# Properties

 Label 380.2.c.b Level $380$ Weight $2$ Character orbit 380.c Analytic conductor $3.034$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(229,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("380.229");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.c (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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.14077504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 14x^{2} + 4$$ x^6 + 9*x^4 + 14*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 - 2) q^{9}+O(q^{10})$$ q + b2 * q^3 - b3 * q^5 + (-b4 - b3) * q^7 + (-b5 + b3 - b1 - 2) * q^9 $$q + \beta_{2} q^{3} - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 - 2) q^{9} + (\beta_1 + 3) q^{11} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{15} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{17} - q^{19} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{21} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{23} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{25} + ( - \beta_{5} + 3 \beta_{4} + \cdots - 4 \beta_{2}) q^{27}+ \cdots + ( - 6 \beta_{5} + 6 \beta_{3} + \cdots - 17) q^{99}+O(q^{100})$$ q + b2 * q^3 - b3 * q^5 + (-b4 - b3) * q^7 + (-b5 + b3 - b1 - 2) * q^9 + (b1 + 3) * q^11 + (-b5 + b4 - b2) * q^13 + (-b5 - b4 + b2 - 2) * q^15 + (-b5 - b3 - 2*b2) * q^17 - q^19 + (-2*b5 + 2*b3) * q^21 + (-b5 + b4 - 2*b2) * q^23 + (b3 + 2*b2 + b1 - 1) * q^25 + (-b5 + 3*b4 + 2*b3 - 4*b2) * q^27 + (-2*b5 + 2*b3 + 2*b1) * q^29 + (2*b5 - 2*b3 - 2*b1 + 2) * q^31 + (b5 - b4 + 6*b2) * q^33 + (b5 + b4 + b3 + 2*b2 - b1 - 2) * q^35 - b2 * q^37 + (b5 - b3 + b1 + 1) * q^39 + (-2*b5 + 2*b3 - 2*b1) * q^41 + (2*b5 - 3*b4 - b3) * q^43 + (-3*b5 + 2*b4 + 2*b3 - 4*b2 - b1 - 5) * q^45 + (2*b5 - 3*b4 - b3 + 4*b2) * q^47 + (3*b5 - 3*b3 - 2*b1 - 1) * q^49 + (2*b1 + 6) * q^51 + (-2*b5 - 2*b3 - b2) * q^53 + (3*b5 - 2*b4 - 4*b3 + 2*b2 + 1) * q^55 - b2 * q^57 + (2*b5 - 2*b3 - 4) * q^59 + (b1 + 5) * q^61 + (b4 + b3 - 4*b2) * q^63 + (-b2 + 2*b1 - 2) * q^65 + (b5 + 3*b4 + 4*b3 - b2) * q^67 + (2*b5 - 2*b3 + 2*b1 + 6) * q^69 + (2*b5 - 2*b3 + 8) * q^71 + (3*b5 + 3*b3 + 2*b2) * q^73 + (2*b3 + b2 - 2*b1 - 8) * q^75 + (-b5 - 4*b4 - 5*b3) * q^77 - 4 * q^79 + (5*b5 - 5*b3 + b1 + 10) * q^81 + (b5 - b4 + 2*b2) * q^83 + (2*b5 + 2*b4 + b3 + b1 - 2) * q^85 + (2*b5 + 2*b4 + 4*b3 + 2*b2) * q^87 + (-2*b5 + 2*b3 - 2) * q^89 + 4 * q^91 + (-2*b5 - 2*b4 - 4*b3) * q^93 + b3 * q^95 + (2*b5 + 2*b3 + b2) * q^97 + (-6*b5 + 6*b3 - 3*b1 - 17) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{5} - 10 q^{9}+O(q^{10})$$ 6 * q - q^5 - 10 * q^9 $$6 q - q^{5} - 10 q^{9} + 18 q^{11} - 10 q^{15} - 6 q^{19} + 4 q^{21} - 5 q^{25} + 4 q^{29} + 8 q^{31} - 13 q^{35} + 4 q^{39} + 4 q^{41} - 27 q^{45} - 12 q^{49} + 36 q^{51} + q^{55} - 28 q^{59} + 30 q^{61} - 12 q^{65} + 32 q^{69} + 44 q^{71} - 46 q^{75} - 24 q^{79} + 50 q^{81} - 15 q^{85} - 8 q^{89} + 24 q^{91} + q^{95} - 90 q^{99}+O(q^{100})$$ 6 * q - q^5 - 10 * q^9 + 18 * q^11 - 10 * q^15 - 6 * q^19 + 4 * q^21 - 5 * q^25 + 4 * q^29 + 8 * q^31 - 13 * q^35 + 4 * q^39 + 4 * q^41 - 27 * q^45 - 12 * q^49 + 36 * q^51 + q^55 - 28 * q^59 + 30 * q^61 - 12 * q^65 + 32 * q^69 + 44 * q^71 - 46 * q^75 - 24 * q^79 + 50 * q^81 - 15 * q^85 - 8 * q^89 + 24 * q^91 + q^95 - 90 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 14x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 9\nu^{3} + 14\nu ) / 2$$ (v^5 + 9*v^3 + 14*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} - \nu^{4} - 8\nu^{3} - 8\nu^{2} - 8\nu - 6 ) / 2$$ (-v^5 - v^4 - 8*v^3 - 8*v^2 - 8*v - 6) / 2 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 4\nu + 6 ) / 2$$ (-v^5 + v^4 - 8*v^3 + 8*v^2 - 4*v + 6) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 8\nu + 6 ) / 2$$ (-v^5 + v^4 - 8*v^3 + 8*v^2 - 8*v + 6) / 2
 $$\nu$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 - 3$$ b1 - 3 $$\nu^{3}$$ $$=$$ $$4\beta_{5} - 3\beta_{4} + \beta_{3} + 2\beta_{2}$$ 4*b5 - 3*b4 + b3 + 2*b2 $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{3} - 8\beta _1 + 18$$ b5 - b3 - 8*b1 + 18 $$\nu^{5}$$ $$=$$ $$-29\beta_{5} + 20\beta_{4} - 9\beta_{3} - 16\beta_{2}$$ -29*b5 + 20*b4 - 9*b3 - 16*b2

## Character values

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

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
229.1
 − 0.608430i − 1.23277i − 2.66648i 2.66648i 1.23277i 0.608430i
0 3.28715i 0 1.58777 1.57448i 0 1.93210i 0 −7.80536 0
229.2 0 1.62236i 0 −1.92411 + 1.13921i 0 4.74397i 0 0.367938 0
229.3 0 0.750054i 0 −0.163664 2.23007i 0 0.872810i 0 2.43742 0
229.4 0 0.750054i 0 −0.163664 + 2.23007i 0 0.872810i 0 2.43742 0
229.5 0 1.62236i 0 −1.92411 1.13921i 0 4.74397i 0 0.367938 0
229.6 0 3.28715i 0 1.58777 + 1.57448i 0 1.93210i 0 −7.80536 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.c.b 6
3.b odd 2 1 3420.2.f.c 6
4.b odd 2 1 1520.2.d.i 6
5.b even 2 1 inner 380.2.c.b 6
5.c odd 4 2 1900.2.a.k 6
15.d odd 2 1 3420.2.f.c 6
20.d odd 2 1 1520.2.d.i 6
20.e even 4 2 7600.2.a.cj 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.b 6 1.a even 1 1 trivial
380.2.c.b 6 5.b even 2 1 inner
1520.2.d.i 6 4.b odd 2 1
1520.2.d.i 6 20.d odd 2 1
1900.2.a.k 6 5.c odd 4 2
3420.2.f.c 6 3.b odd 2 1
3420.2.f.c 6 15.d odd 2 1
7600.2.a.cj 6 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 14T_{3}^{4} + 36T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 14 T^{4} + \cdots + 16$$
$5$ $$T^{6} + T^{5} + \cdots + 125$$
$7$ $$T^{6} + 27 T^{4} + \cdots + 64$$
$11$ $$(T^{3} - 9 T^{2} + 14 T + 28)^{2}$$
$13$ $$T^{6} + 26 T^{4} + \cdots + 64$$
$17$ $$T^{6} + 51 T^{4} + \cdots + 3136$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 44 T^{4} + \cdots + 256$$
$29$ $$(T^{3} - 2 T^{2} - 84 T - 88)^{2}$$
$31$ $$(T^{3} - 4 T^{2} + \cdots + 256)^{2}$$
$37$ $$T^{6} + 14 T^{4} + \cdots + 16$$
$41$ $$(T^{3} - 2 T^{2} + \cdots + 488)^{2}$$
$43$ $$T^{6} + 227 T^{4} + \cdots + 3136$$
$47$ $$T^{6} + 243 T^{4} + \cdots + 118336$$
$53$ $$T^{6} + 114 T^{4} + \cdots + 23104$$
$59$ $$(T^{3} + 14 T^{2} + \cdots - 128)^{2}$$
$61$ $$(T^{3} - 15 T^{2} + \cdots - 44)^{2}$$
$67$ $$T^{6} + 378 T^{4} + \cdots + 222784$$
$71$ $$(T^{3} - 22 T^{2} + \cdots - 32)^{2}$$
$73$ $$T^{6} + 251 T^{4} + \cdots + 118336$$
$79$ $$(T + 4)^{6}$$
$83$ $$T^{6} + 44 T^{4} + \cdots + 256$$
$89$ $$(T^{3} + 4 T^{2} - 44 T - 64)^{2}$$
$97$ $$T^{6} + 114 T^{4} + \cdots + 23104$$