Embedded newform 111.2.q.b.92.9

• Label: 111.2.q.b.92.9
• Level: $111$
• Weight: $2$
• Character: 111.92
• 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			92.9
Character	\(\chi\)	\(=\)	111.92
Dual form			111.2.q.b.35.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.92028 - 0.168002i) q^{2} +(0.456056 + 1.67093i) q^{3} +(1.68962 - 0.297925i) q^{4} +(-0.875519 - 1.87756i) q^{5} +(1.15647 + 3.13203i) q^{6} +(-2.22546 - 0.810002i) q^{7} +(-0.529381 + 0.141847i) q^{8} +(-2.58403 + 1.52408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.92028	−	0.168002i	1.35784	−	0.118796i	0.615139	−	0.788419i	\(-0.289100\pi\)
							0.742701	+	0.669623i	\(0.233544\pi\)
\(3\)	0.456056	+	1.67093i	0.263304	+	0.964713i
							\(5\)	−0.875519	−	1.87756i	−0.391544	−	0.839669i	−0.998974	−	0.0452953i	\(-0.985577\pi\)
0.607430	−	0.794373i	\(-0.292201\pi\)
\(7\)	−2.22546	−	0.810002i	−0.841146	−	0.306152i	−0.114721	−	0.993398i	\(-0.536597\pi\)
							−0.726425	+	0.687246i	\(0.758820\pi\)
\(11\)	1.34602	−	2.33138i	0.405842	−	0.702938i	−0.588577	−	0.808441i	\(-0.700312\pi\)
							0.994419	+	0.105503i	\(0.0336451\pi\)
\(13\)	4.27035	+	2.99013i	1.18438	+	0.829312i	0.988486	−	0.151313i	\(-0.0483500\pi\)
							0.195895	+	0.980625i	\(0.437239\pi\)
\(17\)	0.0267380	−	0.0187221i	0.00648491	−	0.00454078i	−0.570329	−	0.821416i	\(-0.693184\pi\)
							0.576814	+	0.816875i	\(0.304296\pi\)
\(19\)	0.263542	−	3.01230i	0.0604608	−	0.691070i	−0.904119	−	0.427280i	\(-0.859472\pi\)
							0.964580	−	0.263790i	\(-0.0849725\pi\)
\(23\)	−0.797677	+	2.97697i	−0.166327	+	0.620741i	0.831540	+	0.555465i	\(0.187460\pi\)
							−0.997867	+	0.0652765i	\(0.979207\pi\)
\(29\)	2.68631	+	10.0254i	0.498835	+	1.86168i	0.507396	+	0.861713i	\(0.330608\pi\)
							−0.00856133	+	0.999963i	\(0.502725\pi\)
\(31\)	−4.23361	−	4.23361i	−0.760378	−	0.760378i	0.216012	−	0.976391i	\(-0.430695\pi\)
							−0.976391	+	0.216012i	\(0.930695\pi\)
\(37\)	−0.530889	+	6.05955i	−0.0872776	+	0.996184i
							\(41\)	0.222930	+	1.26430i	0.0348159	+	0.197451i	0.997255	−	0.0740481i	\(-0.0235918\pi\)
−0.962439	+	0.271499i	\(0.912481\pi\)
\(43\)	7.05464	−	7.05464i	1.07582	−	1.07582i	0.0789429	−	0.996879i	\(-0.474846\pi\)
							0.996879	−	0.0789429i	\(-0.0251545\pi\)
\(47\)	2.29265	−	1.32366i	0.334418	−	0.193076i	−0.323383	−	0.946268i	\(-0.604820\pi\)
							0.657801	+	0.753192i	\(0.271487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.92.9	yes	120
3.2	odd	2	inner	111.2.q.b.92.2	yes	120
37.35	odd	36	inner	111.2.q.b.35.2	✓	120
111.35	even	36	inner	111.2.q.b.35.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.35.2	✓	120	37.35	odd	36	inner
111.2.q.b.35.9	yes	120	111.35	even	36	inner
111.2.q.b.92.2	yes	120	3.2	odd	2	inner
111.2.q.b.92.9	yes	120	1.1	even	1	trivial