Newform orbit 648.1.b.a

• Label: 648.1.b.a
• Level: $648$
• Weight: $1$
• Character orbit: 648.b
• Self dual: yes
• Analytic conductor: $0.323$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	648.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.323394128186\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.648.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.1259712.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} + q^{4} - q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

163.1

0

−1.00000

0

1.00000

0

0

0

−1.00000

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.1.b.a		1
3.b	odd	2	1		648.1.b.b		1
4.b	odd	2	1		2592.1.b.a		1
8.b	even	2	1		2592.1.b.a		1
8.d	odd	2	1	CM	648.1.b.a		1
9.c	even	3	2		216.1.p.a		2
9.d	odd	6	2		72.1.p.a	✓	2
12.b	even	2	1		2592.1.b.b		1
24.f	even	2	1		648.1.b.b		1
24.h	odd	2	1		2592.1.b.b		1
36.f	odd	6	2		864.1.t.a		2
36.h	even	6	2		288.1.t.a		2
45.h	odd	6	2		1800.1.bk.d		2
45.l	even	12	4		1800.1.ba.b		4
63.i	even	6	2		3528.1.ce.b		2
63.j	odd	6	2		3528.1.ce.a		2
63.n	odd	6	2		3528.1.ba.b		2
63.o	even	6	2		3528.1.cg.a		2
63.s	even	6	2		3528.1.ba.a		2
72.j	odd	6	2		288.1.t.a		2
72.l	even	6	2		72.1.p.a	✓	2
72.n	even	6	2		864.1.t.a		2
72.p	odd	6	2		216.1.p.a		2
144.u	even	12	4		2304.1.o.c		4
144.w	odd	12	4		2304.1.o.c		4
360.bd	even	6	2		1800.1.bk.d		2
360.bt	odd	12	4		1800.1.ba.b		4
504.u	odd	6	2		3528.1.ba.a		2
504.bt	even	6	2		3528.1.ce.a		2
504.cm	odd	6	2		3528.1.ce.b		2
504.co	odd	6	2		3528.1.cg.a		2
504.cy	even	6	2		3528.1.ba.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.1.p.a	✓	2	9.d	odd	6	2
72.1.p.a	✓	2	72.l	even	6	2
216.1.p.a		2	9.c	even	3	2
216.1.p.a		2	72.p	odd	6	2
288.1.t.a		2	36.h	even	6	2
288.1.t.a		2	72.j	odd	6	2
648.1.b.a		1	1.a	even	1	1	trivial
648.1.b.a		1	8.d	odd	2	1	CM
648.1.b.b		1	3.b	odd	2	1
648.1.b.b		1	24.f	even	2	1
864.1.t.a		2	36.f	odd	6	2
864.1.t.a		2	72.n	even	6	2
1800.1.ba.b		4	45.l	even	12	4
1800.1.ba.b		4	360.bt	odd	12	4
1800.1.bk.d		2	45.h	odd	6	2
1800.1.bk.d		2	360.bd	even	6	2
2304.1.o.c		4	144.u	even	12	4
2304.1.o.c		4	144.w	odd	12	4
2592.1.b.a		1	4.b	odd	2	1
2592.1.b.a		1	8.b	even	2	1
2592.1.b.b		1	12.b	even	2	1
2592.1.b.b		1	24.h	odd	2	1
3528.1.ba.a		2	63.s	even	6	2
3528.1.ba.a		2	504.u	odd	6	2
3528.1.ba.b		2	63.n	odd	6	2
3528.1.ba.b		2	504.cy	even	6	2
3528.1.ce.a		2	63.j	odd	6	2
3528.1.ce.a		2	504.bt	even	6	2
3528.1.ce.b		2	63.i	even	6	2
3528.1.ce.b		2	504.cm	odd	6	2
3528.1.cg.a		2	63.o	even	6	2
3528.1.cg.a		2	504.co	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 1 \) acting on \(S_{1}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

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