Embedded newform 210.3.k.b.167.13

• Label: 210.3.k.b.167.13
• Level: $210$
• Weight: $3$
• Character: 210.167
• 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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			167.13
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.b.83.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(2.09499 - 2.14733i) q^{3} -2.00000i q^{4} +(1.13661 + 4.86910i) q^{5} +(-0.0523328 - 4.24232i) q^{6} +(6.60685 + 2.31291i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-0.222012 - 8.99726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{2} - 4 q^{7} - 64 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.00000i	0.500000	−	0.500000i
							\(3\)	2.09499	−	2.14733i	0.698331	−	0.715775i
\(5\)	1.13661	+	4.86910i	0.227322	+	0.973820i
							\(7\)	6.60685	+	2.31291i	0.943836	+	0.330415i
\(11\)		−	16.9733i		−	1.54303i								−0.636213	−	0.771513i	\(-0.719500\pi\)
							0.636213	−	0.771513i	\(-0.280500\pi\)
\(13\)	10.2231	+	10.2231i	0.786392	+	0.786392i	0.980901	−	0.194509i	\(-0.0623114\pi\)
							−0.194509	+	0.980901i	\(0.562311\pi\)
\(17\)	8.79877	+	8.79877i	0.517575	+	0.517575i	0.916837	−	0.399262i	\(-0.130734\pi\)
							−0.399262	+	0.916837i	\(0.630734\pi\)
\(19\)	−24.7369			−1.30194			−0.650972	−	0.759102i	\(-0.725639\pi\)
							−0.650972	+	0.759102i	\(0.725639\pi\)
\(23\)	−19.2569	−	19.2569i	−0.837258	−	0.837258i	0.151239	−	0.988497i	\(-0.451674\pi\)
							−0.988497	+	0.151239i	\(0.951674\pi\)
\(29\)	−1.67978			−0.0579235			−0.0289618	−	0.999581i	\(-0.509220\pi\)
							−0.0289618	+	0.999581i	\(0.509220\pi\)
\(31\)			36.8991i			1.19029i	0.803616	+	0.595147i	\(0.202906\pi\)
							−0.803616	+	0.595147i	\(0.797094\pi\)
\(37\)	40.5381	+	40.5381i	1.09562	+	1.09562i	0.994916	+	0.100708i	\(0.0321107\pi\)
							0.100708	+	0.994916i	\(0.467889\pi\)
\(41\)	0.885911			0.0216076			0.0108038	−	0.999942i	\(-0.496561\pi\)
							0.0108038	+	0.999942i	\(0.496561\pi\)
\(43\)	−9.87427	+	9.87427i	−0.229634	+	0.229634i	−0.812540	−	0.582906i	\(-0.801916\pi\)
							0.582906	+	0.812540i	\(0.301916\pi\)
\(47\)	−33.7538	−	33.7538i	−0.718165	−	0.718165i	0.250064	−	0.968229i	\(-0.419548\pi\)
							−0.968229	+	0.250064i	\(0.919548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.b.167.13	yes	32
3.2	odd	2		210.3.k.a.167.5	yes	32
5.3	odd	4		210.3.k.a.83.12	yes	32
7.6	odd	2	inner	210.3.k.b.167.4	yes	32
15.8	even	4	inner	210.3.k.b.83.4	yes	32
21.20	even	2		210.3.k.a.167.12	yes	32
35.13	even	4		210.3.k.a.83.5	✓	32
105.83	odd	4	inner	210.3.k.b.83.13	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.5	✓	32	35.13	even	4
210.3.k.a.83.12	yes	32	5.3	odd	4
210.3.k.a.167.5	yes	32	3.2	odd	2
210.3.k.a.167.12	yes	32	21.20	even	2
210.3.k.b.83.4	yes	32	15.8	even	4	inner
210.3.k.b.83.13	yes	32	105.83	odd	4	inner
210.3.k.b.167.4	yes	32	7.6	odd	2	inner
210.3.k.b.167.13	yes	32	1.1	even	1	trivial