# Properties

 Label 2736.2.a.bf Level $2736$ Weight $2$ Character orbit 2736.a Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 171) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} + \beta_{3} q^{11} + 2 q^{13} -\beta_{2} q^{17} - q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 2 - \beta_{1} ) q^{25} + ( \beta_{2} - \beta_{3} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( -\beta_{2} - 2 \beta_{3} ) q^{35} + ( 4 - 2 \beta_{1} ) q^{37} + ( -\beta_{2} - \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} ) q^{43} + ( 2 \beta_{2} - \beta_{3} ) q^{47} + ( 2 + \beta_{1} ) q^{49} + ( \beta_{2} + 3 \beta_{3} ) q^{53} + ( 3 + 3 \beta_{1} ) q^{55} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 + 3 \beta_{1} ) q^{61} -2 \beta_{2} q^{65} + 4 q^{67} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 5 + 3 \beta_{1} ) q^{73} + ( 3 \beta_{2} - 2 \beta_{3} ) q^{77} + 4 q^{79} + ( 3 \beta_{2} + \beta_{3} ) q^{83} + ( 7 - \beta_{1} ) q^{85} + ( \beta_{2} + 3 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + \beta_{2} q^{95} + ( -6 - 4 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7} + O(q^{10})$$ $$4 q - 2 q^{7} + 8 q^{13} - 4 q^{19} + 10 q^{25} + 4 q^{31} + 20 q^{37} + 10 q^{43} + 6 q^{49} + 6 q^{55} + 14 q^{61} + 16 q^{67} + 14 q^{73} + 16 q^{79} + 30 q^{85} - 4 q^{91} - 16 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} - x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 3 \nu^{2} - 9 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 5 \beta_{2} - 2 \beta_{1} + 12$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} + 4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.04374 −1.82405 −0.548230 0.328543
0 0 0 −3.22060 0 2.37228 0 0 0
1.2 0 0 0 −2.15121 0 −3.37228 0 0 0
1.3 0 0 0 2.15121 0 −3.37228 0 0 0
1.4 0 0 0 3.22060 0 2.37228 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.a.bf 4
3.b odd 2 1 inner 2736.2.a.bf 4
4.b odd 2 1 171.2.a.e 4
12.b even 2 1 171.2.a.e 4
20.d odd 2 1 4275.2.a.bp 4
28.d even 2 1 8379.2.a.bw 4
60.h even 2 1 4275.2.a.bp 4
76.d even 2 1 3249.2.a.bf 4
84.h odd 2 1 8379.2.a.bw 4
228.b odd 2 1 3249.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.a.e 4 4.b odd 2 1
171.2.a.e 4 12.b even 2 1
2736.2.a.bf 4 1.a even 1 1 trivial
2736.2.a.bf 4 3.b odd 2 1 inner
3249.2.a.bf 4 76.d even 2 1
3249.2.a.bf 4 228.b odd 2 1
4275.2.a.bp 4 20.d odd 2 1
4275.2.a.bp 4 60.h even 2 1
8379.2.a.bw 4 28.d even 2 1
8379.2.a.bw 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

 $$T_{5}^{4} - 15 T_{5}^{2} + 48$$ $$T_{7}^{2} + T_{7} - 8$$ $$T_{11}^{4} - 27 T_{11}^{2} + 108$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$48 - 15 T^{2} + T^{4}$$
$7$ $$( -8 + T + T^{2} )^{2}$$
$11$ $$108 - 27 T^{2} + T^{4}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$48 - 15 T^{2} + T^{4}$$
$19$ $$( 1 + T )^{4}$$
$23$ $$48 - 48 T^{2} + T^{4}$$
$29$ $$48 - 48 T^{2} + T^{4}$$
$31$ $$( -32 - 2 T + T^{2} )^{2}$$
$37$ $$( -8 - 10 T + T^{2} )^{2}$$
$41$ $$192 - 36 T^{2} + T^{4}$$
$43$ $$( -68 - 5 T + T^{2} )^{2}$$
$47$ $$1452 - 99 T^{2} + T^{4}$$
$53$ $$13872 - 240 T^{2} + T^{4}$$
$59$ $$3072 - 144 T^{2} + T^{4}$$
$61$ $$( -62 - 7 T + T^{2} )^{2}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$768 - 192 T^{2} + T^{4}$$
$73$ $$( -62 - 7 T + T^{2} )^{2}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$432 - 144 T^{2} + T^{4}$$
$89$ $$13872 - 240 T^{2} + T^{4}$$
$97$ $$( -116 + 8 T + T^{2} )^{2}$$