# Properties

 Label 2205.4.a.f Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $2$ 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: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 315) 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 $$q - 3 q^{2} + q^{4} + 5 q^{5} + 21 q^{8} - 15 q^{10} - 60 q^{11} - 38 q^{13} - 71 q^{16} - 84 q^{17} - 110 q^{19} + 5 q^{20} + 180 q^{22} - 120 q^{23} + 25 q^{25} + 114 q^{26} - 162 q^{29} - 236 q^{31} + 45 q^{32} + 252 q^{34} - 376 q^{37} + 330 q^{38} + 105 q^{40} - 126 q^{41} - 34 q^{43} - 60 q^{44} + 360 q^{46} - 6 q^{47} - 75 q^{50} - 38 q^{52} - 582 q^{53} - 300 q^{55} + 486 q^{58} + 492 q^{59} + 880 q^{61} + 708 q^{62} + 433 q^{64} - 190 q^{65} - 826 q^{67} - 84 q^{68} + 666 q^{71} + 826 q^{73} + 1128 q^{74} - 110 q^{76} - 592 q^{79} - 355 q^{80} + 378 q^{82} + 792 q^{83} - 420 q^{85} + 102 q^{86} - 1260 q^{88} + 1002 q^{89} - 120 q^{92} + 18 q^{94} - 550 q^{95} - 1442 q^{97}+O(q^{100})$$ q - 3 * q^2 + q^4 + 5 * q^5 + 21 * q^8 - 15 * q^10 - 60 * q^11 - 38 * q^13 - 71 * q^16 - 84 * q^17 - 110 * q^19 + 5 * q^20 + 180 * q^22 - 120 * q^23 + 25 * q^25 + 114 * q^26 - 162 * q^29 - 236 * q^31 + 45 * q^32 + 252 * q^34 - 376 * q^37 + 330 * q^38 + 105 * q^40 - 126 * q^41 - 34 * q^43 - 60 * q^44 + 360 * q^46 - 6 * q^47 - 75 * q^50 - 38 * q^52 - 582 * q^53 - 300 * q^55 + 486 * q^58 + 492 * q^59 + 880 * q^61 + 708 * q^62 + 433 * q^64 - 190 * q^65 - 826 * q^67 - 84 * q^68 + 666 * q^71 + 826 * q^73 + 1128 * q^74 - 110 * q^76 - 592 * q^79 - 355 * q^80 + 378 * q^82 + 792 * q^83 - 420 * q^85 + 102 * q^86 - 1260 * q^88 + 1002 * q^89 - 120 * q^92 + 18 * q^94 - 550 * q^95 - 1442 * 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.f 1
3.b odd 2 1 2205.4.a.p 1
7.b odd 2 1 315.4.a.b 1
21.c even 2 1 315.4.a.e yes 1
35.c odd 2 1 1575.4.a.i 1
105.g even 2 1 1575.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.b 1 7.b odd 2 1
315.4.a.e yes 1 21.c even 2 1
1575.4.a.c 1 105.g even 2 1
1575.4.a.i 1 35.c odd 2 1
2205.4.a.f 1 1.a even 1 1 trivial
2205.4.a.p 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{11} + 60$$ T11 + 60 $$T_{13} + 38$$ T13 + 38 $$T_{17} + 84$$ T17 + 84

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T + 60$$
$13$ $$T + 38$$
$17$ $$T + 84$$
$19$ $$T + 110$$
$23$ $$T + 120$$
$29$ $$T + 162$$
$31$ $$T + 236$$
$37$ $$T + 376$$
$41$ $$T + 126$$
$43$ $$T + 34$$
$47$ $$T + 6$$
$53$ $$T + 582$$
$59$ $$T - 492$$
$61$ $$T - 880$$
$67$ $$T + 826$$
$71$ $$T - 666$$
$73$ $$T - 826$$
$79$ $$T + 592$$
$83$ $$T - 792$$
$89$ $$T - 1002$$
$97$ $$T + 1442$$