# Properties

 Label 570.2.c.a Level $570$ Weight $2$ Character orbit 570.c Analytic conductor $4.551$ Analytic rank $0$ Dimension $4$ 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} - \zeta_{8}^{2} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9} +O(q^{10})$$ q + z^2 * q^2 + (-z^3 + z^2 + z) * q^3 - q^4 + (2*z^3 - z) * q^5 + (z^3 + z - 1) * q^6 - z^2 * q^8 + (2*z^3 + 2*z + 1) * q^9 $$q + \zeta_{8}^{2} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} - \zeta_{8}^{2} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{10} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{11} + (\zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{12} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{13} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8} - 3) q^{15} + q^{16} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} - 2 \zeta_{8}) q^{18} + (3 \zeta_{8}^{3} + 3 \zeta_{8} + 1) q^{19} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{20} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{22} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{23} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{24} + ( - 3 \zeta_{8}^{2} + 4) q^{25} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{26} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} - \zeta_{8}) q^{27} - 6 q^{29} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{30} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{31} + \zeta_{8}^{2} q^{32} + (2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{33} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{36} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{37} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} - 3 \zeta_{8}) q^{38} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{39} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{40} + 6 q^{41} + 12 \zeta_{8}^{2} q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{44} + (2 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - \zeta_{8} - 2) q^{45} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{46} + ( - 9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{47} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{48} + 7 q^{49} + (4 \zeta_{8}^{2} + 3) q^{50} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{52} - 6 \zeta_{8}^{2} q^{53} + ( - \zeta_{8}^{3} - \zeta_{8} - 5) q^{54} + ( - 6 \zeta_{8}^{2} - 2) q^{55} + (2 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 2 \zeta_{8}) q^{57} - 6 \zeta_{8}^{2} q^{58} - 6 q^{59} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8} + 3) q^{60} + 10 q^{61} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{62} - q^{64} + ( - 3 \zeta_{8}^{2} + 9) q^{65} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 4) q^{66} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{69} + 12 q^{71} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8}) q^{72} + 6 \zeta_{8}^{2} q^{73} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{74} + ( - 7 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + \zeta_{8} + 3) q^{75} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 1) q^{76} + ( - 3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 3 \zeta_{8}) q^{78} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{79} + (2 \zeta_{8}^{3} - \zeta_{8}) q^{80} + (4 \zeta_{8}^{3} + 4 \zeta_{8} - 7) q^{81} + 6 \zeta_{8}^{2} q^{82} - 12 q^{86} + (6 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 6 \zeta_{8}) q^{87} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{88} - 12 q^{89} + ( - \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} + 6) q^{90} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{92} + ( - 3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 3 \zeta_{8}) q^{93} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{94} + (2 \zeta_{8}^{3} - 9 \zeta_{8}^{2} - \zeta_{8} - 3) q^{95} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{96} + 7 \zeta_{8}^{2} q^{98} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 8) q^{99} +O(q^{100})$$ q + z^2 * q^2 + (-z^3 + z^2 + z) * q^3 - q^4 + (2*z^3 - z) * q^5 + (z^3 + z - 1) * q^6 - z^2 * q^8 + (2*z^3 + 2*z + 1) * q^9 + (-z^3 - 2*z) * q^10 + (2*z^3 + 2*z) * q^11 + (z^3 - z^2 - z) * q^12 + (3*z^3 - 3*z) * q^13 + (-z^3 + z^2 - 2*z - 3) * q^15 + q^16 + (2*z^3 + z^2 - 2*z) * q^18 + (3*z^3 + 3*z + 1) * q^19 + (-2*z^3 + z) * q^20 + (2*z^3 - 2*z) * q^22 + (3*z^3 - 3*z) * q^23 + (-z^3 - z + 1) * q^24 + (-3*z^2 + 4) * q^25 + (-3*z^3 - 3*z) * q^26 + (z^3 + 5*z^2 - z) * q^27 - 6 * q^29 + (-2*z^3 - 3*z^2 + z - 1) * q^30 + (-3*z^3 - 3*z) * q^31 + z^2 * q^32 + (2*z^3 + 4*z^2 - 2*z) * q^33 + (-2*z^3 - 2*z - 1) * q^36 + (3*z^3 - 3*z) * q^37 + (3*z^3 + z^2 - 3*z) * q^38 + (-3*z^3 - 3*z - 6) * q^39 + (z^3 + 2*z) * q^40 + 6 * q^41 + 12*z^2 * q^43 + (-2*z^3 - 2*z) * q^44 + (2*z^3 - 6*z^2 - z - 2) * q^45 + (-3*z^3 - 3*z) * q^46 + (-9*z^3 + 9*z) * q^47 + (-z^3 + z^2 + z) * q^48 + 7 * q^49 + (4*z^2 + 3) * q^50 + (-3*z^3 + 3*z) * q^52 - 6*z^2 * q^53 + (-z^3 - z - 5) * q^54 + (-6*z^2 - 2) * q^55 + (2*z^3 + 7*z^2 - 2*z) * q^57 - 6*z^2 * q^58 - 6 * q^59 + (z^3 - z^2 + 2*z + 3) * q^60 + 10 * q^61 + (-3*z^3 + 3*z) * q^62 - q^64 + (-3*z^2 + 9) * q^65 + (-2*z^3 - 2*z - 4) * q^66 + (-3*z^3 - 3*z - 6) * q^69 + 12 * q^71 + (-2*z^3 - z^2 + 2*z) * q^72 + 6*z^2 * q^73 + (-3*z^3 - 3*z) * q^74 + (-7*z^3 + 4*z^2 + z + 3) * q^75 + (-3*z^3 - 3*z - 1) * q^76 + (-3*z^3 - 6*z^2 + 3*z) * q^78 + (9*z^3 + 9*z) * q^79 + (2*z^3 - z) * q^80 + (4*z^3 + 4*z - 7) * q^81 + 6*z^2 * q^82 - 12 * q^86 + (6*z^3 - 6*z^2 - 6*z) * q^87 + (-2*z^3 + 2*z) * q^88 - 12 * q^89 + (-z^3 - 2*z^2 - 2*z + 6) * q^90 + (-3*z^3 + 3*z) * q^92 + (-3*z^3 - 6*z^2 + 3*z) * q^93 + (9*z^3 + 9*z) * q^94 + (2*z^3 - 9*z^2 - z - 3) * q^95 + (z^3 + z - 1) * q^96 + 7*z^2 * q^98 + (2*z^3 + 2*z - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 + 4 * q^9 $$4 q - 4 q^{4} - 4 q^{6} + 4 q^{9} - 12 q^{15} + 4 q^{16} + 4 q^{19} + 4 q^{24} + 16 q^{25} - 24 q^{29} - 4 q^{30} - 4 q^{36} - 24 q^{39} + 24 q^{41} - 8 q^{45} + 28 q^{49} + 12 q^{50} - 20 q^{54} - 8 q^{55} - 24 q^{59} + 12 q^{60} + 40 q^{61} - 4 q^{64} + 36 q^{65} - 16 q^{66} - 24 q^{69} + 48 q^{71} + 12 q^{75} - 4 q^{76} - 28 q^{81} - 48 q^{86} - 48 q^{89} + 24 q^{90} - 12 q^{95} - 4 q^{96} - 32 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 + 4 * q^9 - 12 * q^15 + 4 * q^16 + 4 * q^19 + 4 * q^24 + 16 * q^25 - 24 * q^29 - 4 * q^30 - 4 * q^36 - 24 * q^39 + 24 * q^41 - 8 * q^45 + 28 * q^49 + 12 * q^50 - 20 * q^54 - 8 * q^55 - 24 * q^59 + 12 * q^60 + 40 * q^61 - 4 * q^64 + 36 * q^65 - 16 * q^66 - 24 * q^69 + 48 * q^71 + 12 * q^75 - 4 * q^76 - 28 * q^81 - 48 * q^86 - 48 * q^89 + 24 * q^90 - 12 * q^95 - 4 * q^96 - 32 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i −1.41421 1.00000i −1.00000 2.12132 + 0.707107i −1.00000 + 1.41421i 0 1.00000i 1.00000 + 2.82843i 0.707107 2.12132i
569.2 1.00000i 1.41421 1.00000i −1.00000 −2.12132 0.707107i −1.00000 1.41421i 0 1.00000i 1.00000 2.82843i −0.707107 + 2.12132i
569.3 1.00000i −1.41421 + 1.00000i −1.00000 2.12132 0.707107i −1.00000 1.41421i 0 1.00000i 1.00000 2.82843i 0.707107 + 2.12132i
569.4 1.00000i 1.41421 + 1.00000i −1.00000 −2.12132 + 0.707107i −1.00000 + 1.41421i 0 1.00000i 1.00000 + 2.82843i −0.707107 2.12132i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
57.d even 2 1 inner
285.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.c.a 4
3.b odd 2 1 570.2.c.b yes 4
5.b even 2 1 inner 570.2.c.a 4
15.d odd 2 1 570.2.c.b yes 4
19.b odd 2 1 570.2.c.b yes 4
57.d even 2 1 inner 570.2.c.a 4
95.d odd 2 1 570.2.c.b yes 4
285.b even 2 1 inner 570.2.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.a 4 1.a even 1 1 trivial
570.2.c.a 4 5.b even 2 1 inner
570.2.c.a 4 57.d even 2 1 inner
570.2.c.a 4 285.b even 2 1 inner
570.2.c.b yes 4 3.b odd 2 1
570.2.c.b yes 4 15.d odd 2 1
570.2.c.b yes 4 19.b odd 2 1
570.2.c.b yes 4 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} + 8$$ T11^2 + 8 $$T_{29} + 6$$ T29 + 6 $$T_{37}^{2} - 18$$ T37^2 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$T^{4} - 8T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 8)^{2}$$
$13$ $$(T^{2} - 18)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 2 T + 19)^{2}$$
$23$ $$(T^{2} - 18)^{2}$$
$29$ $$(T + 6)^{4}$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$(T^{2} - 18)^{2}$$
$41$ $$(T - 6)^{4}$$
$43$ $$(T^{2} + 144)^{2}$$
$47$ $$(T^{2} - 162)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T + 6)^{4}$$
$61$ $$(T - 10)^{4}$$
$67$ $$T^{4}$$
$71$ $$(T - 12)^{4}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$(T^{2} + 162)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T + 12)^{4}$$
$97$ $$T^{4}$$