# Properties

 Label 5225.2.a.a Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ 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] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 - q^4 + 3 * q^8 - 3 * q^9 $$q - q^{2} - q^{4} + 3 q^{8} - 3 q^{9} - q^{11} - 2 q^{13} - q^{16} + 6 q^{17} + 3 q^{18} + q^{19} + q^{22} + 8 q^{23} + 2 q^{26} - 6 q^{29} + 4 q^{31} - 5 q^{32} - 6 q^{34} + 3 q^{36} + 2 q^{37} - q^{38} - 10 q^{41} - 4 q^{43} + q^{44} - 8 q^{46} - 7 q^{49} + 2 q^{52} + 2 q^{53} + 6 q^{58} - 8 q^{59} + 14 q^{61} - 4 q^{62} + 7 q^{64} - 8 q^{67} - 6 q^{68} - 4 q^{71} - 9 q^{72} - 2 q^{73} - 2 q^{74} - q^{76} - 16 q^{79} + 9 q^{81} + 10 q^{82} + 4 q^{83} + 4 q^{86} - 3 q^{88} + 10 q^{89} - 8 q^{92} - 10 q^{97} + 7 q^{98} + 3 q^{99}+O(q^{100})$$ q - q^2 - q^4 + 3 * q^8 - 3 * q^9 - q^11 - 2 * q^13 - q^16 + 6 * q^17 + 3 * q^18 + q^19 + q^22 + 8 * q^23 + 2 * q^26 - 6 * q^29 + 4 * q^31 - 5 * q^32 - 6 * q^34 + 3 * q^36 + 2 * q^37 - q^38 - 10 * q^41 - 4 * q^43 + q^44 - 8 * q^46 - 7 * q^49 + 2 * q^52 + 2 * q^53 + 6 * q^58 - 8 * q^59 + 14 * q^61 - 4 * q^62 + 7 * q^64 - 8 * q^67 - 6 * q^68 - 4 * q^71 - 9 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 16 * q^79 + 9 * q^81 + 10 * q^82 + 4 * q^83 + 4 * q^86 - 3 * q^88 + 10 * q^89 - 8 * q^92 - 10 * q^97 + 7 * q^98 + 3 * 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 0 −1.00000 0 0 0 3.00000 −3.00000 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
$$5$$ $$+1$$
$$11$$ $$+1$$
$$19$$ $$-1$$

## 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 5225.2.a.a 1
5.b even 2 1 1045.2.a.b 1
15.d odd 2 1 9405.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.b 1 5.b even 2 1
5225.2.a.a 1 1.a even 1 1 trivial
9405.2.a.d 1 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T - 1$$
$23$ $$T - 8$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 2$$
$59$ $$T + 8$$
$61$ $$T - 14$$
$67$ $$T + 8$$
$71$ $$T + 4$$
$73$ $$T + 2$$
$79$ $$T + 16$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T + 10$$