# Properties

 Label 896.2.e.f Level $896$ Weight $2$ Character orbit 896.e Analytic conductor $7.155$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{13} -2 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{3} q^{17} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{19} + ( 5 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{21} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} -4 \zeta_{12}^{3} q^{27} + ( 4 - 8 \zeta_{12}^{2} ) q^{29} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{33} + ( 1 + 4 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} -4 \zeta_{12}^{3} q^{37} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{39} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{43} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{45} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( -4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{51} + ( 4 - 8 \zeta_{12}^{2} ) q^{53} -4 q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 3 - 6 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{59} + ( -3 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{61} + ( -1 + 8 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{63} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{71} + ( 4 - 8 \zeta_{12}^{2} ) q^{73} + ( -3 + 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{75} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( 1 - 2 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{83} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} + ( -12 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{89} + ( 3 - 4 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( 4 - 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{93} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} -12 \zeta_{12}^{3} q^{97} + ( -14 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} + 8q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{5} + 8q^{7} - 4q^{9} + 8q^{11} + 12q^{13} + 12q^{21} - 4q^{25} + 8q^{35} - 24q^{43} + 20q^{45} + 4q^{49} - 16q^{51} - 16q^{55} - 12q^{61} - 8q^{63} + 24q^{67} + 32q^{69} + 16q^{77} + 4q^{81} - 48q^{87} + 24q^{91} - 56q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
447.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 2.73205i 0 −0.732051 0 2.00000 + 1.73205i 0 −4.46410 0
447.2 0 0.732051i 0 2.73205 0 2.00000 + 1.73205i 0 2.46410 0
447.3 0 0.732051i 0 2.73205 0 2.00000 1.73205i 0 2.46410 0
447.4 0 2.73205i 0 −0.732051 0 2.00000 1.73205i 0 −4.46410 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
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.e.f yes 4
4.b odd 2 1 896.2.e.e yes 4
7.b odd 2 1 896.2.e.a 4
8.b even 2 1 896.2.e.b yes 4
8.d odd 2 1 896.2.e.a 4
16.e even 4 1 1792.2.f.a 4
16.e even 4 1 1792.2.f.h 4
16.f odd 4 1 1792.2.f.b 4
16.f odd 4 1 1792.2.f.i 4
28.d even 2 1 896.2.e.b yes 4
56.e even 2 1 inner 896.2.e.f yes 4
56.h odd 2 1 896.2.e.e yes 4
112.j even 4 1 1792.2.f.a 4
112.j even 4 1 1792.2.f.h 4
112.l odd 4 1 1792.2.f.b 4
112.l odd 4 1 1792.2.f.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.a 4 7.b odd 2 1
896.2.e.a 4 8.d odd 2 1
896.2.e.b yes 4 8.b even 2 1
896.2.e.b yes 4 28.d even 2 1
896.2.e.e yes 4 4.b odd 2 1
896.2.e.e yes 4 56.h odd 2 1
896.2.e.f yes 4 1.a even 1 1 trivial
896.2.e.f yes 4 56.e even 2 1 inner
1792.2.f.a 4 16.e even 4 1
1792.2.f.a 4 112.j even 4 1
1792.2.f.b 4 16.f odd 4 1
1792.2.f.b 4 112.l odd 4 1
1792.2.f.h 4 16.e even 4 1
1792.2.f.h 4 112.j even 4 1
1792.2.f.i 4 16.f odd 4 1
1792.2.f.i 4 112.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(896, [\chi])$$:

 $$T_{3}^{4} + 8 T_{3}^{2} + 4$$ $$T_{5}^{2} - 2 T_{5} - 2$$ $$T_{11}^{2} - 4 T_{11} - 8$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 + 8 T^{2} + T^{4}$$
$5$ $$( -2 - 2 T + T^{2} )^{2}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$( -8 - 4 T + T^{2} )^{2}$$
$13$ $$( 6 - 6 T + T^{2} )^{2}$$
$17$ $$( 16 + T^{2} )^{2}$$
$19$ $$36 + 24 T^{2} + T^{4}$$
$23$ $$64 + 32 T^{2} + T^{4}$$
$29$ $$( 48 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$1024 + 128 T^{2} + T^{4}$$
$43$ $$( 24 + 12 T + T^{2} )^{2}$$
$47$ $$( -48 + T^{2} )^{2}$$
$53$ $$( 48 + T^{2} )^{2}$$
$59$ $$2916 + 216 T^{2} + T^{4}$$
$61$ $$( -66 + 6 T + T^{2} )^{2}$$
$67$ $$( 24 - 12 T + T^{2} )^{2}$$
$71$ $$2704 + 152 T^{2} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 12 + T^{2} )^{2}$$
$83$ $$6084 + 168 T^{2} + T^{4}$$
$89$ $$256 + 224 T^{2} + T^{4}$$
$97$ $$( 144 + T^{2} )^{2}$$