Embedded newform 93.2.p.b.17.4

• Label: 93.2.p.b.17.4
• Level: $93$
• Weight: $2$
• Character: 93.17
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			17.4
Character	\(\chi\)	\(=\)	93.17
Dual form			93.2.p.b.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13134 - 0.367594i) q^{2} +(1.09350 - 1.34323i) q^{3} +(-0.473233 - 0.343824i) q^{4} +(-3.25867 - 1.88139i) q^{5} +(-1.73088 + 1.11768i) q^{6} +(-0.121266 + 1.15377i) q^{7} +(1.80741 + 2.48769i) q^{8} +(-0.608530 - 2.93763i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13134	−	0.367594i	−0.799977	−	0.259928i	−0.119630	−	0.992819i	\(-0.538171\pi\)
							−0.680347	+	0.732890i	\(0.738171\pi\)
\(3\)	1.09350	−	1.34323i	0.631331	−	0.775514i
							\(5\)	−3.25867	−	1.88139i	−1.45732	−	0.841385i	−0.458442	−	0.888724i	\(-0.651592\pi\)
−0.998879	+	0.0473395i	\(0.984926\pi\)
\(7\)	−0.121266	+	1.15377i	−0.0458341	+	0.436082i	0.947409	+	0.320027i	\(0.103692\pi\)
							−0.993243	+	0.116056i	\(0.962975\pi\)
\(11\)	1.44229	−	0.642148i	0.434866	−	0.193615i	−0.177615	−	0.984100i	\(-0.556838\pi\)
							0.612481	+	0.790485i	\(0.290172\pi\)
\(13\)	1.12113	−	5.27450i	0.310945	−	1.46288i	−0.493963	−	0.869483i	\(-0.664452\pi\)
							0.804908	−	0.593399i	\(-0.202215\pi\)
\(17\)	3.70486	+	1.64951i	0.898561	+	0.400065i	0.803430	−	0.595400i	\(-0.203006\pi\)
							0.0951312	+	0.995465i	\(0.469673\pi\)
\(19\)	2.07066	−	0.440133i	0.475043	−	0.100973i	0.0358327	−	0.999358i	\(-0.488592\pi\)
							0.439210	+	0.898384i	\(0.355258\pi\)
\(23\)	1.05297	−	0.765029i	0.219560	−	0.159520i	−0.472569	−	0.881294i	\(-0.656673\pi\)
							0.692129	+	0.721774i	\(0.256673\pi\)
\(29\)	−0.285528	+	0.878764i	−0.0530212	+	0.163182i	−0.974061	−	0.226286i	\(-0.927341\pi\)
							0.921040	+	0.389469i	\(0.127341\pi\)
\(31\)	1.59217	+	5.33526i	0.285963	+	0.958241i
							\(37\)	−7.98383	+	4.60947i	−1.31253	+	0.757792i	−0.982515	−	0.186183i	\(-0.940388\pi\)
−0.330019	+	0.943974i	\(0.607055\pi\)
\(41\)	−2.74468	+	2.47132i	−0.428647	+	0.385955i	−0.855022	−	0.518591i	\(-0.826457\pi\)
							0.426376	+	0.904546i	\(0.359790\pi\)
\(43\)	−1.64332	−	7.73123i	−0.250605	−	1.17900i	−0.905854	−	0.423590i	\(-0.860770\pi\)
							0.655250	−	0.755412i	\(-0.272563\pi\)
\(47\)	7.19341	−	2.33728i	1.04927	−	0.340928i	0.266886	−	0.963728i	\(-0.414005\pi\)
							0.782381	+	0.622801i	\(0.214005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.17.4	yes	64
3.2	odd	2	inner	93.2.p.b.17.5	yes	64
31.11	odd	30	inner	93.2.p.b.11.5	yes	64
93.11	even	30	inner	93.2.p.b.11.4	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.4	✓	64	93.11	even	30	inner
93.2.p.b.11.5	yes	64	31.11	odd	30	inner
93.2.p.b.17.4	yes	64	1.1	even	1	trivial
93.2.p.b.17.5	yes	64	3.2	odd	2	inner