# Properties

 Label 390.2.b.d Level $390$ Weight $2$ Character orbit 390.b 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(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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + \beta_1 q^{8} + q^{9}+O(q^{10})$$ q - b1 * q^2 + q^3 - q^4 + b1 * q^5 - b1 * q^6 + (b3 + b2 - b1) * q^7 + b1 * q^8 + q^9 $$q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + \beta_1 q^{8} + q^{9} + q^{10} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} - q^{12} + ( - \beta_{3} + 2 \beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{14} + \beta_1 q^{15} + q^{16} + 2 q^{17} - \beta_1 q^{18} - 6 \beta_1 q^{19} - \beta_1 q^{20} + (\beta_{3} + \beta_{2} - \beta_1) q^{21} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{22} + (\beta_{3} - \beta_{2} - \beta_1) q^{23} + \beta_1 q^{24} - q^{25} + ( - \beta_{2} - 2 \beta_1 + 2) q^{26} + q^{27} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{28} + 2 q^{29} + q^{30} + (\beta_{3} + \beta_{2} - \beta_1) q^{31} - \beta_1 q^{32} + (\beta_{3} + \beta_{2} + \beta_1) q^{33} - 2 \beta_1 q^{34} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{35} - q^{36} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{37} - 6 q^{38} + ( - \beta_{3} + 2 \beta_1 + 1) q^{39} - q^{40} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{41} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{42} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{43}+ \cdots + (\beta_{3} + \beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 + q^3 - q^4 + b1 * q^5 - b1 * q^6 + (b3 + b2 - b1) * q^7 + b1 * q^8 + q^9 + q^10 + (b3 + b2 + b1) * q^11 - q^12 + (-b3 + 2*b1 + 1) * q^13 + (-b3 + b2 + b1 - 2) * q^14 + b1 * q^15 + q^16 + 2 * q^17 - b1 * q^18 - 6*b1 * q^19 - b1 * q^20 + (b3 + b2 - b1) * q^21 + (-b3 + b2 + b1) * q^22 + (b3 - b2 - b1) * q^23 + b1 * q^24 - q^25 + (-b2 - 2*b1 + 2) * q^26 + q^27 + (-b3 - b2 + b1) * q^28 + 2 * q^29 + q^30 + (b3 + b2 - b1) * q^31 - b1 * q^32 + (b3 + b2 + b1) * q^33 - 2*b1 * q^34 + (b3 - b2 - b1 + 2) * q^35 - q^36 + (-b3 - b2 - b1) * q^37 - 6 * q^38 + (-b3 + 2*b1 + 1) * q^39 - q^40 + (-b3 - b2 + 5*b1) * q^41 + (-b3 + b2 + b1 - 2) * q^42 + (-2*b3 + 2*b2 + 2*b1 - 4) * q^43 + (-b3 - b2 - b1) * q^44 + b1 * q^45 + (b3 + b2 + b1) * q^46 + (-2*b3 - 2*b2 + 2*b1) * q^47 + q^48 + (-2*b3 + 2*b2 + 2*b1 - 13) * q^49 + b1 * q^50 + 2 * q^51 + (b3 - 2*b1 - 1) * q^52 + (3*b3 - 3*b2 - 3*b1 + 2) * q^53 - b1 * q^54 + (b3 - b2 - b1) * q^55 + (b3 - b2 - b1 + 2) * q^56 - 6*b1 * q^57 - 2*b1 * q^58 + (-b3 - b2 + 3*b1) * q^59 - b1 * q^60 + 10 * q^61 + (-b3 + b2 + b1 - 2) * q^62 + (b3 + b2 - b1) * q^63 - q^64 + (b2 + 2*b1 - 2) * q^65 + (-b3 + b2 + b1) * q^66 + (-b3 - b2 + 9*b1) * q^67 - 2 * q^68 + (b3 - b2 - b1) * q^69 + (b3 + b2 - b1) * q^70 + (2*b3 + 2*b2 + 2*b1) * q^71 + b1 * q^72 + (-b3 - b2 - 9*b1) * q^73 + (b3 - b2 - b1) * q^74 - q^75 + 6*b1 * q^76 - 16 * q^77 + (-b2 - 2*b1 + 2) * q^78 + 8 * q^79 + b1 * q^80 + q^81 + (b3 - b2 - b1 + 6) * q^82 + (-2*b3 - 2*b2 + 2*b1) * q^83 + (-b3 - b2 + b1) * q^84 + 2*b1 * q^85 + (-2*b3 - 2*b2 + 2*b1) * q^86 + 2 * q^87 + (b3 - b2 - b1) * q^88 + (b3 + b2 - b1) * q^89 + q^90 + (3*b3 - b2 + 5*b1 + 12) * q^91 + (-b3 + b2 + b1) * q^92 + (b3 + b2 - b1) * q^93 + (2*b3 - 2*b2 - 2*b1 + 4) * q^94 + 6 * q^95 - b1 * q^96 + (-b3 - b2 - 9*b1) * q^97 + (-2*b3 - 2*b2 + 11*b1) * q^98 + (b3 + b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 8 q^{17} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} - 24 q^{38} + 6 q^{39} - 4 q^{40} - 4 q^{42} - 8 q^{43} + 4 q^{48} - 44 q^{49} + 8 q^{51} - 6 q^{52} - 4 q^{53} - 4 q^{55} + 4 q^{56} + 40 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} + 4 q^{66} - 8 q^{68} - 4 q^{69} - 4 q^{74} - 4 q^{75} - 64 q^{77} + 6 q^{78} + 32 q^{79} + 4 q^{81} + 20 q^{82} + 8 q^{87} - 4 q^{88} + 4 q^{90} + 40 q^{91} + 4 q^{92} + 8 q^{94} + 24 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 - 4 * q^12 + 6 * q^13 - 4 * q^14 + 4 * q^16 + 8 * q^17 + 4 * q^22 - 4 * q^23 - 4 * q^25 + 6 * q^26 + 4 * q^27 + 8 * q^29 + 4 * q^30 + 4 * q^35 - 4 * q^36 - 24 * q^38 + 6 * q^39 - 4 * q^40 - 4 * q^42 - 8 * q^43 + 4 * q^48 - 44 * q^49 + 8 * q^51 - 6 * q^52 - 4 * q^53 - 4 * q^55 + 4 * q^56 + 40 * q^61 - 4 * q^62 - 4 * q^64 - 6 * q^65 + 4 * q^66 - 8 * q^68 - 4 * q^69 - 4 * q^74 - 4 * q^75 - 64 * q^77 + 6 * q^78 + 32 * q^79 + 4 * q^81 + 20 * q^82 + 8 * q^87 - 4 * q^88 + 4 * q^90 + 40 * q^91 + 4 * q^92 + 8 * q^94 + 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 5$$ v^2 + v + 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 4\nu^{2} + 9\nu - 20 ) / 4$$ (v^3 - 4*v^2 + 9*v - 20) / 4
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta _1 - 10 ) / 2$$ (-b3 + b2 + b1 - 10) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{3} - 5\beta_{2} + 13\beta_1 ) / 2$$ (-5*b3 - 5*b2 + 13*b1) / 2

## Character values

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

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

## 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
 − 2.56155i 1.56155i − 1.56155i 2.56155i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.12311i 1.00000i 1.00000 1.00000
181.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.12311i 1.00000i 1.00000 1.00000
181.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.12311i 1.00000i 1.00000 1.00000
181.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.12311i 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.d 4
3.b odd 2 1 1170.2.b.f 4
4.b odd 2 1 3120.2.g.o 4
5.b even 2 1 1950.2.b.h 4
5.c odd 4 1 1950.2.f.l 4
5.c odd 4 1 1950.2.f.o 4
13.b even 2 1 inner 390.2.b.d 4
13.d odd 4 1 5070.2.a.bd 2
13.d odd 4 1 5070.2.a.bh 2
39.d odd 2 1 1170.2.b.f 4
52.b odd 2 1 3120.2.g.o 4
65.d even 2 1 1950.2.b.h 4
65.h odd 4 1 1950.2.f.l 4
65.h odd 4 1 1950.2.f.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.d 4 1.a even 1 1 trivial
390.2.b.d 4 13.b even 2 1 inner
1170.2.b.f 4 3.b odd 2 1
1170.2.b.f 4 39.d odd 2 1
1950.2.b.h 4 5.b even 2 1
1950.2.b.h 4 65.d even 2 1
1950.2.f.l 4 5.c odd 4 1
1950.2.f.l 4 65.h odd 4 1
1950.2.f.o 4 5.c odd 4 1
1950.2.f.o 4 65.h odd 4 1
3120.2.g.o 4 4.b odd 2 1
3120.2.g.o 4 52.b odd 2 1
5070.2.a.bd 2 13.d odd 4 1
5070.2.a.bh 2 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{4} + 36T_{7}^{2} + 256$$ T7^4 + 36*T7^2 + 256 $$T_{11}^{4} + 36T_{11}^{2} + 256$$ T11^4 + 36*T11^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 36T^{2} + 256$$
$11$ $$T^{4} + 36T^{2} + 256$$
$13$ $$T^{4} - 6 T^{3} + \cdots + 169$$
$17$ $$(T - 2)^{4}$$
$19$ $$(T^{2} + 36)^{2}$$
$23$ $$(T^{2} + 2 T - 16)^{2}$$
$29$ $$(T - 2)^{4}$$
$31$ $$T^{4} + 36T^{2} + 256$$
$37$ $$T^{4} + 36T^{2} + 256$$
$41$ $$T^{4} + 84T^{2} + 64$$
$43$ $$(T^{2} + 4 T - 64)^{2}$$
$47$ $$T^{4} + 144T^{2} + 4096$$
$53$ $$(T^{2} + 2 T - 152)^{2}$$
$59$ $$T^{4} + 52T^{2} + 64$$
$61$ $$(T - 10)^{4}$$
$67$ $$T^{4} + 196T^{2} + 4096$$
$71$ $$T^{4} + 144T^{2} + 4096$$
$73$ $$T^{4} + 196T^{2} + 4096$$
$79$ $$(T - 8)^{4}$$
$83$ $$T^{4} + 144T^{2} + 4096$$
$89$ $$T^{4} + 36T^{2} + 256$$
$97$ $$T^{4} + 196T^{2} + 4096$$