Embedded newform 93.2.p.b.74.2

• Label: 93.2.p.b.74.2
• Level: $93$
• Weight: $2$
• Character: 93.74
• 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			74.2
Character	\(\chi\)	\(=\)	93.74
Dual form			93.2.p.b.44.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13925 + 1.56805i) q^{2} +(1.68300 + 0.409301i) q^{3} +(-0.542842 - 1.67070i) q^{4} +(1.08227 + 0.624852i) q^{5} +(-2.55916 + 2.17272i) q^{6} +(0.706282 - 0.784405i) q^{7} +(-0.448535 - 0.145738i) q^{8} +(2.66495 + 1.37770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13925	+	1.56805i	−0.805574	+	1.10878i	0.186417	+	0.982471i	\(0.440312\pi\)
							−0.991991	+	0.126307i	\(0.959688\pi\)
\(3\)	1.68300	+	0.409301i	0.971678	+	0.236310i
							\(5\)	1.08227	+	0.624852i	0.484008	+	0.279442i	0.722085	−	0.691804i	\(-0.243184\pi\)
−0.238077	+	0.971246i	\(0.576517\pi\)
\(7\)	0.706282	−	0.784405i	0.266949	−	0.296477i	−0.594735	−	0.803922i	\(-0.702743\pi\)
							0.861685	+	0.507444i	\(0.169410\pi\)
\(11\)	−4.69221	−	0.997360i	−1.41475	−	0.300715i	−0.563781	−	0.825924i	\(-0.690654\pi\)
							−0.850973	+	0.525209i	\(0.823987\pi\)
\(13\)	−2.66546	+	0.280152i	−0.739267	+	0.0777001i	−0.466674	−	0.884429i	\(-0.654548\pi\)
							−0.272593	+	0.962129i	\(0.587881\pi\)
\(17\)	3.35151	−	0.712386i	0.812862	−	0.172779i	0.217317	−	0.976101i	\(-0.430270\pi\)
							0.595545	+	0.803322i	\(0.296936\pi\)
\(19\)	−0.155395	+	1.47848i	−0.0356500	+	0.339188i	0.962130	+	0.272591i	\(0.0878806\pi\)
							−0.997780	+	0.0665963i	\(0.978786\pi\)
\(23\)	2.30060	−	7.08050i	0.479707	−	1.47639i	−0.359795	−	0.933031i	\(-0.617153\pi\)
							0.839502	−	0.543356i	\(-0.182847\pi\)
\(29\)	−0.00867334	−	0.00630155i	−0.00161060	−	0.00117017i	0.586980	−	0.809602i	\(-0.300317\pi\)
							−0.588590	+	0.808432i	\(0.700317\pi\)
\(31\)	5.08432	+	2.26930i	0.913170	+	0.407579i
							\(37\)	−3.74869	+	2.16431i	−0.616281	+	0.355810i	−0.775420	−	0.631446i	\(-0.782462\pi\)
0.159139	+	0.987256i	\(0.449128\pi\)
\(41\)	2.03766	+	4.57665i	0.318228	+	0.714752i	0.999856	−	0.0169464i	\(-0.00539445\pi\)
							−0.681628	+	0.731699i	\(0.738728\pi\)
\(43\)	−12.2558	−	1.28814i	−1.86899	−	0.196439i	−0.898732	−	0.438497i	\(-0.855511\pi\)
							−0.970262	+	0.242058i	\(0.922177\pi\)
\(47\)	−4.61204	−	6.34793i	−0.672735	−	0.925940i	0.327083	−	0.944995i	\(-0.393934\pi\)
							−0.999818	+	0.0190552i	\(0.993934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.p.b.74.2	yes	64
3.2	odd	2	inner	93.2.p.b.74.7	yes	64
31.13	odd	30	inner	93.2.p.b.44.7	yes	64
93.44	even	30	inner	93.2.p.b.44.2	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.p.b.44.2	✓	64	93.44	even	30	inner
93.2.p.b.44.7	yes	64	31.13	odd	30	inner
93.2.p.b.74.2	yes	64	1.1	even	1	trivial
93.2.p.b.74.7	yes	64	3.2	odd	2	inner