Embedded newform 588.2.ba.a.283.13

• Label: 588.2.ba.a.283.13
• Level: $588$
• Weight: $2$
• Character: 588.283
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.ba (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			283.13
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.a.187.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.147261 - 1.40653i) q^{2} +(0.988831 - 0.149042i) q^{3} +(-1.95663 + 0.414253i) q^{4} +(-0.849860 - 0.333546i) q^{5} +(-0.355248 - 1.36887i) q^{6} +(0.217261 + 2.63682i) q^{7} +(0.870793 + 2.69104i) q^{8} +(0.955573 - 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.147261	−	1.40653i	−0.104129	−	0.994564i
							\(3\)	0.988831	−	0.149042i	0.570902	−	0.0860496i
\(5\)	−0.849860	−	0.333546i	−0.380069	−	0.149166i								0.167613	−	0.985853i	\(-0.446394\pi\)
							−0.547682	+	0.836687i	\(0.684489\pi\)
\(7\)	0.217261	+	2.63682i	0.0821170	+	0.996623i
							\(11\)	−1.69734	+	5.50265i	−0.511768	+	1.65911i	0.218411	+	0.975857i	\(0.429913\pi\)
−0.730180	+	0.683255i	\(0.760564\pi\)
\(13\)	1.31380	−	0.299866i	0.364382	−	0.0831678i	−0.0364100	−	0.999337i	\(-0.511592\pi\)
							0.400792	+	0.916169i	\(0.368735\pi\)
\(17\)	2.98697	+	4.38108i	0.724446	+	1.06257i	0.995001	+	0.0998628i	\(0.0318404\pi\)
							−0.270555	+	0.962704i	\(0.587207\pi\)
\(19\)	−0.650881	−	1.12736i	−0.149322	−	0.258634i	0.781655	−	0.623711i	\(-0.214376\pi\)
							−0.930977	+	0.365077i	\(0.881043\pi\)
\(23\)	−2.73659	+	4.01384i	−0.570618	+	0.836943i	−0.997736	−	0.0672526i	\(-0.978577\pi\)
							0.427118	+	0.904196i	\(0.359529\pi\)
\(29\)	4.16617	+	2.00632i	0.773639	+	0.372565i	0.778679	−	0.627422i	\(-0.215890\pi\)
							−0.00504004	+	0.999987i	\(0.501604\pi\)
\(31\)	3.03112	−	5.25006i	0.544406	−	0.942938i	−0.454238	−	0.890880i	\(-0.650089\pi\)
							0.998644	−	0.0520580i	\(-0.0165781\pi\)
\(37\)	0.0836554	+	1.11630i	0.0137529	+	0.183519i	0.999850	+	0.0173466i	\(0.00552186\pi\)
							−0.986097	+	0.166173i	\(0.946859\pi\)
\(41\)	0.134652	−	0.107381i	0.0210291	−	0.0167701i	−0.612919	−	0.790146i	\(-0.710005\pi\)
							0.633948	+	0.773376i	\(0.281433\pi\)
\(43\)	3.11463	+	2.48383i	0.474976	+	0.378781i	0.831517	−	0.555499i	\(-0.187473\pi\)
							−0.356541	+	0.934280i	\(0.616044\pi\)
\(47\)	0.0410181	−	0.0380593i	0.00598311	−	0.00555151i	−0.677176	−	0.735822i	\(-0.736796\pi\)
							0.683159	+	0.730270i	\(0.260606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.283.13	yes	336
4.3	odd	2		588.2.ba.b.283.27	yes	336
49.40	odd	42		588.2.ba.b.187.27	yes	336
196.187	even	42	inner	588.2.ba.a.187.13	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.13	✓	336	196.187	even	42	inner
588.2.ba.a.283.13	yes	336	1.1	even	1	trivial
588.2.ba.b.187.27	yes	336	49.40	odd	42
588.2.ba.b.283.27	yes	336	4.3	odd	2