# Properties

 Label 1856.2.e.h Level $1856$ Weight $2$ Character orbit 1856.e Analytic conductor $14.820$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 928) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + 2 q^{9} + 3 \beta_{1} q^{11} + q^{13} -\beta_{1} q^{15} + \beta_{2} q^{17} -4 \beta_{1} q^{19} + \beta_{2} q^{21} + \beta_{3} q^{23} -4 q^{25} + 5 \beta_{1} q^{27} + ( 3 - \beta_{2} ) q^{29} -5 \beta_{1} q^{31} -3 q^{33} -\beta_{3} q^{35} + \beta_{1} q^{39} + \beta_{2} q^{41} -\beta_{1} q^{43} -2 q^{45} -3 \beta_{1} q^{47} + 13 q^{49} -\beta_{3} q^{51} + 9 q^{53} -3 \beta_{1} q^{55} + 4 q^{57} -\beta_{3} q^{59} + 3 \beta_{2} q^{61} + 2 \beta_{3} q^{63} - q^{65} -2 \beta_{3} q^{67} + \beta_{2} q^{69} -2 \beta_{3} q^{71} -2 \beta_{2} q^{73} -4 \beta_{1} q^{75} + 3 \beta_{2} q^{77} -15 \beta_{1} q^{79} + q^{81} + \beta_{3} q^{83} -\beta_{2} q^{85} + ( 3 \beta_{1} + \beta_{3} ) q^{87} + 3 \beta_{2} q^{89} + \beta_{3} q^{91} + 5 q^{93} + 4 \beta_{1} q^{95} + 3 \beta_{2} q^{97} + 6 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} + 8 q^{9} + O(q^{10})$$ $$4 q - 4 q^{5} + 8 q^{9} + 4 q^{13} - 16 q^{25} + 12 q^{29} - 12 q^{33} - 8 q^{45} + 52 q^{49} + 36 q^{53} + 16 q^{57} - 4 q^{65} + 4 q^{81} + 20 q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 4 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$581$$ $$639$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1217.1
 1.61803i − 0.618034i − 1.61803i 0.618034i
0 1.00000i 0 −1.00000 0 −4.47214 0 2.00000 0
1217.2 0 1.00000i 0 −1.00000 0 4.47214 0 2.00000 0
1217.3 0 1.00000i 0 −1.00000 0 −4.47214 0 2.00000 0
1217.4 0 1.00000i 0 −1.00000 0 4.47214 0 2.00000 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
4.b odd 2 1 inner
29.b even 2 1 inner
116.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.e.h 4
4.b odd 2 1 inner 1856.2.e.h 4
8.b even 2 1 928.2.e.b 4
8.d odd 2 1 928.2.e.b 4
29.b even 2 1 inner 1856.2.e.h 4
116.d odd 2 1 inner 1856.2.e.h 4
232.b odd 2 1 928.2.e.b 4
232.g even 2 1 928.2.e.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.e.b 4 8.b even 2 1
928.2.e.b 4 8.d odd 2 1
928.2.e.b 4 232.b odd 2 1
928.2.e.b 4 232.g even 2 1
1856.2.e.h 4 1.a even 1 1 trivial
1856.2.e.h 4 4.b odd 2 1 inner
1856.2.e.h 4 29.b even 2 1 inner
1856.2.e.h 4 116.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1856, [\chi])$$:

 $$T_{3}^{2} + 1$$ $$T_{7}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$( -20 + T^{2} )^{2}$$
$11$ $$( 9 + T^{2} )^{2}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$( 29 - 6 T + T^{2} )^{2}$$
$31$ $$( 25 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( 1 + T^{2} )^{2}$$
$47$ $$( 9 + T^{2} )^{2}$$
$53$ $$( -9 + T )^{4}$$
$59$ $$( -20 + T^{2} )^{2}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( -80 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( 80 + T^{2} )^{2}$$
$79$ $$( 225 + T^{2} )^{2}$$
$83$ $$( -20 + T^{2} )^{2}$$
$89$ $$( 180 + T^{2} )^{2}$$
$97$ $$( 180 + T^{2} )^{2}$$