Newform orbit 637.2.x.b

• Label: 637.2.x.b
• Level: $637$
• Weight: $2$
• Character orbit: 637.x
• 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.x (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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} + 12 q^{4} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−2.28713	−	0.612835i		−	1.20226i	3.12335	+	1.80327i	3.08075	−	0.825486i	−0.736784	+	2.74971i		0		−2.68981	−	2.68981i	1.55458			−7.55197
19.2	−2.28713	−	0.612835i			1.20226i	3.12335	+	1.80327i	−3.08075	+	0.825486i	0.736784	−	2.74971i		0		−2.68981	−	2.68981i	1.55458			7.55197
19.3	−0.813723	−	0.218036i		−	1.43324i	−1.11745	−	0.645157i	1.38386	−	0.370805i	−0.312498	+	1.16626i		0		1.96000	+	1.96000i	0.945832			−1.20693
19.4	−0.813723	−	0.218036i			1.43324i	−1.11745	−	0.645157i	−1.38386	+	0.370805i	0.312498	−	1.16626i		0		1.96000	+	1.96000i	0.945832			1.20693
19.5	1.33353	+	0.357317i		−	1.07207i	−0.0814361	−	0.0470171i	−2.77200	+	0.742756i	0.383068	−	1.42963i		0		−2.04421	−	2.04421i	1.85067			−3.96194
19.6	1.33353	+	0.357317i			1.07207i	−0.0814361	−	0.0470171i	2.77200	−	0.742756i	−0.383068	+	1.42963i		0		−2.04421	−	2.04421i	1.85067			3.96194
19.7	2.26733	+	0.607529i		−	2.76026i	3.03963	+	1.75493i	1.53921	−	0.412430i	1.67694	−	6.25841i		0		2.50608	+	2.50608i	−4.61904			3.74046
19.8	2.26733	+	0.607529i			2.76026i	3.03963	+	1.75493i	−1.53921	+	0.412430i	−1.67694	+	6.25841i		0		2.50608	+	2.50608i	−4.61904			−3.74046
80.1	−0.433802	−	1.61897i		−	0.637748i	−0.700831	+	0.404625i	0.520288	−	1.94174i	−1.03250	+	0.276656i		0		−1.41124	−	1.41124i	2.59328			−3.36932
80.2	−0.433802	−	1.61897i			0.637748i	−0.700831	+	0.404625i	−0.520288	+	1.94174i	1.03250	−	0.276656i		0		−1.41124	−	1.41124i	2.59328			3.36932
80.3	0.0302180	+	0.112775i		−	2.59871i	1.72025	−	0.993184i	−0.456951	+	1.70537i	0.293070	−	0.0785278i		0		0.329103	+	0.329103i	−3.75328			−0.206131
80.4	0.0302180	+	0.112775i			2.59871i	1.72025	−	0.993184i	0.456951	−	1.70537i	−0.293070	+	0.0785278i		0		0.329103	+	0.329103i	−3.75328			0.206131
80.5	0.239080	+	0.892257i		−	1.14107i	0.993087	−	0.573359i	1.02497	−	3.82526i	1.01813	−	0.272807i		0		2.05537	+	2.05537i	1.69796			3.65816
80.6	0.239080	+	0.892257i			1.14107i	0.993087	−	0.573359i	−1.02497	+	3.82526i	−1.01813	+	0.272807i		0		2.05537	+	2.05537i	1.69796			−3.65816
80.7	0.664504	+	2.47996i		−	2.69629i	−3.97660	+	2.29589i	−0.103104	+	0.384791i	6.68671	−	1.79170i		0		−4.70528	−	4.70528i	−4.27000			−1.02278
80.8	0.664504	+	2.47996i			2.69629i	−3.97660	+	2.29589i	0.103104	−	0.384791i	−6.68671	+	1.79170i		0		−4.70528	−	4.70528i	−4.27000			1.02278
215.1	−0.433802	+	1.61897i		−	0.637748i	−0.700831	−	0.404625i	−0.520288	−	1.94174i	1.03250	+	0.276656i		0		−1.41124	+	1.41124i	2.59328			3.36932
215.2	−0.433802	+	1.61897i			0.637748i	−0.700831	−	0.404625i	0.520288	+	1.94174i	−1.03250	−	0.276656i		0		−1.41124	+	1.41124i	2.59328			−3.36932
215.3	0.0302180	−	0.112775i		−	2.59871i	1.72025	+	0.993184i	0.456951	+	1.70537i	−0.293070	−	0.0785278i		0		0.329103	−	0.329103i	−3.75328			0.206131
215.4	0.0302180	−	0.112775i			2.59871i	1.72025	+	0.993184i	−0.456951	−	1.70537i	0.293070	+	0.0785278i		0		0.329103	−	0.329103i	−3.75328			−0.206131

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^16 - 2*T2^15 - T2^14 + 10*T2^13 - 41*T2^12 + 10*T2^11 + 148*T2^10 - 94*T2^9 + 446*T2^8 - 810*T2^7 - 196*T2^6 + 610*T2^5 - 329*T2^4 + 312*T2^3 + 639*T2^2 - 36*T2 + 9