Properties

Label 570.2.bc.a
Level $570$
Weight $2$
Character orbit 570.bc
Analytic conductor $4.551$
Analytic rank $0$
Dimension $60$
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.bc (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60q - 6q^{5} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 6q^{5} - 6q^{10} - 12q^{11} + 24q^{14} + 12q^{15} + 24q^{19} + 12q^{20} - 30q^{21} + 18q^{25} - 6q^{26} + 36q^{29} - 6q^{30} + 36q^{31} + 30q^{35} + 12q^{39} + 6q^{40} + 42q^{41} - 6q^{44} - 6q^{45} + 12q^{46} + 66q^{49} + 18q^{55} - 24q^{56} - 12q^{59} + 6q^{60} + 84q^{61} + 30q^{64} - 24q^{65} + 12q^{66} + 12q^{69} - 72q^{71} - 42q^{74} + 6q^{76} - 72q^{79} - 6q^{80} + 12q^{84} - 138q^{85} - 36q^{86} - 180q^{89} - 12q^{90} - 60q^{94} + 24q^{95} - 60q^{96} - 6q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −0.642788 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −2.14038 + 0.647126i 0.939693 0.342020i −0.536979 0.310025i 0.866025 0.500000i −0.766044 0.642788i 1.87154 + 1.22366i
139.2 −0.642788 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −1.78880 1.34171i 0.939693 0.342020i 3.95160 + 2.28146i 0.866025 0.500000i −0.766044 0.642788i 0.122006 + 2.23274i
139.3 −0.642788 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −1.32288 1.80277i 0.939693 0.342020i −3.17019 1.83031i 0.866025 0.500000i −0.766044 0.642788i −0.530667 + 2.17219i
139.4 −0.642788 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i 2.18336 0.482631i 0.939693 0.342020i −2.55062 1.47260i 0.866025 0.500000i −0.766044 0.642788i −1.77315 1.36232i
139.5 −0.642788 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i 2.22668 + 0.204655i 0.939693 0.342020i 2.92703 + 1.68992i 0.866025 0.500000i −0.766044 0.642788i −1.27451 1.83729i
139.6 0.642788 + 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i −2.21676 + 0.293229i 0.939693 0.342020i 2.55062 + 1.47260i −0.866025 + 0.500000i −0.766044 0.642788i −1.64953 1.50965i
139.7 0.642788 + 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i −2.02240 + 0.953883i 0.939693 0.342020i −2.92703 1.68992i −0.866025 + 0.500000i −0.766044 0.642788i −2.03069 0.936105i
139.8 0.642788 + 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 0.626522 2.14650i 0.939693 0.342020i 3.17019 + 1.83031i −0.866025 + 0.500000i −0.766044 0.642788i 2.04704 0.899801i
139.9 0.642788 + 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 1.22203 1.87260i 0.939693 0.342020i −3.95160 2.28146i −0.866025 + 0.500000i −0.766044 0.642788i 2.22000 0.267559i
139.10 0.642788 + 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 2.23263 0.123954i 0.939693 0.342020i 0.536979 + 0.310025i −0.866025 + 0.500000i −0.766044 0.642788i 1.53006 + 1.63062i
169.1 −0.984808 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i −2.14282 0.639010i −0.766044 + 0.642788i −0.495802 + 0.286252i −0.866025 0.500000i −0.173648 0.984808i 1.99930 + 1.00140i
169.2 −0.984808 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i −0.318910 + 2.21321i −0.766044 + 0.642788i −2.49117 + 1.43828i −0.866025 0.500000i −0.173648 0.984808i 0.698385 2.12421i
169.3 −0.984808 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i −0.174899 2.22922i −0.766044 + 0.642788i −2.19817 + 1.26911i −0.866025 0.500000i −0.173648 0.984808i −0.214857 + 2.22572i
169.4 −0.984808 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i 0.588223 + 2.15731i −0.766044 + 0.642788i 4.50054 2.59839i −0.866025 0.500000i −0.173648 0.984808i −0.204673 2.22668i
169.5 −0.984808 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i 2.19119 + 0.445735i −0.766044 + 0.642788i 0.187519 0.108264i −0.866025 0.500000i −0.173648 0.984808i −2.08050 0.819460i
169.6 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i −1.66692 1.49043i −0.766044 + 0.642788i 2.49117 1.43828i 0.866025 + 0.500000i −0.173648 0.984808i −1.38279 1.75724i
169.7 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i −1.23075 + 1.86689i −0.766044 + 0.642788i 0.495802 0.286252i 0.866025 + 0.500000i −0.173648 0.984808i −1.53623 + 1.62481i
169.8 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i −0.936089 2.03070i −0.766044 + 0.642788i −4.50054 + 2.59839i 0.866025 + 0.500000i −0.173648 0.984808i −0.569240 2.16240i
169.9 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 1.29893 + 1.82010i −0.766044 + 0.642788i 2.19817 1.26911i 0.866025 + 0.500000i −0.173648 0.984808i 0.963141 + 2.01801i
169.10 0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 1.39204 1.74992i −0.766044 + 0.642788i −0.187519 + 0.108264i 0.866025 + 0.500000i −0.173648 0.984808i 1.67476 1.48161i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bc.a 60
5.b even 2 1 inner 570.2.bc.a 60
19.e even 9 1 inner 570.2.bc.a 60
95.p even 18 1 inner 570.2.bc.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bc.a 60 1.a even 1 1 trivial
570.2.bc.a 60 5.b even 2 1 inner
570.2.bc.a 60 19.e even 9 1 inner
570.2.bc.a 60 95.p even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(71\!\cdots\!04\)\( T_{7}^{44} - \)\(10\!\cdots\!17\)\( T_{7}^{42} + \)\(14\!\cdots\!76\)\( T_{7}^{40} - \)\(15\!\cdots\!26\)\( T_{7}^{38} + \)\(14\!\cdots\!01\)\( T_{7}^{36} - \)\(11\!\cdots\!87\)\( T_{7}^{34} + \)\(78\!\cdots\!11\)\( T_{7}^{32} - \)\(44\!\cdots\!02\)\( T_{7}^{30} + \)\(21\!\cdots\!83\)\( T_{7}^{28} - \)\(82\!\cdots\!82\)\( T_{7}^{26} + \)\(25\!\cdots\!35\)\( T_{7}^{24} - \)\(59\!\cdots\!86\)\( T_{7}^{22} + \)\(10\!\cdots\!33\)\( T_{7}^{20} - \)\(11\!\cdots\!99\)\( T_{7}^{18} + \)\(93\!\cdots\!29\)\( T_{7}^{16} - \)\(55\!\cdots\!72\)\( T_{7}^{14} + \)\(24\!\cdots\!71\)\( T_{7}^{12} - \)\(82\!\cdots\!88\)\( T_{7}^{10} + \)\(20\!\cdots\!65\)\( T_{7}^{8} - \)\(34\!\cdots\!92\)\( T_{7}^{6} + \)\(38\!\cdots\!11\)\( T_{7}^{4} - \)\(16\!\cdots\!49\)\( T_{7}^{2} + \)\(50\!\cdots\!21\)\( \)">\(T_{7}^{60} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).