Embedded newform 147.2.k.a.62.9

• Label: 147.2.k.a.62.9
• Level: $147$
• Weight: $2$
• Character: 147.62
• 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			62.9
Character	\(\chi\)	\(=\)	147.62
Dual form			147.2.k.a.83.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.399023 - 0.0910744i) q^{2} +(-0.596197 - 1.62621i) q^{3} +(-1.65101 + 0.795086i) q^{4} +(-1.77155 - 2.22145i) q^{5} +(-0.386002 - 0.594595i) q^{6} +(-2.45493 - 0.986562i) q^{7} +(-1.22636 + 0.977992i) q^{8} +(-2.28910 + 1.93908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{3} + 2 q^{4} + 7 q^{6} - 14 q^{7} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.399023	−	0.0910744i	0.282152	−	0.0643993i	−0.0791030	−	0.996866i	\(-0.525206\pi\)
							0.361255	+	0.932467i	\(0.382348\pi\)
\(3\)	−0.596197	−	1.62621i	−0.344215	−	0.938891i
							\(5\)	−1.77155	−	2.22145i	−0.792259	−	0.993462i	−0.999884	−	0.0152532i	\(-0.995145\pi\)
0.207624	−	0.978209i	\(-0.433427\pi\)
\(7\)	−2.45493	−	0.986562i	−0.927877	−	0.372885i
							\(11\)	3.98455	−	0.909447i	1.20139	−	0.274208i	0.425427	−	0.904993i	\(-0.360124\pi\)
0.775958	+	0.630784i	\(0.217267\pi\)
\(13\)	4.69288	−	1.07112i	1.30157	−	0.297075i	0.485098	−	0.874460i	\(-0.338784\pi\)
							0.816472	+	0.577385i	\(0.195927\pi\)
\(17\)	−4.81022	−	2.31648i	−1.16665	−	0.561828i	−0.252655	−	0.967556i	\(-0.581304\pi\)
							−0.913994	+	0.405728i	\(0.867018\pi\)
\(19\)		−	4.40986i		−	1.01169i	−0.862624	−	0.505845i	\(-0.831181\pi\)
							0.862624	−	0.505845i	\(-0.168819\pi\)
\(23\)	−0.270428	−	0.561550i	−0.0563882	−	0.117091i	0.870871	−	0.491512i	\(-0.163556\pi\)
							−0.927259	+	0.374421i	\(0.877841\pi\)
\(29\)	−2.44930	+	5.08601i	−0.454823	+	0.944449i	0.539888	+	0.841737i	\(0.318466\pi\)
							−0.994711	+	0.102713i	\(0.967248\pi\)
\(31\)		−	0.670225i		−	0.120376i	−0.998187	−	0.0601879i	\(-0.980830\pi\)
							0.998187	−	0.0601879i	\(-0.0191700\pi\)
\(37\)	−6.37330	−	3.06922i	−1.04776	−	0.504577i	−0.170887	−	0.985291i	\(-0.554663\pi\)
							−0.876878	+	0.480714i	\(0.840378\pi\)
\(41\)	4.48934	+	5.62946i	0.701117	+	0.879173i	0.997106	−	0.0760180i	\(-0.0242206\pi\)
							−0.295989	+	0.955191i	\(0.595649\pi\)
\(43\)	3.55967	−	4.46368i	0.542844	−	0.680705i	−0.432439	−	0.901663i	\(-0.642347\pi\)
							0.975283	+	0.220958i	\(0.0709183\pi\)
\(47\)	0.622897	+	2.72909i	0.0908589	+	0.398079i	0.999823	−	0.0187947i	\(-0.00598289\pi\)
							−0.908964	+	0.416874i	\(0.863126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.k.a.62.9	yes	96
3.2	odd	2	inner	147.2.k.a.62.8	✓	96
49.34	odd	14	inner	147.2.k.a.83.8	yes	96
147.83	even	14	inner	147.2.k.a.83.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.k.a.62.8	✓	96	3.2	odd	2	inner
147.2.k.a.62.9	yes	96	1.1	even	1	trivial
147.2.k.a.83.8	yes	96	49.34	odd	14	inner
147.2.k.a.83.9	yes	96	147.83	even	14	inner