Embedded newform 33.6.f.a.2.17

• Label: 33.6.f.a.2.17
• Level: $33$
• Weight: $6$
• Character: 33.2
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.95379 - 9.09083i) q^{2} +(14.4062 - 5.95502i) q^{3} +(-48.0298 - 34.8957i) q^{4} +(-91.6682 + 29.7848i) q^{5} +(-11.5833 - 148.554i) q^{6} +(70.5766 - 97.1403i) q^{7} +(-211.641 + 153.766i) q^{8} +(172.075 - 171.578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.95379	−	9.09083i	0.522161	−	1.60705i	−0.247700	−	0.968837i	\(-0.579675\pi\)
							0.769862	−	0.638211i	\(-0.220325\pi\)
\(3\)	14.4062	−	5.95502i	0.924156	−	0.382015i
							\(5\)	−91.6682	+	29.7848i	−1.63981	+	0.532807i	−0.976496	−	0.215534i	\(-0.930851\pi\)
−0.663315	+	0.748341i	\(0.730851\pi\)
\(7\)	70.5766	−	97.1403i	0.544397	−	0.749298i	−0.444842	−	0.895609i	\(-0.646740\pi\)
							0.989239	+	0.146311i	\(0.0467401\pi\)
\(11\)	277.875	−	289.545i	0.692417	−	0.721498i
							\(13\)	390.347	+	126.831i	0.640608	+	0.208146i	0.611269	−	0.791423i	\(-0.290660\pi\)
0.0293397	+	0.999569i	\(0.490660\pi\)
\(17\)	39.0029	+	120.038i	0.0327321	+	0.100739i	0.966088	−	0.258214i	\(-0.0831339\pi\)
							−0.933356	+	0.358953i	\(0.883134\pi\)
\(19\)	1063.98	+	1464.45i	0.676163	+	0.930658i	0.999880	−	0.0154955i	\(-0.00493256\pi\)
							−0.323717	+	0.946154i	\(0.604933\pi\)
\(23\)			379.653i			0.149647i	0.997197	+	0.0748234i	\(0.0238393\pi\)
							−0.997197	+	0.0748234i	\(0.976161\pi\)
\(29\)	582.330	+	423.088i	0.128580	+	0.0934190i	0.650217	−	0.759749i	\(-0.274678\pi\)
							−0.521636	+	0.853168i	\(0.674678\pi\)
\(31\)	−2348.72	+	7228.62i	−0.438963	+	1.35099i	0.450008	+	0.893024i	\(0.351421\pi\)
							−0.888971	+	0.457963i	\(0.848579\pi\)
\(37\)	−9263.80	−	6730.55i	−1.11246	−	0.808251i	−0.129412	−	0.991591i	\(-0.541309\pi\)
							−0.983049	+	0.183340i	\(0.941309\pi\)
\(41\)	6792.51	−	4935.05i	0.631060	−	0.458492i	−0.225707	−	0.974195i	\(-0.572469\pi\)
							0.856767	+	0.515703i	\(0.172469\pi\)
\(43\)			7794.19i			0.642835i	0.946938	+	0.321418i	\(0.104159\pi\)
							−0.946938	+	0.321418i	\(0.895841\pi\)
\(47\)	−1576.98	−	2170.53i	−0.104131	−	0.143325i	0.753771	−	0.657137i	\(-0.228233\pi\)
							−0.857903	+	0.513812i	\(0.828233\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.17	yes	72
3.2	odd	2	inner	33.6.f.a.2.2	✓	72
11.6	odd	10	inner	33.6.f.a.17.2	yes	72
33.17	even	10	inner	33.6.f.a.17.17	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.2	✓	72	3.2	odd	2	inner
33.6.f.a.2.17	yes	72	1.1	even	1	trivial
33.6.f.a.17.2	yes	72	11.6	odd	10	inner
33.6.f.a.17.17	yes	72	33.17	even	10	inner