# Properties

 Label 3528.1.bp.b Level $3528$ Weight $1$ Character orbit 3528.bp Analytic conductor $1.761$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -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.bp (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.4536.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + \zeta_{6} q^{3} + q^{4} -\zeta_{6} q^{5} + \zeta_{6} q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + q^{2} + \zeta_{6} q^{3} + q^{4} -\zeta_{6} q^{5} + \zeta_{6} q^{6} + q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{10} + \zeta_{6} q^{12} -2 \zeta_{6}^{2} q^{13} -\zeta_{6}^{2} q^{15} + q^{16} + \zeta_{6}^{2} q^{18} + \zeta_{6}^{2} q^{19} -\zeta_{6} q^{20} + \zeta_{6} q^{23} + \zeta_{6} q^{24} -2 \zeta_{6}^{2} q^{26} - q^{27} -\zeta_{6}^{2} q^{30} + q^{32} + \zeta_{6}^{2} q^{36} + \zeta_{6}^{2} q^{38} + 2 q^{39} -\zeta_{6} q^{40} + q^{45} + \zeta_{6} q^{46} + \zeta_{6} q^{48} -2 \zeta_{6}^{2} q^{52} - q^{54} - q^{57} -2 q^{59} -\zeta_{6}^{2} q^{60} + q^{61} + q^{64} -2 q^{65} + \zeta_{6}^{2} q^{69} - q^{71} + \zeta_{6}^{2} q^{72} + \zeta_{6}^{2} q^{76} + 2 q^{78} - q^{79} -\zeta_{6} q^{80} -\zeta_{6} q^{81} + 2 \zeta_{6} q^{83} + q^{90} + \zeta_{6} q^{92} + q^{95} + \zeta_{6} q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + q^{3} + 2q^{4} - q^{5} + q^{6} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q + 2q^{2} + q^{3} + 2q^{4} - q^{5} + q^{6} + 2q^{8} - q^{9} - q^{10} + q^{12} + 2q^{13} + q^{15} + 2q^{16} - q^{18} - q^{19} - q^{20} + q^{23} + q^{24} + 2q^{26} - 2q^{27} + q^{30} + 2q^{32} - q^{36} - q^{38} + 4q^{39} - q^{40} + 2q^{45} + q^{46} + q^{48} + 2q^{52} - 2q^{54} - 2q^{57} - 4q^{59} + q^{60} + 2q^{61} + 2q^{64} - 4q^{65} - q^{69} - 2q^{71} - q^{72} - q^{76} + 4q^{78} - 2q^{79} - q^{80} - q^{81} + 2q^{83} + 2q^{90} + q^{92} + 2q^{95} + q^{96} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$-\zeta_{6}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1501.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
3253.1 1.00000 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
63.h even 3 1 inner
504.bp 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.bp.b 2
7.b odd 2 1 3528.1.bp.a 2
7.c even 3 1 504.1.bn.a 2
7.c even 3 1 3528.1.cw.b 2
7.d odd 6 1 504.1.bn.b yes 2
7.d odd 6 1 3528.1.cw.a 2
8.b even 2 1 3528.1.bp.a 2
9.c even 3 1 3528.1.cw.b 2
21.g even 6 1 1512.1.bn.a 2
21.h odd 6 1 1512.1.bn.b 2
28.f even 6 1 2016.1.bv.a 2
28.g odd 6 1 2016.1.bv.b 2
56.h odd 2 1 CM 3528.1.bp.b 2
56.j odd 6 1 504.1.bn.a 2
56.j odd 6 1 3528.1.cw.b 2
56.k odd 6 1 2016.1.bv.a 2
56.m even 6 1 2016.1.bv.b 2
56.p even 6 1 504.1.bn.b yes 2
56.p even 6 1 3528.1.cw.a 2
63.g even 3 1 504.1.bn.a 2
63.h even 3 1 inner 3528.1.bp.b 2
63.k odd 6 1 504.1.bn.b yes 2
63.l odd 6 1 3528.1.cw.a 2
63.n odd 6 1 1512.1.bn.b 2
63.s even 6 1 1512.1.bn.a 2
63.t odd 6 1 3528.1.bp.a 2
72.n even 6 1 3528.1.cw.a 2
168.s odd 6 1 1512.1.bn.a 2
168.ba even 6 1 1512.1.bn.b 2
252.n even 6 1 2016.1.bv.a 2
252.bl odd 6 1 2016.1.bv.b 2
504.w even 6 1 504.1.bn.b yes 2
504.y even 6 1 1512.1.bn.b 2
504.ba odd 6 1 2016.1.bv.a 2
504.bn odd 6 1 3528.1.cw.b 2
504.bp odd 6 1 inner 3528.1.bp.b 2
504.cq even 6 1 3528.1.bp.a 2
504.cw odd 6 1 504.1.bn.a 2
504.cz even 6 1 2016.1.bv.b 2
504.db odd 6 1 1512.1.bn.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bn.a 2 7.c even 3 1
504.1.bn.a 2 56.j odd 6 1
504.1.bn.a 2 63.g even 3 1
504.1.bn.a 2 504.cw odd 6 1
504.1.bn.b yes 2 7.d odd 6 1
504.1.bn.b yes 2 56.p even 6 1
504.1.bn.b yes 2 63.k odd 6 1
504.1.bn.b yes 2 504.w even 6 1
1512.1.bn.a 2 21.g even 6 1
1512.1.bn.a 2 63.s even 6 1
1512.1.bn.a 2 168.s odd 6 1
1512.1.bn.a 2 504.db odd 6 1
1512.1.bn.b 2 21.h odd 6 1
1512.1.bn.b 2 63.n odd 6 1
1512.1.bn.b 2 168.ba even 6 1
1512.1.bn.b 2 504.y even 6 1
2016.1.bv.a 2 28.f even 6 1
2016.1.bv.a 2 56.k odd 6 1
2016.1.bv.a 2 252.n even 6 1
2016.1.bv.a 2 504.ba odd 6 1
2016.1.bv.b 2 28.g odd 6 1
2016.1.bv.b 2 56.m even 6 1
2016.1.bv.b 2 252.bl odd 6 1
2016.1.bv.b 2 504.cz even 6 1
3528.1.bp.a 2 7.b odd 2 1
3528.1.bp.a 2 8.b even 2 1
3528.1.bp.a 2 63.t odd 6 1
3528.1.bp.a 2 504.cq even 6 1
3528.1.bp.b 2 1.a even 1 1 trivial
3528.1.bp.b 2 56.h odd 2 1 CM
3528.1.bp.b 2 63.h even 3 1 inner
3528.1.bp.b 2 504.bp odd 6 1 inner
3528.1.cw.a 2 7.d odd 6 1
3528.1.cw.a 2 56.p even 6 1
3528.1.cw.a 2 63.l odd 6 1
3528.1.cw.a 2 72.n even 6 1
3528.1.cw.b 2 7.c even 3 1
3528.1.cw.b 2 9.c even 3 1
3528.1.cw.b 2 56.j odd 6 1
3528.1.cw.b 2 504.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3528, [\chi])$$.

## Hecke characteristic polynomials

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