Embedded newform 93.2.p.b.11.5

• Label: 93.2.p.b.11.5
• Level: $93$
• Weight: $2$
• Character: 93.11
• 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			11.5
Character	\(\chi\)	\(=\)	93.11
Dual form			93.2.p.b.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13134 - 0.367594i) q^{2} +(-1.22157 - 1.22791i) q^{3} +(-0.473233 + 0.343824i) q^{4} +(3.25867 - 1.88139i) q^{5} +(-1.83338 - 0.940142i) q^{6} +(-0.121266 - 1.15377i) q^{7} +(-1.80741 + 2.48769i) q^{8} +(-0.0155363 + 2.99996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)

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	\(-0.461829\pi\)
							0.680347	+	0.732890i	\(0.261829\pi\)
\(3\)	−1.22157	−	1.22791i	−0.705273	−	0.708935i
							\(5\)	3.25867	−	1.88139i	1.45732	−	0.841385i	0.458442	−	0.888724i	\(-0.348408\pi\)
0.998879	+	0.0473395i	\(0.0150743\pi\)
\(7\)	−0.121266	−	1.15377i	−0.0458341	−	0.436082i	−0.993243	−	0.116056i	\(-0.962975\pi\)
							0.947409	−	0.320027i	\(-0.103692\pi\)
\(11\)	−1.44229	−	0.642148i	−0.434866	−	0.193615i	0.177615	−	0.984100i	\(-0.443162\pi\)
							−0.612481	+	0.790485i	\(0.709828\pi\)
\(13\)	1.12113	+	5.27450i	0.310945	+	1.46288i	0.804908	+	0.593399i	\(0.202215\pi\)
							−0.493963	+	0.869483i	\(0.664452\pi\)
\(17\)	−3.70486	+	1.64951i	−0.898561	+	0.400065i	−0.803430	−	0.595400i	\(-0.796994\pi\)
							−0.0951312	+	0.995465i	\(0.530327\pi\)
\(19\)	2.07066	+	0.440133i	0.475043	+	0.100973i	0.439210	−	0.898384i	\(-0.355258\pi\)
							0.0358327	+	0.999358i	\(0.488592\pi\)
\(23\)	−1.05297	−	0.765029i	−0.219560	−	0.159520i	0.472569	−	0.881294i	\(-0.343327\pi\)
							−0.692129	+	0.721774i	\(0.743327\pi\)
\(29\)	0.285528	+	0.878764i	0.0530212	+	0.163182i	0.974061	−	0.226286i	\(-0.0726586\pi\)
							−0.921040	+	0.389469i	\(0.872659\pi\)
\(31\)	1.59217	−	5.33526i	0.285963	−	0.958241i
							\(37\)	−7.98383	−	4.60947i	−1.31253	−	0.757792i	−0.330019	−	0.943974i	\(-0.607055\pi\)
−0.982515	+	0.186183i	\(0.940388\pi\)
\(41\)	2.74468	+	2.47132i	0.428647	+	0.385955i	0.855022	−	0.518591i	\(-0.173543\pi\)
							−0.426376	+	0.904546i	\(0.640210\pi\)
\(43\)	−1.64332	+	7.73123i	−0.250605	+	1.17900i	0.655250	+	0.755412i	\(0.272563\pi\)
							−0.905854	+	0.423590i	\(0.860770\pi\)
\(47\)	−7.19341	−	2.33728i	−1.04927	−	0.340928i	−0.266886	−	0.963728i	\(-0.585995\pi\)
							−0.782381	+	0.622801i	\(0.785995\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.4	✓	64	3.2	odd	2	inner
93.2.p.b.11.5	yes	64	1.1	even	1	trivial
93.2.p.b.17.4	yes	64	31.17	odd	30	inner
93.2.p.b.17.5	yes	64	93.17	even	30	inner