Embedded newform 210.3.k.a.167.7

• Label: 210.3.k.a.167.7
• Level: $210$
• Weight: $3$
• Character: 210.167
• 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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			167.7
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.a.83.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(-0.282815 - 2.98664i) q^{3} -2.00000i q^{4} +(-3.28357 - 3.77070i) q^{5} +(3.26945 + 2.70382i) q^{6} +(-3.67639 - 5.95686i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-8.84003 + 1.68933i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.00000i	−0.500000	+	0.500000i
							\(3\)	−0.282815	−	2.98664i	−0.0942716	−	0.995547i
\(5\)	−3.28357	−	3.77070i	−0.656714	−	0.754139i
							\(7\)	−3.67639	−	5.95686i	−0.525199	−	0.850979i
\(11\)			19.5576i			1.77797i								0.457939	+	0.888984i	\(0.348588\pi\)
							−0.457939	+	0.888984i	\(0.651412\pi\)
\(13\)	−2.90656	−	2.90656i	−0.223582	−	0.223582i	0.586423	−	0.810005i	\(-0.300536\pi\)
							−0.810005	+	0.586423i	\(0.800536\pi\)
\(17\)	16.3194	+	16.3194i	0.959967	+	0.959967i	0.999229	−	0.0392622i	\(-0.0125008\pi\)
							−0.0392622	+	0.999229i	\(0.512501\pi\)
\(19\)	−8.66094			−0.455839			−0.227919	−	0.973680i	\(-0.573192\pi\)
							−0.227919	+	0.973680i	\(0.573192\pi\)
\(23\)	−6.73947	−	6.73947i	−0.293021	−	0.293021i	0.545252	−	0.838272i	\(-0.316434\pi\)
							−0.838272	+	0.545252i	\(0.816434\pi\)
\(29\)	−31.3396			−1.08067			−0.540337	−	0.841449i	\(-0.681703\pi\)
							−0.540337	+	0.841449i	\(0.681703\pi\)
\(31\)		−	39.4508i		−	1.27261i	−0.771439	−	0.636304i	\(-0.780463\pi\)
							0.771439	−	0.636304i	\(-0.219537\pi\)
\(37\)	−25.1721	−	25.1721i	−0.680326	−	0.680326i	0.279747	−	0.960074i	\(-0.409749\pi\)
							−0.960074	+	0.279747i	\(0.909749\pi\)
\(41\)	−58.9348			−1.43743			−0.718717	−	0.695303i	\(-0.755270\pi\)
							−0.718717	+	0.695303i	\(0.755270\pi\)
\(43\)	10.5096	−	10.5096i	0.244409	−	0.244409i	−0.574262	−	0.818671i	\(-0.694711\pi\)
							0.818671	+	0.574262i	\(0.194711\pi\)
\(47\)	−29.2211	−	29.2211i	−0.621725	−	0.621725i	0.324247	−	0.945972i	\(-0.394889\pi\)
							−0.945972	+	0.324247i	\(0.894889\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.167.7	yes	32
3.2	odd	2		210.3.k.b.167.2	yes	32
5.3	odd	4		210.3.k.b.83.15	yes	32
7.6	odd	2	inner	210.3.k.a.167.10	yes	32
15.8	even	4	inner	210.3.k.a.83.10	yes	32
21.20	even	2		210.3.k.b.167.15	yes	32
35.13	even	4		210.3.k.b.83.2	yes	32
105.83	odd	4	inner	210.3.k.a.83.7	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.7	✓	32	105.83	odd	4	inner
210.3.k.a.83.10	yes	32	15.8	even	4	inner
210.3.k.a.167.7	yes	32	1.1	even	1	trivial
210.3.k.a.167.10	yes	32	7.6	odd	2	inner
210.3.k.b.83.2	yes	32	35.13	even	4
210.3.k.b.83.15	yes	32	5.3	odd	4
210.3.k.b.167.2	yes	32	3.2	odd	2
210.3.k.b.167.15	yes	32	21.20	even	2