Newform orbit 4212.2.i.f

• Label: 4212.2.i.f
• Level: $4212$
• Weight: $2$
• Character orbit: 4212.i
• Analytic conductor: $33.633$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.6329893314\)
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:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times\).

\(n\)	\(2107\)	\(3485\)	\(3889\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1405.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

0

0

−1.00000

−

1.73205i

0

0

0

2809.1

0

0

0

0

0

−1.00000

+

1.73205i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4212.2.i.f		2
3.b	odd	2	1		4212.2.i.g		2
9.c	even	3	1		156.2.a.b	✓	1
9.c	even	3	1	inner	4212.2.i.f		2
9.d	odd	6	1		468.2.a.c		1
9.d	odd	6	1		4212.2.i.g		2
36.f	odd	6	1		624.2.a.b		1
36.h	even	6	1		1872.2.a.i		1
45.j	even	6	1		3900.2.a.a		1
45.k	odd	12	2		3900.2.h.e		2
63.l	odd	6	1		7644.2.a.a		1
72.j	odd	6	1		7488.2.a.bf		1
72.l	even	6	1		7488.2.a.bb		1
72.n	even	6	1		2496.2.a.h		1
72.p	odd	6	1		2496.2.a.v		1
117.f	even	3	1		2028.2.i.b		2
117.h	even	3	1		2028.2.i.b		2
117.l	even	6	1		2028.2.i.c		2
117.n	odd	6	1		6084.2.a.h		1
117.r	even	6	1		2028.2.i.c		2
117.t	even	6	1		2028.2.a.e		1
117.w	odd	12	2		2028.2.q.d		4
117.y	odd	12	2		2028.2.b.d		2
117.z	even	12	2		6084.2.b.a		2
117.bb	odd	12	2		2028.2.q.d		4
468.bg	odd	6	1		8112.2.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.a.b	✓	1	9.c	even	3	1
468.2.a.c		1	9.d	odd	6	1
624.2.a.b		1	36.f	odd	6	1
1872.2.a.i		1	36.h	even	6	1
2028.2.a.e		1	117.t	even	6	1
2028.2.b.d		2	117.y	odd	12	2
2028.2.i.b		2	117.f	even	3	1
2028.2.i.b		2	117.h	even	3	1
2028.2.i.c		2	117.l	even	6	1
2028.2.i.c		2	117.r	even	6	1
2028.2.q.d		4	117.w	odd	12	2
2028.2.q.d		4	117.bb	odd	12	2
2496.2.a.h		1	72.n	even	6	1
2496.2.a.v		1	72.p	odd	6	1
3900.2.a.a		1	45.j	even	6	1
3900.2.h.e		2	45.k	odd	12	2
4212.2.i.f		2	1.a	even	1	1	trivial
4212.2.i.f		2	9.c	even	3	1	inner
4212.2.i.g		2	3.b	odd	2	1
4212.2.i.g		2	9.d	odd	6	1
6084.2.a.h		1	117.n	odd	6	1
6084.2.b.a		2	117.z	even	12	2
7488.2.a.bb		1	72.l	even	6	1
7488.2.a.bf		1	72.j	odd	6	1
7644.2.a.a		1	63.l	odd	6	1
8112.2.a.i		1	468.bg	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4212, [\chi])\):

\( T_{5} \)

\( T_{7}^{2} + 2T_{7} + 4 \)

\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 2T + 4 \)
$11$	\( T^{2} \)