# Properties

 Label 390.2.p.g Level $390$ Weight $2$ Character orbit 390.p Analytic conductor $3.114$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(161,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("390.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.p (of order $$4$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.40960000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 7x^{4} + 1$$ x^8 + 7*x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} - \beta_{6} q^{3} - \beta_{2} q^{4} - \beta_{7} q^{5} + (\beta_{5} + 1) q^{6} + \beta_1 q^{8} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q - b7 * q^2 - b6 * q^3 - b2 * q^4 - b7 * q^5 + (b5 + 1) * q^6 + b1 * q^8 + (-b5 + b3 + b2 + 2) * q^9 $$q - \beta_{7} q^{2} - \beta_{6} q^{3} - \beta_{2} q^{4} - \beta_{7} q^{5} + (\beta_{5} + 1) q^{6} + \beta_1 q^{8} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{9} - \beta_{2} q^{10} + 2 \beta_1 q^{11} + ( - \beta_{7} + \beta_{4}) q^{12} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 2) q^{13} + (\beta_{5} + 1) q^{15} - q^{16} + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots - 2) q^{17}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots + 6 \beta_1) q^{99}+O(q^{100})$$ q - b7 * q^2 - b6 * q^3 - b2 * q^4 - b7 * q^5 + (b5 + 1) * q^6 + b1 * q^8 + (-b5 + b3 + b2 + 2) * q^9 - b2 * q^10 + 2*b1 * q^11 + (-b7 + b4) * q^12 + (-b6 - b4 - 2*b2 + 2) * q^13 + (b5 + 1) * q^15 - q^16 + (-2*b7 - 2*b5 - 2*b3 - 2*b1 - 2) * q^17 + (-2*b7 - b6 - b4) * q^18 + (-2*b6 - 2*b4 - 2*b2 - 2) * q^19 + b1 * q^20 - 2 * q^22 + (-2*b5 - 2*b3 - 2) * q^23 + (-b3 - b2) * q^24 - b2 * q^25 + (-2*b7 + b5 + b3 + 2*b1 + 1) * q^26 + (3*b7 - b6 - 3*b1) * q^27 + (-2*b5 + 2*b3 + 2*b2) * q^29 + (-b7 + b4) * q^30 + (-3*b2 - 3) * q^31 + b7 * q^32 + (-2*b3 - 2*b2) * q^33 + (2*b7 + 2*b6 - 2*b4 - 2*b2 - 2*b1 + 2) * q^34 + (b5 + b3 - 2*b2 + 1) * q^36 + (4*b2 - 4) * q^37 + (2*b7 + 2*b5 + 2*b3 + 2*b1 + 2) * q^38 + (-2*b7 - 2*b6 - b5 + 2*b4 + 3*b2 + 2) * q^39 - q^40 + (2*b5 - b2 + 1) * q^41 + (b7 - 2*b4 - 4*b2 - b1) * q^43 + 2*b7 * q^44 + (-2*b7 - b6 - b4) * q^45 + (2*b7 + 2*b6 - 2*b4 - 2*b1) * q^46 + (-4*b3 - 2*b2 - 4*b1 - 2) * q^47 + b6 * q^48 + 7*b2 * q^49 + b1 * q^50 + (4*b7 + 2*b5 + 2*b4 + 2*b3 + 2*b2 + 6*b1 + 2) * q^51 + (-b7 - b6 + b4 - 2*b2 + b1 - 2) * q^52 + (b7 - b1) * q^53 + (b5 + 3*b2 + 4) * q^54 - 2 * q^55 + (-2*b7 + 2*b6 - 2*b5 + 2*b4 + 6*b2 + 4) * q^57 + (-2*b6 - 2*b4) * q^58 + (-4*b3 - 2*b2 - 2*b1 - 2) * q^59 + (-b3 - b2) * q^60 + (4*b7 + 8*b6 - 4*b1 - 2) * q^61 + (3*b7 + 3*b1) * q^62 + b2 * q^64 + (-2*b7 + b5 + b3 + 2*b1 + 1) * q^65 + 2*b6 * q^66 + (2*b6 + 2*b4 + 4*b2 + 4) * q^67 + (-2*b7 - 2*b5 + 2*b3 + 2*b2 + 2*b1) * q^68 + (4*b7 + 2*b4 + 6*b1) * q^69 + (-4*b7 + 6*b5 - 3*b2 + 3) * q^71 + (-b7 - b6 + b4 + 3*b1) * q^72 + (-2*b7 - 2*b6 + 2*b4 + 2*b2 + 2*b1 - 2) * q^73 + (4*b7 - 4*b1) * q^74 + (-b7 + b4) * q^75 + (-2*b7 - 2*b6 + 2*b4 + 2*b2 + 2*b1 - 2) * q^76 + (-2*b7 + 2*b5 - b4 - 2*b3 - 2*b2 - 3*b1 + 2) * q^78 - 10 * q^79 + b7 * q^80 + (-4*b5 + 4*b3 + 4*b2 - 1) * q^81 + (-b7 + 2*b4 + b1) * q^82 + 8*b7 * q^83 + (2*b7 + 2*b6 - 2*b4 - 2*b2 - 2*b1 + 2) * q^85 + (2*b3 + b2 + 4*b1 + 1) * q^86 + (6*b7 + 2*b6 - 6*b1) * q^87 + 2*b2 * q^88 + (2*b3 + b2 - 4*b1 + 1) * q^89 + (b5 + b3 - 2*b2 + 1) * q^90 + (-2*b5 + 2*b3 + 2*b2) * q^92 + (-3*b7 + 3*b6 + 3*b4) * q^93 + (2*b7 + 4*b6 - 2*b1 + 4) * q^94 + (2*b7 + 2*b5 + 2*b3 + 2*b1 + 2) * q^95 + (-b5 - 1) * q^96 + (-2*b6 - 2*b4 + 2*b2 + 2) * q^97 - 7*b1 * q^98 + (-2*b7 - 2*b6 + 2*b4 + 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{6} + 16 q^{9}+O(q^{10})$$ 8 * q + 4 * q^6 + 16 * q^9 $$8 q + 4 q^{6} + 16 q^{9} + 16 q^{13} + 4 q^{15} - 8 q^{16} - 16 q^{19} - 16 q^{22} + 4 q^{24} - 24 q^{31} + 8 q^{33} + 16 q^{34} - 32 q^{37} + 20 q^{39} - 8 q^{40} - 16 q^{52} + 28 q^{54} - 16 q^{55} + 40 q^{57} + 4 q^{60} - 16 q^{61} + 32 q^{67} - 16 q^{73} - 16 q^{76} + 16 q^{78} - 80 q^{79} - 8 q^{81} + 16 q^{85} + 32 q^{94} - 4 q^{96} + 16 q^{97}+O(q^{100})$$ 8 * q + 4 * q^6 + 16 * q^9 + 16 * q^13 + 4 * q^15 - 8 * q^16 - 16 * q^19 - 16 * q^22 + 4 * q^24 - 24 * q^31 + 8 * q^33 + 16 * q^34 - 32 * q^37 + 20 * q^39 - 8 * q^40 - 16 * q^52 + 28 * q^54 - 16 * q^55 + 40 * q^57 + 4 * q^60 - 16 * q^61 + 32 * q^67 - 16 * q^73 - 16 * q^76 + 16 * q^78 - 80 * q^79 - 8 * q^81 + 16 * q^85 + 32 * q^94 - 4 * q^96 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 5\nu ) / 3$$ (v^5 + 5*v) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 8\nu^{2} ) / 3$$ (v^6 + 8*v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + \nu^{4} + 5\nu^{2} + 2 ) / 3$$ (v^6 + v^4 + 5*v^2 + 2) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 8\nu^{3} + 3\nu ) / 3$$ (v^7 + 8*v^3 + 3*v) / 3 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + \nu^{4} - 5\nu^{2} + 2 ) / 3$$ (-v^6 + v^4 - 5*v^2 + 2) / 3 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3$$ (-v^7 + v^5 - 8*v^3 + 8*v) / 3 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} - 13\nu^{3} ) / 3$$ (-2*v^7 - 13*v^3) / 3
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{4} - \beta_1 ) / 2$$ (b6 + b4 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{3} + 2\beta_{2} ) / 2$$ (b5 - b3 + 2*b2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{4} + \beta_1$$ b7 - b6 + b4 + b1 $$\nu^{4}$$ $$=$$ $$( 3\beta_{5} + 3\beta_{3} - 4 ) / 2$$ (3*b5 + 3*b3 - 4) / 2 $$\nu^{5}$$ $$=$$ $$( -5\beta_{6} - 5\beta_{4} + 11\beta_1 ) / 2$$ (-5*b6 - 5*b4 + 11*b1) / 2 $$\nu^{6}$$ $$=$$ $$-4\beta_{5} + 4\beta_{3} - 5\beta_{2}$$ -4*b5 + 4*b3 - 5*b2 $$\nu^{7}$$ $$=$$ $$( -16\beta_{7} + 13\beta_{6} - 13\beta_{4} - 13\beta_1 ) / 2$$ (-16*b7 + 13*b6 - 13*b4 - 13*b1) / 2

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.437016 − 0.437016i −1.14412 + 1.14412i 1.14412 − 1.14412i −0.437016 + 0.437016i 0.437016 + 0.437016i −1.14412 − 1.14412i 1.14412 + 1.14412i −0.437016 − 0.437016i
−0.707107 0.707107i −1.58114 + 0.707107i 1.00000i −0.707107 0.707107i 1.61803 + 0.618034i 0 0.707107 0.707107i 2.00000 2.23607i 1.00000i
161.2 −0.707107 0.707107i 1.58114 + 0.707107i 1.00000i −0.707107 0.707107i −0.618034 1.61803i 0 0.707107 0.707107i 2.00000 + 2.23607i 1.00000i
161.3 0.707107 + 0.707107i −1.58114 0.707107i 1.00000i 0.707107 + 0.707107i −0.618034 1.61803i 0 −0.707107 + 0.707107i 2.00000 + 2.23607i 1.00000i
161.4 0.707107 + 0.707107i 1.58114 0.707107i 1.00000i 0.707107 + 0.707107i 1.61803 + 0.618034i 0 −0.707107 + 0.707107i 2.00000 2.23607i 1.00000i
281.1 −0.707107 + 0.707107i −1.58114 0.707107i 1.00000i −0.707107 + 0.707107i 1.61803 0.618034i 0 0.707107 + 0.707107i 2.00000 + 2.23607i 1.00000i
281.2 −0.707107 + 0.707107i 1.58114 0.707107i 1.00000i −0.707107 + 0.707107i −0.618034 + 1.61803i 0 0.707107 + 0.707107i 2.00000 2.23607i 1.00000i
281.3 0.707107 0.707107i −1.58114 + 0.707107i 1.00000i 0.707107 0.707107i −0.618034 + 1.61803i 0 −0.707107 0.707107i 2.00000 2.23607i 1.00000i
281.4 0.707107 0.707107i 1.58114 + 0.707107i 1.00000i 0.707107 0.707107i 1.61803 0.618034i 0 −0.707107 0.707107i 2.00000 + 2.23607i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.p.g 8
3.b odd 2 1 inner 390.2.p.g 8
13.d odd 4 1 inner 390.2.p.g 8
39.f even 4 1 inner 390.2.p.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.p.g 8 1.a even 1 1 trivial
390.2.p.g 8 3.b odd 2 1 inner
390.2.p.g 8 13.d odd 4 1 inner
390.2.p.g 8 39.f even 4 1 inner

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}^{4} + 16$$ T11^4 + 16 $$T_{17}^{4} - 56T_{17}^{2} + 144$$ T17^4 - 56*T17^2 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{2}$$
$3$ $$(T^{4} - 4 T^{2} + 9)^{2}$$
$5$ $$(T^{4} + 1)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{4} + 16)^{2}$$
$13$ $$(T^{4} - 8 T^{3} + \cdots + 169)^{2}$$
$17$ $$(T^{4} - 56 T^{2} + 144)^{2}$$
$19$ $$(T^{4} + 8 T^{3} + \cdots + 144)^{2}$$
$23$ $$(T^{2} - 20)^{4}$$
$29$ $$(T^{2} + 20)^{4}$$
$31$ $$(T^{2} + 6 T + 18)^{4}$$
$37$ $$(T^{2} + 8 T + 32)^{4}$$
$41$ $$(T^{4} + 100)^{2}$$
$43$ $$(T^{4} + 52 T^{2} + 36)^{2}$$
$47$ $$T^{8} + 11392 T^{4} + 331776$$
$53$ $$(T^{2} + 2)^{4}$$
$59$ $$T^{8} + 5152 T^{4} + 1679616$$
$61$ $$(T^{2} + 4 T - 156)^{4}$$
$67$ $$(T^{4} - 16 T^{3} + \cdots + 144)^{2}$$
$71$ $$T^{8} + 33992 T^{4} + 29986576$$
$73$ $$(T^{4} + 8 T^{3} + \cdots + 144)^{2}$$
$79$ $$(T + 10)^{8}$$
$83$ $$(T^{4} + 4096)^{2}$$
$89$ $$T^{8} + 2632 T^{4} + 1296$$
$97$ $$(T^{4} - 8 T^{3} + \cdots + 144)^{2}$$