Embedded newform 111.2.m.a.23.9

• Label: 111.2.m.a.23.9
• Level: $111$
• Weight: $2$
• Character: 111.23
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.m (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.9
Character	\(\chi\)	\(=\)	111.23
Dual form			111.2.m.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.05489 - 0.550605i) q^{2} +(-1.12774 + 1.31461i) q^{3} +(2.18734 - 1.26286i) q^{4} +(1.01486 + 0.271931i) q^{5} +(-1.59354 + 3.32232i) q^{6} +(-0.456855 - 0.791296i) q^{7} +(0.790833 - 0.790833i) q^{8} +(-0.456423 - 2.96508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 6 q^{3} - 12 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.05489	−	0.550605i	1.45302	−	0.389337i	0.555949	−	0.831216i	\(-0.312355\pi\)
							0.897074	+	0.441880i	\(0.145688\pi\)
\(3\)	−1.12774	+	1.31461i	−0.651099	+	0.758993i
							\(5\)	1.01486	+	0.271931i	0.453860	+	0.121611i	0.478505	−	0.878085i	\(-0.341179\pi\)
−0.0246452	+	0.999696i	\(0.507846\pi\)
\(7\)	−0.456855	−	0.791296i	−0.172675	−	0.299082i	0.766679	−	0.642030i	\(-0.221908\pi\)
							−0.939354	+	0.342949i	\(0.888574\pi\)
\(11\)	−4.31253			−1.30028			−0.650138	−	0.759816i	\(-0.725289\pi\)
							−0.650138	+	0.759816i	\(0.725289\pi\)
\(13\)	1.38417	−	5.16578i	0.383899	−	1.43273i	−0.455997	−	0.889981i	\(-0.650717\pi\)
							0.839896	−	0.542748i	\(-0.182616\pi\)
\(17\)	1.19475	+	4.45889i	0.289771	+	1.08144i	0.945282	+	0.326254i	\(0.105786\pi\)
							−0.655512	+	0.755185i	\(0.727547\pi\)
\(19\)	−0.494283	+	1.84469i	−0.113396	+	0.423201i	−0.999162	−	0.0409320i	\(-0.986967\pi\)
							0.885766	+	0.464133i	\(0.153634\pi\)
\(23\)	−0.392323	+	0.392323i	−0.0818049	+	0.0818049i	−0.746825	−	0.665020i	\(-0.768423\pi\)
							0.665020	+	0.746825i	\(0.268423\pi\)
\(29\)	0.841388	+	0.841388i	0.156242	+	0.156242i	0.780899	−	0.624657i	\(-0.214761\pi\)
							−0.624657	+	0.780899i	\(0.714761\pi\)
\(31\)	4.29926	−	4.29926i	0.772170	−	0.772170i	−0.206316	−	0.978485i	\(-0.566147\pi\)
							0.978485	+	0.206316i	\(0.0661475\pi\)
\(37\)	5.83424	+	1.72096i	0.959143	+	0.282923i
							\(41\)	3.89438	+	6.74526i	0.608199	+	1.05343i	0.991537	+	0.129824i	\(0.0414413\pi\)
−0.383338	+	0.923608i	\(0.625225\pi\)
\(43\)	1.76813	+	1.76813i	0.269638	+	0.269638i	0.828954	−	0.559316i	\(-0.188936\pi\)
							−0.559316	+	0.828954i	\(0.688936\pi\)
\(47\)		−	6.56766i		−	0.957992i	−0.877817	−	0.478996i	\(-0.841001\pi\)
							0.877817	−	0.478996i	\(-0.158999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.m.a.23.9	yes	40
3.2	odd	2	inner	111.2.m.a.23.2	✓	40
37.29	odd	12	inner	111.2.m.a.29.2	yes	40
111.29	even	12	inner	111.2.m.a.29.9	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.m.a.23.2	✓	40	3.2	odd	2	inner
111.2.m.a.23.9	yes	40	1.1	even	1	trivial
111.2.m.a.29.2	yes	40	37.29	odd	12	inner
111.2.m.a.29.9	yes	40	111.29	even	12	inner