Embedded newform 605.2.s.b.16.2

• Label: 605.2.s.b.16.2
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.2
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40958 - 0.276474i) q^{2} +(-1.30445 - 0.947741i) q^{3} +(3.78161 + 0.879377i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(2.88116 + 2.64430i) q^{6} +(-2.42714 - 2.49747i) q^{7} +(-4.30030 - 1.53434i) q^{8} +(-0.123665 - 0.380602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40958	−	0.276474i	−1.70383	−	0.195496i	−0.793269	−	0.608871i	\(-0.791623\pi\)
							−0.910561	+	0.413375i	\(0.864350\pi\)
\(3\)	−1.30445	−	0.947741i	−0.753126	−	0.547178i	0.143668	−	0.989626i	\(-0.454110\pi\)
							−0.896794	+	0.442448i	\(0.854110\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	−2.42714	−	2.49747i	−0.917372	−	0.943953i	0.0812364	−	0.996695i	\(-0.474113\pi\)
−0.998608	+	0.0527416i	\(0.983204\pi\)
\(11\)	2.90012	+	1.60913i	0.874419	+	0.485172i
							\(13\)	0.617779	−	1.02447i	0.171341	−	0.284137i	−0.758812	−	0.651310i	\(-0.774220\pi\)
0.930153	+	0.367173i	\(0.119674\pi\)
\(17\)	3.01209	−	2.46298i	0.730540	−	0.597360i	−0.192847	−	0.981229i	\(-0.561772\pi\)
							0.923388	+	0.383869i	\(0.125409\pi\)
\(19\)	−6.97344	−	0.398755i	−1.59982	−	0.0914808i	−0.765747	−	0.643142i	\(-0.777631\pi\)
							−0.834069	+	0.551661i	\(0.813994\pi\)
\(23\)	0.524211	−	3.64597i	0.109306	−	0.760237i	−0.859271	−	0.511521i	\(-0.829082\pi\)
							0.968576	−	0.248716i	\(-0.0800086\pi\)
\(29\)	3.60204	−	9.25176i	0.668882	−	1.71801i	−0.0267824	−	0.999641i	\(-0.508526\pi\)
							0.695664	−	0.718367i	\(-0.255110\pi\)
\(31\)	4.00238	+	1.69142i	0.718849	+	0.303788i	0.717837	−	0.696211i	\(-0.245132\pi\)
							0.00101199	+	0.999999i	\(0.499678\pi\)
\(37\)	0.533639	−	6.21305i	0.0877297	−	1.02142i	−0.811619	−	0.584187i	\(-0.801413\pi\)
							0.899349	−	0.437232i	\(-0.144041\pi\)
\(41\)	−4.36413	+	2.46454i	−0.681563	+	0.384896i	−0.793291	−	0.608842i	\(-0.791634\pi\)
							0.111728	+	0.993739i	\(0.464361\pi\)
\(43\)	−8.17021	+	2.39899i	−1.24595	+	0.365843i	−0.837246	−	0.546826i	\(-0.815836\pi\)
							−0.408700	+	0.912669i	\(0.634018\pi\)
\(47\)	−2.24232	+	11.0663i	−0.327076	+	1.61418i	0.392489	+	0.919757i	\(0.371614\pi\)
							−0.719565	+	0.694425i	\(0.755659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.2	✓	880
121.53	even	55	inner	605.2.s.b.416.2	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.2	✓	880	1.1	even	1	trivial
605.2.s.b.416.2	yes	880	121.53	even	55	inner