Embedded newform 33.6.f.a.2.10

• Label: 33.6.f.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.489986 - 1.50802i) q^{2} +(4.79885 - 14.8314i) q^{3} +(23.8545 + 17.3313i) q^{4} +(16.6412 - 5.40705i) q^{5} +(-20.0147 - 14.5039i) q^{6} +(36.5406 - 50.2938i) q^{7} +(78.8739 - 57.3052i) q^{8} +(-196.942 - 142.348i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.489986	−	1.50802i	0.0866180	−	0.266583i	−0.898361	−	0.439258i	\(-0.855241\pi\)
							0.984979	+	0.172676i	\(0.0552412\pi\)
\(3\)	4.79885	−	14.8314i	0.307846	−	0.951436i
							\(5\)	16.6412	−	5.40705i	0.297687	−	0.0967242i	−0.156366	−	0.987699i	\(-0.549978\pi\)
0.454052	+	0.890975i	\(0.349978\pi\)
\(7\)	36.5406	−	50.2938i	0.281858	−	0.387944i	−0.644490	−	0.764613i	\(-0.722930\pi\)
							0.926348	+	0.376668i	\(0.122930\pi\)
\(11\)	−56.2699	−	397.347i	−0.140215	−	0.990121i
							\(13\)	283.554	+	92.1323i	0.465347	+	0.151201i	0.532300	−	0.846556i	\(-0.321328\pi\)
−0.0669529	+	0.997756i	\(0.521328\pi\)
\(17\)	320.060	+	985.043i	0.268602	+	0.826671i	0.990842	+	0.135029i	\(0.0431126\pi\)
							−0.722240	+	0.691643i	\(0.756887\pi\)
\(19\)	−821.679	−	1130.94i	−0.522177	−	0.718715i	0.463736	−	0.885974i	\(-0.346509\pi\)
							−0.985913	+	0.167258i	\(0.946509\pi\)
\(23\)			3741.39i			1.47473i	0.675494	+	0.737366i	\(0.263931\pi\)
							−0.675494	+	0.737366i	\(0.736069\pi\)
\(29\)	5353.85	+	3889.80i	1.18215	+	0.858880i	0.992412	−	0.122955i	\(-0.0392371\pi\)
							0.189735	+	0.981835i	\(0.439237\pi\)
\(31\)	−153.897	+	473.647i	−0.0287625	+	0.0885218i	−0.964407	−	0.264421i	\(-0.914819\pi\)
							0.935645	+	0.352943i	\(0.114819\pi\)
\(37\)	219.834	+	159.719i	0.0263992	+	0.0191802i	0.600907	−	0.799319i	\(-0.294806\pi\)
							−0.574507	+	0.818499i	\(0.694806\pi\)
\(41\)	−3713.69	+	2698.16i	−0.345022	+	0.250673i	−0.746778	−	0.665074i	\(-0.768400\pi\)
							0.401756	+	0.915747i	\(0.368400\pi\)
\(43\)		−	14178.6i		−	1.16939i	−0.811252	−	0.584697i	\(-0.801213\pi\)
							0.811252	−	0.584697i	\(-0.198787\pi\)
\(47\)	−2154.98	−	2966.08i	−0.142298	−	0.195857i	0.731919	−	0.681391i	\(-0.238625\pi\)
							−0.874217	+	0.485535i	\(0.838625\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.10	yes	72
3.2	odd	2	inner	33.6.f.a.2.9	✓	72
11.6	odd	10	inner	33.6.f.a.17.9	yes	72
33.17	even	10	inner	33.6.f.a.17.10	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.9	✓	72	3.2	odd	2	inner
33.6.f.a.2.10	yes	72	1.1	even	1	trivial
33.6.f.a.17.9	yes	72	11.6	odd	10	inner
33.6.f.a.17.10	yes	72	33.17	even	10	inner