Newform orbit 900.2.i.b

• Label: 900.2.i.b
• Level: $900$
• Weight: $2$
• Character orbit: 900.i
• Analytic conductor: $7.187$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

301.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

1.73205i

0

0

0

−0.500000

−

0.866025i

0

−3.00000

0

601.1

0

−

1.73205i

0

0

0

−0.500000

+

0.866025i

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.i.b		2
3.b	odd	2	1		2700.2.i.b		2
5.b	even	2	1		36.2.e.a	✓	2
5.c	odd	4	2		900.2.s.b		4
9.c	even	3	1	inner	900.2.i.b		2
9.c	even	3	1		8100.2.a.j		1
9.d	odd	6	1		2700.2.i.b		2
9.d	odd	6	1		8100.2.a.g		1
15.d	odd	2	1		108.2.e.a		2
15.e	even	4	2		2700.2.s.b		4
20.d	odd	2	1		144.2.i.a		2
35.c	odd	2	1		1764.2.j.b		2
35.i	odd	6	1		1764.2.i.c		2
35.i	odd	6	1		1764.2.l.a		2
35.j	even	6	1		1764.2.i.a		2
35.j	even	6	1		1764.2.l.c		2
40.e	odd	2	1		576.2.i.e		2
40.f	even	2	1		576.2.i.f		2
45.h	odd	6	1		108.2.e.a		2
45.h	odd	6	1		324.2.a.a		1
45.j	even	6	1		36.2.e.a	✓	2
45.j	even	6	1		324.2.a.c		1
45.k	odd	12	2		900.2.s.b		4
45.k	odd	12	2		8100.2.d.h		2
45.l	even	12	2		2700.2.s.b		4
45.l	even	12	2		8100.2.d.c		2
60.h	even	2	1		432.2.i.c		2
105.g	even	2	1		5292.2.j.a		2
105.o	odd	6	1		5292.2.i.c		2
105.o	odd	6	1		5292.2.l.a		2
105.p	even	6	1		5292.2.i.a		2
105.p	even	6	1		5292.2.l.c		2
120.i	odd	2	1		1728.2.i.d		2
120.m	even	2	1		1728.2.i.c		2
180.n	even	6	1		432.2.i.c		2
180.n	even	6	1		1296.2.a.b		1
180.p	odd	6	1		144.2.i.a		2
180.p	odd	6	1		1296.2.a.k		1
315.q	odd	6	1		1764.2.l.a		2
315.r	even	6	1		1764.2.l.c		2
315.u	even	6	1		5292.2.i.a		2
315.v	odd	6	1		5292.2.i.c		2
315.z	even	6	1		5292.2.j.a		2
315.bg	odd	6	1		1764.2.j.b		2
315.bn	odd	6	1		1764.2.i.c		2
315.bo	even	6	1		1764.2.i.a		2
315.bq	even	6	1		5292.2.l.c		2
315.br	odd	6	1		5292.2.l.a		2
360.z	odd	6	1		576.2.i.e		2
360.z	odd	6	1		5184.2.a.f		1
360.bd	even	6	1		1728.2.i.c		2
360.bd	even	6	1		5184.2.a.bb		1
360.bh	odd	6	1		1728.2.i.d		2
360.bh	odd	6	1		5184.2.a.ba		1
360.bk	even	6	1		576.2.i.f		2
360.bk	even	6	1		5184.2.a.e		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.e.a	✓	2	5.b	even	2	1
36.2.e.a	✓	2	45.j	even	6	1
108.2.e.a		2	15.d	odd	2	1
108.2.e.a		2	45.h	odd	6	1
144.2.i.a		2	20.d	odd	2	1
144.2.i.a		2	180.p	odd	6	1
324.2.a.a		1	45.h	odd	6	1
324.2.a.c		1	45.j	even	6	1
432.2.i.c		2	60.h	even	2	1
432.2.i.c		2	180.n	even	6	1
576.2.i.e		2	40.e	odd	2	1
576.2.i.e		2	360.z	odd	6	1
576.2.i.f		2	40.f	even	2	1
576.2.i.f		2	360.bk	even	6	1
900.2.i.b		2	1.a	even	1	1	trivial
900.2.i.b		2	9.c	even	3	1	inner
900.2.s.b		4	5.c	odd	4	2
900.2.s.b		4	45.k	odd	12	2
1296.2.a.b		1	180.n	even	6	1
1296.2.a.k		1	180.p	odd	6	1
1728.2.i.c		2	120.m	even	2	1
1728.2.i.c		2	360.bd	even	6	1
1728.2.i.d		2	120.i	odd	2	1
1728.2.i.d		2	360.bh	odd	6	1
1764.2.i.a		2	35.j	even	6	1
1764.2.i.a		2	315.bo	even	6	1
1764.2.i.c		2	35.i	odd	6	1
1764.2.i.c		2	315.bn	odd	6	1
1764.2.j.b		2	35.c	odd	2	1
1764.2.j.b		2	315.bg	odd	6	1
1764.2.l.a		2	35.i	odd	6	1
1764.2.l.a		2	315.q	odd	6	1
1764.2.l.c		2	35.j	even	6	1
1764.2.l.c		2	315.r	even	6	1
2700.2.i.b		2	3.b	odd	2	1
2700.2.i.b		2	9.d	odd	6	1
2700.2.s.b		4	15.e	even	4	2
2700.2.s.b		4	45.l	even	12	2
5184.2.a.e		1	360.bk	even	6	1
5184.2.a.f		1	360.z	odd	6	1
5184.2.a.ba		1	360.bh	odd	6	1
5184.2.a.bb		1	360.bd	even	6	1
5292.2.i.a		2	105.p	even	6	1
5292.2.i.a		2	315.u	even	6	1
5292.2.i.c		2	105.o	odd	6	1
5292.2.i.c		2	315.v	odd	6	1
5292.2.j.a		2	105.g	even	2	1
5292.2.j.a		2	315.z	even	6	1
5292.2.l.a		2	105.o	odd	6	1
5292.2.l.a		2	315.br	odd	6	1
5292.2.l.c		2	105.p	even	6	1
5292.2.l.c		2	315.bq	even	6	1
8100.2.a.g		1	9.d	odd	6	1
8100.2.a.j		1	9.c	even	3	1
8100.2.d.c		2	45.l	even	12	2
8100.2.d.h		2	45.k	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \)

\( T_{11}^{2} + 3T_{11} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + T + 1 \)
$11$	\( T^{2} + 3T + 9 \)