Embedded newform 588.2.ba.b.187.18

• Label: 588.2.ba.b.187.18
• 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			187.18
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.427913 + 1.34792i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(-1.63378 + 1.15359i) q^{4} +(0.113995 - 0.0447396i) q^{5} +(-0.222237 - 1.39664i) q^{6} +(2.30833 - 1.29291i) q^{7} +(-2.25406 - 1.70857i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.427913	+	1.34792i	0.302580	+	0.953124i
							\(3\)	−0.988831	−	0.149042i	−0.570902	−	0.0860496i
\(5\)	0.113995	−	0.0447396i	0.0509800	−	0.0200082i								−0.339714	−	0.940529i	\(-0.610330\pi\)
							0.390694	+	0.920521i	\(0.372235\pi\)
\(7\)	2.30833	−	1.29291i	0.872466	−	0.488674i
							\(11\)	0.636255	+	2.06269i	0.191838	+	0.621923i	0.999480	+	0.0322501i	\(0.0102673\pi\)
−0.807642	+	0.589673i	\(0.799256\pi\)
\(13\)	6.14297	+	1.40209i	1.70375	+	0.388871i	0.960092	−	0.279684i	\(-0.0902298\pi\)
							0.743662	+	0.668555i	\(0.233087\pi\)
\(17\)	−1.76553	+	2.58955i	−0.428203	+	0.628059i	−0.978337	−	0.207018i	\(-0.933624\pi\)
							0.550134	+	0.835076i	\(0.314577\pi\)
\(19\)	−1.73821	+	3.01067i	−0.398773	+	0.690695i	−0.993575	−	0.113178i	\(-0.963897\pi\)
							0.594802	+	0.803872i	\(0.297230\pi\)
\(23\)	−0.604697	−	0.886928i	−0.126088	−	0.184937i	0.758004	−	0.652250i	\(-0.226175\pi\)
							−0.884092	+	0.467312i	\(0.845222\pi\)
\(29\)	2.36859	−	1.14065i	0.439837	−	0.211814i	−0.200841	−	0.979624i	\(-0.564368\pi\)
							0.640678	+	0.767810i	\(0.278653\pi\)
\(31\)	5.04800	+	8.74340i	0.906648	+	1.57036i	0.818689	+	0.574237i	\(0.194701\pi\)
							0.0879590	+	0.996124i	\(0.471966\pi\)
\(37\)	−0.0334785	+	0.446740i	−0.00550384	+	0.0734436i	−0.999279	−	0.0379576i	\(-0.987915\pi\)
							0.993776	+	0.111401i	\(0.0355339\pi\)
\(41\)	−9.19989	−	7.33667i	−1.43678	−	1.14580i	−0.964415	−	0.264393i	\(-0.914828\pi\)
							−0.472367	−	0.881402i	\(-0.656600\pi\)
\(43\)	2.90699	−	2.31824i	0.443311	−	0.353529i	−0.376252	−	0.926517i	\(-0.622787\pi\)
							0.819563	+	0.572988i	\(0.194216\pi\)
\(47\)	8.21808	+	7.62526i	1.19873	+	1.11226i	0.990971	+	0.134078i	\(0.0428074\pi\)
							0.207759	+	0.978180i	\(0.433383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.b.187.18	yes	336
4.3	odd	2		588.2.ba.a.187.26	✓	336
49.38	odd	42		588.2.ba.a.283.26	yes	336
196.87	even	42	inner	588.2.ba.b.283.18	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.26	✓	336	4.3	odd	2
588.2.ba.a.283.26	yes	336	49.38	odd	42
588.2.ba.b.187.18	yes	336	1.1	even	1	trivial
588.2.ba.b.283.18	yes	336	196.87	even	42	inner