 − 0.547484i 0 0.809017 + 0.587785i 0
169.6 0 −0.951057 − 0.309017i 0 2.17409 − 0.522809i 0 −3.04726 + 0.990116i 0 0.809017 + 0.587785i 0
169.7 0 0.951057 + 0.309017i 0 −2.21476 − 0.307950i 0 2.37557 − 0.771870i 0 0.809017 + 0.587785i 0
169.8 0 0.951057 + 0.309017i 0 −1.91874 − 1.14824i 0 −1.68498 + 0.547484i 0 0.809017 + 0.587785i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.bm.a ✓ 48
5.b even 2 1 inner 660.2.bm.a ✓ 48
11.c even 5 1 inner 660.2.bm.a ✓ 48
55.j even 10 1 inner 660.2.bm.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.bm.a ✓ 48 1.a even 1 1 trivial
660.2.bm.a ✓ 48 5.b even 2 1 inner
660.2.bm.a ✓ 48 11.c even 5 1 inner
660.2.bm.a ✓ 48 55.j even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(660, [\chi])\).