# Properties

 Label 2790.2.a.h 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: no (minimal twist has level 310) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 2 q^{11} + q^{16} - 2 q^{17} - 4 q^{19} + q^{20} + 2 q^{22} + 4 q^{23} + q^{25} + 4 q^{29} - q^{31} - q^{32} + 2 q^{34} - 8 q^{37} + 4 q^{38} - q^{40} - 6 q^{41} + 2 q^{43} - 2 q^{44} - 4 q^{46} - 7 q^{49} - q^{50} - 8 q^{53} - 2 q^{55} - 4 q^{58} - 8 q^{59} + q^{62} + q^{64} + 4 q^{67} - 2 q^{68} + 6 q^{73} + 8 q^{74} - 4 q^{76} - 4 q^{79} + q^{80} + 6 q^{82} - 6 q^{83} - 2 q^{85} - 2 q^{86} + 2 q^{88} + 6 q^{89} + 4 q^{92} - 4 q^{95} - 2 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^8 - q^10 - 2 * q^11 + q^16 - 2 * q^17 - 4 * q^19 + q^20 + 2 * q^22 + 4 * q^23 + q^25 + 4 * q^29 - q^31 - q^32 + 2 * q^34 - 8 * q^37 + 4 * q^38 - q^40 - 6 * q^41 + 2 * q^43 - 2 * q^44 - 4 * q^46 - 7 * q^49 - q^50 - 8 * q^53 - 2 * q^55 - 4 * q^58 - 8 * q^59 + q^62 + q^64 + 4 * q^67 - 2 * q^68 + 6 * q^73 + 8 * q^74 - 4 * q^76 - 4 * q^79 + q^80 + 6 * q^82 - 6 * q^83 - 2 * q^85 - 2 * q^86 + 2 * q^88 + 6 * q^89 + 4 * q^92 - 4 * q^95 - 2 * q^97 + 7 * 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 0 −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.h 1
3.b odd 2 1 310.2.a.b 1
12.b even 2 1 2480.2.a.c 1
15.d odd 2 1 1550.2.a.a 1
15.e even 4 2 1550.2.b.e 2
24.f even 2 1 9920.2.a.bg 1
24.h odd 2 1 9920.2.a.d 1
93.c even 2 1 9610.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.b 1 3.b odd 2 1
1550.2.a.a 1 15.d odd 2 1
1550.2.b.e 2 15.e even 4 2
2480.2.a.c 1 12.b even 2 1
2790.2.a.h 1 1.a even 1 1 trivial
9610.2.a.a 1 93.c even 2 1
9920.2.a.d 1 24.h odd 2 1
9920.2.a.bg 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7}$$ T7 $$T_{11} + 2$$ T11 + 2 $$T_{13}$$ T13 $$T_{17} + 2$$ T17 + 2 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

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