# Properties

 Label 3528.1.bw.b Level $3528$ Weight $1$ Character orbit 3528.bw Analytic conductor $1.761$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -7, -24, 168 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3528.bw (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-6}, \sqrt{-7})$$ Artin image $C_3\times D_4:C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{8} + 2 \zeta_{12} q^{11} -\zeta_{12}^{2} q^{16} -2 q^{22} -\zeta_{12}^{4} q^{25} + 2 \zeta_{12}^{3} q^{29} + \zeta_{12} q^{32} -2 \zeta_{12}^{5} q^{44} + \zeta_{12}^{3} q^{50} -2 \zeta_{12} q^{53} -2 \zeta_{12}^{2} q^{58} - q^{64} + 2 \zeta_{12}^{2} q^{79} + 2 \zeta_{12}^{4} q^{88} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{16} - 8q^{22} + 2q^{25} - 4q^{58} - 4q^{64} + 4q^{79} - 4q^{88} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
325.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0 0
325.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0 1.00000i 0 0
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0 0
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 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.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
168.i even 2 1 RM by $$\Q(\sqrt{42})$$
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner
168.ba 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.bw.b 4
3.b odd 2 1 inner 3528.1.bw.b 4
7.b odd 2 1 CM 3528.1.bw.b 4
7.c even 3 1 504.1.l.b 2
7.c even 3 1 inner 3528.1.bw.b 4
7.d odd 6 1 504.1.l.b 2
7.d odd 6 1 inner 3528.1.bw.b 4
8.b even 2 1 inner 3528.1.bw.b 4
21.c even 2 1 inner 3528.1.bw.b 4
21.g even 6 1 504.1.l.b 2
21.g even 6 1 inner 3528.1.bw.b 4
21.h odd 6 1 504.1.l.b 2
21.h odd 6 1 inner 3528.1.bw.b 4
24.h odd 2 1 CM 3528.1.bw.b 4
28.f even 6 1 2016.1.l.b 2
28.g odd 6 1 2016.1.l.b 2
56.h odd 2 1 inner 3528.1.bw.b 4
56.j odd 6 1 504.1.l.b 2
56.j odd 6 1 inner 3528.1.bw.b 4
56.k odd 6 1 2016.1.l.b 2
56.m even 6 1 2016.1.l.b 2
56.p even 6 1 504.1.l.b 2
56.p even 6 1 inner 3528.1.bw.b 4
84.j odd 6 1 2016.1.l.b 2
84.n even 6 1 2016.1.l.b 2
168.i even 2 1 RM 3528.1.bw.b 4
168.s odd 6 1 504.1.l.b 2
168.s odd 6 1 inner 3528.1.bw.b 4
168.v even 6 1 2016.1.l.b 2
168.ba even 6 1 504.1.l.b 2
168.ba even 6 1 inner 3528.1.bw.b 4
168.be odd 6 1 2016.1.l.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.l.b 2 7.c even 3 1
504.1.l.b 2 7.d odd 6 1
504.1.l.b 2 21.g even 6 1
504.1.l.b 2 21.h odd 6 1
504.1.l.b 2 56.j odd 6 1
504.1.l.b 2 56.p even 6 1
504.1.l.b 2 168.s odd 6 1
504.1.l.b 2 168.ba even 6 1
2016.1.l.b 2 28.f even 6 1
2016.1.l.b 2 28.g odd 6 1
2016.1.l.b 2 56.k odd 6 1
2016.1.l.b 2 56.m even 6 1
2016.1.l.b 2 84.j odd 6 1
2016.1.l.b 2 84.n even 6 1
2016.1.l.b 2 168.v even 6 1
2016.1.l.b 2 168.be odd 6 1
3528.1.bw.b 4 1.a even 1 1 trivial
3528.1.bw.b 4 3.b odd 2 1 inner
3528.1.bw.b 4 7.b odd 2 1 CM
3528.1.bw.b 4 7.c even 3 1 inner
3528.1.bw.b 4 7.d odd 6 1 inner
3528.1.bw.b 4 8.b even 2 1 inner
3528.1.bw.b 4 21.c even 2 1 inner
3528.1.bw.b 4 21.g even 6 1 inner
3528.1.bw.b 4 21.h odd 6 1 inner
3528.1.bw.b 4 24.h odd 2 1 CM
3528.1.bw.b 4 56.h odd 2 1 inner
3528.1.bw.b 4 56.j odd 6 1 inner
3528.1.bw.b 4 56.p even 6 1 inner
3528.1.bw.b 4 168.i even 2 1 RM
3528.1.bw.b 4 168.s odd 6 1 inner
3528.1.bw.b 4 168.ba 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 $$S_{1}^{\mathrm{new}}(3528, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{4} - 4 T_{11}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$16 - 4 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 4 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$16 - 4 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 4 - 2 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$