Embedded newform 103.2.e.a.14.1

• Label: 103.2.e.a.14.1
• 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			14.1
Character	\(\chi\)	\(=\)	103.14
Dual form			103.2.e.a.81.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48212 + 1.96265i) q^{2} +(0.209968 + 0.421673i) q^{3} +(-1.10797 - 3.89411i) q^{4} +(0.264996 + 2.85977i) q^{5} +(-1.13879 - 0.212877i) q^{6} +(-4.43929 + 1.71979i) q^{7} +(4.69827 + 1.82012i) q^{8} +(1.67418 - 2.21698i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48212	+	1.96265i	−1.04802	+	1.38780i	−0.129886	+	0.991529i	\(0.541461\pi\)
							−0.918133	+	0.396272i	\(0.870304\pi\)
\(3\)	0.209968	+	0.421673i	0.121225	+	0.243453i	0.947461	−	0.319871i	\(-0.103640\pi\)
							−0.826236	+	0.563325i	\(0.809522\pi\)
\(5\)	0.264996	+	2.85977i	0.118510	+	1.27893i	0.822520	+	0.568736i	\(0.192568\pi\)
							−0.704010	+	0.710190i	\(0.748609\pi\)
\(7\)	−4.43929	+	1.71979i	−1.67789	+	0.650019i	−0.996631	−	0.0820203i	\(-0.973863\pi\)
							−0.681262	+	0.732039i	\(0.738569\pi\)
\(11\)	−0.873595	+	1.15683i	−0.263399	+	0.348796i	−0.910427	−	0.413670i	\(-0.864247\pi\)
							0.647028	+	0.762466i	\(0.276012\pi\)
\(13\)	1.46749	−	0.568507i	0.407007	−	0.157675i	−0.149030	−	0.988833i	\(-0.547615\pi\)
							0.556038	+	0.831157i	\(0.312321\pi\)
\(17\)	0.891641	−	0.166677i	0.216255	−	0.0404250i	−0.0745071	−	0.997220i	\(-0.523738\pi\)
							0.290762	+	0.956795i	\(0.406091\pi\)
\(19\)	−2.80269	+	5.62857i	−0.642982	+	1.29128i	0.298562	+	0.954390i	\(0.403493\pi\)
							−0.941544	+	0.336891i	\(0.890625\pi\)
\(23\)	5.10134	+	6.75526i	1.06370	+	1.40857i	0.907104	+	0.420907i	\(0.138288\pi\)
							0.156598	+	0.987662i	\(0.449947\pi\)
\(29\)	0.0262116	+	0.282869i	0.00486738	+	0.0525274i	0.997821	−	0.0659726i	\(-0.0210150\pi\)
							−0.992954	+	0.118500i	\(0.962191\pi\)
\(31\)	7.39541	−	4.57905i	1.32826	−	0.822421i	0.335438	−	0.942062i	\(-0.391116\pi\)
							0.992817	+	0.119641i	\(0.0381744\pi\)
\(37\)	−1.41337	+	1.28845i	−0.232356	+	0.211820i	−0.781497	−	0.623910i	\(-0.785543\pi\)
							0.549141	+	0.835730i	\(0.314955\pi\)
\(41\)	0.426906	−	4.60705i	0.0666716	−	0.719501i	−0.895139	−	0.445786i	\(-0.852924\pi\)
							0.961811	−	0.273715i	\(-0.0882524\pi\)
\(43\)	−1.97387	−	1.79942i	−0.301013	−	0.274409i	0.509070	−	0.860725i	\(-0.329990\pi\)
							−0.810083	+	0.586316i	\(0.800578\pi\)
\(47\)	4.87906			0.711684			0.355842	−	0.934546i	\(-0.384194\pi\)
							0.355842	+	0.934546i	\(0.384194\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.14.1	✓	112
3.2	odd	2		927.2.u.a.838.7		112
103.81	even	17	inner	103.2.e.a.81.1	yes	112
309.287	odd	34		927.2.u.a.802.7		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.14.1	✓	112	1.1	even	1	trivial
103.2.e.a.81.1	yes	112	103.81	even	17	inner
927.2.u.a.802.7		112	309.287	odd	34
927.2.u.a.838.7		112	3.2	odd	2