Embedded newform 605.2.w.a.2.1

• Label: 605.2.w.a.2.1
• Level: $605$
• Weight: $2$
• Character: 605.2
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $5120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.w (of order \(220\), degree \(80\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(5120\)
Relative dimension:	\(64\) over \(\Q(\zeta_{220})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{220}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	605.2
Dual form			605.2.w.a.303.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86668 - 1.97647i) q^{2} +(-0.223065 + 1.40838i) q^{3} +(-0.307761 + 5.38212i) q^{4} +(-1.72207 + 1.42635i) q^{5} +(3.20000 - 2.18810i) q^{6} +(-1.00869 - 1.52134i) q^{7} +(7.05246 - 5.93681i) q^{8} +(0.919406 + 0.298733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86668	−	1.97647i	−1.31994	−	1.39757i	−0.860572	−	0.509328i	\(-0.829894\pi\)
							−0.459369	−	0.888246i	\(-0.651924\pi\)
\(3\)	−0.223065	+	1.40838i	−0.128787	+	0.813126i	0.835737	+	0.549130i	\(0.185041\pi\)
							−0.964524	+	0.263996i	\(0.914959\pi\)
\(5\)	−1.72207	+	1.42635i	−0.770133	+	0.637883i
							\(7\)	−1.00869	−	1.52134i	−0.381249	−	0.575012i	0.591664	−	0.806185i	\(-0.298471\pi\)
−0.972913	+	0.231172i	\(0.925744\pi\)
\(11\)	3.29649	−	0.364864i	0.993930	−	0.110011i
							\(13\)	−1.18472	+	2.02962i	−0.328583	+	0.562917i	−0.980776	−	0.195136i	\(-0.937485\pi\)
0.652194	+	0.758052i	\(0.273849\pi\)
\(17\)	3.13278	−	4.44399i	0.759810	−	1.07783i	−0.234585	−	0.972096i	\(-0.575373\pi\)
							0.994395	−	0.105730i	\(-0.0337178\pi\)
\(19\)	1.57907	+	1.62482i	0.362262	+	0.372759i	0.873620	−	0.486609i	\(-0.161766\pi\)
							−0.511358	+	0.859368i	\(0.670857\pi\)
\(23\)	0.407816	+	1.09340i	0.0850356	+	0.227989i	0.972507	−	0.232873i	\(-0.0748128\pi\)
							−0.887471	+	0.460863i	\(0.847540\pi\)
\(29\)	−3.95448	+	7.49456i	−0.734328	+	1.39171i	0.178739	+	0.983896i	\(0.442798\pi\)
							−0.913067	+	0.407809i	\(0.866293\pi\)
\(31\)	−2.88838	+	2.36182i	−0.518769	+	0.424195i	−0.855807	−	0.517296i	\(-0.826939\pi\)
							0.337038	+	0.941491i	\(0.390575\pi\)
\(37\)	−4.65158	+	10.5829i	−0.764715	+	1.73982i	−0.0971460	+	0.995270i	\(0.530971\pi\)
							−0.667569	+	0.744548i	\(0.732665\pi\)
\(41\)	7.34292	+	5.66223i	1.14677	+	0.884292i	0.994571	−	0.104057i	\(-0.0331826\pi\)
							0.152199	+	0.988350i	\(0.451364\pi\)
\(43\)	−6.53886	+	0.467668i	−0.997167	+	0.0713188i	−0.560371	−	0.828242i	\(-0.689341\pi\)
							−0.436796	+	0.899561i	\(0.643887\pi\)
\(47\)	−1.07267	−	0.508508i	−0.156465	−	0.0741735i	0.348521	−	0.937301i	\(-0.386684\pi\)
							−0.504986	+	0.863128i	\(0.668502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.w.a.2.1	✓	5120
5.3	odd	4	inner	605.2.w.a.123.1	yes	5120
121.61	odd	110	inner	605.2.w.a.182.1	yes	5120
605.303	even	220	inner	605.2.w.a.303.1	yes	5120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.w.a.2.1	✓	5120	1.1	even	1	trivial
605.2.w.a.123.1	yes	5120	5.3	odd	4	inner
605.2.w.a.182.1	yes	5120	121.61	odd	110	inner
605.2.w.a.303.1	yes	5120	605.303	even	220	inner