Embedded newform 33.6.f.a.2.16

• Label: 33.6.f.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.81967 - 8.67805i) q^{2} +(-4.23991 - 15.0008i) q^{3} +(-41.4695 - 30.1293i) q^{4} +(53.2208 - 17.2925i) q^{5} +(-142.133 - 5.50305i) q^{6} +(-14.1075 + 19.4173i) q^{7} +(-142.170 + 103.293i) q^{8} +(-207.046 + 127.204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.81967	−	8.67805i	0.498452	−	1.53408i	−0.313056	−	0.949735i	\(-0.601353\pi\)
							0.811507	−	0.584342i	\(-0.198647\pi\)
\(3\)	−4.23991	−	15.0008i	−0.271990	−	0.962300i
							\(5\)	53.2208	−	17.2925i	0.952043	−	0.309338i	0.208498	−	0.978023i	\(-0.433143\pi\)
0.743546	+	0.668685i	\(0.233143\pi\)
\(7\)	−14.1075	+	19.4173i	−0.108819	+	0.149776i	−0.859953	−	0.510374i	\(-0.829507\pi\)
							0.751134	+	0.660150i	\(0.229507\pi\)
\(11\)	240.584	+	321.202i	0.599494	+	0.800379i
							\(13\)	323.676	+	105.169i	0.531192	+	0.172595i	0.562319	−	0.826921i	\(-0.309909\pi\)
−0.0311266	+	0.999515i	\(0.509909\pi\)
\(17\)	−463.558	−	1426.69i	−0.389029	−	1.19731i	−0.933515	−	0.358538i	\(-0.883275\pi\)
							0.544486	−	0.838770i	\(-0.316725\pi\)
\(19\)	167.174	+	230.095i	0.106239	+	0.146226i	0.858826	−	0.512267i	\(-0.171194\pi\)
							−0.752587	+	0.658493i	\(0.771194\pi\)
\(23\)		−	4346.85i		−	1.71338i	−0.515828	−	0.856692i	\(-0.672516\pi\)
							0.515828	−	0.856692i	\(-0.327484\pi\)
\(29\)	6126.66	+	4451.28i	1.35278	+	0.982856i	0.998867	+	0.0475790i	\(0.0151506\pi\)
							0.353917	+	0.935277i	\(0.384849\pi\)
\(31\)	−937.995	+	2886.85i	−0.175306	+	0.539536i	−0.999647	−	0.0265569i	\(-0.991546\pi\)
							0.824341	+	0.566093i	\(0.191546\pi\)
\(37\)	10155.5	+	7378.41i	1.21954	+	0.886050i	0.996062	−	0.0886584i	\(-0.0282579\pi\)
							0.223481	+	0.974708i	\(0.428258\pi\)
\(41\)	−7654.67	+	5561.45i	−0.711159	+	0.516688i	−0.883548	−	0.468342i	\(-0.844852\pi\)
							0.172388	+	0.985029i	\(0.444852\pi\)
\(43\)		−	8566.64i		−	0.706545i	−0.935521	−	0.353272i	\(-0.885069\pi\)
							0.935521	−	0.353272i	\(-0.114931\pi\)
\(47\)	−7903.45	−	10878.2i	−0.521882	−	0.718309i	0.463984	−	0.885843i	\(-0.346419\pi\)
							−0.985866	+	0.167535i	\(0.946419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.16	yes	72
3.2	odd	2	inner	33.6.f.a.2.3	✓	72
11.6	odd	10	inner	33.6.f.a.17.3	yes	72
33.17	even	10	inner	33.6.f.a.17.16	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.3	✓	72	3.2	odd	2	inner
33.6.f.a.2.16	yes	72	1.1	even	1	trivial
33.6.f.a.17.3	yes	72	11.6	odd	10	inner
33.6.f.a.17.16	yes	72	33.17	even	10	inner