Embedded newform 210.3.k.a.83.13

• Label: 210.3.k.a.83.13
• Level: $210$
• Weight: $3$
• Character: 210.83
• 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			83.13
Character	\(\chi\)	\(=\)	210.83
Dual form			210.3.k.a.167.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +(2.28799 + 1.94039i) q^{3} +2.00000i q^{4} +(-4.58449 + 1.99562i) q^{5} +(-0.347606 - 4.22838i) q^{6} +(-4.23091 - 5.57668i) q^{7} +(2.00000 - 2.00000i) q^{8} +(1.46981 + 8.87917i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{2} - 4 q^{7} + 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.00000i	−0.500000	−	0.500000i
							\(3\)	2.28799	+	1.94039i	0.762664	+	0.646795i
\(5\)	−4.58449	+	1.99562i	−0.916897	+	0.399123i
							\(7\)	−4.23091	−	5.57668i	−0.604416	−	0.796669i
\(11\)			14.6047i			1.32770i								0.747867	+	0.663849i	\(0.231078\pi\)
							−0.747867	+	0.663849i	\(0.768922\pi\)
\(13\)	3.48167	−	3.48167i	0.267821	−	0.267821i	−0.560401	−	0.828222i	\(-0.689353\pi\)
							0.828222	+	0.560401i	\(0.189353\pi\)
\(17\)	−20.1653	+	20.1653i	−1.18619	+	1.18619i	−0.208081	+	0.978112i	\(0.566722\pi\)
							−0.978112	+	0.208081i	\(0.933278\pi\)
\(19\)	−26.4132			−1.39017			−0.695085	−	0.718928i	\(-0.744633\pi\)
							−0.695085	+	0.718928i	\(0.744633\pi\)
\(23\)	−2.68095	+	2.68095i	−0.116563	+	0.116563i	−0.762982	−	0.646419i	\(-0.776266\pi\)
							0.646419	+	0.762982i	\(0.276266\pi\)
\(29\)	−28.5159			−0.983306			−0.491653	−	0.870791i	\(-0.663607\pi\)
							−0.491653	+	0.870791i	\(0.663607\pi\)
\(31\)			15.6511i			0.504875i	0.967613	+	0.252438i	\(0.0812322\pi\)
							−0.967613	+	0.252438i	\(0.918768\pi\)
\(37\)	7.69844	−	7.69844i	0.208066	−	0.208066i	−0.595379	−	0.803445i	\(-0.702998\pi\)
							0.803445	+	0.595379i	\(0.202998\pi\)
\(41\)	37.9832			0.926419			0.463209	−	0.886249i	\(-0.346698\pi\)
							0.463209	+	0.886249i	\(0.346698\pi\)
\(43\)	41.7817	+	41.7817i	0.971667	+	0.971667i	0.999610	−	0.0279426i	\(-0.00889555\pi\)
							−0.0279426	+	0.999610i	\(0.508896\pi\)
\(47\)	21.0822	−	21.0822i	0.448558	−	0.448558i	−0.446317	−	0.894875i	\(-0.647265\pi\)
							0.894875	+	0.446317i	\(0.147265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.k.a.83.13	yes	32
3.2	odd	2		210.3.k.b.83.5	yes	32
5.2	odd	4		210.3.k.b.167.12	yes	32
7.6	odd	2	inner	210.3.k.a.83.4	✓	32
15.2	even	4	inner	210.3.k.a.167.4	yes	32
21.20	even	2		210.3.k.b.83.12	yes	32
35.27	even	4		210.3.k.b.167.5	yes	32
105.62	odd	4	inner	210.3.k.a.167.13	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.k.a.83.4	✓	32	7.6	odd	2	inner
210.3.k.a.83.13	yes	32	1.1	even	1	trivial
210.3.k.a.167.4	yes	32	15.2	even	4	inner
210.3.k.a.167.13	yes	32	105.62	odd	4	inner
210.3.k.b.83.5	yes	32	3.2	odd	2
210.3.k.b.83.12	yes	32	21.20	even	2
210.3.k.b.167.5	yes	32	35.27	even	4
210.3.k.b.167.12	yes	32	5.2	odd	4