# Properties

 Label 441.2.f.a Level $441$ Weight $2$ Character orbit 441.f Analytic conductor $3.521$ Analytic rank $1$ 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.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$1$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i −1.50000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.73205i 0 −3.00000 1.50000 + 2.59808i 1.00000
295.1 −0.500000 + 0.866025i −1.50000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.73205i 0 −3.00000 1.50000 2.59808i 1.00000
 $$n$$: 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.c 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.f.a 2
3.b odd 2 1 1323.2.f.b 2
7.b odd 2 1 441.2.f.b 2
7.c even 3 1 441.2.g.a 2
7.c even 3 1 441.2.h.a 2
7.d odd 6 1 63.2.g.a 2
7.d odd 6 1 63.2.h.a yes 2
9.c even 3 1 inner 441.2.f.a 2
9.c even 3 1 3969.2.a.f 1
9.d odd 6 1 1323.2.f.b 2
9.d odd 6 1 3969.2.a.a 1
21.c even 2 1 1323.2.f.a 2
21.g even 6 1 189.2.g.a 2
21.g even 6 1 189.2.h.a 2
21.h odd 6 1 1323.2.g.a 2
21.h odd 6 1 1323.2.h.a 2
28.f even 6 1 1008.2.q.c 2
28.f even 6 1 1008.2.t.d 2
63.g even 3 1 441.2.h.a 2
63.h even 3 1 441.2.g.a 2
63.i even 6 1 189.2.g.a 2
63.i even 6 1 567.2.e.b 2
63.j odd 6 1 1323.2.g.a 2
63.k odd 6 1 63.2.h.a yes 2
63.k odd 6 1 567.2.e.a 2
63.l odd 6 1 441.2.f.b 2
63.l odd 6 1 3969.2.a.d 1
63.n odd 6 1 1323.2.h.a 2
63.o even 6 1 1323.2.f.a 2
63.o even 6 1 3969.2.a.c 1
63.s even 6 1 189.2.h.a 2
63.s even 6 1 567.2.e.b 2
63.t odd 6 1 63.2.g.a 2
63.t odd 6 1 567.2.e.a 2
84.j odd 6 1 3024.2.q.b 2
84.j odd 6 1 3024.2.t.d 2
252.n even 6 1 1008.2.q.c 2
252.r odd 6 1 3024.2.t.d 2
252.bj even 6 1 1008.2.t.d 2
252.bn odd 6 1 3024.2.q.b 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

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