Embedded newform 603.2.l.a.200.18

• Label: 603.2.l.a.200.18
• Level: $603$
• Weight: $2$
• Character: 603.200
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• 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			200.18
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.762360 + 1.32045i) q^{2} +(0.182202 + 1.72244i) q^{3} +(-0.162384 - 0.281258i) q^{4} +(-0.908294 - 1.57321i) q^{5} +(-2.41329 - 1.07253i) q^{6} +(-2.60798 - 1.50572i) q^{7} -2.55426 q^{8} +(-2.93360 + 0.627664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.762360	+	1.32045i	−0.539070	+	0.933696i	0.459885	+	0.887979i	\(0.347891\pi\)
							−0.998954	+	0.0457175i	\(0.985443\pi\)
\(3\)	0.182202	+	1.72244i	0.105194	+	0.994452i
							\(5\)	−0.908294	−	1.57321i	−0.406201	−	0.703561i	0.588259	−	0.808672i	\(-0.299813\pi\)
−0.994460	+	0.105111i	\(0.966480\pi\)
\(7\)	−2.60798	−	1.50572i	−0.985724	−	0.569108i	−0.0817309	−	0.996654i	\(-0.526045\pi\)
							−0.903994	+	0.427546i	\(0.859378\pi\)
\(11\)	−0.0501649	+	0.0868881i	−0.0151253	+	0.0261978i	−0.873489	−	0.486844i	\(-0.838148\pi\)
							0.858364	+	0.513042i	\(0.171481\pi\)
\(13\)	0.0950627	−	0.0548845i	0.0263657	−	0.0152222i	−0.486759	−	0.873536i	\(-0.661821\pi\)
							0.513125	+	0.858314i	\(0.328488\pi\)
\(17\)		−	0.775491i		−	0.188084i	−0.995568	−	0.0940421i	\(-0.970021\pi\)
							0.995568	−	0.0940421i	\(-0.0299788\pi\)
\(19\)	4.64667			1.06602			0.533010	−	0.846109i	\(-0.321061\pi\)
							0.533010	+	0.846109i	\(0.321061\pi\)
\(23\)	−0.499626	+	0.288459i	−0.104179	+	0.0601479i	−0.551184	−	0.834383i	\(-0.685824\pi\)
							0.447005	+	0.894531i	\(0.352491\pi\)
\(29\)	−5.56159	−	3.21099i	−1.03276	−	0.596265i	−0.114987	−	0.993367i	\(-0.536683\pi\)
							−0.917775	+	0.397102i	\(0.870016\pi\)
\(31\)	3.17176	−	1.83122i	0.569665	−	0.328896i	−0.187351	−	0.982293i	\(-0.559990\pi\)
							0.757015	+	0.653397i	\(0.226657\pi\)
\(37\)	−3.82696			−0.629148			−0.314574	−	0.949233i	\(-0.601862\pi\)
							−0.314574	+	0.949233i	\(0.601862\pi\)
\(41\)	−1.29267	−	2.23897i	−0.201881	−	0.349668i	0.747254	−	0.664539i	\(-0.231372\pi\)
							−0.949134	+	0.314871i	\(0.898039\pi\)
\(43\)	0.564653	+	0.326003i	0.0861088	+	0.0497149i	0.542436	−	0.840097i	\(-0.317502\pi\)
							−0.456327	+	0.889812i	\(0.650835\pi\)
\(47\)	−10.4556	−	6.03655i	−1.52511	−	0.880521i	−0.999557	−	0.0297577i	\(-0.990526\pi\)
							−0.525549	−	0.850763i	\(-0.676140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.18	✓	132
9.5	odd	6	inner	603.2.l.a.401.49	yes	132
67.66	odd	2	inner	603.2.l.a.200.49	yes	132
603.401	even	6	inner	603.2.l.a.401.18	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.18	✓	132	1.1	even	1	trivial
603.2.l.a.200.49	yes	132	67.66	odd	2	inner
603.2.l.a.401.18	yes	132	603.401	even	6	inner
603.2.l.a.401.49	yes	132	9.5	odd	6	inner