Embedded newform 147.2.m.a.4.4

• Label: 147.2.m.a.4.4
• Level: $147$
• Weight: $2$
• Character: 147.4
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.m (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	147.4
Dual form			147.2.m.a.37.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65175 + 1.12614i) q^{2} +(0.733052 + 0.680173i) q^{3} +(0.729391 + 1.85846i) q^{4} +(-0.400180 + 0.123439i) q^{5} +(0.444845 + 1.94899i) q^{6} +(-1.90889 - 1.83197i) q^{7} +(0.00157168 - 0.00688598i) q^{8} +(0.0747301 + 0.997204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 4 q^{3} + 3 q^{4} - 2 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65175	+	1.12614i	1.16796	+	0.796303i	0.982484	−	0.186346i	\(-0.0596644\pi\)
							0.185478	+	0.982648i	\(0.440617\pi\)
\(3\)	0.733052	+	0.680173i	0.423228	+	0.392698i
							\(5\)	−0.400180	+	0.123439i	−0.178966	+	0.0552036i	−0.382943	−	0.923772i	\(-0.625089\pi\)
0.203977	+	0.978976i	\(0.434613\pi\)
\(7\)	−1.90889	−	1.83197i	−0.721493	−	0.692421i
							\(11\)	0.0436333	−	0.582246i	0.0131559	−	0.175554i	−0.986755	−	0.162217i	\(-0.948136\pi\)
0.999911	−	0.0133369i	\(-0.00424539\pi\)
\(13\)	−3.22770	+	1.55438i	−0.895203	+	0.431107i	−0.824154	−	0.566366i	\(-0.808349\pi\)
							−0.0710488	+	0.997473i	\(0.522635\pi\)
\(17\)	−0.518106	+	0.0780919i	−0.125659	+	0.0189401i	−0.211571	−	0.977363i	\(-0.567858\pi\)
							0.0859115	+	0.996303i	\(0.472620\pi\)
\(19\)	−0.603015	−	1.04445i	−0.138341	−	0.239614i	0.788528	−	0.614999i	\(-0.210844\pi\)
							−0.926869	+	0.375385i	\(0.877510\pi\)
\(23\)	8.89988	+	1.34144i	1.85575	+	0.279710i	0.979395	−	0.201952i	\(-0.0647285\pi\)
							0.876357	+	0.481662i	\(0.159967\pi\)
\(29\)	−2.11419	−	2.65111i	−0.392595	−	0.492299i	0.545775	−	0.837932i	\(-0.316235\pi\)
							−0.938370	+	0.345633i	\(0.887664\pi\)
\(31\)	−3.57740	+	6.19624i	−0.642521	+	1.11288i	0.342348	+	0.939573i	\(0.388778\pi\)
							−0.984868	+	0.173305i	\(0.944555\pi\)
\(37\)	1.29416	−	3.29747i	0.212759	−	0.542100i	−0.784159	−	0.620561i	\(-0.786905\pi\)
							0.996917	+	0.0784602i	\(0.0250003\pi\)
\(41\)	0.789835	−	3.46050i	0.123352	−	0.540439i	−0.875056	−	0.484022i	\(-0.839175\pi\)
							0.998407	−	0.0564164i	\(-0.0179674\pi\)
\(43\)	1.23768	+	5.42262i	0.188744	+	0.826941i	0.977280	+	0.211953i	\(0.0679823\pi\)
							−0.788536	+	0.614989i	\(0.789161\pi\)
\(47\)	−4.74838	−	3.23739i	−0.692623	−	0.472222i	0.165139	−	0.986270i	\(-0.447193\pi\)
							−0.857761	+	0.514048i	\(0.828145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.m.a.4.4	✓	48
3.2	odd	2		441.2.bb.c.298.1		48
49.24	odd	42		7203.2.a.k.1.22		24
49.25	even	21		7203.2.a.i.1.22		24
49.37	even	21	inner	147.2.m.a.37.4	yes	48
147.86	odd	42		441.2.bb.c.37.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.m.a.4.4	✓	48	1.1	even	1	trivial
147.2.m.a.37.4	yes	48	49.37	even	21	inner
441.2.bb.c.37.1		48	147.86	odd	42
441.2.bb.c.298.1		48	3.2	odd	2
7203.2.a.i.1.22		24	49.25	even	21
7203.2.a.k.1.22		24	49.24	odd	42