Embedded newform 111.2.q.b.17.3

• Label: 111.2.q.b.17.3
• Level: $111$
• Weight: $2$
• Character: 111.17
• 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			17.3
Character	\(\chi\)	\(=\)	111.17
Dual form			111.2.q.b.98.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976940 + 1.39522i) q^{2} +(0.212604 + 1.71895i) q^{3} +(-0.308173 - 0.846698i) q^{4} +(3.12344 - 0.273266i) q^{5} +(-2.60601 - 1.38269i) q^{6} +(1.23071 + 1.03269i) q^{7} +(-1.80802 - 0.484458i) q^{8} +(-2.90960 + 0.730914i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.976940	+	1.39522i	−0.690801	+	0.986566i	0.308573	+	0.951201i	\(0.400148\pi\)
							−0.999374	+	0.0353656i	\(0.988740\pi\)
\(3\)	0.212604	+	1.71895i	0.122747	+	0.992438i
							\(5\)	3.12344	−	0.273266i	1.39684	−	0.122208i	0.636274	−	0.771463i	\(-0.280475\pi\)
0.760571	+	0.649255i	\(0.224919\pi\)
\(7\)	1.23071	+	1.03269i	0.465165	+	0.390320i	0.845027	−	0.534723i	\(-0.179584\pi\)
							−0.379862	+	0.925043i	\(0.624029\pi\)
\(11\)	−2.35239	−	4.07446i	−0.709273	−	1.22850i	−0.965127	−	0.261782i	\(-0.915690\pi\)
							0.255854	−	0.966715i	\(-0.417643\pi\)
\(13\)	−1.52468	−	0.710972i	−0.422871	−	0.197188i	0.199521	−	0.979894i	\(-0.436061\pi\)
							−0.622392	+	0.782705i	\(0.713839\pi\)
\(17\)	2.71863	−	1.26772i	0.659365	−	0.307467i	−0.0639828	−	0.997951i	\(-0.520380\pi\)
							0.723347	+	0.690484i	\(0.242602\pi\)
\(19\)	1.08911	−	0.762601i	0.249858	−	0.174953i	−0.441937	−	0.897046i	\(-0.645709\pi\)
							0.691796	+	0.722093i	\(0.256820\pi\)
\(23\)	1.78454	+	6.65998i	0.372102	+	1.38870i	0.857534	+	0.514428i	\(0.171996\pi\)
							−0.485432	+	0.874275i	\(0.661338\pi\)
\(29\)	1.93925	−	7.23738i	0.360110	−	1.34395i	−0.513821	−	0.857897i	\(-0.671770\pi\)
							0.873931	−	0.486050i	\(-0.161563\pi\)
\(31\)	−6.54148	+	6.54148i	−1.17488	+	1.17488i	−0.193854	+	0.981030i	\(0.562099\pi\)
							−0.981030	+	0.193854i	\(0.937901\pi\)
\(37\)	3.53781	−	4.94812i	0.581613	−	0.813466i
							\(41\)	−1.36351	+	0.496275i	−0.212944	+	0.0775052i	−0.446289	−	0.894889i	\(-0.647255\pi\)
0.233346	+	0.972394i	\(0.425033\pi\)
\(43\)	1.21826	+	1.21826i	0.185783	+	0.185783i	0.793870	−	0.608087i	\(-0.208063\pi\)
							−0.608087	+	0.793870i	\(0.708063\pi\)
\(47\)	−0.259378	−	0.149752i	−0.0378342	−	0.0218436i	0.480964	−	0.876741i	\(-0.340287\pi\)
							−0.518798	+	0.854897i	\(0.673620\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.17.3	✓	120
3.2	odd	2	inner	111.2.q.b.17.8	yes	120
37.24	odd	36	inner	111.2.q.b.98.8	yes	120
111.98	even	36	inner	111.2.q.b.98.3	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.17.3	✓	120	1.1	even	1	trivial
111.2.q.b.17.8	yes	120	3.2	odd	2	inner
111.2.q.b.98.3	yes	120	111.98	even	36	inner
111.2.q.b.98.8	yes	120	37.24	odd	36	inner