Embedded newform 103.2.e.a.100.1

• Label: 103.2.e.a.100.1
• 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			100.1
Character	\(\chi\)	\(=\)	103.100
Dual form			103.2.e.a.34.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88510 + 1.16721i) q^{2} +(0.0289090 + 0.101605i) q^{3} +(1.29977 - 2.61029i) q^{4} +(2.77975 - 1.07688i) q^{5} +(-0.173090 - 0.157793i) q^{6} +(-0.356063 + 3.84253i) q^{7} +(0.187391 + 2.02227i) q^{8} +(2.54116 - 1.57342i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88510	+	1.16721i	−1.33297	+	0.825340i	−0.993340	−	0.115219i	\(-0.963243\pi\)
							−0.339630	+	0.940559i	\(0.610302\pi\)
\(3\)	0.0289090	+	0.101605i	0.0166906	+	0.0586615i	0.969724	−	0.244205i	\(-0.0785270\pi\)
							−0.953033	+	0.302866i	\(0.902056\pi\)
\(5\)	2.77975	−	1.07688i	1.24314	−	0.481596i	0.352290	−	0.935891i	\(-0.385403\pi\)
							0.890851	+	0.454295i	\(0.150109\pi\)
\(7\)	−0.356063	+	3.84253i	−0.134579	+	1.45234i	0.614007	+	0.789301i	\(0.289557\pi\)
							−0.748586	+	0.663038i	\(0.769267\pi\)
\(11\)	−2.62870	+	1.62762i	−0.792583	+	0.490746i	−0.862041	−	0.506838i	\(-0.830814\pi\)
							0.0694584	+	0.997585i	\(0.477873\pi\)
\(13\)	−0.512930	+	5.53540i	−0.142261	+	1.53524i	0.563096	+	0.826391i	\(0.309610\pi\)
							−0.705357	+	0.708852i	\(0.749213\pi\)
\(17\)	4.16785	−	3.79950i	1.01085	−	0.921514i	0.0139297	−	0.999903i	\(-0.495566\pi\)
							0.996923	+	0.0783892i	\(0.0249777\pi\)
\(19\)	1.02164	−	3.59071i	0.234381	−	0.823765i	−0.751982	−	0.659183i	\(-0.770902\pi\)
							0.986363	−	0.164582i	\(-0.0526274\pi\)
\(23\)	−2.01897	−	1.25009i	−0.420984	−	0.260662i	0.299552	−	0.954080i	\(-0.403163\pi\)
							−0.720536	+	0.693418i	\(0.756104\pi\)
\(29\)	0.818020	−	0.316902i	0.151902	−	0.0588473i	−0.284090	−	0.958798i	\(-0.591692\pi\)
							0.435993	+	0.899950i	\(0.356397\pi\)
\(31\)	−3.23069	−	4.27813i	−0.580249	−	0.768374i	0.409326	−	0.912388i	\(-0.365764\pi\)
							−0.989576	+	0.144014i	\(0.953999\pi\)
\(37\)	−5.95494	−	1.11317i	−0.978987	−	0.183004i	−0.330147	−	0.943930i	\(-0.607098\pi\)
							−0.648840	+	0.760925i	\(0.724745\pi\)
\(41\)	−5.40058	−	2.09219i	−0.843428	−	0.326746i	−0.0995267	−	0.995035i	\(-0.531733\pi\)
							−0.743902	+	0.668289i	\(0.767027\pi\)
\(43\)	10.1001	−	1.88803i	1.54025	−	0.287922i	0.655936	−	0.754817i	\(-0.272274\pi\)
							0.884313	+	0.466894i	\(0.154627\pi\)
\(47\)	−5.92221			−0.863843			−0.431921	−	0.901911i	\(-0.642164\pi\)
							−0.431921	+	0.901911i	\(0.642164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.e.a.100.1	yes	112
3.2	odd	2		927.2.u.a.100.7		112
103.34	even	17	inner	103.2.e.a.34.1	✓	112
309.137	odd	34		927.2.u.a.343.7		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.e.a.34.1	✓	112	103.34	even	17	inner
103.2.e.a.100.1	yes	112	1.1	even	1	trivial
927.2.u.a.100.7		112	3.2	odd	2
927.2.u.a.343.7		112	309.137	odd	34