# Properties

 Label 2736.2.s.bc Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1368) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( 1 - \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( 1 - \beta_{6} ) q^{7} + ( -1 + \beta_{1} ) q^{11} + ( -\beta_{5} - \beta_{7} ) q^{13} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{17} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{23} + ( 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{25} -2 \beta_{7} q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{6} ) q^{31} + ( 1 - \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{35} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} ) q^{37} + ( -2 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{43} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{49} + ( -5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{53} + ( 3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{55} + ( -3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{59} + ( 2 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{61} + ( 5 + \beta_{1} + 2 \beta_{2} + 2 \beta_{6} ) q^{65} + ( \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{67} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - 4 \beta_{1} - \beta_{3} - 4 \beta_{5} ) q^{73} + ( -1 - \beta_{1} ) q^{77} + ( 6 + 3 \beta_{1} + \beta_{2} + 6 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} ) q^{83} + ( 4 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 5 \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{89} + ( -2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} ) q^{91} + ( -3 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{95} + ( 7 + \beta_{1} + \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} + 4q^{7} + O(q^{10})$$ $$8q + 4q^{5} + 4q^{7} - 8q^{11} + 4q^{17} - 8q^{19} - 8q^{23} - 4q^{25} + 4q^{31} + 16q^{37} + 4q^{41} + 6q^{43} - 4q^{47} - 16q^{49} + 16q^{53} + 16q^{55} - 12q^{59} - 8q^{61} + 48q^{65} - 2q^{67} + 4q^{71} - 4q^{73} - 8q^{77} + 22q^{79} - 8q^{83} - 8q^{85} - 12q^{89} + 2q^{91} - 32q^{95} + 24q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-16 \nu^{7} - 199 \nu^{6} - 84 \nu^{5} - 152 \nu^{4} - 4652 \nu^{3} - 272 \nu^{2} - 132 \nu + 3206$$$$)/4243$$ $$\beta_{2}$$ $$=$$ $$($$$$152 \nu^{7} - 231 \nu^{6} + 798 \nu^{5} + 1444 \nu^{4} + 1764 \nu^{3} + 2584 \nu^{2} + 1254 \nu + 11973$$$$)/4243$$ $$\beta_{3}$$ $$=$$ $$($$$$754 \nu^{7} - 1760 \nu^{6} + 6080 \nu^{5} - 1323 \nu^{4} + 13440 \nu^{3} - 12640 \nu^{2} + 16828 \nu - 5760$$$$)/12729$$ $$\beta_{4}$$ $$=$$ $$($$$$-815 \nu^{7} + 3388 \nu^{6} - 11704 \nu^{5} + 15594 \nu^{4} - 25872 \nu^{3} + 24332 \nu^{2} - 56579 \nu + 11088$$$$)/12729$$ $$\beta_{5}$$ $$=$$ $$($$$$1052 \nu^{7} - 2827 \nu^{6} + 9766 \nu^{5} - 6978 \nu^{4} + 21588 \nu^{3} - 20303 \nu^{2} + 4436 \nu - 9252$$$$)/12729$$ $$\beta_{6}$$ $$=$$ $$($$$$-556 \nu^{7} + 510 \nu^{6} - 2919 \nu^{5} - 5282 \nu^{4} - 8909 \nu^{3} - 9452 \nu^{2} - 4587 \nu - 13760$$$$)/4243$$ $$\beta_{7}$$ $$=$$ $$($$$$602 \nu^{7} - 1529 \nu^{6} + 5282 \nu^{5} - 2767 \nu^{4} + 11676 \nu^{3} - 10981 \nu^{2} + 15574 \nu - 5004$$$$)/4243$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - 3 \beta_{3} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{6} + 7 \beta_{2} - 3 \beta_{1} - 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{7} + \beta_{5} - 2 \beta_{4} + 18 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$($$$$-53 \beta_{7} - 16 \beta_{6} + 13 \beta_{5} - 16 \beta_{4} + 89 \beta_{3} - 53 \beta_{2} + 13 \beta_{1} + 89$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-20 \beta_{6} - 72 \beta_{2} + 11 \beta_{1} + 130$$ $$\nu^{7}$$ $$=$$ $$($$$$407 \beta_{7} - 79 \beta_{5} + 122 \beta_{4} - 699 \beta_{3}$$$$)/2$$

## Character values

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

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

## 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
 −0.758290 + 1.31340i 1.39083 − 2.40898i −0.276205 + 0.478401i 0.643668 − 1.11487i −0.758290 − 1.31340i 1.39083 + 2.40898i −0.276205 − 0.478401i 0.643668 + 1.11487i
0 0 0 −1.16659 2.02059i 0 −0.538445 0 0 0
577.2 0 0 0 0.412855 + 0.715087i 0 0.703158 0 0 0
577.3 0 0 0 0.795012 + 1.37700i 0 3.87834 0 0 0
577.4 0 0 0 1.95872 + 3.39260i 0 −2.04306 0 0 0
1873.1 0 0 0 −1.16659 + 2.02059i 0 −0.538445 0 0 0
1873.2 0 0 0 0.412855 0.715087i 0 0.703158 0 0 0
1873.3 0 0 0 0.795012 1.37700i 0 3.87834 0 0 0
1873.4 0 0 0 1.95872 3.39260i 0 −2.04306 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
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.bc 8
3.b odd 2 1 2736.2.s.ba 8
4.b odd 2 1 1368.2.s.m yes 8
12.b even 2 1 1368.2.s.l 8
19.c even 3 1 inner 2736.2.s.bc 8
57.h odd 6 1 2736.2.s.ba 8
76.g odd 6 1 1368.2.s.m yes 8
228.m even 6 1 1368.2.s.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.s.l 8 12.b even 2 1
1368.2.s.l 8 228.m even 6 1
1368.2.s.m yes 8 4.b odd 2 1
1368.2.s.m yes 8 76.g odd 6 1
2736.2.s.ba 8 3.b odd 2 1
2736.2.s.ba 8 57.h odd 6 1
2736.2.s.bc 8 1.a even 1 1 trivial
2736.2.s.bc 8 19.c even 3 1 inner

## Hecke kernels

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

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{4} - 2 T_{7}^{3} - 8 T_{7}^{2} + 2 T_{7} + 3$$ $$T_{11}^{4} + 4 T_{11}^{3} - 4 T_{11}^{2} - 12 T_{11} - 4$$ $$T_{13}^{8} + 30 T_{13}^{6} - 16 T_{13}^{5} + 719 T_{13}^{4} - 240 T_{13}^{3} + 5494 T_{13}^{2} + 1448 T_{13} + 32761$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$144 - 240 T + 352 T^{2} - 176 T^{3} + 108 T^{4} - 24 T^{5} + 20 T^{6} - 4 T^{7} + T^{8}$$
$7$ $$( 3 + 2 T - 8 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( -4 - 12 T - 4 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$13$ $$32761 + 1448 T + 5494 T^{2} - 240 T^{3} + 719 T^{4} - 16 T^{5} + 30 T^{6} + T^{8}$$
$17$ $$4096 - 8192 T + 14336 T^{2} - 4608 T^{3} + 1600 T^{4} - 128 T^{5} + 48 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$130321 + 54872 T + 10108 T^{2} + 1672 T^{3} + 326 T^{4} + 88 T^{5} + 28 T^{6} + 8 T^{7} + T^{8}$$
$23$ $$400 + 9360 T + 217824 T^{2} + 28400 T^{3} + 7364 T^{4} + 456 T^{5} + 124 T^{6} + 8 T^{7} + T^{8}$$
$29$ $$36864 - 43008 T + 37888 T^{2} - 14336 T^{3} + 4288 T^{4} - 448 T^{5} + 64 T^{6} + T^{8}$$
$31$ $$( 27 + 18 T - 24 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$37$ $$( 197 + 320 T - 78 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$41$ $$409600 - 368640 T + 260096 T^{2} - 69632 T^{3} + 15488 T^{4} - 704 T^{5} + 128 T^{6} - 4 T^{7} + T^{8}$$
$43$ $$2455489 + 435626 T + 196376 T^{2} - 2324 T^{3} + 5877 T^{4} - 100 T^{5} + 112 T^{6} - 6 T^{7} + T^{8}$$
$47$ $$409600 + 327680 T + 210944 T^{2} + 46080 T^{3} + 9088 T^{4} + 704 T^{5} + 96 T^{6} + 4 T^{7} + T^{8}$$
$53$ $$26998416 - 8001840 T + 2142976 T^{2} - 234032 T^{3} + 31772 T^{4} - 2376 T^{5} + 300 T^{6} - 16 T^{7} + T^{8}$$
$59$ $$15376 - 11408 T + 9952 T^{2} - 1872 T^{3} + 1124 T^{4} + 40 T^{5} + 156 T^{6} + 12 T^{7} + T^{8}$$
$61$ $$299209 + 323824 T + 312174 T^{2} + 50192 T^{3} + 10183 T^{4} + 624 T^{5} + 134 T^{6} + 8 T^{7} + T^{8}$$
$67$ $$2002225 - 319790 T + 226536 T^{2} + 22364 T^{3} + 14413 T^{4} + 204 T^{5} + 128 T^{6} + 2 T^{7} + T^{8}$$
$71$ $$11505664 - 217088 T + 546816 T^{2} + 37376 T^{3} + 21952 T^{4} + 768 T^{5} + 176 T^{6} - 4 T^{7} + T^{8}$$
$73$ $$25281 + 90948 T + 302698 T^{2} + 89360 T^{3} + 26163 T^{4} + 528 T^{5} + 170 T^{6} + 4 T^{7} + T^{8}$$
$79$ $$139129 - 306606 T + 696572 T^{2} + 29620 T^{3} + 21593 T^{4} - 2876 T^{5} + 428 T^{6} - 22 T^{7} + T^{8}$$
$83$ $$( 3392 + 64 T - 160 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$89$ $$39388176 - 5999856 T + 1692160 T^{2} - 32080 T^{3} + 20572 T^{4} + 424 T^{5} + 268 T^{6} + 12 T^{7} + T^{8}$$
$97$ $$25600 - 43520 T + 50944 T^{2} - 31488 T^{3} + 14048 T^{4} - 2912 T^{5} + 432 T^{6} - 24 T^{7} + T^{8}$$