Embedded newform 147.4.o.a.101.6

• Label: 147.4.o.a.101.6
• 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			101.6
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.32320 - 1.69673i) q^{2} +(5.16240 + 0.591255i) q^{3} +(9.94671 + 9.22920i) q^{4} +(7.56055 + 5.15469i) q^{5} +(-21.3149 - 11.3153i) q^{6} +(16.1590 + 9.04908i) q^{7} +(-11.2217 - 23.3020i) q^{8} +(26.3008 + 6.10459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.32320	−	1.69673i	−1.52848	−	0.599884i	−0.555653	−	0.831415i	\(-0.687532\pi\)
							−0.972828	+	0.231530i	\(0.925627\pi\)
\(3\)	5.16240	+	0.591255i	0.993505	+	0.113787i
							\(5\)	7.56055	+	5.15469i	0.676236	+	0.461050i	0.852110	−	0.523363i	\(-0.175323\pi\)
−0.175874	+	0.984413i	\(0.556275\pi\)
\(7\)	16.1590	+	9.04908i	0.872505	+	0.488605i
							\(11\)	4.18891	+	27.7916i	0.114818	+	0.761771i	0.969940	+	0.243343i	\(0.0782442\pi\)
−0.855122	+	0.518427i	\(0.826518\pi\)
\(13\)	−55.3068	+	44.1057i	−1.17995	+	0.940978i	−0.999095	−	0.0425317i	\(-0.986458\pi\)
							−0.180854	+	0.983510i	\(0.557886\pi\)
\(17\)	−76.7608	−	23.6776i	−1.09513	−	0.337803i	−0.306002	−	0.952031i	\(-0.598991\pi\)
							−0.789129	+	0.614228i	\(0.789468\pi\)
\(19\)	−73.4757	+	42.4212i	−0.887183	+	0.512215i	−0.873020	−	0.487684i	\(-0.837842\pi\)
							−0.0141629	+	0.999900i	\(0.504508\pi\)
\(23\)	41.5542	+	134.715i	0.376724	+	1.22131i	0.923359	+	0.383938i	\(0.125432\pi\)
							−0.546635	+	0.837371i	\(0.684091\pi\)
\(29\)	275.699	−	62.9265i	1.76538	−	0.402936i	0.788211	−	0.615405i	\(-0.211007\pi\)
							0.977168	+	0.212469i	\(0.0681503\pi\)
\(31\)	−99.1330	−	57.2345i	−0.574349	−	0.331601i	0.184535	−	0.982826i	\(-0.440922\pi\)
							−0.758884	+	0.651225i	\(0.774255\pi\)
\(37\)	172.357	−	159.924i	0.765821	−	0.710578i	−0.197093	−	0.980385i	\(-0.563150\pi\)
							0.962915	+	0.269806i	\(0.0869597\pi\)
\(41\)	−205.165	+	98.8021i	−0.781496	+	0.376349i	−0.781703	−	0.623651i	\(-0.785648\pi\)
							0.000206755		1.00000i	\(0.499934\pi\)
\(43\)	224.640	+	108.181i	0.796681	+	0.383661i	0.787514	−	0.616297i	\(-0.211368\pi\)
							0.00916690	+	0.999958i	\(0.497082\pi\)
\(47\)	145.063	−	369.615i	0.450205	−	1.14710i	−0.508086	−	0.861307i	\(-0.669647\pi\)
							0.958290	−	0.285797i	\(-0.0922581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.6	✓	648
3.2	odd	2	inner	147.4.o.a.101.49	yes	648
49.33	odd	42	inner	147.4.o.a.131.49	yes	648
147.131	even	42	inner	147.4.o.a.131.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.6	✓	648	1.1	even	1	trivial
147.4.o.a.101.49	yes	648	3.2	odd	2	inner
147.4.o.a.131.6	yes	648	147.131	even	42	inner
147.4.o.a.131.49	yes	648	49.33	odd	42	inner