Embedded newform 33.3.h.a.5.2

• Label: 33.3.h.a.5.2
• Level: $33$
• Weight: $3$
• Character: 33.5
• Analytic conductor: $0.899$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.2
Root			\(0.951057 - 0.309017i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.5
Dual form			33.3.h.a.20.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.951057 - 0.309017i) q^{2} +(2.93236 - 0.633446i) q^{3} +(-2.42705 + 1.76336i) q^{4} +(-4.16750 - 1.35410i) q^{5} +(2.59310 - 1.50859i) q^{6} +(1.73607 - 1.26133i) q^{7} +(-4.11450 + 5.66312i) q^{8} +(8.19749 - 3.71499i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.951057	−	0.309017i	0.475528	−	0.154508i	−0.0614403	−	0.998111i	\(-0.519569\pi\)
							0.536969	+	0.843602i	\(0.319569\pi\)
\(3\)	2.93236	−	0.633446i	0.977454	−	0.211149i
							\(5\)	−4.16750	−	1.35410i	−0.833499	−	0.270820i	−0.138981	−	0.990295i	\(-0.544383\pi\)
−0.694519	+	0.719475i	\(0.744383\pi\)
\(7\)	1.73607	−	1.26133i	0.248010	−	0.180190i	−0.456835	−	0.889552i	\(-0.651017\pi\)
							0.704844	+	0.709362i	\(0.251017\pi\)
\(11\)	−9.06154	+	6.23607i	−0.823776	+	0.566915i
							\(13\)	−2.70820	−	8.33499i	−0.208323	−	0.641153i	−0.999561	−	0.0296440i	\(-0.990563\pi\)
0.791237	−	0.611509i	\(-0.209437\pi\)
\(17\)	5.70634	+	1.85410i	0.335667	+	0.109065i	0.472001	−	0.881598i	\(-0.343532\pi\)
							−0.136334	+	0.990663i	\(0.543532\pi\)
\(19\)	25.1803	+	18.2946i	1.32528	+	0.962873i	0.999850	+	0.0173072i	\(0.00550933\pi\)
							0.325431	+	0.945566i	\(0.394491\pi\)
\(23\)		−	26.4721i		−	1.15096i	−0.817815	−	0.575481i	\(-0.804815\pi\)
							0.817815	−	0.575481i	\(-0.195185\pi\)
\(29\)	27.8582	+	38.3435i	0.960626	+	1.32219i	0.946642	+	0.322286i	\(0.104451\pi\)
							0.0139836	+	0.999902i	\(0.495549\pi\)
\(31\)	−9.51722	−	29.2910i	−0.307007	−	0.944871i	−0.978921	−	0.204241i	\(-0.934527\pi\)
							0.671913	−	0.740630i	\(-0.265473\pi\)
\(37\)	−23.9443	+	17.3965i	−0.647142	+	0.470177i	−0.862297	−	0.506404i	\(-0.830975\pi\)
							0.215154	+	0.976580i	\(0.430975\pi\)
\(41\)	−25.6255	+	35.2705i	−0.625013	+	0.860256i	−0.997706	−	0.0676984i	\(-0.978434\pi\)
							0.372693	+	0.927955i	\(0.378434\pi\)
\(43\)	−39.7082			−0.923447			−0.461723	−	0.887024i	\(-0.652769\pi\)
							−0.461723	+	0.887024i	\(0.652769\pi\)
\(47\)	−2.52265	+	3.47214i	−0.0536735	+	0.0738752i	−0.835010	−	0.550235i	\(-0.814538\pi\)
							0.781337	+	0.624110i	\(0.214538\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.h.a.5.2	yes	8
3.2	odd	2	inner	33.3.h.a.5.1	✓	8
11.2	odd	10		363.3.h.g.251.2		8
11.3	even	5		363.3.b.f.122.2		4
11.4	even	5		363.3.h.h.245.2		8
11.5	even	5		363.3.h.h.323.1		8
11.6	odd	10		363.3.h.i.323.2		8
11.7	odd	10		363.3.h.i.245.1		8
11.8	odd	10		363.3.b.g.122.4		4
11.9	even	5	inner	33.3.h.a.20.1	yes	8
11.10	odd	2		363.3.h.g.269.1		8
33.2	even	10		363.3.h.g.251.1		8
33.5	odd	10		363.3.h.h.323.2		8
33.8	even	10		363.3.b.g.122.1		4
33.14	odd	10		363.3.b.f.122.3		4
33.17	even	10		363.3.h.i.323.1		8
33.20	odd	10	inner	33.3.h.a.20.2	yes	8
33.26	odd	10		363.3.h.h.245.1		8
33.29	even	10		363.3.h.i.245.2		8
33.32	even	2		363.3.h.g.269.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.a.5.1	✓	8	3.2	odd	2	inner
33.3.h.a.5.2	yes	8	1.1	even	1	trivial
33.3.h.a.20.1	yes	8	11.9	even	5	inner
33.3.h.a.20.2	yes	8	33.20	odd	10	inner
363.3.b.f.122.2		4	11.3	even	5
363.3.b.f.122.3		4	33.14	odd	10
363.3.b.g.122.1		4	33.8	even	10
363.3.b.g.122.4		4	11.8	odd	10
363.3.h.g.251.1		8	33.2	even	10
363.3.h.g.251.2		8	11.2	odd	10
363.3.h.g.269.1		8	11.10	odd	2
363.3.h.g.269.2		8	33.32	even	2
363.3.h.h.245.1		8	33.26	odd	10
363.3.h.h.245.2		8	11.4	even	5
363.3.h.h.323.1		8	11.5	even	5
363.3.h.h.323.2		8	33.5	odd	10
363.3.h.i.245.1		8	11.7	odd	10
363.3.h.i.245.2		8	33.29	even	10
363.3.h.i.323.1		8	33.17	even	10
363.3.h.i.323.2		8	11.6	odd	10