Embedded newform 123.2.o.a.110.12

• Label: 123.2.o.a.110.12
• 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			110.12
Character	\(\chi\)	\(=\)	123.110
Dual form			123.2.o.a.104.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21386 - 2.38233i) q^{2} +(-1.72998 - 0.0846176i) q^{3} +(-3.02647 - 4.16557i) q^{4} +(-0.0992868 + 0.626872i) q^{5} +(-2.30154 + 4.01867i) q^{6} +(1.16438 - 1.36331i) q^{7} +(-8.31579 + 1.31709i) q^{8} +(2.98568 + 0.292774i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21386	−	2.38233i	0.858326	−	1.68456i	0.138548	−	0.990356i	\(-0.455757\pi\)
							0.719778	−	0.694204i	\(-0.244243\pi\)
\(3\)	−1.72998	−	0.0846176i	−0.998806	−	0.0488540i
							\(5\)	−0.0992868	+	0.626872i	−0.0444024	+	0.280346i	−0.999892	−	0.0146781i	\(-0.995328\pi\)
0.955490	+	0.295024i	\(0.0953276\pi\)
\(7\)	1.16438	−	1.36331i	0.440094	−	0.515283i	−0.495382	−	0.868675i	\(-0.664972\pi\)
							0.935475	+	0.353392i	\(0.114972\pi\)
\(11\)	1.80594	+	2.94703i	0.544511	+	0.888562i	0.999999	−	0.00168071i	\(-0.000534988\pi\)
							−0.455487	+	0.890242i	\(0.650535\pi\)
\(13\)	0.481090	−	6.11282i	0.133430	−	1.69539i	−0.457933	−	0.888987i	\(-0.651410\pi\)
							0.591363	−	0.806405i	\(-0.298590\pi\)
\(17\)	0.501697	−	2.08972i	0.121679	−	0.506831i	−0.877927	−	0.478794i	\(-0.841074\pi\)
							0.999607	−	0.0280376i	\(-0.00892582\pi\)
\(19\)	0.0229300	+	0.291353i	0.00526050	+	0.0668410i	0.998987	−	0.0449945i	\(-0.0143270\pi\)
							−0.993727	+	0.111836i	\(0.964327\pi\)
\(23\)	2.60870	+	8.02874i	0.543951	+	1.67411i	0.723472	+	0.690354i	\(0.242545\pi\)
							−0.179521	+	0.983754i	\(0.557455\pi\)
\(29\)	0.356352	+	1.48431i	0.0661730	+	0.275630i	0.995641	−	0.0932736i	\(-0.0297331\pi\)
							−0.929468	+	0.368904i	\(0.879733\pi\)
\(31\)	−2.28079	+	3.13924i	−0.409642	+	0.563824i	−0.963131	−	0.269033i	\(-0.913296\pi\)
							0.553489	+	0.832856i	\(0.313296\pi\)
\(37\)	−2.56163	+	1.86114i	−0.421130	+	0.305969i	−0.778092	−	0.628150i	\(-0.783812\pi\)
							0.356962	+	0.934119i	\(0.383812\pi\)
\(41\)	3.45107	−	5.39353i	0.538966	−	0.842327i
							\(43\)	−2.08816	−	1.06397i	−0.318442	−	0.162254i	0.287462	−	0.957792i	\(-0.407188\pi\)
−0.605904	+	0.795538i	\(0.707188\pi\)
\(47\)	−1.05505	+	0.901098i	−0.153895	+	0.131439i	−0.723062	−	0.690783i	\(-0.757266\pi\)
							0.569167	+	0.822222i	\(0.307266\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.110.12	yes	192
3.2	odd	2	inner	123.2.o.a.110.1	yes	192
41.22	odd	40	inner	123.2.o.a.104.1	✓	192
123.104	even	40	inner	123.2.o.a.104.12	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.104.1	✓	192	41.22	odd	40	inner
123.2.o.a.104.12	yes	192	123.104	even	40	inner
123.2.o.a.110.1	yes	192	3.2	odd	2	inner
123.2.o.a.110.12	yes	192	1.1	even	1	trivial