Embedded newform 588.2.ba.b.187.15

• Label: 588.2.ba.b.187.15
• 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			187.15
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0531662 - 1.41321i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(-1.99435 - 0.150270i) q^{4} +(-0.462080 + 0.181353i) q^{5} +(-0.263201 + 1.38951i) q^{6} +(-2.59558 - 0.512816i) q^{7} +(-0.318396 + 2.81045i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0531662	−	1.41321i	0.0375942	−	0.999293i
							\(3\)	−0.988831	−	0.149042i	−0.570902	−	0.0860496i
\(5\)	−0.462080	+	0.181353i	−0.206648	+	0.0811035i								−0.466408	−	0.884570i	\(-0.654452\pi\)
							0.259759	+	0.965673i	\(0.416357\pi\)
\(7\)	−2.59558	−	0.512816i	−0.981036	−	0.193826i
							\(11\)	1.33192	+	4.31796i	0.401588	+	1.30192i	0.900346	+	0.435174i	\(0.143313\pi\)
−0.498759	+	0.866741i	\(0.666211\pi\)
\(13\)	3.68398	+	0.840845i	1.02175	+	0.233208i	0.700399	−	0.713751i	\(-0.253005\pi\)
							0.321353	+	0.946959i	\(0.395862\pi\)
\(17\)	1.13757	−	1.66851i	0.275902	−	0.404674i	−0.663130	−	0.748504i	\(-0.730772\pi\)
							0.939032	+	0.343831i	\(0.111725\pi\)
\(19\)	0.488877	−	0.846760i	0.112156	−	0.194260i	−0.804483	−	0.593975i	\(-0.797558\pi\)
							0.916639	+	0.399715i	\(0.130891\pi\)
\(23\)	−2.39323	−	3.51023i	−0.499024	−	0.731933i	0.491257	−	0.871014i	\(-0.336537\pi\)
							−0.990281	+	0.139081i	\(0.955585\pi\)
\(29\)	8.70671	−	4.19293i	1.61680	−	0.778608i	0.616832	−	0.787095i	\(-0.288416\pi\)
							0.999964	+	0.00848736i	\(0.00270164\pi\)
\(31\)	4.48423	+	7.76692i	0.805392	+	1.39498i	0.916026	+	0.401119i	\(0.131379\pi\)
							−0.110634	+	0.993861i	\(0.535288\pi\)
\(37\)	−0.392678	+	5.23992i	−0.0645559	+	0.861438i	0.867109	+	0.498119i	\(0.165976\pi\)
							−0.931665	+	0.363319i	\(0.881643\pi\)
\(41\)	2.78734	+	2.22283i	0.435310	+	0.347148i	0.816497	−	0.577350i	\(-0.195913\pi\)
							−0.381187	+	0.924498i	\(0.624485\pi\)
\(43\)	−2.87270	+	2.29090i	−0.438083	+	0.349360i	−0.817562	−	0.575841i	\(-0.804675\pi\)
							0.379479	+	0.925200i	\(0.376103\pi\)
\(47\)	4.41509	+	4.09660i	0.644007	+	0.597551i	0.933042	−	0.359767i	\(-0.117144\pi\)
							−0.289035	+	0.957318i	\(0.593334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.b.187.15	yes	336
4.3	odd	2		588.2.ba.a.187.2	✓	336
49.38	odd	42		588.2.ba.a.283.2	yes	336
196.87	even	42	inner	588.2.ba.b.283.15	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.2	✓	336	4.3	odd	2
588.2.ba.a.283.2	yes	336	49.38	odd	42
588.2.ba.b.187.15	yes	336	1.1	even	1	trivial
588.2.ba.b.283.15	yes	336	196.87	even	42	inner