# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{16}^{4} q^{2} - q^{4} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{5} + \zeta_{16}^{4} q^{8} +O(q^{10})$$ $$q -\zeta_{16}^{4} q^{2} - q^{4} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{5} + \zeta_{16}^{4} q^{8} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{10} + ( \zeta_{16}^{3} + \zeta_{16}^{5} ) q^{13} + q^{16} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{17} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{20} + ( 1 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{25} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{26} -\zeta_{16}^{4} q^{32} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{34} + ( -\zeta_{16}^{2} + \zeta_{16}^{6} ) q^{37} + ( \zeta_{16} + \zeta_{16}^{7} ) q^{40} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{41} + ( \zeta_{16}^{2} - \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{50} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{52} + ( -\zeta_{16}^{2} - \zeta_{16}^{6} ) q^{53} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{61} - q^{64} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{65} + ( -\zeta_{16} + \zeta_{16}^{7} ) q^{68} + ( \zeta_{16} + \zeta_{16}^{7} ) q^{73} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{74} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{80} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{82} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{85} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{89} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{16} + 8q^{25} - 8q^{64} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1763.1
 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 − 0.923880i
1.00000i 0 −1.00000 −1.84776 0 0 1.00000i 0 1.84776i
1763.2 1.00000i 0 −1.00000 −0.765367 0 0 1.00000i 0 0.765367i
1763.3 1.00000i 0 −1.00000 0.765367 0 0 1.00000i 0 0.765367i
1763.4 1.00000i 0 −1.00000 1.84776 0 0 1.00000i 0 1.84776i
1763.5 1.00000i 0 −1.00000 −1.84776 0 0 1.00000i 0 1.84776i
1763.6 1.00000i 0 −1.00000 −0.765367 0 0 1.00000i 0 0.765367i
1763.7 1.00000i 0 −1.00000 0.765367 0 0 1.00000i 0 0.765367i
1763.8 1.00000i 0 −1.00000 1.84776 0 0 1.00000i 0 1.84776i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1763.8 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
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

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