Newform orbit 252.2.be.a

• Label: 252.2.be.a
• Level: $252$
• Weight: $2$
• Character orbit: 252.be
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
107.1	−1.41312	−	0.0556843i		0		1.99380	+	0.157377i	−1.80224	−	1.04052i		0		−1.89429	+	1.84707i	−2.80871	−	0.333415i		0		2.48883	+	1.57073i
107.2	−1.40272	−	0.179921i		0		1.93526	+	0.504757i	−0.604694	−	0.349120i		0		−1.16254	−	2.37666i	−2.62381	−	1.05623i		0		0.785403	+	0.598515i
107.3	−1.30138	+	0.553555i		0		1.38715	−	1.44076i	3.35279	+	1.93573i		0		1.03277	−	2.43585i	−1.00767	+	2.64284i		0		−5.43477	−	0.663164i
107.4	−0.857177	−	1.12483i		0		−0.530496	+	1.92836i	−0.604694	−	0.349120i		0		1.16254	+	2.37666i	2.62381	−	1.05623i		0		0.125628	+	0.979437i
107.5	−0.783636	+	1.17725i		0		−0.771830	−	1.84507i	2.15525	+	1.24433i		0		−2.64453	+	0.0803545i	2.77694	+	0.537226i		0		−3.15382	+	1.56216i
107.6	−0.754782	−	1.19595i		0		−0.860607	+	1.80537i	−1.80224	−	1.04052i		0		1.89429	−	1.84707i	2.80871	−	0.333415i		0		0.115881	+	2.94076i
107.7	−0.627710	+	1.26727i		0		−1.21196	−	1.59096i	−2.15525	−	1.24433i		0		2.64453	−	0.0803545i	2.77694	−	0.537226i		0		2.92978	−	1.95021i
107.8	−0.171295	−	1.40380i		0		−1.94132	+	0.480929i	3.35279	+	1.93573i		0		−1.03277	+	2.43585i	1.00767	+	2.64284i		0		2.14307	−	5.03823i
107.9	0.171295	+	1.40380i		0		−1.94132	+	0.480929i	−3.35279	−	1.93573i		0		−1.03277	+	2.43585i	−1.00767	−	2.64284i		0		2.14307	−	5.03823i
107.10	0.627710	−	1.26727i		0		−1.21196	−	1.59096i	2.15525	+	1.24433i		0		2.64453	−	0.0803545i	−2.77694	+	0.537226i		0		2.92978	−	1.95021i
107.11	0.754782	+	1.19595i		0		−0.860607	+	1.80537i	1.80224	+	1.04052i		0		1.89429	−	1.84707i	−2.80871	+	0.333415i		0		0.115881	+	2.94076i
107.12	0.783636	−	1.17725i		0		−0.771830	−	1.84507i	−2.15525	−	1.24433i		0		−2.64453	+	0.0803545i	−2.77694	−	0.537226i		0		−3.15382	+	1.56216i
107.13	0.857177	+	1.12483i		0		−0.530496	+	1.92836i	0.604694	+	0.349120i		0		1.16254	+	2.37666i	−2.62381	+	1.05623i		0		0.125628	+	0.979437i
107.14	1.30138	−	0.553555i		0		1.38715	−	1.44076i	−3.35279	−	1.93573i		0		1.03277	−	2.43585i	1.00767	−	2.64284i		0		−5.43477	−	0.663164i
107.15	1.40272	+	0.179921i		0		1.93526	+	0.504757i	0.604694	+	0.349120i		0		−1.16254	−	2.37666i	2.62381	+	1.05623i		0		0.785403	+	0.598515i
107.16	1.41312	+	0.0556843i		0		1.99380	+	0.157377i	1.80224	+	1.04052i		0		−1.89429	+	1.84707i	2.80871	+	0.333415i		0		2.48883	+	1.57073i
179.1	−1.41312	+	0.0556843i		0		1.99380	−	0.157377i	−1.80224	+	1.04052i		0		−1.89429	−	1.84707i	−2.80871	+	0.333415i		0		2.48883	−	1.57073i
179.2	−1.40272	+	0.179921i		0		1.93526	−	0.504757i	−0.604694	+	0.349120i		0		−1.16254	+	2.37666i	−2.62381	+	1.05623i		0		0.785403	−	0.598515i
179.3	−1.30138	−	0.553555i		0		1.38715	+	1.44076i	3.35279	−	1.93573i		0		1.03277	+	2.43585i	−1.00767	−	2.64284i		0		−5.43477	+	0.663164i
179.4	−0.857177	+	1.12483i		0		−0.530496	−	1.92836i	−0.604694	+	0.349120i		0		1.16254	−	2.37666i	2.62381	+	1.05623i		0		0.125628	−	0.979437i

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
7.c	even	3	1	inner
12.b	even	2	1	inner
21.h	odd	6	1	inner
28.g	odd	6	1	inner
84.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.2.be.a	✓	32
3.b	odd	2	1	inner	252.2.be.a	✓	32
4.b	odd	2	1	inner	252.2.be.a	✓	32
7.c	even	3	1	inner	252.2.be.a	✓	32
7.c	even	3	1		1764.2.e.i		16
7.d	odd	6	1		1764.2.e.h		16
12.b	even	2	1	inner	252.2.be.a	✓	32
21.g	even	6	1		1764.2.e.h		16
21.h	odd	6	1	inner	252.2.be.a	✓	32
21.h	odd	6	1		1764.2.e.i		16
28.f	even	6	1		1764.2.e.h		16
28.g	odd	6	1	inner	252.2.be.a	✓	32
28.g	odd	6	1		1764.2.e.i		16
84.j	odd	6	1		1764.2.e.h		16
84.n	even	6	1	inner	252.2.be.a	✓	32
84.n	even	6	1		1764.2.e.i		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.be.a	✓	32	1.a	even	1	1	trivial
252.2.be.a	✓	32	3.b	odd	2	1	inner
252.2.be.a	✓	32	4.b	odd	2	1	inner
252.2.be.a	✓	32	7.c	even	3	1	inner
252.2.be.a	✓	32	12.b	even	2	1	inner
252.2.be.a	✓	32	21.h	odd	6	1	inner
252.2.be.a	✓	32	28.g	odd	6	1	inner
252.2.be.a	✓	32	84.n	even	6	1	inner
1764.2.e.h		16	7.d	odd	6	1
1764.2.e.h		16	21.g	even	6	1
1764.2.e.h		16	28.f	even	6	1
1764.2.e.h		16	84.j	odd	6	1
1764.2.e.i		16	7.c	even	3	1
1764.2.e.i		16	21.h	odd	6	1
1764.2.e.i		16	28.g	odd	6	1
1764.2.e.i		16	84.n	even	6	1

Hecke kernels

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