Embedded newform 144.2.u.a.131.15

• Label: 144.2.u.a.131.15
• 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			131.15
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.688374 + 1.23537i) q^{2} +(1.10465 - 1.33407i) q^{3} +(-1.05228 + 1.70079i) q^{4} +(0.664471 - 0.178044i) q^{5} +(2.40849 + 0.446316i) q^{6} +(0.645693 + 1.11837i) q^{7} +(-2.82548 - 0.129179i) q^{8} +(-0.559489 - 2.94737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.688374	+	1.23537i	0.486754	+	0.873539i
							\(3\)	1.10465	−	1.33407i	0.637771	−	0.770226i
\(5\)	0.664471	−	0.178044i	0.297161	−	0.0796239i								−0.107158	−	0.994242i	\(-0.534175\pi\)
							0.404319	+	0.914618i	\(0.367509\pi\)
\(7\)	0.645693	+	1.11837i	0.244049	+	0.422705i	0.961864	−	0.273529i	\(-0.0881909\pi\)
							−0.717815	+	0.696234i	\(0.754858\pi\)
\(11\)	3.21069	+	0.860301i	0.968058	+	0.259390i	0.708008	−	0.706205i	\(-0.249594\pi\)
							0.260051	+	0.965595i	\(0.416261\pi\)
\(13\)	−4.74727	+	1.27203i	−1.31666	+	0.352797i	−0.847725	−	0.530437i	\(-0.822028\pi\)
							−0.468933	+	0.883234i	\(0.655361\pi\)
\(17\)		−	5.58523i		−	1.35462i	−0.735699	−	0.677308i	\(-0.763146\pi\)
							0.735699	−	0.677308i	\(-0.236854\pi\)
\(19\)	−2.49649	+	2.49649i	−0.572733	+	0.572733i	−0.932891	−	0.360158i	\(-0.882723\pi\)
							0.360158	+	0.932891i	\(0.382723\pi\)
\(23\)	−2.36529	−	1.36560i	−0.493197	−	0.284747i	0.232703	−	0.972548i	\(-0.425243\pi\)
							−0.725900	+	0.687801i	\(0.758576\pi\)
\(29\)	−2.95682	−	0.792277i	−0.549067	−	0.147122i	−0.0263884	−	0.999652i	\(-0.508401\pi\)
							−0.522679	+	0.852530i	\(0.675067\pi\)
\(31\)	5.28160	+	3.04933i	0.948604	+	0.547677i	0.892647	−	0.450757i	\(-0.148846\pi\)
							0.0559568	+	0.998433i	\(0.482179\pi\)
\(37\)	0.507420	−	0.507420i	0.0834193	−	0.0834193i	−0.664166	−	0.747585i	\(-0.731213\pi\)
							0.747585	+	0.664166i	\(0.231213\pi\)
\(41\)	4.89892	−	8.48518i	0.765083	−	1.32516i	−0.175120	−	0.984547i	\(-0.556031\pi\)
							0.940203	−	0.340616i	\(-0.110635\pi\)
\(43\)	0.254540	−	0.949956i	0.0388170	−	0.144867i	−0.943798	−	0.330524i	\(-0.892775\pi\)
							0.982615	+	0.185657i	\(0.0594413\pi\)
\(47\)	6.13774	+	10.6309i	0.895281	+	1.55067i	0.833456	+	0.552586i	\(0.186359\pi\)
							0.0618250	+	0.998087i	\(0.480308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.15	yes	88
3.2	odd	2		432.2.v.a.179.8		88
4.3	odd	2		576.2.y.a.239.7		88
9.2	odd	6	inner	144.2.u.a.83.8	yes	88
9.7	even	3		432.2.v.a.35.15		88
12.11	even	2		1728.2.z.a.1583.10		88
16.5	even	4		576.2.y.a.527.5		88
16.11	odd	4	inner	144.2.u.a.59.8	yes	88
36.7	odd	6		1728.2.z.a.1007.10		88
36.11	even	6		576.2.y.a.47.5		88
48.5	odd	4		1728.2.z.a.719.10		88
48.11	even	4		432.2.v.a.395.15		88
144.11	even	12	inner	144.2.u.a.11.15	✓	88
144.43	odd	12		432.2.v.a.251.8		88
144.101	odd	12		576.2.y.a.335.7		88
144.133	even	12		1728.2.z.a.143.10		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.15	✓	88	144.11	even	12	inner
144.2.u.a.59.8	yes	88	16.11	odd	4	inner
144.2.u.a.83.8	yes	88	9.2	odd	6	inner
144.2.u.a.131.15	yes	88	1.1	even	1	trivial
432.2.v.a.35.15		88	9.7	even	3
432.2.v.a.179.8		88	3.2	odd	2
432.2.v.a.251.8		88	144.43	odd	12
432.2.v.a.395.15		88	48.11	even	4
576.2.y.a.47.5		88	36.11	even	6
576.2.y.a.239.7		88	4.3	odd	2
576.2.y.a.335.7		88	144.101	odd	12
576.2.y.a.527.5		88	16.5	even	4
1728.2.z.a.143.10		88	144.133	even	12
1728.2.z.a.719.10		88	48.5	odd	4
1728.2.z.a.1007.10		88	36.7	odd	6
1728.2.z.a.1583.10		88	12.11	even	2