Newform orbit 228.2.f.a

• Label: 228.2.f.a
• Level: $228$
• Weight: $2$
• Character orbit: 228.f
• Analytic conductor: $1.821$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.82058916609\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	10.0.28906271571968.1
Defining polynomial:	\( x^{10} - 2x^{9} + x^{8} + 8x^{2} - 32x + 32 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 8x^{2} - 32x + 32 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{9} - \nu^{7} + 2\nu^{6} + 4\nu^{5} + 4\nu^{4} - 8\nu^{2} - 8\nu + 16 ) / 16 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} + \nu^{7} + 2\nu^{4} - 4\nu^{2} + 8\nu - 16 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{9} + \nu^{7} - 6\nu^{6} + 4\nu^{5} + 8\nu - 16 ) / 16 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{9} + 3\nu^{7} + 2\nu^{6} - 4\nu^{4} + 8\nu^{3} - 8\nu^{2} - 8\nu + 16 ) / 16 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{9} + \nu^{7} - 2\nu^{5} + 2\nu^{4} - 4\nu^{3} + 4\nu^{2} - 8\nu + 16 ) / 8 \)
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{9} + 4\nu^{8} + \nu^{7} + 2\nu^{6} - 4\nu^{4} - 8\nu^{3} + 8\nu^{2} - 24\nu + 80 ) / 16 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)	\(115\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

151.1

1.39467	+	0.234296i
1.39467	−	0.234296i
1.13961	+	0.837430i
1.13961	−	0.837430i
0.391657	+	1.35890i
0.391657	−	1.35890i
−0.610711	+	1.27555i
−0.610711	−	1.27555i
−1.31523	+	0.519787i
−1.31523	−	0.519787i

−1.39467

−

0.234296i

1.00000

1.89021

+

0.653533i

1.25558

−1.39467

−

0.234296i

−

4.64552i

−2.48310

−

1.35433i

1.00000

−1.75112

−

0.294178i

151.2

−1.39467

+

0.234296i

1.00000

1.89021

−

0.653533i

1.25558

−1.39467

+

0.234296i

4.64552i

−2.48310

+

1.35433i

1.00000

−1.75112

+

0.294178i

151.3

−1.13961

−

0.837430i

1.00000

0.597421

+

1.90869i

−3.88360

−1.13961

−

0.837430i

2.45653i

0.917565

−

2.67546i

1.00000

4.42579

+

3.25224i

151.4

−1.13961

+

0.837430i

1.00000

0.597421

−

1.90869i

−3.88360

−1.13961

+

0.837430i

−

2.45653i

0.917565

+

2.67546i

1.00000

4.42579

−

3.25224i

151.5

−0.391657

−

1.35890i

1.00000

−1.69321

+

1.06444i

3.42712

−0.391657

−

1.35890i

1.84067i

2.10963

+

1.88400i

1.00000

−1.34226

−

4.65711i

151.6

−0.391657

+

1.35890i

1.00000

−1.69321

−

1.06444i

3.42712

−0.391657

+

1.35890i

−

1.84067i

2.10963

−

1.88400i

1.00000

−1.34226

+

4.65711i

151.7

0.610711

−

1.27555i

1.00000

−1.25406

−

1.55799i

0.399423

0.610711

−

1.27555i

−

2.63719i

−2.75316

+

0.648146i

1.00000

0.243932

−

0.509484i

151.8

0.610711

+

1.27555i

1.00000

−1.25406

+

1.55799i

0.399423

0.610711

+

1.27555i

2.63719i

−2.75316

−

0.648146i

1.00000

0.243932

+

0.509484i

151.9

1.31523

−

0.519787i

1.00000

1.45964

−

1.36728i

−1.19853

1.31523

−

0.519787i

−

0.204235i

1.20907

−

2.55698i

1.00000

−1.57634

+

0.622981i

151.10

1.31523

+

0.519787i

1.00000

1.45964

+

1.36728i

−1.19853

1.31523

+

0.519787i

0.204235i

1.20907

+

2.55698i

1.00000

−1.57634

−

0.622981i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
76.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.2.f.a	✓	10
3.b	odd	2	1		684.2.f.d		10
4.b	odd	2	1		228.2.f.b	yes	10
8.b	even	2	1		3648.2.k.i		10
8.d	odd	2	1		3648.2.k.j		10
12.b	even	2	1		684.2.f.c		10
19.b	odd	2	1		228.2.f.b	yes	10
57.d	even	2	1		684.2.f.c		10
76.d	even	2	1	inner	228.2.f.a	✓	10
152.b	even	2	1		3648.2.k.i		10
152.g	odd	2	1		3648.2.k.j		10
228.b	odd	2	1		684.2.f.d		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.2.f.a	✓	10	1.a	even	1	1	trivial
228.2.f.a	✓	10	76.d	even	2	1	inner
228.2.f.b	yes	10	4.b	odd	2	1
228.2.f.b	yes	10	19.b	odd	2	1
684.2.f.c		10	12.b	even	2	1
684.2.f.c		10	57.d	even	2	1
684.2.f.d		10	3.b	odd	2	1
684.2.f.d		10	228.b	odd	2	1
3648.2.k.i		10	8.b	even	2	1
3648.2.k.i		10	152.b	even	2	1
3648.2.k.j		10	8.d	odd	2	1
3648.2.k.j		10	152.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{31}^{5} + 6T_{31}^{4} - 56T_{31}^{3} - 208T_{31}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(228, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^10 + 2*T^9 + T^8 + 8*T^2 + 32*T + 32
$3$	\( (T - 1)^{10} \)
$5$	(T^5 - 15*T^3 + 6*T^2 + 20*T - 8)^2
$7$	T^10 + 38*T^8 + 441*T^6 + 2016*T^4 + 3152*T^2 + 128
$11$	T^10 + 64*T^8 + 1509*T^6 + 15882*T^4 + 68032*T^2 + 59168