Embedded newform 144.2.u.a.131.21

• Label: 144.2.u.a.131.21
• Level: $144$
• Weight: $2$
• Character: 144.131
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			131.21
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37902 + 0.313539i) q^{2} +(-0.412428 - 1.68223i) q^{3} +(1.80339 + 0.864752i) q^{4} +(0.289971 - 0.0776974i) q^{5} +(-0.0413015 - 2.44914i) q^{6} +(-0.374023 - 0.647827i) q^{7} +(2.21577 + 1.75794i) q^{8} +(-2.65981 + 1.38760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37902	+	0.313539i	0.975114	+	0.221705i
							\(3\)	−0.412428	−	1.68223i	−0.238116	−	0.971237i
\(5\)	0.289971	−	0.0776974i	0.129679	−	0.0347473i								−0.193396	−	0.981121i	\(-0.561950\pi\)
							0.323075	+	0.946373i	\(0.395283\pi\)
\(7\)	−0.374023	−	0.647827i	−0.141367	−	0.244855i	0.786644	−	0.617406i	\(-0.211817\pi\)
							−0.928012	+	0.372551i	\(0.878483\pi\)
\(11\)	−2.23720	−	0.599457i	−0.674542	−	0.180743i	−0.0947421	−	0.995502i	\(-0.530203\pi\)
							−0.579800	+	0.814759i	\(0.696869\pi\)
\(13\)	1.60318	−	0.429571i	0.444642	−	0.119142i	−0.0295491	−	0.999563i	\(-0.509407\pi\)
							0.474191	+	0.880422i	\(0.342740\pi\)
\(17\)			6.74518i			1.63595i	0.575256	+	0.817973i	\(0.304902\pi\)
							−0.575256	+	0.817973i	\(0.695098\pi\)
\(19\)	0.621335	−	0.621335i	0.142544	−	0.142544i	−0.632234	−	0.774778i	\(-0.717862\pi\)
							0.774778	+	0.632234i	\(0.217862\pi\)
\(23\)	−6.06191	−	3.49985i	−1.26400	−	0.729769i	−0.290151	−	0.956981i	\(-0.593706\pi\)
							−0.973845	+	0.227212i	\(0.927039\pi\)
\(29\)	−5.44240	−	1.45829i	−1.01063	−	0.270797i	−0.284736	−	0.958606i	\(-0.591906\pi\)
							−0.725892	+	0.687809i	\(0.758573\pi\)
\(31\)	3.13647	+	1.81084i	0.563326	+	0.325236i	0.754479	−	0.656324i	\(-0.227889\pi\)
							−0.191153	+	0.981560i	\(0.561223\pi\)
\(37\)	6.74053	−	6.74053i	1.10814	−	1.10814i	0.114740	−	0.993396i	\(-0.463396\pi\)
							0.993396	−	0.114740i	\(-0.0366036\pi\)
\(41\)	−1.39492	+	2.41607i	−0.217850	+	0.377327i	−0.954150	−	0.299328i	\(-0.903238\pi\)
							0.736301	+	0.676655i	\(0.236571\pi\)
\(43\)	1.89907	−	7.08744i	0.289606	−	1.08082i	−0.655801	−	0.754934i	\(-0.727669\pi\)
							0.945407	−	0.325891i	\(-0.105664\pi\)
\(47\)	0.307120	+	0.531947i	0.0447980	+	0.0775924i	0.887555	−	0.460702i	\(-0.152402\pi\)
							−0.842757	+	0.538294i	\(0.819069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.21	yes	88
3.2	odd	2		432.2.v.a.179.2		88
4.3	odd	2		576.2.y.a.239.12		88
9.2	odd	6	inner	144.2.u.a.83.15	yes	88
9.7	even	3		432.2.v.a.35.8		88
12.11	even	2		1728.2.z.a.1583.11		88
16.5	even	4		576.2.y.a.527.2		88
16.11	odd	4	inner	144.2.u.a.59.15	yes	88
36.7	odd	6		1728.2.z.a.1007.11		88
36.11	even	6		576.2.y.a.47.2		88
48.5	odd	4		1728.2.z.a.719.11		88
48.11	even	4		432.2.v.a.395.8		88
144.11	even	12	inner	144.2.u.a.11.21	✓	88
144.43	odd	12		432.2.v.a.251.2		88
144.101	odd	12		576.2.y.a.335.12		88
144.133	even	12		1728.2.z.a.143.11		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.21	✓	88	144.11	even	12	inner
144.2.u.a.59.15	yes	88	16.11	odd	4	inner
144.2.u.a.83.15	yes	88	9.2	odd	6	inner
144.2.u.a.131.21	yes	88	1.1	even	1	trivial
432.2.v.a.35.8		88	9.7	even	3
432.2.v.a.179.2		88	3.2	odd	2
432.2.v.a.251.2		88	144.43	odd	12
432.2.v.a.395.8		88	48.11	even	4
576.2.y.a.47.2		88	36.11	even	6
576.2.y.a.239.12		88	4.3	odd	2
576.2.y.a.335.12		88	144.101	odd	12
576.2.y.a.527.2		88	16.5	even	4
1728.2.z.a.143.11		88	144.133	even	12
1728.2.z.a.719.11		88	48.5	odd	4
1728.2.z.a.1007.11		88	36.7	odd	6
1728.2.z.a.1583.11		88	12.11	even	2