Embedded newform 603.2.h.c.364.9

• Label: 603.2.h.c.364.9
• Level: $603$
• Weight: $2$
• Character: 603.364
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.h (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			364.9
Character	\(\chi\)	\(=\)	603.364
Dual form			603.2.h.c.439.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.24620 q^{2} +(-1.70592 + 0.299744i) q^{3} +3.04541 q^{4} +(-1.53025 + 2.65047i) q^{5} +(3.83183 - 0.673285i) q^{6} +(0.754542 - 1.30690i) q^{7} -2.34821 q^{8} +(2.82031 - 1.02268i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 3 q^{3} + 130 q^{4} + 5 q^{5} - 6 q^{6} + 4 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24620			−1.58830			−0.794152	−	0.607720i	\(-0.792084\pi\)
							−0.794152	+	0.607720i	\(0.792084\pi\)
\(3\)	−1.70592	+	0.299744i	−0.984912	+	0.173057i
							\(5\)	−1.53025	+	2.65047i	−0.684348	+	1.18533i	0.289293	+	0.957241i	\(0.406580\pi\)
−0.973641	+	0.228085i	\(0.926754\pi\)
\(7\)	0.754542	−	1.30690i	0.285190	−	0.493963i	−0.687465	−	0.726217i	\(-0.741277\pi\)
							0.972655	+	0.232254i	\(0.0746100\pi\)
\(11\)	−1.18261	+	2.04835i	−0.356572	+	0.617600i	−0.987386	−	0.158334i	\(-0.949388\pi\)
							0.630814	+	0.775934i	\(0.282721\pi\)
\(13\)	−0.809161	−	1.40151i	−0.224421	−	0.388708i	0.731725	−	0.681600i	\(-0.238716\pi\)
							−0.956146	+	0.292892i	\(0.905382\pi\)
\(17\)	2.55025	−	4.41716i	0.618526	−	1.07132i	−0.371229	−	0.928541i	\(-0.621064\pi\)
							0.989755	−	0.142777i	\(-0.0456032\pi\)
\(19\)	1.18133	−	2.04612i	0.271016	−	0.469413i	−0.698107	−	0.715994i	\(-0.745974\pi\)
							0.969122	+	0.246581i	\(0.0793072\pi\)
\(23\)	−1.73437	−	3.00401i	−0.361641	−	0.626380i	0.626590	−	0.779349i	\(-0.284450\pi\)
							−0.988231	+	0.152969i	\(0.951117\pi\)
\(29\)	−2.14458	+	3.71452i	−0.398238	+	0.689769i	−0.993509	−	0.113757i	\(-0.963712\pi\)
							0.595271	+	0.803525i	\(0.297045\pi\)
\(31\)	−0.555834			−0.0998308			−0.0499154	−	0.998753i	\(-0.515895\pi\)
							−0.0499154	+	0.998753i	\(0.515895\pi\)
\(37\)	3.21565	−	5.56967i	0.528650	−	0.915648i	−0.470792	−	0.882244i	\(-0.656032\pi\)
							0.999442	−	0.0334042i	\(-0.0106349\pi\)
\(41\)	−0.807026			−0.126036			−0.0630181	−	0.998012i	\(-0.520073\pi\)
							−0.0630181	+	0.998012i	\(0.520073\pi\)
\(43\)	3.05313	+	5.28817i	0.465598	+	0.806439i	0.999228	−	0.0392789i	\(-0.0125061\pi\)
							−0.533631	+	0.845718i	\(0.679173\pi\)
\(47\)	3.90899	−	6.77057i	0.570185	−	0.987589i	−0.426362	−	0.904553i	\(-0.640205\pi\)
							0.996547	−	0.0830360i	\(-0.0264616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.h.c.364.9	yes	128
9.7	even	3		603.2.f.c.565.56	yes	128
67.37	even	3		603.2.f.c.238.56	✓	128
603.439	even	3	inner	603.2.h.c.439.9	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.56	✓	128	67.37	even	3
603.2.f.c.565.56	yes	128	9.7	even	3
603.2.h.c.364.9	yes	128	1.1	even	1	trivial
603.2.h.c.439.9	yes	128	603.439	even	3	inner