# Properties

 Label 448.1.bf.a Level $448$ Weight $1$ Character orbit 448.bf Analytic conductor $0.224$ Analytic rank $0$ Dimension $8$ Projective image $D_{16}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 448.bf (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.223581125660$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{16} q^{2} + \zeta_{16}^{2} q^{4} + \zeta_{16}^{5} q^{7} -\zeta_{16}^{3} q^{8} + \zeta_{16}^{3} q^{9} +O(q^{10})$$ $$q -\zeta_{16} q^{2} + \zeta_{16}^{2} q^{4} + \zeta_{16}^{5} q^{7} -\zeta_{16}^{3} q^{8} + \zeta_{16}^{3} q^{9} + ( \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{11} -\zeta_{16}^{6} q^{14} + \zeta_{16}^{4} q^{16} -\zeta_{16}^{4} q^{18} + ( -1 - \zeta_{16}^{7} ) q^{22} + ( -\zeta_{16}^{2} + \zeta_{16}^{4} ) q^{23} + \zeta_{16} q^{25} + \zeta_{16}^{7} q^{28} + ( -\zeta_{16}^{5} - \zeta_{16}^{6} ) q^{29} -\zeta_{16}^{5} q^{32} + \zeta_{16}^{5} q^{36} + ( \zeta_{16}^{2} + \zeta_{16}^{7} ) q^{37} + ( \zeta_{16} - \zeta_{16}^{4} ) q^{43} + ( -1 + \zeta_{16} ) q^{44} + ( \zeta_{16}^{3} - \zeta_{16}^{5} ) q^{46} -\zeta_{16}^{2} q^{49} -\zeta_{16}^{2} q^{50} + ( -\zeta_{16} - \zeta_{16}^{4} ) q^{53} + q^{56} + ( \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{58} - q^{63} + \zeta_{16}^{6} q^{64} + ( -1 - \zeta_{16}^{3} ) q^{67} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{71} -\zeta_{16}^{6} q^{72} + ( 1 - \zeta_{16}^{3} ) q^{74} + ( -\zeta_{16}^{3} + \zeta_{16}^{4} ) q^{77} + ( -\zeta_{16} - \zeta_{16}^{3} ) q^{79} + \zeta_{16}^{6} q^{81} + ( -\zeta_{16}^{2} + \zeta_{16}^{5} ) q^{86} + ( \zeta_{16} - \zeta_{16}^{2} ) q^{88} + ( -\zeta_{16}^{4} + \zeta_{16}^{6} ) q^{92} + \zeta_{16}^{3} q^{98} + ( -\zeta_{16} + \zeta_{16}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{22} - 8q^{44} + 8q^{56} - 8q^{63} - 8q^{67} + 8q^{74} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{16}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 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 0.382683 − 0.923880i
−0.923880 + 0.382683i 0 0.707107 0.707107i 0 0 −0.382683 0.923880i −0.382683 + 0.923880i 0.382683 0.923880i 0
69.1 −0.923880 0.382683i 0 0.707107 + 0.707107i 0 0 −0.382683 + 0.923880i −0.382683 0.923880i 0.382683 + 0.923880i 0
125.1 −0.382683 0.923880i 0 −0.707107 + 0.707107i 0 0 0.923880 0.382683i 0.923880 + 0.382683i −0.923880 0.382683i 0
181.1 0.382683 0.923880i 0 −0.707107 0.707107i 0 0 −0.923880 0.382683i −0.923880 + 0.382683i 0.923880 0.382683i 0
237.1 0.923880 0.382683i 0 0.707107 0.707107i 0 0 0.382683 + 0.923880i 0.382683 0.923880i −0.382683 + 0.923880i 0
293.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i 0 0 0.382683 0.923880i 0.382683 + 0.923880i −0.382683 0.923880i 0
349.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0 0 −0.923880 + 0.382683i −0.923880 0.382683i 0.923880 + 0.382683i 0
405.1 −0.382683 + 0.923880i 0 −0.707107 0.707107i 0 0 0.923880 + 0.382683i 0.923880 0.382683i −0.923880 + 0.382683i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 405.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
64.i even 16 1 inner
448.bf odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.1.bf.a 8
4.b odd 2 1 1792.1.bf.a 8
7.b odd 2 1 CM 448.1.bf.a 8
7.c even 3 2 3136.1.ce.a 16
7.d odd 6 2 3136.1.ce.a 16
8.b even 2 1 3584.1.bf.a 8
8.d odd 2 1 3584.1.bf.b 8
28.d even 2 1 1792.1.bf.a 8
56.e even 2 1 3584.1.bf.b 8
56.h odd 2 1 3584.1.bf.a 8
64.i even 16 1 inner 448.1.bf.a 8
64.i even 16 1 3584.1.bf.a 8
64.j odd 16 1 1792.1.bf.a 8
64.j odd 16 1 3584.1.bf.b 8
448.bd even 16 1 1792.1.bf.a 8
448.bd even 16 1 3584.1.bf.b 8
448.bf odd 16 1 inner 448.1.bf.a 8
448.bf odd 16 1 3584.1.bf.a 8
448.bk odd 48 2 3136.1.ce.a 16
448.bn even 48 2 3136.1.ce.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.1.bf.a 8 1.a even 1 1 trivial
448.1.bf.a 8 7.b odd 2 1 CM
448.1.bf.a 8 64.i even 16 1 inner
448.1.bf.a 8 448.bf odd 16 1 inner
1792.1.bf.a 8 4.b odd 2 1
1792.1.bf.a 8 28.d even 2 1
1792.1.bf.a 8 64.j odd 16 1
1792.1.bf.a 8 448.bd even 16 1
3136.1.ce.a 16 7.c even 3 2
3136.1.ce.a 16 7.d odd 6 2
3136.1.ce.a 16 448.bk odd 48 2
3136.1.ce.a 16 448.bn even 48 2
3584.1.bf.a 8 8.b even 2 1
3584.1.bf.a 8 56.h odd 2 1
3584.1.bf.a 8 64.i even 16 1
3584.1.bf.a 8 448.bf odd 16 1
3584.1.bf.b 8 8.d odd 2 1
3584.1.bf.b 8 56.e even 2 1
3584.1.bf.b 8 64.j odd 16 1
3584.1.bf.b 8 448.bd even 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$1 + T^{8}$$
$11$ $$2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 2 + 4 T + 2 T^{2} + T^{4} )^{2}$$
$29$ $$2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$2 + 8 T + 12 T^{2} + 2 T^{4} - 8 T^{5} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8}$$
$47$ $$T^{8}$$
$53$ $$2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$2 + 8 T + 28 T^{2} + 56 T^{3} + 70 T^{4} + 56 T^{5} + 28 T^{6} + 8 T^{7} + T^{8}$$
$71$ $$16 + T^{8}$$
$73$ $$T^{8}$$
$79$ $$4 + 12 T^{4} + T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$