Embedded newform 605.2.o.a.34.20

• Label: 605.2.o.a.34.20
• 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			34.20
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19871 + 0.172349i) q^{2} -1.61947i q^{3} +(-0.511781 + 0.150273i) q^{4} +(1.81795 + 1.30195i) q^{5} +(0.279113 + 1.94128i) q^{6} +(0.386764 - 0.601816i) q^{7} +(2.79078 - 1.27450i) q^{8} +0.377319 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19871	+	0.172349i	−0.847617	+	0.121869i	−0.552418	−	0.833567i	\(-0.686295\pi\)
							−0.295199	+	0.955436i	\(0.595386\pi\)
\(3\)		−	1.61947i		−	0.935001i	−0.883993	−	0.467501i	\(-0.845155\pi\)
							0.883993	−	0.467501i	\(-0.154845\pi\)
\(5\)	1.81795	+	1.30195i	0.813011	+	0.582248i
							\(7\)	0.386764	−	0.601816i	0.146183	−	0.227465i	−0.760439	−	0.649410i	\(-0.775016\pi\)
0.906622	+	0.421945i	\(0.138652\pi\)
\(11\)	−3.27925	−	0.496516i	−0.988731	−	0.149705i
							\(13\)	1.62901	+	5.54788i	0.451805	+	1.53871i	0.799246	+	0.601004i	\(0.205233\pi\)
−0.347441	+	0.937702i	\(0.612949\pi\)
\(17\)	1.75240	−	1.51846i	0.425019	−	0.368281i	−0.415929	−	0.909397i	\(-0.636543\pi\)
							0.840948	+	0.541116i	\(0.181998\pi\)
\(19\)	−0.158266	+	0.182648i	−0.0363086	+	0.0419024i	−0.773613	−	0.633658i	\(-0.781553\pi\)
							0.737305	+	0.675560i	\(0.236098\pi\)
\(23\)	3.36877	+	5.24190i	0.702436	+	1.09301i	0.990779	+	0.135485i	\(0.0432593\pi\)
							−0.288343	+	0.957527i	\(0.593104\pi\)
\(29\)	−0.971643	+	1.12134i	−0.180429	+	0.208227i	−0.838759	−	0.544504i	\(-0.816718\pi\)
							0.658329	+	0.752730i	\(0.271264\pi\)
\(31\)	−0.937958	−	0.275409i	−0.168462	−	0.0494650i	0.196414	−	0.980521i	\(-0.437070\pi\)
							−0.364876	+	0.931056i	\(0.618889\pi\)
\(37\)	2.82289	−	9.61389i	0.464081	−	1.58051i	−0.312129	−	0.950040i	\(-0.601042\pi\)
							0.776210	−	0.630475i	\(-0.217140\pi\)
\(41\)	1.01692	+	7.07285i	0.158817	+	1.10459i	0.900819	+	0.434196i	\(0.142967\pi\)
							−0.742002	+	0.670398i	\(0.766124\pi\)
\(43\)	6.88797	−	3.14563i	1.05041	−	0.479704i	0.186027	−	0.982545i	\(-0.440439\pi\)
							0.864379	+	0.502840i	\(0.167712\pi\)
\(47\)	11.7348	+	1.68721i	1.71170	+	0.246105i	0.927344	−	0.374211i	\(-0.122086\pi\)
							0.784353	+	0.620315i	\(0.212995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.20	✓	640
5.4	even	2	inner	605.2.o.a.34.45	yes	640
121.89	even	11	inner	605.2.o.a.89.45	yes	640
605.89	even	22	inner	605.2.o.a.89.20	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.20	✓	640	1.1	even	1	trivial
605.2.o.a.34.45	yes	640	5.4	even	2	inner
605.2.o.a.89.20	yes	640	605.89	even	22	inner
605.2.o.a.89.45	yes	640	121.89	even	11	inner