Embedded newform 603.2.ba.a.553.15

• Label: 603.2.ba.a.553.15
• Level: $603$
• Weight: $2$
• Character: 603.553
• 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.ba (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			553.15
Character	\(\chi\)	\(=\)	603.553
Dual form			603.2.ba.a.205.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06266 - 1.49230i) q^{2} +(0.830183 + 1.52013i) q^{3} +(-0.443577 + 1.28163i) q^{4} +(-3.17158 - 2.49416i) q^{5} +(1.38629 - 2.85427i) q^{6} +(-0.875956 + 1.91808i) q^{7} +(-1.13163 + 0.332275i) q^{8} +(-1.62159 + 2.52397i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 14 q^{2} - 19 q^{3} + 52 q^{4} - 14 q^{5} - 25 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{32}{33}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06266	−	1.49230i	−0.751417	−	1.05522i	−0.996324	−	0.0856708i	\(-0.972697\pi\)
							0.244907	−	0.969547i	\(-0.421243\pi\)
\(3\)	0.830183	+	1.52013i	0.479306	+	0.877648i
							\(5\)	−3.17158	−	2.49416i	−1.41837	−	1.11542i	−0.975794	−	0.218691i	\(-0.929821\pi\)
−0.442579	−	0.896730i	\(-0.645936\pi\)
\(7\)	−0.875956	+	1.91808i	−0.331080	+	0.724965i	−0.999828	−	0.0185242i	\(-0.994103\pi\)
							0.668748	+	0.743489i	\(0.266830\pi\)
\(11\)	−0.0625068	+	0.434744i	−0.0188465	+	0.131080i	−0.997072	−	0.0764623i	\(-0.975638\pi\)
							0.978226	+	0.207543i	\(0.0665466\pi\)
\(13\)	3.82529	+	1.12321i	1.06095	+	0.311522i	0.765232	−	0.643755i	\(-0.222624\pi\)
							0.295714	+	0.955277i	\(0.404443\pi\)
\(17\)	4.82859	+	0.930635i	1.17111	+	0.225712i	0.737472	−	0.675377i	\(-0.236019\pi\)
							0.433633	+	0.901089i	\(0.357231\pi\)
\(19\)	3.57019	−	5.01363i	0.819057	−	1.15020i	−0.167352	−	0.985897i	\(-0.553522\pi\)
							0.986409	−	0.164308i	\(-0.0525390\pi\)
\(23\)	2.83974	−	1.82499i	0.592127	−	0.380537i	−0.209990	−	0.977704i	\(-0.567343\pi\)
							0.802117	+	0.597166i	\(0.203707\pi\)
\(29\)	1.71683			0.318807			0.159404	−	0.987213i	\(-0.449043\pi\)
							0.159404	+	0.987213i	\(0.449043\pi\)
\(31\)	1.17892	+	1.12410i	0.211741	+	0.201894i	0.788526	−	0.615001i	\(-0.210845\pi\)
							−0.576785	+	0.816896i	\(0.695693\pi\)
\(37\)	−2.45486	−	4.25195i	−0.403577	−	0.699016i	0.590578	−	0.806981i	\(-0.298900\pi\)
							−0.994155	+	0.107965i	\(0.965567\pi\)
\(41\)	3.19837	+	0.616435i	0.499502	+	0.0962710i	0.432778	−	0.901500i	\(-0.357533\pi\)
							0.0667235	+	0.997772i	\(0.478745\pi\)
\(43\)	4.92036	+	0.948321i	0.750347	+	0.144618i	0.550066	−	0.835121i	\(-0.314602\pi\)
							0.200281	+	0.979739i	\(0.435815\pi\)
\(47\)	−8.85800	+	5.69269i	−1.29207	+	0.830365i	−0.992325	−	0.123654i	\(-0.960539\pi\)
							−0.299747	+	0.954019i	\(0.596902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.ba.a.553.15	yes	1320
9.7	even	3		603.2.y.a.151.15	yes	1320
67.4	even	33		603.2.y.a.4.15	✓	1320
603.205	even	33	inner	603.2.ba.a.205.15	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.15	✓	1320	67.4	even	33
603.2.y.a.151.15	yes	1320	9.7	even	3
603.2.ba.a.205.15	yes	1320	603.205	even	33	inner
603.2.ba.a.553.15	yes	1320	1.1	even	1	trivial