Newform orbit 608.2.b.a

• Label: 608.2.b.a
• Level: $608$
• Weight: $2$
• Character orbit: 608.b
• Analytic conductor: $4.855$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.85490444289\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + 2 \beta q^{3} - 5 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

303.1

	−	1.41421i
		1.41421i

0

−

2.82843i

0

0

0

0

0

−5.00000

0

303.2

0

2.82843i

0

0

0

0

0

−5.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
19.b	odd	2	1	inner
152.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	608.2.b.a		2
3.b	odd	2	1		5472.2.e.a		2
4.b	odd	2	1		152.2.b.a	✓	2
8.b	even	2	1		152.2.b.a	✓	2
8.d	odd	2	1	CM	608.2.b.a		2
12.b	even	2	1		1368.2.e.a		2
19.b	odd	2	1	inner	608.2.b.a		2
24.f	even	2	1		5472.2.e.a		2
24.h	odd	2	1		1368.2.e.a		2
57.d	even	2	1		5472.2.e.a		2
76.d	even	2	1		152.2.b.a	✓	2
152.b	even	2	1	inner	608.2.b.a		2
152.g	odd	2	1		152.2.b.a	✓	2
228.b	odd	2	1		1368.2.e.a		2
456.l	odd	2	1		5472.2.e.a		2
456.p	even	2	1		1368.2.e.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.b.a	✓	2	4.b	odd	2	1
152.2.b.a	✓	2	8.b	even	2	1
152.2.b.a	✓	2	76.d	even	2	1
152.2.b.a	✓	2	152.g	odd	2	1
608.2.b.a		2	1.a	even	1	1	trivial
608.2.b.a		2	8.d	odd	2	1	CM
608.2.b.a		2	19.b	odd	2	1	inner
608.2.b.a		2	152.b	even	2	1	inner
1368.2.e.a		2	12.b	even	2	1
1368.2.e.a		2	24.h	odd	2	1
1368.2.e.a		2	228.b	odd	2	1
1368.2.e.a		2	456.p	even	2	1
5472.2.e.a		2	3.b	odd	2	1
5472.2.e.a		2	24.f	even	2	1
5472.2.e.a		2	57.d	even	2	1
5472.2.e.a		2	456.l	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 8 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( (T + 6)^{2} \)