Embedded newform 103.2.e.a.100.4

• Label: 103.2.e.a.100.4
• Level: $103$
• Weight: $2$
• Character: 103.100
• Analytic conductor: $0.822$
• Analytic rank: $0$
• Dimension: $112$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	103.e (of order \(17\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.822459140819\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(7\) over \(\Q(\zeta_{17})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label			100.4
Character	\(\chi\)	\(=\)	103.100
Dual form			103.2.e.a.34.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.380353 + 0.235505i) q^{2} +(0.0966754 + 0.339779i) q^{3} +(-0.802271 + 1.61118i) q^{4} +(-3.76014 + 1.45669i) q^{5} +(-0.116790 - 0.106468i) q^{6} +(-0.225954 + 2.43843i) q^{7} +(-0.156848 - 1.69266i) q^{8} +(2.44455 - 1.51360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{15}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.380353	+	0.235505i	−0.268950	+	0.166527i	−0.654281	−	0.756252i	\(-0.727029\pi\)
							0.385331	+	0.922779i	\(0.374087\pi\)
\(3\)	0.0966754	+	0.339779i	0.0558156	+	0.196171i	0.984719	−	0.174151i	\(-0.0557179\pi\)
							−0.928903	+	0.370322i	\(0.879247\pi\)
\(5\)	−3.76014	+	1.45669i	−1.68159	+	0.651450i	−0.996994	−	0.0774773i	\(-0.975313\pi\)
							−0.684591	+	0.728927i	\(0.740019\pi\)
\(7\)	−0.225954	+	2.43843i	−0.0854026	+	0.921641i	0.839983	+	0.542613i	\(0.182565\pi\)
							−0.925385	+	0.379028i	\(0.876259\pi\)
\(11\)	3.61018	−	2.23533i	1.08851	−	0.673977i	0.139817	−	0.990177i	\(-0.455349\pi\)
							0.948692	+	0.316201i	\(0.102407\pi\)
\(13\)	−0.474581	+	5.12155i	−0.131625	+	1.42046i	0.632083	+	0.774901i	\(0.282200\pi\)
							−0.763708	+	0.645561i	\(0.776623\pi\)
\(17\)	0.848809	−	0.773792i	0.205866	−	0.187672i	−0.564195	−	0.825641i	\(-0.690814\pi\)
							0.770062	+	0.637969i	\(0.220225\pi\)
\(19\)	−1.09205	+	3.83817i	−0.250534	+	0.880537i	0.729877	+	0.683578i	\(0.239577\pi\)
							−0.980412	+	0.196959i	\(0.936893\pi\)
\(23\)	3.44277	+	2.13167i	0.717866	+	0.444484i	0.836154	−	0.548495i	\(-0.184799\pi\)
							−0.118287	+	0.992979i	\(0.537740\pi\)
\(29\)	−4.69630	+	1.81936i	−0.872081	+	0.337846i	−0.755353	−	0.655318i	\(-0.772535\pi\)
							−0.116728	+	0.993164i	\(0.537241\pi\)
\(31\)	−3.56240	−	4.71738i	−0.639825	−	0.847266i	0.356376	−	0.934343i	\(-0.384012\pi\)
							−0.996202	+	0.0870768i	\(0.972247\pi\)
\(37\)	7.01974	+	1.31222i	1.15404	+	0.215727i	0.725778	−	0.687929i	\(-0.241480\pi\)
							0.428260	+	0.903656i	\(0.359127\pi\)
\(41\)	1.75968	+	0.681704i	0.274816	+	0.106464i	0.494706	−	0.869060i	\(-0.335276\pi\)
							−0.219890	+	0.975525i	\(0.570570\pi\)
\(43\)	−3.78714	+	0.707939i	−0.577533	+	0.107960i	−0.464408	−	0.885621i	\(-0.653733\pi\)
							−0.113125	+	0.993581i	\(0.536086\pi\)
\(47\)	−2.72376			−0.397301			−0.198650	−	0.980070i	\(-0.563656\pi\)
							−0.198650	+	0.980070i	\(0.563656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.100.4	yes	112
3.2	odd	2		927.2.u.a.100.4		112
103.34	even	17	inner	103.2.e.a.34.4	✓	112
309.137	odd	34		927.2.u.a.343.4		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.34.4	✓	112	103.34	even	17	inner
103.2.e.a.100.4	yes	112	1.1	even	1	trivial
927.2.u.a.100.4		112	3.2	odd	2
927.2.u.a.343.4		112	309.137	odd	34