# Properties

 Label 1764.1.q.b Level 1764 Weight 1 Character orbit 1764.q Analytic conductor 0.880 Analytic rank 0 Dimension 16 Projective image $$D_{8}$$ CM discriminant -4 Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.880350682285$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Projective image $$D_{8}$$ Projective field Galois closure of 8.0.38423222208.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{48}^{20} q^{2} -\zeta_{48}^{16} q^{4} + ( -\zeta_{48}^{5} - \zeta_{48}^{11} ) q^{5} -\zeta_{48}^{12} q^{8} +O(q^{10})$$ $$q -\zeta_{48}^{20} q^{2} -\zeta_{48}^{16} q^{4} + ( -\zeta_{48}^{5} - \zeta_{48}^{11} ) q^{5} -\zeta_{48}^{12} q^{8} + ( -\zeta_{48} - \zeta_{48}^{7} ) q^{10} + ( \zeta_{48}^{3} + \zeta_{48}^{21} ) q^{13} -\zeta_{48}^{8} q^{16} + ( \zeta_{48} - \zeta_{48}^{7} ) q^{17} + ( -\zeta_{48}^{3} + \zeta_{48}^{21} ) q^{20} + ( \zeta_{48}^{10} + \zeta_{48}^{16} + \zeta_{48}^{22} ) q^{25} + ( \zeta_{48}^{17} - \zeta_{48}^{23} ) q^{26} -\zeta_{48}^{4} q^{32} + ( -\zeta_{48}^{3} - \zeta_{48}^{21} ) q^{34} + ( -\zeta_{48}^{2} - \zeta_{48}^{14} ) q^{37} + ( \zeta_{48}^{17} + \zeta_{48}^{23} ) q^{40} + ( -\zeta_{48}^{9} + \zeta_{48}^{15} ) q^{41} + ( \zeta_{48}^{6} + \zeta_{48}^{12} + \zeta_{48}^{18} ) q^{50} + ( \zeta_{48}^{13} - \zeta_{48}^{19} ) q^{52} + ( \zeta_{48}^{10} - \zeta_{48}^{22} ) q^{53} + ( -\zeta_{48}^{17} - \zeta_{48}^{23} ) q^{61} - q^{64} + ( \zeta_{48}^{2} - \zeta_{48}^{14} ) q^{65} + ( -\zeta_{48}^{17} + \zeta_{48}^{23} ) q^{68} + ( \zeta_{48} + \zeta_{48}^{7} ) q^{73} + ( -\zeta_{48}^{10} + \zeta_{48}^{22} ) q^{74} + ( \zeta_{48}^{13} + \zeta_{48}^{19} ) q^{80} + ( -\zeta_{48}^{5} + \zeta_{48}^{11} ) q^{82} + ( -\zeta_{48}^{6} + \zeta_{48}^{18} ) q^{85} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{89} + ( \zeta_{48}^{9} + \zeta_{48}^{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + O(q^{10})$$ $$16q + 8q^{4} - 8q^{16} - 8q^{25} - 16q^{64} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{48}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.608761 + 0.793353i −0.793353 − 0.608761i 0.793353 + 0.608761i 0.608761 − 0.793353i 0.991445 − 0.130526i −0.130526 − 0.991445i 0.130526 + 0.991445i −0.991445 + 0.130526i −0.608761 − 0.793353i −0.793353 + 0.608761i 0.793353 − 0.608761i 0.608761 + 0.793353i 0.991445 + 0.130526i −0.130526 + 0.991445i 0.130526 − 0.991445i −0.991445 − 0.130526i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
215.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.382683 0.662827i 0 0 1.00000i 0 0.662827 0.382683i
215.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.923880 1.60021i 0 0 1.00000i 0 1.60021 0.923880i
1403.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
1403.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.5 0.866025 0.500000i 0 0.500000 0.866025i −0.923880 1.60021i 0 0 1.00000i 0 −1.60021 0.923880i
1403.6 0.866025 0.500000i 0 0.500000 0.866025i −0.382683 0.662827i 0 0 1.00000i 0 −0.662827 0.382683i
1403.7 0.866025 0.500000i 0 0.500000 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.8 0.866025 0.500000i 0 0.500000 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1403.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.h odd 2 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.q.b 16
3.b odd 2 1 inner 1764.1.q.b 16
4.b odd 2 1 CM 1764.1.q.b 16
7.b odd 2 1 inner 1764.1.q.b 16
7.c even 3 1 1764.1.h.a 8
7.c even 3 1 inner 1764.1.q.b 16
7.d odd 6 1 1764.1.h.a 8
7.d odd 6 1 inner 1764.1.q.b 16
12.b even 2 1 inner 1764.1.q.b 16
21.c even 2 1 inner 1764.1.q.b 16
21.g even 6 1 1764.1.h.a 8
21.g even 6 1 inner 1764.1.q.b 16
21.h odd 6 1 1764.1.h.a 8
21.h odd 6 1 inner 1764.1.q.b 16
28.d even 2 1 inner 1764.1.q.b 16
28.f even 6 1 1764.1.h.a 8
28.f even 6 1 inner 1764.1.q.b 16
28.g odd 6 1 1764.1.h.a 8
28.g odd 6 1 inner 1764.1.q.b 16
84.h odd 2 1 inner 1764.1.q.b 16
84.j odd 6 1 1764.1.h.a 8
84.j odd 6 1 inner 1764.1.q.b 16
84.n even 6 1 1764.1.h.a 8
84.n even 6 1 inner 1764.1.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.h.a 8 7.c even 3 1
1764.1.h.a 8 7.d odd 6 1
1764.1.h.a 8 21.g even 6 1
1764.1.h.a 8 21.h odd 6 1
1764.1.h.a 8 28.f even 6 1
1764.1.h.a 8 28.g odd 6 1
1764.1.h.a 8 84.j odd 6 1
1764.1.h.a 8 84.n even 6 1
1764.1.q.b 16 1.a even 1 1 trivial
1764.1.q.b 16 3.b odd 2 1 inner
1764.1.q.b 16 4.b odd 2 1 CM
1764.1.q.b 16 7.b odd 2 1 inner
1764.1.q.b 16 7.c even 3 1 inner
1764.1.q.b 16 7.d odd 6 1 inner
1764.1.q.b 16 12.b even 2 1 inner
1764.1.q.b 16 21.c even 2 1 inner
1764.1.q.b 16 21.g even 6 1 inner
1764.1.q.b 16 21.h odd 6 1 inner
1764.1.q.b 16 28.d even 2 1 inner
1764.1.q.b 16 28.f even 6 1 inner
1764.1.q.b 16 28.g odd 6 1 inner
1764.1.q.b 16 84.h odd 2 1 inner
1764.1.q.b 16 84.j odd 6 1 inner
1764.1.q.b 16 84.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 4 T_{5}^{6} + 14 T_{5}^{4} + 8 T_{5}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ 1
$5$ $$( 1 - T^{8} + T^{16} )^{2}$$
$7$ 1
$11$ $$( 1 - T^{2} + T^{4} )^{8}$$
$13$ $$( 1 + T^{8} )^{4}$$
$17$ $$( 1 - T^{8} + T^{16} )^{2}$$
$19$ $$( 1 - T^{2} + T^{4} )^{8}$$
$23$ $$( 1 - T^{2} + T^{4} )^{8}$$
$29$ $$( 1 - T )^{16}( 1 + T )^{16}$$
$31$ $$( 1 - T^{2} + T^{4} )^{8}$$
$37$ $$( 1 - T^{4} + T^{8} )^{4}$$
$41$ $$( 1 + T^{8} )^{4}$$
$43$ $$( 1 - T )^{16}( 1 + T )^{16}$$
$47$ $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
$53$ $$( 1 - T^{4} + T^{8} )^{4}$$
$59$ $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
$61$ $$( 1 - T^{8} + T^{16} )^{2}$$
$67$ $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
$71$ $$( 1 + T^{2} )^{16}$$
$73$ $$( 1 - T^{8} + T^{16} )^{2}$$
$79$ $$( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8}$$
$83$ $$( 1 - T )^{16}( 1 + T )^{16}$$
$89$ $$( 1 - T^{8} + T^{16} )^{2}$$
$97$ $$( 1 + T^{8} )^{4}$$