# Properties

 Label 896.2.b.e Level $896$ Weight $2$ Character orbit 896.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{1} q^{5} - q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{1} q^{5} - q^{7} + ( -3 + \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + \beta_{1} q^{13} + ( -6 + \beta_{3} ) q^{15} -2 q^{17} -\beta_{1} q^{19} -\beta_{1} q^{21} + ( -2 - \beta_{3} ) q^{23} + ( -1 + \beta_{3} ) q^{25} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 3 \beta_{1} - \beta_{2} ) q^{29} + ( 4 - 2 \beta_{3} ) q^{33} -\beta_{1} q^{35} + ( \beta_{1} + \beta_{2} ) q^{37} + ( -6 + \beta_{3} ) q^{39} + 2 q^{41} + ( 3 \beta_{1} + \beta_{2} ) q^{43} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{45} + ( -4 - 2 \beta_{3} ) q^{47} + q^{49} -2 \beta_{1} q^{51} + ( -3 \beta_{1} - \beta_{2} ) q^{53} + ( 4 - 2 \beta_{3} ) q^{55} + ( 6 - \beta_{3} ) q^{57} + ( -\beta_{1} + 2 \beta_{2} ) q^{59} -\beta_{1} q^{61} + ( 3 - \beta_{3} ) q^{63} + ( -6 + \beta_{3} ) q^{65} + ( 3 \beta_{1} - \beta_{2} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{69} + ( -4 + 2 \beta_{3} ) q^{71} + ( -6 - 2 \beta_{3} ) q^{73} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{75} + ( \beta_{1} - \beta_{2} ) q^{77} + ( -4 + 2 \beta_{3} ) q^{79} + ( 11 - 3 \beta_{3} ) q^{81} + ( \beta_{1} - 2 \beta_{2} ) q^{83} -2 \beta_{1} q^{85} + ( -16 + 4 \beta_{3} ) q^{87} -2 q^{89} -\beta_{1} q^{91} + ( 6 - \beta_{3} ) q^{95} + ( 2 + 2 \beta_{3} ) q^{97} + ( 9 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} - 12q^{9} + O(q^{10})$$ $$4q - 4q^{7} - 12q^{9} - 24q^{15} - 8q^{17} - 8q^{23} - 4q^{25} + 16q^{33} - 24q^{39} + 8q^{41} - 16q^{47} + 4q^{49} + 16q^{55} + 24q^{57} + 12q^{63} - 24q^{65} - 16q^{71} - 24q^{73} - 16q^{79} + 44q^{81} - 64q^{87} - 8q^{89} + 24q^{95} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{3} + 10 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} - 5 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
0 3.23607i 0 3.23607i 0 −1.00000 0 −7.47214 0
449.2 0 1.23607i 0 1.23607i 0 −1.00000 0 1.47214 0
449.3 0 1.23607i 0 1.23607i 0 −1.00000 0 1.47214 0
449.4 0 3.23607i 0 3.23607i 0 −1.00000 0 −7.47214 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
8.b 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.b.e 4
4.b odd 2 1 896.2.b.g yes 4
8.b even 2 1 inner 896.2.b.e 4
8.d odd 2 1 896.2.b.g yes 4
16.e even 4 1 1792.2.a.l 2
16.e even 4 1 1792.2.a.r 2
16.f odd 4 1 1792.2.a.j 2
16.f odd 4 1 1792.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.e 4 1.a even 1 1 trivial
896.2.b.e 4 8.b even 2 1 inner
896.2.b.g yes 4 4.b odd 2 1
896.2.b.g yes 4 8.d odd 2 1
1792.2.a.j 2 16.f odd 4 1
1792.2.a.l 2 16.e even 4 1
1792.2.a.r 2 16.e even 4 1
1792.2.a.t 2 16.f odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ $$T_{23}^{2} + 4 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$16 + 12 T^{2} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$( 16 + T^{2} )^{2}$$
$13$ $$16 + 12 T^{2} + T^{4}$$
$17$ $$( 2 + T )^{4}$$
$19$ $$16 + 12 T^{2} + T^{4}$$
$23$ $$( -16 + 4 T + T^{2} )^{2}$$
$29$ $$256 + 112 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$256 + 48 T^{2} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$( 80 + T^{2} )^{2}$$
$47$ $$( -64 + 8 T + T^{2} )^{2}$$
$53$ $$( 80 + T^{2} )^{2}$$
$59$ $$1936 + 108 T^{2} + T^{4}$$
$61$ $$16 + 12 T^{2} + T^{4}$$
$67$ $$256 + 112 T^{2} + T^{4}$$
$71$ $$( -64 + 8 T + T^{2} )^{2}$$
$73$ $$( -44 + 12 T + T^{2} )^{2}$$
$79$ $$( -64 + 8 T + T^{2} )^{2}$$
$83$ $$1936 + 108 T^{2} + T^{4}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( -76 - 4 T + T^{2} )^{2}$$