# Properties

 Label 2736.2.s.u Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{7} + ( 3 + \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -2 - 4 \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{23} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 - \beta_{3} ) q^{31} + ( -9 - 3 \beta_{1} - 9 \beta_{2} ) q^{35} -5 q^{37} + ( 4 + 4 \beta_{2} ) q^{41} + ( 8 - \beta_{1} + 8 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 4 - 4 \beta_{3} ) q^{49} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{53} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{55} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{59} + 5 \beta_{2} q^{61} + ( 13 - \beta_{3} ) q^{65} + ( -3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 10 + 10 \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{73} + ( 1 + \beta_{3} ) q^{77} + ( 6 - \beta_{1} + 6 \beta_{2} ) q^{79} + ( 10 - 2 \beta_{3} ) q^{83} -12 \beta_{2} q^{85} + ( -\beta_{1} - 11 \beta_{2} - \beta_{3} ) q^{89} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 9 - \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{95} -4 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 8q^{7} + O(q^{10})$$ $$4q + 2q^{5} - 8q^{7} + 12q^{11} - 2q^{13} + 4q^{17} + 6q^{23} - 6q^{25} + 4q^{29} - 8q^{31} - 18q^{35} - 20q^{37} + 8q^{41} + 16q^{43} + 16q^{49} - 2q^{53} - 8q^{55} - 10q^{59} - 10q^{61} + 52q^{65} + 12q^{67} + 20q^{71} + 2q^{73} + 4q^{77} + 12q^{79} + 40q^{83} + 24q^{85} + 22q^{89} - 24q^{91} + 26q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0 0 0 −0.822876 1.42526i 0 0.645751 0 0 0
577.2 0 0 0 1.82288 + 3.15731i 0 −4.64575 0 0 0
1873.1 0 0 0 −0.822876 + 1.42526i 0 0.645751 0 0 0
1873.2 0 0 0 1.82288 3.15731i 0 −4.64575 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.u 4
3.b odd 2 1 912.2.q.g 4
4.b odd 2 1 684.2.k.g 4
12.b even 2 1 228.2.i.b 4
19.c even 3 1 inner 2736.2.s.u 4
57.h odd 6 1 912.2.q.g 4
76.g odd 6 1 684.2.k.g 4
228.m even 6 1 228.2.i.b 4
228.m even 6 1 4332.2.a.h 2
228.n odd 6 1 4332.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.i.b 4 12.b even 2 1
228.2.i.b 4 228.m even 6 1
684.2.k.g 4 4.b odd 2 1
684.2.k.g 4 76.g odd 6 1
912.2.q.g 4 3.b odd 2 1
912.2.q.g 4 57.h odd 6 1
2736.2.s.u 4 1.a even 1 1 trivial
2736.2.s.u 4 19.c even 3 1 inner
4332.2.a.h 2 228.m even 6 1
4332.2.a.m 2 228.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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36$$ $$T_{7}^{2} + 4 T_{7} - 3$$ $$T_{11}^{2} - 6 T_{11} + 2$$ $$T_{13}^{4} + 2 T_{13}^{3} + 31 T_{13}^{2} - 54 T_{13} + 729$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$( -3 + 4 T + T^{2} )^{2}$$
$11$ $$( 2 - 6 T + T^{2} )^{2}$$
$13$ $$729 - 54 T + 31 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$361 + 10 T^{2} + T^{4}$$
$23$ $$4 - 12 T + 34 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( -3 + 4 T + T^{2} )^{2}$$
$37$ $$( 5 + T )^{4}$$
$41$ $$( 16 - 4 T + T^{2} )^{2}$$
$43$ $$3249 - 912 T + 199 T^{2} - 16 T^{3} + T^{4}$$
$47$ $$784 + 28 T^{2} + T^{4}$$
$53$ $$3844 - 124 T + 66 T^{2} + 2 T^{3} + T^{4}$$
$59$ $$324 + 180 T + 82 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$( 25 + 5 T + T^{2} )^{2}$$
$67$ $$729 + 324 T + 171 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$( 100 - 10 T + T^{2} )^{2}$$
$73$ $$729 + 54 T + 31 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$841 - 348 T + 115 T^{2} - 12 T^{3} + T^{4}$$
$83$ $$( 72 - 20 T + T^{2} )^{2}$$
$89$ $$12996 - 2508 T + 370 T^{2} - 22 T^{3} + T^{4}$$
$97$ $$12544 + 112 T^{2} + T^{4}$$