Newform orbit 561.2.g.a

• Label: 561.2.g.a
• Level: $561$
• Weight: $2$
• Character orbit: 561.g
• Analytic conductor: $4.480$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 561 = 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	561.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47960755339\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 17x^{10} + 104x^{8} + 288x^{6} + 356x^{4} + 152x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{2} + 1) q^{2} + \beta_{4} q^{3} + ( - \beta_{3} + \beta_{2} + 1) q^{4} + (\beta_{9} - \beta_{4}) q^{5} - \beta_1 q^{6} + \beta_{11} q^{7} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{8} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 10 q^{4} + 18 q^{8} - 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17x^{10} + 104x^{8} + 288x^{6} + 356x^{4} + 152x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -5\nu^{10} - 64\nu^{8} - 241\nu^{6} - 302\nu^{4} - 90\nu^{2} - 42 ) / 34 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{10} - 64\nu^{8} - 241\nu^{6} - 302\nu^{4} - 56\nu^{2} + 60 ) / 34 \)
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{11} + 29\nu^{9} + 144\nu^{7} + 335\nu^{5} + 410\nu^{3} + 214\nu ) / 34 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{10} + 23\nu^{8} + 174\nu^{6} + 482\nu^{4} + 358\nu^{2} - 46 ) / 34 \)
\(\beta_{6}\)	\(=\)	\( ( 7\nu^{10} + 93\nu^{8} + 368\nu^{6} + 450\nu^{4} - 10\nu^{2} - 50 ) / 34 \)
\(\beta_{7}\)	\(=\)	\( ( -3\nu^{11} - 35\nu^{9} - 97\nu^{7} + 33\nu^{5} + 354\nu^{3} + 342\nu ) / 34 \)
\(\beta_{8}\)	\(=\)	\( ( \nu^{11} + 6\nu^{9} - 47\nu^{7} - 368\nu^{5} - 730\nu^{3} - 386\nu ) / 34 \)
\(\beta_{9}\)	\(=\)	\( ( -5\nu^{11} - 64\nu^{9} - 224\nu^{7} - 115\nu^{5} + 420\nu^{3} + 264\nu ) / 34 \)
\(\beta_{10}\)	\(=\)	\( ( 14\nu^{10} + 203\nu^{8} + 957\nu^{6} + 1750\nu^{4} + 1034\nu^{2} + 70 ) / 34 \)
\(\beta_{11}\)	\(=\)	\( ( -9\nu^{11} - 139\nu^{9} - 733\nu^{7} - 1635\nu^{5} - 1454\nu^{3} - 334\nu ) / 34 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/561\mathbb{Z}\right)^\times\).

\(n\)	\(188\)	\(409\)	\(496\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

	−	1.88799i
		1.88799i
	−	0.893504i
		0.893504i
	−	0.167711i
		0.167711i
		1.47553i
	−	1.47553i
		1.77590i
	−	1.77590i
		2.69778i
	−	2.69778i

−1.88799

−

1.00000i

1.56451

1.22459i

1.88799i

−

0.648622i

0.822197

−1.00000

−

2.31202i

67.2

−1.88799

1.00000i

1.56451

−

1.22459i

−

1.88799i

0.648622i

0.822197

−1.00000

2.31202i

67.3

−0.893504

−

1.00000i

−1.20165

2.44539i

0.893504i

−

2.73686i

2.86069

−1.00000

−

2.18497i

67.4

−0.893504

1.00000i

−1.20165

−

2.44539i

−

0.893504i

2.73686i

2.86069

−1.00000

2.18497i

67.5

−0.167711

−

1.00000i

−1.97187

−

0.243530i

0.167711i

1.45208i

0.666126

−1.00000

0.0408427i

67.6

−0.167711

1.00000i

−1.97187

0.243530i

−

0.167711i

−

1.45208i

0.666126

−1.00000

−

0.0408427i

67.7

1.47553

−

1.00000i

0.177179

−

2.35180i

−

1.47553i

−

1.59387i

−2.68962

−1.00000

−

3.47014i

67.8

1.47553

1.00000i

0.177179

2.35180i

1.47553i

1.59387i

−2.68962

−1.00000

3.47014i

67.9

1.77590

−

1.00000i

1.15383

3.64603i

−

1.77590i

2.05306i

−1.50272

−1.00000

6.47499i

67.10

1.77590

1.00000i

1.15383

−

3.64603i

1.77590i

−

2.05306i

−1.50272

−1.00000

−

6.47499i

67.11

2.69778

−

1.00000i

5.27801

1.27931i

−

2.69778i

0.474209i

8.84333

−1.00000

3.45130i

67.12

2.69778

1.00000i

5.27801

−

1.27931i

2.69778i

−

0.474209i

8.84333

−1.00000

−

3.45130i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	561.2.g.a	✓	12
3.b	odd	2	1		1683.2.g.a		12
17.b	even	2	1	inner	561.2.g.a	✓	12
17.c	even	4	1		9537.2.a.bd		6
17.c	even	4	1		9537.2.a.be		6
51.c	odd	2	1		1683.2.g.a		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
561.2.g.a	✓	12	1.a	even	1	1	trivial
561.2.g.a	✓	12	17.b	even	2	1	inner
1683.2.g.a		12	3.b	odd	2	1
1683.2.g.a		12	51.c	odd	2	1
9537.2.a.bd		6	17.c	even	4	1
9537.2.a.be		6	17.c	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 14T_{2}^{3} + 2T_{2}^{2} - 12T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^6 - 3*T^5 - 4*T^4 + 14*T^3 + 2*T^2 - 12*T - 2)^2
$3$	\( (T^{2} + 1)^{6} \)
$5$	T^12 + 28*T^10 + 268*T^8 + 1100*T^6 + 1900*T^4 + 1188*T^2 + 64
$7$	T^12 + 17*T^10 + 102*T^8 + 270*T^6 + 313*T^4 + 129*T^2 + 16
$11$	\( (T^{2} + 1)^{6} \)