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 $2$

Related objects

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.

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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$T_{17}^{24} + 12 T_{17}^{23} - 40 T_{17}^{22} - 1056 T_{17}^{21} + 812 T_{17}^{20} + 66120 T_{17}^{19} + 147584 T_{17}^{18} - 1931112 T_{17}^{17} - 7089220 T_{17}^{16} + 40050480 T_{17}^{15} + 217155696 T_{17}^{14} + \cdots + 18547561529344$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.