Embedded newform 210.3.k.a.167.11

• Label: 210.3.k.a.167.11
• 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			167.11
Character	\(\chi\)	\(=\)	210.167
Dual form			210.3.k.a.83.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.21742 + 2.74188i) q^{3} -2.00000i q^{4} +(-3.32079 - 3.73796i) q^{5} +(-3.95930 - 1.52446i) q^{6} +(2.29543 - 6.61294i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-6.03579 + 6.67602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	1.21742	+	2.74188i	0.405806	+	0.913959i
\(5\)	−3.32079	−	3.73796i	−0.664159	−	0.747592i
							\(7\)	2.29543	−	6.61294i	0.327919	−	0.944706i
\(11\)		−	10.5733i		−	0.961209i								−0.876938	−	0.480604i	\(-0.840417\pi\)
							0.876938	−	0.480604i	\(-0.159583\pi\)
\(13\)	−14.9899	−	14.9899i	−1.15307	−	1.15307i	−0.985934	−	0.167134i	\(-0.946549\pi\)
							−0.167134	−	0.985934i	\(-0.553451\pi\)
\(17\)	−15.4921	−	15.4921i	−0.911302	−	0.911302i	0.0850728	−	0.996375i	\(-0.472888\pi\)
							−0.996375	+	0.0850728i	\(0.972888\pi\)
\(19\)	17.3342			0.912329			0.456164	−	0.889896i	\(-0.349223\pi\)
							0.456164	+	0.889896i	\(0.349223\pi\)
\(23\)	23.1753	+	23.1753i	1.00762	+	1.00762i	0.999971	+	0.00765214i	\(0.00243578\pi\)
							0.00765214	+	0.999971i	\(0.497564\pi\)
\(29\)	23.7038			0.817373			0.408686	−	0.912675i	\(-0.365987\pi\)
							0.408686	+	0.912675i	\(0.365987\pi\)
\(31\)		−	33.1422i		−	1.06910i	−0.845136	−	0.534551i	\(-0.820481\pi\)
							0.845136	−	0.534551i	\(-0.179519\pi\)
\(37\)	−17.6315	−	17.6315i	−0.476526	−	0.476526i	0.427492	−	0.904019i	\(-0.359397\pi\)
							−0.904019	+	0.427492i	\(0.859397\pi\)
\(41\)	11.8368			0.288703			0.144352	−	0.989526i	\(-0.453890\pi\)
							0.144352	+	0.989526i	\(0.453890\pi\)
\(43\)	−22.8095	+	22.8095i	−0.530453	+	0.530453i	−0.920707	−	0.390254i	\(-0.872387\pi\)
							0.390254	+	0.920707i	\(0.372387\pi\)
\(47\)	−12.6291	−	12.6291i	−0.268704	−	0.268704i	0.559874	−	0.828578i	\(-0.310850\pi\)
							−0.828578	+	0.559874i	\(0.810850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.167.11	yes	32
3.2	odd	2		210.3.k.b.167.14	yes	32
5.3	odd	4		210.3.k.b.83.3	yes	32
7.6	odd	2	inner	210.3.k.a.167.6	yes	32
15.8	even	4	inner	210.3.k.a.83.6	✓	32
21.20	even	2		210.3.k.b.167.3	yes	32
35.13	even	4		210.3.k.b.83.14	yes	32
105.83	odd	4	inner	210.3.k.a.83.11	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.6	✓	32	15.8	even	4	inner
210.3.k.a.83.11	yes	32	105.83	odd	4	inner
210.3.k.a.167.6	yes	32	7.6	odd	2	inner
210.3.k.a.167.11	yes	32	1.1	even	1	trivial
210.3.k.b.83.3	yes	32	5.3	odd	4
210.3.k.b.83.14	yes	32	35.13	even	4
210.3.k.b.167.3	yes	32	21.20	even	2
210.3.k.b.167.14	yes	32	3.2	odd	2