Embedded newform 33.8.e.a.4.4

• Label: 33.8.e.a.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.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.14212 - 1.55634i) q^{2} +(-8.34346 + 25.6785i) q^{3} +(-37.3877 - 115.067i) q^{4} +(121.253 - 88.0954i) q^{5} +(57.8373 - 42.0213i) q^{6} +(231.374 + 712.095i) q^{7} +(-203.727 + 627.009i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.14212	−	1.55634i	−0.189339	−	0.137563i	0.489078	−	0.872240i	\(-0.337333\pi\)
							−0.678416	+	0.734678i	\(0.737333\pi\)
\(3\)	−8.34346	+	25.6785i	−0.178411	+	0.549093i
							\(5\)	121.253	−	88.0954i	0.433807	−	0.315180i	−0.349362	−	0.936988i	\(-0.613602\pi\)
0.783170	+	0.621808i	\(0.213602\pi\)
\(7\)	231.374	+	712.095i	0.254959	+	0.784684i	0.993838	+	0.110845i	\(0.0353557\pi\)
							−0.738879	+	0.673839i	\(0.764644\pi\)
\(11\)	−4139.39	+	1533.82i	−0.937697	+	0.347455i
							\(13\)	−12389.2	−	9001.28i	−1.56402	−	1.13633i	−0.932617	−	0.360869i	\(-0.882480\pi\)
−0.631401	−	0.775457i	\(-0.717520\pi\)
\(17\)	−6583.81	+	4783.42i	−0.325017	+	0.236139i	−0.738313	−	0.674458i	\(-0.764377\pi\)
							0.413296	+	0.910597i	\(0.364377\pi\)
\(19\)	−10016.8	+	30828.5i	−0.335035	+	1.03113i	0.631669	+	0.775238i	\(0.282370\pi\)
							−0.966705	+	0.255895i	\(0.917630\pi\)
\(23\)	−19503.7			−0.334248			−0.167124	−	0.985936i	\(-0.553448\pi\)
							−0.167124	+	0.985936i	\(0.553448\pi\)
\(29\)	−79699.4	−	245289.i	−0.606823	−	1.86761i	−0.483739	−	0.875212i	\(-0.660722\pi\)
							−0.123083	−	0.992396i	\(-0.539278\pi\)
\(31\)	118891.	+	86379.3i	0.716775	+	0.520767i	0.885352	−	0.464921i	\(-0.153917\pi\)
							−0.168577	+	0.985688i	\(0.553917\pi\)
\(37\)	99140.9	+	305124.i	0.321771	+	0.990309i	0.972877	+	0.231323i	\(0.0743053\pi\)
							−0.651106	+	0.758987i	\(0.725695\pi\)
\(41\)	−8898.90	+	27388.0i	−0.0201648	+	0.0620607i	−0.960633	−	0.277822i	\(-0.910387\pi\)
							0.940468	+	0.339883i	\(0.110387\pi\)
\(43\)	−457913.			−0.878301			−0.439151	−	0.898413i	\(-0.644721\pi\)
							−0.439151	+	0.898413i	\(0.644721\pi\)
\(47\)	351649.	−	1.08226e6i	0.494045	−	1.52051i	−0.324394	−	0.945922i	\(-0.605160\pi\)
							0.818439	−	0.574593i	\(-0.194840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.4.4	✓	28
3.2	odd	2		99.8.f.c.37.4		28
11.3	even	5	inner	33.8.e.a.25.4	yes	28
11.5	even	5		363.8.a.q.1.8		14
11.6	odd	10		363.8.a.p.1.7		14
33.14	odd	10		99.8.f.c.91.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.4	✓	28	1.1	even	1	trivial
33.8.e.a.25.4	yes	28	11.3	even	5	inner
99.8.f.c.37.4		28	3.2	odd	2
99.8.f.c.91.4		28	33.14	odd	10
363.8.a.p.1.7		14	11.6	odd	10
363.8.a.q.1.8		14	11.5	even	5