Newform orbit 3332.1.be.c

• Label: 3332.1.be.c
• Level: $3332$
• Weight: $1$
• Character orbit: 3332.be
• Analytic conductor: $1.663$
• Analytic rank: $0$
• Dimension: $12$
• Projective image: $D_{14}$
• CM discriminant: -68
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3332.be (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66288462209\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{14}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \zeta_{28}^{8} q^{2} + (\zeta_{28}^{13} + \zeta_{28}^{7}) q^{3} - \zeta_{28}^{2} q^{4} + (\zeta_{28}^{7} + \zeta_{28}) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( - \zeta_{28}^{12} - \zeta_{28}^{6} - 1) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(885\)	\(1667\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{28}^{6}\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

407.1

0.974928	+	0.222521i
−0.974928	−	0.222521i
0.974928	−	0.222521i
−0.974928	+	0.222521i
0.433884	+	0.900969i
−0.433884	−	0.900969i
0.781831	−	0.623490i
−0.781831	+	0.623490i
0.781831	+	0.623490i
−0.781831	−	0.623490i
0.433884	−	0.900969i
−0.433884	+	0.900969i

0.222521

−

0.974928i

−0.974928

+

1.22252i

−0.900969

−

0.433884i

0

0.974928

+

1.22252i

0.781831

+

0.623490i

−0.623490

+

0.781831i

−0.321552

−

1.40881i

0

407.2

0.222521

−

0.974928i

0.974928

−

1.22252i

−0.900969

−

0.433884i

0

−0.974928

−

1.22252i

−0.781831

−

0.623490i

−0.623490

+

0.781831i

−0.321552

−

1.40881i

0

1359.1

0.222521

+

0.974928i

−0.974928

−

1.22252i

−0.900969

+

0.433884i

0

0.974928

−

1.22252i

0.781831

−

0.623490i

−0.623490

−

0.781831i

−0.321552

+

1.40881i

0

1359.2

0.222521

+

0.974928i

0.974928

+

1.22252i

−0.900969

+

0.433884i

0

−0.974928

+

1.22252i

−0.781831

+

0.623490i

−0.623490

−

0.781831i

−0.321552

+

1.40881i

0

1835.1

0.900969

−

0.433884i

−0.433884

+

1.90097i

0.623490

−

0.781831i

0

0.433884

+

1.90097i

−0.974928

−

0.222521i

0.222521

−

0.974928i

−2.52446

−

1.21572i

0

1835.2

0.900969

−

0.433884i

0.433884

−

1.90097i

0.623490

−

0.781831i

0

−0.433884

−

1.90097i

0.974928

+

0.222521i

0.222521

−

0.974928i

−2.52446

−

1.21572i

0

2311.1

−0.623490

−

0.781831i

−0.781831

+

0.376510i

−0.222521

+

0.974928i

0

0.781831

+

0.376510i

−0.433884

−

0.900969i

0.900969

−

0.433884i

−0.153989

+

0.193096i

0

2311.2

−0.623490

−

0.781831i

0.781831

−

0.376510i

−0.222521

+

0.974928i

0

−0.781831

−

0.376510i

0.433884

+

0.900969i

0.900969

−

0.433884i

−0.153989

+

0.193096i

0

2787.1

−0.623490

+

0.781831i

−0.781831

−

0.376510i

−0.222521

−

0.974928i

0

0.781831

−

0.376510i

−0.433884

+

0.900969i

0.900969

+

0.433884i

−0.153989

−

0.193096i

0

2787.2

−0.623490

+

0.781831i

0.781831

+

0.376510i

−0.222521

−

0.974928i

0

−0.781831

+

0.376510i

0.433884

−

0.900969i

0.900969

+

0.433884i

−0.153989

−

0.193096i

0

3263.1

0.900969

+

0.433884i

−0.433884

−

1.90097i

0.623490

+

0.781831i

0

0.433884

−

1.90097i

−0.974928

+

0.222521i

0.222521

+

0.974928i

−2.52446

+

1.21572i

0

3263.2

0.900969

+

0.433884i

0.433884

+

1.90097i

0.623490

+

0.781831i

0

−0.433884

+

1.90097i

0.974928

−

0.222521i

0.222521

+

0.974928i

−2.52446

+

1.21572i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
68.d	odd	2	1	CM by \(\Q(\sqrt{-17}) \)
4.b	odd	2	1	inner
17.b	even	2	1	inner
49.e	even	7	1	inner
196.k	odd	14	1	inner
833.r	even	14	1	inner
3332.be	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3332.1.be.c	✓	12
4.b	odd	2	1	inner	3332.1.be.c	✓	12
17.b	even	2	1	inner	3332.1.be.c	✓	12
49.e	even	7	1	inner	3332.1.be.c	✓	12
68.d	odd	2	1	CM	3332.1.be.c	✓	12
196.k	odd	14	1	inner	3332.1.be.c	✓	12
833.r	even	14	1	inner	3332.1.be.c	✓	12
3332.be	odd	14	1	inner	3332.1.be.c	✓	12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3332.1.be.c	✓	12	1.a	even	1	1	trivial
3332.1.be.c	✓	12	4.b	odd	2	1	inner
3332.1.be.c	✓	12	17.b	even	2	1	inner
3332.1.be.c	✓	12	49.e	even	7	1	inner
3332.1.be.c	✓	12	68.d	odd	2	1	CM
3332.1.be.c	✓	12	196.k	odd	14	1	inner
3332.1.be.c	✓	12	833.r	even	14	1	inner
3332.1.be.c	✓	12	3332.be	odd	14	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 7T_{3}^{10} + 21T_{3}^{8} + 35T_{3}^{6} + 49T_{3}^{4} - 49T_{3}^{2} + 49 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{2} \)
$3$	T^12 + 7*T^10 + 21*T^8 + 35*T^6 + 49*T^4 - 49*T^2 + 49
$5$	\( T^{12} \)
$7$	T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	T^12 + 7*T^10 + 21*T^8 + 35*T^6 + 49*T^4 - 49*T^2 + 49