Newform orbit 441.5.d.a

• Label: 441.5.d.a
• Level: $441$
• Weight: $5$
• Character orbit: 441.d
• Analytic conductor: $45.586$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(45.5861537200\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 q^{2} - 12 q^{4} - 6 \beta q^{5} + 56 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 24 q^{4} + 112 q^{8}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

244.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.00000

0

−12.0000

−

10.3923i

0

0

56.0000

0

20.7846i

244.2

−2.00000

0

−12.0000

10.3923i

0

0

56.0000

0

−

20.7846i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.5.d.a		2
3.b	odd	2	1		147.5.d.b		2
7.b	odd	2	1	inner	441.5.d.a		2
7.c	even	3	1		63.5.m.c		2
7.d	odd	6	1		63.5.m.c		2
21.c	even	2	1		147.5.d.b		2
21.g	even	6	1		21.5.f.a	✓	2
21.g	even	6	1		147.5.f.a		2
21.h	odd	6	1		21.5.f.a	✓	2
21.h	odd	6	1		147.5.f.a		2
84.j	odd	6	1		336.5.bh.b		2
84.n	even	6	1		336.5.bh.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.5.f.a	✓	2	21.g	even	6	1
21.5.f.a	✓	2	21.h	odd	6	1
63.5.m.c		2	7.c	even	3	1
63.5.m.c		2	7.d	odd	6	1
147.5.d.b		2	3.b	odd	2	1
147.5.d.b		2	21.c	even	2	1
147.5.f.a		2	21.g	even	6	1
147.5.f.a		2	21.h	odd	6	1
336.5.bh.b		2	84.j	odd	6	1
336.5.bh.b		2	84.n	even	6	1
441.5.d.a		2	1.a	even	1	1	trivial
441.5.d.a		2	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{5}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 108 \)
$7$	\( T^{2} \)
$11$	\( (T + 194)^{2} \)