Newform orbit 441.2.e.i

• Label: 441.2.e.i
• Level: $441$
• Weight: $2$
• Character orbit: 441.e
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} - 2 \beta_{2} q^{5} - \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{3} + \zeta_{12} \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(-\beta_{1}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

226.1

0.866025	−	0.500000i
−0.866025	+	0.500000i
0.866025	+	0.500000i
−0.866025	−	0.500000i

−0.866025

+

1.50000i

0

−0.500000

−

0.866025i

−1.73205

+

3.00000i

0

0

−1.73205

0

−3.00000

−

5.19615i

226.2

0.866025

−

1.50000i

0

−0.500000

−

0.866025i

1.73205

−

3.00000i

0

0

1.73205

0

−3.00000

−

5.19615i

361.1

−0.866025

−

1.50000i

0

−0.500000

+

0.866025i

−1.73205

−

3.00000i

0

0

−1.73205

0

−3.00000

+

5.19615i

361.2

0.866025

+

1.50000i

0

−0.500000

+

0.866025i

1.73205

+

3.00000i

0

0

1.73205

0

−3.00000

+

5.19615i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.2.e.i		4
3.b	odd	2	1	inner	441.2.e.i		4
7.b	odd	2	1		441.2.e.j		4
7.c	even	3	1		441.2.a.g		2
7.c	even	3	1	inner	441.2.e.i		4
7.d	odd	6	1		63.2.a.b	✓	2
7.d	odd	6	1		441.2.e.j		4
21.c	even	2	1		441.2.e.j		4
21.g	even	6	1		63.2.a.b	✓	2
21.g	even	6	1		441.2.e.j		4
21.h	odd	6	1		441.2.a.g		2
21.h	odd	6	1	inner	441.2.e.i		4
28.f	even	6	1		1008.2.a.n		2
28.g	odd	6	1		7056.2.a.cm		2
35.i	odd	6	1		1575.2.a.q		2
35.k	even	12	2		1575.2.d.i		4
56.j	odd	6	1		4032.2.a.bt		2
56.m	even	6	1		4032.2.a.bq		2
63.i	even	6	1		567.2.f.j		4
63.k	odd	6	1		567.2.f.j		4
63.s	even	6	1		567.2.f.j		4
63.t	odd	6	1		567.2.f.j		4
77.i	even	6	1		7623.2.a.bi		2
84.j	odd	6	1		1008.2.a.n		2
84.n	even	6	1		7056.2.a.cm		2
105.p	even	6	1		1575.2.a.q		2
105.w	odd	12	2		1575.2.d.i		4
168.ba	even	6	1		4032.2.a.bt		2
168.be	odd	6	1		4032.2.a.bq		2
231.k	odd	6	1		7623.2.a.bi		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.a.b	✓	2	7.d	odd	6	1
63.2.a.b	✓	2	21.g	even	6	1
441.2.a.g		2	7.c	even	3	1
441.2.a.g		2	21.h	odd	6	1
441.2.e.i		4	1.a	even	1	1	trivial
441.2.e.i		4	3.b	odd	2	1	inner
441.2.e.i		4	7.c	even	3	1	inner
441.2.e.i		4	21.h	odd	6	1	inner
441.2.e.j		4	7.b	odd	2	1
441.2.e.j		4	7.d	odd	6	1
441.2.e.j		4	21.c	even	2	1
441.2.e.j		4	21.g	even	6	1
567.2.f.j		4	63.i	even	6	1
567.2.f.j		4	63.k	odd	6	1
567.2.f.j		4	63.s	even	6	1
567.2.f.j		4	63.t	odd	6	1
1008.2.a.n		2	28.f	even	6	1
1008.2.a.n		2	84.j	odd	6	1
1575.2.a.q		2	35.i	odd	6	1
1575.2.a.q		2	105.p	even	6	1
1575.2.d.i		4	35.k	even	12	2
1575.2.d.i		4	105.w	odd	12	2
4032.2.a.bq		2	56.m	even	6	1
4032.2.a.bq		2	168.be	odd	6	1
4032.2.a.bt		2	56.j	odd	6	1
4032.2.a.bt		2	168.ba	even	6	1
7056.2.a.cm		2	28.g	odd	6	1
7056.2.a.cm		2	84.n	even	6	1
7623.2.a.bi		2	77.i	even	6	1
7623.2.a.bi		2	231.k	odd	6	1

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{2} + 9 \)

\( T_{5}^{4} + 12T_{5}^{2} + 144 \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 3T^{2} + 9 \)
$3$	\( T^{4} \)
$5$	\( T^{4} + 12T^{2} + 144 \)
$7$	\( T^{4} \)
$11$	\( T^{4} + 12T^{2} + 144 \)