# Properties

 Label 9.3.d.a Level $9$ Weight $3$ Character orbit 9.d Analytic conductor $0.245$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 9.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.245232237924$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{5} + ( 6 - 3 \zeta_{6} ) q^{6} + ( -2 + 2 \zeta_{6} ) q^{7} + ( -5 + 10 \zeta_{6} ) q^{8} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( -3 + 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 4 - 2 \zeta_{6} ) q^{5} + ( 6 - 3 \zeta_{6} ) q^{6} + ( -2 + 2 \zeta_{6} ) q^{7} + ( -5 + 10 \zeta_{6} ) q^{8} -9 \zeta_{6} q^{9} -6 q^{10} + ( -1 - \zeta_{6} ) q^{11} + 3 q^{12} + 4 \zeta_{6} q^{13} + ( 4 - 2 \zeta_{6} ) q^{14} + ( -6 + 12 \zeta_{6} ) q^{15} + ( 11 - 11 \zeta_{6} ) q^{16} + ( 9 - 18 \zeta_{6} ) q^{17} + ( -9 + 18 \zeta_{6} ) q^{18} + 11 q^{19} + ( -2 - 2 \zeta_{6} ) q^{20} -6 \zeta_{6} q^{21} + 3 \zeta_{6} q^{22} + ( -32 + 16 \zeta_{6} ) q^{23} + ( -15 - 15 \zeta_{6} ) q^{24} + ( -13 + 13 \zeta_{6} ) q^{25} + ( 4 - 8 \zeta_{6} ) q^{26} + 27 q^{27} + 2 q^{28} + ( 26 + 26 \zeta_{6} ) q^{29} + ( 18 - 18 \zeta_{6} ) q^{30} -32 \zeta_{6} q^{31} + ( 18 - 9 \zeta_{6} ) q^{32} + ( 6 - 3 \zeta_{6} ) q^{33} + ( -27 + 27 \zeta_{6} ) q^{34} + ( -4 + 8 \zeta_{6} ) q^{35} + ( -9 + 9 \zeta_{6} ) q^{36} -34 q^{37} + ( -11 - 11 \zeta_{6} ) q^{38} -12 q^{39} + 30 \zeta_{6} q^{40} + ( -14 + 7 \zeta_{6} ) q^{41} + ( -6 + 12 \zeta_{6} ) q^{42} + ( 61 - 61 \zeta_{6} ) q^{43} + ( -1 + 2 \zeta_{6} ) q^{44} + ( -18 - 18 \zeta_{6} ) q^{45} + 48 q^{46} + ( -28 - 28 \zeta_{6} ) q^{47} + 33 \zeta_{6} q^{48} + 45 \zeta_{6} q^{49} + ( 26 - 13 \zeta_{6} ) q^{50} + ( 27 + 27 \zeta_{6} ) q^{51} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -27 - 27 \zeta_{6} ) q^{54} -6 q^{55} + ( -10 - 10 \zeta_{6} ) q^{56} + ( -33 + 33 \zeta_{6} ) q^{57} -78 \zeta_{6} q^{58} + ( 58 - 29 \zeta_{6} ) q^{59} + ( 12 - 6 \zeta_{6} ) q^{60} + ( -56 + 56 \zeta_{6} ) q^{61} + ( -32 + 64 \zeta_{6} ) q^{62} + 18 q^{63} -71 q^{64} + ( 8 + 8 \zeta_{6} ) q^{65} -9 q^{66} + 31 \zeta_{6} q^{67} + ( -18 + 9 \zeta_{6} ) q^{68} + ( 48 - 96 \zeta_{6} ) q^{69} + ( 12 - 12 \zeta_{6} ) q^{70} + ( -18 + 36 \zeta_{6} ) q^{71} + ( 90 - 45 \zeta_{6} ) q^{72} + 65 q^{73} + ( 34 + 34 \zeta_{6} ) q^{74} -39 \zeta_{6} q^{75} -11 \zeta_{6} q^{76} + ( 4 - 2 \zeta_{6} ) q^{77} + ( 12 + 12 \zeta_{6} ) q^{78} + ( -38 + 38 \zeta_{6} ) q^{79} + ( 22 - 44 \zeta_{6} ) q^{80} + ( -81 + 81 \zeta_{6} ) q^{81} + 21 q^{82} + ( -28 - 28 \zeta_{6} ) q^{83} + ( -6 + 6 \zeta_{6} ) q^{84} -54 \zeta_{6} q^{85} + ( -122 + 61 \zeta_{6} ) q^{86} + ( -156 + 78 \zeta_{6} ) q^{87} + ( 15 - 15 \zeta_{6} ) q^{88} + ( 72 - 144 \zeta_{6} ) q^{89} + 54 \zeta_{6} q^{90} -8 q^{91} + ( 16 + 16 \zeta_{6} ) q^{92} + 96 q^{93} + 84 \zeta_{6} q^{94} + ( 44 - 22 \zeta_{6} ) q^{95} + ( -27 + 54 \zeta_{6} ) q^{96} + ( 115 - 115 \zeta_{6} ) q^{97} + ( 45 - 90 \zeta_{6} ) q^{98} + ( -9 + 18 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 9 q^{6} - 2 q^{7} - 9 q^{9} + O(q^{10})$$ $$2 q - 3 q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 9 q^{6} - 2 q^{7} - 9 q^{9} - 12 q^{10} - 3 q^{11} + 6 q^{12} + 4 q^{13} + 6 q^{14} + 11 q^{16} + 22 q^{19} - 6 q^{20} - 6 q^{21} + 3 q^{22} - 48 q^{23} - 45 q^{24} - 13 q^{25} + 54 q^{27} + 4 q^{28} + 78 q^{29} + 18 q^{30} - 32 q^{31} + 27 q^{32} + 9 q^{33} - 27 q^{34} - 9 q^{36} - 68 q^{37} - 33 q^{38} - 24 q^{39} + 30 q^{40} - 21 q^{41} + 61 q^{43} - 54 q^{45} + 96 q^{46} - 84 q^{47} + 33 q^{48} + 45 q^{49} + 39 q^{50} + 81 q^{51} + 4 q^{52} - 81 q^{54} - 12 q^{55} - 30 q^{56} - 33 q^{57} - 78 q^{58} + 87 q^{59} + 18 q^{60} - 56 q^{61} + 36 q^{63} - 142 q^{64} + 24 q^{65} - 18 q^{66} + 31 q^{67} - 27 q^{68} + 12 q^{70} + 135 q^{72} + 130 q^{73} + 102 q^{74} - 39 q^{75} - 11 q^{76} + 6 q^{77} + 36 q^{78} - 38 q^{79} - 81 q^{81} + 42 q^{82} - 84 q^{83} - 6 q^{84} - 54 q^{85} - 183 q^{86} - 234 q^{87} + 15 q^{88} + 54 q^{90} - 16 q^{91} + 48 q^{92} + 192 q^{93} + 84 q^{94} + 66 q^{95} + 115 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i −1.50000 + 2.59808i −0.500000 0.866025i 3.00000 1.73205i 4.50000 2.59808i −1.00000 + 1.73205i 8.66025i −4.50000 7.79423i −6.00000
5.1 −1.50000 + 0.866025i −1.50000 2.59808i −0.500000 + 0.866025i 3.00000 + 1.73205i 4.50000 + 2.59808i −1.00000 1.73205i 8.66025i −4.50000 + 7.79423i −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 9.3.d.a 2
3.b odd 2 1 27.3.d.a 2
4.b odd 2 1 144.3.q.a 2
5.b even 2 1 225.3.j.a 2
5.c odd 4 2 225.3.i.a 4
7.b odd 2 1 441.3.r.a 2
7.c even 3 1 441.3.j.a 2
7.c even 3 1 441.3.n.b 2
7.d odd 6 1 441.3.j.b 2
7.d odd 6 1 441.3.n.a 2
8.b even 2 1 576.3.q.b 2
8.d odd 2 1 576.3.q.a 2
9.c even 3 1 27.3.d.a 2
9.c even 3 1 81.3.b.a 2
9.d odd 6 1 inner 9.3.d.a 2
9.d odd 6 1 81.3.b.a 2
12.b even 2 1 432.3.q.a 2
15.d odd 2 1 675.3.j.a 2
15.e even 4 2 675.3.i.a 4
24.f even 2 1 1728.3.q.b 2
24.h odd 2 1 1728.3.q.a 2
36.f odd 6 1 432.3.q.a 2
36.f odd 6 1 1296.3.e.a 2
36.h even 6 1 144.3.q.a 2
36.h even 6 1 1296.3.e.a 2
45.h odd 6 1 225.3.j.a 2
45.j even 6 1 675.3.j.a 2
45.k odd 12 2 675.3.i.a 4
45.l even 12 2 225.3.i.a 4
63.i even 6 1 441.3.n.a 2
63.j odd 6 1 441.3.n.b 2
63.n odd 6 1 441.3.j.a 2
63.o even 6 1 441.3.r.a 2
63.s even 6 1 441.3.j.b 2
72.j odd 6 1 576.3.q.b 2
72.l even 6 1 576.3.q.a 2
72.n even 6 1 1728.3.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 1.a even 1 1 trivial
9.3.d.a 2 9.d odd 6 1 inner
27.3.d.a 2 3.b odd 2 1
27.3.d.a 2 9.c even 3 1
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 4.b odd 2 1
144.3.q.a 2 36.h even 6 1
225.3.i.a 4 5.c odd 4 2
225.3.i.a 4 45.l even 12 2
225.3.j.a 2 5.b even 2 1
225.3.j.a 2 45.h odd 6 1
432.3.q.a 2 12.b even 2 1
432.3.q.a 2 36.f odd 6 1
441.3.j.a 2 7.c even 3 1
441.3.j.a 2 63.n odd 6 1
441.3.j.b 2 7.d odd 6 1
441.3.j.b 2 63.s even 6 1
441.3.n.a 2 7.d odd 6 1
441.3.n.a 2 63.i even 6 1
441.3.n.b 2 7.c even 3 1
441.3.n.b 2 63.j odd 6 1
441.3.r.a 2 7.b odd 2 1
441.3.r.a 2 63.o even 6 1
576.3.q.a 2 8.d odd 2 1
576.3.q.a 2 72.l even 6 1
576.3.q.b 2 8.b even 2 1
576.3.q.b 2 72.j odd 6 1
675.3.i.a 4 15.e even 4 2
675.3.i.a 4 45.k odd 12 2
675.3.j.a 2 15.d odd 2 1
675.3.j.a 2 45.j even 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 24.h odd 2 1
1728.3.q.a 2 72.n even 6 1
1728.3.q.b 2 24.f even 2 1
1728.3.q.b 2 72.p odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(9, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$12 - 6 T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$3 + 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$243 + T^{2}$$
$19$ $$( -11 + T )^{2}$$
$23$ $$768 + 48 T + T^{2}$$
$29$ $$2028 - 78 T + T^{2}$$
$31$ $$1024 + 32 T + T^{2}$$
$37$ $$( 34 + T )^{2}$$
$41$ $$147 + 21 T + T^{2}$$
$43$ $$3721 - 61 T + T^{2}$$
$47$ $$2352 + 84 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 - 87 T + T^{2}$$
$61$ $$3136 + 56 T + T^{2}$$
$67$ $$961 - 31 T + T^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$( -65 + T )^{2}$$
$79$ $$1444 + 38 T + T^{2}$$
$83$ $$2352 + 84 T + T^{2}$$
$89$ $$15552 + T^{2}$$
$97$ $$13225 - 115 T + T^{2}$$