Embedded newform 93.2.p.b.17.1

• Label: 93.2.p.b.17.1
• 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			17.1
Character	\(\chi\)	\(=\)	93.17
Dual form			93.2.p.b.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49541 - 0.810809i) q^{2} +(0.448212 - 1.67305i) q^{3} +(3.95165 + 2.87104i) q^{4} +(2.63849 + 1.52333i) q^{5} +(-2.47500 + 3.81155i) q^{6} +(0.249973 - 2.37834i) q^{7} +(-4.44863 - 6.12302i) q^{8} +(-2.59821 - 1.49976i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49541	−	0.810809i	−1.76452	−	0.573329i	−0.766871	−	0.641801i	\(-0.778188\pi\)
							−0.997653	+	0.0684719i	\(0.978188\pi\)
\(3\)	0.448212	−	1.67305i	0.258775	−	0.965938i
							\(5\)	2.63849	+	1.52333i	1.17997	+	0.681254i	0.956007	−	0.293344i	\(-0.0947681\pi\)
0.223960	+	0.974598i	\(0.428101\pi\)
\(7\)	0.249973	−	2.37834i	0.0944810	−	0.898927i	−0.839922	−	0.542708i	\(-0.817399\pi\)
							0.934403	−	0.356219i	\(-0.115934\pi\)
\(11\)	1.54619	−	0.688406i	0.466193	−	0.207562i	−0.160177	−	0.987088i	\(-0.551207\pi\)
							0.626370	+	0.779526i	\(0.284540\pi\)
\(13\)	−0.291741	+	1.37253i	−0.0809144	+	0.380672i	−0.999907	−	0.0136013i	\(-0.995670\pi\)
							0.918993	+	0.394274i	\(0.129004\pi\)
\(17\)	−1.93865	−	0.863144i	−0.470193	−	0.209343i	0.157941	−	0.987449i	\(-0.449514\pi\)
							−0.628134	+	0.778105i	\(0.716181\pi\)
\(19\)	−2.45568	+	0.521972i	−0.563372	+	0.119748i	−0.480790	−	0.876836i	\(-0.659650\pi\)
							−0.0825823	+	0.996584i	\(0.526317\pi\)
\(23\)	5.55133	−	4.03327i	1.15753	−	0.840996i	0.168067	−	0.985775i	\(-0.446247\pi\)
							0.989464	+	0.144780i	\(0.0462474\pi\)
\(29\)	−2.09702	+	6.45397i	−0.389407	+	1.19847i	0.543825	+	0.839199i	\(0.316976\pi\)
							−0.933232	+	0.359274i	\(0.883024\pi\)
\(31\)	3.23934	+	4.52843i	0.581802	+	0.813330i
							\(37\)	−0.261230	+	0.150821i	−0.0429460	+	0.0247949i	−0.521319	−	0.853362i	\(-0.674560\pi\)
0.478373	+	0.878157i	\(0.341227\pi\)
\(41\)	−2.86386	+	2.57863i	−0.447260	+	0.402715i	−0.861745	−	0.507341i	\(-0.830628\pi\)
							0.414485	+	0.910056i	\(0.363962\pi\)
\(43\)	0.0590713	+	0.277909i	0.00900829	+	0.0423807i	0.982422	−	0.186673i	\(-0.0597703\pi\)
							−0.973414	+	0.229053i	\(0.926437\pi\)
\(47\)	−7.84369	+	2.54857i	−1.14412	+	0.371747i	−0.818926	−	0.573900i	\(-0.805430\pi\)
							−0.325195	+	0.945647i	\(0.605430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.17.1	yes	64
3.2	odd	2	inner	93.2.p.b.17.8	yes	64
31.11	odd	30	inner	93.2.p.b.11.8	yes	64
93.11	even	30	inner	93.2.p.b.11.1	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.1	✓	64	93.11	even	30	inner
93.2.p.b.11.8	yes	64	31.11	odd	30	inner
93.2.p.b.17.1	yes	64	1.1	even	1	trivial
93.2.p.b.17.8	yes	64	3.2	odd	2	inner