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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Newspace parameters

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

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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 8q^{5} - 20q^{6} - 8q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 8q^{5} - 20q^{6} - 8q^{7} - 12q^{10} + 8q^{11} - 24q^{13} + 20q^{16} + 8q^{17} + 12q^{21} + 24q^{22} + 16q^{23} - 16q^{25} + 32q^{26} + 4q^{28} + 16q^{30} - 24q^{33} - 4q^{35} - 20q^{36} - 8q^{38} - 36q^{41} + 4q^{42} - 4q^{43} + 8q^{47} - 24q^{52} - 16q^{55} - 8q^{57} + 16q^{58} + 12q^{60} - 8q^{61} - 16q^{62} - 4q^{63} - 4q^{66} - 36q^{67} - 16q^{68} + 48q^{70} + 72q^{71} - 16q^{73} - 8q^{76} + 24q^{77} + 24q^{78} + 8q^{80} + 20q^{81} - 24q^{82} - 40q^{83} - 48q^{85} + 16q^{87} - 72q^{91} - 16q^{92} + 16q^{93} - 16q^{95} - 40q^{96} + 36q^{97} - 48q^{98} + O(q^{100}) \)

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.

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 487.10
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 \(T_{7}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).