Newform orbit 441.2.i.a

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

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.i (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:	\(\Z[a_1, \ldots, a_{5}]\)
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.

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

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(\zeta_{6}\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

68.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.73205i

−

1.73205i

−1.00000

1.50000

+

2.59808i

3.00000

0

1.73205i

−3.00000

−4.50000

+

2.59808i

227.1

−

1.73205i

1.73205i

−1.00000

1.50000

−

2.59808i

3.00000

0

−

1.73205i

−3.00000

−4.50000

−

2.59808i

Inner twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3 \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} - 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 3T + 3 \)