Newform orbit 666.2.a.i

• Label: 666.2.a.i
• Level: $666$
• Weight: $2$
• Character orbit: 666.a
• Self dual: yes
• Analytic conductor: $5.318$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + q^{4} + ( - 3 \beta + 1) q^{5} - 2 \beta q^{7} - q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

1.61803
−0.618034

−1.00000

0

1.00000

−3.85410

0

−3.23607

−1.00000

0

3.85410

1.2

−1.00000

0

1.00000

2.85410

0

1.23607

−1.00000

0

−2.85410

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(37\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.a.i		2
3.b	odd	2	1		74.2.a.b	✓	2
4.b	odd	2	1		5328.2.a.bc		2
12.b	even	2	1		592.2.a.g		2
15.d	odd	2	1		1850.2.a.t		2
15.e	even	4	2		1850.2.b.j		4
21.c	even	2	1		3626.2.a.s		2
24.f	even	2	1		2368.2.a.u		2
24.h	odd	2	1		2368.2.a.y		2
33.d	even	2	1		8954.2.a.j		2
111.d	odd	2	1		2738.2.a.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
74.2.a.b	✓	2	3.b	odd	2	1
592.2.a.g		2	12.b	even	2	1
666.2.a.i		2	1.a	even	1	1	trivial
1850.2.a.t		2	15.d	odd	2	1
1850.2.b.j		4	15.e	even	4	2
2368.2.a.u		2	24.f	even	2	1
2368.2.a.y		2	24.h	odd	2	1
2738.2.a.g		2	111.d	odd	2	1
3626.2.a.s		2	21.c	even	2	1
5328.2.a.bc		2	4.b	odd	2	1
8954.2.a.j		2	33.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5}^{2} + T_{5} - 11 \)

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

Hecke characteristic polynomials

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