Embedded newform 588.2.ba.a.103.14

• Label: 588.2.ba.a.103.14
• 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			103.14
Character	\(\chi\)	\(=\)	588.103
Dual form			588.2.ba.a.451.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0925486 + 1.41118i) q^{2} +(-0.826239 - 0.563320i) q^{3} +(-1.98287 - 0.261206i) q^{4} +(-2.91108 - 0.218156i) q^{5} +(0.871414 - 1.11384i) q^{6} +(2.63369 + 0.252375i) q^{7} +(0.552121 - 2.77402i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0925486	+	1.41118i	−0.0654417	+	0.997856i
							\(3\)	−0.826239	−	0.563320i	−0.477029	−	0.325233i
\(5\)	−2.91108	−	0.218156i	−1.30188	−	0.0975621i								−0.594325	−	0.804225i	\(-0.702581\pi\)
							−0.707551	+	0.706662i	\(0.750200\pi\)
\(7\)	2.63369	+	0.252375i	0.995440	+	0.0953886i
							\(11\)	−2.98312	−	1.17079i	−0.899445	−	0.353006i	−0.129846	−	0.991534i	\(-0.541448\pi\)
−0.769599	+	0.638528i	\(0.779544\pi\)
\(13\)	3.94266	−	3.14417i	1.09350	−	0.872035i	0.101072	−	0.994879i	\(-0.467773\pi\)
							0.992426	+	0.122844i	\(0.0392015\pi\)
\(17\)	4.72234	+	5.08948i	1.14534	+	1.23438i	0.968196	+	0.250194i	\(0.0804944\pi\)
							0.177141	+	0.984185i	\(0.443315\pi\)
\(19\)	−3.27302	+	5.66903i	−0.750882	+	1.30057i	0.196514	+	0.980501i	\(0.437038\pi\)
							−0.947396	+	0.320064i	\(0.896295\pi\)
\(23\)	0.146909	−	0.158330i	0.0306326	−	0.0330141i	−0.717550	−	0.696507i	\(-0.754736\pi\)
							0.748182	+	0.663493i	\(0.230927\pi\)
\(29\)	−0.804901	−	3.52650i	−0.149466	−	0.654855i	−0.993034	−	0.117831i	\(-0.962406\pi\)
							0.843567	−	0.537023i	\(-0.180451\pi\)
\(31\)	3.43753	+	5.95398i	0.617399	+	1.06937i	0.989959	+	0.141358i	\(0.0451468\pi\)
							−0.372560	+	0.928008i	\(0.621520\pi\)
\(37\)	10.8558	+	3.34857i	1.78468	+	0.550501i	0.997781	−	0.0665800i	\(-0.0212088\pi\)
							0.786900	+	0.617081i	\(0.211685\pi\)
\(41\)	3.15759	+	6.55681i	0.493133	+	1.02400i	0.987917	+	0.154985i	\(0.0495330\pi\)
							−0.494784	+	0.869016i	\(0.664753\pi\)
\(43\)	0.367586	−	0.763299i	0.0560563	−	0.116402i	−0.871059	−	0.491179i	\(-0.836566\pi\)
							0.927115	+	0.374776i	\(0.122281\pi\)
\(47\)	−1.86848	+	0.281629i	−0.272546	+	0.0410798i	−0.283893	−	0.958856i	\(-0.591626\pi\)
							0.0113470	+	0.999936i	\(0.496388\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.103.14	✓	336
4.3	odd	2		588.2.ba.b.103.11	yes	336
49.10	odd	42		588.2.ba.b.451.11	yes	336
196.59	even	42	inner	588.2.ba.a.451.14	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.103.14	✓	336	1.1	even	1	trivial
588.2.ba.a.451.14	yes	336	196.59	even	42	inner
588.2.ba.b.103.11	yes	336	4.3	odd	2
588.2.ba.b.451.11	yes	336	49.10	odd	42