Properties

 Label 1728.3.q.b Level $1728$ Weight $3$ Character orbit 1728.q 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.q (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 4) q^{5} + ( - 2 \zeta_{6} + 2) q^{7}+O(q^{10})$$ q + (-2*z + 4) * q^5 + (-2*z + 2) * q^7 $$q + ( - 2 \zeta_{6} + 4) q^{5} + ( - 2 \zeta_{6} + 2) q^{7} + (\zeta_{6} + 1) q^{11} - 4 \zeta_{6} q^{13} + (18 \zeta_{6} - 9) q^{17} + 11 q^{19} + (16 \zeta_{6} - 32) q^{23} + (13 \zeta_{6} - 13) q^{25} + (26 \zeta_{6} + 26) q^{29} + 32 \zeta_{6} q^{31} + ( - 8 \zeta_{6} + 4) q^{35} + 34 q^{37} + ( - 7 \zeta_{6} + 14) q^{41} + ( - 61 \zeta_{6} + 61) q^{43} + ( - 28 \zeta_{6} - 28) q^{47} + 45 \zeta_{6} q^{49} + 6 q^{55} + (29 \zeta_{6} - 58) q^{59} + ( - 56 \zeta_{6} + 56) q^{61} + ( - 8 \zeta_{6} - 8) q^{65} + 31 \zeta_{6} q^{67} + (36 \zeta_{6} - 18) q^{71} + 65 q^{73} + ( - 2 \zeta_{6} + 4) q^{77} + ( - 38 \zeta_{6} + 38) q^{79} + (28 \zeta_{6} + 28) q^{83} + 54 \zeta_{6} q^{85} + (144 \zeta_{6} - 72) q^{89} - 8 q^{91} + ( - 22 \zeta_{6} + 44) q^{95} + ( - 115 \zeta_{6} + 115) q^{97} +O(q^{100})$$ q + (-2*z + 4) * q^5 + (-2*z + 2) * q^7 + (z + 1) * q^11 - 4*z * q^13 + (18*z - 9) * q^17 + 11 * q^19 + (16*z - 32) * q^23 + (13*z - 13) * q^25 + (26*z + 26) * q^29 + 32*z * q^31 + (-8*z + 4) * q^35 + 34 * q^37 + (-7*z + 14) * q^41 + (-61*z + 61) * q^43 + (-28*z - 28) * q^47 + 45*z * q^49 + 6 * q^55 + (29*z - 58) * q^59 + (-56*z + 56) * q^61 + (-8*z - 8) * q^65 + 31*z * q^67 + (36*z - 18) * q^71 + 65 * q^73 + (-2*z + 4) * q^77 + (-38*z + 38) * q^79 + (28*z + 28) * q^83 + 54*z * q^85 + (144*z - 72) * q^89 - 8 * q^91 + (-22*z + 44) * q^95 + (-115*z + 115) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 6 * q^5 + 2 * q^7 $$2 q + 6 q^{5} + 2 q^{7} + 3 q^{11} - 4 q^{13} + 22 q^{19} - 48 q^{23} - 13 q^{25} + 78 q^{29} + 32 q^{31} + 68 q^{37} + 21 q^{41} + 61 q^{43} - 84 q^{47} + 45 q^{49} + 12 q^{55} - 87 q^{59} + 56 q^{61} - 24 q^{65} + 31 q^{67} + 130 q^{73} + 6 q^{77} + 38 q^{79} + 84 q^{83} + 54 q^{85} - 16 q^{91} + 66 q^{95} + 115 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 + 2 * q^7 + 3 * q^11 - 4 * q^13 + 22 * q^19 - 48 * q^23 - 13 * q^25 + 78 * q^29 + 32 * q^31 + 68 * q^37 + 21 * q^41 + 61 * q^43 - 84 * q^47 + 45 * q^49 + 12 * q^55 - 87 * q^59 + 56 * q^61 - 24 * q^65 + 31 * q^67 + 130 * q^73 + 6 * q^77 + 38 * q^79 + 84 * q^83 + 54 * q^85 - 16 * q^91 + 66 * q^95 + 115 * q^97

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 3.00000 + 1.73205i 0 1.00000 + 1.73205i 0 0 0
1601.1 0 0 0 3.00000 1.73205i 0 1.00000 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
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.b 2
3.b odd 2 1 576.3.q.a 2
4.b odd 2 1 1728.3.q.a 2
8.b even 2 1 432.3.q.a 2
8.d odd 2 1 27.3.d.a 2
9.c even 3 1 576.3.q.a 2
9.d odd 6 1 inner 1728.3.q.b 2
12.b even 2 1 576.3.q.b 2
24.f even 2 1 9.3.d.a 2
24.h odd 2 1 144.3.q.a 2
36.f odd 6 1 576.3.q.b 2
36.h even 6 1 1728.3.q.a 2
40.e odd 2 1 675.3.j.a 2
40.k even 4 2 675.3.i.a 4
72.j odd 6 1 432.3.q.a 2
72.j odd 6 1 1296.3.e.a 2
72.l even 6 1 27.3.d.a 2
72.l even 6 1 81.3.b.a 2
72.n even 6 1 144.3.q.a 2
72.n even 6 1 1296.3.e.a 2
72.p odd 6 1 9.3.d.a 2
72.p odd 6 1 81.3.b.a 2
120.m even 2 1 225.3.j.a 2
120.q odd 4 2 225.3.i.a 4
168.e odd 2 1 441.3.r.a 2
168.v even 6 1 441.3.j.a 2
168.v even 6 1 441.3.n.b 2
168.be odd 6 1 441.3.j.b 2
168.be odd 6 1 441.3.n.a 2
360.z odd 6 1 225.3.j.a 2
360.bd even 6 1 675.3.j.a 2
360.bo even 12 2 225.3.i.a 4
360.bt odd 12 2 675.3.i.a 4
504.ba odd 6 1 441.3.j.a 2
504.be even 6 1 441.3.r.a 2
504.bf even 6 1 441.3.n.a 2
504.ce odd 6 1 441.3.n.b 2
504.cz even 6 1 441.3.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 24.f even 2 1
9.3.d.a 2 72.p odd 6 1
27.3.d.a 2 8.d odd 2 1
27.3.d.a 2 72.l even 6 1
81.3.b.a 2 72.l even 6 1
81.3.b.a 2 72.p odd 6 1
144.3.q.a 2 24.h odd 2 1
144.3.q.a 2 72.n even 6 1
225.3.i.a 4 120.q odd 4 2
225.3.i.a 4 360.bo even 12 2
225.3.j.a 2 120.m even 2 1
225.3.j.a 2 360.z odd 6 1
432.3.q.a 2 8.b even 2 1
432.3.q.a 2 72.j odd 6 1
441.3.j.a 2 168.v even 6 1
441.3.j.a 2 504.ba odd 6 1
441.3.j.b 2 168.be odd 6 1
441.3.j.b 2 504.cz even 6 1
441.3.n.a 2 168.be odd 6 1
441.3.n.a 2 504.bf even 6 1
441.3.n.b 2 168.v even 6 1
441.3.n.b 2 504.ce odd 6 1
441.3.r.a 2 168.e odd 2 1
441.3.r.a 2 504.be even 6 1
576.3.q.a 2 3.b odd 2 1
576.3.q.a 2 9.c even 3 1
576.3.q.b 2 12.b even 2 1
576.3.q.b 2 36.f odd 6 1
675.3.i.a 4 40.k even 4 2
675.3.i.a 4 360.bt odd 12 2
675.3.j.a 2 40.e odd 2 1
675.3.j.a 2 360.bd even 6 1
1296.3.e.a 2 72.j odd 6 1
1296.3.e.a 2 72.n even 6 1
1728.3.q.a 2 4.b odd 2 1
1728.3.q.a 2 36.h even 6 1
1728.3.q.b 2 1.a even 1 1 trivial
1728.3.q.b 2 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 $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{2} - 6T_{5} + 12$$ T5^2 - 6*T5 + 12 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 12$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} - 3T + 3$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$T^{2} + 243$$
$19$ $$(T - 11)^{2}$$
$23$ $$T^{2} + 48T + 768$$
$29$ $$T^{2} - 78T + 2028$$
$31$ $$T^{2} - 32T + 1024$$
$37$ $$(T - 34)^{2}$$
$41$ $$T^{2} - 21T + 147$$
$43$ $$T^{2} - 61T + 3721$$
$47$ $$T^{2} + 84T + 2352$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 87T + 2523$$
$61$ $$T^{2} - 56T + 3136$$
$67$ $$T^{2} - 31T + 961$$
$71$ $$T^{2} + 972$$
$73$ $$(T - 65)^{2}$$
$79$ $$T^{2} - 38T + 1444$$
$83$ $$T^{2} - 84T + 2352$$
$89$ $$T^{2} + 15552$$
$97$ $$T^{2} - 115T + 13225$$