Embedded newform 210.3.k.a.167.2

• Label: 210.3.k.a.167.2
• 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			167.2
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.a.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(-2.84642 + 0.947561i) q^{3} -2.00000i q^{4} +(-4.64638 + 1.84693i) q^{5} +(1.89886 - 3.79398i) q^{6} +(-6.25771 - 3.13705i) q^{7} +(2.00000 + 2.00000i) q^{8} +(7.20426 - 5.39432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.84642	+	0.947561i	−0.948808	+	0.315854i
\(5\)	−4.64638	+	1.84693i	−0.929276	+	0.369386i
							\(7\)	−6.25771	−	3.13705i	−0.893958	−	0.448150i
\(11\)		−	2.08576i		−	0.189615i								−0.995496	−	0.0948073i	\(-0.969777\pi\)
							0.995496	−	0.0948073i	\(-0.0302235\pi\)
\(13\)	8.39517	+	8.39517i	0.645782	+	0.645782i	0.951971	−	0.306189i	\(-0.0990538\pi\)
							−0.306189	+	0.951971i	\(0.599054\pi\)
\(17\)	−4.96522	−	4.96522i	−0.292072	−	0.292072i	0.545826	−	0.837898i	\(-0.316216\pi\)
							−0.837898	+	0.545826i	\(0.816216\pi\)
\(19\)	17.3668			0.914041			0.457021	−	0.889456i	\(-0.348917\pi\)
							0.457021	+	0.889456i	\(0.348917\pi\)
\(23\)	3.08467	+	3.08467i	0.134116	+	0.134116i	0.770978	−	0.636862i	\(-0.219768\pi\)
							−0.636862	+	0.770978i	\(0.719768\pi\)
\(29\)	39.1891			1.35135			0.675674	−	0.737201i	\(-0.263853\pi\)
							0.675674	+	0.737201i	\(0.263853\pi\)
\(31\)		−	42.3954i		−	1.36759i	−0.729673	−	0.683796i	\(-0.760328\pi\)
							0.729673	−	0.683796i	\(-0.239672\pi\)
\(37\)	36.7464	+	36.7464i	0.993146	+	0.993146i	0.999977	−	0.00683066i	\(-0.00217428\pi\)
							−0.00683066	+	0.999977i	\(0.502174\pi\)
\(41\)	−15.5827			−0.380065			−0.190032	−	0.981778i	\(-0.560859\pi\)
							−0.190032	+	0.981778i	\(0.560859\pi\)
\(43\)	22.8274	−	22.8274i	0.530869	−	0.530869i	−0.389962	−	0.920831i	\(-0.627512\pi\)
							0.920831	+	0.389962i	\(0.127512\pi\)
\(47\)	−33.4161	−	33.4161i	−0.710980	−	0.710980i	0.255760	−	0.966740i	\(-0.417674\pi\)
							−0.966740	+	0.255760i	\(0.917674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.167.2	yes	32
3.2	odd	2		210.3.k.b.167.10	yes	32
5.3	odd	4		210.3.k.b.83.7	yes	32
7.6	odd	2	inner	210.3.k.a.167.15	yes	32
15.8	even	4	inner	210.3.k.a.83.15	yes	32
21.20	even	2		210.3.k.b.167.7	yes	32
35.13	even	4		210.3.k.b.83.10	yes	32
105.83	odd	4	inner	210.3.k.a.83.2	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.2	✓	32	105.83	odd	4	inner
210.3.k.a.83.15	yes	32	15.8	even	4	inner
210.3.k.a.167.2	yes	32	1.1	even	1	trivial
210.3.k.a.167.15	yes	32	7.6	odd	2	inner
210.3.k.b.83.7	yes	32	5.3	odd	4
210.3.k.b.83.10	yes	32	35.13	even	4
210.3.k.b.167.7	yes	32	21.20	even	2
210.3.k.b.167.10	yes	32	3.2	odd	2