Embedded newform 123.2.o.a.101.6

• Label: 123.2.o.a.101.6
• Level: $123$
• Weight: $2$
• Character: 123.101
• 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			101.6
Character	\(\chi\)	\(=\)	123.101
Dual form			123.2.o.a.95.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.210104 - 0.412353i) q^{2} +(1.44294 - 0.958091i) q^{3} +(1.04968 - 1.44476i) q^{4} +(0.654630 + 4.13317i) q^{5} +(-0.698239 - 0.393700i) q^{6} +(-0.663206 + 0.566432i) q^{7} +(-1.73049 - 0.274082i) q^{8} +(1.16412 - 2.76493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.210104	−	0.412353i	−0.148566	−	0.291578i	0.804715	−	0.593661i	\(-0.202318\pi\)
							−0.953282	+	0.302083i	\(0.902318\pi\)
\(3\)	1.44294	−	0.958091i	0.833079	−	0.553154i
							\(5\)	0.654630	+	4.13317i	0.292760	+	1.84841i	0.494764	+	0.869027i	\(0.335254\pi\)
−0.202005	+	0.979385i	\(0.564746\pi\)
\(7\)	−0.663206	+	0.566432i	−0.250668	+	0.214091i	−0.765900	−	0.642960i	\(-0.777706\pi\)
							0.515231	+	0.857051i	\(0.327706\pi\)
\(11\)	−2.71979	−	1.66669i	−0.820048	−	0.502526i	0.0480935	−	0.998843i	\(-0.484685\pi\)
							−0.868142	+	0.496317i	\(0.834685\pi\)
\(13\)	−4.33422	+	0.341111i	−1.20210	+	0.0946071i	−0.663644	−	0.748048i	\(-0.730991\pi\)
							−0.538453	+	0.842656i	\(0.680991\pi\)
\(17\)	−0.884964	+	0.212461i	−0.214635	+	0.0515293i	−0.339337	−	0.940665i	\(-0.610203\pi\)
							0.124702	+	0.992194i	\(0.460203\pi\)
\(19\)	4.24407	+	0.334015i	0.973655	+	0.0766283i	0.555296	−	0.831653i	\(-0.312605\pi\)
							0.418360	+	0.908281i	\(0.362605\pi\)
\(23\)	−0.0470905	+	0.144930i	−0.00981906	+	0.0302200i	−0.955846	−	0.293868i	\(-0.905057\pi\)
							0.946027	+	0.324088i	\(0.105057\pi\)
\(29\)	−2.47195	−	0.593462i	−0.459029	−	0.110203i	−0.00266524	−	0.999996i	\(-0.500848\pi\)
							−0.456364	+	0.889793i	\(0.650848\pi\)
\(31\)	2.54716	+	3.50587i	0.457484	+	0.629673i	0.973985	−	0.226614i	\(-0.0727655\pi\)
							−0.516500	+	0.856287i	\(0.672766\pi\)
\(37\)	7.30591	+	5.30805i	1.20108	+	0.872639i	0.994391	−	0.105767i	\(-0.0337298\pi\)
							0.206693	+	0.978406i	\(0.433730\pi\)
\(41\)	1.19102	−	6.29138i	0.186005	−	0.982549i
							\(43\)	4.24467	−	2.16277i	0.647306	−	0.329819i	−0.0993271	−	0.995055i	\(-0.531669\pi\)
0.746633	+	0.665236i	\(0.231669\pi\)
\(47\)	2.90001	−	3.39548i	0.423010	−	0.495281i	−0.507447	−	0.861683i	\(-0.669411\pi\)
							0.930457	+	0.366402i	\(0.119411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.101.6	yes	192
3.2	odd	2	inner	123.2.o.a.101.7	yes	192
41.13	odd	40	inner	123.2.o.a.95.7	yes	192
123.95	even	40	inner	123.2.o.a.95.6	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.95.6	✓	192	123.95	even	40	inner
123.2.o.a.95.7	yes	192	41.13	odd	40	inner
123.2.o.a.101.6	yes	192	1.1	even	1	trivial
123.2.o.a.101.7	yes	192	3.2	odd	2	inner