Embedded newform 93.4.g.a.26.2

• Label: 93.4.g.a.26.2
• Level: $93$
• Weight: $4$
• Character: 93.26
• Analytic conductor: $5.487$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	93.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.48717763053\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.2
Character	\(\chi\)	\(=\)	93.26
Dual form			93.4.g.a.68.29

$q$-expansion

\(f(q)\)	\(=\)	\(q-5.02024i q^{2} +(-4.97128 - 1.51209i) q^{3} -17.2028 q^{4} +(-9.62201 + 5.55527i) q^{5} +(-7.59107 + 24.9570i) q^{6} +(13.9460 - 24.1553i) q^{7} +46.2002i q^{8} +(22.4272 + 15.0341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 3 q^{3} - 220 q^{4} + 66 q^{6} + 24 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	5.02024i		−	1.77492i	−0.460883	−	0.887461i	\(-0.652467\pi\)
							0.460883	−	0.887461i	\(-0.347533\pi\)
\(3\)	−4.97128	−	1.51209i	−0.956722	−	0.291002i
							\(5\)	−9.62201	+	5.55527i	−0.860619	+	0.496879i	−0.864220	−	0.503115i	\(-0.832187\pi\)
0.00360042	+	0.999994i	\(0.498854\pi\)
\(7\)	13.9460	−	24.1553i	0.753016	−	1.30426i	−0.193339	−	0.981132i	\(-0.561932\pi\)
							0.946355	−	0.323130i	\(-0.104735\pi\)
\(11\)	2.73610	+	4.73907i	0.0749969	+	0.129898i	0.901085	−	0.433643i	\(-0.142772\pi\)
							−0.826088	+	0.563541i	\(0.809439\pi\)
\(13\)	−21.4234	+	12.3688i	−0.457060	+	0.263884i	−0.710807	−	0.703387i	\(-0.751670\pi\)
							0.253747	+	0.967271i	\(0.418337\pi\)
\(17\)	−61.0345	+	105.715i	−0.870767	+	1.50821i	−0.00956284	+	0.999954i	\(0.503044\pi\)
							−0.861204	+	0.508259i	\(0.830289\pi\)
\(19\)	36.2525	−	62.7912i	0.437731	−	0.758173i	−0.559783	−	0.828639i	\(-0.689115\pi\)
							0.997514	+	0.0704665i	\(0.0224488\pi\)
\(23\)	−162.303			−1.47142			−0.735708	−	0.677299i	\(-0.763150\pi\)
							−0.735708	+	0.677299i	\(0.763150\pi\)
\(29\)	92.6579			0.593315			0.296658	−	0.954984i	\(-0.404128\pi\)
							0.296658	+	0.954984i	\(0.404128\pi\)
\(31\)	128.484	+	115.251i	0.744400	+	0.667734i
							\(37\)	−209.736	−	121.091i	−0.931902	−	0.538034i	−0.0444894	−	0.999010i	\(-0.514166\pi\)
−0.887413	+	0.460976i	\(0.847499\pi\)
\(41\)	−4.72265	+	2.72662i	−0.0179891	+	0.0103860i	−0.508968	−	0.860786i	\(-0.669973\pi\)
							0.490978	+	0.871172i	\(0.336639\pi\)
\(43\)	−402.042	−	232.119i	−1.42583	−	0.823205i	−0.429044	−	0.903283i	\(-0.641150\pi\)
							−0.996789	+	0.0800785i	\(0.974483\pi\)
\(47\)			183.304i			0.568886i	0.958693	+	0.284443i	\(0.0918086\pi\)
							−0.958693	+	0.284443i	\(0.908191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.4.g.a.26.2	✓	60
3.2	odd	2	inner	93.4.g.a.26.29	yes	60
31.6	odd	6	inner	93.4.g.a.68.2	yes	60
93.68	even	6	inner	93.4.g.a.68.29	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.4.g.a.26.2	✓	60	1.1	even	1	trivial
93.4.g.a.26.29	yes	60	3.2	odd	2	inner
93.4.g.a.68.2	yes	60	31.6	odd	6	inner
93.4.g.a.68.29	yes	60	93.68	even	6	inner