# Properties

 Label 8550.2.a.ct Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( - \beta_1 + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (-b1 + 1) * q^7 + q^8 $$q + q^{2} + q^{4} + ( - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{11} + ( - \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_1 + 1) q^{14} + q^{16} + ( - \beta_1 + 1) q^{17} - q^{19} + (\beta_{2} + \beta_1 + 1) q^{22} + (\beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{2} - \beta_1 + 2) q^{26} + ( - \beta_1 + 1) q^{28} + ( - 2 \beta_{2} + \beta_1 + 2) q^{29} + (2 \beta_1 + 1) q^{31} + q^{32} + ( - \beta_1 + 1) q^{34} + (2 \beta_{2} + 2 \beta_1) q^{37} - q^{38} + (\beta_{2} + 3) q^{41} + (3 \beta_{2} - \beta_1) q^{43} + (\beta_{2} + \beta_1 + 1) q^{44} + (\beta_{2} - 2 \beta_1) q^{46} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{47} + (\beta_{2} - 2) q^{49} + ( - \beta_{2} - \beta_1 + 2) q^{52} + ( - 3 \beta_{2} - \beta_1 + 3) q^{53} + ( - \beta_1 + 1) q^{56} + ( - 2 \beta_{2} + \beta_1 + 2) q^{58} + ( - \beta_{2} + \beta_1 + 4) q^{59} + (3 \beta_{2} + \beta_1 - 1) q^{61} + (2 \beta_1 + 1) q^{62} + q^{64} + ( - 4 \beta_{2} + \beta_1 + 4) q^{67} + ( - \beta_1 + 1) q^{68} + ( - 2 \beta_{2} - 3 \beta_1 + 7) q^{71} + ( - 3 \beta_{2} + 6) q^{73} + (2 \beta_{2} + 2 \beta_1) q^{74} - q^{76} + (\beta_{2} - 3 \beta_1 - 2) q^{77} + (2 \beta_{2} + 4 \beta_1 - 5) q^{79} + (\beta_{2} + 3) q^{82} + ( - 2 \beta_{2} - \beta_1) q^{83} + (3 \beta_{2} - \beta_1) q^{86} + (\beta_{2} + \beta_1 + 1) q^{88} + (3 \beta_{2} + 2) q^{89} + ( - \beta_{2} + 5) q^{91} + (\beta_{2} - 2 \beta_1) q^{92} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{94} + (3 \beta_1 - 3) q^{97} + (\beta_{2} - 2) q^{98}+O(q^{100})$$ q + q^2 + q^4 + (-b1 + 1) * q^7 + q^8 + (b2 + b1 + 1) * q^11 + (-b2 - b1 + 2) * q^13 + (-b1 + 1) * q^14 + q^16 + (-b1 + 1) * q^17 - q^19 + (b2 + b1 + 1) * q^22 + (b2 - 2*b1) * q^23 + (-b2 - b1 + 2) * q^26 + (-b1 + 1) * q^28 + (-2*b2 + b1 + 2) * q^29 + (2*b1 + 1) * q^31 + q^32 + (-b1 + 1) * q^34 + (2*b2 + 2*b1) * q^37 - q^38 + (b2 + 3) * q^41 + (3*b2 - b1) * q^43 + (b2 + b1 + 1) * q^44 + (b2 - 2*b1) * q^46 + (-3*b2 - 2*b1 + 3) * q^47 + (b2 - 2) * q^49 + (-b2 - b1 + 2) * q^52 + (-3*b2 - b1 + 3) * q^53 + (-b1 + 1) * q^56 + (-2*b2 + b1 + 2) * q^58 + (-b2 + b1 + 4) * q^59 + (3*b2 + b1 - 1) * q^61 + (2*b1 + 1) * q^62 + q^64 + (-4*b2 + b1 + 4) * q^67 + (-b1 + 1) * q^68 + (-2*b2 - 3*b1 + 7) * q^71 + (-3*b2 + 6) * q^73 + (2*b2 + 2*b1) * q^74 - q^76 + (b2 - 3*b1 - 2) * q^77 + (2*b2 + 4*b1 - 5) * q^79 + (b2 + 3) * q^82 + (-2*b2 - b1) * q^83 + (3*b2 - b1) * q^86 + (b2 + b1 + 1) * q^88 + (3*b2 + 2) * q^89 + (-b2 + 5) * q^91 + (b2 - 2*b1) * q^92 + (-3*b2 - 2*b1 + 3) * q^94 + (3*b1 - 3) * q^97 + (b2 - 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8} + 5 q^{11} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} + 5 q^{22} - q^{23} + 4 q^{26} + 2 q^{28} + 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{37} - 3 q^{38} + 10 q^{41} + 2 q^{43} + 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} + 4 q^{52} + 5 q^{53} + 2 q^{56} + 5 q^{58} + 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{68} + 16 q^{71} + 15 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} + 10 q^{82} - 3 q^{83} + 2 q^{86} + 5 q^{88} + 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} - 6 q^{97} - 5 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 + 5 * q^11 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 + 5 * q^22 - q^23 + 4 * q^26 + 2 * q^28 + 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^37 - 3 * q^38 + 10 * q^41 + 2 * q^43 + 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 + 4 * q^52 + 5 * q^53 + 2 * q^56 + 5 * q^58 + 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^68 + 16 * q^71 + 15 * q^73 + 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 + 10 * q^82 - 3 * q^83 + 2 * q^86 + 5 * q^88 + 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 - 6 * q^97 - 5 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## 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
 3.35386 −0.476452 −1.87740
1.00000 0 1.00000 0 0 −2.35386 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 1.47645 1.00000 0 0
1.3 1.00000 0 1.00000 0 0 2.87740 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.ct yes 3
3.b odd 2 1 8550.2.a.ch yes 3
5.b even 2 1 8550.2.a.cc 3
15.d odd 2 1 8550.2.a.cm yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cc 3 5.b even 2 1
8550.2.a.ch yes 3 3.b odd 2 1
8550.2.a.cm yes 3 15.d odd 2 1
8550.2.a.ct yes 3 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{7}^{3} - 2T_{7}^{2} - 6T_{7} + 10$$ T7^3 - 2*T7^2 - 6*T7 + 10 $$T_{11}^{3} - 5T_{11}^{2} - 5T_{11} + 27$$ T11^3 - 5*T11^2 - 5*T11 + 27 $$T_{13}^{3} - 4T_{13}^{2} - 8T_{13} + 6$$ T13^3 - 4*T13^2 - 8*T13 + 6 $$T_{17}^{3} - 2T_{17}^{2} - 6T_{17} + 10$$ T17^3 - 2*T17^2 - 6*T17 + 10 $$T_{23}^{3} + T_{23}^{2} - 45T_{23} - 81$$ T23^3 + T23^2 - 45*T23 - 81 $$T_{53}^{3} - 5T_{53}^{2} - 73T_{53} - 117$$ T53^3 - 5*T53^2 - 73*T53 - 117

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} - 6 T + 10$$
$11$ $$T^{3} - 5 T^{2} - 5 T + 27$$
$13$ $$T^{3} - 4 T^{2} - 8 T + 6$$
$17$ $$T^{3} - 2 T^{2} - 6 T + 10$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + T^{2} - 45 T - 81$$
$29$ $$T^{3} - 5 T^{2} - 43 T + 197$$
$31$ $$T^{3} - 5 T^{2} - 21 T + 1$$
$37$ $$T^{3} - 4 T^{2} - 48 T + 144$$
$41$ $$T^{3} - 10 T^{2} + 24 T - 4$$
$43$ $$T^{3} - 2 T^{2} - 100 T - 162$$
$47$ $$T^{3} - 4 T^{2} - 88 T - 204$$
$53$ $$T^{3} - 5 T^{2} - 73 T - 117$$
$59$ $$T^{3} - 12 T^{2} + 28 T + 50$$
$61$ $$T^{3} - T^{2} - 81 T + 275$$
$67$ $$T^{3} - 9 T^{2} - 143 T + 845$$
$71$ $$T^{3} - 16 T^{2} + 2 T + 354$$
$73$ $$T^{3} - 15 T^{2} - 9 T + 243$$
$79$ $$T^{3} + 9 T^{2} - 101 T - 709$$
$83$ $$T^{3} + 3 T^{2} - 35 T - 127$$
$89$ $$T^{3} - 9 T^{2} - 57 T + 277$$
$97$ $$T^{3} + 6 T^{2} - 54 T - 270$$