Embedded newform 33.8.e.b.16.5

• Label: 33.8.e.b.16.5
• Level: $33$
• Weight: $8$
• Character: 33.16
• 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			16.5
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.b.31.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.813458 + 2.50357i) q^{2} +(-21.8435 - 15.8702i) q^{3} +(97.9481 - 71.1634i) q^{4} +(78.7070 - 242.235i) q^{5} +(21.9634 - 67.5963i) q^{6} +(-1135.08 + 824.681i) q^{7} +(530.435 + 385.384i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.813458	+	2.50357i	0.0719002	+	0.221286i	0.980549	−	0.196276i	\(-0.0628848\pi\)
							−0.908649	+	0.417562i	\(0.862885\pi\)
\(3\)	−21.8435	−	15.8702i	−0.467086	−	0.339358i
							\(5\)	78.7070	−	242.235i	0.281591	−	0.866647i	−0.705809	−	0.708402i	\(-0.749416\pi\)
0.987400	−	0.158245i	\(-0.0505836\pi\)
\(7\)	−1135.08	+	824.681i	−1.25078	+	0.908747i	−0.998267	−	0.0588455i	\(-0.981258\pi\)
							−0.252516	+	0.967593i	\(0.581258\pi\)
\(11\)	−3175.50	−	3066.49i	−0.719346	−	0.694652i
							\(13\)	−4283.83	−	13184.3i	−0.540792	−	1.66439i	−0.730790	−	0.682602i	\(-0.760848\pi\)
0.189998	−	0.981784i	\(-0.439152\pi\)
\(17\)	−5268.59	+	16215.1i	−0.260090	+	0.800475i	0.732694	+	0.680558i	\(0.238263\pi\)
							−0.992784	+	0.119916i	\(0.961737\pi\)
\(19\)	−12568.3	−	9131.40i	−0.420377	−	0.305422i	0.357413	−	0.933947i	\(-0.383659\pi\)
							−0.777789	+	0.628525i	\(0.783659\pi\)
\(23\)	−58901.9			−1.00944			−0.504721	−	0.863282i	\(-0.668405\pi\)
							−0.504721	+	0.863282i	\(0.668405\pi\)
\(29\)	−24930.5	+	18113.0i	−0.189818	+	0.137911i	−0.678635	−	0.734475i	\(-0.737428\pi\)
							0.488818	+	0.872386i	\(0.337428\pi\)
\(31\)	−60252.9	−	185439.i	−0.363256	−	1.11799i	−0.951066	−	0.308987i	\(-0.900010\pi\)
							0.587811	−	0.808999i	\(-0.299990\pi\)
\(37\)	345677.	−	251149.i	1.12193	−	0.815127i	0.137426	−	0.990512i	\(-0.456117\pi\)
							0.984500	+	0.175385i	\(0.0561170\pi\)
\(41\)	637965.	+	463509.i	1.44562	+	1.05030i	0.986831	+	0.161756i	\(0.0517157\pi\)
							0.458787	+	0.888547i	\(0.348284\pi\)
\(43\)	20485.0			0.0392914			0.0196457	−	0.999807i	\(-0.493746\pi\)
							0.0196457	+	0.999807i	\(0.493746\pi\)
\(47\)	−461937.	−	335617.i	−0.648994	−	0.471522i	0.213935	−	0.976848i	\(-0.431372\pi\)
							−0.862928	+	0.505326i	\(0.831372\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.16.5	✓	28
3.2	odd	2		99.8.f.b.82.3		28
11.3	even	5		363.8.a.o.1.8		14
11.8	odd	10		363.8.a.r.1.7		14
11.9	even	5	inner	33.8.e.b.31.5	yes	28
33.20	odd	10		99.8.f.b.64.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.5	✓	28	1.1	even	1	trivial
33.8.e.b.31.5	yes	28	11.9	even	5	inner
99.8.f.b.64.3		28	33.20	odd	10
99.8.f.b.82.3		28	3.2	odd	2
363.8.a.o.1.8		14	11.3	even	5
363.8.a.r.1.7		14	11.8	odd	10