Embedded newform 103.2.e.a.14.7

• Label: 103.2.e.a.14.7
• 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			14.7
Character	\(\chi\)	\(=\)	103.14
Dual form			103.2.e.a.81.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29292 - 1.71210i) q^{2} +(-0.635278 - 1.27581i) q^{3} +(-0.712323 - 2.50355i) q^{4} +(0.176519 + 1.90495i) q^{5} +(-3.00567 - 0.561858i) q^{6} +(-0.635064 + 0.246025i) q^{7} +(-1.20618 - 0.467276i) q^{8} +(0.583791 - 0.773065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29292	−	1.71210i	0.914231	−	1.21064i	−0.0629049	−	0.998020i	\(-0.520036\pi\)
							0.977136	−	0.212617i	\(-0.0681988\pi\)
\(3\)	−0.635278	−	1.27581i	−0.366778	−	0.736589i	0.632501	−	0.774559i	\(-0.282028\pi\)
							−0.999279	+	0.0379704i	\(0.987911\pi\)
\(5\)	0.176519	+	1.90495i	0.0789419	+	0.851919i	0.939496	+	0.342560i	\(0.111294\pi\)
							−0.860554	+	0.509359i	\(0.829883\pi\)
\(7\)	−0.635064	+	0.246025i	−0.240032	+	0.0929887i	−0.478253	−	0.878222i	\(-0.658730\pi\)
							0.238221	+	0.971211i	\(0.423436\pi\)
\(11\)	−2.22569	+	2.94729i	−0.671070	+	0.888640i	−0.998416	−	0.0562645i	\(-0.982081\pi\)
							0.327346	+	0.944904i	\(0.393846\pi\)
\(13\)	−2.07766	+	0.804890i	−0.576239	+	0.223236i	−0.631685	−	0.775225i	\(-0.717636\pi\)
							0.0554454	+	0.998462i	\(0.482342\pi\)
\(17\)	5.10596	−	0.954468i	1.23838	−	0.231493i	0.476476	−	0.879187i	\(-0.341914\pi\)
							0.761900	+	0.647695i	\(0.224267\pi\)
\(19\)	−1.60254	+	3.21834i	−0.367648	+	0.738337i	−0.999319	−	0.0368995i	\(-0.988252\pi\)
							0.631671	+	0.775237i	\(0.282370\pi\)
\(23\)	−2.48301	−	3.28804i	−0.517744	−	0.685604i	0.461766	−	0.887002i	\(-0.347216\pi\)
							−0.979509	+	0.201398i	\(0.935451\pi\)
\(29\)	−0.144067	−	1.55473i	−0.0267525	−	0.288705i	−0.998591	−	0.0530591i	\(-0.983103\pi\)
							0.971839	−	0.235646i	\(-0.0757207\pi\)
\(31\)	−5.08077	+	3.14588i	−0.912532	+	0.565016i	−0.900473	−	0.434911i	\(-0.856780\pi\)
							−0.0120591	+	0.999927i	\(0.503839\pi\)
\(37\)	−6.89052	+	6.28154i	−1.13279	+	1.03268i	−0.133662	+	0.991027i	\(0.542674\pi\)
							−0.999132	+	0.0416514i	\(0.986738\pi\)
\(41\)	−0.265465	+	2.86482i	−0.0414586	+	0.447410i	0.949550	+	0.313615i	\(0.101540\pi\)
							−0.991009	+	0.133795i	\(0.957284\pi\)
\(43\)	−3.68983	−	3.36372i	−0.562693	−	0.512963i	0.341616	−	0.939839i	\(-0.389026\pi\)
							−0.904310	+	0.426877i	\(0.859614\pi\)
\(47\)	0.100444			0.0146513			0.00732563	−	0.999973i	\(-0.497668\pi\)
							0.00732563	+	0.999973i	\(0.497668\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.14.7	✓	112
3.2	odd	2		927.2.u.a.838.1		112
103.81	even	17	inner	103.2.e.a.81.7	yes	112
309.287	odd	34		927.2.u.a.802.1		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.14.7	✓	112	1.1	even	1	trivial
103.2.e.a.81.7	yes	112	103.81	even	17	inner
927.2.u.a.802.1		112	309.287	odd	34
927.2.u.a.838.1		112	3.2	odd	2