Embedded newform 588.2.ba.a.103.2

• Label: 588.2.ba.a.103.2
• Level: $588$
• Weight: $2$
• Character: 588.103
• 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			103.2
Character	\(\chi\)	\(=\)	588.103
Dual form			588.2.ba.a.451.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36468 + 0.371026i) q^{2} +(-0.826239 - 0.563320i) q^{3} +(1.72468 - 1.01266i) q^{4} +(-0.793563 - 0.0594693i) q^{5} +(1.33655 + 0.462193i) q^{6} +(0.251406 + 2.63378i) q^{7} +(-1.97791 + 2.02185i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36468	+	0.371026i	−0.964971	+	0.262355i
							\(3\)	−0.826239	−	0.563320i	−0.477029	−	0.325233i
\(5\)	−0.793563	−	0.0594693i	−0.354892	−	0.0265955i								−0.103909	−	0.994587i	\(-0.533135\pi\)
							−0.250984	+	0.967991i	\(0.580754\pi\)
\(7\)	0.251406	+	2.63378i	0.0950225	+	0.995475i
							\(11\)	0.289085	+	0.113458i	0.0871624	+	0.0342087i	0.408522	−	0.912749i	\(-0.366044\pi\)
−0.321359	+	0.946957i	\(0.604140\pi\)
\(13\)	1.76635	−	1.40862i	0.489898	−	0.390681i	−0.347150	−	0.937809i	\(-0.612851\pi\)
							0.837048	+	0.547129i	\(0.184279\pi\)
\(17\)	−4.13748	−	4.45915i	−1.00349	−	1.08150i	−0.996671	−	0.0815250i	\(-0.974021\pi\)
							−0.00681565	−	0.999977i	\(-0.502170\pi\)
\(19\)	−1.37289	+	2.37792i	−0.314963	+	0.545532i	−0.979430	−	0.201786i	\(-0.935325\pi\)
							0.664467	+	0.747318i	\(0.268659\pi\)
\(23\)	−3.50508	+	3.77758i	−0.730859	+	0.787679i	−0.983698	−	0.179830i	\(-0.942445\pi\)
							0.252839	+	0.967508i	\(0.418636\pi\)
\(29\)	0.640483	+	2.80614i	0.118935	+	0.521087i	0.998936	+	0.0461159i	\(0.0146844\pi\)
							−0.880001	+	0.474971i	\(0.842458\pi\)
\(31\)	1.70520	+	2.95350i	0.306264	+	0.530464i	0.977542	−	0.210741i	\(-0.0675877\pi\)
							−0.671278	+	0.741206i	\(0.734254\pi\)
\(37\)	5.89005	+	1.81684i	0.968318	+	0.298686i	0.738279	−	0.674495i	\(-0.235639\pi\)
							0.230038	+	0.973182i	\(0.426115\pi\)
\(41\)	1.04291	+	2.16563i	0.162875	+	0.338214i	0.966394	−	0.257065i	\(-0.0827556\pi\)
							−0.803519	+	0.595280i	\(0.797041\pi\)
\(43\)	−4.97849	+	10.3379i	−0.759212	+	1.57652i	0.0567320	+	0.998389i	\(0.481932\pi\)
							−0.815944	+	0.578131i	\(0.803782\pi\)
\(47\)	−8.84333	+	1.33292i	−1.28993	+	0.194426i	−0.757940	−	0.652324i	\(-0.773794\pi\)
							−0.531992	+	0.846750i	\(0.678556\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.103.2	✓	336
4.3	odd	2		588.2.ba.b.103.1	yes	336
49.10	odd	42		588.2.ba.b.451.1	yes	336
196.59	even	42	inner	588.2.ba.a.451.2	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.103.2	✓	336	1.1	even	1	trivial
588.2.ba.a.451.2	yes	336	196.59	even	42	inner
588.2.ba.b.103.1	yes	336	4.3	odd	2
588.2.ba.b.451.1	yes	336	49.10	odd	42