Embedded newform 33.6.f.a.2.18

• Label: 33.6.f.a.2.18
• 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			2.18
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.08173 - 9.48458i) q^{2} +(-12.6871 + 9.05745i) q^{3} +(-54.5716 - 39.6486i) q^{4} +(-16.3931 + 5.32643i) q^{5} +(46.8079 + 148.244i) q^{6} +(-74.1699 + 102.086i) q^{7} +(-286.047 + 207.825i) q^{8} +(78.9251 - 229.826i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.08173	−	9.48458i	0.544777	−	1.67665i	−0.176742	−	0.984257i	\(-0.556556\pi\)
							0.721519	−	0.692395i	\(-0.243444\pi\)
\(3\)	−12.6871	+	9.05745i	−0.813878	+	0.581036i
							\(5\)	−16.3931	+	5.32643i	−0.293248	+	0.0952821i	−0.451947	−	0.892045i	\(-0.649270\pi\)
0.158698	+	0.987327i	\(0.449270\pi\)
\(7\)	−74.1699	+	102.086i	−0.572114	+	0.787448i	−0.992803	−	0.119757i	\(-0.961788\pi\)
							0.420689	+	0.907205i	\(0.361788\pi\)
\(11\)	−293.152	−	274.067i	−0.730484	−	0.682929i
							\(13\)	−289.976	−	94.2191i	−0.475888	−	0.154625i	0.0612456	−	0.998123i	\(-0.480493\pi\)
−0.537133	+	0.843498i	\(0.680493\pi\)
\(17\)	415.883	+	1279.96i	0.349019	+	1.07417i	0.959397	+	0.282060i	\(0.0910176\pi\)
							−0.610378	+	0.792110i	\(0.708982\pi\)
\(19\)	−1592.36	−	2191.69i	−1.01195	−	1.39282i	−0.917700	−	0.397274i	\(-0.869956\pi\)
							−0.0942452	−	0.995549i	\(-0.530044\pi\)
\(23\)		−	3564.43i		−	1.40498i	−0.711694	−	0.702490i	\(-0.752072\pi\)
							0.711694	−	0.702490i	\(-0.247928\pi\)
\(29\)	3069.44	+	2230.08i	0.677742	+	0.492408i	0.872608	−	0.488422i	\(-0.162427\pi\)
							−0.194866	+	0.980830i	\(0.562427\pi\)
\(31\)	848.134	−	2610.29i	0.158511	−	0.487847i	−0.839989	−	0.542604i	\(-0.817438\pi\)
							0.998500	+	0.0547568i	\(0.0174384\pi\)
\(37\)	−159.754	−	116.068i	−0.0191843	−	0.0139382i	0.578152	−	0.815929i	\(-0.303774\pi\)
							−0.597336	+	0.801991i	\(0.703774\pi\)
\(41\)	13619.3	−	9894.99i	1.26530	−	0.919297i	0.266299	−	0.963891i	\(-0.414199\pi\)
							0.999005	+	0.0445934i	\(0.0141992\pi\)
\(43\)			6557.27i			0.540819i	0.962745	+	0.270410i	\(0.0871591\pi\)
							−0.962745	+	0.270410i	\(0.912841\pi\)
\(47\)	6782.17	+	9334.86i	0.447841	+	0.616401i	0.971932	−	0.235262i	\(-0.0755947\pi\)
							−0.524091	+	0.851663i	\(0.675595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.18	yes	72
3.2	odd	2	inner	33.6.f.a.2.1	✓	72
11.6	odd	10	inner	33.6.f.a.17.1	yes	72
33.17	even	10	inner	33.6.f.a.17.18	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.1	✓	72	3.2	odd	2	inner
33.6.f.a.2.18	yes	72	1.1	even	1	trivial
33.6.f.a.17.1	yes	72	11.6	odd	10	inner
33.6.f.a.17.18	yes	72	33.17	even	10	inner