Embedded newform 588.2.ba.b.187.25

• Label: 588.2.ba.b.187.25
• 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			187.25
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32790 + 0.486507i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(1.52662 + 1.29206i) q^{4} +(1.78497 - 0.700548i) q^{5} +(-1.24056 - 0.678986i) q^{6} +(-0.922480 - 2.47972i) q^{7} +(1.39860 + 2.45844i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32790	+	0.486507i	0.938965	+	0.344012i
							\(3\)	−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	0.982333	−	0.187140i	\(-0.0599219\pi\)
−0.917062	+	0.398745i	\(0.869446\pi\)
\(13\)	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	\(-0.640579\pi\)
							0.996643	−	0.0818643i	\(-0.0260874\pi\)
\(23\)	0.444279	+	0.651637i	0.0926385	+	0.135876i	0.869760	−	0.493475i	\(-0.164273\pi\)
							−0.777122	+	0.629350i	\(0.783321\pi\)
\(29\)	−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	0.992756	−	0.120147i	\(-0.0383365\pi\)
							−0.600428	+	0.799679i	\(0.705003\pi\)
\(37\)	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	−0.883550	−	0.468338i	\(-0.844853\pi\)
							0.259987	+	0.965612i	\(0.416282\pi\)
\(47\)	−1.29820	−	1.20456i	−0.189363	−	0.175703i	0.579794	−	0.814763i	\(-0.303133\pi\)
							−0.769157	+	0.639060i	\(0.779323\pi\)

(See \(a_n\) instead)

Twists

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

    

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