Embedded newform 33.6.f.a.29.2

• Label: 33.6.f.a.29.2
• Level: $33$
• Weight: $6$
• Character: 33.29
• 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			29.2
Character	\(\chi\)	\(=\)	33.29
Dual form			33.6.f.a.8.2

$q$-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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.8.2	✓	72	33.8	even	10	inner
33.6.f.a.8.17	yes	72	11.8	odd	10	inner
33.6.f.a.29.2	yes	72	1.1	even	1	trivial
33.6.f.a.29.17	yes	72	3.2	odd	2	inner