# Properties

 Label 2736.2.bm.m Level $2736$ Weight $2$ Character orbit 2736.bm 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.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.31726512.1 Defining polynomial: $$x^{6} - x^{5} + 10 x^{4} + 3 x^{3} + 84 x^{2} - 27 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 4 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{17} + ( 3 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{19} -\beta_{4} q^{23} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{31} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{35} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{37} + ( \beta_{1} - 2 \beta_{2} - 3 \beta_{5} ) q^{41} + ( 2 - 2 \beta_{3} ) q^{43} + ( 4 + 2 \beta_{3} + \beta_{4} ) q^{47} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{49} + ( 8 + 4 \beta_{3} ) q^{53} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{55} + ( -9 - 9 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{59} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{67} + ( -6 - 6 \beta_{3} ) q^{71} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{73} + ( 7 - \beta_{2} ) q^{77} + ( -6 + 2 \beta_{1} - 6 \beta_{3} ) q^{79} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 4 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 6 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{89} + ( 2 \beta_{4} - 4 \beta_{5} ) q^{91} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} + ( 3 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{5} + O(q^{10})$$ $$6q - 2q^{5} + 18q^{13} + 2q^{17} + 17q^{19} - 5q^{25} + 12q^{29} + 4q^{31} + 6q^{35} - 3q^{41} + 18q^{43} + 18q^{47} + 2q^{49} + 36q^{53} - 12q^{55} - 27q^{59} - 10q^{61} - 11q^{67} - 18q^{71} - 5q^{73} + 40q^{77} - 16q^{79} + 26q^{85} + 24q^{89} + 6q^{95} + 21q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 10 x^{4} + 3 x^{3} + 84 x^{2} - 27 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 10 \nu^{4} - 100 \nu^{3} + 84 \nu^{2} - 27 \nu + 270$$$$)/813$$ $$\beta_{3}$$ $$=$$ $$($$$$30 \nu^{5} - 29 \nu^{4} + 290 \nu^{3} + 190 \nu^{2} + 2436 \nu - 783$$$$)/813$$ $$\beta_{4}$$ $$=$$ $$($$$$-161 \nu^{5} - 16 \nu^{4} - 1466 \nu^{3} - 1923 \nu^{2} - 14916 \nu - 5310$$$$)/2439$$ $$\beta_{5}$$ $$=$$ $$($$$$191 \nu^{5} - 284 \nu^{4} + 2027 \nu^{3} - 597 \nu^{2} + 15726 \nu - 10107$$$$)/2439$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - \beta_{4} - 10 \beta_{2} - 3$$ $$\nu^{4}$$ $$=$$ $$10 \beta_{5} - 20 \beta_{4} - 57 \beta_{3} - 16 \beta_{1} - 57$$ $$\nu^{5}$$ $$=$$ $$32 \beta_{5} - 16 \beta_{4} - 66 \beta_{3} + 103 \beta_{2} - 103 \beta_{1}$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −1.35887 + 2.35363i 0.162698 − 0.281802i 1.69617 − 2.93786i −1.35887 − 2.35363i 0.162698 + 0.281802i 1.69617 + 2.93786i
0 0 0 −1.85887 + 3.21966i 0 2.36936i 0 0 0
559.2 0 0 0 −0.337302 + 0.584224i 0 3.59084i 0 0 0
559.3 0 0 0 1.19617 2.07183i 0 1.22147i 0 0 0
1855.1 0 0 0 −1.85887 3.21966i 0 2.36936i 0 0 0
1855.2 0 0 0 −0.337302 0.584224i 0 3.59084i 0 0 0
1855.3 0 0 0 1.19617 + 2.07183i 0 1.22147i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1855.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.m 6
3.b odd 2 1 304.2.n.e yes 6
4.b odd 2 1 2736.2.bm.l 6
12.b even 2 1 304.2.n.d 6
19.d odd 6 1 2736.2.bm.l 6
24.f even 2 1 1216.2.n.e 6
24.h odd 2 1 1216.2.n.d 6
57.f even 6 1 304.2.n.d 6
76.f even 6 1 inner 2736.2.bm.m 6
228.n odd 6 1 304.2.n.e yes 6
456.s odd 6 1 1216.2.n.d 6
456.v even 6 1 1216.2.n.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.d 6 12.b even 2 1
304.2.n.d 6 57.f even 6 1
304.2.n.e yes 6 3.b odd 2 1
304.2.n.e yes 6 228.n odd 6 1
1216.2.n.d 6 24.h odd 2 1
1216.2.n.d 6 456.s odd 6 1
1216.2.n.e 6 24.f even 2 1
1216.2.n.e 6 456.v even 6 1
2736.2.bm.l 6 4.b odd 2 1
2736.2.bm.l 6 19.d odd 6 1
2736.2.bm.m 6 1.a even 1 1 trivial
2736.2.bm.m 6 76.f even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{6} + 2 T_{5}^{5} + 12 T_{5}^{4} - 4 T_{5}^{3} + 76 T_{5}^{2} + 48 T_{5} + 36$$ $$T_{7}^{6} + 20 T_{7}^{4} + 100 T_{7}^{2} + 108$$ $$T_{11}^{6} + 29 T_{11}^{4} + 235 T_{11}^{2} + 507$$ $$T_{23}^{6} - 10 T_{23}^{4} + 100 T_{23}^{2} + 180 T_{23} + 108$$ $$T_{31}^{3} - 2 T_{31}^{2} - 54 T_{31} - 66$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$36 + 48 T + 76 T^{2} - 4 T^{3} + 12 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$108 + 100 T^{2} + 20 T^{4} + T^{6}$$
$11$ $$507 + 235 T^{2} + 29 T^{4} + T^{6}$$
$13$ $$( 12 - 6 T + T^{2} )^{3}$$
$17$ $$144 + 528 T + 1912 T^{2} + 112 T^{3} + 48 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$6859 - 6137 T + 2736 T^{2} - 775 T^{3} + 144 T^{4} - 17 T^{5} + T^{6}$$
$23$ $$108 + 180 T + 100 T^{2} - 10 T^{4} + T^{6}$$
$29$ $$2700 - 1080 T - 216 T^{2} + 144 T^{3} + 36 T^{4} - 12 T^{5} + T^{6}$$
$31$ $$( -66 - 54 T - 2 T^{2} + T^{3} )^{2}$$
$37$ $$2700 + 864 T^{2} + 72 T^{4} + T^{6}$$
$41$ $$54675 + 40095 T + 9396 T^{2} - 297 T^{3} - 96 T^{4} + 3 T^{5} + T^{6}$$
$43$ $$( 12 - 6 T + T^{2} )^{3}$$
$47$ $$12 + 156 T + 640 T^{2} - 468 T^{3} + 134 T^{4} - 18 T^{5} + T^{6}$$
$53$ $$( 48 - 12 T + T^{2} )^{3}$$
$59$ $$263169 + 109269 T + 31518 T^{2} + 4725 T^{3} + 516 T^{4} + 27 T^{5} + T^{6}$$
$61$ $$36 + 144 T + 516 T^{2} + 228 T^{3} + 76 T^{4} + 10 T^{5} + T^{6}$$
$67$ $$245025 + 19305 T + 6966 T^{2} + 561 T^{3} + 160 T^{4} + 11 T^{5} + T^{6}$$
$71$ $$( 36 + 6 T + T^{2} )^{3}$$
$73$ $$289 + 425 T + 710 T^{2} - 91 T^{3} + 50 T^{4} + 5 T^{5} + T^{6}$$
$79$ $$2304 - 2304 T + 3072 T^{2} + 864 T^{3} + 208 T^{4} + 16 T^{5} + T^{6}$$
$83$ $$3 + 3031 T^{2} + 113 T^{4} + T^{6}$$
$89$ $$97200 - 38880 T + 864 T^{2} + 1728 T^{3} + 120 T^{4} - 24 T^{5} + T^{6}$$
$97$ $$151875 - 42525 T - 756 T^{2} + 1323 T^{3} + 84 T^{4} - 21 T^{5} + T^{6}$$