# Properties

 Label 380.2.i.b Level $380$ Weight $2$ Character orbit 380.i Analytic conductor $3.034$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(121,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("380.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.i (of order $$3$$, 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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1783323.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9$$ x^6 - x^5 + 5*x^4 - 2*x^3 + 19*x^2 - 12*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{3}) q^{3} + (\beta_{4} + 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10})$$ q + (-b5 + b3) * q^3 + (b4 + 1) * q^5 + (-b3 + 1) * q^7 + (-2*b5 + 3*b4 + b3 + 2*b2 + b1) * q^9 $$q + ( - \beta_{5} + \beta_{3}) q^{3} + (\beta_{4} + 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + (3 \beta_{5} + 3 \beta_{4} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q + (-b5 + b3) * q^3 + (b4 + 1) * q^5 + (-b3 + 1) * q^7 + (-2*b5 + 3*b4 + b3 + 2*b2 + b1) * q^9 + (-b3 + 2) * q^11 - b4 * q^13 - b5 * q^15 + (b3 - b2 - 2*b1) * q^17 + (b5 + b4 + b3 - b2 - b1) * q^19 + (b5 - 6*b4 - b2 - 2*b1 - 6) * q^21 + (2*b5 - 5*b4 - b3 - 2*b2 - b1) * q^23 + b4 * q^25 + (-2*b3 + b2 - b1 - 3) * q^27 + (-3*b5 + 3*b4 + 2*b3 + 4*b2 + 2*b1) * q^29 + (-b2 + b1 + 4) * q^31 + (-6*b4 + b3 - b2 - 2*b1 - 6) * q^33 + (b5 + b4 - b3 + 1) * q^35 + (-b3 + 1) * q^37 + b3 * q^39 + (-3*b5 - 3*b4 + 3*b3 - 3) * q^41 + (b5 + 3*b4 - 2*b3 + b2 + 2*b1 + 3) * q^43 + (b2 - b1 - 3) * q^45 + (3*b5 - 3*b4 - 2*b3 - 4*b2 - 2*b1) * q^47 + (-2*b3 - b2 + b1) * q^49 + (9*b4 + b3 + 2*b2 + b1) * q^51 + (-2*b5 - 4*b4 + b3 + 2*b2 + b1) * q^53 + (b5 + 2*b4 - b3 + 2) * q^55 + (-2*b5 + 9*b4 + b3 + b2 + 3*b1 + 9) * q^57 + (-3*b5 + 3*b4 + b3 + 2*b2 + 4*b1 + 3) * q^59 + (2*b5 + 3*b4) * q^61 + 5*b5 * q^63 + q^65 + (2*b5 - 2*b3 - 4*b2 - 2*b1) * q^67 + (7*b3 - b2 + b1 + 3) * q^69 + (-3*b5 + 3*b4 + 2*b3 + b2 + 2*b1 + 3) * q^71 + (7*b4 - b3 + b2 + 2*b1 + 7) * q^73 - b3 * q^75 + (-3*b3 - b2 + b1 + 8) * q^77 + (2*b5 - 10*b4 - 2*b3 - 10) * q^79 + (5*b5 - 6*b4 - 5*b3 - 6) * q^81 + (-b2 + b1 - 10) * q^83 + (-b3 - 2*b2 - b1) * q^85 + (-7*b3 + b2 - b1) * q^87 + (4*b5 + 2*b4) * q^89 + (-b5 - b4) * q^91 + (-8*b5 + 3*b4 + 7*b3 + b2 + 2*b1 + 3) * q^93 + (-b5 + b3 - b2 - 1) * q^95 + (5*b5 - 4*b4 - 3*b3 - 2*b2 - 4*b1 - 4) * q^97 + (3*b5 + 3*b4 + b3 + 2*b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} + 3 q^{5} + 4 q^{7} - 8 q^{9}+O(q^{10})$$ 6 * q + q^3 + 3 * q^5 + 4 * q^7 - 8 * q^9 $$6 q + q^{3} + 3 q^{5} + 4 q^{7} - 8 q^{9} + 10 q^{11} + 3 q^{13} - q^{15} + 3 q^{17} - 16 q^{21} + 14 q^{23} - 3 q^{25} - 20 q^{27} - 6 q^{29} + 22 q^{31} - 15 q^{33} + 2 q^{35} + 4 q^{37} + 2 q^{39} - 6 q^{41} + 5 q^{43} - 16 q^{45} + 6 q^{47} - 6 q^{49} - 24 q^{51} + 13 q^{53} + 5 q^{55} + 25 q^{57} + 6 q^{59} - 7 q^{61} + 5 q^{63} + 6 q^{65} - 4 q^{67} + 30 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{75} + 40 q^{77} - 32 q^{79} - 23 q^{81} - 62 q^{83} - 3 q^{85} - 12 q^{87} - 2 q^{89} + 2 q^{91} + 14 q^{93} - 6 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100})$$ 6 * q + q^3 + 3 * q^5 + 4 * q^7 - 8 * q^9 + 10 * q^11 + 3 * q^13 - q^15 + 3 * q^17 - 16 * q^21 + 14 * q^23 - 3 * q^25 - 20 * q^27 - 6 * q^29 + 22 * q^31 - 15 * q^33 + 2 * q^35 + 4 * q^37 + 2 * q^39 - 6 * q^41 + 5 * q^43 - 16 * q^45 + 6 * q^47 - 6 * q^49 - 24 * q^51 + 13 * q^53 + 5 * q^55 + 25 * q^57 + 6 * q^59 - 7 * q^61 + 5 * q^63 + 6 * q^65 - 4 * q^67 + 30 * q^69 + 9 * q^71 + 18 * q^73 - 2 * q^75 + 40 * q^77 - 32 * q^79 - 23 * q^81 - 62 * q^83 - 3 * q^85 - 12 * q^87 - 2 * q^89 + 2 * q^91 + 14 * q^93 - 6 * q^95 - 11 * q^97 - 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + 15\nu^{4} + 8\nu^{3} + 57\nu^{2} + 47\nu + 180 ) / 83$$ (-3*v^5 + 15*v^4 + 8*v^3 + 57*v^2 + 47*v + 180) / 83 $$\beta_{2}$$ $$=$$ $$( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 131\nu - 240 ) / 83$$ (4*v^5 - 20*v^4 + 17*v^3 - 76*v^2 + 131*v - 240) / 83 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 25\nu^{4} - 42\nu^{3} + 95\nu^{2} - 60\nu + 300 ) / 83$$ (-5*v^5 + 25*v^4 - 42*v^3 + 95*v^2 - 60*v + 300) / 83 $$\beta_{4}$$ $$=$$ $$( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu - 45 ) / 249$$ (-20*v^5 + 17*v^4 - 85*v^3 - 35*v^2 - 323*v - 45) / 249 $$\beta_{5}$$ $$=$$ $$( 16\nu^{5} + 3\nu^{4} + 68\nu^{3} + 28\nu^{2} + 358\nu + 36 ) / 83$$ (16*v^5 + 3*v^4 + 68*v^3 + 28*v^2 + 358*v + 36) / 83
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + \beta_1 ) / 3$$ (b3 + 2*b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( -3\beta_{5} - 9\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 9 ) / 3$$ (-3*b5 - 9*b4 + 2*b3 + b2 + 2*b1 - 9) / 3 $$\nu^{3}$$ $$=$$ $$( -7\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 3$$ (-7*b3 - 5*b2 + 5*b1) / 3 $$\nu^{4}$$ $$=$$ $$5\beta_{5} + 12\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta_1$$ 5*b5 + 12*b4 - 2*b3 - 4*b2 - 2*b1 $$\nu^{5}$$ $$=$$ $$( 18\beta_{5} + 9\beta_{4} + 5\beta_{3} - 23\beta_{2} - 46\beta _1 + 9 ) / 3$$ (18*b5 + 9*b4 + 5*b3 - 23*b2 - 46*b1 + 9) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −0.956115 + 1.65604i 1.09935 − 1.90412i 0.356769 − 0.617942i −0.956115 − 1.65604i 1.09935 + 1.90412i 0.356769 + 0.617942i
0 −1.28442 2.22469i 0 0.500000 + 0.866025i 0 3.56885 0 −1.79949 + 3.11682i 0
121.2 0 0.182224 + 0.315621i 0 0.500000 + 0.866025i 0 0.635552 0 1.43359 2.48305i 0
121.3 0 1.60220 + 2.77509i 0 0.500000 + 0.866025i 0 −2.20440 0 −3.63409 + 6.29444i 0
201.1 0 −1.28442 + 2.22469i 0 0.500000 0.866025i 0 3.56885 0 −1.79949 3.11682i 0
201.2 0 0.182224 0.315621i 0 0.500000 0.866025i 0 0.635552 0 1.43359 + 2.48305i 0
201.3 0 1.60220 2.77509i 0 0.500000 0.866025i 0 −2.20440 0 −3.63409 6.29444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.i.b 6
3.b odd 2 1 3420.2.t.v 6
4.b odd 2 1 1520.2.q.i 6
5.b even 2 1 1900.2.i.c 6
5.c odd 4 2 1900.2.s.c 12
19.c even 3 1 inner 380.2.i.b 6
19.c even 3 1 7220.2.a.n 3
19.d odd 6 1 7220.2.a.o 3
57.h odd 6 1 3420.2.t.v 6
76.g odd 6 1 1520.2.q.i 6
95.i even 6 1 1900.2.i.c 6
95.m odd 12 2 1900.2.s.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.b 6 1.a even 1 1 trivial
380.2.i.b 6 19.c even 3 1 inner
1520.2.q.i 6 4.b odd 2 1
1520.2.q.i 6 76.g odd 6 1
1900.2.i.c 6 5.b even 2 1
1900.2.i.c 6 95.i even 6 1
1900.2.s.c 12 5.c odd 4 2
1900.2.s.c 12 95.m odd 12 2
3420.2.t.v 6 3.b odd 2 1
3420.2.t.v 6 57.h odd 6 1
7220.2.a.n 3 19.c even 3 1
7220.2.a.o 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - T_{3}^{5} + 9T_{3}^{4} + 2T_{3}^{3} + 67T_{3}^{2} - 24T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - T^{5} + 9 T^{4} + \cdots + 9$$
$5$ $$(T^{2} - T + 1)^{3}$$
$7$ $$(T^{3} - 2 T^{2} - 7 T + 5)^{2}$$
$11$ $$(T^{3} - 5 T^{2} + 9)^{2}$$
$13$ $$(T^{2} - T + 1)^{3}$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 6561$$
$19$ $$T^{6} - 18 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 14 T^{5} + \cdots + 2025$$
$29$ $$T^{6} + 6 T^{5} + \cdots + 263169$$
$31$ $$(T^{3} - 11 T^{2} + \cdots + 71)^{2}$$
$37$ $$(T^{3} - 2 T^{2} - 7 T + 5)^{2}$$
$41$ $$T^{6} + 6 T^{5} + \cdots + 18225$$
$43$ $$T^{6} - 5 T^{5} + \cdots + 11025$$
$47$ $$T^{6} - 6 T^{5} + \cdots + 263169$$
$53$ $$T^{6} - 13 T^{5} + \cdots + 81$$
$59$ $$T^{6} - 6 T^{5} + \cdots + 263169$$
$61$ $$T^{6} + 7 T^{5} + \cdots + 3969$$
$67$ $$T^{6} + 4 T^{5} + \cdots + 28224$$
$71$ $$T^{6} - 9 T^{5} + \cdots + 729$$
$73$ $$T^{6} - 18 T^{5} + \cdots + 625$$
$79$ $$T^{6} + 32 T^{5} + \cdots + 732736$$
$83$ $$(T^{3} + 31 T^{2} + \cdots + 855)^{2}$$
$89$ $$T^{6} + 2 T^{5} + \cdots + 5184$$
$97$ $$T^{6} + 11 T^{5} + \cdots + 93025$$