# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} - 30 x^{5} - 5 x^{4} + 114 x^{3} + 300 x^{2} + 116 x + 19$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( 1 - \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{7} + ( 1 - \beta_{5} + \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{13} + ( -1 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{25} + ( 2 + 2 \beta_{1} - 2 \beta_{5} ) q^{29} + ( 4 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( 3 + \beta_{1} - \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{35} + ( -1 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{37} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{41} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{43} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{47} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{49} + ( 1 + \beta_{2} ) q^{53} + ( -\beta_{1} - \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{55} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{61} + ( -2 - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{65} + ( 1 - \beta_{2} + 2 \beta_{4} + 5 \beta_{6} - 4 \beta_{7} ) q^{67} + ( -1 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{71} + ( -1 - 2 \beta_{3} - 3 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -2 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{79} + ( -4 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} ) q^{83} + ( 2 - 2 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -10 + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{6} - \beta_{7} ) q^{89} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{91} + ( 7 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{95} + ( 5 - \beta_{1} - \beta_{4} - 5 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{5} + O(q^{10})$$ $$8q + 2q^{5} - 4q^{17} + 6q^{23} - 12q^{25} + 12q^{29} + 28q^{31} + 18q^{35} + 12q^{41} + 18q^{43} + 12q^{47} - 24q^{49} + 6q^{53} - 12q^{55} + 10q^{59} - 4q^{61} - 6q^{67} - 8q^{71} - 8q^{73} - 28q^{77} - 14q^{79} - 8q^{85} - 54q^{89} - 26q^{91} + 38q^{95} + 60q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 30 x^{5} - 5 x^{4} + 114 x^{3} + 300 x^{2} + 116 x + 19$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$44 \nu^{7} - 2059 \nu^{6} + 6647 \nu^{5} - 15007 \nu^{4} + 50938 \nu^{3} + 80360 \nu^{2} - 278981 \nu + 154035$$$$)/307713$$ $$\beta_{2}$$ $$=$$ $$($$$$-88 \nu^{7} + 4118 \nu^{6} - 13294 \nu^{5} + 30014 \nu^{4} - 101876 \nu^{3} + 146993 \nu^{2} + 250249 \nu - 308070$$$$)/307713$$ $$\beta_{3}$$ $$=$$ $$($$$$3500 \nu^{7} - 15922 \nu^{6} + 21212 \nu^{5} - 114745 \nu^{4} + 251431 \nu^{3} + 373886 \nu^{2} + 131509 \nu - 1214655$$$$)/307713$$ $$\beta_{4}$$ $$=$$ $$($$$$-3772 \nu^{7} + 8669 \nu^{6} - 10351 \nu^{5} + 111605 \nu^{4} - 2846 \nu^{3} - 315175 \nu^{2} - 309125 \nu - 420924$$$$)/307713$$ $$\beta_{5}$$ $$=$$ $$($$$$5106 \nu^{7} - 10484 \nu^{6} - 7253 \nu^{5} - 142319 \nu^{4} - 28670 \nu^{3} + 830669 \nu^{2} + 1282798 \nu + 414680$$$$)/307713$$ $$\beta_{6}$$ $$=$$ $$($$$$-1002 \nu^{7} + 2264 \nu^{6} - 178 \nu^{5} + 29375 \nu^{4} + 254 \nu^{3} - 124940 \nu^{2} - 273328 \nu - 78329$$$$)/43959$$ $$\beta_{7}$$ $$=$$ $$($$$$-32081 \nu^{7} + 65584 \nu^{6} - 5933 \nu^{5} + 958120 \nu^{4} + 99677 \nu^{3} - 3586874 \nu^{2} - 9255799 \nu - 2690259$$$$)/307713$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 4 \beta_{2} + 7 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-6 \beta_{7} + 35 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - \beta_{1} + 47$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 15 \beta_{6} - 12 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 14 \beta_{2} - 2 \beta_{1} + 35$$ $$\nu^{5}$$ $$=$$ $$($$$$-133 \beta_{6} - 187 \beta_{5} - 7 \beta_{4} + 146 \beta_{2} + 191 \beta_{1} + 56$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-198 \beta_{7} + 992 \beta_{6} + 47 \beta_{5} + 38 \beta_{4} + 150 \beta_{3} + 335 \beta_{2} + 311 \beta_{1} + 797$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-273 \beta_{7} + 1618 \beta_{6} - 119 \beta_{5} - 485 \beta_{4} + 693 \beta_{3} + 1192 \beta_{2} - 653 \beta_{1} + 4249$$$$)/3$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{6}$$ $$1$$ $$-1$$ $$1$$

## 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
 −0.213988 + 0.172868i −0.654220 + 2.95767i −1.27736 − 1.04884i 3.14556 − 0.349646i −0.213988 − 0.172868i −0.654220 − 2.95767i −1.27736 + 1.04884i 3.14556 + 0.349646i
0 0 0 −2.00488 + 3.47255i 0 1.92982i 0 0 0
559.2 0 0 0 0.353597 0.612447i 0 0.0924751i 0 0 0
559.3 0 0 0 0.912850 1.58110i 0 4.99333i 0 0 0
559.4 0 0 0 1.73843 3.01105i 0 3.36658i 0 0 0
1855.1 0 0 0 −2.00488 3.47255i 0 1.92982i 0 0 0
1855.2 0 0 0 0.353597 + 0.612447i 0 0.0924751i 0 0 0
1855.3 0 0 0 0.912850 + 1.58110i 0 4.99333i 0 0 0
1855.4 0 0 0 1.73843 + 3.01105i 0 3.36658i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1855.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.s 8
3.b odd 2 1 912.2.bb.g 8
4.b odd 2 1 2736.2.bm.r 8
12.b even 2 1 912.2.bb.h yes 8
19.d odd 6 1 2736.2.bm.r 8
57.f even 6 1 912.2.bb.h yes 8
76.f even 6 1 inner 2736.2.bm.s 8
228.n odd 6 1 912.2.bb.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.g 8 3.b odd 2 1
912.2.bb.g 8 228.n odd 6 1
912.2.bb.h yes 8 12.b even 2 1
912.2.bb.h yes 8 57.f even 6 1
2736.2.bm.r 8 4.b odd 2 1
2736.2.bm.r 8 19.d odd 6 1
2736.2.bm.s 8 1.a even 1 1 trivial
2736.2.bm.s 8 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{8} + 40 T_{7}^{6} + 418 T_{7}^{4} + 1056 T_{7}^{2} + 9$$ $$T_{11}^{8} + 40 T_{11}^{6} + 520 T_{11}^{4} + 2376 T_{11}^{2} + 2916$$ $$T_{23}^{8} - \cdots$$ $$T_{31}^{4} - 14 T_{31}^{3} + 10 T_{31}^{2} + 282 T_{31} - 45$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$324 - 648 T + 1044 T^{2} - 576 T^{3} + 286 T^{4} - 44 T^{5} + 18 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$9 + 1056 T^{2} + 418 T^{4} + 40 T^{6} + T^{8}$$
$11$ $$2916 + 2376 T^{2} + 520 T^{4} + 40 T^{6} + T^{8}$$
$13$ $$225 - 1080 T + 2118 T^{2} - 1872 T^{3} + 691 T^{4} - 26 T^{6} + T^{8}$$
$17$ $$129600 + 95040 T + 53856 T^{2} + 14496 T^{3} + 3352 T^{4} + 352 T^{5} + 60 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$130321 - 1444 T^{2} + 1368 T^{3} + 258 T^{4} + 72 T^{5} - 4 T^{6} + T^{8}$$
$23$ $$12996 - 2736 T - 4140 T^{2} + 912 T^{3} + 1378 T^{4} + 228 T^{5} - 26 T^{6} - 6 T^{7} + T^{8}$$
$29$ $$746496 - 746496 T + 297216 T^{2} - 48384 T^{3} + 544 T^{4} + 672 T^{5} - 8 T^{6} - 12 T^{7} + T^{8}$$
$31$ $$( -45 + 282 T + 10 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$37$ $$81225 + 42372 T^{2} + 6046 T^{4} + 148 T^{6} + T^{8}$$
$41$ $$129600 - 233280 T + 161568 T^{2} - 38880 T^{3} + 1368 T^{4} + 720 T^{5} - 12 T^{6} - 12 T^{7} + T^{8}$$
$43$ $$1946025 + 1330830 T + 211302 T^{2} - 62964 T^{3} - 2763 T^{4} + 1188 T^{5} + 42 T^{6} - 18 T^{7} + T^{8}$$
$47$ $$576 + 9792 T + 53856 T^{2} - 27744 T^{3} + 2968 T^{4} + 816 T^{5} - 20 T^{6} - 12 T^{7} + T^{8}$$
$53$ $$36 + 288 T + 684 T^{2} - 672 T^{3} + 94 T^{4} + 84 T^{5} - 2 T^{6} - 6 T^{7} + T^{8}$$
$59$ $$14152644 + 2302344 T + 743220 T^{2} + 15264 T^{3} + 11962 T^{4} - 244 T^{5} + 198 T^{6} - 10 T^{7} + T^{8}$$
$61$ $$1590121 - 630500 T + 424018 T^{2} + 58912 T^{3} + 19783 T^{4} + 448 T^{5} + 154 T^{6} + 4 T^{7} + T^{8}$$
$67$ $$693889 + 1154538 T + 1749398 T^{2} + 295512 T^{3} + 51585 T^{4} + 1536 T^{5} + 242 T^{6} + 6 T^{7} + T^{8}$$
$71$ $$627264 - 95040 T + 68256 T^{2} - 4512 T^{3} + 4792 T^{4} - 304 T^{5} + 132 T^{6} + 8 T^{7} + T^{8}$$
$73$ $$393129 + 240768 T + 123630 T^{2} + 24624 T^{3} + 5143 T^{4} + 464 T^{5} + 102 T^{6} + 8 T^{7} + T^{8}$$
$79$ $$5331481 + 2683058 T + 1225558 T^{2} + 127400 T^{3} + 21493 T^{4} + 1568 T^{5} + 250 T^{6} + 14 T^{7} + T^{8}$$
$83$ $$13483584 + 2229120 T^{2} + 72288 T^{4} + 504 T^{6} + T^{8}$$
$89$ $$61496964 + 14774328 T - 400932 T^{2} - 380568 T^{3} + 14734 T^{4} + 10908 T^{5} + 1174 T^{6} + 54 T^{7} + T^{8}$$
$97$ $$10890000 - 12038400 T + 5835168 T^{2} - 1546752 T^{3} + 249436 T^{4} - 25440 T^{5} + 1624 T^{6} - 60 T^{7} + T^{8}$$