Newform orbit 210.3.p.a

• Label: 210.3.p.a
• Level: $210$
• Weight: $3$
• Character orbit: 210.p
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 12 q^{5} - 48 q^{9}+O(q^{10}) \)

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	−1.22474	−	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	−3.49700	+	3.57365i			2.44949i	0.180689	−	6.99767i		−	2.82843i	−1.50000	+	2.59808i	6.80989	−	1.90405i
19.2	−1.22474	−	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	−0.214021	−	4.99542i			2.44949i	5.88548	−	3.78961i		−	2.82843i	−1.50000	+	2.59808i	−3.27017	+	6.26945i
19.3	−1.22474	−	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	2.84271	+	4.11327i			2.44949i	−6.72066	−	1.95772i		−	2.82843i	−1.50000	+	2.59808i	−0.573066	−	7.04781i
19.4	−1.22474	−	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	4.84836	−	1.22205i			2.44949i	−1.07756	+	6.91657i		−	2.82843i	−1.50000	+	2.59808i	−6.80213	−	1.93160i
19.5	−1.22474	−	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	−3.86586	−	3.17098i		−	2.44949i	6.18245	+	3.28288i		−	2.82843i	−1.50000	+	2.59808i	2.49248	+	6.61722i
19.6	−1.22474	−	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	−2.13749	−	4.52008i		−	2.44949i	−6.07868	−	3.47127i		−	2.82843i	−1.50000	+	2.59808i	−0.578308	+	7.04738i
19.7	−1.22474	−	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	2.51052	+	4.32404i		−	2.44949i	−0.686090	+	6.96630i		−	2.82843i	−1.50000	+	2.59808i	−0.0171898	−	7.07105i
19.8	−1.22474	−	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	4.96228	−	0.613015i		−	2.44949i	2.31437	−	6.60634i		−	2.82843i	−1.50000	+	2.59808i	−6.51099	−	2.75807i
19.9	1.22474	+	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	−4.98325	+	0.408925i		−	2.44949i	6.07868	+	3.47127i			2.82843i	−1.50000	+	2.59808i	−6.39236	−	3.02286i
19.10	1.22474	+	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	−4.67908	−	1.76245i		−	2.44949i	−6.18245	−	3.28288i			2.82843i	−1.50000	+	2.59808i	−4.48444	−	5.46716i
19.11	1.22474	+	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	1.95025	+	4.60397i		−	2.44949i	−2.31437	+	6.60634i			2.82843i	−1.50000	+	2.59808i	−0.866933	+	7.01772i
19.12	1.22474	+	0.707107i	−0.866025	−	1.50000i	1.00000	+	1.73205i	4.99999	+	0.0121550i		−	2.44949i	0.686090	−	6.96630i			2.82843i	−1.50000	+	2.59808i	6.11511	+	3.55041i
19.13	1.22474	+	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	−4.43317	+	2.31236i			2.44949i	−5.88548	+	3.78961i			2.82843i	−1.50000	+	2.59808i	−7.06459	−	0.302672i
19.14	1.22474	+	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	1.34637	−	4.81532i			2.44949i	−0.180689	+	6.99767i			2.82843i	−1.50000	+	2.59808i	5.05390	−	4.94551i
19.15	1.22474	+	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	1.36585	+	4.80983i			2.44949i	1.07756	−	6.91657i			2.82843i	−1.50000	+	2.59808i	−1.72825	+	6.85661i
19.16	1.22474	+	0.707107i	0.866025	+	1.50000i	1.00000	+	1.73205i	4.98355	+	0.405219i			2.44949i	6.72066	+	1.95772i			2.82843i	−1.50000	+	2.59808i	5.81705	+	4.02019i
199.1	−1.22474	+	0.707107i	−0.866025	+	1.50000i	1.00000	−	1.73205i	−3.49700	−	3.57365i		−	2.44949i	0.180689	+	6.99767i			2.82843i	−1.50000	−	2.59808i	6.80989	+	1.90405i
199.2	−1.22474	+	0.707107i	−0.866025	+	1.50000i	1.00000	−	1.73205i	−0.214021	+	4.99542i		−	2.44949i	5.88548	+	3.78961i			2.82843i	−1.50000	−	2.59808i	−3.27017	−	6.26945i
199.3	−1.22474	+	0.707107i	−0.866025	+	1.50000i	1.00000	−	1.73205i	2.84271	−	4.11327i		−	2.44949i	−6.72066	+	1.95772i			2.82843i	−1.50000	−	2.59808i	−0.573066	+	7.04781i
199.4	−1.22474	+	0.707107i	−0.866025	+	1.50000i	1.00000	−	1.73205i	4.84836	+	1.22205i		−	2.44949i	−1.07756	−	6.91657i			2.82843i	−1.50000	−	2.59808i	−6.80213	+	1.93160i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.d	odd	6	1	inner
35.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	210.3.p.a	✓	32
3.b	odd	2	1		630.3.bc.b		32
5.b	even	2	1	inner	210.3.p.a	✓	32
5.c	odd	4	1		1050.3.p.g		16
5.c	odd	4	1		1050.3.p.h		16
7.c	even	3	1		1470.3.h.a		32
7.d	odd	6	1	inner	210.3.p.a	✓	32
7.d	odd	6	1		1470.3.h.a		32
15.d	odd	2	1		630.3.bc.b		32
21.g	even	6	1		630.3.bc.b		32
35.i	odd	6	1	inner	210.3.p.a	✓	32
35.i	odd	6	1		1470.3.h.a		32
35.j	even	6	1		1470.3.h.a		32
35.k	even	12	1		1050.3.p.g		16
35.k	even	12	1		1050.3.p.h		16
105.p	even	6	1		630.3.bc.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.3.p.a	✓	32	1.a	even	1	1	trivial
210.3.p.a	✓	32	5.b	even	2	1	inner
210.3.p.a	✓	32	7.d	odd	6	1	inner
210.3.p.a	✓	32	35.i	odd	6	1	inner
630.3.bc.b		32	3.b	odd	2	1
630.3.bc.b		32	15.d	odd	2	1
630.3.bc.b		32	21.g	even	6	1
630.3.bc.b		32	105.p	even	6	1
1050.3.p.g		16	5.c	odd	4	1
1050.3.p.g		16	35.k	even	12	1
1050.3.p.h		16	5.c	odd	4	1
1050.3.p.h		16	35.k	even	12	1
1470.3.h.a		32	7.c	even	3	1
1470.3.h.a		32	7.d	odd	6	1
1470.3.h.a		32	35.i	odd	6	1
1470.3.h.a		32	35.j	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).