Embedded newform 210.3.w.a.17.15

• Label: 210.3.w.a.17.15
• 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			17.15
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.a.173.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(2.86958 - 0.874940i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-4.99957 + 0.0656422i) q^{5} +(-4.24017 + 0.144852i) q^{6} +(-2.70652 + 6.45560i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(7.46896 - 5.02142i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	2.86958	−	0.874940i	0.956526	−	0.291647i
\(5\)	−4.99957	+	0.0656422i	−0.999914	+	0.0131284i
							\(7\)	−2.70652	+	6.45560i	−0.386646	+	0.922228i
\(11\)	17.7852	+	10.2683i	1.61684	+	0.933482i								0.987731	+	0.156164i	\(0.0499130\pi\)
							0.629108	+	0.777318i	\(0.283420\pi\)
\(13\)	10.2742	+	10.2742i	0.790321	+	0.790321i	0.981546	−	0.191225i	\(-0.0612460\pi\)
							−0.191225	+	0.981546i	\(0.561246\pi\)
\(17\)	16.4097	−	4.39696i	0.965276	−	0.258645i	0.258444	−	0.966026i	\(-0.416790\pi\)
							0.706832	+	0.707381i	\(0.250124\pi\)
\(19\)	−11.5679	−	20.0362i	−0.608838	−	1.05454i	−0.991432	−	0.130622i	\(-0.958302\pi\)
							0.382594	−	0.923917i	\(-0.375031\pi\)
\(23\)	−5.50900	+	20.5599i	−0.239522	+	0.893907i	0.736537	+	0.676398i	\(0.236460\pi\)
							−0.976058	+	0.217509i	\(0.930207\pi\)
\(29\)	2.30614			0.0795221			0.0397610	−	0.999209i	\(-0.487340\pi\)
							0.0397610	+	0.999209i	\(0.487340\pi\)
\(31\)	3.09559	+	1.78724i	0.0998577	+	0.0576529i	0.549097	−	0.835758i	\(-0.314972\pi\)
							−0.449239	+	0.893411i	\(0.648305\pi\)
\(37\)	−10.7499	+	40.1190i	−0.290536	+	1.08430i	0.654161	+	0.756355i	\(0.273022\pi\)
							−0.944698	+	0.327942i	\(0.893645\pi\)
\(41\)	−0.0268075			−0.000653841			−0.000326921	−	1.00000i	\(-0.500104\pi\)
							−0.000326921		1.00000i	\(0.500104\pi\)
\(43\)	−6.00590	+	6.00590i	−0.139672	+	0.139672i	−0.773486	−	0.633814i	\(-0.781489\pi\)
							0.633814	+	0.773486i	\(0.281489\pi\)
\(47\)	−3.18676	+	11.8931i	−0.0678033	+	0.253046i	−0.991505	−	0.130066i	\(-0.958481\pi\)
							0.923702	+	0.383112i	\(0.125148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.17.15	✓	64
3.2	odd	2		210.3.w.b.17.15	yes	64
5.3	odd	4		210.3.w.b.143.9	yes	64
7.5	odd	6	inner	210.3.w.a.47.11	yes	64
15.8	even	4	inner	210.3.w.a.143.11	yes	64
21.5	even	6		210.3.w.b.47.9	yes	64
35.33	even	12		210.3.w.b.173.15	yes	64
105.68	odd	12	inner	210.3.w.a.173.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.15	✓	64	1.1	even	1	trivial
210.3.w.a.47.11	yes	64	7.5	odd	6	inner
210.3.w.a.143.11	yes	64	15.8	even	4	inner
210.3.w.a.173.15	yes	64	105.68	odd	12	inner
210.3.w.b.17.15	yes	64	3.2	odd	2
210.3.w.b.47.9	yes	64	21.5	even	6
210.3.w.b.143.9	yes	64	5.3	odd	4
210.3.w.b.173.15	yes	64	35.33	even	12