Embedded newform 123.2.o.a.101.4

• Label: 123.2.o.a.101.4
• 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			101.4
Character	\(\chi\)	\(=\)	123.101
Dual form			123.2.o.a.95.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.595655 - 1.16904i) q^{2} +(-0.981205 - 1.42732i) q^{3} +(0.163723 - 0.225346i) q^{4} +(-0.0620289 - 0.391635i) q^{5} +(-1.08413 + 1.99726i) q^{6} +(0.908544 - 0.775970i) q^{7} +(-2.95274 - 0.467668i) q^{8} +(-1.07447 + 2.80098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.595655	−	1.16904i	−0.421192	−	0.826635i	−0.999938	−	0.0111278i	\(-0.996458\pi\)
							0.578746	−	0.815508i	\(-0.303542\pi\)
\(3\)	−0.981205	−	1.42732i	−0.566499	−	0.824063i
							\(5\)	−0.0620289	−	0.391635i	−0.0277402	−	0.175145i	0.969928	−	0.243391i	\(-0.0782597\pi\)
−0.997669	+	0.0682461i	\(0.978260\pi\)
\(7\)	0.908544	−	0.775970i	0.343397	−	0.293289i	−0.460977	−	0.887412i	\(-0.652501\pi\)
							0.804374	+	0.594123i	\(0.202501\pi\)
\(11\)	−1.43876	−	0.881676i	−0.433804	−	0.265835i	0.288408	−	0.957508i	\(-0.406874\pi\)
							−0.722211	+	0.691672i	\(0.756874\pi\)
\(13\)	−1.10082	+	0.0866365i	−0.305313	+	0.0240286i	−0.230190	−	0.973146i	\(-0.573935\pi\)
							−0.0751226	+	0.997174i	\(0.523935\pi\)
\(17\)	7.17148	−	1.72172i	1.73934	−	0.417579i	0.765249	−	0.643735i	\(-0.222616\pi\)
							0.974091	+	0.226156i	\(0.0726160\pi\)
\(19\)	−2.37573	−	0.186974i	−0.545029	−	0.0428947i	−0.197044	−	0.980395i	\(-0.563134\pi\)
							−0.347985	+	0.937500i	\(0.613134\pi\)
\(23\)	1.46757	−	4.51671i	0.306009	−	0.941799i	−0.673290	−	0.739379i	\(-0.735119\pi\)
							0.979299	−	0.202420i	\(-0.0648807\pi\)
\(29\)	4.31741	+	1.03652i	0.801722	+	0.192476i	0.613546	−	0.789659i	\(-0.289742\pi\)
							0.188176	+	0.982135i	\(0.439742\pi\)
\(31\)	0.811450	+	1.11686i	0.145741	+	0.200595i	0.875646	−	0.482953i	\(-0.160436\pi\)
							−0.729905	+	0.683548i	\(0.760436\pi\)
\(37\)	−2.82710	−	2.05401i	−0.464773	−	0.337677i	0.330628	−	0.943761i	\(-0.392740\pi\)
							−0.795401	+	0.606084i	\(0.792740\pi\)
\(41\)	−4.37586	+	4.67459i	−0.683395	+	0.730049i
							\(43\)	5.92702	−	3.01997i	0.903862	−	0.460541i	0.0606731	−	0.998158i	\(-0.480675\pi\)
0.843189	+	0.537617i	\(0.180675\pi\)
\(47\)	4.59725	−	5.38268i	0.670577	−	0.785145i	−0.315944	−	0.948778i	\(-0.602321\pi\)
							0.986521	+	0.163633i	\(0.0523212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.101.4	yes	192
3.2	odd	2	inner	123.2.o.a.101.9	yes	192
41.13	odd	40	inner	123.2.o.a.95.9	yes	192
123.95	even	40	inner	123.2.o.a.95.4	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.95.4	✓	192	123.95	even	40	inner
123.2.o.a.95.9	yes	192	41.13	odd	40	inner
123.2.o.a.101.4	yes	192	1.1	even	1	trivial
123.2.o.a.101.9	yes	192	3.2	odd	2	inner