Newform orbit 155.2.e.b

• Label: 155.2.e.b
• Level: $155$
• Weight: $2$
• Character orbit: 155.e
• Analytic conductor: $1.238$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 155 = 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	155.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.23768123133\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(96\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

36.1

0.500000	+	0.866025i
0.500000	−	0.866025i

2.00000

−1.00000

+

1.73205i

2.00000

0.500000

+

0.866025i

−2.00000

+

3.46410i

1.00000

−

1.73205i

0

−0.500000

−

0.866025i

1.00000

+

1.73205i

56.1

2.00000

−1.00000

−

1.73205i

2.00000

0.500000

−

0.866025i

−2.00000

−

3.46410i

1.00000

+

1.73205i

0

−0.500000

+

0.866025i

1.00000

−

1.73205i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	155.2.e.b	✓	2
5.b	even	2	1		775.2.e.a		2
5.c	odd	4	2		775.2.o.a		4
31.c	even	3	1	inner	155.2.e.b	✓	2
31.c	even	3	1		4805.2.a.g		1
31.e	odd	6	1		4805.2.a.f		1
155.j	even	6	1		775.2.e.a		2
155.o	odd	12	2		775.2.o.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
155.2.e.b	✓	2	1.a	even	1	1	trivial
155.2.e.b	✓	2	31.c	even	3	1	inner
775.2.e.a		2	5.b	even	2	1
775.2.e.a		2	155.j	even	6	1
775.2.o.a		4	5.c	odd	4	2
775.2.o.a		4	155.o	odd	12	2
4805.2.a.f		1	31.e	odd	6	1
4805.2.a.g		1	31.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\).

Hecke characteristic polynomials

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