Embedded newform 58.3.f.b.11.3

• Label: 58.3.f.b.11.3
• Level: $58$
• Weight: $3$
• Character: 58.11
• Analytic conductor: $1.580$
• Analytic rank: $0$
• Dimension: $36$
• 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			11.3
Character	\(\chi\)	\(=\)	58.11
Dual form			58.3.f.b.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.33485 - 0.467085i) q^{2} +(0.397504 + 3.52794i) q^{3} +(1.56366 - 1.24698i) q^{4} +(-0.990768 + 2.05735i) q^{5} +(2.17846 + 4.52361i) q^{6} +(1.68871 - 2.11758i) q^{7} +(1.50481 - 2.39490i) q^{8} +(-3.51401 + 0.802050i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.33485	−	0.467085i	0.667426	−	0.233543i
							\(3\)	0.397504	+	3.52794i	0.132501	+	1.17598i	0.867332	+	0.497730i	\(0.165833\pi\)
−0.734831	+	0.678250i	\(0.762738\pi\)
\(5\)	−0.990768	+	2.05735i	−0.198154	+	0.411470i	−0.976242	−	0.216685i	\(-0.930476\pi\)
							0.778088	+	0.628155i	\(0.216190\pi\)
\(7\)	1.68871	−	2.11758i	0.241245	−	0.302511i	−0.646438	−	0.762966i	\(-0.723742\pi\)
							0.887683	+	0.460455i	\(0.152314\pi\)
\(11\)	−8.26427	−	13.1525i	−0.751297	−	1.19568i	−0.975255	−	0.221084i	\(-0.929040\pi\)
							0.223957	−	0.974599i	\(-0.428102\pi\)
\(13\)	2.24005	+	0.511276i	0.172311	+	0.0393289i	0.307806	−	0.951449i	\(-0.400405\pi\)
							−0.135495	+	0.990778i	\(0.543262\pi\)
\(17\)	7.53643	−	7.53643i	0.443319	−	0.443319i	−0.449807	−	0.893126i	\(-0.648507\pi\)
							0.893126	+	0.449807i	\(0.148507\pi\)
\(19\)	−18.6721	−	2.10384i	−0.982742	−	0.110728i	−0.394053	−	0.919088i	\(-0.628927\pi\)
							−0.588689	+	0.808359i	\(0.700356\pi\)
\(23\)	−23.4171	+	11.2771i	−1.01814	+	0.490308i	−0.867054	−	0.498213i	\(-0.833990\pi\)
							−0.151081	+	0.988521i	\(0.548275\pi\)
\(29\)	−28.9994	+	0.180801i	−0.999981	+	0.00623453i
							\(31\)	26.6754	−	9.33412i	0.860496	−	0.301101i	0.136277	−	0.990671i	\(-0.456486\pi\)
0.724219	+	0.689570i	\(0.242201\pi\)
\(37\)	29.0552	−	46.2411i	0.785276	−	1.24976i	−0.178735	−	0.983897i	\(-0.557201\pi\)
							0.964011	−	0.265862i	\(-0.0856566\pi\)
\(41\)	39.4715	+	39.4715i	0.962719	+	0.962719i	0.999330	−	0.0366106i	\(-0.0116561\pi\)
							−0.0366106	+	0.999330i	\(0.511656\pi\)
\(43\)	−18.7484	+	53.5798i	−0.436009	+	1.24604i	0.491052	+	0.871130i	\(0.336613\pi\)
							−0.927061	+	0.374911i	\(0.877673\pi\)
\(47\)	15.8446	−	9.95582i	0.337119	−	0.211826i	−0.352824	−	0.935690i	\(-0.614779\pi\)
							0.689943	+	0.723864i	\(0.257636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.3.f.b.11.3	✓	36
29.8	odd	28	inner	58.3.f.b.37.3	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.3.f.b.11.3	✓	36	1.1	even	1	trivial
58.3.f.b.37.3	yes	36	29.8	odd	28	inner