# Properties

 Label 550.2.a.k 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,2,Mod(1,550)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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: no (minimal twist has level 110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^8 + q^9 $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{8} + q^{9} + q^{11} + 2 q^{12} - 2 q^{13} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + q^{22} - 2 q^{23} + 2 q^{24} - 2 q^{26} - 4 q^{27} - 10 q^{29} - 8 q^{31} + q^{32} + 2 q^{33} + 6 q^{34} + q^{36} + 8 q^{37} + 4 q^{38} - 4 q^{39} - 2 q^{41} + q^{44} - 2 q^{46} + 2 q^{47} + 2 q^{48} - 7 q^{49} + 12 q^{51} - 2 q^{52} - 4 q^{54} + 8 q^{57} - 10 q^{58} - 12 q^{59} - 10 q^{61} - 8 q^{62} + q^{64} + 2 q^{66} - 6 q^{67} + 6 q^{68} - 4 q^{69} + q^{72} + 6 q^{73} + 8 q^{74} + 4 q^{76} - 4 q^{78} + 12 q^{79} - 11 q^{81} - 2 q^{82} + 16 q^{83} - 20 q^{87} + q^{88} + 18 q^{89} - 2 q^{92} - 16 q^{93} + 2 q^{94} + 2 q^{96} - 12 q^{97} - 7 q^{98} + q^{99}+O(q^{100})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^8 + q^9 + q^11 + 2 * q^12 - 2 * q^13 + q^16 + 6 * q^17 + q^18 + 4 * q^19 + q^22 - 2 * q^23 + 2 * q^24 - 2 * q^26 - 4 * q^27 - 10 * q^29 - 8 * q^31 + q^32 + 2 * q^33 + 6 * q^34 + q^36 + 8 * q^37 + 4 * q^38 - 4 * q^39 - 2 * q^41 + q^44 - 2 * q^46 + 2 * q^47 + 2 * q^48 - 7 * q^49 + 12 * q^51 - 2 * q^52 - 4 * q^54 + 8 * q^57 - 10 * q^58 - 12 * q^59 - 10 * q^61 - 8 * q^62 + q^64 + 2 * q^66 - 6 * q^67 + 6 * q^68 - 4 * q^69 + q^72 + 6 * q^73 + 8 * q^74 + 4 * q^76 - 4 * q^78 + 12 * q^79 - 11 * q^81 - 2 * q^82 + 16 * q^83 - 20 * q^87 + q^88 + 18 * q^89 - 2 * q^92 - 16 * q^93 + 2 * q^94 + 2 * q^96 - 12 * q^97 - 7 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0 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.k 1
3.b odd 2 1 4950.2.a.j 1
4.b odd 2 1 4400.2.a.f 1
5.b even 2 1 550.2.a.c 1
5.c odd 4 2 110.2.b.b 2
11.b odd 2 1 6050.2.a.q 1
15.d odd 2 1 4950.2.a.bj 1
15.e even 4 2 990.2.c.c 2
20.d odd 2 1 4400.2.a.ba 1
20.e even 4 2 880.2.b.e 2
55.d odd 2 1 6050.2.a.x 1
55.e even 4 2 1210.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.b 2 5.c odd 4 2
550.2.a.c 1 5.b even 2 1
550.2.a.k 1 1.a even 1 1 trivial
880.2.b.e 2 20.e even 4 2
990.2.c.c 2 15.e even 4 2
1210.2.b.d 2 55.e even 4 2
4400.2.a.f 1 4.b odd 2 1
4400.2.a.ba 1 20.d odd 2 1
4950.2.a.j 1 3.b odd 2 1
4950.2.a.bj 1 15.d odd 2 1
6050.2.a.q 1 11.b odd 2 1
6050.2.a.x 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 $$T_{7}$$ T7 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T - 4$$
$23$ $$T + 2$$
$29$ $$T + 10$$
$31$ $$T + 8$$
$37$ $$T - 8$$
$41$ $$T + 2$$
$43$ $$T$$
$47$ $$T - 2$$
$53$ $$T$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T + 6$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T - 12$$
$83$ $$T - 16$$
$89$ $$T - 18$$
$97$ $$T + 12$$