Embedded newform 123.2.o.a.11.10

• Label: 123.2.o.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	123.11
Dual form			123.2.o.a.56.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.301761 + 1.90525i) q^{2} +(1.57802 + 0.714033i) q^{3} +(-1.63679 + 0.531826i) q^{4} +(-1.77427 - 0.904038i) q^{5} +(-0.884223 + 3.22199i) q^{6} +(-1.07322 + 0.257656i) q^{7} +(0.244312 + 0.479489i) q^{8} +(1.98031 + 2.25352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.301761	+	1.90525i	0.213378	+	1.34721i	0.829034	+	0.559198i	\(0.188891\pi\)
							−0.615657	+	0.788015i	\(0.711109\pi\)
\(3\)	1.57802	+	0.714033i	0.911072	+	0.412247i
							\(5\)	−1.77427	−	0.904038i	−0.793480	−	0.404298i	0.00976119	−	0.999952i	\(-0.496893\pi\)
−0.803241	+	0.595654i	\(0.796893\pi\)
\(7\)	−1.07322	+	0.257656i	−0.405637	+	0.0973849i	−0.431129	−	0.902290i	\(-0.641885\pi\)
							0.0254921	+	0.999675i	\(0.491885\pi\)
\(11\)	2.12627	−	2.48954i	0.641094	−	0.750625i	−0.340783	−	0.940142i	\(-0.610692\pi\)
							0.981877	+	0.189517i	\(0.0606923\pi\)
\(13\)	−2.82122	−	4.60382i	−0.782467	−	1.27687i	−0.956385	−	0.292109i	\(-0.905643\pi\)
							0.173918	−	0.984760i	\(-0.444357\pi\)
\(17\)	4.73071	−	0.372315i	1.14736	−	0.0902995i	0.509516	−	0.860461i	\(-0.329825\pi\)
							0.637849	+	0.770162i	\(0.279825\pi\)
\(19\)	0.0743523	−	0.121332i	0.0170576	−	0.0278355i	−0.843998	−	0.536347i	\(-0.819804\pi\)
							0.861055	+	0.508511i	\(0.169804\pi\)
\(23\)	−1.03955	−	0.755278i	−0.216761	−	0.157486i	0.474106	−	0.880468i	\(-0.342771\pi\)
							−0.690868	+	0.722981i	\(0.742771\pi\)
\(29\)	−8.15747	−	0.642007i	−1.51480	−	0.119218i	−0.706386	−	0.707827i	\(-0.749676\pi\)
							−0.808418	+	0.588609i	\(0.799676\pi\)
\(31\)	5.61688	+	1.82504i	1.00882	+	0.327786i	0.766385	−	0.642381i	\(-0.222053\pi\)
							0.242437	+	0.970167i	\(0.422053\pi\)
\(37\)	0.568938	+	1.75101i	0.0935328	+	0.287864i	0.986869	−	0.161525i	\(-0.0516414\pi\)
							−0.893336	+	0.449390i	\(0.851641\pi\)
\(41\)	1.21494	+	6.28681i	0.189741	+	0.981834i
							\(43\)	−1.98570	+	0.314504i	−0.302817	+	0.0479615i	−0.305994	−	0.952033i	\(-0.598989\pi\)
0.00317703	+	0.999995i	\(0.498989\pi\)
\(47\)	1.75268	−	7.30045i	0.255655	−	1.06488i	−0.685813	−	0.727778i	\(-0.740553\pi\)
							0.941468	−	0.337102i	\(-0.109447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.11.10	yes	192
3.2	odd	2	inner	123.2.o.a.11.3	✓	192
41.15	odd	40	inner	123.2.o.a.56.3	yes	192
123.56	even	40	inner	123.2.o.a.56.10	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.11.3	✓	192	3.2	odd	2	inner
123.2.o.a.11.10	yes	192	1.1	even	1	trivial
123.2.o.a.56.3	yes	192	41.15	odd	40	inner
123.2.o.a.56.10	yes	192	123.56	even	40	inner