Embedded newform 210.3.w.a.173.1

• Label: 210.3.w.a.173.1
• Level: $210$
• Weight: $3$
• Character: 210.173
• 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			173.1
Character	\(\chi\)	\(=\)	210.173
Dual form			210.3.w.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-2.99585 + 0.157742i) q^{3} +(1.73205 - 1.00000i) q^{4} +(2.05899 - 4.55638i) q^{5} +(4.03467 - 1.31204i) q^{6} +(-6.80611 + 1.63612i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(8.95023 - 0.945145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	−2.99585	+	0.157742i	−0.998617	+	0.0525808i
\(5\)	2.05899	−	4.55638i	0.411798	−	0.911275i
							\(7\)	−6.80611	+	1.63612i	−0.972301	+	0.233732i
\(11\)	7.66474	−	4.42524i	0.696794	−	0.402294i								−0.109358	−	0.994002i	\(-0.534880\pi\)
							0.806152	+	0.591708i	\(0.201546\pi\)
\(13\)	−5.26652	+	5.26652i	−0.405117	+	0.405117i	−0.880032	−	0.474915i	\(-0.842479\pi\)
							0.474915	+	0.880032i	\(0.342479\pi\)
\(17\)	−9.09260	−	2.43636i	−0.534859	−	0.143315i	−0.0187279	−	0.999825i	\(-0.505962\pi\)
							−0.516131	+	0.856510i	\(0.672628\pi\)
\(19\)	−16.3775	+	28.3666i	−0.861971	+	1.49298i	0.00805170	+	0.999968i	\(0.497437\pi\)
							−0.870023	+	0.493011i	\(0.835896\pi\)
\(23\)	3.23126	+	12.0592i	0.140490	+	0.524314i	0.999915	+	0.0130527i	\(0.00415493\pi\)
							−0.859425	+	0.511261i	\(0.829178\pi\)
\(29\)	−48.1807			−1.66140			−0.830701	−	0.556719i	\(-0.812060\pi\)
							−0.830701	+	0.556719i	\(0.812060\pi\)
\(31\)	−42.0786	+	24.2941i	−1.35737	+	0.783680i	−0.989269	−	0.146105i	\(-0.953326\pi\)
							−0.368104	+	0.929785i	\(0.619993\pi\)
\(37\)	4.02200	+	15.0103i	0.108703	+	0.405684i	0.998739	−	0.0502056i	\(-0.0159877\pi\)
							−0.890036	+	0.455890i	\(0.849321\pi\)
\(41\)	45.6603			1.11367			0.556833	−	0.830625i	\(-0.312016\pi\)
							0.556833	+	0.830625i	\(0.312016\pi\)
\(43\)	−17.0633	−	17.0633i	−0.396821	−	0.396821i	0.480289	−	0.877110i	\(-0.340532\pi\)
							−0.877110	+	0.480289i	\(0.840532\pi\)
\(47\)	18.2007	+	67.9259i	0.387249	+	1.44523i	0.834592	+	0.550869i	\(0.185704\pi\)
							−0.447343	+	0.894362i	\(0.647630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.173.1	yes	64
3.2	odd	2		210.3.w.b.173.3	yes	64
5.2	odd	4		210.3.w.b.47.8	yes	64
7.3	odd	6	inner	210.3.w.a.143.7	yes	64
15.2	even	4	inner	210.3.w.a.47.7	yes	64
21.17	even	6		210.3.w.b.143.8	yes	64
35.17	even	12		210.3.w.b.17.3	yes	64
105.17	odd	12	inner	210.3.w.a.17.1	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.1	✓	64	105.17	odd	12	inner
210.3.w.a.47.7	yes	64	15.2	even	4	inner
210.3.w.a.143.7	yes	64	7.3	odd	6	inner
210.3.w.a.173.1	yes	64	1.1	even	1	trivial
210.3.w.b.17.3	yes	64	35.17	even	12
210.3.w.b.47.8	yes	64	5.2	odd	4
210.3.w.b.143.8	yes	64	21.17	even	6
210.3.w.b.173.3	yes	64	3.2	odd	2