Embedded newform 210.3.v.b.193.5

• Label: 210.3.v.b.193.5
• Level: $210$
• Weight: $3$
• Character: 210.193
• 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.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			193.5
Character	\(\chi\)	\(=\)	210.193
Dual form			210.3.v.b.37.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 + 0.366025i) q^{2} +(1.67303 - 0.448288i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-3.99375 + 3.00831i) q^{5} +2.44949 q^{6} +(6.54111 + 2.49277i) q^{7} +(2.00000 + 2.00000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 16 q^{2} - 8 q^{5} + 24 q^{7} + 64 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−3.99375	+	3.00831i	−0.798750	+	0.601662i
							\(7\)	6.54111	+	2.49277i	0.934444	+	0.356110i
\(11\)	1.59310	−	2.75932i	0.144827	−	0.250848i								−0.784481	−	0.620152i	\(-0.787071\pi\)
							0.929308	+	0.369305i	\(0.120404\pi\)
\(13\)	16.7146	+	16.7146i	1.28574	+	1.28574i	0.937351	+	0.348386i	\(0.113270\pi\)
							0.348386	+	0.937351i	\(0.386730\pi\)
\(17\)	−3.89785	−	14.5470i	−0.229285	−	0.855704i	−0.980642	−	0.195808i	\(-0.937267\pi\)
							0.751357	−	0.659896i	\(-0.229400\pi\)
\(19\)	−13.5028	+	7.79585i	−0.710674	+	0.410308i	−0.811311	−	0.584615i	\(-0.801245\pi\)
							0.100636	+	0.994923i	\(0.467912\pi\)
\(23\)	−1.45196	+	5.41879i	−0.0631287	+	0.235600i	−0.990280	−	0.139087i	\(-0.955583\pi\)
							0.927151	+	0.374687i	\(0.122250\pi\)
\(29\)		−	47.4699i		−	1.63689i	−0.574583	−	0.818446i	\(-0.694836\pi\)
							0.574583	−	0.818446i	\(-0.305164\pi\)
\(31\)	0.731778	−	1.26748i	0.0236057	−	0.0408863i	−0.853981	−	0.520304i	\(-0.825819\pi\)
							0.877587	+	0.479417i	\(0.159152\pi\)
\(37\)	−53.3715	−	14.3009i	−1.44247	−	0.386510i	−0.549074	−	0.835773i	\(-0.685020\pi\)
							−0.893399	+	0.449264i	\(0.851686\pi\)
\(41\)	−27.2740			−0.665219			−0.332610	−	0.943065i	\(-0.607929\pi\)
							−0.332610	+	0.943065i	\(0.607929\pi\)
\(43\)	−16.8709	−	16.8709i	−0.392346	−	0.392346i	0.483177	−	0.875523i	\(-0.339483\pi\)
							−0.875523	+	0.483177i	\(0.839483\pi\)
\(47\)	26.2915	+	7.04479i	0.559394	+	0.149889i	0.527428	−	0.849600i	\(-0.323156\pi\)
							0.0319661	+	0.999489i	\(0.489823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.b.193.5	yes	32
5.2	odd	4	inner	210.3.v.b.67.4	yes	32
7.2	even	3	inner	210.3.v.b.163.4	yes	32
35.2	odd	12	inner	210.3.v.b.37.5	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.b.37.5	✓	32	35.2	odd	12	inner
210.3.v.b.67.4	yes	32	5.2	odd	4	inner
210.3.v.b.163.4	yes	32	7.2	even	3	inner
210.3.v.b.193.5	yes	32	1.1	even	1	trivial