# Properties

 Label 570.2.k.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 4 q^{3} + 4 q^{6} + 20 q^{7}+O(q^{10})$$ 36 * q + 4 * q^3 + 4 * q^6 + 20 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 4 q^{3} + 4 q^{6} + 20 q^{7} - 4 q^{10} - 4 q^{12} + 8 q^{13} + 4 q^{15} - 36 q^{16} + 16 q^{21} - 4 q^{22} + 16 q^{25} - 44 q^{27} + 20 q^{28} + 32 q^{30} - 24 q^{31} - 4 q^{33} + 4 q^{36} - 8 q^{40} + 12 q^{42} - 8 q^{43} + 28 q^{45} - 16 q^{46} - 4 q^{48} + 40 q^{51} - 8 q^{52} - 36 q^{55} - 4 q^{57} + 44 q^{58} + 16 q^{60} - 120 q^{61} - 12 q^{63} + 80 q^{67} - 36 q^{70} + 44 q^{73} + 4 q^{75} - 36 q^{76} - 64 q^{78} + 36 q^{81} + 8 q^{82} - 24 q^{85} - 28 q^{87} - 4 q^{88} + 44 q^{90} - 72 q^{93} - 4 q^{96} + 92 q^{97}+O(q^{100})$$ 36 * q + 4 * q^3 + 4 * q^6 + 20 * q^7 - 4 * q^10 - 4 * q^12 + 8 * q^13 + 4 * q^15 - 36 * q^16 + 16 * q^21 - 4 * q^22 + 16 * q^25 - 44 * q^27 + 20 * q^28 + 32 * q^30 - 24 * q^31 - 4 * q^33 + 4 * q^36 - 8 * q^40 + 12 * q^42 - 8 * q^43 + 28 * q^45 - 16 * q^46 - 4 * q^48 + 40 * q^51 - 8 * q^52 - 36 * q^55 - 4 * q^57 + 44 * q^58 + 16 * q^60 - 120 * q^61 - 12 * q^63 + 80 * q^67 - 36 * q^70 + 44 * q^73 + 4 * q^75 - 36 * q^76 - 64 * q^78 + 36 * q^81 + 8 * q^82 - 24 * q^85 - 28 * q^87 - 4 * q^88 + 44 * q^90 - 72 * q^93 - 4 * q^96 + 92 * 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 −1.69896 0.336929i 1.00000i 2.14526 0.630765i 1.43959 0.963104i 0.804978 + 0.804978i 0.707107 + 0.707107i 2.77296 + 1.14486i −1.07091 + 1.96295i
77.2 −0.707107 + 0.707107i −1.63140 + 0.581841i 1.00000i −1.36872 + 1.76822i 0.742149 1.56500i 0.102829 + 0.102829i 0.707107 + 0.707107i 2.32292 1.89843i −0.282491 2.21815i
77.3 −0.707107 + 0.707107i −1.32879 1.11099i 1.00000i −1.28568 + 1.82949i 1.72519 0.154007i 1.30605 + 1.30605i 0.707107 + 0.707107i 0.531382 + 2.95256i −0.384532 2.20276i
77.4 −0.707107 + 0.707107i −0.0916341 1.72963i 1.00000i −2.17182 0.532166i 1.28782 + 1.15823i 1.32508 + 1.32508i 0.707107 + 0.707107i −2.98321 + 0.316985i 1.91201 1.15941i
77.5 −0.707107 + 0.707107i 0.492817 + 1.66046i 1.00000i −1.86278 + 1.23695i −1.52260 0.825649i −3.33681 3.33681i 0.707107 + 0.707107i −2.51426 + 1.63661i 0.442526 2.19184i
77.6 −0.707107 + 0.707107i 0.577604 + 1.63290i 1.00000i 2.23603 0.0135681i −1.56306 0.746209i 3.38538 + 3.38538i 0.707107 + 0.707107i −2.33275 + 1.88634i −1.57152 + 1.59070i
77.7 −0.707107 + 0.707107i 0.901679 1.47884i 1.00000i 1.61979 1.54152i 0.408116 + 1.68328i −1.95232 1.95232i 0.707107 + 0.707107i −1.37395 2.66688i −0.0553424 + 2.23538i
77.8 −0.707107 + 0.707107i 1.34719 1.08861i 1.00000i −0.539236 2.17007i −0.182842 + 1.72237i 2.86820 + 2.86820i 0.707107 + 0.707107i 0.629843 2.93314i 1.91577 + 1.15318i
77.9 −0.707107 + 0.707107i 1.72439 + 0.162692i 1.00000i 0.520052 + 2.17475i −1.33437 + 1.10429i 0.496606 + 0.496606i 0.707107 + 0.707107i 2.94706 + 0.561090i −1.90551 1.17005i
77.10 0.707107 0.707107i −1.66046 0.492817i 1.00000i 1.86278 1.23695i −1.52260 + 0.825649i −3.33681 3.33681i −0.707107 0.707107i 2.51426 + 1.63661i 0.442526 2.19184i
77.11 0.707107 0.707107i −1.63290 0.577604i 1.00000i −2.23603 + 0.0135681i −1.56306 + 0.746209i 3.38538 + 3.38538i −0.707107 0.707107i 2.33275 + 1.88634i −1.57152 + 1.59070i
77.12 0.707107 0.707107i −0.581841 + 1.63140i 1.00000i 1.36872 1.76822i 0.742149 + 1.56500i 0.102829 + 0.102829i −0.707107 0.707107i −2.32292 1.89843i −0.282491 2.21815i
77.13 0.707107 0.707107i −0.162692 1.72439i 1.00000i −0.520052 2.17475i −1.33437 1.10429i 0.496606 + 0.496606i −0.707107 0.707107i −2.94706 + 0.561090i −1.90551 1.17005i
77.14 0.707107 0.707107i 0.336929 + 1.69896i 1.00000i −2.14526 + 0.630765i 1.43959 + 0.963104i 0.804978 + 0.804978i −0.707107 0.707107i −2.77296 + 1.14486i −1.07091 + 1.96295i
77.15 0.707107 0.707107i 1.08861 1.34719i 1.00000i 0.539236 + 2.17007i −0.182842 1.72237i 2.86820 + 2.86820i −0.707107 0.707107i −0.629843 2.93314i 1.91577 + 1.15318i
77.16 0.707107 0.707107i 1.11099 + 1.32879i 1.00000i 1.28568 1.82949i 1.72519 + 0.154007i 1.30605 + 1.30605i −0.707107 0.707107i −0.531382 + 2.95256i −0.384532 2.20276i
77.17 0.707107 0.707107i 1.47884 0.901679i 1.00000i −1.61979 + 1.54152i 0.408116 1.68328i −1.95232 1.95232i −0.707107 0.707107i 1.37395 2.66688i −0.0553424 + 2.23538i
77.18 0.707107 0.707107i 1.72963 + 0.0916341i 1.00000i 2.17182 + 0.532166i 1.28782 1.15823i 1.32508 + 1.32508i −0.707107 0.707107i 2.98321 + 0.316985i 1.91201 1.15941i
533.1 −0.707107 0.707107i −1.69896 + 0.336929i 1.00000i 2.14526 + 0.630765i 1.43959 + 0.963104i 0.804978 0.804978i 0.707107 0.707107i 2.77296 1.14486i −1.07091 1.96295i
533.2 −0.707107 0.707107i −1.63140 0.581841i 1.00000i −1.36872 1.76822i 0.742149 + 1.56500i 0.102829 0.102829i 0.707107 0.707107i 2.32292 + 1.89843i −0.282491 + 2.21815i
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.b 36
3.b odd 2 1 inner 570.2.k.b 36
5.c odd 4 1 inner 570.2.k.b 36
15.e even 4 1 inner 570.2.k.b 36

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{18} - 10 T_{7}^{17} + 50 T_{7}^{16} - 136 T_{7}^{15} + 798 T_{7}^{14} - 5852 T_{7}^{13} + 27868 T_{7}^{12} - 75632 T_{7}^{11} + 164009 T_{7}^{10} - 458282 T_{7}^{9} + 1552610 T_{7}^{8} - 4009000 T_{7}^{7} + \cdots + 10368$$ acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$.