# Properties

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

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## 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−1.00000 0 1.00000 0 0 −5.12311 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 3.12311 −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.bp 2
3.b odd 2 1 8550.2.a.bx 2
5.b even 2 1 1710.2.a.x yes 2
15.d odd 2 1 1710.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.v 2 15.d odd 2 1
1710.2.a.x yes 2 5.b even 2 1
8550.2.a.bp 2 1.a even 1 1 trivial
8550.2.a.bx 2 3.b odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 2 T_{7} - 16$$ $$T_{11} - 2$$ $$T_{13} + 4$$ $$T_{17}^{2} - 2 T_{17} - 16$$ $$T_{23}$$ $$T_{53}^{2} + 8 T_{53} - 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-16 + 2 T + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$-16 - 2 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$8 - 10 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-16 + 2 T + T^{2}$$
$43$ $$-64 + 4 T + T^{2}$$
$47$ $$-64 + 4 T + T^{2}$$
$53$ $$-52 + 8 T + T^{2}$$
$59$ $$-16 - 2 T + T^{2}$$
$61$ $$-52 - 8 T + T^{2}$$
$67$ $$-64 - 4 T + T^{2}$$
$71$ $$-64 + 4 T + T^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$8 - 10 T + T^{2}$$
$83$ $$104 + 22 T + T^{2}$$
$89$ $$-152 - 2 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$
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