Properties

Label 570.2.s.a
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.

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

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 −1.62233 + 0.606673i −0.500000 + 0.866025i 0.866025 0.500000i 1.33656 + 1.10164i −1.76552 1.00000 2.26390 1.96845i −0.866025 0.500000i
221.2 −0.500000 0.866025i −1.22385 + 1.22564i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.67336 + 0.447064i 3.20940 1.00000 −0.00438801 3.00000i 0.866025 + 0.500000i
221.3 −0.500000 0.866025i −1.01463 1.40375i −0.500000 + 0.866025i −0.866025 + 0.500000i −0.708367 + 1.58057i 2.73284 1.00000 −0.941036 + 2.84859i 0.866025 + 0.500000i
221.4 −0.500000 0.866025i −0.929852 1.46129i −0.500000 + 0.866025i 0.866025 0.500000i −0.800591 + 1.53592i −4.66317 1.00000 −1.27075 + 2.71757i −0.866025 0.500000i
221.5 −0.500000 0.866025i −0.641070 + 1.60905i −0.500000 + 0.866025i −0.866025 + 0.500000i 1.71401 0.249340i −2.43208 1.00000 −2.17806 2.06302i 0.866025 + 0.500000i
221.6 −0.500000 0.866025i 0.0903420 1.72969i −0.500000 + 0.866025i 0.866025 0.500000i −1.54313 + 0.786608i 2.34168 1.00000 −2.98368 0.312528i −0.866025 0.500000i
221.7 −0.500000 0.866025i 0.362568 + 1.69368i −0.500000 + 0.866025i 0.866025 0.500000i 1.28548 1.16083i −0.535070 1.00000 −2.73709 + 1.22815i −0.866025 0.500000i
221.8 −0.500000 0.866025i 0.691758 1.58791i −0.500000 + 0.866025i −0.866025 + 0.500000i −1.72105 + 0.194877i −1.96058 1.00000 −2.04294 2.19691i 0.866025 + 0.500000i
221.9 −0.500000 0.866025i 1.37489 + 1.05342i −0.500000 + 0.866025i 0.866025 0.500000i 0.224845 1.71739i −1.74360 1.00000 0.780619 + 2.89666i −0.866025 0.500000i
221.10 −0.500000 0.866025i 1.56078 + 0.750973i −0.500000 + 0.866025i −0.866025 + 0.500000i −0.130029 1.72716i −4.16200 1.00000 1.87208 + 2.34421i 0.866025 + 0.500000i
221.11 −0.500000 0.866025i 1.62701 0.593994i −0.500000 + 0.866025i −0.866025 + 0.500000i −1.32792 1.11204i −0.387589 1.00000 2.29434 1.93287i 0.866025 + 0.500000i
221.12 −0.500000 0.866025i 1.72438 0.162784i −0.500000 + 0.866025i 0.866025 0.500000i −1.00317 1.41197i 3.36569 1.00000 2.94700 0.561404i −0.866025 0.500000i
521.1 −0.500000 + 0.866025i −1.62233 0.606673i −0.500000 0.866025i 0.866025 + 0.500000i 1.33656 1.10164i −1.76552 1.00000 2.26390 + 1.96845i −0.866025 + 0.500000i
521.2 −0.500000 + 0.866025i −1.22385 1.22564i −0.500000 0.866025i −0.866025 0.500000i 1.67336 0.447064i 3.20940 1.00000 −0.00438801 + 3.00000i 0.866025 0.500000i
521.3 −0.500000 + 0.866025i −1.01463 + 1.40375i −0.500000 0.866025i −0.866025 0.500000i −0.708367 1.58057i 2.73284 1.00000 −0.941036 2.84859i 0.866025 0.500000i
521.4 −0.500000 + 0.866025i −0.929852 + 1.46129i −0.500000 0.866025i 0.866025 + 0.500000i −0.800591 1.53592i −4.66317 1.00000 −1.27075 2.71757i −0.866025 + 0.500000i
521.5 −0.500000 + 0.866025i −0.641070 1.60905i −0.500000 0.866025i −0.866025 0.500000i 1.71401 + 0.249340i −2.43208 1.00000 −2.17806 + 2.06302i 0.866025 0.500000i
521.6 −0.500000 + 0.866025i 0.0903420 + 1.72969i −0.500000 0.866025i 0.866025 + 0.500000i −1.54313 0.786608i 2.34168 1.00000 −2.98368 + 0.312528i −0.866025 + 0.500000i
521.7 −0.500000 + 0.866025i 0.362568 1.69368i −0.500000 0.866025i 0.866025 + 0.500000i 1.28548 + 1.16083i −0.535070 1.00000 −2.73709 1.22815i −0.866025 + 0.500000i
521.8 −0.500000 + 0.866025i 0.691758 + 1.58791i −0.500000 0.866025i −0.866025 0.500000i −1.72105 0.194877i −1.96058 1.00000 −2.04294 + 2.19691i 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:
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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.a 24
3.b odd 2 1 570.2.s.b yes 24
19.d odd 6 1 570.2.s.b yes 24
57.f even 6 1 inner 570.2.s.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.s.a 24 1.a even 1 1 trivial
570.2.s.a 24 57.f even 6 1 inner
570.2.s.b yes 24 3.b odd 2 1
570.2.s.b yes 24 19.d odd 6 1

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 \(S_{2}^{\mathrm{new}}(570, [\chi])\).