# Properties

 Label 2736.2.k.f Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 4 \zeta_{6} ) q^{7} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -2 + 4 \zeta_{6} ) q^{13} + 6 q^{17} + ( 3 + 2 \zeta_{6} ) q^{19} -5 q^{25} + ( -4 + 8 \zeta_{6} ) q^{29} -10 q^{31} + ( -2 + 4 \zeta_{6} ) q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( 6 - 12 \zeta_{6} ) q^{43} + ( -4 + 8 \zeta_{6} ) q^{47} -5 q^{49} + ( -8 + 16 \zeta_{6} ) q^{53} + 12 q^{59} + 10 q^{61} -4 q^{67} + 12 q^{71} -2 q^{73} + 12 q^{77} + 10 q^{79} + ( 2 - 4 \zeta_{6} ) q^{83} + ( -4 + 8 \zeta_{6} ) q^{89} + 12 q^{91} + ( -4 + 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 12q^{17} + 8q^{19} - 10q^{25} - 20q^{31} - 10q^{49} + 24q^{59} + 20q^{61} - 8q^{67} + 24q^{71} - 4q^{73} + 24q^{77} + 20q^{79} + 24q^{91} + O(q^{100})$$

## 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 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 3.46410i 0 0 0
2431.2 0 0 0 0 0 3.46410i 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
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.f 2
3.b odd 2 1 912.2.k.b 2
4.b odd 2 1 2736.2.k.e 2
12.b even 2 1 912.2.k.e yes 2
19.b odd 2 1 2736.2.k.e 2
24.f even 2 1 3648.2.k.b 2
24.h odd 2 1 3648.2.k.e 2
57.d even 2 1 912.2.k.e yes 2
76.d even 2 1 inner 2736.2.k.f 2
228.b odd 2 1 912.2.k.b 2
456.l odd 2 1 3648.2.k.e 2
456.p even 2 1 3648.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.b 2 3.b odd 2 1
912.2.k.b 2 228.b odd 2 1
912.2.k.e yes 2 12.b even 2 1
912.2.k.e yes 2 57.d even 2 1
2736.2.k.e 2 4.b odd 2 1
2736.2.k.e 2 19.b odd 2 1
2736.2.k.f 2 1.a even 1 1 trivial
2736.2.k.f 2 76.d even 2 1 inner
3648.2.k.b 2 24.f even 2 1
3648.2.k.b 2 456.p even 2 1
3648.2.k.e 2 24.h odd 2 1
3648.2.k.e 2 456.l odd 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 12$$ $$T_{11}^{2} + 12$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$12 + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$48 + T^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$12 + T^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$108 + T^{2}$$
$47$ $$48 + T^{2}$$
$53$ $$192 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$48 + T^{2}$$
$97$ $$48 + T^{2}$$
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