Embedded newform 144.2.u.a.131.9

• Label: 144.2.u.a.131.9
• 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			131.9
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.665270 + 1.24796i) q^{2} +(0.117756 - 1.72804i) q^{3} +(-1.11483 - 1.66047i) q^{4} +(-3.58858 + 0.961558i) q^{5} +(2.07820 + 1.29657i) q^{6} +(-1.29216 - 2.23809i) q^{7} +(2.81387 - 0.286608i) q^{8} +(-2.97227 - 0.406975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.665270	+	1.24796i	−0.470417	+	0.882444i
							\(3\)	0.117756	−	1.72804i	0.0679864	−	0.997686i
\(5\)	−3.58858	+	0.961558i	−1.60486	+	0.430022i								−0.946507	−	0.322684i	\(-0.895415\pi\)
							−0.658357	+	0.752706i	\(0.728748\pi\)
\(7\)	−1.29216	−	2.23809i	−0.488391	−	0.845919i	0.511520	−	0.859272i	\(-0.329083\pi\)
							−0.999911	+	0.0133531i	\(0.995749\pi\)
\(11\)	−2.02011	−	0.541286i	−0.609085	−	0.163204i	−0.0589240	−	0.998262i	\(-0.518767\pi\)
							−0.550161	+	0.835059i	\(0.685434\pi\)
\(13\)	−0.998074	+	0.267433i	−0.276816	+	0.0741726i	−0.394556	−	0.918872i	\(-0.629102\pi\)
							0.117740	+	0.993044i	\(0.462435\pi\)
\(17\)		−	4.13788i		−	1.00358i	−0.864988	−	0.501792i	\(-0.832674\pi\)
							0.864988	−	0.501792i	\(-0.167326\pi\)
\(19\)	2.49279	−	2.49279i	0.571886	−	0.571886i	−0.360769	−	0.932655i	\(-0.617486\pi\)
							0.932655	+	0.360769i	\(0.117486\pi\)
\(23\)	7.01915	+	4.05251i	1.46359	+	0.845006i	0.999175	−	0.0406102i	\(-0.0129302\pi\)
							0.464418	+	0.885616i	\(0.346264\pi\)
\(29\)	−7.02439	−	1.88218i	−1.30440	−	0.349512i	−0.461286	−	0.887252i	\(-0.652612\pi\)
							−0.843111	+	0.537740i	\(0.819278\pi\)
\(31\)	−2.03331	−	1.17393i	−0.365194	−	0.210845i	0.306163	−	0.951979i	\(-0.400955\pi\)
							−0.671357	+	0.741134i	\(0.734288\pi\)
\(37\)	−4.75590	+	4.75590i	−0.781865	+	0.781865i	−0.980145	−	0.198280i	\(-0.936464\pi\)
							0.198280	+	0.980145i	\(0.436464\pi\)
\(41\)	0.636217	−	1.10196i	0.0993604	−	0.172097i	−0.812060	−	0.583574i	\(-0.801654\pi\)
							0.911420	+	0.411477i	\(0.134987\pi\)
\(43\)	0.389880	−	1.45505i	0.0594562	−	0.221894i	−0.929805	−	0.368053i	\(-0.880025\pi\)
							0.989261	+	0.146159i	\(0.0466912\pi\)
\(47\)	−3.60814	−	6.24948i	−0.526301	−	0.911580i	−0.999530	−	0.0306407i	\(-0.990245\pi\)
							0.473230	−	0.880939i	\(-0.343088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.9	yes	88
3.2	odd	2		432.2.v.a.179.14		88
4.3	odd	2		576.2.y.a.239.10		88
9.2	odd	6	inner	144.2.u.a.83.2	yes	88
9.7	even	3		432.2.v.a.35.21		88
12.11	even	2		1728.2.z.a.1583.21		88
16.5	even	4		576.2.y.a.527.1		88
16.11	odd	4	inner	144.2.u.a.59.2	yes	88
36.7	odd	6		1728.2.z.a.1007.21		88
36.11	even	6		576.2.y.a.47.1		88
48.5	odd	4		1728.2.z.a.719.21		88
48.11	even	4		432.2.v.a.395.21		88
144.11	even	12	inner	144.2.u.a.11.9	✓	88
144.43	odd	12		432.2.v.a.251.14		88
144.101	odd	12		576.2.y.a.335.10		88
144.133	even	12		1728.2.z.a.143.21		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.9	✓	88	144.11	even	12	inner
144.2.u.a.59.2	yes	88	16.11	odd	4	inner
144.2.u.a.83.2	yes	88	9.2	odd	6	inner
144.2.u.a.131.9	yes	88	1.1	even	1	trivial
432.2.v.a.35.21		88	9.7	even	3
432.2.v.a.179.14		88	3.2	odd	2
432.2.v.a.251.14		88	144.43	odd	12
432.2.v.a.395.21		88	48.11	even	4
576.2.y.a.47.1		88	36.11	even	6
576.2.y.a.239.10		88	4.3	odd	2
576.2.y.a.335.10		88	144.101	odd	12
576.2.y.a.527.1		88	16.5	even	4
1728.2.z.a.143.21		88	144.133	even	12
1728.2.z.a.719.21		88	48.5	odd	4
1728.2.z.a.1007.21		88	36.7	odd	6
1728.2.z.a.1583.21		88	12.11	even	2