Embedded newform 603.2.y.a.4.19

• Label: 603.2.y.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.607142 - 1.32945i) q^{2} +(1.69190 + 0.370779i) q^{3} +(-0.0891071 + 0.102835i) q^{4} +(2.40251 + 0.961822i) q^{5} +(-0.534289 - 2.47442i) q^{6} +(3.57597 + 0.341463i) q^{7} +(-2.61384 - 0.767493i) q^{8} +(2.72505 + 1.25464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.607142	−	1.32945i	−0.429314	−	0.940066i	−0.993438	−	0.114376i	\(-0.963513\pi\)
							0.564124	−	0.825690i	\(-0.309214\pi\)
\(3\)	1.69190	+	0.370779i	0.976818	+	0.214069i
							\(5\)	2.40251	+	0.961822i	1.07444	+	0.430140i	0.840332	−	0.542072i	\(-0.182360\pi\)
0.234105	+	0.972211i	\(0.424784\pi\)
\(7\)	3.57597	+	0.341463i	1.35159	+	0.129061i	0.745590	−	0.666405i	\(-0.232168\pi\)
							0.605998	+	0.795466i	\(0.292774\pi\)
\(11\)	−3.68684	−	1.47599i	−1.11162	−	0.445027i	−0.258120	−	0.966113i	\(-0.583103\pi\)
							−0.853504	+	0.521086i	\(0.825527\pi\)
\(13\)	−0.124004	−	0.118238i	−0.0343925	−	0.0327932i	0.672688	−	0.739926i	\(-0.265140\pi\)
							−0.707080	+	0.707133i	\(0.749988\pi\)
\(17\)	0.840637	−	0.162020i	0.203884	−	0.0392955i	−0.0862863	−	0.996270i	\(-0.527500\pi\)
							0.290171	+	0.956975i	\(0.406288\pi\)
\(19\)	−2.58580	−	3.63124i	−0.593222	−	0.833064i	0.403414	−	0.915018i	\(-0.367823\pi\)
							−0.996636	+	0.0819537i	\(0.973884\pi\)
\(23\)	−0.309464	+	6.49644i	−0.0645276	+	1.35460i	0.701902	+	0.712274i	\(0.252334\pi\)
							−0.766430	+	0.642328i	\(0.777969\pi\)
\(29\)	−0.954330	+	1.65295i	−0.177215	+	0.306945i	−0.940925	−	0.338614i	\(-0.890042\pi\)
							0.763711	+	0.645559i	\(0.223375\pi\)
\(31\)	3.88298	+	1.14015i	0.697404	+	0.204776i	0.611171	−	0.791499i	\(-0.290699\pi\)
							0.0862333	+	0.996275i	\(0.472517\pi\)
\(37\)	−1.20534	+	2.08771i	−0.198157	+	0.343218i	−0.947931	−	0.318476i	\(-0.896829\pi\)
							0.749774	+	0.661694i	\(0.230162\pi\)
\(41\)	−7.87014	−	9.08262i	−1.22911	−	1.41847i	−0.875600	−	0.483037i	\(-0.839534\pi\)
							−0.353509	−	0.935431i	\(-0.615012\pi\)
\(43\)	−0.0721752	+	0.208537i	−0.0110066	+	0.0318015i	−0.950370	−	0.311121i	\(-0.899296\pi\)
							0.939364	+	0.342922i	\(0.111417\pi\)
\(47\)	3.98847	−	2.05620i	0.581779	−	0.299928i	−0.142103	−	0.989852i	\(-0.545386\pi\)
							0.723882	+	0.689924i	\(0.242356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.19	✓	1320
9.7	even	3		603.2.ba.a.205.19	yes	1320
67.17	even	33		603.2.ba.a.553.19	yes	1320
603.151	even	33	inner	603.2.y.a.151.19	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.19	✓	1320	1.1	even	1	trivial
603.2.y.a.151.19	yes	1320	603.151	even	33	inner
603.2.ba.a.205.19	yes	1320	9.7	even	3
603.2.ba.a.553.19	yes	1320	67.17	even	33