Embedded newform 93.2.p.b.11.1

• Label: 93.2.p.b.11.1
• Level: $93$
• Weight: $2$
• Character: 93.11
• 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			11.1
Character	\(\chi\)	\(=\)	93.11
Dual form			93.2.p.b.17.1

$q$-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 \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.11.1	✓	64	1.1	even	1	trivial
93.2.p.b.11.8	yes	64	3.2	odd	2	inner
93.2.p.b.17.1	yes	64	93.17	even	30	inner
93.2.p.b.17.8	yes	64	31.17	odd	30	inner