Embedded newform 115.5.h.a.11.11

• Label: 115.5.h.a.11.11
• Level: $115$
• Weight: $5$
• Character: 115.11
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.11
Character	\(\chi\)	\(=\)	115.11
Dual form			115.5.h.a.21.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63066 + 1.69062i) q^{2} +(0.551076 - 3.83282i) q^{3} +(-2.58448 + 5.65923i) q^{4} +(3.14987 + 10.7275i) q^{5} +(5.03015 + 11.0145i) q^{6} +(-52.9290 + 45.8632i) q^{7} +(-9.88917 - 68.7807i) q^{8} +(63.3321 + 18.5960i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63066	+	1.69062i	−0.657664	+	0.422655i	−0.826460	−	0.562996i	\(-0.809649\pi\)
							0.168795	+	0.985651i	\(0.446012\pi\)
\(3\)	0.551076	−	3.83282i	0.0612306	−	0.425868i	−0.936031	−	0.351917i	\(-0.885530\pi\)
							0.997262	−	0.0739514i	\(-0.0235610\pi\)
\(5\)	3.14987	+	10.7275i	0.125995	+	0.429098i
							\(7\)	−52.9290	+	45.8632i	−1.08018	+	0.935984i	−0.998159	−	0.0606462i	\(-0.980684\pi\)
−0.0820240	+	0.996630i	\(0.526138\pi\)
\(11\)	71.1988	−	110.788i	0.588420	−	0.915600i	−0.411571	−	0.911378i	\(-0.635020\pi\)
							0.999991	−	0.00422189i	\(-0.00134387\pi\)
\(13\)	−179.673	+	207.354i	−1.06315	+	1.22695i	−0.0902062	+	0.995923i	\(0.528753\pi\)
							−0.972948	+	0.231023i	\(0.925793\pi\)
\(17\)	256.492	−	117.136i	0.887515	−	0.405314i	0.0811285	−	0.996704i	\(-0.474148\pi\)
							0.806386	+	0.591389i	\(0.201420\pi\)
\(19\)	−581.773	−	265.687i	−1.61156	−	0.735975i	−0.613025	−	0.790064i	\(-0.710047\pi\)
							−0.998536	+	0.0540887i	\(0.982775\pi\)
\(23\)	−498.424	−	177.242i	−0.942200	−	0.335051i
							\(29\)	−313.846	−	687.227i	−0.373182	−	0.817155i	−0.999299	−	0.0374262i	\(-0.988084\pi\)
0.626117	−	0.779729i	\(-0.284643\pi\)
\(31\)	−189.706	−	1319.44i	−0.197405	−	1.37298i	−0.811778	−	0.583967i	\(-0.801500\pi\)
							0.614372	−	0.789016i	\(-0.289409\pi\)
\(37\)	71.0613	−	242.013i	0.0519075	−	0.176781i	−0.929459	−	0.368925i	\(-0.879726\pi\)
							0.981367	+	0.192144i	\(0.0615442\pi\)
\(41\)	−367.368	+	107.869i	−0.218541	+	0.0641695i	−0.389170	−	0.921166i	\(-0.627238\pi\)
							0.170629	+	0.985335i	\(0.445420\pi\)
\(43\)	−522.317	−	75.0979i	−0.282486	−	0.0406154i	−0.000384646	−	1.00000i	\(-0.500122\pi\)
							−0.282102	+	0.959385i	\(0.591032\pi\)
\(47\)	−2576.60			−1.16641			−0.583205	−	0.812325i	\(-0.698202\pi\)
							−0.583205	+	0.812325i	\(0.698202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.h.a.11.11	✓	320
23.21	odd	22	inner	115.5.h.a.21.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.h.a.11.11	✓	320	1.1	even	1	trivial
115.5.h.a.21.11	yes	320	23.21	odd	22	inner