Embedded newform 129.2.j.a.2.10

• Label: 129.2.j.a.2.10
• Level: $129$
• Weight: $2$
• Character: 129.2
• Analytic conductor: $1.030$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03007018607\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	129.2
Dual form			129.2.j.a.65.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.44652 - 0.696607i) q^{2} +(0.786439 - 1.54322i) q^{3} +(0.360178 - 0.451649i) q^{4} +(-0.245120 + 1.07394i) q^{5} +(0.0625841 - 2.78013i) q^{6} -0.328256i q^{7} +(-0.508139 + 2.22630i) q^{8} +(-1.76303 - 2.42729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{3} - 18 q^{4} - 14 q^{6} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\)	\(44\)	\(46\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.44652	−	0.696607i	1.02284	−	0.492576i	0.154214	−	0.988037i	\(-0.450715\pi\)
							0.868630	+	0.495462i	\(0.165001\pi\)
\(3\)	0.786439	−	1.54322i	0.454051	−	0.890976i
							\(5\)	−0.245120	+	1.07394i	−0.109621	+	0.480281i	0.890080	+	0.455805i	\(0.150649\pi\)
−0.999700	+	0.0244753i	\(0.992208\pi\)
\(7\)		−	0.328256i		−	0.124069i	−0.998074	−	0.0620346i	\(-0.980241\pi\)
							0.998074	−	0.0620346i	\(-0.0197589\pi\)
\(11\)	−2.24313	+	1.78884i	−0.676330	+	0.539355i	−0.900316	−	0.435237i	\(-0.856664\pi\)
							0.223986	+	0.974592i	\(0.428093\pi\)
\(13\)	0.503434	−	2.20569i	0.139628	−	0.611748i	−0.855889	−	0.517160i	\(-0.826989\pi\)
							0.995516	−	0.0945885i	\(-0.0301535\pi\)
\(17\)	−2.54006	+	0.579753i	−0.616056	+	0.140611i	−0.519150	−	0.854683i	\(-0.673752\pi\)
							−0.0969054	+	0.995294i	\(0.530894\pi\)
\(19\)	0.684905	+	0.546193i	0.157128	+	0.125305i	0.698898	−	0.715221i	\(-0.253674\pi\)
							−0.541770	+	0.840527i	\(0.682246\pi\)
\(23\)	0.977772	−	0.779747i	0.203880	−	0.162589i	−0.516227	−	0.856452i	\(-0.672664\pi\)
							0.720107	+	0.693863i	\(0.244093\pi\)
\(29\)	7.99114	−	3.84833i	1.48392	−	0.714617i	0.495817	−	0.868427i	\(-0.334869\pi\)
							0.988100	+	0.153810i	\(0.0491544\pi\)
\(31\)	−3.81550	+	1.83745i	−0.685285	+	0.330016i	−0.743925	−	0.668263i	\(-0.767038\pi\)
							0.0586400	+	0.998279i	\(0.481324\pi\)
\(37\)		−	3.53596i		−	0.581308i	−0.956828	−	0.290654i	\(-0.906127\pi\)
							0.956828	−	0.290654i	\(-0.0938729\pi\)
\(41\)	−4.20677	−	8.73545i	−0.656987	−	1.36425i	−0.917095	−	0.398669i	\(-0.869472\pi\)
							0.260108	−	0.965580i	\(-0.416242\pi\)
\(43\)	−4.76502	+	4.50495i	−0.726658	+	0.686999i
							\(47\)	−4.75344	−	3.79074i	−0.693360	−	0.552936i	0.212163	−	0.977234i	\(-0.431949\pi\)
−0.905522	+	0.424298i	\(0.860521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.j.a.2.10	yes	72
3.2	odd	2	inner	129.2.j.a.2.3	✓	72
43.22	odd	14	inner	129.2.j.a.65.3	yes	72
129.65	even	14	inner	129.2.j.a.65.10	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.j.a.2.3	✓	72	3.2	odd	2	inner
129.2.j.a.2.10	yes	72	1.1	even	1	trivial
129.2.j.a.65.3	yes	72	43.22	odd	14	inner
129.2.j.a.65.10	yes	72	129.65	even	14	inner