# Properties

 Label 1728.2.a.d Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{5} + q^{7} + O(q^{10})$$ $$q - 3 q^{5} + q^{7} - 3 q^{11} + 4 q^{13} + 2 q^{19} + 6 q^{23} + 4 q^{25} - 6 q^{29} - 5 q^{31} - 3 q^{35} - 2 q^{37} - 6 q^{41} - 10 q^{43} - 6 q^{47} - 6 q^{49} - 9 q^{53} + 9 q^{55} + 12 q^{59} - 8 q^{61} - 12 q^{65} + 14 q^{67} - 7 q^{73} - 3 q^{77} - 8 q^{79} - 3 q^{83} - 18 q^{89} + 4 q^{91} - 6 q^{95} - q^{97} + O(q^{100})$$

## 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
 0
0 0 0 −3.00000 0 1.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.d 1
3.b odd 2 1 1728.2.a.z 1
4.b odd 2 1 1728.2.a.c 1
8.b even 2 1 432.2.a.g 1
8.d odd 2 1 54.2.a.a 1
12.b even 2 1 1728.2.a.y 1
24.f even 2 1 54.2.a.b yes 1
24.h odd 2 1 432.2.a.b 1
40.e odd 2 1 1350.2.a.r 1
40.k even 4 2 1350.2.c.b 2
56.e even 2 1 2646.2.a.a 1
72.j odd 6 2 1296.2.i.o 2
72.l even 6 2 162.2.c.b 2
72.n even 6 2 1296.2.i.c 2
72.p odd 6 2 162.2.c.c 2
88.g even 2 1 6534.2.a.bc 1
104.h odd 2 1 9126.2.a.u 1
120.m even 2 1 1350.2.a.h 1
120.q odd 4 2 1350.2.c.k 2
168.e odd 2 1 2646.2.a.bd 1
264.p odd 2 1 6534.2.a.b 1
312.h even 2 1 9126.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 8.d odd 2 1
54.2.a.b yes 1 24.f even 2 1
162.2.c.b 2 72.l even 6 2
162.2.c.c 2 72.p odd 6 2
432.2.a.b 1 24.h odd 2 1
432.2.a.g 1 8.b even 2 1
1296.2.i.c 2 72.n even 6 2
1296.2.i.o 2 72.j odd 6 2
1350.2.a.h 1 120.m even 2 1
1350.2.a.r 1 40.e odd 2 1
1350.2.c.b 2 40.k even 4 2
1350.2.c.k 2 120.q odd 4 2
1728.2.a.c 1 4.b odd 2 1
1728.2.a.d 1 1.a even 1 1 trivial
1728.2.a.y 1 12.b even 2 1
1728.2.a.z 1 3.b odd 2 1
2646.2.a.a 1 56.e even 2 1
2646.2.a.bd 1 168.e odd 2 1
6534.2.a.b 1 264.p odd 2 1
6534.2.a.bc 1 88.g even 2 1
9126.2.a.r 1 312.h even 2 1
9126.2.a.u 1 104.h odd 2 1

## Hecke kernels

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

 $$T_{5} + 3$$ $$T_{7} - 1$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$3 + T$$
$7$ $$-1 + T$$
$11$ $$3 + T$$
$13$ $$-4 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$-6 + T$$
$29$ $$6 + T$$
$31$ $$5 + T$$
$37$ $$2 + T$$
$41$ $$6 + T$$
$43$ $$10 + T$$
$47$ $$6 + T$$
$53$ $$9 + T$$
$59$ $$-12 + T$$
$61$ $$8 + T$$
$67$ $$-14 + T$$
$71$ $$T$$
$73$ $$7 + T$$
$79$ $$8 + T$$
$83$ $$3 + T$$
$89$ $$18 + T$$
$97$ $$1 + T$$