Embedded newform 574.2.s.a.387.4

• Label: 574.2.s.a.387.4
• Level: $574$
• Weight: $2$
• Character: 574.387
• Analytic conductor: $4.583$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 574 = 2 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	574.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.58341307602\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			387.4
Character	\(\chi\)	\(=\)	574.387
Dual form			574.2.s.a.221.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(-1.02427 - 1.77408i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.131135 + 1.24766i) q^{5} +(-0.633031 - 1.94827i) q^{6} +(-2.58503 - 0.563558i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-0.598242 + 1.03619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{2} + 2 q^{3} + 14 q^{4} + 4 q^{6} + 2 q^{7} + 28 q^{8} - 58 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(493\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.978148	+	0.207912i	0.691655	+	0.147016i
							\(3\)	−1.02427	−	1.77408i	−0.591360	−	1.02427i	−0.994049	−	0.108929i	\(-0.965258\pi\)
0.402689	−	0.915337i	\(-0.368076\pi\)
\(5\)	0.131135	+	1.24766i	0.0586453	+	0.557972i	0.983912	+	0.178655i	\(0.0571745\pi\)
							−0.925267	+	0.379318i	\(0.876159\pi\)
\(7\)	−2.58503	−	0.563558i	−0.977051	−	0.213005i
							\(11\)	0.512030	−	4.87164i	0.154383	−	1.46886i	−0.593397	−	0.804910i	\(-0.702214\pi\)
0.747780	−	0.663946i	\(-0.231120\pi\)
\(13\)	−0.169775	−	0.522512i	−0.0470870	−	0.144919i	0.924749	−	0.380578i	\(-0.124275\pi\)
							−0.971836	+	0.235659i	\(0.924275\pi\)
\(17\)	0.727952	−	6.92600i	0.176554	−	1.67980i	−0.444302	−	0.895877i	\(-0.646548\pi\)
							0.620856	−	0.783925i	\(-0.286785\pi\)
\(19\)	−1.45023	−	1.61064i	−0.332705	−	0.369506i	0.553460	−	0.832875i	\(-0.313307\pi\)
							−0.886165	+	0.463369i	\(0.846640\pi\)
\(23\)	−1.68758	−	0.358706i	−0.351884	−	0.0747953i	0.0285784	−	0.999592i	\(-0.490902\pi\)
							−0.380463	+	0.924796i	\(0.624235\pi\)
\(29\)	1.13246	−	0.822781i	0.210293	−	0.152787i	−0.477653	−	0.878548i	\(-0.658513\pi\)
							0.687946	+	0.725762i	\(0.258513\pi\)
\(31\)	−0.162384	+	1.54498i	−0.0291651	+	0.277487i	0.970214	+	0.242250i	\(0.0778853\pi\)
							−0.999379	+	0.0352372i	\(0.988781\pi\)
\(37\)	0.563979	+	5.36590i	0.0927175	+	0.882148i	0.937721	+	0.347388i	\(0.112931\pi\)
							−0.845004	+	0.534760i	\(0.820402\pi\)
\(41\)	−3.15637	+	5.57111i	−0.492943	+	0.870062i
							\(43\)	−1.79525	−	5.52521i	−0.273773	−	0.842587i	−0.989541	−	0.144249i	\(-0.953923\pi\)
0.715768	−	0.698338i	\(-0.246077\pi\)
\(47\)	−5.48684	−	1.16626i	−0.800337	−	0.170117i	−0.210452	−	0.977604i	\(-0.567494\pi\)
							−0.589885	+	0.807487i	\(0.700827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.s.a.387.4	yes	112
7.4	even	3	inner	574.2.s.a.305.4	yes	112
41.16	even	5	inner	574.2.s.a.303.4	yes	112
287.221	even	15	inner	574.2.s.a.221.4	✓	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.s.a.221.4	✓	112	287.221	even	15	inner
574.2.s.a.303.4	yes	112	41.16	even	5	inner
574.2.s.a.305.4	yes	112	7.4	even	3	inner
574.2.s.a.387.4	yes	112	1.1	even	1	trivial