# Properties

 Label 2736.1.o.a Level $2736$ Weight $1$ Character orbit 2736.o Self dual yes Analytic conductor $1.365$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -19, 57 Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 171) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-19})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.8208.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{7} + O(q^{10})$$ $$q + 2q^{7} + q^{19} - q^{25} - 2q^{43} + 3q^{49} - 2q^{61} + 2q^{73} + O(q^{100})$$

## 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
 0
0 0 0 0 0 2.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
57.d even 2 1 RM by $$\Q(\sqrt{57})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.o.a 1
3.b odd 2 1 CM 2736.1.o.a 1
4.b odd 2 1 171.1.c.a 1
12.b even 2 1 171.1.c.a 1
19.b odd 2 1 CM 2736.1.o.a 1
36.f odd 6 2 1539.1.o.b 2
36.h even 6 2 1539.1.o.b 2
57.d even 2 1 RM 2736.1.o.a 1
76.d even 2 1 171.1.c.a 1
76.f even 6 2 3249.1.p.a 2
76.g odd 6 2 3249.1.p.a 2
76.k even 18 6 3249.1.ba.c 6
76.l odd 18 6 3249.1.ba.c 6
228.b odd 2 1 171.1.c.a 1
228.m even 6 2 3249.1.p.a 2
228.n odd 6 2 3249.1.p.a 2
228.u odd 18 6 3249.1.ba.c 6
228.v even 18 6 3249.1.ba.c 6
684.w even 6 2 1539.1.o.b 2
684.bh odd 6 2 1539.1.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.c.a 1 4.b odd 2 1
171.1.c.a 1 12.b even 2 1
171.1.c.a 1 76.d even 2 1
171.1.c.a 1 228.b odd 2 1
1539.1.o.b 2 36.f odd 6 2
1539.1.o.b 2 36.h even 6 2
1539.1.o.b 2 684.w even 6 2
1539.1.o.b 2 684.bh odd 6 2
2736.1.o.a 1 1.a even 1 1 trivial
2736.1.o.a 1 3.b odd 2 1 CM
2736.1.o.a 1 19.b odd 2 1 CM
2736.1.o.a 1 57.d even 2 1 RM
3249.1.p.a 2 76.f even 6 2
3249.1.p.a 2 76.g odd 6 2
3249.1.p.a 2 228.m even 6 2
3249.1.p.a 2 228.n odd 6 2
3249.1.ba.c 6 76.k even 18 6
3249.1.ba.c 6 76.l odd 18 6
3249.1.ba.c 6 228.u odd 18 6
3249.1.ba.c 6 228.v even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$-1 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$2 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$2 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$
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