# Properties

 Label 1856.1.h.d Level $1856$ Weight $1$ Character orbit 1856.h Self dual yes Analytic conductor $0.926$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -116 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.926264663447$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 464) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.53824.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + q^{5} + 2 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + q^{5} + 2 q^{9} + \beta q^{11} - q^{13} -\beta q^{15} -\beta q^{27} + q^{29} -\beta q^{31} -3 q^{33} + \beta q^{39} + \beta q^{43} + 2 q^{45} + \beta q^{47} + q^{49} - q^{53} + \beta q^{55} - q^{65} + \beta q^{79} + q^{81} -\beta q^{87} + 3 q^{93} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 4 q^{9} + O(q^{10})$$ $$2 q + 2 q^{5} + 4 q^{9} - 2 q^{13} + 2 q^{29} - 6 q^{33} + 4 q^{45} + 2 q^{49} - 2 q^{53} - 2 q^{65} + 2 q^{81} + 6 q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1855.1
 1.73205 −1.73205
0 −1.73205 0 1.00000 0 0 0 2.00000 0
1855.2 0 1.73205 0 1.00000 0 0 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
116.d odd 2 1 CM by $$\Q(\sqrt{-29})$$
4.b odd 2 1 inner
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.1.h.d 2
4.b odd 2 1 inner 1856.1.h.d 2
8.b even 2 1 464.1.h.b 2
8.d odd 2 1 464.1.h.b 2
29.b even 2 1 inner 1856.1.h.d 2
116.d odd 2 1 CM 1856.1.h.d 2
232.b odd 2 1 464.1.h.b 2
232.g even 2 1 464.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.1.h.b 2 8.b even 2 1
464.1.h.b 2 8.d odd 2 1
464.1.h.b 2 232.b odd 2 1
464.1.h.b 2 232.g even 2 1
1856.1.h.d 2 1.a even 1 1 trivial
1856.1.h.d 2 4.b odd 2 1 inner
1856.1.h.d 2 29.b even 2 1 inner
1856.1.h.d 2 116.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(1856, [\chi])$$.

## Hecke characteristic polynomials

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