Embedded newform 588.2.ba.a.103.8

• Label: 588.2.ba.a.103.8
• 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			103.8
Character	\(\chi\)	\(=\)	588.103
Dual form			588.2.ba.a.451.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.839211 + 1.13830i) q^{2} +(-0.826239 - 0.563320i) q^{3} +(-0.591451 - 1.91055i) q^{4} +(-4.04500 - 0.303131i) q^{5} +(1.33462 - 0.467763i) q^{6} +(-2.59666 + 0.507311i) q^{7} +(2.67113 + 0.930102i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.839211	+	1.13830i	−0.593412	+	0.804899i
							\(3\)	−0.826239	−	0.563320i	−0.477029	−	0.325233i
\(5\)	−4.04500	−	0.303131i	−1.80898	−	0.135564i								−0.873023	−	0.487680i	\(-0.837843\pi\)
							−0.935957	+	0.352115i	\(0.885462\pi\)
\(7\)	−2.59666	+	0.507311i	−0.981445	+	0.191746i
							\(11\)	−1.73642	−	0.681494i	−0.523550	−	0.205478i	0.0888229	−	0.996047i	\(-0.471689\pi\)
−0.612373	+	0.790569i	\(0.709785\pi\)
\(13\)	−0.689679	+	0.550001i	−0.191283	+	0.152543i	−0.714448	−	0.699688i	\(-0.753322\pi\)
							0.523166	+	0.852231i	\(0.324751\pi\)
\(17\)	−0.345445	−	0.372301i	−0.0837826	−	0.0902962i	0.689769	−	0.724030i	\(-0.257712\pi\)
							−0.773552	+	0.633733i	\(0.781522\pi\)
\(19\)	3.71820	−	6.44011i	0.853013	−	1.47746i	−0.0254637	−	0.999676i	\(-0.508106\pi\)
							0.878476	−	0.477786i	\(-0.158560\pi\)
\(23\)	−2.31261	+	2.49240i	−0.482212	+	0.519701i	−0.926557	−	0.376155i	\(-0.877246\pi\)
							0.444345	+	0.895856i	\(0.353437\pi\)
\(29\)	2.06696	+	9.05596i	0.383825	+	1.68165i	0.685368	+	0.728196i	\(0.259641\pi\)
							−0.301543	+	0.953453i	\(0.597502\pi\)
\(31\)	−0.368176	−	0.637699i	−0.0661263	−	0.114534i	0.831067	−	0.556173i	\(-0.187731\pi\)
							−0.897193	+	0.441639i	\(0.854397\pi\)
\(37\)	6.20752	+	1.91476i	1.02051	+	0.314785i	0.759470	−	0.650543i	\(-0.225459\pi\)
							0.261040	+	0.965328i	\(0.415935\pi\)
\(41\)	3.11365	+	6.46557i	0.486271	+	1.00975i	0.989356	+	0.145515i	\(0.0464839\pi\)
							−0.503085	+	0.864237i	\(0.667802\pi\)
\(43\)	2.26877	−	4.71115i	0.345984	−	0.718443i	−0.653268	−	0.757127i	\(-0.726602\pi\)
							0.999251	+	0.0386841i	\(0.0123166\pi\)
\(47\)	2.48732	−	0.374903i	0.362813	−	0.0546852i	0.0348935	−	0.999391i	\(-0.488891\pi\)
							0.327919	+	0.944706i	\(0.393653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.103.8	✓	336
4.3	odd	2		588.2.ba.b.103.5	yes	336
49.10	odd	42		588.2.ba.b.451.5	yes	336
196.59	even	42	inner	588.2.ba.a.451.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.103.8	✓	336	1.1	even	1	trivial
588.2.ba.a.451.8	yes	336	196.59	even	42	inner
588.2.ba.b.103.5	yes	336	4.3	odd	2
588.2.ba.b.451.5	yes	336	49.10	odd	42