# Properties

 Label 3528.1.g.a Level $3528$ Weight $1$ Character orbit 3528.g Self dual yes Analytic conductor $1.761$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -7, -8, 56 Inner twists $4$

# 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: yes Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 392) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{-7})$$ Artin image $D_4$ Artin field Galois closure of 4.0.24696.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} + 2q^{11} + q^{16} - 2q^{22} + q^{25} - q^{32} - 2q^{43} + 2q^{44} - q^{50} + q^{64} + 2q^{67} + 2q^{86} - 2q^{88} + O(q^{100})$$

## 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
 0
−1.00000 0 1.00000 0 0 0 −1.00000 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 $$\Q(\sqrt{-7})$$
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
56.e even 2 1 RM by $$\Q(\sqrt{14})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.g.a 1
3.b odd 2 1 392.1.g.a 1
7.b odd 2 1 CM 3528.1.g.a 1
7.c even 3 2 3528.1.bx.a 2
7.d odd 6 2 3528.1.bx.a 2
8.d odd 2 1 CM 3528.1.g.a 1
12.b even 2 1 1568.1.g.a 1
21.c even 2 1 392.1.g.a 1
21.g even 6 2 392.1.k.a 2
21.h odd 6 2 392.1.k.a 2
24.f even 2 1 392.1.g.a 1
24.h odd 2 1 1568.1.g.a 1
56.e even 2 1 RM 3528.1.g.a 1
56.k odd 6 2 3528.1.bx.a 2
56.m even 6 2 3528.1.bx.a 2
84.h odd 2 1 1568.1.g.a 1
84.j odd 6 2 1568.1.o.a 2
84.n even 6 2 1568.1.o.a 2
168.e odd 2 1 392.1.g.a 1
168.i even 2 1 1568.1.g.a 1
168.s odd 6 2 1568.1.o.a 2
168.v even 6 2 392.1.k.a 2
168.ba even 6 2 1568.1.o.a 2
168.be odd 6 2 392.1.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.a 1 3.b odd 2 1
392.1.g.a 1 21.c even 2 1
392.1.g.a 1 24.f even 2 1
392.1.g.a 1 168.e odd 2 1
392.1.k.a 2 21.g even 6 2
392.1.k.a 2 21.h odd 6 2
392.1.k.a 2 168.v even 6 2
392.1.k.a 2 168.be odd 6 2
1568.1.g.a 1 12.b even 2 1
1568.1.g.a 1 24.h odd 2 1
1568.1.g.a 1 84.h odd 2 1
1568.1.g.a 1 168.i even 2 1
1568.1.o.a 2 84.j odd 6 2
1568.1.o.a 2 84.n even 6 2
1568.1.o.a 2 168.s odd 6 2
1568.1.o.a 2 168.ba even 6 2
3528.1.g.a 1 1.a even 1 1 trivial
3528.1.g.a 1 7.b odd 2 1 CM
3528.1.g.a 1 8.d odd 2 1 CM
3528.1.g.a 1 56.e even 2 1 RM
3528.1.bx.a 2 7.c even 3 2
3528.1.bx.a 2 7.d odd 6 2
3528.1.bx.a 2 56.k odd 6 2
3528.1.bx.a 2 56.m even 6 2

## Hecke kernels

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

 $$T_{11} - 2$$ $$T_{17}$$

## Hecke characteristic polynomials

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