# Properties

 Label 570.2.k.a Level $570$ Weight $2$ Character orbit 570.k Analytic conductor $4.551$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 4 q^{3} + 4 q^{6} - 12 q^{7}+O(q^{10})$$ 36 * q + 4 * q^3 + 4 * q^6 - 12 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 4 q^{3} + 4 q^{6} - 12 q^{7} - 4 q^{10} - 4 q^{12} + 8 q^{13} + 4 q^{15} - 36 q^{16} - 32 q^{21} - 4 q^{22} + 32 q^{25} + 28 q^{27} - 12 q^{28} - 8 q^{30} + 8 q^{31} + 36 q^{33} + 4 q^{36} - 32 q^{37} - 8 q^{40} + 12 q^{42} - 24 q^{43} - 28 q^{45} - 16 q^{46} - 4 q^{48} - 40 q^{51} - 8 q^{52} - 4 q^{55} + 4 q^{57} - 4 q^{58} - 24 q^{60} + 200 q^{61} + 28 q^{63} + 12 q^{70} - 68 q^{73} - 36 q^{75} + 36 q^{76} + 24 q^{78} - 92 q^{81} + 24 q^{82} + 24 q^{85} + 28 q^{87} - 4 q^{88} - 68 q^{90} + 64 q^{91} + 16 q^{93} - 4 q^{96} - 148 q^{97}+O(q^{100})$$ 36 * q + 4 * q^3 + 4 * q^6 - 12 * q^7 - 4 * q^10 - 4 * q^12 + 8 * q^13 + 4 * q^15 - 36 * q^16 - 32 * q^21 - 4 * q^22 + 32 * q^25 + 28 * q^27 - 12 * q^28 - 8 * q^30 + 8 * q^31 + 36 * q^33 + 4 * q^36 - 32 * q^37 - 8 * q^40 + 12 * q^42 - 24 * q^43 - 28 * q^45 - 16 * q^46 - 4 * q^48 - 40 * q^51 - 8 * q^52 - 4 * q^55 + 4 * q^57 - 4 * q^58 - 24 * q^60 + 200 * q^61 + 28 * q^63 + 12 * q^70 - 68 * q^73 - 36 * q^75 + 36 * q^76 + 24 * q^78 - 92 * q^81 + 24 * q^82 + 24 * q^85 + 28 * q^87 - 4 * q^88 - 68 * q^90 + 64 * q^91 + 16 * q^93 - 4 * q^96 - 148 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1 −0.707107 + 0.707107i −1.49908 0.867616i 1.00000i −0.850241 2.06811i 1.67351 0.446512i 0.811234 + 0.811234i 0.707107 + 0.707107i 1.49448 + 2.60125i 2.06359 + 0.861165i
77.2 −0.707107 + 0.707107i −1.35438 1.07966i 1.00000i 1.25473 + 1.85086i 1.72112 0.194257i −3.56316 3.56316i 0.707107 + 0.707107i 0.668679 + 2.92453i −2.19598 0.421528i
77.3 −0.707107 + 0.707107i −1.10477 + 1.33397i 1.00000i −2.20748 0.356413i −0.162066 1.72445i 1.29452 + 1.29452i 0.707107 + 0.707107i −0.558949 2.94747i 1.81295 1.30890i
77.4 −0.707107 + 0.707107i −0.752859 + 1.55987i 1.00000i 1.50962 + 1.64956i −0.570645 1.63535i 0.306664 + 0.306664i 0.707107 + 0.707107i −1.86641 2.34873i −2.23388 0.0989549i
77.5 −0.707107 + 0.707107i −0.274986 1.71008i 1.00000i 2.05001 + 0.893014i 1.40366 + 1.01477i 2.76585 + 2.76585i 0.707107 + 0.707107i −2.84877 + 0.940499i −2.08103 + 0.818117i
77.6 −0.707107 + 0.707107i 0.668751 1.59774i 1.00000i −0.595769 + 2.15524i 0.656894 + 1.60265i −0.0702822 0.0702822i 0.707107 + 0.707107i −2.10554 2.13698i −1.10271 1.94526i
77.7 −0.707107 + 0.707107i 1.43759 + 0.966096i 1.00000i −1.74702 1.39568i −1.69966 + 0.333395i 0.371042 + 0.371042i 0.707107 + 0.707107i 1.13332 + 2.77770i 2.22222 0.248430i
77.8 −0.707107 + 0.707107i 1.52000 0.830420i 1.00000i −2.23234 + 0.129088i −0.487608 + 1.66200i −2.47472 2.47472i 0.707107 + 0.707107i 1.62081 2.52448i 1.48722 1.66978i
77.9 −0.707107 + 0.707107i 1.65263 + 0.518471i 1.00000i 2.11139 0.736229i −1.53520 + 0.801972i −2.44114 2.44114i 0.707107 + 0.707107i 2.46238 + 1.71368i −0.972386 + 2.01357i
77.10 0.707107 0.707107i −1.55987 + 0.752859i 1.00000i −1.50962 1.64956i −0.570645 + 1.63535i 0.306664 + 0.306664i −0.707107 0.707107i 1.86641 2.34873i −2.23388 0.0989549i
77.11 0.707107 0.707107i −1.33397 + 1.10477i 1.00000i 2.20748 + 0.356413i −0.162066 + 1.72445i 1.29452 + 1.29452i −0.707107 0.707107i 0.558949 2.94747i 1.81295 1.30890i
77.12 0.707107 0.707107i −0.966096 1.43759i 1.00000i 1.74702 + 1.39568i −1.69966 0.333395i 0.371042 + 0.371042i −0.707107 0.707107i −1.13332 + 2.77770i 2.22222 0.248430i
77.13 0.707107 0.707107i −0.518471 1.65263i 1.00000i −2.11139 + 0.736229i −1.53520 0.801972i −2.44114 2.44114i −0.707107 0.707107i −2.46238 + 1.71368i −0.972386 + 2.01357i
77.14 0.707107 0.707107i 0.830420 1.52000i 1.00000i 2.23234 0.129088i −0.487608 1.66200i −2.47472 2.47472i −0.707107 0.707107i −1.62081 2.52448i 1.48722 1.66978i
77.15 0.707107 0.707107i 0.867616 + 1.49908i 1.00000i 0.850241 + 2.06811i 1.67351 + 0.446512i 0.811234 + 0.811234i −0.707107 0.707107i −1.49448 + 2.60125i 2.06359 + 0.861165i
77.16 0.707107 0.707107i 1.07966 + 1.35438i 1.00000i −1.25473 1.85086i 1.72112 + 0.194257i −3.56316 3.56316i −0.707107 0.707107i −0.668679 + 2.92453i −2.19598 0.421528i
77.17 0.707107 0.707107i 1.59774 0.668751i 1.00000i 0.595769 2.15524i 0.656894 1.60265i −0.0702822 0.0702822i −0.707107 0.707107i 2.10554 2.13698i −1.10271 1.94526i
77.18 0.707107 0.707107i 1.71008 + 0.274986i 1.00000i −2.05001 0.893014i 1.40366 1.01477i 2.76585 + 2.76585i −0.707107 0.707107i 2.84877 + 0.940499i −2.08103 + 0.818117i
533.1 −0.707107 0.707107i −1.49908 + 0.867616i 1.00000i −0.850241 + 2.06811i 1.67351 + 0.446512i 0.811234 0.811234i 0.707107 0.707107i 1.49448 2.60125i 2.06359 0.861165i
533.2 −0.707107 0.707107i −1.35438 + 1.07966i 1.00000i 1.25473 1.85086i 1.72112 + 0.194257i −3.56316 + 3.56316i 0.707107 0.707107i 0.668679 2.92453i −2.19598 + 0.421528i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 533.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.k.a 36
3.b odd 2 1 inner 570.2.k.a 36
5.c odd 4 1 inner 570.2.k.a 36
15.e even 4 1 inner 570.2.k.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.k.a 36 1.a even 1 1 trivial
570.2.k.a 36 3.b odd 2 1 inner
570.2.k.a 36 5.c odd 4 1 inner
570.2.k.a 36 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{18} + 6 T_{7}^{17} + 18 T_{7}^{16} - 32 T_{7}^{15} + 238 T_{7}^{14} + 1452 T_{7}^{13} + 4940 T_{7}^{12} - 9320 T_{7}^{11} + 2505 T_{7}^{10} + 9918 T_{7}^{9} + 191090 T_{7}^{8} - 558328 T_{7}^{7} + 784912 T_{7}^{6} + \cdots + 128$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.