Newform orbit 420.2.ba.a

• Label: 420.2.ba.a
• Level: $420$
• Weight: $2$
• Character orbit: 420.ba
• Analytic conductor: $3.354$
• Analytic rank: $0$
• Dimension: $8$
• CM discriminant: -20
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35371688489\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.3317760000.8
Defining polynomial:	\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + (\beta_{7} + \beta_{5}) q^{3} + 2 \beta_{3} q^{4} - \beta_{6} q^{5} + (\beta_{3} + \beta_{2}) q^{6} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1) q^{7} - 2 \beta_{4} q^{8} + ( - 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \)
\(\beta_{3}\)	\(=\)	\( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \)
\(\beta_{4}\)	\(=\)	\( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \)
\(\beta_{5}\)	\(=\)	\( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \)
\(\beta_{6}\)	\(=\)	\( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} - 4\nu^{5} - 7\nu^{3} - 36\nu ) / 27 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(241\)	\(281\)	\(337\)
\(\chi(n)\)	\(-1\)	\(-1 + \beta_{3}\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

179.1

−0.178197	+	1.72286i
1.40294	−	1.01575i
−1.40294	+	1.01575i
0.178197	−	1.72286i
−0.178197	−	1.72286i
1.40294	+	1.01575i
−1.40294	−	1.01575i
0.178197	+	1.72286i

−1.22474

+

0.707107i

−1.40294

−

1.01575i

1.00000

−

1.73205i

1.93649

−

1.11803i

2.43649

+

0.252009i

2.62769

−

0.308646i

2.82843i

0.936492

+

2.85008i

−1.58114

+

2.73861i

179.2

−1.22474

+

0.707107i

0.178197

+

1.72286i

1.00000

−

1.73205i

−1.93649

+

1.11803i

−1.43649

−

1.98406i

1.04655

+

2.42997i

2.82843i

−2.93649

+

0.614017i

1.58114

−

2.73861i

179.3

1.22474

−

0.707107i

−0.178197

−

1.72286i

1.00000

−

1.73205i

−1.93649

+

1.11803i

−1.43649

−

1.98406i

−1.04655

−

2.42997i

−

2.82843i

−2.93649

+

0.614017i

−1.58114

+

2.73861i

179.4

1.22474

−

0.707107i

1.40294

+

1.01575i

1.00000

−

1.73205i

1.93649

−

1.11803i

2.43649

+

0.252009i

−2.62769

+

0.308646i

−

2.82843i

0.936492

+

2.85008i

1.58114

−

2.73861i

359.1

−1.22474

−

0.707107i

−1.40294

+

1.01575i

1.00000

+

1.73205i

1.93649

+

1.11803i

2.43649

−

0.252009i

2.62769

+

0.308646i

−

2.82843i

0.936492

−

2.85008i

−1.58114

−

2.73861i

359.2

−1.22474

−

0.707107i

0.178197

−

1.72286i

1.00000

+

1.73205i

−1.93649

−

1.11803i

−1.43649

+

1.98406i

1.04655

−

2.42997i

−

2.82843i

−2.93649

−

0.614017i

1.58114

+

2.73861i

359.3

1.22474

+

0.707107i

−0.178197

+

1.72286i

1.00000

+

1.73205i

−1.93649

−

1.11803i

−1.43649

+

1.98406i

−1.04655

+

2.42997i

2.82843i

−2.93649

−

0.614017i

−1.58114

−

2.73861i

359.4

1.22474

+

0.707107i

1.40294

−

1.01575i

1.00000

+

1.73205i

1.93649

+

1.11803i

2.43649

−

0.252009i

−2.62769

−

0.308646i

2.82843i

0.936492

−

2.85008i

1.58114

+

2.73861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
4.b	odd	2	1	inner
5.b	even	2	1	inner
21.h	odd	6	1	inner
84.n	even	6	1	inner
105.o	odd	6	1	inner
420.ba	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	420.2.ba.a	✓	8
3.b	odd	2	1		420.2.ba.b	yes	8
4.b	odd	2	1	inner	420.2.ba.a	✓	8
5.b	even	2	1	inner	420.2.ba.a	✓	8
7.c	even	3	1		420.2.ba.b	yes	8
12.b	even	2	1		420.2.ba.b	yes	8
15.d	odd	2	1		420.2.ba.b	yes	8
20.d	odd	2	1	CM	420.2.ba.a	✓	8
21.h	odd	6	1	inner	420.2.ba.a	✓	8
28.g	odd	6	1		420.2.ba.b	yes	8
35.j	even	6	1		420.2.ba.b	yes	8
60.h	even	2	1		420.2.ba.b	yes	8
84.n	even	6	1	inner	420.2.ba.a	✓	8
105.o	odd	6	1	inner	420.2.ba.a	✓	8
140.p	odd	6	1		420.2.ba.b	yes	8
420.ba	even	6	1	inner	420.2.ba.a	✓	8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.ba.a	✓	8	1.a	even	1	1	trivial
420.2.ba.a	✓	8	4.b	odd	2	1	inner
420.2.ba.a	✓	8	5.b	even	2	1	inner
420.2.ba.a	✓	8	20.d	odd	2	1	CM
420.2.ba.a	✓	8	21.h	odd	6	1	inner
420.2.ba.a	✓	8	84.n	even	6	1	inner
420.2.ba.a	✓	8	105.o	odd	6	1	inner
420.2.ba.a	✓	8	420.ba	even	6	1	inner
420.2.ba.b	yes	8	3.b	odd	2	1
420.2.ba.b	yes	8	7.c	even	3	1
420.2.ba.b	yes	8	12.b	even	2	1
420.2.ba.b	yes	8	15.d	odd	2	1
420.2.ba.b	yes	8	28.g	odd	6	1
420.2.ba.b	yes	8	35.j	even	6	1
420.2.ba.b	yes	8	60.h	even	2	1
420.2.ba.b	yes	8	140.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11} \)

\( T_{89}^{4} + 18T_{89}^{3} + 55T_{89}^{2} - 954T_{89} + 2809 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 2 T^{2} + 4)^{2} \)
$3$	T^8 + 4*T^6 + 7*T^4 + 36*T^2 + 81
$5$	\( (T^{4} - 5 T^{2} + 25)^{2} \)
$7$	T^8 - 4*T^6 - 33*T^4 - 196*T^2 + 2401
$11$	\( T^{8} \)