# Properties

 Label 2736.2.k.l Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM discriminant -228 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-6}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 26 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q -\beta_{1} q^{11} + \beta_{2} q^{19} + 3 \beta_{1} q^{23} -5 q^{25} + \beta_{3} q^{29} + 2 \beta_{2} q^{31} -\beta_{3} q^{41} -5 \beta_{1} q^{47} + 7 q^{49} -\beta_{3} q^{53} + 4 q^{61} + 2 \beta_{2} q^{67} -8 q^{73} + 4 \beta_{2} q^{79} + 7 \beta_{1} q^{83} + \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 20q^{25} + 28q^{49} + 16q^{61} - 32q^{73} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 26 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/50$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 51 \nu$$$$)/50$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{2} - 13$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{3} + 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 51 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 −4.35890 + 2.44949i 4.35890 + 2.44949i −4.35890 − 2.44949i 4.35890 − 2.44949i
0 0 0 0 0 0 0 0 0
2431.2 0 0 0 0 0 0 0 0 0
2431.3 0 0 0 0 0 0 0 0 0
2431.4 0 0 0 0 0 0 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
228.b odd 2 1 CM by $$\Q(\sqrt{-57})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner
76.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.k.l 4
3.b odd 2 1 inner 2736.2.k.l 4
4.b odd 2 1 inner 2736.2.k.l 4
12.b even 2 1 inner 2736.2.k.l 4
19.b odd 2 1 inner 2736.2.k.l 4
57.d even 2 1 inner 2736.2.k.l 4
76.d even 2 1 inner 2736.2.k.l 4
228.b odd 2 1 CM 2736.2.k.l 4

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}$$ $$T_{11}^{2} + 6$$ $$T_{31}^{2} - 76$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 6 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$( -19 + T^{2} )^{2}$$
$23$ $$( 54 + T^{2} )^{2}$$
$29$ $$( 114 + T^{2} )^{2}$$
$31$ $$( -76 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 114 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 150 + T^{2} )^{2}$$
$53$ $$( 114 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -4 + T )^{4}$$
$67$ $$( -76 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 8 + T )^{4}$$
$79$ $$( -304 + T^{2} )^{2}$$
$83$ $$( 294 + T^{2} )^{2}$$
$89$ $$( 114 + T^{2} )^{2}$$
$97$ $$T^{4}$$