# Properties

 Label 8550.2.a.cf Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_{2} q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + b2 * q^7 - q^8 $$q - q^{2} + q^{4} + \beta_{2} q^{7} - q^{8} + ( - \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{2} + 2) q^{13} - \beta_{2} q^{14} + q^{16} + (\beta_1 - 3) q^{17} - q^{19} + (\beta_{2} + \beta_1 - 1) q^{22} + (\beta_{2} - 2) q^{23} + (\beta_{2} - 2) q^{26} + \beta_{2} q^{28} + (\beta_{2} + \beta_1 - 3) q^{29} + ( - \beta_{2} + \beta_1 + 5) q^{31} - q^{32} + ( - \beta_1 + 3) q^{34} + (\beta_{2} - 2 \beta_1) q^{37} + q^{38} + \beta_{2} q^{41} + ( - \beta_{2} + \beta_1 - 3) q^{43} + ( - \beta_{2} - \beta_1 + 1) q^{44} + ( - \beta_{2} + 2) q^{46} + ( - \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 2) q^{52} + (2 \beta_1 - 4) q^{53} - \beta_{2} q^{56} + ( - \beta_{2} - \beta_1 + 3) q^{58} + ( - \beta_1 - 5) q^{59} + 2 q^{61} + (\beta_{2} - \beta_1 - 5) q^{62} + q^{64} + (2 \beta_1 - 2) q^{67} + (\beta_1 - 3) q^{68} + (2 \beta_{2} + 2 \beta_1 + 2) q^{71} + (2 \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - \beta_{2} + 2 \beta_1) q^{74} - q^{76} + (4 \beta_{2} - 8) q^{77} + (\beta_{2} - \beta_1 + 7) q^{79} - \beta_{2} q^{82} + (3 \beta_1 + 1) q^{83} + (\beta_{2} - \beta_1 + 3) q^{86} + (\beta_{2} + \beta_1 - 1) q^{88} + ( - \beta_{2} - 2 \beta_1 - 2) q^{89} + (4 \beta_{2} + 2 \beta_1 - 10) q^{91} + (\beta_{2} - 2) q^{92} + (\beta_{2} + 2 \beta_1) q^{94} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{97} + (2 \beta_{2} + 2 \beta_1 - 3) q^{98}+O(q^{100})$$ q - q^2 + q^4 + b2 * q^7 - q^8 + (-b2 - b1 + 1) * q^11 + (-b2 + 2) * q^13 - b2 * q^14 + q^16 + (b1 - 3) * q^17 - q^19 + (b2 + b1 - 1) * q^22 + (b2 - 2) * q^23 + (b2 - 2) * q^26 + b2 * q^28 + (b2 + b1 - 3) * q^29 + (-b2 + b1 + 5) * q^31 - q^32 + (-b1 + 3) * q^34 + (b2 - 2*b1) * q^37 + q^38 + b2 * q^41 + (-b2 + b1 - 3) * q^43 + (-b2 - b1 + 1) * q^44 + (-b2 + 2) * q^46 + (-b2 - 2*b1) * q^47 + (-2*b2 - 2*b1 + 3) * q^49 + (-b2 + 2) * q^52 + (2*b1 - 4) * q^53 - b2 * q^56 + (-b2 - b1 + 3) * q^58 + (-b1 - 5) * q^59 + 2 * q^61 + (b2 - b1 - 5) * q^62 + q^64 + (2*b1 - 2) * q^67 + (b1 - 3) * q^68 + (2*b2 + 2*b1 + 2) * q^71 + (2*b2 + 2*b1 - 2) * q^73 + (-b2 + 2*b1) * q^74 - q^76 + (4*b2 - 8) * q^77 + (b2 - b1 + 7) * q^79 - b2 * q^82 + (3*b1 + 1) * q^83 + (b2 - b1 + 3) * q^86 + (b2 + b1 - 1) * q^88 + (-b2 - 2*b1 - 2) * q^89 + (4*b2 + 2*b1 - 10) * q^91 + (b2 - 2) * q^92 + (b2 + 2*b1) * q^94 + (-3*b2 - 3*b1 - 3) * q^97 + (2*b2 + 2*b1 - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 4 * q^11 + 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^19 - 4 * q^22 - 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 - 3 * q^32 + 10 * q^34 + 2 * q^37 + 3 * q^38 - 10 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 11 * q^49 + 6 * q^52 - 14 * q^53 + 10 * q^58 - 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 8 * q^67 - 10 * q^68 + 4 * q^71 - 8 * q^73 - 2 * q^74 - 3 * q^76 - 24 * q^77 + 22 * q^79 + 10 * q^86 - 4 * q^88 - 4 * q^89 - 32 * q^91 - 6 * q^92 - 2 * q^94 - 6 * q^97 - 11 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 2\nu - 4$$ 2*v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 5 ) / 2$$ (b2 + b1 + 5) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.311108 2.17009 −1.48119
−1.00000 0 1.00000 0 0 −4.42864 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 1.07838 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 3.35026 −1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8550.2.a.cf 3
3.b odd 2 1 2850.2.a.bn 3
5.b even 2 1 8550.2.a.cr 3
5.c odd 4 2 1710.2.d.e 6
15.d odd 2 1 2850.2.a.bk 3
15.e even 4 2 570.2.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.d 6 15.e even 4 2
1710.2.d.e 6 5.c odd 4 2
2850.2.a.bk 3 15.d odd 2 1
2850.2.a.bn 3 3.b odd 2 1
8550.2.a.cf 3 1.a even 1 1 trivial
8550.2.a.cr 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{3} - 16T_{7} + 16$$ T7^3 - 16*T7 + 16 $$T_{11}^{3} - 4T_{11}^{2} - 16T_{11} + 32$$ T11^3 - 4*T11^2 - 16*T11 + 32 $$T_{13}^{3} - 6T_{13}^{2} - 4T_{13} + 8$$ T13^3 - 6*T13^2 - 4*T13 + 8 $$T_{17}^{3} + 10T_{17}^{2} + 20T_{17} - 8$$ T17^3 + 10*T17^2 + 20*T17 - 8 $$T_{23}^{3} + 6T_{23}^{2} - 4T_{23} - 8$$ T23^3 + 6*T23^2 - 4*T23 - 8 $$T_{53}^{3} + 14T_{53}^{2} + 12T_{53} - 152$$ T53^3 + 14*T53^2 + 12*T53 - 152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 16T + 16$$
$11$ $$T^{3} - 4 T^{2} - 16 T + 32$$
$13$ $$T^{3} - 6 T^{2} - 4 T + 8$$
$17$ $$T^{3} + 10 T^{2} + 20 T - 8$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 6 T^{2} - 4 T - 8$$
$29$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$31$ $$T^{3} - 14 T^{2} + 28 T + 152$$
$37$ $$T^{3} - 2 T^{2} - 84 T - 232$$
$41$ $$T^{3} - 16T + 16$$
$43$ $$T^{3} + 10 T^{2} - 4 T - 8$$
$47$ $$T^{3} - 2 T^{2} - 52 T + 184$$
$53$ $$T^{3} + 14 T^{2} + 12 T - 152$$
$59$ $$T^{3} + 14 T^{2} + 52 T + 40$$
$61$ $$(T - 2)^{3}$$
$67$ $$T^{3} + 8 T^{2} - 32 T - 128$$
$71$ $$T^{3} - 4 T^{2} - 80 T + 64$$
$73$ $$T^{3} + 8 T^{2} - 64 T - 256$$
$79$ $$T^{3} - 22 T^{2} + 124 T - 200$$
$83$ $$T^{3} - 120T - 16$$
$89$ $$T^{3} + 4 T^{2} - 48 T + 80$$
$97$ $$T^{3} + 6 T^{2} - 180 T - 216$$