# Properties

 Label 5550.2.a.bk Level $5550$ Weight $2$ Character orbit 5550.a Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} - q^{11} + q^{12} - 2q^{13} - q^{14} + q^{16} + 7q^{17} + q^{18} - 2q^{19} - q^{21} - q^{22} + q^{24} - 2q^{26} + q^{27} - q^{28} + 9q^{29} + 7q^{31} + q^{32} - q^{33} + 7q^{34} + q^{36} + q^{37} - 2q^{38} - 2q^{39} - 11q^{41} - q^{42} + 11q^{43} - q^{44} - 8q^{47} + q^{48} - 6q^{49} + 7q^{51} - 2q^{52} + 11q^{53} + q^{54} - q^{56} - 2q^{57} + 9q^{58} - 10q^{59} - q^{61} + 7q^{62} - q^{63} + q^{64} - q^{66} + 8q^{67} + 7q^{68} + q^{72} - 4q^{73} + q^{74} - 2q^{76} + q^{77} - 2q^{78} + 12q^{79} + q^{81} - 11q^{82} + 6q^{83} - q^{84} + 11q^{86} + 9q^{87} - q^{88} + 6q^{89} + 2q^{91} + 7q^{93} - 8q^{94} + q^{96} + 19q^{97} - 6q^{98} - q^{99} + O(q^{100})$$

## 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 1.00000 1.00000 0 1.00000 −1.00000 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$37$$ $$-1$$

## 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 5550.2.a.bk 1
5.b even 2 1 1110.2.a.b 1
15.d odd 2 1 3330.2.a.x 1
20.d odd 2 1 8880.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.b 1 5.b even 2 1
3330.2.a.x 1 15.d odd 2 1
5550.2.a.bk 1 1.a even 1 1 trivial
8880.2.a.s 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5550))$$:

 $$T_{7} + 1$$ $$T_{11} + 1$$ $$T_{13} + 2$$ $$T_{17} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$1 + T$$
$13$ $$2 + T$$
$17$ $$-7 + T$$
$19$ $$2 + T$$
$23$ $$T$$
$29$ $$-9 + T$$
$31$ $$-7 + T$$
$37$ $$-1 + T$$
$41$ $$11 + T$$
$43$ $$-11 + T$$
$47$ $$8 + T$$
$53$ $$-11 + T$$
$59$ $$10 + T$$
$61$ $$1 + T$$
$67$ $$-8 + T$$
$71$ $$T$$
$73$ $$4 + T$$
$79$ $$-12 + T$$
$83$ $$-6 + T$$
$89$ $$-6 + T$$
$97$ $$-19 + T$$
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