Embedded newform 605.2.u.a.4.7

• Label: 605.2.u.a.4.7
• Level: $605$
• Weight: $2$
• Character: 605.4
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $2560$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.u (of order \(110\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(2560\)
Relative dimension:	\(64\) over \(\Q(\zeta_{110})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{110}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	605.4
Dual form			605.2.u.a.454.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.133191 + 2.32924i) q^{2} +(-2.85319 - 0.927057i) q^{3} +(-3.42066 - 0.392484i) q^{4} +(-0.141192 - 2.23161i) q^{5} +(2.53936 - 6.52229i) q^{6} +(-0.386813 + 0.915309i) q^{7} +(0.574118 - 3.31751i) q^{8} +(4.85420 + 3.52678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2560 q - 144 q^{4} - 43 q^{5} - 114 q^{6} + 534 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.133191	+	2.32924i	−0.0941801	+	1.64702i	0.517331	+	0.855786i	\(0.326926\pi\)
							−0.611511	+	0.791236i	\(0.709438\pi\)
\(3\)	−2.85319	−	0.927057i	−1.64729	−	0.535237i	−0.669139	−	0.743137i	\(-0.733337\pi\)
							−0.978150	+	0.207900i	\(0.933337\pi\)
\(5\)	−0.141192	−	2.23161i	−0.0631428	−	0.998005i
							\(7\)	−0.386813	+	0.915309i	−0.146201	+	0.345954i	−0.977837	−	0.209368i	\(-0.932859\pi\)
0.831636	+	0.555322i	\(0.187405\pi\)
\(11\)	1.98540	−	2.65673i	0.598620	−	0.801033i
							\(13\)	−0.570443	−	1.01012i	−0.158213	−	0.280158i	0.780299	−	0.625407i	\(-0.215067\pi\)
−0.938511	+	0.345249i	\(0.887794\pi\)
\(17\)	1.52230	+	4.26654i	0.369212	+	1.03479i	0.971609	+	0.236592i	\(0.0760306\pi\)
							−0.602397	+	0.798197i	\(0.705788\pi\)
\(19\)	0.0211928	−	0.741844i	0.00486195	−	0.170191i	−0.993533	−	0.113548i	\(-0.963779\pi\)
							0.998395	−	0.0566429i	\(-0.0180397\pi\)
\(23\)	−2.10026	+	1.81989i	−0.437935	+	0.379473i	−0.845739	−	0.533597i	\(-0.820840\pi\)
							0.407804	+	0.913069i	\(0.366295\pi\)
\(29\)	−2.68754	−	3.93041i	−0.499064	−	0.729858i	0.490894	−	0.871219i	\(-0.336670\pi\)
							−0.989958	+	0.141361i	\(0.954852\pi\)
\(31\)	−1.68078	+	8.29497i	−0.301877	+	1.48982i	0.485984	+	0.873968i	\(0.338461\pi\)
							−0.787861	+	0.615854i	\(0.788811\pi\)
\(37\)	−0.277759	−	0.302638i	−0.0456632	−	0.0497534i	0.713489	−	0.700666i	\(-0.247114\pi\)
							−0.759153	+	0.650913i	\(0.774386\pi\)
\(41\)	2.11956	+	8.06365i	0.331020	+	1.25933i	0.899604	+	0.436706i	\(0.143855\pi\)
							−0.568584	+	0.822625i	\(0.692509\pi\)
\(43\)	−6.16877	+	0.886936i	−0.940729	+	0.135256i	−0.595584	−	0.803293i	\(-0.703079\pi\)
							−0.345145	+	0.938549i	\(0.612170\pi\)
\(47\)	2.18586	+	2.67319i	0.318840	+	0.389925i	0.908610	−	0.417647i	\(-0.137145\pi\)
							−0.589769	+	0.807572i	\(0.700781\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.u.a.4.7	✓	2560
5.4	even	2	inner	605.2.u.a.4.58	yes	2560
121.91	even	55	inner	605.2.u.a.454.58	yes	2560
605.454	even	110	inner	605.2.u.a.454.7	yes	2560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.u.a.4.7	✓	2560	1.1	even	1	trivial
605.2.u.a.4.58	yes	2560	5.4	even	2	inner
605.2.u.a.454.7	yes	2560	605.454	even	110	inner
605.2.u.a.454.58	yes	2560	121.91	even	55	inner