Embedded newform 210.3.k.a.167.9

• Label: 210.3.k.a.167.9
• 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			167.9
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.a.83.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(0.00829838 - 2.99999i) q^{3} -2.00000i q^{4} +(3.67015 - 3.39558i) q^{5} +(2.99169 + 3.00829i) q^{6} +(6.84640 + 1.45834i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-8.99986 - 0.0497901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	0.00829838	−	2.99999i	0.00276613	−	0.999996i
\(5\)	3.67015	−	3.39558i	0.734030	−	0.679117i
							\(7\)	6.84640	+	1.45834i	0.978058	+	0.208334i
\(11\)		−	6.08610i		−	0.553282i								−0.960973	−	0.276641i	\(-0.910779\pi\)
							0.960973	−	0.276641i	\(-0.0892213\pi\)
\(13\)	4.00045	+	4.00045i	0.307727	+	0.307727i	0.844027	−	0.536300i	\(-0.180179\pi\)
							−0.536300	+	0.844027i	\(0.680179\pi\)
\(17\)	−14.8174	−	14.8174i	−0.871610	−	0.871610i	0.121037	−	0.992648i	\(-0.461378\pi\)
							−0.992648	+	0.121037i	\(0.961378\pi\)
\(19\)	20.4190			1.07468			0.537341	−	0.843365i	\(-0.319429\pi\)
							0.537341	+	0.843365i	\(0.319429\pi\)
\(23\)	−20.6285	−	20.6285i	−0.896892	−	0.896892i	0.0982681	−	0.995160i	\(-0.468670\pi\)
							−0.995160	+	0.0982681i	\(0.968670\pi\)
\(29\)	19.5317			0.673508			0.336754	−	0.941593i	\(-0.390671\pi\)
							0.336754	+	0.941593i	\(0.390671\pi\)
\(31\)		−	4.36235i		−	0.140721i	−0.997522	−	0.0703605i	\(-0.977585\pi\)
							0.997522	−	0.0703605i	\(-0.0224150\pi\)
\(37\)	−1.64351	−	1.64351i	−0.0444191	−	0.0444191i	0.684548	−	0.728967i	\(-0.259999\pi\)
							−0.728967	+	0.684548i	\(0.759999\pi\)
\(41\)	42.2693			1.03096			0.515480	−	0.856902i	\(-0.327614\pi\)
							0.515480	+	0.856902i	\(0.327614\pi\)
\(43\)	−45.0034	+	45.0034i	−1.04659	+	1.04659i	−0.0477300	+	0.998860i	\(0.515199\pi\)
							−0.998860	+	0.0477300i	\(0.984801\pi\)
\(47\)	−36.6983	−	36.6983i	−0.780815	−	0.780815i	0.199153	−	0.979968i	\(-0.436181\pi\)
							−0.979968	+	0.199153i	\(0.936181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.167.9	yes	32
3.2	odd	2		210.3.k.b.167.1	yes	32
5.3	odd	4		210.3.k.b.83.16	yes	32
7.6	odd	2	inner	210.3.k.a.167.8	yes	32
15.8	even	4	inner	210.3.k.a.83.8	✓	32
21.20	even	2		210.3.k.b.167.16	yes	32
35.13	even	4		210.3.k.b.83.1	yes	32
105.83	odd	4	inner	210.3.k.a.83.9	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.8	✓	32	15.8	even	4	inner
210.3.k.a.83.9	yes	32	105.83	odd	4	inner
210.3.k.a.167.8	yes	32	7.6	odd	2	inner
210.3.k.a.167.9	yes	32	1.1	even	1	trivial
210.3.k.b.83.1	yes	32	35.13	even	4
210.3.k.b.83.16	yes	32	5.3	odd	4
210.3.k.b.167.1	yes	32	3.2	odd	2
210.3.k.b.167.16	yes	32	21.20	even	2