Embedded newform 144.2.u.a.59.12

• Label: 144.2.u.a.59.12
• Level: $144$
• Weight: $2$
• Character: 144.59
• 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			59.12
Character	\(\chi\)	\(=\)	144.59
Dual form			144.2.u.a.83.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0381759 + 1.41370i) q^{2} +(-1.59321 - 0.679468i) q^{3} +(-1.99709 + 0.107938i) q^{4} +(-0.642590 - 2.39818i) q^{5} +(0.899740 - 2.27826i) q^{6} +(-1.93190 - 3.34616i) q^{7} +(-0.228833 - 2.81916i) q^{8} +(2.07665 + 2.16507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0381759	+	1.41370i	0.0269945	+	0.999636i
							\(3\)	−1.59321	−	0.679468i	−0.919841	−	0.392291i
\(5\)	−0.642590	−	2.39818i	−0.287375	−	1.07250i								−0.947086	−	0.320979i	\(-0.895988\pi\)
							0.659711	−	0.751519i	\(-0.270679\pi\)
\(7\)	−1.93190	−	3.34616i	−0.730191	−	1.26473i	−0.956801	−	0.290742i	\(-0.906098\pi\)
							0.226610	−	0.973985i	\(-0.427236\pi\)
\(11\)	−1.07665	+	4.01810i	−0.324621	+	1.21150i	0.590071	+	0.807351i	\(0.299100\pi\)
							−0.914692	+	0.404151i	\(0.867567\pi\)
\(13\)	−0.850284	−	3.17330i	−0.235826	−	0.880116i	−0.977775	−	0.209659i	\(-0.932765\pi\)
							0.741948	−	0.670457i	\(-0.233902\pi\)
\(17\)			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	−0.807981	−	0.589208i	\(-0.799440\pi\)
							−0.589208	−	0.807981i	\(-0.700560\pi\)
\(23\)	0.521462	+	0.301066i	0.108732	+	0.0627767i	0.553380	−	0.832929i	\(-0.313338\pi\)
							−0.444648	+	0.895706i	\(0.646671\pi\)
\(29\)	0.0730837	−	0.272752i	0.0135713	−	0.0506488i	−0.958808	−	0.284054i	\(-0.908320\pi\)
							0.972379	+	0.233406i	\(0.0749870\pi\)
\(31\)	5.84901	+	3.37693i	1.05051	+	0.606514i	0.922793	−	0.385296i	\(-0.125901\pi\)
							0.127721	+	0.991810i	\(0.459234\pi\)
\(37\)	0.00346351	+	0.00346351i	0.000569398	+	0.000569398i	0.707391	−	0.706822i	\(-0.249872\pi\)
							−0.706822	+	0.707391i	\(0.749872\pi\)
\(41\)	0.614318	−	1.06403i	0.0959403	−	0.166174i	−0.814060	−	0.580780i	\(-0.802748\pi\)
							0.910001	+	0.414607i	\(0.136081\pi\)
\(43\)	0.563698	+	0.151042i	0.0859631	+	0.0230337i	0.301544	−	0.953452i	\(-0.402498\pi\)
							−0.215581	+	0.976486i	\(0.569165\pi\)
\(47\)	−1.24972	−	2.16458i	−0.182290	−	0.315736i	0.760370	−	0.649490i	\(-0.225018\pi\)
							−0.942660	+	0.333754i	\(0.891684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.59.12	yes	88
3.2	odd	2		432.2.v.a.395.11		88
4.3	odd	2		576.2.y.a.527.19		88
9.2	odd	6	inner	144.2.u.a.11.3	✓	88
9.7	even	3		432.2.v.a.251.20		88
12.11	even	2		1728.2.z.a.719.17		88
16.3	odd	4	inner	144.2.u.a.131.3	yes	88
16.13	even	4		576.2.y.a.239.14		88
36.7	odd	6		1728.2.z.a.143.17		88
36.11	even	6		576.2.y.a.335.14		88
48.29	odd	4		1728.2.z.a.1583.17		88
48.35	even	4		432.2.v.a.179.20		88
144.29	odd	12		576.2.y.a.47.19		88
144.61	even	12		1728.2.z.a.1007.17		88
144.83	even	12	inner	144.2.u.a.83.12	yes	88
144.115	odd	12		432.2.v.a.35.11		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.3	✓	88	9.2	odd	6	inner
144.2.u.a.59.12	yes	88	1.1	even	1	trivial
144.2.u.a.83.12	yes	88	144.83	even	12	inner
144.2.u.a.131.3	yes	88	16.3	odd	4	inner
432.2.v.a.35.11		88	144.115	odd	12
432.2.v.a.179.20		88	48.35	even	4
432.2.v.a.251.20		88	9.7	even	3
432.2.v.a.395.11		88	3.2	odd	2
576.2.y.a.47.19		88	144.29	odd	12
576.2.y.a.239.14		88	16.13	even	4
576.2.y.a.335.14		88	36.11	even	6
576.2.y.a.527.19		88	4.3	odd	2
1728.2.z.a.143.17		88	36.7	odd	6
1728.2.z.a.719.17		88	12.11	even	2
1728.2.z.a.1007.17		88	144.61	even	12
1728.2.z.a.1583.17		88	48.29	odd	4