Embedded newform 210.3.w.b.17.16

• Label: 210.3.w.b.17.16
• Level: $210$
• Weight: $3$
• Character: 210.17
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			17.16
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.b.173.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(2.95132 - 0.538273i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-2.52260 - 4.31700i) q^{5} +(4.22859 + 0.344962i) q^{6} +(0.692615 - 6.96565i) q^{7} +(2.00000 + 2.00000i) q^{8} +(8.42052 - 3.17723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	+	0.366025i	0.683013	+	0.183013i
							\(3\)	2.95132	−	0.538273i	0.983772	−	0.179424i
\(5\)	−2.52260	−	4.31700i	−0.504521	−	0.863400i
							\(7\)	0.692615	−	6.96565i	0.0989450	−	0.995093i
\(11\)	−3.24428	−	1.87308i	−0.294934	−	0.170280i								0.345231	−	0.938518i	\(-0.387801\pi\)
							−0.640165	+	0.768237i	\(0.721134\pi\)
\(13\)	5.63880	+	5.63880i	0.433754	+	0.433754i	0.889903	−	0.456149i	\(-0.150772\pi\)
							−0.456149	+	0.889903i	\(0.650772\pi\)
\(17\)	4.24716	−	1.13802i	0.249833	−	0.0669425i	−0.131729	−	0.991286i	\(-0.542053\pi\)
							0.381562	+	0.924343i	\(0.375386\pi\)
\(19\)	8.60142	+	14.8981i	0.452707	+	0.784111i	0.998553	−	0.0537744i	\(-0.0171252\pi\)
							−0.545847	+	0.837885i	\(0.683792\pi\)
\(23\)	−9.56743	+	35.7061i	−0.415975	+	1.55244i	0.366899	+	0.930261i	\(0.380420\pi\)
							−0.782874	+	0.622180i	\(0.786247\pi\)
\(29\)	−29.5157			−1.01778			−0.508892	−	0.860830i	\(-0.669945\pi\)
							−0.508892	+	0.860830i	\(0.669945\pi\)
\(31\)	−11.0230	−	6.36411i	−0.355579	−	0.205294i	0.311561	−	0.950226i	\(-0.399148\pi\)
							−0.667140	+	0.744933i	\(0.732482\pi\)
\(37\)	−6.31767	+	23.5779i	−0.170748	+	0.637240i	0.826489	+	0.562953i	\(0.190335\pi\)
							−0.997237	+	0.0742870i	\(0.976332\pi\)
\(41\)	26.2318			0.639800			0.319900	−	0.947451i	\(-0.396351\pi\)
							0.319900	+	0.947451i	\(0.396351\pi\)
\(43\)	38.5379	−	38.5379i	0.896231	−	0.896231i	−0.0988691	−	0.995100i	\(-0.531522\pi\)
							0.995100	+	0.0988691i	\(0.0315225\pi\)
\(47\)	7.31893	−	27.3146i	0.155722	−	0.581162i	−0.843321	−	0.537411i	\(-0.819402\pi\)
							0.999042	−	0.0437512i	\(-0.0139309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.17.16	yes	64
3.2	odd	2		210.3.w.a.17.14	✓	64
5.3	odd	4		210.3.w.a.143.10	yes	64
7.5	odd	6	inner	210.3.w.b.47.10	yes	64
15.8	even	4	inner	210.3.w.b.143.10	yes	64
21.5	even	6		210.3.w.a.47.10	yes	64
35.33	even	12		210.3.w.a.173.14	yes	64
105.68	odd	12	inner	210.3.w.b.173.16	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.14	✓	64	3.2	odd	2
210.3.w.a.47.10	yes	64	21.5	even	6
210.3.w.a.143.10	yes	64	5.3	odd	4
210.3.w.a.173.14	yes	64	35.33	even	12
210.3.w.b.17.16	yes	64	1.1	even	1	trivial
210.3.w.b.47.10	yes	64	7.5	odd	6	inner
210.3.w.b.143.10	yes	64	15.8	even	4	inner
210.3.w.b.173.16	yes	64	105.68	odd	12	inner