# Properties

 Label 1728.4.a.bi Level $1728$ Weight $4$ Character orbit 1728.a Self dual yes Analytic conductor $101.955$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 216) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta ) q^{5} + ( -3 + 2 \beta ) q^{7} +O(q^{10})$$ $$q + ( -2 - \beta ) q^{5} + ( -3 + 2 \beta ) q^{7} + ( -26 + 3 \beta ) q^{11} + ( -13 + 2 \beta ) q^{13} + ( 94 - \beta ) q^{17} + ( 37 + 8 \beta ) q^{19} + ( 74 + 5 \beta ) q^{23} + ( 59 + 4 \beta ) q^{25} + ( -144 - 8 \beta ) q^{29} + ( -124 + 8 \beta ) q^{31} + ( -354 - \beta ) q^{35} + ( -171 - 10 \beta ) q^{37} -32 \beta q^{41} + ( 128 + 4 \beta ) q^{43} + ( 66 + \beta ) q^{47} + ( 386 - 12 \beta ) q^{49} + ( -476 - 14 \beta ) q^{53} + ( -488 + 20 \beta ) q^{55} + ( 502 - 21 \beta ) q^{59} + ( 17 + 34 \beta ) q^{61} + ( -334 + 9 \beta ) q^{65} + ( 433 + 16 \beta ) q^{67} + ( 388 - 30 \beta ) q^{71} + ( 937 - 12 \beta ) q^{73} + ( 1158 - 61 \beta ) q^{77} + ( 91 - 26 \beta ) q^{79} + ( 668 + 46 \beta ) q^{83} + ( -8 - 92 \beta ) q^{85} + ( 438 - 21 \beta ) q^{89} + ( 759 - 32 \beta ) q^{91} + ( -1514 - 53 \beta ) q^{95} + ( -19 + 88 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 6q^{7} - 52q^{11} - 26q^{13} + 188q^{17} + 74q^{19} + 148q^{23} + 118q^{25} - 288q^{29} - 248q^{31} - 708q^{35} - 342q^{37} + 256q^{43} + 132q^{47} + 772q^{49} - 952q^{53} - 976q^{55} + 1004q^{59} + 34q^{61} - 668q^{65} + 866q^{67} + 776q^{71} + 1874q^{73} + 2316q^{77} + 182q^{79} + 1336q^{83} - 16q^{85} + 876q^{89} + 1518q^{91} - 3028q^{95} - 38q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 0 0 −15.4164 0 23.8328 0 0 0
1.2 0 0 0 11.4164 0 −29.8328 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 1728.4.a.bi 2
3.b odd 2 1 1728.4.a.bq 2
4.b odd 2 1 1728.4.a.bj 2
8.b even 2 1 216.4.a.g yes 2
8.d odd 2 1 432.4.a.r 2
12.b even 2 1 1728.4.a.br 2
24.f even 2 1 432.4.a.p 2
24.h odd 2 1 216.4.a.f 2
72.j odd 6 2 648.4.i.r 4
72.n even 6 2 648.4.i.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.f 2 24.h odd 2 1
216.4.a.g yes 2 8.b even 2 1
432.4.a.p 2 24.f even 2 1
432.4.a.r 2 8.d odd 2 1
648.4.i.o 4 72.n even 6 2
648.4.i.r 4 72.j odd 6 2
1728.4.a.bi 2 1.a even 1 1 trivial
1728.4.a.bj 2 4.b odd 2 1
1728.4.a.bq 2 3.b odd 2 1
1728.4.a.br 2 12.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} - 176$$ $$T_{7}^{2} + 6 T_{7} - 711$$ $$T_{11}^{2} + 52 T_{11} - 944$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-176 + 4 T + T^{2}$$
$7$ $$-711 + 6 T + T^{2}$$
$11$ $$-944 + 52 T + T^{2}$$
$13$ $$-551 + 26 T + T^{2}$$
$17$ $$8656 - 188 T + T^{2}$$
$19$ $$-10151 - 74 T + T^{2}$$
$23$ $$976 - 148 T + T^{2}$$
$29$ $$9216 + 288 T + T^{2}$$
$31$ $$3856 + 248 T + T^{2}$$
$37$ $$11241 + 342 T + T^{2}$$
$41$ $$-184320 + T^{2}$$
$43$ $$13504 - 256 T + T^{2}$$
$47$ $$4176 - 132 T + T^{2}$$
$53$ $$191296 + 952 T + T^{2}$$
$59$ $$172624 - 1004 T + T^{2}$$
$61$ $$-207791 - 34 T + T^{2}$$
$67$ $$141409 - 866 T + T^{2}$$
$71$ $$-11456 - 776 T + T^{2}$$
$73$ $$852049 - 1874 T + T^{2}$$
$79$ $$-113399 - 182 T + T^{2}$$
$83$ $$65344 - 1336 T + T^{2}$$
$89$ $$112464 - 876 T + T^{2}$$
$97$ $$-1393559 + 38 T + T^{2}$$