# Properties

 Label 896.1.bm.a Level $896$ Weight $1$ Character orbit 896.bm Analytic conductor $0.447$ Analytic rank $0$ Dimension $16$ Projective image $D_{32}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 896.bm (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{32}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{32} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{32}^{13} q^{2} -\zeta_{32}^{10} q^{4} -\zeta_{32}^{9} q^{7} -\zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} +O(q^{10})$$ $$q -\zeta_{32}^{13} q^{2} -\zeta_{32}^{10} q^{4} -\zeta_{32}^{9} q^{7} -\zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} + ( -\zeta_{32}^{11} + \zeta_{32}^{14} ) q^{11} -\zeta_{32}^{6} q^{14} -\zeta_{32}^{4} q^{16} + \zeta_{32}^{12} q^{18} + ( -\zeta_{32}^{8} + \zeta_{32}^{11} ) q^{22} + ( \zeta_{32}^{2} + \zeta_{32}^{4} ) q^{23} + \zeta_{32}^{5} q^{25} -\zeta_{32}^{3} q^{28} + ( \zeta_{32} + \zeta_{32}^{6} ) q^{29} -\zeta_{32} q^{32} + \zeta_{32}^{9} q^{36} + ( -\zeta_{32}^{3} + \zeta_{32}^{10} ) q^{37} + ( -\zeta_{32}^{5} - \zeta_{32}^{12} ) q^{43} + ( -\zeta_{32}^{5} + \zeta_{32}^{8} ) q^{44} + ( \zeta_{32} - \zeta_{32}^{15} ) q^{46} -\zeta_{32}^{2} q^{49} + \zeta_{32}^{2} q^{50} + ( \zeta_{32}^{12} + \zeta_{32}^{13} ) q^{53} - q^{56} + ( \zeta_{32}^{3} - \zeta_{32}^{14} ) q^{58} + \zeta_{32}^{8} q^{63} + \zeta_{32}^{14} q^{64} + ( \zeta_{32}^{7} - \zeta_{32}^{8} ) q^{67} + ( \zeta_{32}^{3} - \zeta_{32}^{15} ) q^{71} + \zeta_{32}^{6} q^{72} + ( -1 + \zeta_{32}^{7} ) q^{74} + ( -\zeta_{32}^{4} + \zeta_{32}^{7} ) q^{77} + ( -\zeta_{32}^{7} + \zeta_{32}^{13} ) q^{79} -\zeta_{32}^{14} q^{81} + ( -\zeta_{32}^{2} - \zeta_{32}^{9} ) q^{86} + ( -\zeta_{32}^{2} + \zeta_{32}^{5} ) q^{88} + ( -\zeta_{32}^{12} - \zeta_{32}^{14} ) q^{92} + \zeta_{32}^{15} q^{98} + ( \zeta_{32}^{10} - \zeta_{32}^{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{56} - 16q^{74} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{32}^{5}$$

## 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
 0.831470 + 0.555570i 0.831470 − 0.555570i 0.195090 + 0.980785i 0.980785 + 0.195090i −0.555570 + 0.831470i 0.555570 + 0.831470i −0.980785 + 0.195090i −0.195090 + 0.980785i −0.831470 − 0.555570i −0.831470 + 0.555570i −0.195090 − 0.980785i −0.980785 − 0.195090i 0.555570 − 0.831470i −0.555570 − 0.831470i 0.980785 − 0.195090i 0.195090 − 0.980785i
−0.195090 0.980785i 0 −0.923880 + 0.382683i 0 0 −0.555570 + 0.831470i 0.555570 + 0.831470i −0.831470 + 0.555570i 0
69.1 −0.195090 + 0.980785i 0 −0.923880 0.382683i 0 0 −0.555570 0.831470i 0.555570 0.831470i −0.831470 0.555570i 0
125.1 −0.555570 + 0.831470i 0 −0.382683 0.923880i 0 0 −0.980785 + 0.195090i 0.980785 + 0.195090i −0.195090 + 0.980785i 0
181.1 0.831470 0.555570i 0 0.382683 0.923880i 0 0 0.195090 0.980785i −0.195090 0.980785i −0.980785 + 0.195090i 0
237.1 0.980785 0.195090i 0 0.923880 0.382683i 0 0 −0.831470 0.555570i 0.831470 0.555570i 0.555570 + 0.831470i 0
293.1 −0.980785 0.195090i 0 0.923880 + 0.382683i 0 0 0.831470 0.555570i −0.831470 0.555570i −0.555570 + 0.831470i 0
349.1 −0.831470 0.555570i 0 0.382683 + 0.923880i 0 0 −0.195090 0.980785i 0.195090 0.980785i 0.980785 + 0.195090i 0
405.1 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0 0 0.980785 + 0.195090i −0.980785 + 0.195090i 0.195090 + 0.980785i 0
461.1 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0 0 0.555570 0.831470i −0.555570 0.831470i 0.831470 0.555570i 0
517.1 0.195090 0.980785i 0 −0.923880 0.382683i 0 0 0.555570 + 0.831470i −0.555570 + 0.831470i 0.831470 + 0.555570i 0
573.1 0.555570 0.831470i 0 −0.382683 0.923880i 0 0 0.980785 0.195090i −0.980785 0.195090i 0.195090 0.980785i 0
629.1 −0.831470 + 0.555570i 0 0.382683 0.923880i 0 0 −0.195090 + 0.980785i 0.195090 + 0.980785i 0.980785 0.195090i 0
685.1 −0.980785 + 0.195090i 0 0.923880 0.382683i 0 0 0.831470 + 0.555570i −0.831470 + 0.555570i −0.555570 0.831470i 0
741.1 0.980785 + 0.195090i 0 0.923880 + 0.382683i 0 0 −0.831470 + 0.555570i 0.831470 + 0.555570i 0.555570 0.831470i 0
797.1 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0 0 0.195090 + 0.980785i −0.195090 + 0.980785i −0.980785 0.195090i 0
853.1 −0.555570 0.831470i 0 −0.382683 + 0.923880i 0 0 −0.980785 0.195090i 0.980785 0.195090i −0.195090 0.980785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 853.1 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 $$\Q(\sqrt{-7})$$
128.k even 32 1 inner
896.bm odd 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.bm.a 16
4.b odd 2 1 3584.1.bm.a 16
7.b odd 2 1 CM 896.1.bm.a 16
28.d even 2 1 3584.1.bm.a 16
128.k even 32 1 inner 896.1.bm.a 16
128.l odd 32 1 3584.1.bm.a 16
896.bk even 32 1 3584.1.bm.a 16
896.bm odd 32 1 inner 896.1.bm.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.bm.a 16 1.a even 1 1 trivial
896.1.bm.a 16 7.b odd 2 1 CM
896.1.bm.a 16 128.k even 32 1 inner
896.1.bm.a 16 896.bm odd 32 1 inner
3584.1.bm.a 16 4.b odd 2 1
3584.1.bm.a 16 28.d even 2 1
3584.1.bm.a 16 128.l odd 32 1
3584.1.bm.a 16 896.bk even 32 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$1 + T^{16}$$
$11$ $$2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} )^{2}$$
$29$ $$2 + 16 T + 88 T^{2} + 192 T^{3} + 140 T^{4} + 16 T^{5} + 2 T^{8} - 48 T^{9} + 40 T^{10} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$2 + 16 T + 40 T^{2} + 140 T^{4} - 48 T^{5} + 192 T^{7} + 2 T^{8} + 88 T^{10} + 16 T^{13} + T^{16}$$
$41$ $$T^{16}$$
$43$ $$2 + 16 T + 72 T^{2} + 80 T^{3} + 4 T^{4} + 56 T^{6} - 160 T^{7} + 6 T^{8} + 16 T^{11} + 4 T^{12} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$2 - 16 T + 72 T^{2} - 80 T^{3} + 4 T^{4} + 56 T^{6} + 160 T^{7} + 6 T^{8} - 16 T^{11} + 4 T^{12} + T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16}$$
$67$ $$2 - 16 T + 8 T^{2} + 112 T^{3} + 28 T^{4} - 112 T^{5} + 56 T^{6} + 16 T^{7} + 70 T^{8} + 56 T^{10} + 28 T^{12} + 8 T^{14} + T^{16}$$
$71$ $$16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16}$$
$73$ $$T^{16}$$
$79$ $$4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$