Embedded newform 123.2.o.a.110.3

• Label: 123.2.o.a.110.3
• Level: $123$
• Weight: $2$
• Character: 123.110
• Analytic conductor: $0.982$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	123.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.982159944862\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(12\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			110.3
Character	\(\chi\)	\(=\)	123.110
Dual form			123.2.o.a.104.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.750800 + 1.47353i) q^{2} +(-1.38799 + 1.03609i) q^{3} +(-0.432012 - 0.594614i) q^{4} +(-0.103220 + 0.651706i) q^{5} +(-0.484597 - 2.82314i) q^{6} +(-1.17916 + 1.38062i) q^{7} +(-2.06630 + 0.327270i) q^{8} +(0.853051 - 2.87616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.750800	+	1.47353i	−0.530896	+	1.04194i	0.457382	+	0.889271i	\(0.348787\pi\)
							−0.988277	+	0.152671i	\(0.951213\pi\)
\(3\)	−1.38799	+	1.03609i	−0.801358	+	0.598185i
							\(5\)	−0.103220	+	0.651706i	−0.0461614	+	0.291452i	−0.999960	−	0.00899662i	\(-0.997136\pi\)
0.953798	+	0.300448i	\(0.0971362\pi\)
\(7\)	−1.17916	+	1.38062i	−0.445679	+	0.521824i	−0.937082	−	0.349109i	\(-0.886484\pi\)
							0.491403	+	0.870932i	\(0.336484\pi\)
\(11\)	−0.889901	−	1.45219i	−0.268315	−	0.437851i	0.689897	−	0.723907i	\(-0.257656\pi\)
							−0.958213	+	0.286056i	\(0.907656\pi\)
\(13\)	−0.0823816	+	1.04676i	−0.0228485	+	0.290318i	0.974868	+	0.222782i	\(0.0715138\pi\)
							−0.997717	+	0.0675362i	\(0.978486\pi\)
\(17\)	−0.540868	+	2.25288i	−0.131180	+	0.546403i	0.867611	+	0.497243i	\(0.165654\pi\)
							−0.998791	+	0.0491598i	\(0.984346\pi\)
\(19\)	0.405271	+	5.14945i	0.0929754	+	1.18136i	0.849568	+	0.527479i	\(0.176863\pi\)
							−0.756593	+	0.653886i	\(0.773137\pi\)
\(23\)	1.68717	+	5.19256i	0.351798	+	1.08272i	0.957843	+	0.287293i	\(0.0927554\pi\)
							−0.606044	+	0.795431i	\(0.707245\pi\)
\(29\)	−0.192019	−	0.799816i	−0.0356570	−	0.148522i	0.951702	−	0.307025i	\(-0.0993334\pi\)
							−0.987359	+	0.158502i	\(0.949333\pi\)
\(31\)	−3.67231	+	5.05450i	−0.659566	+	0.907815i	−0.999467	−	0.0326463i	\(-0.989607\pi\)
							0.339901	+	0.940461i	\(0.389607\pi\)
\(37\)	4.58599	−	3.33192i	0.753932	−	0.547764i	−0.143111	−	0.989707i	\(-0.545711\pi\)
							0.897043	+	0.441943i	\(0.145711\pi\)
\(41\)	6.39472	−	0.327972i	0.998687	−	0.0512207i
							\(43\)	−7.88106	−	4.01560i	−1.20185	−	0.612373i	−0.265728	−	0.964048i	\(-0.585612\pi\)
−0.936122	+	0.351675i	\(0.885612\pi\)
\(47\)	5.81802	−	4.96906i	0.848646	−	0.724812i	−0.114606	−	0.993411i	\(-0.536561\pi\)
							0.963252	+	0.268599i	\(0.0865606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.110.3	yes	192
3.2	odd	2	inner	123.2.o.a.110.10	yes	192
41.22	odd	40	inner	123.2.o.a.104.10	yes	192
123.104	even	40	inner	123.2.o.a.104.3	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.104.3	✓	192	123.104	even	40	inner
123.2.o.a.104.10	yes	192	41.22	odd	40	inner
123.2.o.a.110.3	yes	192	1.1	even	1	trivial
123.2.o.a.110.10	yes	192	3.2	odd	2	inner