Embedded newform 605.2.o.a.34.18

• Label: 605.2.o.a.34.18
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.18
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46552 + 0.210710i) q^{2} -2.35971i q^{3} +(0.184359 - 0.0541326i) q^{4} +(-1.20996 + 1.88042i) q^{5} +(0.497215 + 3.45820i) q^{6} +(0.230168 - 0.358148i) q^{7} +(2.43481 - 1.11194i) q^{8} -2.56825 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46552	+	0.210710i	−1.03628	+	0.148994i	−0.639398	−	0.768876i	\(-0.720816\pi\)
							−0.396880	+	0.917870i	\(0.629907\pi\)
\(3\)		−	2.35971i		−	1.36238i	−0.732106	−	0.681191i	\(-0.761462\pi\)
							0.732106	−	0.681191i	\(-0.238538\pi\)
\(5\)	−1.20996	+	1.88042i	−0.541111	+	0.840951i
							\(7\)	0.230168	−	0.358148i	0.0869953	−	0.135367i	−0.795029	−	0.606572i	\(-0.792544\pi\)
0.882024	+	0.471204i	\(0.156181\pi\)
\(11\)	−0.607045	−	3.26060i	−0.183031	−	0.983107i
							\(13\)	−0.479025	−	1.63141i	−0.132858	−	0.452472i	0.866011	−	0.500025i	\(-0.166676\pi\)
−0.998869	+	0.0475532i	\(0.984858\pi\)
\(17\)	−3.02964	+	2.62520i	−0.734797	+	0.636705i	−0.939669	−	0.342084i	\(-0.888867\pi\)
							0.204873	+	0.978789i	\(0.434322\pi\)
\(19\)	−1.57166	+	1.81379i	−0.360564	+	0.416113i	−0.906829	−	0.421500i	\(-0.861504\pi\)
							0.546265	+	0.837613i	\(0.316049\pi\)
\(23\)	−1.91671	−	2.98246i	−0.399661	−	0.621885i	0.581848	−	0.813297i	\(-0.302330\pi\)
							−0.981510	+	0.191412i	\(0.938693\pi\)
\(29\)	−4.47364	+	5.16286i	−0.830734	+	0.958719i	−0.999638	−	0.0269141i	\(-0.991432\pi\)
							0.168903	+	0.985633i	\(0.445977\pi\)
\(31\)	2.87390	+	0.843854i	0.516168	+	0.151561i	0.529434	−	0.848351i	\(-0.322404\pi\)
							−0.0132655	+	0.999912i	\(0.504223\pi\)
\(37\)	−1.85275	+	6.30990i	−0.304591	+	1.03734i	0.654927	+	0.755692i	\(0.272699\pi\)
							−0.959518	+	0.281649i	\(0.909119\pi\)
\(41\)	1.00134	+	6.96449i	0.156383	+	1.08767i	0.905228	+	0.424925i	\(0.139700\pi\)
							−0.748845	+	0.662745i	\(0.769391\pi\)
\(43\)	4.31275	−	1.96957i	0.657689	−	0.300356i	−0.0584748	−	0.998289i	\(-0.518624\pi\)
							0.716164	+	0.697933i	\(0.245896\pi\)
\(47\)	−9.88425	−	1.42114i	−1.44177	−	0.207295i	−0.623360	−	0.781935i	\(-0.714233\pi\)
							−0.818406	+	0.574640i	\(0.805142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.18	✓	640
5.4	even	2	inner	605.2.o.a.34.47	yes	640
121.89	even	11	inner	605.2.o.a.89.47	yes	640
605.89	even	22	inner	605.2.o.a.89.18	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.18	✓	640	1.1	even	1	trivial
605.2.o.a.34.47	yes	640	5.4	even	2	inner
605.2.o.a.89.18	yes	640	605.89	even	22	inner
605.2.o.a.89.47	yes	640	121.89	even	11	inner