Embedded newform 93.2.p.b.17.6

• Label: 93.2.p.b.17.6
• 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			17.6
Character	\(\chi\)	\(=\)	93.17
Dual form			93.2.p.b.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25999 + 0.409396i) q^{2} +(1.36749 + 1.06300i) q^{3} +(-0.198061 - 0.143900i) q^{4} +(-1.61219 - 0.930798i) q^{5} +(1.28784 + 1.89921i) q^{6} +(0.00741560 - 0.0705547i) q^{7} +(-1.74808 - 2.40602i) q^{8} +(0.740080 + 2.90728i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

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	0.718496	−	0.695531i	\(-0.244831\pi\)
							0.172452	+	0.985018i	\(0.444831\pi\)
\(3\)	1.36749	+	1.06300i	0.789523	+	0.613721i
							\(5\)	−1.61219	−	0.930798i	−0.720993	−	0.416266i	0.0941248	−	0.995560i	\(-0.469995\pi\)
−0.815118	+	0.579295i	\(0.803328\pi\)
\(7\)	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	−0.817878	−	0.575391i	\(-0.804850\pi\)
							−0.119668	+	0.992814i	\(0.538183\pi\)
\(13\)	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	0.916880	−	0.399164i	\(-0.130700\pi\)
							0.316876	+	0.948467i	\(0.397366\pi\)
\(19\)	−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	0.0820087	−	0.996632i	\(-0.473866\pi\)
							0.973195	+	0.229981i	\(0.0738665\pi\)
\(29\)	−2.61086	+	8.03541i	−0.484825	+	1.49214i	0.347409	+	0.937714i	\(0.387062\pi\)
							−0.832234	+	0.554425i	\(0.812938\pi\)
\(31\)	−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	−0.511571	−	0.859241i	\(-0.670936\pi\)
							0.801061	+	0.598583i	\(0.204269\pi\)
\(43\)	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	−0.772646	−	0.634837i	\(-0.781067\pi\)
							−0.251936	+	0.967744i	\(0.581067\pi\)

(See \(a_n\) instead)

Twists

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

    

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