Embedded newform 605.2.s.b.16.16

• Label: 605.2.s.b.16.16
• 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			16.16
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.b.416.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46553 + 0.168154i) q^{2} +(-2.14433 - 1.55794i) q^{3} +(0.171477 + 0.0398753i) q^{4} +(-0.564443 + 0.825472i) q^{5} +(-2.88060 - 2.64379i) q^{6} +(0.537054 + 0.552615i) q^{7} +(-2.53411 - 0.904170i) q^{8} +(1.24390 + 3.82832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 4 q^{2} - q^{3} + 24 q^{4} + 22 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46553	+	0.168154i	1.03629	+	0.118903i	0.615403	−	0.788212i	\(-0.288993\pi\)
							0.420883	+	0.907115i	\(0.361720\pi\)
\(3\)	−2.14433	−	1.55794i	−1.23803	−	0.899480i	−0.240562	−	0.970634i	\(-0.577332\pi\)
							−0.997465	+	0.0711538i	\(0.977332\pi\)
\(5\)	−0.564443	+	0.825472i	−0.252427	+	0.369162i
							\(7\)	0.537054	+	0.552615i	0.202987	+	0.208869i	0.810980	−	0.585073i	\(-0.198934\pi\)
−0.607993	+	0.793942i	\(0.708025\pi\)
\(11\)	1.91913	+	2.70498i	0.578640	+	0.815583i
							\(13\)	−2.98709	+	4.95354i	−0.828471	+	1.37386i	0.0970939	+	0.995275i	\(0.469045\pi\)
−0.925565	+	0.378589i	\(0.876409\pi\)
\(17\)	2.21013	−	1.80721i	0.536034	−	0.438313i	−0.325848	−	0.945422i	\(-0.605650\pi\)
							0.861882	+	0.507109i	\(0.169286\pi\)
\(19\)	4.31778	+	0.246900i	0.990567	+	0.0566427i	0.544852	−	0.838532i	\(-0.316586\pi\)
							0.445715	+	0.895175i	\(0.352950\pi\)
\(23\)	−0.597021	+	4.15237i	−0.124487	+	0.865829i	0.827886	+	0.560896i	\(0.189543\pi\)
							−0.952374	+	0.304933i	\(0.901366\pi\)
\(29\)	−0.773829	+	1.98756i	−0.143696	+	0.369081i	−0.985257	−	0.171082i	\(-0.945274\pi\)
							0.841560	+	0.540163i	\(0.181637\pi\)
\(31\)	5.19895	+	2.19709i	0.933759	+	0.394610i	0.802442	−	0.596731i	\(-0.203534\pi\)
							0.131317	+	0.991340i	\(0.458079\pi\)
\(37\)	−0.684525	+	7.96977i	−0.112535	+	1.31022i	0.694403	+	0.719587i	\(0.255669\pi\)
							−0.806938	+	0.590636i	\(0.798877\pi\)
\(41\)	−5.15630	+	2.91189i	−0.805278	+	0.454761i	−0.838643	−	0.544682i	\(-0.816650\pi\)
							0.0333647	+	0.999443i	\(0.489378\pi\)
\(43\)	−11.8141	+	3.46892i	−1.80163	+	0.529005i	−0.997827	−	0.0658941i	\(-0.979010\pi\)
							−0.803800	+	0.594899i	\(0.797192\pi\)
\(47\)	−0.123716	+	0.610562i	−0.0180458	+	0.0890596i	−0.988093	−	0.153856i	\(-0.950831\pi\)
							0.970048	+	0.242915i	\(0.0781037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.b.16.16	✓	880
121.53	even	55	inner	605.2.s.b.416.16	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.b.16.16	✓	880	1.1	even	1	trivial
605.2.s.b.416.16	yes	880	121.53	even	55	inner