Embedded newform 144.2.u.a.131.3

• Label: 144.2.u.a.131.3
• 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			131.3
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20521 - 0.739910i) q^{2} +(-0.679468 + 1.59321i) q^{3} +(0.905065 + 1.78350i) q^{4} +(-2.39818 + 0.642590i) q^{5} +(1.99774 - 1.41741i) q^{6} +(-1.93190 - 3.34616i) q^{7} +(0.228833 - 2.81916i) q^{8} +(-2.07665 - 2.16507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20521	−	0.739910i	−0.852213	−	0.523196i
							\(3\)	−0.679468	+	1.59321i	−0.392291	+	0.919841i
\(5\)	−2.39818	+	0.642590i	−1.07250	+	0.287375i								−0.751519	−	0.659711i	\(-0.770679\pi\)
							−0.320979	+	0.947086i	\(0.604012\pi\)
\(7\)	−1.93190	−	3.34616i	−0.730191	−	1.26473i	−0.956801	−	0.290742i	\(-0.906098\pi\)
							0.226610	−	0.973985i	\(-0.427236\pi\)
\(11\)	−4.01810	−	1.07665i	−1.21150	−	0.324621i	−0.404151	−	0.914692i	\(-0.632433\pi\)
							−0.807351	+	0.590071i	\(0.799100\pi\)
\(13\)	3.17330	−	0.850284i	0.880116	−	0.235826i	0.209659	−	0.977775i	\(-0.432765\pi\)
							0.670457	+	0.741948i	\(0.266098\pi\)
\(17\)			1.33161i			0.322963i	0.986876	+	0.161482i	\(0.0516272\pi\)
							−0.986876	+	0.161482i	\(0.948373\pi\)
\(19\)	−6.09021	+	6.09021i	−1.39719	+	1.39719i	−0.589208	+	0.807981i	\(0.700560\pi\)
							−0.807981	+	0.589208i	\(0.799440\pi\)
\(23\)	0.521462	+	0.301066i	0.108732	+	0.0627767i	0.553380	−	0.832929i	\(-0.313338\pi\)
							−0.444648	+	0.895706i	\(0.646671\pi\)
\(29\)	0.272752	+	0.0730837i	0.0506488	+	0.0135713i	0.284054	−	0.958808i	\(-0.408320\pi\)
							−0.233406	+	0.972379i	\(0.574987\pi\)
\(31\)	−5.84901	−	3.37693i	−1.05051	−	0.606514i	−0.127721	−	0.991810i	\(-0.540766\pi\)
							−0.922793	+	0.385296i	\(0.874099\pi\)
\(37\)	0.00346351	−	0.00346351i	0.000569398	−	0.000569398i	−0.706822	−	0.707391i	\(-0.749872\pi\)
							0.707391	+	0.706822i	\(0.249872\pi\)
\(41\)	−0.614318	+	1.06403i	−0.0959403	+	0.166174i	−0.910001	−	0.414607i	\(-0.863919\pi\)
							0.814060	+	0.580780i	\(0.197252\pi\)
\(43\)	−0.151042	+	0.563698i	−0.0230337	+	0.0859631i	−0.976486	−	0.215581i	\(-0.930835\pi\)
							0.953452	+	0.301544i	\(0.0975021\pi\)
\(47\)	1.24972	+	2.16458i	0.182290	+	0.315736i	0.942660	−	0.333754i	\(-0.108316\pi\)
							−0.760370	+	0.649490i	\(0.774982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.3	yes	88
3.2	odd	2		432.2.v.a.179.20		88
4.3	odd	2		576.2.y.a.239.14		88
9.2	odd	6	inner	144.2.u.a.83.12	yes	88
9.7	even	3		432.2.v.a.35.11		88
12.11	even	2		1728.2.z.a.1583.17		88
16.5	even	4		576.2.y.a.527.19		88
16.11	odd	4	inner	144.2.u.a.59.12	yes	88
36.7	odd	6		1728.2.z.a.1007.17		88
36.11	even	6		576.2.y.a.47.19		88
48.5	odd	4		1728.2.z.a.719.17		88
48.11	even	4		432.2.v.a.395.11		88
144.11	even	12	inner	144.2.u.a.11.3	✓	88
144.43	odd	12		432.2.v.a.251.20		88
144.101	odd	12		576.2.y.a.335.14		88
144.133	even	12		1728.2.z.a.143.17		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.3	✓	88	144.11	even	12	inner
144.2.u.a.59.12	yes	88	16.11	odd	4	inner
144.2.u.a.83.12	yes	88	9.2	odd	6	inner
144.2.u.a.131.3	yes	88	1.1	even	1	trivial
432.2.v.a.35.11		88	9.7	even	3
432.2.v.a.179.20		88	3.2	odd	2
432.2.v.a.251.20		88	144.43	odd	12
432.2.v.a.395.11		88	48.11	even	4
576.2.y.a.47.19		88	36.11	even	6
576.2.y.a.239.14		88	4.3	odd	2
576.2.y.a.335.14		88	144.101	odd	12
576.2.y.a.527.19		88	16.5	even	4
1728.2.z.a.143.17		88	144.133	even	12
1728.2.z.a.719.17		88	48.5	odd	4
1728.2.z.a.1007.17		88	36.7	odd	6
1728.2.z.a.1583.17		88	12.11	even	2