Newform orbit 6664.2.a.h

• Label: 6664.2.a.h
• Level: $6664$
• Weight: $2$
• Character orbit: 6664.a
• Self dual: yes
• Analytic conductor: $53.212$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(53.2123079070\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 952)
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 + b * q^3 + (b + 1) * q^5 + (b - 2) * q^9 + (4*b - 2) * q^11 + (-4*b + 2) * q^13 + (2*b + 1) * q^15 - q^17 + 4*b * q^19 + (2*b - 8) * q^23 + (3*b - 3) * q^25 + (-4*b + 1) * q^27 + 2 * q^29 + (5*b + 1) * q^31 + (2*b + 4) * q^33 + 4 * q^37 + (-2*b - 4) * q^39 + (5*b + 2) * q^41 + (-5*b - 2) * q^43 - q^45 + (2*b + 8) * q^47 - b * q^51 + (-5*b + 5) * q^53 + (6*b + 2) * q^55 + (4*b + 4) * q^57 + 4 * q^59 + (-b - 6) * q^61 + (-6*b - 2) * q^65 + (5*b + 3) * q^67 + (-6*b + 2) * q^69 + (-9*b + 6) * q^73 + 3 * q^75 + (10*b - 8) * q^79 + (-6*b + 2) * q^81 + (2*b + 8) * q^83 + (-b - 1) * q^85 + 2*b * q^87 + (-6*b + 8) * q^89 + (6*b + 5) * q^93 + (8*b + 4) * q^95 + (13*b - 7) * q^97 + (-6*b + 8) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^3 + 3 * q^5 - 3 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^19 - 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 + 10 * q^33 + 8 * q^37 - 10 * q^39 + 9 * q^41 - 9 * q^43 - 2 * q^45 + 18 * q^47 - q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 - 13 * q^61 - 10 * q^65 + 11 * q^67 - 2 * q^69 + 3 * q^73 + 6 * q^75 - 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 + 10 * q^89 + 16 * q^93 + 16 * q^95 - q^97 + 10 * q^99

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

−0.618034
1.61803

0

−0.618034

0

0.381966

0

0

0

−2.61803

0

1.2

0

1.61803

0

2.61803

0

0

0

−0.381966

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(7\)	\( -1 \)
\(17\)	\( +1 \)

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	6664.2.a.h		2
7.b	odd	2	1		952.2.a.a	✓	2
21.c	even	2	1		8568.2.a.v		2
28.d	even	2	1		1904.2.a.i		2
56.e	even	2	1		7616.2.a.r		2
56.h	odd	2	1		7616.2.a.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
952.2.a.a	✓	2	7.b	odd	2	1
1904.2.a.i		2	28.d	even	2	1
6664.2.a.h		2	1.a	even	1	1	trivial
7616.2.a.r		2	56.e	even	2	1
7616.2.a.w		2	56.h	odd	2	1
8568.2.a.v		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6664))\):

\( T_{3}^{2} - T_{3} - 1 \)

\( T_{5}^{2} - 3T_{5} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T - 1 \)
$5$	\( T^{2} - 3T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 20 \)