Embedded newform 33.8.e.b.25.3

• Label: 33.8.e.b.25.3
• Level: $33$
• Weight: $8$
• Character: 33.25
• 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			25.3
Character	\(\chi\)	\(=\)	33.25
Dual form			33.8.e.b.4.3

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

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	−0.747823	−	0.663899i	\(-0.768901\pi\)
							0.400315	+	0.916378i	\(0.368901\pi\)
\(3\)	8.34346	+	25.6785i	0.178411	+	0.549093i
							\(5\)	−221.480	−	160.915i	−0.792393	−	0.575707i	0.116280	−	0.993216i	\(-0.462903\pi\)
−0.908673	+	0.417510i	\(0.862903\pi\)
\(7\)	−146.721	+	451.561i	−0.161677	+	0.497592i	−0.998776	−	0.0494599i	\(-0.984250\pi\)
							0.837099	+	0.547052i	\(0.184250\pi\)
\(11\)	3317.03	−	2912.81i	0.751408	−	0.659838i
							\(13\)	592.129	−	430.207i	0.0747506	−	0.0543095i	−0.549782	−	0.835308i	\(-0.685289\pi\)
0.624533	+	0.780998i	\(0.285289\pi\)
\(17\)	−18715.0	−	13597.2i	−0.923885	−	0.671242i	0.0206026	−	0.999788i	\(-0.493442\pi\)
							−0.944488	+	0.328546i	\(0.893442\pi\)
\(19\)	−13394.8	−	41224.9i	−0.448021	−	1.37887i	−0.879137	−	0.476570i	\(-0.841880\pi\)
							0.431116	−	0.902297i	\(-0.358120\pi\)
\(23\)	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	0.698308	+	0.715798i	\(0.253937\pi\)
							−0.985678	+	0.168638i	\(0.946063\pi\)
\(31\)	−235199.	+	170882.i	−1.41798	+	1.03022i	−0.425875	+	0.904782i	\(0.640034\pi\)
							−0.992101	+	0.125438i	\(0.959966\pi\)
\(37\)	−93323.0	+	287219.i	−0.302888	+	0.932195i	0.677568	+	0.735460i	\(0.263034\pi\)
							−0.980457	+	0.196735i	\(0.936966\pi\)
\(41\)	−91985.1	−	283101.i	−0.208437	−	0.641502i	−0.999555	−	0.0298386i	\(-0.990501\pi\)
							0.791118	−	0.611663i	\(-0.209499\pi\)
\(43\)	−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	−0.926290	−	0.376810i	\(-0.877021\pi\)
							0.527901	−	0.849306i	\(-0.322979\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.4.3	✓	28	11.4	even	5	inner
33.8.e.b.25.3	yes	28	1.1	even	1	trivial
99.8.f.b.37.5		28	33.26	odd	10
99.8.f.b.91.5		28	3.2	odd	2
363.8.a.o.1.10		14	11.9	even	5
363.8.a.r.1.5		14	11.2	odd	10