Embedded newform 603.2.l.a.200.11

• Label: 603.2.l.a.200.11
• Level: $603$
• Weight: $2$
• Character: 603.200
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			200.11
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00472 + 1.74023i) q^{2} +(-1.02992 - 1.39258i) q^{3} +(-1.01894 - 1.76486i) q^{4} +(-0.0100438 - 0.0173963i) q^{5} +(3.45819 - 0.393147i) q^{6} +(-1.41764 - 0.818474i) q^{7} +0.0761201 q^{8} +(-0.878531 + 2.86848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00472	+	1.74023i	−0.710447	+	1.23053i	0.254242	+	0.967141i	\(0.418174\pi\)
							−0.964689	+	0.263390i	\(0.915159\pi\)
\(3\)	−1.02992	−	1.39258i	−0.594624	−	0.804004i
							\(5\)	−0.0100438	−	0.0173963i	−0.00449171	−	0.00777987i	0.863771	−	0.503885i	\(-0.168096\pi\)
−0.868263	+	0.496105i	\(0.834763\pi\)
\(7\)	−1.41764	−	0.818474i	−0.535817	−	0.309354i	0.207565	−	0.978221i	\(-0.433446\pi\)
							−0.743382	+	0.668867i	\(0.766779\pi\)
\(11\)	−0.470956	+	0.815719i	−0.141998	+	0.245948i	−0.928249	−	0.371959i	\(-0.878686\pi\)
							0.786251	+	0.617908i	\(0.212019\pi\)
\(13\)	0.237653	−	0.137209i	0.0659129	−	0.0380549i	−0.466681	−	0.884426i	\(-0.654551\pi\)
							0.532594	+	0.846371i	\(0.321217\pi\)
\(17\)			2.80676i			0.680739i	0.940292	+	0.340369i	\(0.110552\pi\)
							−0.940292	+	0.340369i	\(0.889448\pi\)
\(19\)	−0.780266			−0.179005			−0.0895027	−	0.995987i	\(-0.528528\pi\)
							−0.0895027	+	0.995987i	\(0.528528\pi\)
\(23\)	4.33114	−	2.50058i	0.903105	−	0.521408i	0.0248987	−	0.999690i	\(-0.492074\pi\)
							0.878206	+	0.478282i	\(0.158740\pi\)
\(29\)	5.71522	+	3.29968i	1.06129	+	0.612736i	0.925789	−	0.378042i	\(-0.123402\pi\)
							0.135501	+	0.990777i	\(0.456736\pi\)
\(31\)	8.04375	−	4.64406i	1.44470	−	0.834098i	0.446542	−	0.894763i	\(-0.352655\pi\)
							0.998158	+	0.0606648i	\(0.0193221\pi\)
\(37\)	−3.48778			−0.573387			−0.286693	−	0.958022i	\(-0.592556\pi\)
							−0.286693	+	0.958022i	\(0.592556\pi\)
\(41\)	3.70462	+	6.41659i	0.578564	+	1.00210i	0.995644	+	0.0932331i	\(0.0297202\pi\)
							−0.417080	+	0.908870i	\(0.636946\pi\)
\(43\)	2.44686	+	1.41270i	0.373143	+	0.215434i	0.674831	−	0.737973i	\(-0.264217\pi\)
							−0.301688	+	0.953407i	\(0.597550\pi\)
\(47\)	−1.02384	−	0.591117i	−0.149343	−	0.0862233i	0.423466	−	0.905912i	\(-0.360813\pi\)
							−0.572809	+	0.819689i	\(0.694147\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.11	✓	132
9.5	odd	6	inner	603.2.l.a.401.56	yes	132
67.66	odd	2	inner	603.2.l.a.200.56	yes	132
603.401	even	6	inner	603.2.l.a.401.11	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.11	✓	132	1.1	even	1	trivial
603.2.l.a.200.56	yes	132	67.66	odd	2	inner
603.2.l.a.401.11	yes	132	603.401	even	6	inner
603.2.l.a.401.56	yes	132	9.5	odd	6	inner