Embedded newform 603.2.y.a.4.10

• Label: 603.2.y.a.4.10
• Level: $603$
• Weight: $2$
• Character: 603.4
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $1320$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.10
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.929661 - 2.03567i) q^{2} +(-0.481843 - 1.66368i) q^{3} +(-1.96997 + 2.27347i) q^{4} +(2.90470 + 1.16287i) q^{5} +(-2.93875 + 2.52753i) q^{6} +(2.26585 + 0.216363i) q^{7} +(2.16493 + 0.635681i) q^{8} +(-2.53565 + 1.60327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.929661	−	2.03567i	−0.657369	−	1.43944i	−0.884953	−	0.465679i	\(-0.845810\pi\)
							0.227584	−	0.973758i	\(-0.426917\pi\)
\(3\)	−0.481843	−	1.66368i	−0.278192	−	0.960525i
							\(5\)	2.90470	+	1.16287i	1.29902	+	0.520050i	0.915067	−	0.403303i	\(-0.132138\pi\)
0.383954	+	0.923352i	\(0.374562\pi\)
\(7\)	2.26585	+	0.216363i	0.856412	+	0.0817774i	0.514018	−	0.857779i	\(-0.328156\pi\)
							0.342394	+	0.939557i	\(0.388762\pi\)
\(11\)	1.25985	+	0.504367i	0.379859	+	0.152072i	0.553731	−	0.832696i	\(-0.313204\pi\)
							−0.173872	+	0.984768i	\(0.555628\pi\)
\(13\)	4.08355	+	3.89366i	1.13257	+	1.07991i	0.996007	+	0.0892763i	\(0.0284554\pi\)
							0.136567	+	0.990631i	\(0.456393\pi\)
\(17\)	4.32861	−	0.834272i	1.04984	−	0.202341i	0.364982	−	0.931015i	\(-0.381075\pi\)
							0.684861	+	0.728674i	\(0.259863\pi\)
\(19\)	−2.45923	−	3.45351i	−0.564187	−	0.792290i	0.429667	−	0.902987i	\(-0.358631\pi\)
							−0.993854	+	0.110697i	\(0.964692\pi\)
\(23\)	0.265152	−	5.56622i	0.0552879	−	1.16064i	−0.785355	−	0.619046i	\(-0.787519\pi\)
							0.840643	−	0.541590i	\(-0.182178\pi\)
\(29\)	2.85910	−	4.95211i	0.530922	−	0.919585i	−0.468426	−	0.883503i	\(-0.655179\pi\)
							0.999349	−	0.0360821i	\(-0.0114878\pi\)
\(31\)	−2.20575	−	0.647667i	−0.396164	−	0.116324i	0.0775814	−	0.996986i	\(-0.475280\pi\)
							−0.473746	+	0.880662i	\(0.657098\pi\)
\(37\)	−1.93431	+	3.35033i	−0.317999	+	0.550790i	−0.980070	−	0.198650i	\(-0.936344\pi\)
							0.662072	+	0.749441i	\(0.269677\pi\)
\(41\)	5.06035	+	5.83995i	0.790293	+	0.912047i	0.997807	−	0.0661850i	\(-0.0210828\pi\)
							−0.207514	+	0.978232i	\(0.566537\pi\)
\(43\)	−1.91599	+	5.53589i	−0.292186	+	0.844216i	0.699396	+	0.714734i	\(0.253452\pi\)
							−0.991582	+	0.129481i	\(0.958669\pi\)
\(47\)	−5.80498	+	2.99267i	−0.846743	+	0.436526i	−0.826277	−	0.563264i	\(-0.809545\pi\)
							−0.0204660	+	0.999791i	\(0.506515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.10	✓	1320
9.7	even	3		603.2.ba.a.205.10	yes	1320
67.17	even	33		603.2.ba.a.553.10	yes	1320
603.151	even	33	inner	603.2.y.a.151.10	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.10	✓	1320	1.1	even	1	trivial
603.2.y.a.151.10	yes	1320	603.151	even	33	inner
603.2.ba.a.205.10	yes	1320	9.7	even	3
603.2.ba.a.553.10	yes	1320	67.17	even	33