Embedded newform 102.3.g.a.83.6

• Label: 102.3.g.a.83.6
• Level: $102$
• Weight: $3$
• Character: 102.83
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	102.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77929869648\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			83.6
Character	\(\chi\)	\(=\)	102.83
Dual form			102.3.g.a.59.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(2.79886 - 1.07998i) q^{3} -2.00000i q^{4} +(2.45376 + 5.92389i) q^{5} +(-1.71888 + 3.87885i) q^{6} +(-2.42737 - 1.00545i) q^{7} +(2.00000 + 2.00000i) q^{8} +(6.66728 - 6.04544i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 4 q^{3} + 8 q^{5} - 4 q^{6} + 48 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)

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
							\(3\)	2.79886	−	1.07998i	0.932955	−	0.359994i
\(5\)	2.45376	+	5.92389i	0.490752	+	1.18478i								0.954338	+	0.298728i	\(0.0965622\pi\)
							−0.463587	+	0.886051i	\(0.653438\pi\)
\(7\)	−2.42737	−	1.00545i	−0.346766	−	0.143635i	0.202501	−	0.979282i	\(-0.435093\pi\)
							−0.549267	+	0.835647i	\(0.685093\pi\)
\(11\)	14.2208	+	5.89045i	1.29280	+	0.535496i	0.919819	−	0.392343i	\(-0.128335\pi\)
							0.372982	+	0.927839i	\(0.378335\pi\)
\(13\)		−	3.81184i		−	0.293218i	−0.989195	−	0.146609i	\(-0.953164\pi\)
							0.989195	−	0.146609i	\(-0.0468359\pi\)
\(17\)	−15.4293	+	7.13697i	−0.907607	+	0.419822i
							\(19\)	9.95630	+	9.95630i	0.524016	+	0.524016i	0.918782	−	0.394766i	\(-0.129174\pi\)
−0.394766	+	0.918782i	\(0.629174\pi\)
\(23\)	−11.4276	−	4.73345i	−0.496850	−	0.205802i	0.120164	−	0.992754i	\(-0.461658\pi\)
							−0.617014	+	0.786952i	\(0.711658\pi\)
\(29\)	−15.7748	−	38.0836i	−0.543957	−	1.31323i	−0.921910	−	0.387404i	\(-0.873372\pi\)
							0.377953	−	0.925825i	\(-0.376628\pi\)
\(31\)	−10.0183	−	24.1863i	−0.323171	−	0.780204i	−0.999066	−	0.0432061i	\(-0.986243\pi\)
							0.675895	−	0.736998i	\(-0.263757\pi\)
\(37\)	−19.7992	−	47.7996i	−0.535115	−	1.29188i	−0.928098	−	0.372336i	\(-0.878557\pi\)
							0.392983	−	0.919546i	\(-0.371443\pi\)
\(41\)	−24.3029	+	58.6725i	−0.592754	+	1.43104i	0.288078	+	0.957607i	\(0.406984\pi\)
							−0.880832	+	0.473429i	\(0.843016\pi\)
\(43\)	−4.47973	+	4.47973i	−0.104180	+	0.104180i	−0.757275	−	0.653096i	\(-0.773470\pi\)
							0.653096	+	0.757275i	\(0.273470\pi\)
\(47\)	−71.4076			−1.51931			−0.759656	−	0.650325i	\(-0.774633\pi\)
							−0.759656	+	0.650325i	\(0.774633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.3.g.a.83.6	yes	24
3.2	odd	2		102.3.g.b.83.1	yes	24
17.8	even	8		102.3.g.b.59.1	yes	24
51.8	odd	8	inner	102.3.g.a.59.6	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.3.g.a.59.6	✓	24	51.8	odd	8	inner
102.3.g.a.83.6	yes	24	1.1	even	1	trivial
102.3.g.b.59.1	yes	24	17.8	even	8
102.3.g.b.83.1	yes	24	3.2	odd	2