Embedded newform 93.2.p.b.44.3

• Label: 93.2.p.b.44.3
• Level: $93$
• Weight: $2$
• Character: 93.44
• 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			44.3
Character	\(\chi\)	\(=\)	93.44
Dual form			93.2.p.b.74.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.888042 - 1.22228i) q^{2} +(-1.27658 + 1.17061i) q^{3} +(-0.0873279 + 0.268768i) q^{4} +(3.07140 - 1.77328i) q^{5} +(2.56448 + 0.520794i) q^{6} +(-2.50044 - 2.77703i) q^{7} +(-2.46770 + 0.801805i) q^{8} +(0.259328 - 2.98877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.888042	−	1.22228i	−0.627941	−	0.864286i	0.369960	−	0.929048i	\(-0.379371\pi\)
							−0.997901	+	0.0647616i	\(0.979371\pi\)
\(3\)	−1.27658	+	1.17061i	−0.737035	+	0.675854i
							\(5\)	3.07140	−	1.77328i	1.37357	−	0.793033i	0.382197	−	0.924081i	\(-0.375167\pi\)
0.991376	+	0.131048i	\(0.0418341\pi\)
\(7\)	−2.50044	−	2.77703i	−0.945079	−	1.04962i	−0.998696	−	0.0510553i	\(-0.983742\pi\)
							0.0536165	−	0.998562i	\(-0.482925\pi\)
\(11\)	3.58707	−	0.762455i	1.08154	−	0.229889i	0.367522	−	0.930015i	\(-0.380206\pi\)
							0.714019	+	0.700126i	\(0.246873\pi\)
\(13\)	0.782875	+	0.0822835i	0.217130	+	0.0228213i	0.212469	−	0.977168i	\(-0.431850\pi\)
							0.00466146	+	0.999989i	\(0.498516\pi\)
\(17\)	−1.02799	−	0.218506i	−0.249324	−	0.0529954i	0.0815543	−	0.996669i	\(-0.474012\pi\)
							−0.330878	+	0.943673i	\(0.607345\pi\)
\(19\)	0.576916	+	5.48899i	0.132354	+	1.25926i	0.836008	+	0.548718i	\(0.184884\pi\)
							−0.703654	+	0.710543i	\(0.748450\pi\)
\(23\)	1.07117	+	3.29673i	0.223355	+	0.687415i	0.998454	+	0.0555764i	\(0.0176996\pi\)
							−0.775100	+	0.631839i	\(0.782300\pi\)
\(29\)	−0.886968	+	0.644420i	−0.164706	+	0.119666i	−0.667085	−	0.744982i	\(-0.732458\pi\)
							0.502379	+	0.864647i	\(0.332458\pi\)
\(31\)	−5.51729	−	0.748014i	−0.990934	−	0.134347i
							\(37\)	6.93625	+	4.00465i	1.14031	+	0.658360i	0.946508	−	0.322680i	\(-0.104584\pi\)
0.193805	+	0.981040i	\(0.437917\pi\)
\(41\)	−0.494781	+	1.11130i	−0.0772718	+	0.173555i	−0.948051	−	0.318120i	\(-0.896949\pi\)
							0.870779	+	0.491675i	\(0.163615\pi\)
\(43\)	−0.597850	+	0.0628366i	−0.0911713	+	0.00958249i	−0.150004	−	0.988685i	\(-0.547929\pi\)
							0.0588330	+	0.998268i	\(0.481262\pi\)
\(47\)	2.87970	−	3.96356i	0.420047	−	0.578145i	−0.545586	−	0.838055i	\(-0.683693\pi\)
							0.965633	+	0.259910i	\(0.0836929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.44.3	✓	64
3.2	odd	2	inner	93.2.p.b.44.6	yes	64
31.12	odd	30	inner	93.2.p.b.74.6	yes	64
93.74	even	30	inner	93.2.p.b.74.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.44.3	✓	64	1.1	even	1	trivial
93.2.p.b.44.6	yes	64	3.2	odd	2	inner
93.2.p.b.74.3	yes	64	93.74	even	30	inner
93.2.p.b.74.6	yes	64	31.12	odd	30	inner