# Properties

 Label 3528.1.d.b Level $3528$ Weight $1$ Character orbit 3528.d Analytic conductor $1.761$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3528.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Projective image $$S_{4}$$ Projective field Galois closure of 4.2.21168.2 Artin image $\GL(2,3)$ Artin field Galois closure of 8.2.263473523712.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} -\beta q^{11} + q^{13} + q^{19} - q^{25} + q^{31} + q^{37} + \beta q^{41} - q^{43} -\beta q^{47} + 2 q^{55} + \beta q^{65} - q^{67} + \beta q^{71} - q^{73} + q^{79} + \beta q^{83} + \beta q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{13} + 2q^{19} - 2q^{25} + 2q^{31} + 2q^{37} - 2q^{43} + 4q^{55} - 2q^{67} - 2q^{73} + 2q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1961.1
 − 1.41421i 1.41421i
0 0 0 1.41421i 0 0 0 0 0
1961.2 0 0 0 1.41421i 0 0 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.d.b 2
3.b odd 2 1 inner 3528.1.d.b 2
7.b odd 2 1 3528.1.d.a 2
7.c even 3 2 3528.1.cu.a 4
7.d odd 6 2 504.1.cu.a 4
21.c even 2 1 3528.1.d.a 2
21.g even 6 2 504.1.cu.a 4
21.h odd 6 2 3528.1.cu.a 4
28.f even 6 2 1008.1.dc.a 4
84.j odd 6 2 1008.1.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a 4 7.d odd 6 2
504.1.cu.a 4 21.g even 6 2
1008.1.dc.a 4 28.f even 6 2
1008.1.dc.a 4 84.j odd 6 2
3528.1.d.a 2 7.b odd 2 1
3528.1.d.a 2 21.c even 2 1
3528.1.d.b 2 1.a even 1 1 trivial
3528.1.d.b 2 3.b odd 2 1 inner
3528.1.cu.a 4 7.c even 3 2
3528.1.cu.a 4 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(3528, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( -1 + T )^{2}$$
$41$ $$2 + T^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$2 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$( 1 + T )^{2}$$
$71$ $$2 + T^{2}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$( -1 + T )^{2}$$
$83$ $$2 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$