Newform orbit 252.2.b.a

• Label: 252.2.b.a
• Level: $252$
• Weight: $2$
• Character orbit: 252.b
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (\beta - 2) q^{4} + (2 \beta - 1) q^{7} + ( - \beta - 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 3 q^{4} - 5 q^{8}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

55.1

0.500000	−	1.32288i
0.500000	+	1.32288i

0.500000

−

1.32288i

0

−1.50000

−

1.32288i

0

0

−

2.64575i

−2.50000

+

1.32288i

0

0

55.2

0.500000

+

1.32288i

0

−1.50000

+

1.32288i

0

0

2.64575i

−2.50000

−

1.32288i

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
4.b	odd	2	1	inner
28.d	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.b.a		2
3.b	odd	2	1		28.2.d.a	✓	2
4.b	odd	2	1	inner	252.2.b.a		2
7.b	odd	2	1	CM	252.2.b.a		2
8.b	even	2	1		4032.2.b.e		2
8.d	odd	2	1		4032.2.b.e		2
12.b	even	2	1		28.2.d.a	✓	2
15.d	odd	2	1		700.2.g.a		2
15.e	even	4	2		700.2.c.d		4
21.c	even	2	1		28.2.d.a	✓	2
21.g	even	6	2		196.2.f.b		4
21.h	odd	6	2		196.2.f.b		4
24.f	even	2	1		448.2.f.b		2
24.h	odd	2	1		448.2.f.b		2
28.d	even	2	1	inner	252.2.b.a		2
48.i	odd	4	2		1792.2.e.b		4
48.k	even	4	2		1792.2.e.b		4
56.e	even	2	1		4032.2.b.e		2
56.h	odd	2	1		4032.2.b.e		2
60.h	even	2	1		700.2.g.a		2
60.l	odd	4	2		700.2.c.d		4
84.h	odd	2	1		28.2.d.a	✓	2
84.j	odd	6	2		196.2.f.b		4
84.n	even	6	2		196.2.f.b		4
105.g	even	2	1		700.2.g.a		2
105.k	odd	4	2		700.2.c.d		4
168.e	odd	2	1		448.2.f.b		2
168.i	even	2	1		448.2.f.b		2
336.v	odd	4	2		1792.2.e.b		4
336.y	even	4	2		1792.2.e.b		4
420.o	odd	2	1		700.2.g.a		2
420.w	even	4	2		700.2.c.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.d.a	✓	2	3.b	odd	2	1
28.2.d.a	✓	2	12.b	even	2	1
28.2.d.a	✓	2	21.c	even	2	1
28.2.d.a	✓	2	84.h	odd	2	1
196.2.f.b		4	21.g	even	6	2
196.2.f.b		4	21.h	odd	6	2
196.2.f.b		4	84.j	odd	6	2
196.2.f.b		4	84.n	even	6	2
252.2.b.a		2	1.a	even	1	1	trivial
252.2.b.a		2	4.b	odd	2	1	inner
252.2.b.a		2	7.b	odd	2	1	CM
252.2.b.a		2	28.d	even	2	1	inner
448.2.f.b		2	24.f	even	2	1
448.2.f.b		2	24.h	odd	2	1
448.2.f.b		2	168.e	odd	2	1
448.2.f.b		2	168.i	even	2	1
700.2.c.d		4	15.e	even	4	2
700.2.c.d		4	60.l	odd	4	2
700.2.c.d		4	105.k	odd	4	2
700.2.c.d		4	420.w	even	4	2
700.2.g.a		2	15.d	odd	2	1
700.2.g.a		2	60.h	even	2	1
700.2.g.a		2	105.g	even	2	1
700.2.g.a		2	420.o	odd	2	1
1792.2.e.b		4	48.i	odd	4	2
1792.2.e.b		4	48.k	even	4	2
1792.2.e.b		4	336.v	odd	4	2
1792.2.e.b		4	336.y	even	4	2
4032.2.b.e		2	8.b	even	2	1
4032.2.b.e		2	8.d	odd	2	1
4032.2.b.e		2	56.e	even	2	1
4032.2.b.e		2	56.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5} \)

\( T_{11}^{2} + 28 \)

\( T_{19} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 2 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 7 \)
$11$	\( T^{2} + 28 \)