Properties

 Label 27.3.d.a Level $27$ Weight $3$ Character orbit 27.d Analytic conductor $0.736$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 27.d (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -6 q^{10} + ( 1 + \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( -4 + 2 \zeta_{6} ) q^{14} + ( 11 - 11 \zeta_{6} ) q^{16} + ( -9 + 18 \zeta_{6} ) q^{17} + 11 q^{19} + ( 2 + 2 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + ( 32 - 16 \zeta_{6} ) q^{23} + ( -13 + 13 \zeta_{6} ) q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} + 2 q^{28} + ( -26 - 26 \zeta_{6} ) q^{29} -32 \zeta_{6} q^{31} + ( -18 + 9 \zeta_{6} ) q^{32} + ( -27 + 27 \zeta_{6} ) q^{34} + ( 4 - 8 \zeta_{6} ) q^{35} -34 q^{37} + ( 11 + 11 \zeta_{6} ) q^{38} + 30 \zeta_{6} q^{40} + ( 14 - 7 \zeta_{6} ) q^{41} + ( 61 - 61 \zeta_{6} ) q^{43} + ( 1 - 2 \zeta_{6} ) q^{44} + 48 q^{46} + ( 28 + 28 \zeta_{6} ) q^{47} + 45 \zeta_{6} q^{49} + ( -26 + 13 \zeta_{6} ) q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} -6 q^{55} + ( 10 + 10 \zeta_{6} ) q^{56} -78 \zeta_{6} q^{58} + ( -58 + 29 \zeta_{6} ) q^{59} + ( -56 + 56 \zeta_{6} ) q^{61} + ( 32 - 64 \zeta_{6} ) q^{62} -71 q^{64} + ( -8 - 8 \zeta_{6} ) q^{65} + 31 \zeta_{6} q^{67} + ( 18 - 9 \zeta_{6} ) q^{68} + ( 12 - 12 \zeta_{6} ) q^{70} + ( 18 - 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( -34 - 34 \zeta_{6} ) q^{74} -11 \zeta_{6} q^{76} + ( -4 + 2 \zeta_{6} ) q^{77} + ( -38 + 38 \zeta_{6} ) q^{79} + ( -22 + 44 \zeta_{6} ) q^{80} + 21 q^{82} + ( 28 + 28 \zeta_{6} ) q^{83} -54 \zeta_{6} q^{85} + ( 122 - 61 \zeta_{6} ) q^{86} + ( 15 - 15 \zeta_{6} ) q^{88} + ( -72 + 144 \zeta_{6} ) q^{89} -8 q^{91} + ( -16 - 16 \zeta_{6} ) q^{92} + 84 \zeta_{6} q^{94} + ( -44 + 22 \zeta_{6} ) q^{95} + ( 115 - 115 \zeta_{6} ) q^{97} + ( -45 + 90 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - q^{4} - 6 q^{5} - 2 q^{7} + O(q^{10})$$ $$2 q + 3 q^{2} - q^{4} - 6 q^{5} - 2 q^{7} - 12 q^{10} + 3 q^{11} + 4 q^{13} - 6 q^{14} + 11 q^{16} + 22 q^{19} + 6 q^{20} + 3 q^{22} + 48 q^{23} - 13 q^{25} + 4 q^{28} - 78 q^{29} - 32 q^{31} - 27 q^{32} - 27 q^{34} - 68 q^{37} + 33 q^{38} + 30 q^{40} + 21 q^{41} + 61 q^{43} + 96 q^{46} + 84 q^{47} + 45 q^{49} - 39 q^{50} + 4 q^{52} - 12 q^{55} + 30 q^{56} - 78 q^{58} - 87 q^{59} - 56 q^{61} - 142 q^{64} - 24 q^{65} + 31 q^{67} + 27 q^{68} + 12 q^{70} + 130 q^{73} - 102 q^{74} - 11 q^{76} - 6 q^{77} - 38 q^{79} + 42 q^{82} + 84 q^{83} - 54 q^{85} + 183 q^{86} + 15 q^{88} - 16 q^{91} - 48 q^{92} + 84 q^{94} - 66 q^{95} + 115 q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 + 0.866025i 0 −0.500000 0.866025i −3.00000 + 1.73205i 0 −1.00000 + 1.73205i 8.66025i 0 −6.00000
17.1 1.50000 0.866025i 0 −0.500000 + 0.866025i −3.00000 1.73205i 0 −1.00000 1.73205i 8.66025i 0 −6.00000
 $$n$$: 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 27.3.d.a 2
3.b odd 2 1 9.3.d.a 2
4.b odd 2 1 432.3.q.a 2
5.b even 2 1 675.3.j.a 2
5.c odd 4 2 675.3.i.a 4
8.b even 2 1 1728.3.q.a 2
8.d odd 2 1 1728.3.q.b 2
9.c even 3 1 9.3.d.a 2
9.c even 3 1 81.3.b.a 2
9.d odd 6 1 inner 27.3.d.a 2
9.d odd 6 1 81.3.b.a 2
12.b even 2 1 144.3.q.a 2
15.d odd 2 1 225.3.j.a 2
15.e even 4 2 225.3.i.a 4
21.c even 2 1 441.3.r.a 2
21.g even 6 1 441.3.j.b 2
21.g even 6 1 441.3.n.a 2
21.h odd 6 1 441.3.j.a 2
21.h odd 6 1 441.3.n.b 2
24.f even 2 1 576.3.q.a 2
24.h odd 2 1 576.3.q.b 2
36.f odd 6 1 144.3.q.a 2
36.f odd 6 1 1296.3.e.a 2
36.h even 6 1 432.3.q.a 2
36.h even 6 1 1296.3.e.a 2
45.h odd 6 1 675.3.j.a 2
45.j even 6 1 225.3.j.a 2
45.k odd 12 2 225.3.i.a 4
45.l even 12 2 675.3.i.a 4
63.g even 3 1 441.3.j.a 2
63.h even 3 1 441.3.n.b 2
63.k odd 6 1 441.3.j.b 2
63.l odd 6 1 441.3.r.a 2
63.t odd 6 1 441.3.n.a 2
72.j odd 6 1 1728.3.q.a 2
72.l even 6 1 1728.3.q.b 2
72.n even 6 1 576.3.q.b 2
72.p odd 6 1 576.3.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 3.b odd 2 1
9.3.d.a 2 9.c even 3 1
27.3.d.a 2 1.a even 1 1 trivial
27.3.d.a 2 9.d odd 6 1 inner
81.3.b.a 2 9.c even 3 1
81.3.b.a 2 9.d odd 6 1
144.3.q.a 2 12.b even 2 1
144.3.q.a 2 36.f odd 6 1
225.3.i.a 4 15.e even 4 2
225.3.i.a 4 45.k odd 12 2
225.3.j.a 2 15.d odd 2 1
225.3.j.a 2 45.j even 6 1
432.3.q.a 2 4.b odd 2 1
432.3.q.a 2 36.h even 6 1
441.3.j.a 2 21.h odd 6 1
441.3.j.a 2 63.g even 3 1
441.3.j.b 2 21.g even 6 1
441.3.j.b 2 63.k odd 6 1
441.3.n.a 2 21.g even 6 1
441.3.n.a 2 63.t odd 6 1
441.3.n.b 2 21.h odd 6 1
441.3.n.b 2 63.h even 3 1
441.3.r.a 2 21.c even 2 1
441.3.r.a 2 63.l odd 6 1
576.3.q.a 2 24.f even 2 1
576.3.q.a 2 72.p odd 6 1
576.3.q.b 2 24.h odd 2 1
576.3.q.b 2 72.n even 6 1
675.3.i.a 4 5.c odd 4 2
675.3.i.a 4 45.l even 12 2
675.3.j.a 2 5.b even 2 1
675.3.j.a 2 45.h odd 6 1
1296.3.e.a 2 36.f odd 6 1
1296.3.e.a 2 36.h even 6 1
1728.3.q.a 2 8.b even 2 1
1728.3.q.a 2 72.j odd 6 1
1728.3.q.b 2 8.d odd 2 1
1728.3.q.b 2 72.l even 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(27, [\chi])$$.

Hecke characteristic polynomials

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