Newform orbit 900.3.c.e

• Label: 900.3.c.e
• Level: $900$
• Weight: $3$
• Character orbit: 900.c
• Analytic conductor: $24.523$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.5232237924\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + ( - 2 \beta - 2) q^{4} - 4 \beta q^{7} + 8 q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

451.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

0

−2.00000

+

3.46410i

0

0

6.92820i

8.00000

0

0

451.2

−1.00000

+

1.73205i

0

−2.00000

−

3.46410i

0

0

−

6.92820i

8.00000

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.3.c.e		2
3.b	odd	2	1		300.3.c.b		2
4.b	odd	2	1	inner	900.3.c.e		2
5.b	even	2	1		36.3.d.c		2
5.c	odd	4	2		900.3.f.c		4
12.b	even	2	1		300.3.c.b		2
15.d	odd	2	1		12.3.d.a	✓	2
15.e	even	4	2		300.3.f.a		4
20.d	odd	2	1		36.3.d.c		2
20.e	even	4	2		900.3.f.c		4
40.e	odd	2	1		576.3.g.e		2
40.f	even	2	1		576.3.g.e		2
45.h	odd	6	1		324.3.f.d		2
45.h	odd	6	1		324.3.f.j		2
45.j	even	6	1		324.3.f.a		2
45.j	even	6	1		324.3.f.g		2
60.h	even	2	1		12.3.d.a	✓	2
60.l	odd	4	2		300.3.f.a		4
80.k	odd	4	2		2304.3.b.l		4
80.q	even	4	2		2304.3.b.l		4
105.g	even	2	1		588.3.g.b		2
120.i	odd	2	1		192.3.g.b		2
120.m	even	2	1		192.3.g.b		2
180.n	even	6	1		324.3.f.d		2
180.n	even	6	1		324.3.f.j		2
180.p	odd	6	1		324.3.f.a		2
180.p	odd	6	1		324.3.f.g		2
240.t	even	4	2		768.3.b.c		4
240.bm	odd	4	2		768.3.b.c		4
420.o	odd	2	1		588.3.g.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.3.d.a	✓	2	15.d	odd	2	1
12.3.d.a	✓	2	60.h	even	2	1
36.3.d.c		2	5.b	even	2	1
36.3.d.c		2	20.d	odd	2	1
192.3.g.b		2	120.i	odd	2	1
192.3.g.b		2	120.m	even	2	1
300.3.c.b		2	3.b	odd	2	1
300.3.c.b		2	12.b	even	2	1
300.3.f.a		4	15.e	even	4	2
300.3.f.a		4	60.l	odd	4	2
324.3.f.a		2	45.j	even	6	1
324.3.f.a		2	180.p	odd	6	1
324.3.f.d		2	45.h	odd	6	1
324.3.f.d		2	180.n	even	6	1
324.3.f.g		2	45.j	even	6	1
324.3.f.g		2	180.p	odd	6	1
324.3.f.j		2	45.h	odd	6	1
324.3.f.j		2	180.n	even	6	1
576.3.g.e		2	40.e	odd	2	1
576.3.g.e		2	40.f	even	2	1
588.3.g.b		2	105.g	even	2	1
588.3.g.b		2	420.o	odd	2	1
768.3.b.c		4	240.t	even	4	2
768.3.b.c		4	240.bm	odd	4	2
900.3.c.e		2	1.a	even	1	1	trivial
900.3.c.e		2	4.b	odd	2	1	inner
900.3.f.c		4	5.c	odd	4	2
900.3.f.c		4	20.e	even	4	2
2304.3.b.l		4	80.k	odd	4	2
2304.3.b.l		4	80.q	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 48 \)

\( T_{13} + 2 \)

\( T_{17} - 10 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 48 \)
$11$	\( T^{2} + 48 \)