Embedded newform 33.8.e.a.25.2

• Label: 33.8.e.a.25.2
• 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			25.2
Character	\(\chi\)	\(=\)	33.25
Dual form			33.8.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.1414 + 8.82122i) q^{2} +(-8.34346 - 25.6785i) q^{3} +(30.0447 - 92.4681i) q^{4} +(158.638 + 115.257i) q^{5} +(327.817 + 238.173i) q^{6} +(-372.740 + 1147.18i) q^{7} +(-142.714 - 439.227i) q^{8} +(-589.773 + 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.1414	+	8.82122i	−1.07316	+	0.779693i	−0.976477	−	0.215623i	\(-0.930822\pi\)
							−0.0966787	+	0.995316i	\(0.530822\pi\)
\(3\)	−8.34346	−	25.6785i	−0.178411	−	0.549093i
							\(5\)	158.638	+	115.257i	0.567562	+	0.412358i	0.834219	−	0.551434i	\(-0.185919\pi\)
−0.266657	+	0.963792i	\(0.585919\pi\)
\(7\)	−372.740	+	1147.18i	−0.410736	+	1.26412i	0.505273	+	0.862959i	\(0.331392\pi\)
							−0.916009	+	0.401157i	\(0.868608\pi\)
\(11\)	−3527.62	+	2653.88i	−0.799112	+	0.601183i
							\(13\)	9804.60	−	7123.46i	1.23774	−	0.899268i	0.240291	−	0.970701i	\(-0.422757\pi\)
0.997445	+	0.0714325i	\(0.0227570\pi\)
\(17\)	−24416.2	−	17739.4i	−1.20533	−	0.875725i	−0.210534	−	0.977587i	\(-0.567520\pi\)
							−0.994799	+	0.101861i	\(0.967520\pi\)
\(19\)	−10909.2	−	33574.9i	−0.364883	−	1.12299i	−0.950054	−	0.312085i	\(-0.898973\pi\)
							0.585171	−	0.810910i	\(-0.301027\pi\)
\(23\)	−70550.7			−1.20908			−0.604539	−	0.796576i	\(-0.706643\pi\)
							−0.604539	+	0.796576i	\(0.706643\pi\)
\(29\)	24198.9	−	74476.7i	0.184248	−	0.567058i	−0.815686	−	0.578494i	\(-0.803640\pi\)
							0.999935	+	0.0114366i	\(0.00364045\pi\)
\(31\)	−140559.	+	102122.i	−0.847409	+	0.615679i	−0.924430	−	0.381351i	\(-0.875459\pi\)
							0.0770215	+	0.997029i	\(0.475459\pi\)
\(37\)	−44586.9	+	137225.i	−0.144711	+	0.445375i	−0.996974	−	0.0777393i	\(-0.975230\pi\)
							0.852263	+	0.523114i	\(0.175230\pi\)
\(41\)	−161876.	−	498202.i	−0.366807	−	1.12892i	−0.948842	−	0.315752i	\(-0.897743\pi\)
							0.582034	−	0.813164i	\(-0.302257\pi\)
\(43\)	630218.			1.20879			0.604395	−	0.796685i	\(-0.293415\pi\)
							0.604395	+	0.796685i	\(0.293415\pi\)
\(47\)	−133242.	−	410078.i	−0.187198	−	0.576135i	0.812782	−	0.582568i	\(-0.197952\pi\)
							−0.999979	+	0.00643349i	\(0.997952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.25.2	yes	28
3.2	odd	2		99.8.f.c.91.6		28
11.2	odd	10		363.8.a.p.1.3		14
11.4	even	5	inner	33.8.e.a.4.2	✓	28
11.9	even	5		363.8.a.q.1.12		14
33.26	odd	10		99.8.f.c.37.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.2	✓	28	11.4	even	5	inner
33.8.e.a.25.2	yes	28	1.1	even	1	trivial
99.8.f.c.37.6		28	33.26	odd	10
99.8.f.c.91.6		28	3.2	odd	2
363.8.a.p.1.3		14	11.2	odd	10
363.8.a.q.1.12		14	11.9	even	5