# Properties

 Label 3528.1.cg.e Level $3528$ Weight $1$ Character orbit 3528.cg Analytic conductor $1.761$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ 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.cg (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{12}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{8} q^{2} -\zeta_{24}^{5} q^{3} -\zeta_{24}^{4} q^{4} -\zeta_{24} q^{6} - q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{8} q^{2} -\zeta_{24}^{5} q^{3} -\zeta_{24}^{4} q^{4} -\zeta_{24} q^{6} - q^{8} + \zeta_{24}^{10} q^{9} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{11} + \zeta_{24}^{9} q^{12} + \zeta_{24}^{8} q^{16} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{6} q^{18} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{19} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{22} + \zeta_{24}^{5} q^{24} + \zeta_{24}^{8} q^{25} + \zeta_{24}^{3} q^{27} + \zeta_{24}^{4} q^{32} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{33} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{34} + \zeta_{24}^{2} q^{36} + ( \zeta_{24} - \zeta_{24}^{3} ) q^{38} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{41} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{43} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{44} + \zeta_{24} q^{48} + \zeta_{24}^{4} q^{50} + ( \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{51} -\zeta_{24}^{11} q^{54} + ( -1 - \zeta_{24}^{10} ) q^{57} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{59} + q^{64} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{66} -\zeta_{24}^{4} q^{67} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{68} -\zeta_{24}^{10} q^{72} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{73} + \zeta_{24} q^{75} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{76} -\zeta_{24}^{8} q^{81} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{82} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{83} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{86} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{88} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{89} -\zeta_{24}^{9} q^{96} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{97} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{16} - 4q^{25} + 4q^{32} + 4q^{50} + 4q^{51} - 8q^{57} + 8q^{64} - 4q^{67} + 4q^{81} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$-\zeta_{24}^{4}$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2059.1
 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i
0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 0 −0.258819 + 0.965926i 0 −1.00000 0.866025 0.500000i 0
2059.2 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i 0 −0.965926 0.258819i 0 −1.00000 −0.866025 + 0.500000i 0
2059.3 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 0 0.965926 + 0.258819i 0 −1.00000 −0.866025 + 0.500000i 0
2059.4 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i 0 0.258819 0.965926i 0 −1.00000 0.866025 0.500000i 0
3235.1 0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 0 −0.258819 0.965926i 0 −1.00000 0.866025 + 0.500000i 0
3235.2 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i 0 −0.965926 + 0.258819i 0 −1.00000 −0.866025 0.500000i 0
3235.3 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i 0 0.965926 0.258819i 0 −1.00000 −0.866025 0.500000i 0
3235.4 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i 0 0.258819 + 0.965926i 0 −1.00000 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3235.4 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
9.c even 3 1 inner
56.e even 2 1 inner
63.l odd 6 1 inner
72.p odd 6 1 inner
504.be 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.cg.e 8
7.b odd 2 1 inner 3528.1.cg.e 8
7.c even 3 1 3528.1.ba.e 8
7.c even 3 1 3528.1.ce.e 8
7.d odd 6 1 3528.1.ba.e 8
7.d odd 6 1 3528.1.ce.e 8
8.d odd 2 1 CM 3528.1.cg.e 8
9.c even 3 1 inner 3528.1.cg.e 8
56.e even 2 1 inner 3528.1.cg.e 8
56.k odd 6 1 3528.1.ba.e 8
56.k odd 6 1 3528.1.ce.e 8
56.m even 6 1 3528.1.ba.e 8
56.m even 6 1 3528.1.ce.e 8
63.g even 3 1 3528.1.ce.e 8
63.h even 3 1 3528.1.ba.e 8
63.k odd 6 1 3528.1.ce.e 8
63.l odd 6 1 inner 3528.1.cg.e 8
63.t odd 6 1 3528.1.ba.e 8
72.p odd 6 1 inner 3528.1.cg.e 8
504.ba odd 6 1 3528.1.ce.e 8
504.be even 6 1 inner 3528.1.cg.e 8
504.bf even 6 1 3528.1.ba.e 8
504.ce odd 6 1 3528.1.ba.e 8
504.cz even 6 1 3528.1.ce.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.ba.e 8 7.c even 3 1
3528.1.ba.e 8 7.d odd 6 1
3528.1.ba.e 8 56.k odd 6 1
3528.1.ba.e 8 56.m even 6 1
3528.1.ba.e 8 63.h even 3 1
3528.1.ba.e 8 63.t odd 6 1
3528.1.ba.e 8 504.bf even 6 1
3528.1.ba.e 8 504.ce odd 6 1
3528.1.ce.e 8 7.c even 3 1
3528.1.ce.e 8 7.d odd 6 1
3528.1.ce.e 8 56.k odd 6 1
3528.1.ce.e 8 56.m even 6 1
3528.1.ce.e 8 63.g even 3 1
3528.1.ce.e 8 63.k odd 6 1
3528.1.ce.e 8 504.ba odd 6 1
3528.1.ce.e 8 504.cz even 6 1
3528.1.cg.e 8 1.a even 1 1 trivial
3528.1.cg.e 8 7.b odd 2 1 inner
3528.1.cg.e 8 8.d odd 2 1 CM
3528.1.cg.e 8 9.c even 3 1 inner
3528.1.cg.e 8 56.e even 2 1 inner
3528.1.cg.e 8 63.l odd 6 1 inner
3528.1.cg.e 8 72.p odd 6 1 inner
3528.1.cg.e 8 504.be 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 $$S_{1}^{\mathrm{new}}(3528, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{4}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$19$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$
$43$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 1 + T + T^{2} )^{4}$$
$71$ $$T^{8}$$
$73$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$89$ $$( -2 + T^{2} )^{4}$$
$97$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$