Embedded newform 33.6.f.a.8.2

• Label: 33.6.f.a.8.2
• Level: $33$
• Weight: $6$
• Character: 33.8
• 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			8.2
Character	\(\chi\)	\(=\)	33.8
Dual form			33.6.f.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.80447 + 5.67028i) q^{2} +(14.8651 - 4.69358i) q^{3} +(18.8692 - 58.0733i) q^{4} +(-46.3038 + 63.7318i) q^{5} +(-89.4001 + 120.920i) q^{6} +(-135.347 - 43.9769i) q^{7} +(86.6348 + 266.635i) q^{8} +(198.941 - 139.541i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.80447	+	5.67028i	−1.37965	+	1.00237i	−0.382736	+	0.923858i	\(0.625018\pi\)
							−0.996913	+	0.0785160i	\(0.974982\pi\)
\(3\)	14.8651	−	4.69358i	0.953595	−	0.301093i
							\(5\)	−46.3038	+	63.7318i	−0.828308	+	1.14007i	0.159927	+	0.987129i	\(0.448874\pi\)
−0.988236	+	0.152940i	\(0.951126\pi\)
\(7\)	−135.347	−	43.9769i	−1.04401	−	0.339219i	−0.263693	−	0.964607i	\(-0.584941\pi\)
							−0.780314	+	0.625388i	\(0.784941\pi\)
\(11\)	−281.284	−	286.234i	−0.700913	−	0.713247i
							\(13\)	−87.0414	−	119.802i	−0.142846	−	0.196610i	0.731599	−	0.681735i	\(-0.238774\pi\)
−0.874445	+	0.485125i	\(0.838774\pi\)
\(17\)	−1754.90	−	1275.01i	−1.47275	−	1.07002i	−0.979805	−	0.199954i	\(-0.935921\pi\)
							−0.492944	−	0.870061i	\(-0.664079\pi\)
\(19\)	−95.9970	+	31.1913i	−0.0610061	+	0.0198221i	−0.339361	−	0.940656i	\(-0.610211\pi\)
							0.278355	+	0.960478i	\(0.410211\pi\)
\(23\)			368.216i			0.145139i	0.997363	+	0.0725694i	\(0.0231199\pi\)
							−0.997363	+	0.0725694i	\(0.976880\pi\)
\(29\)	−1429.99	+	4401.07i	−0.315747	+	0.971768i	0.659699	+	0.751530i	\(0.270684\pi\)
							−0.975446	+	0.220239i	\(0.929316\pi\)
\(31\)	−3090.40	+	2245.31i	−0.577578	+	0.419635i	−0.837850	−	0.545900i	\(-0.816188\pi\)
							0.260272	+	0.965535i	\(0.416188\pi\)
\(37\)	−2571.90	+	7915.49i	−0.308851	+	0.950547i	0.669360	+	0.742938i	\(0.266568\pi\)
							−0.978212	+	0.207609i	\(0.933432\pi\)
\(41\)	4267.35	+	13133.5i	0.396459	+	1.22018i	0.927819	+	0.373030i	\(0.121681\pi\)
							−0.531360	+	0.847146i	\(0.678319\pi\)
\(43\)			488.950i			0.0403268i	0.999797	+	0.0201634i	\(0.00641864\pi\)
							−0.999797	+	0.0201634i	\(0.993581\pi\)
\(47\)	10651.2	−	3460.77i	0.703319	−	0.228522i	0.0645427	−	0.997915i	\(-0.479441\pi\)
							0.638776	+	0.769393i	\(0.279441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.8.2	✓	72
3.2	odd	2	inner	33.6.f.a.8.17	yes	72
11.7	odd	10	inner	33.6.f.a.29.17	yes	72
33.29	even	10	inner	33.6.f.a.29.2	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.8.2	✓	72	1.1	even	1	trivial
33.6.f.a.8.17	yes	72	3.2	odd	2	inner
33.6.f.a.29.2	yes	72	33.29	even	10	inner
33.6.f.a.29.17	yes	72	11.7	odd	10	inner