# Properties

 Label 3528.1.u.a Level $3528$ Weight $1$ Character orbit 3528.u Analytic conductor $1.761$ Analytic rank $0$ Dimension $16$ Projective image $D_{24}$ 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.u (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{24}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{24} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{48}^{4} q^{2} -\zeta_{48}^{9} q^{3} + \zeta_{48}^{8} q^{4} + \zeta_{48}^{13} q^{6} -\zeta_{48}^{12} q^{8} + \zeta_{48}^{18} q^{9} +O(q^{10})$$ $$q -\zeta_{48}^{4} q^{2} -\zeta_{48}^{9} q^{3} + \zeta_{48}^{8} q^{4} + \zeta_{48}^{13} q^{6} -\zeta_{48}^{12} q^{8} + \zeta_{48}^{18} q^{9} + ( \zeta_{48}^{2} + \zeta_{48}^{22} ) q^{11} -\zeta_{48}^{17} q^{12} + \zeta_{48}^{16} q^{16} + ( \zeta_{48}^{3} - \zeta_{48}^{5} ) q^{17} -\zeta_{48}^{22} q^{18} + ( -\zeta_{48}^{3} + \zeta_{48}^{13} ) q^{19} + ( \zeta_{48}^{2} - \zeta_{48}^{6} ) q^{22} + \zeta_{48}^{21} q^{24} + q^{25} + \zeta_{48}^{3} q^{27} -\zeta_{48}^{20} q^{32} + ( \zeta_{48}^{7} - \zeta_{48}^{11} ) q^{33} + ( -\zeta_{48}^{7} + \zeta_{48}^{9} ) q^{34} -\zeta_{48}^{2} q^{36} + ( \zeta_{48}^{7} - \zeta_{48}^{17} ) q^{38} + ( -\zeta_{48}^{15} - \zeta_{48}^{17} ) q^{41} + ( \zeta_{48}^{6} + \zeta_{48}^{10} ) q^{43} + ( -\zeta_{48}^{6} + \zeta_{48}^{10} ) q^{44} + \zeta_{48} q^{48} -\zeta_{48}^{4} q^{50} + ( -\zeta_{48}^{12} + \zeta_{48}^{14} ) q^{51} -\zeta_{48}^{7} q^{54} + ( \zeta_{48}^{12} - \zeta_{48}^{22} ) q^{57} + ( -\zeta_{48}^{3} - \zeta_{48}^{13} ) q^{59} - q^{64} + ( -\zeta_{48}^{11} + \zeta_{48}^{15} ) q^{66} + ( \zeta_{48}^{4} + \zeta_{48}^{12} ) q^{67} + ( \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{68} + \zeta_{48}^{6} q^{72} + ( \zeta_{48}^{9} - \zeta_{48}^{23} ) q^{73} -\zeta_{48}^{9} q^{75} + ( -\zeta_{48}^{11} + \zeta_{48}^{21} ) q^{76} -\zeta_{48}^{12} q^{81} + ( \zeta_{48}^{19} + \zeta_{48}^{21} ) q^{82} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{83} + ( -\zeta_{48}^{10} - \zeta_{48}^{14} ) q^{86} + ( \zeta_{48}^{10} - \zeta_{48}^{14} ) q^{88} + ( \zeta_{48}^{17} - \zeta_{48}^{23} ) q^{89} -\zeta_{48}^{5} q^{96} + ( -\zeta_{48}^{19} - \zeta_{48}^{21} ) q^{97} + ( -\zeta_{48}^{16} + \zeta_{48}^{20} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + O(q^{10})$$ $$16q + 8q^{4} - 8q^{16} + 16q^{25} - 16q^{64} + 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$-\zeta_{48}^{16}$$ $$\zeta_{48}^{8}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
803.1
 0.130526 + 0.991445i 0.991445 − 0.130526i −0.991445 + 0.130526i −0.130526 − 0.991445i 0.793353 + 0.608761i −0.608761 + 0.793353i 0.608761 − 0.793353i −0.793353 − 0.608761i 0.130526 − 0.991445i 0.991445 + 0.130526i −0.991445 − 0.130526i −0.130526 + 0.991445i 0.793353 − 0.608761i −0.608761 − 0.793353i 0.608761 + 0.793353i −0.793353 + 0.608761i
−0.866025 + 0.500000i −0.923880 0.382683i 0.500000 0.866025i 0 0.991445 0.130526i 0 1.00000i 0.707107 + 0.707107i 0
803.2 −0.866025 + 0.500000i −0.382683 + 0.923880i 0.500000 0.866025i 0 −0.130526 0.991445i 0 1.00000i −0.707107 0.707107i 0
803.3 −0.866025 + 0.500000i 0.382683 0.923880i 0.500000 0.866025i 0 0.130526 + 0.991445i 0 1.00000i −0.707107 0.707107i 0
803.4 −0.866025 + 0.500000i 0.923880 + 0.382683i 0.500000 0.866025i 0 −0.991445 + 0.130526i 0 1.00000i 0.707107 + 0.707107i 0
803.5 0.866025 0.500000i −0.923880 + 0.382683i 0.500000 0.866025i 0 −0.608761 + 0.793353i 0 1.00000i 0.707107 0.707107i 0
803.6 0.866025 0.500000i −0.382683 0.923880i 0.500000 0.866025i 0 −0.793353 0.608761i 0 1.00000i −0.707107 + 0.707107i 0
803.7 0.866025 0.500000i 0.382683 + 0.923880i 0.500000 0.866025i 0 0.793353 + 0.608761i 0 1.00000i −0.707107 + 0.707107i 0
803.8 0.866025 0.500000i 0.923880 0.382683i 0.500000 0.866025i 0 0.608761 0.793353i 0 1.00000i 0.707107 0.707107i 0
2579.1 −0.866025 0.500000i −0.923880 + 0.382683i 0.500000 + 0.866025i 0 0.991445 + 0.130526i 0 1.00000i 0.707107 0.707107i 0
2579.2 −0.866025 0.500000i −0.382683 0.923880i 0.500000 + 0.866025i 0 −0.130526 + 0.991445i 0 1.00000i −0.707107 + 0.707107i 0
2579.3 −0.866025 0.500000i 0.382683 + 0.923880i 0.500000 + 0.866025i 0 0.130526 0.991445i 0 1.00000i −0.707107 + 0.707107i 0
2579.4 −0.866025 0.500000i 0.923880 0.382683i 0.500000 + 0.866025i 0 −0.991445 0.130526i 0 1.00000i 0.707107 0.707107i 0
2579.5 0.866025 + 0.500000i −0.923880 0.382683i 0.500000 + 0.866025i 0 −0.608761 0.793353i 0 1.00000i 0.707107 + 0.707107i 0
2579.6 0.866025 + 0.500000i −0.382683 + 0.923880i 0.500000 + 0.866025i 0 −0.793353 + 0.608761i 0 1.00000i −0.707107 0.707107i 0
2579.7 0.866025 + 0.500000i 0.382683 0.923880i 0.500000 + 0.866025i 0 0.793353 0.608761i 0 1.00000i −0.707107 0.707107i 0
2579.8 0.866025 + 0.500000i 0.923880 + 0.382683i 0.500000 + 0.866025i 0 0.608761 + 0.793353i 0 1.00000i 0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2579.8 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.n odd 6 1 inner
63.s even 6 1 inner
504.u odd 6 1 inner
504.cy 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.u.a 16
7.b odd 2 1 inner 3528.1.u.a 16
7.c even 3 1 3528.1.cm.a 16
7.c even 3 1 3528.1.co.a 16
7.d odd 6 1 3528.1.cm.a 16
7.d odd 6 1 3528.1.co.a 16
8.d odd 2 1 CM 3528.1.u.a 16
9.d odd 6 1 3528.1.cm.a 16
56.e even 2 1 inner 3528.1.u.a 16
56.k odd 6 1 3528.1.cm.a 16
56.k odd 6 1 3528.1.co.a 16
56.m even 6 1 3528.1.cm.a 16
56.m even 6 1 3528.1.co.a 16
63.i even 6 1 3528.1.co.a 16
63.j odd 6 1 3528.1.co.a 16
63.n odd 6 1 inner 3528.1.u.a 16
63.o even 6 1 3528.1.cm.a 16
63.s even 6 1 inner 3528.1.u.a 16
72.l even 6 1 3528.1.cm.a 16
504.u odd 6 1 inner 3528.1.u.a 16
504.bt even 6 1 3528.1.co.a 16
504.cm odd 6 1 3528.1.co.a 16
504.co odd 6 1 3528.1.cm.a 16
504.cy even 6 1 inner 3528.1.u.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.u.a 16 1.a even 1 1 trivial
3528.1.u.a 16 7.b odd 2 1 inner
3528.1.u.a 16 8.d odd 2 1 CM
3528.1.u.a 16 56.e even 2 1 inner
3528.1.u.a 16 63.n odd 6 1 inner
3528.1.u.a 16 63.s even 6 1 inner
3528.1.u.a 16 504.u odd 6 1 inner
3528.1.u.a 16 504.cy even 6 1 inner
3528.1.cm.a 16 7.c even 3 1
3528.1.cm.a 16 7.d odd 6 1
3528.1.cm.a 16 9.d odd 6 1
3528.1.cm.a 16 56.k odd 6 1
3528.1.cm.a 16 56.m even 6 1
3528.1.cm.a 16 63.o even 6 1
3528.1.cm.a 16 72.l even 6 1
3528.1.cm.a 16 504.co odd 6 1
3528.1.co.a 16 7.c even 3 1
3528.1.co.a 16 7.d odd 6 1
3528.1.co.a 16 56.k odd 6 1
3528.1.co.a 16 56.m even 6 1
3528.1.co.a 16 63.i even 6 1
3528.1.co.a 16 63.j odd 6 1
3528.1.co.a 16 504.bt even 6 1
3528.1.co.a 16 504.cm odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$( 1 + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16}$$
$11$ $$( 1 + 4 T^{2} + T^{4} )^{4}$$
$13$ $$T^{16}$$
$17$ $$1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16}$$
$19$ $$1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16}$$
$43$ $$( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16}$$
$61$ $$T^{16}$$
$67$ $$( 9 + 3 T^{2} + T^{4} )^{4}$$
$71$ $$T^{16}$$
$73$ $$1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$89$ $$( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$97$ $$1 - 16 T^{2} + 236 T^{4} - 304 T^{6} + 271 T^{8} - 128 T^{10} + 44 T^{12} - 8 T^{14} + T^{16}$$