Embedded newform 333.3.bn.a.86.15

• Label: 333.3.bn.a.86.15
• Level: $333$
• Weight: $3$
• Character: 333.86
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $444$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bn (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(444\)
Relative dimension:	\(74\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			86.15
Character	\(\chi\)	\(=\)	333.86
Dual form			333.3.bn.a.182.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.924319 + 2.53955i) q^{2} +(-0.934435 + 2.85076i) q^{3} +(-2.53075 - 2.12355i) q^{4} +(4.08474 + 0.720250i) q^{5} +(-6.37592 - 5.00805i) q^{6} +(-0.00592698 - 0.00497333i) q^{7} +(-1.62976 + 0.940944i) q^{8} +(-7.25366 - 5.32770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 444 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} - 12 q^{6} - 6 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.924319	+	2.53955i	−0.462159	+	1.26977i	0.461698	+	0.887037i	\(0.347240\pi\)
							−0.923858	+	0.382736i	\(0.874982\pi\)
\(3\)	−0.934435	+	2.85076i	−0.311478	+	0.950253i
							\(5\)	4.08474	+	0.720250i	0.816948	+	0.144050i	0.566481	−	0.824075i	\(-0.308305\pi\)
0.250468	+	0.968125i	\(0.419416\pi\)
\(7\)	−0.00592698	−	0.00497333i	−0.000846712	−	0.000710476i	0.642364	−	0.766400i	\(-0.277954\pi\)
							−0.643211	+	0.765689i	\(0.722398\pi\)
\(11\)	1.01879	+	0.588198i	0.0926172	+	0.0534726i	0.545593	−	0.838050i	\(-0.316304\pi\)
							−0.452976	+	0.891523i	\(0.649638\pi\)
\(13\)	−4.50597	−	3.78096i	−0.346613	−	0.290843i	0.452815	−	0.891604i	\(-0.350420\pi\)
							−0.799428	+	0.600761i	\(0.794864\pi\)
\(17\)	−27.7032	+	4.88482i	−1.62960	+	0.287342i	−0.912330	−	0.409456i	\(-0.865719\pi\)
							−0.717268	+	0.696798i	\(0.754608\pi\)
\(19\)	−21.7424	−	18.2440i	−1.14434	−	0.960213i	−0.144765	−	0.989466i	\(-0.546243\pi\)
							−0.999572	+	0.0292534i	\(0.990687\pi\)
\(23\)	−9.03627	+	5.21709i	−0.392881	+	0.226830i	−0.683408	−	0.730037i	\(-0.739503\pi\)
							0.290526	+	0.956867i	\(0.406170\pi\)
\(29\)	31.9211	+	18.4297i	1.10073	+	0.635506i	0.936412	−	0.350902i	\(-0.114125\pi\)
							0.164316	+	0.986408i	\(0.447458\pi\)
\(31\)	−27.6025	+	47.8089i	−0.890403	+	1.54222i	−0.0510096	+	0.998698i	\(0.516244\pi\)
							−0.839393	+	0.543525i	\(0.817089\pi\)
\(37\)	−12.5993	+	34.7888i	−0.340520	+	0.940237i
							\(41\)	42.5189	−	50.6720i	1.03705	−	1.23590i	0.0657971	−	0.997833i	\(-0.479041\pi\)
0.971248	−	0.238069i	\(-0.0765146\pi\)
\(43\)	18.7434	+	32.4646i	0.435894	+	0.754990i	0.997368	−	0.0725039i	\(-0.0230990\pi\)
							−0.561474	+	0.827494i	\(0.689766\pi\)
\(47\)		−	21.1223i		−	0.449411i	−0.974427	−	0.224706i	\(-0.927858\pi\)
							0.974427	−	0.224706i	\(-0.0721421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bn.a.86.15	yes	444
9.2	odd	6		333.3.bk.a.308.15	yes	444
37.34	even	9		333.3.bk.a.293.15	✓	444
333.182	odd	18	inner	333.3.bn.a.182.15	yes	444

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.bk.a.293.15	✓	444	37.34	even	9
333.3.bk.a.308.15	yes	444	9.2	odd	6
333.3.bn.a.86.15	yes	444	1.1	even	1	trivial
333.3.bn.a.182.15	yes	444	333.182	odd	18	inner