Embedded newform 147.4.o.a.101.17

• Label: 147.4.o.a.101.17
• Level: $147$
• Weight: $4$
• Character: 147.101
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.17
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.43321 - 0.954966i) q^{2} +(2.84355 + 4.34905i) q^{3} +(-0.855841 - 0.794104i) q^{4} +(-3.49881 - 2.38545i) q^{5} +(-2.76578 - 13.2977i) q^{6} +(5.62890 - 17.6441i) q^{7} +(10.3971 + 21.5899i) q^{8} +(-10.8284 + 24.7335i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.43321	−	0.954966i	−0.860271	−	0.337632i	−0.106137	−	0.994352i	\(-0.533848\pi\)
							−0.754135	+	0.656720i	\(0.771943\pi\)
\(3\)	2.84355	+	4.34905i	0.547242	+	0.836974i
							\(5\)	−3.49881	−	2.38545i	−0.312943	−	0.213361i	0.396655	−	0.917968i	\(-0.370171\pi\)
−0.709598	+	0.704607i	\(0.751123\pi\)
\(7\)	5.62890	−	17.6441i	0.303932	−	0.952694i
							\(11\)	0.00994148	+	0.0659574i	0.000272497	+	0.00180790i	0.988967	−	0.148138i	\(-0.0473281\pi\)
−0.988694	+	0.149946i	\(0.952090\pi\)
\(13\)	−36.9675	+	29.4806i	−0.788688	+	0.628958i	−0.932714	−	0.360616i	\(-0.882566\pi\)
							0.144026	+	0.989574i	\(0.453995\pi\)
\(17\)	−108.031	−	33.3232i	−1.54126	−	0.475416i	−0.596345	−	0.802728i	\(-0.703381\pi\)
							−0.944916	+	0.327312i	\(0.893857\pi\)
\(19\)	38.3055	−	22.1157i	0.462520	−	0.267036i	−0.250583	−	0.968095i	\(-0.580622\pi\)
							0.713103	+	0.701059i	\(0.247289\pi\)
\(23\)	−40.9407	−	132.726i	−0.371162	−	1.20328i	−0.927976	−	0.372640i	\(-0.878453\pi\)
							0.556814	−	0.830637i	\(-0.312024\pi\)
\(29\)	−223.233	+	50.9514i	−1.42942	+	0.326256i	−0.866055	−	0.499950i	\(-0.833352\pi\)
							−0.563368	+	0.826206i	\(0.690495\pi\)
\(31\)	−70.3677	−	40.6268i	−0.407691	−	0.235380i	0.282106	−	0.959383i	\(-0.408967\pi\)
							−0.689797	+	0.724003i	\(0.742300\pi\)
\(37\)	−294.106	+	272.890i	−1.30677	+	1.21251i	−0.345021	+	0.938595i	\(0.612128\pi\)
							−0.961754	+	0.273915i	\(0.911681\pi\)
\(41\)	−196.392	+	94.5776i	−0.748081	+	0.360257i	−0.768767	−	0.639529i	\(-0.779129\pi\)
							0.0206857	+	0.999786i	\(0.493415\pi\)
\(43\)	437.904	+	210.883i	1.55302	+	0.747893i	0.996551	−	0.0829789i	\(-0.0264434\pi\)
							0.556464	+	0.830872i	\(0.312158\pi\)
\(47\)	165.472	−	421.616i	0.513545	−	1.30849i	−0.405413	−	0.914134i	\(-0.632872\pi\)
							0.918958	−	0.394357i	\(-0.129033\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.17	✓	648
3.2	odd	2	inner	147.4.o.a.101.38	yes	648
49.33	odd	42	inner	147.4.o.a.131.38	yes	648
147.131	even	42	inner	147.4.o.a.131.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.17	✓	648	1.1	even	1	trivial
147.4.o.a.101.38	yes	648	3.2	odd	2	inner
147.4.o.a.131.17	yes	648	147.131	even	42	inner
147.4.o.a.131.38	yes	648	49.33	odd	42	inner