Newform orbit 900.2.bf.a

• Label: 900.2.bf.a
• Level: $900$
• Weight: $2$
• Character orbit: 900.bf
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $8$
• CM discriminant: -20
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.12960000.1
Defining polynomial:	\( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b7 + b4 - 1) * q^2 + (b3 + b1) * q^3 - 2*b1 * q^4 + (b7 + b6 - b5 - b3 - b1) * q^6 + (2*b7 - b5 + 2*b4 - b2 - b1 - 2) * q^7 + (-2*b7 + 2*b1 - 2) * q^8 + (2*b7 - b6 - b5 - b4 + b2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 2 q^{3} + 8 q^{6} - 6 q^{7} - 16 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 13\nu ) / 8 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} - 4\nu^{4} + 8\nu^{2} + 4\nu - 3 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} + 4\nu^{4} - 8\nu^{2} + 4\nu + 3 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{7} - \nu^{6} - 8\nu^{5} + 24\nu^{3} - \nu - 13 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( -3\nu^{7} - 3\nu^{6} + 8\nu^{5} + 8\nu^{4} - 24\nu^{3} - 16\nu^{2} + 9\nu + 1 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 15\nu ) / 8 \)

Character values

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

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(-\beta_{4}\)	\(-1\)	\(-\beta_{1} + \beta_{7}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

−1.40126	+	0.809017i
0.535233	−	0.309017i
−0.535233	+	0.309017i
1.40126	−	0.809017i
−0.535233	−	0.309017i
1.40126	+	0.809017i
−1.40126	−	0.809017i
0.535233	+	0.309017i

0.366025

+

1.36603i

−1.34425

−

1.09224i

−1.73205

+

1.00000i

0

1.00000

−

2.23607i

−0.159767

−

0.596259i

−2.00000

−

2.00000i

0.614017

+

2.93649i

0

7.2

0.366025

+

1.36603i

1.71028

−

0.273784i

−1.73205

+

1.00000i

0

1.00000

+

2.23607i

1.25784

+

4.69433i

−2.00000

−

2.00000i

2.85008

−

0.936492i

0

43.1

−1.36603

+

0.366025i

−1.09224

+

1.34425i

1.73205

−

1.00000i

0

1.00000

−

2.23607i

0.596259

−

0.159767i

−2.00000

+

2.00000i

−0.614017

−

2.93649i

0

43.2

−1.36603

+

0.366025i

−0.273784

−

1.71028i

1.73205

−

1.00000i

0

1.00000

+

2.23607i

−4.69433

+

1.25784i

−2.00000

+

2.00000i

−2.85008

+

0.936492i

0

607.1

−1.36603

−

0.366025i

−1.09224

−

1.34425i

1.73205

+

1.00000i

0

1.00000

+

2.23607i

0.596259

+

0.159767i

−2.00000

−

2.00000i

−0.614017

+

2.93649i

0

607.2

−1.36603

−

0.366025i

−0.273784

+

1.71028i

1.73205

+

1.00000i

0

1.00000

−

2.23607i

−4.69433

−

1.25784i

−2.00000

−

2.00000i

−2.85008

−

0.936492i

0

643.1

0.366025

−

1.36603i

−1.34425

+

1.09224i

−1.73205

−

1.00000i

0

1.00000

+

2.23607i

−0.159767

+

0.596259i

−2.00000

+

2.00000i

0.614017

−

2.93649i

0

643.2

0.366025

−

1.36603i

1.71028

+

0.273784i

−1.73205

−

1.00000i

0

1.00000

−

2.23607i

1.25784

−

4.69433i

−2.00000

+

2.00000i

2.85008

+

0.936492i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
5.c	odd	4	1	inner
9.c	even	3	1	inner
20.e	even	4	1	inner
45.k	odd	12	1	inner
180.p	odd	6	1	inner
180.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.bf.a	✓	8
4.b	odd	2	1		900.2.bf.b	yes	8
5.b	even	2	1		900.2.bf.b	yes	8
5.c	odd	4	1	inner	900.2.bf.a	✓	8
5.c	odd	4	1		900.2.bf.b	yes	8
9.c	even	3	1	inner	900.2.bf.a	✓	8
20.d	odd	2	1	CM	900.2.bf.a	✓	8
20.e	even	4	1	inner	900.2.bf.a	✓	8
20.e	even	4	1		900.2.bf.b	yes	8
36.f	odd	6	1		900.2.bf.b	yes	8
45.j	even	6	1		900.2.bf.b	yes	8
45.k	odd	12	1	inner	900.2.bf.a	✓	8
45.k	odd	12	1		900.2.bf.b	yes	8
180.p	odd	6	1	inner	900.2.bf.a	✓	8
180.x	even	12	1	inner	900.2.bf.a	✓	8
180.x	even	12	1		900.2.bf.b	yes	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.bf.a	✓	8	1.a	even	1	1	trivial
900.2.bf.a	✓	8	5.c	odd	4	1	inner
900.2.bf.a	✓	8	9.c	even	3	1	inner
900.2.bf.a	✓	8	20.d	odd	2	1	CM
900.2.bf.a	✓	8	20.e	even	4	1	inner
900.2.bf.a	✓	8	45.k	odd	12	1	inner
900.2.bf.a	✓	8	180.p	odd	6	1	inner
900.2.bf.a	✓	8	180.x	even	12	1	inner
900.2.bf.b	yes	8	4.b	odd	2	1
900.2.bf.b	yes	8	5.b	even	2	1
900.2.bf.b	yes	8	5.c	odd	4	1
900.2.bf.b	yes	8	20.e	even	4	1
900.2.bf.b	yes	8	36.f	odd	6	1
900.2.bf.b	yes	8	45.j	even	6	1
900.2.bf.b	yes	8	45.k	odd	12	1
900.2.bf.b	yes	8	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 6T_{7}^{7} + 18T_{7}^{6} + 144T_{7}^{5} + 423T_{7}^{4} - 432T_{7}^{3} + 162T_{7}^{2} - 162T_{7} + 81 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^4 + 2*T^3 + 2*T^2 + 4*T + 4)^2
$3$	T^8 + 2*T^7 + 2*T^6 - 8*T^5 - 17*T^4 - 24*T^3 + 18*T^2 + 54*T + 81
$5$	\( T^{8} \)
$7$	T^8 + 6*T^7 + 18*T^6 + 144*T^5 + 423*T^4 - 432*T^3 + 162*T^2 - 162*T + 81
$11$	\( T^{8} \)