Properties

 Label 528.2.o.a Level $528$ Weight $2$ Character orbit 528.o Analytic conductor $4.216$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,2,Mod(175,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("528.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 528.o (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: $$4.21610122672$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 5$$ (v^3 + 5*v) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 5\nu ) / 5$$ (-v^3 + 5*v) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$5\beta_1$$ 5*b1 $$\nu^{3}$$ $$=$$ $$( -5\beta_{3} + 5\beta_{2} ) / 2$$ (-5*b3 + 5*b2) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
175.1
 1.58114 − 1.58114i −1.58114 + 1.58114i 1.58114 + 1.58114i −1.58114 − 1.58114i
0 1.00000i 0 −2.00000 0 −3.16228 0 −1.00000 0
175.2 0 1.00000i 0 −2.00000 0 3.16228 0 −1.00000 0
175.3 0 1.00000i 0 −2.00000 0 −3.16228 0 −1.00000 0
175.4 0 1.00000i 0 −2.00000 0 3.16228 0 −1.00000 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
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.o.a 4
3.b odd 2 1 1584.2.o.f 4
4.b odd 2 1 inner 528.2.o.a 4
8.b even 2 1 2112.2.o.c 4
8.d odd 2 1 2112.2.o.c 4
11.b odd 2 1 inner 528.2.o.a 4
12.b even 2 1 1584.2.o.f 4
33.d even 2 1 1584.2.o.f 4
44.c even 2 1 inner 528.2.o.a 4
88.b odd 2 1 2112.2.o.c 4
88.g even 2 1 2112.2.o.c 4
132.d odd 2 1 1584.2.o.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.o.a 4 1.a even 1 1 trivial
528.2.o.a 4 4.b odd 2 1 inner
528.2.o.a 4 11.b odd 2 1 inner
528.2.o.a 4 44.c even 2 1 inner
1584.2.o.f 4 3.b odd 2 1
1584.2.o.f 4 12.b even 2 1
1584.2.o.f 4 33.d even 2 1
1584.2.o.f 4 132.d odd 2 1
2112.2.o.c 4 8.b even 2 1
2112.2.o.c 4 8.d odd 2 1
2112.2.o.c 4 88.b odd 2 1
2112.2.o.c 4 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(528, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T + 2)^{4}$$
$7$ $$(T^{2} - 10)^{2}$$
$11$ $$T^{4} - 18T^{2} + 121$$
$13$ $$(T^{2} + 40)^{2}$$
$17$ $$(T^{2} + 10)^{2}$$
$19$ $$(T^{2} - 10)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 90)^{2}$$
$31$ $$(T^{2} + 64)^{2}$$
$37$ $$(T + 8)^{4}$$
$41$ $$(T^{2} + 10)^{2}$$
$43$ $$(T^{2} - 10)^{2}$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T - 6)^{4}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 144)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 40)^{2}$$
$79$ $$(T^{2} - 250)^{2}$$
$83$ $$(T^{2} - 160)^{2}$$
$89$ $$(T - 6)^{4}$$
$97$ $$(T - 8)^{4}$$