Embedded newform 58.3.f.b.27.2

• Label: 58.3.f.b.27.2
• Level: $58$
• Weight: $3$
• Character: 58.27
• Analytic conductor: $1.580$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	58.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.58038553329\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(3\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.2
Character	\(\chi\)	\(=\)	58.27
Dual form			58.3.f.b.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.158342 - 1.40532i) q^{2} +(-0.332478 + 0.529136i) q^{3} +(-1.94986 - 0.445042i) q^{4} +(5.90652 - 4.71029i) q^{5} +(0.690961 + 0.551023i) q^{6} +(-1.94275 - 8.51175i) q^{7} +(-0.934170 + 2.66971i) q^{8} +(3.73551 + 7.75687i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{15}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.158342	−	1.40532i	0.0791708	−	0.702661i
							\(3\)	−0.332478	+	0.529136i	−0.110826	+	0.176379i	−0.897532	−	0.440949i	\(-0.854642\pi\)
0.786706	+	0.617328i	\(0.211785\pi\)
\(5\)	5.90652	−	4.71029i	1.18130	−	0.942059i	0.182154	−	0.983270i	\(-0.441693\pi\)
							0.999150	+	0.0412114i	\(0.0131217\pi\)
\(7\)	−1.94275	−	8.51175i	−0.277536	−	1.21596i	−0.900898	−	0.434031i	\(-0.857091\pi\)
							0.623362	−	0.781934i	\(-0.285766\pi\)
\(11\)	−0.845163	−	2.41534i	−0.0768330	−	0.219576i	0.899060	−	0.437825i	\(-0.144251\pi\)
							−0.975893	+	0.218249i	\(0.929965\pi\)
\(13\)	−9.34762	+	19.4105i	−0.719047	+	1.49312i	0.144852	+	0.989453i	\(0.453729\pi\)
							−0.863900	+	0.503664i	\(0.831985\pi\)
\(17\)	9.76109	+	9.76109i	0.574182	+	0.574182i	0.933294	−	0.359113i	\(-0.116921\pi\)
							−0.359113	+	0.933294i	\(0.616921\pi\)
\(19\)	19.1319	−	12.0214i	1.00694	−	0.632704i	0.0757103	−	0.997130i	\(-0.475878\pi\)
							0.931232	+	0.364426i	\(0.118735\pi\)
\(23\)	−22.6369	+	28.3858i	−0.984215	+	1.23417i	−0.0120357	+	0.999928i	\(0.503831\pi\)
							−0.972179	+	0.234239i	\(0.924740\pi\)
\(29\)	−25.8706	+	13.1039i	−0.892090	+	0.451857i
							\(31\)	0.850049	−	7.54439i	0.0274209	−	0.243367i	−0.972523	−	0.232806i	\(-0.925209\pi\)
0.999944	−	0.0105619i	\(-0.00336200\pi\)
\(37\)	5.18298	−	14.8121i	0.140080	−	0.400327i	−0.852375	−	0.522931i	\(-0.824839\pi\)
							0.992456	+	0.122604i	\(0.0391244\pi\)
\(41\)	38.8852	−	38.8852i	0.948419	−	0.948419i	−0.0503147	−	0.998733i	\(-0.516022\pi\)
							0.998733	+	0.0503147i	\(0.0160224\pi\)
\(43\)	43.6179	−	4.91456i	1.01437	−	0.114292i	0.410902	−	0.911680i	\(-0.365214\pi\)
							0.603468	+	0.797388i	\(0.293785\pi\)
\(47\)	−55.2922	+	19.3476i	−1.17643	+	0.411651i	−0.846518	−	0.532361i	\(-0.821305\pi\)
							−0.329913	+	0.944011i	\(0.607019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.27.2	✓	36
29.14	odd	28	inner	58.3.f.b.43.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.27.2	✓	36	1.1	even	1	trivial
58.3.f.b.43.2	yes	36	29.14	odd	28	inner