Properties

Label 570.2.bc.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 60q + 6q^{5} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q + 6q^{5} + 6q^{10} - 12q^{11} + 12q^{15} + 24q^{19} + 12q^{20} - 18q^{21} - 6q^{25} - 6q^{26} + 12q^{29} + 6q^{30} - 36q^{31} - 6q^{35} - 12q^{39} - 6q^{40} - 30q^{41} - 6q^{44} - 6q^{45} - 36q^{46} + 18q^{49} - 90q^{55} + 24q^{56} + 60q^{59} + 6q^{60} + 36q^{61} + 30q^{64} + 24q^{65} - 12q^{66} + 36q^{69} - 72q^{70} - 72q^{71} + 6q^{74} + 6q^{76} + 6q^{80} + 12q^{84} + 18q^{85} - 12q^{86} + 60q^{89} + 12q^{90} - 48q^{91} - 156q^{94} - 84q^{95} + 60q^{96} - 6q^{99} + O(q^{100}) \)

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 −1.70452 1.44728i −0.939693 + 0.342020i 0.727975 + 0.420297i 0.866025 0.500000i −0.766044 0.642788i −0.0130335 + 2.23603i
139.2 −0.642788 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i −1.66564 + 1.49186i −0.939693 + 0.342020i −0.828729 0.478467i 0.866025 0.500000i −0.766044 0.642788i 2.21348 + 0.317005i
139.3 −0.642788 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 0.752231 2.10574i −0.939693 + 0.342020i 1.31266 + 0.757865i 0.866025 0.500000i −0.766044 0.642788i −2.09662 + 0.777303i
139.4 −0.642788 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 1.06634 + 1.96543i −0.939693 + 0.342020i −4.20443 2.42743i 0.866025 0.500000i −0.766044 0.642788i 0.820177 2.08022i
139.5 −0.642788 0.766044i 0.342020 0.939693i −0.173648 + 0.984808i 2.04631 + 0.901446i −0.939693 + 0.342020i 1.77015 + 1.02199i 0.866025 0.500000i −0.766044 0.642788i −0.624797 2.14700i
139.6 0.642788 + 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −1.61459 + 1.54696i −0.939693 + 0.342020i −1.77015 1.02199i −0.866025 + 0.500000i −0.766044 0.642788i −2.22288 0.242482i
139.7 0.642788 + 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −1.42707 1.72147i −0.939693 + 0.342020i −1.31266 0.757865i −0.866025 + 0.500000i −0.766044 0.642788i 0.401420 2.19974i
139.8 0.642788 + 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i −0.329815 + 2.21161i −0.939693 + 0.342020i 4.20443 + 2.42743i −0.866025 + 0.500000i −0.766044 0.642788i −1.90619 + 1.16894i
139.9 0.642788 + 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i 1.10673 1.94298i −0.939693 + 0.342020i −0.727975 0.420297i −0.866025 + 0.500000i −0.766044 0.642788i 2.19980 0.401118i
139.10 0.642788 + 0.766044i −0.342020 + 0.939693i −0.173648 + 0.984808i 2.07544 + 0.832208i −0.939693 + 0.342020i 0.828729 + 0.478467i −0.866025 + 0.500000i −0.766044 0.642788i 0.696556 + 2.12481i
169.1 −0.984808 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i −2.23600 0.0180231i 0.766044 0.642788i −3.41402 + 1.97108i −0.866025 0.500000i −0.173648 0.984808i 2.19890 + 0.406026i
169.2 −0.984808 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i −0.834883 2.07436i 0.766044 0.642788i 2.83436 1.63642i −0.866025 0.500000i −0.173648 0.984808i 0.461990 + 2.18782i
169.3 −0.984808 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 1.26599 1.84317i 0.766044 0.642788i −2.40081 + 1.38611i −0.866025 0.500000i −0.173648 0.984808i −1.56682 + 1.59533i
169.4 −0.984808 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 1.35630 + 1.77777i 0.766044 0.642788i −2.04388 + 1.18003i −0.866025 0.500000i −0.173648 0.984808i −1.02699 1.98628i
169.5 −0.984808 0.173648i −0.642788 + 0.766044i 0.939693 + 0.342020i 2.18519 0.474287i 0.766044 0.642788i 2.26623 1.30841i −0.866025 0.500000i −0.173648 0.984808i −2.23435 + 0.0876270i
169.6 0.984808 + 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i −1.70129 + 1.45108i 0.766044 0.642788i 3.41402 1.97108i 0.866025 + 0.500000i −0.173648 0.984808i −1.92742 + 1.13361i
169.7 0.984808 + 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i −0.103744 2.23366i 0.766044 0.642788i 2.04388 1.18003i 0.866025 + 0.500000i −0.173648 0.984808i 0.285703 2.21774i
169.8 0.984808 + 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i 0.693816 + 2.12570i 0.766044 0.642788i −2.83436 + 1.63642i 0.866025 + 0.500000i −0.173648 0.984808i 0.314150 + 2.21389i
169.9 0.984808 + 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i 1.97882 1.04129i 0.766044 0.642788i −2.26623 + 1.30841i 0.866025 + 0.500000i −0.173648 0.984808i 2.12957 0.681850i
169.10 0.984808 + 0.173648i 0.642788 0.766044i 0.939693 + 0.342020i 2.15457 + 0.598186i 0.766044 0.642788i 2.40081 1.38611i 0.866025 + 0.500000i −0.173648 0.984808i 2.01796 + 0.963235i
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.b 60
5.b even 2 1 inner 570.2.bc.b 60
19.e even 9 1 inner 570.2.bc.b 60
95.p even 18 1 inner 570.2.bc.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bc.b 60 1.a even 1 1 trivial
570.2.bc.b 60 5.b even 2 1 inner
570.2.bc.b 60 19.e even 9 1 inner
570.2.bc.b 60 95.p even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!20\)\( T_{7}^{44} - \)\(18\!\cdots\!29\)\( T_{7}^{42} + \)\(19\!\cdots\!60\)\( T_{7}^{40} - \)\(17\!\cdots\!18\)\( T_{7}^{38} + \)\(13\!\cdots\!53\)\( T_{7}^{36} - \)\(92\!\cdots\!63\)\( T_{7}^{34} + \)\(53\!\cdots\!27\)\( T_{7}^{32} - \)\(26\!\cdots\!18\)\( T_{7}^{30} + \)\(11\!\cdots\!63\)\( T_{7}^{28} - \)\(40\!\cdots\!90\)\( T_{7}^{26} + \)\(12\!\cdots\!71\)\( T_{7}^{24} - \)\(31\!\cdots\!86\)\( T_{7}^{22} + \)\(63\!\cdots\!93\)\( T_{7}^{20} - \)\(10\!\cdots\!51\)\( T_{7}^{18} + \)\(13\!\cdots\!81\)\( T_{7}^{16} - \)\(13\!\cdots\!92\)\( T_{7}^{14} + \)\(10\!\cdots\!23\)\( T_{7}^{12} - \)\(58\!\cdots\!64\)\( T_{7}^{10} + \)\(24\!\cdots\!05\)\( T_{7}^{8} - \)\(63\!\cdots\!48\)\( T_{7}^{6} + \)\(10\!\cdots\!19\)\( T_{7}^{4} - \)\(93\!\cdots\!13\)\( T_{7}^{2} + \)\(80\!\cdots\!81\)\( \)">\(T_{7}^{60} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).