Embedded newform 115.3.h.a.11.10

• Label: 115.3.h.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.291588 - 0.187393i) q^{2} +(-0.754507 + 5.24771i) q^{3} +(-1.61175 + 3.52924i) q^{4} +(-0.629973 - 2.14549i) q^{5} +(0.763376 + 1.67156i) q^{6} +(4.68392 - 4.05864i) q^{7} +(0.388698 + 2.70345i) q^{8} +(-18.3338 - 5.38328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.291588	−	0.187393i	0.145794	−	0.0936963i	−0.465710	−	0.884937i	\(-0.654201\pi\)
							0.611504	+	0.791241i	\(0.290565\pi\)
\(3\)	−0.754507	+	5.24771i	−0.251502	+	1.74924i	0.337702	+	0.941253i	\(0.390350\pi\)
							−0.589204	+	0.807984i	\(0.700559\pi\)
\(5\)	−0.629973	−	2.14549i	−0.125995	−	0.429098i
							\(7\)	4.68392	−	4.05864i	0.669131	−	0.579805i	−0.252630	−	0.967563i	\(-0.581296\pi\)
0.921761	+	0.387758i	\(0.126750\pi\)
\(11\)	−4.00839	+	6.23718i	−0.364399	+	0.567016i	−0.974242	−	0.225506i	\(-0.927596\pi\)
							0.609843	+	0.792523i	\(0.291233\pi\)
\(13\)	−11.6501	+	13.4450i	−0.896164	+	1.03423i	0.103054	+	0.994676i	\(0.467139\pi\)
							−0.999218	+	0.0395521i	\(0.987407\pi\)
\(17\)	8.78585	−	4.01236i	0.516815	−	0.236021i	−0.139893	−	0.990167i	\(-0.544676\pi\)
							0.656708	+	0.754145i	\(0.271949\pi\)
\(19\)	23.8354	+	10.8853i	1.25450	+	0.572909i	0.928103	−	0.372324i	\(-0.121439\pi\)
							0.326394	+	0.945234i	\(0.394166\pi\)
\(23\)	7.96796	−	21.5757i	0.346433	−	0.938075i
							\(29\)	19.4190	+	42.5217i	0.669621	+	1.46626i	0.873278	+	0.487223i	\(0.161990\pi\)
−0.203657	+	0.979042i	\(0.565283\pi\)
\(31\)	6.34350	+	44.1200i	0.204629	+	1.42323i	0.790321	+	0.612693i	\(0.209914\pi\)
							−0.585692	+	0.810534i	\(0.699177\pi\)
\(37\)	−2.80805	+	9.56334i	−0.0758932	+	0.258469i	−0.988696	−	0.149931i	\(-0.952095\pi\)
							0.912803	+	0.408400i	\(0.133913\pi\)
\(41\)	41.0691	−	12.0590i	1.00168	−	0.294121i	0.260538	−	0.965464i	\(-0.416100\pi\)
							0.741147	+	0.671342i	\(0.234282\pi\)
\(43\)	13.6554	+	1.96335i	0.317567	+	0.0456592i	0.299255	−	0.954173i	\(-0.403262\pi\)
							0.0183117	+	0.999832i	\(0.494171\pi\)
\(47\)	−54.5694			−1.16105			−0.580525	−	0.814242i	\(-0.697153\pi\)
							−0.580525	+	0.814242i	\(0.697153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.10	✓	160
23.21	odd	22	inner	115.3.h.a.21.10	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.10	✓	160	1.1	even	1	trivial
115.3.h.a.21.10	yes	160	23.21	odd	22	inner