Embedded newform 147.2.o.b.101.8

• Label: 147.2.o.b.101.8
• Level: $147$
• Weight: $2$
• Character: 147.101
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.8
Character	\(\chi\)	\(=\)	147.101
Dual form			147.2.o.b.131.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.236609 - 0.0928620i) q^{2} +(-0.145446 + 1.72593i) q^{3} +(-1.41874 - 1.31640i) q^{4} +(1.66272 + 1.13362i) q^{5} +(0.194687 - 0.394864i) q^{6} +(1.28490 + 2.31280i) q^{7} +(0.434011 + 0.901234i) q^{8} +(-2.95769 - 0.502060i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.236609	−	0.0928620i	−0.167308	−	0.0656634i	0.280209	−	0.959939i	\(-0.409596\pi\)
							−0.447517	+	0.894276i	\(0.647691\pi\)
\(3\)	−0.145446	+	1.72593i	−0.0839733	+	0.996468i
							\(5\)	1.66272	+	1.13362i	0.743589	+	0.506971i	0.874860	−	0.484376i	\(-0.160953\pi\)
−0.131271	+	0.991347i	\(0.541906\pi\)
\(7\)	1.28490	+	2.31280i	0.485645	+	0.874156i
							\(11\)	0.730006	+	4.84327i	0.220105	+	1.46030i	0.776885	+	0.629642i	\(0.216799\pi\)
−0.556780	+	0.830660i	\(0.687963\pi\)
\(13\)	1.15368	−	0.920028i	0.319973	−	0.255170i	−0.450310	−	0.892872i	\(-0.648686\pi\)
							0.770283	+	0.637702i	\(0.220115\pi\)
\(17\)	−4.96351	−	1.53104i	−1.20383	−	0.371332i	−0.372985	−	0.927837i	\(-0.621666\pi\)
							−0.830844	+	0.556505i	\(0.812142\pi\)
\(19\)	6.46990	−	3.73540i	1.48430	−	0.856959i	0.484455	−	0.874816i	\(-0.339018\pi\)
							0.999841	+	0.0178572i	\(0.00568444\pi\)
\(23\)	−1.23562	−	4.00578i	−0.257644	−	0.835263i	−0.988301	−	0.152517i	\(-0.951262\pi\)
							0.730656	−	0.682745i	\(-0.239214\pi\)
\(29\)	5.37434	−	1.22666i	0.997990	−	0.227785i	0.307830	−	0.951441i	\(-0.400397\pi\)
							0.690160	+	0.723657i	\(0.257540\pi\)
\(31\)	0.635505	+	0.366909i	0.114140	+	0.0658988i	0.555983	−	0.831194i	\(-0.312342\pi\)
							−0.441843	+	0.897092i	\(0.645675\pi\)
\(37\)	2.23228	−	2.07125i	0.366984	−	0.340512i	−0.475027	−	0.879971i	\(-0.657562\pi\)
							0.842012	+	0.539459i	\(0.181371\pi\)
\(41\)	−5.15291	+	2.48151i	−0.804750	+	0.387547i	−0.790585	−	0.612353i	\(-0.790223\pi\)
							−0.0141650	+	0.999900i	\(0.504509\pi\)
\(43\)	−3.97258	−	1.91309i	−0.605813	−	0.291744i	0.105721	−	0.994396i	\(-0.466285\pi\)
							−0.711534	+	0.702652i	\(0.751999\pi\)
\(47\)	3.45128	−	8.79371i	0.503421	−	1.28269i	−0.422975	−	0.906142i	\(-0.639014\pi\)
							0.926395	−	0.376553i	\(-0.122891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.101.8	✓	192
3.2	odd	2	inner	147.2.o.b.101.9	yes	192
49.33	odd	42	inner	147.2.o.b.131.9	yes	192
147.131	even	42	inner	147.2.o.b.131.8	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.101.8	✓	192	1.1	even	1	trivial
147.2.o.b.101.9	yes	192	3.2	odd	2	inner
147.2.o.b.131.8	yes	192	147.131	even	42	inner
147.2.o.b.131.9	yes	192	49.33	odd	42	inner