# Properties

 Label 2736.2.k.p Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{5} q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} + ( -\beta_{6} + \beta_{7} ) q^{11} + \beta_{1} q^{17} + ( -\beta_{2} - 2 \beta_{3} ) q^{19} -\beta_{7} q^{23} + ( 1 - \beta_{4} ) q^{25} + ( \beta_{6} - \beta_{7} ) q^{35} + ( -\beta_{2} - 3 \beta_{3} ) q^{43} + ( \beta_{6} - 2 \beta_{7} ) q^{47} + ( -3 - 3 \beta_{4} ) q^{49} + ( -3 \beta_{2} + \beta_{3} ) q^{55} + ( 8 - \beta_{4} ) q^{61} + ( -4 - 3 \beta_{4} ) q^{73} + ( -3 \beta_{1} + 4 \beta_{5} ) q^{77} + 4 \beta_{7} q^{83} + ( 2 - 3 \beta_{4} ) q^{85} + ( -\beta_{6} - 2 \beta_{7} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{25} - 36q^{49} + 60q^{61} - 44q^{73} + 4q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 84 \nu^{5} + 356 \nu^{3} + 925 \nu$$$$)/500$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{6} - 56 \nu^{4} - 504 \nu^{2} - 1325$$$$)/700$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} + 63 \nu^{2} + 250$$$$)/175$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 100$$$$)/25$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} - 36 \nu^{3} - 45 \nu$$$$)/100$$ $$\beta_{6}$$ $$=$$ $$($$$$-23 \nu^{7} - 182 \nu^{5} - 938 \nu^{3} - 5525 \nu$$$$)/1750$$ $$\beta_{7}$$ $$=$$ $$($$$$-18 \nu^{7} - 112 \nu^{5} - 308 \nu^{3} - 1250 \nu$$$$)/875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 4 \beta_{6} + 3 \beta_{5} - \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 4 \beta_{2} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{7} - 14 \beta_{5} + 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{4} + 9 \beta_{3} + 20 \beta_{2} - 11$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-67 \beta_{7} + 44 \beta_{6} + 89 \beta_{5} + 45 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$-56 \beta_{3} - 28 \beta_{2} + 27$$ $$\nu^{7}$$ $$=$$ $$($$$$-281 \beta_{7} + 4 \beta_{6} - 283 \beta_{5} - 279 \beta_{1}$$$$)/8$$

## 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
 −0.656712 − 2.13746i −0.656712 + 2.13746i −1.52274 − 1.63746i −1.52274 + 1.63746i 1.52274 + 1.63746i 1.52274 − 1.63746i 0.656712 + 2.13746i 0.656712 − 2.13746i
0 0 0 −3.04547 0 0.418627i 0 0 0
2431.2 0 0 0 −3.04547 0 0.418627i 0 0 0
2431.3 0 0 0 −1.31342 0 4.77753i 0 0 0
2431.4 0 0 0 −1.31342 0 4.77753i 0 0 0
2431.5 0 0 0 1.31342 0 4.77753i 0 0 0
2431.6 0 0 0 1.31342 0 4.77753i 0 0 0
2431.7 0 0 0 3.04547 0 0.418627i 0 0 0
2431.8 0 0 0 3.04547 0 0.418627i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2431.8 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.p 8
3.b odd 2 1 inner 2736.2.k.p 8
4.b odd 2 1 inner 2736.2.k.p 8
12.b even 2 1 inner 2736.2.k.p 8
19.b odd 2 1 CM 2736.2.k.p 8
57.d even 2 1 inner 2736.2.k.p 8
76.d even 2 1 inner 2736.2.k.p 8
228.b odd 2 1 inner 2736.2.k.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.k.p 8 1.a even 1 1 trivial
2736.2.k.p 8 3.b odd 2 1 inner
2736.2.k.p 8 4.b odd 2 1 inner
2736.2.k.p 8 12.b even 2 1 inner
2736.2.k.p 8 19.b odd 2 1 CM
2736.2.k.p 8 57.d even 2 1 inner
2736.2.k.p 8 76.d even 2 1 inner
2736.2.k.p 8 228.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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{4} - 11 T_{5}^{2} + 16$$ $$T_{7}^{4} + 23 T_{7}^{2} + 4$$ $$T_{11}^{4} + 41 T_{11}^{2} + 64$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 16 - 11 T^{2} + T^{4} )^{2}$$
$7$ $$( 4 + 23 T^{2} + T^{4} )^{2}$$
$11$ $$( 64 + 41 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 1024 - 83 T^{2} + T^{4} )^{2}$$
$19$ $$( 19 + T^{2} )^{4}$$
$23$ $$( 16 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 1764 + 87 T^{2} + T^{4} )^{2}$$
$47$ $$( 784 + 113 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 42 - 15 T + T^{2} )^{4}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( -98 + 11 T + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$( 256 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$