Embedded newform 147.4.o.a.131.9

• Label: 147.4.o.a.131.9
• Level: $147$
• Weight: $4$
• Character: 147.131
• 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			131.9
Character	\(\chi\)	\(=\)	147.131
Dual form			147.4.o.a.101.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.81031 + 1.49544i) q^{2} +(-2.12806 + 4.74040i) q^{3} +(6.41775 - 5.95480i) q^{4} +(16.4115 - 11.1892i) q^{5} +(1.01959 - 21.2448i) q^{6} +(-10.6461 + 15.1545i) q^{7} +(-1.34059 + 2.78376i) q^{8} +(-17.9428 - 20.1757i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.81031	+	1.49544i	−1.34715	+	0.528717i	−0.925743	−	0.378154i	\(-0.876559\pi\)
							−0.421407	+	0.906872i	\(0.638464\pi\)
\(3\)	−2.12806	+	4.74040i	−0.409545	+	0.912290i
							\(5\)	16.4115	−	11.1892i	1.46789	−	1.00079i	0.475226	−	0.879864i	\(-0.342366\pi\)
0.992662	−	0.120925i	\(-0.0385862\pi\)
\(7\)	−10.6461	+	15.1545i	−0.574836	+	0.818269i
							\(11\)	−7.93468	+	52.6432i	−0.217491	+	1.44296i	0.567461	+	0.823400i	\(0.307926\pi\)
−0.784952	+	0.619556i	\(0.787313\pi\)
\(13\)	−50.8769	−	40.5730i	−1.08544	−	0.865609i	−0.0939219	−	0.995580i	\(-0.529940\pi\)
							−0.991518	+	0.129970i	\(0.958512\pi\)
\(17\)	−38.2642	+	11.8029i	−0.545907	+	0.168390i	−0.555428	−	0.831565i	\(-0.687446\pi\)
							0.00952091	+	0.999955i	\(0.496969\pi\)
\(19\)	−44.3254	−	25.5913i	−0.535208	−	0.309003i	0.207927	−	0.978144i	\(-0.433329\pi\)
							−0.743135	+	0.669142i	\(0.766662\pi\)
\(23\)	22.6546	−	73.4443i	0.205383	−	0.665835i	−0.793012	−	0.609205i	\(-0.791488\pi\)
							0.998395	−	0.0566294i	\(-0.0180354\pi\)
\(29\)	−77.9880	−	17.8002i	−0.499379	−	0.113980i	−0.0345904	−	0.999402i	\(-0.511013\pi\)
							−0.464789	+	0.885421i	\(0.653870\pi\)
\(31\)	−241.902	+	139.662i	−1.40151	+	0.809164i	−0.994548	−	0.104280i	\(-0.966746\pi\)
							−0.406965	+	0.913444i	\(0.633413\pi\)
\(37\)	−93.7837	−	87.0185i	−0.416701	−	0.386642i	0.443770	−	0.896140i	\(-0.353641\pi\)
							−0.860472	+	0.509498i	\(0.829831\pi\)
\(41\)	162.048	+	78.0381i	0.617259	+	0.297256i	0.716262	−	0.697831i	\(-0.245851\pi\)
							−0.0990039	+	0.995087i	\(0.531566\pi\)
\(43\)	−40.4011	+	19.4561i	−0.143282	+	0.0690008i	−0.504151	−	0.863616i	\(-0.668194\pi\)
							0.360869	+	0.932616i	\(0.382480\pi\)
\(47\)	−151.249	−	385.377i	−0.469404	−	1.19602i	−0.948081	−	0.318030i	\(-0.896979\pi\)
							0.478677	−	0.877991i	\(-0.341117\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.131.9	yes	648
3.2	odd	2	inner	147.4.o.a.131.46	yes	648
49.3	odd	42	inner	147.4.o.a.101.46	yes	648
147.101	even	42	inner	147.4.o.a.101.9	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.9	✓	648	147.101	even	42	inner
147.4.o.a.101.46	yes	648	49.3	odd	42	inner
147.4.o.a.131.9	yes	648	1.1	even	1	trivial
147.4.o.a.131.46	yes	648	3.2	odd	2	inner