# Properties

 Label 5550.2.a.bd 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

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

## 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 4.00000 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.bd 1
5.b even 2 1 1110.2.a.h 1
15.d odd 2 1 3330.2.a.m 1
20.d odd 2 1 8880.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.h 1 5.b even 2 1
3330.2.a.m 1 15.d odd 2 1
5550.2.a.bd 1 1.a even 1 1 trivial
8880.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5550))$$:

 $$T_{7} - 4$$ $$T_{11} - 6$$ $$T_{13} + 2$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$-6 + T$$
$13$ $$2 + T$$
$17$ $$-6 + T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$-8 + T$$
$37$ $$1 + T$$
$41$ $$6 + T$$
$43$ $$8 + T$$
$47$ $$6 + T$$
$53$ $$6 + T$$
$59$ $$12 + T$$
$61$ $$-8 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$14 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$8 + T$$