# Properties

 Label 8550.2.a.cm 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: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7 x - 3$$ 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.

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

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

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

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cc 3 3.b odd 2 1
8550.2.a.ch yes 3 5.b even 2 1
8550.2.a.cm yes 3 1.a even 1 1 trivial
8550.2.a.ct yes 3 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 6 T_{7} - 10$$ $$T_{11}^{3} + 5 T_{11}^{2} - 5 T_{11} - 27$$ $$T_{13}^{3} + 4 T_{13}^{2} - 8 T_{13} - 6$$ $$T_{17}^{3} - 2 T_{17}^{2} - 6 T_{17} + 10$$ $$T_{23}^{3} + T_{23}^{2} - 45 T_{23} - 81$$ $$T_{53}^{3} - 5 T_{53}^{2} - 73 T_{53} - 117$$

## Hecke characteristic polynomials

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