# Properties

 Label 3528.1.g.b Level $3528$ Weight $1$ Character orbit 3528.g Analytic conductor $1.761$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -7, -168, 24 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{6}, \sqrt{-7})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.5489031744.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{2} - q^{4} + i q^{8} +O(q^{10})$$ $$q -i q^{2} - q^{4} + i q^{8} + q^{16} -2 i q^{23} + q^{25} + 2 i q^{29} -i q^{32} + 2 q^{43} -2 q^{46} -i q^{50} -2 i q^{53} + 2 q^{58} - q^{64} + 2 q^{67} -2 i q^{71} -2 i q^{86} + 2 i q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{16} + 2q^{25} + 4q^{43} - 4q^{46} + 4q^{58} - 2q^{64} + 4q^{67} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
883.2 1.00000i 0 −1.00000 0 0 0 1.00000i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$
168.e odd 2 1 CM by $$\Q(\sqrt{-42})$$
3.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.g.b 2
3.b odd 2 1 inner 3528.1.g.b 2
7.b odd 2 1 CM 3528.1.g.b 2
7.c even 3 2 3528.1.bx.c 4
7.d odd 6 2 3528.1.bx.c 4
8.d odd 2 1 inner 3528.1.g.b 2
21.c even 2 1 inner 3528.1.g.b 2
21.g even 6 2 3528.1.bx.c 4
21.h odd 6 2 3528.1.bx.c 4
24.f even 2 1 RM 3528.1.g.b 2
56.e even 2 1 inner 3528.1.g.b 2
56.k odd 6 2 3528.1.bx.c 4
56.m even 6 2 3528.1.bx.c 4
168.e odd 2 1 CM 3528.1.g.b 2
168.v even 6 2 3528.1.bx.c 4
168.be odd 6 2 3528.1.bx.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.g.b 2 1.a even 1 1 trivial
3528.1.g.b 2 3.b odd 2 1 inner
3528.1.g.b 2 7.b odd 2 1 CM
3528.1.g.b 2 8.d odd 2 1 inner
3528.1.g.b 2 21.c even 2 1 inner
3528.1.g.b 2 24.f even 2 1 RM
3528.1.g.b 2 56.e even 2 1 inner
3528.1.g.b 2 168.e odd 2 1 CM
3528.1.bx.c 4 7.c even 3 2
3528.1.bx.c 4 7.d odd 6 2
3528.1.bx.c 4 21.g even 6 2
3528.1.bx.c 4 21.h odd 6 2
3528.1.bx.c 4 56.k odd 6 2
3528.1.bx.c 4 56.m even 6 2
3528.1.bx.c 4 168.v even 6 2
3528.1.bx.c 4 168.be odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3528, [\chi])$$:

 $$T_{11}$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$