# Properties

 Label 2736.2.s.t 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 228) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} + \beta_{3} q^{11} + ( 1 - \beta_{1} ) q^{13} -2 \beta_{2} q^{17} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{19} + ( 6 - 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{1} ) q^{25} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 7 - \beta_{3} ) q^{31} + ( 6 \beta_{1} - \beta_{2} ) q^{35} + ( -1 - 4 \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} -2 \beta_{3} q^{49} + ( -\beta_{2} + \beta_{3} ) q^{53} -6 \beta_{1} q^{55} -3 \beta_{2} q^{59} + ( 7 - 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} -\beta_{3} q^{65} + ( -7 + 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} -6 \beta_{1} q^{71} + ( 7 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -6 + \beta_{3} ) q^{77} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -12 + 2 \beta_{3} ) q^{83} + ( -12 + 12 \beta_{1} ) q^{85} + ( -6 + 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 12 - 6 \beta_{1} + \beta_{2} ) q^{95} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 2q^{13} - 4q^{19} + 12q^{23} - 2q^{25} + 28q^{31} + 12q^{35} - 4q^{37} - 2q^{43} - 12q^{47} - 12q^{55} + 14q^{61} - 14q^{67} - 12q^{71} + 14q^{73} - 24q^{77} + 10q^{79} - 48q^{83} - 24q^{85} - 12q^{89} + 2q^{91} + 36q^{95} + 8q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 −1.44949 0 0 0
577.2 0 0 0 1.22474 + 2.12132i 0 3.44949 0 0 0
1873.1 0 0 0 −1.22474 + 2.12132i 0 −1.44949 0 0 0
1873.2 0 0 0 1.22474 2.12132i 0 3.44949 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.t 4
3.b odd 2 1 912.2.q.i 4
4.b odd 2 1 684.2.k.f 4
12.b even 2 1 228.2.i.a 4
19.c even 3 1 inner 2736.2.s.t 4
57.h odd 6 1 912.2.q.i 4
76.g odd 6 1 684.2.k.f 4
228.m even 6 1 228.2.i.a 4
228.m even 6 1 4332.2.a.l 2
228.n odd 6 1 4332.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.i.a 4 12.b even 2 1
228.2.i.a 4 228.m even 6 1
684.2.k.f 4 4.b odd 2 1
684.2.k.f 4 76.g odd 6 1
912.2.q.i 4 3.b odd 2 1
912.2.q.i 4 57.h odd 6 1
2736.2.s.t 4 1.a even 1 1 trivial
2736.2.s.t 4 19.c even 3 1 inner
4332.2.a.g 2 228.n odd 6 1
4332.2.a.l 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{4} + 6 T_{5}^{2} + 36$$ $$T_{7}^{2} - 2 T_{7} - 5$$ $$T_{11}^{2} - 6$$ $$T_{13}^{2} - T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 6 T^{2} + T^{4}$$
$7$ $$( -5 - 2 T + T^{2} )^{2}$$
$11$ $$( -6 + T^{2} )^{2}$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$576 + 24 T^{2} + T^{4}$$
$19$ $$( 19 + 2 T + T^{2} )^{2}$$
$23$ $$900 - 360 T + 114 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$576 + 24 T^{2} + T^{4}$$
$31$ $$( 43 - 14 T + T^{2} )^{2}$$
$37$ $$( -95 + 2 T + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$25 - 10 T + 9 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$36 + 6 T^{2} + T^{4}$$
$59$ $$2916 + 54 T^{2} + T^{4}$$
$61$ $$625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$1849 + 602 T + 153 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{2}$$
$73$ $$625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4}$$
$83$ $$( 120 + 24 T + T^{2} )^{2}$$
$89$ $$900 + 360 T + 114 T^{2} + 12 T^{3} + T^{4}$$
$97$ $$40000 + 1600 T + 264 T^{2} - 8 T^{3} + T^{4}$$