# Properties

 Label 2736.2.k.j Level $2736$ Weight $2$ Character orbit 2736.k 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.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{7} + ( 2 \beta_{1} - \beta_{3} ) q^{11} + ( 3 + \beta_{2} ) q^{17} + ( \beta_{1} + 2 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{23} + ( 10 - \beta_{2} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{3} ) q^{35} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{43} + ( 6 \beta_{1} - \beta_{3} ) q^{47} + ( -6 + 3 \beta_{2} ) q^{49} -7 \beta_{3} q^{55} + ( -7 - \beta_{2} ) q^{61} + ( -7 + 3 \beta_{2} ) q^{73} + ( -7 + \beta_{2} ) q^{77} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 11 + 3 \beta_{2} ) q^{85} + ( -10 \beta_{1} - \beta_{3} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} + 14q^{17} + 38q^{25} - 18q^{49} - 30q^{61} - 22q^{73} - 26q^{77} + 50q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 15$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} - 2 \beta_{1} + 7$$

## 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
 −1.63746 + 1.52274i −1.63746 − 1.52274i 2.13746 − 0.656712i 2.13746 + 0.656712i
0 0 0 −4.27492 0 4.77753i 0 0 0
2431.2 0 0 0 −4.27492 0 4.77753i 0 0 0
2431.3 0 0 0 3.27492 0 0.418627i 0 0 0
2431.4 0 0 0 3.27492 0 0.418627i 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 $$\Q(\sqrt{-19})$$
4.b odd 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.j 4
3.b odd 2 1 304.2.h.c 4
4.b odd 2 1 inner 2736.2.k.j 4
12.b even 2 1 304.2.h.c 4
19.b odd 2 1 CM 2736.2.k.j 4
24.f even 2 1 1216.2.h.b 4
24.h odd 2 1 1216.2.h.b 4
57.d even 2 1 304.2.h.c 4
76.d even 2 1 inner 2736.2.k.j 4
228.b odd 2 1 304.2.h.c 4
456.l odd 2 1 1216.2.h.b 4
456.p even 2 1 1216.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.c 4 3.b odd 2 1
304.2.h.c 4 12.b even 2 1
304.2.h.c 4 57.d even 2 1
304.2.h.c 4 228.b odd 2 1
1216.2.h.b 4 24.f even 2 1
1216.2.h.b 4 24.h odd 2 1
1216.2.h.b 4 456.l odd 2 1
1216.2.h.b 4 456.p even 2 1
2736.2.k.j 4 1.a even 1 1 trivial
2736.2.k.j 4 4.b odd 2 1 inner
2736.2.k.j 4 19.b odd 2 1 CM
2736.2.k.j 4 76.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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{2} + T_{5} - 14$$ $$T_{7}^{4} + 23 T_{7}^{2} + 4$$ $$T_{11}^{4} + 47 T_{11}^{2} + 196$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -14 + T + T^{2} )^{2}$$
$7$ $$4 + 23 T^{2} + T^{4}$$
$11$ $$196 + 47 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -2 - 7 T + T^{2} )^{2}$$
$19$ $$( 19 + T^{2} )^{2}$$
$23$ $$( 76 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$1764 + 87 T^{2} + T^{4}$$
$47$ $$14884 + 263 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 42 + 15 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -98 + 11 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 76 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$