Embedded newform 333.3.bg.a.88.13

• Label: 333.3.bg.a.88.13
• Level: $333$
• Weight: $3$
• Character: 333.88
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $296$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			88.13
Character	\(\chi\)	\(=\)	333.88
Dual form			333.3.bg.a.193.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.745183 + 2.78106i) q^{2} +(1.99542 - 2.24015i) q^{3} +(-3.71491 - 2.14480i) q^{4} +(-0.720689 - 2.68965i) q^{5} +(4.74305 + 7.21872i) q^{6} +2.95625 q^{7} +(0.589600 - 0.589600i) q^{8} +(-1.03658 - 8.94011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 296 q - 2 q^{2} - 6 q^{3} - 6 q^{4} + 4 q^{5} + 12 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.745183	+	2.78106i	−0.372592	+	1.39053i	0.484240	+	0.874935i	\(0.339096\pi\)
							−0.856832	+	0.515596i	\(0.827571\pi\)
\(3\)	1.99542	−	2.24015i	0.665141	−	0.746718i
							\(5\)	−0.720689	−	2.68965i	−0.144138	−	0.537929i	−0.999792	−	0.0203819i	\(-0.993512\pi\)
0.855655	−	0.517547i	\(-0.173155\pi\)
\(7\)	2.95625			0.422322			0.211161	−	0.977451i	\(-0.432276\pi\)
							0.211161	+	0.977451i	\(0.432276\pi\)
\(11\)	5.47774	+	3.16257i	0.497976	+	0.287507i	0.727877	−	0.685707i	\(-0.240507\pi\)
							−0.229901	+	0.973214i	\(0.573840\pi\)
\(13\)	−3.24658	−	12.1164i	−0.249737	−	0.932032i	−0.970943	−	0.239311i	\(-0.923079\pi\)
							0.721206	−	0.692721i	\(-0.243588\pi\)
\(17\)	−3.49370	−	13.0387i	−0.205512	−	0.766980i	−0.989293	−	0.145943i	\(-0.953378\pi\)
							0.783781	−	0.621037i	\(-0.213288\pi\)
\(19\)	35.2738	+	9.45158i	1.85651	+	0.497452i	0.999831	−	0.0183908i	\(-0.00585432\pi\)
							0.856684	+	0.515842i	\(0.172521\pi\)
\(23\)	−28.3756	−	7.60322i	−1.23372	−	0.330575i	−0.417694	−	0.908588i	\(-0.637162\pi\)
							−0.816028	+	0.578013i	\(0.803828\pi\)
\(29\)	40.9697	−	10.9778i	1.41275	−	0.378545i	0.529843	−	0.848096i	\(-0.322251\pi\)
							0.882905	+	0.469551i	\(0.155584\pi\)
\(31\)	8.35509	−	31.1816i	0.269519	−	1.00586i	−0.689907	−	0.723898i	\(-0.742349\pi\)
							0.959426	−	0.281961i	\(-0.0909848\pi\)
\(37\)	27.6888	+	24.5424i	0.748347	+	0.663308i
							\(41\)	30.0517	+	17.3504i	0.732968	+	0.423179i	0.819507	−	0.573069i	\(-0.194247\pi\)
−0.0865387	+	0.996248i	\(0.527581\pi\)
\(43\)	−18.5285	−	69.1494i	−0.430896	−	1.60813i	−0.750701	−	0.660642i	\(-0.770284\pi\)
							0.319805	−	0.947483i	\(-0.396383\pi\)
\(47\)	−35.3340	+	61.2003i	−0.751788	+	1.30213i	0.195168	+	0.980770i	\(0.437475\pi\)
							−0.946956	+	0.321365i	\(0.895858\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bg.a.88.13	yes	296
9.4	even	3		333.3.ba.a.310.13	yes	296
37.8	odd	12		333.3.ba.a.304.13	✓	296
333.193	odd	12	inner	333.3.bg.a.193.13	yes	296

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.ba.a.304.13	✓	296	37.8	odd	12
333.3.ba.a.310.13	yes	296	9.4	even	3
333.3.bg.a.88.13	yes	296	1.1	even	1	trivial
333.3.bg.a.193.13	yes	296	333.193	odd	12	inner