Newform orbit 450.3.g.b

• Label: 450.3.g.b
• Level: $450$
• Weight: $3$
• Character orbit: 450.g
• Analytic conductor: $12.262$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2616118962\)
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 10)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

307.1

		1.00000i
	−	1.00000i

−1.00000

−

1.00000i

0

2.00000i

0

0

−2.00000

−

2.00000i

2.00000

−

2.00000i

0

0

343.1

−1.00000

+

1.00000i

0

−

2.00000i

0

0

−2.00000

+

2.00000i

2.00000

+

2.00000i

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	450.3.g.b		2
3.b	odd	2	1		50.3.c.c		2
5.b	even	2	1		90.3.g.b		2
5.c	odd	4	1		90.3.g.b		2
5.c	odd	4	1	inner	450.3.g.b		2
12.b	even	2	1		400.3.p.b		2
15.d	odd	2	1		10.3.c.a	✓	2
15.e	even	4	1		10.3.c.a	✓	2
15.e	even	4	1		50.3.c.c		2
20.d	odd	2	1		720.3.bh.c		2
20.e	even	4	1		720.3.bh.c		2
60.h	even	2	1		80.3.p.c		2
60.l	odd	4	1		80.3.p.c		2
60.l	odd	4	1		400.3.p.b		2
105.g	even	2	1		490.3.f.b		2
105.k	odd	4	1		490.3.f.b		2
120.i	odd	2	1		320.3.p.h		2
120.m	even	2	1		320.3.p.a		2
120.q	odd	4	1		320.3.p.a		2
120.w	even	4	1		320.3.p.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.3.c.a	✓	2	15.d	odd	2	1
10.3.c.a	✓	2	15.e	even	4	1
50.3.c.c		2	3.b	odd	2	1
50.3.c.c		2	15.e	even	4	1
80.3.p.c		2	60.h	even	2	1
80.3.p.c		2	60.l	odd	4	1
90.3.g.b		2	5.b	even	2	1
90.3.g.b		2	5.c	odd	4	1
320.3.p.a		2	120.m	even	2	1
320.3.p.a		2	120.q	odd	4	1
320.3.p.h		2	120.i	odd	2	1
320.3.p.h		2	120.w	even	4	1
400.3.p.b		2	12.b	even	2	1
400.3.p.b		2	60.l	odd	4	1
450.3.g.b		2	1.a	even	1	1	trivial
450.3.g.b		2	5.c	odd	4	1	inner
490.3.f.b		2	105.g	even	2	1
490.3.f.b		2	105.k	odd	4	1
720.3.bh.c		2	20.d	odd	2	1
720.3.bh.c		2	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 4T_{7} + 8 \)

\( T_{11} - 8 \)

\( T_{17}^{2} - 14T_{17} + 98 \)

Hecke characteristic polynomials

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