Newform orbit 34.2.c.a

• Label: 34.2.c.a
• Level: $34$
• Weight: $2$
• Character orbit: 34.c
• Analytic conductor: $0.271$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 34 = 2 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	34.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.271491366872\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

13.1

		1.00000i
	−	1.00000i

−

1.00000i

−1.00000

−

1.00000i

−1.00000

2.00000

+

2.00000i

−1.00000

+

1.00000i

−2.00000

+

2.00000i

1.00000i

−

1.00000i

2.00000

−

2.00000i

21.1

1.00000i

−1.00000

+

1.00000i

−1.00000

2.00000

−

2.00000i

−1.00000

−

1.00000i

−2.00000

−

2.00000i

−

1.00000i

1.00000i

2.00000

+

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	34.2.c.a	✓	2
3.b	odd	2	1		306.2.g.b		2
4.b	odd	2	1		272.2.o.f		2
5.b	even	2	1		850.2.h.f		2
5.c	odd	4	1		850.2.g.b		2
5.c	odd	4	1		850.2.g.c		2
8.b	even	2	1		1088.2.o.l		2
8.d	odd	2	1		1088.2.o.d		2
12.b	even	2	1		2448.2.be.c		2
17.b	even	2	1		578.2.c.c		2
17.c	even	4	1	inner	34.2.c.a	✓	2
17.c	even	4	1		578.2.c.c		2
17.d	even	8	2		578.2.a.c		2
17.d	even	8	2		578.2.b.b		2
17.e	odd	16	8		578.2.d.d		8
51.f	odd	4	1		306.2.g.b		2
51.g	odd	8	2		5202.2.a.v		2
68.f	odd	4	1		272.2.o.f		2
68.g	odd	8	2		4624.2.a.r		2
85.f	odd	4	1		850.2.g.b		2
85.i	odd	4	1		850.2.g.c		2
85.j	even	4	1		850.2.h.f		2
136.i	even	4	1		1088.2.o.l		2
136.j	odd	4	1		1088.2.o.d		2
204.l	even	4	1		2448.2.be.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
34.2.c.a	✓	2	1.a	even	1	1	trivial
34.2.c.a	✓	2	17.c	even	4	1	inner
272.2.o.f		2	4.b	odd	2	1
272.2.o.f		2	68.f	odd	4	1
306.2.g.b		2	3.b	odd	2	1
306.2.g.b		2	51.f	odd	4	1
578.2.a.c		2	17.d	even	8	2
578.2.b.b		2	17.d	even	8	2
578.2.c.c		2	17.b	even	2	1
578.2.c.c		2	17.c	even	4	1
578.2.d.d		8	17.e	odd	16	8
850.2.g.b		2	5.c	odd	4	1
850.2.g.b		2	85.f	odd	4	1
850.2.g.c		2	5.c	odd	4	1
850.2.g.c		2	85.i	odd	4	1
850.2.h.f		2	5.b	even	2	1
850.2.h.f		2	85.j	even	4	1
1088.2.o.d		2	8.d	odd	2	1
1088.2.o.d		2	136.j	odd	4	1
1088.2.o.l		2	8.b	even	2	1
1088.2.o.l		2	136.i	even	4	1
2448.2.be.c		2	12.b	even	2	1
2448.2.be.c		2	204.l	even	4	1
4624.2.a.r		2	68.g	odd	8	2
5202.2.a.v		2	51.g	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(34, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 1 \)
$3$	\( T^{2} + 2T + 2 \)
$5$	\( T^{2} - 4T + 8 \)
$7$	\( T^{2} + 4T + 8 \)
$11$	\( T^{2} - 2T + 2 \)