# Properties

 Label 2736.2.k.m Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM discriminant -19 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: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} -\beta_{3} q^{7} + 5 \beta_{1} q^{11} + \beta_{2} q^{17} + \beta_{3} q^{19} + 4 \beta_{1} q^{23} + 14 q^{25} + 19 \beta_{1} q^{35} -3 \beta_{3} q^{43} -13 \beta_{1} q^{47} -12 q^{49} -5 \beta_{3} q^{55} -15 q^{61} + 11 q^{73} + 5 \beta_{2} q^{77} -16 \beta_{1} q^{83} -19 q^{85} -19 \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 56q^{25} - 48q^{49} - 60q^{61} + 44q^{73} - 76q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 14 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 7 \beta_{1}$$

## 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
 2.17945 + 0.500000i 2.17945 − 0.500000i −2.17945 − 0.500000i −2.17945 + 0.500000i
0 0 0 −4.35890 0 4.35890i 0 0 0
2431.2 0 0 0 −4.35890 0 4.35890i 0 0 0
2431.3 0 0 0 4.35890 0 4.35890i 0 0 0
2431.4 0 0 0 4.35890 0 4.35890i 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
4.b odd 2 1 inner
12.b even 2 1 inner
57.d even 2 1 inner
76.d even 2 1 inner
228.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.k.m 4
3.b odd 2 1 inner 2736.2.k.m 4
4.b odd 2 1 inner 2736.2.k.m 4
12.b even 2 1 inner 2736.2.k.m 4
19.b odd 2 1 CM 2736.2.k.m 4
57.d even 2 1 inner 2736.2.k.m 4
76.d even 2 1 inner 2736.2.k.m 4
228.b odd 2 1 inner 2736.2.k.m 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 19$$ $$T_{7}^{2} + 19$$ $$T_{11}^{2} + 25$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -19 + T^{2} )^{2}$$
$7$ $$( 19 + T^{2} )^{2}$$
$11$ $$( 25 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -19 + T^{2} )^{2}$$
$19$ $$( 19 + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 171 + T^{2} )^{2}$$
$47$ $$( 169 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 15 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -11 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( 256 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$