Embedded newform 144.2.u.a.131.2

• Label: 144.2.u.a.131.2
• 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			131.2
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39618 - 0.225141i) q^{2} +(-1.36987 - 1.05993i) q^{3} +(1.89862 + 0.628673i) q^{4} +(-2.78704 + 0.746784i) q^{5} +(1.67395 + 1.78827i) q^{6} +(1.16672 + 2.02082i) q^{7} +(-2.50928 - 1.30520i) q^{8} +(0.753089 + 2.90394i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39618	−	0.225141i	−0.987247	−	0.159198i
							\(3\)	−1.36987	−	1.05993i	−0.790895	−	0.611952i
\(5\)	−2.78704	+	0.746784i	−1.24640	+	0.333972i								−0.820945	−	0.571007i	\(-0.806553\pi\)
							−0.425456	+	0.904979i	\(0.639886\pi\)
\(7\)	1.16672	+	2.02082i	0.440979	+	0.763799i	0.997762	−	0.0668592i	\(-0.0212978\pi\)
							−0.556783	+	0.830658i	\(0.687964\pi\)
\(11\)	5.53982	+	1.48439i	1.67032	+	0.447560i	0.965196	−	0.261527i	\(-0.0842259\pi\)
							0.705121	+	0.709087i	\(0.250893\pi\)
\(13\)	−3.90572	+	1.04654i	−1.08325	+	0.290257i	−0.755928	−	0.654655i	\(-0.772814\pi\)
							−0.327325	+	0.944912i	\(0.606147\pi\)
\(17\)			6.45489i			1.56554i	0.622310	+	0.782771i	\(0.286194\pi\)
							−0.622310	+	0.782771i	\(0.713806\pi\)
\(19\)	−1.50499	+	1.50499i	−0.345267	+	0.345267i	−0.858343	−	0.513076i	\(-0.828506\pi\)
							0.513076	+	0.858343i	\(0.328506\pi\)
\(23\)	−0.0418190	−	0.0241442i	−0.00871987	−	0.00503442i	0.495634	−	0.868532i	\(-0.334936\pi\)
							−0.504354	+	0.863497i	\(0.668269\pi\)
\(29\)	−5.08507	−	1.36254i	−0.944274	−	0.253018i	−0.246343	−	0.969183i	\(-0.579229\pi\)
							−0.697931	+	0.716165i	\(0.745896\pi\)
\(31\)	−1.65029	−	0.952797i	−0.296401	−	0.171127i	0.344424	−	0.938814i	\(-0.388074\pi\)
							−0.640825	+	0.767687i	\(0.721408\pi\)
\(37\)	0.489763	−	0.489763i	0.0805165	−	0.0805165i	−0.665702	−	0.746218i	\(-0.731868\pi\)
							0.746218	+	0.665702i	\(0.231868\pi\)
\(41\)	0.0155357	−	0.0269087i	0.00242628	−	0.00420243i	−0.864810	−	0.502100i	\(-0.832561\pi\)
							0.867236	+	0.497897i	\(0.165894\pi\)
\(43\)	−1.01892	+	3.80267i	−0.155384	+	0.579901i	0.843688	+	0.536834i	\(0.180380\pi\)
							−0.999072	+	0.0430675i	\(0.986287\pi\)
\(47\)	0.0913288	+	0.158186i	0.0133217	+	0.0230738i	0.872609	−	0.488419i	\(-0.162426\pi\)
							−0.859288	+	0.511493i	\(0.829093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.2	yes	88
3.2	odd	2		432.2.v.a.179.21		88
4.3	odd	2		576.2.y.a.239.18		88
9.2	odd	6	inner	144.2.u.a.83.10	yes	88
9.7	even	3		432.2.v.a.35.13		88
12.11	even	2		1728.2.z.a.1583.19		88
16.5	even	4		576.2.y.a.527.7		88
16.11	odd	4	inner	144.2.u.a.59.10	yes	88
36.7	odd	6		1728.2.z.a.1007.19		88
36.11	even	6		576.2.y.a.47.7		88
48.5	odd	4		1728.2.z.a.719.19		88
48.11	even	4		432.2.v.a.395.13		88
144.11	even	12	inner	144.2.u.a.11.2	✓	88
144.43	odd	12		432.2.v.a.251.21		88
144.101	odd	12		576.2.y.a.335.18		88
144.133	even	12		1728.2.z.a.143.19		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.2	✓	88	144.11	even	12	inner
144.2.u.a.59.10	yes	88	16.11	odd	4	inner
144.2.u.a.83.10	yes	88	9.2	odd	6	inner
144.2.u.a.131.2	yes	88	1.1	even	1	trivial
432.2.v.a.35.13		88	9.7	even	3
432.2.v.a.179.21		88	3.2	odd	2
432.2.v.a.251.21		88	144.43	odd	12
432.2.v.a.395.13		88	48.11	even	4
576.2.y.a.47.7		88	36.11	even	6
576.2.y.a.239.18		88	4.3	odd	2
576.2.y.a.335.18		88	144.101	odd	12
576.2.y.a.527.7		88	16.5	even	4
1728.2.z.a.143.19		88	144.133	even	12
1728.2.z.a.719.19		88	48.5	odd	4
1728.2.z.a.1007.19		88	36.7	odd	6
1728.2.z.a.1583.19		88	12.11	even	2