Embedded newform 603.2.l.a.200.15

• Label: 603.2.l.a.200.15
• 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			200.15
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.838601 + 1.45250i) q^{2} +(1.47894 + 0.901512i) q^{3} +(-0.406503 - 0.704084i) q^{4} +(-1.89359 - 3.27979i) q^{5} +(-2.54969 + 1.39216i) q^{6} +(2.42055 + 1.39750i) q^{7} -1.99083 q^{8} +(1.37455 + 2.66657i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.838601	+	1.45250i	−0.592980	+	1.02707i	0.400848	+	0.916145i	\(0.368716\pi\)
							−0.993828	+	0.110928i	\(0.964618\pi\)
\(3\)	1.47894	+	0.901512i	0.853869	+	0.520488i
							\(5\)	−1.89359	−	3.27979i	−0.846839	−	1.46677i	−0.884015	−	0.467459i	\(-0.845170\pi\)
0.0371759	−	0.999309i	\(-0.488164\pi\)
\(7\)	2.42055	+	1.39750i	0.914882	+	0.528207i	0.881999	−	0.471252i	\(-0.156198\pi\)
							0.0328831	+	0.999459i	\(0.489531\pi\)
\(11\)	2.88410	−	4.99541i	0.869589	−	1.50617i	0.00717083	−	0.999974i	\(-0.497717\pi\)
							0.862418	−	0.506197i	\(-0.168949\pi\)
\(13\)	5.23722	−	3.02371i	1.45254	−	0.838627i	0.453919	−	0.891043i	\(-0.350026\pi\)
							0.998625	+	0.0524166i	\(0.0166924\pi\)
\(17\)			6.31104i			1.53065i	0.643643	+	0.765326i	\(0.277422\pi\)
							−0.643643	+	0.765326i	\(0.722578\pi\)
\(19\)	−3.27709			−0.751816			−0.375908	−	0.926657i	\(-0.622669\pi\)
							−0.375908	+	0.926657i	\(0.622669\pi\)
\(23\)	5.67771	−	3.27803i	1.18388	−	0.683516i	0.226974	−	0.973901i	\(-0.427117\pi\)
							0.956910	+	0.290385i	\(0.0937834\pi\)
\(29\)	−2.07196	−	1.19625i	−0.384753	−	0.222137i	0.295131	−	0.955457i	\(-0.404637\pi\)
							−0.679884	+	0.733320i	\(0.737970\pi\)
\(31\)	−2.63600	+	1.52190i	−0.473440	+	0.273341i	−0.717679	−	0.696374i	\(-0.754795\pi\)
							0.244238	+	0.969715i	\(0.421462\pi\)
\(37\)	6.86919			1.12929			0.564644	−	0.825335i	\(-0.309014\pi\)
							0.564644	+	0.825335i	\(0.309014\pi\)
\(41\)	4.07331	+	7.05517i	0.636144	+	1.10183i	0.986272	+	0.165131i	\(0.0528048\pi\)
							−0.350128	+	0.936702i	\(0.613862\pi\)
\(43\)	−2.01764	−	1.16488i	−0.307687	−	0.177643i	0.338204	−	0.941073i	\(-0.390181\pi\)
							−0.645891	+	0.763430i	\(0.723514\pi\)
\(47\)	−1.07031	−	0.617945i	−0.156121	−	0.0901366i	0.419904	−	0.907568i	\(-0.362064\pi\)
							−0.576025	+	0.817432i	\(0.695397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.15	✓	132
9.5	odd	6	inner	603.2.l.a.401.52	yes	132
67.66	odd	2	inner	603.2.l.a.200.52	yes	132
603.401	even	6	inner	603.2.l.a.401.15	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.15	✓	132	1.1	even	1	trivial
603.2.l.a.200.52	yes	132	67.66	odd	2	inner
603.2.l.a.401.15	yes	132	603.401	even	6	inner
603.2.l.a.401.52	yes	132	9.5	odd	6	inner