Embedded newform 605.2.s.b.16.15

• Label: 605.2.s.b.16.15
• 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			16.15
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27289 + 0.146051i) q^{2} +(-1.27498 - 0.926331i) q^{3} +(-0.349093 - 0.0811782i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(-1.48763 - 1.36533i) q^{6} +(1.90619 + 1.96142i) q^{7} +(-2.84597 - 1.01544i) q^{8} +(-0.159553 - 0.491055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27289	+	0.146051i	0.900073	+	0.103274i	0.551661	−	0.834068i	\(-0.313994\pi\)
							0.348411	+	0.937342i	\(0.386721\pi\)
\(3\)	−1.27498	−	0.926331i	−0.736113	−	0.534817i	0.155379	−	0.987855i	\(-0.450340\pi\)
							−0.891491	+	0.453038i	\(0.850340\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	1.90619	+	1.96142i	0.720470	+	0.741346i	0.974140	−	0.225944i	\(-0.0725468\pi\)
−0.253670	+	0.967291i	\(0.581638\pi\)
\(11\)	−3.25961	−	0.612318i	−0.982810	−	0.184621i
							\(13\)	0.611190	−	1.01354i	0.169513	−	0.281107i	−0.759979	−	0.649948i	\(-0.774791\pi\)
0.929492	+	0.368841i	\(0.120245\pi\)
\(17\)	−5.16855	+	4.22630i	−1.25356	+	1.02503i	−0.255250	+	0.966875i	\(0.582158\pi\)
							−0.998308	+	0.0581539i	\(0.981479\pi\)
\(19\)	−4.96239	−	0.283760i	−1.13845	−	0.0650990i	−0.522325	−	0.852746i	\(-0.674935\pi\)
							−0.616126	+	0.787648i	\(0.711299\pi\)
\(23\)	0.180859	−	1.25790i	0.0377116	−	0.262290i	−0.962239	−	0.272205i	\(-0.912247\pi\)
							0.999951	+	0.00991515i	\(0.00315614\pi\)
\(29\)	−3.21753	+	8.26416i	−0.597481	+	1.53462i	0.230040	+	0.973181i	\(0.426114\pi\)
							−0.827521	+	0.561435i	\(0.810249\pi\)
\(31\)	−4.08251	−	1.72528i	−0.733241	−	0.309870i	−0.00951197	−	0.999955i	\(-0.503028\pi\)
							−0.723729	+	0.690085i	\(0.757573\pi\)
\(37\)	0.590710	−	6.87751i	0.0971121	−	1.13066i	−0.771859	−	0.635793i	\(-0.780673\pi\)
							0.868971	−	0.494862i	\(-0.164782\pi\)
\(41\)	8.52192	−	4.81255i	1.33090	−	0.751594i	0.348310	−	0.937379i	\(-0.386756\pi\)
							0.982590	+	0.185785i	\(0.0594829\pi\)
\(43\)	−5.41645	+	1.59041i	−0.826000	+	0.242536i	−0.667298	−	0.744791i	\(-0.732549\pi\)
							−0.158702	+	0.987326i	\(0.550731\pi\)
\(47\)	−0.287985	+	1.42126i	−0.0420069	+	0.207312i	−0.995589	−	0.0938206i	\(-0.970092\pi\)
							0.953582	+	0.301133i	\(0.0973647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.15	✓	880
121.53	even	55	inner	605.2.s.b.416.15	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.15	✓	880	1.1	even	1	trivial
605.2.s.b.416.15	yes	880	121.53	even	55	inner