Embedded newform 210.3.v.a.67.8

• Label: 210.3.v.a.67.8
• Level: $210$
• Weight: $3$
• Character: 210.67
• 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			67.8
Character	\(\chi\)	\(=\)	210.67
Dual form			210.3.v.a.163.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(0.448288 + 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(4.81996 + 1.32965i) q^{5} +2.44949 q^{6} +(4.03230 + 5.72194i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	−	1.36603i	0.183013	−	0.683013i
							\(3\)	0.448288	+	1.67303i	0.149429	+	0.557678i
\(5\)	4.81996	+	1.32965i	0.963993	+	0.265929i
							\(7\)	4.03230	+	5.72194i	0.576043	+	0.817419i
\(11\)	−5.44212	+	9.42602i	−0.494738	+	0.856911i								−0.999982	−	0.00606548i	\(-0.998069\pi\)
							0.505244	+	0.862977i	\(0.331403\pi\)
\(13\)	−4.13640	+	4.13640i	−0.318184	+	0.318184i	−0.848069	−	0.529885i	\(-0.822235\pi\)
							0.529885	+	0.848069i	\(0.322235\pi\)
\(17\)	1.82667	−	0.489455i	0.107451	−	0.0287915i	−0.204693	−	0.978826i	\(-0.565620\pi\)
							0.312144	+	0.950035i	\(0.398953\pi\)
\(19\)	27.1044	−	15.6487i	1.42655	−	0.823618i	0.429701	−	0.902971i	\(-0.358619\pi\)
							0.996847	+	0.0793537i	\(0.0252856\pi\)
\(23\)	21.8434	+	5.85291i	0.949711	+	0.254474i	0.700240	−	0.713908i	\(-0.253077\pi\)
							0.249471	+	0.968382i	\(0.419743\pi\)
\(29\)		−	35.4354i		−	1.22191i	−0.791665	−	0.610956i	\(-0.790785\pi\)
							0.791665	−	0.610956i	\(-0.209215\pi\)
\(31\)	−7.10988	+	12.3147i	−0.229351	+	0.397248i	−0.957616	−	0.288048i	\(-0.906994\pi\)
							0.728265	+	0.685296i	\(0.240327\pi\)
\(37\)	−3.60515	+	13.4546i	−0.0974365	+	0.363638i	−0.997378	−	0.0723735i	\(-0.976943\pi\)
							0.899941	+	0.436011i	\(0.143609\pi\)
\(41\)	−75.8676			−1.85043			−0.925215	−	0.379443i	\(-0.876115\pi\)
							−0.925215	+	0.379443i	\(0.876115\pi\)
\(43\)	−5.54001	+	5.54001i	−0.128837	+	0.128837i	−0.768585	−	0.639748i	\(-0.779039\pi\)
							0.639748	+	0.768585i	\(0.279039\pi\)
\(47\)	23.9991	−	89.5659i	0.510619	−	1.90566i	0.0967860	−	0.995305i	\(-0.469144\pi\)
							0.413833	−	0.910353i	\(-0.364190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.67.8	yes	32
5.3	odd	4	inner	210.3.v.a.193.2	yes	32
7.2	even	3	inner	210.3.v.a.37.2	✓	32
35.23	odd	12	inner	210.3.v.a.163.8	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.2	✓	32	7.2	even	3	inner
210.3.v.a.67.8	yes	32	1.1	even	1	trivial
210.3.v.a.163.8	yes	32	35.23	odd	12	inner
210.3.v.a.193.2	yes	32	5.3	odd	4	inner