# Properties

 Label 8550.2.a.bw Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1710) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
1.00000 0 1.00000 0 0 −2.00000 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −2.00000 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.bw 2
3.b odd 2 1 8550.2.a.bo 2
5.b even 2 1 1710.2.a.u 2
15.d odd 2 1 1710.2.a.y yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.u 2 5.b even 2 1
1710.2.a.y yes 2 15.d odd 2 1
8550.2.a.bo 2 3.b odd 2 1
8550.2.a.bw 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7} + 2$$ $$T_{11}^{2} - 12$$ $$T_{13}^{2} + 4 T_{13} - 8$$ $$T_{17}^{2} - 12$$ $$T_{23}^{2} - 48$$ $$T_{53}^{2} + 12 T_{53} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-48 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-8 - 4 T + T^{2}$$
$37$ $$-8 + 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$-8 + 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-12 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$16 + 16 T + T^{2}$$
$71$ $$-48 + T^{2}$$
$73$ $$-44 + 4 T + T^{2}$$
$79$ $$-8 - 4 T + T^{2}$$
$83$ $$24 + 12 T + T^{2}$$
$89$ $$-48 + T^{2}$$
$97$ $$-44 + 16 T + T^{2}$$