Embedded newform 588.2.ba.a.187.19

• Label: 588.2.ba.a.187.19
• 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			187.19
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.584381 + 1.28783i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-1.31700 + 1.50516i) q^{4} +(1.78497 - 0.700548i) q^{5} +(0.385913 + 1.36054i) q^{6} +(0.922480 + 2.47972i) q^{7} +(-2.70802 - 0.816479i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.584381	+	1.28783i	0.413219	+	0.910631i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	1.78497	−	0.700548i	0.798262	−	0.313295i								0.0690769	−	0.997611i	\(-0.477995\pi\)
							0.729185	+	0.684317i	\(0.239899\pi\)
\(7\)	0.922480	+	2.47972i	0.348665	+	0.937247i
							\(11\)	−0.216481	−	0.701815i	−0.0652715	−	0.211605i	0.917062	−	0.398745i	\(-0.130554\pi\)
−0.982333	+	0.187140i	\(0.940078\pi\)
\(13\)	3.31769	+	0.757241i	0.920161	+	0.210021i	0.656265	−	0.754530i	\(-0.272135\pi\)
							0.263896	+	0.964551i	\(0.414992\pi\)
\(17\)	1.16427	−	1.70767i	0.282377	−	0.414171i	−0.658678	−	0.752425i	\(-0.728884\pi\)
							0.941055	+	0.338254i	\(0.109836\pi\)
\(19\)	−2.48117	+	4.29750i	−0.569218	+	0.985915i	0.427425	+	0.904051i	\(0.359421\pi\)
							−0.996643	+	0.0818643i	\(0.973913\pi\)
\(23\)	−0.444279	−	0.651637i	−0.0926385	−	0.135876i	0.777122	−	0.629350i	\(-0.216679\pi\)
							−0.869760	+	0.493475i	\(0.835727\pi\)
\(29\)	−2.50189	+	1.20485i	−0.464589	+	0.223734i	−0.651509	−	0.758641i	\(-0.725864\pi\)
							0.186920	+	0.982375i	\(0.440149\pi\)
\(31\)	−2.18439	−	3.78347i	−0.392328	−	0.679532i	0.600428	−	0.799679i	\(-0.294997\pi\)
							−0.992756	+	0.120147i	\(0.961664\pi\)
\(37\)	0.356465	−	4.75669i	0.0586024	−	0.781995i	−0.888052	−	0.459744i	\(-0.847941\pi\)
							0.946654	−	0.322251i	\(-0.104440\pi\)
\(41\)	4.70628	+	3.75313i	0.734997	+	0.586141i	0.917816	−	0.397006i	\(-0.129951\pi\)
							−0.182819	+	0.983147i	\(0.558522\pi\)
\(43\)	4.08897	−	3.26085i	0.623563	−	0.497275i	−0.259987	−	0.965612i	\(-0.583718\pi\)
							0.883550	+	0.468338i	\(0.155147\pi\)
\(47\)	1.29820	+	1.20456i	0.189363	+	0.175703i	0.769157	−	0.639060i	\(-0.220677\pi\)
							−0.579794	+	0.814763i	\(0.696867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.19	✓	336
4.3	odd	2		588.2.ba.b.187.25	yes	336
49.38	odd	42		588.2.ba.b.283.25	yes	336
196.87	even	42	inner	588.2.ba.a.283.19	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.19	✓	336	1.1	even	1	trivial
588.2.ba.a.283.19	yes	336	196.87	even	42	inner
588.2.ba.b.187.25	yes	336	4.3	odd	2
588.2.ba.b.283.25	yes	336	49.38	odd	42