Embedded newform 210.3.v.a.163.7

• Label: 210.3.v.a.163.7
• Level: $210$
• Weight: $3$
• Character: 210.163
• 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			163.7
Character	\(\chi\)	\(=\)	210.163
Dual form			210.3.v.a.67.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(0.448288 - 1.67303i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(3.99034 + 3.01284i) q^{5} +2.44949 q^{6} +(-5.53495 + 4.28537i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 16 q^{2} - 8 q^{7} - 64 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	+	1.36603i	0.183013	+	0.683013i
							\(3\)	0.448288	−	1.67303i	0.149429	−	0.557678i
\(5\)	3.99034	+	3.01284i	0.798068	+	0.602567i
							\(7\)	−5.53495	+	4.28537i	−0.790707	+	0.612195i
\(11\)	8.47882	+	14.6857i	0.770802	+	1.33507i								0.937124	+	0.348996i	\(0.113477\pi\)
							−0.166322	+	0.986071i	\(0.553189\pi\)
\(13\)	3.05181	+	3.05181i	0.234754	+	0.234754i	0.814674	−	0.579920i	\(-0.196916\pi\)
							−0.579920	+	0.814674i	\(0.696916\pi\)
\(17\)	10.7449	+	2.87910i	0.632055	+	0.169359i	0.560602	−	0.828085i	\(-0.310570\pi\)
							0.0714531	+	0.997444i	\(0.477236\pi\)
\(19\)	5.74405	+	3.31633i	0.302318	+	0.174544i	0.643484	−	0.765460i	\(-0.277488\pi\)
							−0.341166	+	0.940003i	\(0.610822\pi\)
\(23\)	−16.7817	+	4.49666i	−0.729641	+	0.195507i	−0.604469	−	0.796628i	\(-0.706615\pi\)
							−0.125172	+	0.992135i	\(0.539948\pi\)
\(29\)		−	35.8861i		−	1.23745i	−0.785607	−	0.618725i	\(-0.787649\pi\)
							0.785607	−	0.618725i	\(-0.212351\pi\)
\(31\)	22.7901	+	39.4736i	0.735165	+	1.27334i	0.954651	+	0.297727i	\(0.0962286\pi\)
							−0.219486	+	0.975616i	\(0.570438\pi\)
\(37\)	−4.74524	−	17.7095i	−0.128250	−	0.478635i	0.871685	−	0.490067i	\(-0.163028\pi\)
							−0.999935	+	0.0114320i	\(0.996361\pi\)
\(41\)	−42.9609			−1.04783			−0.523913	−	0.851772i	\(-0.675528\pi\)
							−0.523913	+	0.851772i	\(0.675528\pi\)
\(43\)	−1.98628	−	1.98628i	−0.0461926	−	0.0461926i	0.683633	−	0.729826i	\(-0.260399\pi\)
							−0.729826	+	0.683633i	\(0.760399\pi\)
\(47\)	−11.3192	−	42.2437i	−0.240833	−	0.898801i	−0.975432	−	0.220300i	\(-0.929296\pi\)
							0.734599	−	0.678501i	\(-0.237370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.v.a.163.7	yes	32
5.2	odd	4	inner	210.3.v.a.37.1	✓	32
7.4	even	3	inner	210.3.v.a.193.1	yes	32
35.32	odd	12	inner	210.3.v.a.67.7	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.v.a.37.1	✓	32	5.2	odd	4	inner
210.3.v.a.67.7	yes	32	35.32	odd	12	inner
210.3.v.a.163.7	yes	32	1.1	even	1	trivial
210.3.v.a.193.1	yes	32	7.4	even	3	inner