Embedded newform 588.2.ba.a.187.13

• Label: 588.2.ba.a.187.13
• Level: $588$
• Weight: $2$
• Character: 588.187
• 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			187.13
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.13

$q$-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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{23}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.13	✓	336	1.1	even	1	trivial
588.2.ba.a.283.13	yes	336	196.87	even	42	inner
588.2.ba.b.187.27	yes	336	4.3	odd	2
588.2.ba.b.283.27	yes	336	49.38	odd	42