Newform orbit 468.2.b.a

• Label: 468.2.b.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.b
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
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:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{5} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(209\)	\(235\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

181.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

−

3.46410i

0

0

0

0

0

181.2

0

0

0

3.46410i

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.b.a		2
3.b	odd	2	1		156.2.b.a	✓	2
4.b	odd	2	1		1872.2.c.c		2
12.b	even	2	1		624.2.c.f		2
13.b	even	2	1	inner	468.2.b.a		2
13.d	odd	4	2		6084.2.a.v		2
15.d	odd	2	1		3900.2.c.c		2
15.e	even	4	2		3900.2.j.h		4
21.c	even	2	1		7644.2.e.g		2
24.f	even	2	1		2496.2.c.e		2
24.h	odd	2	1		2496.2.c.l		2
39.d	odd	2	1		156.2.b.a	✓	2
39.f	even	4	2		2028.2.a.g		2
39.h	odd	6	1		2028.2.q.b		2
39.h	odd	6	1		2028.2.q.c		2
39.i	odd	6	1		2028.2.q.b		2
39.i	odd	6	1		2028.2.q.c		2
39.k	even	12	4		2028.2.i.i		4
52.b	odd	2	1		1872.2.c.c		2
156.h	even	2	1		624.2.c.f		2
156.l	odd	4	2		8112.2.a.bs		2
195.e	odd	2	1		3900.2.c.c		2
195.s	even	4	2		3900.2.j.h		4
273.g	even	2	1		7644.2.e.g		2
312.b	odd	2	1		2496.2.c.l		2
312.h	even	2	1		2496.2.c.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.b.a	✓	2	3.b	odd	2	1
156.2.b.a	✓	2	39.d	odd	2	1
468.2.b.a		2	1.a	even	1	1	trivial
468.2.b.a		2	13.b	even	2	1	inner
624.2.c.f		2	12.b	even	2	1
624.2.c.f		2	156.h	even	2	1
1872.2.c.c		2	4.b	odd	2	1
1872.2.c.c		2	52.b	odd	2	1
2028.2.a.g		2	39.f	even	4	2
2028.2.i.i		4	39.k	even	12	4
2028.2.q.b		2	39.h	odd	6	1
2028.2.q.b		2	39.i	odd	6	1
2028.2.q.c		2	39.h	odd	6	1
2028.2.q.c		2	39.i	odd	6	1
2496.2.c.e		2	24.f	even	2	1
2496.2.c.e		2	312.h	even	2	1
2496.2.c.l		2	24.h	odd	2	1
2496.2.c.l		2	312.b	odd	2	1
3900.2.c.c		2	15.d	odd	2	1
3900.2.c.c		2	195.e	odd	2	1
3900.2.j.h		4	15.e	even	4	2
3900.2.j.h		4	195.s	even	4	2
6084.2.a.v		2	13.d	odd	4	2
7644.2.e.g		2	21.c	even	2	1
7644.2.e.g		2	273.g	even	2	1
8112.2.a.bs		2	156.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

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