# Properties

 Label 2790.2.a.x Level $2790$ Weight $2$ Character orbit 2790.a Self dual yes Analytic conductor $22.278$ 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] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, 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("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2790.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: $$22.2782621639$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - 3 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 - 3 * q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} - 3 q^{7} + q^{8} + q^{10} - 5 q^{11} - 3 q^{14} + q^{16} - 2 q^{17} - q^{19} + q^{20} - 5 q^{22} + 7 q^{23} + q^{25} - 3 q^{28} + 4 q^{29} - q^{31} + q^{32} - 2 q^{34} - 3 q^{35} - 8 q^{37} - q^{38} + q^{40} - 12 q^{41} - q^{43} - 5 q^{44} + 7 q^{46} - 12 q^{47} + 2 q^{49} + q^{50} - 5 q^{53} - 5 q^{55} - 3 q^{56} + 4 q^{58} - 8 q^{59} - 6 q^{61} - q^{62} + q^{64} - 2 q^{67} - 2 q^{68} - 3 q^{70} - 3 q^{71} - 15 q^{73} - 8 q^{74} - q^{76} + 15 q^{77} - q^{79} + q^{80} - 12 q^{82} + 6 q^{83} - 2 q^{85} - q^{86} - 5 q^{88} + 15 q^{89} + 7 q^{92} - 12 q^{94} - q^{95} + 16 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 - 3 * q^7 + q^8 + q^10 - 5 * q^11 - 3 * q^14 + q^16 - 2 * q^17 - q^19 + q^20 - 5 * q^22 + 7 * q^23 + q^25 - 3 * q^28 + 4 * q^29 - q^31 + q^32 - 2 * q^34 - 3 * q^35 - 8 * q^37 - q^38 + q^40 - 12 * q^41 - q^43 - 5 * q^44 + 7 * q^46 - 12 * q^47 + 2 * q^49 + q^50 - 5 * q^53 - 5 * q^55 - 3 * q^56 + 4 * q^58 - 8 * q^59 - 6 * q^61 - q^62 + q^64 - 2 * q^67 - 2 * q^68 - 3 * q^70 - 3 * q^71 - 15 * q^73 - 8 * q^74 - q^76 + 15 * q^77 - q^79 + q^80 - 12 * q^82 + 6 * q^83 - 2 * q^85 - q^86 - 5 * q^88 + 15 * q^89 + 7 * q^92 - 12 * q^94 - q^95 + 16 * q^97 + 2 * q^98

## 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
1.00000 0 1.00000 1.00000 0 −3.00000 1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$31$$ $$1$$

## 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 2790.2.a.x yes 1
3.b odd 2 1 2790.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2790.2.a.a 1 3.b odd 2 1
2790.2.a.x yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 5$$ T11 + 5 $$T_{13}$$ T13 $$T_{17} + 2$$ T17 + 2 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T + 2$$
$19$ $$T + 1$$
$23$ $$T - 7$$
$29$ $$T - 4$$
$31$ $$T + 1$$
$37$ $$T + 8$$
$41$ $$T + 12$$
$43$ $$T + 1$$
$47$ $$T + 12$$
$53$ $$T + 5$$
$59$ $$T + 8$$
$61$ $$T + 6$$
$67$ $$T + 2$$
$71$ $$T + 3$$
$73$ $$T + 15$$
$79$ $$T + 1$$
$83$ $$T - 6$$
$89$ $$T - 15$$
$97$ $$T - 16$$