# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -6 + 3 \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + ( 3 + 2 \zeta_{6} ) q^{19} + ( -6 + 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( -10 + 5 \zeta_{6} ) q^{29} -4 q^{31} + ( 6 + 6 \zeta_{6} ) q^{35} + ( -5 - 5 \zeta_{6} ) q^{41} + ( -7 - 7 \zeta_{6} ) q^{43} + ( 2 - \zeta_{6} ) q^{47} -5 q^{49} + ( -2 + \zeta_{6} ) q^{53} + ( 6 + 6 \zeta_{6} ) q^{55} + ( 3 - 3 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} + ( -9 + 18 \zeta_{6} ) q^{65} -5 \zeta_{6} q^{67} + ( -9 + 9 \zeta_{6} ) q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} -12 q^{77} + ( -7 + 7 \zeta_{6} ) q^{79} + ( 2 - 4 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} + ( -10 + 5 \zeta_{6} ) q^{89} -18 \zeta_{6} q^{91} + ( 15 - 9 \zeta_{6} ) q^{95} + ( 5 + 5 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{5} + O(q^{10})$$ $$2q + 3q^{5} - 9q^{13} + 3q^{17} + 8q^{19} - 9q^{23} - 4q^{25} - 15q^{29} - 8q^{31} + 18q^{35} - 15q^{41} - 21q^{43} + 3q^{47} - 10q^{49} - 3q^{53} + 18q^{55} + 3q^{59} - 7q^{61} - 5q^{67} - 9q^{71} - 7q^{73} - 24q^{77} - 7q^{79} - 9q^{85} - 15q^{89} - 18q^{91} + 21q^{95} + 15q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.50000 2.59808i 0 3.46410i 0 0 0
1855.1 0 0 0 1.50000 + 2.59808i 0 3.46410i 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
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.h 2
3.b odd 2 1 304.2.n.b yes 2
4.b odd 2 1 2736.2.bm.g 2
12.b even 2 1 304.2.n.a 2
19.d odd 6 1 2736.2.bm.g 2
24.f even 2 1 1216.2.n.b 2
24.h odd 2 1 1216.2.n.a 2
57.f even 6 1 304.2.n.a 2
76.f even 6 1 inner 2736.2.bm.h 2
228.n odd 6 1 304.2.n.b yes 2
456.s odd 6 1 1216.2.n.a 2
456.v even 6 1 1216.2.n.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.a 2 12.b even 2 1
304.2.n.a 2 57.f even 6 1
304.2.n.b yes 2 3.b odd 2 1
304.2.n.b yes 2 228.n odd 6 1
1216.2.n.a 2 24.h odd 2 1
1216.2.n.a 2 456.s odd 6 1
1216.2.n.b 2 24.f even 2 1
1216.2.n.b 2 456.v even 6 1
2736.2.bm.g 2 4.b odd 2 1
2736.2.bm.g 2 19.d odd 6 1
2736.2.bm.h 2 1.a even 1 1 trivial
2736.2.bm.h 2 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{7}^{2} + 12$$ $$T_{11}^{2} + 12$$ $$T_{23}^{2} + 9 T_{23} + 27$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$27 + 9 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$27 + 9 T + T^{2}$$
$29$ $$75 + 15 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$T^{2}$$
$41$ $$75 + 15 T + T^{2}$$
$43$ $$147 + 21 T + T^{2}$$
$47$ $$3 - 3 T + T^{2}$$
$53$ $$3 + 3 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$81 + 9 T + T^{2}$$
$73$ $$49 + 7 T + T^{2}$$
$79$ $$49 + 7 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$75 + 15 T + T^{2}$$
$97$ $$75 - 15 T + T^{2}$$
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