# Properties

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

# 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} - \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{7} + \cdots + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^2 - z * q^3 + z^2 * q^4 + (z^2 - 2*z - 1) * q^5 - z^2 * q^6 + (z^3 + z^2 - z - 2) * q^7 + z^3 * q^8 + z^2 * q^9 $$q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} - \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{7} + \cdots + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{99} +O(q^{100})$$ q + z * q^2 - z * q^3 + z^2 * q^4 + (z^2 - 2*z - 1) * q^5 - z^2 * q^6 + (z^3 + z^2 - z - 2) * q^7 + z^3 * q^8 + z^2 * q^9 + (z^3 - 2*z^2 - z) * q^10 + (2*z^3 - z^2 - z + 1) * q^11 - z^3 * q^12 + (2*z^2 - 3*z - 2) * q^13 + (z^3 - 2*z - 1) * q^14 + (-z^3 + 2*z^2 + z) * q^15 + (z^2 - 1) * q^16 + (z^3 + 2*z^2 - z - 4) * q^17 + z^3 * q^18 + (-z^3 + 3*z^2 - z) * q^19 + (-2*z^3 - 1) * q^20 + (-z^3 + 2*z + 1) * q^21 + (-z^3 + z^2 + z - 2) * q^22 + (-3*z^2 - z - 3) * q^23 + (-z^2 + 1) * q^24 + (-4*z^3 + 3*z^2 + 4*z) * q^25 + (2*z^3 - 3*z^2 - 2*z) * q^26 - z^3 * q^27 + (-z^2 - z - 1) * q^28 + (4*z^3 + 3*z^2 - 2*z - 3) * q^29 + (2*z^3 + 1) * q^30 + 4 * q^31 + (z^3 - z) * q^32 + (z^3 - z^2 - z + 2) * q^33 + (2*z^3 - 4*z - 1) * q^34 + (-3*z^3 - 2*z^2 + 4*z + 3) * q^35 + (z^2 - 1) * q^36 + (4*z^2 + z + 4) * q^37 + (3*z^3 - 2*z^2 + 1) * q^38 + (-2*z^3 + 3*z^2 + 2*z) * q^39 + (-2*z^2 - z + 2) * q^40 + (-6*z^3 - 4*z^2 + 3*z + 4) * q^41 + (z^2 + z + 1) * q^42 + (5*z^3 - z^2 - 5*z + 2) * q^43 + (z^3 - 2*z + 1) * q^44 + (-2*z^3 - 1) * q^45 + (-3*z^3 - z^2 - 3*z) * q^46 + (z^3 - 10*z^2 + 5) * q^47 + (-z^3 + z) * q^48 + (-4*z^3 + 3*z^2 + 2*z - 3) * q^49 + (3*z^3 + 4) * q^50 + (-2*z^3 + 4*z + 1) * q^51 + (-3*z^3 - 2) * q^52 + (6*z^3 + 2*z^2 - 1) * q^53 + (-z^2 + 1) * q^54 + (z^3 - z^2 - 3*z + 4) * q^55 + (-z^3 - z^2 - z) * q^56 + (-3*z^3 + 2*z^2 - 1) * q^57 + (3*z^3 + 2*z^2 - 3*z - 4) * q^58 + (-6*z^3 - 2*z^2 - 6*z) * q^59 + (2*z^2 + z - 2) * q^60 + (7*z^3 - 2*z^2 + 7*z) * q^61 + 4*z * q^62 + (-z^2 - z - 1) * q^63 - q^64 + (-7*z^3 + 4*z^2 + 7*z) * q^65 + (-z^3 + 2*z - 1) * q^66 + (5*z^2 - z + 5) * q^67 + (-2*z^2 - z - 2) * q^68 + (3*z^3 + z^2 + 3*z) * q^69 + (-2*z^3 + z^2 + 3*z + 3) * q^70 + (z^3 - 3*z^2 + z) * q^71 + (z^3 - z) * q^72 + (4*z^3 - 10*z^2 + 5) * q^73 + (4*z^3 + z^2 + 4*z) * q^74 + (-3*z^3 - 4) * q^75 + (-2*z^3 + 3*z^2 + z - 3) * q^76 - 2*z^3 * q^77 + (3*z^3 + 2) * q^78 - 12 * q^79 + (-2*z^3 - z^2 + 2*z) * q^80 + (z^2 - 1) * q^81 + (-4*z^3 - 3*z^2 + 4*z + 6) * q^82 + (7*z^3 - 2*z^2 + 1) * q^83 + (z^3 + z^2 + z) * q^84 + (-5*z^3 - 4*z^2 + 8*z + 4) * q^85 + (-z^3 + 2*z - 5) * q^86 + (-3*z^3 - 2*z^2 + 3*z + 4) * q^87 + (-z^2 + z - 1) * q^88 + (8*z^3 - 2*z^2 - 4*z + 2) * q^89 + (-2*z^2 - z + 2) * q^90 + (-5*z^3 - 4*z^2 + 6*z + 5) * q^91 + (-z^3 - 6*z^2 + 3) * q^92 - 4*z * q^93 + (-10*z^3 + z^2 + 5*z - 1) * q^94 + (-7*z^3 + 4*z^2 + 2*z - 5) * q^95 + q^96 + (-10*z^3 + 10*z) * q^97 + (3*z^3 - 2*z^2 - 3*z + 4) * q^98 + (z^3 - 2*z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^7 + 2 * q^9 $$4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{9} - 4 q^{10} + 2 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{15} - 2 q^{16} - 12 q^{17} + 6 q^{19} - 4 q^{20} + 4 q^{21} - 6 q^{22} - 18 q^{23} + 2 q^{24} + 6 q^{25} - 6 q^{26} - 6 q^{28} - 6 q^{29} + 4 q^{30} + 16 q^{31} + 6 q^{33} - 4 q^{34} + 8 q^{35} - 2 q^{36} + 24 q^{37} + 6 q^{39} + 4 q^{40} + 8 q^{41} + 6 q^{42} + 6 q^{43} + 4 q^{44} - 4 q^{45} - 2 q^{46} - 6 q^{49} + 16 q^{50} + 4 q^{51} - 8 q^{52} + 2 q^{54} + 14 q^{55} - 2 q^{56} - 12 q^{58} - 4 q^{59} - 4 q^{60} - 4 q^{61} - 6 q^{63} - 4 q^{64} + 8 q^{65} - 4 q^{66} + 30 q^{67} - 12 q^{68} + 2 q^{69} + 14 q^{70} - 6 q^{71} + 2 q^{74} - 16 q^{75} - 6 q^{76} + 8 q^{78} - 48 q^{79} - 2 q^{80} - 2 q^{81} + 18 q^{82} + 2 q^{84} + 8 q^{85} - 20 q^{86} + 12 q^{87} - 6 q^{88} + 4 q^{89} + 4 q^{90} + 12 q^{91} - 2 q^{94} - 12 q^{95} + 4 q^{96} + 12 q^{98} + 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^7 + 2 * q^9 - 4 * q^10 + 2 * q^11 - 4 * q^13 - 4 * q^14 + 4 * q^15 - 2 * q^16 - 12 * q^17 + 6 * q^19 - 4 * q^20 + 4 * q^21 - 6 * q^22 - 18 * q^23 + 2 * q^24 + 6 * q^25 - 6 * q^26 - 6 * q^28 - 6 * q^29 + 4 * q^30 + 16 * q^31 + 6 * q^33 - 4 * q^34 + 8 * q^35 - 2 * q^36 + 24 * q^37 + 6 * q^39 + 4 * q^40 + 8 * q^41 + 6 * q^42 + 6 * q^43 + 4 * q^44 - 4 * q^45 - 2 * q^46 - 6 * q^49 + 16 * q^50 + 4 * q^51 - 8 * q^52 + 2 * q^54 + 14 * q^55 - 2 * q^56 - 12 * q^58 - 4 * q^59 - 4 * q^60 - 4 * q^61 - 6 * q^63 - 4 * q^64 + 8 * q^65 - 4 * q^66 + 30 * q^67 - 12 * q^68 + 2 * q^69 + 14 * q^70 - 6 * q^71 + 2 * q^74 - 16 * q^75 - 6 * q^76 + 8 * q^78 - 48 * q^79 - 2 * q^80 - 2 * q^81 + 18 * q^82 + 2 * q^84 + 8 * q^85 - 20 * q^86 + 12 * q^87 - 6 * q^88 + 4 * q^89 + 4 * q^90 + 12 * q^91 - 2 * q^94 - 12 * q^95 + 4 * q^96 + 12 * q^98 + 4 * 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 1.23205 + 1.86603i −0.500000 0.866025i −0.633975 + 0.366025i 1.00000i 0.500000 + 0.866025i −0.133975 2.23205i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.23205 0.133975i −0.500000 0.866025i −2.36603 + 1.36603i 1.00000i 0.500000 + 0.866025i −1.86603 1.23205i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.23205 1.86603i −0.500000 + 0.866025i −0.633975 0.366025i 1.00000i 0.500000 0.866025i −0.133975 + 2.23205i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.23205 + 0.133975i −0.500000 + 0.866025i −2.36603 1.36603i 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
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.b 4
3.b odd 2 1 1170.2.bp.d 4
5.b even 2 1 390.2.y.c yes 4
5.c odd 4 1 1950.2.i.y 4
5.c odd 4 1 1950.2.i.bh 4
13.c even 3 1 390.2.y.c yes 4
15.d odd 2 1 1170.2.bp.e 4
39.i odd 6 1 1170.2.bp.e 4
65.n even 6 1 inner 390.2.y.b 4
65.q odd 12 1 1950.2.i.y 4
65.q odd 12 1 1950.2.i.bh 4
195.x odd 6 1 1170.2.bp.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 6T_{7}^{3} + 14T_{7}^{2} + 12T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 25$$
$7$ $$T^{4} + 6 T^{3} + \cdots + 4$$
$11$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$13$ $$T^{4} + 4 T^{3} + \cdots + 169$$
$17$ $$T^{4} + 12 T^{3} + \cdots + 121$$
$19$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$23$ $$T^{4} + 18 T^{3} + \cdots + 676$$
$29$ $$T^{4} + 6 T^{3} + \cdots + 9$$
$31$ $$(T - 4)^{4}$$
$37$ $$T^{4} - 24 T^{3} + \cdots + 2209$$
$41$ $$T^{4} - 8 T^{3} + \cdots + 121$$
$43$ $$T^{4} - 6 T^{3} + \cdots + 484$$
$47$ $$T^{4} + 152T^{2} + 5476$$
$53$ $$T^{4} + 78T^{2} + 1089$$
$59$ $$T^{4} + 4 T^{3} + \cdots + 10816$$
$61$ $$T^{4} + 4 T^{3} + \cdots + 20449$$
$67$ $$T^{4} - 30 T^{3} + \cdots + 5476$$
$71$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$73$ $$T^{4} + 182T^{2} + 3481$$
$79$ $$(T + 12)^{4}$$
$83$ $$T^{4} + 104T^{2} + 2116$$
$89$ $$T^{4} - 4 T^{3} + \cdots + 1936$$
$97$ $$T^{4} - 100 T^{2} + 10000$$