Embedded newform 588.2.ba.a.187.28

• Label: 588.2.ba.a.187.28
• 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			187.28
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41414 - 0.0148433i) q^{2} +(0.988831 + 0.149042i) q^{3} +(1.99956 - 0.0419808i) q^{4} +(2.69617 - 1.05817i) q^{5} +(1.40055 + 0.196089i) q^{6} +(-2.61576 - 0.397253i) q^{7} +(2.82703 - 0.0890465i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41414	−	0.0148433i	0.999945	−	0.0104958i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	2.69617	−	1.05817i	1.20577	−	0.473228i								0.324517	−	0.945880i	\(-0.394798\pi\)
							0.881249	+	0.472652i	\(0.156703\pi\)
\(7\)	−2.61576	−	0.397253i	−0.988664	−	0.150148i
							\(11\)	1.39197	+	4.51267i	0.419696	+	1.36062i	0.880890	+	0.473322i	\(0.156945\pi\)
−0.461194	+	0.887300i	\(0.652579\pi\)
\(13\)	−6.59678	−	1.50567i	−1.82962	−	0.417598i	−0.837876	−	0.545860i	\(-0.816203\pi\)
							−0.991740	+	0.128262i	\(0.959060\pi\)
\(17\)	3.43363	−	5.03621i	0.832778	−	1.22146i	−0.140099	−	0.990137i	\(-0.544742\pi\)
							0.972877	−	0.231323i	\(-0.0743055\pi\)
\(19\)	−2.96767	+	5.14015i	−0.680830	+	1.17923i	0.293898	+	0.955837i	\(0.405047\pi\)
							−0.974728	+	0.223396i	\(0.928286\pi\)
\(23\)	−2.40805	−	3.53196i	−0.502113	−	0.736465i	0.488589	−	0.872514i	\(-0.337512\pi\)
							−0.990702	+	0.136049i	\(0.956560\pi\)
\(29\)	−5.60179	+	2.69768i	−1.04023	+	0.500947i	−0.874399	−	0.485208i	\(-0.838744\pi\)
							−0.165828	+	0.986155i	\(0.553030\pi\)
\(31\)	−0.948993	−	1.64370i	−0.170444	−	0.295218i	0.768131	−	0.640293i	\(-0.221187\pi\)
							−0.938575	+	0.345075i	\(0.887854\pi\)
\(37\)	0.487943	−	6.51115i	0.0802173	−	1.07043i	−0.801296	−	0.598267i	\(-0.795856\pi\)
							0.881514	−	0.472158i	\(-0.156525\pi\)
\(41\)	5.49727	+	4.38392i	0.858529	+	0.684654i	0.950371	−	0.311120i	\(-0.100704\pi\)
							−0.0918421	+	0.995774i	\(0.529275\pi\)
\(43\)	−1.40424	+	1.11985i	−0.214145	+	0.170775i	−0.724688	−	0.689078i	\(-0.758016\pi\)
							0.510543	+	0.859852i	\(0.329444\pi\)
\(47\)	−4.26845	−	3.96054i	−0.622617	−	0.577704i	0.304483	−	0.952518i	\(-0.401516\pi\)
							−0.927100	+	0.374813i	\(0.877707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.28	✓	336
4.3	odd	2		588.2.ba.b.187.16	yes	336
49.38	odd	42		588.2.ba.b.283.16	yes	336
196.87	even	42	inner	588.2.ba.a.283.28	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.28	✓	336	1.1	even	1	trivial
588.2.ba.a.283.28	yes	336	196.87	even	42	inner
588.2.ba.b.187.16	yes	336	4.3	odd	2
588.2.ba.b.283.16	yes	336	49.38	odd	42