Embedded newform 93.2.p.b.53.3

• Label: 93.2.p.b.53.3
• Level: $93$
• Weight: $2$
• Character: 93.53
• 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			53.3
Character	\(\chi\)	\(=\)	93.53
Dual form			93.2.p.b.86.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16443 - 0.378346i) q^{2} +(-1.32867 + 1.11114i) q^{3} +(-0.405283 - 0.294455i) q^{4} +(2.19606 - 1.26789i) q^{5} +(1.96754 - 0.791144i) q^{6} +(4.16768 - 1.85557i) q^{7} +(1.79983 + 2.47726i) q^{8} +(0.530744 - 2.95268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16443	−	0.378346i	−0.823376	−	0.267531i	−0.133123	−	0.991099i	\(-0.542501\pi\)
							−0.690253	+	0.723568i	\(0.742501\pi\)
\(3\)	−1.32867	+	1.11114i	−0.767110	+	0.641516i
							\(5\)	2.19606	−	1.26789i	0.982106	−	0.567019i	0.0792006	−	0.996859i	\(-0.474763\pi\)
0.902905	+	0.429840i	\(0.141430\pi\)
\(7\)	4.16768	−	1.85557i	1.57523	−	0.701340i	0.581546	−	0.813514i	\(-0.302448\pi\)
							0.993689	+	0.112174i	\(0.0357814\pi\)
\(11\)	−0.275736	+	2.62345i	−0.0831375	+	0.791000i	0.870930	+	0.491408i	\(0.163517\pi\)
							−0.954067	+	0.299593i	\(0.903149\pi\)
\(13\)	−2.55380	−	2.29945i	−0.708296	−	0.637752i	0.234115	−	0.972209i	\(-0.424781\pi\)
							−0.942411	+	0.334456i	\(0.891447\pi\)
\(17\)	−0.337016	−	3.20649i	−0.0817383	−	0.777688i	−0.956224	−	0.292636i	\(-0.905467\pi\)
							0.874486	−	0.485052i	\(-0.161199\pi\)
\(19\)	1.53882	+	1.70904i	0.353030	+	0.392080i	0.893336	−	0.449389i	\(-0.148358\pi\)
							−0.540306	+	0.841469i	\(0.681691\pi\)
\(23\)	3.78210	−	2.74786i	0.788623	−	0.572968i	−0.118932	−	0.992902i	\(-0.537947\pi\)
							0.907554	+	0.419935i	\(0.137947\pi\)
\(29\)	−2.05840	+	6.33509i	−0.382235	+	1.17640i	0.556232	+	0.831027i	\(0.312247\pi\)
							−0.938467	+	0.345370i	\(0.887753\pi\)
\(31\)	−1.29544	+	5.41496i	−0.232668	+	0.972556i
							\(37\)	−1.86974	−	1.07950i	−0.307383	−	0.177468i	0.338372	−	0.941013i	\(-0.390124\pi\)
−0.645755	+	0.763545i	\(0.723457\pi\)
\(41\)	−0.643328	−	3.02662i	−0.100471	−	0.472679i	−0.999402	−	0.0345879i	\(-0.988988\pi\)
							0.898931	−	0.438091i	\(-0.144345\pi\)
\(43\)	−3.49461	+	3.14656i	−0.532924	+	0.479846i	−0.891100	−	0.453808i	\(-0.850065\pi\)
							0.358176	+	0.933654i	\(0.383399\pi\)
\(47\)	−9.44560	+	3.06906i	−1.37778	+	0.447669i	−0.901940	−	0.431862i	\(-0.857857\pi\)
							−0.475842	+	0.879531i	\(0.657857\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.53.3	✓	64
3.2	odd	2	inner	93.2.p.b.53.6	yes	64
31.24	odd	30	inner	93.2.p.b.86.6	yes	64
93.86	even	30	inner	93.2.p.b.86.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.53.3	✓	64	1.1	even	1	trivial
93.2.p.b.53.6	yes	64	3.2	odd	2	inner
93.2.p.b.86.3	yes	64	93.86	even	30	inner
93.2.p.b.86.6	yes	64	31.24	odd	30	inner