Embedded newform 115.3.h.a.21.3

• Label: 115.3.h.a.21.3
• Level: $115$
• Weight: $3$
• Character: 115.21
• 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			21.3
Character	\(\chi\)	\(=\)	115.21
Dual form			115.3.h.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.76404 - 1.77634i) q^{2} +(-0.745160 - 5.18270i) q^{3} +(2.82287 + 6.18121i) q^{4} +(-0.629973 + 2.14549i) q^{5} +(-7.14658 + 15.6488i) q^{6} +(-7.41885 - 6.42847i) q^{7} +(1.30706 - 9.09078i) q^{8} +(-17.6697 + 5.18828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.76404	−	1.77634i	−1.38202	−	0.888170i	−0.382657	−	0.923890i	\(-0.624991\pi\)
							−0.999362	+	0.0357205i	\(0.988627\pi\)
\(3\)	−0.745160	−	5.18270i	−0.248387	−	1.72757i	−0.607541	−	0.794288i	\(-0.707844\pi\)
							0.359155	−	0.933278i	\(-0.383065\pi\)
\(5\)	−0.629973	+	2.14549i	−0.125995	+	0.429098i
							\(7\)	−7.41885	−	6.42847i	−1.05984	−	0.918353i	−0.0630134	−	0.998013i	\(-0.520071\pi\)
−0.996822	+	0.0796600i	\(0.974617\pi\)
\(11\)	−2.08201	−	3.23967i	−0.189273	−	0.294515i	0.733628	−	0.679551i	\(-0.237825\pi\)
							−0.922902	+	0.385036i	\(0.874189\pi\)
\(13\)	14.4775	+	16.7079i	1.11365	+	1.28522i	0.954579	+	0.297958i	\(0.0963057\pi\)
							0.159075	+	0.987267i	\(0.449149\pi\)
\(17\)	−12.6930	−	5.79672i	−0.746649	−	0.340983i	0.00549215	−	0.999985i	\(-0.498252\pi\)
							−0.752142	+	0.659002i	\(0.770979\pi\)
\(19\)	15.9118	−	7.26665i	0.837461	−	0.382455i	0.0499463	−	0.998752i	\(-0.484095\pi\)
							0.787514	+	0.616296i	\(0.211368\pi\)
\(23\)	−21.0023	−	9.37569i	−0.913143	−	0.407639i
							\(29\)	−3.61144	+	7.90795i	−0.124532	+	0.272688i	−0.961622	−	0.274378i	\(-0.911528\pi\)
0.837089	+	0.547066i	\(0.184255\pi\)
\(31\)	−3.03038	+	21.0767i	−0.0977541	+	0.679895i	0.880737	+	0.473606i	\(0.157048\pi\)
							−0.978491	+	0.206289i	\(0.933861\pi\)
\(37\)	1.37818	+	4.69365i	0.0372481	+	0.126855i	0.976018	−	0.217689i	\(-0.0698517\pi\)
							−0.938770	+	0.344544i	\(0.888034\pi\)
\(41\)	−50.5799	−	14.8516i	−1.23366	−	0.362234i	−0.401028	−	0.916066i	\(-0.631347\pi\)
							−0.832629	+	0.553831i	\(0.813165\pi\)
\(43\)	−28.1559	+	4.04821i	−0.654789	+	0.0941445i	−0.461699	−	0.887037i	\(-0.652760\pi\)
							−0.193090	+	0.981181i	\(0.561851\pi\)
\(47\)	−71.4174			−1.51952			−0.759759	−	0.650204i	\(-0.774683\pi\)
							−0.759759	+	0.650204i	\(0.774683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.21.3	yes	160
23.11	odd	22	inner	115.3.h.a.11.3	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.3	✓	160	23.11	odd	22	inner
115.3.h.a.21.3	yes	160	1.1	even	1	trivial