Embedded newform 333.3.bg.a.88.15

• Label: 333.3.bg.a.88.15
• 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			88.15
Character	\(\chi\)	\(=\)	333.88
Dual form			333.3.bg.a.193.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.721912 + 2.69421i) q^{2} +(-2.93419 - 0.624946i) q^{3} +(-3.27352 - 1.88997i) q^{4} +(0.101754 + 0.379751i) q^{5} +(3.80196 - 7.45416i) q^{6} -4.41460 q^{7} +(-0.434034 + 0.434034i) q^{8} +(8.21888 + 3.66742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 296 q - 2 q^{2} - 6 q^{3} - 6 q^{4} + 4 q^{5} + 12 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.721912	+	2.69421i	−0.360956	+	1.34711i	0.511865	+	0.859066i	\(0.328955\pi\)
							−0.872821	+	0.488040i	\(0.837712\pi\)
\(3\)	−2.93419	−	0.624946i	−0.978062	−	0.208315i
							\(5\)	0.101754	+	0.379751i	0.0203508	+	0.0759502i	0.975354	−	0.220644i	\(-0.0708158\pi\)
−0.955004	+	0.296594i	\(0.904149\pi\)
\(7\)	−4.41460			−0.630657			−0.315329	−	0.948983i	\(-0.602115\pi\)
							−0.315329	+	0.948983i	\(0.602115\pi\)
\(11\)	−5.46748	−	3.15665i	−0.497043	−	0.286968i	0.230448	−	0.973085i	\(-0.425981\pi\)
							−0.727492	+	0.686116i	\(0.759314\pi\)
\(13\)	−4.95703	−	18.4999i	−0.381310	−	1.42307i	−0.843902	−	0.536497i	\(-0.819747\pi\)
							0.462593	−	0.886571i	\(-0.346919\pi\)
\(17\)	−0.0161114	−	0.0601286i	−0.000947730	−	0.00353698i	0.965450	−	0.260587i	\(-0.0839161\pi\)
							−0.966398	+	0.257050i	\(0.917249\pi\)
\(19\)	16.4835	+	4.41675i	0.867554	+	0.232460i	0.665030	−	0.746817i	\(-0.268419\pi\)
							0.202524	+	0.979277i	\(0.435085\pi\)
\(23\)	38.8736	+	10.4161i	1.69016	+	0.452876i	0.970430	−	0.241382i	\(-0.0776007\pi\)
							0.719726	+	0.694258i	\(0.244267\pi\)
\(29\)	−16.5360	+	4.43080i	−0.570206	+	0.152786i	−0.532391	−	0.846499i	\(-0.678706\pi\)
							−0.0378147	+	0.999285i	\(0.512040\pi\)
\(31\)	−13.5909	+	50.7221i	−0.438417	+	1.63620i	0.294337	+	0.955702i	\(0.404901\pi\)
							−0.732754	+	0.680494i	\(0.761765\pi\)
\(37\)	−10.1269	−	35.5872i	−0.273699	−	0.961815i
							\(41\)	56.5457	+	32.6467i	1.37916	+	0.796261i	0.992059	−	0.125775i	\(-0.0401418\pi\)
0.387105	+	0.922036i	\(0.373475\pi\)
\(43\)	−9.59498	−	35.8089i	−0.223139	−	0.832766i	−0.983142	−	0.182846i	\(-0.941469\pi\)
							0.760003	−	0.649920i	\(-0.225198\pi\)
\(47\)	12.2859	−	21.2798i	0.261402	−	0.452762i	−0.705213	−	0.708996i	\(-0.749148\pi\)
							0.966615	+	0.256234i	\(0.0824818\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bg.a.88.15	yes	296
9.4	even	3		333.3.ba.a.310.15	yes	296
37.8	odd	12		333.3.ba.a.304.15	✓	296
333.193	odd	12	inner	333.3.bg.a.193.15	yes	296

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.ba.a.304.15	✓	296	37.8	odd	12
333.3.ba.a.310.15	yes	296	9.4	even	3
333.3.bg.a.88.15	yes	296	1.1	even	1	trivial
333.3.bg.a.193.15	yes	296	333.193	odd	12	inner