Embedded newform 210.3.k.b.83.5

• Label: 210.3.k.b.83.5
• 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			83.5
Character	\(\chi\)	\(=\)	210.83
Dual form			210.3.k.b.167.5

$q$-expansion

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

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
							\(3\)	−1.94039	−	2.28799i	−0.646795	−	0.762664i
\(5\)	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	\(-0.768922\pi\)
							0.747867	−	0.663849i	\(-0.231078\pi\)
\(13\)	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	\(-0.433278\pi\)
							0.978112	−	0.208081i	\(-0.0667219\pi\)
\(19\)	−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	−0.646419	−	0.762982i	\(-0.723734\pi\)
							0.762982	+	0.646419i	\(0.223734\pi\)
\(29\)	28.5159			0.983306			0.491653	−	0.870791i	\(-0.336393\pi\)
							0.491653	+	0.870791i	\(0.336393\pi\)
\(31\)			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	\(-0.653302\pi\)
							−0.463209	+	0.886249i	\(0.653302\pi\)
\(43\)	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	−0.894875	−	0.446317i	\(-0.852735\pi\)
							0.446317	+	0.894875i	\(0.352735\pi\)

(See \(a_n\) instead)

Twists

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

    

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