Embedded newform 103.2.e.a.13.7

• Label: 103.2.e.a.13.7
• Level: $103$
• Weight: $2$
• Character: 103.13
• 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			13.7
Character	\(\chi\)	\(=\)	103.13
Dual form			103.2.e.a.8.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.80079 + 0.336626i) q^{2} +(0.0591724 - 0.638572i) q^{3} +(1.26458 + 0.489899i) q^{4} +(-0.551389 + 0.730157i) q^{5} +(0.321517 - 1.13001i) q^{6} +(-0.608782 + 0.376942i) q^{7} +(-1.00284 - 0.620934i) q^{8} +(2.54465 + 0.475677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{12}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.80079	+	0.336626i	1.27335	+	0.238030i	0.776672	−	0.629905i	\(-0.216906\pi\)
							0.496677	+	0.867935i	\(0.334553\pi\)
\(3\)	0.0591724	−	0.638572i	0.0341632	−	0.368680i	−0.961440	−	0.275013i	\(-0.911318\pi\)
							0.995604	−	0.0936664i	\(-0.0298587\pi\)
\(5\)	−0.551389	+	0.730157i	−0.246589	+	0.326536i	−0.904431	−	0.426619i	\(-0.859704\pi\)
							0.657843	+	0.753155i	\(0.271469\pi\)
\(7\)	−0.608782	+	0.376942i	−0.230098	+	0.142471i	−0.636639	−	0.771162i	\(-0.719676\pi\)
							0.406541	+	0.913632i	\(0.366735\pi\)
\(11\)	−4.30417	−	0.804588i	−1.29776	−	0.242593i	−0.510852	−	0.859669i	\(-0.670670\pi\)
							−0.786903	+	0.617076i	\(0.788317\pi\)
\(13\)	−0.590629	+	0.365702i	−0.163811	+	0.101428i	−0.605889	−	0.795549i	\(-0.707183\pi\)
							0.442078	+	0.896977i	\(0.354241\pi\)
\(17\)	0.124467	+	0.437455i	0.0301876	+	0.106098i	0.975455	−	0.220197i	\(-0.0706701\pi\)
							−0.945268	+	0.326296i	\(0.894200\pi\)
\(19\)	0.187174	+	2.01993i	0.0429406	+	0.463403i	0.989872	+	0.141966i	\(0.0453424\pi\)
							−0.946931	+	0.321437i	\(0.895834\pi\)
\(23\)	1.85403	−	0.346579i	0.386592	−	0.0722666i	0.0131343	−	0.999914i	\(-0.495819\pi\)
							0.373458	+	0.927647i	\(0.378172\pi\)
\(29\)	3.61364	−	4.78524i	0.671036	−	0.888596i	−0.327378	−	0.944894i	\(-0.606165\pi\)
							0.998414	+	0.0562977i	\(0.0179296\pi\)
\(31\)	2.66344	+	2.42804i	0.478367	+	0.436089i	0.876618	−	0.481187i	\(-0.159794\pi\)
							−0.398250	+	0.917277i	\(0.630383\pi\)
\(37\)	4.15165	−	8.33765i	0.682528	−	1.37070i	−0.234594	−	0.972093i	\(-0.575376\pi\)
							0.917122	−	0.398607i	\(-0.130506\pi\)
\(41\)	1.58181	+	2.09466i	0.247037	+	0.327130i	0.904594	−	0.426275i	\(-0.140174\pi\)
							−0.657556	+	0.753406i	\(0.728410\pi\)
\(43\)	0.342733	+	0.688301i	0.0522663	+	0.104965i	0.919757	−	0.392489i	\(-0.128386\pi\)
							−0.867490	+	0.497454i	\(0.834268\pi\)
\(47\)	−4.36803			−0.637142			−0.318571	−	0.947899i	\(-0.603203\pi\)
							−0.318571	+	0.947899i	\(0.603203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.13.7	yes	112
3.2	odd	2		927.2.u.a.631.1		112
103.8	even	17	inner	103.2.e.a.8.7	✓	112
309.8	odd	34		927.2.u.a.523.1		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.8.7	✓	112	103.8	even	17	inner
103.2.e.a.13.7	yes	112	1.1	even	1	trivial
927.2.u.a.523.1		112	309.8	odd	34
927.2.u.a.631.1		112	3.2	odd	2