Newform orbit 252.2.e.a

• Label: 252.2.e.a
• Level: $252$
• Weight: $2$
• Character orbit: 252.e
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.653473922154496.1
Defining polynomial:	\( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{4} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 1 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{9} - 2\nu^{7} + \nu^{5} + 6\nu^{3} - 8\nu ) / 16 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{10} - 2\nu^{8} + 23\nu^{6} - 58\nu^{4} + 96\nu^{2} - 160 ) / 64 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{8} + 2\nu^{6} - 5\nu^{4} + 6\nu^{2} - 8 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 34\nu^{4} - 64\nu^{2} + 32 ) / 64 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{11} - 4\nu^{9} + 5\nu^{7} - 12\nu^{5} + 12\nu^{3} - 16\nu ) / 32 \)
\(\beta_{8}\)	\(=\)	\( ( -3\nu^{11} + 6\nu^{9} - 19\nu^{7} + 30\nu^{5} - 24\nu^{3} ) / 64 \)
\(\beta_{9}\)	\(=\)	\( ( -\nu^{11} + 10\nu^{9} - 33\nu^{7} + 82\nu^{5} - 120\nu^{3} + 128\nu ) / 64 \)
\(\beta_{10}\)	\(=\)	\( ( 3\nu^{11} - 10\nu^{9} + 27\nu^{7} - 34\nu^{5} + 64\nu^{3} - 32\nu ) / 64 \)
\(\beta_{11}\)	\(=\)	\( ( -3\nu^{10} + 10\nu^{8} - 27\nu^{6} + 66\nu^{4} - 96\nu^{2} + 96 ) / 32 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

71.1

−1.35489	−	0.405301i
−1.35489	+	0.405301i
−1.16947	−	0.795191i
−1.16947	+	0.795191i
−0.892524	−	1.09700i
−0.892524	+	1.09700i
0.892524	−	1.09700i
0.892524	+	1.09700i
1.16947	−	0.795191i
1.16947	+	0.795191i
1.35489	−	0.405301i
1.35489	+	0.405301i

−1.35489

−

0.405301i

0

1.67146

+

1.09828i

3.31339i

0

1.00000i

−1.81951

−

2.16549i

0

1.34292

−

4.48929i

71.2

−1.35489

+

0.405301i

0

1.67146

−

1.09828i

−

3.31339i

0

−

1.00000i

−1.81951

+

2.16549i

0

1.34292

+

4.48929i

71.3

−1.16947

−

0.795191i

0

0.735342

+

1.85991i

−

0.665647i

0

1.00000i

0.619022

−

2.75986i

0

−0.529317

+

0.778457i

71.4

−1.16947

+

0.795191i

0

0.735342

−

1.85991i

0.665647i

0

−

1.00000i

0.619022

+

2.75986i

0

−0.529317

−

0.778457i

71.5

−0.892524

−

1.09700i

0

−0.406803

+

1.95819i

−

2.56483i

0

−

1.00000i

2.51121

−

1.30147i

0

−2.81361

+

2.28917i

71.6

−0.892524

+

1.09700i

0

−0.406803

−

1.95819i

2.56483i

0

1.00000i

2.51121

+

1.30147i

0

−2.81361

−

2.28917i

71.7

0.892524

−

1.09700i

0

−0.406803

−

1.95819i

−

2.56483i

0

1.00000i

−2.51121

−

1.30147i

0

−2.81361

−

2.28917i

71.8

0.892524

+

1.09700i

0

−0.406803

+

1.95819i

2.56483i

0

−

1.00000i

−2.51121

+

1.30147i

0

−2.81361

+

2.28917i

71.9

1.16947

−

0.795191i

0

0.735342

−

1.85991i

−

0.665647i

0

−

1.00000i

−0.619022

−

2.75986i

0

−0.529317

−

0.778457i

71.10

1.16947

+

0.795191i

0

0.735342

+

1.85991i

0.665647i

0

1.00000i

−0.619022

+

2.75986i

0

−0.529317

+

0.778457i

71.11

1.35489

−

0.405301i

0

1.67146

−

1.09828i

3.31339i

0

−

1.00000i

1.81951

−

2.16549i

0

1.34292

+

4.48929i

71.12

1.35489

+

0.405301i

0

1.67146

+

1.09828i

−

3.31339i

0

1.00000i

1.81951

+

2.16549i

0

1.34292

−

4.48929i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.2.e.a	✓	12
3.b	odd	2	1	inner	252.2.e.a	✓	12
4.b	odd	2	1	inner	252.2.e.a	✓	12
7.b	odd	2	1		1764.2.e.g		12
8.b	even	2	1		4032.2.h.h		12
8.d	odd	2	1		4032.2.h.h		12
12.b	even	2	1	inner	252.2.e.a	✓	12
21.c	even	2	1		1764.2.e.g		12
24.f	even	2	1		4032.2.h.h		12
24.h	odd	2	1		4032.2.h.h		12
28.d	even	2	1		1764.2.e.g		12
84.h	odd	2	1		1764.2.e.g		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.e.a	✓	12	1.a	even	1	1	trivial
252.2.e.a	✓	12	3.b	odd	2	1	inner
252.2.e.a	✓	12	4.b	odd	2	1	inner
252.2.e.a	✓	12	12.b	even	2	1	inner
1764.2.e.g		12	7.b	odd	2	1
1764.2.e.g		12	21.c	even	2	1
1764.2.e.g		12	28.d	even	2	1
1764.2.e.g		12	84.h	odd	2	1
4032.2.h.h		12	8.b	even	2	1
4032.2.h.h		12	8.d	odd	2	1
4032.2.h.h		12	24.f	even	2	1
4032.2.h.h		12	24.h	odd	2	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 - 4*T^10 + 13*T^8 - 28*T^6 + 52*T^4 - 64*T^2 + 64
$3$	\( T^{12} \)
$5$	(T^6 + 18*T^4 + 80*T^2 + 32)^2
$7$	\( (T^{2} + 1)^{6} \)
$11$	(T^6 - 28*T^4 + 132*T^2 - 128)^2