Embedded newform 93.3.o.b.20.1

• Label: 93.3.o.b.20.1
• Level: $93$
• Weight: $3$
• Character: 93.20
• Analytic conductor: $2.534$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	93.o (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53406645855\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			20.1
Character	\(\chi\)	\(=\)	93.20
Dual form			93.3.o.b.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.53230 + 1.14771i) q^{2} +(0.0100012 - 2.99998i) q^{3} +(7.92385 - 5.75701i) q^{4} +(-0.947269 + 0.546906i) q^{5} +(3.40780 + 10.6083i) q^{6} +(-0.0471289 - 0.448402i) q^{7} +(-12.6497 + 17.4108i) q^{8} +(-8.99980 - 0.0600068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 8 q^{3} + 64 q^{4} - 5 q^{6} + 14 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.53230	+	1.14771i	−1.76615	+	0.573857i	−0.997808	−	0.0661800i	\(-0.978919\pi\)
							−0.768344	+	0.640037i	\(0.778919\pi\)
\(3\)	0.0100012	−	2.99998i	0.00333373	−	0.999994i
							\(5\)	−0.947269	+	0.546906i	−0.189454	+	0.109381i	−0.591727	−	0.806139i	\(-0.701554\pi\)
0.402273	+	0.915520i	\(0.368220\pi\)
\(7\)	−0.0471289	−	0.448402i	−0.00673270	−	0.0640574i	0.990640	−	0.136503i	\(-0.0435863\pi\)
							−0.997372	+	0.0724456i	\(0.976920\pi\)
\(11\)	3.68334	−	8.27291i	0.334849	−	0.752083i	−0.665137	−	0.746722i	\(-0.731627\pi\)
							0.999986	−	0.00536133i	\(-0.00170657\pi\)
\(13\)	−15.1972	+	3.23027i	−1.16902	+	0.248483i	−0.751217	−	0.660055i	\(-0.770533\pi\)
							−0.417802	+	0.908538i	\(0.637199\pi\)
\(17\)	−13.3171	−	29.9108i	−0.783361	−	1.75946i	−0.637939	−	0.770087i	\(-0.720213\pi\)
							−0.145422	−	0.989370i	\(-0.546454\pi\)
\(19\)	−10.8635	−	2.30912i	−0.571765	−	0.121532i	−0.0870492	−	0.996204i	\(-0.527744\pi\)
							−0.484716	+	0.874672i	\(0.661077\pi\)
\(23\)	−18.5890	+	25.5856i	−0.808219	+	1.11242i	0.183377	+	0.983043i	\(0.441297\pi\)
							−0.991596	+	0.129375i	\(0.958703\pi\)
\(29\)	6.56401	−	2.13278i	0.226345	−	0.0735440i	−0.193649	−	0.981071i	\(-0.562032\pi\)
							0.419994	+	0.907527i	\(0.362032\pi\)
\(31\)	2.44460	−	30.9035i	0.0788581	−	0.996886i
							\(37\)	28.5792	−	49.5007i	0.772412	−	1.33786i	−0.163825	−	0.986489i	\(-0.552383\pi\)
0.936238	−	0.351368i	\(-0.114283\pi\)
\(41\)	3.09781	+	2.78928i	0.0755564	+	0.0680313i	0.706044	−	0.708168i	\(-0.250478\pi\)
							−0.630487	+	0.776200i	\(0.717145\pi\)
\(43\)	−47.4278	−	10.0811i	−1.10297	−	0.234444i	−0.379770	−	0.925081i	\(-0.623997\pi\)
							−0.723201	+	0.690637i	\(0.757330\pi\)
\(47\)	41.1352	+	13.3656i	0.875217	+	0.284375i	0.711970	−	0.702210i	\(-0.247803\pi\)
							0.163247	+	0.986585i	\(0.447803\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.3.o.b.20.1	yes	144
3.2	odd	2	inner	93.3.o.b.20.18	yes	144
31.14	even	15	inner	93.3.o.b.14.18	yes	144
93.14	odd	30	inner	93.3.o.b.14.1	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.3.o.b.14.1	✓	144	93.14	odd	30	inner
93.3.o.b.14.18	yes	144	31.14	even	15	inner
93.3.o.b.20.1	yes	144	1.1	even	1	trivial
93.3.o.b.20.18	yes	144	3.2	odd	2	inner