# Properties

 Label 3528.1.bx.b Level $3528$ Weight $1$ Character orbit 3528.bx Analytic conductor $1.761$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -8 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.bx (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 392) Projective image $$D_{4}$$ Projective field Galois closure of 4.0.2744.1 Artin image $C_6\times D_8$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{48} - \cdots)$$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + q^{8} + \beta_{2} q^{16} -\beta_{1} q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( -1 - \beta_{2} ) q^{25} + ( -1 - \beta_{2} ) q^{32} -\beta_{3} q^{34} + \beta_{1} q^{38} + \beta_{3} q^{41} + q^{50} -\beta_{1} q^{59} + q^{64} + ( 2 + 2 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{3} ) q^{68} -\beta_{1} q^{73} + \beta_{3} q^{76} + ( -\beta_{1} - \beta_{3} ) q^{82} -\beta_{3} q^{83} + ( -\beta_{1} - \beta_{3} ) q^{89} + \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 4q^{8} - 2q^{16} - 2q^{25} - 2q^{32} + 4q^{50} + 4q^{64} + 4q^{67} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
667.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 0 0
667.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 0 0
1243.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 0 0
1243.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
56.e even 2 1 inner
56.k odd 6 1 inner
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.bx.b 4
3.b odd 2 1 392.1.k.b 4
7.b odd 2 1 inner 3528.1.bx.b 4
7.c even 3 1 3528.1.g.c 2
7.c even 3 1 inner 3528.1.bx.b 4
7.d odd 6 1 3528.1.g.c 2
7.d odd 6 1 inner 3528.1.bx.b 4
8.d odd 2 1 CM 3528.1.bx.b 4
12.b even 2 1 1568.1.o.b 4
21.c even 2 1 392.1.k.b 4
21.g even 6 1 392.1.g.b 2
21.g even 6 1 392.1.k.b 4
21.h odd 6 1 392.1.g.b 2
21.h odd 6 1 392.1.k.b 4
24.f even 2 1 392.1.k.b 4
24.h odd 2 1 1568.1.o.b 4
56.e even 2 1 inner 3528.1.bx.b 4
56.k odd 6 1 3528.1.g.c 2
56.k odd 6 1 inner 3528.1.bx.b 4
56.m even 6 1 3528.1.g.c 2
56.m even 6 1 inner 3528.1.bx.b 4
84.h odd 2 1 1568.1.o.b 4
84.j odd 6 1 1568.1.g.b 2
84.j odd 6 1 1568.1.o.b 4
84.n even 6 1 1568.1.g.b 2
84.n even 6 1 1568.1.o.b 4
168.e odd 2 1 392.1.k.b 4
168.i even 2 1 1568.1.o.b 4
168.s odd 6 1 1568.1.g.b 2
168.s odd 6 1 1568.1.o.b 4
168.v even 6 1 392.1.g.b 2
168.v even 6 1 392.1.k.b 4
168.ba even 6 1 1568.1.g.b 2
168.ba even 6 1 1568.1.o.b 4
168.be odd 6 1 392.1.g.b 2
168.be odd 6 1 392.1.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.b 2 21.g even 6 1
392.1.g.b 2 21.h odd 6 1
392.1.g.b 2 168.v even 6 1
392.1.g.b 2 168.be odd 6 1
392.1.k.b 4 3.b odd 2 1
392.1.k.b 4 21.c even 2 1
392.1.k.b 4 21.g even 6 1
392.1.k.b 4 21.h odd 6 1
392.1.k.b 4 24.f even 2 1
392.1.k.b 4 168.e odd 2 1
392.1.k.b 4 168.v even 6 1
392.1.k.b 4 168.be odd 6 1
1568.1.g.b 2 84.j odd 6 1
1568.1.g.b 2 84.n even 6 1
1568.1.g.b 2 168.s odd 6 1
1568.1.g.b 2 168.ba even 6 1
1568.1.o.b 4 12.b even 2 1
1568.1.o.b 4 24.h odd 2 1
1568.1.o.b 4 84.h odd 2 1
1568.1.o.b 4 84.j odd 6 1
1568.1.o.b 4 84.n even 6 1
1568.1.o.b 4 168.i even 2 1
1568.1.o.b 4 168.s odd 6 1
1568.1.o.b 4 168.ba even 6 1
3528.1.g.c 2 7.c even 3 1
3528.1.g.c 2 7.d odd 6 1
3528.1.g.c 2 56.k odd 6 1
3528.1.g.c 2 56.m even 6 1
3528.1.bx.b 4 1.a even 1 1 trivial
3528.1.bx.b 4 7.b odd 2 1 inner
3528.1.bx.b 4 7.c even 3 1 inner
3528.1.bx.b 4 7.d odd 6 1 inner
3528.1.bx.b 4 8.d odd 2 1 CM
3528.1.bx.b 4 56.e even 2 1 inner
3528.1.bx.b 4 56.k odd 6 1 inner
3528.1.bx.b 4 56.m even 6 1 inner

## Hecke kernels

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

 $$T_{11}$$ $$T_{17}^{4} + 2 T_{17}^{2} + 4$$

## Hecke characteristic polynomials

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