# Properties

 Label 550.2.a.i 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - 5 q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 - 5 * q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} - q^{6} - 5 q^{7} + q^{8} - 2 q^{9} + q^{11} - q^{12} - 2 q^{13} - 5 q^{14} + q^{16} - 3 q^{17} - 2 q^{18} - 7 q^{19} + 5 q^{21} + q^{22} + 6 q^{23} - q^{24} - 2 q^{26} + 5 q^{27} - 5 q^{28} - 3 q^{29} - 7 q^{31} + q^{32} - q^{33} - 3 q^{34} - 2 q^{36} + 7 q^{37} - 7 q^{38} + 2 q^{39} + 6 q^{41} + 5 q^{42} - 8 q^{43} + q^{44} + 6 q^{46} - 6 q^{47} - q^{48} + 18 q^{49} + 3 q^{51} - 2 q^{52} + 3 q^{53} + 5 q^{54} - 5 q^{56} + 7 q^{57} - 3 q^{58} - 6 q^{59} - q^{61} - 7 q^{62} + 10 q^{63} + q^{64} - q^{66} - 8 q^{67} - 3 q^{68} - 6 q^{69} + 3 q^{71} - 2 q^{72} - 2 q^{73} + 7 q^{74} - 7 q^{76} - 5 q^{77} + 2 q^{78} - 10 q^{79} + q^{81} + 6 q^{82} + 6 q^{83} + 5 q^{84} - 8 q^{86} + 3 q^{87} + q^{88} + 9 q^{89} + 10 q^{91} + 6 q^{92} + 7 q^{93} - 6 q^{94} - q^{96} + 4 q^{97} + 18 q^{98} - 2 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 - 5 * q^7 + q^8 - 2 * q^9 + q^11 - q^12 - 2 * q^13 - 5 * q^14 + q^16 - 3 * q^17 - 2 * q^18 - 7 * q^19 + 5 * q^21 + q^22 + 6 * q^23 - q^24 - 2 * q^26 + 5 * q^27 - 5 * q^28 - 3 * q^29 - 7 * q^31 + q^32 - q^33 - 3 * q^34 - 2 * q^36 + 7 * q^37 - 7 * q^38 + 2 * q^39 + 6 * q^41 + 5 * q^42 - 8 * q^43 + q^44 + 6 * q^46 - 6 * q^47 - q^48 + 18 * q^49 + 3 * q^51 - 2 * q^52 + 3 * q^53 + 5 * q^54 - 5 * q^56 + 7 * q^57 - 3 * q^58 - 6 * q^59 - q^61 - 7 * q^62 + 10 * q^63 + q^64 - q^66 - 8 * q^67 - 3 * q^68 - 6 * q^69 + 3 * q^71 - 2 * q^72 - 2 * q^73 + 7 * q^74 - 7 * q^76 - 5 * q^77 + 2 * q^78 - 10 * q^79 + q^81 + 6 * q^82 + 6 * q^83 + 5 * q^84 - 8 * q^86 + 3 * q^87 + q^88 + 9 * q^89 + 10 * q^91 + 6 * q^92 + 7 * q^93 - 6 * q^94 - q^96 + 4 * q^97 + 18 * q^98 - 2 * 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 −1.00000 1.00000 0 −1.00000 −5.00000 1.00000 −2.00000 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.i 1
3.b odd 2 1 4950.2.a.a 1
4.b odd 2 1 4400.2.a.w 1
5.b even 2 1 110.2.a.a 1
5.c odd 4 2 550.2.b.b 2
11.b odd 2 1 6050.2.a.i 1
15.d odd 2 1 990.2.a.l 1
15.e even 4 2 4950.2.c.a 2
20.d odd 2 1 880.2.a.c 1
20.e even 4 2 4400.2.b.g 2
35.c odd 2 1 5390.2.a.h 1
40.e odd 2 1 3520.2.a.z 1
40.f even 2 1 3520.2.a.l 1
55.d odd 2 1 1210.2.a.k 1
60.h even 2 1 7920.2.a.s 1
220.g even 2 1 9680.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 5.b even 2 1
550.2.a.i 1 1.a even 1 1 trivial
550.2.b.b 2 5.c odd 4 2
880.2.a.c 1 20.d odd 2 1
990.2.a.l 1 15.d odd 2 1
1210.2.a.k 1 55.d odd 2 1
3520.2.a.l 1 40.f even 2 1
3520.2.a.z 1 40.e odd 2 1
4400.2.a.w 1 4.b odd 2 1
4400.2.b.g 2 20.e even 4 2
4950.2.a.a 1 3.b odd 2 1
4950.2.c.a 2 15.e even 4 2
5390.2.a.h 1 35.c odd 2 1
6050.2.a.i 1 11.b odd 2 1
7920.2.a.s 1 60.h even 2 1
9680.2.a.j 1 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(550))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} + 5$$ T7 + 5 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 5$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T + 7$$
$23$ $$T - 6$$
$29$ $$T + 3$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T - 6$$
$43$ $$T + 8$$
$47$ $$T + 6$$
$53$ $$T - 3$$
$59$ $$T + 6$$
$61$ $$T + 1$$
$67$ $$T + 8$$
$71$ $$T - 3$$
$73$ $$T + 2$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T - 9$$
$97$ $$T - 4$$