# Properties

 Label 2736.3.o.d Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$74.5506003290$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 q^{5} + 5 q^{7} +O(q^{10})$$ $$q -4 q^{5} + 5 q^{7} -10 q^{11} -\beta q^{13} -15 q^{17} + ( 6 + 5 \beta ) q^{19} + 35 q^{23} -9 q^{25} + 5 \beta q^{29} -10 \beta q^{31} -20 q^{35} + 6 \beta q^{37} + 10 \beta q^{41} + 20 q^{43} + 10 q^{47} -24 q^{49} -21 \beta q^{53} + 40 q^{55} -5 \beta q^{59} -40 q^{61} + 4 \beta q^{65} + 11 \beta q^{67} -30 \beta q^{71} + 105 q^{73} -50 q^{77} -10 \beta q^{79} -40 q^{83} + 60 q^{85} -5 \beta q^{91} + ( -24 - 20 \beta ) q^{95} -34 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + 10q^{7} + O(q^{10})$$ $$2q - 8q^{5} + 10q^{7} - 20q^{11} - 30q^{17} + 12q^{19} + 70q^{23} - 18q^{25} - 40q^{35} + 40q^{43} + 20q^{47} - 48q^{49} + 80q^{55} - 80q^{61} + 210q^{73} - 100q^{77} - 80q^{83} + 120q^{85} - 48q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 3.60555i − 3.60555i
0 0 0 −4.00000 0 5.00000 0 0 0
721.2 0 0 0 −4.00000 0 5.00000 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.d 2
3.b odd 2 1 304.3.e.d 2
4.b odd 2 1 171.3.c.b 2
12.b even 2 1 19.3.b.b 2
19.b odd 2 1 inner 2736.3.o.d 2
24.f even 2 1 1216.3.e.g 2
24.h odd 2 1 1216.3.e.h 2
57.d even 2 1 304.3.e.d 2
60.h even 2 1 475.3.c.b 2
60.l odd 4 2 475.3.d.b 4
76.d even 2 1 171.3.c.b 2
228.b odd 2 1 19.3.b.b 2
228.m even 6 2 361.3.d.b 4
228.n odd 6 2 361.3.d.b 4
228.u odd 18 6 361.3.f.d 12
228.v even 18 6 361.3.f.d 12
456.l odd 2 1 1216.3.e.g 2
456.p even 2 1 1216.3.e.h 2
1140.p odd 2 1 475.3.c.b 2
1140.w even 4 2 475.3.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 12.b even 2 1
19.3.b.b 2 228.b odd 2 1
171.3.c.b 2 4.b odd 2 1
171.3.c.b 2 76.d even 2 1
304.3.e.d 2 3.b odd 2 1
304.3.e.d 2 57.d even 2 1
361.3.d.b 4 228.m even 6 2
361.3.d.b 4 228.n odd 6 2
361.3.f.d 12 228.u odd 18 6
361.3.f.d 12 228.v even 18 6
475.3.c.b 2 60.h even 2 1
475.3.c.b 2 1140.p odd 2 1
475.3.d.b 4 60.l odd 4 2
475.3.d.b 4 1140.w even 4 2
1216.3.e.g 2 24.f even 2 1
1216.3.e.g 2 456.l odd 2 1
1216.3.e.h 2 24.h odd 2 1
1216.3.e.h 2 456.p even 2 1
2736.3.o.d 2 1.a even 1 1 trivial
2736.3.o.d 2 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} + 4$$ $$T_{7} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$( -5 + T )^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$13 + T^{2}$$
$17$ $$( 15 + T )^{2}$$
$19$ $$361 - 12 T + T^{2}$$
$23$ $$( -35 + T )^{2}$$
$29$ $$325 + T^{2}$$
$31$ $$1300 + T^{2}$$
$37$ $$468 + T^{2}$$
$41$ $$1300 + T^{2}$$
$43$ $$( -20 + T )^{2}$$
$47$ $$( -10 + T )^{2}$$
$53$ $$5733 + T^{2}$$
$59$ $$325 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$1573 + T^{2}$$
$71$ $$11700 + T^{2}$$
$73$ $$( -105 + T )^{2}$$
$79$ $$1300 + T^{2}$$
$83$ $$( 40 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$15028 + T^{2}$$