Embedded newform 603.2.f.c.238.8

• Label: 603.2.f.c.238.8
• Level: $603$
• Weight: $2$
• Character: 603.238
• 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.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			238.8
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17727 + 2.03909i) q^{2} +(1.28378 + 1.16272i) q^{3} +(-1.77193 - 3.06907i) q^{4} +(0.679676 - 1.17723i) q^{5} +(-3.88224 + 1.24890i) q^{6} +1.20666 q^{7} +3.63508 q^{8} +(0.296168 + 2.98534i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17727	+	2.03909i	−0.832456	+	1.44186i	0.0636297	+	0.997974i	\(0.479732\pi\)
							−0.896085	+	0.443882i	\(0.853601\pi\)
\(3\)	1.28378	+	1.16272i	0.741189	+	0.671296i
							\(5\)	0.679676	−	1.17723i	0.303960	−	0.526475i	−0.673069	−	0.739580i	\(-0.735024\pi\)
0.977029	+	0.213105i	\(0.0683576\pi\)
\(7\)	1.20666			0.456076			0.228038	−	0.973652i	\(-0.426769\pi\)
							0.228038	+	0.973652i	\(0.426769\pi\)
\(11\)	0.763201			0.230114			0.115057	−	0.993359i	\(-0.463295\pi\)
							0.115057	+	0.993359i	\(0.463295\pi\)
\(13\)	6.91230			1.91713			0.958563	−	0.284880i	\(-0.0919538\pi\)
							0.958563	+	0.284880i	\(0.0919538\pi\)
\(17\)	−1.67053	−	2.89343i	−0.405162	−	0.701761i	0.589178	−	0.808003i	\(-0.299451\pi\)
							−0.994340	+	0.106242i	\(0.966118\pi\)
\(19\)	1.28108	+	2.21890i	0.293901	+	0.509051i	0.974729	−	0.223392i	\(-0.0717131\pi\)
							−0.680828	+	0.732444i	\(0.738380\pi\)
\(23\)	1.47212			0.306958			0.153479	−	0.988152i	\(-0.450952\pi\)
							0.153479	+	0.988152i	\(0.450952\pi\)
\(29\)	−4.78378			−0.888326			−0.444163	−	0.895946i	\(-0.646499\pi\)
							−0.444163	+	0.895946i	\(0.646499\pi\)
\(31\)	0.104327	+	0.180700i	0.0187377	+	0.0324546i	0.875242	−	0.483685i	\(-0.160702\pi\)
							−0.856505	+	0.516140i	\(0.827369\pi\)
\(37\)	−1.35073	−	2.33953i	−0.222058	−	0.384616i	0.733375	−	0.679825i	\(-0.237944\pi\)
							−0.955433	+	0.295209i	\(0.904611\pi\)
\(41\)	3.54165	+	6.13431i	0.553112	+	0.958019i	0.998048	+	0.0624560i	\(0.0198933\pi\)
							−0.444935	+	0.895563i	\(0.646773\pi\)
\(43\)	0.346126	+	0.599508i	0.0527838	+	0.0914242i	0.891210	−	0.453591i	\(-0.149857\pi\)
							−0.838426	+	0.545015i	\(0.816524\pi\)
\(47\)	−11.8922			−1.73466			−0.867328	−	0.497738i	\(-0.834164\pi\)
							−0.867328	+	0.497738i	\(0.834164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.8	✓	128
9.7	even	3		603.2.h.c.439.57	yes	128
67.29	even	3		603.2.h.c.364.57	yes	128
603.565	even	3	inner	603.2.f.c.565.8	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.8	✓	128	1.1	even	1	trivial
603.2.f.c.565.8	yes	128	603.565	even	3	inner
603.2.h.c.364.57	yes	128	67.29	even	3
603.2.h.c.439.57	yes	128	9.7	even	3