# Properties

 Label 441.3.n.b Level $441$ Weight $3$ Character orbit 441.n 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.n (of order $$6$$, degree $$2$$, not 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{2} + 3 q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( -2 + 4 \zeta_{6} ) q^{5} + ( 6 - 3 \zeta_{6} ) q^{6} + ( -5 + 10 \zeta_{6} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{2} + 3 q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( -2 + 4 \zeta_{6} ) q^{5} + ( 6 - 3 \zeta_{6} ) q^{6} + ( -5 + 10 \zeta_{6} ) q^{8} + 9 q^{9} + 6 \zeta_{6} q^{10} + ( -1 + 2 \zeta_{6} ) q^{11} + ( -3 + 3 \zeta_{6} ) q^{12} + 4 \zeta_{6} q^{13} + ( -6 + 12 \zeta_{6} ) q^{15} + 11 \zeta_{6} q^{16} + ( -18 + 9 \zeta_{6} ) q^{17} + ( 18 - 9 \zeta_{6} ) q^{18} + ( -11 + 11 \zeta_{6} ) q^{19} + ( -2 - 2 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + ( 16 - 32 \zeta_{6} ) q^{23} + ( -15 + 30 \zeta_{6} ) q^{24} + 13 q^{25} + ( 4 + 4 \zeta_{6} ) q^{26} + 27 q^{27} + ( 26 + 26 \zeta_{6} ) q^{29} + 18 \zeta_{6} q^{30} + ( -32 + 32 \zeta_{6} ) q^{31} + ( -9 - 9 \zeta_{6} ) q^{32} + ( -3 + 6 \zeta_{6} ) q^{33} + ( -27 + 27 \zeta_{6} ) q^{34} + ( -9 + 9 \zeta_{6} ) q^{36} + ( 34 - 34 \zeta_{6} ) q^{37} + ( -11 + 22 \zeta_{6} ) q^{38} + 12 \zeta_{6} q^{39} -30 q^{40} + ( -14 + 7 \zeta_{6} ) q^{41} + ( 61 - 61 \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} ) q^{44} + ( -18 + 36 \zeta_{6} ) q^{45} -48 \zeta_{6} q^{46} + ( 56 - 28 \zeta_{6} ) q^{47} + 33 \zeta_{6} q^{48} + ( 26 - 13 \zeta_{6} ) q^{50} + ( -54 + 27 \zeta_{6} ) q^{51} -4 q^{52} + ( 54 - 27 \zeta_{6} ) q^{54} -6 q^{55} + ( -33 + 33 \zeta_{6} ) q^{57} + 78 q^{58} + ( -29 - 29 \zeta_{6} ) q^{59} + ( -6 - 6 \zeta_{6} ) q^{60} -56 \zeta_{6} q^{61} + ( -32 + 64 \zeta_{6} ) q^{62} -71 q^{64} + ( -16 + 8 \zeta_{6} ) q^{65} + 9 \zeta_{6} q^{66} + ( 31 - 31 \zeta_{6} ) q^{67} + ( 9 - 18 \zeta_{6} ) q^{68} + ( 48 - 96 \zeta_{6} ) q^{69} + ( -18 + 36 \zeta_{6} ) q^{71} + ( -45 + 90 \zeta_{6} ) q^{72} -65 \zeta_{6} q^{73} + ( 34 - 68 \zeta_{6} ) q^{74} + 39 q^{75} -11 \zeta_{6} q^{76} + ( 12 + 12 \zeta_{6} ) q^{78} -38 \zeta_{6} q^{79} + ( -44 + 22 \zeta_{6} ) q^{80} + 81 q^{81} + ( -21 + 21 \zeta_{6} ) q^{82} + ( -28 - 28 \zeta_{6} ) q^{83} -54 \zeta_{6} q^{85} + ( 61 - 122 \zeta_{6} ) q^{86} + ( 78 + 78 \zeta_{6} ) q^{87} -15 q^{88} + ( 72 + 72 \zeta_{6} ) q^{89} + 54 \zeta_{6} q^{90} + ( 16 + 16 \zeta_{6} ) q^{92} + ( -96 + 96 \zeta_{6} ) q^{93} + ( 84 - 84 \zeta_{6} ) q^{94} + ( -22 - 22 \zeta_{6} ) q^{95} + ( -27 - 27 \zeta_{6} ) q^{96} + ( 115 - 115 \zeta_{6} ) q^{97} + ( -9 + 18 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} + 6q^{3} - q^{4} + 9q^{6} + 18q^{9} + O(q^{10})$$ $$2q + 3q^{2} + 6q^{3} - q^{4} + 9q^{6} + 18q^{9} + 6q^{10} - 3q^{12} + 4q^{13} + 11q^{16} - 27q^{17} + 27q^{18} - 11q^{19} - 6q^{20} + 3q^{22} + 26q^{25} + 12q^{26} + 54q^{27} + 78q^{29} + 18q^{30} - 32q^{31} - 27q^{32} - 27q^{34} - 9q^{36} + 34q^{37} + 12q^{39} - 60q^{40} - 21q^{41} + 61q^{43} - 3q^{44} - 48q^{46} + 84q^{47} + 33q^{48} + 39q^{50} - 81q^{51} - 8q^{52} + 81q^{54} - 12q^{55} - 33q^{57} + 156q^{58} - 87q^{59} - 18q^{60} - 56q^{61} - 142q^{64} - 24q^{65} + 9q^{66} + 31q^{67} - 65q^{73} + 78q^{75} - 11q^{76} + 36q^{78} - 38q^{79} - 66q^{80} + 162q^{81} - 21q^{82} - 84q^{83} - 54q^{85} + 234q^{87} - 30q^{88} + 216q^{89} + 54q^{90} + 48q^{92} - 96q^{93} + 84q^{94} - 66q^{95} - 81q^{96} + 115q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
128.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 0.866025i 3.00000 −0.500000 + 0.866025i 3.46410i 4.50000 2.59808i 0 8.66025i 9.00000 3.00000 + 5.19615i
410.1 1.50000 + 0.866025i 3.00000 −0.500000 0.866025i 3.46410i 4.50000 + 2.59808i 0 8.66025i 9.00000 3.00000 5.19615i
 $$n$$: 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
63.n 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.n.b 2
7.b odd 2 1 441.3.n.a 2
7.c even 3 1 9.3.d.a 2
7.c even 3 1 441.3.j.a 2
7.d odd 6 1 441.3.j.b 2
7.d odd 6 1 441.3.r.a 2
9.d odd 6 1 441.3.j.a 2
21.h odd 6 1 27.3.d.a 2
28.g odd 6 1 144.3.q.a 2
35.j even 6 1 225.3.j.a 2
35.l odd 12 2 225.3.i.a 4
56.k odd 6 1 576.3.q.a 2
56.p even 6 1 576.3.q.b 2
63.g even 3 1 81.3.b.a 2
63.h even 3 1 27.3.d.a 2
63.i even 6 1 441.3.r.a 2
63.j odd 6 1 9.3.d.a 2
63.n odd 6 1 81.3.b.a 2
63.n odd 6 1 inner 441.3.n.b 2
63.o even 6 1 441.3.j.b 2
63.s even 6 1 441.3.n.a 2
84.n even 6 1 432.3.q.a 2
105.o odd 6 1 675.3.j.a 2
105.x even 12 2 675.3.i.a 4
168.s odd 6 1 1728.3.q.a 2
168.v even 6 1 1728.3.q.b 2
252.o even 6 1 1296.3.e.a 2
252.u odd 6 1 432.3.q.a 2
252.bb even 6 1 144.3.q.a 2
252.bl odd 6 1 1296.3.e.a 2
315.r even 6 1 675.3.j.a 2
315.br odd 6 1 225.3.j.a 2
315.bt odd 12 2 675.3.i.a 4
315.bv even 12 2 225.3.i.a 4
504.bi odd 6 1 576.3.q.b 2
504.bt even 6 1 576.3.q.a 2
504.ce odd 6 1 1728.3.q.b 2
504.cq even 6 1 1728.3.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 7.c even 3 1
9.3.d.a 2 63.j odd 6 1
27.3.d.a 2 21.h odd 6 1
27.3.d.a 2 63.h even 3 1
81.3.b.a 2 63.g even 3 1
81.3.b.a 2 63.n odd 6 1
144.3.q.a 2 28.g odd 6 1
144.3.q.a 2 252.bb even 6 1
225.3.i.a 4 35.l odd 12 2
225.3.i.a 4 315.bv even 12 2
225.3.j.a 2 35.j even 6 1
225.3.j.a 2 315.br odd 6 1
432.3.q.a 2 84.n even 6 1
432.3.q.a 2 252.u odd 6 1
441.3.j.a 2 7.c even 3 1
441.3.j.a 2 9.d odd 6 1
441.3.j.b 2 7.d odd 6 1
441.3.j.b 2 63.o even 6 1
441.3.n.a 2 7.b odd 2 1
441.3.n.a 2 63.s even 6 1
441.3.n.b 2 1.a even 1 1 trivial
441.3.n.b 2 63.n odd 6 1 inner
441.3.r.a 2 7.d odd 6 1
441.3.r.a 2 63.i even 6 1
576.3.q.a 2 56.k odd 6 1
576.3.q.a 2 504.bt even 6 1
576.3.q.b 2 56.p even 6 1
576.3.q.b 2 504.bi odd 6 1
675.3.i.a 4 105.x even 12 2
675.3.i.a 4 315.bt odd 12 2
675.3.j.a 2 105.o odd 6 1
675.3.j.a 2 315.r even 6 1
1296.3.e.a 2 252.o even 6 1
1296.3.e.a 2 252.bl odd 6 1
1728.3.q.a 2 168.s odd 6 1
1728.3.q.a 2 504.cq even 6 1
1728.3.q.b 2 168.v even 6 1
1728.3.q.b 2 504.ce odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 3 T_{2} + 3$$ $$T_{13}^{2} - 4 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 3 T + T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$243 + 27 T + T^{2}$$
$19$ $$121 + 11 T + T^{2}$$
$23$ $$768 + T^{2}$$
$29$ $$2028 - 78 T + T^{2}$$
$31$ $$1024 + 32 T + T^{2}$$
$37$ $$1156 - 34 T + T^{2}$$
$41$ $$147 + 21 T + T^{2}$$
$43$ $$3721 - 61 T + T^{2}$$
$47$ $$2352 - 84 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 + 87 T + T^{2}$$
$61$ $$3136 + 56 T + T^{2}$$
$67$ $$961 - 31 T + T^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$4225 + 65 T + T^{2}$$
$79$ $$1444 + 38 T + T^{2}$$
$83$ $$2352 + 84 T + T^{2}$$
$89$ $$15552 - 216 T + T^{2}$$
$97$ $$13225 - 115 T + T^{2}$$