Embedded newform 574.2.s.a.387.1

• Label: 574.2.s.a.387.1
• 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			387.1
Character	\(\chi\)	\(=\)	574.387
Dual form			574.2.s.a.221.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(-1.53894 - 2.66552i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.318178 + 3.02726i) q^{5} +(-0.951115 - 2.92723i) q^{6} +(2.44263 + 1.01663i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-3.23665 + 5.60605i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{2} + 2 q^{3} + 14 q^{4} + 4 q^{6} + 2 q^{7} + 28 q^{8} - 58 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.53894	−	2.66552i	−0.888506	−	1.53894i	−0.841642	−	0.540036i	\(-0.818411\pi\)
−0.0468636	−	0.998901i	\(-0.514923\pi\)
\(5\)	0.318178	+	3.02726i	0.142294	+	1.35383i	0.799747	+	0.600337i	\(0.204967\pi\)
							−0.657453	+	0.753495i	\(0.728366\pi\)
\(7\)	2.44263	+	1.01663i	0.923229	+	0.384251i
							\(11\)	0.257558	−	2.45050i	0.0776565	−	0.738852i	−0.884534	−	0.466475i	\(-0.845524\pi\)
0.962191	−	0.272377i	\(-0.0878097\pi\)
\(13\)	1.44633	+	4.45135i	0.401140	+	1.23458i	0.924075	+	0.382210i	\(0.124837\pi\)
							−0.522935	+	0.852373i	\(0.675163\pi\)
\(17\)	−0.482620	+	4.59182i	−0.117052	+	1.11368i	0.765490	+	0.643447i	\(0.222496\pi\)
							−0.882543	+	0.470232i	\(0.844170\pi\)
\(19\)	1.24687	+	1.38479i	0.286051	+	0.317692i	0.868995	−	0.494820i	\(-0.164766\pi\)
							−0.582944	+	0.812512i	\(0.698099\pi\)
\(23\)	−0.154533	−	0.0328470i	−0.0322224	−	0.00684908i	0.191772	−	0.981439i	\(-0.438576\pi\)
							−0.223995	+	0.974590i	\(0.571910\pi\)
\(29\)	6.72158	−	4.88351i	1.24817	−	0.906845i	0.250051	−	0.968233i	\(-0.419553\pi\)
							0.998114	+	0.0613875i	\(0.0195526\pi\)
\(31\)	0.865391	−	8.23364i	0.155429	−	1.47881i	−0.587385	−	0.809308i	\(-0.699842\pi\)
							0.742814	−	0.669498i	\(-0.233491\pi\)
\(37\)	−0.616855	−	5.86898i	−0.101410	−	0.964854i	−0.920382	−	0.391020i	\(-0.872122\pi\)
							0.818972	−	0.573834i	\(-0.194544\pi\)
\(41\)	−6.33526	+	0.929775i	−0.989401	+	0.145207i
							\(43\)	−0.201123	−	0.618992i	−0.0306709	−	0.0943953i	0.934549	−	0.355834i	\(-0.115803\pi\)
−0.965220	+	0.261439i	\(0.915803\pi\)
\(47\)	−1.74864	−	0.371685i	−0.255065	−	0.0542158i	0.0786042	−	0.996906i	\(-0.474954\pi\)
							−0.333670	+	0.942690i	\(0.608287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	574.2.s.a.387.1	yes	112
7.4	even	3	inner	574.2.s.a.305.1	yes	112
41.16	even	5	inner	574.2.s.a.303.1	yes	112
287.221	even	15	inner	574.2.s.a.221.1	✓	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.s.a.221.1	✓	112	287.221	even	15	inner
574.2.s.a.303.1	yes	112	41.16	even	5	inner
574.2.s.a.305.1	yes	112	7.4	even	3	inner
574.2.s.a.387.1	yes	112	1.1	even	1	trivial