# 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 $2$

# 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} + 4 q^{3} - 12 q^{4} - 2 q^{6} - 12 q^{7} + 24 q^{8} - 4 q^{9}+O(q^{10})$$ 24 * q - 12 * q^2 + 4 * q^3 - 12 * q^4 - 2 * q^6 - 12 * q^7 + 24 * q^8 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} + 4 q^{3} - 12 q^{4} - 2 q^{6} - 12 q^{7} + 24 q^{8} - 4 q^{9} - 2 q^{12} + 18 q^{13} + 6 q^{14} - 12 q^{16} + 12 q^{17} + 2 q^{18} + 6 q^{19} - 6 q^{21} + 18 q^{22} + 4 q^{24} + 12 q^{25} + 28 q^{27} + 6 q^{28} - 12 q^{32} - 22 q^{33} - 12 q^{34} + 2 q^{36} + 6 q^{38} + 40 q^{39} + 6 q^{41} - 6 q^{42} - 22 q^{43} - 18 q^{44} + 8 q^{45} + 12 q^{47} - 2 q^{48} + 12 q^{49} - 24 q^{50} - 20 q^{51} - 18 q^{52} + 8 q^{53} + 4 q^{54} - 12 q^{56} + 26 q^{59} + 22 q^{61} - 18 q^{62} + 6 q^{63} + 24 q^{64} + 8 q^{65} + 8 q^{66} - 48 q^{67} - 64 q^{69} + 24 q^{71} - 4 q^{72} - 8 q^{73} + 30 q^{74} + 2 q^{75} - 12 q^{76} - 38 q^{78} + 18 q^{79} - 12 q^{81} + 6 q^{82} + 12 q^{84} - 22 q^{86} - 24 q^{87} + 28 q^{89} + 8 q^{90} + 18 q^{91} + 2 q^{93} - 2 q^{96} + 6 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100})$$ 24 * q - 12 * q^2 + 4 * q^3 - 12 * q^4 - 2 * q^6 - 12 * q^7 + 24 * q^8 - 4 * 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 + 4 * q^24 + 12 * q^25 + 28 * q^27 + 6 * q^28 - 12 * q^32 - 22 * q^33 - 12 * q^34 + 2 * 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 - 2 * q^48 + 12 * q^49 - 24 * q^50 - 20 * q^51 - 18 * q^52 + 8 * q^53 + 4 * q^54 - 12 * q^56 + 26 * q^59 + 22 * q^61 - 18 * q^62 + 6 * q^63 + 24 * q^64 + 8 * q^65 + 8 * q^66 - 48 * q^67 - 64 * q^69 + 24 * q^71 - 4 * q^72 - 8 * q^73 + 30 * q^74 + 2 * q^75 - 12 * q^76 - 38 * q^78 + 18 * q^79 - 12 * q^81 + 6 * q^82 + 12 * q^84 - 22 * q^86 - 24 * q^87 + 28 * q^89 + 8 * q^90 + 18 * q^91 + 2 * q^93 - 2 * q^96 + 6 * q^97 - 6 * q^98 + 2 * 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 −1.62233 + 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: 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.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 $$T_{17}^{24} - 12 T_{17}^{23} - 40 T_{17}^{22} + 1056 T_{17}^{21} + 812 T_{17}^{20} - 66120 T_{17}^{19} + 147584 T_{17}^{18} + 1931112 T_{17}^{17} - 7089220 T_{17}^{16} - 40050480 T_{17}^{15} + 217155696 T_{17}^{14} + \cdots + 18547561529344$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.