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Properties

Label 570.2.x.a
Level $570$
Weight $2$
Character orbit 570.x
Analytic conductor $4.551$
Analytic rank $0$
Dimension $40$
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] = [570,2,Mod(103,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.x (of order \(12\), 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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{5} - 20 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{5} - 20 q^{6} - 8 q^{7} - 12 q^{10} + 8 q^{11} - 24 q^{13} + 20 q^{16} + 8 q^{17} + 12 q^{21} + 24 q^{22} + 16 q^{23} - 16 q^{25} + 32 q^{26} + 4 q^{28} + 16 q^{30} - 24 q^{33} - 4 q^{35} - 20 q^{36} - 8 q^{38} - 36 q^{41} + 4 q^{42} - 4 q^{43} + 8 q^{47} - 24 q^{52} - 16 q^{55} - 8 q^{57} + 16 q^{58} + 12 q^{60} - 8 q^{61} - 16 q^{62} - 4 q^{63} - 4 q^{66} - 36 q^{67} - 16 q^{68} + 48 q^{70} + 72 q^{71} - 16 q^{73} - 8 q^{76} + 24 q^{77} + 24 q^{78} + 8 q^{80} + 20 q^{81} - 24 q^{82} - 40 q^{83} - 48 q^{85} + 16 q^{87} - 72 q^{91} - 16 q^{92} + 16 q^{93} - 16 q^{95} - 40 q^{96} + 36 q^{97} - 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i −2.02908 − 0.939602i −0.500000 − 0.866025i 0.555770 − 0.555770i −0.707107 + 0.707107i −0.866025 + 0.500000i 2.20312 + 0.382422i
103.2 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i −0.991672 + 2.00414i −0.500000 − 0.866025i 1.93610 − 1.93610i −0.707107 + 0.707107i −0.866025 + 0.500000i 0.439172 − 2.19252i
103.3 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i −0.0684861 + 2.23502i −0.500000 − 0.866025i −0.515842 + 0.515842i −0.707107 + 0.707107i −0.866025 + 0.500000i −0.512313 − 2.17659i
103.4 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i 1.56592 − 1.59622i −0.500000 − 0.866025i 0.819066 − 0.819066i −0.707107 + 0.707107i −0.866025 + 0.500000i −1.09943 + 1.94711i
103.5 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i 2.19635 − 0.419581i −0.500000 − 0.866025i −2.57035 + 2.57035i −0.707107 + 0.707107i −0.866025 + 0.500000i −2.01292 + 0.973741i
103.6 0.965926 − 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −2.19222 − 0.440644i −0.500000 − 0.866025i −1.44846 + 1.44846i 0.707107 − 0.707107i −0.866025 + 0.500000i −2.23157 + 0.141759i
103.7 0.965926 − 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −1.40065 + 1.74303i −0.500000 − 0.866025i −3.60529 + 3.60529i 0.707107 − 0.707107i −0.866025 + 0.500000i −0.901798 + 2.04616i
103.8 0.965926 − 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −0.743478 + 2.10885i −0.500000 − 0.866025i 2.46264 − 2.46264i 0.707107 − 0.707107i −0.866025 + 0.500000i −0.172335 + 2.22942i
103.9 0.965926 − 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −0.347530 − 2.20890i −0.500000 − 0.866025i 0.475663 − 0.475663i 0.707107 − 0.707107i −0.866025 + 0.500000i −0.907393 − 2.04368i
103.10 0.965926 − 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i 2.01085 + 0.977997i −0.500000 − 0.866025i −0.109293 + 0.109293i 0.707107 − 0.707107i −0.866025 + 0.500000i 2.19546 + 0.424226i
217.1 −0.258819 − 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −1.90134 + 1.17682i −0.500000 − 0.866025i −0.515842 − 0.515842i 0.707107 + 0.707107i 0.866025 − 0.500000i 1.62882 + 1.53197i
217.2 −0.258819 − 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −1.23980 + 1.86088i −0.500000 − 0.866025i 1.93610 + 1.93610i 0.707107 + 0.707107i 0.866025 − 0.500000i 2.11836 + 0.715924i
217.3 −0.258819 − 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −0.734807 − 2.11188i −0.500000 − 0.866025i −2.57035 − 2.57035i 0.707107 + 0.707107i 0.866025 − 0.500000i −1.84974 + 1.25637i
217.4 −0.258819 − 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i 0.599405 − 2.15423i −0.500000 − 0.866025i 0.819066 + 0.819066i 0.707107 + 0.707107i 0.866025 − 0.500000i −2.23597 − 0.0214243i
217.5 −0.258819 − 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i 1.82826 + 1.28743i −0.500000 − 0.866025i 0.555770 + 0.555770i 0.707107 + 0.707107i 0.866025 − 0.500000i 0.770374 − 2.09917i
217.6 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i −1.85240 − 1.25245i −0.500000 − 0.866025i −0.109293 − 0.109293i −0.707107 − 0.707107i 0.866025 − 0.500000i 0.730338 − 2.11343i
217.7 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i −1.45458 + 1.69830i −0.500000 − 0.866025i 2.46264 + 2.46264i −0.707107 − 0.707107i 0.866025 − 0.500000i −2.01690 − 0.965462i
217.8 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i −0.809185 + 2.08452i −0.500000 − 0.866025i −3.60529 − 3.60529i −0.707107 − 0.707107i 0.866025 − 0.500000i −2.22292 − 0.242099i
217.9 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 1.47772 + 1.67820i −0.500000 − 0.866025i −1.44846 − 1.44846i −0.707107 − 0.707107i 0.866025 − 0.500000i −1.23855 + 1.86172i
217.10 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 2.08673 − 0.803478i −0.500000 − 0.866025i 0.475663 + 0.475663i −0.707107 − 0.707107i 0.866025 − 0.500000i 1.31618 + 1.80767i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.x.a ✓ 40
5.c odd 4 1 inner 570.2.x.a ✓ 40
19.d odd 6 1 inner 570.2.x.a ✓ 40
95.l even 12 1 inner 570.2.x.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.x.a ✓ 40 1.a even 1 1 trivial
570.2.x.a ✓ 40 5.c odd 4 1 inner
570.2.x.a ✓ 40 19.d odd 6 1 inner
570.2.x.a ✓ 40 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} - 48 T_{7}^{17} + 313 T_{7}^{16} + 712 T_{7}^{15} + 1496 T_{7}^{14} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display