# Properties

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

# Learn more

## 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{5} + 5 q^{7} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{5} + 5 q^{7} -4 q^{11} -5 \zeta_{6} q^{13} + ( -5 + 2 \zeta_{6} ) q^{19} + 6 \zeta_{6} q^{23} + \zeta_{6} q^{25} + 8 \zeta_{6} q^{29} + q^{31} + ( -10 + 10 \zeta_{6} ) q^{35} + 7 q^{37} + ( -11 + 11 \zeta_{6} ) q^{43} -10 \zeta_{6} q^{47} + 18 q^{49} + 6 \zeta_{6} q^{53} + ( 8 - 8 \zeta_{6} ) q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + 10 q^{65} + 5 \zeta_{6} q^{67} + ( -6 + 6 \zeta_{6} ) q^{71} + ( -1 + \zeta_{6} ) q^{73} -20 q^{77} + ( -13 + 13 \zeta_{6} ) q^{79} -4 q^{83} -12 \zeta_{6} q^{89} -25 \zeta_{6} q^{91} + ( 6 - 10 \zeta_{6} ) q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 10q^{7} + O(q^{10})$$ $$2q - 2q^{5} + 10q^{7} - 8q^{11} - 5q^{13} - 8q^{19} + 6q^{23} + q^{25} + 8q^{29} + 2q^{31} - 10q^{35} + 14q^{37} - 11q^{43} - 10q^{47} + 36q^{49} + 6q^{53} + 8q^{55} - 8q^{59} + q^{61} + 20q^{65} + 5q^{67} - 6q^{71} - q^{73} - 40q^{77} - 13q^{79} - 8q^{83} - 12q^{89} - 25q^{91} + 2q^{95} - 2q^{97} + O(q^{100})$$

## 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 −1.00000 1.73205i 0 5.00000 0 0 0
1873.1 0 0 0 −1.00000 + 1.73205i 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
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.f 2
3.b odd 2 1 912.2.q.c 2
4.b odd 2 1 1368.2.s.b 2
12.b even 2 1 456.2.q.c 2
19.c even 3 1 inner 2736.2.s.f 2
57.h odd 6 1 912.2.q.c 2
76.g odd 6 1 1368.2.s.b 2
228.m even 6 1 456.2.q.c 2
228.m even 6 1 8664.2.a.b 1
228.n odd 6 1 8664.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.c 2 12.b even 2 1
456.2.q.c 2 228.m even 6 1
912.2.q.c 2 3.b odd 2 1
912.2.q.c 2 57.h odd 6 1
1368.2.s.b 2 4.b odd 2 1
1368.2.s.b 2 76.g odd 6 1
2736.2.s.f 2 1.a even 1 1 trivial
2736.2.s.f 2 19.c even 3 1 inner
8664.2.a.b 1 228.m even 6 1
8664.2.a.h 1 228.n odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{7} - 5$$ $$T_{11} + 4$$ $$T_{13}^{2} + 5 T_{13} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$( -5 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$64 - 8 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$100 + 10 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$1 + T + T^{2}$$
$79$ $$169 + 13 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$144 + 12 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$
show more
show less