Newform orbit 441.2.f.g

• Label: 441.2.f.g
• Level: $441$
• Weight: $2$
• Character orbit: 441.f
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\beta_{6} - \beta_1) q^{2} + \beta_{10} q^{3} + (\beta_{7} - \beta_{6} - \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8}) q^{6} + (\beta_{4} + \beta_1 + 2) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 6 q^{4} + 24 q^{8} + 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) :

\(\nu\)	\(=\)	\( ( -\beta_{10} - \beta_{8} + 2\beta_{5} - 2\beta_{3} ) / 3 \)
\(\nu^{2}\)	\(=\)	\( \beta_{6} + 2\beta_{2} + 2 \)
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + 3\beta_{10} - \beta_{9} + 4\beta_{8} + 4\beta_{5} - 5\beta_{3} ) / 3 \)
\(\nu^{4}\)	\(=\)	\( -\beta_{7} + 5\beta_{6} - \beta_{4} + 7\beta_{2} - 5\beta_1 \)
\(\nu^{5}\)	\(=\)	\( ( 11\beta_{11} + 42\beta_{10} + 4\beta_{9} + 32\beta_{8} - 16\beta_{5} + 2\beta_{3} ) / 3 \)
\(\nu^{6}\)	\(=\)	\( -7\beta_{4} - 23\beta _1 - 28 \)
\(\nu^{7}\)	\(=\)	\( ( 37\beta_{11} + 141\beta_{10} + 53\beta_{9} + 67\beta_{8} - 134\beta_{5} + 118\beta_{3} ) / 3 \)
\(\nu^{8}\)	\(=\)	\( 37\beta_{7} - 104\beta_{6} - 118\beta_{2} - 118 \)
\(\nu^{9}\)	\(=\)	\( ( -67\beta_{11} - 222\beta_{10} + 178\beta_{9} - 289\beta_{8} - 289\beta_{5} + 578\beta_{3} ) / 3 \)
\(\nu^{10}\)	\(=\)	\( 178\beta_{7} - 467\beta_{6} + 178\beta_{4} - 511\beta_{2} + 467\beta_1 \)
\(\nu^{11}\)	\(=\)	\( ( -1112\beta_{11} - 3891\beta_{10} - 289\beta_{9} - 2534\beta_{8} + 1267\beta_{5} + 379\beta_{3} ) / 3 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

148.1

1.82904	−	1.05600i
−1.82904	+	1.05600i
−1.29589	+	0.748185i
1.29589	−	0.748185i
0.474636	−	0.274031i
−0.474636	+	0.274031i
1.82904	+	1.05600i
−1.82904	−	1.05600i
−1.29589	−	0.748185i
1.29589	+	0.748185i
0.474636	+	0.274031i
−0.474636	−	0.274031i

−1.23025

−

2.13086i

−1.66238

+

0.486291i

−2.02704

+

3.51094i

−1.82904

+

3.16799i

3.08137

+

2.94405i

0

5.05408

2.52704

−

1.61680i

9.00071

148.2

−1.23025

−

2.13086i

1.66238

−

0.486291i

−2.02704

+

3.51094i

1.82904

−

3.16799i

−3.08137

−

2.94405i

0

5.05408

2.52704

−

1.61680i

−9.00071

148.3

−0.119562

−

0.207087i

−1.12441

+

1.31746i

0.971410

−

1.68253i

1.29589

−

2.24456i

0.407265

+

0.0753324i

0

−0.942820

−0.471410

−

2.96273i

−0.619757

148.4

−0.119562

−

0.207087i

1.12441

−

1.31746i

0.971410

−

1.68253i

−1.29589

+

2.24456i

−0.407265

−

0.0753324i

0

−0.942820

−0.471410

−

2.96273i

0.619757

148.5

0.849814

+

1.47192i

−1.40434

−

1.01381i

−0.444368

+

0.769668i

−0.474636

+

0.822093i

0.298820

−

2.92864i

0

1.88874

0.944368

+

2.84748i

−1.61341

148.6

0.849814

+

1.47192i

1.40434

+

1.01381i

−0.444368

+

0.769668i

0.474636

−

0.822093i

−0.298820

+

2.92864i

0

1.88874

0.944368

+

2.84748i

1.61341

295.1

−1.23025

+

2.13086i

−1.66238

−

0.486291i

−2.02704

−

3.51094i

−1.82904

−

3.16799i

3.08137

−

2.94405i

0

5.05408

2.52704

+

1.61680i

9.00071

295.2

−1.23025

+

2.13086i

1.66238

+

0.486291i

−2.02704

−

3.51094i

1.82904

+

3.16799i

−3.08137

+

2.94405i

0

5.05408

2.52704

+

1.61680i

−9.00071

295.3

−0.119562

+

0.207087i

−1.12441

−

1.31746i

0.971410

+

1.68253i

1.29589

+

2.24456i

0.407265

−

0.0753324i

0

−0.942820

−0.471410

+

2.96273i

−0.619757

295.4

−0.119562

+

0.207087i

1.12441

+

1.31746i

0.971410

+

1.68253i

−1.29589

−

2.24456i

−0.407265

+

0.0753324i

0

−0.942820

−0.471410

+

2.96273i

0.619757

295.5

0.849814

−

1.47192i

−1.40434

+

1.01381i

−0.444368

−

0.769668i

−0.474636

−

0.822093i

0.298820

+

2.92864i

0

1.88874

0.944368

−

2.84748i

−1.61341

295.6

0.849814

−

1.47192i

1.40434

−

1.01381i

−0.444368

−

0.769668i

0.474636

+

0.822093i

−0.298820

−

2.92864i

0

1.88874

0.944368

−

2.84748i

1.61341

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.c	even	3	1	inner
63.l	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.2.f.g	✓	12
3.b	odd	2	1		1323.2.f.g		12
7.b	odd	2	1	inner	441.2.f.g	✓	12
7.c	even	3	1		441.2.g.g		12
7.c	even	3	1		441.2.h.g		12
7.d	odd	6	1		441.2.g.g		12
7.d	odd	6	1		441.2.h.g		12
9.c	even	3	1	inner	441.2.f.g	✓	12
9.c	even	3	1		3969.2.a.be		6
9.d	odd	6	1		1323.2.f.g		12
9.d	odd	6	1		3969.2.a.bd		6
21.c	even	2	1		1323.2.f.g		12
21.g	even	6	1		1323.2.g.g		12
21.g	even	6	1		1323.2.h.g		12
21.h	odd	6	1		1323.2.g.g		12
21.h	odd	6	1		1323.2.h.g		12
63.g	even	3	1		441.2.h.g		12
63.h	even	3	1		441.2.g.g		12
63.i	even	6	1		1323.2.g.g		12
63.j	odd	6	1		1323.2.g.g		12
63.k	odd	6	1		441.2.h.g		12
63.l	odd	6	1	inner	441.2.f.g	✓	12
63.l	odd	6	1		3969.2.a.be		6
63.n	odd	6	1		1323.2.h.g		12
63.o	even	6	1		1323.2.f.g		12
63.o	even	6	1		3969.2.a.bd		6
63.s	even	6	1		1323.2.h.g		12
63.t	odd	6	1		441.2.g.g		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.f.g	✓	12	1.a	even	1	1	trivial
441.2.f.g	✓	12	7.b	odd	2	1	inner
441.2.f.g	✓	12	9.c	even	3	1	inner
441.2.f.g	✓	12	63.l	odd	6	1	inner
441.2.g.g		12	7.c	even	3	1
441.2.g.g		12	7.d	odd	6	1
441.2.g.g		12	63.h	even	3	1
441.2.g.g		12	63.t	odd	6	1
441.2.h.g		12	7.c	even	3	1
441.2.h.g		12	7.d	odd	6	1
441.2.h.g		12	63.g	even	3	1
441.2.h.g		12	63.k	odd	6	1
1323.2.f.g		12	3.b	odd	2	1
1323.2.f.g		12	9.d	odd	6	1
1323.2.f.g		12	21.c	even	2	1
1323.2.f.g		12	63.o	even	6	1
1323.2.g.g		12	21.g	even	6	1
1323.2.g.g		12	21.h	odd	6	1
1323.2.g.g		12	63.i	even	6	1
1323.2.g.g		12	63.j	odd	6	1
1323.2.h.g		12	21.g	even	6	1
1323.2.h.g		12	21.h	odd	6	1
1323.2.h.g		12	63.n	odd	6	1
1323.2.h.g		12	63.s	even	6	1
3969.2.a.bd		6	9.d	odd	6	1
3969.2.a.bd		6	63.o	even	6	1
3969.2.a.be		6	9.c	even	3	1
3969.2.a.be		6	63.l	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1 \)

\( T_{5}^{12} + 21T_{5}^{10} + 333T_{5}^{8} + 2106T_{5}^{6} + 9963T_{5}^{4} + 8748T_{5}^{2} + 6561 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^6 + T^5 + 5*T^4 - 2*T^3 + 17*T^2 + 4*T + 1)^2
$3$	T^12 - 6*T^10 + 30*T^8 - 99*T^6 + 270*T^4 - 486*T^2 + 729
$5$	T^12 + 21*T^10 + 333*T^8 + 2106*T^6 + 9963*T^4 + 8748*T^2 + 6561
$7$	\( T^{12} \)
$11$	(T^6 + 4*T^5 + 17*T^4 - 2*T^3 + 5*T^2 + T + 1)^2