# Properties

 Label 2736.2.s.h Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.s (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 684) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -5 q^{7} +O(q^{10})$$ $$q -5 q^{7} -5 \zeta_{6} q^{13} + ( -3 - 2 \zeta_{6} ) q^{19} + 5 \zeta_{6} q^{25} + 7 q^{31} + 11 q^{37} + ( -13 + 13 \zeta_{6} ) q^{43} + 18 q^{49} + \zeta_{6} q^{61} + 11 \zeta_{6} q^{67} + ( -17 + 17 \zeta_{6} ) q^{73} + ( 17 - 17 \zeta_{6} ) q^{79} + 25 \zeta_{6} q^{91} + ( -14 + 14 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{7} + O(q^{10})$$ $$2q - 10q^{7} - 5q^{13} - 8q^{19} + 5q^{25} + 14q^{31} + 22q^{37} - 13q^{43} + 36q^{49} + q^{61} + 11q^{67} - 17q^{73} + 17q^{79} + 25q^{91} - 14q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −5.00000 0 0 0
1873.1 0 0 0 0 0 −5.00000 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.c even 3 1 inner
57.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.h 2
3.b odd 2 1 CM 2736.2.s.h 2
4.b odd 2 1 684.2.k.d 2
12.b even 2 1 684.2.k.d 2
19.c even 3 1 inner 2736.2.s.h 2
57.h odd 6 1 inner 2736.2.s.h 2
76.g odd 6 1 684.2.k.d 2
228.m even 6 1 684.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.k.d 2 4.b odd 2 1
684.2.k.d 2 12.b even 2 1
684.2.k.d 2 76.g odd 6 1
684.2.k.d 2 228.m even 6 1
2736.2.s.h 2 1.a even 1 1 trivial
2736.2.s.h 2 3.b odd 2 1 CM
2736.2.s.h 2 19.c even 3 1 inner
2736.2.s.h 2 57.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}$$ $$T_{7} + 5$$ $$T_{11}$$ $$T_{13}^{2} + 5 T_{13} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -7 + T )^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$169 + 13 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$121 - 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$289 + 17 T + T^{2}$$
$79$ $$289 - 17 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$196 + 14 T + T^{2}$$