Embedded newform 103.2.e.a.100.3

• Label: 103.2.e.a.100.3
• 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			100.3
Character	\(\chi\)	\(=\)	103.100
Dual form			103.2.e.a.34.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12094 + 0.694055i) q^{2} +(0.934687 + 3.28508i) q^{3} +(-0.116689 + 0.234342i) q^{4} +(1.26875 - 0.491515i) q^{5} +(-3.32775 - 3.03365i) q^{6} +(0.179817 - 1.94053i) q^{7} +(-0.275141 - 2.96925i) q^{8} +(-7.36748 + 4.56175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12094	+	0.694055i	−0.792622	+	0.490771i	−0.862054	−	0.506816i	\(-0.830822\pi\)
							0.0694320	+	0.997587i	\(0.477881\pi\)
\(3\)	0.934687	+	3.28508i	0.539642	+	1.89664i	0.430427	+	0.902625i	\(0.358363\pi\)
							0.109214	+	0.994018i	\(0.465167\pi\)
\(5\)	1.26875	−	0.491515i	0.567401	−	0.219812i	−0.0604162	−	0.998173i	\(-0.519243\pi\)
							0.627817	+	0.778361i	\(0.283949\pi\)
\(7\)	0.179817	−	1.94053i	0.0679644	−	0.733453i	−0.891760	−	0.452509i	\(-0.850529\pi\)
							0.959724	−	0.280944i	\(-0.0906475\pi\)
\(11\)	3.98848	−	2.46956i	1.20257	−	0.744601i	0.229099	−	0.973403i	\(-0.426422\pi\)
							0.973473	+	0.228802i	\(0.0734808\pi\)
\(13\)	−0.157995	+	1.70503i	−0.0438198	+	0.472891i	0.945344	+	0.326075i	\(0.105726\pi\)
							−0.989164	+	0.146816i	\(0.953097\pi\)
\(17\)	0.580274	−	0.528990i	0.140737	−	0.128299i	−0.600266	−	0.799800i	\(-0.704939\pi\)
							0.741003	+	0.671502i	\(0.234350\pi\)
\(19\)	−0.370957	+	1.30378i	−0.0851034	+	0.299108i	−0.992679	−	0.120781i	\(-0.961460\pi\)
							0.907576	+	0.419888i	\(0.137931\pi\)
\(23\)	−3.48439	−	2.15744i	−0.726545	−	0.449858i	0.112687	−	0.993630i	\(-0.464054\pi\)
							−0.839232	+	0.543773i	\(0.816995\pi\)
\(29\)	8.85188	−	3.42924i	1.64375	−	0.636793i	0.651233	−	0.758878i	\(-0.274252\pi\)
							0.992520	+	0.122085i	\(0.0389579\pi\)
\(31\)	1.17469	+	1.55555i	0.210981	+	0.279384i	0.891186	−	0.453638i	\(-0.149874\pi\)
							−0.680205	+	0.733022i	\(0.738109\pi\)
\(37\)	2.04922	+	0.383065i	0.336889	+	0.0629755i	0.349476	−	0.936945i	\(-0.386360\pi\)
							−0.0125870	+	0.999921i	\(0.504007\pi\)
\(41\)	−4.55209	−	1.76349i	−0.710917	−	0.275411i	−0.0214934	−	0.999769i	\(-0.506842\pi\)
							−0.689424	+	0.724358i	\(0.742136\pi\)
\(43\)	−5.63012	+	1.05245i	−0.858586	+	0.160497i	−0.594593	−	0.804027i	\(-0.702687\pi\)
							−0.263993	+	0.964525i	\(0.585040\pi\)
\(47\)	0.973851			0.142051			0.0710254	−	0.997475i	\(-0.477373\pi\)
							0.0710254	+	0.997475i	\(0.477373\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.100.3	yes	112
3.2	odd	2		927.2.u.a.100.5		112
103.34	even	17	inner	103.2.e.a.34.3	✓	112
309.137	odd	34		927.2.u.a.343.5		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.34.3	✓	112	103.34	even	17	inner
103.2.e.a.100.3	yes	112	1.1	even	1	trivial
927.2.u.a.100.5		112	3.2	odd	2
927.2.u.a.343.5		112	309.137	odd	34