Embedded newform 147.2.k.a.41.12

• Label: 147.2.k.a.41.12
• Level: $147$
• Weight: $2$
• Character: 147.41
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			41.12
Character	\(\chi\)	\(=\)	147.41
Dual form			147.2.k.a.104.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.853995 - 0.681038i) q^{2} +(-0.522370 + 1.65140i) q^{3} +(-0.179548 + 0.786650i) q^{4} +(2.05139 - 0.987900i) q^{5} +(0.678567 + 1.76604i) q^{6} +(-1.78522 + 1.95269i) q^{7} +(1.33027 + 2.76233i) q^{8} +(-2.45426 - 1.72528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.853995	−	0.681038i	0.603866	−	0.481567i	−0.273187	−	0.961961i	\(-0.588078\pi\)
							0.877053	+	0.480394i	\(0.159506\pi\)
\(3\)	−0.522370	+	1.65140i	−0.301590	+	0.953438i
							\(5\)	2.05139	−	0.987900i	0.917412	−	0.441802i	0.0852653	−	0.996358i	\(-0.472826\pi\)
0.832146	+	0.554556i	\(0.187112\pi\)
\(7\)	−1.78522	+	1.95269i	−0.674750	+	0.738046i
							\(11\)	0.472840	−	0.377077i	0.142567	−	0.113693i	−0.549610	−	0.835422i	\(-0.685224\pi\)
0.692176	+	0.721729i	\(0.256652\pi\)
\(13\)	4.60769	−	3.67451i	1.27794	−	1.01913i	0.279688	−	0.960091i	\(-0.409769\pi\)
							0.998256	−	0.0590353i	\(-0.0188025\pi\)
\(17\)	−1.04497	−	4.57829i	−0.253441	−	1.11040i	−0.928118	−	0.372286i	\(-0.878574\pi\)
							0.674677	−	0.738113i	\(-0.264283\pi\)
\(19\)			1.41690i			0.325058i	0.986704	+	0.162529i	\(0.0519651\pi\)
							−0.986704	+	0.162529i	\(0.948035\pi\)
\(23\)	−5.00433	−	1.14221i	−1.04348	−	0.238166i	−0.333762	−	0.942657i	\(-0.608318\pi\)
							−0.709713	+	0.704491i	\(0.751175\pi\)
\(29\)	0.134934	−	0.0307978i	0.0250566	−	0.00571901i	−0.209974	−	0.977707i	\(-0.567338\pi\)
							0.235031	+	0.971988i	\(0.424481\pi\)
\(31\)		−	9.28275i		−	1.66723i	−0.552345	−	0.833615i	\(-0.686267\pi\)
							0.552345	−	0.833615i	\(-0.313733\pi\)
\(37\)	0.542403	+	2.37642i	0.0891706	+	0.390682i	0.999743	−	0.0226676i	\(-0.00721594\pi\)
							−0.910572	+	0.413349i	\(0.864359\pi\)
\(41\)	−8.06490	+	3.88385i	−1.25953	+	0.606556i	−0.940049	−	0.341039i	\(-0.889221\pi\)
							−0.319477	+	0.947594i	\(0.603507\pi\)
\(43\)	3.18417	+	1.53342i	0.485582	+	0.233844i	0.660624	−	0.750717i	\(-0.270292\pi\)
							−0.175042	+	0.984561i	\(0.556006\pi\)
\(47\)	3.12720	+	3.92139i	0.456149	+	0.571993i	0.955719	−	0.294280i	\(-0.0950800\pi\)
							−0.499570	+	0.866274i	\(0.666509\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.41.12	yes	96
3.2	odd	2	inner	147.2.k.a.41.5	✓	96
49.6	odd	14	inner	147.2.k.a.104.5	yes	96
147.104	even	14	inner	147.2.k.a.104.12	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.41.5	✓	96	3.2	odd	2	inner
147.2.k.a.41.12	yes	96	1.1	even	1	trivial
147.2.k.a.104.5	yes	96	49.6	odd	14	inner
147.2.k.a.104.12	yes	96	147.104	even	14	inner