# Properties

 Label 8550.2.a.cu Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.93185 −0.517638 0.517638 1.93185
−1.00000 0 1.00000 0 0 −3.86370 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −1.03528 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 1.03528 −1.00000 0 0
1.4 −1.00000 0 1.00000 0 0 3.86370 −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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8550.2.a.cu 4
3.b odd 2 1 8550.2.a.cv 4
5.b even 2 1 8550.2.a.cv 4
5.c odd 4 2 1710.2.d.g 8
15.d odd 2 1 inner 8550.2.a.cu 4
15.e even 4 2 1710.2.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.d.g 8 5.c odd 4 2
1710.2.d.g 8 15.e even 4 2
8550.2.a.cu 4 1.a even 1 1 trivial
8550.2.a.cu 4 15.d odd 2 1 inner
8550.2.a.cv 4 3.b odd 2 1
8550.2.a.cv 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{4} - 16 T_{7}^{2} + 16$$ $$T_{11}^{4} - 16 T_{11}^{2} + 16$$ $$T_{13}^{4} - 16 T_{13}^{2} + 16$$ $$T_{17}^{2} - 8 T_{17} + 4$$ $$T_{23}^{2} - 4 T_{23} - 8$$ $$T_{53} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 16 T^{2} + T^{4}$$
$11$ $$16 - 16 T^{2} + T^{4}$$
$13$ $$16 - 16 T^{2} + T^{4}$$
$17$ $$( 4 - 8 T + T^{2} )^{2}$$
$19$ $$( 1 + T )^{4}$$
$23$ $$( -8 - 4 T + T^{2} )^{2}$$
$29$ $$1936 - 112 T^{2} + T^{4}$$
$31$ $$( -32 + 8 T + T^{2} )^{2}$$
$37$ $$144 - 48 T^{2} + T^{4}$$
$41$ $$144 - 48 T^{2} + T^{4}$$
$43$ $$144 - 48 T^{2} + T^{4}$$
$47$ $$( 24 - 12 T + T^{2} )^{2}$$
$53$ $$( -6 + T )^{4}$$
$59$ $$7744 - 208 T^{2} + T^{4}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$2304 - 192 T^{2} + T^{4}$$
$71$ $$4096 - 256 T^{2} + T^{4}$$
$73$ $$2304 - 192 T^{2} + T^{4}$$
$79$ $$( 4 + T )^{4}$$
$83$ $$( -72 - 12 T + T^{2} )^{2}$$
$89$ $$16 - 16 T^{2} + T^{4}$$
$97$ $$1936 - 304 T^{2} + T^{4}$$