# Properties

 Label 8550.2.a.bv 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: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2850) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.44949 2.44949
1.00000 0 1.00000 0 0 −4.44949 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 0.449490 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.bv 2
3.b odd 2 1 2850.2.a.bc 2
5.b even 2 1 8550.2.a.bu 2
15.d odd 2 1 2850.2.a.bj yes 2
15.e even 4 2 2850.2.d.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bc 2 3.b odd 2 1
2850.2.a.bj yes 2 15.d odd 2 1
2850.2.d.w 4 15.e even 4 2
8550.2.a.bu 2 5.b even 2 1
8550.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{2} + 4 T_{7} - 2$$ $$T_{11}^{2} + 2 T_{11} - 5$$ $$T_{13}^{2} - 6$$ $$T_{17}^{2} - 4 T_{17} - 2$$ $$T_{23} + 1$$ $$T_{53}^{2} + 10 T_{53} + 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 + 4 T + T^{2}$$
$11$ $$-5 + 2 T + T^{2}$$
$13$ $$-6 + T^{2}$$
$17$ $$-2 - 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-45 + 6 T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$-92 + 4 T + T^{2}$$
$41$ $$-8 + 8 T + T^{2}$$
$43$ $$-6 + T^{2}$$
$47$ $$-92 - 4 T + T^{2}$$
$53$ $$19 + 10 T + T^{2}$$
$59$ $$10 + 8 T + T^{2}$$
$61$ $$43 + 14 T + T^{2}$$
$67$ $$-141 + 6 T + T^{2}$$
$71$ $$10 - 8 T + T^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$( 5 + T )^{2}$$
$83$ $$-53 - 2 T + T^{2}$$
$89$ $$25 + 14 T + T^{2}$$
$97$ $$10 + 8 T + T^{2}$$