# Properties

 Label 896.2.b.h 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: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{1} q^{5} + q^{7} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{1} q^{5} + q^{7} + ( -1 - \beta_{3} ) q^{9} + \beta_{2} q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( -4 - \beta_{3} ) q^{15} + ( -2 - 2 \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( 4 + \beta_{3} ) q^{23} + ( 1 - \beta_{3} ) q^{25} -\beta_{2} q^{27} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + 2 \beta_{3} q^{31} -2 \beta_{3} q^{33} + \beta_{1} q^{35} + 2 \beta_{1} q^{37} + ( 4 - \beta_{3} ) q^{39} + ( 2 - 2 \beta_{3} ) q^{41} -\beta_{2} q^{43} + ( -3 \beta_{1} - \beta_{2} ) q^{45} -8 q^{47} + q^{49} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{51} -4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + ( -4 - 3 \beta_{3} ) q^{57} + ( -\beta_{1} + \beta_{2} ) q^{59} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -1 - \beta_{3} ) q^{63} + ( 4 - \beta_{3} ) q^{65} + ( 2 \beta_{1} - \beta_{2} ) q^{67} + ( 6 \beta_{1} + \beta_{2} ) q^{69} + ( 8 - 2 \beta_{3} ) q^{71} + ( 6 + 2 \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{2} ) q^{75} + \beta_{2} q^{77} + ( -8 + 2 \beta_{3} ) q^{79} + ( -3 - \beta_{3} ) q^{81} -5 \beta_{1} q^{83} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 8 + 6 \beta_{3} ) q^{87} -2 q^{89} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -4 - 3 \beta_{3} ) q^{95} -10 q^{97} + ( -4 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{7} - 4q^{9} - 16q^{15} - 8q^{17} + 16q^{23} + 4q^{25} + 16q^{39} + 8q^{41} - 32q^{47} + 4q^{49} - 16q^{57} - 4q^{63} + 16q^{65} + 32q^{71} + 24q^{73} - 32q^{79} - 12q^{81} + 32q^{87} - 8q^{89} - 16q^{95} - 40q^{97} + O(q^{100})$$

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

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

## 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
 − 0.765367i − 1.84776i 1.84776i 0.765367i
0 2.61313i 0 2.61313i 0 1.00000 0 −3.82843 0
449.2 0 1.08239i 0 1.08239i 0 1.00000 0 1.82843 0
449.3 0 1.08239i 0 1.08239i 0 1.00000 0 1.82843 0
449.4 0 2.61313i 0 2.61313i 0 1.00000 0 −3.82843 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.h yes 4
4.b odd 2 1 896.2.b.f 4
8.b even 2 1 inner 896.2.b.h yes 4
8.d odd 2 1 896.2.b.f 4
16.e even 4 2 1792.2.a.u 4
16.f odd 4 2 1792.2.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.f 4 4.b odd 2 1
896.2.b.f 4 8.d odd 2 1
896.2.b.h yes 4 1.a even 1 1 trivial
896.2.b.h yes 4 8.b even 2 1 inner
1792.2.a.u 4 16.e even 4 2
1792.2.a.w 4 16.f odd 4 2

## Hecke kernels

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

 $$T_{3}^{4} + 8 T_{3}^{2} + 8$$ $$T_{23}^{2} - 8 T_{23} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$8 + 8 T^{2} + T^{4}$$
$5$ $$8 + 8 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$128 + 32 T^{2} + T^{4}$$
$13$ $$8 + 40 T^{2} + T^{4}$$
$17$ $$( -28 + 4 T + T^{2} )^{2}$$
$19$ $$392 + 40 T^{2} + T^{4}$$
$23$ $$( 8 - 8 T + T^{2} )^{2}$$
$29$ $$6272 + 160 T^{2} + T^{4}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$128 + 32 T^{2} + T^{4}$$
$41$ $$( -28 - 4 T + T^{2} )^{2}$$
$43$ $$128 + 32 T^{2} + T^{4}$$
$47$ $$( 8 + T )^{4}$$
$53$ $$2048 + 128 T^{2} + T^{4}$$
$59$ $$8 + 40 T^{2} + T^{4}$$
$61$ $$7688 + 200 T^{2} + T^{4}$$
$67$ $$512 + 64 T^{2} + T^{4}$$
$71$ $$( 32 - 16 T + T^{2} )^{2}$$
$73$ $$( 4 - 12 T + T^{2} )^{2}$$
$79$ $$( 32 + 16 T + T^{2} )^{2}$$
$83$ $$5000 + 200 T^{2} + T^{4}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( 10 + T )^{4}$$