Properties

 Label 585.2.h.e Level $585$ Weight $2$ Character orbit 585.h Analytic conductor $4.671$ Analytic rank $0$ Dimension $4$ CM discriminant -195 Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(64,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("585.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.h (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: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-5}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 5x^{2} - 4x + 69$$ x^4 - 2*x^3 + 5*x^2 - 4*x + 69 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q - 2 * q^4 - b1 * q^5 + b3 * q^7 $$q - 2 q^{4} - \beta_1 q^{5} + \beta_{3} q^{7} + \beta_1 q^{11} - \beta_{3} q^{13} + 4 q^{16} - \beta_{2} q^{17} + 2 \beta_1 q^{20} - \beta_{2} q^{23} - 5 q^{25} - 2 \beta_{3} q^{28} - \beta_{2} q^{35} + \beta_{3} q^{37} - 5 \beta_1 q^{41} - 2 \beta_1 q^{44} + 6 q^{49} + 2 \beta_{3} q^{52} - \beta_{2} q^{53} + 5 q^{55} + 4 \beta_1 q^{59} - 7 q^{61} - 8 q^{64} + \beta_{2} q^{65} + 4 \beta_{3} q^{67} + 2 \beta_{2} q^{68} + 7 \beta_1 q^{71} - 2 \beta_{3} q^{73} + \beta_{2} q^{77} + 11 q^{79} - 4 \beta_1 q^{80} - 5 \beta_{3} q^{85} + \beta_1 q^{89} - 13 q^{91} + 2 \beta_{2} q^{92} + \beta_{3} q^{97}+O(q^{100})$$ q - 2 * q^4 - b1 * q^5 + b3 * q^7 + b1 * q^11 - b3 * q^13 + 4 * q^16 - b2 * q^17 + 2*b1 * q^20 - b2 * q^23 - 5 * q^25 - 2*b3 * q^28 - b2 * q^35 + b3 * q^37 - 5*b1 * q^41 - 2*b1 * q^44 + 6 * q^49 + 2*b3 * q^52 - b2 * q^53 + 5 * q^55 + 4*b1 * q^59 - 7 * q^61 - 8 * q^64 + b2 * q^65 + 4*b3 * q^67 + 2*b2 * q^68 + 7*b1 * q^71 - 2*b3 * q^73 + b2 * q^77 + 11 * q^79 - 4*b1 * q^80 - 5*b3 * q^85 + b1 * q^89 - 13 * q^91 + 2*b2 * q^92 + b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100})$$ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 24 * q^49 + 20 * q^55 - 28 * q^61 - 32 * q^64 + 44 * q^79 - 52 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 5x^{2} - 4x + 69$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} + 25\nu - 12 ) / 33$$ (2*v^3 - 3*v^2 + 25*v - 12) / 33 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} + 16\nu - 9 ) / 33$$ (-4*v^3 + 6*v^2 + 16*v - 9) / 33
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta _1 + 1 ) / 2$$ (b3 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 2\beta _1 - 3 ) / 2$$ (b3 + 2*b2 + 2*b1 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -11\beta_{3} + 3\beta_{2} + 11\beta _1 - 5 ) / 2$$ (-11*b3 + 3*b2 + 11*b1 - 5) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −1.30278 + 2.23607i 2.30278 + 2.23607i −1.30278 − 2.23607i 2.30278 − 2.23607i
0 0 −2.00000 2.23607i 0 −3.60555 0 0 0
64.2 0 0 −2.00000 2.23607i 0 3.60555 0 0 0
64.3 0 0 −2.00000 2.23607i 0 −3.60555 0 0 0
64.4 0 0 −2.00000 2.23607i 0 3.60555 0 0 0
 $$n$$: 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
195.e odd 2 1 CM by $$\Q(\sqrt{-195})$$
3.b odd 2 1 inner
5.b even 2 1 inner
13.b even 2 1 inner
15.d odd 2 1 inner
39.d odd 2 1 inner
65.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.h.e 4
3.b odd 2 1 inner 585.2.h.e 4
5.b even 2 1 inner 585.2.h.e 4
13.b even 2 1 inner 585.2.h.e 4
15.d odd 2 1 inner 585.2.h.e 4
39.d odd 2 1 inner 585.2.h.e 4
65.d even 2 1 inner 585.2.h.e 4
195.e odd 2 1 CM 585.2.h.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.h.e 4 1.a even 1 1 trivial
585.2.h.e 4 3.b odd 2 1 inner
585.2.h.e 4 5.b even 2 1 inner
585.2.h.e 4 13.b even 2 1 inner
585.2.h.e 4 15.d odd 2 1 inner
585.2.h.e 4 39.d odd 2 1 inner
585.2.h.e 4 65.d even 2 1 inner
585.2.h.e 4 195.e odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} - 13)^{2}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} - 13)^{2}$$
$17$ $$(T^{2} + 65)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 65)^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 13)^{2}$$
$41$ $$(T^{2} + 125)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 65)^{2}$$
$59$ $$(T^{2} + 80)^{2}$$
$61$ $$(T + 7)^{4}$$
$67$ $$(T^{2} - 208)^{2}$$
$71$ $$(T^{2} + 245)^{2}$$
$73$ $$(T^{2} - 52)^{2}$$
$79$ $$(T - 11)^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 5)^{2}$$
$97$ $$(T^{2} - 13)^{2}$$