# Properties

 Label 896.1.v.a Level $896$ Weight $1$ Character orbit 896.v Analytic conductor $0.447$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 896.v (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.5156108238848.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{9} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{11} + ( 1 + \zeta_{8}^{2} ) q^{23} -\zeta_{8}^{3} q^{25} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{29} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{37} + ( 1 + \zeta_{8}^{3} ) q^{43} -\zeta_{8}^{2} q^{49} + ( -1 - \zeta_{8}^{3} ) q^{53} - q^{63} + ( -1 + \zeta_{8} ) q^{67} + ( -1 - \zeta_{8} ) q^{77} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{79} + \zeta_{8}^{2} q^{81} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{23} + 4 q^{43} - 4 q^{53} - 4 q^{63} - 4 q^{67} - 4 q^{77} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{8}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 0 0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0
433.1 0 0 0 0 0 −0.707107 0.707107i 0 0.707107 0.707107i 0
657.1 0 0 0 0 0 0.707107 0.707107i 0 −0.707107 0.707107i 0
881.1 0 0 0 0 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$\Q(\sqrt{-7})$$
32.g even 8 1 inner
224.v odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.v.a 4
4.b odd 2 1 224.1.v.a 4
7.b odd 2 1 CM 896.1.v.a 4
8.b even 2 1 1792.1.v.b 4
8.d odd 2 1 1792.1.v.a 4
12.b even 2 1 2016.1.dp.b 4
16.e even 4 1 3584.1.v.a 4
16.e even 4 1 3584.1.v.c 4
16.f odd 4 1 3584.1.v.b 4
16.f odd 4 1 3584.1.v.d 4
28.d even 2 1 224.1.v.a 4
28.f even 6 2 1568.1.bl.a 8
28.g odd 6 2 1568.1.bl.a 8
32.g even 8 1 inner 896.1.v.a 4
32.g even 8 1 1792.1.v.b 4
32.g even 8 1 3584.1.v.a 4
32.g even 8 1 3584.1.v.c 4
32.h odd 8 1 224.1.v.a 4
32.h odd 8 1 1792.1.v.a 4
32.h odd 8 1 3584.1.v.b 4
32.h odd 8 1 3584.1.v.d 4
56.e even 2 1 1792.1.v.a 4
56.h odd 2 1 1792.1.v.b 4
84.h odd 2 1 2016.1.dp.b 4
96.o even 8 1 2016.1.dp.b 4
112.j even 4 1 3584.1.v.b 4
112.j even 4 1 3584.1.v.d 4
112.l odd 4 1 3584.1.v.a 4
112.l odd 4 1 3584.1.v.c 4
224.v odd 8 1 inner 896.1.v.a 4
224.v odd 8 1 1792.1.v.b 4
224.v odd 8 1 3584.1.v.a 4
224.v odd 8 1 3584.1.v.c 4
224.x even 8 1 224.1.v.a 4
224.x even 8 1 1792.1.v.a 4
224.x even 8 1 3584.1.v.b 4
224.x even 8 1 3584.1.v.d 4
224.be even 24 2 1568.1.bl.a 8
224.bf odd 24 2 1568.1.bl.a 8
672.br odd 8 1 2016.1.dp.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 4.b odd 2 1
224.1.v.a 4 28.d even 2 1
224.1.v.a 4 32.h odd 8 1
224.1.v.a 4 224.x even 8 1
896.1.v.a 4 1.a even 1 1 trivial
896.1.v.a 4 7.b odd 2 1 CM
896.1.v.a 4 32.g even 8 1 inner
896.1.v.a 4 224.v odd 8 1 inner
1568.1.bl.a 8 28.f even 6 2
1568.1.bl.a 8 28.g odd 6 2
1568.1.bl.a 8 224.be even 24 2
1568.1.bl.a 8 224.bf odd 24 2
1792.1.v.a 4 8.d odd 2 1
1792.1.v.a 4 32.h odd 8 1
1792.1.v.a 4 56.e even 2 1
1792.1.v.a 4 224.x even 8 1
1792.1.v.b 4 8.b even 2 1
1792.1.v.b 4 32.g even 8 1
1792.1.v.b 4 56.h odd 2 1
1792.1.v.b 4 224.v odd 8 1
2016.1.dp.b 4 12.b even 2 1
2016.1.dp.b 4 84.h odd 2 1
2016.1.dp.b 4 96.o even 8 1
2016.1.dp.b 4 672.br odd 8 1
3584.1.v.a 4 16.e even 4 1
3584.1.v.a 4 32.g even 8 1
3584.1.v.a 4 112.l odd 4 1
3584.1.v.a 4 224.v odd 8 1
3584.1.v.b 4 16.f odd 4 1
3584.1.v.b 4 32.h odd 8 1
3584.1.v.b 4 112.j even 4 1
3584.1.v.b 4 224.x even 8 1
3584.1.v.c 4 16.e even 4 1
3584.1.v.c 4 32.g even 8 1
3584.1.v.c 4 112.l odd 4 1
3584.1.v.c 4 224.v odd 8 1
3584.1.v.d 4 16.f odd 4 1
3584.1.v.d 4 32.h odd 8 1
3584.1.v.d 4 112.j even 4 1
3584.1.v.d 4 224.x even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(896, [\chi])$$.

## Hecke characteristic polynomials

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