Newform orbit 3969.1.t.c

• Label: 3969.1.t.c
• Level: $3969$
• Weight: $1$
• Character orbit: 3969.t
• Analytic conductor: $1.981$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -3, -7, 21
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3969.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.98078903514\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image:	$C_3\times D_4$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - q^{4} + q^{16} + \zeta_{6}^{2} q^{25} - 2 \zeta_{6}^{2} q^{37} + 2 \zeta_{6} q^{43} - q^{64} + 2 q^{67} + 2 q^{79} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{16} - q^{25} + 2 q^{37} + 2 q^{43} - 2 q^{64} + 4 q^{67} + 4 q^{79}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).

\(n\)	\(2108\)	\(3727\)
\(\chi(n)\)	\(\zeta_{6}^{2}\)	\(-\zeta_{6}^{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2971.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

−1.00000

0

0

0

0

0

0

3106.1

0

0

−1.00000

0

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
21.c	even	2	1	RM by \(\Q(\sqrt{21}) \)
63.h	even	3	1	inner
63.i	even	6	1	inner
63.j	odd	6	1	inner
63.t	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3969.1.t.c		2
3.b	odd	2	1	CM	3969.1.t.c		2
7.b	odd	2	1	CM	3969.1.t.c		2
7.c	even	3	1		567.1.l.b		2
7.c	even	3	1		3969.1.k.b		2
7.d	odd	6	1		567.1.l.b		2
7.d	odd	6	1		3969.1.k.b		2
9.c	even	3	1		441.1.m.a		2
9.c	even	3	1		3969.1.k.b		2
9.d	odd	6	1		441.1.m.a		2
9.d	odd	6	1		3969.1.k.b		2
21.c	even	2	1	RM	3969.1.t.c		2
21.g	even	6	1		567.1.l.b		2
21.g	even	6	1		3969.1.k.b		2
21.h	odd	6	1		567.1.l.b		2
21.h	odd	6	1		3969.1.k.b		2
63.g	even	3	1		441.1.m.a		2
63.g	even	3	1		567.1.l.b		2
63.h	even	3	1		63.1.d.a	✓	1
63.h	even	3	1	inner	3969.1.t.c		2
63.i	even	6	1		63.1.d.a	✓	1
63.i	even	6	1	inner	3969.1.t.c		2
63.j	odd	6	1		63.1.d.a	✓	1
63.j	odd	6	1	inner	3969.1.t.c		2
63.k	odd	6	1		441.1.m.a		2
63.k	odd	6	1		567.1.l.b		2
63.l	odd	6	1		441.1.m.a		2
63.l	odd	6	1		3969.1.k.b		2
63.n	odd	6	1		441.1.m.a		2
63.n	odd	6	1		567.1.l.b		2
63.o	even	6	1		441.1.m.a		2
63.o	even	6	1		3969.1.k.b		2
63.s	even	6	1		441.1.m.a		2
63.s	even	6	1		567.1.l.b		2
63.t	odd	6	1		63.1.d.a	✓	1
63.t	odd	6	1	inner	3969.1.t.c		2
252.r	odd	6	1		1008.1.f.a		1
252.u	odd	6	1		1008.1.f.a		1
252.bb	even	6	1		1008.1.f.a		1
252.bj	even	6	1		1008.1.f.a		1
315.q	odd	6	1		1575.1.h.b		1
315.r	even	6	1		1575.1.h.b		1
315.bq	even	6	1		1575.1.h.b		1
315.br	odd	6	1		1575.1.h.b		1
315.bs	even	12	2		1575.1.e.b		2
315.bt	odd	12	2		1575.1.e.b		2
315.bu	odd	12	2		1575.1.e.b		2
315.bv	even	12	2		1575.1.e.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.1.d.a	✓	1	63.h	even	3	1
63.1.d.a	✓	1	63.i	even	6	1
63.1.d.a	✓	1	63.j	odd	6	1
63.1.d.a	✓	1	63.t	odd	6	1
441.1.m.a		2	9.c	even	3	1
441.1.m.a		2	9.d	odd	6	1
441.1.m.a		2	63.g	even	3	1
441.1.m.a		2	63.k	odd	6	1
441.1.m.a		2	63.l	odd	6	1
441.1.m.a		2	63.n	odd	6	1
441.1.m.a		2	63.o	even	6	1
441.1.m.a		2	63.s	even	6	1
567.1.l.b		2	7.c	even	3	1
567.1.l.b		2	7.d	odd	6	1
567.1.l.b		2	21.g	even	6	1
567.1.l.b		2	21.h	odd	6	1
567.1.l.b		2	63.g	even	3	1
567.1.l.b		2	63.k	odd	6	1
567.1.l.b		2	63.n	odd	6	1
567.1.l.b		2	63.s	even	6	1
1008.1.f.a		1	252.r	odd	6	1
1008.1.f.a		1	252.u	odd	6	1
1008.1.f.a		1	252.bb	even	6	1
1008.1.f.a		1	252.bj	even	6	1
1575.1.e.b		2	315.bs	even	12	2
1575.1.e.b		2	315.bt	odd	12	2
1575.1.e.b		2	315.bu	odd	12	2
1575.1.e.b		2	315.bv	even	12	2
1575.1.h.b		1	315.q	odd	6	1
1575.1.h.b		1	315.r	even	6	1
1575.1.h.b		1	315.bq	even	6	1
1575.1.h.b		1	315.br	odd	6	1
3969.1.k.b		2	7.c	even	3	1
3969.1.k.b		2	7.d	odd	6	1
3969.1.k.b		2	9.c	even	3	1
3969.1.k.b		2	9.d	odd	6	1
3969.1.k.b		2	21.g	even	6	1
3969.1.k.b		2	21.h	odd	6	1
3969.1.k.b		2	63.l	odd	6	1
3969.1.k.b		2	63.o	even	6	1
3969.1.t.c		2	1.a	even	1	1	trivial
3969.1.t.c		2	3.b	odd	2	1	CM
3969.1.t.c		2	7.b	odd	2	1	CM
3969.1.t.c		2	21.c	even	2	1	RM
3969.1.t.c		2	63.h	even	3	1	inner
3969.1.t.c		2	63.i	even	6	1	inner
3969.1.t.c		2	63.j	odd	6	1	inner
3969.1.t.c		2	63.t	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3969, [\chi])\):

\( T_{2} \)

\( T_{13} \)

Hecke characteristic polynomials

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