Embedded newform 115.3.h.a.11.13

• Label: 115.3.h.a.11.13
• 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			11.13
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.25259 - 1.44765i) q^{2} +(-0.677841 + 4.71449i) q^{3} +(1.31681 - 2.88342i) q^{4} +(0.629973 + 2.14549i) q^{5} +(5.29805 + 11.6011i) q^{6} +(-1.25341 + 1.08608i) q^{7} +(0.316337 + 2.20017i) q^{8} +(-13.1315 - 3.85576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.25259	−	1.44765i	1.12630	−	0.723827i	0.161512	−	0.986871i	\(-0.448363\pi\)
							0.964784	+	0.263044i	\(0.0847265\pi\)
\(3\)	−0.677841	+	4.71449i	−0.225947	+	1.57150i	0.488978	+	0.872296i	\(0.337370\pi\)
							−0.714925	+	0.699201i	\(0.753539\pi\)
\(5\)	0.629973	+	2.14549i	0.125995	+	0.429098i
							\(7\)	−1.25341	+	1.08608i	−0.179058	+	0.155155i	−0.739778	−	0.672850i	\(-0.765070\pi\)
0.560720	+	0.828005i	\(0.310524\pi\)
\(11\)	2.89710	−	4.50798i	0.263373	−	0.409816i	−0.684230	−	0.729266i	\(-0.739862\pi\)
							0.947603	+	0.319450i	\(0.103498\pi\)
\(13\)	7.81187	−	9.01537i	0.600913	−	0.693490i	−0.371053	−	0.928612i	\(-0.621003\pi\)
							0.971966	+	0.235121i	\(0.0755487\pi\)
\(17\)	29.5986	−	13.5172i	1.74110	−	0.795132i	0.750086	−	0.661341i	\(-0.230012\pi\)
							0.991010	−	0.133791i	\(-0.0427151\pi\)
\(19\)	−14.1750	−	6.47352i	−0.746055	−	0.340712i	0.00585062	−	0.999983i	\(-0.498138\pi\)
							−0.751905	+	0.659271i	\(0.770865\pi\)
\(23\)	−19.3293	−	12.4651i	−0.840404	−	0.541961i
							\(29\)	5.69822	+	12.4774i	0.196490	+	0.430254i	0.982073	−	0.188503i	\(-0.0603637\pi\)
−0.785582	+	0.618757i	\(0.787636\pi\)
\(31\)	−4.60692	−	32.0418i	−0.148610	−	1.03361i	−0.918498	−	0.395427i	\(-0.870597\pi\)
							0.769887	−	0.638180i	\(-0.220312\pi\)
\(37\)	−5.34762	+	18.2123i	−0.144530	+	0.492225i	−0.999657	−	0.0261944i	\(-0.991661\pi\)
							0.855127	+	0.518419i	\(0.173479\pi\)
\(41\)	−30.1804	+	8.86176i	−0.736107	+	0.216140i	−0.628234	−	0.778024i	\(-0.716222\pi\)
							−0.107872	+	0.994165i	\(0.534404\pi\)
\(43\)	7.46289	+	1.07300i	0.173556	+	0.0249535i	0.228545	−	0.973533i	\(-0.426603\pi\)
							−0.0549891	+	0.998487i	\(0.517512\pi\)
\(47\)	−35.7232			−0.760069			−0.380034	−	0.924972i	\(-0.624088\pi\)
							−0.380034	+	0.924972i	\(0.624088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.13	✓	160
23.21	odd	22	inner	115.3.h.a.21.13	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.13	✓	160	1.1	even	1	trivial
115.3.h.a.21.13	yes	160	23.21	odd	22	inner