Newform orbit 252.3.z.b

• Label: 252.3.z.b
• Level: $252$
• Weight: $3$
• Character orbit: 252.z
• Analytic conductor: $6.867$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.86650266188\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 2) q^{5} + (3 \zeta_{6} + 5) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 13 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(1\)	\(\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

73.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

−1.50000

+

0.866025i

0

6.50000

+

2.59808i

0

0

0

145.1

0

0

0

−1.50000

−

0.866025i

0

6.50000

−

2.59808i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.3.z.b		2
3.b	odd	2	1		84.3.m.a	✓	2
4.b	odd	2	1		1008.3.cg.b		2
7.b	odd	2	1		1764.3.z.e		2
7.c	even	3	1		1764.3.d.c		2
7.c	even	3	1		1764.3.z.e		2
7.d	odd	6	1	inner	252.3.z.b		2
7.d	odd	6	1		1764.3.d.c		2
12.b	even	2	1		336.3.bh.b		2
15.d	odd	2	1		2100.3.bd.b		2
15.e	even	4	2		2100.3.be.c		4
21.c	even	2	1		588.3.m.a		2
21.g	even	6	1		84.3.m.a	✓	2
21.g	even	6	1		588.3.d.a		2
21.h	odd	6	1		588.3.d.a		2
21.h	odd	6	1		588.3.m.a		2
28.f	even	6	1		1008.3.cg.b		2
84.j	odd	6	1		336.3.bh.b		2
84.j	odd	6	1		2352.3.f.c		2
84.n	even	6	1		2352.3.f.c		2
105.p	even	6	1		2100.3.bd.b		2
105.w	odd	12	2		2100.3.be.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.m.a	✓	2	3.b	odd	2	1
84.3.m.a	✓	2	21.g	even	6	1
252.3.z.b		2	1.a	even	1	1	trivial
252.3.z.b		2	7.d	odd	6	1	inner
336.3.bh.b		2	12.b	even	2	1
336.3.bh.b		2	84.j	odd	6	1
588.3.d.a		2	21.g	even	6	1
588.3.d.a		2	21.h	odd	6	1
588.3.m.a		2	21.c	even	2	1
588.3.m.a		2	21.h	odd	6	1
1008.3.cg.b		2	4.b	odd	2	1
1008.3.cg.b		2	28.f	even	6	1
1764.3.d.c		2	7.c	even	3	1
1764.3.d.c		2	7.d	odd	6	1
1764.3.z.e		2	7.b	odd	2	1
1764.3.z.e		2	7.c	even	3	1
2100.3.bd.b		2	15.d	odd	2	1
2100.3.bd.b		2	105.p	even	6	1
2100.3.be.c		4	15.e	even	4	2
2100.3.be.c		4	105.w	odd	12	2
2352.3.f.c		2	84.j	odd	6	1
2352.3.f.c		2	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 3 \)

\( T_{11}^{2} + 3T_{11} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 3 \)
$7$	\( T^{2} - 13T + 49 \)
$11$	\( T^{2} + 3T + 9 \)