Embedded newform 33.5.h.a.14.5

• Label: 33.5.h.a.14.5
• 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			14.5
Character	\(\chi\)	\(=\)	33.14
Dual form			33.5.h.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95750 - 2.69427i) q^{2} +(-3.92492 - 8.09908i) q^{3} +(1.51699 - 4.66881i) q^{4} +(-11.1246 + 15.3116i) q^{5} +(-14.1381 + 26.4288i) q^{6} +(0.0531312 - 0.163521i) q^{7} +(-66.2255 + 21.5180i) q^{8} +(-50.1900 + 63.5764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95750	−	2.69427i	−0.489376	−	0.673568i	0.490897	−	0.871218i	\(-0.336669\pi\)
							−0.980273	+	0.197650i	\(0.936669\pi\)
\(3\)	−3.92492	−	8.09908i	−0.436102	−	0.899897i
							\(5\)	−11.1246	+	15.3116i	−0.444982	+	0.612466i	−0.971310	−	0.237816i	\(-0.923568\pi\)
0.526328	+	0.850282i	\(0.323568\pi\)
\(7\)	0.0531312	−	0.163521i	0.00108431	−	0.00333717i	−0.950513	−	0.310685i	\(-0.899442\pi\)
							0.951597	+	0.307348i	\(0.0994415\pi\)
\(11\)	−58.0937	−	106.142i	−0.480113	−	0.877207i
							\(13\)	17.9248	−	13.0231i	0.106064	−	0.0770598i	−0.533489	−	0.845807i	\(-0.679120\pi\)
0.639553	+	0.768747i	\(0.279120\pi\)
\(17\)	100.941	−	138.934i	0.349278	−	0.480740i	−0.597845	−	0.801612i	\(-0.703976\pi\)
							0.947122	+	0.320872i	\(0.103976\pi\)
\(19\)	−134.050	−	412.562i	−0.371329	−	1.14283i	−0.945922	−	0.324393i	\(-0.894840\pi\)
							0.574594	−	0.818439i	\(-0.305160\pi\)
\(23\)		−	779.919i		−	1.47433i	−0.675714	−	0.737164i	\(-0.736165\pi\)
							0.675714	−	0.737164i	\(-0.263835\pi\)
\(29\)	−959.815	−	311.863i	−1.14128	−	0.370824i	−0.323427	−	0.946253i	\(-0.604835\pi\)
							−0.817851	+	0.575430i	\(0.804835\pi\)
\(31\)	−587.598	+	426.915i	−0.611445	+	0.444240i	−0.849923	−	0.526907i	\(-0.823351\pi\)
							0.238478	+	0.971148i	\(0.423351\pi\)
\(37\)	428.134	−	1317.66i	0.312735	−	0.962499i	−0.663942	−	0.747784i	\(-0.731118\pi\)
							0.976677	−	0.214715i	\(-0.0688822\pi\)
\(41\)	−664.681	+	215.968i	−0.395408	+	0.128476i	−0.499971	−	0.866042i	\(-0.666656\pi\)
							0.104562	+	0.994518i	\(0.466656\pi\)
\(43\)	932.343			0.504242			0.252121	−	0.967696i	\(-0.418872\pi\)
							0.252121	+	0.967696i	\(0.418872\pi\)
\(47\)	3538.47	−	1149.72i	1.60184	−	0.520471i	0.634281	−	0.773103i	\(-0.281296\pi\)
							0.967562	+	0.252632i	\(0.0812962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.14.5	✓	56
3.2	odd	2	inner	33.5.h.a.14.10	yes	56
11.4	even	5	inner	33.5.h.a.26.10	yes	56
33.26	odd	10	inner	33.5.h.a.26.5	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.14.5	✓	56	1.1	even	1	trivial
33.5.h.a.14.10	yes	56	3.2	odd	2	inner
33.5.h.a.26.5	yes	56	33.26	odd	10	inner
33.5.h.a.26.10	yes	56	11.4	even	5	inner