Properties

 Label 2736.2.s.k Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - q^{7} +O(q^{10})$$ $$q - q^{7} -2 q^{11} + 3 \zeta_{6} q^{13} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -3 - 2 \zeta_{6} ) q^{19} -4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} + 3 q^{31} -5 q^{37} + ( 4 - 4 \zeta_{6} ) q^{41} + ( -9 + 9 \zeta_{6} ) q^{43} -10 \zeta_{6} q^{47} -6 q^{49} -4 \zeta_{6} q^{53} + ( 14 - 14 \zeta_{6} ) q^{59} -11 \zeta_{6} q^{61} + 3 \zeta_{6} q^{67} + ( -14 + 14 \zeta_{6} ) q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + 2 q^{77} + ( 1 - \zeta_{6} ) q^{79} + 8 q^{83} -14 \zeta_{6} q^{89} -3 \zeta_{6} q^{91} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} - 4q^{11} + 3q^{13} + 4q^{17} - 8q^{19} - 4q^{23} + 5q^{25} + 6q^{31} - 10q^{37} + 4q^{41} - 9q^{43} - 10q^{47} - 12q^{49} - 4q^{53} + 14q^{59} - 11q^{61} + 3q^{67} - 14q^{71} + 11q^{73} + 4q^{77} + q^{79} + 16q^{83} - 14q^{89} - 3q^{91} - 2q^{97} + O(q^{100})$$

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 −1.00000 0 0 0
1873.1 0 0 0 0 0 −1.00000 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
19.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.k 2
3.b odd 2 1 912.2.q.b 2
4.b odd 2 1 342.2.g.c 2
12.b even 2 1 114.2.e.b 2
19.c even 3 1 inner 2736.2.s.k 2
57.h odd 6 1 912.2.q.b 2
76.f even 6 1 6498.2.a.g 1
76.g odd 6 1 342.2.g.c 2
76.g odd 6 1 6498.2.a.u 1
228.m even 6 1 114.2.e.b 2
228.m even 6 1 2166.2.a.b 1
228.n odd 6 1 2166.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 12.b even 2 1
114.2.e.b 2 228.m even 6 1
342.2.g.c 2 4.b odd 2 1
342.2.g.c 2 76.g odd 6 1
912.2.q.b 2 3.b odd 2 1
912.2.q.b 2 57.h odd 6 1
2166.2.a.b 1 228.m even 6 1
2166.2.a.h 1 228.n odd 6 1
2736.2.s.k 2 1.a even 1 1 trivial
2736.2.s.k 2 19.c even 3 1 inner
6498.2.a.g 1 76.f even 6 1
6498.2.a.u 1 76.g odd 6 1

Hecke kernels

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

 $$T_{5}$$ $$T_{7} + 1$$ $$T_{11} + 2$$ $$T_{13}^{2} - 3 T_{13} + 9$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$9 - 3 T + T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$16 - 4 T + T^{2}$$
$43$ $$81 + 9 T + T^{2}$$
$47$ $$100 + 10 T + T^{2}$$
$53$ $$16 + 4 T + T^{2}$$
$59$ $$196 - 14 T + T^{2}$$
$61$ $$121 + 11 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$196 + 14 T + T^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$196 + 14 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$