Properties

Label 570.2.s.b
Level $570$
Weight $2$
Character orbit 570.s
Analytic conductor $4.551$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{2} + 2q^{3} - 12q^{4} + 4q^{6} - 12q^{7} - 24q^{8} + 2q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{2} + 2q^{3} - 12q^{4} + 4q^{6} - 12q^{7} - 24q^{8} + 2q^{9} + 2q^{12} + 18q^{13} - 6q^{14} - 12q^{16} - 12q^{17} - 2q^{18} + 6q^{19} + 6q^{21} + 18q^{22} - 2q^{24} + 12q^{25} - 28q^{27} + 6q^{28} + 12q^{32} - 8q^{33} - 12q^{34} - 4q^{36} - 6q^{38} + 40q^{39} - 6q^{41} - 6q^{42} - 22q^{43} + 18q^{44} + 8q^{45} - 12q^{47} - 4q^{48} + 12q^{49} + 24q^{50} - 4q^{51} - 18q^{52} - 8q^{53} - 32q^{54} + 12q^{56} - 20q^{57} - 26q^{59} + 22q^{61} + 18q^{62} + 30q^{63} + 24q^{64} - 8q^{65} - 22q^{66} - 48q^{67} + 64q^{69} - 24q^{71} - 2q^{72} - 8q^{73} - 30q^{74} - 2q^{75} - 12q^{76} + 2q^{78} + 18q^{79} - 6q^{81} + 6q^{82} - 12q^{84} + 22q^{86} - 24q^{87} - 28q^{89} + 16q^{90} + 18q^{91} + 14q^{93} - 2q^{96} + 6q^{97} + 6q^{98} - 52q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 0.500000 + 0.866025i −1.71401 0.249340i −0.500000 + 0.866025i 0.866025 0.500000i −0.641070 1.60905i −2.43208 −1.00000 2.87566 + 0.854742i 0.866025 + 0.500000i
221.2 0.500000 + 0.866025i −1.67336 + 0.447064i −0.500000 + 0.866025i 0.866025 0.500000i −1.22385 1.22564i 3.20940 −1.00000 2.60027 1.49620i 0.866025 + 0.500000i
221.3 0.500000 + 0.866025i −1.33656 + 1.10164i −0.500000 + 0.866025i −0.866025 + 0.500000i −1.62233 0.606673i −1.76552 −1.00000 0.572776 2.94481i −0.866025 0.500000i
221.4 0.500000 + 0.866025i −1.28548 1.16083i −0.500000 + 0.866025i −0.866025 + 0.500000i 0.362568 1.69368i −0.535070 −1.00000 0.304938 + 2.98446i −0.866025 0.500000i
221.5 0.500000 + 0.866025i −0.224845 1.71739i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.37489 1.05342i −1.74360 −1.00000 −2.89889 + 0.772294i −0.866025 0.500000i
221.6 0.500000 + 0.866025i 0.130029 1.72716i −0.500000 + 0.866025i 0.866025 0.500000i 1.56078 0.750973i −4.16200 −1.00000 −2.96618 0.449163i 0.866025 + 0.500000i
221.7 0.500000 + 0.866025i 0.708367 + 1.58057i −0.500000 + 0.866025i 0.866025 0.500000i −1.01463 + 1.40375i 2.73284 −1.00000 −1.99643 + 2.23925i 0.866025 + 0.500000i
221.8 0.500000 + 0.866025i 0.800591 + 1.53592i −0.500000 + 0.866025i −0.866025 + 0.500000i −0.929852 + 1.46129i −4.66317 −1.00000 −1.71811 + 2.45929i −0.866025 0.500000i
221.9 0.500000 + 0.866025i 1.00317 1.41197i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.72438 + 0.162784i 3.36569 −1.00000 −0.987311 2.83288i −0.866025 0.500000i
221.10 0.500000 + 0.866025i 1.32792 1.11204i −0.500000 + 0.866025i 0.866025 0.500000i 1.62701 + 0.593994i −0.387589 −1.00000 0.526745 2.95339i 0.866025 + 0.500000i
221.11 0.500000 + 0.866025i 1.54313 + 0.786608i −0.500000 + 0.866025i −0.866025 + 0.500000i 0.0903420 + 1.72969i 2.34168 −1.00000 1.76250 + 2.42768i −0.866025 0.500000i
221.12 0.500000 + 0.866025i 1.72105 + 0.194877i −0.500000 + 0.866025i 0.866025 0.500000i 0.691758 + 1.58791i −1.96058 −1.00000 2.92405 + 0.670786i 0.866025 + 0.500000i
521.1 0.500000 0.866025i −1.71401 + 0.249340i −0.500000 0.866025i 0.866025 + 0.500000i −0.641070 + 1.60905i −2.43208 −1.00000 2.87566 0.854742i 0.866025 0.500000i
521.2 0.500000 0.866025i −1.67336 0.447064i −0.500000 0.866025i 0.866025 + 0.500000i −1.22385 + 1.22564i 3.20940 −1.00000 2.60027 + 1.49620i 0.866025 0.500000i
521.3 0.500000 0.866025i −1.33656 1.10164i −0.500000 0.866025i −0.866025 0.500000i −1.62233 + 0.606673i −1.76552 −1.00000 0.572776 + 2.94481i −0.866025 + 0.500000i
521.4 0.500000 0.866025i −1.28548 + 1.16083i −0.500000 0.866025i −0.866025 0.500000i 0.362568 + 1.69368i −0.535070 −1.00000 0.304938 2.98446i −0.866025 + 0.500000i
521.5 0.500000 0.866025i −0.224845 + 1.71739i −0.500000 0.866025i −0.866025 0.500000i 1.37489 + 1.05342i −1.74360 −1.00000 −2.89889 0.772294i −0.866025 + 0.500000i
521.6 0.500000 0.866025i 0.130029 + 1.72716i −0.500000 0.866025i 0.866025 + 0.500000i 1.56078 + 0.750973i −4.16200 −1.00000 −2.96618 + 0.449163i 0.866025 0.500000i
521.7 0.500000 0.866025i 0.708367 1.58057i −0.500000 0.866025i 0.866025 + 0.500000i −1.01463 1.40375i 2.73284 −1.00000 −1.99643 2.23925i 0.866025 0.500000i
521.8 0.500000 0.866025i 0.800591 1.53592i −0.500000 0.866025i −0.866025 0.500000i −0.929852 1.46129i −4.66317 −1.00000 −1.71811 2.45929i −0.866025 + 0.500000i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.s.b yes 24
3.b odd 2 1 570.2.s.a 24
19.d odd 6 1 570.2.s.a 24
57.f even 6 1 inner 570.2.s.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.s.a 24 3.b odd 2 1
570.2.s.a 24 19.d odd 6 1
570.2.s.b yes 24 1.a even 1 1 trivial
570.2.s.b yes 24 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!44\)\( T_{17}^{6} + \)\(36\!\cdots\!36\)\( T_{17}^{5} - \)\(19\!\cdots\!48\)\( T_{17}^{4} - \)\(10\!\cdots\!60\)\( T_{17}^{3} + 666028847104 T_{17}^{2} + \)\(22\!\cdots\!40\)\( T_{17} + \)\(18\!\cdots\!44\)\( \)">\(T_{17}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).