Embedded newform 588.2.ba.b.187.26

• Label: 588.2.ba.b.187.26
• 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			187.26
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38449 - 0.288431i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(1.83362 - 0.798658i) q^{4} +(-2.36872 + 0.929654i) q^{5} +(-1.41201 + 0.0788620i) q^{6} +(-2.64453 - 0.0802313i) q^{7} +(2.30826 - 1.63460i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38449	−	0.288431i	0.978981	−	0.203951i
							\(3\)	−0.988831	−	0.149042i	−0.570902	−	0.0860496i
\(5\)	−2.36872	+	0.929654i	−1.05932	+	0.415754i								−0.830097	−	0.557619i	\(-0.811715\pi\)
							−0.229227	+	0.973373i	\(0.573620\pi\)
\(7\)	−2.64453	−	0.0802313i	−0.999540	−	0.0303246i
							\(11\)	−1.59456	−	5.16943i	−0.480777	−	1.55864i	−0.793361	−	0.608751i	\(-0.791671\pi\)
0.312584	−	0.949890i	\(-0.398805\pi\)
\(13\)	−5.52320	−	1.26063i	−1.53186	−	0.349637i	−0.628256	−	0.778006i	\(-0.716231\pi\)
							−0.903604	+	0.428369i	\(0.859088\pi\)
\(17\)	−1.00305	+	1.47121i	−0.243276	+	0.356820i	−0.928385	−	0.371621i	\(-0.878802\pi\)
							0.685109	+	0.728441i	\(0.259755\pi\)
\(19\)	3.25367	−	5.63552i	0.746443	−	1.29288i	−0.203075	−	0.979163i	\(-0.565093\pi\)
							0.949518	−	0.313714i	\(-0.101573\pi\)
\(23\)	−1.62162	−	2.37848i	−0.338131	−	0.495948i	0.619084	−	0.785325i	\(-0.287504\pi\)
							−0.957215	+	0.289377i	\(0.906552\pi\)
\(29\)	−2.37784	+	1.14511i	−0.441554	+	0.212641i	−0.641433	−	0.767179i	\(-0.721660\pi\)
							0.199878	+	0.979821i	\(0.435945\pi\)
\(31\)	5.27677	+	9.13963i	0.947735	+	1.64153i	0.750180	+	0.661234i	\(0.229967\pi\)
							0.197556	+	0.980292i	\(0.436700\pi\)
\(37\)	−0.00446569	+	0.0595905i	−0.000734155	+	0.00979662i	−0.997559	−	0.0698309i	\(-0.977754\pi\)
							0.996825	+	0.0796275i	\(0.0253731\pi\)
\(41\)	−3.22570	−	2.57241i	−0.503770	−	0.401743i	0.338361	−	0.941016i	\(-0.390128\pi\)
							−0.842131	+	0.539273i	\(0.818699\pi\)
\(43\)	−3.96988	+	3.16587i	−0.605401	+	0.482791i	−0.877564	−	0.479460i	\(-0.840832\pi\)
							0.272163	+	0.962251i	\(0.412261\pi\)
\(47\)	−6.84217	−	6.34861i	−0.998034	−	0.926040i	−0.000772315	−	1.00000i	\(-0.500246\pi\)
							−0.997261	+	0.0739599i	\(0.976436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.b.187.26	yes	336
4.3	odd	2		588.2.ba.a.187.12	✓	336
49.38	odd	42		588.2.ba.a.283.12	yes	336
196.87	even	42	inner	588.2.ba.b.283.26	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.12	✓	336	4.3	odd	2
588.2.ba.a.283.12	yes	336	49.38	odd	42
588.2.ba.b.187.26	yes	336	1.1	even	1	trivial
588.2.ba.b.283.26	yes	336	196.87	even	42	inner