# Properties

 Label 1856.2.a.z Level $1856$ Weight $2$ Character orbit 1856.a Self dual yes Analytic conductor $14.820$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5 x^{2} + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 928) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{11} + ( 4 + \beta_{3} ) q^{13} + ( \beta_{1} - 4 \beta_{2} ) q^{15} + ( -3 + \beta_{3} ) q^{17} + 2 \beta_{2} q^{19} + ( 5 - \beta_{3} ) q^{21} + ( -\beta_{1} + \beta_{2} ) q^{23} + ( 4 - 4 \beta_{3} ) q^{25} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{27} - q^{29} + \beta_{1} q^{31} + ( -5 + 4 \beta_{3} ) q^{33} + ( \beta_{1} - 3 \beta_{2} ) q^{35} + ( 2 + 2 \beta_{3} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} ) q^{39} + ( -5 + \beta_{3} ) q^{41} + ( 4 \beta_{1} - \beta_{2} ) q^{43} + ( 14 - 6 \beta_{3} ) q^{45} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + 2 \beta_{3} ) q^{49} + ( -\beta_{1} + 5 \beta_{2} ) q^{51} + ( 6 + \beta_{3} ) q^{53} + ( -\beta_{1} + 6 \beta_{2} ) q^{55} + ( -10 + 4 \beta_{3} ) q^{57} + ( -5 \beta_{1} + \beta_{2} ) q^{59} + ( 5 + \beta_{3} ) q^{61} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{63} + ( 3 - 2 \beta_{3} ) q^{65} -4 \beta_{2} q^{67} + ( -5 + 3 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + 4 \beta_{3} ) q^{73} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{75} + ( 5 + 7 \beta_{3} ) q^{77} -3 \beta_{1} q^{79} + ( 9 - 2 \beta_{3} ) q^{81} + ( -\beta_{1} + 5 \beta_{2} ) q^{83} + ( -11 + 5 \beta_{3} ) q^{85} + \beta_{2} q^{87} + ( -5 - 3 \beta_{3} ) q^{89} + ( -7 \beta_{1} - 3 \beta_{2} ) q^{91} -\beta_{3} q^{93} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{95} + ( 3 - 3 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{5} + 8 q^{9} + O(q^{10})$$ $$4 q + 8 q^{5} + 8 q^{9} + 16 q^{13} - 12 q^{17} + 20 q^{21} + 16 q^{25} - 4 q^{29} - 20 q^{33} + 8 q^{37} - 20 q^{41} + 56 q^{45} + 12 q^{49} + 24 q^{53} - 40 q^{57} + 20 q^{61} + 12 q^{65} - 20 q^{69} + 16 q^{73} + 20 q^{77} + 36 q^{81} - 44 q^{85} - 20 q^{89} + 12 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.17557 1.90211 −1.90211 −1.17557
0 −3.07768 0 4.23607 0 −2.35114 0 6.47214 0
1.2 0 −0.726543 0 −0.236068 0 −3.80423 0 −2.47214 0
1.3 0 0.726543 0 −0.236068 0 3.80423 0 −2.47214 0
1.4 0 3.07768 0 4.23607 0 2.35114 0 6.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$29$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.a.z 4
4.b odd 2 1 inner 1856.2.a.z 4
8.b even 2 1 928.2.a.f 4
8.d odd 2 1 928.2.a.f 4
24.f even 2 1 8352.2.a.ba 4
24.h odd 2 1 8352.2.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.a.f 4 8.b even 2 1
928.2.a.f 4 8.d odd 2 1
1856.2.a.z 4 1.a even 1 1 trivial
1856.2.a.z 4 4.b odd 2 1 inner
8352.2.a.ba 4 24.f even 2 1
8352.2.a.ba 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{4} - 10 T_{3}^{2} + 5$$ $$T_{5}^{2} - 4 T_{5} - 1$$ $$T_{17}^{2} + 6 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$5 - 10 T^{2} + T^{4}$$
$5$ $$( -1 - 4 T + T^{2} )^{2}$$
$7$ $$80 - 20 T^{2} + T^{4}$$
$11$ $$605 - 50 T^{2} + T^{4}$$
$13$ $$( 11 - 8 T + T^{2} )^{2}$$
$17$ $$( 4 + 6 T + T^{2} )^{2}$$
$19$ $$80 - 40 T^{2} + T^{4}$$
$23$ $$80 - 20 T^{2} + T^{4}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$5 - 10 T^{2} + T^{4}$$
$37$ $$( -16 - 4 T + T^{2} )^{2}$$
$41$ $$( 20 + 10 T + T^{2} )^{2}$$
$43$ $$4805 - 170 T^{2} + T^{4}$$
$47$ $$605 - 50 T^{2} + T^{4}$$
$53$ $$( 31 - 12 T + T^{2} )^{2}$$
$59$ $$9680 - 260 T^{2} + T^{4}$$
$61$ $$( 20 - 10 T + T^{2} )^{2}$$
$67$ $$1280 - 160 T^{2} + T^{4}$$
$71$ $$2000 - 200 T^{2} + T^{4}$$
$73$ $$( -64 - 8 T + T^{2} )^{2}$$
$79$ $$405 - 90 T^{2} + T^{4}$$
$83$ $$80 - 260 T^{2} + T^{4}$$
$89$ $$( -20 + 10 T + T^{2} )^{2}$$
$97$ $$( -36 - 6 T + T^{2} )^{2}$$