# Properties

 Label 1856.2.a.q Level $1856$ Weight $2$ Character orbit 1856.a Self dual yes Analytic conductor $14.820$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
0 −2.41421 0 −1.00000 0 2.00000 0 2.82843 0
1.2 0 0.414214 0 −1.00000 0 2.00000 0 −2.82843 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.a.q 2
4.b odd 2 1 1856.2.a.u 2
8.b even 2 1 928.2.a.e yes 2
8.d odd 2 1 928.2.a.c 2
24.f even 2 1 8352.2.a.l 2
24.h odd 2 1 8352.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.a.c 2 8.d odd 2 1
928.2.a.e yes 2 8.b even 2 1
1856.2.a.q 2 1.a even 1 1 trivial
1856.2.a.u 2 4.b odd 2 1
8352.2.a.l 2 24.f even 2 1
8352.2.a.o 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{5} + 1$$ $$T_{17}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$-1 + 2 T + T^{2}$$
$13$ $$-7 + 2 T + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$-28 + 4 T + T^{2}$$
$23$ $$8 - 8 T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$-1 - 2 T + T^{2}$$
$37$ $$-32 + T^{2}$$
$41$ $$-4 + 4 T + T^{2}$$
$43$ $$-17 + 2 T + T^{2}$$
$47$ $$7 - 6 T + T^{2}$$
$53$ $$-63 + 6 T + T^{2}$$
$59$ $$-56 + 8 T + T^{2}$$
$61$ $$-56 + 8 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$28 - 12 T + T^{2}$$
$73$ $$( 8 + T )^{2}$$
$79$ $$-97 - 2 T + T^{2}$$
$83$ $$-56 + 8 T + T^{2}$$
$89$ $$28 + 12 T + T^{2}$$
$97$ $$28 + 12 T + T^{2}$$