# Properties

 Label 2205.4.a.g Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ 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] = [2205,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$130.099211563$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 5 q^{5} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 5 * q^5 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 5 q^{5} + 21 q^{8} - 15 q^{10} + 45 q^{11} - 59 q^{13} - 71 q^{16} - 54 q^{17} + 121 q^{19} + 5 q^{20} - 135 q^{22} - 69 q^{23} + 25 q^{25} + 177 q^{26} + 162 q^{29} + 88 q^{31} + 45 q^{32} + 162 q^{34} - 259 q^{37} - 363 q^{38} + 105 q^{40} + 195 q^{41} - 286 q^{43} + 45 q^{44} + 207 q^{46} + 45 q^{47} - 75 q^{50} - 59 q^{52} - 597 q^{53} + 225 q^{55} - 486 q^{58} - 360 q^{59} - 392 q^{61} - 264 q^{62} + 433 q^{64} - 295 q^{65} - 280 q^{67} - 54 q^{68} - 48 q^{71} - 668 q^{73} + 777 q^{74} + 121 q^{76} + 782 q^{79} - 355 q^{80} - 585 q^{82} + 768 q^{83} - 270 q^{85} + 858 q^{86} + 945 q^{88} - 1194 q^{89} - 69 q^{92} - 135 q^{94} + 605 q^{95} - 902 q^{97}+O(q^{100})$$ q - 3 * q^2 + q^4 + 5 * q^5 + 21 * q^8 - 15 * q^10 + 45 * q^11 - 59 * q^13 - 71 * q^16 - 54 * q^17 + 121 * q^19 + 5 * q^20 - 135 * q^22 - 69 * q^23 + 25 * q^25 + 177 * q^26 + 162 * q^29 + 88 * q^31 + 45 * q^32 + 162 * q^34 - 259 * q^37 - 363 * q^38 + 105 * q^40 + 195 * q^41 - 286 * q^43 + 45 * q^44 + 207 * q^46 + 45 * q^47 - 75 * q^50 - 59 * q^52 - 597 * q^53 + 225 * q^55 - 486 * q^58 - 360 * q^59 - 392 * q^61 - 264 * q^62 + 433 * q^64 - 295 * q^65 - 280 * q^67 - 54 * q^68 - 48 * q^71 - 668 * q^73 + 777 * q^74 + 121 * q^76 + 782 * q^79 - 355 * q^80 - 585 * q^82 + 768 * q^83 - 270 * q^85 + 858 * q^86 + 945 * q^88 - 1194 * q^89 - 69 * q^92 - 135 * q^94 + 605 * q^95 - 902 * 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
−3.00000 0 1.00000 5.00000 0 0 21.0000 0 −15.0000
 $$n$$: 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 2205.4.a.g 1
3.b odd 2 1 245.4.a.f 1
7.b odd 2 1 2205.4.a.e 1
7.d odd 6 2 315.4.j.b 2
15.d odd 2 1 1225.4.a.a 1
21.c even 2 1 245.4.a.e 1
21.g even 6 2 35.4.e.a 2
21.h odd 6 2 245.4.e.a 2
84.j odd 6 2 560.4.q.b 2
105.g even 2 1 1225.4.a.b 1
105.p even 6 2 175.4.e.b 2
105.w odd 12 4 175.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 21.g even 6 2
175.4.e.b 2 105.p even 6 2
175.4.k.b 4 105.w odd 12 4
245.4.a.e 1 21.c even 2 1
245.4.a.f 1 3.b odd 2 1
245.4.e.a 2 21.h odd 6 2
315.4.j.b 2 7.d odd 6 2
560.4.q.b 2 84.j odd 6 2
1225.4.a.a 1 15.d odd 2 1
1225.4.a.b 1 105.g even 2 1
2205.4.a.e 1 7.b odd 2 1
2205.4.a.g 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{11} - 45$$ T11 - 45 $$T_{13} + 59$$ T13 + 59 $$T_{17} + 54$$ T17 + 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T - 45$$
$13$ $$T + 59$$
$17$ $$T + 54$$
$19$ $$T - 121$$
$23$ $$T + 69$$
$29$ $$T - 162$$
$31$ $$T - 88$$
$37$ $$T + 259$$
$41$ $$T - 195$$
$43$ $$T + 286$$
$47$ $$T - 45$$
$53$ $$T + 597$$
$59$ $$T + 360$$
$61$ $$T + 392$$
$67$ $$T + 280$$
$71$ $$T + 48$$
$73$ $$T + 668$$
$79$ $$T - 782$$
$83$ $$T - 768$$
$89$ $$T + 1194$$
$97$ $$T + 902$$