# Properties

 Label 1728.2.bc.d Level 1728 Weight 2 Character orbit 1728.bc Analytic conductor 13.798 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.bc (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( -3 + 2 \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{11} + ( -1 + \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{13} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{17} + ( 1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( 1 - 3 \zeta_{12} + \zeta_{12}^{2} ) q^{23} + ( 8 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{25} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{31} + ( 2 - 2 \zeta_{12}^{3} ) q^{35} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + 3 \zeta_{12} q^{41} + ( 3 - 5 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + ( -5 \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{49} + ( 6 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{53} + ( -2 + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{55} + ( 4 + 4 \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{59} + ( -6 + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( -6 \zeta_{12} - 14 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{65} + ( -3 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( -1 + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( -7 + 7 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{77} + 12 \zeta_{12}^{2} q^{79} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{83} + ( 4 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{85} -2 \zeta_{12}^{3} q^{89} + ( -6 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{95} + ( \zeta_{12} - 10 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{5} + 6q^{7} + O(q^{10})$$ $$4q + 8q^{5} + 6q^{7} - 10q^{11} - 10q^{13} + 16q^{17} + 6q^{19} + 6q^{23} + 24q^{25} + 6q^{29} + 8q^{31} + 8q^{35} + 12q^{37} + 16q^{43} - 2q^{47} - 6q^{49} + 16q^{53} + 6q^{59} - 12q^{61} - 28q^{65} - 16q^{67} - 22q^{77} + 24q^{79} - 2q^{83} + 20q^{85} - 20q^{91} - 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 0 0 0.267949 + 1.00000i 0 2.36603 1.36603i 0 0 0
721.1 0 0 0 3.73205 + 1.00000i 0 0.633975 + 0.366025i 0 0 0
1009.1 0 0 0 3.73205 1.00000i 0 0.633975 0.366025i 0 0 0
1585.1 0 0 0 0.267949 1.00000i 0 2.36603 + 1.36603i 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
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.bc.d 4
3.b odd 2 1 576.2.bb.d 4
4.b odd 2 1 432.2.y.c 4
9.c even 3 1 1728.2.bc.a 4
9.d odd 6 1 576.2.bb.c 4
12.b even 2 1 144.2.x.b 4
16.e even 4 1 1728.2.bc.a 4
16.f odd 4 1 432.2.y.b 4
36.f odd 6 1 432.2.y.b 4
36.h even 6 1 144.2.x.c yes 4
48.i odd 4 1 576.2.bb.c 4
48.k even 4 1 144.2.x.c yes 4
144.u even 12 1 144.2.x.b 4
144.v odd 12 1 432.2.y.c 4
144.w odd 12 1 576.2.bb.d 4
144.x even 12 1 inner 1728.2.bc.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b 4 12.b even 2 1
144.2.x.b 4 144.u even 12 1
144.2.x.c yes 4 36.h even 6 1
144.2.x.c yes 4 48.k even 4 1
432.2.y.b 4 16.f odd 4 1
432.2.y.b 4 36.f odd 6 1
432.2.y.c 4 4.b odd 2 1
432.2.y.c 4 144.v odd 12 1
576.2.bb.c 4 9.d odd 6 1
576.2.bb.c 4 48.i odd 4 1
576.2.bb.d 4 3.b odd 2 1
576.2.bb.d 4 144.w odd 12 1
1728.2.bc.a 4 9.c even 3 1
1728.2.bc.a 4 16.e even 4 1
1728.2.bc.d 4 1.a even 1 1 trivial
1728.2.bc.d 4 144.x even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 8 T_{5}^{3} + 20 T_{5}^{2} - 16 T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 8 T + 20 T^{2} + 4 T^{3} - 89 T^{4} + 20 T^{5} + 500 T^{6} - 1000 T^{7} + 625 T^{8}$$
$7$ $$1 - 6 T + 28 T^{2} - 96 T^{3} + 291 T^{4} - 672 T^{5} + 1372 T^{6} - 2058 T^{7} + 2401 T^{8}$$
$11$ $$1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 902 T^{5} + 4961 T^{6} + 13310 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 5 T + 13 T^{2} )^{2}( 1 + 23 T^{2} + 169 T^{4} )$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 2508 T^{5} + 6498 T^{6} - 41154 T^{7} + 130321 T^{8}$$
$23$ $$1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 5520 T^{5} + 27508 T^{6} - 73002 T^{7} + 279841 T^{8}$$
$29$ $$1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 1044 T^{5} + 15138 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 992 T^{5} - 1922 T^{6} - 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 21756 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8}$$
$43$ $$1 - 16 T + 65 T^{2} + 624 T^{3} - 8092 T^{4} + 26832 T^{5} + 120185 T^{6} - 1272112 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 6956 T^{5} - 35344 T^{6} + 207646 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 51728 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 35046 T^{5} + 156645 T^{6} - 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 97356 T^{5} + 669780 T^{6} + 2723772 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 16 T + 113 T^{2} + 384 T^{3} - 172 T^{4} + 25728 T^{5} + 507257 T^{6} + 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 158 T^{2} + 16131 T^{4} - 841982 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 174 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$1 + 20 T + 109 T^{2} + 1940 T^{3} + 36472 T^{4} + 188180 T^{5} + 1025581 T^{6} + 18253460 T^{7} + 88529281 T^{8}$$