Embedded newform 574.2.s.a.387.2

• Label: 574.2.s.a.387.2
• Level: $574$
• Weight: $2$
• Character: 574.387
• Analytic conductor: $4.583$
• Analytic rank: $0$
• Dimension: $112$
• 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			387.2
Character	\(\chi\)	\(=\)	574.387
Dual form			574.2.s.a.221.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(-1.19230 - 2.06512i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-0.424372 - 4.03763i) q^{5} +(-0.736880 - 2.26788i) q^{6} +(2.58013 - 0.585583i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-1.34315 + 2.32640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{2} + 2 q^{3} + 14 q^{4} + 4 q^{6} + 2 q^{7} + 28 q^{8} - 58 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.19230	−	2.06512i	−0.688373	−	1.19230i	−0.972364	−	0.233470i	\(-0.924992\pi\)
0.283991	−	0.958827i	\(-0.408341\pi\)
\(5\)	−0.424372	−	4.03763i	−0.189785	−	1.80568i	−0.511967	−	0.859005i	\(-0.671083\pi\)
							0.322182	−	0.946678i	\(-0.395584\pi\)
\(7\)	2.58013	−	0.585583i	0.975199	−	0.221330i
							\(11\)	0.0562918	−	0.535581i	0.0169726	−	0.161484i	−0.982753	−	0.184925i	\(-0.940796\pi\)
0.999725	+	0.0234415i	\(0.00746236\pi\)
\(13\)	1.81955	+	5.59999i	0.504651	+	1.55316i	0.801356	+	0.598188i	\(0.204112\pi\)
							−0.296705	+	0.954969i	\(0.595888\pi\)
\(17\)	0.549719	−	5.23022i	0.133326	−	1.26852i	−0.699360	−	0.714770i	\(-0.746531\pi\)
							0.832686	−	0.553745i	\(-0.186802\pi\)
\(19\)	−3.26800	−	3.62948i	−0.749731	−	0.832661i	0.240710	−	0.970597i	\(-0.422620\pi\)
							−0.990441	+	0.137936i	\(0.955953\pi\)
\(23\)	3.20651	+	0.681565i	0.668604	+	0.142116i	0.529694	−	0.848189i	\(-0.322307\pi\)
							0.138910	+	0.990305i	\(0.455640\pi\)
\(29\)	−6.03091	+	4.38171i	−1.11991	+	0.813663i	−0.984196	−	0.177084i	\(-0.943333\pi\)
							−0.135716	+	0.990748i	\(0.543333\pi\)
\(31\)	−0.627881	+	5.97389i	−0.112771	+	1.07294i	0.781035	+	0.624488i	\(0.214692\pi\)
							−0.893805	+	0.448455i	\(0.851974\pi\)
\(37\)	0.410089	+	3.90173i	0.0674182	+	0.641441i	0.975097	+	0.221777i	\(0.0711856\pi\)
							−0.907679	+	0.419665i	\(0.862148\pi\)
\(41\)	6.15921	−	1.75048i	0.961906	−	0.273379i
							\(43\)	−1.77153	−	5.45222i	−0.270156	−	0.831455i	−0.990460	−	0.137797i	\(-0.955998\pi\)
0.720304	−	0.693658i	\(-0.244002\pi\)
\(47\)	7.81016	+	1.66010i	1.13923	+	0.242151i	0.738635	−	0.674105i	\(-0.235471\pi\)
							0.400594	+	0.916256i	\(0.368804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.s.a.387.2	yes	112
7.4	even	3	inner	574.2.s.a.305.2	yes	112
41.16	even	5	inner	574.2.s.a.303.2	yes	112
287.221	even	15	inner	574.2.s.a.221.2	✓	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.s.a.221.2	✓	112	287.221	even	15	inner
574.2.s.a.303.2	yes	112	41.16	even	5	inner
574.2.s.a.305.2	yes	112	7.4	even	3	inner
574.2.s.a.387.2	yes	112	1.1	even	1	trivial