Embedded newform 123.2.o.a.11.5

• Label: 123.2.o.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	123.11
Dual form			123.2.o.a.56.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.161468 - 1.01947i) q^{2} +(-1.38568 - 1.03918i) q^{3} +(0.888868 - 0.288811i) q^{4} +(-2.26856 - 1.15589i) q^{5} +(-0.835670 + 1.58045i) q^{6} +(-1.08656 + 0.260860i) q^{7} +(-1.37515 - 2.69889i) q^{8} +(0.840207 + 2.87994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.161468	−	1.01947i	−0.114175	−	0.720873i	−0.976660	−	0.214791i	\(-0.931093\pi\)
							0.862485	−	0.506083i	\(-0.168907\pi\)
\(3\)	−1.38568	−	1.03918i	−0.800022	−	0.599971i
							\(5\)	−2.26856	−	1.15589i	−1.01453	−	0.516929i	−0.134032	−	0.990977i	\(-0.542792\pi\)
−0.880499	+	0.474048i	\(0.842792\pi\)
\(7\)	−1.08656	+	0.260860i	−0.410681	+	0.0985959i	−0.433521	−	0.901143i	\(-0.642729\pi\)
							0.0228401	+	0.999739i	\(0.492729\pi\)
\(11\)	1.53792	−	1.80067i	0.463700	−	0.542923i	−0.478450	−	0.878115i	\(-0.658801\pi\)
							0.942150	+	0.335192i	\(0.108801\pi\)
\(13\)	−0.726295	−	1.18521i	−0.201438	−	0.328717i	0.735930	−	0.677058i	\(-0.236745\pi\)
							−0.937368	+	0.348341i	\(0.886745\pi\)
\(17\)	−1.52684	+	0.120165i	−0.370314	+	0.0291443i	−0.262251	−	0.965000i	\(-0.584465\pi\)
							−0.108063	+	0.994144i	\(0.534465\pi\)
\(19\)	3.34563	−	5.45957i	0.767540	−	1.25251i	−0.194849	−	0.980833i	\(-0.562422\pi\)
							0.962388	−	0.271678i	\(-0.0875784\pi\)
\(23\)	7.14455	+	5.19082i	1.48974	+	1.08236i	0.974251	+	0.225465i	\(0.0723900\pi\)
							0.515490	+	0.856896i	\(0.327610\pi\)
\(29\)	6.38040	+	0.502148i	1.18481	+	0.0932466i	0.655509	−	0.755187i	\(-0.272454\pi\)
							0.529301	+	0.848434i	\(0.322454\pi\)
\(31\)	5.33542	+	1.73358i	0.958271	+	0.311361i	0.746072	−	0.665866i	\(-0.231938\pi\)
							0.212199	+	0.977227i	\(0.431938\pi\)
\(37\)	−0.788497	−	2.42675i	−0.129628	−	0.398955i	0.865088	−	0.501621i	\(-0.167263\pi\)
							−0.994716	+	0.102666i	\(0.967263\pi\)
\(41\)	−4.94656	+	4.06590i	−0.772523	+	0.634986i
							\(43\)	−6.72522	+	1.06517i	−1.02559	+	0.162437i	−0.646492	−	0.762921i	\(-0.723765\pi\)
−0.379095	+	0.925358i	\(0.623765\pi\)
\(47\)	1.86741	−	7.77832i	0.272390	−	1.13459i	−0.653541	−	0.756891i	\(-0.726717\pi\)
							0.925931	−	0.377694i	\(-0.123283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.11.5	✓	192
3.2	odd	2	inner	123.2.o.a.11.8	yes	192
41.15	odd	40	inner	123.2.o.a.56.8	yes	192
123.56	even	40	inner	123.2.o.a.56.5	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.11.5	✓	192	1.1	even	1	trivial
123.2.o.a.11.8	yes	192	3.2	odd	2	inner
123.2.o.a.56.5	yes	192	123.56	even	40	inner
123.2.o.a.56.8	yes	192	41.15	odd	40	inner