Embedded newform 33.6.f.a.2.14

• Label: 33.6.f.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.97194 - 6.06902i) q^{2} +(15.5882 + 0.0966769i) q^{3} +(-7.05585 - 5.12637i) q^{4} +(59.6930 - 19.3954i) q^{5} +(31.3257 - 94.4141i) q^{6} +(-97.0552 + 133.585i) q^{7} +(120.178 - 87.3143i) q^{8} +(242.981 + 3.01403i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.97194	−	6.06902i	0.348593	−	1.07286i	−0.611038	−	0.791601i	\(-0.709248\pi\)
							0.959632	−	0.281259i	\(-0.0907521\pi\)
\(3\)	15.5882	+	0.0966769i	0.999981	+	0.00620183i
							\(5\)	59.6930	−	19.3954i	1.06782	−	0.346956i	0.278183	−	0.960528i	\(-0.410268\pi\)
0.789639	+	0.613572i	\(0.210268\pi\)
\(7\)	−97.0552	+	133.585i	−0.748641	+	1.03042i	0.249434	+	0.968392i	\(0.419756\pi\)
							−0.998075	+	0.0620242i	\(0.980244\pi\)
\(11\)	−390.430	−	92.8174i	−0.972886	−	0.231285i
							\(13\)	−1021.69	−	331.966i	−1.67671	−	0.544797i	−0.692445	−	0.721471i	\(-0.743466\pi\)
−0.984270	+	0.176673i	\(0.943466\pi\)
\(17\)	−77.1051	−	237.305i	−0.0647084	−	0.199152i	0.913475	−	0.406895i	\(-0.133388\pi\)
							−0.978184	+	0.207743i	\(0.933388\pi\)
\(19\)	1751.76	+	2411.09i	1.11324	+	1.53225i	0.816548	+	0.577277i	\(0.195885\pi\)
							0.296696	+	0.954972i	\(0.404115\pi\)
\(23\)			1763.53i			0.695126i	0.937657	+	0.347563i	\(0.112991\pi\)
							−0.937657	+	0.347563i	\(0.887009\pi\)
\(29\)	−2059.70	−	1496.46i	−0.454789	−	0.330423i	0.336695	−	0.941614i	\(-0.390691\pi\)
							−0.791484	+	0.611191i	\(0.790691\pi\)
\(31\)	547.442	−	1684.85i	0.102314	−	0.314889i	−0.886777	−	0.462198i	\(-0.847061\pi\)
							0.989091	+	0.147309i	\(0.0470610\pi\)
\(37\)	1794.75	+	1303.96i	0.215526	+	0.156589i	0.690310	−	0.723513i	\(-0.257474\pi\)
							−0.474784	+	0.880102i	\(0.657474\pi\)
\(41\)	−69.9620	+	50.8304i	−0.00649984	+	0.00472241i	−0.591030	−	0.806649i	\(-0.701279\pi\)
							0.584531	+	0.811372i	\(0.301279\pi\)
\(43\)			461.539i			0.0380660i	0.999819	+	0.0190330i	\(0.00605876\pi\)
							−0.999819	+	0.0190330i	\(0.993941\pi\)
\(47\)	−9135.16	−	12573.5i	−0.603214	−	0.830253i	0.392784	−	0.919631i	\(-0.371512\pi\)
							−0.995998	+	0.0893782i	\(0.971512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.14	yes	72
3.2	odd	2	inner	33.6.f.a.2.5	✓	72
11.6	odd	10	inner	33.6.f.a.17.5	yes	72
33.17	even	10	inner	33.6.f.a.17.14	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.5	✓	72	3.2	odd	2	inner
33.6.f.a.2.14	yes	72	1.1	even	1	trivial
33.6.f.a.17.5	yes	72	11.6	odd	10	inner
33.6.f.a.17.14	yes	72	33.17	even	10	inner