# Properties

 Label 8550.2.a.s Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 4 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 4 * q^7 + q^8 $$q + q^{2} + q^{4} - 4 q^{7} + q^{8} + 6 q^{13} - 4 q^{14} + q^{16} + 2 q^{17} + q^{19} + 8 q^{23} + 6 q^{26} - 4 q^{28} - 2 q^{29} - 8 q^{31} + q^{32} + 2 q^{34} - 10 q^{37} + q^{38} - 6 q^{41} + 8 q^{43} + 8 q^{46} + 9 q^{49} + 6 q^{52} - 2 q^{53} - 4 q^{56} - 2 q^{58} + 12 q^{59} - 2 q^{61} - 8 q^{62} + q^{64} + 12 q^{67} + 2 q^{68} + 8 q^{71} - 2 q^{73} - 10 q^{74} + q^{76} - 16 q^{79} - 6 q^{82} - 4 q^{83} + 8 q^{86} - 6 q^{89} - 24 q^{91} + 8 q^{92} + 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 4 * q^7 + q^8 + 6 * q^13 - 4 * q^14 + q^16 + 2 * q^17 + q^19 + 8 * q^23 + 6 * q^26 - 4 * q^28 - 2 * q^29 - 8 * q^31 + q^32 + 2 * q^34 - 10 * q^37 + q^38 - 6 * q^41 + 8 * q^43 + 8 * q^46 + 9 * q^49 + 6 * q^52 - 2 * q^53 - 4 * q^56 - 2 * q^58 + 12 * q^59 - 2 * q^61 - 8 * q^62 + q^64 + 12 * q^67 + 2 * q^68 + 8 * q^71 - 2 * q^73 - 10 * q^74 + q^76 - 16 * q^79 - 6 * q^82 - 4 * q^83 + 8 * q^86 - 6 * q^89 - 24 * q^91 + 8 * q^92 + 10 * q^97 + 9 * q^98

## 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 0 1.00000 0 0 −4.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 8550.2.a.s 1
3.b odd 2 1 2850.2.a.h 1
5.b even 2 1 1710.2.a.e 1
15.d odd 2 1 570.2.a.i 1
15.e even 4 2 2850.2.d.c 2
60.h even 2 1 4560.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.i 1 15.d odd 2 1
1710.2.a.e 1 5.b even 2 1
2850.2.a.h 1 3.b odd 2 1
2850.2.d.c 2 15.e even 4 2
4560.2.a.x 1 60.h even 2 1
8550.2.a.s 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 2$$ T17 - 2 $$T_{23} - 8$$ T23 - 8 $$T_{53} + 2$$ T53 + 2

## Hecke characteristic polynomials

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