# Properties

 Label 2736.2.f.e Level $2736$ Weight $2$ Character orbit 2736.f Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{19})$$ Defining polynomial: $$x^{4} + 20 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 171) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -\beta_{3} q^{7} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{3} q^{19} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + ( -5 + \beta_{3} ) q^{25} + ( \beta_{1} - 3 \beta_{2} ) q^{35} + q^{43} + ( -\beta_{1} + 2 \beta_{2} ) q^{47} + 12 q^{49} + ( 7 + 2 \beta_{3} ) q^{55} + \beta_{3} q^{61} + 11 q^{73} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{77} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -13 + 4 \beta_{3} ) q^{85} + ( -\beta_{1} + 3 \beta_{2} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 20q^{25} + 4q^{43} + 48q^{49} + 28q^{55} + 44q^{73} - 52q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 20 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 10$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} - 11 \beta_{1}$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 3.78931i − 2.37510i 2.37510i 3.78931i
0 0 0 3.78931i 0 4.35890 0 0 0
1025.2 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.3 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.4 0 0 0 3.78931i 0 4.35890 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.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
3.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.f.e 4
3.b odd 2 1 inner 2736.2.f.e 4
4.b odd 2 1 171.2.d.a 4
12.b even 2 1 171.2.d.a 4
19.b odd 2 1 CM 2736.2.f.e 4
57.d even 2 1 inner 2736.2.f.e 4
76.d even 2 1 171.2.d.a 4
228.b odd 2 1 171.2.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.d.a 4 4.b odd 2 1
171.2.d.a 4 12.b even 2 1
171.2.d.a 4 76.d even 2 1
171.2.d.a 4 228.b odd 2 1
2736.2.f.e 4 1.a even 1 1 trivial
2736.2.f.e 4 3.b odd 2 1 inner
2736.2.f.e 4 19.b odd 2 1 CM
2736.2.f.e 4 57.d even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 20 T_{5}^{2} + 81$$ $$T_{7}^{2} - 19$$ $$T_{11}^{4} + 44 T_{11}^{2} + 9$$ $$T_{29}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$81 + 20 T^{2} + T^{4}$$
$7$ $$( -19 + T^{2} )^{2}$$
$11$ $$9 + 44 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$225 + 68 T^{2} + T^{4}$$
$19$ $$( -19 + T^{2} )^{2}$$
$23$ $$900 + 92 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$( -1 + T )^{4}$$
$47$ $$5625 + 188 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -19 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -11 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$8100 + 332 T^{2} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$