Embedded newform 115.3.h.a.11.15

• Label: 115.3.h.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.92018 - 1.87668i) q^{2} +(-0.393934 + 2.73987i) q^{3} +(3.34383 - 7.32197i) q^{4} +(-0.629973 - 2.14549i) q^{5} +(3.99152 + 8.74020i) q^{6} +(4.90483 - 4.25006i) q^{7} +(-2.00042 - 13.9132i) q^{8} +(1.28373 + 0.376936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.92018	−	1.87668i	1.46009	−	0.938342i	0.461398	−	0.887193i	\(-0.347348\pi\)
							0.998691	−	0.0511486i	\(-0.0162882\pi\)
\(3\)	−0.393934	+	2.73987i	−0.131311	+	0.913290i	0.812537	+	0.582910i	\(0.198086\pi\)
							−0.943848	+	0.330380i	\(0.892823\pi\)
\(5\)	−0.629973	−	2.14549i	−0.125995	−	0.429098i
							\(7\)	4.90483	−	4.25006i	0.700690	−	0.607151i	−0.229900	−	0.973214i	\(-0.573840\pi\)
0.930590	+	0.366063i	\(0.119295\pi\)
\(11\)	−6.88339	+	10.7108i	−0.625762	+	0.973705i	0.373181	+	0.927758i	\(0.378267\pi\)
							−0.998944	+	0.0459470i	\(0.985369\pi\)
\(13\)	0.199287	−	0.229990i	0.0153298	−	0.0176915i	−0.748033	−	0.663662i	\(-0.769001\pi\)
							0.763363	+	0.645970i	\(0.223547\pi\)
\(17\)	−18.7777	+	8.57547i	−1.10457	+	0.504440i	−0.882367	−	0.470561i	\(-0.844051\pi\)
							−0.222201	+	0.975001i	\(0.571324\pi\)
\(19\)	−11.4868	−	5.24583i	−0.604566	−	0.276096i	0.0895234	−	0.995985i	\(-0.471466\pi\)
							−0.694090	+	0.719889i	\(0.744193\pi\)
\(23\)	−13.8051	+	18.3962i	−0.600220	+	0.799835i
							\(29\)	−9.63701	−	21.1021i	−0.332311	−	0.727659i	0.667546	−	0.744568i	\(-0.267345\pi\)
−0.999857	+	0.0169091i	\(0.994617\pi\)
\(31\)	−1.89878	−	13.2063i	−0.0612509	−	0.426010i	−0.997256	−	0.0740243i	\(-0.976416\pi\)
							0.936006	−	0.351985i	\(-0.114493\pi\)
\(37\)	12.6991	−	43.2491i	0.343219	−	1.16890i	−0.589343	−	0.807883i	\(-0.700613\pi\)
							0.932561	−	0.361012i	\(-0.117569\pi\)
\(41\)	13.7419	−	4.03498i	0.335167	−	0.0984141i	−0.109818	−	0.993952i	\(-0.535027\pi\)
							0.444986	+	0.895538i	\(0.353209\pi\)
\(43\)	48.1517	+	6.92317i	1.11981	+	0.161004i	0.677262	−	0.735742i	\(-0.263167\pi\)
							0.442545	+	0.896746i	\(0.354076\pi\)
\(47\)	85.2045			1.81286			0.906430	−	0.422355i	\(-0.138797\pi\)
							0.906430	+	0.422355i	\(0.138797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.15	✓	160
23.21	odd	22	inner	115.3.h.a.21.15	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.15	✓	160	1.1	even	1	trivial
115.3.h.a.21.15	yes	160	23.21	odd	22	inner