Newform orbit 210.3.s.a

• Label: 210.3.s.a
• Level: $210$
• Weight: $3$
• Character orbit: 210.s
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{4} - 8 q^{6} + 20 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.22474	+	0.707107i	−2.97598	−	0.378837i	1.00000	−	1.73205i	1.93649	−	1.11803i	3.91270	−	1.64036i	−2.08959	+	6.68084i			2.82843i	8.71297	+	2.25482i	−1.58114	+	2.73861i
11.2	−1.22474	+	0.707107i	−2.87585	−	0.854111i	1.00000	−	1.73205i	−1.93649	+	1.11803i	4.12613	−	0.987462i	6.85537	−	1.41561i			2.82843i	7.54099	+	4.91259i	1.58114	−	2.73861i
11.3	−1.22474	+	0.707107i	−1.55909	+	2.56305i	1.00000	−	1.73205i	−1.93649	+	1.11803i	0.0971326	−	4.24153i	1.98659	−	6.71219i			2.82843i	−4.13849	−	7.99205i	1.58114	−	2.73861i
11.4	−1.22474	+	0.707107i	−0.690423	−	2.91947i	1.00000	−	1.73205i	−1.93649	+	1.11803i	2.90997	+	3.08741i	0.399356	+	6.98860i			2.82843i	−8.04663	+	4.03134i	1.58114	−	2.73861i
11.5	−1.22474	+	0.707107i	−0.328980	+	2.98191i	1.00000	−	1.73205i	1.93649	−	1.11803i	−1.70561	−	3.88470i	−5.45202	−	4.39039i			2.82843i	−8.78354	−	1.96197i	−1.58114	+	2.73861i
11.6	−1.22474	+	0.707107i	1.03277	+	2.81663i	1.00000	−	1.73205i	1.93649	−	1.11803i	−3.25654	−	2.71937i	5.52236	+	4.30158i			2.82843i	−6.86677	+	5.81786i	−1.58114	+	2.73861i
11.7	−1.22474	+	0.707107i	1.03973	−	2.81407i	1.00000	−	1.73205i	1.93649	−	1.11803i	0.716446	+	4.18171i	6.69226	−	2.05274i			2.82843i	−6.83794	−	5.85172i	−1.58114	+	2.73861i
11.8	−1.22474	+	0.707107i	1.92630	−	2.29987i	1.00000	−	1.73205i	−1.93649	+	1.11803i	−0.732971	+	4.17885i	−5.85650	−	3.83424i			2.82843i	−1.57876	−	8.86045i	1.58114	−	2.73861i
11.9	−1.22474	+	0.707107i	2.66551	+	1.37661i	1.00000	−	1.73205i	−1.93649	+	1.11803i	−4.23798	+	0.198807i	3.85860	−	5.84048i			2.82843i	5.20990	+	7.33873i	1.58114	−	2.73861i
11.10	−1.22474	+	0.707107i	2.99076	+	0.235263i	1.00000	−	1.73205i	1.93649	−	1.11803i	−3.82928	+	1.82665i	−6.91642	+	1.07847i			2.82843i	8.88930	+	1.40723i	−1.58114	+	2.73861i
11.11	1.22474	−	0.707107i	−2.95566	+	0.513907i	1.00000	−	1.73205i	−1.93649	+	1.11803i	−3.25654	+	2.71937i	5.52236	+	4.30158i		−	2.82843i	8.47180	−	3.03786i	−1.58114	+	2.73861i
11.12	1.22474	−	0.707107i	−2.52493	−	1.62010i	1.00000	−	1.73205i	1.93649	−	1.11803i	−4.23798	−	0.198807i	3.85860	−	5.84048i		−	2.82843i	3.75058	+	8.18127i	1.58114	−	2.73861i
11.13	1.22474	−	0.707107i	−2.41792	+	1.77586i	1.00000	−	1.73205i	−1.93649	+	1.11803i	−1.70561	+	3.88470i	−5.45202	−	4.39039i		−	2.82843i	2.69265	−	8.58776i	−1.58114	+	2.73861i
11.14	1.22474	−	0.707107i	−1.69912	−	2.47244i	1.00000	−	1.73205i	−1.93649	+	1.11803i	−3.82928	−	1.82665i	−6.91642	+	1.07847i		−	2.82843i	−3.22595	+	8.40198i	−1.58114	+	2.73861i
11.15	1.22474	−	0.707107i	−1.44013	+	2.63174i	1.00000	−	1.73205i	1.93649	−	1.11803i	0.0971326	+	4.24153i	1.98659	−	6.71219i		−	2.82843i	−4.85208	−	7.58006i	1.58114	−	2.73861i
11.16	1.22474	−	0.707107i	1.02859	−	2.81815i	1.00000	−	1.73205i	1.93649	−	1.11803i	−0.732971	−	4.17885i	−5.85650	−	3.83424i		−	2.82843i	−6.88399	−	5.79747i	1.58114	−	2.73861i
11.17	1.22474	−	0.707107i	1.81607	+	2.38786i	1.00000	−	1.73205i	−1.93649	+	1.11803i	3.91270	+	1.64036i	−2.08959	+	6.68084i		−	2.82843i	−2.40375	+	8.67306i	−1.58114	+	2.73861i
11.18	1.22474	−	0.707107i	1.91719	−	2.30746i	1.00000	−	1.73205i	−1.93649	+	1.11803i	0.716446	−	4.18171i	6.69226	−	2.05274i		−	2.82843i	−1.64877	−	8.84769i	−1.58114	+	2.73861i
11.19	1.22474	−	0.707107i	2.17761	+	2.06350i	1.00000	−	1.73205i	1.93649	−	1.11803i	4.12613	+	0.987462i	6.85537	−	1.41561i		−	2.82843i	0.483930	+	8.98698i	1.58114	−	2.73861i
11.20	1.22474	−	0.707107i	2.87355	−	0.861812i	1.00000	−	1.73205i	1.93649	−	1.11803i	2.90997	−	3.08741i	0.399356	+	6.98860i		−	2.82843i	7.51456	−	4.95292i	1.58114	−	2.73861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	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.s.a	✓	40
3.b	odd	2	1	inner	210.3.s.a	✓	40
7.c	even	3	1	inner	210.3.s.a	✓	40
21.h	odd	6	1	inner	210.3.s.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.3.s.a	✓	40	1.a	even	1	1	trivial
210.3.s.a	✓	40	3.b	odd	2	1	inner
210.3.s.a	✓	40	7.c	even	3	1	inner
210.3.s.a	✓	40	21.h	odd	6	1	inner

Hecke kernels

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