Embedded newform 103.2.e.a.14.3

• Label: 103.2.e.a.14.3
• Level: $103$
• Weight: $2$
• Character: 103.14
• 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			14.3
Character	\(\chi\)	\(=\)	103.14
Dual form			103.2.e.a.81.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.512089 + 0.678116i) q^{2} +(0.835617 + 1.67814i) q^{3} +(0.349720 + 1.22914i) q^{4} +(-0.121677 - 1.31310i) q^{5} +(-1.56589 - 0.292715i) q^{6} +(-0.469286 + 0.181802i) q^{7} +(-2.59733 - 1.00621i) q^{8} +(-0.310011 + 0.410521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{8}{17}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.512089	+	0.678116i	−0.362102	+	0.479500i	−0.942202	−	0.335045i	\(-0.891249\pi\)
							0.580100	+	0.814545i	\(0.303013\pi\)
\(3\)	0.835617	+	1.67814i	0.482444	+	0.968877i	0.993974	+	0.109612i	\(0.0349610\pi\)
							−0.511531	+	0.859265i	\(0.670921\pi\)
\(5\)	−0.121677	−	1.31310i	−0.0544156	−	0.587238i	−0.978667	−	0.205453i	\(-0.934133\pi\)
							0.924251	−	0.381785i	\(-0.124690\pi\)
\(7\)	−0.469286	+	0.181802i	−0.177373	+	0.0687148i	−0.448291	−	0.893888i	\(-0.647967\pi\)
							0.270917	+	0.962603i	\(0.412673\pi\)
\(11\)	−1.17175	+	1.55165i	−0.353296	+	0.467839i	−0.939611	−	0.342244i	\(-0.888813\pi\)
							0.586316	+	0.810083i	\(0.300578\pi\)
\(13\)	3.25380	−	1.26053i	0.902441	−	0.349607i	0.135089	−	0.990833i	\(-0.456868\pi\)
							0.767352	+	0.641226i	\(0.221574\pi\)
\(17\)	3.46901	−	0.648471i	0.841360	−	0.157277i	0.254608	−	0.967044i	\(-0.418054\pi\)
							0.586752	+	0.809767i	\(0.300407\pi\)
\(19\)	−0.388558	+	0.780330i	−0.0891413	+	0.179020i	−0.935248	−	0.353994i	\(-0.884823\pi\)
							0.846106	+	0.533014i	\(0.178941\pi\)
\(23\)	−4.38406	−	5.80543i	−0.914139	−	1.21052i	−0.977160	−	0.212504i	\(-0.931838\pi\)
							0.0630209	−	0.998012i	\(-0.479927\pi\)
\(29\)	−0.446324	−	4.81660i	−0.0828802	−	0.894420i	−0.931097	−	0.364771i	\(-0.881147\pi\)
							0.848217	−	0.529649i	\(-0.177676\pi\)
\(31\)	−5.11379	+	3.16632i	−0.918464	+	0.568689i	−0.902266	−	0.431180i	\(-0.858097\pi\)
							−0.0161977	+	0.999869i	\(0.505156\pi\)
\(37\)	0.511085	−	0.465916i	0.0840219	−	0.0765961i	−0.630595	−	0.776112i	\(-0.717189\pi\)
							0.714617	+	0.699516i	\(0.246601\pi\)
\(41\)	−0.120661	+	1.30214i	−0.0188441	+	0.203361i	0.981107	+	0.193466i	\(0.0619730\pi\)
							−0.999951	+	0.00989423i	\(0.996851\pi\)
\(43\)	3.31514	+	3.02215i	0.505554	+	0.460873i	0.885837	−	0.463997i	\(-0.153585\pi\)
							−0.380283	+	0.924870i	\(0.624173\pi\)
\(47\)	−2.32599			−0.339280			−0.169640	−	0.985506i	\(-0.554261\pi\)
							−0.169640	+	0.985506i	\(0.554261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.14.3	✓	112
3.2	odd	2		927.2.u.a.838.5		112
103.81	even	17	inner	103.2.e.a.81.3	yes	112
309.287	odd	34		927.2.u.a.802.5		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.14.3	✓	112	1.1	even	1	trivial
103.2.e.a.81.3	yes	112	103.81	even	17	inner
927.2.u.a.802.5		112	309.287	odd	34
927.2.u.a.838.5		112	3.2	odd	2