Embedded newform 103.2.g.a.15.3

• Label: 103.2.g.a.15.3
• Level: $103$
• Weight: $2$
• Character: 103.15
• 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			15.3
Character	\(\chi\)	\(=\)	103.15
Dual form			103.2.g.a.55.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23467 - 1.27330i) q^{2} +(-1.25677 - 1.66423i) q^{3} +(-0.0352822 + 1.14517i) q^{4} +(-0.347823 - 0.987049i) q^{5} +(-0.567366 + 3.65503i) q^{6} +(-0.176988 + 0.808152i) q^{7} +(-1.11973 + 1.02077i) q^{8} +(-0.369211 + 1.29764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 33 q^{2} - 30 q^{3} - 25 q^{4} - 37 q^{5} - 31 q^{6} - 29 q^{7} - 40 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{20}{51}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23467	−	1.27330i	−0.873046	−	0.900359i	0.122705	−	0.992443i	\(-0.460843\pi\)
							−0.995751	+	0.0920845i	\(0.970647\pi\)
\(3\)	−1.25677	−	1.66423i	−0.725597	−	0.960846i	0.274403	−	0.961615i	\(-0.411520\pi\)
							−1.00000		0.000769320i	\(0.999755\pi\)
\(5\)	−0.347823	−	0.987049i	−0.155551	−	0.441422i	0.839191	−	0.543837i	\(-0.183029\pi\)
							−0.994742	+	0.102415i	\(0.967343\pi\)
\(7\)	−0.176988	+	0.808152i	−0.0668952	+	0.305453i	−0.998283	−	0.0585758i	\(-0.981344\pi\)
							0.931388	+	0.364029i	\(0.118599\pi\)
\(11\)	−1.80261	+	0.453374i	−0.543507	+	0.136697i	−0.505895	−	0.862595i	\(-0.668838\pi\)
							−0.0376116	+	0.999292i	\(0.511975\pi\)
\(13\)	−0.382171	−	0.348395i	−0.105995	−	0.0966274i	0.618963	−	0.785420i	\(-0.287553\pi\)
							−0.724959	+	0.688792i	\(0.758141\pi\)
\(17\)	−1.04714	−	6.74578i	−0.253968	−	1.63609i	−0.681305	−	0.731999i	\(-0.738587\pi\)
							0.427337	−	0.904092i	\(-0.359452\pi\)
\(19\)	0.198484	+	0.468937i	0.0455353	+	0.107581i	0.942101	−	0.335330i	\(-0.108848\pi\)
							−0.896566	+	0.442911i	\(0.853946\pi\)
\(23\)	−1.22670	−	4.31140i	−0.255784	−	0.898989i	−0.978224	−	0.207554i	\(-0.933450\pi\)
							0.722439	−	0.691434i	\(-0.243021\pi\)
\(29\)	3.58810	−	4.18805i	0.666294	−	0.777702i	−0.319149	−	0.947704i	\(-0.603397\pi\)
							0.985444	+	0.170003i	\(0.0543776\pi\)
\(31\)	1.20874	+	2.42748i	0.217096	+	0.435987i	0.976762	−	0.214326i	\(-0.0687554\pi\)
							−0.759666	+	0.650313i	\(0.774638\pi\)
\(37\)	−0.394470	−	4.25701i	−0.0648505	−	0.699848i	−0.964648	−	0.263541i	\(-0.915110\pi\)
							0.899798	−	0.436307i	\(-0.143714\pi\)
\(41\)	−2.97504	+	8.44255i	−0.464623	+	1.31851i	0.442171	+	0.896931i	\(0.354209\pi\)
							−0.906794	+	0.421574i	\(0.861478\pi\)
\(43\)	−4.07358	−	2.88363i	−0.621216	−	0.439749i	0.223693	−	0.974660i	\(-0.428189\pi\)
							−0.844909	+	0.534910i	\(0.820345\pi\)
\(47\)	2.44099	+	4.22793i	0.356056	+	0.616707i	0.987298	−	0.158879i	\(-0.0507880\pi\)
							−0.631242	+	0.775586i	\(0.717455\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	103.2.g.a.15.3	✓	256
3.2	odd	2		927.2.ba.c.118.6		256
103.55	even	51	inner	103.2.g.a.55.3	yes	256
309.158	odd	102		927.2.ba.c.55.6		256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
103.2.g.a.15.3	✓	256	1.1	even	1	trivial
103.2.g.a.55.3	yes	256	103.55	even	51	inner
927.2.ba.c.55.6		256	309.158	odd	102
927.2.ba.c.118.6		256	3.2	odd	2