Embedded newform 605.2.w.a.2.12

• Label: 605.2.w.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	605.2
Dual form			605.2.w.a.303.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34858 - 1.42789i) q^{2} +(-0.353163 + 2.22978i) q^{3} +(-0.106045 + 1.85451i) q^{4} +(-2.16972 + 0.540671i) q^{5} +(3.66016 - 2.50276i) q^{6} +(0.961173 + 1.44967i) q^{7} +(-0.214051 + 0.180190i) q^{8} +(-1.99404 - 0.647904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34858	−	1.42789i	−0.953588	−	1.00967i	−0.999942	−	0.0107487i	\(-0.996579\pi\)
							0.0463544	−	0.998925i	\(-0.485240\pi\)
\(3\)	−0.353163	+	2.22978i	−0.203899	+	1.28737i	0.647184	+	0.762334i	\(0.275947\pi\)
							−0.851082	+	0.525032i	\(0.824053\pi\)
\(5\)	−2.16972	+	0.540671i	−0.970327	+	0.241796i
							\(7\)	0.961173	+	1.44967i	0.363289	+	0.547925i	0.968778	−	0.247928i	\(-0.0797498\pi\)
−0.605489	+	0.795854i	\(0.707023\pi\)
\(11\)	−0.722596	+	3.23695i	−0.217871	+	0.975978i
							\(13\)	0.122171	−	0.209300i	0.0338842	−	0.0580493i	−0.846206	−	0.532856i	\(-0.821119\pi\)
0.880090	+	0.474807i	\(0.157482\pi\)
\(17\)	−0.652698	+	0.925882i	−0.158303	+	0.224559i	−0.888730	−	0.458431i	\(-0.848412\pi\)
							0.730428	+	0.682990i	\(0.239321\pi\)
\(19\)	−6.03704	−	6.21197i	−1.38499	−	1.42512i	−0.773325	−	0.634009i	\(-0.781408\pi\)
							−0.611668	−	0.791115i	\(-0.709501\pi\)
\(23\)	1.76015	+	4.71913i	0.367016	+	0.984007i	0.980883	+	0.194598i	\(0.0623402\pi\)
							−0.613867	+	0.789409i	\(0.710387\pi\)
\(29\)	1.13118	−	2.14382i	0.210054	−	0.398097i	−0.756716	−	0.653744i	\(-0.773197\pi\)
							0.966770	+	0.255647i	\(0.0822883\pi\)
\(31\)	2.70873	−	2.21492i	0.486503	−	0.397812i	−0.357699	−	0.933837i	\(-0.616439\pi\)
							0.844202	+	0.536025i	\(0.180075\pi\)
\(37\)	1.49939	−	3.41129i	0.246498	−	0.560813i	−0.748221	−	0.663450i	\(-0.769092\pi\)
							0.994719	+	0.102637i	\(0.0327280\pi\)
\(41\)	−7.89942	−	6.09137i	−1.23368	−	0.951312i	−0.233902	−	0.972260i	\(-0.575150\pi\)
							−0.999781	+	0.0209485i	\(0.993331\pi\)
\(43\)	−11.9539	+	0.854961i	−1.82296	+	0.130380i	−0.940452	−	0.339926i	\(-0.889598\pi\)
							−0.882503	+	0.470306i	\(0.844143\pi\)
\(47\)	−7.06609	−	3.34973i	−1.03069	−	0.488608i	−0.163482	−	0.986546i	\(-0.552273\pi\)
							−0.867213	+	0.497938i	\(0.834091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.w.a.2.12	✓	5120
5.3	odd	4	inner	605.2.w.a.123.12	yes	5120
121.61	odd	110	inner	605.2.w.a.182.12	yes	5120
605.303	even	220	inner	605.2.w.a.303.12	yes	5120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.w.a.2.12	✓	5120	1.1	even	1	trivial
605.2.w.a.123.12	yes	5120	5.3	odd	4	inner
605.2.w.a.182.12	yes	5120	121.61	odd	110	inner
605.2.w.a.303.12	yes	5120	605.303	even	220	inner