# Properties

 Label 3528.1.bi.a Level $3528$ Weight $1$ Character orbit 3528.bi Analytic conductor $1.761$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -56 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.bi (of order $$6$$, degree $$2$$, not 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{6} q^{2} -\zeta_{24}^{7} q^{3} - q^{4} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} -\zeta_{24} q^{6} + \zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{6} q^{2} -\zeta_{24}^{7} q^{3} - q^{4} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} -\zeta_{24} q^{6} + \zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{9} + ( \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{10} + \zeta_{24}^{7} q^{12} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{13} + ( -1 + \zeta_{24}^{10} ) q^{15} + q^{16} + \zeta_{24}^{8} q^{18} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{19} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{20} + ( 1 - \zeta_{24}^{8} ) q^{23} + \zeta_{24} q^{24} + ( \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{25} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{26} + \zeta_{24}^{9} q^{27} + ( \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{30} -\zeta_{24}^{6} q^{32} + \zeta_{24}^{2} q^{36} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{38} + ( -1 + \zeta_{24}^{6} ) q^{39} + ( -\zeta_{24}^{9} - \zeta_{24}^{11} ) q^{40} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{45} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{46} -\zeta_{24}^{7} q^{48} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{50} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{52} + \zeta_{24}^{3} q^{54} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} ) q^{57} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{59} + ( 1 - \zeta_{24}^{10} ) q^{60} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{61} - q^{64} + ( \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{65} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{69} -\zeta_{24}^{6} q^{71} -\zeta_{24}^{8} q^{72} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{75} + ( \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{76} + ( 1 + \zeta_{24}^{6} ) q^{78} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{79} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{80} + \zeta_{24}^{4} q^{81} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{83} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{90} + ( -1 + \zeta_{24}^{8} ) q^{92} + ( \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{95} -\zeta_{24} q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} - 8q^{15} + 8q^{16} - 4q^{18} + 12q^{23} - 4q^{25} + 4q^{30} - 8q^{39} + 12q^{50} + 4q^{57} + 8q^{60} - 8q^{64} + 4q^{72} + 8q^{78} + 4q^{81} - 12q^{92} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$-\zeta_{24}^{8}$$ $$\zeta_{24}^{8}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1157.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i
1.00000i −0.258819 + 0.965926i −1.00000 0.965926 + 1.67303i 0.965926 + 0.258819i 0 1.00000i −0.866025 0.500000i 1.67303 0.965926i
1157.2 1.00000i 0.258819 0.965926i −1.00000 −0.965926 1.67303i −0.965926 0.258819i 0 1.00000i −0.866025 0.500000i −1.67303 + 0.965926i
1157.3 1.00000i −0.965926 0.258819i −1.00000 0.258819 + 0.448288i 0.258819 0.965926i 0 1.00000i 0.866025 + 0.500000i −0.448288 + 0.258819i
1157.4 1.00000i 0.965926 + 0.258819i −1.00000 −0.258819 0.448288i −0.258819 + 0.965926i 0 1.00000i 0.866025 + 0.500000i 0.448288 0.258819i
2909.1 1.00000i −0.965926 + 0.258819i −1.00000 0.258819 0.448288i 0.258819 + 0.965926i 0 1.00000i 0.866025 0.500000i −0.448288 0.258819i
2909.2 1.00000i 0.965926 0.258819i −1.00000 −0.258819 + 0.448288i −0.258819 0.965926i 0 1.00000i 0.866025 0.500000i 0.448288 + 0.258819i
2909.3 1.00000i −0.258819 0.965926i −1.00000 0.965926 1.67303i 0.965926 0.258819i 0 1.00000i −0.866025 + 0.500000i 1.67303 + 0.965926i
2909.4 1.00000i 0.258819 + 0.965926i −1.00000 −0.965926 + 1.67303i −0.965926 + 0.258819i 0 1.00000i −0.866025 + 0.500000i −1.67303 0.965926i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2909.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner
504.bi odd 6 1 inner
504.ca 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.bi.a 8
7.b odd 2 1 inner 3528.1.bi.a 8
7.c even 3 1 3528.1.bg.a 8
7.c even 3 1 3528.1.db.a 8
7.d odd 6 1 3528.1.bg.a 8
7.d odd 6 1 3528.1.db.a 8
8.b even 2 1 inner 3528.1.bi.a 8
9.d odd 6 1 3528.1.db.a 8
56.h odd 2 1 CM 3528.1.bi.a 8
56.j odd 6 1 3528.1.bg.a 8
56.j odd 6 1 3528.1.db.a 8
56.p even 6 1 3528.1.bg.a 8
56.p even 6 1 3528.1.db.a 8
63.i even 6 1 inner 3528.1.bi.a 8
63.j odd 6 1 inner 3528.1.bi.a 8
63.n odd 6 1 3528.1.bg.a 8
63.o even 6 1 3528.1.db.a 8
63.s even 6 1 3528.1.bg.a 8
72.j odd 6 1 3528.1.db.a 8
504.y even 6 1 3528.1.bg.a 8
504.bi odd 6 1 inner 3528.1.bi.a 8
504.ca even 6 1 inner 3528.1.bi.a 8
504.cc even 6 1 3528.1.db.a 8
504.db odd 6 1 3528.1.bg.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.bg.a 8 7.c even 3 1
3528.1.bg.a 8 7.d odd 6 1
3528.1.bg.a 8 56.j odd 6 1
3528.1.bg.a 8 56.p even 6 1
3528.1.bg.a 8 63.n odd 6 1
3528.1.bg.a 8 63.s even 6 1
3528.1.bg.a 8 504.y even 6 1
3528.1.bg.a 8 504.db odd 6 1
3528.1.bi.a 8 1.a even 1 1 trivial
3528.1.bi.a 8 7.b odd 2 1 inner
3528.1.bi.a 8 8.b even 2 1 inner
3528.1.bi.a 8 56.h odd 2 1 CM
3528.1.bi.a 8 63.i even 6 1 inner
3528.1.bi.a 8 63.j odd 6 1 inner
3528.1.bi.a 8 504.bi odd 6 1 inner
3528.1.bi.a 8 504.ca even 6 1 inner
3528.1.db.a 8 7.c even 3 1
3528.1.db.a 8 7.d odd 6 1
3528.1.db.a 8 9.d odd 6 1
3528.1.db.a 8 56.j odd 6 1
3528.1.db.a 8 56.p even 6 1
3528.1.db.a 8 63.o even 6 1
3528.1.db.a 8 72.j odd 6 1
3528.1.db.a 8 504.cc even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3528, [\chi])$$.

## Hecke characteristic polynomials

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