Properties

 Label 5225.2.a.b Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} + 4 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 + 4 * q^7 - 2 * q^9 $$q - q^{3} - 2 q^{4} + 4 q^{7} - 2 q^{9} + q^{11} + 2 q^{12} - 2 q^{13} + 4 q^{16} + q^{19} - 4 q^{21} - 3 q^{23} + 5 q^{27} - 8 q^{28} - 6 q^{29} - 7 q^{31} - q^{33} + 4 q^{36} + 7 q^{37} + 2 q^{39} + 10 q^{43} - 2 q^{44} - 4 q^{48} + 9 q^{49} + 4 q^{52} - 6 q^{53} - q^{57} + 3 q^{59} - 10 q^{61} - 8 q^{63} - 8 q^{64} - 11 q^{67} + 3 q^{69} + 15 q^{71} - 8 q^{73} - 2 q^{76} + 4 q^{77} - 16 q^{79} + q^{81} + 8 q^{84} + 6 q^{87} + 9 q^{89} - 8 q^{91} + 6 q^{92} + 7 q^{93} + q^{97} - 2 q^{99}+O(q^{100})$$ q - q^3 - 2 * q^4 + 4 * q^7 - 2 * q^9 + q^11 + 2 * q^12 - 2 * q^13 + 4 * q^16 + q^19 - 4 * q^21 - 3 * q^23 + 5 * q^27 - 8 * q^28 - 6 * q^29 - 7 * q^31 - q^33 + 4 * q^36 + 7 * q^37 + 2 * q^39 + 10 * q^43 - 2 * q^44 - 4 * q^48 + 9 * q^49 + 4 * q^52 - 6 * q^53 - q^57 + 3 * q^59 - 10 * q^61 - 8 * q^63 - 8 * q^64 - 11 * q^67 + 3 * q^69 + 15 * q^71 - 8 * q^73 - 2 * q^76 + 4 * q^77 - 16 * q^79 + q^81 + 8 * q^84 + 6 * q^87 + 9 * q^89 - 8 * q^91 + 6 * q^92 + 7 * q^93 + q^97 - 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
0 −1.00000 −2.00000 0 0 4.00000 0 −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
$$5$$ $$1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

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.b 1
5.b even 2 1 209.2.a.a 1
15.d odd 2 1 1881.2.a.c 1
20.d odd 2 1 3344.2.a.d 1
55.d odd 2 1 2299.2.a.c 1
95.d odd 2 1 3971.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.a 1 5.b even 2 1
1881.2.a.c 1 15.d odd 2 1
2299.2.a.c 1 55.d odd 2 1
3344.2.a.d 1 20.d odd 2 1
3971.2.a.a 1 95.d odd 2 1
5225.2.a.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}$$ T2 $$T_{7} - 4$$ T7 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T - 1$$
$23$ $$T + 3$$
$29$ $$T + 6$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T$$
$43$ $$T - 10$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T + 10$$
$67$ $$T + 11$$
$71$ $$T - 15$$
$73$ $$T + 8$$
$79$ $$T + 16$$
$83$ $$T$$
$89$ $$T - 9$$
$97$ $$T - 1$$