Embedded newform 210.3.k.a.167.1

• Label: 210.3.k.a.167.1
• 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			167.1
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.a.83.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(-2.99336 - 0.199427i) q^{3} -2.00000i q^{4} +(4.37611 - 2.41861i) q^{5} +(3.19279 - 2.79394i) q^{6} +(-5.12625 + 4.76671i) q^{7} +(2.00000 + 2.00000i) q^{8} +(8.92046 + 1.19391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.99336	−	0.199427i	−0.997788	−	0.0664755i
\(5\)	4.37611	−	2.41861i	0.875222	−	0.483722i
							\(7\)	−5.12625	+	4.76671i	−0.732322	+	0.680959i
\(11\)			6.70149i			0.609226i								0.952476	+	0.304613i	\(0.0985271\pi\)
							−0.952476	+	0.304613i	\(0.901473\pi\)
\(13\)	−16.0066	−	16.0066i	−1.23128	−	1.23128i	−0.963474	−	0.267802i	\(-0.913703\pi\)
							−0.267802	−	0.963474i	\(-0.586297\pi\)
\(17\)	−7.21482	−	7.21482i	−0.424401	−	0.424401i	0.462315	−	0.886716i	\(-0.347019\pi\)
							−0.886716	+	0.462315i	\(0.847019\pi\)
\(19\)	−8.06294			−0.424365			−0.212183	−	0.977230i	\(-0.568057\pi\)
							−0.212183	+	0.977230i	\(0.568057\pi\)
\(23\)	−11.7195	−	11.7195i	−0.509543	−	0.509543i	0.404843	−	0.914386i	\(-0.367326\pi\)
							−0.914386	+	0.404843i	\(0.867326\pi\)
\(29\)	−6.17789			−0.213031			−0.106515	−	0.994311i	\(-0.533969\pi\)
							−0.106515	+	0.994311i	\(0.533969\pi\)
\(31\)		−	41.4735i		−	1.33785i	−0.743328	−	0.668927i	\(-0.766754\pi\)
							0.743328	−	0.668927i	\(-0.233246\pi\)
\(37\)	−37.8620	−	37.8620i	−1.02330	−	1.02330i	−0.999722	−	0.0235744i	\(-0.992495\pi\)
							−0.0235744	−	0.999722i	\(-0.507505\pi\)
\(41\)	−74.2121			−1.81005			−0.905025	−	0.425358i	\(-0.860148\pi\)
							−0.905025	+	0.425358i	\(0.860148\pi\)
\(43\)	−42.3069	+	42.3069i	−0.983882	+	0.983882i	−0.999872	−	0.0159899i	\(-0.994910\pi\)
							0.0159899	+	0.999872i	\(0.494910\pi\)
\(47\)	39.4156	+	39.4156i	0.838630	+	0.838630i	0.988679	−	0.150049i	\(-0.0479431\pi\)
							−0.150049	+	0.988679i	\(0.547943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.167.1	yes	32
3.2	odd	2		210.3.k.b.167.8	yes	32
5.3	odd	4		210.3.k.b.83.9	yes	32
7.6	odd	2	inner	210.3.k.a.167.16	yes	32
15.8	even	4	inner	210.3.k.a.83.16	yes	32
21.20	even	2		210.3.k.b.167.9	yes	32
35.13	even	4		210.3.k.b.83.8	yes	32
105.83	odd	4	inner	210.3.k.a.83.1	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.1	✓	32	105.83	odd	4	inner
210.3.k.a.83.16	yes	32	15.8	even	4	inner
210.3.k.a.167.1	yes	32	1.1	even	1	trivial
210.3.k.a.167.16	yes	32	7.6	odd	2	inner
210.3.k.b.83.8	yes	32	35.13	even	4
210.3.k.b.83.9	yes	32	5.3	odd	4
210.3.k.b.167.8	yes	32	3.2	odd	2
210.3.k.b.167.9	yes	32	21.20	even	2