Embedded newform 605.2.r.a.32.3

• Label: 605.2.r.a.32.3
• Level: $605$
• Weight: $2$
• Character: 605.32
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			32.3
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60536 - 2.14451i) q^{2} +(-1.49754 + 1.49754i) q^{3} +(-1.45828 + 4.96643i) q^{4} +(1.95274 - 1.08941i) q^{5} +(5.61559 + 0.807400i) q^{6} +(0.692711 - 3.18434i) q^{7} +(7.97176 - 2.97331i) q^{8} -1.48527i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60536	−	2.14451i	−1.13516	−	1.51640i	−0.824488	−	0.565880i	\(-0.808537\pi\)
							−0.310673	−	0.950517i	\(-0.600554\pi\)
\(3\)	−1.49754	+	1.49754i	−0.864607	+	0.864607i	−0.991869	−	0.127262i	\(-0.959381\pi\)
							0.127262	+	0.991869i	\(0.459381\pi\)
\(5\)	1.95274	−	1.08941i	0.873292	−	0.487197i
							\(7\)	0.692711	−	3.18434i	0.261820	−	1.20357i	−0.638942	−	0.769255i	\(-0.720628\pi\)
0.900762	−	0.434313i	\(-0.143009\pi\)
\(11\)	−3.23810	+	0.717425i	−0.976324	+	0.216312i
							\(13\)	2.28894	+	4.19189i	0.634839	+	1.16262i	0.974990	+	0.222250i	\(0.0713400\pi\)
−0.340151	+	0.940371i	\(0.610478\pi\)
\(17\)	−0.142775	−	1.99625i	−0.0346279	−	0.484162i	−0.984944	−	0.172875i	\(-0.944694\pi\)
							0.950316	−	0.311287i	\(-0.100760\pi\)
\(19\)	4.26544	+	4.92258i	0.978560	+	1.12932i	0.991592	+	0.129403i	\(0.0413061\pi\)
							−0.0130324	+	0.999915i	\(0.504148\pi\)
\(23\)	−1.78806	+	0.388969i	−0.372837	+	0.0811057i	−0.395079	−	0.918647i	\(-0.629283\pi\)
							0.0222428	+	0.999753i	\(0.492919\pi\)
\(29\)	−2.17787	−	2.51340i	−0.404420	−	0.466726i	0.516608	−	0.856222i	\(-0.327195\pi\)
							−0.921028	+	0.389496i	\(0.872649\pi\)
\(31\)	−6.48097	+	1.90299i	−1.16402	+	0.341786i	−0.805992	−	0.591926i	\(-0.798368\pi\)
							−0.358025	+	0.933712i	\(0.616550\pi\)
\(37\)	10.2906	+	5.61907i	1.69176	+	0.923770i	0.973714	+	0.227773i	\(0.0731444\pi\)
							0.718045	+	0.695997i	\(0.245037\pi\)
\(41\)	9.95893	+	1.43188i	1.55532	+	0.223622i	0.865706	−	0.500552i	\(-0.166870\pi\)
							0.689617	+	0.724174i	\(0.257779\pi\)
\(43\)	7.11769	−	2.65476i	1.08544	−	0.404847i	0.257850	−	0.966185i	\(-0.416986\pi\)
							0.827587	+	0.561338i	\(0.189713\pi\)
\(47\)	3.82979	−	5.11600i	0.558633	−	0.746246i	−0.428953	−	0.903327i	\(-0.641117\pi\)
							0.987586	+	0.157081i	\(0.0502083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.3	✓	1280
5.3	odd	4	inner	605.2.r.a.153.3	yes	1280
121.87	odd	22	inner	605.2.r.a.87.3	yes	1280
605.208	even	44	inner	605.2.r.a.208.3	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.3	✓	1280	1.1	even	1	trivial
605.2.r.a.87.3	yes	1280	121.87	odd	22	inner
605.2.r.a.153.3	yes	1280	5.3	odd	4	inner
605.2.r.a.208.3	yes	1280	605.208	even	44	inner