Embedded newform 115.3.h.a.11.9

• Label: 115.3.h.a.11.9
• Level: $115$
• Weight: $3$
• Character: 115.11
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.9
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.235723 - 0.151490i) q^{2} +(0.0812454 - 0.565074i) q^{3} +(-1.62904 + 3.56711i) q^{4} +(0.629973 + 2.14549i) q^{5} +(-0.0664516 - 0.145509i) q^{6} +(-3.04533 + 2.63880i) q^{7} +(0.315887 + 2.19704i) q^{8} +(8.32273 + 2.44377i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.235723	−	0.151490i	0.117861	−	0.0757449i	−0.480382	−	0.877059i	\(-0.659502\pi\)
							0.598243	+	0.801314i	\(0.295866\pi\)
\(3\)	0.0812454	−	0.565074i	0.0270818	−	0.188358i	−0.971790	−	0.235848i	\(-0.924213\pi\)
							0.998872	+	0.0474902i	\(0.0151223\pi\)
\(5\)	0.629973	+	2.14549i	0.125995	+	0.429098i
							\(7\)	−3.04533	+	2.63880i	−0.435048	+	0.376971i	−0.844673	−	0.535283i	\(-0.820205\pi\)
0.409625	+	0.912254i	\(0.365659\pi\)
\(11\)	−9.09049	+	14.1451i	−0.826408	+	1.28592i	0.129302	+	0.991605i	\(0.458726\pi\)
							−0.955710	+	0.294311i	\(0.904910\pi\)
\(13\)	−0.630384	+	0.727502i	−0.0484911	+	0.0559617i	−0.779477	−	0.626431i	\(-0.784515\pi\)
							0.730986	+	0.682393i	\(0.239061\pi\)
\(17\)	20.9128	−	9.55055i	1.23016	−	0.561797i	0.309033	−	0.951051i	\(-0.399995\pi\)
							0.921130	+	0.389254i	\(0.127267\pi\)
\(19\)	23.8310	+	10.8833i	1.25426	+	0.572803i	0.928037	−	0.372489i	\(-0.121496\pi\)
							0.326227	+	0.945292i	\(0.394223\pi\)
\(23\)	−22.9977	+	0.326060i	−0.999900	+	0.0141765i
							\(29\)	−14.6504	−	32.0799i	−0.505186	−	1.10620i	−0.974749	−	0.223304i	\(-0.928316\pi\)
0.469563	−	0.882899i	\(-0.344411\pi\)
\(31\)	0.766092	+	5.32829i	0.0247127	+	0.171880i	0.998440	−	0.0558373i	\(-0.0177828\pi\)
							−0.973727	+	0.227718i	\(0.926874\pi\)
\(37\)	13.4289	−	45.7346i	0.362943	−	1.23607i	−0.552462	−	0.833538i	\(-0.686312\pi\)
							0.915406	−	0.402533i	\(-0.131870\pi\)
\(41\)	−17.5763	+	5.16086i	−0.428690	+	0.125875i	−0.488958	−	0.872308i	\(-0.662623\pi\)
							0.0602678	+	0.998182i	\(0.480805\pi\)
\(43\)	30.5781	+	4.39647i	0.711119	+	0.102244i	0.488376	−	0.872634i	\(-0.337590\pi\)
							0.222744	+	0.974877i	\(0.428499\pi\)
\(47\)	66.9553			1.42458			0.712290	−	0.701885i	\(-0.247658\pi\)
							0.712290	+	0.701885i	\(0.247658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.9	✓	160
23.21	odd	22	inner	115.3.h.a.21.9	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.9	✓	160	1.1	even	1	trivial
115.3.h.a.21.9	yes	160	23.21	odd	22	inner