Embedded newform 603.2.y.a.4.5

• Label: 603.2.y.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08130 - 2.36771i) q^{2} +(-1.72152 + 0.190672i) q^{3} +(-3.12714 + 3.60892i) q^{4} +(2.07131 + 0.829229i) q^{5} +(2.31294 + 3.86990i) q^{6} +(-0.856451 - 0.0817811i) q^{7} +(6.93126 + 2.03520i) q^{8} +(2.92729 - 0.656494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08130	−	2.36771i	−0.764594	−	1.67423i	−0.738206	−	0.674575i	\(-0.764327\pi\)
							−0.0263880	−	0.999652i	\(-0.508401\pi\)
\(3\)	−1.72152	+	0.190672i	−0.993922	+	0.110085i
							\(5\)	2.07131	+	0.829229i	0.926320	+	0.370843i	0.785265	−	0.619159i	\(-0.212526\pi\)
0.141054	+	0.990002i	\(0.454951\pi\)
\(7\)	−0.856451	−	0.0817811i	−0.323708	−	0.0309104i	−0.0680628	−	0.997681i	\(-0.521682\pi\)
							−0.255645	+	0.966771i	\(0.582288\pi\)
\(11\)	−1.57369	−	0.630009i	−0.474484	−	0.189955i	0.122080	−	0.992520i	\(-0.461044\pi\)
							−0.596564	+	0.802565i	\(0.703468\pi\)
\(13\)	−0.169679	−	0.161789i	−0.0470605	−	0.0448721i	0.666184	−	0.745787i	\(-0.267927\pi\)
							−0.713244	+	0.700915i	\(0.752775\pi\)
\(17\)	−0.859132	+	0.165584i	−0.208370	+	0.0401600i	−0.292368	−	0.956306i	\(-0.594443\pi\)
							0.0839979	+	0.996466i	\(0.473231\pi\)
\(19\)	−4.41008	−	6.19309i	−1.01174	−	1.42079i	−0.904458	−	0.426563i	\(-0.859724\pi\)
							−0.107283	−	0.994229i	\(-0.534215\pi\)
\(23\)	−0.137450	+	2.88543i	−0.0286603	+	0.601654i	0.938161	+	0.346200i	\(0.112528\pi\)
							−0.966821	+	0.255454i	\(0.917775\pi\)
\(29\)	−2.61056	+	4.52162i	−0.484769	+	0.839644i	−0.999847	−	0.0174990i	\(-0.994430\pi\)
							0.515078	+	0.857143i	\(0.327763\pi\)
\(31\)	−1.08585	−	0.318835i	−0.195025	−	0.0572645i	0.182761	−	0.983157i	\(-0.441496\pi\)
							−0.377786	+	0.925893i	\(0.623315\pi\)
\(37\)	3.93992	−	6.82414i	0.647719	−	1.12188i	−0.335948	−	0.941881i	\(-0.609057\pi\)
							0.983666	−	0.180001i	\(-0.0576101\pi\)
\(41\)	−5.42677	−	6.26282i	−0.847519	−	0.978089i	0.152429	−	0.988314i	\(-0.451290\pi\)
							−0.999948	+	0.0102256i	\(0.996745\pi\)
\(43\)	−1.55359	+	4.48880i	−0.236920	+	0.684535i	0.762380	+	0.647129i	\(0.224031\pi\)
							−0.999300	+	0.0374057i	\(0.988091\pi\)
\(47\)	−0.0255272	+	0.0131602i	−0.00372352	+	0.00191961i	−0.460087	−	0.887874i	\(-0.652182\pi\)
							0.456364	+	0.889793i	\(0.349152\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.5	✓	1320
9.7	even	3		603.2.ba.a.205.5	yes	1320
67.17	even	33		603.2.ba.a.553.5	yes	1320
603.151	even	33	inner	603.2.y.a.151.5	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.5	✓	1320	1.1	even	1	trivial
603.2.y.a.151.5	yes	1320	603.151	even	33	inner
603.2.ba.a.205.5	yes	1320	9.7	even	3
603.2.ba.a.553.5	yes	1320	67.17	even	33