Newform orbit 441.7.b.a

• Label: 441.7.b.a
• Level: $441$
• Weight: $7$
• Character orbit: 441.b
• Analytic conductor: $101.454$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	441.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(101.453850876\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 98 q^{4} - 5 \beta q^{5} - 34 \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 196 q^{4}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

197.1

	−	1.41421i
		1.41421i

−

12.7279i

0

−98.0000

63.6396i

0

0

432.749i

0

810.000

197.2

12.7279i

0

−98.0000

−

63.6396i

0

0

−

432.749i

0

810.000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.7.b.a		2
3.b	odd	2	1	inner	441.7.b.a		2
7.b	odd	2	1		9.7.b.a	✓	2
21.c	even	2	1		9.7.b.a	✓	2
28.d	even	2	1		144.7.e.a		2
35.c	odd	2	1		225.7.c.a		2
35.f	even	4	2		225.7.d.a		4
56.e	even	2	1		576.7.e.a		2
56.h	odd	2	1		576.7.e.l		2
63.l	odd	6	2		81.7.d.d		4
63.o	even	6	2		81.7.d.d		4
84.h	odd	2	1		144.7.e.a		2
105.g	even	2	1		225.7.c.a		2
105.k	odd	4	2		225.7.d.a		4
168.e	odd	2	1		576.7.e.a		2
168.i	even	2	1		576.7.e.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.7.b.a	✓	2	7.b	odd	2	1
9.7.b.a	✓	2	21.c	even	2	1
81.7.d.d		4	63.l	odd	6	2
81.7.d.d		4	63.o	even	6	2
144.7.e.a		2	28.d	even	2	1
144.7.e.a		2	84.h	odd	2	1
225.7.c.a		2	35.c	odd	2	1
225.7.c.a		2	105.g	even	2	1
225.7.d.a		4	35.f	even	4	2
225.7.d.a		4	105.k	odd	4	2
441.7.b.a		2	1.a	even	1	1	trivial
441.7.b.a		2	3.b	odd	2	1	inner
576.7.e.a		2	56.e	even	2	1
576.7.e.a		2	168.e	odd	2	1
576.7.e.l		2	56.h	odd	2	1
576.7.e.l		2	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} + 162 \)

\( T_{13} + 344 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 162 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4050 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 749088 \)