# Properties

 Label 390.2.a.a Level $390$ Weight $2$ Character orbit 390.a Self dual yes Analytic conductor $3.114$ 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] = [390,2,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.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: $$3.11416567883$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{12} - q^{13} + q^{15} + q^{16} - 6 q^{17} - q^{18} - q^{20} - 4 q^{23} + q^{24} + q^{25} + q^{26} - q^{27} - 10 q^{29} - q^{30} - q^{32} + 6 q^{34} + q^{36} - 6 q^{37} + q^{39} + q^{40} + 2 q^{41} - 4 q^{43} - q^{45} + 4 q^{46} - q^{48} - 7 q^{49} - q^{50} + 6 q^{51} - q^{52} - 6 q^{53} + q^{54} + 10 q^{58} + q^{60} + 6 q^{61} + q^{64} + q^{65} + 4 q^{67} - 6 q^{68} + 4 q^{69} + 16 q^{71} - q^{72} - 2 q^{73} + 6 q^{74} - q^{75} - q^{78} - q^{80} + q^{81} - 2 q^{82} + 4 q^{83} + 6 q^{85} + 4 q^{86} + 10 q^{87} - 6 q^{89} + q^{90} - 4 q^{92} + q^{96} + 14 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - q^8 + q^9 + q^10 - q^12 - q^13 + q^15 + q^16 - 6 * q^17 - q^18 - q^20 - 4 * q^23 + q^24 + q^25 + q^26 - q^27 - 10 * q^29 - q^30 - q^32 + 6 * q^34 + q^36 - 6 * q^37 + q^39 + q^40 + 2 * q^41 - 4 * q^43 - q^45 + 4 * q^46 - q^48 - 7 * q^49 - q^50 + 6 * q^51 - q^52 - 6 * q^53 + q^54 + 10 * q^58 + q^60 + 6 * q^61 + q^64 + q^65 + 4 * q^67 - 6 * q^68 + 4 * q^69 + 16 * q^71 - q^72 - 2 * q^73 + 6 * q^74 - q^75 - q^78 - q^80 + q^81 - 2 * q^82 + 4 * q^83 + 6 * q^85 + 4 * q^86 + 10 * q^87 - 6 * q^89 + q^90 - 4 * q^92 + q^96 + 14 * 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 −1.00000 1.00000 −1.00000 1.00000 0 −1.00000 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 390.2.a.a 1
3.b odd 2 1 1170.2.a.m 1
4.b odd 2 1 3120.2.a.q 1
5.b even 2 1 1950.2.a.y 1
5.c odd 4 2 1950.2.e.l 2
12.b even 2 1 9360.2.a.bn 1
13.b even 2 1 5070.2.a.s 1
13.d odd 4 2 5070.2.b.c 2
15.d odd 2 1 5850.2.a.m 1
15.e even 4 2 5850.2.e.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.a 1 1.a even 1 1 trivial
1170.2.a.m 1 3.b odd 2 1
1950.2.a.y 1 5.b even 2 1
1950.2.e.l 2 5.c odd 4 2
3120.2.a.q 1 4.b odd 2 1
5070.2.a.s 1 13.b even 2 1
5070.2.b.c 2 13.d odd 4 2
5850.2.a.m 1 15.d odd 2 1
5850.2.e.p 2 15.e even 4 2
9360.2.a.bn 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T + 10$$
$31$ $$T$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 6$$
$67$ $$T - 4$$
$71$ $$T - 16$$
$73$ $$T + 2$$
$79$ $$T$$
$83$ $$T - 4$$
$89$ $$T + 6$$
$97$ $$T - 14$$