# Properties

 Label 2205.2.a.t Level $2205$ Weight $2$ Character orbit 2205.a Self dual yes Analytic conductor $17.607$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{5} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 - q^5 - 2*b * q^8 $$q + \beta q^{2} - q^{5} - 2 \beta q^{8} - \beta q^{10} + ( - 2 \beta + 3) q^{11} + (\beta + 3) q^{13} - 4 q^{16} + ( - 3 \beta + 1) q^{17} + 6 q^{19} + (3 \beta - 4) q^{22} + ( - \beta - 6) q^{23} + q^{25} + (3 \beta + 2) q^{26} + (4 \beta + 3) q^{29} + (3 \beta + 6) q^{31} + (\beta - 6) q^{34} + (3 \beta - 2) q^{37} + 6 \beta q^{38} + 2 \beta q^{40} + (3 \beta + 2) q^{41} + 2 q^{43} + ( - 6 \beta - 2) q^{46} + (3 \beta + 3) q^{47} + \beta q^{50} - 3 \beta q^{53} + (2 \beta - 3) q^{55} + (3 \beta + 8) q^{58} + (3 \beta - 2) q^{59} - 2 \beta q^{61} + (6 \beta + 6) q^{62} + 8 q^{64} + ( - \beta - 3) q^{65} + ( - 3 \beta - 4) q^{67} + ( - 2 \beta + 6) q^{71} - 6 \beta q^{73} + ( - 2 \beta + 6) q^{74} + (6 \beta - 7) q^{79} + 4 q^{80} + (2 \beta + 6) q^{82} + (3 \beta - 1) q^{85} + 2 \beta q^{86} + ( - 6 \beta + 8) q^{88} + 8 q^{89} + (3 \beta + 6) q^{94} - 6 q^{95} + (3 \beta + 9) q^{97} +O(q^{100})$$ q + b * q^2 - q^5 - 2*b * q^8 - b * q^10 + (-2*b + 3) * q^11 + (b + 3) * q^13 - 4 * q^16 + (-3*b + 1) * q^17 + 6 * q^19 + (3*b - 4) * q^22 + (-b - 6) * q^23 + q^25 + (3*b + 2) * q^26 + (4*b + 3) * q^29 + (3*b + 6) * q^31 + (b - 6) * q^34 + (3*b - 2) * q^37 + 6*b * q^38 + 2*b * q^40 + (3*b + 2) * q^41 + 2 * q^43 + (-6*b - 2) * q^46 + (3*b + 3) * q^47 + b * q^50 - 3*b * q^53 + (2*b - 3) * q^55 + (3*b + 8) * q^58 + (3*b - 2) * q^59 - 2*b * q^61 + (6*b + 6) * q^62 + 8 * q^64 + (-b - 3) * q^65 + (-3*b - 4) * q^67 + (-2*b + 6) * q^71 - 6*b * q^73 + (-2*b + 6) * q^74 + (6*b - 7) * q^79 + 4 * q^80 + (2*b + 6) * q^82 + (3*b - 1) * q^85 + 2*b * q^86 + (-6*b + 8) * q^88 + 8 * q^89 + (3*b + 6) * q^94 - 6 * q^95 + (3*b + 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} + 6 q^{11} + 6 q^{13} - 8 q^{16} + 2 q^{17} + 12 q^{19} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 6 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 4 q^{46} + 6 q^{47} - 6 q^{55} + 16 q^{58} - 4 q^{59} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} - 14 q^{79} + 8 q^{80} + 12 q^{82} - 2 q^{85} + 16 q^{88} + 16 q^{89} + 12 q^{94} - 12 q^{95} + 18 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^11 + 6 * q^13 - 8 * q^16 + 2 * q^17 + 12 * q^19 - 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 6 * q^29 + 12 * q^31 - 12 * q^34 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 4 * q^46 + 6 * q^47 - 6 * q^55 + 16 * q^58 - 4 * q^59 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 - 14 * q^79 + 8 * q^80 + 12 * q^82 - 2 * q^85 + 16 * q^88 + 16 * q^89 + 12 * q^94 - 12 * q^95 + 18 * 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
 −1.41421 1.41421
−1.41421 0 0 −1.00000 0 0 2.82843 0 1.41421
1.2 1.41421 0 0 −1.00000 0 0 −2.82843 0 −1.41421
 $$n$$: 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.t 2
3.b odd 2 1 245.2.a.f yes 2
7.b odd 2 1 2205.2.a.v 2
12.b even 2 1 3920.2.a.br 2
15.d odd 2 1 1225.2.a.p 2
15.e even 4 2 1225.2.b.j 4
21.c even 2 1 245.2.a.e 2
21.g even 6 2 245.2.e.g 4
21.h odd 6 2 245.2.e.f 4
84.h odd 2 1 3920.2.a.bw 2
105.g even 2 1 1225.2.a.r 2
105.k odd 4 2 1225.2.b.i 4

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

## Hecke kernels

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

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{11}^{2} - 6T_{11} + 1$$ T11^2 - 6*T11 + 1 $$T_{13}^{2} - 6T_{13} + 7$$ T13^2 - 6*T13 + 7 $$T_{17}^{2} - 2T_{17} - 17$$ T17^2 - 2*T17 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 1$$
$13$ $$T^{2} - 6T + 7$$
$17$ $$T^{2} - 2T - 17$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 12T + 34$$
$29$ $$T^{2} - 6T - 23$$
$31$ $$T^{2} - 12T + 18$$
$37$ $$T^{2} + 4T - 14$$
$41$ $$T^{2} - 4T - 14$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 6T - 9$$
$53$ $$T^{2} - 18$$
$59$ $$T^{2} + 4T - 14$$
$61$ $$T^{2} - 8$$
$67$ $$T^{2} + 8T - 2$$
$71$ $$T^{2} - 12T + 28$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} + 14T - 23$$
$83$ $$T^{2}$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} - 18T + 63$$