Embedded newform 123.2.o.a.11.2

• Label: 123.2.o.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	123.11
Dual form			123.2.o.a.56.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.359723 - 2.27120i) q^{2} +(0.374897 - 1.69099i) q^{3} +(-3.12684 + 1.01597i) q^{4} +(0.136866 + 0.0697367i) q^{5} +(-3.97544 - 0.243178i) q^{6} +(1.38834 - 0.333310i) q^{7} +(1.34435 + 2.63844i) q^{8} +(-2.71890 - 1.26790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{3} - 40 q^{4} - 4 q^{6} - 32 q^{7} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.359723	−	2.27120i	−0.254362	−	1.60598i	−0.702277	−	0.711904i	\(-0.747833\pi\)
							0.447914	−	0.894077i	\(-0.352167\pi\)
\(3\)	0.374897	−	1.69099i	0.216447	−	0.976294i
							\(5\)	0.136866	+	0.0697367i	0.0612083	+	0.0311872i	0.484327	−	0.874887i	\(-0.339065\pi\)
−0.423118	+	0.906074i	\(0.639065\pi\)
\(7\)	1.38834	−	0.333310i	0.524742	−	0.125979i	0.0375846	−	0.999293i	\(-0.488034\pi\)
							0.487158	+	0.873314i	\(0.338034\pi\)
\(11\)	−0.659760	+	0.772480i	−0.198925	+	0.232912i	−0.850898	−	0.525331i	\(-0.823942\pi\)
							0.651973	+	0.758242i	\(0.273942\pi\)
\(13\)	2.99551	+	4.88822i	0.830804	+	1.35575i	0.932588	+	0.360942i	\(0.117545\pi\)
							−0.101784	+	0.994806i	\(0.532455\pi\)
\(17\)	4.60624	−	0.362519i	1.11718	−	0.0879238i	0.493603	−	0.869687i	\(-0.335680\pi\)
							0.623575	+	0.781764i	\(0.285680\pi\)
\(19\)	2.40069	−	3.91757i	0.550756	−	0.898751i	−0.449230	−	0.893416i	\(-0.648302\pi\)
							0.999986	−	0.00533536i	\(-0.00169831\pi\)
\(23\)	−5.59085	−	4.06199i	−1.16577	−	0.846983i	−0.175276	−	0.984519i	\(-0.556082\pi\)
							−0.990497	+	0.137536i	\(0.956082\pi\)
\(29\)	0.0606223	+	0.00477108i	0.0112573	+	0.000885968i	0.0840865	−	0.996458i	\(-0.473203\pi\)
							−0.0728292	+	0.997344i	\(0.523203\pi\)
\(31\)	4.62927	+	1.50414i	0.831442	+	0.270152i	0.693652	−	0.720310i	\(-0.256000\pi\)
							0.137789	+	0.990462i	\(0.456000\pi\)
\(37\)	0.402856	+	1.23986i	0.0662291	+	0.203832i	0.978695	−	0.205322i	\(-0.0658240\pi\)
							−0.912466	+	0.409154i	\(0.865824\pi\)
\(41\)	−0.530223	+	6.38113i	−0.0828070	+	0.996566i
							\(43\)	8.28994	−	1.31300i	1.26420	−	0.200230i	0.511899	−	0.859046i	\(-0.328942\pi\)
0.752305	+	0.658815i	\(0.228942\pi\)
\(47\)	−1.46050	+	6.08340i	−0.213035	+	0.887355i	0.758730	+	0.651405i	\(0.225820\pi\)
							−0.971765	+	0.235950i	\(0.924180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.o.a.11.2	✓	192
3.2	odd	2	inner	123.2.o.a.11.11	yes	192
41.15	odd	40	inner	123.2.o.a.56.11	yes	192
123.56	even	40	inner	123.2.o.a.56.2	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.o.a.11.2	✓	192	1.1	even	1	trivial
123.2.o.a.11.11	yes	192	3.2	odd	2	inner
123.2.o.a.56.2	yes	192	123.56	even	40	inner
123.2.o.a.56.11	yes	192	41.15	odd	40	inner