Embedded newform 144.3.v.a.139.25

• Label: 144.3.v.a.139.25
• Level: $144$
• Weight: $3$
• Character: 144.139
• Analytic conductor: $3.924$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92371580679\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			139.25
Character	\(\chi\)	\(=\)	144.139
Dual form			144.3.v.a.115.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.232149 - 1.98648i) q^{2} +(0.171846 + 2.99507i) q^{3} +(-3.89221 - 0.922317i) q^{4} +(0.828901 - 0.222103i) q^{5} +(5.98955 + 0.353933i) q^{6} +(5.23163 + 9.06144i) q^{7} +(-2.73574 + 7.51769i) q^{8} +(-8.94094 + 1.02938i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.232149	−	1.98648i	0.116074	−	0.993241i
							\(3\)	0.171846	+	2.99507i	0.0572821	+	0.998358i
\(5\)	0.828901	−	0.222103i	0.165780	−	0.0444207i								−0.174974	−	0.984573i	\(-0.555984\pi\)
							0.340754	+	0.940152i	\(0.389318\pi\)
\(7\)	5.23163	+	9.06144i	0.747375	+	1.29449i	0.949077	+	0.315045i	\(0.102020\pi\)
							−0.201701	+	0.979447i	\(0.564647\pi\)
\(11\)	7.04151	+	1.88677i	0.640137	+	0.171524i	0.564266	−	0.825593i	\(-0.309159\pi\)
							0.0758716	+	0.997118i	\(0.475826\pi\)
\(13\)	1.87144	+	6.98432i	0.143957	+	0.537255i	0.999800	+	0.0200186i	\(0.00637255\pi\)
							−0.855843	+	0.517236i	\(0.826961\pi\)
\(17\)	10.9027			0.641336			0.320668	−	0.947192i	\(-0.396093\pi\)
							0.320668	+	0.947192i	\(0.396093\pi\)
\(19\)	5.61772	+	5.61772i	0.295669	+	0.295669i	0.839315	−	0.543646i	\(-0.182956\pi\)
							−0.543646	+	0.839315i	\(0.682956\pi\)
\(23\)	8.49079	−	14.7065i	0.369165	−	0.639412i	−0.620271	−	0.784388i	\(-0.712977\pi\)
							0.989435	+	0.144976i	\(0.0463105\pi\)
\(29\)	−8.49945	−	2.27742i	−0.293084	−	0.0785317i	0.109281	−	0.994011i	\(-0.465145\pi\)
							−0.402365	+	0.915479i	\(0.631812\pi\)
\(31\)	−6.67353	−	3.85297i	−0.215275	−	0.124289i	0.388485	−	0.921455i	\(-0.372998\pi\)
							−0.603761	+	0.797166i	\(0.706332\pi\)
\(37\)	−36.7870	−	36.7870i	−0.994244	−	0.994244i	0.00573969	−	0.999984i	\(-0.498173\pi\)
							−0.999984	+	0.00573969i	\(0.998173\pi\)
\(41\)	70.5661	+	40.7414i	1.72112	+	0.993692i	0.916635	+	0.399726i	\(0.130895\pi\)
							0.804490	+	0.593966i	\(0.202439\pi\)
\(43\)	−44.8228	−	12.0102i	−1.04239	−	0.279308i	−0.303287	−	0.952899i	\(-0.598084\pi\)
							−0.739104	+	0.673591i	\(0.764751\pi\)
\(47\)	21.0600	−	12.1590i	0.448085	−	0.258702i	−0.258936	−	0.965894i	\(-0.583372\pi\)
							0.707021	+	0.707192i	\(0.250039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.v.a.139.25	yes	184
3.2	odd	2		432.3.w.a.91.22		184
9.2	odd	6		432.3.w.a.235.40		184
9.7	even	3	inner	144.3.v.a.43.7	✓	184
16.3	odd	4	inner	144.3.v.a.67.7	yes	184
48.35	even	4		432.3.w.a.307.40		184
144.83	even	12		432.3.w.a.19.22		184
144.115	odd	12	inner	144.3.v.a.115.25	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.v.a.43.7	✓	184	9.7	even	3	inner
144.3.v.a.67.7	yes	184	16.3	odd	4	inner
144.3.v.a.115.25	yes	184	144.115	odd	12	inner
144.3.v.a.139.25	yes	184	1.1	even	1	trivial
432.3.w.a.19.22		184	144.83	even	12
432.3.w.a.91.22		184	3.2	odd	2
432.3.w.a.235.40		184	9.2	odd	6
432.3.w.a.307.40		184	48.35	even	4