# Properties

 Label 2205.4.a.o 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 105) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} + 5 q^{5}+O(q^{10})$$ q - 8 * q^4 + 5 * q^5 $$q - 8 q^{4} + 5 q^{5} - 42 q^{11} - 20 q^{13} + 64 q^{16} + 66 q^{17} - 38 q^{19} - 40 q^{20} - 12 q^{23} + 25 q^{25} + 258 q^{29} - 146 q^{31} + 434 q^{37} - 282 q^{41} + 20 q^{43} + 336 q^{44} - 72 q^{47} + 160 q^{52} - 336 q^{53} - 210 q^{55} - 360 q^{59} + 682 q^{61} - 512 q^{64} - 100 q^{65} + 812 q^{67} - 528 q^{68} - 810 q^{71} + 124 q^{73} + 304 q^{76} + 1136 q^{79} + 320 q^{80} + 156 q^{83} + 330 q^{85} - 1038 q^{89} + 96 q^{92} - 190 q^{95} - 1208 q^{97}+O(q^{100})$$ q - 8 * q^4 + 5 * q^5 - 42 * q^11 - 20 * q^13 + 64 * q^16 + 66 * q^17 - 38 * q^19 - 40 * q^20 - 12 * q^23 + 25 * q^25 + 258 * q^29 - 146 * q^31 + 434 * q^37 - 282 * q^41 + 20 * q^43 + 336 * q^44 - 72 * q^47 + 160 * q^52 - 336 * q^53 - 210 * q^55 - 360 * q^59 + 682 * q^61 - 512 * q^64 - 100 * q^65 + 812 * q^67 - 528 * q^68 - 810 * q^71 + 124 * q^73 + 304 * q^76 + 1136 * q^79 + 320 * q^80 + 156 * q^83 + 330 * q^85 - 1038 * q^89 + 96 * q^92 - 190 * q^95 - 1208 * 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
0 0 −8.00000 5.00000 0 0 0 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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.o 1
3.b odd 2 1 735.4.a.c 1
7.b odd 2 1 315.4.a.d 1
21.c even 2 1 105.4.a.a 1
35.c odd 2 1 1575.4.a.f 1
84.h odd 2 1 1680.4.a.s 1
105.g even 2 1 525.4.a.e 1
105.k odd 4 2 525.4.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 21.c even 2 1
315.4.a.d 1 7.b odd 2 1
525.4.a.e 1 105.g even 2 1
525.4.d.f 2 105.k odd 4 2
735.4.a.c 1 3.b odd 2 1
1575.4.a.f 1 35.c odd 2 1
1680.4.a.s 1 84.h odd 2 1
2205.4.a.o 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}$$ T2 $$T_{11} + 42$$ T11 + 42 $$T_{13} + 20$$ T13 + 20 $$T_{17} - 66$$ T17 - 66

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T + 42$$
$13$ $$T + 20$$
$17$ $$T - 66$$
$19$ $$T + 38$$
$23$ $$T + 12$$
$29$ $$T - 258$$
$31$ $$T + 146$$
$37$ $$T - 434$$
$41$ $$T + 282$$
$43$ $$T - 20$$
$47$ $$T + 72$$
$53$ $$T + 336$$
$59$ $$T + 360$$
$61$ $$T - 682$$
$67$ $$T - 812$$
$71$ $$T + 810$$
$73$ $$T - 124$$
$79$ $$T - 1136$$
$83$ $$T - 156$$
$89$ $$T + 1038$$
$97$ $$T + 1208$$