Embedded newform 588.2.ba.a.283.7

• Label: 588.2.ba.a.283.7
• Level: $588$
• Weight: $2$
• Character: 588.283
• 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			283.7
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.a.187.7

$q$-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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.7	✓	336	196.187	even	42	inner
588.2.ba.a.283.7	yes	336	1.1	even	1	trivial
588.2.ba.b.187.21	yes	336	49.40	odd	42
588.2.ba.b.283.21	yes	336	4.3	odd	2