Embedded newform 603.2.f.c.238.12

• Label: 603.2.f.c.238.12
• 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			238.12
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.943426 + 1.63406i) q^{2} +(0.958630 - 1.44258i) q^{3} +(-0.780103 - 1.35118i) q^{4} +(0.159032 - 0.275451i) q^{5} +(1.45286 + 2.92742i) q^{6} -1.45939 q^{7} -0.829824 q^{8} +(-1.16206 - 2.76579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.943426	+	1.63406i	−0.667103	+	1.15546i	0.311608	+	0.950211i	\(0.399132\pi\)
							−0.978711	+	0.205245i	\(0.934201\pi\)
\(3\)	0.958630	−	1.44258i	0.553465	−	0.832872i
							\(5\)	0.159032	−	0.275451i	0.0711213	−	0.123186i	−0.828272	−	0.560327i	\(-0.810676\pi\)
0.899393	+	0.437141i	\(0.144009\pi\)
\(7\)	−1.45939			−0.551596			−0.275798	−	0.961216i	\(-0.588942\pi\)
							−0.275798	+	0.961216i	\(0.588942\pi\)
\(11\)	−3.03244			−0.914315			−0.457158	−	0.889386i	\(-0.651132\pi\)
							−0.457158	+	0.889386i	\(0.651132\pi\)
\(13\)	−1.46001			−0.404933			−0.202467	−	0.979289i	\(-0.564896\pi\)
							−0.202467	+	0.979289i	\(0.564896\pi\)
\(17\)	−1.94051	−	3.36106i	−0.470642	−	0.815176i	0.528794	−	0.848750i	\(-0.322644\pi\)
							−0.999436	+	0.0335739i	\(0.989311\pi\)
\(19\)	1.37446	+	2.38064i	0.315324	+	0.546157i	0.979506	−	0.201414i	\(-0.0645535\pi\)
							−0.664182	+	0.747571i	\(0.731220\pi\)
\(23\)	−7.44385			−1.55215			−0.776075	−	0.630640i	\(-0.782792\pi\)
							−0.776075	+	0.630640i	\(0.782792\pi\)
\(29\)	−1.85606			−0.344662			−0.172331	−	0.985039i	\(-0.555130\pi\)
							−0.172331	+	0.985039i	\(0.555130\pi\)
\(31\)	−3.54417	−	6.13868i	−0.636552	−	1.10254i	−0.986184	−	0.165653i	\(-0.947027\pi\)
							0.349632	−	0.936887i	\(-0.386306\pi\)
\(37\)	−1.18608	−	2.05436i	−0.194991	−	0.337734i	0.751907	−	0.659270i	\(-0.229134\pi\)
							−0.946898	+	0.321535i	\(0.895801\pi\)
\(41\)	−0.0648970	−	0.112405i	−0.0101352	−	0.0175547i	0.860913	−	0.508752i	\(-0.169893\pi\)
							−0.871049	+	0.491197i	\(0.836560\pi\)
\(43\)	−0.906506	−	1.57011i	−0.138241	−	0.239440i	0.788590	−	0.614919i	\(-0.210811\pi\)
							−0.926831	+	0.375479i	\(0.877478\pi\)
\(47\)	1.62555			0.237110			0.118555	−	0.992947i	\(-0.462174\pi\)
							0.118555	+	0.992947i	\(0.462174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.12	✓	128
9.7	even	3		603.2.h.c.439.53	yes	128
67.29	even	3		603.2.h.c.364.53	yes	128
603.565	even	3	inner	603.2.f.c.565.12	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.12	✓	128	1.1	even	1	trivial
603.2.f.c.565.12	yes	128	603.565	even	3	inner
603.2.h.c.364.53	yes	128	67.29	even	3
603.2.h.c.439.53	yes	128	9.7	even	3