Embedded newform 210.3.w.b.17.12

• Label: 210.3.w.b.17.12
• 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			17.12
Character	\(\chi\)	\(=\)	210.17
Dual form			210.3.w.b.173.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(2.23964 + 1.99599i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-2.72423 + 4.19268i) q^{5} +(2.32883 + 3.54635i) q^{6} +(-5.35730 - 4.50548i) q^{7} +(2.00000 + 2.00000i) q^{8} +(1.03201 + 8.94064i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{2} + 6 q^{3} + 12 q^{5} + 4 q^{7} + 128 q^{8} + 16 q^{9}+O(q^{10}) \)

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
							\(3\)	2.23964	+	1.99599i	0.746548	+	0.665332i
\(5\)	−2.72423	+	4.19268i	−0.544847	+	0.838536i
							\(7\)	−5.35730	−	4.50548i	−0.765329	−	0.643640i
\(11\)	8.15539	+	4.70851i	0.741399	+	0.428047i								0.822578	−	0.568653i	\(-0.192535\pi\)
							−0.0811789	+	0.996700i	\(0.525869\pi\)
\(13\)	16.1515	+	16.1515i	1.24243	+	1.24243i	0.958991	+	0.283436i	\(0.0914742\pi\)
							0.283436	+	0.958991i	\(0.408526\pi\)
\(17\)	−9.03315	+	2.42042i	−0.531362	+	0.142378i	−0.514517	−	0.857480i	\(-0.672029\pi\)
							−0.0168447	+	0.999858i	\(0.505362\pi\)
\(19\)	−13.1762	−	22.8218i	−0.693482	−	1.20115i	−0.970690	−	0.240336i	\(-0.922742\pi\)
							0.277208	−	0.960810i	\(-0.410591\pi\)
\(23\)	6.16910	−	23.0234i	0.268222	−	1.00102i	−0.692027	−	0.721872i	\(-0.743282\pi\)
							0.960249	−	0.279146i	\(-0.0900513\pi\)
\(29\)	34.0152			1.17294			0.586470	−	0.809971i	\(-0.300517\pi\)
							0.586470	+	0.809971i	\(0.300517\pi\)
\(31\)	2.88724	+	1.66695i	0.0931366	+	0.0537724i	0.545845	−	0.837886i	\(-0.316209\pi\)
							−0.452708	+	0.891659i	\(0.649542\pi\)
\(37\)	12.3909	−	46.2436i	0.334890	−	1.24983i	−0.569098	−	0.822270i	\(-0.692708\pi\)
							0.903988	−	0.427557i	\(-0.140626\pi\)
\(41\)	3.31953			0.0809641			0.0404820	−	0.999180i	\(-0.487111\pi\)
							0.0404820	+	0.999180i	\(0.487111\pi\)
\(43\)	28.9282	−	28.9282i	0.672749	−	0.672749i	−0.285600	−	0.958349i	\(-0.592193\pi\)
							0.958349	+	0.285600i	\(0.0921928\pi\)
\(47\)	0.830153	−	3.09817i	0.0176628	−	0.0659186i	−0.956532	−	0.291628i	\(-0.905803\pi\)
							0.974195	+	0.225709i	\(0.0724699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.b.17.12	yes	64
3.2	odd	2		210.3.w.a.17.10	✓	64
5.3	odd	4		210.3.w.a.143.5	yes	64
7.5	odd	6	inner	210.3.w.b.47.15	yes	64
15.8	even	4	inner	210.3.w.b.143.15	yes	64
21.5	even	6		210.3.w.a.47.5	yes	64
35.33	even	12		210.3.w.a.173.10	yes	64
105.68	odd	12	inner	210.3.w.b.173.12	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.10	✓	64	3.2	odd	2
210.3.w.a.47.5	yes	64	21.5	even	6
210.3.w.a.143.5	yes	64	5.3	odd	4
210.3.w.a.173.10	yes	64	35.33	even	12
210.3.w.b.17.12	yes	64	1.1	even	1	trivial
210.3.w.b.47.15	yes	64	7.5	odd	6	inner
210.3.w.b.143.15	yes	64	15.8	even	4	inner
210.3.w.b.173.12	yes	64	105.68	odd	12	inner