# Properties

 Label 2736.2.s.y Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.2696112.1 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + ( 3 + 2 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{23} + ( -\beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{29} + ( 3 + 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -4 + 4 \beta_{4} - 2 \beta_{5} ) q^{35} + ( -5 - 3 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -7 + 2 \beta_{1} + 7 \beta_{4} - 2 \beta_{5} ) q^{41} + ( 3 - 4 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{43} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} + ( 3 + \beta_{1} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( -4 - \beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{59} + ( -3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{61} + ( -7 - 3 \beta_{3} ) q^{65} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( 1 - \beta_{4} - \beta_{5} ) q^{71} + ( 3 + 2 \beta_{1} - 3 \beta_{4} ) q^{73} + ( -3 + \beta_{2} - 3 \beta_{3} ) q^{77} + ( -7 + 2 \beta_{1} + 7 \beta_{4} + \beta_{5} ) q^{79} + ( 5 + 3 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{85} + ( 6 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 14 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 4 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{5} + 4q^{7} + O(q^{10})$$ $$6q + q^{5} + 4q^{7} + 8q^{11} + q^{13} + 11q^{17} - q^{23} + 6q^{25} - 3q^{29} + 12q^{31} - 12q^{35} - 24q^{37} - 19q^{41} + 5q^{43} - 17q^{47} + 22q^{49} - 5q^{53} + 10q^{55} - 13q^{59} + 3q^{61} - 42q^{65} - 9q^{67} + 3q^{71} + 11q^{73} - 20q^{77} - 19q^{79} + 24q^{83} - 17q^{85} + 3q^{89} + 44q^{91} + 13q^{95} - q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} + 8 \nu - 40$$$$)/82$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 10 \nu^{4} + 9 \nu^{3} - 36 \nu^{2} + 16 \nu - 121$$$$)/41$$ $$\beta_{4}$$ $$=$$ $$($$$$-10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72$$$$)/82$$ $$\beta_{5}$$ $$=$$ $$($$$$-26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 454 \nu - 26$$$$)/82$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} + 13 \beta_{4} - 2 \beta_{1} - 13$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 18 \beta_{2} - 18 \beta_{1}$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$1$$ $$1$$

## 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
 −0.906803 − 1.57063i 0.235342 + 0.407624i 1.17146 + 2.02903i −0.906803 + 1.57063i 0.235342 − 0.407624i 1.17146 − 2.02903i
0 0 0 −0.906803 1.57063i 0 2.52444 0 0 0
577.2 0 0 0 0.235342 + 0.407624i 0 3.30777 0 0 0
577.3 0 0 0 1.17146 + 2.02903i 0 −3.83221 0 0 0
1873.1 0 0 0 −0.906803 + 1.57063i 0 2.52444 0 0 0
1873.2 0 0 0 0.235342 0.407624i 0 3.30777 0 0 0
1873.3 0 0 0 1.17146 2.02903i 0 −3.83221 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1873.3 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.y 6
3.b odd 2 1 304.2.i.f 6
4.b odd 2 1 1368.2.s.k 6
12.b even 2 1 152.2.i.c 6
19.c even 3 1 inner 2736.2.s.y 6
24.f even 2 1 1216.2.i.n 6
24.h odd 2 1 1216.2.i.m 6
57.f even 6 1 5776.2.a.bq 3
57.h odd 6 1 304.2.i.f 6
57.h odd 6 1 5776.2.a.bk 3
76.g odd 6 1 1368.2.s.k 6
228.m even 6 1 152.2.i.c 6
228.m even 6 1 2888.2.a.r 3
228.n odd 6 1 2888.2.a.n 3
456.u even 6 1 1216.2.i.n 6
456.x odd 6 1 1216.2.i.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.c 6 12.b even 2 1
152.2.i.c 6 228.m even 6 1
304.2.i.f 6 3.b odd 2 1
304.2.i.f 6 57.h odd 6 1
1216.2.i.m 6 24.h odd 2 1
1216.2.i.m 6 456.x odd 6 1
1216.2.i.n 6 24.f even 2 1
1216.2.i.n 6 456.u even 6 1
1368.2.s.k 6 4.b odd 2 1
1368.2.s.k 6 76.g odd 6 1
2736.2.s.y 6 1.a even 1 1 trivial
2736.2.s.y 6 19.c even 3 1 inner
2888.2.a.n 3 228.n odd 6 1
2888.2.a.r 3 228.m even 6 1
5776.2.a.bk 3 57.h odd 6 1
5776.2.a.bq 3 57.f even 6 1

## Hecke kernels

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

 $$T_{5}^{6} - T_{5}^{5} + 5 T_{5}^{4} + 18 T_{5}^{2} - 8 T_{5} + 4$$ $$T_{7}^{3} - 2 T_{7}^{2} - 14 T_{7} + 32$$ $$T_{11}^{3} - 4 T_{11}^{2} + T_{11} + 4$$ $$T_{13}^{6} - T_{13}^{5} + 33 T_{13}^{4} - 120 T_{13}^{3} + 1100 T_{13}^{2} - 2432 T_{13} + 5776$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$4 - 8 T + 18 T^{2} + 5 T^{4} - T^{5} + T^{6}$$
$7$ $$( 32 - 14 T - 2 T^{2} + T^{3} )^{2}$$
$11$ $$( 4 + T - 4 T^{2} + T^{3} )^{2}$$
$13$ $$5776 - 2432 T + 1100 T^{2} - 120 T^{3} + 33 T^{4} - T^{5} + T^{6}$$
$17$ $$1024 + 768 T + 928 T^{2} - 328 T^{3} + 97 T^{4} - 11 T^{5} + T^{6}$$
$19$ $$6859 + 342 T^{2} + 16 T^{3} + 18 T^{4} + T^{6}$$
$23$ $$4 + 8 T + 18 T^{2} + 5 T^{4} + T^{5} + T^{6}$$
$29$ $$4 - 80 T + 1594 T^{2} - 124 T^{3} + 49 T^{4} + 3 T^{5} + T^{6}$$
$31$ $$( 216 - 54 T - 6 T^{2} + T^{3} )^{2}$$
$37$ $$( -292 - 18 T + 12 T^{2} + T^{3} )^{2}$$
$41$ $$1681 + 3731 T + 7502 T^{2} + 1647 T^{3} + 270 T^{4} + 19 T^{5} + T^{6}$$
$43$ $$118336 - 28896 T + 8776 T^{2} - 268 T^{3} + 109 T^{4} - 5 T^{5} + T^{6}$$
$47$ $$521284 + 12274 T^{2} + 1444 T^{3} + 289 T^{4} + 17 T^{5} + T^{6}$$
$53$ $$1936 + 352 T + 284 T^{2} + 48 T^{3} + 33 T^{4} + 5 T^{5} + T^{6}$$
$59$ $$2809 + 2597 T + 1712 T^{2} + 531 T^{3} + 120 T^{4} + 13 T^{5} + T^{6}$$
$61$ $$58564 + 24200 T + 9274 T^{2} + 784 T^{3} + 109 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$529 + 713 T + 1168 T^{2} - 233 T^{3} + 112 T^{4} + 9 T^{5} + T^{6}$$
$71$ $$16 - 16 T + 28 T^{2} + 4 T^{3} + 13 T^{4} - 3 T^{5} + T^{6}$$
$73$ $$361 + 437 T + 738 T^{2} - 291 T^{3} + 98 T^{4} - 11 T^{5} + T^{6}$$
$79$ $$256 - 1408 T + 8048 T^{2} + 1704 T^{3} + 273 T^{4} + 19 T^{5} + T^{6}$$
$83$ $$( 632 - 43 T - 12 T^{2} + T^{3} )^{2}$$
$89$ $$295936 - 100096 T + 35488 T^{2} - 536 T^{3} + 193 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$1 + 17 T + 290 T^{2} - 15 T^{3} + 18 T^{4} + T^{5} + T^{6}$$