Embedded newform 123.2.o.a.110.2

• Label: 123.2.o.a.110.2
• 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			110.2
Character	\(\chi\)	\(=\)	123.110
Dual form			123.2.o.a.104.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07682 + 2.11337i) q^{2} +(1.72725 + 0.128921i) q^{3} +(-2.13123 - 2.93338i) q^{4} +(-0.134202 + 0.847316i) q^{5} +(-2.13238 + 3.51148i) q^{6} +(-2.44657 + 2.86457i) q^{7} +(3.80888 - 0.603267i) q^{8} +(2.96676 + 0.445357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07682	+	2.11337i	−0.761423	+	1.49438i	0.104677	+	0.994506i	\(0.466619\pi\)
							−0.866100	+	0.499871i	\(0.833381\pi\)
\(3\)	1.72725	+	0.128921i	0.997226	+	0.0744326i
							\(5\)	−0.134202	+	0.847316i	−0.0600168	+	0.378931i	0.939337	+	0.342995i	\(0.111441\pi\)
−0.999354	+	0.0359363i	\(0.988559\pi\)
\(7\)	−2.44657	+	2.86457i	−0.924717	+	1.08270i	0.0716403	+	0.997431i	\(0.477177\pi\)
							−0.996358	+	0.0852743i	\(0.972823\pi\)
\(11\)	1.58913	+	2.59322i	0.479141	+	0.781886i	0.997303	−	0.0733960i	\(-0.0233837\pi\)
							−0.518162	+	0.855282i	\(0.673384\pi\)
\(13\)	0.0226925	−	0.288336i	0.00629378	−	0.0799700i	−0.992968	−	0.118381i	\(-0.962230\pi\)
							0.999262	+	0.0384109i	\(0.0122296\pi\)
\(17\)	1.61821	−	6.74032i	0.392473	−	1.63477i	−0.330440	−	0.943827i	\(-0.607197\pi\)
							0.722913	−	0.690939i	\(-0.242803\pi\)
\(19\)	−0.263306	−	3.34562i	−0.0604065	−	0.767538i	−0.950293	−	0.311356i	\(-0.899217\pi\)
							0.889887	−	0.456182i	\(-0.150783\pi\)
\(23\)	1.00735	+	3.10029i	0.210046	+	0.646455i	0.999468	+	0.0326048i	\(0.0103803\pi\)
							−0.789422	+	0.613851i	\(0.789620\pi\)
\(29\)	−0.519535	−	2.16402i	−0.0964752	−	0.401848i	0.903147	−	0.429331i	\(-0.141251\pi\)
							−0.999622	+	0.0274831i	\(0.991251\pi\)
\(31\)	5.21200	−	7.17370i	0.936102	−	1.28843i	−0.0213286	−	0.999773i	\(-0.506790\pi\)
							0.957431	−	0.288662i	\(-0.0932104\pi\)
\(37\)	−5.49355	+	3.99130i	−0.903135	+	0.656166i	−0.939269	−	0.343181i	\(-0.888496\pi\)
							0.0361346	+	0.999347i	\(0.488496\pi\)
\(41\)	−6.35380	−	0.793250i	−0.992297	−	0.123885i
							\(43\)	−1.78148	−	0.907709i	−0.271673	−	0.138424i	0.312847	−	0.949803i	\(-0.398717\pi\)
−0.584520	+	0.811379i	\(0.698717\pi\)
\(47\)	−8.66100	+	7.39719i	−1.26334	+	1.07899i	−0.270106	+	0.962831i	\(0.587059\pi\)
							−0.993230	+	0.116160i	\(0.962941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.110.2	yes	192
3.2	odd	2	inner	123.2.o.a.110.11	yes	192
41.22	odd	40	inner	123.2.o.a.104.11	yes	192
123.104	even	40	inner	123.2.o.a.104.2	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.104.2	✓	192	123.104	even	40	inner
123.2.o.a.104.11	yes	192	41.22	odd	40	inner
123.2.o.a.110.2	yes	192	1.1	even	1	trivial
123.2.o.a.110.11	yes	192	3.2	odd	2	inner