# Properties

 Label 5550.2.a.t Level $5550$ Weight $2$ Character orbit 5550.a Self dual yes Analytic conductor $44.317$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 3 q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 3 q^{7} - q^{8} + q^{9} - 5 q^{11} + q^{12} + 2 q^{13} - 3 q^{14} + q^{16} - 3 q^{17} - q^{18} - 6 q^{19} + 3 q^{21} + 5 q^{22} + 4 q^{23} - q^{24} - 2 q^{26} + q^{27} + 3 q^{28} - q^{29} - 3 q^{31} - q^{32} - 5 q^{33} + 3 q^{34} + q^{36} + q^{37} + 6 q^{38} + 2 q^{39} - 7 q^{41} - 3 q^{42} - 3 q^{43} - 5 q^{44} - 4 q^{46} + q^{48} + 2 q^{49} - 3 q^{51} + 2 q^{52} - 5 q^{53} - q^{54} - 3 q^{56} - 6 q^{57} + q^{58} + 6 q^{59} + 5 q^{61} + 3 q^{62} + 3 q^{63} + q^{64} + 5 q^{66} + 4 q^{67} - 3 q^{68} + 4 q^{69} - 12 q^{71} - q^{72} - q^{74} - 6 q^{76} - 15 q^{77} - 2 q^{78} - 4 q^{79} + q^{81} + 7 q^{82} - 6 q^{83} + 3 q^{84} + 3 q^{86} - q^{87} + 5 q^{88} - 18 q^{89} + 6 q^{91} + 4 q^{92} - 3 q^{93} - q^{96} + 13 q^{97} - 2 q^{98} - 5 q^{99} + O(q^{100})$$

## 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 3.00000 −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.t 1
5.b even 2 1 1110.2.a.j 1
15.d odd 2 1 3330.2.a.b 1
20.d odd 2 1 8880.2.a.bc 1

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

 $$T_{7} - 3$$ $$T_{11} + 5$$ $$T_{13} - 2$$ $$T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-3 + T$$
$11$ $$5 + T$$
$13$ $$-2 + T$$
$17$ $$3 + T$$
$19$ $$6 + T$$
$23$ $$-4 + T$$
$29$ $$1 + T$$
$31$ $$3 + T$$
$37$ $$-1 + T$$
$41$ $$7 + T$$
$43$ $$3 + T$$
$47$ $$T$$
$53$ $$5 + T$$
$59$ $$-6 + T$$
$61$ $$-5 + T$$
$67$ $$-4 + T$$
$71$ $$12 + T$$
$73$ $$T$$
$79$ $$4 + T$$
$83$ $$6 + T$$
$89$ $$18 + T$$
$97$ $$-13 + T$$