Newform orbit 234.2.b.a

• Label: 234.2.b.a
• Level: $234$
• Weight: $2$
• Character orbit: 234.b
• Analytic conductor: $1.868$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.86849940730\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + i q^{2} - q^{4} + 2 i q^{5} + 2 i q^{7} - i q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

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

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

	−	1.00000i
		1.00000i

−

1.00000i

0

−1.00000

−

2.00000i

0

−

2.00000i

1.00000i

0

−2.00000

181.2

1.00000i

0

−1.00000

2.00000i

0

2.00000i

−

1.00000i

0

−2.00000

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	234.2.b.a		2
3.b	odd	2	1		78.2.b.a	✓	2
4.b	odd	2	1		1872.2.c.b		2
12.b	even	2	1		624.2.c.a		2
13.b	even	2	1	inner	234.2.b.a		2
13.d	odd	4	1		3042.2.a.c		1
13.d	odd	4	1		3042.2.a.n		1
15.d	odd	2	1		1950.2.b.c		2
15.e	even	4	1		1950.2.f.d		2
15.e	even	4	1		1950.2.f.g		2
21.c	even	2	1		3822.2.c.d		2
24.f	even	2	1		2496.2.c.m		2
24.h	odd	2	1		2496.2.c.f		2
39.d	odd	2	1		78.2.b.a	✓	2
39.f	even	4	1		1014.2.a.b		1
39.f	even	4	1		1014.2.a.g		1
39.h	odd	6	2		1014.2.i.c		4
39.i	odd	6	2		1014.2.i.c		4
39.k	even	12	2		1014.2.e.b		2
39.k	even	12	2		1014.2.e.e		2
52.b	odd	2	1		1872.2.c.b		2
156.h	even	2	1		624.2.c.a		2
156.l	odd	4	1		8112.2.a.g		1
156.l	odd	4	1		8112.2.a.j		1
195.e	odd	2	1		1950.2.b.c		2
195.s	even	4	1		1950.2.f.d		2
195.s	even	4	1		1950.2.f.g		2
273.g	even	2	1		3822.2.c.d		2
312.b	odd	2	1		2496.2.c.f		2
312.h	even	2	1		2496.2.c.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.b.a	✓	2	3.b	odd	2	1
78.2.b.a	✓	2	39.d	odd	2	1
234.2.b.a		2	1.a	even	1	1	trivial
234.2.b.a		2	13.b	even	2	1	inner
624.2.c.a		2	12.b	even	2	1
624.2.c.a		2	156.h	even	2	1
1014.2.a.b		1	39.f	even	4	1
1014.2.a.g		1	39.f	even	4	1
1014.2.e.b		2	39.k	even	12	2
1014.2.e.e		2	39.k	even	12	2
1014.2.i.c		4	39.h	odd	6	2
1014.2.i.c		4	39.i	odd	6	2
1872.2.c.b		2	4.b	odd	2	1
1872.2.c.b		2	52.b	odd	2	1
1950.2.b.c		2	15.d	odd	2	1
1950.2.b.c		2	195.e	odd	2	1
1950.2.f.d		2	15.e	even	4	1
1950.2.f.d		2	195.s	even	4	1
1950.2.f.g		2	15.e	even	4	1
1950.2.f.g		2	195.s	even	4	1
2496.2.c.f		2	24.h	odd	2	1
2496.2.c.f		2	312.b	odd	2	1
2496.2.c.m		2	24.f	even	2	1
2496.2.c.m		2	312.h	even	2	1
3042.2.a.c		1	13.d	odd	4	1
3042.2.a.n		1	13.d	odd	4	1
3822.2.c.d		2	21.c	even	2	1
3822.2.c.d		2	273.g	even	2	1
8112.2.a.g		1	156.l	odd	4	1
8112.2.a.j		1	156.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\).

Hecke characteristic polynomials

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