Embedded newform 603.2.t.a.164.8

• Label: 603.2.t.a.164.8
• 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			164.8
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.8

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.34276 q^{2} +(1.66573 - 0.474716i) q^{3} +3.48853 q^{4} +(-1.98526 - 3.43857i) q^{5} +(-3.90240 + 1.11215i) q^{6} +(-1.93666 + 1.11813i) q^{7} -3.48727 q^{8} +(2.54929 - 1.58149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.34276			−1.65658			−0.828291	−	0.560298i	\(-0.810687\pi\)
							−0.828291	+	0.560298i	\(0.810687\pi\)
\(3\)	1.66573	−	0.474716i	0.961708	−	0.274077i
							\(5\)	−1.98526	−	3.43857i	−0.887836	−	1.53778i	−0.842428	−	0.538809i	\(-0.818875\pi\)
−0.0454077	−	0.998969i	\(-0.514459\pi\)
\(7\)	−1.93666	+	1.11813i	−0.731989	+	0.422614i	−0.819149	−	0.573580i	\(-0.805554\pi\)
							0.0871605	+	0.996194i	\(0.472221\pi\)
\(11\)	−0.629405	−	1.09016i	−0.189773	−	0.328696i	0.755402	−	0.655262i	\(-0.227442\pi\)
							−0.945174	+	0.326566i	\(0.894108\pi\)
\(13\)	0.574637	+	0.331767i	0.159376	+	0.0920156i	0.577567	−	0.816344i	\(-0.304002\pi\)
							−0.418191	+	0.908359i	\(0.637336\pi\)
\(17\)	3.65750	−	2.11166i	0.887074	−	0.512152i	0.0140898	−	0.999901i	\(-0.495515\pi\)
							0.872984	+	0.487748i	\(0.162182\pi\)
\(19\)	−1.41410	−	2.44929i	−0.324416	−	0.561905i	0.656978	−	0.753910i	\(-0.271834\pi\)
							−0.981394	+	0.192005i	\(0.938501\pi\)
\(23\)	−3.91513	−	2.26040i	−0.816362	−	0.471327i	0.0327986	−	0.999462i	\(-0.489558\pi\)
							−0.849160	+	0.528135i	\(0.822891\pi\)
\(29\)	−9.15197	+	5.28389i	−1.69948	+	0.981194i	−0.753228	+	0.657759i	\(0.771504\pi\)
							−0.946250	+	0.323435i	\(0.895162\pi\)
\(31\)		−	2.16583i		−	0.388995i	−0.980903	−	0.194497i	\(-0.937692\pi\)
							0.980903	−	0.194497i	\(-0.0623076\pi\)
\(37\)	−3.24082	−	5.61326i	−0.532787	−	0.922814i	−0.999267	−	0.0382826i	\(-0.987811\pi\)
							0.466480	−	0.884532i	\(-0.345522\pi\)
\(41\)	3.58381			0.559698			0.279849	−	0.960044i	\(-0.409716\pi\)
							0.279849	+	0.960044i	\(0.409716\pi\)
\(43\)	6.19894	+	3.57896i	0.945329	+	0.545786i	0.891627	−	0.452771i	\(-0.149565\pi\)
							0.0537021	+	0.998557i	\(0.482898\pi\)
\(47\)	−4.87725	+	2.81588i	−0.711419	+	0.410738i	−0.811586	−	0.584232i	\(-0.801396\pi\)
							0.100167	+	0.994971i	\(0.468062\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.t.a.164.8	yes	132
9.5	odd	6		603.2.k.a.365.8	yes	132
67.38	odd	6		603.2.k.a.38.8	✓	132
603.239	even	6	inner	603.2.t.a.239.8	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.8	✓	132	67.38	odd	6
603.2.k.a.365.8	yes	132	9.5	odd	6
603.2.t.a.164.8	yes	132	1.1	even	1	trivial
603.2.t.a.239.8	yes	132	603.239	even	6	inner