Embedded newform 93.2.p.b.17.3

• Label: 93.2.p.b.17.3
• 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			17.3
Character	\(\chi\)	\(=\)	93.17
Dual form			93.2.p.b.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25999 - 0.409396i) q^{2} +(1.20011 + 1.24889i) q^{3} +(-0.198061 - 0.143900i) q^{4} +(1.61219 + 0.930798i) q^{5} +(-1.00084 - 2.06491i) q^{6} +(0.00741560 - 0.0705547i) q^{7} +(1.74808 + 2.40602i) q^{8} +(-0.119450 + 2.99762i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25999	−	0.409396i	−0.890948	−	0.289487i	−0.172452	−	0.985018i	\(-0.555169\pi\)
							−0.718496	+	0.695531i	\(0.755169\pi\)
\(3\)	1.20011	+	1.24889i	0.692887	+	0.721047i
							\(5\)	1.61219	+	0.930798i	0.720993	+	0.416266i	0.815118	−	0.579295i	\(-0.196672\pi\)
−0.0941248	+	0.995560i	\(0.530005\pi\)
\(7\)	0.00741560	−	0.0705547i	0.00280283	−	0.0266672i	−0.993031	−	0.117853i	\(-0.962399\pi\)
							0.995834	+	0.0911855i	\(0.0290656\pi\)
\(11\)	3.10949	−	1.38443i	0.937547	−	0.417423i	0.119668	−	0.992814i	\(-0.461817\pi\)
							0.817878	+	0.575391i	\(0.195150\pi\)
\(13\)	0.288783	−	1.35862i	0.0800940	−	0.376813i	−0.919784	−	0.392425i	\(-0.871636\pi\)
							0.999878	+	0.0156120i	\(0.00496964\pi\)
\(17\)	−5.08690	−	2.26484i	−1.23376	−	0.549303i	−0.316876	−	0.948467i	\(-0.602634\pi\)
							−0.916880	+	0.399164i	\(0.869300\pi\)
\(19\)	−0.796188	+	0.169235i	−0.182658	+	0.0388252i	−0.298332	−	0.954462i	\(-0.596430\pi\)
							0.115674	+	0.993287i	\(0.463097\pi\)
\(23\)	−5.06058	+	3.67673i	−1.05520	+	0.766650i	−0.973195	−	0.229981i	\(-0.926134\pi\)
							−0.0820087	+	0.996632i	\(0.526134\pi\)
\(29\)	2.61086	−	8.03541i	0.484825	−	1.49214i	−0.347409	−	0.937714i	\(-0.612938\pi\)
							0.832234	−	0.554425i	\(-0.187062\pi\)
\(31\)	−5.41298	−	1.30369i	−0.972201	−	0.234150i
							\(37\)	4.79602	−	2.76898i	0.788460	−	0.455218i	−0.0509600	−	0.998701i	\(-0.516228\pi\)
0.839420	+	0.543483i	\(0.182895\pi\)
\(41\)	−1.85364	+	1.66903i	−0.289490	+	0.260658i	−0.801061	−	0.598583i	\(-0.795731\pi\)
							0.511571	+	0.859241i	\(0.329064\pi\)
\(43\)	1.23168	+	5.79461i	0.187830	+	0.883670i	0.966586	+	0.256343i	\(0.0825177\pi\)
							−0.778756	+	0.627327i	\(0.784149\pi\)
\(47\)	7.02418	−	2.28230i	1.02458	−	0.332907i	0.251936	−	0.967744i	\(-0.418933\pi\)
							0.772646	+	0.634837i	\(0.218933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.17.3	yes	64
3.2	odd	2	inner	93.2.p.b.17.6	yes	64
31.11	odd	30	inner	93.2.p.b.11.6	yes	64
93.11	even	30	inner	93.2.p.b.11.3	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.3	✓	64	93.11	even	30	inner
93.2.p.b.11.6	yes	64	31.11	odd	30	inner
93.2.p.b.17.3	yes	64	1.1	even	1	trivial
93.2.p.b.17.6	yes	64	3.2	odd	2	inner