Embedded newform 147.3.l.a.8.5

• Label: 147.3.l.a.8.5
• Level: $147$
• Weight: $3$
• Character: 147.8
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			8.5
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46298 + 1.96416i) q^{2} +(-0.664950 + 2.92538i) q^{3} +(1.31826 - 5.77568i) q^{4} +(1.01622 + 2.11020i) q^{5} +(-4.10816 - 8.51123i) q^{6} +(-5.04359 - 4.85409i) q^{7} +(2.63011 + 5.46149i) q^{8} +(-8.11568 - 3.89046i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46298	+	1.96416i	−1.23149	+	0.982081i	−0.231534	+	0.972827i	\(0.574374\pi\)
							−0.999957	+	0.00925436i	\(0.997054\pi\)
\(3\)	−0.664950	+	2.92538i	−0.221650	+	0.975126i
							\(5\)	1.01622	+	2.11020i	0.203243	+	0.422039i	0.977530	−	0.210796i	\(-0.0676055\pi\)
−0.774287	+	0.632835i	\(0.781891\pi\)
\(7\)	−5.04359	−	4.85409i	−0.720513	−	0.693441i
							\(11\)	−2.12655	+	1.69587i	−0.193323	+	0.154170i	−0.715368	−	0.698748i	\(-0.753741\pi\)
0.522045	+	0.852918i	\(0.325169\pi\)
\(13\)	−9.81089	−	12.3025i	−0.754684	−	0.946343i	0.245048	−	0.969511i	\(-0.421196\pi\)
							−0.999732	+	0.0231675i	\(0.992625\pi\)
\(17\)	−6.03401	+	1.37722i	−0.354942	+	0.0810131i	−0.396274	−	0.918132i	\(-0.629697\pi\)
							0.0413323	+	0.999145i	\(0.486840\pi\)
\(19\)	13.9735			0.735445			0.367723	−	0.929936i	\(-0.380138\pi\)
							0.367723	+	0.929936i	\(0.380138\pi\)
\(23\)	1.30975	+	0.298943i	0.0569458	+	0.0129975i	0.250899	−	0.968013i	\(-0.419274\pi\)
							−0.193953	+	0.981011i	\(0.562131\pi\)
\(29\)	−35.3348	+	8.06493i	−1.21844	+	0.278101i	−0.782946	−	0.622090i	\(-0.786284\pi\)
							−0.435495	+	0.900191i	\(0.643427\pi\)
\(31\)	0.0406953			0.00131275			0.000656376	−	1.00000i	\(-0.499791\pi\)
							0.000656376		1.00000i	\(0.499791\pi\)
\(37\)	8.05992	+	35.3128i	0.217836	+	0.954400i	0.959073	+	0.283158i	\(0.0913819\pi\)
							−0.741238	+	0.671242i	\(0.765761\pi\)
\(41\)	−19.1319	−	39.7277i	−0.466631	−	0.968970i	−0.992934	−	0.118672i	\(-0.962136\pi\)
							0.526302	−	0.850297i	\(-0.323578\pi\)
\(43\)	−74.8165	−	36.0297i	−1.73992	−	0.837901i	−0.982812	−	0.184611i	\(-0.940897\pi\)
							−0.757108	−	0.653290i	\(-0.773388\pi\)
\(47\)	−11.2882	+	9.00203i	−0.240174	+	0.191533i	−0.736178	−	0.676788i	\(-0.763372\pi\)
							0.496004	+	0.868320i	\(0.334800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.5	✓	216
3.2	odd	2	inner	147.3.l.a.8.32	yes	216
49.43	even	7	inner	147.3.l.a.92.32	yes	216
147.92	odd	14	inner	147.3.l.a.92.5	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.5	✓	216	1.1	even	1	trivial
147.3.l.a.8.32	yes	216	3.2	odd	2	inner
147.3.l.a.92.5	yes	216	147.92	odd	14	inner
147.3.l.a.92.32	yes	216	49.43	even	7	inner