Newform orbit 612.2.b.a

• Label: 612.2.b.a
• Level: $612$
• Weight: $2$
• Character orbit: 612.b
• Analytic conductor: $4.887$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	612.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.88684460370\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + 2 \beta q^{5} + 3 \beta q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(137\)	\(307\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

577.1

	−	1.41421i
		1.41421i

0

0

0

−

2.82843i

0

−

4.24264i

0

0

0

577.2

0

0

0

2.82843i

0

4.24264i

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	612.2.b.a		2
3.b	odd	2	1		68.2.b.a	✓	2
4.b	odd	2	1		2448.2.c.d		2
12.b	even	2	1		272.2.b.c		2
15.d	odd	2	1		1700.2.c.a		2
15.e	even	4	2		1700.2.g.a		4
17.b	even	2	1	inner	612.2.b.a		2
21.c	even	2	1		3332.2.b.a		2
24.f	even	2	1		1088.2.b.f		2
24.h	odd	2	1		1088.2.b.e		2
51.c	odd	2	1		68.2.b.a	✓	2
51.f	odd	4	2		1156.2.a.c		2
51.g	odd	8	2		1156.2.e.a		2
51.g	odd	8	2		1156.2.e.b		2
51.i	even	16	8		1156.2.h.d		8
68.d	odd	2	1		2448.2.c.d		2
204.h	even	2	1		272.2.b.c		2
204.l	even	4	2		4624.2.a.n		2
255.h	odd	2	1		1700.2.c.a		2
255.o	even	4	2		1700.2.g.a		4
357.c	even	2	1		3332.2.b.a		2
408.b	odd	2	1		1088.2.b.e		2
408.h	even	2	1		1088.2.b.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.2.b.a	✓	2	3.b	odd	2	1
68.2.b.a	✓	2	51.c	odd	2	1
272.2.b.c		2	12.b	even	2	1
272.2.b.c		2	204.h	even	2	1
612.2.b.a		2	1.a	even	1	1	trivial
612.2.b.a		2	17.b	even	2	1	inner
1088.2.b.e		2	24.h	odd	2	1
1088.2.b.e		2	408.b	odd	2	1
1088.2.b.f		2	24.f	even	2	1
1088.2.b.f		2	408.h	even	2	1
1156.2.a.c		2	51.f	odd	4	2
1156.2.e.a		2	51.g	odd	8	2
1156.2.e.b		2	51.g	odd	8	2
1156.2.h.d		8	51.i	even	16	8
1700.2.c.a		2	15.d	odd	2	1
1700.2.c.a		2	255.h	odd	2	1
1700.2.g.a		4	15.e	even	4	2
1700.2.g.a		4	255.o	even	4	2
2448.2.c.d		2	4.b	odd	2	1
2448.2.c.d		2	68.d	odd	2	1
3332.2.b.a		2	21.c	even	2	1
3332.2.b.a		2	357.c	even	2	1
4624.2.a.n		2	204.l	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(612, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 8 \)
$7$	\( T^{2} + 18 \)
$11$	\( T^{2} + 2 \)