# Properties

 Label 2736.2.s.s Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 456) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + ( -2 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + ( -2 - \beta_{3} ) q^{7} + ( -1 - \beta_{3} ) q^{11} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -4 - 4 \beta_{1} ) q^{17} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 + 3 \beta_{3} ) q^{31} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{35} + 7 q^{37} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{43} -2 \beta_{1} q^{47} + ( 2 + 4 \beta_{3} ) q^{49} + ( \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{53} + ( -4 - 4 \beta_{1} ) q^{55} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{59} + 11 \beta_{1} q^{61} + ( 9 - \beta_{3} ) q^{65} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} -6 \beta_{2} q^{71} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{3} ) q^{77} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{79} -8 q^{83} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 12 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{91} + ( -7 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} - 8q^{7} + O(q^{10})$$ $$4q - 2q^{5} - 8q^{7} - 4q^{11} + 2q^{13} - 8q^{17} + 8q^{19} + 2q^{23} - 2q^{25} - 12q^{29} + 8q^{31} - 6q^{35} + 28q^{37} + 4q^{41} - 8q^{43} + 4q^{47} + 8q^{49} - 2q^{53} - 8q^{55} + 6q^{59} - 22q^{61} + 36q^{65} - 4q^{67} + 2q^{73} + 28q^{77} + 4q^{79} - 32q^{83} - 8q^{85} - 6q^{89} - 24q^{91} - 34q^{95} - 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} + 6 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
0 0 0 −1.61803 2.80252i 0 0.236068 0 0 0
577.2 0 0 0 0.618034 + 1.07047i 0 −4.23607 0 0 0
1873.1 0 0 0 −1.61803 + 2.80252i 0 0.236068 0 0 0
1873.2 0 0 0 0.618034 1.07047i 0 −4.23607 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.s 4
3.b odd 2 1 912.2.q.j 4
4.b odd 2 1 1368.2.s.h 4
12.b even 2 1 456.2.q.d 4
19.c even 3 1 inner 2736.2.s.s 4
57.h odd 6 1 912.2.q.j 4
76.g odd 6 1 1368.2.s.h 4
228.m even 6 1 456.2.q.d 4
228.m even 6 1 8664.2.a.u 2
228.n odd 6 1 8664.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.d 4 12.b even 2 1
456.2.q.d 4 228.m even 6 1
912.2.q.j 4 3.b odd 2 1
912.2.q.j 4 57.h odd 6 1
1368.2.s.h 4 4.b odd 2 1
1368.2.s.h 4 76.g odd 6 1
2736.2.s.s 4 1.a even 1 1 trivial
2736.2.s.s 4 19.c even 3 1 inner
8664.2.a.q 2 228.n odd 6 1
8664.2.a.u 2 228.m even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 2 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 16$$ $$T_{7}^{2} + 4 T_{7} - 1$$ $$T_{11}^{2} + 2 T_{11} - 4$$ $$T_{13}^{4} - 2 T_{13}^{3} + 23 T_{13}^{2} + 38 T_{13} + 361$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$( -1 + 4 T + T^{2} )^{2}$$
$11$ $$( -4 + 2 T + T^{2} )^{2}$$
$13$ $$361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$( 16 + 4 T + T^{2} )^{2}$$
$19$ $$( 19 - 4 T + T^{2} )^{2}$$
$23$ $$16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$256 + 192 T + 128 T^{2} + 12 T^{3} + T^{4}$$
$31$ $$( -41 - 4 T + T^{2} )^{2}$$
$37$ $$( -7 + T )^{4}$$
$41$ $$256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$121 + 88 T + 53 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$15376 - 248 T + 128 T^{2} + 2 T^{3} + T^{4}$$
$59$ $$1296 + 216 T + 72 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$( 121 + 11 T + T^{2} )^{2}$$
$67$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$32400 + 180 T^{2} + T^{4}$$
$73$ $$361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4}$$
$83$ $$( 8 + T )^{4}$$
$89$ $$1296 - 216 T + 72 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$5776 - 304 T + 92 T^{2} + 4 T^{3} + T^{4}$$