# Properties

 Label 550.2.a.h Level $550$ Weight $2$ Character orbit 550.a Self dual yes Analytic conductor $4.392$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} + q^{4} - 3 q^{6} - q^{7} + q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 + q^4 - 3 * q^6 - q^7 + q^8 + 6 * q^9 $$q + q^{2} - 3 q^{3} + q^{4} - 3 q^{6} - q^{7} + q^{8} + 6 q^{9} - q^{11} - 3 q^{12} - q^{14} + q^{16} - 5 q^{17} + 6 q^{18} - 7 q^{19} + 3 q^{21} - q^{22} - 8 q^{23} - 3 q^{24} - 9 q^{27} - q^{28} + 3 q^{29} - 5 q^{31} + q^{32} + 3 q^{33} - 5 q^{34} + 6 q^{36} - q^{37} - 7 q^{38} - 8 q^{41} + 3 q^{42} + 10 q^{43} - q^{44} - 8 q^{46} - 3 q^{48} - 6 q^{49} + 15 q^{51} - q^{53} - 9 q^{54} - q^{56} + 21 q^{57} + 3 q^{58} + 12 q^{59} + 5 q^{61} - 5 q^{62} - 6 q^{63} + q^{64} + 3 q^{66} - 4 q^{67} - 5 q^{68} + 24 q^{69} - 7 q^{71} + 6 q^{72} + 2 q^{73} - q^{74} - 7 q^{76} + q^{77} - 4 q^{79} + 9 q^{81} - 8 q^{82} + 3 q^{84} + 10 q^{86} - 9 q^{87} - q^{88} - 7 q^{89} - 8 q^{92} + 15 q^{93} - 3 q^{96} + 8 q^{97} - 6 q^{98} - 6 q^{99}+O(q^{100})$$ q + q^2 - 3 * q^3 + q^4 - 3 * q^6 - q^7 + q^8 + 6 * q^9 - q^11 - 3 * q^12 - q^14 + q^16 - 5 * q^17 + 6 * q^18 - 7 * q^19 + 3 * q^21 - q^22 - 8 * q^23 - 3 * q^24 - 9 * q^27 - q^28 + 3 * q^29 - 5 * q^31 + q^32 + 3 * q^33 - 5 * q^34 + 6 * q^36 - q^37 - 7 * q^38 - 8 * q^41 + 3 * q^42 + 10 * q^43 - q^44 - 8 * q^46 - 3 * q^48 - 6 * q^49 + 15 * q^51 - q^53 - 9 * q^54 - q^56 + 21 * q^57 + 3 * q^58 + 12 * q^59 + 5 * q^61 - 5 * q^62 - 6 * q^63 + q^64 + 3 * q^66 - 4 * q^67 - 5 * q^68 + 24 * q^69 - 7 * q^71 + 6 * q^72 + 2 * q^73 - q^74 - 7 * q^76 + q^77 - 4 * q^79 + 9 * q^81 - 8 * q^82 + 3 * q^84 + 10 * q^86 - 9 * q^87 - q^88 - 7 * q^89 - 8 * q^92 + 15 * q^93 - 3 * q^96 + 8 * q^97 - 6 * q^98 - 6 * 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 −3.00000 1.00000 0 −3.00000 −1.00000 1.00000 6.00000 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.h 1
3.b odd 2 1 4950.2.a.h 1
4.b odd 2 1 4400.2.a.bd 1
5.b even 2 1 550.2.a.g 1
5.c odd 4 2 110.2.b.c 2
11.b odd 2 1 6050.2.a.b 1
15.d odd 2 1 4950.2.a.bn 1
15.e even 4 2 990.2.c.b 2
20.d odd 2 1 4400.2.a.b 1
20.e even 4 2 880.2.b.f 2
55.d odd 2 1 6050.2.a.bo 1
55.e even 4 2 1210.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.c 2 5.c odd 4 2
550.2.a.g 1 5.b even 2 1
550.2.a.h 1 1.a even 1 1 trivial
880.2.b.f 2 20.e even 4 2
990.2.c.b 2 15.e even 4 2
1210.2.b.e 2 55.e even 4 2
4400.2.a.b 1 20.d odd 2 1
4400.2.a.bd 1 4.b odd 2 1
4950.2.a.h 1 3.b odd 2 1
4950.2.a.bn 1 15.d odd 2 1
6050.2.a.b 1 11.b odd 2 1
6050.2.a.bo 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(550))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{7} + 1$$ T7 + 1 $$T_{13}$$ T13

## Hecke characteristic polynomials

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