Embedded newform 603.2.h.c.364.12

• Label: 603.2.h.c.364.12
• 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			364.12
Character	\(\chi\)	\(=\)	603.364
Dual form			603.2.h.c.439.12

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.05440 q^{2} +(-0.358844 - 1.69447i) q^{3} +2.22058 q^{4} +(1.67810 - 2.90655i) q^{5} +(0.737211 + 3.48113i) q^{6} +(-0.853884 + 1.47897i) q^{7} -0.453151 q^{8} +(-2.74246 + 1.21610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 3 q^{3} + 130 q^{4} + 5 q^{5} - 6 q^{6} + 4 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.05440			−1.45268			−0.726341	−	0.687334i	\(-0.758781\pi\)
							−0.726341	+	0.687334i	\(0.758781\pi\)
\(3\)	−0.358844	−	1.69447i	−0.207179	−	0.978303i
							\(5\)	1.67810	−	2.90655i	0.750468	−	1.29985i	−0.197128	−	0.980378i	\(-0.563162\pi\)
0.947596	−	0.319471i	\(-0.103505\pi\)
\(7\)	−0.853884	+	1.47897i	−0.322738	+	0.558998i	−0.981052	−	0.193745i	\(-0.937937\pi\)
							0.658314	+	0.752743i	\(0.271270\pi\)
\(11\)	1.34277	−	2.32574i	0.404859	−	0.701237i	−0.589446	−	0.807808i	\(-0.700654\pi\)
							0.994305	+	0.106571i	\(0.0339871\pi\)
\(13\)	−1.74297	−	3.01891i	−0.483412	−	0.837294i	0.516406	−	0.856344i	\(-0.327269\pi\)
							−0.999819	+	0.0190492i	\(0.993936\pi\)
\(17\)	1.19544	−	2.07056i	0.289937	−	0.502185i	−0.683858	−	0.729616i	\(-0.739699\pi\)
							0.973794	+	0.227430i	\(0.0730324\pi\)
\(19\)	3.33198	−	5.77116i	0.764408	−	1.32399i	−0.176150	−	0.984363i	\(-0.556365\pi\)
							0.940559	−	0.339631i	\(-0.110302\pi\)
\(23\)	2.87417	+	4.97821i	0.599307	+	1.03803i	0.992924	+	0.118755i	\(0.0378903\pi\)
							−0.393617	+	0.919274i	\(0.628776\pi\)
\(29\)	1.04205	−	1.80488i	0.193503	−	0.335158i	−0.752905	−	0.658129i	\(-0.771348\pi\)
							0.946409	+	0.322971i	\(0.104682\pi\)
\(31\)	−9.17803			−1.64842			−0.824211	−	0.566283i	\(-0.808381\pi\)
							−0.824211	+	0.566283i	\(0.808381\pi\)
\(37\)	1.04527	−	1.81047i	0.171842	−	0.297639i	−0.767222	−	0.641382i	\(-0.778361\pi\)
							0.939064	+	0.343743i	\(0.111695\pi\)
\(41\)	−7.25386			−1.13286			−0.566431	−	0.824109i	\(-0.691676\pi\)
							−0.566431	+	0.824109i	\(0.691676\pi\)
\(43\)	2.25289	+	3.90212i	0.343563	+	0.595068i	0.985092	−	0.172031i	\(-0.0550329\pi\)
							−0.641529	+	0.767099i	\(0.721700\pi\)
\(47\)	5.36195	−	9.28717i	0.782121	−	1.35467i	−0.148583	−	0.988900i	\(-0.547471\pi\)
							0.930704	−	0.365773i	\(-0.119196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.h.c.364.12	yes	128
9.7	even	3		603.2.f.c.565.53	yes	128
67.37	even	3		603.2.f.c.238.53	✓	128
603.439	even	3	inner	603.2.h.c.439.12	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.53	✓	128	67.37	even	3
603.2.f.c.565.53	yes	128	9.7	even	3
603.2.h.c.364.12	yes	128	1.1	even	1	trivial
603.2.h.c.439.12	yes	128	603.439	even	3	inner