# Properties

 Label 390.2.l.a Level $390$ Weight $2$ Character orbit 390.l 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(53,390)]

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, 3, 0]))

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

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

chi := DirichletCharacter("390.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{6} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9}+O(q^{10})$$ q + z * q^2 + (-z^3 + z^2 + z) * q^3 + z^2 * q^4 + (-z^3 - 2*z) * q^5 + (z^3 + z^2 + 1) * q^6 + (-2*z^3 - 2*z^2 + 2) * q^7 + z^3 * q^8 + (2*z^3 + 2*z + 1) * q^9 $$q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{6} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{9} + ( - 2 \zeta_{8}^{2} + 1) q^{10} + 4 \zeta_{8}^{2} q^{11} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{12} + \zeta_{8} q^{13} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 2) q^{14} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \cdots - 1) q^{15}+ \cdots + (8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 8 \zeta_{8}) q^{99}+O(q^{100})$$ q + z * q^2 + (-z^3 + z^2 + z) * q^3 + z^2 * q^4 + (-z^3 - 2*z) * q^5 + (z^3 + z^2 + 1) * q^6 + (-2*z^3 - 2*z^2 + 2) * q^7 + z^3 * q^8 + (2*z^3 + 2*z + 1) * q^9 + (-2*z^2 + 1) * q^10 + 4*z^2 * q^11 + (z^3 + z - 1) * q^12 + z * q^13 + (-2*z^3 + 2*z + 2) * q^14 + (-2*z^3 - 3*z^2 + z - 1) * q^15 - q^16 + (-3*z^2 - 3) * q^17 + (2*z^2 + z - 2) * q^18 + (-2*z^3 + 4*z^2 - 2*z) * q^19 + (-2*z^3 + z) * q^20 + (-4*z^3 + 2*z + 4) * q^21 + 4*z^3 * q^22 - 4*z^3 * q^23 + (z^2 - z - 1) * q^24 + (3*z^2 - 4) * q^25 + z^2 * q^26 + (z^3 + 5*z^2 - z) * q^27 + (2*z^2 + 2*z + 2) * q^28 - 6 * q^29 + (-3*z^3 + z^2 - z + 2) * q^30 + (z^3 - z + 2) * q^31 - z * q^32 + (4*z^3 + 4*z - 4) * q^33 + (-3*z^3 - 3*z) * q^34 + (2*z^3 - 2*z^2 - 6*z - 4) * q^35 + (2*z^3 + z^2 - 2*z) * q^36 + (2*z^3 + 4*z^2 - 4) * q^37 + (4*z^3 - 2*z^2 + 2) * q^38 + (z^3 + z^2 + 1) * q^39 + (z^2 + 2) * q^40 + 2*z^2 * q^41 + (2*z^2 + 4*z + 4) * q^42 + (-3*z^2 - 6*z - 3) * q^43 - 4 * q^44 + (-z^3 - 2*z^2 - 2*z + 6) * q^45 + 4 * q^46 + (2*z^2 + 6*z + 2) * q^47 + (z^3 - z^2 - z) * q^48 + (-8*z^3 - 5*z^2 - 8*z) * q^49 + (3*z^3 - 4*z) * q^50 + (-3*z^2 - 6*z + 3) * q^51 + z^3 * q^52 + (-2*z^3 - 2*z^2 + 2) * q^53 + (5*z^3 - z^2 - 1) * q^54 + (-8*z^3 + 4*z) * q^55 + (2*z^3 + 2*z^2 + 2*z) * q^56 + (2*z^3 - 4*z^2 + 6*z - 4) * q^57 - 6*z * q^58 + (6*z^3 - 6*z - 4) * q^59 + (z^3 - z^2 + 2*z + 3) * q^60 + (2*z^3 - 2*z + 2) * q^61 + (-z^2 + 2*z - 1) * q^62 + (-2*z^3 + 2*z^2 + 8*z + 6) * q^63 - z^2 * q^64 + (-2*z^2 + 1) * q^65 + (4*z^2 - 4*z - 4) * q^66 + (-4*z^3 - 2*z^2 + 2) * q^67 + (-3*z^2 + 3) * q^68 + (-4*z^2 + 4*z + 4) * q^69 + (-2*z^3 - 6*z^2 - 4*z - 2) * q^70 + (3*z^3 - 10*z^2 + 3*z) * q^71 + (z^3 - 2*z^2 - 2) * q^72 + (-4*z^2 + 10*z - 4) * q^73 + (4*z^3 - 4*z - 2) * q^74 + (7*z^3 - 4*z^2 - z - 3) * q^75 + (-2*z^3 + 2*z - 4) * q^76 + (8*z^2 + 8*z + 8) * q^77 + (z^3 + z - 1) * q^78 + (-6*z^3 + 4*z^2 - 6*z) * q^79 + (z^3 + 2*z) * q^80 + (4*z^3 + 4*z - 7) * q^81 + 2*z^3 * q^82 + (-4*z^3 + 4*z^2 - 4) * q^83 + (2*z^3 + 4*z^2 + 4*z) * q^84 + (9*z^3 + 3*z) * q^85 + (-3*z^3 - 6*z^2 - 3*z) * q^86 + (6*z^3 - 6*z^2 - 6*z) * q^87 - 4*z * q^88 + (-2*z^3 + 2*z + 14) * q^89 + (-2*z^3 - 2*z^2 + 6*z + 1) * q^90 + (-2*z^3 + 2*z + 2) * q^91 + 4*z * q^92 + (-3*z^3 + 2*z^2 + z - 2) * q^93 + (2*z^3 + 6*z^2 + 2*z) * q^94 + (-8*z^3 + 2*z^2 + 4*z - 6) * q^95 + (-z^3 - z^2 - 1) * q^96 + (10*z^3 - 2*z^2 + 2) * q^97 + (-5*z^3 - 8*z^2 + 8) * q^98 + (8*z^3 + 4*z^2 - 8*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{6} + 8 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9 $$4 q + 4 q^{6} + 8 q^{7} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 8 q^{14} - 4 q^{15} - 4 q^{16} - 12 q^{17} - 8 q^{18} + 16 q^{21} - 4 q^{24} - 16 q^{25} + 8 q^{28} - 24 q^{29} + 8 q^{30} + 8 q^{31} - 16 q^{33} - 16 q^{35} - 16 q^{37} + 8 q^{38} + 4 q^{39} + 8 q^{40} + 16 q^{42} - 12 q^{43} - 16 q^{44} + 24 q^{45} + 16 q^{46} + 8 q^{47} + 12 q^{51} + 8 q^{53} - 4 q^{54} - 16 q^{57} - 16 q^{59} + 12 q^{60} + 8 q^{61} - 4 q^{62} + 24 q^{63} + 4 q^{65} - 16 q^{66} + 8 q^{67} + 12 q^{68} + 16 q^{69} - 8 q^{70} - 8 q^{72} - 16 q^{73} - 8 q^{74} - 12 q^{75} - 16 q^{76} + 32 q^{77} - 4 q^{78} - 28 q^{81} - 16 q^{83} + 56 q^{89} + 4 q^{90} + 8 q^{91} - 8 q^{93} - 24 q^{95} - 4 q^{96} + 8 q^{97} + 32 q^{98}+O(q^{100})$$ 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9 + 4 * q^10 - 4 * q^12 + 8 * q^14 - 4 * q^15 - 4 * q^16 - 12 * q^17 - 8 * q^18 + 16 * q^21 - 4 * q^24 - 16 * q^25 + 8 * q^28 - 24 * q^29 + 8 * q^30 + 8 * q^31 - 16 * q^33 - 16 * q^35 - 16 * q^37 + 8 * q^38 + 4 * q^39 + 8 * q^40 + 16 * q^42 - 12 * q^43 - 16 * q^44 + 24 * q^45 + 16 * q^46 + 8 * q^47 + 12 * q^51 + 8 * q^53 - 4 * q^54 - 16 * q^57 - 16 * q^59 + 12 * q^60 + 8 * q^61 - 4 * q^62 + 24 * q^63 + 4 * q^65 - 16 * q^66 + 8 * q^67 + 12 * q^68 + 16 * q^69 - 8 * q^70 - 8 * q^72 - 16 * q^73 - 8 * q^74 - 12 * q^75 - 16 * q^76 + 32 * q^77 - 4 * q^78 - 28 * q^81 - 16 * q^83 + 56 * q^89 + 4 * q^90 + 8 * q^91 - 8 * q^93 - 24 * q^95 - 4 * q^96 + 8 * q^97 + 32 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 0.707107i −1.41421 + 1.00000i 1.00000i 0.707107 + 2.12132i 1.70711 + 0.292893i 0.585786 0.585786i 0.707107 0.707107i 1.00000 2.82843i 1.00000 2.00000i
53.2 0.707107 + 0.707107i 1.41421 + 1.00000i 1.00000i −0.707107 2.12132i 0.292893 + 1.70711i 3.41421 3.41421i −0.707107 + 0.707107i 1.00000 + 2.82843i 1.00000 2.00000i
287.1 −0.707107 + 0.707107i −1.41421 1.00000i 1.00000i 0.707107 2.12132i 1.70711 0.292893i 0.585786 + 0.585786i 0.707107 + 0.707107i 1.00000 + 2.82843i 1.00000 + 2.00000i
287.2 0.707107 0.707107i 1.41421 1.00000i 1.00000i −0.707107 + 2.12132i 0.292893 1.70711i 3.41421 + 3.41421i −0.707107 0.707107i 1.00000 2.82843i 1.00000 + 2.00000i
 $$n$$: 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
15.e 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.l.a 4
3.b odd 2 1 390.2.l.b yes 4
5.c odd 4 1 390.2.l.b yes 4
15.e even 4 1 inner 390.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.l.a 4 1.a even 1 1 trivial
390.2.l.a 4 15.e even 4 1 inner
390.2.l.b yes 4 3.b odd 2 1
390.2.l.b yes 4 5.c odd 4 1

## Hecke kernels

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

 $$T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} - 32T_{7} + 16$$ T7^4 - 8*T7^3 + 32*T7^2 - 32*T7 + 16 $$T_{17}^{2} + 6T_{17} + 18$$ T17^2 + 6*T17 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$T^{4} + 8T^{2} + 25$$
$7$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$11$ $$(T^{2} + 16)^{2}$$
$13$ $$T^{4} + 1$$
$17$ $$(T^{2} + 6 T + 18)^{2}$$
$19$ $$T^{4} + 48T^{2} + 64$$
$23$ $$T^{4} + 256$$
$29$ $$(T + 6)^{4}$$
$31$ $$(T^{2} - 4 T + 2)^{2}$$
$37$ $$T^{4} + 16 T^{3} + \cdots + 784$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$T^{4} + 12 T^{3} + \cdots + 324$$
$47$ $$T^{4} - 8 T^{3} + \cdots + 784$$
$53$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$59$ $$(T^{2} + 8 T - 56)^{2}$$
$61$ $$(T^{2} - 4 T - 4)^{2}$$
$67$ $$T^{4} - 8 T^{3} + \cdots + 64$$
$71$ $$T^{4} + 236T^{2} + 6724$$
$73$ $$T^{4} + 16 T^{3} + \cdots + 4624$$
$79$ $$T^{4} + 176T^{2} + 3136$$
$83$ $$T^{4} + 16 T^{3} + \cdots + 256$$
$89$ $$(T^{2} - 28 T + 188)^{2}$$
$97$ $$T^{4} - 8 T^{3} + \cdots + 8464$$