# Properties

 Label 441.2.g.a Level $441$ Weight $2$ Character orbit 441.g Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ 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 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} -3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( -1 + \zeta_{6} ) q^{10} + 5 q^{11} + ( 1 + \zeta_{6} ) q^{12} + ( -5 + 5 \zeta_{6} ) q^{13} + ( 2 - \zeta_{6} ) q^{15} + ( 1 - \zeta_{6} ) q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{18} + \zeta_{6} q^{19} + \zeta_{6} q^{20} + ( -5 + 5 \zeta_{6} ) q^{22} + 3 q^{23} + ( -6 + 3 \zeta_{6} ) q^{24} -4 q^{25} -5 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + \zeta_{6} q^{29} + ( -1 + 2 \zeta_{6} ) q^{30} -5 \zeta_{6} q^{32} + ( 10 - 5 \zeta_{6} ) q^{33} + 3 \zeta_{6} q^{34} + 3 q^{36} -3 \zeta_{6} q^{37} - q^{38} + ( -5 + 10 \zeta_{6} ) q^{39} -3 q^{40} + ( -5 + 5 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + 5 \zeta_{6} q^{44} + ( 3 - 3 \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{46} + ( 1 - 2 \zeta_{6} ) q^{48} + ( 4 - 4 \zeta_{6} ) q^{50} + ( 3 - 6 \zeta_{6} ) q^{51} -5 q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + ( 3 + 3 \zeta_{6} ) q^{54} + 5 q^{55} + ( 1 + \zeta_{6} ) q^{57} - q^{58} + ( 1 + \zeta_{6} ) q^{60} + ( -14 + 14 \zeta_{6} ) q^{61} + 7 q^{64} + ( -5 + 5 \zeta_{6} ) q^{65} + ( -5 + 10 \zeta_{6} ) q^{66} -4 \zeta_{6} q^{67} + 3 q^{68} + ( 6 - 3 \zeta_{6} ) q^{69} -12 q^{71} + ( -9 + 9 \zeta_{6} ) q^{72} + ( 3 - 3 \zeta_{6} ) q^{73} + 3 q^{74} + ( -8 + 4 \zeta_{6} ) q^{75} + ( -1 + \zeta_{6} ) q^{76} + ( -5 - 5 \zeta_{6} ) q^{78} + ( -8 + 8 \zeta_{6} ) q^{79} + ( 1 - \zeta_{6} ) q^{80} -9 \zeta_{6} q^{81} -5 \zeta_{6} q^{82} -9 \zeta_{6} q^{83} + ( 3 - 3 \zeta_{6} ) q^{85} - q^{86} + ( 1 + \zeta_{6} ) q^{87} -15 q^{88} -13 \zeta_{6} q^{89} + 3 \zeta_{6} q^{90} + 3 \zeta_{6} q^{92} + \zeta_{6} q^{95} + ( -5 - 5 \zeta_{6} ) q^{96} -9 \zeta_{6} q^{97} + ( 15 - 15 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} + q^{4} + 2q^{5} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} + q^{4} + 2q^{5} - 6q^{8} + 3q^{9} - q^{10} + 10q^{11} + 3q^{12} - 5q^{13} + 3q^{15} + q^{16} + 3q^{17} + 3q^{18} + q^{19} + q^{20} - 5q^{22} + 6q^{23} - 9q^{24} - 8q^{25} - 5q^{26} + q^{29} - 5q^{32} + 15q^{33} + 3q^{34} + 6q^{36} - 3q^{37} - 2q^{38} - 6q^{40} - 5q^{41} + q^{43} + 5q^{44} + 3q^{45} - 3q^{46} + 4q^{50} - 10q^{52} + 9q^{53} + 9q^{54} + 10q^{55} + 3q^{57} - 2q^{58} + 3q^{60} - 14q^{61} + 14q^{64} - 5q^{65} - 4q^{67} + 6q^{68} + 9q^{69} - 24q^{71} - 9q^{72} + 3q^{73} + 6q^{74} - 12q^{75} - q^{76} - 15q^{78} - 8q^{79} + q^{80} - 9q^{81} - 5q^{82} - 9q^{83} + 3q^{85} - 2q^{86} + 3q^{87} - 30q^{88} - 13q^{89} + 3q^{90} + 3q^{92} + q^{95} - 15q^{96} - 9q^{97} + 15q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 −3.00000 1.50000 2.59808i −0.500000 + 0.866025i
79.1 −0.500000 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i 1.00000 1.73205i 0 −3.00000 1.50000 + 2.59808i −0.500000 0.866025i
 $$n$$: 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.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.g.a 2
3.b odd 2 1 1323.2.g.a 2
7.b odd 2 1 63.2.g.a 2
7.c even 3 1 441.2.f.a 2
7.c even 3 1 441.2.h.a 2
7.d odd 6 1 63.2.h.a yes 2
7.d odd 6 1 441.2.f.b 2
9.c even 3 1 441.2.h.a 2
9.d odd 6 1 1323.2.h.a 2
21.c even 2 1 189.2.g.a 2
21.g even 6 1 189.2.h.a 2
21.g even 6 1 1323.2.f.a 2
21.h odd 6 1 1323.2.f.b 2
21.h odd 6 1 1323.2.h.a 2
28.d even 2 1 1008.2.t.d 2
28.f even 6 1 1008.2.q.c 2
63.g even 3 1 inner 441.2.g.a 2
63.g even 3 1 3969.2.a.f 1
63.h even 3 1 441.2.f.a 2
63.i even 6 1 567.2.e.b 2
63.i even 6 1 1323.2.f.a 2
63.j odd 6 1 1323.2.f.b 2
63.k odd 6 1 63.2.g.a 2
63.k odd 6 1 3969.2.a.d 1
63.l odd 6 1 63.2.h.a yes 2
63.l odd 6 1 567.2.e.a 2
63.n odd 6 1 1323.2.g.a 2
63.n odd 6 1 3969.2.a.a 1
63.o even 6 1 189.2.h.a 2
63.o even 6 1 567.2.e.b 2
63.s even 6 1 189.2.g.a 2
63.s even 6 1 3969.2.a.c 1
63.t odd 6 1 441.2.f.b 2
63.t odd 6 1 567.2.e.a 2
84.h odd 2 1 3024.2.t.d 2
84.j odd 6 1 3024.2.q.b 2
252.n even 6 1 1008.2.t.d 2
252.s odd 6 1 3024.2.q.b 2
252.bi even 6 1 1008.2.q.c 2
252.bn odd 6 1 3024.2.t.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 7.b odd 2 1
63.2.g.a 2 63.k odd 6 1
63.2.h.a yes 2 7.d odd 6 1
63.2.h.a yes 2 63.l odd 6 1
189.2.g.a 2 21.c even 2 1
189.2.g.a 2 63.s even 6 1
189.2.h.a 2 21.g even 6 1
189.2.h.a 2 63.o even 6 1
441.2.f.a 2 7.c even 3 1
441.2.f.a 2 63.h even 3 1
441.2.f.b 2 7.d odd 6 1
441.2.f.b 2 63.t odd 6 1
441.2.g.a 2 1.a even 1 1 trivial
441.2.g.a 2 63.g even 3 1 inner
441.2.h.a 2 7.c even 3 1
441.2.h.a 2 9.c even 3 1
567.2.e.a 2 63.l odd 6 1
567.2.e.a 2 63.t odd 6 1
567.2.e.b 2 63.i even 6 1
567.2.e.b 2 63.o even 6 1
1008.2.q.c 2 28.f even 6 1
1008.2.q.c 2 252.bi even 6 1
1008.2.t.d 2 28.d even 2 1
1008.2.t.d 2 252.n even 6 1
1323.2.f.a 2 21.g even 6 1
1323.2.f.a 2 63.i even 6 1
1323.2.f.b 2 21.h odd 6 1
1323.2.f.b 2 63.j odd 6 1
1323.2.g.a 2 3.b odd 2 1
1323.2.g.a 2 63.n odd 6 1
1323.2.h.a 2 9.d odd 6 1
1323.2.h.a 2 21.h odd 6 1
3024.2.q.b 2 84.j odd 6 1
3024.2.q.b 2 252.s odd 6 1
3024.2.t.d 2 84.h odd 2 1
3024.2.t.d 2 252.bn odd 6 1
3969.2.a.a 1 63.n odd 6 1
3969.2.a.c 1 63.s even 6 1
3969.2.a.d 1 63.k odd 6 1
3969.2.a.f 1 63.g even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{5} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$( -3 + T )^{2}$$
$29$ $$1 - T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$25 + 5 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$169 + 13 T + T^{2}$$
$97$ $$81 + 9 T + T^{2}$$
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