Embedded newform 147.4.o.a.101.36

• Label: 147.4.o.a.101.36
• Level: $147$
• Weight: $4$
• Character: 147.101
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.36
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.86568 + 0.732225i) q^{2} +(5.06292 - 1.16913i) q^{3} +(-2.91981 - 2.70919i) q^{4} +(7.00045 + 4.77283i) q^{5} +(10.3018 + 1.52597i) q^{6} +(10.3324 + 15.3702i) q^{7} +(-10.4205 - 21.6384i) q^{8} +(24.2663 - 11.8384i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.86568	+	0.732225i	0.659617	+	0.258881i	0.671450	−	0.741049i	\(-0.265672\pi\)
							−0.0118336	+	0.999930i	\(0.503767\pi\)
\(3\)	5.06292	−	1.16913i	0.974359	−	0.224999i
							\(5\)	7.00045	+	4.77283i	0.626139	+	0.426895i	0.834385	−	0.551182i	\(-0.185823\pi\)
−0.208246	+	0.978077i	\(0.566775\pi\)
\(7\)	10.3324	+	15.3702i	0.557895	+	0.829911i
							\(11\)	4.91362	+	32.5997i	0.134683	+	0.893563i	0.949483	+	0.313820i	\(0.101609\pi\)
−0.814800	+	0.579743i	\(0.803153\pi\)
\(13\)	46.1982	−	36.8418i	0.985621	−	0.786007i	0.00877776	−	0.999961i	\(-0.497206\pi\)
							0.976844	+	0.213955i	\(0.0686345\pi\)
\(17\)	−46.7745	−	14.4280i	−0.667322	−	0.205842i	−0.0574585	−	0.998348i	\(-0.518300\pi\)
							−0.609864	+	0.792506i	\(0.708776\pi\)
\(19\)	57.6394	−	33.2781i	0.695967	−	0.401817i	−0.109876	−	0.993945i	\(-0.535045\pi\)
							0.805844	+	0.592128i	\(0.201712\pi\)
\(23\)	7.82005	+	25.3520i	0.0708954	+	0.229837i	0.984085	−	0.177697i	\(-0.0568648\pi\)
							−0.913190	+	0.407535i	\(0.866389\pi\)
\(29\)	−260.075	+	59.3604i	−1.66533	+	0.380102i	−0.948409	−	0.317048i	\(-0.897308\pi\)
							−0.716925	+	0.697150i	\(0.754451\pi\)
\(31\)	39.4109	+	22.7539i	0.228336	+	0.131830i	0.609804	−	0.792552i	\(-0.291248\pi\)
							−0.381468	+	0.924382i	\(0.624581\pi\)
\(37\)	−182.343	+	169.190i	−0.810189	+	0.751746i	−0.971976	−	0.235080i	\(-0.924465\pi\)
							0.161787	+	0.986826i	\(0.448274\pi\)
\(41\)	−367.455	+	176.957i	−1.39968	+	0.674049i	−0.973096	−	0.230400i	\(-0.925997\pi\)
							−0.426581	+	0.904449i	\(0.640282\pi\)
\(43\)	82.4286	+	39.6955i	0.292331	+	0.140779i	0.574302	−	0.818644i	\(-0.305274\pi\)
							−0.281970	+	0.959423i	\(0.590988\pi\)
\(47\)	32.5135	−	82.8431i	0.100906	−	0.257104i	−0.871502	−	0.490393i	\(-0.836853\pi\)
							0.972408	+	0.233288i	\(0.0749486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.36	yes	648
3.2	odd	2	inner	147.4.o.a.101.19	✓	648
49.33	odd	42	inner	147.4.o.a.131.19	yes	648
147.131	even	42	inner	147.4.o.a.131.36	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.19	✓	648	3.2	odd	2	inner
147.4.o.a.101.36	yes	648	1.1	even	1	trivial
147.4.o.a.131.19	yes	648	49.33	odd	42	inner
147.4.o.a.131.36	yes	648	147.131	even	42	inner