Embedded newform 588.2.ba.a.283.28

• Label: 588.2.ba.a.283.28
• Level: $588$
• Weight: $2$
• Character: 588.283
• 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			283.28
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.a.187.28

$q$-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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.28	✓	336	196.187	even	42	inner
588.2.ba.a.283.28	yes	336	1.1	even	1	trivial
588.2.ba.b.187.16	yes	336	49.40	odd	42
588.2.ba.b.283.16	yes	336	4.3	odd	2