# Properties

 Label 2736.2.s.bb Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.764411904.5 Defining polynomial: $$x^{8} - 6 x^{6} + 21 x^{4} - 54 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 171) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + ( 1 + \beta_{5} ) q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{3} ) q^{13} + 2 \beta_{1} q^{17} + ( -1 + 2 \beta_{4} - \beta_{5} ) q^{19} + ( -\beta_{2} + \beta_{7} ) q^{23} + ( -1 - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{25} + ( -2 \beta_{2} + 2 \beta_{7} ) q^{29} + ( 7 + \beta_{5} ) q^{31} + ( \beta_{1} - \beta_{6} ) q^{35} + ( -1 + 2 \beta_{5} ) q^{37} -2 \beta_{6} q^{41} + ( \beta_{3} + 3 \beta_{4} ) q^{43} + 2 \beta_{7} q^{47} + 2 \beta_{5} q^{49} -\beta_{2} q^{53} + 6 \beta_{3} q^{55} + ( -3 \beta_{1} - \beta_{6} ) q^{59} + ( -5 - 5 \beta_{3} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -7 - 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{71} + 5 \beta_{3} q^{73} + ( \beta_{1} - \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 7 \beta_{3} + 3 \beta_{4} ) q^{79} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 12 + 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{85} + 5 \beta_{2} q^{89} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{91} + ( -\beta_{1} + 4 \beta_{2} + \beta_{6} - 2 \beta_{7} ) q^{95} + ( 8 \beta_{3} + 2 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{7} + O(q^{10})$$ $$8q + 8q^{7} + 4q^{13} - 8q^{19} - 4q^{25} + 56q^{31} - 8q^{37} - 4q^{43} - 24q^{55} - 20q^{61} - 28q^{67} - 20q^{73} - 28q^{79} + 48q^{85} + 4q^{91} - 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 6 x^{6} + 21 x^{4} - 54 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 21 \nu^{5} + 87 \nu^{3} - 162 \nu$$$$)/135$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 24 \nu^{5} + 48 \nu^{3} + 27 \nu$$$$)/135$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} + 3 \nu^{4} - 6 \nu^{2} - 9$$$$)/45$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{6} - 3 \nu^{4} + 51 \nu^{2} - 81$$$$)/45$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 6 \nu^{4} - 12 \nu^{2} + 27$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 12 \nu^{5} - 114 \nu^{3} + 54 \nu$$$$)/135$$ $$\beta_{7}$$ $$=$$ $$($$$$-22 \nu^{7} + 78 \nu^{5} - 246 \nu^{3} + 486 \nu$$$$)/135$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} + 4 \beta_{2} - 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{6} + 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{7} - 7 \beta_{6} - 10 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{5} - 30 \beta_{3} - 15$$ $$\nu^{7}$$ $$=$$ $$($$$$-9 \beta_{7} - 6 \beta_{6} - 6 \beta_{2} - 30 \beta_{1}$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −1.27970 + 1.16721i −1.69185 − 0.370982i 1.69185 + 0.370982i 1.27970 − 1.16721i −1.27970 − 1.16721i −1.69185 + 0.370982i 1.69185 − 0.370982i 1.27970 + 1.16721i
0 0 0 −1.65068 2.85906i 0 −1.44949 0 0 0
577.2 0 0 0 −0.524648 0.908716i 0 3.44949 0 0 0
577.3 0 0 0 0.524648 + 0.908716i 0 3.44949 0 0 0
577.4 0 0 0 1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.1 0 0 0 −1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.2 0 0 0 −0.524648 + 0.908716i 0 3.44949 0 0 0
1873.3 0 0 0 0.524648 0.908716i 0 3.44949 0 0 0
1873.4 0 0 0 1.65068 2.85906i 0 −1.44949 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1873.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.bb 8
3.b odd 2 1 inner 2736.2.s.bb 8
4.b odd 2 1 171.2.f.c 8
12.b even 2 1 171.2.f.c 8
19.c even 3 1 inner 2736.2.s.bb 8
57.h odd 6 1 inner 2736.2.s.bb 8
76.f even 6 1 3249.2.a.be 4
76.g odd 6 1 171.2.f.c 8
76.g odd 6 1 3249.2.a.bd 4
228.m even 6 1 171.2.f.c 8
228.m even 6 1 3249.2.a.bd 4
228.n odd 6 1 3249.2.a.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.f.c 8 4.b odd 2 1
171.2.f.c 8 12.b even 2 1
171.2.f.c 8 76.g odd 6 1
171.2.f.c 8 228.m even 6 1
2736.2.s.bb 8 1.a even 1 1 trivial
2736.2.s.bb 8 3.b odd 2 1 inner
2736.2.s.bb 8 19.c even 3 1 inner
2736.2.s.bb 8 57.h odd 6 1 inner
3249.2.a.bd 4 76.g odd 6 1
3249.2.a.bd 4 228.m even 6 1
3249.2.a.be 4 76.f even 6 1
3249.2.a.be 4 228.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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{8} + 12 T_{5}^{6} + 132 T_{5}^{4} + 144 T_{5}^{2} + 144$$ $$T_{7}^{2} - 2 T_{7} - 5$$ $$T_{11}^{4} - 36 T_{11}^{2} + 108$$ $$T_{13}^{2} - T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$144 + 144 T^{2} + 132 T^{4} + 12 T^{6} + T^{8}$$
$7$ $$( -5 - 2 T + T^{2} )^{4}$$
$11$ $$( 108 - 36 T^{2} + T^{4} )^{2}$$
$13$ $$( 1 - T + T^{2} )^{4}$$
$17$ $$36864 + 9216 T^{2} + 2112 T^{4} + 48 T^{6} + T^{8}$$
$19$ $$( 19 + 2 T + T^{2} )^{4}$$
$23$ $$90000 + 10800 T^{2} + 996 T^{4} + 36 T^{6} + T^{8}$$
$29$ $$23040000 + 691200 T^{2} + 15936 T^{4} + 144 T^{6} + T^{8}$$
$31$ $$( 43 - 14 T + T^{2} )^{4}$$
$37$ $$( -23 + 2 T + T^{2} )^{4}$$
$41$ $$589824 + 73728 T^{2} + 8448 T^{4} + 96 T^{6} + T^{8}$$
$43$ $$( 2809 - 106 T + 57 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$47$ $$589824 + 73728 T^{2} + 8448 T^{4} + 96 T^{6} + T^{8}$$
$53$ $$144 + 144 T^{2} + 132 T^{4} + 12 T^{6} + T^{8}$$
$59$ $$18766224 + 571824 T^{2} + 13092 T^{4} + 132 T^{6} + T^{8}$$
$61$ $$( 25 + 5 T + T^{2} )^{4}$$
$67$ $$( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$71$ $$23040000 + 691200 T^{2} + 15936 T^{4} + 144 T^{6} + T^{8}$$
$73$ $$( 25 + 5 T + T^{2} )^{4}$$
$79$ $$( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$83$ $$( 1728 - 144 T^{2} + T^{4} )^{2}$$
$89$ $$56250000 + 2250000 T^{2} + 82500 T^{4} + 300 T^{6} + T^{8}$$
$97$ $$( 1600 + 640 T + 216 T^{2} + 16 T^{3} + T^{4} )^{2}$$
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