Embedded newform 33.5.h.a.14.6

• Label: 33.5.h.a.14.6
• Level: $33$
• Weight: $5$
• Character: 33.14
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			14.6
Character	\(\chi\)	\(=\)	33.14
Dual form			33.5.h.a.26.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.966669 - 1.33051i) q^{2} +(-3.49528 + 8.29355i) q^{3} +(4.10848 - 12.6446i) q^{4} +(-16.0026 + 22.0258i) q^{5} +(14.4134 - 3.36663i) q^{6} +(-23.7358 + 73.0514i) q^{7} +(-45.8209 + 14.8881i) q^{8} +(-56.5660 - 57.9766i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.966669	−	1.33051i	−0.241667	−	0.332626i	0.670904	−	0.741544i	\(-0.265906\pi\)
							−0.912571	+	0.408918i	\(0.865906\pi\)
\(3\)	−3.49528	+	8.29355i	−0.388365	+	0.921506i
							\(5\)	−16.0026	+	22.0258i	−0.640106	+	0.881030i	−0.998621	−	0.0524936i	\(-0.983283\pi\)
0.358515	+	0.933524i	\(0.383283\pi\)
\(7\)	−23.7358	+	73.0514i	−0.484405	+	1.49084i	0.348436	+	0.937333i	\(0.386713\pi\)
							−0.832841	+	0.553512i	\(0.813287\pi\)
\(11\)	63.4457	+	103.032i	0.524345	+	0.851506i
							\(13\)	−38.0897	+	27.6738i	−0.225383	+	0.163750i	−0.694746	−	0.719255i	\(-0.744483\pi\)
0.469364	+	0.883005i	\(0.344483\pi\)
\(17\)	223.825	−	308.069i	0.774482	−	1.06598i	−0.221387	−	0.975186i	\(-0.571059\pi\)
							0.995869	−	0.0907971i	\(-0.0289415\pi\)
\(19\)	−86.2718	−	265.517i	−0.238980	−	0.735505i	−0.996568	−	0.0827723i	\(-0.973623\pi\)
							0.757589	−	0.652732i	\(-0.226377\pi\)
\(23\)			649.260i			1.22733i	0.789565	+	0.613667i	\(0.210306\pi\)
							−0.789565	+	0.613667i	\(0.789694\pi\)
\(29\)	403.621	+	131.144i	0.479930	+	0.155939i	0.538984	−	0.842316i	\(-0.318808\pi\)
							−0.0590538	+	0.998255i	\(0.518808\pi\)
\(31\)	−8.00846	+	5.81849i	−0.00833347	+	0.00605462i	−0.591944	−	0.805979i	\(-0.701639\pi\)
							0.583611	+	0.812034i	\(0.301639\pi\)
\(37\)	−710.605	+	2187.02i	−0.519069	+	1.59753i	0.256685	+	0.966495i	\(0.417370\pi\)
							−0.775754	+	0.631035i	\(0.782630\pi\)
\(41\)	−518.632	+	168.514i	−0.308526	+	0.100246i	−0.459188	−	0.888339i	\(-0.651859\pi\)
							0.150662	+	0.988585i	\(0.451859\pi\)
\(43\)	2166.70			1.17182			0.585912	−	0.810375i	\(-0.300737\pi\)
							0.585912	+	0.810375i	\(0.300737\pi\)
\(47\)	−871.260	+	283.090i	−0.394414	+	0.128153i	−0.499508	−	0.866309i	\(-0.666486\pi\)
							0.105094	+	0.994462i	\(0.466486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.14.6	✓	56
3.2	odd	2	inner	33.5.h.a.14.9	yes	56
11.4	even	5	inner	33.5.h.a.26.9	yes	56
33.26	odd	10	inner	33.5.h.a.26.6	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.14.6	✓	56	1.1	even	1	trivial
33.5.h.a.14.9	yes	56	3.2	odd	2	inner
33.5.h.a.26.6	yes	56	33.26	odd	10	inner
33.5.h.a.26.9	yes	56	11.4	even	5	inner