# Properties

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

# Related objects

## Newspace parameters

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

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + 3 q^{5} -3 q^{6} + ( -1 + 2 \zeta_{6} ) q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + 3 q^{5} -3 q^{6} + ( -1 + 2 \zeta_{6} ) q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( -3 - 3 \zeta_{6} ) q^{10} + ( 1 - 2 \zeta_{6} ) q^{11} + ( 1 + \zeta_{6} ) q^{12} + ( 1 + \zeta_{6} ) q^{13} + ( 6 - 3 \zeta_{6} ) q^{15} + ( 5 - 5 \zeta_{6} ) q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -6 + 3 \zeta_{6} ) q^{18} + ( 6 - 3 \zeta_{6} ) q^{19} + 3 \zeta_{6} q^{20} + ( -3 + 3 \zeta_{6} ) q^{22} + ( -3 + 6 \zeta_{6} ) q^{23} + 3 \zeta_{6} q^{24} + 4 q^{25} -3 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -6 + 3 \zeta_{6} ) q^{29} -9 q^{30} + ( 4 - 2 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} -3 \zeta_{6} q^{33} + ( 6 - 3 \zeta_{6} ) q^{34} + 3 q^{36} -7 \zeta_{6} q^{37} -9 q^{38} + 3 q^{39} + ( -3 + 6 \zeta_{6} ) q^{40} + ( -3 + 3 \zeta_{6} ) q^{41} -\zeta_{6} q^{43} + ( 2 - \zeta_{6} ) q^{44} + ( 9 - 9 \zeta_{6} ) q^{45} + ( 9 - 9 \zeta_{6} ) q^{46} + ( 5 - 10 \zeta_{6} ) q^{48} + ( -4 - 4 \zeta_{6} ) q^{50} + ( -3 + 6 \zeta_{6} ) q^{51} + ( -1 + 2 \zeta_{6} ) q^{52} + ( -5 - 5 \zeta_{6} ) q^{53} + ( -9 + 9 \zeta_{6} ) q^{54} + ( 3 - 6 \zeta_{6} ) q^{55} + ( 9 - 9 \zeta_{6} ) q^{57} + 9 q^{58} + ( 3 + 3 \zeta_{6} ) q^{60} + ( -8 - 8 \zeta_{6} ) q^{61} -6 q^{62} - q^{64} + ( 3 + 3 \zeta_{6} ) q^{65} + ( -3 + 6 \zeta_{6} ) q^{66} + 4 \zeta_{6} q^{67} -3 q^{68} + 9 \zeta_{6} q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( 3 + 3 \zeta_{6} ) q^{72} + ( 3 + 3 \zeta_{6} ) q^{73} + ( -7 + 14 \zeta_{6} ) q^{74} + ( 8 - 4 \zeta_{6} ) q^{75} + ( 3 + 3 \zeta_{6} ) q^{76} + ( -3 - 3 \zeta_{6} ) q^{78} + ( -8 + 8 \zeta_{6} ) q^{79} + ( 15 - 15 \zeta_{6} ) q^{80} -9 \zeta_{6} q^{81} + ( 6 - 3 \zeta_{6} ) q^{82} + 15 \zeta_{6} q^{83} + ( -9 + 9 \zeta_{6} ) q^{85} + ( -1 + 2 \zeta_{6} ) q^{86} + ( -9 + 9 \zeta_{6} ) q^{87} + 3 q^{88} -3 \zeta_{6} q^{89} + ( -18 + 9 \zeta_{6} ) q^{90} + ( -6 + 3 \zeta_{6} ) q^{92} + ( 6 - 6 \zeta_{6} ) q^{93} + ( 18 - 9 \zeta_{6} ) q^{95} + ( -9 + 9 \zeta_{6} ) q^{96} + ( 2 - \zeta_{6} ) q^{97} + ( -3 - 3 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} + 3q^{3} + q^{4} + 6q^{5} - 6q^{6} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{2} + 3q^{3} + q^{4} + 6q^{5} - 6q^{6} + 3q^{9} - 9q^{10} + 3q^{12} + 3q^{13} + 9q^{15} + 5q^{16} - 3q^{17} - 9q^{18} + 9q^{19} + 3q^{20} - 3q^{22} + 3q^{24} + 8q^{25} - 3q^{26} - 9q^{29} - 18q^{30} + 6q^{31} - 9q^{32} - 3q^{33} + 9q^{34} + 6q^{36} - 7q^{37} - 18q^{38} + 6q^{39} - 3q^{41} - q^{43} + 3q^{44} + 9q^{45} + 9q^{46} - 12q^{50} - 15q^{53} - 9q^{54} + 9q^{57} + 18q^{58} + 9q^{60} - 24q^{61} - 12q^{62} - 2q^{64} + 9q^{65} + 4q^{67} - 6q^{68} + 9q^{69} + 9q^{72} + 9q^{73} + 12q^{75} + 9q^{76} - 9q^{78} - 8q^{79} + 15q^{80} - 9q^{81} + 9q^{82} + 15q^{83} - 9q^{85} - 9q^{87} + 6q^{88} - 3q^{89} - 27q^{90} - 9q^{92} + 6q^{93} + 27q^{95} - 9q^{96} + 3q^{97} - 9q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i 3.00000 −3.00000 0 1.73205i 1.50000 + 2.59808i −4.50000 + 2.59808i
374.1 −1.50000 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i 3.00000 −3.00000 0 1.73205i 1.50000 2.59808i −4.50000 2.59808i
 $$n$$: 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.s even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + T^{2}$$
$13$ $$3 - 3 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$27 - 9 T + T^{2}$$
$23$ $$27 + T^{2}$$
$29$ $$27 + 9 T + T^{2}$$
$31$ $$12 - 6 T + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$75 + 15 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$192 + 24 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$27 - 9 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$225 - 15 T + T^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$3 - 3 T + T^{2}$$