Embedded newform 33.8.e.b.16.1

• Label: 33.8.e.b.16.1
• 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			16.1
Character	\(\chi\)	\(=\)	33.16
Dual form			33.8.e.b.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.47025 - 19.9134i) q^{2} +(-21.8435 - 15.8702i) q^{3} +(-251.124 + 182.452i) q^{4} +(-11.5844 + 35.6531i) q^{5} +(-174.697 + 537.661i) q^{6} +(80.4609 - 58.4583i) q^{7} +(3089.84 + 2244.90i) q^{8} +(225.273 + 693.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.47025	−	19.9134i	−0.571894	−	1.76011i	−0.646518	−	0.762898i	\(-0.723776\pi\)
							0.0746240	−	0.997212i	\(-0.476224\pi\)
\(3\)	−21.8435	−	15.8702i	−0.467086	−	0.339358i
							\(5\)	−11.5844	+	35.6531i	−0.0414456	+	0.127557i	−0.969638	−	0.244543i	\(-0.921362\pi\)
0.928193	+	0.372100i	\(0.121362\pi\)
\(7\)	80.4609	−	58.4583i	0.0886628	−	0.0644173i	−0.542571	−	0.840010i	\(-0.682549\pi\)
							0.631234	+	0.775593i	\(0.282549\pi\)
\(11\)	−1385.93	+	4191.22i	−0.313955	+	0.949438i
							\(13\)	−2827.46	−	8702.04i	−0.356940	−	1.09855i	−0.954876	−	0.297005i	\(-0.904012\pi\)
0.597936	−	0.801544i	\(-0.295988\pi\)
\(17\)	−3563.05	+	10965.9i	−0.175894	+	0.541345i	−0.999673	−	0.0255653i	\(-0.991861\pi\)
							0.823780	+	0.566910i	\(0.191861\pi\)
\(19\)	−10084.4	−	7326.72i	−0.337296	−	0.245060i	0.406224	−	0.913773i	\(-0.366845\pi\)
							−0.743520	+	0.668714i	\(0.766845\pi\)
\(23\)	97738.3			1.67501			0.837505	−	0.546429i	\(-0.184013\pi\)
							0.837505	+	0.546429i	\(0.184013\pi\)
\(29\)	−183848.	+	133574.i	−1.39980	+	1.01702i	−0.405095	+	0.914275i	\(0.632762\pi\)
							−0.994708	+	0.102742i	\(0.967238\pi\)
\(31\)	55814.9	+	171781.i	0.336499	+	1.03564i	0.965979	+	0.258621i	\(0.0832682\pi\)
							−0.629479	+	0.777017i	\(0.716732\pi\)
\(37\)	147200.	−	106947.i	0.477751	−	0.347107i	−0.322703	−	0.946500i	\(-0.604592\pi\)
							0.800454	+	0.599394i	\(0.204592\pi\)
\(41\)	508392.	+	369368.i	1.15201	+	0.836982i	0.988746	−	0.149601i	\(-0.0477990\pi\)
							0.163260	+	0.986583i	\(0.447799\pi\)
\(43\)	−549233.			−1.05346			−0.526729	−	0.850034i	\(-0.676582\pi\)
							−0.526729	+	0.850034i	\(0.676582\pi\)
\(47\)	176295.	+	128086.i	0.247684	+	0.179953i	0.704700	−	0.709505i	\(-0.251082\pi\)
							−0.457016	+	0.889459i	\(0.651082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.16.1	✓	28
3.2	odd	2		99.8.f.b.82.7		28
11.3	even	5		363.8.a.o.1.2		14
11.8	odd	10		363.8.a.r.1.13		14
11.9	even	5	inner	33.8.e.b.31.1	yes	28
33.20	odd	10		99.8.f.b.64.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.1	✓	28	1.1	even	1	trivial
33.8.e.b.31.1	yes	28	11.9	even	5	inner
99.8.f.b.64.7		28	33.20	odd	10
99.8.f.b.82.7		28	3.2	odd	2
363.8.a.o.1.2		14	11.3	even	5
363.8.a.r.1.13		14	11.8	odd	10