Newform orbit 416.2.i.b

• Label: 416.2.i.b
• Level: $416$
• Weight: $2$
• Character orbit: 416.i
• Analytic conductor: $3.322$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 2) q^{3} + q^{5} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

289.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

1.00000

+

1.73205i

0

1.00000

0

0

0

−0.500000

+

0.866025i

0

321.1

0

1.00000

−

1.73205i

0

1.00000

0

0

0

−0.500000

−

0.866025i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.i.b	yes	2
4.b	odd	2	1		416.2.i.a	✓	2
8.b	even	2	1		832.2.i.b		2
8.d	odd	2	1		832.2.i.h		2
13.c	even	3	1	inner	416.2.i.b	yes	2
13.c	even	3	1		5408.2.a.d		1
13.e	even	6	1		5408.2.a.c		1
52.i	odd	6	1		5408.2.a.j		1
52.j	odd	6	1		416.2.i.a	✓	2
52.j	odd	6	1		5408.2.a.k		1
104.n	odd	6	1		832.2.i.h		2
104.r	even	6	1		832.2.i.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.i.a	✓	2	4.b	odd	2	1
416.2.i.a	✓	2	52.j	odd	6	1
416.2.i.b	yes	2	1.a	even	1	1	trivial
416.2.i.b	yes	2	13.c	even	3	1	inner
832.2.i.b		2	8.b	even	2	1
832.2.i.b		2	104.r	even	6	1
832.2.i.h		2	8.d	odd	2	1
832.2.i.h		2	104.n	odd	6	1
5408.2.a.c		1	13.e	even	6	1
5408.2.a.d		1	13.c	even	3	1
5408.2.a.j		1	52.i	odd	6	1
5408.2.a.k		1	52.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T + 16 \)