Newform orbit 210.3.q.a

• Label: 210.3.q.a
• Level: $210$
• Weight: $3$
• Character orbit: 210.q
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
149.1	−0.707107	+	1.22474i	−2.99996	+	0.0151730i	−1.00000	−	1.73205i	−3.19601	−	3.84519i	2.10271	−	3.68492i	−6.79135	+	1.69633i	2.82843			8.99954	−	0.0910371i	6.96930	−	1.19534i
149.2	−0.707107	+	1.22474i	−2.97672	+	0.373050i	−1.00000	−	1.73205i	−2.48660	+	4.33784i	1.64796	−	3.90950i	1.32164	+	6.87410i	2.82843			8.72167	−	2.22093i	−3.55445	−	6.11276i
149.3	−0.707107	+	1.22474i	−2.59977	−	1.49707i	−1.00000	−	1.73205i	2.43105	+	4.36921i	3.67184	−	2.12546i	−1.98724	−	6.71199i	2.82843			4.51756	+	7.78406i	−7.07018	−	0.112086i
149.4	−0.707107	+	1.22474i	−2.31822	+	1.90417i	−1.00000	−	1.73205i	4.61270	+	1.92950i	−0.692887	−	4.18568i	6.90570	+	1.14510i	2.82843			1.74830	−	8.82856i	−5.62482	+	4.28502i
149.5	−0.707107	+	1.22474i	−1.64845	−	2.50651i	−1.00000	−	1.73205i	4.68918	−	1.73539i	4.23547	−	0.246562i	−3.35661	+	6.14273i	2.82843			−3.56521	+	8.26373i	−1.19034	+	6.97016i
149.6	−0.707107	+	1.22474i	−1.34648	−	2.68086i	−1.00000	−	1.73205i	−0.841696	−	4.92865i	4.23547	+	0.246562i	3.35661	−	6.14273i	2.82843			−5.37400	+	7.21943i	6.63150	+	2.45422i
149.7	−0.707107	+	1.22474i	−1.10765	+	2.78803i	−1.00000	−	1.73205i	−4.37658	−	2.41777i	−2.63140	−	3.32803i	5.67547	−	4.09745i	2.82843			−6.54621	−	6.17633i	6.05585	−	3.65057i
149.8	−0.707107	+	1.22474i	−0.355071	+	2.97891i	−1.00000	−	1.73205i	−1.67341	+	4.71166i	−3.39734	−	2.54128i	−7.00000	+	0.00780758i	2.82843			−8.74785	−	2.11545i	−4.58730	−	5.38114i
149.9	−0.707107	+	1.22474i	0.00338259	−	3.00000i	−1.00000	−	1.73205i	−4.99937	+	0.0792566i	3.67184	+	2.12546i	1.98724	+	6.71199i	2.82843			−8.99998	−	0.0202955i	3.43802	−	6.17900i
149.10	−0.707107	+	1.22474i	0.422498	+	2.97010i	−1.00000	−	1.73205i	1.86933	−	4.63741i	−3.93637	−	1.58273i	−0.123332	+	6.99891i	2.82843			−8.64299	+	2.50972i	4.35784	+	5.56860i
149.11	−0.707107	+	1.22474i	1.51312	−	2.59046i	−1.00000	−	1.73205i	4.92804	+	0.845231i	2.10271	+	3.68492i	6.79135	−	1.69633i	2.82843			−4.42093	−	7.83935i	−4.51984	+	5.43792i
149.12	−0.707107	+	1.22474i	1.81143	−	2.39139i	−1.00000	−	1.73205i	−2.51338	+	4.32238i	1.64796	+	3.90950i	−1.32164	−	6.87410i	2.82843			−2.43745	−	8.66365i	−3.51658	−	6.13463i
149.13	−0.707107	+	1.22474i	2.36093	+	1.85094i	−1.00000	−	1.73205i	3.08146	−	3.93759i	−3.93637	+	1.58273i	0.123332	−	6.99891i	2.82843			2.14801	+	8.73991i	2.64363	+	6.55830i
149.14	−0.707107	+	1.22474i	2.75735	+	1.18196i	−1.00000	−	1.73205i	−3.24371	+	3.80504i	−3.39734	+	2.54128i	7.00000	−	0.00780758i	2.82843			6.20596	+	6.51813i	−2.36656	−	6.66329i
149.15	−0.707107	+	1.22474i	2.80817	−	1.05556i	−1.00000	−	1.73205i	−3.97735	−	3.02996i	−0.692887	+	4.18568i	−6.90570	−	1.14510i	2.82843			6.77160	−	5.92835i	6.52334	−	2.72873i
149.16	−0.707107	+	1.22474i	2.96833	+	0.434759i	−1.00000	−	1.73205i	4.28213	+	2.58134i	−2.63140	+	3.32803i	−5.67547	+	4.09745i	2.82843			8.62197	+	2.58102i	−6.18941	+	3.41924i
149.17	0.707107	−	1.22474i	−2.96833	−	0.434759i	−1.00000	−	1.73205i	4.37658	+	2.41777i	−2.63140	+	3.32803i	5.67547	−	4.09745i	−2.82843			8.62197	+	2.58102i	6.05585	−	3.65057i
149.18	0.707107	−	1.22474i	−2.80817	+	1.05556i	−1.00000	−	1.73205i	−4.61270	−	1.92950i	−0.692887	+	4.18568i	6.90570	+	1.14510i	−2.82843			6.77160	−	5.92835i	−5.62482	+	4.28502i
149.19	0.707107	−	1.22474i	−2.75735	−	1.18196i	−1.00000	−	1.73205i	1.67341	−	4.71166i	−3.39734	+	2.54128i	−7.00000	+	0.00780758i	−2.82843			6.20596	+	6.51813i	−4.58730	−	5.38114i
149.20	0.707107	−	1.22474i	−2.36093	−	1.85094i	−1.00000	−	1.73205i	−1.86933	+	4.63741i	−3.93637	+	1.58273i	−0.123332	+	6.99891i	−2.82843			2.14801	+	8.73991i	4.35784	+	5.56860i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.c	even	3	1	inner
15.d	odd	2	1	inner
21.h	odd	6	1	inner
35.j	even	6	1	inner
105.o	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.q.a	✓	64
3.b	odd	2	1	inner	210.3.q.a	✓	64
5.b	even	2	1	inner	210.3.q.a	✓	64
7.c	even	3	1	inner	210.3.q.a	✓	64
15.d	odd	2	1	inner	210.3.q.a	✓	64
21.h	odd	6	1	inner	210.3.q.a	✓	64
35.j	even	6	1	inner	210.3.q.a	✓	64
105.o	odd	6	1	inner	210.3.q.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.3.q.a	✓	64	1.a	even	1	1	trivial
210.3.q.a	✓	64	3.b	odd	2	1	inner
210.3.q.a	✓	64	5.b	even	2	1	inner
210.3.q.a	✓	64	7.c	even	3	1	inner
210.3.q.a	✓	64	15.d	odd	2	1	inner
210.3.q.a	✓	64	21.h	odd	6	1	inner
210.3.q.a	✓	64	35.j	even	6	1	inner
210.3.q.a	✓	64	105.o	odd	6	1	inner

Hecke kernels

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