# Properties

 Label 2205.2.a.j Level $2205$ Weight $2$ Character orbit 2205.a Self dual yes Analytic conductor $17.607$ 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] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$17.6070136457$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} - q^{5}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 - q^5 $$q + 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{10} - q^{11} - 3 q^{13} - 4 q^{16} - 3 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} + q^{29} - 6 q^{31} - 8 q^{32} - 6 q^{34} - 12 q^{38} + 6 q^{41} - 6 q^{43} - 2 q^{44} + 8 q^{46} - 9 q^{47} + 2 q^{50} - 6 q^{52} + 10 q^{53} + q^{55} + 2 q^{58} - 6 q^{59} - 12 q^{62} - 8 q^{64} + 3 q^{65} - 14 q^{67} - 6 q^{68} + 8 q^{71} - 6 q^{73} - 12 q^{76} - q^{79} + 4 q^{80} + 12 q^{82} + 12 q^{83} + 3 q^{85} - 12 q^{86} + 12 q^{89} + 8 q^{92} - 18 q^{94} + 6 q^{95} + 15 q^{97}+O(q^{100})$$ q + 2 * q^2 + 2 * q^4 - q^5 - 2 * q^10 - q^11 - 3 * q^13 - 4 * q^16 - 3 * q^17 - 6 * q^19 - 2 * q^20 - 2 * q^22 + 4 * q^23 + q^25 - 6 * q^26 + q^29 - 6 * q^31 - 8 * q^32 - 6 * q^34 - 12 * q^38 + 6 * q^41 - 6 * q^43 - 2 * q^44 + 8 * q^46 - 9 * q^47 + 2 * q^50 - 6 * q^52 + 10 * q^53 + q^55 + 2 * q^58 - 6 * q^59 - 12 * q^62 - 8 * q^64 + 3 * q^65 - 14 * q^67 - 6 * q^68 + 8 * q^71 - 6 * q^73 - 12 * q^76 - q^79 + 4 * q^80 + 12 * q^82 + 12 * q^83 + 3 * q^85 - 12 * q^86 + 12 * q^89 + 8 * q^92 - 18 * q^94 + 6 * q^95 + 15 * q^97

## 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
2.00000 0 2.00000 −1.00000 0 0 0 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 2205.2.a.j 1
3.b odd 2 1 245.2.a.a 1
7.b odd 2 1 2205.2.a.l 1
12.b even 2 1 3920.2.a.bj 1
15.d odd 2 1 1225.2.a.j 1
15.e even 4 2 1225.2.b.b 2
21.c even 2 1 245.2.a.b yes 1
21.g even 6 2 245.2.e.c 2
21.h odd 6 2 245.2.e.d 2
84.h odd 2 1 3920.2.a.a 1
105.g even 2 1 1225.2.a.h 1
105.k odd 4 2 1225.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 3.b odd 2 1
245.2.a.b yes 1 21.c even 2 1
245.2.e.c 2 21.g even 6 2
245.2.e.d 2 21.h odd 6 2
1225.2.a.h 1 105.g even 2 1
1225.2.a.j 1 15.d odd 2 1
1225.2.b.a 2 105.k odd 4 2
1225.2.b.b 2 15.e even 4 2
2205.2.a.j 1 1.a even 1 1 trivial
2205.2.a.l 1 7.b odd 2 1
3920.2.a.a 1 84.h odd 2 1
3920.2.a.bj 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{11} + 1$$ T11 + 1 $$T_{13} + 3$$ T13 + 3 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 3$$
$17$ $$T + 3$$
$19$ $$T + 6$$
$23$ $$T - 4$$
$29$ $$T - 1$$
$31$ $$T + 6$$
$37$ $$T$$
$41$ $$T - 6$$
$43$ $$T + 6$$
$47$ $$T + 9$$
$53$ $$T - 10$$
$59$ $$T + 6$$
$61$ $$T$$
$67$ $$T + 14$$
$71$ $$T - 8$$
$73$ $$T + 6$$
$79$ $$T + 1$$
$83$ $$T - 12$$
$89$ $$T - 12$$
$97$ $$T - 15$$