Embedded newform 33.6.f.a.2.4

• Label: 33.6.f.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.32473 + 7.15479i) q^{2} +(-12.7649 - 8.94746i) q^{3} +(-19.8981 - 14.4568i) q^{4} +(-28.0945 + 9.12845i) q^{5} +(93.6922 - 70.5298i) q^{6} +(146.291 - 201.352i) q^{7} +(-45.0663 + 32.7426i) q^{8} +(82.8858 + 228.427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.32473	+	7.15479i	−0.410958	+	1.26480i	0.504858	+	0.863202i	\(0.331545\pi\)
							−0.915816	+	0.401597i	\(0.868455\pi\)
\(3\)	−12.7649	−	8.94746i	−0.818869	−	0.573980i
							\(5\)	−28.0945	+	9.12845i	−0.502569	+	0.163295i	−0.549320	−	0.835612i	\(-0.685113\pi\)
0.0467509	+	0.998907i	\(0.485113\pi\)
\(7\)	146.291	−	201.352i	1.12842	−	1.55314i	0.337398	−	0.941362i	\(-0.390453\pi\)
							0.791027	−	0.611781i	\(-0.209547\pi\)
\(11\)	183.551	−	356.876i	0.457377	−	0.889273i
							\(13\)	277.377	+	90.1254i	0.455211	+	0.147907i	0.527643	−	0.849466i	\(-0.323076\pi\)
−0.0724318	+	0.997373i	\(0.523076\pi\)
\(17\)	−322.736	−	993.279i	−0.270848	−	0.833583i	−0.990288	−	0.139029i	\(-0.955602\pi\)
							0.719441	−	0.694554i	\(-0.244398\pi\)
\(19\)	−986.663	−	1358.02i	−0.627025	−	0.863025i	0.370816	−	0.928706i	\(-0.379078\pi\)
							−0.997841	+	0.0656809i	\(0.979078\pi\)
\(23\)		−	3499.82i		−	1.37952i	−0.724040	−	0.689758i	\(-0.757717\pi\)
							0.724040	−	0.689758i	\(-0.242283\pi\)
\(29\)	2333.85	+	1695.64i	0.515321	+	0.374403i	0.814838	−	0.579688i	\(-0.196826\pi\)
							−0.299517	+	0.954091i	\(0.596826\pi\)
\(31\)	−577.959	+	1778.78i	−0.108017	+	0.332443i	−0.990427	−	0.138039i	\(-0.955920\pi\)
							0.882410	+	0.470482i	\(0.155920\pi\)
\(37\)	2271.81	+	1650.56i	0.272814	+	0.198211i	0.715777	−	0.698329i	\(-0.246073\pi\)
							−0.442963	+	0.896540i	\(0.646073\pi\)
\(41\)	902.939	−	656.024i	0.0838878	−	0.0609481i	−0.545051	−	0.838403i	\(-0.683490\pi\)
							0.628939	+	0.777455i	\(0.283490\pi\)
\(43\)			9314.82i			0.768251i	0.923281	+	0.384126i	\(0.125497\pi\)
							−0.923281	+	0.384126i	\(0.874503\pi\)
\(47\)	−293.328	−	403.731i	−0.0193691	−	0.0266592i	0.799223	−	0.601035i	\(-0.205245\pi\)
							−0.818592	+	0.574376i	\(0.805245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.4	✓	72
3.2	odd	2	inner	33.6.f.a.2.15	yes	72
11.6	odd	10	inner	33.6.f.a.17.15	yes	72
33.17	even	10	inner	33.6.f.a.17.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.4	✓	72	1.1	even	1	trivial
33.6.f.a.2.15	yes	72	3.2	odd	2	inner
33.6.f.a.17.4	yes	72	33.17	even	10	inner
33.6.f.a.17.15	yes	72	11.6	odd	10	inner