# Properties

 Label 8550.2.a.cd 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: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -\beta_{1} + \beta_{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -\beta_{1} + \beta_{2} ) q^{7} - q^{8} + ( 2 - \beta_{1} ) q^{11} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{2} ) q^{14} + q^{16} + ( 2 - \beta_{1} - \beta_{2} ) q^{17} + q^{19} + ( -2 + \beta_{1} ) q^{22} + \beta_{2} q^{23} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{26} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 5 - \beta_{1} - \beta_{2} ) q^{29} + ( 1 - 4 \beta_{1} ) q^{31} - q^{32} + ( -2 + \beta_{1} + \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} ) q^{37} - q^{38} + ( 3 + 3 \beta_{2} ) q^{41} + ( -3 - \beta_{1} ) q^{43} + ( 2 - \beta_{1} ) q^{44} -\beta_{2} q^{46} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{47} -3 \beta_{2} q^{49} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{52} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( \beta_{1} - \beta_{2} ) q^{56} + ( -5 + \beta_{1} + \beta_{2} ) q^{58} + ( 1 + 7 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -1 + 4 \beta_{1} ) q^{62} + q^{64} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} ) q^{68} + ( 8 - \beta_{1} - \beta_{2} ) q^{71} + ( -4 + 4 \beta_{1} + \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} ) q^{74} + q^{76} + ( 3 - 3 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 1 + 2 \beta_{1} ) q^{79} + ( -3 - 3 \beta_{2} ) q^{82} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{83} + ( 3 + \beta_{1} ) q^{86} + ( -2 + \beta_{1} ) q^{88} + ( 8 - 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -11 + 5 \beta_{2} ) q^{91} + \beta_{2} q^{92} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{94} + ( -6 + 5 \beta_{1} + \beta_{2} ) q^{97} + 3 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 5 q^{11} + 2 q^{14} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 5 q^{22} - q^{23} - 2 q^{28} + 15 q^{29} - q^{31} - 3 q^{32} - 6 q^{34} + 4 q^{37} - 3 q^{38} + 6 q^{41} - 10 q^{43} + 5 q^{44} + q^{46} + 3 q^{49} - 5 q^{53} + 2 q^{56} - 15 q^{58} + 12 q^{59} + q^{61} + q^{62} + 3 q^{64} - q^{67} + 6 q^{68} + 24 q^{71} - 9 q^{73} - 4 q^{74} + 3 q^{76} + 4 q^{77} + 5 q^{79} - 6 q^{82} - 11 q^{83} + 10 q^{86} - 5 q^{88} + 23 q^{89} - 38 q^{91} - q^{92} - 14 q^{97} - 3 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 0.571993 2.51414 −2.08613
−1.00000 0 1.00000 0 0 −4.24482 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −0.193252 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 2.43807 −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.cd 3
3.b odd 2 1 8550.2.a.cn yes 3
5.b even 2 1 8550.2.a.cs yes 3
15.d odd 2 1 8550.2.a.cg yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cd 3 1.a even 1 1 trivial
8550.2.a.cg yes 3 15.d odd 2 1
8550.2.a.cn yes 3 3.b odd 2 1
8550.2.a.cs yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 10 T_{7} - 2$$ $$T_{11}^{3} - 5 T_{11}^{2} + 3 T_{11} + 3$$ $$T_{13}^{3} - 36 T_{13} - 82$$ $$T_{17}^{3} - 6 T_{17}^{2} - 6 T_{17} + 54$$ $$T_{23}^{3} + T_{23}^{2} - 9 T_{23} + 3$$ $$T_{53}^{3} + 5 T_{53}^{2} - 57 T_{53} - 183$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$-2 - 10 T + 2 T^{2} + T^{3}$$
$11$ $$3 + 3 T - 5 T^{2} + T^{3}$$
$13$ $$-82 - 36 T + T^{3}$$
$17$ $$54 - 6 T - 6 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$3 - 9 T + T^{2} + T^{3}$$
$29$ $$-9 + 57 T - 15 T^{2} + T^{3}$$
$31$ $$-109 - 85 T + T^{2} + T^{3}$$
$37$ $$16 - 16 T - 4 T^{2} + T^{3}$$
$41$ $$324 - 72 T - 6 T^{2} + T^{3}$$
$43$ $$18 + 28 T + 10 T^{2} + T^{3}$$
$47$ $$36 - 24 T + T^{3}$$
$53$ $$-183 - 57 T + 5 T^{2} + T^{3}$$
$59$ $$2466 - 204 T - 12 T^{2} + T^{3}$$
$61$ $$639 - 137 T - T^{2} + T^{3}$$
$67$ $$-141 - 47 T + T^{2} + T^{3}$$
$71$ $$-342 + 174 T - 24 T^{2} + T^{3}$$
$73$ $$-541 - 81 T + 9 T^{2} + T^{3}$$
$79$ $$41 - 13 T - 5 T^{2} + T^{3}$$
$83$ $$-39 - 27 T + 11 T^{2} + T^{3}$$
$89$ $$2307 - 33 T - 23 T^{2} + T^{3}$$
$97$ $$-974 - 94 T + 14 T^{2} + T^{3}$$