Properties

 Label 8550.2.a.o Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + 2q^{7} - q^{8} + 2q^{11} - 2q^{14} + q^{16} - 2q^{17} + q^{19} - 2q^{22} - 8q^{23} + 2q^{28} - q^{32} + 2q^{34} - 4q^{37} - q^{38} + 8q^{41} + 6q^{43} + 2q^{44} + 8q^{46} - 8q^{47} - 3q^{49} - 10q^{53} - 2q^{56} + 8q^{59} + 2q^{61} + q^{64} - 2q^{68} - 8q^{71} + 2q^{73} + 4q^{74} + q^{76} + 4q^{77} - 8q^{79} - 8q^{82} - 16q^{83} - 6q^{86} - 2q^{88} - 16q^{89} - 8q^{92} + 8q^{94} - 8q^{97} + 3q^{98} + O(q^{100})$$

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
 0
−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.o 1
3.b odd 2 1 2850.2.a.ba 1
5.b even 2 1 1710.2.a.n 1
15.d odd 2 1 570.2.a.c 1
15.e even 4 2 2850.2.d.n 2
60.h even 2 1 4560.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.c 1 15.d odd 2 1
1710.2.a.n 1 5.b even 2 1
2850.2.a.ba 1 3.b odd 2 1
2850.2.d.n 2 15.e even 4 2
4560.2.a.bd 1 60.h even 2 1
8550.2.a.o 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7} - 2$$ $$T_{11} - 2$$ $$T_{13}$$ $$T_{17} + 2$$ $$T_{23} + 8$$ $$T_{53} + 10$$

Hecke characteristic polynomials

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