# Properties

 Label 1728.2.a.bd Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} -\beta q^{7} +O(q^{10})$$ $$q -\beta q^{5} -\beta q^{7} + q^{11} -4 q^{13} -2 \beta q^{19} + 6 q^{23} + 8 q^{25} + 2 \beta q^{29} + \beta q^{31} + 13 q^{35} -10 q^{37} -2 \beta q^{41} + 2 \beta q^{43} + 10 q^{47} + 6 q^{49} + \beta q^{53} -\beta q^{55} -4 q^{59} + 4 \beta q^{65} + 2 \beta q^{67} -8 q^{71} -3 q^{73} -\beta q^{77} + 4 \beta q^{79} + 9 q^{83} + 2 \beta q^{89} + 4 \beta q^{91} + 26 q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{11} - 8q^{13} + 12q^{23} + 16q^{25} + 26q^{35} - 20q^{37} + 20q^{47} + 12q^{49} - 8q^{59} - 16q^{71} - 6q^{73} + 18q^{83} + 52q^{95} + 14q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.30278 −1.30278
0 0 0 −3.60555 0 −3.60555 0 0 0
1.2 0 0 0 3.60555 0 3.60555 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.bd 2
3.b odd 2 1 1728.2.a.bc 2
4.b odd 2 1 1728.2.a.bc 2
8.b even 2 1 864.2.a.m 2
8.d odd 2 1 864.2.a.n yes 2
12.b even 2 1 inner 1728.2.a.bd 2
24.f even 2 1 864.2.a.m 2
24.h odd 2 1 864.2.a.n yes 2
72.j odd 6 2 2592.2.i.bb 4
72.l even 6 2 2592.2.i.bc 4
72.n even 6 2 2592.2.i.bc 4
72.p odd 6 2 2592.2.i.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 8.b even 2 1
864.2.a.m 2 24.f even 2 1
864.2.a.n yes 2 8.d odd 2 1
864.2.a.n yes 2 24.h odd 2 1
1728.2.a.bc 2 3.b odd 2 1
1728.2.a.bc 2 4.b odd 2 1
1728.2.a.bd 2 1.a even 1 1 trivial
1728.2.a.bd 2 12.b even 2 1 inner
2592.2.i.bb 4 72.j odd 6 2
2592.2.i.bb 4 72.p odd 6 2
2592.2.i.bc 4 72.l even 6 2
2592.2.i.bc 4 72.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5}^{2} - 13$$ $$T_{7}^{2} - 13$$ $$T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-13 + T^{2}$$
$7$ $$-13 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$-52 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$-52 + T^{2}$$
$31$ $$-13 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$-52 + T^{2}$$
$43$ $$-52 + T^{2}$$
$47$ $$( -10 + T )^{2}$$
$53$ $$-13 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$-52 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( 3 + T )^{2}$$
$79$ $$-208 + T^{2}$$
$83$ $$( -9 + T )^{2}$$
$89$ $$-52 + T^{2}$$
$97$ $$( -7 + T )^{2}$$