# Properties

 Label 1728.3.q.d Level $1728$ Weight $3$ Character orbit 1728.q Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - 3 \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -3 - 3 \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 6 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{11} + ( 5 + 2 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{13} + ( 6 - 12 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 10 - 4 \beta_{1} + 2 \beta_{3} ) q^{19} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{23} + 2 \beta_{2} q^{25} + ( 6 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 19 + 3 \beta_{1} - 19 \beta_{2} - 6 \beta_{3} ) q^{31} + ( 3 - 6 \beta_{2} + 9 \beta_{3} ) q^{35} + ( -32 + 4 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 21 + 6 \beta_{1} + 21 \beta_{2} ) q^{41} + ( -3 \beta_{1} + 23 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -18 + 7 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} ) q^{47} + ( -6 + 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -30 + 60 \beta_{2} - 10 \beta_{3} ) q^{53} + ( -27 - 6 \beta_{1} + 3 \beta_{3} ) q^{55} + ( 21 + 13 \beta_{1} + 21 \beta_{2} ) q^{59} + ( -6 \beta_{1} - 31 \beta_{2} - 6 \beta_{3} ) q^{61} + ( -30 - 18 \beta_{1} + 15 \beta_{2} + 18 \beta_{3} ) q^{65} + ( 53 - 3 \beta_{1} - 53 \beta_{2} + 6 \beta_{3} ) q^{67} + ( -30 + 60 \beta_{2} + 8 \beta_{3} ) q^{71} + ( -52 - 12 \beta_{1} + 6 \beta_{3} ) q^{73} + ( -15 - 8 \beta_{1} - 15 \beta_{2} ) q^{77} + ( -5 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{79} + ( -126 + 5 \beta_{1} + 63 \beta_{2} - 5 \beta_{3} ) q^{83} + ( -54 + 6 \beta_{1} + 54 \beta_{2} - 12 \beta_{3} ) q^{85} + ( -30 + 60 \beta_{2} - 22 \beta_{3} ) q^{89} + ( -103 - 6 \beta_{1} + 3 \beta_{3} ) q^{91} + ( -30 + 18 \beta_{1} - 30 \beta_{2} ) q^{95} + ( -14 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{5} + 2 q^{7} + O(q^{10})$$ $$4 q - 18 q^{5} + 2 q^{7} + 18 q^{11} + 10 q^{13} + 40 q^{19} - 18 q^{23} + 4 q^{25} + 18 q^{29} + 38 q^{31} - 128 q^{37} + 126 q^{41} + 46 q^{43} - 54 q^{47} - 12 q^{49} - 108 q^{55} + 126 q^{59} - 62 q^{61} - 90 q^{65} + 106 q^{67} - 208 q^{73} - 90 q^{77} + 14 q^{79} - 378 q^{83} - 108 q^{85} - 412 q^{91} - 180 q^{95} + 14 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$$$/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −4.50000 2.59808i 0 −3.17423 5.49794i 0 0 0
449.2 0 0 0 −4.50000 2.59808i 0 4.17423 + 7.22999i 0 0 0
1601.1 0 0 0 −4.50000 + 2.59808i 0 −3.17423 + 5.49794i 0 0 0
1601.2 0 0 0 −4.50000 + 2.59808i 0 4.17423 7.22999i 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
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.q.d 4
3.b odd 2 1 576.3.q.f 4
4.b odd 2 1 1728.3.q.c 4
8.b even 2 1 54.3.d.a 4
8.d odd 2 1 432.3.q.d 4
9.c even 3 1 576.3.q.f 4
9.d odd 6 1 inner 1728.3.q.d 4
12.b even 2 1 576.3.q.e 4
24.f even 2 1 144.3.q.c 4
24.h odd 2 1 18.3.d.a 4
36.f odd 6 1 576.3.q.e 4
36.h even 6 1 1728.3.q.c 4
40.f even 2 1 1350.3.i.b 4
40.i odd 4 2 1350.3.k.a 8
72.j odd 6 1 54.3.d.a 4
72.j odd 6 1 162.3.b.a 4
72.l even 6 1 432.3.q.d 4
72.l even 6 1 1296.3.e.g 4
72.n even 6 1 18.3.d.a 4
72.n even 6 1 162.3.b.a 4
72.p odd 6 1 144.3.q.c 4
72.p odd 6 1 1296.3.e.g 4
120.i odd 2 1 450.3.i.b 4
120.w even 4 2 450.3.k.a 8
360.bh odd 6 1 1350.3.i.b 4
360.bk even 6 1 450.3.i.b 4
360.br even 12 2 1350.3.k.a 8
360.bu odd 12 2 450.3.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 24.h odd 2 1
18.3.d.a 4 72.n even 6 1
54.3.d.a 4 8.b even 2 1
54.3.d.a 4 72.j odd 6 1
144.3.q.c 4 24.f even 2 1
144.3.q.c 4 72.p odd 6 1
162.3.b.a 4 72.j odd 6 1
162.3.b.a 4 72.n even 6 1
432.3.q.d 4 8.d odd 2 1
432.3.q.d 4 72.l even 6 1
450.3.i.b 4 120.i odd 2 1
450.3.i.b 4 360.bk even 6 1
450.3.k.a 8 120.w even 4 2
450.3.k.a 8 360.bu odd 12 2
576.3.q.e 4 12.b even 2 1
576.3.q.e 4 36.f odd 6 1
576.3.q.f 4 3.b odd 2 1
576.3.q.f 4 9.c even 3 1
1296.3.e.g 4 72.l even 6 1
1296.3.e.g 4 72.p odd 6 1
1350.3.i.b 4 40.f even 2 1
1350.3.i.b 4 360.bh odd 6 1
1350.3.k.a 8 40.i odd 4 2
1350.3.k.a 8 360.br even 12 2
1728.3.q.c 4 4.b odd 2 1
1728.3.q.c 4 36.h even 6 1
1728.3.q.d 4 1.a even 1 1 trivial
1728.3.q.d 4 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} + 9 T_{5} + 27$$ $$T_{7}^{4} - 2 T_{7}^{3} + 57 T_{7}^{2} + 106 T_{7} + 2809$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 27 + 9 T + T^{2} )^{2}$$
$7$ $$2809 + 106 T + 57 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$81 - 162 T + 117 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$36481 + 1910 T + 291 T^{2} - 10 T^{3} + T^{4}$$
$17$ $$1296 + 360 T^{2} + T^{4}$$
$19$ $$( -116 - 20 T + T^{2} )^{2}$$
$23$ $$81 + 162 T + 117 T^{2} + 18 T^{3} + T^{4}$$
$29$ $$2025 + 810 T + 63 T^{2} - 18 T^{3} + T^{4}$$
$31$ $$15625 + 4750 T + 1569 T^{2} - 38 T^{3} + T^{4}$$
$37$ $$( 808 + 64 T + T^{2} )^{2}$$
$41$ $$455625 - 85050 T + 5967 T^{2} - 126 T^{3} + T^{4}$$
$43$ $$1849 - 1978 T + 2073 T^{2} - 46 T^{3} + T^{4}$$
$47$ $$408321 - 34506 T + 333 T^{2} + 54 T^{3} + T^{4}$$
$53$ $$810000 + 9000 T^{2} + T^{4}$$
$59$ $$2954961 + 216594 T + 3573 T^{2} - 126 T^{3} + T^{4}$$
$61$ $$966289 - 60946 T + 4827 T^{2} + 62 T^{3} + T^{4}$$
$67$ $$5396329 - 246238 T + 8913 T^{2} - 106 T^{3} + T^{4}$$
$71$ $$2396304 + 7704 T^{2} + T^{4}$$
$73$ $$( 760 + 104 T + T^{2} )^{2}$$
$79$ $$1692601 + 18214 T + 1497 T^{2} - 14 T^{3} + T^{4}$$
$83$ $$131262849 + 4330746 T + 59085 T^{2} + 378 T^{3} + T^{4}$$
$89$ $$36144144 + 22824 T^{2} + T^{4}$$
$97$ $$110986225 + 147490 T + 10731 T^{2} - 14 T^{3} + T^{4}$$