Embedded newform 33.6.f.a.2.9

• Label: 33.6.f.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.489986 + 1.50802i) q^{2} +(15.5884 + 0.0191829i) q^{3} +(23.8545 + 17.3313i) q^{4} +(-16.6412 + 5.40705i) q^{5} +(-7.66704 + 23.4983i) q^{6} +(36.5406 - 50.2938i) q^{7} +(-78.8739 + 57.3052i) q^{8} +(242.999 + 0.598063i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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	−0.984979	−	0.172676i	\(-0.944759\pi\)
							0.898361	+	0.439258i	\(0.144759\pi\)
\(3\)	15.5884	+	0.0191829i	0.999999	+	0.00123058i
							\(5\)	−16.6412	+	5.40705i	−0.297687	+	0.0967242i	−0.454052	−	0.890975i	\(-0.650022\pi\)
0.156366	+	0.987699i	\(0.450022\pi\)
\(7\)	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	\(-0.956887\pi\)
							0.722240	−	0.691643i	\(-0.243113\pi\)
\(19\)	−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	\(-0.736069\pi\)
							0.675494	−	0.737366i	\(-0.263931\pi\)
\(29\)	−5353.85	−	3889.80i	−1.18215	−	0.858880i	−0.189735	−	0.981835i	\(-0.560763\pi\)
							−0.992412	+	0.122955i	\(0.960763\pi\)
\(31\)	−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	−0.401756	−	0.915747i	\(-0.631600\pi\)
							0.746778	+	0.665074i	\(0.231600\pi\)
\(43\)		−	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	0.874217	−	0.485535i	\(-0.161375\pi\)
							−0.731919	+	0.681391i	\(0.761375\pi\)

(See \(a_n\) instead)

Twists

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

    

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