# Properties

 Label 2736.2.s.v 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + ( -2 - \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 3 + \beta_{3} ) q^{31} + ( 6 + 6 \beta_{2} ) q^{35} + ( 3 - \beta_{3} ) q^{37} + ( 5 + 2 \beta_{1} + 5 \beta_{2} ) q^{41} + ( 6 - 2 \beta_{1} + 6 \beta_{2} ) q^{43} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + 2 \beta_{3} ) q^{49} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 5 - \beta_{1} + 5 \beta_{2} ) q^{55} -3 \beta_{1} q^{59} + ( -3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -2 + 2 \beta_{3} ) q^{65} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{67} + ( 8 + 2 \beta_{1} + 8 \beta_{2} ) q^{71} + ( -7 - 2 \beta_{1} - 7 \beta_{2} ) q^{73} + ( 9 + 3 \beta_{3} ) q^{77} + ( -4 - 4 \beta_{2} ) q^{79} + 3 \beta_{3} q^{83} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -9 - 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{95} + ( 9 + 2 \beta_{1} + 9 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 4q^{7} + O(q^{10})$$ $$4q + 2q^{5} - 4q^{7} - 8q^{11} - 4q^{13} - 12q^{19} - 2q^{23} - 6q^{25} - 2q^{29} + 12q^{31} + 12q^{35} + 12q^{37} + 10q^{41} + 12q^{43} - 14q^{47} + 4q^{49} - 4q^{53} + 10q^{55} + 14q^{61} - 8q^{65} - 4q^{67} + 16q^{71} - 14q^{73} + 36q^{77} - 8q^{79} + 4q^{91} - 28q^{95} + 18q^{97} + O(q^{100})$$

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 −3.64575 0 0 0
577.2 0 0 0 1.82288 + 3.15731i 0 1.64575 0 0 0
1873.1 0 0 0 −0.822876 + 1.42526i 0 −3.64575 0 0 0
1873.2 0 0 0 1.82288 3.15731i 0 1.64575 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.v 4
3.b odd 2 1 304.2.i.e 4
4.b odd 2 1 342.2.g.f 4
12.b even 2 1 38.2.c.b 4
19.c even 3 1 inner 2736.2.s.v 4
24.f even 2 1 1216.2.i.l 4
24.h odd 2 1 1216.2.i.k 4
57.f even 6 1 5776.2.a.z 2
57.h odd 6 1 304.2.i.e 4
57.h odd 6 1 5776.2.a.ba 2
60.h even 2 1 950.2.e.k 4
60.l odd 4 2 950.2.j.g 8
76.f even 6 1 6498.2.a.bg 2
76.g odd 6 1 342.2.g.f 4
76.g odd 6 1 6498.2.a.ba 2
228.b odd 2 1 722.2.c.j 4
228.m even 6 1 38.2.c.b 4
228.m even 6 1 722.2.a.j 2
228.n odd 6 1 722.2.a.g 2
228.n odd 6 1 722.2.c.j 4
228.u odd 18 6 722.2.e.o 12
228.v even 18 6 722.2.e.n 12
456.u even 6 1 1216.2.i.l 4
456.x odd 6 1 1216.2.i.k 4
1140.bn even 6 1 950.2.e.k 4
1140.bu odd 12 2 950.2.j.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 12.b even 2 1
38.2.c.b 4 228.m even 6 1
304.2.i.e 4 3.b odd 2 1
304.2.i.e 4 57.h odd 6 1
342.2.g.f 4 4.b odd 2 1
342.2.g.f 4 76.g odd 6 1
722.2.a.g 2 228.n odd 6 1
722.2.a.j 2 228.m even 6 1
722.2.c.j 4 228.b odd 2 1
722.2.c.j 4 228.n odd 6 1
722.2.e.n 12 228.v even 18 6
722.2.e.o 12 228.u odd 18 6
950.2.e.k 4 60.h even 2 1
950.2.e.k 4 1140.bn even 6 1
950.2.j.g 8 60.l odd 4 2
950.2.j.g 8 1140.bu odd 12 2
1216.2.i.k 4 24.h odd 2 1
1216.2.i.k 4 456.x odd 6 1
1216.2.i.l 4 24.f even 2 1
1216.2.i.l 4 456.u even 6 1
2736.2.s.v 4 1.a even 1 1 trivial
2736.2.s.v 4 19.c even 3 1 inner
5776.2.a.z 2 57.f even 6 1
5776.2.a.ba 2 57.h odd 6 1
6498.2.a.ba 2 76.g odd 6 1
6498.2.a.bg 2 76.f even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$( -6 + 2 T + T^{2} )^{2}$$
$11$ $$( -3 + 4 T + T^{2} )^{2}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$31$ $$( 2 - 6 T + T^{2} )^{2}$$
$37$ $$( 2 - 6 T + T^{2} )^{2}$$
$41$ $$9 + 30 T + 103 T^{2} - 10 T^{3} + T^{4}$$
$43$ $$64 - 96 T + 136 T^{2} - 12 T^{3} + T^{4}$$
$47$ $$1764 + 588 T + 154 T^{2} + 14 T^{3} + T^{4}$$
$53$ $$11664 - 432 T + 124 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$3969 + 63 T^{2} + T^{4}$$
$61$ $$196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$441 + 294 T + 175 T^{2} + 14 T^{3} + T^{4}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$( -63 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$2809 - 954 T + 271 T^{2} - 18 T^{3} + T^{4}$$