# Properties

 Label 550.2.a.m Level $550$ Weight $2$ Character orbit 550.a Self dual yes Analytic conductor $4.392$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.39177211117$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + 4 * q^7 + q^8 + q^9 $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + 4 q^{7} + q^{8} + q^{9} - q^{11} + 2 q^{12} - 5 q^{13} + 4 q^{14} + q^{16} + q^{18} - 7 q^{19} + 8 q^{21} - q^{22} - 3 q^{23} + 2 q^{24} - 5 q^{26} - 4 q^{27} + 4 q^{28} + 3 q^{29} + 5 q^{31} + q^{32} - 2 q^{33} + q^{36} + 4 q^{37} - 7 q^{38} - 10 q^{39} + 12 q^{41} + 8 q^{42} - 5 q^{43} - q^{44} - 3 q^{46} + 2 q^{48} + 9 q^{49} - 5 q^{52} - 6 q^{53} - 4 q^{54} + 4 q^{56} - 14 q^{57} + 3 q^{58} + 12 q^{59} - 10 q^{61} + 5 q^{62} + 4 q^{63} + q^{64} - 2 q^{66} - 14 q^{67} - 6 q^{69} + 3 q^{71} + q^{72} - 8 q^{73} + 4 q^{74} - 7 q^{76} - 4 q^{77} - 10 q^{78} - 4 q^{79} - 11 q^{81} + 12 q^{82} + 15 q^{83} + 8 q^{84} - 5 q^{86} + 6 q^{87} - q^{88} + 3 q^{89} - 20 q^{91} - 3 q^{92} + 10 q^{93} + 2 q^{96} + 13 q^{97} + 9 q^{98} - q^{99}+O(q^{100})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + 4 * q^7 + q^8 + q^9 - q^11 + 2 * q^12 - 5 * q^13 + 4 * q^14 + q^16 + q^18 - 7 * q^19 + 8 * q^21 - q^22 - 3 * q^23 + 2 * q^24 - 5 * q^26 - 4 * q^27 + 4 * q^28 + 3 * q^29 + 5 * q^31 + q^32 - 2 * q^33 + q^36 + 4 * q^37 - 7 * q^38 - 10 * q^39 + 12 * q^41 + 8 * q^42 - 5 * q^43 - q^44 - 3 * q^46 + 2 * q^48 + 9 * q^49 - 5 * q^52 - 6 * q^53 - 4 * q^54 + 4 * q^56 - 14 * q^57 + 3 * q^58 + 12 * q^59 - 10 * q^61 + 5 * q^62 + 4 * q^63 + q^64 - 2 * q^66 - 14 * q^67 - 6 * q^69 + 3 * q^71 + q^72 - 8 * q^73 + 4 * q^74 - 7 * q^76 - 4 * q^77 - 10 * q^78 - 4 * q^79 - 11 * q^81 + 12 * q^82 + 15 * q^83 + 8 * q^84 - 5 * q^86 + 6 * q^87 - q^88 + 3 * q^89 - 20 * q^91 - 3 * q^92 + 10 * q^93 + 2 * q^96 + 13 * q^97 + 9 * q^98 - q^99

## 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 2.00000 1.00000 0 2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 550.2.a.m yes 1
3.b odd 2 1 4950.2.a.u 1
4.b odd 2 1 4400.2.a.d 1
5.b even 2 1 550.2.a.a 1
5.c odd 4 2 550.2.b.d 2
11.b odd 2 1 6050.2.a.n 1
15.d odd 2 1 4950.2.a.y 1
15.e even 4 2 4950.2.c.ba 2
20.d odd 2 1 4400.2.a.bc 1
20.e even 4 2 4400.2.b.e 2
55.d odd 2 1 6050.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 5.b even 2 1
550.2.a.m yes 1 1.a even 1 1 trivial
550.2.b.d 2 5.c odd 4 2
4400.2.a.d 1 4.b odd 2 1
4400.2.a.bc 1 20.d odd 2 1
4400.2.b.e 2 20.e even 4 2
4950.2.a.u 1 3.b odd 2 1
4950.2.a.y 1 15.d odd 2 1
4950.2.c.ba 2 15.e even 4 2
6050.2.a.n 1 11.b odd 2 1
6050.2.a.bb 1 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(550))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} - 4$$ T7 - 4 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T + 1$$
$13$ $$T + 5$$
$17$ $$T$$
$19$ $$T + 7$$
$23$ $$T + 3$$
$29$ $$T - 3$$
$31$ $$T - 5$$
$37$ $$T - 4$$
$41$ $$T - 12$$
$43$ $$T + 5$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T + 14$$
$71$ $$T - 3$$
$73$ $$T + 8$$
$79$ $$T + 4$$
$83$ $$T - 15$$
$89$ $$T - 3$$
$97$ $$T - 13$$