Embedded newform 603.2.y.a.4.15

• Label: 603.2.y.a.4.15
• Level: $603$
• Weight: $2$
• Character: 603.4
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• 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			4.15
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.761040 - 1.66645i) q^{2} +(-1.03789 + 1.38665i) q^{3} +(-0.888137 + 1.02496i) q^{4} +(3.74579 + 1.49959i) q^{5} +(3.10064 + 0.674293i) q^{6} +(2.09908 + 0.200438i) q^{7} +(-1.13163 - 0.332275i) q^{8} +(-0.845572 - 2.87837i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.761040	−	1.66645i	−0.538137	−	1.17835i	−0.962105	−	0.272678i	\(-0.912091\pi\)
							0.423969	−	0.905677i	\(-0.360637\pi\)
\(3\)	−1.03789	+	1.38665i	−0.599226	+	0.800580i
							\(5\)	3.74579	+	1.49959i	1.67517	+	0.670637i	0.997880	−	0.0650805i	\(-0.0207304\pi\)
0.677289	+	0.735717i	\(0.263155\pi\)
\(7\)	2.09908	+	0.200438i	0.793378	+	0.0757584i	0.483872	−	0.875139i	\(-0.339230\pi\)
							0.309506	+	0.950897i	\(0.399836\pi\)
\(11\)	0.407753	+	0.163240i	0.122942	+	0.0492186i	0.432318	−	0.901721i	\(-0.357696\pi\)
							−0.309376	+	0.950940i	\(0.600120\pi\)
\(13\)	−2.88537	−	2.75120i	−0.800258	−	0.763045i	0.174892	−	0.984588i	\(-0.444042\pi\)
							−0.975151	+	0.221543i	\(0.928891\pi\)
\(17\)	4.82859	−	0.930635i	1.17111	−	0.225712i	0.433633	−	0.901089i	\(-0.357231\pi\)
							0.737472	+	0.675377i	\(0.236019\pi\)
\(19\)	3.57019	+	5.01363i	0.819057	+	1.15020i	0.986409	+	0.164308i	\(0.0525390\pi\)
							−0.167352	+	0.985897i	\(0.553522\pi\)
\(23\)	0.160618	−	3.37179i	0.0334912	−	0.703066i	−0.918220	−	0.396071i	\(-0.870373\pi\)
							0.951711	−	0.306995i	\(-0.0993235\pi\)
\(29\)	−0.858415	+	1.48682i	−0.159404	+	0.276095i	−0.934654	−	0.355559i	\(-0.884290\pi\)
							0.775250	+	0.631654i	\(0.217624\pi\)
\(31\)	−1.56296	−	0.458927i	−0.280716	−	0.0824257i	0.138344	−	0.990384i	\(-0.455822\pi\)
							−0.419060	+	0.907959i	\(0.637640\pi\)
\(37\)	−2.45486	+	4.25195i	−0.403577	+	0.699016i	−0.994155	−	0.107965i	\(-0.965567\pi\)
							0.590578	+	0.806981i	\(0.298900\pi\)
\(41\)	−2.13303	−	2.46165i	−0.333124	−	0.384446i	0.564333	−	0.825547i	\(-0.309133\pi\)
							−0.897457	+	0.441102i	\(0.854588\pi\)
\(43\)	−1.63891	+	4.73531i	−0.249931	+	0.722129i	0.748338	+	0.663318i	\(0.230852\pi\)
							−0.998269	+	0.0588111i	\(0.981269\pi\)
\(47\)	9.35902	−	4.82491i	1.36515	−	0.703785i	0.389075	−	0.921206i	\(-0.372795\pi\)
							0.976078	+	0.217420i	\(0.0697643\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.15	✓	1320	1.1	even	1	trivial
603.2.y.a.151.15	yes	1320	603.151	even	33	inner
603.2.ba.a.205.15	yes	1320	9.7	even	3
603.2.ba.a.553.15	yes	1320	67.17	even	33