Properties

 Label 81.3.b.a Level $81$ Weight $3$ Character orbit 81.b Analytic conductor $2.207$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 81.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.20709014132$$ 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: $$2$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + 2 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + 2 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -6 q^{10} + ( 1 - 2 \zeta_{6} ) q^{11} -4 q^{13} + ( 2 - 4 \zeta_{6} ) q^{14} -11 q^{16} + ( -9 + 18 \zeta_{6} ) q^{17} + 11 q^{19} + ( 2 - 4 \zeta_{6} ) q^{20} -3 q^{22} + ( -16 + 32 \zeta_{6} ) q^{23} + 13 q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} + 2 q^{28} + ( -26 + 52 \zeta_{6} ) q^{29} + 32 q^{31} + ( 9 - 18 \zeta_{6} ) q^{32} + 27 q^{34} + ( 4 - 8 \zeta_{6} ) q^{35} -34 q^{37} + ( 11 - 22 \zeta_{6} ) q^{38} -30 q^{40} + ( -7 + 14 \zeta_{6} ) q^{41} -61 q^{43} + ( 1 - 2 \zeta_{6} ) q^{44} + 48 q^{46} + ( 28 - 56 \zeta_{6} ) q^{47} -45 q^{49} + ( 13 - 26 \zeta_{6} ) q^{50} -4 q^{52} -6 q^{55} + ( 10 - 20 \zeta_{6} ) q^{56} + 78 q^{58} + ( 29 - 58 \zeta_{6} ) q^{59} + 56 q^{61} + ( 32 - 64 \zeta_{6} ) q^{62} -71 q^{64} + ( -8 + 16 \zeta_{6} ) q^{65} -31 q^{67} + ( -9 + 18 \zeta_{6} ) q^{68} -12 q^{70} + ( 18 - 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( -34 + 68 \zeta_{6} ) q^{74} + 11 q^{76} + ( 2 - 4 \zeta_{6} ) q^{77} + 38 q^{79} + ( -22 + 44 \zeta_{6} ) q^{80} + 21 q^{82} + ( 28 - 56 \zeta_{6} ) q^{83} + 54 q^{85} + ( -61 + 122 \zeta_{6} ) q^{86} -15 q^{88} + ( -72 + 144 \zeta_{6} ) q^{89} -8 q^{91} + ( -16 + 32 \zeta_{6} ) q^{92} -84 q^{94} + ( 22 - 44 \zeta_{6} ) q^{95} -115 q^{97} + ( -45 + 90 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 4 q^{7} + O(q^{10})$$ $$2 q + 2 q^{4} + 4 q^{7} - 12 q^{10} - 8 q^{13} - 22 q^{16} + 22 q^{19} - 6 q^{22} + 26 q^{25} + 4 q^{28} + 64 q^{31} + 54 q^{34} - 68 q^{37} - 60 q^{40} - 122 q^{43} + 96 q^{46} - 90 q^{49} - 8 q^{52} - 12 q^{55} + 156 q^{58} + 112 q^{61} - 142 q^{64} - 62 q^{67} - 24 q^{70} + 130 q^{73} + 22 q^{76} + 76 q^{79} + 42 q^{82} + 108 q^{85} - 30 q^{88} - 16 q^{91} - 168 q^{94} - 230 q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0 1.00000 3.46410i 0 2.00000 8.66025i 0 −6.00000
80.2 1.73205i 0 1.00000 3.46410i 0 2.00000 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
3.b odd 2 1 inner

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{3}^{\mathrm{new}}(81, [\chi])$$.

Hecke characteristic polynomials

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