Embedded newform 111.2.q.b.56.10

• Label: 111.2.q.b.56.10
• Level: $111$
• Weight: $2$
• Character: 111.56
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			56.10
Character	\(\chi\)	\(=\)	111.56
Dual form			111.2.q.b.2.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.236515 + 2.70338i) q^{2} +(-1.48045 - 0.899037i) q^{3} +(-5.28271 + 0.931484i) q^{4} +(-2.06833 + 0.964480i) q^{5} +(2.08029 - 4.21485i) q^{6} +(1.79445 + 0.653127i) q^{7} +(-2.36288 - 8.81837i) q^{8} +(1.38347 + 2.66196i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{35}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.236515	+	2.70338i	0.167241	+	1.91158i	0.364461	+	0.931219i	\(0.381253\pi\)
							−0.197220	+	0.980359i	\(0.563191\pi\)
\(3\)	−1.48045	−	0.899037i	−0.854738	−	0.519059i
							\(5\)	−2.06833	+	0.964480i	−0.924987	+	0.431328i	−0.825942	−	0.563755i	\(-0.809356\pi\)
−0.0990444	+	0.995083i	\(0.531579\pi\)
\(7\)	1.79445	+	0.653127i	0.678239	+	0.246859i	0.658091	−	0.752938i	\(-0.271364\pi\)
							0.0201477	+	0.999797i	\(0.493586\pi\)
\(11\)	−1.78148	+	3.08562i	−0.537137	+	0.930348i	0.461920	+	0.886922i	\(0.347161\pi\)
							−0.999057	+	0.0434266i	\(0.986173\pi\)
\(13\)	−1.71815	+	2.45377i	−0.476528	+	0.680553i	−0.983396	−	0.181471i	\(-0.941914\pi\)
							0.506868	+	0.862024i	\(0.330803\pi\)
\(17\)	0.742402	+	1.06026i	0.180059	+	0.257151i	0.899028	−	0.437891i	\(-0.144275\pi\)
							−0.718969	+	0.695042i	\(0.755386\pi\)
\(19\)	1.14170	+	0.0998855i	0.261923	+	0.0229153i	0.217361	−	0.976091i	\(-0.430255\pi\)
							0.0445622	+	0.999007i	\(0.485811\pi\)
\(23\)	4.25357	+	1.13974i	0.886930	+	0.237652i	0.673395	−	0.739283i	\(-0.264836\pi\)
							0.213535	+	0.976935i	\(0.431502\pi\)
\(29\)	−2.33426	+	0.625462i	−0.433461	+	0.116145i	−0.468950	−	0.883225i	\(-0.655368\pi\)
							0.0354895	+	0.999370i	\(0.488701\pi\)
\(31\)	5.97281	−	5.97281i	1.07275	−	1.07275i	0.0756111	−	0.997137i	\(-0.475909\pi\)
							0.997137	−	0.0756111i	\(-0.0240908\pi\)
\(37\)	−1.22308	+	5.95853i	−0.201073	+	0.979576i
							\(41\)	−0.817769	−	4.63780i	−0.127714	−	0.724302i	−0.979659	−	0.200671i	\(-0.935688\pi\)
0.851945	−	0.523632i	\(-0.175423\pi\)
\(43\)	6.64208	+	6.64208i	1.01291	+	1.01291i	0.999916	+	0.0129913i	\(0.00413539\pi\)
							0.0129913	+	0.999916i	\(0.495865\pi\)
\(47\)	−3.72007	+	2.14778i	−0.542628	+	0.313286i	−0.746143	−	0.665785i	\(-0.768097\pi\)
							0.203516	+	0.979072i	\(0.434763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.56.10	yes	120
3.2	odd	2	inner	111.2.q.b.56.1	yes	120
37.2	odd	36	inner	111.2.q.b.2.1	✓	120
111.2	even	36	inner	111.2.q.b.2.10	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.2.1	✓	120	37.2	odd	36	inner
111.2.q.b.2.10	yes	120	111.2	even	36	inner
111.2.q.b.56.1	yes	120	3.2	odd	2	inner
111.2.q.b.56.10	yes	120	1.1	even	1	trivial