# Properties

 Label 550.2.a.a 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: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ 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.

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 1.a even 1 1 trivial
550.2.a.m yes 1 5.b even 2 1
550.2.b.d 2 5.c odd 4 2
4400.2.a.d 1 20.d odd 2 1
4400.2.a.bc 1 4.b odd 2 1
4400.2.b.e 2 20.e even 4 2
4950.2.a.u 1 15.d odd 2 1
4950.2.a.y 1 3.b odd 2 1
4950.2.c.ba 2 15.e even 4 2
6050.2.a.n 1 55.d odd 2 1
6050.2.a.bb 1 11.b 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

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