# Properties

 Label 2736.2.k.g Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ Defining polynomial: $$x^{2} - x + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + \beta q^{7} +O(q^{10})$$ $$q + q^{5} + \beta q^{7} -\beta q^{11} -7 q^{17} -\beta q^{19} -2 \beta q^{23} -4 q^{25} + \beta q^{35} -3 \beta q^{43} -\beta q^{47} -12 q^{49} -\beta q^{55} + 15 q^{61} + 11 q^{73} + 19 q^{77} + 2 \beta q^{83} -7 q^{85} -\beta q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} - 14q^{17} - 8q^{25} - 24q^{49} + 30q^{61} + 22q^{73} + 38q^{77} - 14q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 0.5 − 2.17945i 0.5 + 2.17945i
0 0 0 1.00000 0 4.35890i 0 0 0
2431.2 0 0 0 1.00000 0 4.35890i 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 CM by $$\Q(\sqrt{-19})$$
4.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.g 2
3.b odd 2 1 304.2.h.a 2
4.b odd 2 1 inner 2736.2.k.g 2
12.b even 2 1 304.2.h.a 2
19.b odd 2 1 CM 2736.2.k.g 2
24.f even 2 1 1216.2.h.a 2
24.h odd 2 1 1216.2.h.a 2
57.d even 2 1 304.2.h.a 2
76.d even 2 1 inner 2736.2.k.g 2
228.b odd 2 1 304.2.h.a 2
456.l odd 2 1 1216.2.h.a 2
456.p even 2 1 1216.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.a 2 3.b odd 2 1
304.2.h.a 2 12.b even 2 1
304.2.h.a 2 57.d even 2 1
304.2.h.a 2 228.b odd 2 1
1216.2.h.a 2 24.f even 2 1
1216.2.h.a 2 24.h odd 2 1
1216.2.h.a 2 456.l odd 2 1
1216.2.h.a 2 456.p even 2 1
2736.2.k.g 2 1.a even 1 1 trivial
2736.2.k.g 2 4.b odd 2 1 inner
2736.2.k.g 2 19.b odd 2 1 CM
2736.2.k.g 2 76.d even 2 1 inner

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{7}^{2} + 19$$ $$T_{11}^{2} + 19$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$19 + T^{2}$$
$11$ $$19 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 7 + T )^{2}$$
$19$ $$19 + T^{2}$$
$23$ $$76 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$171 + T^{2}$$
$47$ $$19 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -15 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -11 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$76 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$