Embedded newform 588.2.ba.b.187.12

• Label: 588.2.ba.b.187.12
• 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			187.12
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.541252 + 1.30654i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(-1.41409 - 1.41433i) q^{4} +(1.46529 - 0.575084i) q^{5} +(0.729936 - 1.21128i) q^{6} +(-1.59045 - 2.11435i) q^{7} +(2.61327 - 1.08206i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.541252	+	1.30654i	−0.382723	+	0.923863i
							\(3\)	−0.988831	−	0.149042i	−0.570902	−	0.0860496i
\(5\)	1.46529	−	0.575084i	0.655298	−	0.257186i								−0.0143129	−	0.999898i	\(-0.504556\pi\)
							0.669611	+	0.742712i	\(0.266461\pi\)
\(7\)	−1.59045	−	2.11435i	−0.601133	−	0.799149i
							\(11\)	0.356094	+	1.15443i	0.107366	+	0.348073i	0.993274	−	0.115789i	\(-0.0369396\pi\)
−0.885908	+	0.463862i	\(0.846463\pi\)
\(13\)	−4.55756	−	1.04023i	−1.26404	−	0.288509i	−0.462591	−	0.886572i	\(-0.653080\pi\)
							−0.801449	+	0.598063i	\(0.795937\pi\)
\(17\)	−3.40097	+	4.98831i	−0.824856	+	1.20984i	0.150369	+	0.988630i	\(0.451954\pi\)
							−0.975226	+	0.221212i	\(0.928999\pi\)
\(19\)	−1.30804	+	2.26558i	−0.300084	+	0.519761i	−0.976155	−	0.217076i	\(-0.930348\pi\)
							0.676071	+	0.736837i	\(0.263681\pi\)
\(23\)	−3.84038	−	5.63281i	−0.800776	−	1.17452i	−0.981589	−	0.191005i	\(-0.938825\pi\)
							0.180814	−	0.983517i	\(-0.442127\pi\)
\(29\)	−1.17079	+	0.563822i	−0.217410	+	0.104699i	−0.539419	−	0.842037i	\(-0.681356\pi\)
							0.322009	+	0.946737i	\(0.395642\pi\)
\(31\)	1.77120	+	3.06781i	0.318117	+	0.550995i	0.980095	−	0.198528i	\(-0.0636161\pi\)
							−0.661978	+	0.749523i	\(0.730283\pi\)
\(37\)	0.292609	−	3.90459i	0.0481046	−	0.641911i	−0.920089	−	0.391709i	\(-0.871884\pi\)
							0.968194	−	0.250202i	\(-0.0804969\pi\)
\(41\)	−3.78087	−	3.01514i	−0.590473	−	0.470886i	0.282090	−	0.959388i	\(-0.408972\pi\)
							−0.872563	+	0.488501i	\(0.837544\pi\)
\(43\)	−3.36309	+	2.68197i	−0.512866	+	0.408997i	−0.845432	−	0.534083i	\(-0.820657\pi\)
							0.332566	+	0.943080i	\(0.392085\pi\)
\(47\)	0.449594	+	0.417162i	0.0655800	+	0.0608493i	0.712282	−	0.701893i	\(-0.247662\pi\)
							−0.646702	+	0.762743i	\(0.723852\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.b.187.12	yes	336
4.3	odd	2		588.2.ba.a.187.25	✓	336
49.38	odd	42		588.2.ba.a.283.25	yes	336
196.87	even	42	inner	588.2.ba.b.283.12	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.25	✓	336	4.3	odd	2
588.2.ba.a.283.25	yes	336	49.38	odd	42
588.2.ba.b.187.12	yes	336	1.1	even	1	trivial
588.2.ba.b.283.12	yes	336	196.87	even	42	inner