Newform orbit 600.3.l.b

• Label: 600.3.l.b
• Level: $600$
• Weight: $3$
• Character orbit: 600.l
• Analytic conductor: $16.349$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.3488158616\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{3} + 6 q^{7} + ( - 2 \beta - 7) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 12 q^{7} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\)	\(151\)	\(301\)	\(401\)	\(577\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

401.1

	−	1.41421i
		1.41421i

0

−1.00000

−

2.82843i

0

0

0

6.00000

0

−7.00000

+

5.65685i

0

401.2

0

−1.00000

+

2.82843i

0

0

0

6.00000

0

−7.00000

−

5.65685i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.3.l.b		2
3.b	odd	2	1	inner	600.3.l.b		2
4.b	odd	2	1		1200.3.l.n		2
5.b	even	2	1		24.3.e.a	✓	2
5.c	odd	4	2		600.3.c.a		4
12.b	even	2	1		1200.3.l.n		2
15.d	odd	2	1		24.3.e.a	✓	2
15.e	even	4	2		600.3.c.a		4
20.d	odd	2	1		48.3.e.b		2
20.e	even	4	2		1200.3.c.i		4
35.c	odd	2	1		1176.3.d.a		2
40.e	odd	2	1		192.3.e.d		2
40.f	even	2	1		192.3.e.c		2
45.h	odd	6	2		648.3.m.d		4
45.j	even	6	2		648.3.m.d		4
60.h	even	2	1		48.3.e.b		2
60.l	odd	4	2		1200.3.c.i		4
80.k	odd	4	2		768.3.h.c		4
80.q	even	4	2		768.3.h.d		4
105.g	even	2	1		1176.3.d.a		2
120.i	odd	2	1		192.3.e.c		2
120.m	even	2	1		192.3.e.d		2
180.n	even	6	2		1296.3.q.e		4
180.p	odd	6	2		1296.3.q.e		4
240.t	even	4	2		768.3.h.c		4
240.bm	odd	4	2		768.3.h.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.3.e.a	✓	2	5.b	even	2	1
24.3.e.a	✓	2	15.d	odd	2	1
48.3.e.b		2	20.d	odd	2	1
48.3.e.b		2	60.h	even	2	1
192.3.e.c		2	40.f	even	2	1
192.3.e.c		2	120.i	odd	2	1
192.3.e.d		2	40.e	odd	2	1
192.3.e.d		2	120.m	even	2	1
600.3.c.a		4	5.c	odd	4	2
600.3.c.a		4	15.e	even	4	2
600.3.l.b		2	1.a	even	1	1	trivial
600.3.l.b		2	3.b	odd	2	1	inner
648.3.m.d		4	45.h	odd	6	2
648.3.m.d		4	45.j	even	6	2
768.3.h.c		4	80.k	odd	4	2
768.3.h.c		4	240.t	even	4	2
768.3.h.d		4	80.q	even	4	2
768.3.h.d		4	240.bm	odd	4	2
1176.3.d.a		2	35.c	odd	2	1
1176.3.d.a		2	105.g	even	2	1
1200.3.c.i		4	20.e	even	4	2
1200.3.c.i		4	60.l	odd	4	2
1200.3.l.n		2	4.b	odd	2	1
1200.3.l.n		2	12.b	even	2	1
1296.3.q.e		4	180.n	even	6	2
1296.3.q.e		4	180.p	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 6 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2T + 9 \)
$5$	\( T^{2} \)
$7$	\( (T - 6)^{2} \)
$11$	\( T^{2} + 32 \)