# Properties

 Label 390.2.y.a Level $390$ Weight $2$ Character orbit 390.y Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(139,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("390.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.y (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - 1) q^{5} - \zeta_{12}^{2} q^{6} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^2 - z * q^3 + z^2 * q^4 + (2*z^3 - 1) * q^5 - z^2 * q^6 + (-4*z^3 + 4*z) * q^7 + z^3 * q^8 + z^2 * q^9 $$q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - 1) q^{5} - \zeta_{12}^{2} q^{6} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{10} + (5 \zeta_{12}^{2} - 5) q^{11} - \zeta_{12}^{3} q^{12} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + 4 q^{14} + ( - 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{15} + (\zeta_{12}^{2} - 1) q^{16} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + \zeta_{12}^{3} q^{18} + 8 \zeta_{12}^{2} q^{19} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{20} + \cdots - 5 q^{99} +O(q^{100})$$ q + z * q^2 - z * q^3 + z^2 * q^4 + (2*z^3 - 1) * q^5 - z^2 * q^6 + (-4*z^3 + 4*z) * q^7 + z^3 * q^8 + z^2 * q^9 + (2*z^2 - z - 2) * q^10 + (5*z^2 - 5) * q^11 - z^3 * q^12 + (-3*z^3 + 4*z) * q^13 + 4 * q^14 + (-2*z^2 + z + 2) * q^15 + (z^2 - 1) * q^16 + (2*z^3 - 2*z) * q^17 + z^3 * q^18 + 8*z^2 * q^19 + (2*z^3 - z^2 - 2*z) * q^20 - 4 * q^21 + (5*z^3 - 5*z) * q^22 + 3*z * q^23 + (-z^2 + 1) * q^24 + (-4*z^3 - 3) * q^25 + (z^2 + 3) * q^26 - z^3 * q^27 + 4*z * q^28 + (-7*z^2 + 7) * q^29 + (-2*z^3 + z^2 + 2*z) * q^30 - 7 * q^31 + (z^3 - z) * q^32 + (-5*z^3 + 5*z) * q^33 - 2 * q^34 + (4*z^3 + 8*z^2 - 4*z) * q^35 + (z^2 - 1) * q^36 + 3*z * q^37 + 8*z^3 * q^38 + (-z^2 - 3) * q^39 + (-z^3 - 2) * q^40 - 4*z * q^42 + (-z^3 + z) * q^43 - 5 * q^44 + (2*z^3 - z^2 - 2*z) * q^45 + 3*z^2 * q^46 - 7*z^3 * q^47 + (-z^3 + z) * q^48 + (-9*z^2 + 9) * q^49 + (-4*z^2 - 3*z + 4) * q^50 + 2 * q^51 + (z^3 + 3*z) * q^52 - 10*z^3 * q^53 + (-z^2 + 1) * q^54 + (-5*z^2 - 10*z + 5) * q^55 + 4*z^2 * q^56 - 8*z^3 * q^57 + (-7*z^3 + 7*z) * q^58 - 5*z^2 * q^59 + (z^3 + 2) * q^60 - 4*z^2 * q^61 - 7*z * q^62 + 4*z * q^63 - q^64 + (3*z^3 + 8*z^2 - 4*z - 2) * q^65 + 5 * q^66 - 4*z * q^67 - 2*z * q^68 - 3*z^2 * q^69 + (8*z^3 - 4) * q^70 - 8*z^2 * q^71 + (z^3 - z) * q^72 - 4*z^3 * q^73 + 3*z^2 * q^74 + (4*z^2 + 3*z - 4) * q^75 + (8*z^2 - 8) * q^76 + 20*z^3 * q^77 + (-z^3 - 3*z) * q^78 - q^79 + (-z^2 - 2*z + 1) * q^80 + (z^2 - 1) * q^81 + 12*z^3 * q^83 - 4*z^2 * q^84 + (-2*z^3 - 4*z^2 + 2*z) * q^85 + q^86 + (7*z^3 - 7*z) * q^87 - 5*z * q^88 + (-z^3 - 2) * q^90 + (-12*z^2 + 16) * q^91 + 3*z^3 * q^92 + 7*z * q^93 + (-7*z^2 + 7) * q^94 + (16*z^3 - 8*z^2 - 16*z) * q^95 + q^96 + (12*z^3 - 12*z) * q^97 + (-9*z^3 + 9*z) * q^98 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^9 $$4 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{9} - 4 q^{10} - 10 q^{11} + 16 q^{14} + 4 q^{15} - 2 q^{16} + 16 q^{19} - 2 q^{20} - 16 q^{21} + 2 q^{24} - 12 q^{25} + 14 q^{26} + 14 q^{29} + 2 q^{30} - 28 q^{31} - 8 q^{34} + 16 q^{35} - 2 q^{36} - 14 q^{39} - 8 q^{40} - 20 q^{44} - 2 q^{45} + 6 q^{46} + 18 q^{49} + 8 q^{50} + 8 q^{51} + 2 q^{54} + 10 q^{55} + 8 q^{56} - 10 q^{59} + 8 q^{60} - 8 q^{61} - 4 q^{64} + 8 q^{65} + 20 q^{66} - 6 q^{69} - 16 q^{70} - 16 q^{71} + 6 q^{74} - 8 q^{75} - 16 q^{76} - 4 q^{79} + 2 q^{80} - 2 q^{81} - 8 q^{84} - 8 q^{85} + 4 q^{86} - 8 q^{90} + 40 q^{91} + 14 q^{94} - 16 q^{95} + 4 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^9 - 4 * q^10 - 10 * q^11 + 16 * q^14 + 4 * q^15 - 2 * q^16 + 16 * q^19 - 2 * q^20 - 16 * q^21 + 2 * q^24 - 12 * q^25 + 14 * q^26 + 14 * q^29 + 2 * q^30 - 28 * q^31 - 8 * q^34 + 16 * q^35 - 2 * q^36 - 14 * q^39 - 8 * q^40 - 20 * q^44 - 2 * q^45 + 6 * q^46 + 18 * q^49 + 8 * q^50 + 8 * q^51 + 2 * q^54 + 10 * q^55 + 8 * q^56 - 10 * q^59 + 8 * q^60 - 8 * q^61 - 4 * q^64 + 8 * q^65 + 20 * q^66 - 6 * q^69 - 16 * q^70 - 16 * q^71 + 6 * q^74 - 8 * q^75 - 16 * q^76 - 4 * q^79 + 2 * q^80 - 2 * q^81 - 8 * q^84 - 8 * q^85 + 4 * q^86 - 8 * q^90 + 40 * q^91 + 14 * q^94 - 16 * q^95 + 4 * q^96 - 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 2.00000i −0.500000 0.866025i −3.46410 + 2.00000i 1.00000i 0.500000 + 0.866025i −0.133975 + 2.23205i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.00000 + 2.00000i −0.500000 0.866025i 3.46410 2.00000i 1.00000i 0.500000 + 0.866025i −1.86603 + 1.23205i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.00000 + 2.00000i −0.500000 + 0.866025i −3.46410 2.00000i 1.00000i 0.500000 0.866025i −0.133975 2.23205i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.00000 2.00000i −0.500000 + 0.866025i 3.46410 + 2.00000i 1.00000i 0.500000 0.866025i −1.86603 1.23205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.y.a 4
3.b odd 2 1 1170.2.bp.f 4
5.b even 2 1 inner 390.2.y.a 4
5.c odd 4 1 1950.2.i.g 2
5.c odd 4 1 1950.2.i.t 2
13.c even 3 1 inner 390.2.y.a 4
15.d odd 2 1 1170.2.bp.f 4
39.i odd 6 1 1170.2.bp.f 4
65.n even 6 1 inner 390.2.y.a 4
65.q odd 12 1 1950.2.i.g 2
65.q odd 12 1 1950.2.i.t 2
195.x odd 6 1 1170.2.bp.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.a 4 1.a even 1 1 trivial
390.2.y.a 4 5.b even 2 1 inner
390.2.y.a 4 13.c even 3 1 inner
390.2.y.a 4 65.n even 6 1 inner
1170.2.bp.f 4 3.b odd 2 1
1170.2.bp.f 4 15.d odd 2 1
1170.2.bp.f 4 39.i odd 6 1
1170.2.bp.f 4 195.x odd 6 1
1950.2.i.g 2 5.c odd 4 1
1950.2.i.g 2 65.q odd 12 1
1950.2.i.t 2 5.c odd 4 1
1950.2.i.t 2 65.q odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 16T_{7}^{2} + 256$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$(T^{2} + 2 T + 5)^{2}$$
$7$ $$T^{4} - 16T^{2} + 256$$
$11$ $$(T^{2} + 5 T + 25)^{2}$$
$13$ $$T^{4} - 22T^{2} + 169$$
$17$ $$T^{4} - 4T^{2} + 16$$
$19$ $$(T^{2} - 8 T + 64)^{2}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$(T^{2} - 7 T + 49)^{2}$$
$31$ $$(T + 7)^{4}$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$T^{4}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$(T^{2} + 49)^{2}$$
$53$ $$(T^{2} + 100)^{2}$$
$59$ $$(T^{2} + 5 T + 25)^{2}$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} - 16T^{2} + 256$$
$71$ $$(T^{2} + 8 T + 64)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T + 1)^{4}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 144 T^{2} + 20736$$