Embedded newform 33.8.e.b.4.3

• Label: 33.8.e.b.4.3
• Level: $33$
• Weight: $8$
• Character: 33.4
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	33.4
Dual form			33.8.e.b.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.93160 - 2.85647i) q^{2} +(8.34346 - 25.6785i) q^{3} +(-32.2561 - 99.2742i) q^{4} +(-221.480 + 160.915i) q^{5} +(-106.153 + 77.1248i) q^{6} +(-146.721 - 451.561i) q^{7} +(-348.978 + 1074.04i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.93160	−	2.85647i	−0.347508	−	0.252479i	0.400315	−	0.916378i	\(-0.368901\pi\)
							−0.747823	+	0.663899i	\(0.768901\pi\)
\(3\)	8.34346	−	25.6785i	0.178411	−	0.549093i
							\(5\)	−221.480	+	160.915i	−0.792393	+	0.575707i	−0.908673	−	0.417510i	\(-0.862903\pi\)
0.116280	+	0.993216i	\(0.462903\pi\)
\(7\)	−146.721	−	451.561i	−0.161677	−	0.497592i	0.837099	−	0.547052i	\(-0.184250\pi\)
							−0.998776	+	0.0494599i	\(0.984250\pi\)
\(11\)	3317.03	+	2912.81i	0.751408	+	0.659838i
							\(13\)	592.129	+	430.207i	0.0747506	+	0.0543095i	0.624533	−	0.780998i	\(-0.285289\pi\)
−0.549782	+	0.835308i	\(0.685289\pi\)
\(17\)	−18715.0	+	13597.2i	−0.923885	+	0.671242i	−0.944488	−	0.328546i	\(-0.893442\pi\)
							0.0206026	+	0.999788i	\(0.493442\pi\)
\(19\)	−13394.8	+	41224.9i	−0.448021	+	1.37887i	0.431116	+	0.902297i	\(0.358120\pi\)
							−0.879137	+	0.476570i	\(0.841880\pi\)
\(23\)	37742.3			0.646816			0.323408	−	0.946260i	\(-0.395171\pi\)
							0.323408	+	0.946260i	\(0.395171\pi\)
\(29\)	−37742.9	−	116161.i	−0.287371	−	0.884436i	−0.985678	−	0.168638i	\(-0.946063\pi\)
							0.698308	−	0.715798i	\(-0.253937\pi\)
\(31\)	−235199.	−	170882.i	−1.41798	−	1.03022i	−0.992101	−	0.125438i	\(-0.959966\pi\)
							−0.425875	−	0.904782i	\(-0.640034\pi\)
\(37\)	−93323.0	−	287219.i	−0.302888	−	0.932195i	−0.980457	−	0.196735i	\(-0.936966\pi\)
							0.677568	−	0.735460i	\(-0.263034\pi\)
\(41\)	−91985.1	+	283101.i	−0.208437	+	0.641502i	0.791118	+	0.611663i	\(0.209499\pi\)
							−0.999555	+	0.0298386i	\(0.990501\pi\)
\(43\)	−274853.			−0.527182			−0.263591	−	0.964635i	\(-0.584907\pi\)
							−0.263591	+	0.964635i	\(0.584907\pi\)
\(47\)	−283563.	+	872718.i	−0.398389	+	1.22612i	0.527901	+	0.849306i	\(0.322979\pi\)
							−0.926290	+	0.376810i	\(0.877021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.4.3	✓	28
3.2	odd	2		99.8.f.b.37.5		28
11.3	even	5	inner	33.8.e.b.25.3	yes	28
11.5	even	5		363.8.a.o.1.10		14
11.6	odd	10		363.8.a.r.1.5		14
33.14	odd	10		99.8.f.b.91.5		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.4.3	✓	28	1.1	even	1	trivial
33.8.e.b.25.3	yes	28	11.3	even	5	inner
99.8.f.b.37.5		28	3.2	odd	2
99.8.f.b.91.5		28	33.14	odd	10
363.8.a.o.1.10		14	11.5	even	5
363.8.a.r.1.5		14	11.6	odd	10