Embedded newform 210.3.v.a.37.2

• Label: 210.3.v.a.37.2
• Level: $210$
• Weight: $3$
• Character: 210.37
• 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			37.2
Character	\(\chi\)	\(=\)	210.37
Dual form			210.3.v.a.193.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-1.67303 - 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-1.25847 - 4.83903i) q^{5} +2.44949 q^{6} +(5.72194 + 4.03230i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	+	0.366025i	−0.683013	+	0.183013i
							\(3\)	−1.67303	−	0.448288i	−0.557678	−	0.149429i
\(5\)	−1.25847	−	4.83903i	−0.251695	−	0.967807i
							\(7\)	5.72194	+	4.03230i	0.817419	+	0.576043i
\(11\)	−5.44212	−	9.42602i	−0.494738	−	0.856911i								0.505244	−	0.862977i	\(-0.331403\pi\)
							−0.999982	+	0.00606548i	\(0.998069\pi\)
\(13\)	−4.13640	+	4.13640i	−0.318184	+	0.318184i	−0.848069	−	0.529885i	\(-0.822235\pi\)
							0.529885	+	0.848069i	\(0.322235\pi\)
\(17\)	−0.489455	+	1.82667i	−0.0287915	+	0.107451i	−0.978826	−	0.204693i	\(-0.934380\pi\)
							0.950035	+	0.312144i	\(0.101047\pi\)
\(19\)	−27.1044	−	15.6487i	−1.42655	−	0.823618i	−0.429701	−	0.902971i	\(-0.641381\pi\)
							−0.996847	+	0.0793537i	\(0.974714\pi\)
\(23\)	−5.85291	−	21.8434i	−0.254474	−	0.949711i	−0.968382	−	0.249471i	\(-0.919743\pi\)
							0.713908	−	0.700240i	\(-0.246923\pi\)
\(29\)		−	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	0.728265	−	0.685296i	\(-0.240327\pi\)
							−0.957616	+	0.288048i	\(0.906994\pi\)
\(37\)	13.4546	−	3.60515i	0.363638	−	0.0974365i	−0.0723735	−	0.997378i	\(-0.523057\pi\)
							0.436011	+	0.899941i	\(0.356391\pi\)
\(41\)	−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	\(0.279039\pi\)
\(47\)	−89.5659	+	23.9991i	−1.90566	+	0.510619i	−0.910353	+	0.413833i	\(0.864190\pi\)
							−0.995305	+	0.0967860i	\(0.969144\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.2	✓	32	1.1	even	1	trivial
210.3.v.a.67.8	yes	32	7.4	even	3	inner
210.3.v.a.163.8	yes	32	5.3	odd	4	inner
210.3.v.a.193.2	yes	32	35.18	odd	12	inner