# Properties

 Label 1728.4.a.z Level 1728 Weight 4 Character orbit 1728.a Self dual yes Analytic conductor 101.955 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$101.955300490$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 9q^{5} + q^{7} + O(q^{10})$$ $$q + 9q^{5} + q^{7} - 63q^{11} + 28q^{13} - 72q^{17} + 98q^{19} + 126q^{23} - 44q^{25} - 126q^{29} + 259q^{31} + 9q^{35} - 386q^{37} + 450q^{41} - 34q^{43} - 54q^{47} - 342q^{49} - 693q^{53} - 567q^{55} - 180q^{59} + 280q^{61} + 252q^{65} - 586q^{67} + 504q^{71} + 161q^{73} - 63q^{77} - 440q^{79} - 999q^{83} - 648q^{85} - 882q^{89} + 28q^{91} + 882q^{95} - 721q^{97} + O(q^{100})$$

## 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 9.00000 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

## 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 1728.4.a.z 1
3.b odd 2 1 1728.4.a.h 1
4.b odd 2 1 1728.4.a.y 1
8.b even 2 1 432.4.a.c 1
8.d odd 2 1 108.4.a.a 1
12.b even 2 1 1728.4.a.g 1
24.f even 2 1 108.4.a.d yes 1
24.h odd 2 1 432.4.a.l 1
72.l even 6 2 324.4.e.b 2
72.p odd 6 2 324.4.e.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 8.d odd 2 1
108.4.a.d yes 1 24.f even 2 1
324.4.e.b 2 72.l even 6 2
324.4.e.g 2 72.p odd 6 2
432.4.a.c 1 8.b even 2 1
432.4.a.l 1 24.h odd 2 1
1728.4.a.g 1 12.b even 2 1
1728.4.a.h 1 3.b odd 2 1
1728.4.a.y 1 4.b odd 2 1
1728.4.a.z 1 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5} - 9$$ $$T_{7} - 1$$ $$T_{11} + 63$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 9 T + 125 T^{2}$$
$7$ $$1 - T + 343 T^{2}$$
$11$ $$1 + 63 T + 1331 T^{2}$$
$13$ $$1 - 28 T + 2197 T^{2}$$
$17$ $$1 + 72 T + 4913 T^{2}$$
$19$ $$1 - 98 T + 6859 T^{2}$$
$23$ $$1 - 126 T + 12167 T^{2}$$
$29$ $$1 + 126 T + 24389 T^{2}$$
$31$ $$1 - 259 T + 29791 T^{2}$$
$37$ $$1 + 386 T + 50653 T^{2}$$
$41$ $$1 - 450 T + 68921 T^{2}$$
$43$ $$1 + 34 T + 79507 T^{2}$$
$47$ $$1 + 54 T + 103823 T^{2}$$
$53$ $$1 + 693 T + 148877 T^{2}$$
$59$ $$1 + 180 T + 205379 T^{2}$$
$61$ $$1 - 280 T + 226981 T^{2}$$
$67$ $$1 + 586 T + 300763 T^{2}$$
$71$ $$1 - 504 T + 357911 T^{2}$$
$73$ $$1 - 161 T + 389017 T^{2}$$
$79$ $$1 + 440 T + 493039 T^{2}$$
$83$ $$1 + 999 T + 571787 T^{2}$$
$89$ $$1 + 882 T + 704969 T^{2}$$
$97$ $$1 + 721 T + 912673 T^{2}$$
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