# Properties

 Label 2736.3.o.h 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: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} -5 q^{7} +O(q^{10})$$ $$q + q^{5} -5 q^{7} + 5 q^{11} -2 \beta q^{13} + 25 q^{17} -19 q^{19} -10 q^{23} -24 q^{25} -5 \beta q^{29} + 5 \beta q^{31} -5 q^{35} -3 \beta q^{37} + 5 \beta q^{41} -5 q^{43} + 5 q^{47} -24 q^{49} + 3 \beta q^{53} + 5 q^{55} + 10 \beta q^{59} + 95 q^{61} -2 \beta q^{65} -13 \beta q^{67} -25 q^{73} -25 q^{77} -5 \beta q^{79} -130 q^{83} + 25 q^{85} + 15 \beta q^{89} + 10 \beta q^{91} -19 q^{95} + 2 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 10q^{7} + O(q^{10})$$ $$2q + 2q^{5} - 10q^{7} + 10q^{11} + 50q^{17} - 38q^{19} - 20q^{23} - 48q^{25} - 10q^{35} - 10q^{43} + 10q^{47} - 48q^{49} + 10q^{55} + 190q^{61} - 50q^{73} - 50q^{77} - 260q^{83} + 50q^{85} - 38q^{95} + O(q^{100})$$

## 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
 1.41421i − 1.41421i
0 0 0 1.00000 0 −5.00000 0 0 0
721.2 0 0 0 1.00000 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.h 2
3.b odd 2 1 304.3.e.c 2
4.b odd 2 1 342.3.d.a 2
12.b even 2 1 38.3.b.a 2
19.b odd 2 1 inner 2736.3.o.h 2
24.f even 2 1 1216.3.e.j 2
24.h odd 2 1 1216.3.e.i 2
57.d even 2 1 304.3.e.c 2
60.h even 2 1 950.3.c.a 2
60.l odd 4 2 950.3.d.a 4
76.d even 2 1 342.3.d.a 2
228.b odd 2 1 38.3.b.a 2
456.l odd 2 1 1216.3.e.j 2
456.p even 2 1 1216.3.e.i 2
1140.p odd 2 1 950.3.c.a 2
1140.w even 4 2 950.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 12.b even 2 1
38.3.b.a 2 228.b odd 2 1
304.3.e.c 2 3.b odd 2 1
304.3.e.c 2 57.d even 2 1
342.3.d.a 2 4.b odd 2 1
342.3.d.a 2 76.d even 2 1
950.3.c.a 2 60.h even 2 1
950.3.c.a 2 1140.p odd 2 1
950.3.d.a 4 60.l odd 4 2
950.3.d.a 4 1140.w even 4 2
1216.3.e.i 2 24.h odd 2 1
1216.3.e.i 2 456.p even 2 1
1216.3.e.j 2 24.f even 2 1
1216.3.e.j 2 456.l odd 2 1
2736.3.o.h 2 1.a even 1 1 trivial
2736.3.o.h 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 $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 1$$ $$T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$288 + T^{2}$$
$17$ $$( -25 + T )^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$( 10 + T )^{2}$$
$29$ $$1800 + T^{2}$$
$31$ $$1800 + T^{2}$$
$37$ $$648 + T^{2}$$
$41$ $$1800 + T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$( -5 + T )^{2}$$
$53$ $$648 + T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$( -95 + T )^{2}$$
$67$ $$12168 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 25 + T )^{2}$$
$79$ $$1800 + T^{2}$$
$83$ $$( 130 + T )^{2}$$
$89$ $$16200 + T^{2}$$
$97$ $$288 + T^{2}$$