# Properties

 Label 528.4.b.a Level $528$ Weight $4$ Character orbit 528.b Analytic conductor $31.153$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,4,Mod(65,528)]

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([0, 0, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("528.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.b (of order $$2$$, degree $$1$$, not 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: $$31.1530084830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) 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 $$\beta = \sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{3} - 4 \beta q^{5} + (8 \beta + 5) q^{9} +O(q^{10})$$ q + (b + 4) * q^3 - 4*b * q^5 + (8*b + 5) * q^9 $$q + (\beta + 4) q^{3} - 4 \beta q^{5} + (8 \beta + 5) q^{9} + 11 \beta q^{11} + ( - 16 \beta + 44) q^{15} + 58 \beta q^{23} - 51 q^{25} + (37 \beta - 68) q^{27} + 340 q^{31} + (44 \beta - 121) q^{33} + 434 q^{37} + ( - 20 \beta + 352) q^{45} - 194 \beta q^{47} + 343 q^{49} + 68 \beta q^{53} + 484 q^{55} + 166 \beta q^{59} - 416 q^{67} + (232 \beta - 638) q^{69} + 310 \beta q^{71} + ( - 51 \beta - 204) q^{75} + (80 \beta - 679) q^{81} - 40 \beta q^{89} + (340 \beta + 1360) q^{93} - 34 q^{97} + (55 \beta - 968) q^{99} +O(q^{100})$$ q + (b + 4) * q^3 - 4*b * q^5 + (8*b + 5) * q^9 + 11*b * q^11 + (-16*b + 44) * q^15 + 58*b * q^23 - 51 * q^25 + (37*b - 68) * q^27 + 340 * q^31 + (44*b - 121) * q^33 + 434 * q^37 + (-20*b + 352) * q^45 - 194*b * q^47 + 343 * q^49 + 68*b * q^53 + 484 * q^55 + 166*b * q^59 - 416 * q^67 + (232*b - 638) * q^69 + 310*b * q^71 + (-51*b - 204) * q^75 + (80*b - 679) * q^81 - 40*b * q^89 + (340*b + 1360) * q^93 - 34 * q^97 + (55*b - 968) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} + 10 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 + 10 * q^9 $$2 q + 8 q^{3} + 10 q^{9} + 88 q^{15} - 102 q^{25} - 136 q^{27} + 680 q^{31} - 242 q^{33} + 868 q^{37} + 704 q^{45} + 686 q^{49} + 968 q^{55} - 832 q^{67} - 1276 q^{69} - 408 q^{75} - 1358 q^{81} + 2720 q^{93} - 68 q^{97} - 1936 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 + 10 * q^9 + 88 * q^15 - 102 * q^25 - 136 * q^27 + 680 * q^31 - 242 * q^33 + 868 * q^37 + 704 * q^45 + 686 * q^49 + 968 * q^55 - 832 * q^67 - 1276 * q^69 - 408 * q^75 - 1358 * q^81 + 2720 * q^93 - 68 * q^97 - 1936 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 1.65831i 0.5 + 1.65831i
0 4.00000 3.31662i 0 13.2665i 0 0 0 5.00000 26.5330i 0
65.2 0 4.00000 + 3.31662i 0 13.2665i 0 0 0 5.00000 + 26.5330i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.4.b.a 2
3.b odd 2 1 inner 528.4.b.a 2
4.b odd 2 1 33.4.d.a 2
11.b odd 2 1 CM 528.4.b.a 2
12.b even 2 1 33.4.d.a 2
33.d even 2 1 inner 528.4.b.a 2
44.c even 2 1 33.4.d.a 2
132.d odd 2 1 33.4.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.a 2 4.b odd 2 1
33.4.d.a 2 12.b even 2 1
33.4.d.a 2 44.c even 2 1
33.4.d.a 2 132.d odd 2 1
528.4.b.a 2 1.a even 1 1 trivial
528.4.b.a 2 3.b odd 2 1 inner
528.4.b.a 2 11.b odd 2 1 CM
528.4.b.a 2 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(528, [\chi])$$:

 $$T_{5}^{2} + 176$$ T5^2 + 176 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8T + 27$$
$5$ $$T^{2} + 176$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 1331$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 37004$$
$29$ $$T^{2}$$
$31$ $$(T - 340)^{2}$$
$37$ $$(T - 434)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 413996$$
$53$ $$T^{2} + 50864$$
$59$ $$T^{2} + 303116$$
$61$ $$T^{2}$$
$67$ $$(T + 416)^{2}$$
$71$ $$T^{2} + 1057100$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 17600$$
$97$ $$(T + 34)^{2}$$