Properties

 Label 8550.2.a.bs 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: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 1 + \beta ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 1 + \beta ) q^{7} - q^{8} + ( -2 + \beta ) q^{11} + ( 1 - \beta ) q^{13} + ( -1 - \beta ) q^{14} + q^{16} + ( -1 + 3 \beta ) q^{17} - q^{19} + ( 2 - \beta ) q^{22} + ( -1 - 4 \beta ) q^{23} + ( -1 + \beta ) q^{26} + ( 1 + \beta ) q^{28} -\beta q^{29} + ( 1 + 2 \beta ) q^{31} - q^{32} + ( 1 - 3 \beta ) q^{34} + 2 q^{37} + q^{38} + ( -4 - 4 \beta ) q^{41} + ( 3 - 3 \beta ) q^{43} + ( -2 + \beta ) q^{44} + ( 1 + 4 \beta ) q^{46} -2 \beta q^{47} + ( -3 + 2 \beta ) q^{49} + ( 1 - \beta ) q^{52} + \beta q^{53} + ( -1 - \beta ) q^{56} + \beta q^{58} + ( 3 - 3 \beta ) q^{59} + ( 2 - 5 \beta ) q^{61} + ( -1 - 2 \beta ) q^{62} + q^{64} + ( 2 + \beta ) q^{67} + ( -1 + 3 \beta ) q^{68} + ( -7 + 3 \beta ) q^{71} + ( -1 - 2 \beta ) q^{73} -2 q^{74} - q^{76} + ( 1 - \beta ) q^{77} + ( -9 - 2 \beta ) q^{79} + ( 4 + 4 \beta ) q^{82} + ( 2 - \beta ) q^{83} + ( -3 + 3 \beta ) q^{86} + ( 2 - \beta ) q^{88} + ( 3 + 8 \beta ) q^{89} -2 q^{91} + ( -1 - 4 \beta ) q^{92} + 2 \beta q^{94} + ( 3 - 7 \beta ) q^{97} + ( 3 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} - 4q^{11} + 2q^{13} - 2q^{14} + 2q^{16} - 2q^{17} - 2q^{19} + 4q^{22} - 2q^{23} - 2q^{26} + 2q^{28} + 2q^{31} - 2q^{32} + 2q^{34} + 4q^{37} + 2q^{38} - 8q^{41} + 6q^{43} - 4q^{44} + 2q^{46} - 6q^{49} + 2q^{52} - 2q^{56} + 6q^{59} + 4q^{61} - 2q^{62} + 2q^{64} + 4q^{67} - 2q^{68} - 14q^{71} - 2q^{73} - 4q^{74} - 2q^{76} + 2q^{77} - 18q^{79} + 8q^{82} + 4q^{83} - 6q^{86} + 4q^{88} + 6q^{89} - 4q^{91} - 2q^{92} + 6q^{97} + 6q^{98} + O(q^{100})$$

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
 −1.73205 1.73205
−1.00000 0 1.00000 0 0 −0.732051 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.73205 −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.bs 2
3.b odd 2 1 2850.2.a.bi yes 2
5.b even 2 1 8550.2.a.by 2
15.d odd 2 1 2850.2.a.bd 2
15.e even 4 2 2850.2.d.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bd 2 15.d odd 2 1
2850.2.a.bi yes 2 3.b odd 2 1
2850.2.d.x 4 15.e even 4 2
8550.2.a.bs 2 1.a even 1 1 trivial
8550.2.a.by 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11}^{2} + 4 T_{11} + 1$$ $$T_{13}^{2} - 2 T_{13} - 2$$ $$T_{17}^{2} + 2 T_{17} - 26$$ $$T_{23}^{2} + 2 T_{23} - 47$$ $$T_{53}^{2} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 - 2 T + T^{2}$$
$11$ $$1 + 4 T + T^{2}$$
$13$ $$-2 - 2 T + T^{2}$$
$17$ $$-26 + 2 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-47 + 2 T + T^{2}$$
$29$ $$-3 + T^{2}$$
$31$ $$-11 - 2 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-32 + 8 T + T^{2}$$
$43$ $$-18 - 6 T + T^{2}$$
$47$ $$-12 + T^{2}$$
$53$ $$-3 + T^{2}$$
$59$ $$-18 - 6 T + T^{2}$$
$61$ $$-71 - 4 T + T^{2}$$
$67$ $$1 - 4 T + T^{2}$$
$71$ $$22 + 14 T + T^{2}$$
$73$ $$-11 + 2 T + T^{2}$$
$79$ $$69 + 18 T + T^{2}$$
$83$ $$1 - 4 T + T^{2}$$
$89$ $$-183 - 6 T + T^{2}$$
$97$ $$-138 - 6 T + T^{2}$$