Embedded newform 123.2.o.a.11.9

• Label: 123.2.o.a.11.9
• Level: $123$
• Weight: $2$
• Character: 123.11
• 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			11.9
Character	\(\chi\)	\(=\)	123.11
Dual form			123.2.o.a.56.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.204191 + 1.28921i) q^{2} +(-0.191951 - 1.72138i) q^{3} +(0.281736 - 0.0915416i) q^{4} +(-1.07673 - 0.548623i) q^{5} +(2.18003 - 0.598957i) q^{6} +(3.44592 - 0.827292i) q^{7} +(1.36072 + 2.67056i) q^{8} +(-2.92631 + 0.660841i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.204191	+	1.28921i	0.144385	+	0.911611i	0.948417	+	0.317025i	\(0.102684\pi\)
							−0.804032	+	0.594586i	\(0.797316\pi\)
\(3\)	−0.191951	−	1.72138i	−0.110823	−	0.993840i
							\(5\)	−1.07673	−	0.548623i	−0.481529	−	0.245351i	0.196337	−	0.980536i	\(-0.437095\pi\)
−0.677867	+	0.735185i	\(0.737095\pi\)
\(7\)	3.44592	−	0.827292i	1.30244	−	0.312687i	0.477859	−	0.878437i	\(-0.341413\pi\)
							0.824577	+	0.565750i	\(0.191413\pi\)
\(11\)	0.784299	−	0.918296i	0.236475	−	0.276877i	−0.629460	−	0.777033i	\(-0.716724\pi\)
							0.865935	+	0.500156i	\(0.166724\pi\)
\(13\)	−0.178534	−	0.291342i	−0.0495165	−	0.0808037i	0.826924	−	0.562313i	\(-0.190089\pi\)
							−0.876441	+	0.481510i	\(0.840089\pi\)
\(17\)	−4.10341	+	0.322945i	−0.995222	+	0.0783257i	−0.565597	−	0.824682i	\(-0.691354\pi\)
							−0.429625	+	0.903007i	\(0.641354\pi\)
\(19\)	−3.81150	+	6.21981i	−0.874419	+	1.42692i	0.0296619	+	0.999560i	\(0.490557\pi\)
							−0.904081	+	0.427362i	\(0.859443\pi\)
\(23\)	−1.27070	−	0.923220i	−0.264960	−	0.192505i	0.447371	−	0.894349i	\(-0.352360\pi\)
							−0.712331	+	0.701844i	\(0.752360\pi\)
\(29\)	5.66170	+	0.445586i	1.05135	+	0.0827432i	0.592341	−	0.805688i	\(-0.298204\pi\)
							0.459011	+	0.888431i	\(0.348204\pi\)
\(31\)	−3.68244	−	1.19650i	−0.661387	−	0.214898i	−0.0409585	−	0.999161i	\(-0.513041\pi\)
							−0.620428	+	0.784263i	\(0.713041\pi\)
\(37\)	1.78523	+	5.49438i	0.293491	+	0.903271i	0.983724	+	0.179685i	\(0.0575077\pi\)
							−0.690234	+	0.723587i	\(0.742492\pi\)
\(41\)	2.95400	−	5.68101i	0.461338	−	0.887224i
							\(43\)	−7.41911	+	1.17507i	−1.13140	+	0.179197i	−0.693936	−	0.720037i	\(-0.744125\pi\)
−0.437468	+	0.899234i	\(0.644125\pi\)
\(47\)	0.345172	−	1.43775i	0.0503485	−	0.209717i	−0.941525	−	0.336942i	\(-0.890607\pi\)
							0.991874	+	0.127226i	\(0.0406072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.11.9	yes	192
3.2	odd	2	inner	123.2.o.a.11.4	✓	192
41.15	odd	40	inner	123.2.o.a.56.4	yes	192
123.56	even	40	inner	123.2.o.a.56.9	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.11.4	✓	192	3.2	odd	2	inner
123.2.o.a.11.9	yes	192	1.1	even	1	trivial
123.2.o.a.56.4	yes	192	41.15	odd	40	inner
123.2.o.a.56.9	yes	192	123.56	even	40	inner