# Properties

 Label 390.2.s.a Level $390$ Weight $2$ Character orbit 390.s Analytic conductor $3.114$ Analytic rank $0$ Dimension $8$ Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(77,390)]

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, 1, 2]))

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

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

chi := DirichletCharacter("390.77");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + \beta_{2} q^{4} + (\beta_{7} - 2 \beta_1) q^{5} + ( - \beta_{4} - \beta_1) q^{6} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b3 + 1) * q^3 + b2 * q^4 + (b7 - 2*b1) * q^5 + (-b4 - b1) * q^6 + (-2*b7 - 2*b6 + 2*b4 + 2*b1) * q^7 + b7 * q^8 + (b5 + b3 + 2*b2 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + \beta_{2} q^{4} + (\beta_{7} - 2 \beta_1) q^{5} + ( - \beta_{4} - \beta_1) q^{6} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 4 \beta_{7} + 8 \beta_{6} + 4 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b3 + 1) * q^3 + b2 * q^4 + (b7 - 2*b1) * q^5 + (-b4 - b1) * q^6 + (-2*b7 - 2*b6 + 2*b4 + 2*b1) * q^7 + b7 * q^8 + (b5 + b3 + 2*b2 + 1) * q^9 + (2*b2 - 1) * q^10 + (4*b7 + 4*b1) * q^11 + (-b5 + b2) * q^12 + (b6 + b4 - 2*b2 + 2) * q^13 + (2*b5 + 2*b3 + 2) * q^14 + (b7 + b6 - 2*b4 - 3*b1) * q^15 - q^16 + (2*b5 - b2 + 1) * q^17 + (2*b7 - b6 - b4) * q^18 + (2*b7 + 4*b6 - 2*b1) * q^19 + (2*b7 + b1) * q^20 + (-6*b7 + 2*b4 - 4*b1) * q^21 + (-4*b2 - 4) * q^22 + (b7 + b6 - b1) * q^24 + (3*b2 - 4) * q^25 + (-2*b7 + b5 - b3 - b2 - 2*b1) * q^26 + (-b5 + 4*b2 + 3) * q^27 + (-2*b6 - 2*b4) * q^28 + (-2*b5 - 2*b3 - 2) * q^29 + (-2*b5 - b3 + 2*b2 - 1) * q^30 + (b7 - 2*b4 - b1) * q^31 + b1 * q^32 + (4*b7 + 4*b6 + 4*b4) * q^33 + (-b7 - 2*b6 + b1) * q^34 + (2*b5 + 6*b3 + 2*b2 + 4) * q^35 + (-b5 + b3 + b2 - 2) * q^36 + (-2*b7 - 2*b6 + 2*b4 + 2*b1) * q^37 + (-4*b3 - 2*b2 - 2) * q^38 + (-2*b7 + b6 + 2*b5 + 2*b3 - 2*b2 + 2*b1 + 2) * q^39 + (-b2 - 2) * q^40 + (-4*b7 - 4*b1) * q^41 + (2*b5 + 4*b2 + 6) * q^42 + (5*b2 - 5) * q^43 + (-4*b7 + 4*b1) * q^44 + (5*b7 - b6 - 3*b4 + b1) * q^45 - 2*b1 * q^47 + (-b3 - 1) * q^48 - 13*b2 * q^49 + (3*b7 + 4*b1) * q^50 + (b5 - b3 - b2 + 5) * q^51 + (-b7 - b6 + b4 + 2*b2 + b1 + 2) * q^52 + (-4*b3 - 2*b2 - 2) * q^53 + (4*b7 + b6 - 4*b1) * q^54 + (-12*b2 - 4) * q^55 + (-2*b5 + 2*b3 + 2*b2) * q^56 + (2*b7 + 2*b6 - 2*b4 + 8*b1) * q^57 + (2*b6 + 2*b4) * q^58 + (-2*b7 + 2*b1) * q^59 + (2*b7 + 2*b6 + b4 - b1) * q^60 + 10 * q^61 + (-2*b5 + b2 - 1) * q^62 + (-10*b7 - 4*b6 - 4*b4) * q^63 - b2 * q^64 + (-2*b7 + 3*b5 - b3 - 2*b2 - 6*b1 + 1) * q^65 + (4*b5 - 4*b3 - 4*b2 - 4) * q^66 + (2*b3 + b2 + 1) * q^68 + (2*b7 - 2*b6 - 6*b4 - 2*b1) * q^70 + (b7 + b1) * q^71 + (b7 + b6 - b4 + b1) * q^72 + (-2*b6 - 2*b4) * q^73 + (2*b5 + 2*b3 + 2) * q^74 + (-3*b5 - 4*b3 + 3*b2 - 4) * q^75 + (-2*b7 + 4*b4 + 2*b1) * q^76 + (-16*b5 + 8*b2 - 8) * q^77 + (-2*b7 - 2*b6 - 2*b4 - b3 - 3*b2 + 2) * q^78 + (-b7 + 2*b1) * q^80 + (-4*b5 + 4*b3 + 4*b2 + 1) * q^81 + (4*b2 + 4) * q^82 + 4*b7 * q^83 + (4*b7 - 2*b6 - 4*b1) * q^84 + (-b7 - 4*b6 - 2*b4 + b1) * q^85 + (5*b7 + 5*b1) * q^86 + (-2*b5 - 4*b2 - 6) * q^87 + (-4*b2 + 4) * q^88 + (2*b7 - 2*b1) * q^89 + (-3*b5 + b3 - 5) * q^90 + (-4*b7 + 8*b4 + 4*b1 - 10) * q^91 + (5*b7 - b6 - b4) * q^93 + 2*b2 * q^94 + (4*b5 - 8*b3 - 6*b2 - 2) * q^95 + (b4 + b1) * q^96 + (-6*b7 - 6*b6 + 6*b4 + 6*b1) * q^97 - 13*b7 * q^98 + (-4*b7 + 8*b6 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3}+O(q^{10})$$ 8 * q + 4 * q^3 $$8 q + 4 q^{3} - 8 q^{10} + 4 q^{12} + 16 q^{13} - 8 q^{16} - 32 q^{22} - 32 q^{25} + 28 q^{27} + 4 q^{30} - 16 q^{36} - 16 q^{40} + 40 q^{42} - 40 q^{43} - 4 q^{48} + 40 q^{51} + 16 q^{52} - 32 q^{55} + 80 q^{61} - 32 q^{66} - 4 q^{75} + 20 q^{78} + 8 q^{81} + 32 q^{82} - 40 q^{87} + 32 q^{88} - 32 q^{90} - 80 q^{91}+O(q^{100})$$ 8 * q + 4 * q^3 - 8 * q^10 + 4 * q^12 + 16 * q^13 - 8 * q^16 - 32 * q^22 - 32 * q^25 + 28 * q^27 + 4 * q^30 - 16 * q^36 - 16 * q^40 + 40 * q^42 - 40 * q^43 - 4 * q^48 + 40 * q^51 + 16 * q^52 - 32 * q^55 + 80 * q^61 - 32 * q^66 - 4 * q^75 + 20 * q^78 + 8 * q^81 + 32 * q^82 - 40 * q^87 + 32 * q^88 - 32 * q^90 - 80 * q^91

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 $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1
 −1.14412 − 1.14412i 0.437016 + 0.437016i 1.14412 + 1.14412i −0.437016 − 0.437016i −1.14412 + 1.14412i 0.437016 − 0.437016i 1.14412 − 1.14412i −0.437016 + 0.437016i
−0.707107 0.707107i −0.618034 1.61803i 1.00000i −0.707107 2.12132i −0.707107 + 1.58114i 3.16228 3.16228i 0.707107 0.707107i −2.23607 + 2.00000i −1.00000 + 2.00000i
77.2 −0.707107 0.707107i 1.61803 + 0.618034i 1.00000i −0.707107 2.12132i −0.707107 1.58114i −3.16228 + 3.16228i 0.707107 0.707107i 2.23607 + 2.00000i −1.00000 + 2.00000i
77.3 0.707107 + 0.707107i −0.618034 1.61803i 1.00000i 0.707107 + 2.12132i 0.707107 1.58114i −3.16228 + 3.16228i −0.707107 + 0.707107i −2.23607 + 2.00000i −1.00000 + 2.00000i
77.4 0.707107 + 0.707107i 1.61803 + 0.618034i 1.00000i 0.707107 + 2.12132i 0.707107 + 1.58114i 3.16228 3.16228i −0.707107 + 0.707107i 2.23607 + 2.00000i −1.00000 + 2.00000i
233.1 −0.707107 + 0.707107i −0.618034 + 1.61803i 1.00000i −0.707107 + 2.12132i −0.707107 1.58114i 3.16228 + 3.16228i 0.707107 + 0.707107i −2.23607 2.00000i −1.00000 2.00000i
233.2 −0.707107 + 0.707107i 1.61803 0.618034i 1.00000i −0.707107 + 2.12132i −0.707107 + 1.58114i −3.16228 3.16228i 0.707107 + 0.707107i 2.23607 2.00000i −1.00000 2.00000i
233.3 0.707107 0.707107i −0.618034 + 1.61803i 1.00000i 0.707107 2.12132i 0.707107 + 1.58114i −3.16228 3.16228i −0.707107 0.707107i −2.23607 2.00000i −1.00000 2.00000i
233.4 0.707107 0.707107i 1.61803 0.618034i 1.00000i 0.707107 2.12132i 0.707107 1.58114i 3.16228 + 3.16228i −0.707107 0.707107i 2.23607 2.00000i −1.00000 2.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.4 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
5.c odd 4 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
39.d odd 2 1 inner
65.h odd 4 1 inner
195.s 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.s.a 8
3.b odd 2 1 inner 390.2.s.a 8
5.c odd 4 1 inner 390.2.s.a 8
13.b even 2 1 inner 390.2.s.a 8
15.e even 4 1 inner 390.2.s.a 8
39.d odd 2 1 inner 390.2.s.a 8
65.h odd 4 1 inner 390.2.s.a 8
195.s even 4 1 inner 390.2.s.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.s.a 8 1.a even 1 1 trivial
390.2.s.a 8 3.b odd 2 1 inner
390.2.s.a 8 5.c odd 4 1 inner
390.2.s.a 8 13.b even 2 1 inner
390.2.s.a 8 15.e even 4 1 inner
390.2.s.a 8 39.d odd 2 1 inner
390.2.s.a 8 65.h odd 4 1 inner
390.2.s.a 8 195.s 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{4} + 400$$ T7^4 + 400 $$T_{17}^{4} + 100$$ T17^4 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{2}$$
$3$ $$(T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 9)^{2}$$
$5$ $$(T^{4} + 8 T^{2} + 25)^{2}$$
$7$ $$(T^{4} + 400)^{2}$$
$11$ $$(T^{2} - 32)^{4}$$
$13$ $$(T^{4} - 8 T^{3} + \cdots + 169)^{2}$$
$17$ $$(T^{4} + 100)^{2}$$
$19$ $$(T^{2} - 40)^{4}$$
$23$ $$T^{8}$$
$29$ $$(T^{2} - 20)^{4}$$
$31$ $$(T^{2} + 10)^{4}$$
$37$ $$(T^{4} + 400)^{2}$$
$41$ $$(T^{2} - 32)^{4}$$
$43$ $$(T^{2} + 10 T + 50)^{4}$$
$47$ $$(T^{4} + 16)^{2}$$
$53$ $$(T^{4} + 1600)^{2}$$
$59$ $$(T^{2} + 8)^{4}$$
$61$ $$(T - 10)^{8}$$
$67$ $$T^{8}$$
$71$ $$(T^{2} - 2)^{4}$$
$73$ $$(T^{4} + 400)^{2}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} + 256)^{2}$$
$89$ $$(T^{2} + 8)^{4}$$
$97$ $$(T^{4} + 32400)^{2}$$