Embedded newform 588.2.ba.a.187.17

• Label: 588.2.ba.a.187.17
• 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			187.17
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.450362 - 1.34059i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-1.59435 - 1.20750i) q^{4} +(-2.67539 + 1.05002i) q^{5} +(0.645136 - 1.25849i) q^{6} +(0.927902 + 2.47770i) q^{7} +(-2.33679 + 1.59355i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.450362	−	1.34059i	0.318454	−	0.947938i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	−2.67539	+	1.05002i	−1.19647	+	0.469581i								−0.878143	−	0.478399i	\(-0.841217\pi\)
							−0.318330	+	0.947980i	\(0.603122\pi\)
\(7\)	0.927902	+	2.47770i	0.350714	+	0.936483i
							\(11\)	1.21319	+	3.93306i	0.365790	+	1.18586i	0.932263	+	0.361782i	\(0.117831\pi\)
−0.566473	+	0.824080i	\(0.691692\pi\)
\(13\)	4.15433	+	0.948198i	1.15220	+	0.262983i	0.755616	−	0.655015i	\(-0.227338\pi\)
							0.396587	+	0.917997i	\(0.370195\pi\)
\(17\)	−1.41274	+	2.07211i	−0.342640	+	0.502560i	−0.958421	−	0.285357i	\(-0.907888\pi\)
							0.615782	+	0.787917i	\(0.288840\pi\)
\(19\)	1.77717	−	3.07815i	0.407711	−	0.706176i	−0.586922	−	0.809644i	\(-0.699660\pi\)
							0.994633	+	0.103467i	\(0.0329937\pi\)
\(23\)	0.0806465	+	0.118287i	0.0168160	+	0.0246645i	0.834555	−	0.550925i	\(-0.185725\pi\)
							−0.817739	+	0.575590i	\(0.804773\pi\)
\(29\)	−1.88067	+	0.905681i	−0.349231	+	0.168181i	−0.600273	−	0.799796i	\(-0.704941\pi\)
							0.251042	+	0.967976i	\(0.419227\pi\)
\(31\)	−2.09249	−	3.62430i	−0.375823	−	0.650944i	0.614627	−	0.788818i	\(-0.289306\pi\)
							−0.990450	+	0.137874i	\(0.955973\pi\)
\(37\)	−0.213032	+	2.84272i	−0.0350223	+	0.467340i	0.951929	+	0.306319i	\(0.0990975\pi\)
							−0.986951	+	0.161020i	\(0.948522\pi\)
\(41\)	5.74932	+	4.58493i	0.897893	+	0.716045i	0.959396	−	0.282061i	\(-0.0910182\pi\)
							−0.0615037	+	0.998107i	\(0.519590\pi\)
\(43\)	−0.530190	+	0.422812i	−0.0808532	+	0.0644783i	−0.663082	−	0.748546i	\(-0.730752\pi\)
							0.582229	+	0.813025i	\(0.302181\pi\)
\(47\)	−4.03381	−	3.74283i	−0.588392	−	0.545948i	0.328797	−	0.944401i	\(-0.393357\pi\)
							−0.917189	+	0.398453i	\(0.869547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.17	✓	336
4.3	odd	2		588.2.ba.b.187.4	yes	336
49.38	odd	42		588.2.ba.b.283.4	yes	336
196.87	even	42	inner	588.2.ba.a.283.17	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.17	✓	336	1.1	even	1	trivial
588.2.ba.a.283.17	yes	336	196.87	even	42	inner
588.2.ba.b.187.4	yes	336	4.3	odd	2
588.2.ba.b.283.4	yes	336	49.38	odd	42