Embedded newform 147.4.o.a.101.5

• Label: 147.4.o.a.101.5
• 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			101.5
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.33721 - 1.70223i) q^{2} +(0.419963 + 5.17915i) q^{3} +(10.0494 + 9.32446i) q^{4} +(-10.6270 - 7.24539i) q^{5} +(6.99464 - 23.1779i) q^{6} +(3.74797 + 18.1371i) q^{7} +(-11.5412 - 23.9655i) q^{8} +(-26.6473 + 4.35010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.33721	−	1.70223i	−1.53343	−	0.601829i	−0.559535	−	0.828807i	\(-0.689020\pi\)
							−0.973900	+	0.226978i	\(0.927115\pi\)
\(3\)	0.419963	+	5.17915i	0.0808218	+	0.996729i
							\(5\)	−10.6270	−	7.24539i	−0.950511	−	0.648047i	−0.0144536	−	0.999896i	\(-0.504601\pi\)
−0.936057	+	0.351848i	\(0.885553\pi\)
\(7\)	3.74797	+	18.1371i	0.202372	+	0.979309i
							\(11\)	4.49673	+	29.8339i	0.123256	+	0.817750i	0.961965	+	0.273171i	\(0.0880725\pi\)
−0.838709	+	0.544579i	\(0.816689\pi\)
\(13\)	−2.70312	+	2.15567i	−0.0576700	+	0.0459903i	−0.651901	−	0.758304i	\(-0.726028\pi\)
							0.594231	+	0.804295i	\(0.297457\pi\)
\(17\)	28.7565	+	8.87020i	0.410263	+	0.126549i	0.493015	−	0.870021i	\(-0.335895\pi\)
							−0.0827521	+	0.996570i	\(0.526371\pi\)
\(19\)	14.1514	−	8.17030i	0.170871	−	0.0986524i	−0.412126	−	0.911127i	\(-0.635213\pi\)
							0.582996	+	0.812475i	\(0.301880\pi\)
\(23\)	−29.1095	−	94.3707i	−0.263902	−	0.855550i	−0.986426	−	0.164204i	\(-0.947494\pi\)
							0.722524	−	0.691346i	\(-0.242982\pi\)
\(29\)	−179.469	+	40.9627i	−1.14919	+	0.262296i	−0.754362	−	0.656458i	\(-0.772054\pi\)
							−0.394830	+	0.918754i	\(0.629197\pi\)
\(31\)	−257.744	−	148.809i	−1.49330	−	0.862155i	−0.493326	−	0.869845i	\(-0.664219\pi\)
							−0.999970	+	0.00768957i	\(0.997552\pi\)
\(37\)	130.022	−	120.642i	0.577714	−	0.536040i	−0.336283	−	0.941761i	\(-0.609170\pi\)
							0.913997	+	0.405721i	\(0.132979\pi\)
\(41\)	305.492	−	147.117i	1.16366	−	0.560387i	0.250547	−	0.968104i	\(-0.419389\pi\)
							0.913108	+	0.407717i	\(0.133675\pi\)
\(43\)	−411.644	−	198.238i	−1.45989	−	0.703045i	−0.475608	−	0.879658i	\(-0.657772\pi\)
							−0.984280	+	0.176612i	\(0.943486\pi\)
\(47\)	138.240	−	352.231i	0.429031	−	1.09315i	−0.538997	−	0.842308i	\(-0.681197\pi\)
							0.968027	−	0.250844i	\(-0.0807082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.5	✓	648
3.2	odd	2	inner	147.4.o.a.101.50	yes	648
49.33	odd	42	inner	147.4.o.a.131.50	yes	648
147.131	even	42	inner	147.4.o.a.131.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.5	✓	648	1.1	even	1	trivial
147.4.o.a.101.50	yes	648	3.2	odd	2	inner
147.4.o.a.131.5	yes	648	147.131	even	42	inner
147.4.o.a.131.50	yes	648	49.33	odd	42	inner