Properties

 Label 2736.2.s.w 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} + ( 3 - \beta_{3} ) q^{11} -3 \beta_{1} q^{13} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 2 + 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{23} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + 4 \beta_{1} q^{29} -\beta_{3} q^{31} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{35} + ( -3 + 2 \beta_{3} ) q^{37} + ( -6 - 6 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 - 4 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{47} -2 q^{49} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 8 + 8 \beta_{1} + 4 \beta_{2} ) q^{55} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 3 - 3 \beta_{3} ) q^{65} + ( 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 6 + 6 \beta_{1} ) q^{71} + ( -11 - 11 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 5 - 3 \beta_{3} ) q^{77} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -10 - 2 \beta_{3} ) q^{83} + ( 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{91} + ( -3 + 7 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -6 - 6 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + O(q^{10})$$ $$4q + 2q^{5} + 12q^{11} + 6q^{13} + 4q^{17} + 8q^{19} - 2q^{23} - 2q^{25} - 8q^{29} + 10q^{35} - 12q^{37} - 12q^{41} - 8q^{43} + 4q^{47} - 8q^{49} + 6q^{53} + 16q^{55} + 2q^{59} + 6q^{61} + 12q^{65} - 12q^{67} + 12q^{71} - 22q^{73} + 20q^{77} - 8q^{79} - 40q^{83} - 24q^{85} - 6q^{89} - 26q^{95} - 12q^{97} + O(q^{100})$$

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
 −0.309017 − 0.535233i 0.809017 + 1.40126i −0.309017 + 0.535233i 0.809017 − 1.40126i
0 0 0 −0.618034 1.07047i 0 −2.23607 0 0 0
577.2 0 0 0 1.61803 + 2.80252i 0 2.23607 0 0 0
1873.1 0 0 0 −0.618034 + 1.07047i 0 −2.23607 0 0 0
1873.2 0 0 0 1.61803 2.80252i 0 2.23607 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.w 4
3.b odd 2 1 912.2.q.h 4
4.b odd 2 1 1368.2.s.i 4
12.b even 2 1 456.2.q.e 4
19.c even 3 1 inner 2736.2.s.w 4
57.h odd 6 1 912.2.q.h 4
76.g odd 6 1 1368.2.s.i 4
228.m even 6 1 456.2.q.e 4
228.m even 6 1 8664.2.a.s 2
228.n odd 6 1 8664.2.a.w 2

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

Hecke kernels

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

 $$T_{5}^{4} - 2 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 16$$ $$T_{7}^{2} - 5$$ $$T_{11}^{2} - 6 T_{11} + 4$$ $$T_{13}^{2} - 3 T_{13} + 9$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$( -5 + T^{2} )^{2}$$
$11$ $$( 4 - 6 T + T^{2} )^{2}$$
$13$ $$( 9 - 3 T + T^{2} )^{2}$$
$17$ $$256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$( 19 - 4 T + T^{2} )^{2}$$
$23$ $$1936 - 88 T + 48 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( 16 + 4 T + T^{2} )^{2}$$
$31$ $$( -5 + T^{2} )^{2}$$
$37$ $$( -11 + 6 T + T^{2} )^{2}$$
$41$ $$256 + 192 T + 128 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$5776 + 304 T + 92 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$1296 + 216 T + 72 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$1936 + 88 T + 48 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$81 - 108 T + 153 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$( 36 - 6 T + T^{2} )^{2}$$
$73$ $$10201 + 2222 T + 383 T^{2} + 22 T^{3} + T^{4}$$
$79$ $$121 + 88 T + 53 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$( 80 + 20 T + T^{2} )^{2}$$
$89$ $$16 + 24 T + 32 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 36 + 6 T + T^{2} )^{2}$$