Newform orbit 637.2.bb.b

• Label: 637.2.bb.b
• Level: $637$
• Weight: $2$
• Character orbit: 637.bb
• Analytic conductor: $5.086$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.08647060876\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
227.1	−1.65980	+	1.65980i	−2.39046	+	1.38013i		−	3.50987i	−0.412430	+	1.53921i	1.67694	−	6.25841i		0		2.50608	+	2.50608i	2.30952	−	4.00020i	−1.87023	−	3.23933i
227.2	−1.65980	+	1.65980i	2.39046	−	1.38013i		−	3.50987i	0.412430	−	1.53921i	−1.67694	+	6.25841i		0		2.50608	+	2.50608i	2.30952	−	4.00020i	1.87023	+	3.23933i
227.3	−0.976208	+	0.976208i	−0.928437	+	0.536034i			0.0940343i	0.742756	−	2.77200i	0.383068	−	1.42963i		0		−2.04421	−	2.04421i	−0.925336	+	1.60273i	1.98097	+	3.43114i
227.4	−0.976208	+	0.976208i	0.928437	−	0.536034i			0.0940343i	−0.742756	+	2.77200i	−0.383068	+	1.42963i		0		−2.04421	−	2.04421i	−0.925336	+	1.60273i	−1.98097	−	3.43114i
227.5	0.595687	−	0.595687i	−1.24122	+	0.716618i			1.29031i	−0.370805	+	1.38386i	−0.312498	+	1.16626i		0		1.96000	+	1.96000i	−0.472916	+	0.819114i	0.603466	+	1.04523i
227.6	0.595687	−	0.595687i	1.24122	−	0.716618i			1.29031i	0.370805	−	1.38386i	0.312498	−	1.16626i		0		1.96000	+	1.96000i	−0.472916	+	0.819114i	−0.603466	−	1.04523i
227.7	1.67430	−	1.67430i	−1.04118	+	0.601128i		−	3.60653i	−0.825486	+	3.08075i	−0.736784	+	2.74971i		0		−2.68981	−	2.68981i	−0.777291	+	1.34631i	3.77599	+	6.54020i
227.8	1.67430	−	1.67430i	1.04118	−	0.601128i		−	3.60653i	0.825486	−	3.08075i	0.736784	−	2.74971i		0		−2.68981	−	2.68981i	−0.777291	+	1.34631i	−3.77599	−	6.54020i
362.1	−1.65980	−	1.65980i	−2.39046	−	1.38013i			3.50987i	−0.412430	−	1.53921i	1.67694	+	6.25841i		0		2.50608	−	2.50608i	2.30952	+	4.00020i	−1.87023	+	3.23933i
362.2	−1.65980	−	1.65980i	2.39046	+	1.38013i			3.50987i	0.412430	+	1.53921i	−1.67694	−	6.25841i		0		2.50608	−	2.50608i	2.30952	+	4.00020i	1.87023	−	3.23933i
362.3	−0.976208	−	0.976208i	−0.928437	−	0.536034i		−	0.0940343i	0.742756	+	2.77200i	0.383068	+	1.42963i		0		−2.04421	+	2.04421i	−0.925336	−	1.60273i	1.98097	−	3.43114i
362.4	−0.976208	−	0.976208i	0.928437	+	0.536034i		−	0.0940343i	−0.742756	−	2.77200i	−0.383068	−	1.42963i		0		−2.04421	+	2.04421i	−0.925336	−	1.60273i	−1.98097	+	3.43114i
362.5	0.595687	+	0.595687i	−1.24122	−	0.716618i		−	1.29031i	−0.370805	−	1.38386i	−0.312498	−	1.16626i		0		1.96000	−	1.96000i	−0.472916	−	0.819114i	0.603466	−	1.04523i
362.6	0.595687	+	0.595687i	1.24122	+	0.716618i		−	1.29031i	0.370805	+	1.38386i	0.312498	+	1.16626i		0		1.96000	−	1.96000i	−0.472916	−	0.819114i	−0.603466	+	1.04523i
362.7	1.67430	+	1.67430i	−1.04118	−	0.601128i			3.60653i	−0.825486	−	3.08075i	−0.736784	−	2.74971i		0		−2.68981	+	2.68981i	−0.777291	−	1.34631i	3.77599	−	6.54020i
362.8	1.67430	+	1.67430i	1.04118	+	0.601128i			3.60653i	0.825486	+	3.08075i	0.736784	+	2.74971i		0		−2.68981	+	2.68981i	−0.777291	−	1.34631i	−3.77599	+	6.54020i
423.1	−1.18517	−	1.18517i	−0.552306	+	0.318874i			0.809250i	1.94174	+	0.520288i	1.03250	+	0.276656i		0		−1.41124	+	1.41124i	−1.29664	+	2.24584i	−1.68466	−	2.91792i
423.2	−1.18517	−	1.18517i	0.552306	−	0.318874i			0.809250i	−1.94174	−	0.520288i	−1.03250	−	0.276656i		0		−1.41124	+	1.41124i	−1.29664	+	2.24584i	1.68466	+	2.91792i
423.3	0.0825572	+	0.0825572i	−2.25055	+	1.29935i		−	1.98637i	−1.70537	−	0.456951i	−0.293070	−	0.0785278i		0		0.329103	−	0.329103i	1.87664	−	3.25044i	−0.103066	−	0.178515i
423.4	0.0825572	+	0.0825572i	2.25055	−	1.29935i		−	1.98637i	1.70537	+	0.456951i	0.293070	+	0.0785278i		0		0.329103	−	0.329103i	1.87664	−	3.25044i	0.103066	+	0.178515i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
91.x	odd	12	1	inner
91.ba	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	637.2.bb.b		32
7.b	odd	2	1	inner	637.2.bb.b		32
7.c	even	3	1		91.2.bc.a	✓	32
7.c	even	3	1		637.2.x.b		32
7.d	odd	6	1		91.2.bc.a	✓	32
7.d	odd	6	1		637.2.x.b		32
13.f	odd	12	1		637.2.x.b		32
21.g	even	6	1		819.2.fm.g		32
21.h	odd	6	1		819.2.fm.g		32
91.w	even	12	1		91.2.bc.a	✓	32
91.x	odd	12	1	inner	637.2.bb.b		32
91.ba	even	12	1	inner	637.2.bb.b		32
91.bc	even	12	1		637.2.x.b		32
91.bd	odd	12	1		91.2.bc.a	✓	32
273.bw	even	12	1		819.2.fm.g		32
273.ch	odd	12	1		819.2.fm.g		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.bc.a	✓	32	7.c	even	3	1
91.2.bc.a	✓	32	7.d	odd	6	1
91.2.bc.a	✓	32	91.w	even	12	1
91.2.bc.a	✓	32	91.bd	odd	12	1
637.2.x.b		32	7.c	even	3	1
637.2.x.b		32	7.d	odd	6	1
637.2.x.b		32	13.f	odd	12	1
637.2.x.b		32	91.bc	even	12	1
637.2.bb.b		32	1.a	even	1	1	trivial
637.2.bb.b		32	7.b	odd	2	1	inner
637.2.bb.b		32	91.x	odd	12	1	inner
637.2.bb.b		32	91.ba	even	12	1	inner
819.2.fm.g		32	21.g	even	6	1
819.2.fm.g		32	21.h	odd	6	1
819.2.fm.g		32	273.bw	even	12	1
819.2.fm.g		32	273.ch	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^16 - 2*T2^15 + 2*T2^14 + 4*T2^13 + 49*T2^12 - 80*T2^11 + 70*T2^10 + 122*T2^9 + 608*T2^8 - 630*T2^7 + 302*T2^6 - 32*T2^5 + 1177*T2^4 - 1440*T2^3 + 882*T2^2 - 126*T2 + 9