# Properties

 Label 3528.1.ce.d Level $3528$ Weight $1$ Character orbit 3528.ce Analytic conductor $1.761$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ 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.ce (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.1152216576.12

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + \zeta_{12}^{5} q^{3} + q^{4} + \zeta_{12}^{5} q^{6} + q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})$$ $$q + q^{2} + \zeta_{12}^{5} q^{3} + q^{4} + \zeta_{12}^{5} q^{6} + q^{8} -\zeta_{12}^{4} q^{9} -\zeta_{12}^{2} q^{11} + \zeta_{12}^{5} q^{12} + q^{16} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{17} -\zeta_{12}^{4} q^{18} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{19} -\zeta_{12}^{2} q^{22} + \zeta_{12}^{5} q^{24} -\zeta_{12}^{2} q^{25} + \zeta_{12}^{3} q^{27} + q^{32} + \zeta_{12} q^{33} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{34} -\zeta_{12}^{4} q^{36} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{38} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{41} + \zeta_{12}^{4} q^{43} -\zeta_{12}^{2} q^{44} + \zeta_{12}^{5} q^{48} -\zeta_{12}^{2} q^{50} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{51} + \zeta_{12}^{3} q^{54} + ( 1 + \zeta_{12}^{2} ) q^{57} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{59} + q^{64} + \zeta_{12} q^{66} - q^{67} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{68} -\zeta_{12}^{4} q^{72} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{73} + \zeta_{12} q^{75} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{76} -\zeta_{12}^{2} q^{81} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{82} + \zeta_{12}^{4} q^{86} -\zeta_{12}^{2} q^{88} + \zeta_{12}^{5} q^{96} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} + 4q^{8} + 2q^{9} - 2q^{11} + 4q^{16} + 2q^{18} - 2q^{22} - 2q^{25} + 4q^{32} + 2q^{36} - 2q^{43} - 2q^{44} - 2q^{50} + 6q^{57} + 4q^{64} - 4q^{67} + 2q^{72} - 2q^{81} - 2q^{86} - 2q^{88} - 4q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2419.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000 −0.866025 + 0.500000i 1.00000 0 −0.866025 + 0.500000i 0 1.00000 0.500000 0.866025i 0
2419.2 1.00000 0.866025 0.500000i 1.00000 0 0.866025 0.500000i 0 1.00000 0.500000 0.866025i 0
3019.1 1.00000 −0.866025 0.500000i 1.00000 0 −0.866025 0.500000i 0 1.00000 0.500000 + 0.866025i 0
3019.2 1.00000 0.866025 + 0.500000i 1.00000 0 0.866025 + 0.500000i 0 1.00000 0.500000 + 0.866025i 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
56.e even 2 1 inner
63.h even 3 1 inner
63.t odd 6 1 inner
504.bf even 6 1 inner
504.ce odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.ce.d 4
7.b odd 2 1 inner 3528.1.ce.d 4
7.c even 3 1 3528.1.ba.c 4
7.c even 3 1 3528.1.cg.b 4
7.d odd 6 1 3528.1.ba.c 4
7.d odd 6 1 3528.1.cg.b 4
8.d odd 2 1 CM 3528.1.ce.d 4
9.c even 3 1 3528.1.ba.c 4
56.e even 2 1 inner 3528.1.ce.d 4
56.k odd 6 1 3528.1.ba.c 4
56.k odd 6 1 3528.1.cg.b 4
56.m even 6 1 3528.1.ba.c 4
56.m even 6 1 3528.1.cg.b 4
63.g even 3 1 3528.1.cg.b 4
63.h even 3 1 inner 3528.1.ce.d 4
63.k odd 6 1 3528.1.cg.b 4
63.l odd 6 1 3528.1.ba.c 4
63.t odd 6 1 inner 3528.1.ce.d 4
72.p odd 6 1 3528.1.ba.c 4
504.ba odd 6 1 3528.1.cg.b 4
504.be even 6 1 3528.1.ba.c 4
504.bf even 6 1 inner 3528.1.ce.d 4
504.ce odd 6 1 inner 3528.1.ce.d 4
504.cz even 6 1 3528.1.cg.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.ba.c 4 7.c even 3 1
3528.1.ba.c 4 7.d odd 6 1
3528.1.ba.c 4 9.c even 3 1
3528.1.ba.c 4 56.k odd 6 1
3528.1.ba.c 4 56.m even 6 1
3528.1.ba.c 4 63.l odd 6 1
3528.1.ba.c 4 72.p odd 6 1
3528.1.ba.c 4 504.be even 6 1
3528.1.ce.d 4 1.a even 1 1 trivial
3528.1.ce.d 4 7.b odd 2 1 inner
3528.1.ce.d 4 8.d odd 2 1 CM
3528.1.ce.d 4 56.e even 2 1 inner
3528.1.ce.d 4 63.h even 3 1 inner
3528.1.ce.d 4 63.t odd 6 1 inner
3528.1.ce.d 4 504.bf even 6 1 inner
3528.1.ce.d 4 504.ce odd 6 1 inner
3528.1.cg.b 4 7.c even 3 1
3528.1.cg.b 4 7.d odd 6 1
3528.1.cg.b 4 56.k odd 6 1
3528.1.cg.b 4 56.m even 6 1
3528.1.cg.b 4 63.g even 3 1
3528.1.cg.b 4 63.k odd 6 1
3528.1.cg.b 4 504.ba odd 6 1
3528.1.cg.b 4 504.cz even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{17}^{4} + 3 T_{17}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$9 + 3 T^{2} + T^{4}$$
$19$ $$9 + 3 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$9 + 3 T^{2} + T^{4}$$
$43$ $$( 1 + T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( -3 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 1 + T )^{4}$$
$71$ $$T^{4}$$
$73$ $$9 + 3 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$9 + 3 T^{2} + T^{4}$$