# Properties

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

# Related objects

## 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + 5q^{11} + q^{12} + q^{14} + q^{16} + q^{17} + q^{18} + q^{21} + 5q^{22} + 4q^{23} + q^{24} + q^{27} + q^{28} - 3q^{29} + q^{31} + q^{32} + 5q^{33} + q^{34} + q^{36} - q^{37} - q^{41} + q^{42} + 7q^{43} + 5q^{44} + 4q^{46} - 4q^{47} + q^{48} - 6q^{49} + q^{51} + 3q^{53} + q^{54} + q^{56} - 3q^{58} - 8q^{59} + 5q^{61} + q^{62} + q^{63} + q^{64} + 5q^{66} - 4q^{67} + q^{68} + 4q^{69} - 6q^{71} + q^{72} + 10q^{73} - q^{74} + 5q^{77} + q^{81} - q^{82} - 8q^{83} + q^{84} + 7q^{86} - 3q^{87} + 5q^{88} + 8q^{89} + 4q^{92} + q^{93} - 4q^{94} + q^{96} - q^{97} - 6q^{98} + 5q^{99} + O(q^{100})$$

## 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.bo 1
5.b even 2 1 5550.2.a.d 1
5.c odd 4 2 1110.2.d.c 2
15.e even 4 2 3330.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.c 2 5.c odd 4 2
3330.2.d.d 2 15.e even 4 2
5550.2.a.d 1 5.b even 2 1
5550.2.a.bo 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5550))$$:

 $$T_{7} - 1$$ $$T_{11} - 5$$ $$T_{13}$$ $$T_{17} - 1$$

## Hecke characteristic polynomials

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