Embedded newform 603.2.f.c.238.19

• Label: 603.2.f.c.238.19
• Level: $603$
• Weight: $2$
• Character: 603.238
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			238.19
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.689435 + 1.19414i) q^{2} +(-1.28239 - 1.16424i) q^{3} +(0.0493577 + 0.0854900i) q^{4} +(0.438766 - 0.759964i) q^{5} +(2.27440 - 0.728685i) q^{6} +3.67012 q^{7} -2.89386 q^{8} +(0.289073 + 2.98604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.689435	+	1.19414i	−0.487504	+	0.844382i	−0.999897	−	0.0143690i	\(-0.995426\pi\)
							0.512392	+	0.858751i	\(0.328759\pi\)
\(3\)	−1.28239	−	1.16424i	−0.740391	−	0.672176i
							\(5\)	0.438766	−	0.759964i	0.196222	−	0.339866i	−0.751078	−	0.660213i	\(-0.770466\pi\)
0.947300	+	0.320347i	\(0.103799\pi\)
\(7\)	3.67012			1.38718			0.693588	−	0.720372i	\(-0.256029\pi\)
							0.693588	+	0.720372i	\(0.256029\pi\)
\(11\)	0.109270			0.0329462			0.0164731	−	0.999864i	\(-0.494756\pi\)
							0.0164731	+	0.999864i	\(0.494756\pi\)
\(13\)	1.17188			0.325020			0.162510	−	0.986707i	\(-0.448041\pi\)
							0.162510	+	0.986707i	\(0.448041\pi\)
\(17\)	−3.32708	−	5.76268i	−0.806937	−	1.39766i	−0.914976	−	0.403508i	\(-0.867791\pi\)
							0.108039	−	0.994147i	\(-0.465543\pi\)
\(19\)	0.171184	+	0.296499i	0.0392723	+	0.0680216i	0.884993	−	0.465603i	\(-0.154163\pi\)
							−0.845721	+	0.533625i	\(0.820829\pi\)
\(23\)	3.57133			0.744673			0.372337	−	0.928098i	\(-0.378557\pi\)
							0.372337	+	0.928098i	\(0.378557\pi\)
\(29\)	9.26594			1.72064			0.860321	−	0.509753i	\(-0.170263\pi\)
							0.860321	+	0.509753i	\(0.170263\pi\)
\(31\)	−0.213419	−	0.369652i	−0.0383311	−	0.0663914i	0.846223	−	0.532828i	\(-0.178871\pi\)
							−0.884555	+	0.466437i	\(0.845537\pi\)
\(37\)	0.762475	+	1.32065i	0.125350	+	0.217113i	0.921870	−	0.387500i	\(-0.126661\pi\)
							−0.796520	+	0.604613i	\(0.793328\pi\)
\(41\)	5.33289	+	9.23684i	0.832858	+	1.44255i	0.895763	+	0.444533i	\(0.146630\pi\)
							−0.0629049	+	0.998020i	\(0.520036\pi\)
\(43\)	0.444420	+	0.769758i	0.0677734	+	0.117387i	0.897921	−	0.440157i	\(-0.145077\pi\)
							−0.830147	+	0.557544i	\(0.811744\pi\)
\(47\)	−4.30659			−0.628181			−0.314090	−	0.949393i	\(-0.601699\pi\)
							−0.314090	+	0.949393i	\(0.601699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.19	✓	128
9.7	even	3		603.2.h.c.439.46	yes	128
67.29	even	3		603.2.h.c.364.46	yes	128
603.565	even	3	inner	603.2.f.c.565.19	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.19	✓	128	1.1	even	1	trivial
603.2.f.c.565.19	yes	128	603.565	even	3	inner
603.2.h.c.364.46	yes	128	67.29	even	3
603.2.h.c.439.46	yes	128	9.7	even	3