Newform orbit 665.2.i.d

• Label: 665.2.i.d
• Level: $665$
• Weight: $2$
• Character orbit: 665.i
• Analytic conductor: $5.310$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 665 = 5 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	665.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31005173442\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.954288.1
Defining polynomial:	\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} + \beta_{2} - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} + 1) q^{5} + q^{7} + ( - \beta_{5} + \beta_{4} - 4 \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{4} + 3 q^{5} + 6 q^{7} - 11 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{5} - 4\nu^{4} + \nu^{3} - 9\nu^{2} + 21\nu + 9 ) / 27 \)
\(\beta_{2}\)	\(=\)	\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{5} + 8\nu^{4} - 2\nu^{3} - 9\nu^{2} + 12\nu - 18 ) / 27 \)
\(\beta_{4}\)	\(=\)	\( ( -4\nu^{5} - 2\nu^{4} + 14\nu^{3} + 18\nu^{2} + 24\nu + 45 ) / 27 \)
\(\beta_{5}\)	\(=\)	\( ( 10\nu^{5} + 5\nu^{4} - 8\nu^{3} + 36\nu^{2} - 6\nu - 153 ) / 27 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/665\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(267\)	\(381\)
\(\chi(n)\)	\(-\beta_{2}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

106.1

0.403374	+	1.68443i
1.71903	−	0.211943i
−1.62241	−	0.606458i
0.403374	−	1.68443i
1.71903	+	0.211943i
−1.62241	+	0.606458i

0

−1.66044

+

2.87597i

1.00000

+

1.73205i

0.500000

−

0.866025i

0

1.00000

0

−4.01414

−

6.95269i

0

106.2

0

−0.675970

+

1.17081i

1.00000

+

1.73205i

0.500000

−

0.866025i

0

1.00000

0

0.586130

+

1.01521i

0

106.3

0

1.33641

−

2.31473i

1.00000

+

1.73205i

0.500000

−

0.866025i

0

1.00000

0

−2.07199

−

3.58880i

0

596.1

0

−1.66044

−

2.87597i

1.00000

−

1.73205i

0.500000

+

0.866025i

0

1.00000

0

−4.01414

+

6.95269i

0

596.2

0

−0.675970

−

1.17081i

1.00000

−

1.73205i

0.500000

+

0.866025i

0

1.00000

0

0.586130

−

1.01521i

0

596.3

0

1.33641

+

2.31473i

1.00000

−

1.73205i

0.500000

+

0.866025i

0

1.00000

0

−2.07199

+

3.58880i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	665.2.i.d	✓	6
19.c	even	3	1	inner	665.2.i.d	✓	6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
665.2.i.d	✓	6	1.a	even	1	1	trivial
665.2.i.d	✓	6	19.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(665, [\chi])\):

\( T_{2} \)

\( T_{3}^{6} + 2T_{3}^{5} + 12T_{3}^{4} + 8T_{3}^{3} + 88T_{3}^{2} + 96T_{3} + 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} \)
$3$	T^6 + 2*T^5 + 12*T^4 + 8*T^3 + 88*T^2 + 96*T + 144
$5$	\( (T^{2} - T + 1)^{3} \)
$7$	\( (T - 1)^{6} \)
$11$	\( (T^{3} - T^{2} - 21 T - 3)^{2} \)