Embedded newform 603.2.t.a.164.5

• Label: 603.2.t.a.164.5
• Level: $603$
• Weight: $2$
• Character: 603.164
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.t (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			164.5
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.48405 q^{2} +(-1.30166 - 1.14266i) q^{3} +4.17049 q^{4} +(0.106943 + 0.185231i) q^{5} +(3.23339 + 2.83842i) q^{6} +(1.82321 - 1.05263i) q^{7} -5.39159 q^{8} +(0.388652 + 2.97472i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48405			−1.75649			−0.878243	−	0.478215i	\(-0.841284\pi\)
							−0.878243	+	0.478215i	\(0.841284\pi\)
\(3\)	−1.30166	−	1.14266i	−0.751515	−	0.659716i
							\(5\)	0.106943	+	0.185231i	0.0478263	+	0.0828376i	0.888948	−	0.458009i	\(-0.151437\pi\)
−0.841121	+	0.540847i	\(0.818104\pi\)
\(7\)	1.82321	−	1.05263i	0.689109	−	0.397857i	−0.114169	−	0.993461i	\(-0.536421\pi\)
							0.803278	+	0.595604i	\(0.203087\pi\)
\(11\)	−3.22282	−	5.58209i	−0.971717	−	1.68306i	−0.690368	−	0.723459i	\(-0.742551\pi\)
							−0.281350	−	0.959605i	\(-0.590782\pi\)
\(13\)	−0.249281	−	0.143922i	−0.0691381	−	0.0399169i	0.465032	−	0.885294i	\(-0.346043\pi\)
							−0.534171	+	0.845377i	\(0.679376\pi\)
\(17\)	3.48565	−	2.01244i	0.845394	−	0.488089i	−0.0136998	−	0.999906i	\(-0.504361\pi\)
							0.859094	+	0.511817i	\(0.171028\pi\)
\(19\)	3.47889	+	6.02562i	0.798113	+	1.38237i	0.920844	+	0.389932i	\(0.127501\pi\)
							−0.122731	+	0.992440i	\(0.539165\pi\)
\(23\)	5.97967	+	3.45237i	1.24685	+	0.719868i	0.970480	−	0.241183i	\(-0.0775356\pi\)
							0.276369	+	0.961052i	\(0.410869\pi\)
\(29\)	−6.31066	+	3.64346i	−1.17186	+	0.676574i	−0.954117	−	0.299433i	\(-0.903203\pi\)
							−0.217743	+	0.976006i	\(0.569869\pi\)
\(31\)		−	4.59350i		−	0.825017i	−0.910954	−	0.412509i	\(-0.864653\pi\)
							0.910954	−	0.412509i	\(-0.135347\pi\)
\(37\)	−4.44216	−	7.69404i	−0.730286	−	1.26489i	−0.956761	−	0.290876i	\(-0.906053\pi\)
							0.226475	−	0.974017i	\(-0.427280\pi\)
\(41\)	−2.73797			−0.427600			−0.213800	−	0.976877i	\(-0.568584\pi\)
							−0.213800	+	0.976877i	\(0.568584\pi\)
\(43\)	4.14428	+	2.39270i	0.631997	+	0.364884i	0.781525	−	0.623874i	\(-0.214442\pi\)
							−0.149528	+	0.988757i	\(0.547775\pi\)
\(47\)	5.94330	−	3.43137i	0.866920	−	0.500517i	0.000596587	−	1.00000i	\(-0.499810\pi\)
							0.866324	+	0.499483i	\(0.166477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.t.a.164.5	yes	132
9.5	odd	6		603.2.k.a.365.5	yes	132
67.38	odd	6		603.2.k.a.38.5	✓	132
603.239	even	6	inner	603.2.t.a.239.5	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.5	✓	132	67.38	odd	6
603.2.k.a.365.5	yes	132	9.5	odd	6
603.2.t.a.164.5	yes	132	1.1	even	1	trivial
603.2.t.a.239.5	yes	132	603.239	even	6	inner