Embedded newform 144.2.u.a.131.10

• Label: 144.2.u.a.131.10
• 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			131.10
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.354608 - 1.36903i) q^{2} +(-0.874828 + 1.49488i) q^{3} +(-1.74851 + 0.970941i) q^{4} +(1.76649 - 0.473330i) q^{5} +(2.35677 + 0.667570i) q^{6} +(1.40613 + 2.43549i) q^{7} +(1.94929 + 2.04946i) q^{8} +(-1.46935 - 2.61553i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.354608	−	1.36903i	−0.250746	−	0.968053i
							\(3\)	−0.874828	+	1.49488i	−0.505082	+	0.863071i
\(5\)	1.76649	−	0.473330i	0.789999	−	0.211679i								0.158810	−	0.987309i	\(-0.449234\pi\)
							0.631188	+	0.775630i	\(0.282568\pi\)
\(7\)	1.40613	+	2.43549i	0.531468	+	0.920530i	0.999325	+	0.0367260i	\(0.0116929\pi\)
							−0.467857	+	0.883804i	\(0.654974\pi\)
\(11\)	5.79446	+	1.55262i	1.74710	+	0.468133i	0.984002	−	0.178156i	\(-0.0570130\pi\)
							0.763093	+	0.646288i	\(0.223680\pi\)
\(13\)	−1.10810	+	0.296913i	−0.307330	+	0.0823489i	−0.409188	−	0.912450i	\(-0.634188\pi\)
							0.101858	+	0.994799i	\(0.467521\pi\)
\(17\)		−	0.699378i		−	0.169624i	−0.996397	−	0.0848120i	\(-0.972971\pi\)
							0.996397	−	0.0848120i	\(-0.0270290\pi\)
\(19\)	−2.01481	+	2.01481i	−0.462228	+	0.462228i	−0.899385	−	0.437157i	\(-0.855985\pi\)
							0.437157	+	0.899385i	\(0.355985\pi\)
\(23\)	−4.91498	−	2.83767i	−1.02485	−	0.591695i	−0.109341	−	0.994004i	\(-0.534874\pi\)
							−0.915504	+	0.402310i	\(0.868208\pi\)
\(29\)	5.41802	+	1.45175i	1.00610	+	0.269584i	0.723999	−	0.689801i	\(-0.242302\pi\)
							0.282101	+	0.959385i	\(0.408969\pi\)
\(31\)	−5.15522	−	2.97637i	−0.925905	−	0.534572i	−0.0403909	−	0.999184i	\(-0.512860\pi\)
							−0.885514	+	0.464612i	\(0.846194\pi\)
\(37\)	3.48490	−	3.48490i	0.572913	−	0.572913i	−0.360028	−	0.932941i	\(-0.617233\pi\)
							0.932941	+	0.360028i	\(0.117233\pi\)
\(41\)	1.55998	−	2.70196i	0.243628	−	0.421975i	−0.718117	−	0.695922i	\(-0.754996\pi\)
							0.961745	+	0.273947i	\(0.0883292\pi\)
\(43\)	−1.90129	+	7.09570i	−0.289944	+	1.08208i	0.655207	+	0.755450i	\(0.272581\pi\)
							−0.945151	+	0.326635i	\(0.894085\pi\)
\(47\)	−2.83634	−	4.91269i	−0.413723	−	0.716589i	0.581570	−	0.813496i	\(-0.302438\pi\)
							−0.995293	+	0.0969068i	\(0.969105\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.10	yes	88
3.2	odd	2		432.2.v.a.179.13		88
4.3	odd	2		576.2.y.a.239.16		88
9.2	odd	6	inner	144.2.u.a.83.17	yes	88
9.7	even	3		432.2.v.a.35.6		88
12.11	even	2		1728.2.z.a.1583.6		88
16.5	even	4		576.2.y.a.527.18		88
16.11	odd	4	inner	144.2.u.a.59.17	yes	88
36.7	odd	6		1728.2.z.a.1007.6		88
36.11	even	6		576.2.y.a.47.18		88
48.5	odd	4		1728.2.z.a.719.6		88
48.11	even	4		432.2.v.a.395.6		88
144.11	even	12	inner	144.2.u.a.11.10	✓	88
144.43	odd	12		432.2.v.a.251.13		88
144.101	odd	12		576.2.y.a.335.16		88
144.133	even	12		1728.2.z.a.143.6		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.10	✓	88	144.11	even	12	inner
144.2.u.a.59.17	yes	88	16.11	odd	4	inner
144.2.u.a.83.17	yes	88	9.2	odd	6	inner
144.2.u.a.131.10	yes	88	1.1	even	1	trivial
432.2.v.a.35.6		88	9.7	even	3
432.2.v.a.179.13		88	3.2	odd	2
432.2.v.a.251.13		88	144.43	odd	12
432.2.v.a.395.6		88	48.11	even	4
576.2.y.a.47.18		88	36.11	even	6
576.2.y.a.239.16		88	4.3	odd	2
576.2.y.a.335.16		88	144.101	odd	12
576.2.y.a.527.18		88	16.5	even	4
1728.2.z.a.143.6		88	144.133	even	12
1728.2.z.a.719.6		88	48.5	odd	4
1728.2.z.a.1007.6		88	36.7	odd	6
1728.2.z.a.1583.6		88	12.11	even	2