Embedded newform 33.8.e.b.4.4

• Label: 33.8.e.b.4.4
• Level: $33$
• Weight: $8$
• Character: 33.4
• 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			4.4
Character	\(\chi\)	\(=\)	33.4
Dual form			33.8.e.b.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.24396 - 2.35687i) q^{2} +(8.34346 - 25.6785i) q^{3} +(-34.5858 - 106.444i) q^{4} +(354.050 - 257.233i) q^{5} +(-87.5868 + 63.6355i) q^{6} +(116.729 + 359.256i) q^{7} +(-297.283 + 914.942i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.24396	−	2.35687i	−0.286728	−	0.208320i	0.435119	−	0.900373i	\(-0.356706\pi\)
							−0.721847	+	0.692053i	\(0.756706\pi\)
\(3\)	8.34346	−	25.6785i	0.178411	−	0.549093i
							\(5\)	354.050	−	257.233i	1.26669	−	0.920303i	0.267623	−	0.963524i	\(-0.413762\pi\)
0.999066	+	0.0432204i	\(0.0137618\pi\)
\(7\)	116.729	+	359.256i	0.128628	+	0.395878i	0.994545	−	0.104311i	\(-0.0332639\pi\)
							−0.865916	+	0.500189i	\(0.833264\pi\)
\(11\)	1255.02	−	4232.27i	0.284299	−	0.958736i
							\(13\)	−2377.18	−	1727.12i	−0.300096	−	0.218032i	0.427539	−	0.903997i	\(-0.359381\pi\)
−0.727635	+	0.685964i	\(0.759381\pi\)
\(17\)	−10347.2	+	7517.70i	−0.510802	+	0.371120i	−0.813128	−	0.582085i	\(-0.802237\pi\)
							0.302326	+	0.953205i	\(0.402237\pi\)
\(19\)	8794.46	−	27066.6i	0.294152	−	0.905307i	−0.689353	−	0.724426i	\(-0.742105\pi\)
							0.983505	−	0.180881i	\(-0.0578949\pi\)
\(23\)	−77217.1			−1.32332			−0.661662	−	0.749802i	\(-0.730149\pi\)
							−0.661662	+	0.749802i	\(0.730149\pi\)
\(29\)	68034.8	+	209389.i	0.518010	+	1.59427i	0.777738	+	0.628588i	\(0.216367\pi\)
							−0.259729	+	0.965682i	\(0.583633\pi\)
\(31\)	134843.	+	97969.0i	0.812946	+	0.590640i	0.914683	−	0.404171i	\(-0.132440\pi\)
							−0.101737	+	0.994811i	\(0.532440\pi\)
\(37\)	−113272.	−	348614.i	−0.367634	−	1.13146i	−0.948315	−	0.317329i	\(-0.897214\pi\)
							0.580682	−	0.814131i	\(-0.302786\pi\)
\(41\)	82856.1	−	255005.i	0.187750	−	0.577837i	−0.812234	−	0.583331i	\(-0.801749\pi\)
							0.999985	+	0.00549462i	\(0.00174900\pi\)
\(43\)	821075.			1.57486			0.787432	−	0.616401i	\(-0.211410\pi\)
							0.787432	+	0.616401i	\(0.211410\pi\)
\(47\)	−95620.3	+	294289.i	−0.134341	+	0.413458i	−0.995487	−	0.0948992i	\(-0.969747\pi\)
							0.861146	+	0.508357i	\(0.169747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.b.4.4	✓	28
3.2	odd	2		99.8.f.b.37.4		28
11.3	even	5	inner	33.8.e.b.25.4	yes	28
11.5	even	5		363.8.a.o.1.9		14
11.6	odd	10		363.8.a.r.1.6		14
33.14	odd	10		99.8.f.b.91.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.4.4	✓	28	1.1	even	1	trivial
33.8.e.b.25.4	yes	28	11.3	even	5	inner
99.8.f.b.37.4		28	3.2	odd	2
99.8.f.b.91.4		28	33.14	odd	10
363.8.a.o.1.9		14	11.5	even	5
363.8.a.r.1.6		14	11.6	odd	10