Embedded newform 103.2.g.a.17.2

• Label: 103.2.g.a.17.2
• Level: $103$
• Weight: $2$
• Character: 103.17
• Analytic conductor: $0.822$
• Analytic rank: $0$
• Dimension: $256$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	103.g (of order \(51\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.822459140819\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(8\) over \(\Q(\zeta_{51})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{51}]$

Embedding invariants

Embedding label			17.2
Character	\(\chi\)	\(=\)	103.17
Dual form			103.2.g.a.97.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67700 - 1.18713i) q^{2} +(-1.91316 + 1.74408i) q^{3} +(0.738362 + 2.09532i) q^{4} +(0.374374 - 0.564986i) q^{5} +(5.27881 - 0.653659i) q^{6} +(3.47745 - 3.58624i) q^{7} +(0.124606 - 0.437946i) q^{8} +(0.341576 - 3.68619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{35}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67700	−	1.18713i	−1.18582	−	0.839425i	−0.195841	−	0.980636i	\(-0.562744\pi\)
							−0.989980	+	0.141211i	\(0.954900\pi\)
\(3\)	−1.91316	+	1.74408i	−1.10456	+	1.00694i	−0.104641	+	0.994510i	\(0.533369\pi\)
							−0.999923	+	0.0124326i	\(0.996042\pi\)
\(5\)	0.374374	−	0.564986i	0.167425	−	0.252670i	−0.740261	−	0.672320i	\(-0.765298\pi\)
							0.907686	+	0.419650i	\(0.137847\pi\)
\(7\)	3.47745	−	3.58624i	1.31435	−	1.35547i	0.420379	−	0.907349i	\(-0.361897\pi\)
							0.893973	−	0.448121i	\(-0.147907\pi\)
\(11\)	3.13674	−	1.44313i	0.945761	−	0.435119i	0.116007	−	0.993248i	\(-0.462990\pi\)
							0.829754	+	0.558129i	\(0.188481\pi\)
\(13\)	0.578262	+	2.03238i	0.160381	+	0.563681i	0.999796	+	0.0201901i	\(0.00642716\pi\)
							−0.839415	+	0.543491i	\(0.817102\pi\)
\(17\)	−3.82528	−	0.473672i	−0.927766	−	0.114882i	−0.355246	−	0.934773i	\(-0.615603\pi\)
							−0.572520	+	0.819891i	\(0.694034\pi\)
\(19\)	6.28693	−	2.00002i	1.44232	−	0.458835i	0.522996	−	0.852335i	\(-0.324814\pi\)
							0.919325	+	0.393500i	\(0.128736\pi\)
\(23\)	−0.195625	−	2.11113i	−0.0407907	−	0.440202i	−0.991499	−	0.130115i	\(-0.958465\pi\)
							0.950708	−	0.310087i	\(-0.100358\pi\)
\(29\)	3.48102	+	0.214702i	0.646410	+	0.0398692i	0.381449	−	0.924390i	\(-0.375425\pi\)
							0.264961	+	0.964259i	\(0.414641\pi\)
\(31\)	−6.28432	−	2.43456i	−1.12870	−	0.437260i	−0.276779	−	0.960934i	\(-0.589267\pi\)
							−0.851919	+	0.523674i	\(0.824561\pi\)
\(37\)	−1.64953	+	1.02135i	−0.271181	+	0.167908i	−0.655285	−	0.755381i	\(-0.727452\pi\)
							0.384104	+	0.923290i	\(0.374510\pi\)
\(41\)	1.23353	+	1.86159i	0.192645	+	0.290731i	0.917153	−	0.398536i	\(-0.130482\pi\)
							−0.724507	+	0.689267i	\(0.757933\pi\)
\(43\)	−0.755862	+	0.405890i	−0.115268	+	0.0618976i	−0.529714	−	0.848176i	\(-0.677701\pi\)
							0.414446	+	0.910074i	\(0.363975\pi\)
\(47\)	−0.872936	−	1.51197i	−0.127331	−	0.220543i	0.795311	−	0.606202i	\(-0.207308\pi\)
							−0.922642	+	0.385658i	\(0.873974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.17.2	✓	256
3.2	odd	2		927.2.ba.c.532.7		256
103.97	even	51	inner	103.2.g.a.97.2	yes	256
309.200	odd	102		927.2.ba.c.406.7		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.17.2	✓	256	1.1	even	1	trivial
103.2.g.a.97.2	yes	256	103.97	even	51	inner
927.2.ba.c.406.7		256	309.200	odd	102
927.2.ba.c.532.7		256	3.2	odd	2