# Properties

 Label 441.3.r.a Level $441$ Weight $3$ Character orbit 441.r Analytic conductor $12.016$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 441.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0163796583$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
50.1
 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 0 8.66025i −4.50000 + 7.79423i 6.00000
344.1 −1.50000 0.866025i 1.50000 2.59808i −0.500000 0.866025i −3.00000 + 1.73205i −4.50000 + 2.59808i 0 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 441.3.r.a 2
7.b odd 2 1 9.3.d.a 2
7.c even 3 1 441.3.j.b 2
7.c even 3 1 441.3.n.a 2
7.d odd 6 1 441.3.j.a 2
7.d odd 6 1 441.3.n.b 2
9.d odd 6 1 inner 441.3.r.a 2
21.c even 2 1 27.3.d.a 2
28.d even 2 1 144.3.q.a 2
35.c odd 2 1 225.3.j.a 2
35.f even 4 2 225.3.i.a 4
56.e even 2 1 576.3.q.a 2
56.h odd 2 1 576.3.q.b 2
63.i even 6 1 441.3.n.b 2
63.j odd 6 1 441.3.n.a 2
63.l odd 6 1 27.3.d.a 2
63.l odd 6 1 81.3.b.a 2
63.n odd 6 1 441.3.j.b 2
63.o even 6 1 9.3.d.a 2
63.o even 6 1 81.3.b.a 2
63.s even 6 1 441.3.j.a 2
84.h odd 2 1 432.3.q.a 2
105.g even 2 1 675.3.j.a 2
105.k odd 4 2 675.3.i.a 4
168.e odd 2 1 1728.3.q.b 2
168.i even 2 1 1728.3.q.a 2
252.s odd 6 1 144.3.q.a 2
252.s odd 6 1 1296.3.e.a 2
252.bi even 6 1 432.3.q.a 2
252.bi even 6 1 1296.3.e.a 2
315.z even 6 1 225.3.j.a 2
315.bg odd 6 1 675.3.j.a 2
315.cb even 12 2 675.3.i.a 4
315.cf odd 12 2 225.3.i.a 4
504.be even 6 1 1728.3.q.b 2
504.bn odd 6 1 1728.3.q.a 2
504.cc even 6 1 576.3.q.b 2
504.co 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 7.b odd 2 1
9.3.d.a 2 63.o even 6 1
27.3.d.a 2 21.c even 2 1
27.3.d.a 2 63.l odd 6 1
81.3.b.a 2 63.l odd 6 1
81.3.b.a 2 63.o even 6 1
144.3.q.a 2 28.d even 2 1
144.3.q.a 2 252.s odd 6 1
225.3.i.a 4 35.f even 4 2
225.3.i.a 4 315.cf odd 12 2
225.3.j.a 2 35.c odd 2 1
225.3.j.a 2 315.z even 6 1
432.3.q.a 2 84.h odd 2 1
432.3.q.a 2 252.bi even 6 1
441.3.j.a 2 7.d odd 6 1
441.3.j.a 2 63.s even 6 1
441.3.j.b 2 7.c even 3 1
441.3.j.b 2 63.n odd 6 1
441.3.n.a 2 7.c even 3 1
441.3.n.a 2 63.j odd 6 1
441.3.n.b 2 7.d odd 6 1
441.3.n.b 2 63.i even 6 1
441.3.r.a 2 1.a even 1 1 trivial
441.3.r.a 2 9.d odd 6 1 inner
576.3.q.a 2 56.e even 2 1
576.3.q.a 2 504.co odd 6 1
576.3.q.b 2 56.h odd 2 1
576.3.q.b 2 504.cc even 6 1
675.3.i.a 4 105.k odd 4 2
675.3.i.a 4 315.cb even 12 2
675.3.j.a 2 105.g even 2 1
675.3.j.a 2 315.bg odd 6 1
1296.3.e.a 2 252.s odd 6 1
1296.3.e.a 2 252.bi even 6 1
1728.3.q.a 2 168.i even 2 1
1728.3.q.a 2 504.bn odd 6 1
1728.3.q.b 2 168.e odd 2 1
1728.3.q.b 2 504.be even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + 3 T_{2} + 3$$ $$T_{5}^{2} + 6 T_{5} + 12$$

## Hecke characteristic polynomials

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