Newform orbit 441.3.n.a

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

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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 2) q^{2} - 3 q^{3} + (\zeta_{6} - 1) q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + (3 \zeta_{6} - 6) q^{6} + (10 \zeta_{6} - 5) q^{8} + 9 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 6 q^{3} - q^{4} - 9 q^{6} + 18 q^{9}+O(q^{10}) \)

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

0.500000	+	0.866025i
0.500000	−	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

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.a		2
7.b	odd	2	1		441.3.n.b		2
7.c	even	3	1		441.3.j.b		2
7.c	even	3	1		441.3.r.a		2
7.d	odd	6	1		9.3.d.a	✓	2
7.d	odd	6	1		441.3.j.a		2
9.d	odd	6	1		441.3.j.b		2
21.g	even	6	1		27.3.d.a		2
28.f	even	6	1		144.3.q.a		2
35.i	odd	6	1		225.3.j.a		2
35.k	even	12	2		225.3.i.a		4
56.j	odd	6	1		576.3.q.b		2
56.m	even	6	1		576.3.q.a		2
63.i	even	6	1		9.3.d.a	✓	2
63.j	odd	6	1		441.3.r.a		2
63.k	odd	6	1		81.3.b.a		2
63.n	odd	6	1	inner	441.3.n.a		2
63.o	even	6	1		441.3.j.a		2
63.s	even	6	1		81.3.b.a		2
63.s	even	6	1		441.3.n.b		2
63.t	odd	6	1		27.3.d.a		2
84.j	odd	6	1		432.3.q.a		2
105.p	even	6	1		675.3.j.a		2
105.w	odd	12	2		675.3.i.a		4
168.ba	even	6	1		1728.3.q.a		2
168.be	odd	6	1		1728.3.q.b		2
252.n	even	6	1		1296.3.e.a		2
252.r	odd	6	1		144.3.q.a		2
252.bj	even	6	1		432.3.q.a		2
252.bn	odd	6	1		1296.3.e.a		2
315.q	odd	6	1		675.3.j.a		2
315.bq	even	6	1		225.3.j.a		2
315.bs	even	12	2		675.3.i.a		4
315.bu	odd	12	2		225.3.i.a		4
504.bf	even	6	1		1728.3.q.b		2
504.bp	odd	6	1		1728.3.q.a		2
504.ca	even	6	1		576.3.q.b		2
504.cm	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.d	odd	6	1
9.3.d.a	✓	2	63.i	even	6	1
27.3.d.a		2	21.g	even	6	1
27.3.d.a		2	63.t	odd	6	1
81.3.b.a		2	63.k	odd	6	1
81.3.b.a		2	63.s	even	6	1
144.3.q.a		2	28.f	even	6	1
144.3.q.a		2	252.r	odd	6	1
225.3.i.a		4	35.k	even	12	2
225.3.i.a		4	315.bu	odd	12	2
225.3.j.a		2	35.i	odd	6	1
225.3.j.a		2	315.bq	even	6	1
432.3.q.a		2	84.j	odd	6	1
432.3.q.a		2	252.bj	even	6	1
441.3.j.a		2	7.d	odd	6	1
441.3.j.a		2	63.o	even	6	1
441.3.j.b		2	7.c	even	3	1
441.3.j.b		2	9.d	odd	6	1
441.3.n.a		2	1.a	even	1	1	trivial
441.3.n.a		2	63.n	odd	6	1	inner
441.3.n.b		2	7.b	odd	2	1
441.3.n.b		2	63.s	even	6	1
441.3.r.a		2	7.c	even	3	1
441.3.r.a		2	63.j	odd	6	1
576.3.q.a		2	56.m	even	6	1
576.3.q.a		2	504.cm	odd	6	1
576.3.q.b		2	56.j	odd	6	1
576.3.q.b		2	504.ca	even	6	1
675.3.i.a		4	105.w	odd	12	2
675.3.i.a		4	315.bs	even	12	2
675.3.j.a		2	105.p	even	6	1
675.3.j.a		2	315.q	odd	6	1
1296.3.e.a		2	252.n	even	6	1
1296.3.e.a		2	252.bn	odd	6	1
1728.3.q.a		2	168.ba	even	6	1
1728.3.q.a		2	504.bp	odd	6	1
1728.3.q.b		2	168.be	odd	6	1
1728.3.q.b		2	504.bf	even	6	1

Hecke kernels

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

\( T_{2}^{2} - 3T_{2} + 3 \)

\( T_{13}^{2} + 4T_{13} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T + 3 \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} + 12 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 3 \)