Embedded newform 603.2.l.a.200.14

• Label: 603.2.l.a.200.14
• 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			200.14
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.930733 + 1.61208i) q^{2} +(1.58684 + 0.694223i) q^{3} +(-0.732528 - 1.26878i) q^{4} +(-0.136455 - 0.236347i) q^{5} +(-2.59606 + 1.91197i) q^{6} +(-2.43107 - 1.40358i) q^{7} -0.995781 q^{8} +(2.03611 + 2.20324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.930733	+	1.61208i	−0.658128	+	1.13991i	0.322972	+	0.946408i	\(0.395318\pi\)
							−0.981100	+	0.193502i	\(0.938015\pi\)
\(3\)	1.58684	+	0.694223i	0.916161	+	0.400810i
							\(5\)	−0.136455	−	0.236347i	−0.0610245	−	0.105697i	0.833899	−	0.551917i	\(-0.186103\pi\)
−0.894924	+	0.446219i	\(0.852770\pi\)
\(7\)	−2.43107	−	1.40358i	−0.918858	−	0.530503i	−0.0355872	−	0.999367i	\(-0.511330\pi\)
							−0.883270	+	0.468864i	\(0.844663\pi\)
\(11\)	−0.400135	+	0.693055i	−0.120645	+	0.208964i	−0.920022	−	0.391866i	\(-0.871830\pi\)
							0.799377	+	0.600830i	\(0.205163\pi\)
\(13\)	−4.94878	+	2.85718i	−1.37255	+	0.792440i	−0.991248	−	0.132013i	\(-0.957856\pi\)
							−0.381298	+	0.924452i	\(0.624523\pi\)
\(17\)			4.65218i			1.12832i	0.825666	+	0.564160i	\(0.190800\pi\)
							−0.825666	+	0.564160i	\(0.809200\pi\)
\(19\)	−5.47623			−1.25633			−0.628167	−	0.778079i	\(-0.716194\pi\)
							−0.628167	+	0.778079i	\(0.716194\pi\)
\(23\)	−3.37705	+	1.94974i	−0.704165	+	0.406550i	−0.808897	−	0.587951i	\(-0.799935\pi\)
							0.104732	+	0.994500i	\(0.466602\pi\)
\(29\)	2.40373	+	1.38779i	0.446361	+	0.257707i	0.706292	−	0.707921i	\(-0.250367\pi\)
							−0.259931	+	0.965627i	\(0.583700\pi\)
\(31\)	−6.42152	+	3.70746i	−1.15334	+	0.665880i	−0.949699	−	0.313166i	\(-0.898611\pi\)
							−0.203640	+	0.979046i	\(0.565277\pi\)
\(37\)	−5.97328			−0.982001			−0.491000	−	0.871159i	\(-0.663369\pi\)
							−0.491000	+	0.871159i	\(0.663369\pi\)
\(41\)	−0.699267	−	1.21117i	−0.109207	−	0.189152i	0.806242	−	0.591586i	\(-0.201498\pi\)
							−0.915449	+	0.402433i	\(0.868165\pi\)
\(43\)	8.51136	+	4.91403i	1.29797	+	0.749383i	0.980053	−	0.198736i	\(-0.0636835\pi\)
							0.317916	+	0.948119i	\(0.397017\pi\)
\(47\)	11.6617	+	6.73287i	1.70103	+	0.982090i	0.944721	+	0.327876i	\(0.106333\pi\)
							0.756310	+	0.654214i	\(0.227000\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.14	✓	132
9.5	odd	6	inner	603.2.l.a.401.53	yes	132
67.66	odd	2	inner	603.2.l.a.200.53	yes	132
603.401	even	6	inner	603.2.l.a.401.14	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.14	✓	132	1.1	even	1	trivial
603.2.l.a.200.53	yes	132	67.66	odd	2	inner
603.2.l.a.401.14	yes	132	603.401	even	6	inner
603.2.l.a.401.53	yes	132	9.5	odd	6	inner