Embedded newform 588.2.ba.a.187.7

• Label: 588.2.ba.a.187.7
• 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			187.7
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.944869 + 1.05225i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-0.214444 - 1.98847i) q^{4} +(2.70941 - 1.06337i) q^{5} +(-1.09115 + 0.899668i) q^{6} +(2.19729 + 1.47375i) q^{7} +(2.29498 + 1.65320i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.944869	+	1.05225i	−0.668124	+	0.744050i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	2.70941	−	1.06337i	1.21169	−	0.475552i								0.328469	−	0.944515i	\(-0.393467\pi\)
							0.883218	+	0.468963i	\(0.155372\pi\)
\(7\)	2.19729	+	1.47375i	0.830496	+	0.557024i
							\(11\)	0.192138	+	0.622896i	0.0579317	+	0.187810i	0.979913	−	0.199423i	\(-0.0639069\pi\)
−0.921982	+	0.387234i	\(0.873431\pi\)
\(13\)	−1.67169	−	0.381552i	−0.463644	−	0.105824i	−0.0156831	−	0.999877i	\(-0.504992\pi\)
							−0.447960	+	0.894053i	\(0.647849\pi\)
\(17\)	1.43366	−	2.10279i	0.347713	−	0.510002i	−0.612047	−	0.790821i	\(-0.709654\pi\)
							0.959761	+	0.280819i	\(0.0906062\pi\)
\(19\)	−0.182971	+	0.316916i	−0.0419765	+	0.0727055i	−0.886250	−	0.463207i	\(-0.846699\pi\)
							0.844274	+	0.535912i	\(0.180032\pi\)
\(23\)	−2.51988	−	3.69599i	−0.525432	−	0.770667i	0.468166	−	0.883640i	\(-0.344915\pi\)
							−0.993598	+	0.112973i	\(0.963962\pi\)
\(29\)	3.73489	−	1.79863i	0.693551	−	0.333997i	−0.0536826	−	0.998558i	\(-0.517096\pi\)
							0.747234	+	0.664562i	\(0.231382\pi\)
\(31\)	3.29773	+	5.71184i	0.592290	+	1.02588i	0.993923	+	0.110076i	\(0.0351093\pi\)
							−0.401633	+	0.915801i	\(0.631557\pi\)
\(37\)	−0.0988249	+	1.31873i	−0.0162467	+	0.216797i	0.983171	+	0.182686i	\(0.0584791\pi\)
							−0.999418	+	0.0341114i	\(0.989140\pi\)
\(41\)	−5.90993	−	4.71302i	−0.922977	−	0.736049i	0.0417982	−	0.999126i	\(-0.486691\pi\)
							−0.964775	+	0.263077i	\(0.915263\pi\)
\(43\)	−3.72336	+	2.96928i	−0.567807	+	0.452811i	−0.864834	−	0.502058i	\(-0.832576\pi\)
							0.297027	+	0.954869i	\(0.404005\pi\)
\(47\)	−1.94447	−	1.80421i	−0.283631	−	0.263171i	0.525508	−	0.850789i	\(-0.323875\pi\)
							−0.809139	+	0.587618i	\(0.800066\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.7	✓	336
4.3	odd	2		588.2.ba.b.187.21	yes	336
49.38	odd	42		588.2.ba.b.283.21	yes	336
196.87	even	42	inner	588.2.ba.a.283.7	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.7	✓	336	1.1	even	1	trivial
588.2.ba.a.283.7	yes	336	196.87	even	42	inner
588.2.ba.b.187.21	yes	336	4.3	odd	2
588.2.ba.b.283.21	yes	336	49.38	odd	42