# Properties

 Label 390.2.p.b Level $390$ Weight $2$ Character orbit 390.p 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(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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} - 1) q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{6} + ( - 3 \zeta_{8}^{2} - 3) q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 1) q^{9} +O(q^{10})$$ q + z * q^2 + (-z^3 - z - 1) * q^3 + z^2 * q^4 - z * q^5 + (-z^2 - z + 1) * q^6 + (-3*z^2 - 3) * q^7 + z^3 * q^8 + (2*z^3 + 2*z - 1) * q^9 $$q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} - 1) q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{6} + ( - 3 \zeta_{8}^{2} - 3) q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 1) q^{9} - \zeta_{8}^{2} q^{10} + 4 \zeta_{8}^{3} q^{11} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{12} + (3 \zeta_{8}^{2} - 2) q^{13} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{14} + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{15} - q^{16} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{17} + (2 \zeta_{8}^{2} - \zeta_{8} - 2) q^{18} + ( - 5 \zeta_{8}^{2} + 5) q^{19} - \zeta_{8}^{3} q^{20} + (6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 3) q^{21} - 4 q^{22} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{23} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{24} + \zeta_{8}^{2} q^{25} + (3 \zeta_{8}^{3} - 2 \zeta_{8}) q^{26} + ( - \zeta_{8}^{3} - \zeta_{8} + 5) q^{27} + ( - 3 \zeta_{8}^{2} + 3) q^{28} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{29} + (\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{30} + ( - 3 \zeta_{8}^{2} + 3) q^{31} - \zeta_{8} q^{32} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4) q^{33} + ( - 4 \zeta_{8}^{2} - 4) q^{34} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{35} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{36} + ( - 7 \zeta_{8}^{2} - 7) q^{37} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{38} + ( - \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \cdots + 2) q^{39} + \cdots + ( - 4 \zeta_{8}^{3} - 8 \zeta_{8}^{2} - 8) q^{99} +O(q^{100})$$ q + z * q^2 + (-z^3 - z - 1) * q^3 + z^2 * q^4 - z * q^5 + (-z^2 - z + 1) * q^6 + (-3*z^2 - 3) * q^7 + z^3 * q^8 + (2*z^3 + 2*z - 1) * q^9 - z^2 * q^10 + 4*z^3 * q^11 + (-z^3 - z^2 + z) * q^12 + (3*z^2 - 2) * q^13 + (-3*z^3 - 3*z) * q^14 + (z^2 + z - 1) * q^15 - q^16 + (4*z^3 - 4*z) * q^17 + (2*z^2 - z - 2) * q^18 + (-5*z^2 + 5) * q^19 - z^3 * q^20 + (6*z^3 + 3*z^2 + 3) * q^21 - 4 * q^22 + (2*z^3 - 2*z) * q^23 + (-z^3 + z^2 + 1) * q^24 + z^2 * q^25 + (3*z^3 - 2*z) * q^26 + (-z^3 - z + 5) * q^27 + (-3*z^2 + 3) * q^28 + (-2*z^3 - 2*z) * q^29 + (z^3 + z^2 - z) * q^30 + (-3*z^2 + 3) * q^31 - z * q^32 + (-4*z^3 + 4*z^2 + 4) * q^33 + (-4*z^2 - 4) * q^34 + (3*z^3 + 3*z) * q^35 + (2*z^3 - z^2 - 2*z) * q^36 + (-7*z^2 - 7) * q^37 + (-5*z^3 + 5*z) * q^38 + (-z^3 - 3*z^2 + 5*z + 2) * q^39 + q^40 + 2*z * q^41 + (3*z^3 + 3*z - 6) * q^42 - 6*z^2 * q^43 - 4*z * q^44 + (-2*z^2 + z + 2) * q^45 + (-2*z^2 - 2) * q^46 - 4*z^3 * q^47 + (z^3 + z + 1) * q^48 + 11*z^2 * q^49 + z^3 * q^50 + (-4*z^3 + 8*z^2 + 4*z) * q^51 + (-2*z^2 - 3) * q^52 + (4*z^3 + 4*z) * q^53 + (-z^2 + 5*z + 1) * q^54 + 4 * q^55 + (-3*z^3 + 3*z) * q^56 + (5*z^2 - 10*z - 5) * q^57 + (-2*z^2 + 2) * q^58 + 12*z^3 * q^59 + (z^3 - z^2 - 1) * q^60 + (-3*z^3 + 3*z) * q^62 + (-12*z^3 + 3*z^2 + 3) * q^63 - z^2 * q^64 + (-3*z^3 + 2*z) * q^65 + (4*z^3 + 4*z + 4) * q^66 + (-3*z^2 + 3) * q^67 + (-4*z^3 - 4*z) * q^68 + (-2*z^3 + 4*z^2 + 2*z) * q^69 + (3*z^2 - 3) * q^70 - 4*z * q^71 + (-z^3 - 2*z^2 - 2) * q^72 + (5*z^2 + 5) * q^73 + (-7*z^3 - 7*z) * q^74 + (-z^3 - z^2 + z) * q^75 + (5*z^2 + 5) * q^76 + (-12*z^3 + 12*z) * q^77 + (-3*z^3 + 5*z^2 + 2*z + 1) * q^78 - 10 * q^79 + z * q^80 + (-4*z^3 - 4*z - 7) * q^81 + 2*z^2 * q^82 + 8*z * q^83 + (3*z^2 - 6*z - 3) * q^84 + (4*z^2 + 4) * q^85 - 6*z^3 * q^86 + (2*z^3 + 2*z - 4) * q^87 - 4*z^2 * q^88 - 10*z^3 * q^89 + (-2*z^3 + z^2 + 2*z) * q^90 + (-3*z^2 + 15) * q^91 + (-2*z^3 - 2*z) * q^92 + (3*z^2 - 6*z - 3) * q^93 + 4 * q^94 + (5*z^3 - 5*z) * q^95 + (z^2 + z - 1) * q^96 + (-3*z^2 + 3) * q^97 + 11*z^3 * q^98 + (-4*z^3 - 8*z^2 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{6} - 12 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^6 - 12 * q^7 - 4 * q^9 $$4 q - 4 q^{3} + 4 q^{6} - 12 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{15} - 4 q^{16} - 8 q^{18} + 20 q^{19} + 12 q^{21} - 16 q^{22} + 4 q^{24} + 20 q^{27} + 12 q^{28} + 12 q^{31} + 16 q^{33} - 16 q^{34} - 28 q^{37} + 8 q^{39} + 4 q^{40} - 24 q^{42} + 8 q^{45} - 8 q^{46} + 4 q^{48} - 12 q^{52} + 4 q^{54} + 16 q^{55} - 20 q^{57} + 8 q^{58} - 4 q^{60} + 12 q^{63} + 16 q^{66} + 12 q^{67} - 12 q^{70} - 8 q^{72} + 20 q^{73} + 20 q^{76} + 4 q^{78} - 40 q^{79} - 28 q^{81} - 12 q^{84} + 16 q^{85} - 16 q^{87} + 60 q^{91} - 12 q^{93} + 16 q^{94} - 4 q^{96} + 12 q^{97} - 32 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^6 - 12 * q^7 - 4 * q^9 - 8 * q^13 - 4 * q^15 - 4 * q^16 - 8 * q^18 + 20 * q^19 + 12 * q^21 - 16 * q^22 + 4 * q^24 + 20 * q^27 + 12 * q^28 + 12 * q^31 + 16 * q^33 - 16 * q^34 - 28 * q^37 + 8 * q^39 + 4 * q^40 - 24 * q^42 + 8 * q^45 - 8 * q^46 + 4 * q^48 - 12 * q^52 + 4 * q^54 + 16 * q^55 - 20 * q^57 + 8 * q^58 - 4 * q^60 + 12 * q^63 + 16 * q^66 + 12 * q^67 - 12 * q^70 - 8 * q^72 + 20 * q^73 + 20 * q^76 + 4 * q^78 - 40 * q^79 - 28 * q^81 - 12 * q^84 + 16 * q^85 - 16 * q^87 + 60 * q^91 - 12 * q^93 + 16 * q^94 - 4 * q^96 + 12 * q^97 - 32 * q^99

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 0.707107i −1.00000 + 1.41421i 1.00000i 0.707107 + 0.707107i 1.70711 0.292893i −3.00000 3.00000i 0.707107 0.707107i −1.00000 2.82843i 1.00000i
161.2 0.707107 + 0.707107i −1.00000 1.41421i 1.00000i −0.707107 0.707107i 0.292893 1.70711i −3.00000 3.00000i −0.707107 + 0.707107i −1.00000 + 2.82843i 1.00000i
281.1 −0.707107 + 0.707107i −1.00000 1.41421i 1.00000i 0.707107 0.707107i 1.70711 + 0.292893i −3.00000 + 3.00000i 0.707107 + 0.707107i −1.00000 + 2.82843i 1.00000i
281.2 0.707107 0.707107i −1.00000 + 1.41421i 1.00000i −0.707107 + 0.707107i 0.292893 + 1.70711i −3.00000 + 3.00000i −0.707107 0.707107i −1.00000 2.82843i 1.00000i
 $$n$$: 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
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.b 4
3.b odd 2 1 inner 390.2.p.b 4
13.d odd 4 1 inner 390.2.p.b 4
39.f even 4 1 inner 390.2.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.p.b 4 1.a even 1 1 trivial
390.2.p.b 4 3.b odd 2 1 inner
390.2.p.b 4 13.d odd 4 1 inner
390.2.p.b 4 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18 $$T_{11}^{4} + 256$$ T11^4 + 256 $$T_{17}^{2} - 32$$ T17^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$(T^{2} + 2 T + 3)^{2}$$
$5$ $$T^{4} + 1$$
$7$ $$(T^{2} + 6 T + 18)^{2}$$
$11$ $$T^{4} + 256$$
$13$ $$(T^{2} + 4 T + 13)^{2}$$
$17$ $$(T^{2} - 32)^{2}$$
$19$ $$(T^{2} - 10 T + 50)^{2}$$
$23$ $$(T^{2} - 8)^{2}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} - 6 T + 18)^{2}$$
$37$ $$(T^{2} + 14 T + 98)^{2}$$
$41$ $$T^{4} + 16$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} + 256$$
$53$ $$(T^{2} + 32)^{2}$$
$59$ $$T^{4} + 20736$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 6 T + 18)^{2}$$
$71$ $$T^{4} + 256$$
$73$ $$(T^{2} - 10 T + 50)^{2}$$
$79$ $$(T + 10)^{4}$$
$83$ $$T^{4} + 4096$$
$89$ $$T^{4} + 10000$$
$97$ $$(T^{2} - 6 T + 18)^{2}$$