Embedded newform 93.2.p.b.65.1

• Label: 93.2.p.b.65.1
• Level: $93$
• Weight: $2$
• Character: 93.65
• 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			65.1
Character	\(\chi\)	\(=\)	93.65
Dual form			93.2.p.b.83.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45425 - 2.00160i) q^{2} +(1.04609 + 1.38047i) q^{3} +(-1.27353 + 3.91953i) q^{4} +(3.27845 + 1.89282i) q^{5} +(1.24186 - 4.10140i) q^{6} +(-0.799480 + 0.169935i) q^{7} +(4.99132 - 1.62178i) q^{8} +(-0.811379 + 2.88819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45425	−	2.00160i	−1.02831	−	1.41534i	−0.906211	−	0.422827i	\(-0.861038\pi\)
							−0.122098	−	0.992518i	\(-0.538962\pi\)
\(3\)	1.04609	+	1.38047i	0.603962	+	0.797013i
							\(5\)	3.27845	+	1.89282i	1.46617	+	0.846493i	0.999284	−	0.0378246i	\(-0.0120428\pi\)
0.466885	+	0.884318i	\(0.345376\pi\)
\(7\)	−0.799480	+	0.169935i	−0.302175	+	0.0642293i	−0.356504	−	0.934294i	\(-0.616031\pi\)
							0.0543289	+	0.998523i	\(0.482698\pi\)
\(11\)	−2.47663	−	2.75058i	−0.746733	−	0.829331i	0.243332	−	0.969943i	\(-0.421760\pi\)
							−0.990065	+	0.140612i	\(0.955093\pi\)
\(13\)	−1.11340	−	2.50074i	−0.308802	−	0.693582i	0.690762	−	0.723082i	\(-0.257275\pi\)
							−0.999565	+	0.0295002i	\(0.990608\pi\)
\(17\)	2.87511	−	3.19314i	0.697317	−	0.774449i	−0.285629	−	0.958340i	\(-0.592203\pi\)
							0.982947	+	0.183891i	\(0.0588693\pi\)
\(19\)	0.573829	+	0.255485i	0.131645	+	0.0586123i	0.471502	−	0.881865i	\(-0.343712\pi\)
							−0.339857	+	0.940477i	\(0.610379\pi\)
\(23\)	−0.825062	−	2.53928i	−0.172037	−	0.529477i	0.827448	−	0.561542i	\(-0.189792\pi\)
							−0.999486	+	0.0320651i	\(0.989792\pi\)
\(29\)	−3.68053	+	2.67406i	−0.683457	+	0.496560i	−0.874503	−	0.485021i	\(-0.838812\pi\)
							0.191046	+	0.981581i	\(0.438812\pi\)
\(31\)	−1.89586	−	5.23505i	−0.340507	−	0.940242i
							\(37\)	2.46077	−	1.42073i	0.404548	−	0.233566i	−0.283897	−	0.958855i	\(-0.591627\pi\)
0.688444	+	0.725289i	\(0.258294\pi\)
\(41\)	−2.41300	+	0.253617i	−0.376847	+	0.0396083i	−0.291059	−	0.956705i	\(-0.594008\pi\)
							−0.0857884	+	0.996313i	\(0.527341\pi\)
\(43\)	−0.690329	+	1.55050i	−0.105274	+	0.236450i	−0.958500	−	0.285091i	\(-0.907976\pi\)
							0.853226	+	0.521541i	\(0.174643\pi\)
\(47\)	0.839636	−	1.15566i	0.122473	−	0.168570i	−0.743378	−	0.668872i	\(-0.766778\pi\)
							0.865851	+	0.500302i	\(0.166778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.65.1	✓	64
3.2	odd	2	inner	93.2.p.b.65.8	yes	64
31.21	odd	30	inner	93.2.p.b.83.8	yes	64
93.83	even	30	inner	93.2.p.b.83.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.65.1	✓	64	1.1	even	1	trivial
93.2.p.b.65.8	yes	64	3.2	odd	2	inner
93.2.p.b.83.1	yes	64	93.83	even	30	inner
93.2.p.b.83.8	yes	64	31.21	odd	30	inner