# Properties

 Label 2736.2.bm.k Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{5} -2 \zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{5} -2 \zeta_{12}^{3} q^{7} + 2 \zeta_{12}^{3} q^{11} + ( -2 + \zeta_{12}^{2} ) q^{13} + ( 1 - \zeta_{12}^{2} ) q^{17} + ( 2 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{19} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{2} q^{25} + ( 2 - \zeta_{12}^{2} ) q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + 2 \zeta_{12} q^{35} + ( 2 - 4 \zeta_{12}^{2} ) q^{37} + ( -3 - 3 \zeta_{12}^{2} ) q^{41} + 9 \zeta_{12} q^{43} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{47} + 3 q^{49} + ( -10 + 5 \zeta_{12}^{2} ) q^{53} -2 \zeta_{12} q^{55} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{59} + 9 \zeta_{12}^{2} q^{61} + ( 1 - 2 \zeta_{12}^{2} ) q^{65} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{67} + ( 9 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{71} + ( 1 - \zeta_{12}^{2} ) q^{73} + 4 q^{77} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} + 8 \zeta_{12}^{3} q^{83} + \zeta_{12}^{2} q^{85} + ( 18 - 9 \zeta_{12}^{2} ) q^{89} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} + ( 3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{95} + ( -9 - 9 \zeta_{12}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} - 6q^{13} + 2q^{17} + 8q^{25} + 6q^{29} - 18q^{41} + 12q^{49} - 30q^{53} + 18q^{61} + 2q^{73} + 16q^{77} + 2q^{85} + 54q^{89} - 54q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 −0.500000 + 0.866025i 0 2.00000i 0 0 0
559.2 0 0 0 −0.500000 + 0.866025i 0 2.00000i 0 0 0
1855.1 0 0 0 −0.500000 0.866025i 0 2.00000i 0 0 0
1855.2 0 0 0 −0.500000 0.866025i 0 2.00000i 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
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.k 4
3.b odd 2 1 304.2.n.c 4
4.b odd 2 1 inner 2736.2.bm.k 4
12.b even 2 1 304.2.n.c 4
19.d odd 6 1 inner 2736.2.bm.k 4
24.f even 2 1 1216.2.n.c 4
24.h odd 2 1 1216.2.n.c 4
57.f even 6 1 304.2.n.c 4
76.f even 6 1 inner 2736.2.bm.k 4
228.n odd 6 1 304.2.n.c 4
456.s odd 6 1 1216.2.n.c 4
456.v even 6 1 1216.2.n.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.c 4 3.b odd 2 1
304.2.n.c 4 12.b even 2 1
304.2.n.c 4 57.f even 6 1
304.2.n.c 4 228.n odd 6 1
1216.2.n.c 4 24.f even 2 1
1216.2.n.c 4 24.h odd 2 1
1216.2.n.c 4 456.s odd 6 1
1216.2.n.c 4 456.v even 6 1
2736.2.bm.k 4 1.a even 1 1 trivial
2736.2.bm.k 4 4.b odd 2 1 inner
2736.2.bm.k 4 19.d odd 6 1 inner
2736.2.bm.k 4 76.f even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 4$$ $$T_{23}^{4} - T_{23}^{2} + 1$$ $$T_{31}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 3 + 3 T + T^{2} )^{2}$$
$17$ $$( 1 - T + T^{2} )^{2}$$
$19$ $$361 + 26 T^{2} + T^{4}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( 3 - 3 T + T^{2} )^{2}$$
$31$ $$( -12 + T^{2} )^{2}$$
$37$ $$( 12 + T^{2} )^{2}$$
$41$ $$( 27 + 9 T + T^{2} )^{2}$$
$43$ $$6561 - 81 T^{2} + T^{4}$$
$47$ $$14641 - 121 T^{2} + T^{4}$$
$53$ $$( 75 + 15 T + T^{2} )^{2}$$
$59$ $$5625 + 75 T^{2} + T^{4}$$
$61$ $$( 81 - 9 T + T^{2} )^{2}$$
$67$ $$729 + 27 T^{2} + T^{4}$$
$71$ $$59049 + 243 T^{2} + T^{4}$$
$73$ $$( 1 - T + T^{2} )^{2}$$
$79$ $$9 + 3 T^{2} + T^{4}$$
$83$ $$( 64 + T^{2} )^{2}$$
$89$ $$( 243 - 27 T + T^{2} )^{2}$$
$97$ $$( 243 + 27 T + T^{2} )^{2}$$