Newform orbit 423.1.d.a

• Label: 423.1.d.a
• Level: $423$
• Weight: $1$
• Character orbit: 423.d
• Self dual: yes
• Analytic conductor: $0.211$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -47
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 423 = 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	423.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.211104500344\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 47)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$D_{10}$
Artin field:	Galois closure of 10.0.1185762483.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{4} + (\beta - 1) q^{7} + q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/423\mathbb{Z}\right)^\times\).

\(n\)	\(236\)	\(334\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

46.1

1.61803
−0.618034

−0.618034

0

−0.618034

0

0

0.618034

1.00000

0

0

46.2

1.61803

0

1.61803

0

0

−1.61803

1.00000

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
47.b	odd	2	1	CM by \(\Q(\sqrt{-47}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	423.1.d.a		2
3.b	odd	2	1		47.1.b.a	✓	2
9.c	even	3	2		3807.1.f.a		4
9.d	odd	6	2		3807.1.f.b		4
12.b	even	2	1		752.1.g.a		2
15.d	odd	2	1		1175.1.d.c		2
15.e	even	4	2		1175.1.b.b		4
21.c	even	2	1		2303.1.d.c		2
21.g	even	6	2		2303.1.f.b		4
21.h	odd	6	2		2303.1.f.c		4
24.f	even	2	1		3008.1.g.a		2
24.h	odd	2	1		3008.1.g.b		2
47.b	odd	2	1	CM	423.1.d.a		2
141.c	even	2	1		47.1.b.a	✓	2
141.g	even	46	22		2209.1.d.a		44
141.h	odd	46	22		2209.1.d.a		44
423.f	odd	6	2		3807.1.f.a		4
423.g	even	6	2		3807.1.f.b		4
564.f	odd	2	1		752.1.g.a		2
705.g	even	2	1		1175.1.d.c		2
705.l	odd	4	2		1175.1.b.b		4
987.c	odd	2	1		2303.1.d.c		2
987.m	even	6	2		2303.1.f.c		4
987.o	odd	6	2		2303.1.f.b		4
1128.d	odd	2	1		3008.1.g.a		2
1128.i	even	2	1		3008.1.g.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
47.1.b.a	✓	2	3.b	odd	2	1
47.1.b.a	✓	2	141.c	even	2	1
423.1.d.a		2	1.a	even	1	1	trivial
423.1.d.a		2	47.b	odd	2	1	CM
752.1.g.a		2	12.b	even	2	1
752.1.g.a		2	564.f	odd	2	1
1175.1.b.b		4	15.e	even	4	2
1175.1.b.b		4	705.l	odd	4	2
1175.1.d.c		2	15.d	odd	2	1
1175.1.d.c		2	705.g	even	2	1
2209.1.d.a		44	141.g	even	46	22
2209.1.d.a		44	141.h	odd	46	22
2303.1.d.c		2	21.c	even	2	1
2303.1.d.c		2	987.c	odd	2	1
2303.1.f.b		4	21.g	even	6	2
2303.1.f.b		4	987.o	odd	6	2
2303.1.f.c		4	21.h	odd	6	2
2303.1.f.c		4	987.m	even	6	2
3008.1.g.a		2	24.f	even	2	1
3008.1.g.a		2	1128.d	odd	2	1
3008.1.g.b		2	24.h	odd	2	1
3008.1.g.b		2	1128.i	even	2	1
3807.1.f.a		4	9.c	even	3	2
3807.1.f.a		4	423.f	odd	6	2
3807.1.f.b		4	9.d	odd	6	2
3807.1.f.b		4	423.g	even	6	2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(423, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + T - 1 \)
$11$	\( T^{2} \)