Embedded newform 33.8.e.a.16.6

• Label: 33.8.e.a.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.a.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.92793 + 12.0889i) q^{2} +(21.8435 + 15.8702i) q^{3} +(-27.1593 + 19.7324i) q^{4} +(-17.6507 + 54.3232i) q^{5} +(-106.054 + 326.401i) q^{6} +(-962.124 + 699.024i) q^{7} +(971.059 + 705.516i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.92793	+	12.0889i	0.347183	+	1.06852i	0.960405	+	0.278609i	\(0.0898733\pi\)
							−0.613221	+	0.789911i	\(0.710127\pi\)
\(3\)	21.8435	+	15.8702i	0.467086	+	0.339358i
							\(5\)	−17.6507	+	54.3232i	−0.0631490	+	0.194353i	−0.977653	−	0.210223i	\(-0.932581\pi\)
0.914504	+	0.404576i	\(0.132581\pi\)
\(7\)	−962.124	+	699.024i	−1.06020	+	0.770280i	−0.974125	−	0.226008i	\(-0.927432\pi\)
							−0.0860747	+	0.996289i	\(0.527432\pi\)
\(11\)	−1592.19	+	4117.29i	−0.360678	+	0.932690i
							\(13\)	−158.820	−	488.797i	−0.0200495	−	0.0617059i	0.940531	−	0.339707i	\(-0.110328\pi\)
−0.960581	+	0.278001i	\(0.910328\pi\)
\(17\)	5671.09	−	17453.8i	0.279960	−	0.861627i	−0.707905	−	0.706308i	\(-0.750359\pi\)
							0.987864	−	0.155319i	\(-0.0496406\pi\)
\(19\)	−11294.6	−	8206.03i	−0.377776	−	0.274470i	0.382652	−	0.923893i	\(-0.375011\pi\)
							−0.760428	+	0.649422i	\(0.775011\pi\)
\(23\)	−20508.4			−0.351467			−0.175733	−	0.984438i	\(-0.556230\pi\)
							−0.175733	+	0.984438i	\(0.556230\pi\)
\(29\)	167115.	−	121416.i	1.27240	−	0.924453i	0.273104	−	0.961984i	\(-0.411950\pi\)
							0.999295	+	0.0375319i	\(0.0119496\pi\)
\(31\)	78485.0	+	241552.i	0.473174	+	1.45628i	0.848405	+	0.529348i	\(0.177564\pi\)
							−0.375231	+	0.926932i	\(0.622436\pi\)
\(37\)	235343.	−	170987.i	0.763829	−	0.554954i	−0.136254	−	0.990674i	\(-0.543506\pi\)
							0.900082	+	0.435720i	\(0.143506\pi\)
\(41\)	3826.59	+	2780.18i	0.00867099	+	0.00629984i	0.592112	−	0.805855i	\(-0.298294\pi\)
							−0.583441	+	0.812155i	\(0.698294\pi\)
\(43\)	764405.			1.46617			0.733084	−	0.680138i	\(-0.238080\pi\)
							0.733084	+	0.680138i	\(0.238080\pi\)
\(47\)	−325123.	−	236216.i	−0.456779	−	0.331869i	0.335488	−	0.942045i	\(-0.391099\pi\)
							−0.792266	+	0.610175i	\(0.791099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.16.6	✓	28
3.2	odd	2		99.8.f.c.82.2		28
11.3	even	5		363.8.a.q.1.11		14
11.8	odd	10		363.8.a.p.1.4		14
11.9	even	5	inner	33.8.e.a.31.6	yes	28
33.20	odd	10		99.8.f.c.64.2		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.16.6	✓	28	1.1	even	1	trivial
33.8.e.a.31.6	yes	28	11.9	even	5	inner
99.8.f.c.64.2		28	33.20	odd	10
99.8.f.c.82.2		28	3.2	odd	2
363.8.a.p.1.4		14	11.8	odd	10
363.8.a.q.1.11		14	11.3	even	5