Newform orbit 146.2.e.a

• Label: 146.2.e.a
• Level: $146$
• Weight: $2$
• Character orbit: 146.e
• Analytic conductor: $1.166$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 146 = 2 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	146.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.16581586951\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} - q^{3} + (\zeta_{6} - 1) q^{4} + ( - 2 \zeta_{6} - 2) q^{5} + \zeta_{6} q^{6} + (4 \zeta_{6} - 2) q^{7} + q^{8} - 2 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/146\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

−1.00000

−0.500000

+

0.866025i

−3.00000

−

1.73205i

0.500000

+

0.866025i

3.46410i

1.00000

−2.00000

3.46410i

65.1

−0.500000

+

0.866025i

−1.00000

−0.500000

−

0.866025i

−3.00000

+

1.73205i

0.500000

−

0.866025i

−

3.46410i

1.00000

−2.00000

−

3.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
73.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	146.2.e.a	✓	2
3.b	odd	2	1		1314.2.p.g		2
4.b	odd	2	1		1168.2.bb.b		2
73.e	even	6	1	inner	146.2.e.a	✓	2
219.i	odd	6	1		1314.2.p.g		2
292.i	odd	6	1		1168.2.bb.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
146.2.e.a	✓	2	1.a	even	1	1	trivial
146.2.e.a	✓	2	73.e	even	6	1	inner
1168.2.bb.b		2	4.b	odd	2	1
1168.2.bb.b		2	292.i	odd	6	1
1314.2.p.g		2	3.b	odd	2	1
1314.2.p.g		2	219.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(146, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + 6T + 12 \)
$7$	\( T^{2} + 12 \)
$11$	\( T^{2} + 3T + 3 \)