# Properties

 Label 1728.3.o.a Level 1728 Weight 3 Character orbit 1728.o Analytic conductor 47.085 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 + 4 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -4 + 4 \zeta_{6} ) q^{5} + ( -4 + 2 \zeta_{6} ) q^{7} + ( -14 + 7 \zeta_{6} ) q^{11} + ( -22 + 22 \zeta_{6} ) q^{13} + 11 q^{17} + ( -9 + 18 \zeta_{6} ) q^{19} + ( -14 - 14 \zeta_{6} ) q^{23} + 9 \zeta_{6} q^{25} -34 \zeta_{6} q^{29} + ( 4 + 4 \zeta_{6} ) q^{31} + ( 8 - 16 \zeta_{6} ) q^{35} + 16 q^{37} + ( 13 - 13 \zeta_{6} ) q^{41} + ( 58 - 29 \zeta_{6} ) q^{43} + ( -4 + 2 \zeta_{6} ) q^{47} + ( -37 + 37 \zeta_{6} ) q^{49} + 52 q^{53} + ( 28 - 56 \zeta_{6} ) q^{55} + ( -31 - 31 \zeta_{6} ) q^{59} -16 \zeta_{6} q^{61} -88 \zeta_{6} q^{65} + ( -67 - 67 \zeta_{6} ) q^{67} -25 q^{73} + ( 42 - 42 \zeta_{6} ) q^{77} + ( 32 - 16 \zeta_{6} ) q^{79} + ( 40 - 20 \zeta_{6} ) q^{83} + ( -44 + 44 \zeta_{6} ) q^{85} + 2 q^{89} + ( 44 - 88 \zeta_{6} ) q^{91} + ( -36 - 36 \zeta_{6} ) q^{95} + 43 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 6q^{7} - 21q^{11} - 22q^{13} + 22q^{17} - 42q^{23} + 9q^{25} - 34q^{29} + 12q^{31} + 32q^{37} + 13q^{41} + 87q^{43} - 6q^{47} - 37q^{49} + 104q^{53} - 93q^{59} - 16q^{61} - 88q^{65} - 201q^{67} - 50q^{73} + 42q^{77} + 48q^{79} + 60q^{83} - 44q^{85} + 4q^{89} - 108q^{95} + 43q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −2.00000 3.46410i 0 −3.00000 1.73205i 0 0 0
1279.1 0 0 0 −2.00000 + 3.46410i 0 −3.00000 + 1.73205i 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
36.f 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.o.a 2
3.b odd 2 1 576.3.o.b 2
4.b odd 2 1 1728.3.o.b 2
8.b even 2 1 108.3.f.a 2
8.d odd 2 1 108.3.f.b 2
9.c even 3 1 1728.3.o.b 2
9.d odd 6 1 576.3.o.a 2
12.b even 2 1 576.3.o.a 2
24.f even 2 1 36.3.f.a 2
24.h odd 2 1 36.3.f.b yes 2
36.f odd 6 1 inner 1728.3.o.a 2
36.h even 6 1 576.3.o.b 2
72.j odd 6 1 36.3.f.a 2
72.j odd 6 1 324.3.d.b 2
72.l even 6 1 36.3.f.b yes 2
72.l even 6 1 324.3.d.b 2
72.n even 6 1 108.3.f.b 2
72.n even 6 1 324.3.d.c 2
72.p odd 6 1 108.3.f.a 2
72.p odd 6 1 324.3.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 24.f even 2 1
36.3.f.a 2 72.j odd 6 1
36.3.f.b yes 2 24.h odd 2 1
36.3.f.b yes 2 72.l even 6 1
108.3.f.a 2 8.b even 2 1
108.3.f.a 2 72.p odd 6 1
108.3.f.b 2 8.d odd 2 1
108.3.f.b 2 72.n even 6 1
324.3.d.b 2 72.j odd 6 1
324.3.d.b 2 72.l even 6 1
324.3.d.c 2 72.n even 6 1
324.3.d.c 2 72.p odd 6 1
576.3.o.a 2 9.d odd 6 1
576.3.o.a 2 12.b even 2 1
576.3.o.b 2 3.b odd 2 1
576.3.o.b 2 36.h even 6 1
1728.3.o.a 2 1.a even 1 1 trivial
1728.3.o.a 2 36.f odd 6 1 inner
1728.3.o.b 2 4.b odd 2 1
1728.3.o.b 2 9.c even 3 1

## Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{7}^{2} + 6 T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T - 9 T^{2} + 100 T^{3} + 625 T^{4}$$
$7$ $$1 + 6 T + 61 T^{2} + 294 T^{3} + 2401 T^{4}$$
$11$ $$1 + 21 T + 268 T^{2} + 2541 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - T + 169 T^{2} )( 1 + 23 T + 169 T^{2} )$$
$17$ $$( 1 - 11 T + 289 T^{2} )^{2}$$
$19$ $$1 - 479 T^{2} + 130321 T^{4}$$
$23$ $$1 + 42 T + 1117 T^{2} + 22218 T^{3} + 279841 T^{4}$$
$29$ $$1 + 34 T + 315 T^{2} + 28594 T^{3} + 707281 T^{4}$$
$31$ $$1 - 12 T + 1009 T^{2} - 11532 T^{3} + 923521 T^{4}$$
$37$ $$( 1 - 16 T + 1369 T^{2} )^{2}$$
$41$ $$1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4}$$
$43$ $$1 - 87 T + 4372 T^{2} - 160863 T^{3} + 3418801 T^{4}$$
$47$ $$1 + 6 T + 2221 T^{2} + 13254 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 - 52 T + 2809 T^{2} )^{2}$$
$59$ $$1 + 93 T + 6364 T^{2} + 323733 T^{3} + 12117361 T^{4}$$
$61$ $$1 + 16 T - 3465 T^{2} + 59536 T^{3} + 13845841 T^{4}$$
$67$ $$( 1 + 67 T )^{2}( 1 + 67 T + 4489 T^{2} )$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 25 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 48 T + 7009 T^{2} - 299568 T^{3} + 38950081 T^{4}$$
$83$ $$1 - 60 T + 8089 T^{2} - 413340 T^{3} + 47458321 T^{4}$$
$89$ $$( 1 - 2 T + 7921 T^{2} )^{2}$$
$97$ $$1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4}$$