# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(181,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("390.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 1.00000
181.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.b.a 2
3.b odd 2 1 1170.2.b.b 2
4.b odd 2 1 3120.2.g.f 2
5.b even 2 1 1950.2.b.g 2
5.c odd 4 1 1950.2.f.b 2
5.c odd 4 1 1950.2.f.i 2
13.b even 2 1 inner 390.2.b.a 2
13.d odd 4 1 5070.2.a.g 1
13.d odd 4 1 5070.2.a.m 1
39.d odd 2 1 1170.2.b.b 2
52.b odd 2 1 3120.2.g.f 2
65.d even 2 1 1950.2.b.g 2
65.h odd 4 1 1950.2.f.b 2
65.h odd 4 1 1950.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.a 2 1.a even 1 1 trivial
390.2.b.a 2 13.b even 2 1 inner
1170.2.b.b 2 3.b odd 2 1
1170.2.b.b 2 39.d odd 2 1
1950.2.b.g 2 5.b even 2 1
1950.2.b.g 2 65.d even 2 1
1950.2.f.b 2 5.c odd 4 1
1950.2.f.b 2 65.h odd 4 1
1950.2.f.i 2 5.c odd 4 1
1950.2.f.i 2 65.h odd 4 1
3120.2.g.f 2 4.b odd 2 1
3120.2.g.f 2 52.b odd 2 1
5070.2.a.g 1 13.d odd 4 1
5070.2.a.m 1 13.d odd 4 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}^{2} + 16$$ T11^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 16$$
$41$ $$T^{2} + 4$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$T^{2} + 144$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 100$$
$97$ $$T^{2} + 196$$